%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP078^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n005.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 : Wed Feb 25 08:55:22 AM UTC 2026
% Result : Theorem 2.72s 1.74s
% Output : Refutation 2.72s
% Verified :
% SZS Type : Refutation
% Derivation depth : 9
% Number of leaves : 6
% Syntax : Number of formulae : 26 ( 20 unt; 0 typ; 0 def)
% Number of atoms : 44 ( 27 equ; 0 cnn)
% Maximal formula atoms : 2 ( 1 avg)
% Number of connectives : 265 ( 7 ~; 3 |; 0 &; 252 @)
% ( 0 <=>; 3 =>; 0 <=; 0 <~>)
% Maximal formula depth : 6 ( 2 avg)
% Number of types : 15 ( 14 usr)
% Number of type conns : 5 ( 5 >; 0 *; 0 +; 0 <<)
% Number of symbols : 114 ( 111 usr; 9 con; 0-3 aty)
% Number of variables : 10 ( 0 ^; 10 !; 0 ?; 10 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
real: $tType ).
thf(type_def_7,type,
nat: $tType ).
thf(type_def_8,type,
int: $tType ).
thf(type_def_9,type,
set_nat: $tType ).
thf(type_def_10,type,
set_int: $tType ).
thf(type_def_11,type,
set_real: $tType ).
thf(type_def_12,type,
a: $tType ).
thf(type_def_13,type,
set_a: $tType ).
thf(type_def_14,type,
list_nat: $tType ).
thf(type_def_15,type,
list_a: $tType ).
thf(type_def_16,type,
multiset_nat: $tType ).
thf(type_def_17,type,
multiset_a: $tType ).
thf(type_def_18,type,
multiset_int: $tType ).
thf(type_def_19,type,
set_list_a: $tType ).
thf(func_def_0,type,
set_list_a: $tType ).
thf(func_def_1,type,
multiset_nat: $tType ).
thf(func_def_2,type,
multiset_int: $tType ).
thf(func_def_3,type,
multiset_a: $tType ).
thf(func_def_4,type,
set_real: $tType ).
thf(func_def_5,type,
list_nat: $tType ).
thf(func_def_6,type,
set_nat: $tType ).
thf(func_def_7,type,
set_int: $tType ).
thf(func_def_8,type,
list_a: $tType ).
thf(func_def_9,type,
set_a: $tType ).
thf(func_def_10,type,
real: $tType ).
thf(func_def_11,type,
nat: $tType ).
thf(func_def_12,type,
int: $tType ).
thf(func_def_13,type,
a: $tType ).
thf(func_def_14,type,
inverse_inverse_real: real > real ).
thf(func_def_15,type,
bij_betw_nat_int: ( nat > int ) > set_nat > set_int > $o ).
thf(func_def_16,type,
bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).
thf(func_def_17,type,
bij_betw_nat_real: ( nat > real ) > set_nat > set_real > $o ).
thf(func_def_18,type,
bij_betw_nat_a: ( nat > a ) > set_nat > set_a > $o ).
thf(func_def_19,type,
comp_int_int_nat: ( int > int ) > ( nat > int ) > nat > int ).
thf(func_def_20,type,
comp_int_nat_nat: ( int > nat ) > ( nat > int ) > nat > nat ).
thf(func_def_21,type,
comp_l728830689st_nat: ( list_nat > list_a ) > ( list_nat > list_nat ) > list_nat > list_a ).
thf(func_def_22,type,
comp_l1234327363st_nat: ( list_a > list_a ) > ( list_nat > list_a ) > list_nat > list_a ).
thf(func_def_23,type,
comp_m1386984005et_nat: ( multiset_nat > multiset_nat ) > ( multiset_nat > multiset_nat ) > multiset_nat > multiset_nat ).
thf(func_def_24,type,
comp_m1844659105et_nat: ( multiset_nat > multiset_a ) > ( multiset_nat > multiset_nat ) > multiset_nat > multiset_a ).
thf(func_def_25,type,
comp_m346803701et_nat: ( multiset_nat > nat ) > ( multiset_nat > multiset_nat ) > multiset_nat > nat ).
thf(func_def_26,type,
comp_m1962190563et_nat: ( multiset_a > multiset_nat ) > ( multiset_nat > multiset_a ) > multiset_nat > multiset_nat ).
thf(func_def_27,type,
comp_m540523139et_nat: ( multiset_a > multiset_a ) > ( multiset_nat > multiset_a ) > multiset_nat > multiset_a ).
thf(func_def_28,type,
comp_m1512105363et_nat: ( multiset_a > nat ) > ( multiset_nat > multiset_a ) > multiset_nat > nat ).
thf(func_def_29,type,
comp_nat_int_nat: ( nat > int ) > ( nat > nat ) > nat > int ).
thf(func_def_30,type,
comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).
thf(func_def_31,type,
comp_nat_a_nat: ( nat > a ) > ( nat > nat ) > nat > a ).
thf(func_def_32,type,
comp_a_nat_nat: ( a > nat ) > ( nat > a ) > nat > nat ).
thf(func_def_33,type,
comp_a_a_nat: ( a > a ) > ( nat > a ) > nat > a ).
thf(func_def_34,type,
inj_on_int_int: ( int > int ) > set_int > $o ).
thf(func_def_35,type,
inj_on_int_nat: ( int > nat ) > set_int > $o ).
thf(func_def_36,type,
inj_on_nat_int: ( nat > int ) > set_nat > $o ).
thf(func_def_37,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(func_def_38,type,
inj_on_nat_real: ( nat > real ) > set_nat > $o ).
thf(func_def_39,type,
inj_on_nat_a: ( nat > a ) > set_nat > $o ).
thf(func_def_40,type,
uminus_uminus_int: int > int ).
thf(func_def_41,type,
uminus_uminus_real: real > real ).
thf(func_def_42,type,
zero_zero_int: int ).
thf(func_def_43,type,
zero_z2005226153et_nat: multiset_nat ).
thf(func_def_44,type,
zero_zero_multiset_a: multiset_a ).
thf(func_def_45,type,
zero_zero_nat: nat ).
thf(func_def_46,type,
zero_zero_real: real ).
thf(func_def_47,type,
if_int: $o > int > int > int ).
thf(func_def_48,type,
if_nat: $o > nat > nat > nat ).
thf(func_def_49,type,
if_real: $o > real > real > real ).
thf(func_def_50,type,
nat2: int > nat ).
thf(func_def_51,type,
ring_1_of_int_int: int > int ).
thf(func_def_52,type,
ring_1_of_int_real: int > real ).
thf(func_def_53,type,
gen_length_a: nat > list_a > nat ).
thf(func_def_54,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(func_def_55,type,
map_nat_a: ( nat > a ) > list_nat > list_a ).
thf(func_def_56,type,
map_a_nat: ( a > nat ) > list_a > list_nat ).
thf(func_def_57,type,
map_a_a: ( a > a ) > list_a > list_a ).
thf(func_def_58,type,
list_ex_a: ( a > $o ) > list_a > $o ).
thf(func_def_59,type,
map_tailrec_nat_a: ( nat > a ) > list_nat > list_a ).
thf(func_def_60,type,
nth_nat: list_nat > nat > nat ).
thf(func_def_61,type,
nth_a: list_a > nat > a ).
thf(func_def_62,type,
upt: nat > nat > list_nat ).
thf(func_def_63,type,
image_mset_nat_int: ( nat > int ) > multiset_nat > multiset_int ).
thf(func_def_64,type,
image_mset_nat_nat: ( nat > nat ) > multiset_nat > multiset_nat ).
thf(func_def_65,type,
image_mset_nat_a: ( nat > a ) > multiset_nat > multiset_a ).
thf(func_def_66,type,
image_mset_a_nat: ( a > nat ) > multiset_a > multiset_nat ).
thf(func_def_67,type,
image_mset_a_a: ( a > a ) > multiset_a > multiset_a ).
thf(func_def_68,type,
mset_nat: list_nat > multiset_nat ).
thf(func_def_69,type,
mset_a: list_a > multiset_a ).
thf(func_def_70,type,
mset_set_int: set_int > multiset_int ).
thf(func_def_71,type,
mset_set_nat: set_nat > multiset_nat ).
thf(func_def_72,type,
mset_set_a: set_a > multiset_a ).
thf(func_def_73,type,
size_multiset_nat: ( nat > nat ) > multiset_nat > nat ).
thf(func_def_74,type,
size_multiset_a: ( a > nat ) > multiset_a > nat ).
thf(func_def_75,type,
semiri2019852685at_int: nat > int ).
thf(func_def_76,type,
semiri1382578993at_nat: nat > nat ).
thf(func_def_77,type,
semiri2110766477t_real: nat > real ).
thf(func_def_78,type,
size_size_list_nat: list_nat > nat ).
thf(func_def_79,type,
size_size_list_a: list_a > nat ).
thf(func_def_80,type,
size_s1679505588et_nat: multiset_nat > nat ).
thf(func_def_81,type,
size_size_multiset_a: multiset_a > nat ).
thf(func_def_82,type,
ord_less_int: int > int > $o ).
thf(func_def_83,type,
ord_less_nat: nat > nat > $o ).
thf(func_def_84,type,
ord_less_real: real > real > $o ).
thf(func_def_85,type,
ord_less_set_int: set_int > set_int > $o ).
thf(func_def_86,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(func_def_87,type,
ord_less_set_real: set_real > set_real > $o ).
thf(func_def_88,type,
ord_less_eq_int: int > int > $o ).
thf(func_def_89,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_90,type,
ord_less_eq_real: real > real > $o ).
thf(func_def_91,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(func_def_92,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(func_def_93,type,
image_int_int: ( int > int ) > set_int > set_int ).
thf(func_def_94,type,
image_int_nat: ( int > nat ) > set_int > set_nat ).
thf(func_def_95,type,
image_nat_int: ( nat > int ) > set_nat > set_int ).
thf(func_def_96,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(func_def_97,type,
image_nat_real: ( nat > real ) > set_nat > set_real ).
thf(func_def_98,type,
image_nat_a: ( nat > a ) > set_nat > set_a ).
thf(func_def_99,type,
set_or1199280219an_int: int > int > set_int ).
thf(func_def_100,type,
set_or562006527an_nat: nat > nat > set_nat ).
thf(func_def_101,type,
set_or2075149659n_real: real > real > set_real ).
thf(func_def_102,type,
set_ord_lessThan_int: int > set_int ).
thf(func_def_103,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(func_def_104,type,
set_or1211449801n_real: real > set_real ).
thf(func_def_105,type,
member_int: int > set_int > $o ).
thf(func_def_106,type,
member_list_a: list_a > set_list_a > $o ).
thf(func_def_107,type,
member_nat: nat > set_nat > $o ).
thf(func_def_108,type,
member_real: real > set_real > $o ).
thf(func_def_109,type,
f: nat > nat ).
thf(func_def_110,type,
xs: list_a ).
thf(func_def_111,type,
ys: list_a ).
thf(func_def_126,type,
sK0: list_a > list_a > nat ).
thf(func_def_127,type,
sK1: nat > list_a ).
thf(func_def_128,type,
sK2: ( nat > a > $o ) > nat > list_a ).
thf(func_def_129,type,
sK3: ( nat > a > $o ) > nat > nat ).
thf(func_def_130,type,
sK4: nat > ( nat > a > $o ) > a ).
thf(func_def_131,type,
sK5: list_a > ( nat > a > $o ) > nat > nat ).
thf(func_def_132,type,
sK6: nat > ( nat > $o ) > nat ).
thf(func_def_133,type,
sK7: nat > ( nat > $o ) > nat ).
thf(func_def_134,type,
sK8: ( list_a > $o ) > list_a ).
thf(func_def_135,type,
sK9: nat > ( nat > $o ) > nat ).
thf(func_def_136,type,
sK10: nat > ( nat > $o ) > nat ).
thf(func_def_137,type,
sK11: ( nat > $o ) > nat ).
thf(func_def_140,type,
ph13:
!>[X0: $tType] : X0 ).
thf(f2147,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f2146,f936]) ).
thf(f936,plain,
( ( mset_a @ xs )
!= ( image_mset_nat_a @ ( nth_a @ ys ) @ ( mset_set_nat @ ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ ys ) ) ) ) ),
inference(backward_demodulation,[status(thm)],[f910,f935]) ).
thf(f935,plain,
( ( mset_a @ xs )
= ( image_mset_nat_a @ ( nth_a @ ys ) @ ( image_mset_nat_nat @ f @ ( mset_set_nat @ ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ xs ) ) ) ) ) ),
inference(forward_demodulation,[status(thm)],[f886,f853]) ).
thf(f853,plain,
( xs
= ( map_nat_a @ ( comp_nat_a_nat @ ( nth_a @ ys ) @ f ) @ ( upt @ zero_zero_nat @ ( size_size_list_a @ xs ) ) ) ),
inference(cnf_transformation,[status(thm)],[f7]) ).
thf(f7,axiom,
( xs
= ( map_nat_a @ ( comp_nat_a_nat @ ( nth_a @ ys ) @ f ) @ ( upt @ zero_zero_nat @ ( size_size_list_a @ xs ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.VwIcuknk1l/DTF2THF_3941.p',fact_6__092_060open_062xs_A_061_Amap_A_I_I_B_J_Ays_A_092_060circ_062_Af_J_A_0910_O_O_060length_Axs_093_092_060close_062) ).
thf(f886,plain,
( ( image_mset_nat_a @ ( nth_a @ ys ) @ ( image_mset_nat_nat @ f @ ( mset_set_nat @ ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ xs ) ) ) ) )
= ( mset_a @ ( map_nat_a @ ( comp_nat_a_nat @ ( nth_a @ ys ) @ f ) @ ( upt @ zero_zero_nat @ ( size_size_list_a @ xs ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f6]) ).
thf(f6,axiom,
( ( image_mset_nat_a @ ( nth_a @ ys ) @ ( image_mset_nat_nat @ f @ ( mset_set_nat @ ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ xs ) ) ) ) )
= ( mset_a @ ( map_nat_a @ ( comp_nat_a_nat @ ( nth_a @ ys ) @ f ) @ ( upt @ zero_zero_nat @ ( size_size_list_a @ xs ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.VwIcuknk1l/DTF2THF_3941.p',fact_5__092_060open_062mset_A_Imap_A_I_I_B_J_Ays_A_092_060circ_062_Af_J_A_0910_O_O_060length_Axs_093_J_A_061_Aimage__mset_A_I_I_B_J_Ays_J_A_Iimage__mset_Af_A_Imset__set_A_1230_O_O_060length_Axs_125_J_J_092_060close_062) ).
thf(f910,plain,
( ( image_mset_nat_a @ ( nth_a @ ys ) @ ( image_mset_nat_nat @ f @ ( mset_set_nat @ ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ xs ) ) ) ) )
!= ( image_mset_nat_a @ ( nth_a @ ys ) @ ( mset_set_nat @ ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ ys ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f781]) ).
thf(f781,plain,
( ( image_mset_nat_a @ ( nth_a @ ys ) @ ( image_mset_nat_nat @ f @ ( mset_set_nat @ ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ xs ) ) ) ) )
!= ( image_mset_nat_a @ ( nth_a @ ys ) @ ( mset_set_nat @ ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ ys ) ) ) ) ),
inference(flattening,[status(thm)],[f358]) ).
thf(f358,negated_conjecture,
( ( image_mset_nat_a @ ( nth_a @ ys ) @ ( image_mset_nat_nat @ f @ ( mset_set_nat @ ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ xs ) ) ) ) )
!= ( image_mset_nat_a @ ( nth_a @ ys ) @ ( mset_set_nat @ ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ ys ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f357]) ).
thf(f357,conjecture,
( ( image_mset_nat_a @ ( nth_a @ ys ) @ ( image_mset_nat_nat @ f @ ( mset_set_nat @ ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ xs ) ) ) ) )
= ( image_mset_nat_a @ ( nth_a @ ys ) @ ( mset_set_nat @ ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ ys ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.VwIcuknk1l/DTF2THF_3941.p',conj_0) ).
thf(f2146,plain,
( ( mset_a @ xs )
= ( image_mset_nat_a @ ( nth_a @ ys ) @ ( mset_set_nat @ ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ ys ) ) ) ) ),
inference(backward_demodulation,[status(thm)],[f935,f2145]) ).
thf(f2145,plain,
( ( image_mset_nat_nat @ f @ ( mset_set_nat @ ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ xs ) ) ) )
= ( mset_set_nat @ ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ ys ) ) ) ),
inference(forward_demodulation,[status(thm)],[f2134,f854]) ).
thf(f854,plain,
( ( image_nat_nat @ f @ ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ xs ) ) )
= ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ ys ) ) ),
inference(cnf_transformation,[status(thm)],[f4]) ).
thf(f4,axiom,
( ( image_nat_nat @ f @ ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ xs ) ) )
= ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ ys ) ) ),
file('/export/starexec/sandbox/tmp/tmp.VwIcuknk1l/DTF2THF_3941.p',fact_3__C2_C) ).
thf(f2134,plain,
( ( image_mset_nat_nat @ f @ ( mset_set_nat @ ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ xs ) ) ) )
= ( mset_set_nat @ ( image_nat_nat @ f @ ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ xs ) ) ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f2126]) ).
thf(f2126,plain,
( ( $true != $true )
| ( ( image_mset_nat_nat @ f @ ( mset_set_nat @ ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ xs ) ) ) )
= ( mset_set_nat @ ( image_nat_nat @ f @ ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ xs ) ) ) ) ) ),
inference(superposition,[status(thm)],[f869,f866]) ).
thf(f866,plain,
( ( inj_on_nat_nat @ f @ ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ xs ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f756]) ).
thf(f756,plain,
( ( inj_on_nat_nat @ f @ ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ xs ) ) )
= $true ),
inference(fool_elimination,[status(thm)],[f755]) ).
thf(f755,plain,
inj_on_nat_nat @ f @ ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ xs ) ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
inj_on_nat_nat @ f @ ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ xs ) ),
file('/export/starexec/sandbox/tmp/tmp.VwIcuknk1l/DTF2THF_3941.p',fact_0__C1_C) ).
thf(f869,plain,
! [X0: nat > nat,X1: set_nat] :
( ( ( inj_on_nat_nat @ X0 @ X1 )
!= $true )
| ( ( mset_set_nat @ ( image_nat_nat @ X0 @ X1 ) )
= ( image_mset_nat_nat @ X0 @ ( mset_set_nat @ X1 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f818]) ).
thf(f818,plain,
! [X0: nat > nat,X1: set_nat] :
( ( ( mset_set_nat @ ( image_nat_nat @ X0 @ X1 ) )
= ( image_mset_nat_nat @ X0 @ ( mset_set_nat @ X1 ) ) )
| ( ( inj_on_nat_nat @ X0 @ X1 )
!= $true ) ),
inference(ennf_transformation,[status(thm)],[f562]) ).
thf(f562,plain,
! [X0: nat > nat,X1: set_nat] :
( ( ( inj_on_nat_nat @ X0 @ X1 )
= $true )
=> ( ( mset_set_nat @ ( image_nat_nat @ X0 @ X1 ) )
= ( image_mset_nat_nat @ X0 @ ( mset_set_nat @ X1 ) ) ) ),
inference(fool_elimination,[status(thm)],[f561]) ).
thf(f561,plain,
! [X0: nat > nat,X1: set_nat] :
( ( inj_on_nat_nat @ X0 @ X1 )
=> ( ( mset_set_nat @ ( image_nat_nat @ X0 @ X1 ) )
= ( image_mset_nat_nat @ X0 @ ( mset_set_nat @ X1 ) ) ) ),
inference(rectify,[status(thm)],[f72]) ).
thf(f72,axiom,
! [X8: nat > nat,X29: set_nat] :
( ( inj_on_nat_nat @ X8 @ X29 )
=> ( ( image_mset_nat_nat @ X8 @ ( mset_set_nat @ X29 ) )
= ( mset_set_nat @ ( image_nat_nat @ X8 @ X29 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.VwIcuknk1l/DTF2THF_3941.p',fact_71_image__mset__mset__set) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP078^1 : TPTP v9.2.1. Released v7.5.0.
% 0.13/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n005.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 08:25:14 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.29/0.46 ---- Original DTF file ---
% 0.29/0.46 thf(spec,logic,$$dhol).
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 % File : ITP078^1 : TPTP v9.2.1. Released v7.5.0.
% 0.29/0.46 % Domain : Interactive Theorem Proving
% 0.29/0.46 % Problem : Sledgehammer Hilbert_Function problem prob_85__11621656_1
% 0.29/0.46 % Version : Especial.
% 0.29/0.46 % English :
% 0.29/0.46
% 0.29/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.29/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.29/0.46 % Source : [Des21]
% 0.29/0.46 % Names : Hilbert_Function/prob_85__11621656_1 [Des21]
% 0.29/0.46
% 0.29/0.46 % Status : Theorem
% 0.29/0.46 % Rating : 0.22 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% 0.29/0.46 % Syntax : Number of formulae : 469 ( 228 unt; 112 typ; 0 def)
% 0.29/0.46 % Number of atoms : 857 ( 438 equ; 0 cnn)
% 0.29/0.46 % Maximal formula atoms : 5 ( 2 avg)
% 0.29/0.46 % Number of connectives : 2404 ( 69 ~; 4 |; 37 &;2049 @)
% 0.29/0.46 % ( 0 <=>; 245 =>; 0 <=; 0 <~>)
% 0.29/0.46 % Maximal formula depth : 13 ( 5 avg)
% 0.29/0.46 % Number of types : 15 ( 14 usr)
% 0.29/0.46 % Number of type conns : 493 ( 493 >; 0 *; 0 +; 0 <<)
% 0.29/0.46 % Number of symbols : 101 ( 98 usr; 9 con; 0-3 aty)
% 0.29/0.46 % Number of variables : 818 ( 31 ^; 762 !; 25 ?; 818 :)
% 0.29/0.46 % SPC : TH0_THM_EQU_NAR
% 0.29/0.46
% 0.29/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 15:36:44.233
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 % Could-be-implicit typings (14)
% 0.29/0.46 thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
% 0.29/0.46 set_list_a: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Multiset__Omultiset_It__Nat__Onat_J,type,
% 0.29/0.46 multiset_nat: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Multiset__Omultiset_It__Int__Oint_J,type,
% 0.29/0.46 multiset_int: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Multiset__Omultiset_Itf__a_J,type,
% 0.29/0.46 multiset_a: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
% 0.29/0.46 set_real: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
% 0.29/0.46 list_nat: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 0.29/0.46 set_nat: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
% 0.29/0.46 set_int: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__List__Olist_Itf__a_J,type,
% 0.29/0.46 list_a: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 0.29/0.46 set_a: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Real__Oreal,type,
% 0.29/0.46 real: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Nat__Onat,type,
% 0.29/0.46 nat: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_t__Int__Oint,type,
% 0.29/0.46 int: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_n_tf__a,type,
% 0.29/0.46 a: $tType ).
% 0.29/0.46
% 0.29/0.46 % Explicit typings (98)
% 0.29/0.46 thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
% 0.29/0.46 inverse_inverse_real: real > real ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Int__Oint,type,
% 0.29/0.46 bij_betw_nat_int: ( nat > int ) > set_nat > set_int > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.29/0.46 bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Real__Oreal,type,
% 0.29/0.46 bij_betw_nat_real: ( nat > real ) > set_nat > set_real > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001tf__a,type,
% 0.29/0.46 bij_betw_nat_a: ( nat > a ) > set_nat > set_a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Int__Oint_001t__Nat__Onat,type,
% 0.29/0.46 comp_int_int_nat: ( int > int ) > ( nat > int ) > nat > int ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.29/0.46 comp_int_nat_nat: ( int > nat ) > ( nat > int ) > nat > nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Fun_Ocomp_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_Itf__a_J_001t__List__Olist_It__Nat__Onat_J,type,
% 0.29/0.46 comp_l728830689st_nat: ( list_nat > list_a ) > ( list_nat > list_nat ) > list_nat > list_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Fun_Ocomp_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J_001t__List__Olist_It__Nat__Onat_J,type,
% 0.29/0.46 comp_l1234327363st_nat: ( list_a > list_a ) > ( list_nat > list_a ) > list_nat > list_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Fun_Ocomp_001t__Multiset__Omultiset_It__Nat__Onat_J_001t__Multiset__Omultiset_It__Nat__Onat_J_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
% 0.29/0.46 comp_m1386984005et_nat: ( multiset_nat > multiset_nat ) > ( multiset_nat > multiset_nat ) > multiset_nat > multiset_nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Fun_Ocomp_001t__Multiset__Omultiset_It__Nat__Onat_J_001t__Multiset__Omultiset_Itf__a_J_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
% 0.29/0.46 comp_m1844659105et_nat: ( multiset_nat > multiset_a ) > ( multiset_nat > multiset_nat ) > multiset_nat > multiset_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Fun_Ocomp_001t__Multiset__Omultiset_It__Nat__Onat_J_001t__Nat__Onat_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
% 0.29/0.46 comp_m346803701et_nat: ( multiset_nat > nat ) > ( multiset_nat > multiset_nat ) > multiset_nat > nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Fun_Ocomp_001t__Multiset__Omultiset_Itf__a_J_001t__Multiset__Omultiset_It__Nat__Onat_J_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
% 0.29/0.46 comp_m1962190563et_nat: ( multiset_a > multiset_nat ) > ( multiset_nat > multiset_a ) > multiset_nat > multiset_nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Fun_Ocomp_001t__Multiset__Omultiset_Itf__a_J_001t__Multiset__Omultiset_Itf__a_J_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
% 0.29/0.46 comp_m540523139et_nat: ( multiset_a > multiset_a ) > ( multiset_nat > multiset_a ) > multiset_nat > multiset_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Fun_Ocomp_001t__Multiset__Omultiset_Itf__a_J_001t__Nat__Onat_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
% 0.29/0.46 comp_m1512105363et_nat: ( multiset_a > nat ) > ( multiset_nat > multiset_a ) > multiset_nat > nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Int__Oint_001t__Nat__Onat,type,
% 0.29/0.46 comp_nat_int_nat: ( nat > int ) > ( nat > nat ) > nat > int ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.29/0.46 comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001tf__a_001t__Nat__Onat,type,
% 0.29/0.46 comp_nat_a_nat: ( nat > a ) > ( nat > nat ) > nat > a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Fun_Ocomp_001tf__a_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.29/0.46 comp_a_nat_nat: ( a > nat ) > ( nat > a ) > nat > nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001t__Nat__Onat,type,
% 0.29/0.46 comp_a_a_nat: ( a > a ) > ( nat > a ) > nat > a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Fun_Oinj__on_001t__Int__Oint_001t__Int__Oint,type,
% 0.29/0.46 inj_on_int_int: ( int > int ) > set_int > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Fun_Oinj__on_001t__Int__Oint_001t__Nat__Onat,type,
% 0.29/0.46 inj_on_int_nat: ( int > nat ) > set_int > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Int__Oint,type,
% 0.29/0.46 inj_on_nat_int: ( nat > int ) > set_nat > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.29/0.46 inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Real__Oreal,type,
% 0.29/0.46 inj_on_nat_real: ( nat > real ) > set_nat > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001tf__a,type,
% 0.29/0.46 inj_on_nat_a: ( nat > a ) > set_nat > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
% 0.29/0.46 uminus_uminus_int: int > int ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
% 0.29/0.46 uminus_uminus_real: real > real ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
% 0.29/0.46 zero_zero_int: int ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
% 0.29/0.46 zero_z2005226153et_nat: multiset_nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_Itf__a_J,type,
% 0.29/0.46 zero_zero_multiset_a: multiset_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 0.29/0.46 zero_zero_nat: nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
% 0.29/0.46 zero_zero_real: real ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_If_001t__Int__Oint,type,
% 0.29/0.46 if_int: $o > int > int > int ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_If_001t__Nat__Onat,type,
% 0.29/0.46 if_nat: $o > nat > nat > nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_If_001t__Real__Oreal,type,
% 0.29/0.46 if_real: $o > real > real > real ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Int_Onat,type,
% 0.29/0.46 nat2: int > nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
% 0.29/0.46 ring_1_of_int_int: int > int ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
% 0.29/0.46 ring_1_of_int_real: int > real ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_List_Ogen__length_001tf__a,type,
% 0.29/0.46 gen_length_a: nat > list_a > nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.29/0.46 map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001tf__a,type,
% 0.29/0.46 map_nat_a: ( nat > a ) > list_nat > list_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_List_Olist_Omap_001tf__a_001t__Nat__Onat,type,
% 0.29/0.46 map_a_nat: ( a > nat ) > list_a > list_nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_List_Olist_Omap_001tf__a_001tf__a,type,
% 0.29/0.46 map_a_a: ( a > a ) > list_a > list_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_List_Olist__ex_001tf__a,type,
% 0.29/0.46 list_ex_a: ( a > $o ) > list_a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_List_Omap__tailrec_001t__Nat__Onat_001tf__a,type,
% 0.29/0.46 map_tailrec_nat_a: ( nat > a ) > list_nat > list_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_List_Onth_001t__Nat__Onat,type,
% 0.29/0.46 nth_nat: list_nat > nat > nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_List_Onth_001tf__a,type,
% 0.29/0.46 nth_a: list_a > nat > a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_List_Oupt,type,
% 0.29/0.46 upt: nat > nat > list_nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Multiset_Oimage__mset_001t__Nat__Onat_001t__Int__Oint,type,
% 0.29/0.46 image_mset_nat_int: ( nat > int ) > multiset_nat > multiset_int ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Multiset_Oimage__mset_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.29/0.46 image_mset_nat_nat: ( nat > nat ) > multiset_nat > multiset_nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Multiset_Oimage__mset_001t__Nat__Onat_001tf__a,type,
% 0.29/0.46 image_mset_nat_a: ( nat > a ) > multiset_nat > multiset_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Multiset_Oimage__mset_001tf__a_001t__Nat__Onat,type,
% 0.29/0.46 image_mset_a_nat: ( a > nat ) > multiset_a > multiset_nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Multiset_Oimage__mset_001tf__a_001tf__a,type,
% 0.29/0.46 image_mset_a_a: ( a > a ) > multiset_a > multiset_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Multiset_Omset_001t__Nat__Onat,type,
% 0.29/0.46 mset_nat: list_nat > multiset_nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Multiset_Omset_001tf__a,type,
% 0.29/0.46 mset_a: list_a > multiset_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Multiset_Omset__set_001t__Int__Oint,type,
% 0.29/0.46 mset_set_int: set_int > multiset_int ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Multiset_Omset__set_001t__Nat__Onat,type,
% 0.29/0.46 mset_set_nat: set_nat > multiset_nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Multiset_Omset__set_001tf__a,type,
% 0.29/0.46 mset_set_a: set_a > multiset_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Multiset_Osize__multiset_001t__Nat__Onat,type,
% 0.29/0.46 size_multiset_nat: ( nat > nat ) > multiset_nat > nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Multiset_Osize__multiset_001tf__a,type,
% 0.29/0.46 size_multiset_a: ( a > nat ) > multiset_a > nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
% 0.29/0.46 semiri2019852685at_int: nat > int ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
% 0.29/0.46 semiri1382578993at_nat: nat > nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
% 0.29/0.46 semiri2110766477t_real: nat > real ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
% 0.29/0.46 size_size_list_nat: list_nat > nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
% 0.29/0.46 size_size_list_a: list_a > nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
% 0.29/0.46 size_s1679505588et_nat: multiset_nat > nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_Itf__a_J,type,
% 0.29/0.46 size_size_multiset_a: multiset_a > nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
% 0.29/0.46 ord_less_int: int > int > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 0.29/0.46 ord_less_nat: nat > nat > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
% 0.29/0.46 ord_less_real: real > real > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
% 0.29/0.46 ord_less_set_int: set_int > set_int > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.29/0.46 ord_less_set_nat: set_nat > set_nat > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.29/0.46 ord_less_set_real: set_real > set_real > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
% 0.29/0.46 ord_less_eq_int: int > int > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 0.29/0.46 ord_less_eq_nat: nat > nat > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
% 0.29/0.46 ord_less_eq_real: real > real > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
% 0.29/0.46 ord_less_eq_set_int: set_int > set_int > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.29/0.46 ord_less_eq_set_nat: set_nat > set_nat > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
% 0.29/0.46 image_int_int: ( int > int ) > set_int > set_int ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Nat__Onat,type,
% 0.29/0.46 image_int_nat: ( int > nat ) > set_int > set_nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
% 0.29/0.46 image_nat_int: ( nat > int ) > set_nat > set_int ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.29/0.46 image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Real__Oreal,type,
% 0.29/0.46 image_nat_real: ( nat > real ) > set_nat > set_real ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__a,type,
% 0.29/0.46 image_nat_a: ( nat > a ) > set_nat > set_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
% 0.29/0.46 set_or1199280219an_int: int > int > set_int ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
% 0.29/0.46 set_or562006527an_nat: nat > nat > set_nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Real__Oreal,type,
% 0.29/0.46 set_or2075149659n_real: real > real > set_real ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
% 0.29/0.46 set_ord_lessThan_int: int > set_int ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
% 0.29/0.46 set_ord_lessThan_nat: nat > set_nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
% 0.29/0.46 set_or1211449801n_real: real > set_real ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_member_001t__Int__Oint,type,
% 0.29/0.46 member_int: int > set_int > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
% 0.29/0.46 member_list_a: list_a > set_list_a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_member_001t__Nat__Onat,type,
% 0.29/0.46 member_nat: nat > set_nat > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_member_001t__Real__Oreal,type,
% 0.29/0.46 member_real: real > set_real > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_v_f,type,
% 0.29/0.46 f: nat > nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_v_xs,type,
% 0.29/0.46 xs: list_a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_v_ys,type,
% 0.29/0.46 ys: list_a ).
% 0.29/0.46
% 0.29/0.46 % Relevant facts (349)
% 0.29/0.46 thf(fact_0__C1_C,axiom,
% 0.29/0.46 inj_on_nat_nat @ f @ ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ xs ) ) ).
% 0.29/0.46
% 0.29/0.46 % "1"
% 0.29/0.46 thf(fact_1_calculation,axiom,
% 0.29/0.46 ( ( mset_a @ xs )
% 0.29/0.46 = ( image_mset_nat_a @ ( nth_a @ ys ) @ ( image_mset_nat_nat @ f @ ( mset_set_nat @ ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ xs ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % calculation
% 0.29/0.46 thf(fact_2_assms_I2_J,axiom,
% 0.29/0.46 ! [I: nat] :
% 0.29/0.46 ( ( ord_less_nat @ I @ ( size_size_list_a @ xs ) )
% 0.29/0.46 => ( ( nth_a @ xs @ I )
% 0.29/0.46 = ( nth_a @ ys @ ( f @ I ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % assms(2)
% 0.29/0.46 thf(fact_3__C2_C,axiom,
% 0.29/0.46 ( ( image_nat_nat @ f @ ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ xs ) ) )
% 0.29/0.46 = ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ ys ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % "2"
% 0.29/0.46 thf(fact_4_assms_I1_J,axiom,
% 0.29/0.46 bij_betw_nat_nat @ f @ ( set_ord_lessThan_nat @ ( size_size_list_a @ xs ) ) @ ( set_ord_lessThan_nat @ ( size_size_list_a @ ys ) ) ).
% 0.29/0.46
% 0.29/0.46 % assms(1)
% 0.29/0.46 thf(fact_5__092_060open_062mset_A_Imap_A_I_I_B_J_Ays_A_092_060circ_062_Af_J_A_0910_O_O_060length_Axs_093_J_A_061_Aimage__mset_A_I_I_B_J_Ays_J_A_Iimage__mset_Af_A_Imset__set_A_1230_O_O_060length_Axs_125_J_J_092_060close_062,axiom,
% 0.29/0.46 ( ( mset_a @ ( map_nat_a @ ( comp_nat_a_nat @ ( nth_a @ ys ) @ f ) @ ( upt @ zero_zero_nat @ ( size_size_list_a @ xs ) ) ) )
% 0.29/0.46 = ( image_mset_nat_a @ ( nth_a @ ys ) @ ( image_mset_nat_nat @ f @ ( mset_set_nat @ ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ xs ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % \<open>mset (map ((!) ys \<circ> f) [0..<length xs]) = image_mset ((!) ys) (image_mset f (mset_set {0..<length xs}))\<close>
% 0.29/0.46 thf(fact_6__092_060open_062xs_A_061_Amap_A_I_I_B_J_Ays_A_092_060circ_062_Af_J_A_0910_O_O_060length_Axs_093_092_060close_062,axiom,
% 0.29/0.46 ( xs
% 0.29/0.46 = ( map_nat_a @ ( comp_nat_a_nat @ ( nth_a @ ys ) @ f ) @ ( upt @ zero_zero_nat @ ( size_size_list_a @ xs ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % \<open>xs = map ((!) ys \<circ> f) [0..<length xs]\<close>
% 0.29/0.46 thf(fact_7_Ex__list__of__length,axiom,
% 0.29/0.46 ! [N: nat] :
% 0.29/0.46 ? [Xs: list_a] :
% 0.29/0.46 ( ( size_size_list_a @ Xs )
% 0.29/0.46 = N ) ).
% 0.29/0.46
% 0.29/0.46 % Ex_list_of_length
% 0.29/0.46 thf(fact_8_neq__if__length__neq,axiom,
% 0.29/0.46 ! [Xs2: list_a,Ys: list_a] :
% 0.29/0.46 ( ( ( size_size_list_a @ Xs2 )
% 0.29/0.46 != ( size_size_list_a @ Ys ) )
% 0.29/0.46 => ( Xs2 != Ys ) ) ).
% 0.29/0.46
% 0.29/0.46 % neq_if_length_neq
% 0.29/0.46 thf(fact_9_size__neq__size__imp__neq,axiom,
% 0.29/0.46 ! [X: list_a,Y: list_a] :
% 0.29/0.46 ( ( ( size_size_list_a @ X )
% 0.29/0.46 != ( size_size_list_a @ Y ) )
% 0.29/0.46 => ( X != Y ) ) ).
% 0.29/0.46
% 0.29/0.46 % size_neq_size_imp_neq
% 0.29/0.46 thf(fact_10_not__gr__zero,axiom,
% 0.29/0.46 ! [N: nat] :
% 0.29/0.46 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.29/0.46 = ( N = zero_zero_nat ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_gr_zero
% 0.29/0.46 thf(fact_11_bot__nat__0_Onot__eq__extremum,axiom,
% 0.29/0.46 ! [A: nat] :
% 0.29/0.46 ( ( A != zero_zero_nat )
% 0.29/0.46 = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot_nat_0.not_eq_extremum
% 0.29/0.46 thf(fact_12_less__nat__zero__code,axiom,
% 0.29/0.46 ! [N: nat] :
% 0.29/0.46 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.29/0.46
% 0.29/0.46 % less_nat_zero_code
% 0.29/0.46 thf(fact_13_neq0__conv,axiom,
% 0.29/0.46 ! [N: nat] :
% 0.29/0.46 ( ( N != zero_zero_nat )
% 0.29/0.46 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % neq0_conv
% 0.29/0.46 thf(fact_14_map__map,axiom,
% 0.29/0.46 ! [F: a > a,G: nat > a,Xs2: list_nat] :
% 0.29/0.46 ( ( map_a_a @ F @ ( map_nat_a @ G @ Xs2 ) )
% 0.29/0.46 = ( map_nat_a @ ( comp_a_a_nat @ F @ G ) @ Xs2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % map_map
% 0.29/0.46 thf(fact_15_map__map,axiom,
% 0.29/0.46 ! [F: nat > a,G: nat > nat,Xs2: list_nat] :
% 0.29/0.46 ( ( map_nat_a @ F @ ( map_nat_nat @ G @ Xs2 ) )
% 0.29/0.46 = ( map_nat_a @ ( comp_nat_a_nat @ F @ G ) @ Xs2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % map_map
% 0.29/0.46 thf(fact_16_List_Omap_Ocomp,axiom,
% 0.29/0.47 ! [F: a > a,G: nat > a] :
% 0.29/0.47 ( ( comp_l1234327363st_nat @ ( map_a_a @ F ) @ ( map_nat_a @ G ) )
% 0.29/0.47 = ( map_nat_a @ ( comp_a_a_nat @ F @ G ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % List.map.comp
% 0.29/0.47 thf(fact_17_List_Omap_Ocomp,axiom,
% 0.29/0.47 ! [F: nat > a,G: nat > nat] :
% 0.29/0.47 ( ( comp_l728830689st_nat @ ( map_nat_a @ F ) @ ( map_nat_nat @ G ) )
% 0.29/0.47 = ( map_nat_a @ ( comp_nat_a_nat @ F @ G ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % List.map.comp
% 0.29/0.47 thf(fact_18_map__comp__map,axiom,
% 0.29/0.47 ! [F: a > a,G: nat > a] :
% 0.29/0.47 ( ( comp_l1234327363st_nat @ ( map_a_a @ F ) @ ( map_nat_a @ G ) )
% 0.29/0.47 = ( map_nat_a @ ( comp_a_a_nat @ F @ G ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % map_comp_map
% 0.29/0.47 thf(fact_19_map__comp__map,axiom,
% 0.29/0.47 ! [F: nat > a,G: nat > nat] :
% 0.29/0.47 ( ( comp_l728830689st_nat @ ( map_nat_a @ F ) @ ( map_nat_nat @ G ) )
% 0.29/0.47 = ( map_nat_a @ ( comp_nat_a_nat @ F @ G ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % map_comp_map
% 0.29/0.47 thf(fact_20_list_Omap__comp,axiom,
% 0.29/0.47 ! [G: a > a,F: nat > a,V: list_nat] :
% 0.29/0.47 ( ( map_a_a @ G @ ( map_nat_a @ F @ V ) )
% 0.29/0.47 = ( map_nat_a @ ( comp_a_a_nat @ G @ F ) @ V ) ) ).
% 0.29/0.47
% 0.29/0.47 % list.map_comp
% 0.29/0.47 thf(fact_21_list_Omap__comp,axiom,
% 0.29/0.47 ! [G: nat > a,F: nat > nat,V: list_nat] :
% 0.29/0.47 ( ( map_nat_a @ G @ ( map_nat_nat @ F @ V ) )
% 0.29/0.47 = ( map_nat_a @ ( comp_nat_a_nat @ G @ F ) @ V ) ) ).
% 0.29/0.47
% 0.29/0.47 % list.map_comp
% 0.29/0.47 thf(fact_22_List_Omap_Ocompositionality,axiom,
% 0.29/0.47 ! [F: a > a,G: nat > a,List: list_nat] :
% 0.29/0.47 ( ( map_a_a @ F @ ( map_nat_a @ G @ List ) )
% 0.29/0.47 = ( map_nat_a @ ( comp_a_a_nat @ F @ G ) @ List ) ) ).
% 0.29/0.47
% 0.29/0.47 % List.map.compositionality
% 0.29/0.47 thf(fact_23_List_Omap_Ocompositionality,axiom,
% 0.29/0.47 ! [F: nat > a,G: nat > nat,List: list_nat] :
% 0.29/0.47 ( ( map_nat_a @ F @ ( map_nat_nat @ G @ List ) )
% 0.29/0.47 = ( map_nat_a @ ( comp_nat_a_nat @ F @ G ) @ List ) ) ).
% 0.29/0.47
% 0.29/0.47 % List.map.compositionality
% 0.29/0.47 thf(fact_24_length__map,axiom,
% 0.29/0.47 ! [F: nat > a,Xs2: list_nat] :
% 0.29/0.47 ( ( size_size_list_a @ ( map_nat_a @ F @ Xs2 ) )
% 0.29/0.47 = ( size_size_list_nat @ Xs2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % length_map
% 0.29/0.47 thf(fact_25_length__map,axiom,
% 0.29/0.47 ! [F: a > a,Xs2: list_a] :
% 0.29/0.47 ( ( size_size_list_a @ ( map_a_a @ F @ Xs2 ) )
% 0.29/0.47 = ( size_size_list_a @ Xs2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % length_map
% 0.29/0.47 thf(fact_26_nth__map,axiom,
% 0.29/0.47 ! [N: nat,Xs2: list_nat,F: nat > a] :
% 0.29/0.47 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
% 0.29/0.47 => ( ( nth_a @ ( map_nat_a @ F @ Xs2 ) @ N )
% 0.29/0.47 = ( F @ ( nth_nat @ Xs2 @ N ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % nth_map
% 0.29/0.47 thf(fact_27_nth__map,axiom,
% 0.29/0.47 ! [N: nat,Xs2: list_a,F: a > a] :
% 0.29/0.47 ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs2 ) )
% 0.29/0.47 => ( ( nth_a @ ( map_a_a @ F @ Xs2 ) @ N )
% 0.29/0.47 = ( F @ ( nth_a @ Xs2 @ N ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % nth_map
% 0.29/0.47 thf(fact_28__092_060open_062mset_Axs_A_061_Amset_A_Imap_A_I_I_B_J_Ays_A_092_060circ_062_Af_J_A_0910_O_O_060length_Axs_093_J_092_060close_062,axiom,
% 0.29/0.47 ( ( mset_a @ xs )
% 0.29/0.47 = ( mset_a @ ( map_nat_a @ ( comp_nat_a_nat @ ( nth_a @ ys ) @ f ) @ ( upt @ zero_zero_nat @ ( size_size_list_a @ xs ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % \<open>mset xs = mset (map ((!) ys \<circ> f) [0..<length xs])\<close>
% 0.29/0.47 thf(fact_29_nat__neq__iff,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( M != N )
% 0.29/0.47 = ( ( ord_less_nat @ M @ N )
% 0.29/0.47 | ( ord_less_nat @ N @ M ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % nat_neq_iff
% 0.29/0.47 thf(fact_30_less__not__refl,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ~ ( ord_less_nat @ N @ N ) ).
% 0.29/0.47
% 0.29/0.47 % less_not_refl
% 0.29/0.47 thf(fact_31_less__not__refl2,axiom,
% 0.29/0.47 ! [N: nat,M: nat] :
% 0.29/0.47 ( ( ord_less_nat @ N @ M )
% 0.29/0.47 => ( M != N ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_not_refl2
% 0.29/0.47 thf(fact_32_less__not__refl3,axiom,
% 0.29/0.47 ! [S: nat,T: nat] :
% 0.29/0.47 ( ( ord_less_nat @ S @ T )
% 0.29/0.47 => ( S != T ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_not_refl3
% 0.29/0.47 thf(fact_33_less__irrefl__nat,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ~ ( ord_less_nat @ N @ N ) ).
% 0.29/0.47
% 0.29/0.47 % less_irrefl_nat
% 0.29/0.47 thf(fact_34_nat__less__induct,axiom,
% 0.29/0.47 ! [P: nat > $o,N: nat] :
% 0.29/0.47 ( ! [N2: nat] :
% 0.29/0.47 ( ! [M2: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M2 @ N2 )
% 0.29/0.47 => ( P @ M2 ) )
% 0.29/0.47 => ( P @ N2 ) )
% 0.29/0.47 => ( P @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % nat_less_induct
% 0.29/0.47 thf(fact_35_infinite__descent,axiom,
% 0.29/0.47 ! [P: nat > $o,N: nat] :
% 0.29/0.47 ( ! [N2: nat] :
% 0.29/0.47 ( ~ ( P @ N2 )
% 0.29/0.47 => ? [M2: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M2 @ N2 )
% 0.29/0.47 & ~ ( P @ M2 ) ) )
% 0.29/0.47 => ( P @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % infinite_descent
% 0.29/0.47 thf(fact_36_linorder__neqE__nat,axiom,
% 0.29/0.47 ! [X: nat,Y: nat] :
% 0.29/0.47 ( ( X != Y )
% 0.29/0.47 => ( ~ ( ord_less_nat @ X @ Y )
% 0.29/0.47 => ( ord_less_nat @ Y @ X ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % linorder_neqE_nat
% 0.29/0.47 thf(fact_37_map__eq__imp__length__eq,axiom,
% 0.29/0.47 ! [F: nat > a,Xs2: list_nat,G: nat > a,Ys: list_nat] :
% 0.29/0.47 ( ( ( map_nat_a @ F @ Xs2 )
% 0.29/0.47 = ( map_nat_a @ G @ Ys ) )
% 0.29/0.47 => ( ( size_size_list_nat @ Xs2 )
% 0.29/0.47 = ( size_size_list_nat @ Ys ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % map_eq_imp_length_eq
% 0.29/0.47 thf(fact_38_map__eq__imp__length__eq,axiom,
% 0.29/0.47 ! [F: nat > a,Xs2: list_nat,G: a > a,Ys: list_a] :
% 0.29/0.47 ( ( ( map_nat_a @ F @ Xs2 )
% 0.29/0.47 = ( map_a_a @ G @ Ys ) )
% 0.29/0.47 => ( ( size_size_list_nat @ Xs2 )
% 0.29/0.47 = ( size_size_list_a @ Ys ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % map_eq_imp_length_eq
% 0.29/0.47 thf(fact_39_map__eq__imp__length__eq,axiom,
% 0.29/0.47 ! [F: a > a,Xs2: list_a,G: nat > a,Ys: list_nat] :
% 0.29/0.47 ( ( ( map_a_a @ F @ Xs2 )
% 0.29/0.47 = ( map_nat_a @ G @ Ys ) )
% 0.29/0.47 => ( ( size_size_list_a @ Xs2 )
% 0.29/0.47 = ( size_size_list_nat @ Ys ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % map_eq_imp_length_eq
% 0.29/0.47 thf(fact_40_zero__less__iff__neq__zero,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.47 = ( N != zero_zero_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % zero_less_iff_neq_zero
% 0.29/0.47 thf(fact_41_gr__implies__not__zero,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M @ N )
% 0.29/0.47 => ( N != zero_zero_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % gr_implies_not_zero
% 0.29/0.47 thf(fact_42_not__less__zero,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 % not_less_zero
% 0.29/0.47 thf(fact_43_gr__zeroI,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( N != zero_zero_nat )
% 0.29/0.47 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % gr_zeroI
% 0.29/0.47 thf(fact_44_bot__nat__0_Oextremum__strict,axiom,
% 0.29/0.47 ! [A: nat] :
% 0.29/0.47 ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 % bot_nat_0.extremum_strict
% 0.29/0.47 thf(fact_45_infinite__descent0,axiom,
% 0.29/0.47 ! [P: nat > $o,N: nat] :
% 0.29/0.47 ( ( P @ zero_zero_nat )
% 0.29/0.47 => ( ! [N2: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.47 => ( ~ ( P @ N2 )
% 0.29/0.47 => ? [M2: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M2 @ N2 )
% 0.29/0.47 & ~ ( P @ M2 ) ) ) )
% 0.29/0.47 => ( P @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % infinite_descent0
% 0.29/0.47 thf(fact_46_gr__implies__not0,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M @ N )
% 0.29/0.47 => ( N != zero_zero_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % gr_implies_not0
% 0.29/0.47 thf(fact_47_less__zeroE,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 % less_zeroE
% 0.29/0.47 thf(fact_48_not__less0,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 % not_less0
% 0.29/0.47 thf(fact_49_not__gr0,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.29/0.47 = ( N = zero_zero_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % not_gr0
% 0.29/0.47 thf(fact_50_gr0I,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( N != zero_zero_nat )
% 0.29/0.47 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % gr0I
% 0.29/0.47 thf(fact_51_length__induct,axiom,
% 0.29/0.47 ! [P: list_a > $o,Xs2: list_a] :
% 0.29/0.47 ( ! [Xs: list_a] :
% 0.29/0.47 ( ! [Ys2: list_a] :
% 0.29/0.47 ( ( ord_less_nat @ ( size_size_list_a @ Ys2 ) @ ( size_size_list_a @ Xs ) )
% 0.29/0.47 => ( P @ Ys2 ) )
% 0.29/0.47 => ( P @ Xs ) )
% 0.29/0.47 => ( P @ Xs2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % length_induct
% 0.29/0.47 thf(fact_52_all__nat__less__eq,axiom,
% 0.29/0.47 ! [N: nat,P: nat > $o] :
% 0.29/0.47 ( ( ! [M3: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M3 @ N )
% 0.29/0.47 => ( P @ M3 ) ) )
% 0.29/0.47 = ( ! [X2: nat] :
% 0.29/0.47 ( ( member_nat @ X2 @ ( set_or562006527an_nat @ zero_zero_nat @ N ) )
% 0.29/0.47 => ( P @ X2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % all_nat_less_eq
% 0.29/0.47 thf(fact_53_ex__nat__less__eq,axiom,
% 0.29/0.47 ! [N: nat,P: nat > $o] :
% 0.29/0.47 ( ( ? [M3: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M3 @ N )
% 0.29/0.47 & ( P @ M3 ) ) )
% 0.29/0.47 = ( ? [X2: nat] :
% 0.29/0.47 ( ( member_nat @ X2 @ ( set_or562006527an_nat @ zero_zero_nat @ N ) )
% 0.29/0.47 & ( P @ X2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ex_nat_less_eq
% 0.29/0.47 thf(fact_54_list__eq__iff__nth__eq,axiom,
% 0.29/0.47 ( ( ^ [Y2: list_a,Z: list_a] : ( Y2 = Z ) )
% 0.29/0.47 = ( ^ [Xs3: list_a,Ys3: list_a] :
% 0.29/0.47 ( ( ( size_size_list_a @ Xs3 )
% 0.29/0.47 = ( size_size_list_a @ Ys3 ) )
% 0.29/0.47 & ! [I2: nat] :
% 0.29/0.47 ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs3 ) )
% 0.29/0.47 => ( ( nth_a @ Xs3 @ I2 )
% 0.29/0.47 = ( nth_a @ Ys3 @ I2 ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % list_eq_iff_nth_eq
% 0.29/0.47 thf(fact_55_Skolem__list__nth,axiom,
% 0.29/0.47 ! [K: nat,P: nat > a > $o] :
% 0.29/0.47 ( ( ! [I2: nat] :
% 0.29/0.47 ( ( ord_less_nat @ I2 @ K )
% 0.29/0.47 => ? [X3: a] : ( P @ I2 @ X3 ) ) )
% 0.29/0.47 = ( ? [Xs3: list_a] :
% 0.29/0.47 ( ( ( size_size_list_a @ Xs3 )
% 0.29/0.47 = K )
% 0.29/0.47 & ! [I2: nat] :
% 0.29/0.47 ( ( ord_less_nat @ I2 @ K )
% 0.29/0.47 => ( P @ I2 @ ( nth_a @ Xs3 @ I2 ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Skolem_list_nth
% 0.29/0.47 thf(fact_56_nth__equalityI,axiom,
% 0.29/0.47 ! [Xs2: list_a,Ys: list_a] :
% 0.29/0.47 ( ( ( size_size_list_a @ Xs2 )
% 0.29/0.47 = ( size_size_list_a @ Ys ) )
% 0.29/0.47 => ( ! [I3: nat] :
% 0.29/0.47 ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs2 ) )
% 0.29/0.47 => ( ( nth_a @ Xs2 @ I3 )
% 0.29/0.47 = ( nth_a @ Ys @ I3 ) ) )
% 0.29/0.47 => ( Xs2 = Ys ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % nth_equalityI
% 0.29/0.47 thf(fact_57_zero__reorient,axiom,
% 0.29/0.47 ! [X: nat] :
% 0.29/0.47 ( ( zero_zero_nat = X )
% 0.29/0.47 = ( X = zero_zero_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % zero_reorient
% 0.29/0.47 thf(fact_58_zero__reorient,axiom,
% 0.29/0.47 ! [X: int] :
% 0.29/0.47 ( ( zero_zero_int = X )
% 0.29/0.47 = ( X = zero_zero_int ) ) ).
% 0.29/0.47
% 0.29/0.47 % zero_reorient
% 0.29/0.47 thf(fact_59_zero__reorient,axiom,
% 0.29/0.47 ! [X: real] :
% 0.29/0.47 ( ( zero_zero_real = X )
% 0.29/0.47 = ( X = zero_zero_real ) ) ).
% 0.29/0.47
% 0.29/0.47 % zero_reorient
% 0.29/0.47 thf(fact_60_mset__map,axiom,
% 0.29/0.47 ! [F: a > a,Xs2: list_a] :
% 0.29/0.47 ( ( mset_a @ ( map_a_a @ F @ Xs2 ) )
% 0.29/0.47 = ( image_mset_a_a @ F @ ( mset_a @ Xs2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mset_map
% 0.29/0.47 thf(fact_61_mset__map,axiom,
% 0.29/0.47 ! [F: a > nat,Xs2: list_a] :
% 0.29/0.47 ( ( mset_nat @ ( map_a_nat @ F @ Xs2 ) )
% 0.29/0.47 = ( image_mset_a_nat @ F @ ( mset_a @ Xs2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mset_map
% 0.29/0.47 thf(fact_62_mset__map,axiom,
% 0.29/0.47 ! [F: nat > a,Xs2: list_nat] :
% 0.29/0.47 ( ( mset_a @ ( map_nat_a @ F @ Xs2 ) )
% 0.29/0.47 = ( image_mset_nat_a @ F @ ( mset_nat @ Xs2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mset_map
% 0.29/0.47 thf(fact_63_mset__map,axiom,
% 0.29/0.47 ! [F: nat > nat,Xs2: list_nat] :
% 0.29/0.47 ( ( mset_nat @ ( map_nat_nat @ F @ Xs2 ) )
% 0.29/0.47 = ( image_mset_nat_nat @ F @ ( mset_nat @ Xs2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mset_map
% 0.29/0.47 thf(fact_64_size__mset,axiom,
% 0.29/0.47 ! [Xs2: list_nat] :
% 0.29/0.47 ( ( size_s1679505588et_nat @ ( mset_nat @ Xs2 ) )
% 0.29/0.47 = ( size_size_list_nat @ Xs2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % size_mset
% 0.29/0.47 thf(fact_65_size__mset,axiom,
% 0.29/0.47 ! [Xs2: list_a] :
% 0.29/0.47 ( ( size_size_multiset_a @ ( mset_a @ Xs2 ) )
% 0.29/0.47 = ( size_size_list_a @ Xs2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % size_mset
% 0.29/0.47 thf(fact_66_lessThan__iff,axiom,
% 0.29/0.47 ! [I: int,K: int] :
% 0.29/0.47 ( ( member_int @ I @ ( set_ord_lessThan_int @ K ) )
% 0.29/0.47 = ( ord_less_int @ I @ K ) ) ).
% 0.29/0.47
% 0.29/0.47 % lessThan_iff
% 0.29/0.47 thf(fact_67_lessThan__iff,axiom,
% 0.29/0.47 ! [I: real,K: real] :
% 0.29/0.47 ( ( member_real @ I @ ( set_or1211449801n_real @ K ) )
% 0.29/0.47 = ( ord_less_real @ I @ K ) ) ).
% 0.29/0.47
% 0.29/0.47 % lessThan_iff
% 0.29/0.47 thf(fact_68_lessThan__iff,axiom,
% 0.29/0.47 ! [I: nat,K: nat] :
% 0.29/0.47 ( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
% 0.29/0.47 = ( ord_less_nat @ I @ K ) ) ).
% 0.29/0.47
% 0.29/0.47 % lessThan_iff
% 0.29/0.47 thf(fact_69_image__mset__mset__set,axiom,
% 0.29/0.47 ! [F: nat > int,A2: set_nat] :
% 0.29/0.47 ( ( inj_on_nat_int @ F @ A2 )
% 0.29/0.47 => ( ( image_mset_nat_int @ F @ ( mset_set_nat @ A2 ) )
% 0.29/0.47 = ( mset_set_int @ ( image_nat_int @ F @ A2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % image_mset_mset_set
% 0.29/0.47 thf(fact_70_image__mset__mset__set,axiom,
% 0.29/0.47 ! [F: nat > a,A2: set_nat] :
% 0.29/0.47 ( ( inj_on_nat_a @ F @ A2 )
% 0.29/0.47 => ( ( image_mset_nat_a @ F @ ( mset_set_nat @ A2 ) )
% 0.29/0.47 = ( mset_set_a @ ( image_nat_a @ F @ A2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % image_mset_mset_set
% 0.29/0.47 thf(fact_71_image__mset__mset__set,axiom,
% 0.29/0.47 ! [F: nat > nat,A2: set_nat] :
% 0.29/0.47 ( ( inj_on_nat_nat @ F @ A2 )
% 0.29/0.47 => ( ( image_mset_nat_nat @ F @ ( mset_set_nat @ A2 ) )
% 0.29/0.47 = ( mset_set_nat @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % image_mset_mset_set
% 0.29/0.47 thf(fact_72_lessThan__atLeast0,axiom,
% 0.29/0.47 ( set_ord_lessThan_nat
% 0.29/0.47 = ( set_or562006527an_nat @ zero_zero_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % lessThan_atLeast0
% 0.29/0.47 thf(fact_73_bij__betw__def,axiom,
% 0.29/0.47 ( bij_betw_nat_int
% 0.29/0.47 = ( ^ [F2: nat > int,A3: set_nat,B: set_int] :
% 0.29/0.47 ( ( inj_on_nat_int @ F2 @ A3 )
% 0.29/0.47 & ( ( image_nat_int @ F2 @ A3 )
% 0.29/0.47 = B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % bij_betw_def
% 0.29/0.47 thf(fact_74_bij__betw__def,axiom,
% 0.29/0.47 ( bij_betw_nat_nat
% 0.29/0.47 = ( ^ [F2: nat > nat,A3: set_nat,B: set_nat] :
% 0.29/0.47 ( ( inj_on_nat_nat @ F2 @ A3 )
% 0.29/0.47 & ( ( image_nat_nat @ F2 @ A3 )
% 0.29/0.47 = B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % bij_betw_def
% 0.29/0.47 thf(fact_75_bij__betw__imageI,axiom,
% 0.29/0.47 ! [F: nat > int,A2: set_nat,B2: set_int] :
% 0.29/0.47 ( ( inj_on_nat_int @ F @ A2 )
% 0.29/0.47 => ( ( ( image_nat_int @ F @ A2 )
% 0.29/0.47 = B2 )
% 0.29/0.47 => ( bij_betw_nat_int @ F @ A2 @ B2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % bij_betw_imageI
% 0.29/0.47 thf(fact_76_bij__betw__imageI,axiom,
% 0.29/0.47 ! [F: nat > nat,A2: set_nat,B2: set_nat] :
% 0.29/0.47 ( ( inj_on_nat_nat @ F @ A2 )
% 0.29/0.47 => ( ( ( image_nat_nat @ F @ A2 )
% 0.29/0.47 = B2 )
% 0.29/0.47 => ( bij_betw_nat_nat @ F @ A2 @ B2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % bij_betw_imageI
% 0.29/0.47 thf(fact_77_inj__on__imp__bij__betw,axiom,
% 0.29/0.47 ! [F: nat > int,A2: set_nat] :
% 0.29/0.47 ( ( inj_on_nat_int @ F @ A2 )
% 0.29/0.47 => ( bij_betw_nat_int @ F @ A2 @ ( image_nat_int @ F @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inj_on_imp_bij_betw
% 0.29/0.47 thf(fact_78_inj__on__imp__bij__betw,axiom,
% 0.29/0.47 ! [F: nat > nat,A2: set_nat] :
% 0.29/0.47 ( ( inj_on_nat_nat @ F @ A2 )
% 0.29/0.47 => ( bij_betw_nat_nat @ F @ A2 @ ( image_nat_nat @ F @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inj_on_imp_bij_betw
% 0.29/0.47 thf(fact_79_comp__inj__on,axiom,
% 0.29/0.47 ! [F: nat > int,A2: set_nat,G: int > nat] :
% 0.29/0.47 ( ( inj_on_nat_int @ F @ A2 )
% 0.29/0.47 => ( ( inj_on_int_nat @ G @ ( image_nat_int @ F @ A2 ) )
% 0.29/0.47 => ( inj_on_nat_nat @ ( comp_int_nat_nat @ G @ F ) @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % comp_inj_on
% 0.29/0.47 thf(fact_80_comp__inj__on,axiom,
% 0.29/0.47 ! [F: nat > nat,A2: set_nat,G: nat > a] :
% 0.29/0.47 ( ( inj_on_nat_nat @ F @ A2 )
% 0.29/0.47 => ( ( inj_on_nat_a @ G @ ( image_nat_nat @ F @ A2 ) )
% 0.29/0.47 => ( inj_on_nat_a @ ( comp_nat_a_nat @ G @ F ) @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % comp_inj_on
% 0.29/0.47 thf(fact_81_comp__inj__on,axiom,
% 0.29/0.47 ! [F: nat > nat,A2: set_nat,G: nat > nat] :
% 0.29/0.47 ( ( inj_on_nat_nat @ F @ A2 )
% 0.29/0.47 => ( ( inj_on_nat_nat @ G @ ( image_nat_nat @ F @ A2 ) )
% 0.29/0.47 => ( inj_on_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % comp_inj_on
% 0.29/0.47 thf(fact_82_inj__on__imageI,axiom,
% 0.29/0.47 ! [G: nat > a,F: nat > nat,A2: set_nat] :
% 0.29/0.47 ( ( inj_on_nat_a @ ( comp_nat_a_nat @ G @ F ) @ A2 )
% 0.29/0.47 => ( inj_on_nat_a @ G @ ( image_nat_nat @ F @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inj_on_imageI
% 0.29/0.47 thf(fact_83_inj__on__imageI,axiom,
% 0.29/0.47 ! [G: int > nat,F: nat > int,A2: set_nat] :
% 0.29/0.47 ( ( inj_on_nat_nat @ ( comp_int_nat_nat @ G @ F ) @ A2 )
% 0.29/0.47 => ( inj_on_int_nat @ G @ ( image_nat_int @ F @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inj_on_imageI
% 0.29/0.47 thf(fact_84_inj__on__imageI,axiom,
% 0.29/0.47 ! [G: nat > nat,F: nat > nat,A2: set_nat] :
% 0.29/0.47 ( ( inj_on_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A2 )
% 0.29/0.47 => ( inj_on_nat_nat @ G @ ( image_nat_nat @ F @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inj_on_imageI
% 0.29/0.47 thf(fact_85_comp__inj__on__iff,axiom,
% 0.29/0.47 ! [F: nat > int,A2: set_nat,F3: int > nat] :
% 0.29/0.47 ( ( inj_on_nat_int @ F @ A2 )
% 0.29/0.47 => ( ( inj_on_int_nat @ F3 @ ( image_nat_int @ F @ A2 ) )
% 0.29/0.47 = ( inj_on_nat_nat @ ( comp_int_nat_nat @ F3 @ F ) @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % comp_inj_on_iff
% 0.29/0.47 thf(fact_86_comp__inj__on__iff,axiom,
% 0.29/0.47 ! [F: nat > nat,A2: set_nat,F3: nat > a] :
% 0.29/0.47 ( ( inj_on_nat_nat @ F @ A2 )
% 0.29/0.47 => ( ( inj_on_nat_a @ F3 @ ( image_nat_nat @ F @ A2 ) )
% 0.29/0.47 = ( inj_on_nat_a @ ( comp_nat_a_nat @ F3 @ F ) @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % comp_inj_on_iff
% 0.29/0.47 thf(fact_87_comp__inj__on__iff,axiom,
% 0.29/0.47 ! [F: nat > nat,A2: set_nat,F3: nat > nat] :
% 0.29/0.47 ( ( inj_on_nat_nat @ F @ A2 )
% 0.29/0.47 => ( ( inj_on_nat_nat @ F3 @ ( image_nat_nat @ F @ A2 ) )
% 0.29/0.47 = ( inj_on_nat_nat @ ( comp_nat_nat_nat @ F3 @ F ) @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % comp_inj_on_iff
% 0.29/0.47 thf(fact_88_comp__apply,axiom,
% 0.29/0.47 ( comp_nat_a_nat
% 0.29/0.47 = ( ^ [F2: nat > a,G2: nat > nat,X2: nat] : ( F2 @ ( G2 @ X2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % comp_apply
% 0.29/0.47 thf(fact_89_lessThan__eq__iff,axiom,
% 0.29/0.47 ! [X: nat,Y: nat] :
% 0.29/0.47 ( ( ( set_ord_lessThan_nat @ X )
% 0.29/0.47 = ( set_ord_lessThan_nat @ Y ) )
% 0.29/0.47 = ( X = Y ) ) ).
% 0.29/0.47
% 0.29/0.47 % lessThan_eq_iff
% 0.29/0.47 thf(fact_90_image__mset__empty,axiom,
% 0.29/0.47 ! [F: nat > a] :
% 0.29/0.47 ( ( image_mset_nat_a @ F @ zero_z2005226153et_nat )
% 0.29/0.47 = zero_zero_multiset_a ) ).
% 0.29/0.47
% 0.29/0.47 % image_mset_empty
% 0.29/0.47 thf(fact_91_image__mset__empty,axiom,
% 0.29/0.47 ! [F: nat > nat] :
% 0.29/0.47 ( ( image_mset_nat_nat @ F @ zero_z2005226153et_nat )
% 0.29/0.47 = zero_z2005226153et_nat ) ).
% 0.29/0.47
% 0.29/0.47 % image_mset_empty
% 0.29/0.47 thf(fact_92_image__mset__is__empty__iff,axiom,
% 0.29/0.47 ! [F: nat > a,M4: multiset_nat] :
% 0.29/0.47 ( ( ( image_mset_nat_a @ F @ M4 )
% 0.29/0.47 = zero_zero_multiset_a )
% 0.29/0.47 = ( M4 = zero_z2005226153et_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % image_mset_is_empty_iff
% 0.29/0.47 thf(fact_93_image__mset__is__empty__iff,axiom,
% 0.29/0.47 ! [F: nat > nat,M4: multiset_nat] :
% 0.29/0.47 ( ( ( image_mset_nat_nat @ F @ M4 )
% 0.29/0.47 = zero_z2005226153et_nat )
% 0.29/0.47 = ( M4 = zero_z2005226153et_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % image_mset_is_empty_iff
% 0.29/0.47 thf(fact_94_size__image__mset,axiom,
% 0.29/0.47 ! [F: nat > a,M4: multiset_nat] :
% 0.29/0.47 ( ( size_size_multiset_a @ ( image_mset_nat_a @ F @ M4 ) )
% 0.29/0.47 = ( size_s1679505588et_nat @ M4 ) ) ).
% 0.29/0.47
% 0.29/0.47 % size_image_mset
% 0.29/0.47 thf(fact_95_size__image__mset,axiom,
% 0.29/0.47 ! [F: nat > nat,M4: multiset_nat] :
% 0.29/0.47 ( ( size_s1679505588et_nat @ ( image_mset_nat_nat @ F @ M4 ) )
% 0.29/0.47 = ( size_s1679505588et_nat @ M4 ) ) ).
% 0.29/0.47
% 0.29/0.47 % size_image_mset
% 0.29/0.47 thf(fact_96_mset__upt,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( mset_nat @ ( upt @ M @ N ) )
% 0.29/0.47 = ( mset_set_nat @ ( set_or562006527an_nat @ M @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mset_upt
% 0.29/0.47 thf(fact_97_inj__on__strict__subset,axiom,
% 0.29/0.47 ! [F: nat > int,B2: set_nat,A2: set_nat] :
% 0.29/0.47 ( ( inj_on_nat_int @ F @ B2 )
% 0.29/0.47 => ( ( ord_less_set_nat @ A2 @ B2 )
% 0.29/0.47 => ( ord_less_set_int @ ( image_nat_int @ F @ A2 ) @ ( image_nat_int @ F @ B2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inj_on_strict_subset
% 0.29/0.47 thf(fact_98_inj__on__strict__subset,axiom,
% 0.29/0.47 ! [F: nat > nat,B2: set_nat,A2: set_nat] :
% 0.29/0.47 ( ( inj_on_nat_nat @ F @ B2 )
% 0.29/0.47 => ( ( ord_less_set_nat @ A2 @ B2 )
% 0.29/0.47 => ( ord_less_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inj_on_strict_subset
% 0.29/0.47 thf(fact_99_comp__eq__dest__lhs,axiom,
% 0.29/0.47 ! [A: nat > a,B3: nat > nat,C: nat > a,V: nat] :
% 0.29/0.47 ( ( ( comp_nat_a_nat @ A @ B3 )
% 0.29/0.47 = C )
% 0.29/0.47 => ( ( A @ ( B3 @ V ) )
% 0.29/0.47 = ( C @ V ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % comp_eq_dest_lhs
% 0.29/0.47 thf(fact_100_comp__eq__elim,axiom,
% 0.29/0.47 ! [A: nat > a,B3: nat > nat,C: nat > a,D: nat > nat] :
% 0.29/0.47 ( ( ( comp_nat_a_nat @ A @ B3 )
% 0.29/0.47 = ( comp_nat_a_nat @ C @ D ) )
% 0.29/0.47 => ! [V2: nat] :
% 0.29/0.47 ( ( A @ ( B3 @ V2 ) )
% 0.29/0.47 = ( C @ ( D @ V2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % comp_eq_elim
% 0.29/0.47 thf(fact_101_comp__eq__dest,axiom,
% 0.29/0.47 ! [A: nat > a,B3: nat > nat,C: nat > a,D: nat > nat,V: nat] :
% 0.29/0.47 ( ( ( comp_nat_a_nat @ A @ B3 )
% 0.29/0.47 = ( comp_nat_a_nat @ C @ D ) )
% 0.29/0.47 => ( ( A @ ( B3 @ V ) )
% 0.29/0.47 = ( C @ ( D @ V ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % comp_eq_dest
% 0.29/0.47 thf(fact_102_comp__assoc,axiom,
% 0.29/0.47 ! [F: a > a,G: nat > a,H: nat > nat] :
% 0.29/0.47 ( ( comp_nat_a_nat @ ( comp_a_a_nat @ F @ G ) @ H )
% 0.29/0.47 = ( comp_a_a_nat @ F @ ( comp_nat_a_nat @ G @ H ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % comp_assoc
% 0.29/0.47 thf(fact_103_comp__assoc,axiom,
% 0.29/0.47 ! [F: nat > a,G: nat > nat,H: nat > nat] :
% 0.29/0.47 ( ( comp_nat_a_nat @ ( comp_nat_a_nat @ F @ G ) @ H )
% 0.29/0.47 = ( comp_nat_a_nat @ F @ ( comp_nat_nat_nat @ G @ H ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % comp_assoc
% 0.29/0.47 thf(fact_104_comp__def,axiom,
% 0.29/0.47 ( comp_nat_a_nat
% 0.29/0.47 = ( ^ [F2: nat > a,G2: nat > nat,X2: nat] : ( F2 @ ( G2 @ X2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % comp_def
% 0.29/0.47 thf(fact_105_inj__on__inverseI,axiom,
% 0.29/0.47 ! [A2: set_nat,G: nat > nat,F: nat > nat] :
% 0.29/0.47 ( ! [X4: nat] :
% 0.29/0.47 ( ( member_nat @ X4 @ A2 )
% 0.29/0.47 => ( ( G @ ( F @ X4 ) )
% 0.29/0.47 = X4 ) )
% 0.29/0.47 => ( inj_on_nat_nat @ F @ A2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % inj_on_inverseI
% 0.29/0.47 thf(fact_106_inj__on__contraD,axiom,
% 0.29/0.47 ! [F: nat > nat,A2: set_nat,X: nat,Y: nat] :
% 0.29/0.47 ( ( inj_on_nat_nat @ F @ A2 )
% 0.29/0.47 => ( ( X != Y )
% 0.29/0.47 => ( ( member_nat @ X @ A2 )
% 0.29/0.47 => ( ( member_nat @ Y @ A2 )
% 0.29/0.47 => ( ( F @ X )
% 0.29/0.47 != ( F @ Y ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inj_on_contraD
% 0.29/0.47 thf(fact_107_inj__on__eq__iff,axiom,
% 0.29/0.47 ! [F: nat > nat,A2: set_nat,X: nat,Y: nat] :
% 0.29/0.47 ( ( inj_on_nat_nat @ F @ A2 )
% 0.29/0.47 => ( ( member_nat @ X @ A2 )
% 0.29/0.47 => ( ( member_nat @ Y @ A2 )
% 0.29/0.47 => ( ( ( F @ X )
% 0.29/0.47 = ( F @ Y ) )
% 0.29/0.47 = ( X = Y ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inj_on_eq_iff
% 0.29/0.47 thf(fact_108_inj__on__cong,axiom,
% 0.29/0.47 ! [A2: set_nat,F: nat > nat,G: nat > nat] :
% 0.29/0.47 ( ! [A4: nat] :
% 0.29/0.47 ( ( member_nat @ A4 @ A2 )
% 0.29/0.47 => ( ( F @ A4 )
% 0.29/0.47 = ( G @ A4 ) ) )
% 0.29/0.47 => ( ( inj_on_nat_nat @ F @ A2 )
% 0.29/0.47 = ( inj_on_nat_nat @ G @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inj_on_cong
% 0.29/0.47 thf(fact_109_inj__on__def,axiom,
% 0.29/0.47 ( inj_on_nat_nat
% 0.29/0.47 = ( ^ [F2: nat > nat,A3: set_nat] :
% 0.29/0.47 ! [X2: nat] :
% 0.29/0.47 ( ( member_nat @ X2 @ A3 )
% 0.29/0.47 => ! [Y3: nat] :
% 0.29/0.47 ( ( member_nat @ Y3 @ A3 )
% 0.29/0.47 => ( ( ( F2 @ X2 )
% 0.29/0.47 = ( F2 @ Y3 ) )
% 0.29/0.47 => ( X2 = Y3 ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inj_on_def
% 0.29/0.47 thf(fact_110_inj__onI,axiom,
% 0.29/0.47 ! [A2: set_nat,F: nat > nat] :
% 0.29/0.47 ( ! [X4: nat,Y4: nat] :
% 0.29/0.47 ( ( member_nat @ X4 @ A2 )
% 0.29/0.47 => ( ( member_nat @ Y4 @ A2 )
% 0.29/0.47 => ( ( ( F @ X4 )
% 0.29/0.47 = ( F @ Y4 ) )
% 0.29/0.47 => ( X4 = Y4 ) ) ) )
% 0.29/0.47 => ( inj_on_nat_nat @ F @ A2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % inj_onI
% 0.29/0.47 thf(fact_111_inj__onD,axiom,
% 0.29/0.47 ! [F: nat > nat,A2: set_nat,X: nat,Y: nat] :
% 0.29/0.47 ( ( inj_on_nat_nat @ F @ A2 )
% 0.29/0.47 => ( ( ( F @ X )
% 0.29/0.47 = ( F @ Y ) )
% 0.29/0.47 => ( ( member_nat @ X @ A2 )
% 0.29/0.47 => ( ( member_nat @ Y @ A2 )
% 0.29/0.47 => ( X = Y ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inj_onD
% 0.29/0.47 thf(fact_112_bij__betw__iff__bijections,axiom,
% 0.29/0.47 ( bij_betw_nat_nat
% 0.29/0.47 = ( ^ [F2: nat > nat,A3: set_nat,B: set_nat] :
% 0.29/0.47 ? [G2: nat > nat] :
% 0.29/0.47 ( ! [X2: nat] :
% 0.29/0.47 ( ( member_nat @ X2 @ A3 )
% 0.29/0.47 => ( ( member_nat @ ( F2 @ X2 ) @ B )
% 0.29/0.47 & ( ( G2 @ ( F2 @ X2 ) )
% 0.29/0.47 = X2 ) ) )
% 0.29/0.47 & ! [X2: nat] :
% 0.29/0.47 ( ( member_nat @ X2 @ B )
% 0.29/0.47 => ( ( member_nat @ ( G2 @ X2 ) @ A3 )
% 0.29/0.47 & ( ( F2 @ ( G2 @ X2 ) )
% 0.29/0.47 = X2 ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % bij_betw_iff_bijections
% 0.29/0.47 thf(fact_113_bij__betw__apply,axiom,
% 0.29/0.47 ! [F: nat > nat,A2: set_nat,B2: set_nat,A: nat] :
% 0.29/0.47 ( ( bij_betw_nat_nat @ F @ A2 @ B2 )
% 0.29/0.47 => ( ( member_nat @ A @ A2 )
% 0.29/0.47 => ( member_nat @ ( F @ A ) @ B2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % bij_betw_apply
% 0.29/0.47 thf(fact_114_bij__betw__cong,axiom,
% 0.29/0.47 ! [A2: set_nat,F: nat > nat,G: nat > nat,A5: set_nat] :
% 0.29/0.47 ( ! [A4: nat] :
% 0.29/0.47 ( ( member_nat @ A4 @ A2 )
% 0.29/0.47 => ( ( F @ A4 )
% 0.29/0.47 = ( G @ A4 ) ) )
% 0.29/0.47 => ( ( bij_betw_nat_nat @ F @ A2 @ A5 )
% 0.29/0.47 = ( bij_betw_nat_nat @ G @ A2 @ A5 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % bij_betw_cong
% 0.29/0.47 thf(fact_115_bij__betw__inv,axiom,
% 0.29/0.47 ! [F: nat > nat,A2: set_nat,B2: set_nat] :
% 0.29/0.47 ( ( bij_betw_nat_nat @ F @ A2 @ B2 )
% 0.29/0.47 => ? [G3: nat > nat] : ( bij_betw_nat_nat @ G3 @ B2 @ A2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % bij_betw_inv
% 0.29/0.47 thf(fact_116_bij__betwE,axiom,
% 0.29/0.47 ! [F: nat > nat,A2: set_nat,B2: set_nat] :
% 0.29/0.47 ( ( bij_betw_nat_nat @ F @ A2 @ B2 )
% 0.29/0.47 => ! [X5: nat] :
% 0.29/0.47 ( ( member_nat @ X5 @ A2 )
% 0.29/0.47 => ( member_nat @ ( F @ X5 ) @ B2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % bij_betwE
% 0.29/0.47 thf(fact_117_ex__mset,axiom,
% 0.29/0.47 ! [X6: multiset_a] :
% 0.29/0.47 ? [Xs: list_a] :
% 0.29/0.47 ( ( mset_a @ Xs )
% 0.29/0.47 = X6 ) ).
% 0.29/0.47
% 0.29/0.47 % ex_mset
% 0.29/0.47 thf(fact_118_ex__mset,axiom,
% 0.29/0.47 ! [X6: multiset_nat] :
% 0.29/0.47 ? [Xs: list_nat] :
% 0.29/0.47 ( ( mset_nat @ Xs )
% 0.29/0.47 = X6 ) ).
% 0.29/0.47
% 0.29/0.47 % ex_mset
% 0.29/0.47 thf(fact_119_image__eq__imp__comp,axiom,
% 0.29/0.47 ! [F: nat > nat,A2: set_nat,G: nat > nat,B2: set_nat,H: nat > a] :
% 0.29/0.47 ( ( ( image_nat_nat @ F @ A2 )
% 0.29/0.47 = ( image_nat_nat @ G @ B2 ) )
% 0.29/0.47 => ( ( image_nat_a @ ( comp_nat_a_nat @ H @ F ) @ A2 )
% 0.29/0.47 = ( image_nat_a @ ( comp_nat_a_nat @ H @ G ) @ B2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % image_eq_imp_comp
% 0.29/0.47 thf(fact_120_image__eq__imp__comp,axiom,
% 0.29/0.47 ! [F: nat > nat,A2: set_nat,G: nat > nat,B2: set_nat,H: nat > nat] :
% 0.29/0.47 ( ( ( image_nat_nat @ F @ A2 )
% 0.29/0.47 = ( image_nat_nat @ G @ B2 ) )
% 0.29/0.47 => ( ( image_nat_nat @ ( comp_nat_nat_nat @ H @ F ) @ A2 )
% 0.29/0.47 = ( image_nat_nat @ ( comp_nat_nat_nat @ H @ G ) @ B2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % image_eq_imp_comp
% 0.29/0.47 thf(fact_121_image__eq__imp__comp,axiom,
% 0.29/0.47 ! [F: nat > nat,A2: set_nat,G: nat > nat,B2: set_nat,H: nat > int] :
% 0.29/0.47 ( ( ( image_nat_nat @ F @ A2 )
% 0.29/0.47 = ( image_nat_nat @ G @ B2 ) )
% 0.29/0.47 => ( ( image_nat_int @ ( comp_nat_int_nat @ H @ F ) @ A2 )
% 0.29/0.47 = ( image_nat_int @ ( comp_nat_int_nat @ H @ G ) @ B2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % image_eq_imp_comp
% 0.29/0.47 thf(fact_122_image__eq__imp__comp,axiom,
% 0.29/0.47 ! [F: nat > int,A2: set_nat,G: nat > int,B2: set_nat,H: int > nat] :
% 0.29/0.47 ( ( ( image_nat_int @ F @ A2 )
% 0.29/0.47 = ( image_nat_int @ G @ B2 ) )
% 0.29/0.47 => ( ( image_nat_nat @ ( comp_int_nat_nat @ H @ F ) @ A2 )
% 0.29/0.47 = ( image_nat_nat @ ( comp_int_nat_nat @ H @ G ) @ B2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % image_eq_imp_comp
% 0.29/0.47 thf(fact_123_image__eq__imp__comp,axiom,
% 0.29/0.47 ! [F: nat > int,A2: set_nat,G: nat > int,B2: set_nat,H: int > int] :
% 0.29/0.47 ( ( ( image_nat_int @ F @ A2 )
% 0.29/0.47 = ( image_nat_int @ G @ B2 ) )
% 0.29/0.47 => ( ( image_nat_int @ ( comp_int_int_nat @ H @ F ) @ A2 )
% 0.29/0.47 = ( image_nat_int @ ( comp_int_int_nat @ H @ G ) @ B2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % image_eq_imp_comp
% 0.29/0.47 thf(fact_124_image__comp,axiom,
% 0.29/0.47 ! [F: nat > a,G: nat > nat,R: set_nat] :
% 0.29/0.47 ( ( image_nat_a @ F @ ( image_nat_nat @ G @ R ) )
% 0.29/0.47 = ( image_nat_a @ ( comp_nat_a_nat @ F @ G ) @ R ) ) ).
% 0.29/0.47
% 0.29/0.47 % image_comp
% 0.29/0.47 thf(fact_125_image__comp,axiom,
% 0.29/0.47 ! [F: int > nat,G: nat > int,R: set_nat] :
% 0.29/0.47 ( ( image_int_nat @ F @ ( image_nat_int @ G @ R ) )
% 0.29/0.47 = ( image_nat_nat @ ( comp_int_nat_nat @ F @ G ) @ R ) ) ).
% 0.29/0.47
% 0.29/0.47 % image_comp
% 0.29/0.47 thf(fact_126_image__comp,axiom,
% 0.29/0.47 ! [F: int > int,G: nat > int,R: set_nat] :
% 0.29/0.47 ( ( image_int_int @ F @ ( image_nat_int @ G @ R ) )
% 0.29/0.47 = ( image_nat_int @ ( comp_int_int_nat @ F @ G ) @ R ) ) ).
% 0.29/0.47
% 0.29/0.47 % image_comp
% 0.29/0.47 thf(fact_127_image__comp,axiom,
% 0.29/0.47 ! [F: nat > nat,G: nat > nat,R: set_nat] :
% 0.29/0.47 ( ( image_nat_nat @ F @ ( image_nat_nat @ G @ R ) )
% 0.29/0.47 = ( image_nat_nat @ ( comp_nat_nat_nat @ F @ G ) @ R ) ) ).
% 0.29/0.47
% 0.29/0.47 % image_comp
% 0.29/0.47 thf(fact_128_image__comp,axiom,
% 0.29/0.47 ! [F: nat > int,G: nat > nat,R: set_nat] :
% 0.29/0.47 ( ( image_nat_int @ F @ ( image_nat_nat @ G @ R ) )
% 0.29/0.47 = ( image_nat_int @ ( comp_nat_int_nat @ F @ G ) @ R ) ) ).
% 0.29/0.47
% 0.29/0.47 % image_comp
% 0.29/0.47 thf(fact_129_atLeastLessThan__inj_I2_J,axiom,
% 0.29/0.47 ! [A: real,B3: real,C: real,D: real] :
% 0.29/0.47 ( ( ( set_or2075149659n_real @ A @ B3 )
% 0.29/0.47 = ( set_or2075149659n_real @ C @ D ) )
% 0.29/0.47 => ( ( ord_less_real @ A @ B3 )
% 0.29/0.47 => ( ( ord_less_real @ C @ D )
% 0.29/0.47 => ( B3 = D ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % atLeastLessThan_inj(2)
% 0.29/0.47 thf(fact_130_atLeastLessThan__inj_I2_J,axiom,
% 0.29/0.47 ! [A: nat,B3: nat,C: nat,D: nat] :
% 0.29/0.47 ( ( ( set_or562006527an_nat @ A @ B3 )
% 0.29/0.47 = ( set_or562006527an_nat @ C @ D ) )
% 0.29/0.47 => ( ( ord_less_nat @ A @ B3 )
% 0.29/0.47 => ( ( ord_less_nat @ C @ D )
% 0.29/0.47 => ( B3 = D ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % atLeastLessThan_inj(2)
% 0.29/0.47 thf(fact_131_atLeastLessThan__inj_I2_J,axiom,
% 0.29/0.47 ! [A: int,B3: int,C: int,D: int] :
% 0.29/0.47 ( ( ( set_or1199280219an_int @ A @ B3 )
% 0.29/0.47 = ( set_or1199280219an_int @ C @ D ) )
% 0.29/0.47 => ( ( ord_less_int @ A @ B3 )
% 0.29/0.47 => ( ( ord_less_int @ C @ D )
% 0.29/0.47 => ( B3 = D ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % atLeastLessThan_inj(2)
% 0.29/0.47 thf(fact_132_atLeastLessThan__inj_I1_J,axiom,
% 0.29/0.47 ! [A: real,B3: real,C: real,D: real] :
% 0.29/0.47 ( ( ( set_or2075149659n_real @ A @ B3 )
% 0.29/0.47 = ( set_or2075149659n_real @ C @ D ) )
% 0.29/0.47 => ( ( ord_less_real @ A @ B3 )
% 0.29/0.47 => ( ( ord_less_real @ C @ D )
% 0.29/0.47 => ( A = C ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % atLeastLessThan_inj(1)
% 0.29/0.47 thf(fact_133_atLeastLessThan__inj_I1_J,axiom,
% 0.29/0.47 ! [A: nat,B3: nat,C: nat,D: nat] :
% 0.29/0.47 ( ( ( set_or562006527an_nat @ A @ B3 )
% 0.29/0.47 = ( set_or562006527an_nat @ C @ D ) )
% 0.29/0.47 => ( ( ord_less_nat @ A @ B3 )
% 0.29/0.47 => ( ( ord_less_nat @ C @ D )
% 0.29/0.47 => ( A = C ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % atLeastLessThan_inj(1)
% 0.29/0.47 thf(fact_134_atLeastLessThan__inj_I1_J,axiom,
% 0.29/0.47 ! [A: int,B3: int,C: int,D: int] :
% 0.29/0.47 ( ( ( set_or1199280219an_int @ A @ B3 )
% 0.29/0.47 = ( set_or1199280219an_int @ C @ D ) )
% 0.29/0.47 => ( ( ord_less_int @ A @ B3 )
% 0.29/0.47 => ( ( ord_less_int @ C @ D )
% 0.29/0.47 => ( A = C ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % atLeastLessThan_inj(1)
% 0.29/0.47 thf(fact_135_atLeastLessThan__eq__iff,axiom,
% 0.29/0.47 ! [A: real,B3: real,C: real,D: real] :
% 0.29/0.47 ( ( ord_less_real @ A @ B3 )
% 0.29/0.47 => ( ( ord_less_real @ C @ D )
% 0.29/0.47 => ( ( ( set_or2075149659n_real @ A @ B3 )
% 0.29/0.47 = ( set_or2075149659n_real @ C @ D ) )
% 0.29/0.47 = ( ( A = C )
% 0.29/0.47 & ( B3 = D ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % atLeastLessThan_eq_iff
% 0.29/0.47 thf(fact_136_atLeastLessThan__eq__iff,axiom,
% 0.29/0.47 ! [A: nat,B3: nat,C: nat,D: nat] :
% 0.29/0.47 ( ( ord_less_nat @ A @ B3 )
% 0.29/0.47 => ( ( ord_less_nat @ C @ D )
% 0.29/0.47 => ( ( ( set_or562006527an_nat @ A @ B3 )
% 0.29/0.47 = ( set_or562006527an_nat @ C @ D ) )
% 0.29/0.47 = ( ( A = C )
% 0.29/0.47 & ( B3 = D ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % atLeastLessThan_eq_iff
% 0.29/0.47 thf(fact_137_atLeastLessThan__eq__iff,axiom,
% 0.29/0.47 ! [A: int,B3: int,C: int,D: int] :
% 0.29/0.47 ( ( ord_less_int @ A @ B3 )
% 0.29/0.47 => ( ( ord_less_int @ C @ D )
% 0.29/0.47 => ( ( ( set_or1199280219an_int @ A @ B3 )
% 0.29/0.47 = ( set_or1199280219an_int @ C @ D ) )
% 0.29/0.47 = ( ( A = C )
% 0.29/0.47 & ( B3 = D ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % atLeastLessThan_eq_iff
% 0.29/0.47 thf(fact_138_lessThan__strict__subset__iff,axiom,
% 0.29/0.47 ! [M: int,N: int] :
% 0.29/0.47 ( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N ) )
% 0.29/0.47 = ( ord_less_int @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % lessThan_strict_subset_iff
% 0.29/0.47 thf(fact_139_lessThan__strict__subset__iff,axiom,
% 0.29/0.47 ! [M: real,N: real] :
% 0.29/0.47 ( ( ord_less_set_real @ ( set_or1211449801n_real @ M ) @ ( set_or1211449801n_real @ N ) )
% 0.29/0.47 = ( ord_less_real @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % lessThan_strict_subset_iff
% 0.29/0.47 thf(fact_140_lessThan__strict__subset__iff,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
% 0.29/0.47 = ( ord_less_nat @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % lessThan_strict_subset_iff
% 0.29/0.47 thf(fact_141_inj__on__image__iff,axiom,
% 0.29/0.47 ! [A2: set_nat,G: nat > nat,F: nat > nat] :
% 0.29/0.47 ( ! [X4: nat] :
% 0.29/0.47 ( ( member_nat @ X4 @ A2 )
% 0.29/0.47 => ! [Xa: nat] :
% 0.29/0.47 ( ( member_nat @ Xa @ A2 )
% 0.29/0.47 => ( ( ( G @ ( F @ X4 ) )
% 0.29/0.47 = ( G @ ( F @ Xa ) ) )
% 0.29/0.47 = ( ( G @ X4 )
% 0.29/0.47 = ( G @ Xa ) ) ) ) )
% 0.29/0.47 => ( ( inj_on_nat_nat @ F @ A2 )
% 0.29/0.47 => ( ( inj_on_nat_nat @ G @ ( image_nat_nat @ F @ A2 ) )
% 0.29/0.47 = ( inj_on_nat_nat @ G @ A2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inj_on_image_iff
% 0.29/0.47 thf(fact_142_inj__on__imageI2,axiom,
% 0.29/0.47 ! [F3: nat > a,F: nat > nat,A2: set_nat] :
% 0.29/0.47 ( ( inj_on_nat_a @ ( comp_nat_a_nat @ F3 @ F ) @ A2 )
% 0.29/0.47 => ( inj_on_nat_nat @ F @ A2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % inj_on_imageI2
% 0.29/0.47 thf(fact_143_inj__on__imageI2,axiom,
% 0.29/0.47 ! [F3: nat > nat,F: nat > nat,A2: set_nat] :
% 0.29/0.47 ( ( inj_on_nat_nat @ ( comp_nat_nat_nat @ F3 @ F ) @ A2 )
% 0.29/0.47 => ( inj_on_nat_nat @ F @ A2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % inj_on_imageI2
% 0.29/0.47 thf(fact_144_bij__betw__imp__surj__on,axiom,
% 0.29/0.47 ! [F: nat > int,A2: set_nat,B2: set_int] :
% 0.29/0.47 ( ( bij_betw_nat_int @ F @ A2 @ B2 )
% 0.29/0.47 => ( ( image_nat_int @ F @ A2 )
% 0.29/0.47 = B2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % bij_betw_imp_surj_on
% 0.29/0.47 thf(fact_145_bij__betw__imp__surj__on,axiom,
% 0.29/0.47 ! [F: nat > nat,A2: set_nat,B2: set_nat] :
% 0.29/0.47 ( ( bij_betw_nat_nat @ F @ A2 @ B2 )
% 0.29/0.47 => ( ( image_nat_nat @ F @ A2 )
% 0.29/0.47 = B2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % bij_betw_imp_surj_on
% 0.29/0.47 thf(fact_146_bij__betw__comp__iff,axiom,
% 0.29/0.47 ! [F: nat > nat,A2: set_nat,A5: set_nat,F3: nat > a,A6: set_a] :
% 0.29/0.47 ( ( bij_betw_nat_nat @ F @ A2 @ A5 )
% 0.29/0.47 => ( ( bij_betw_nat_a @ F3 @ A5 @ A6 )
% 0.29/0.47 = ( bij_betw_nat_a @ ( comp_nat_a_nat @ F3 @ F ) @ A2 @ A6 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % bij_betw_comp_iff
% 0.29/0.47 thf(fact_147_bij__betw__comp__iff,axiom,
% 0.29/0.47 ! [F: nat > nat,A2: set_nat,A5: set_nat,F3: nat > nat,A6: set_nat] :
% 0.29/0.47 ( ( bij_betw_nat_nat @ F @ A2 @ A5 )
% 0.29/0.47 => ( ( bij_betw_nat_nat @ F3 @ A5 @ A6 )
% 0.29/0.47 = ( bij_betw_nat_nat @ ( comp_nat_nat_nat @ F3 @ F ) @ A2 @ A6 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % bij_betw_comp_iff
% 0.29/0.47 thf(fact_148_bij__betw__trans,axiom,
% 0.29/0.47 ! [F: nat > nat,A2: set_nat,B2: set_nat,G: nat > a,C2: set_a] :
% 0.29/0.47 ( ( bij_betw_nat_nat @ F @ A2 @ B2 )
% 0.29/0.47 => ( ( bij_betw_nat_a @ G @ B2 @ C2 )
% 0.29/0.47 => ( bij_betw_nat_a @ ( comp_nat_a_nat @ G @ F ) @ A2 @ C2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % bij_betw_trans
% 0.29/0.47 thf(fact_149_bij__betw__trans,axiom,
% 0.29/0.47 ! [F: nat > nat,A2: set_nat,B2: set_nat,G: nat > nat,C2: set_nat] :
% 0.29/0.47 ( ( bij_betw_nat_nat @ F @ A2 @ B2 )
% 0.29/0.47 => ( ( bij_betw_nat_nat @ G @ B2 @ C2 )
% 0.29/0.47 => ( bij_betw_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A2 @ C2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % bij_betw_trans
% 0.29/0.47 thf(fact_150_multiset_Omap__comp,axiom,
% 0.29/0.47 ! [G: a > a,F: nat > a,V: multiset_nat] :
% 0.29/0.47 ( ( image_mset_a_a @ G @ ( image_mset_nat_a @ F @ V ) )
% 0.29/0.47 = ( image_mset_nat_a @ ( comp_a_a_nat @ G @ F ) @ V ) ) ).
% 0.29/0.47
% 0.29/0.47 % multiset.map_comp
% 0.29/0.47 thf(fact_151_multiset_Omap__comp,axiom,
% 0.29/0.47 ! [G: a > nat,F: nat > a,V: multiset_nat] :
% 0.29/0.47 ( ( image_mset_a_nat @ G @ ( image_mset_nat_a @ F @ V ) )
% 0.29/0.47 = ( image_mset_nat_nat @ ( comp_a_nat_nat @ G @ F ) @ V ) ) ).
% 0.29/0.47
% 0.29/0.47 % multiset.map_comp
% 0.29/0.47 thf(fact_152_multiset_Omap__comp,axiom,
% 0.29/0.47 ! [G: nat > a,F: nat > nat,V: multiset_nat] :
% 0.29/0.47 ( ( image_mset_nat_a @ G @ ( image_mset_nat_nat @ F @ V ) )
% 0.29/0.47 = ( image_mset_nat_a @ ( comp_nat_a_nat @ G @ F ) @ V ) ) ).
% 0.29/0.47
% 0.29/0.47 % multiset.map_comp
% 0.29/0.47 thf(fact_153_multiset_Omap__comp,axiom,
% 0.29/0.47 ! [G: nat > nat,F: nat > nat,V: multiset_nat] :
% 0.29/0.47 ( ( image_mset_nat_nat @ G @ ( image_mset_nat_nat @ F @ V ) )
% 0.29/0.47 = ( image_mset_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ V ) ) ).
% 0.29/0.47
% 0.29/0.47 % multiset.map_comp
% 0.29/0.47 thf(fact_154_image__mset_Ocompositionality,axiom,
% 0.29/0.47 ! [F: a > a,G: nat > a,Multiset: multiset_nat] :
% 0.29/0.47 ( ( image_mset_a_a @ F @ ( image_mset_nat_a @ G @ Multiset ) )
% 0.29/0.47 = ( image_mset_nat_a @ ( comp_a_a_nat @ F @ G ) @ Multiset ) ) ).
% 0.29/0.47
% 0.29/0.47 % image_mset.compositionality
% 0.29/0.47 thf(fact_155_image__mset_Ocompositionality,axiom,
% 0.29/0.47 ! [F: a > nat,G: nat > a,Multiset: multiset_nat] :
% 0.29/0.47 ( ( image_mset_a_nat @ F @ ( image_mset_nat_a @ G @ Multiset ) )
% 0.29/0.47 = ( image_mset_nat_nat @ ( comp_a_nat_nat @ F @ G ) @ Multiset ) ) ).
% 0.29/0.47
% 0.29/0.47 % image_mset.compositionality
% 0.29/0.47 thf(fact_156_image__mset_Ocompositionality,axiom,
% 0.29/0.47 ! [F: nat > a,G: nat > nat,Multiset: multiset_nat] :
% 0.29/0.47 ( ( image_mset_nat_a @ F @ ( image_mset_nat_nat @ G @ Multiset ) )
% 0.29/0.47 = ( image_mset_nat_a @ ( comp_nat_a_nat @ F @ G ) @ Multiset ) ) ).
% 0.29/0.47
% 0.29/0.47 % image_mset.compositionality
% 0.29/0.47 thf(fact_157_image__mset_Ocompositionality,axiom,
% 0.29/0.47 ! [F: nat > nat,G: nat > nat,Multiset: multiset_nat] :
% 0.29/0.47 ( ( image_mset_nat_nat @ F @ ( image_mset_nat_nat @ G @ Multiset ) )
% 0.29/0.47 = ( image_mset_nat_nat @ ( comp_nat_nat_nat @ F @ G ) @ Multiset ) ) ).
% 0.29/0.47
% 0.29/0.47 % image_mset.compositionality
% 0.29/0.47 thf(fact_158_image__mset_Ocomp,axiom,
% 0.29/0.47 ! [F: a > a,G: nat > a] :
% 0.29/0.47 ( ( comp_m540523139et_nat @ ( image_mset_a_a @ F ) @ ( image_mset_nat_a @ G ) )
% 0.29/0.47 = ( image_mset_nat_a @ ( comp_a_a_nat @ F @ G ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % image_mset.comp
% 0.29/0.47 thf(fact_159_image__mset_Ocomp,axiom,
% 0.29/0.47 ! [F: a > nat,G: nat > a] :
% 0.29/0.47 ( ( comp_m1962190563et_nat @ ( image_mset_a_nat @ F ) @ ( image_mset_nat_a @ G ) )
% 0.29/0.47 = ( image_mset_nat_nat @ ( comp_a_nat_nat @ F @ G ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % image_mset.comp
% 0.29/0.47 thf(fact_160_image__mset_Ocomp,axiom,
% 0.29/0.47 ! [F: nat > a,G: nat > nat] :
% 0.29/0.47 ( ( comp_m1844659105et_nat @ ( image_mset_nat_a @ F ) @ ( image_mset_nat_nat @ G ) )
% 0.29/0.47 = ( image_mset_nat_a @ ( comp_nat_a_nat @ F @ G ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % image_mset.comp
% 0.29/0.47 thf(fact_161_image__mset_Ocomp,axiom,
% 0.29/0.47 ! [F: nat > nat,G: nat > nat] :
% 0.29/0.47 ( ( comp_m1386984005et_nat @ ( image_mset_nat_nat @ F ) @ ( image_mset_nat_nat @ G ) )
% 0.29/0.47 = ( image_mset_nat_nat @ ( comp_nat_nat_nat @ F @ G ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % image_mset.comp
% 0.29/0.47 thf(fact_162_bij__betw__imp__inj__on,axiom,
% 0.29/0.47 ! [F: nat > nat,A2: set_nat,B2: set_nat] :
% 0.29/0.47 ( ( bij_betw_nat_nat @ F @ A2 @ B2 )
% 0.29/0.47 => ( inj_on_nat_nat @ F @ A2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % bij_betw_imp_inj_on
% 0.29/0.47 thf(fact_163_mset__eq__length,axiom,
% 0.29/0.47 ! [Xs2: list_nat,Ys: list_nat] :
% 0.29/0.47 ( ( ( mset_nat @ Xs2 )
% 0.29/0.47 = ( mset_nat @ Ys ) )
% 0.29/0.47 => ( ( size_size_list_nat @ Xs2 )
% 0.29/0.47 = ( size_size_list_nat @ Ys ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mset_eq_length
% 0.29/0.47 thf(fact_164_mset__eq__length,axiom,
% 0.29/0.47 ! [Xs2: list_a,Ys: list_a] :
% 0.29/0.47 ( ( ( mset_a @ Xs2 )
% 0.29/0.47 = ( mset_a @ Ys ) )
% 0.29/0.47 => ( ( size_size_list_a @ Xs2 )
% 0.29/0.47 = ( size_size_list_a @ Ys ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mset_eq_length
% 0.29/0.47 thf(fact_165_image__eqI,axiom,
% 0.29/0.47 ! [B3: nat,F: nat > nat,X: nat,A2: set_nat] :
% 0.29/0.47 ( ( B3
% 0.29/0.47 = ( F @ X ) )
% 0.29/0.47 => ( ( member_nat @ X @ A2 )
% 0.29/0.47 => ( member_nat @ B3 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % image_eqI
% 0.29/0.47 thf(fact_166_image__eqI,axiom,
% 0.29/0.47 ! [B3: int,F: nat > int,X: nat,A2: set_nat] :
% 0.29/0.47 ( ( B3
% 0.29/0.47 = ( F @ X ) )
% 0.29/0.47 => ( ( member_nat @ X @ A2 )
% 0.29/0.47 => ( member_int @ B3 @ ( image_nat_int @ F @ A2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % image_eqI
% 0.29/0.47 thf(fact_167_mbs_Oless__not__eq,axiom,
% 0.29/0.47 ! [X: list_a,A2: set_list_a,Y: list_a] :
% 0.29/0.47 ( ( member_list_a @ X @ A2 )
% 0.29/0.47 => ( ( ord_less_nat @ ( size_size_list_a @ X ) @ ( size_size_list_a @ Y ) )
% 0.29/0.47 => ( X != Y ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mbs.less_not_eq
% 0.29/0.47 thf(fact_168_Sup_OSUP__image,axiom,
% 0.29/0.47 ! [Sup: set_a > a,G: nat > a,F: nat > nat,A2: set_nat] :
% 0.29/0.47 ( ( Sup @ ( image_nat_a @ G @ ( image_nat_nat @ F @ A2 ) ) )
% 0.29/0.47 = ( Sup @ ( image_nat_a @ ( comp_nat_a_nat @ G @ F ) @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Sup.SUP_image
% 0.29/0.47 thf(fact_169_Sup_OSUP__image,axiom,
% 0.29/0.47 ! [Sup: set_nat > nat,G: int > nat,F: nat > int,A2: set_nat] :
% 0.29/0.47 ( ( Sup @ ( image_int_nat @ G @ ( image_nat_int @ F @ A2 ) ) )
% 0.29/0.47 = ( Sup @ ( image_nat_nat @ ( comp_int_nat_nat @ G @ F ) @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Sup.SUP_image
% 0.29/0.47 thf(fact_170_Sup_OSUP__image,axiom,
% 0.29/0.47 ! [Sup: set_int > int,G: int > int,F: nat > int,A2: set_nat] :
% 0.29/0.47 ( ( Sup @ ( image_int_int @ G @ ( image_nat_int @ F @ A2 ) ) )
% 0.29/0.47 = ( Sup @ ( image_nat_int @ ( comp_int_int_nat @ G @ F ) @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Sup.SUP_image
% 0.29/0.47 thf(fact_171_Sup_OSUP__image,axiom,
% 0.29/0.47 ! [Sup: set_nat > nat,G: nat > nat,F: nat > nat,A2: set_nat] :
% 0.29/0.47 ( ( Sup @ ( image_nat_nat @ G @ ( image_nat_nat @ F @ A2 ) ) )
% 0.29/0.47 = ( Sup @ ( image_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Sup.SUP_image
% 0.29/0.47 thf(fact_172_Sup_OSUP__image,axiom,
% 0.29/0.47 ! [Sup: set_int > int,G: nat > int,F: nat > nat,A2: set_nat] :
% 0.29/0.47 ( ( Sup @ ( image_nat_int @ G @ ( image_nat_nat @ F @ A2 ) ) )
% 0.29/0.47 = ( Sup @ ( image_nat_int @ ( comp_nat_int_nat @ G @ F ) @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Sup.SUP_image
% 0.29/0.47 thf(fact_173_Inf_OINF__image,axiom,
% 0.29/0.47 ! [Inf: set_a > a,G: nat > a,F: nat > nat,A2: set_nat] :
% 0.29/0.47 ( ( Inf @ ( image_nat_a @ G @ ( image_nat_nat @ F @ A2 ) ) )
% 0.29/0.47 = ( Inf @ ( image_nat_a @ ( comp_nat_a_nat @ G @ F ) @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Inf.INF_image
% 0.29/0.47 thf(fact_174_Inf_OINF__image,axiom,
% 0.29/0.47 ! [Inf: set_nat > nat,G: int > nat,F: nat > int,A2: set_nat] :
% 0.29/0.47 ( ( Inf @ ( image_int_nat @ G @ ( image_nat_int @ F @ A2 ) ) )
% 0.29/0.47 = ( Inf @ ( image_nat_nat @ ( comp_int_nat_nat @ G @ F ) @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Inf.INF_image
% 0.29/0.47 thf(fact_175_Inf_OINF__image,axiom,
% 0.29/0.47 ! [Inf: set_int > int,G: int > int,F: nat > int,A2: set_nat] :
% 0.29/0.47 ( ( Inf @ ( image_int_int @ G @ ( image_nat_int @ F @ A2 ) ) )
% 0.29/0.47 = ( Inf @ ( image_nat_int @ ( comp_int_int_nat @ G @ F ) @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Inf.INF_image
% 0.29/0.47 thf(fact_176_Inf_OINF__image,axiom,
% 0.29/0.47 ! [Inf: set_nat > nat,G: nat > nat,F: nat > nat,A2: set_nat] :
% 0.29/0.47 ( ( Inf @ ( image_nat_nat @ G @ ( image_nat_nat @ F @ A2 ) ) )
% 0.29/0.47 = ( Inf @ ( image_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Inf.INF_image
% 0.29/0.47 thf(fact_177_Inf_OINF__image,axiom,
% 0.29/0.47 ! [Inf: set_int > int,G: nat > int,F: nat > nat,A2: set_nat] :
% 0.29/0.47 ( ( Inf @ ( image_nat_int @ G @ ( image_nat_nat @ F @ A2 ) ) )
% 0.29/0.47 = ( Inf @ ( image_nat_int @ ( comp_nat_int_nat @ G @ F ) @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Inf.INF_image
% 0.29/0.47 thf(fact_178_less__numeral__extra_I3_J,axiom,
% 0.29/0.47 ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 % less_numeral_extra(3)
% 0.29/0.47 thf(fact_179_less__numeral__extra_I3_J,axiom,
% 0.29/0.47 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 0.29/0.47
% 0.29/0.47 % less_numeral_extra(3)
% 0.29/0.47 thf(fact_180_less__numeral__extra_I3_J,axiom,
% 0.29/0.47 ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% 0.29/0.47
% 0.29/0.47 % less_numeral_extra(3)
% 0.29/0.47 thf(fact_181_field__lbound__gt__zero,axiom,
% 0.29/0.47 ! [D1: real,D2: real] :
% 0.29/0.47 ( ( ord_less_real @ zero_zero_real @ D1 )
% 0.29/0.47 => ( ( ord_less_real @ zero_zero_real @ D2 )
% 0.29/0.47 => ? [E: real] :
% 0.29/0.47 ( ( ord_less_real @ zero_zero_real @ E )
% 0.29/0.47 & ( ord_less_real @ E @ D1 )
% 0.29/0.47 & ( ord_less_real @ E @ D2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % field_lbound_gt_zero
% 0.29/0.47 thf(fact_182_length__code,axiom,
% 0.29/0.47 ( size_size_list_a
% 0.29/0.47 = ( gen_length_a @ zero_zero_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % length_code
% 0.29/0.47 thf(fact_183_imageI,axiom,
% 0.29/0.47 ! [X: nat,A2: set_nat,F: nat > nat] :
% 0.29/0.47 ( ( member_nat @ X @ A2 )
% 0.29/0.47 => ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % imageI
% 0.29/0.47 thf(fact_184_imageI,axiom,
% 0.29/0.47 ! [X: nat,A2: set_nat,F: nat > int] :
% 0.29/0.47 ( ( member_nat @ X @ A2 )
% 0.29/0.47 => ( member_int @ ( F @ X ) @ ( image_nat_int @ F @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % imageI
% 0.29/0.47 thf(fact_185_image__iff,axiom,
% 0.29/0.47 ! [Z2: nat,F: nat > nat,A2: set_nat] :
% 0.29/0.47 ( ( member_nat @ Z2 @ ( image_nat_nat @ F @ A2 ) )
% 0.29/0.47 = ( ? [X2: nat] :
% 0.29/0.47 ( ( member_nat @ X2 @ A2 )
% 0.29/0.47 & ( Z2
% 0.29/0.47 = ( F @ X2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % image_iff
% 0.29/0.47 thf(fact_186_image__iff,axiom,
% 0.29/0.47 ! [Z2: int,F: nat > int,A2: set_nat] :
% 0.29/0.47 ( ( member_int @ Z2 @ ( image_nat_int @ F @ A2 ) )
% 0.29/0.47 = ( ? [X2: nat] :
% 0.29/0.47 ( ( member_nat @ X2 @ A2 )
% 0.29/0.47 & ( Z2
% 0.29/0.47 = ( F @ X2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % image_iff
% 0.29/0.47 thf(fact_187_bex__imageD,axiom,
% 0.29/0.47 ! [F: nat > nat,A2: set_nat,P: nat > $o] :
% 0.29/0.47 ( ? [X5: nat] :
% 0.29/0.47 ( ( member_nat @ X5 @ ( image_nat_nat @ F @ A2 ) )
% 0.29/0.47 & ( P @ X5 ) )
% 0.29/0.47 => ? [X4: nat] :
% 0.29/0.47 ( ( member_nat @ X4 @ A2 )
% 0.29/0.47 & ( P @ ( F @ X4 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % bex_imageD
% 0.29/0.47 thf(fact_188_bex__imageD,axiom,
% 0.29/0.47 ! [F: nat > int,A2: set_nat,P: int > $o] :
% 0.29/0.47 ( ? [X5: int] :
% 0.29/0.47 ( ( member_int @ X5 @ ( image_nat_int @ F @ A2 ) )
% 0.29/0.47 & ( P @ X5 ) )
% 0.29/0.47 => ? [X4: nat] :
% 0.29/0.47 ( ( member_nat @ X4 @ A2 )
% 0.29/0.47 & ( P @ ( F @ X4 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % bex_imageD
% 0.29/0.47 thf(fact_189_image__cong,axiom,
% 0.29/0.47 ! [M4: set_nat,N3: set_nat,F: nat > nat,G: nat > nat] :
% 0.29/0.47 ( ( M4 = N3 )
% 0.29/0.47 => ( ! [X4: nat] :
% 0.29/0.47 ( ( member_nat @ X4 @ N3 )
% 0.29/0.47 => ( ( F @ X4 )
% 0.29/0.47 = ( G @ X4 ) ) )
% 0.29/0.47 => ( ( image_nat_nat @ F @ M4 )
% 0.29/0.47 = ( image_nat_nat @ G @ N3 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % image_cong
% 0.29/0.47 thf(fact_190_image__cong,axiom,
% 0.29/0.47 ! [M4: set_nat,N3: set_nat,F: nat > int,G: nat > int] :
% 0.29/0.47 ( ( M4 = N3 )
% 0.29/0.47 => ( ! [X4: nat] :
% 0.29/0.47 ( ( member_nat @ X4 @ N3 )
% 0.29/0.47 => ( ( F @ X4 )
% 0.29/0.47 = ( G @ X4 ) ) )
% 0.29/0.47 => ( ( image_nat_int @ F @ M4 )
% 0.29/0.47 = ( image_nat_int @ G @ N3 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % image_cong
% 0.29/0.47 thf(fact_191_ball__imageD,axiom,
% 0.29/0.47 ! [F: nat > nat,A2: set_nat,P: nat > $o] :
% 0.29/0.47 ( ! [X4: nat] :
% 0.29/0.47 ( ( member_nat @ X4 @ ( image_nat_nat @ F @ A2 ) )
% 0.29/0.47 => ( P @ X4 ) )
% 0.29/0.47 => ! [X5: nat] :
% 0.29/0.47 ( ( member_nat @ X5 @ A2 )
% 0.29/0.47 => ( P @ ( F @ X5 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ball_imageD
% 0.29/0.47 thf(fact_192_ball__imageD,axiom,
% 0.29/0.47 ! [F: nat > int,A2: set_nat,P: int > $o] :
% 0.29/0.47 ( ! [X4: int] :
% 0.29/0.47 ( ( member_int @ X4 @ ( image_nat_int @ F @ A2 ) )
% 0.29/0.47 => ( P @ X4 ) )
% 0.29/0.47 => ! [X5: nat] :
% 0.29/0.47 ( ( member_nat @ X5 @ A2 )
% 0.29/0.47 => ( P @ ( F @ X5 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ball_imageD
% 0.29/0.47 thf(fact_193_rev__image__eqI,axiom,
% 0.29/0.47 ! [X: nat,A2: set_nat,B3: nat,F: nat > nat] :
% 0.29/0.47 ( ( member_nat @ X @ A2 )
% 0.29/0.47 => ( ( B3
% 0.29/0.47 = ( F @ X ) )
% 0.29/0.47 => ( member_nat @ B3 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rev_image_eqI
% 0.29/0.47 thf(fact_194_rev__image__eqI,axiom,
% 0.29/0.47 ! [X: nat,A2: set_nat,B3: int,F: nat > int] :
% 0.29/0.47 ( ( member_nat @ X @ A2 )
% 0.29/0.47 => ( ( B3
% 0.29/0.47 = ( F @ X ) )
% 0.29/0.47 => ( member_int @ B3 @ ( image_nat_int @ F @ A2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rev_image_eqI
% 0.29/0.47 thf(fact_195_Inf_OINF__cong,axiom,
% 0.29/0.47 ! [A2: set_nat,B2: set_nat,C2: nat > nat,D3: nat > nat,Inf: set_nat > nat] :
% 0.29/0.47 ( ( A2 = B2 )
% 0.29/0.47 => ( ! [X4: nat] :
% 0.29/0.47 ( ( member_nat @ X4 @ B2 )
% 0.29/0.47 => ( ( C2 @ X4 )
% 0.29/0.47 = ( D3 @ X4 ) ) )
% 0.29/0.47 => ( ( Inf @ ( image_nat_nat @ C2 @ A2 ) )
% 0.29/0.47 = ( Inf @ ( image_nat_nat @ D3 @ B2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Inf.INF_cong
% 0.29/0.47 thf(fact_196_Inf_OINF__cong,axiom,
% 0.29/0.47 ! [A2: set_nat,B2: set_nat,C2: nat > int,D3: nat > int,Inf: set_int > int] :
% 0.29/0.47 ( ( A2 = B2 )
% 0.29/0.47 => ( ! [X4: nat] :
% 0.29/0.47 ( ( member_nat @ X4 @ B2 )
% 0.29/0.47 => ( ( C2 @ X4 )
% 0.29/0.47 = ( D3 @ X4 ) ) )
% 0.29/0.47 => ( ( Inf @ ( image_nat_int @ C2 @ A2 ) )
% 0.29/0.47 = ( Inf @ ( image_nat_int @ D3 @ B2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Inf.INF_cong
% 0.29/0.47 thf(fact_197_Sup_OSUP__cong,axiom,
% 0.29/0.47 ! [A2: set_nat,B2: set_nat,C2: nat > nat,D3: nat > nat,Sup: set_nat > nat] :
% 0.29/0.47 ( ( A2 = B2 )
% 0.29/0.47 => ( ! [X4: nat] :
% 0.29/0.47 ( ( member_nat @ X4 @ B2 )
% 0.29/0.47 => ( ( C2 @ X4 )
% 0.29/0.47 = ( D3 @ X4 ) ) )
% 0.29/0.47 => ( ( Sup @ ( image_nat_nat @ C2 @ A2 ) )
% 0.29/0.47 = ( Sup @ ( image_nat_nat @ D3 @ B2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Sup.SUP_cong
% 0.29/0.47 thf(fact_198_Sup_OSUP__cong,axiom,
% 0.29/0.47 ! [A2: set_nat,B2: set_nat,C2: nat > int,D3: nat > int,Sup: set_int > int] :
% 0.29/0.47 ( ( A2 = B2 )
% 0.29/0.47 => ( ! [X4: nat] :
% 0.29/0.47 ( ( member_nat @ X4 @ B2 )
% 0.29/0.47 => ( ( C2 @ X4 )
% 0.29/0.47 = ( D3 @ X4 ) ) )
% 0.29/0.47 => ( ( Sup @ ( image_nat_int @ C2 @ A2 ) )
% 0.29/0.47 = ( Sup @ ( image_nat_int @ D3 @ B2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Sup.SUP_cong
% 0.29/0.47 thf(fact_199_map__eq__map__tailrec,axiom,
% 0.29/0.47 map_nat_a = map_tailrec_nat_a ).
% 0.29/0.47
% 0.29/0.47 % map_eq_map_tailrec
% 0.29/0.47 thf(fact_200_list__ex__length,axiom,
% 0.29/0.47 ( list_ex_a
% 0.29/0.47 = ( ^ [P2: a > $o,Xs3: list_a] :
% 0.29/0.47 ? [N4: nat] :
% 0.29/0.47 ( ( ord_less_nat @ N4 @ ( size_size_list_a @ Xs3 ) )
% 0.29/0.47 & ( P2 @ ( nth_a @ Xs3 @ N4 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % list_ex_length
% 0.29/0.47 thf(fact_201_of__nat__0__less__iff,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ ( semiri1382578993at_nat @ N ) )
% 0.29/0.47 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % of_nat_0_less_iff
% 0.29/0.47 thf(fact_202_of__nat__0__less__iff,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( ord_less_int @ zero_zero_int @ ( semiri2019852685at_int @ N ) )
% 0.29/0.47 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % of_nat_0_less_iff
% 0.29/0.47 thf(fact_203_of__nat__0__less__iff,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( ord_less_real @ zero_zero_real @ ( semiri2110766477t_real @ N ) )
% 0.29/0.47 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % of_nat_0_less_iff
% 0.29/0.47 thf(fact_204_of__nat__eq__iff,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ( semiri2019852685at_int @ M )
% 0.29/0.47 = ( semiri2019852685at_int @ N ) )
% 0.29/0.47 = ( M = N ) ) ).
% 0.29/0.47
% 0.29/0.47 % of_nat_eq_iff
% 0.29/0.47 thf(fact_205_of__nat__eq__iff,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ( semiri2110766477t_real @ M )
% 0.29/0.47 = ( semiri2110766477t_real @ N ) )
% 0.29/0.47 = ( M = N ) ) ).
% 0.29/0.47
% 0.29/0.47 % of_nat_eq_iff
% 0.29/0.47 thf(fact_206_of__nat__0,axiom,
% 0.29/0.47 ( ( semiri1382578993at_nat @ zero_zero_nat )
% 0.29/0.47 = zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 % of_nat_0
% 0.29/0.47 thf(fact_207_of__nat__0,axiom,
% 0.29/0.47 ( ( semiri2019852685at_int @ zero_zero_nat )
% 0.29/0.47 = zero_zero_int ) ).
% 0.29/0.47
% 0.29/0.47 % of_nat_0
% 0.29/0.47 thf(fact_208_of__nat__0,axiom,
% 0.29/0.47 ( ( semiri2110766477t_real @ zero_zero_nat )
% 0.29/0.47 = zero_zero_real ) ).
% 0.29/0.47
% 0.29/0.47 % of_nat_0
% 0.29/0.47 thf(fact_209_of__nat__0__eq__iff,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( zero_zero_nat
% 0.29/0.47 = ( semiri1382578993at_nat @ N ) )
% 0.29/0.47 = ( zero_zero_nat = N ) ) ).
% 0.29/0.47
% 0.29/0.47 % of_nat_0_eq_iff
% 0.29/0.47 thf(fact_210_of__nat__0__eq__iff,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( zero_zero_int
% 0.29/0.47 = ( semiri2019852685at_int @ N ) )
% 0.29/0.47 = ( zero_zero_nat = N ) ) ).
% 0.29/0.47
% 0.29/0.47 % of_nat_0_eq_iff
% 0.29/0.47 thf(fact_211_of__nat__0__eq__iff,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( zero_zero_real
% 0.29/0.47 = ( semiri2110766477t_real @ N ) )
% 0.29/0.47 = ( zero_zero_nat = N ) ) ).
% 0.29/0.47
% 0.29/0.47 % of_nat_0_eq_iff
% 0.29/0.47 thf(fact_212_of__nat__eq__0__iff,axiom,
% 0.29/0.47 ! [M: nat] :
% 0.29/0.47 ( ( ( semiri1382578993at_nat @ M )
% 0.29/0.47 = zero_zero_nat )
% 0.29/0.47 = ( M = zero_zero_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % of_nat_eq_0_iff
% 0.29/0.47 thf(fact_213_of__nat__eq__0__iff,axiom,
% 0.29/0.47 ! [M: nat] :
% 0.29/0.47 ( ( ( semiri2019852685at_int @ M )
% 0.29/0.47 = zero_zero_int )
% 0.29/0.47 = ( M = zero_zero_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % of_nat_eq_0_iff
% 0.29/0.47 thf(fact_214_of__nat__eq__0__iff,axiom,
% 0.29/0.47 ! [M: nat] :
% 0.29/0.47 ( ( ( semiri2110766477t_real @ M )
% 0.29/0.47 = zero_zero_real )
% 0.29/0.47 = ( M = zero_zero_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % of_nat_eq_0_iff
% 0.29/0.47 thf(fact_215_of__nat__less__iff,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ ( semiri1382578993at_nat @ M ) @ ( semiri1382578993at_nat @ N ) )
% 0.29/0.47 = ( ord_less_nat @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % of_nat_less_iff
% 0.29/0.47 thf(fact_216_of__nat__less__iff,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_int @ ( semiri2019852685at_int @ M ) @ ( semiri2019852685at_int @ N ) )
% 0.29/0.47 = ( ord_less_nat @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % of_nat_less_iff
% 0.29/0.47 thf(fact_217_of__nat__less__iff,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_real @ ( semiri2110766477t_real @ M ) @ ( semiri2110766477t_real @ N ) )
% 0.29/0.47 = ( ord_less_nat @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % of_nat_less_iff
% 0.29/0.47 thf(fact_218_bij__betw__of__nat,axiom,
% 0.29/0.47 ! [N3: set_nat,A2: set_nat] :
% 0.29/0.47 ( ( bij_betw_nat_nat @ semiri1382578993at_nat @ N3 @ A2 )
% 0.29/0.47 = ( ( image_nat_nat @ semiri1382578993at_nat @ N3 )
% 0.29/0.47 = A2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % bij_betw_of_nat
% 0.29/0.47 thf(fact_219_bij__betw__of__nat,axiom,
% 0.29/0.47 ! [N3: set_nat,A2: set_int] :
% 0.29/0.47 ( ( bij_betw_nat_int @ semiri2019852685at_int @ N3 @ A2 )
% 0.29/0.47 = ( ( image_nat_int @ semiri2019852685at_int @ N3 )
% 0.29/0.47 = A2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % bij_betw_of_nat
% 0.29/0.47 thf(fact_220_bij__betw__of__nat,axiom,
% 0.29/0.47 ! [N3: set_nat,A2: set_real] :
% 0.29/0.47 ( ( bij_betw_nat_real @ semiri2110766477t_real @ N3 @ A2 )
% 0.29/0.47 = ( ( image_nat_real @ semiri2110766477t_real @ N3 )
% 0.29/0.47 = A2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % bij_betw_of_nat
% 0.29/0.47 thf(fact_221_image__int__atLeastLessThan,axiom,
% 0.29/0.47 ! [A: nat,B3: nat] :
% 0.29/0.47 ( ( image_nat_int @ semiri2019852685at_int @ ( set_or562006527an_nat @ A @ B3 ) )
% 0.29/0.47 = ( set_or1199280219an_int @ ( semiri2019852685at_int @ A ) @ ( semiri2019852685at_int @ B3 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % image_int_atLeastLessThan
% 0.29/0.47 thf(fact_222_inj__on__of__nat,axiom,
% 0.29/0.47 ! [N3: set_nat] : ( inj_on_nat_nat @ semiri1382578993at_nat @ N3 ) ).
% 0.29/0.47
% 0.29/0.47 % inj_on_of_nat
% 0.29/0.47 thf(fact_223_inj__on__of__nat,axiom,
% 0.29/0.47 ! [N3: set_nat] : ( inj_on_nat_int @ semiri2019852685at_int @ N3 ) ).
% 0.29/0.47
% 0.29/0.47 % inj_on_of_nat
% 0.29/0.47 thf(fact_224_inj__on__of__nat,axiom,
% 0.29/0.47 ! [N3: set_nat] : ( inj_on_nat_real @ semiri2110766477t_real @ N3 ) ).
% 0.29/0.47
% 0.29/0.47 % inj_on_of_nat
% 0.29/0.47 thf(fact_225_of__nat__less__0__iff,axiom,
% 0.29/0.47 ! [M: nat] :
% 0.29/0.47 ~ ( ord_less_nat @ ( semiri1382578993at_nat @ M ) @ zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 % of_nat_less_0_iff
% 0.29/0.47 thf(fact_226_of__nat__less__0__iff,axiom,
% 0.29/0.47 ! [M: nat] :
% 0.29/0.47 ~ ( ord_less_int @ ( semiri2019852685at_int @ M ) @ zero_zero_int ) ).
% 0.29/0.47
% 0.29/0.47 % of_nat_less_0_iff
% 0.29/0.47 thf(fact_227_of__nat__less__0__iff,axiom,
% 0.29/0.47 ! [M: nat] :
% 0.29/0.47 ~ ( ord_less_real @ ( semiri2110766477t_real @ M ) @ zero_zero_real ) ).
% 0.29/0.47
% 0.29/0.47 % of_nat_less_0_iff
% 0.29/0.47 thf(fact_228_of__nat__less__imp__less,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ ( semiri1382578993at_nat @ M ) @ ( semiri1382578993at_nat @ N ) )
% 0.29/0.47 => ( ord_less_nat @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % of_nat_less_imp_less
% 0.29/0.47 thf(fact_229_of__nat__less__imp__less,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_int @ ( semiri2019852685at_int @ M ) @ ( semiri2019852685at_int @ N ) )
% 0.29/0.47 => ( ord_less_nat @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % of_nat_less_imp_less
% 0.29/0.47 thf(fact_230_of__nat__less__imp__less,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_real @ ( semiri2110766477t_real @ M ) @ ( semiri2110766477t_real @ N ) )
% 0.29/0.47 => ( ord_less_nat @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % of_nat_less_imp_less
% 0.29/0.47 thf(fact_231_less__imp__of__nat__less,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M @ N )
% 0.29/0.47 => ( ord_less_nat @ ( semiri1382578993at_nat @ M ) @ ( semiri1382578993at_nat @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_imp_of_nat_less
% 0.29/0.47 thf(fact_232_less__imp__of__nat__less,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M @ N )
% 0.29/0.47 => ( ord_less_int @ ( semiri2019852685at_int @ M ) @ ( semiri2019852685at_int @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_imp_of_nat_less
% 0.29/0.47 thf(fact_233_less__imp__of__nat__less,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M @ N )
% 0.29/0.47 => ( ord_less_real @ ( semiri2110766477t_real @ M ) @ ( semiri2110766477t_real @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_imp_of_nat_less
% 0.29/0.47 thf(fact_234_pos__int__cases,axiom,
% 0.29/0.47 ! [K: int] :
% 0.29/0.47 ( ( ord_less_int @ zero_zero_int @ K )
% 0.29/0.47 => ~ ! [N2: nat] :
% 0.29/0.47 ( ( K
% 0.29/0.47 = ( semiri2019852685at_int @ N2 ) )
% 0.29/0.47 => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % pos_int_cases
% 0.29/0.47 thf(fact_235_zero__less__imp__eq__int,axiom,
% 0.29/0.47 ! [K: int] :
% 0.29/0.47 ( ( ord_less_int @ zero_zero_int @ K )
% 0.29/0.47 => ? [N2: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.47 & ( K
% 0.29/0.47 = ( semiri2019852685at_int @ N2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % zero_less_imp_eq_int
% 0.29/0.47 thf(fact_236_reals__Archimedean2,axiom,
% 0.29/0.47 ! [X: real] :
% 0.29/0.47 ? [N2: nat] : ( ord_less_real @ X @ ( semiri2110766477t_real @ N2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % reals_Archimedean2
% 0.29/0.47 thf(fact_237_int__int__eq,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ( semiri2019852685at_int @ M )
% 0.29/0.47 = ( semiri2019852685at_int @ N ) )
% 0.29/0.47 = ( M = N ) ) ).
% 0.29/0.47
% 0.29/0.47 % int_int_eq
% 0.29/0.47 thf(fact_238_less__int__code_I1_J,axiom,
% 0.29/0.47 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 0.29/0.47
% 0.29/0.47 % less_int_code(1)
% 0.29/0.47 thf(fact_239_size__multiset__o__map,axiom,
% 0.29/0.47 ! [G: a > nat,F: nat > a] :
% 0.29/0.47 ( ( comp_m1512105363et_nat @ ( size_multiset_a @ G ) @ ( image_mset_nat_a @ F ) )
% 0.29/0.47 = ( size_multiset_nat @ ( comp_a_nat_nat @ G @ F ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % size_multiset_o_map
% 0.29/0.47 thf(fact_240_size__multiset__o__map,axiom,
% 0.29/0.47 ! [G: nat > nat,F: nat > nat] :
% 0.29/0.47 ( ( comp_m346803701et_nat @ ( size_multiset_nat @ G ) @ ( image_mset_nat_nat @ F ) )
% 0.29/0.47 = ( size_multiset_nat @ ( comp_nat_nat_nat @ G @ F ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % size_multiset_o_map
% 0.29/0.47 thf(fact_241_nat__int__comparison_I2_J,axiom,
% 0.29/0.47 ( ord_less_nat
% 0.29/0.47 = ( ^ [A7: nat,B4: nat] : ( ord_less_int @ ( semiri2019852685at_int @ A7 ) @ ( semiri2019852685at_int @ B4 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % nat_int_comparison(2)
% 0.29/0.47 thf(fact_242_int__ops_I1_J,axiom,
% 0.29/0.47 ( ( semiri2019852685at_int @ zero_zero_nat )
% 0.29/0.47 = zero_zero_int ) ).
% 0.29/0.47
% 0.29/0.47 % int_ops(1)
% 0.29/0.47 thf(fact_243_neg__int__cases,axiom,
% 0.29/0.47 ! [K: int] :
% 0.29/0.47 ( ( ord_less_int @ K @ zero_zero_int )
% 0.29/0.47 => ~ ! [N2: nat] :
% 0.29/0.47 ( ( K
% 0.29/0.47 = ( uminus_uminus_int @ ( semiri2019852685at_int @ N2 ) ) )
% 0.29/0.47 => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % neg_int_cases
% 0.29/0.47 thf(fact_244_add_Oinverse__inverse,axiom,
% 0.29/0.47 ! [A: int] :
% 0.29/0.47 ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
% 0.29/0.47 = A ) ).
% 0.29/0.47
% 0.29/0.47 % add.inverse_inverse
% 0.29/0.47 thf(fact_245_neg__equal__iff__equal,axiom,
% 0.29/0.47 ! [A: int,B3: int] :
% 0.29/0.47 ( ( ( uminus_uminus_int @ A )
% 0.29/0.47 = ( uminus_uminus_int @ B3 ) )
% 0.29/0.47 = ( A = B3 ) ) ).
% 0.29/0.47
% 0.29/0.47 % neg_equal_iff_equal
% 0.29/0.47 thf(fact_246_add_Oinverse__neutral,axiom,
% 0.29/0.47 ( ( uminus_uminus_real @ zero_zero_real )
% 0.29/0.47 = zero_zero_real ) ).
% 0.29/0.47
% 0.29/0.47 % add.inverse_neutral
% 0.29/0.47 thf(fact_247_add_Oinverse__neutral,axiom,
% 0.29/0.47 ( ( uminus_uminus_int @ zero_zero_int )
% 0.29/0.47 = zero_zero_int ) ).
% 0.29/0.47
% 0.29/0.47 % add.inverse_neutral
% 0.29/0.47 thf(fact_248_neg__0__equal__iff__equal,axiom,
% 0.29/0.47 ! [A: real] :
% 0.29/0.47 ( ( zero_zero_real
% 0.29/0.47 = ( uminus_uminus_real @ A ) )
% 0.29/0.47 = ( zero_zero_real = A ) ) ).
% 0.29/0.47
% 0.29/0.47 % neg_0_equal_iff_equal
% 0.29/0.47 thf(fact_249_neg__0__equal__iff__equal,axiom,
% 0.29/0.47 ! [A: int] :
% 0.29/0.47 ( ( zero_zero_int
% 0.29/0.47 = ( uminus_uminus_int @ A ) )
% 0.29/0.47 = ( zero_zero_int = A ) ) ).
% 0.29/0.47
% 0.29/0.47 % neg_0_equal_iff_equal
% 0.29/0.47 thf(fact_250_neg__equal__0__iff__equal,axiom,
% 0.29/0.47 ! [A: real] :
% 0.29/0.47 ( ( ( uminus_uminus_real @ A )
% 0.29/0.47 = zero_zero_real )
% 0.29/0.47 = ( A = zero_zero_real ) ) ).
% 0.29/0.47
% 0.29/0.47 % neg_equal_0_iff_equal
% 0.29/0.47 thf(fact_251_neg__equal__0__iff__equal,axiom,
% 0.29/0.47 ! [A: int] :
% 0.29/0.47 ( ( ( uminus_uminus_int @ A )
% 0.29/0.47 = zero_zero_int )
% 0.29/0.47 = ( A = zero_zero_int ) ) ).
% 0.29/0.47
% 0.29/0.47 % neg_equal_0_iff_equal
% 0.29/0.47 thf(fact_252_equal__neg__zero,axiom,
% 0.29/0.47 ! [A: real] :
% 0.29/0.47 ( ( A
% 0.29/0.47 = ( uminus_uminus_real @ A ) )
% 0.29/0.47 = ( A = zero_zero_real ) ) ).
% 0.29/0.47
% 0.29/0.47 % equal_neg_zero
% 0.29/0.47 thf(fact_253_equal__neg__zero,axiom,
% 0.29/0.47 ! [A: int] :
% 0.29/0.47 ( ( A
% 0.29/0.47 = ( uminus_uminus_int @ A ) )
% 0.29/0.47 = ( A = zero_zero_int ) ) ).
% 0.29/0.47
% 0.29/0.47 % equal_neg_zero
% 0.29/0.47 thf(fact_254_neg__equal__zero,axiom,
% 0.29/0.47 ! [A: real] :
% 0.29/0.47 ( ( ( uminus_uminus_real @ A )
% 0.29/0.47 = A )
% 0.29/0.47 = ( A = zero_zero_real ) ) ).
% 0.29/0.47
% 0.29/0.47 % neg_equal_zero
% 0.29/0.47 thf(fact_255_neg__equal__zero,axiom,
% 0.29/0.47 ! [A: int] :
% 0.29/0.47 ( ( ( uminus_uminus_int @ A )
% 0.29/0.47 = A )
% 0.29/0.47 = ( A = zero_zero_int ) ) ).
% 0.29/0.47
% 0.29/0.47 % neg_equal_zero
% 0.29/0.47 thf(fact_256_neg__less__iff__less,axiom,
% 0.29/0.47 ! [B3: real,A: real] :
% 0.29/0.47 ( ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A ) )
% 0.29/0.47 = ( ord_less_real @ A @ B3 ) ) ).
% 0.29/0.47
% 0.29/0.47 % neg_less_iff_less
% 0.29/0.47 thf(fact_257_neg__less__iff__less,axiom,
% 0.29/0.47 ! [B3: int,A: int] :
% 0.29/0.47 ( ( ord_less_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A ) )
% 0.29/0.47 = ( ord_less_int @ A @ B3 ) ) ).
% 0.29/0.47
% 0.29/0.47 % neg_less_iff_less
% 0.29/0.47 thf(fact_258_inj__uminus,axiom,
% 0.29/0.47 ! [A2: set_int] : ( inj_on_int_int @ uminus_uminus_int @ A2 ) ).
% 0.29/0.47
% 0.29/0.47 % inj_uminus
% 0.29/0.47 thf(fact_259_neg__less__0__iff__less,axiom,
% 0.29/0.47 ! [A: real] :
% 0.29/0.47 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 0.29/0.47 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 % neg_less_0_iff_less
% 0.29/0.47 thf(fact_260_neg__less__0__iff__less,axiom,
% 0.29/0.47 ! [A: int] :
% 0.29/0.47 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 0.29/0.47 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 % neg_less_0_iff_less
% 0.29/0.47 thf(fact_261_neg__0__less__iff__less,axiom,
% 0.29/0.47 ! [A: real] :
% 0.29/0.47 ( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 0.29/0.47 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.29/0.47
% 0.29/0.47 % neg_0_less_iff_less
% 0.29/0.47 thf(fact_262_neg__0__less__iff__less,axiom,
% 0.29/0.47 ! [A: int] :
% 0.29/0.47 ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 0.29/0.47 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.29/0.47
% 0.29/0.47 % neg_0_less_iff_less
% 0.29/0.47 thf(fact_263_neg__less__pos,axiom,
% 0.29/0.47 ! [A: real] :
% 0.29/0.47 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
% 0.29/0.47 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 % neg_less_pos
% 0.29/0.47 thf(fact_264_neg__less__pos,axiom,
% 0.29/0.47 ! [A: int] :
% 0.29/0.47 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
% 0.29/0.47 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 % neg_less_pos
% 0.29/0.47 thf(fact_265_less__neg__neg,axiom,
% 0.29/0.47 ! [A: real] :
% 0.29/0.47 ( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
% 0.29/0.47 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_neg_neg
% 0.29/0.47 thf(fact_266_less__neg__neg,axiom,
% 0.29/0.47 ! [A: int] :
% 0.29/0.47 ( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
% 0.29/0.47 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_neg_neg
% 0.29/0.47 thf(fact_267_negative__eq__positive,axiom,
% 0.29/0.47 ! [N: nat,M: nat] :
% 0.29/0.47 ( ( ( uminus_uminus_int @ ( semiri2019852685at_int @ N ) )
% 0.29/0.47 = ( semiri2019852685at_int @ M ) )
% 0.29/0.47 = ( ( N = zero_zero_nat )
% 0.29/0.47 & ( M = zero_zero_nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % negative_eq_positive
% 0.29/0.47 thf(fact_268_int__cases2,axiom,
% 0.29/0.47 ! [Z2: int] :
% 0.29/0.47 ( ! [N2: nat] :
% 0.29/0.47 ( Z2
% 0.29/0.47 != ( semiri2019852685at_int @ N2 ) )
% 0.29/0.47 => ~ ! [N2: nat] :
% 0.29/0.47 ( Z2
% 0.29/0.47 != ( uminus_uminus_int @ ( semiri2019852685at_int @ N2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % int_cases2
% 0.29/0.47 thf(fact_269_minus__less__iff,axiom,
% 0.29/0.47 ! [A: real,B3: real] :
% 0.29/0.47 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B3 )
% 0.29/0.47 = ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 % minus_less_iff
% 0.29/0.47 thf(fact_270_minus__less__iff,axiom,
% 0.29/0.47 ! [A: int,B3: int] :
% 0.29/0.47 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B3 )
% 0.29/0.47 = ( ord_less_int @ ( uminus_uminus_int @ B3 ) @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 % minus_less_iff
% 0.29/0.47 thf(fact_271_less__minus__iff,axiom,
% 0.29/0.47 ! [A: real,B3: real] :
% 0.29/0.47 ( ( ord_less_real @ A @ ( uminus_uminus_real @ B3 ) )
% 0.29/0.47 = ( ord_less_real @ B3 @ ( uminus_uminus_real @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_minus_iff
% 0.29/0.47 thf(fact_272_less__minus__iff,axiom,
% 0.29/0.47 ! [A: int,B3: int] :
% 0.29/0.47 ( ( ord_less_int @ A @ ( uminus_uminus_int @ B3 ) )
% 0.29/0.47 = ( ord_less_int @ B3 @ ( uminus_uminus_int @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_minus_iff
% 0.29/0.47 thf(fact_273_uminus__int__code_I1_J,axiom,
% 0.29/0.47 ( ( uminus_uminus_int @ zero_zero_int )
% 0.29/0.47 = zero_zero_int ) ).
% 0.29/0.47
% 0.29/0.47 % uminus_int_code(1)
% 0.29/0.47 thf(fact_274_verit__negate__coefficient_I2_J,axiom,
% 0.29/0.47 ! [A: real,B3: real] :
% 0.29/0.47 ( ( ord_less_real @ A @ B3 )
% 0.29/0.47 => ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % verit_negate_coefficient(2)
% 0.29/0.47 thf(fact_275_verit__negate__coefficient_I2_J,axiom,
% 0.29/0.47 ! [A: int,B3: int] :
% 0.29/0.47 ( ( ord_less_int @ A @ B3 )
% 0.29/0.47 => ( ord_less_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % verit_negate_coefficient(2)
% 0.29/0.47 thf(fact_276_equation__minus__iff,axiom,
% 0.29/0.47 ! [A: int,B3: int] :
% 0.29/0.47 ( ( A
% 0.29/0.47 = ( uminus_uminus_int @ B3 ) )
% 0.29/0.47 = ( B3
% 0.29/0.47 = ( uminus_uminus_int @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % equation_minus_iff
% 0.29/0.47 thf(fact_277_minus__equation__iff,axiom,
% 0.29/0.47 ! [A: int,B3: int] :
% 0.29/0.47 ( ( ( uminus_uminus_int @ A )
% 0.29/0.47 = B3 )
% 0.29/0.47 = ( ( uminus_uminus_int @ B3 )
% 0.29/0.47 = A ) ) ).
% 0.29/0.47
% 0.29/0.47 % minus_equation_iff
% 0.29/0.47 thf(fact_278_not__int__zless__negative,axiom,
% 0.29/0.47 ! [N: nat,M: nat] :
% 0.29/0.47 ~ ( ord_less_int @ ( semiri2019852685at_int @ N ) @ ( uminus_uminus_int @ ( semiri2019852685at_int @ M ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % not_int_zless_negative
% 0.29/0.47 thf(fact_279_int__cases4,axiom,
% 0.29/0.47 ! [M: int] :
% 0.29/0.47 ( ! [N2: nat] :
% 0.29/0.47 ( M
% 0.29/0.47 != ( semiri2019852685at_int @ N2 ) )
% 0.29/0.47 => ~ ! [N2: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.47 => ( M
% 0.29/0.47 != ( uminus_uminus_int @ ( semiri2019852685at_int @ N2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % int_cases4
% 0.29/0.47 thf(fact_280_verit__comp__simplify1_I1_J,axiom,
% 0.29/0.47 ! [A: nat] :
% 0.29/0.47 ~ ( ord_less_nat @ A @ A ) ).
% 0.29/0.47
% 0.29/0.47 % verit_comp_simplify1(1)
% 0.29/0.47 thf(fact_281_verit__comp__simplify1_I1_J,axiom,
% 0.29/0.47 ! [A: int] :
% 0.29/0.47 ~ ( ord_less_int @ A @ A ) ).
% 0.29/0.47
% 0.29/0.47 % verit_comp_simplify1(1)
% 0.29/0.47 thf(fact_282_verit__comp__simplify1_I1_J,axiom,
% 0.29/0.47 ! [A: real] :
% 0.29/0.47 ~ ( ord_less_real @ A @ A ) ).
% 0.29/0.47
% 0.29/0.47 % verit_comp_simplify1(1)
% 0.29/0.47 thf(fact_283_nat__int__comparison_I1_J,axiom,
% 0.29/0.47 ( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
% 0.29/0.47 = ( ^ [A7: nat,B4: nat] :
% 0.29/0.47 ( ( semiri2019852685at_int @ A7 )
% 0.29/0.47 = ( semiri2019852685at_int @ B4 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % nat_int_comparison(1)
% 0.29/0.47 thf(fact_284_int__if,axiom,
% 0.29/0.47 ! [P: $o,A: nat,B3: nat] :
% 0.29/0.47 ( ( P
% 0.29/0.47 => ( ( semiri2019852685at_int @ ( if_nat @ P @ A @ B3 ) )
% 0.29/0.47 = ( semiri2019852685at_int @ A ) ) )
% 0.29/0.47 & ( ~ P
% 0.29/0.47 => ( ( semiri2019852685at_int @ ( if_nat @ P @ A @ B3 ) )
% 0.29/0.47 = ( semiri2019852685at_int @ B3 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % int_if
% 0.29/0.47 thf(fact_285_int__cases3,axiom,
% 0.29/0.47 ! [K: int] :
% 0.29/0.47 ( ( K != zero_zero_int )
% 0.29/0.47 => ( ! [N2: nat] :
% 0.29/0.47 ( ( K
% 0.29/0.47 = ( semiri2019852685at_int @ N2 ) )
% 0.29/0.47 => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
% 0.29/0.47 => ~ ! [N2: nat] :
% 0.29/0.47 ( ( K
% 0.29/0.47 = ( uminus_uminus_int @ ( semiri2019852685at_int @ N2 ) ) )
% 0.29/0.47 => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % int_cases3
% 0.29/0.47 thf(fact_286_norm__frac_Oinduct,axiom,
% 0.29/0.47 ! [P: int > int > $o,A0: int,A1: int] :
% 0.29/0.47 ( ! [A4: int,B5: int] :
% 0.29/0.47 ( ( ( ord_less_int @ B5 @ zero_zero_int )
% 0.29/0.47 => ( P @ ( uminus_uminus_int @ A4 ) @ ( uminus_uminus_int @ B5 ) ) )
% 0.29/0.47 => ( P @ A4 @ B5 ) )
% 0.29/0.47 => ( P @ A0 @ A1 ) ) ).
% 0.29/0.47
% 0.29/0.47 % norm_frac.induct
% 0.29/0.47 thf(fact_287_ex__inverse__of__nat__less,axiom,
% 0.29/0.47 ! [X: real] :
% 0.29/0.47 ( ( ord_less_real @ zero_zero_real @ X )
% 0.29/0.47 => ? [N2: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.47 & ( ord_less_real @ ( inverse_inverse_real @ ( semiri2110766477t_real @ N2 ) ) @ X ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ex_inverse_of_nat_less
% 0.29/0.47 thf(fact_288_zero__less__nat__eq,axiom,
% 0.29/0.47 ! [Z2: int] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z2 ) )
% 0.29/0.47 = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % zero_less_nat_eq
% 0.29/0.47 thf(fact_289_nat__int,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( nat2 @ ( semiri2019852685at_int @ N ) )
% 0.29/0.47 = N ) ).
% 0.29/0.47
% 0.29/0.47 % nat_int
% 0.29/0.47 thf(fact_290_zless__nat__conj,axiom,
% 0.29/0.47 ! [W: int,Z2: int] :
% 0.29/0.47 ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
% 0.29/0.47 = ( ( ord_less_int @ zero_zero_int @ Z2 )
% 0.29/0.47 & ( ord_less_int @ W @ Z2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % zless_nat_conj
% 0.29/0.47 thf(fact_291_nat__zminus__int,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( nat2 @ ( uminus_uminus_int @ ( semiri2019852685at_int @ N ) ) )
% 0.29/0.47 = zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 % nat_zminus_int
% 0.29/0.47 thf(fact_292_nat__zero__as__int,axiom,
% 0.29/0.47 ( zero_zero_nat
% 0.29/0.47 = ( nat2 @ zero_zero_int ) ) ).
% 0.29/0.47
% 0.29/0.47 % nat_zero_as_int
% 0.29/0.47 thf(fact_293_forall__pos__mono,axiom,
% 0.29/0.47 ! [P: real > $o,E2: real] :
% 0.29/0.47 ( ! [D4: real,E: real] :
% 0.29/0.47 ( ( ord_less_real @ D4 @ E )
% 0.29/0.47 => ( ( P @ D4 )
% 0.29/0.47 => ( P @ E ) ) )
% 0.29/0.47 => ( ! [N2: nat] :
% 0.29/0.47 ( ( N2 != zero_zero_nat )
% 0.29/0.47 => ( P @ ( inverse_inverse_real @ ( semiri2110766477t_real @ N2 ) ) ) )
% 0.29/0.47 => ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.29/0.47 => ( P @ E2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % forall_pos_mono
% 0.29/0.47 thf(fact_294_real__arch__inverse,axiom,
% 0.29/0.47 ! [E2: real] :
% 0.29/0.47 ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.29/0.47 = ( ? [N4: nat] :
% 0.29/0.47 ( ( N4 != zero_zero_nat )
% 0.29/0.47 & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri2110766477t_real @ N4 ) ) )
% 0.29/0.47 & ( ord_less_real @ ( inverse_inverse_real @ ( semiri2110766477t_real @ N4 ) ) @ E2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % real_arch_inverse
% 0.29/0.47 thf(fact_295_nat__mono__iff,axiom,
% 0.29/0.47 ! [Z2: int,W: int] :
% 0.29/0.47 ( ( ord_less_int @ zero_zero_int @ Z2 )
% 0.29/0.47 => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
% 0.29/0.47 = ( ord_less_int @ W @ Z2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % nat_mono_iff
% 0.29/0.47 thf(fact_296_zless__nat__eq__int__zless,axiom,
% 0.29/0.47 ! [M: nat,Z2: int] :
% 0.29/0.47 ( ( ord_less_nat @ M @ ( nat2 @ Z2 ) )
% 0.29/0.47 = ( ord_less_int @ ( semiri2019852685at_int @ M ) @ Z2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % zless_nat_eq_int_zless
% 0.29/0.47 thf(fact_297_split__nat,axiom,
% 0.29/0.47 ! [P: nat > $o,I: int] :
% 0.29/0.47 ( ( P @ ( nat2 @ I ) )
% 0.29/0.47 = ( ! [N4: nat] :
% 0.29/0.47 ( ( I
% 0.29/0.47 = ( semiri2019852685at_int @ N4 ) )
% 0.29/0.47 => ( P @ N4 ) )
% 0.29/0.47 & ( ( ord_less_int @ I @ zero_zero_int )
% 0.29/0.47 => ( P @ zero_zero_nat ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % split_nat
% 0.29/0.47 thf(fact_298_inverse__positive__iff__positive,axiom,
% 0.29/0.47 ! [A: real] :
% 0.29/0.47 ( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 0.29/0.47 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 % inverse_positive_iff_positive
% 0.29/0.47 thf(fact_299_inverse__negative__iff__negative,axiom,
% 0.29/0.47 ! [A: real] :
% 0.29/0.47 ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 0.29/0.47 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.29/0.47
% 0.29/0.47 % inverse_negative_iff_negative
% 0.29/0.47 thf(fact_300_inverse__zero,axiom,
% 0.29/0.47 ( ( inverse_inverse_real @ zero_zero_real )
% 0.29/0.47 = zero_zero_real ) ).
% 0.29/0.47
% 0.29/0.47 % inverse_zero
% 0.29/0.47 thf(fact_301_inverse__nonzero__iff__nonzero,axiom,
% 0.29/0.47 ! [A: real] :
% 0.29/0.47 ( ( ( inverse_inverse_real @ A )
% 0.29/0.47 = zero_zero_real )
% 0.29/0.47 = ( A = zero_zero_real ) ) ).
% 0.29/0.47
% 0.29/0.47 % inverse_nonzero_iff_nonzero
% 0.29/0.47 thf(fact_302_inverse__less__iff__less,axiom,
% 0.29/0.47 ! [A: real,B3: real] :
% 0.29/0.47 ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.47 => ( ( ord_less_real @ zero_zero_real @ B3 )
% 0.29/0.47 => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B3 ) )
% 0.29/0.47 = ( ord_less_real @ B3 @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inverse_less_iff_less
% 0.29/0.47 thf(fact_303_inverse__less__iff__less__neg,axiom,
% 0.29/0.47 ! [A: real,B3: real] :
% 0.29/0.47 ( ( ord_less_real @ A @ zero_zero_real )
% 0.29/0.47 => ( ( ord_less_real @ B3 @ zero_zero_real )
% 0.29/0.47 => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B3 ) )
% 0.29/0.47 = ( ord_less_real @ B3 @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inverse_less_iff_less_neg
% 0.29/0.47 thf(fact_304_linordered__field__no__lb,axiom,
% 0.29/0.47 ! [X5: real] :
% 0.29/0.47 ? [Y4: real] : ( ord_less_real @ Y4 @ X5 ) ).
% 0.29/0.47
% 0.29/0.47 % linordered_field_no_lb
% 0.29/0.47 thf(fact_305_linordered__field__no__ub,axiom,
% 0.29/0.47 ! [X5: real] :
% 0.29/0.47 ? [X_1: real] : ( ord_less_real @ X5 @ X_1 ) ).
% 0.29/0.47
% 0.29/0.47 % linordered_field_no_ub
% 0.29/0.47 thf(fact_306_field__class_Ofield__inverse__zero,axiom,
% 0.29/0.47 ( ( inverse_inverse_real @ zero_zero_real )
% 0.29/0.47 = zero_zero_real ) ).
% 0.29/0.47
% 0.29/0.47 % field_class.field_inverse_zero
% 0.29/0.47 thf(fact_307_inverse__zero__imp__zero,axiom,
% 0.29/0.47 ! [A: real] :
% 0.29/0.47 ( ( ( inverse_inverse_real @ A )
% 0.29/0.47 = zero_zero_real )
% 0.29/0.47 => ( A = zero_zero_real ) ) ).
% 0.29/0.47
% 0.29/0.47 % inverse_zero_imp_zero
% 0.29/0.47 thf(fact_308_nonzero__inverse__eq__imp__eq,axiom,
% 0.29/0.47 ! [A: real,B3: real] :
% 0.29/0.47 ( ( ( inverse_inverse_real @ A )
% 0.29/0.47 = ( inverse_inverse_real @ B3 ) )
% 0.29/0.47 => ( ( A != zero_zero_real )
% 0.29/0.47 => ( ( B3 != zero_zero_real )
% 0.29/0.47 => ( A = B3 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % nonzero_inverse_eq_imp_eq
% 0.29/0.47 thf(fact_309_nonzero__inverse__inverse__eq,axiom,
% 0.29/0.47 ! [A: real] :
% 0.29/0.47 ( ( A != zero_zero_real )
% 0.29/0.47 => ( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
% 0.29/0.47 = A ) ) ).
% 0.29/0.47
% 0.29/0.47 % nonzero_inverse_inverse_eq
% 0.29/0.47 thf(fact_310_nonzero__imp__inverse__nonzero,axiom,
% 0.29/0.47 ! [A: real] :
% 0.29/0.47 ( ( A != zero_zero_real )
% 0.29/0.47 => ( ( inverse_inverse_real @ A )
% 0.29/0.47 != zero_zero_real ) ) ).
% 0.29/0.47
% 0.29/0.47 % nonzero_imp_inverse_nonzero
% 0.29/0.47 thf(fact_311_inverse__less__imp__less,axiom,
% 0.29/0.47 ! [A: real,B3: real] :
% 0.29/0.47 ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B3 ) )
% 0.29/0.47 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.47 => ( ord_less_real @ B3 @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inverse_less_imp_less
% 0.29/0.47 thf(fact_312_less__imp__inverse__less,axiom,
% 0.29/0.47 ! [A: real,B3: real] :
% 0.29/0.47 ( ( ord_less_real @ A @ B3 )
% 0.29/0.47 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.47 => ( ord_less_real @ ( inverse_inverse_real @ B3 ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_imp_inverse_less
% 0.29/0.47 thf(fact_313_inverse__less__imp__less__neg,axiom,
% 0.29/0.47 ! [A: real,B3: real] :
% 0.29/0.47 ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B3 ) )
% 0.29/0.47 => ( ( ord_less_real @ B3 @ zero_zero_real )
% 0.29/0.47 => ( ord_less_real @ B3 @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inverse_less_imp_less_neg
% 0.29/0.47 thf(fact_314_less__imp__inverse__less__neg,axiom,
% 0.29/0.47 ! [A: real,B3: real] :
% 0.29/0.47 ( ( ord_less_real @ A @ B3 )
% 0.29/0.47 => ( ( ord_less_real @ B3 @ zero_zero_real )
% 0.29/0.47 => ( ord_less_real @ ( inverse_inverse_real @ B3 ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_imp_inverse_less_neg
% 0.29/0.47 thf(fact_315_inverse__negative__imp__negative,axiom,
% 0.29/0.47 ! [A: real] :
% 0.29/0.47 ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 0.29/0.47 => ( ( A != zero_zero_real )
% 0.29/0.47 => ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inverse_negative_imp_negative
% 0.29/0.47 thf(fact_316_inverse__positive__imp__positive,axiom,
% 0.29/0.47 ! [A: real] :
% 0.29/0.47 ( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 0.29/0.47 => ( ( A != zero_zero_real )
% 0.29/0.47 => ( ord_less_real @ zero_zero_real @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % inverse_positive_imp_positive
% 0.29/0.47 thf(fact_317_negative__imp__inverse__negative,axiom,
% 0.29/0.47 ! [A: real] :
% 0.29/0.47 ( ( ord_less_real @ A @ zero_zero_real )
% 0.29/0.47 => ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real ) ) ).
% 0.29/0.47
% 0.29/0.47 % negative_imp_inverse_negative
% 0.29/0.47 thf(fact_318_positive__imp__inverse__positive,axiom,
% 0.29/0.47 ! [A: real] :
% 0.29/0.47 ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.47 => ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % positive_imp_inverse_positive
% 0.29/0.47 thf(fact_319_nonzero__inverse__minus__eq,axiom,
% 0.29/0.47 ! [A: real] :
% 0.29/0.47 ( ( A != zero_zero_real )
% 0.29/0.47 => ( ( inverse_inverse_real @ ( uminus_uminus_real @ A ) )
% 0.29/0.47 = ( uminus_uminus_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % nonzero_inverse_minus_eq
% 0.29/0.47 thf(fact_320_image__atLeastZeroLessThan__int,axiom,
% 0.29/0.47 ! [U: int] :
% 0.29/0.47 ( ( ord_less_eq_int @ zero_zero_int @ U )
% 0.29/0.47 => ( ( set_or1199280219an_int @ zero_zero_int @ U )
% 0.29/0.47 = ( image_nat_int @ semiri2019852685at_int @ ( set_ord_lessThan_nat @ ( nat2 @ U ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % image_atLeastZeroLessThan_int
% 0.29/0.47 thf(fact_321_of__int__of__nat,axiom,
% 0.29/0.47 ( ring_1_of_int_int
% 0.29/0.47 = ( ^ [K2: int] : ( if_int @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_int @ ( semiri2019852685at_int @ ( nat2 @ ( uminus_uminus_int @ K2 ) ) ) ) @ ( semiri2019852685at_int @ ( nat2 @ K2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % of_int_of_nat
% 0.29/0.47 thf(fact_322_of__int__of__nat,axiom,
% 0.29/0.47 ( ring_1_of_int_real
% 0.29/0.47 = ( ^ [K2: int] : ( if_real @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_real @ ( semiri2110766477t_real @ ( nat2 @ ( uminus_uminus_int @ K2 ) ) ) ) @ ( semiri2110766477t_real @ ( nat2 @ K2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % of_int_of_nat
% 0.29/0.47 thf(fact_323_le__zero__eq,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.29/0.47 = ( N = zero_zero_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_zero_eq
% 0.29/0.47 thf(fact_324_neg__le__iff__le,axiom,
% 0.29/0.47 ! [B3: int,A: int] :
% 0.29/0.47 ( ( ord_less_eq_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A ) )
% 0.29/0.47 = ( ord_less_eq_int @ A @ B3 ) ) ).
% 0.29/0.47
% 0.29/0.47 % neg_le_iff_le
% 0.29/0.47 thf(fact_325_of__nat__le__iff,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_eq_real @ ( semiri2110766477t_real @ M ) @ ( semiri2110766477t_real @ N ) )
% 0.29/0.47 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % of_nat_le_iff
% 0.29/0.47 thf(fact_326_of__nat__le__iff,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_eq_int @ ( semiri2019852685at_int @ M ) @ ( semiri2019852685at_int @ N ) )
% 0.29/0.47 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % of_nat_le_iff
% 0.29/0.47 thf(fact_327_ivl__subset,axiom,
% 0.29/0.47 ! [I: nat,J: nat,M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_eq_set_nat @ ( set_or562006527an_nat @ I @ J ) @ ( set_or562006527an_nat @ M @ N ) )
% 0.29/0.47 = ( ( ord_less_eq_nat @ J @ I )
% 0.29/0.47 | ( ( ord_less_eq_nat @ M @ I )
% 0.29/0.47 & ( ord_less_eq_nat @ J @ N ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ivl_subset
% 0.29/0.47 thf(fact_328_ivl__subset,axiom,
% 0.29/0.47 ! [I: int,J: int,M: int,N: int] :
% 0.29/0.47 ( ( ord_less_eq_set_int @ ( set_or1199280219an_int @ I @ J ) @ ( set_or1199280219an_int @ M @ N ) )
% 0.29/0.47 = ( ( ord_less_eq_int @ J @ I )
% 0.29/0.47 | ( ( ord_less_eq_int @ M @ I )
% 0.29/0.47 & ( ord_less_eq_int @ J @ N ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ivl_subset
% 0.29/0.47 thf(fact_329_lessThan__subset__iff,axiom,
% 0.29/0.47 ! [X: int,Y: int] :
% 0.29/0.47 ( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X ) @ ( set_ord_lessThan_int @ Y ) )
% 0.29/0.47 = ( ord_less_eq_int @ X @ Y ) ) ).
% 0.29/0.47
% 0.29/0.47 % lessThan_subset_iff
% 0.29/0.47 thf(fact_330_lessThan__subset__iff,axiom,
% 0.29/0.47 ! [X: nat,Y: nat] :
% 0.29/0.47 ( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y ) )
% 0.29/0.47 = ( ord_less_eq_nat @ X @ Y ) ) ).
% 0.29/0.47
% 0.29/0.47 % lessThan_subset_iff
% 0.29/0.47 thf(fact_331_neg__0__le__iff__le,axiom,
% 0.29/0.47 ! [A: real] :
% 0.29/0.47 ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 0.29/0.47 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.29/0.47
% 0.29/0.47 % neg_0_le_iff_le
% 0.29/0.47 thf(fact_332_neg__0__le__iff__le,axiom,
% 0.29/0.47 ! [A: int] :
% 0.29/0.47 ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 0.29/0.47 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.29/0.47
% 0.29/0.47 % neg_0_le_iff_le
% 0.29/0.47 thf(fact_333_neg__le__0__iff__le,axiom,
% 0.29/0.47 ! [A: real] :
% 0.29/0.47 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 0.29/0.47 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 % neg_le_0_iff_le
% 0.29/0.47 thf(fact_334_neg__le__0__iff__le,axiom,
% 0.29/0.47 ! [A: int] :
% 0.29/0.47 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 0.29/0.47 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 % neg_le_0_iff_le
% 0.29/0.47 thf(fact_335_less__eq__neg__nonpos,axiom,
% 0.29/0.47 ! [A: real] :
% 0.29/0.47 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
% 0.29/0.47 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_eq_neg_nonpos
% 0.29/0.47 thf(fact_336_less__eq__neg__nonpos,axiom,
% 0.29/0.47 ! [A: int] :
% 0.29/0.47 ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
% 0.29/0.47 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_eq_neg_nonpos
% 0.29/0.47 thf(fact_337_neg__less__eq__nonneg,axiom,
% 0.29/0.47 ! [A: real] :
% 0.29/0.47 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
% 0.29/0.47 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 % neg_less_eq_nonneg
% 0.29/0.47 thf(fact_338_neg__less__eq__nonneg,axiom,
% 0.29/0.47 ! [A: int] :
% 0.29/0.47 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
% 0.29/0.47 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 % neg_less_eq_nonneg
% 0.29/0.47 thf(fact_339_inverse__nonpositive__iff__nonpositive,axiom,
% 0.29/0.47 ! [A: real] :
% 0.29/0.47 ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 0.29/0.47 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.29/0.47
% 0.29/0.47 % inverse_nonpositive_iff_nonpositive
% 0.29/0.47 thf(fact_340_inverse__nonnegative__iff__nonnegative,axiom,
% 0.29/0.47 ! [A: real] :
% 0.29/0.47 ( ( ord_less_eq_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 0.29/0.47 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 % inverse_nonnegative_iff_nonnegative
% 0.29/0.47 thf(fact_341_atLeastLessThan__iff,axiom,
% 0.29/0.47 ! [I: real,L: real,U: real] :
% 0.29/0.47 ( ( member_real @ I @ ( set_or2075149659n_real @ L @ U ) )
% 0.29/0.47 = ( ( ord_less_eq_real @ L @ I )
% 0.29/0.47 & ( ord_less_real @ I @ U ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % atLeastLessThan_iff
% 0.29/0.47 thf(fact_342_atLeastLessThan__iff,axiom,
% 0.29/0.47 ! [I: nat,L: nat,U: nat] :
% 0.29/0.47 ( ( member_nat @ I @ ( set_or562006527an_nat @ L @ U ) )
% 0.29/0.47 = ( ( ord_less_eq_nat @ L @ I )
% 0.29/0.47 & ( ord_less_nat @ I @ U ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % atLeastLessThan_iff
% 0.29/0.47 thf(fact_343_atLeastLessThan__iff,axiom,
% 0.29/0.47 ! [I: int,L: int,U: int] :
% 0.29/0.47 ( ( member_int @ I @ ( set_or1199280219an_int @ L @ U ) )
% 0.29/0.47 = ( ( ord_less_eq_int @ L @ I )
% 0.29/0.47 & ( ord_less_int @ I @ U ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % atLeastLessThan_iff
% 0.29/0.47 thf(fact_344_of__int__eq__0__iff,axiom,
% 0.29/0.47 ! [Z2: int] :
% 0.29/0.47 ( ( ( ring_1_of_int_real @ Z2 )
% 0.29/0.47 = zero_zero_real )
% 0.29/0.47 = ( Z2 = zero_zero_int ) ) ).
% 0.29/0.47
% 0.29/0.47 % of_int_eq_0_iff
% 0.29/0.47 thf(fact_345_negative__zle,axiom,
% 0.29/0.47 ! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri2019852685at_int @ N ) ) @ ( semiri2019852685at_int @ M ) ) ).
% 0.29/0.47
% 0.29/0.47 % negative_zle
% 0.29/0.47 thf(fact_346_nat__le__0,axiom,
% 0.29/0.47 ! [Z2: int] :
% 0.29/0.47 ( ( ord_less_eq_int @ Z2 @ zero_zero_int )
% 0.29/0.47 => ( ( nat2 @ Z2 )
% 0.29/0.47 = zero_zero_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % nat_le_0
% 0.29/0.47 thf(fact_347_nat__0__iff,axiom,
% 0.29/0.47 ! [I: int] :
% 0.29/0.47 ( ( ( nat2 @ I )
% 0.29/0.47 = zero_zero_nat )
% 0.29/0.47 = ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% 0.29/0.47
% 0.29/0.47 % nat_0_iff
% 0.29/0.47 thf(fact_348_int__nat__eq,axiom,
% 0.29/0.47 ! [Z2: int] :
% 0.29/0.47 ( ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.29/0.47 => ( ( semiri2019852685at_int @ ( nat2 @ Z2 ) )
% 0.29/0.47 = Z2 ) )
% 0.29/0.47 & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.29/0.47 => ( ( semiri2019852685at_int @ ( nat2 @ Z2 ) )
% 0.29/0.47 = zero_zero_int ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % int_nat_eq
% 0.29/0.47
% 0.29/0.47 % Helper facts (7)
% 0.29/0.47 thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
% 0.29/0.47 ! [X: int,Y: int] :
% 0.29/0.47 ( ( if_int @ $false @ X @ Y )
% 0.29/0.47 = Y ) ).
% 0.29/0.47
% 0.29/0.47 thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
% 0.29/0.47 ! [X: int,Y: int] :
% 0.29/0.47 ( ( if_int @ $true @ X @ Y )
% 0.29/0.47 = X ) ).
% 0.29/0.47
% 0.29/0.47 thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 0.29/0.47 ! [X: nat,Y: nat] :
% 0.29/0.47 ( ( if_nat @ $false @ X @ Y )
% 0.29/0.47 = Y ) ).
% 0.29/0.47
% 0.29/0.47 thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 0.29/0.47 ! [X: nat,Y: nat] :
% 0.29/0.47 ( ( if_nat @ $true @ X @ Y )
% 0.29/0.47 = X ) ).
% 0.29/0.47
% 0.29/0.47 thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
% 0.29/0.47 ! [P: $o] :
% 0.29/0.47 ( ( P = $true )
% 0.29/0.47 | ( P = $false ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
% 0.29/0.47 ! [X: real,Y: real] :
% 0.29/0.47 ( ( if_real @ $false @ X @ Y )
% 0.29/0.47 = Y ) ).
% 0.29/0.47
% 0.29/0.47 thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
% 0.29/0.47 ! [X: real,Y: real] :
% 0.29/0.47 ( ( if_real @ $true @ X @ Y )
% 0.29/0.47 = X ) ).
% 0.29/0.47
% 0.29/0.47 % Conjectures (1)
% 0.29/0.47 thf(conj_0,conjecture,
% 0.29/0.47 ( ( image_mset_nat_a @ ( nth_a @ ys ) @ ( image_mset_nat_nat @ f @ ( mset_set_nat @ ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ xs ) ) ) ) )
% 0.29/0.47 = ( image_mset_nat_a @ ( nth_a @ ys ) @ ( mset_set_nat @ ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ ys ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 %------------------------------------------------------------------------------
% 0.29/0.47 ------------------------
% 2.39/1.49 ---- Embedded in THF ---
% 2.39/1.50 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.39/1.50 %%% Generated on Tue Feb 24 08:25:15 EST 2026
% 2.39/1.50 %%% using '$$dhol' embedding, version 1.3.0.
% 2.39/1.50 %%% Logic specification used:
% 2.39/1.50 %%% thf(spec, logic, $$dhol).
% 2.39/1.50
% 2.39/1.50 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.VwIcuknk1l/DTF2DTF3941.p
% See solution above
% 2.68/1.50 ------------------------
% 2.68/1.53 ---- Cleaned THF ---
% 2.68/1.53 thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
% 2.68/1.53 set_list_a: $tType ).
% 2.68/1.53
% 2.68/1.53 thf(ty_n_t__Multiset__Omultiset_It__Nat__Onat_J,type,
% 2.68/1.53 multiset_nat: $tType ).
% 2.68/1.53
% 2.68/1.53 thf(ty_n_t__Multiset__Omultiset_It__Int__Oint_J,type,
% 2.68/1.53 multiset_int: $tType ).
% 2.68/1.53
% 2.68/1.53 thf(ty_n_t__Multiset__Omultiset_Itf__a_J,type,
% 2.68/1.53 multiset_a: $tType ).
% 2.68/1.53
% 2.68/1.53 thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
% 2.68/1.53 set_real: $tType ).
% 2.68/1.53
% 2.68/1.53 thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
% 2.68/1.53 list_nat: $tType ).
% 2.68/1.53
% 2.68/1.53 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 2.68/1.53 set_nat: $tType ).
% 2.68/1.53
% 2.68/1.53 thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
% 2.68/1.53 set_int: $tType ).
% 2.68/1.53
% 2.68/1.53 thf(ty_n_t__List__Olist_Itf__a_J,type,
% 2.68/1.53 list_a: $tType ).
% 2.68/1.53
% 2.68/1.53 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 2.68/1.53 set_a: $tType ).
% 2.68/1.53
% 2.68/1.53 thf(ty_n_t__Real__Oreal,type,
% 2.68/1.53 real: $tType ).
% 2.68/1.53
% 2.68/1.53 thf(ty_n_t__Nat__Onat,type,
% 2.68/1.53 nat: $tType ).
% 2.68/1.53
% 2.68/1.53 thf(ty_n_t__Int__Oint,type,
% 2.68/1.53 int: $tType ).
% 2.68/1.53
% 2.68/1.53 thf(ty_n_tf__a,type,
% 2.68/1.53 a: $tType ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
% 2.68/1.53 inverse_inverse_real: real > real ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Int__Oint,type,
% 2.68/1.53 bij_betw_nat_int: ( nat > int ) > set_nat > set_int > $o ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
% 2.68/1.53 bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Real__Oreal,type,
% 2.68/1.53 bij_betw_nat_real: ( nat > real ) > set_nat > set_real > $o ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001tf__a,type,
% 2.68/1.53 bij_betw_nat_a: ( nat > a ) > set_nat > set_a > $o ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Int__Oint_001t__Nat__Onat,type,
% 2.68/1.53 comp_int_int_nat: ( int > int ) > ( nat > int ) > nat > int ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Nat__Onat_001t__Nat__Onat,type,
% 2.68/1.53 comp_int_nat_nat: ( int > nat ) > ( nat > int ) > nat > nat ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Fun_Ocomp_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_Itf__a_J_001t__List__Olist_It__Nat__Onat_J,type,
% 2.68/1.53 comp_l728830689st_nat: ( list_nat > list_a ) > ( list_nat > list_nat ) > list_nat > list_a ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Fun_Ocomp_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J_001t__List__Olist_It__Nat__Onat_J,type,
% 2.68/1.53 comp_l1234327363st_nat: ( list_a > list_a ) > ( list_nat > list_a ) > list_nat > list_a ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Fun_Ocomp_001t__Multiset__Omultiset_It__Nat__Onat_J_001t__Multiset__Omultiset_It__Nat__Onat_J_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
% 2.68/1.53 comp_m1386984005et_nat: ( multiset_nat > multiset_nat ) > ( multiset_nat > multiset_nat ) > multiset_nat > multiset_nat ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Fun_Ocomp_001t__Multiset__Omultiset_It__Nat__Onat_J_001t__Multiset__Omultiset_Itf__a_J_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
% 2.68/1.53 comp_m1844659105et_nat: ( multiset_nat > multiset_a ) > ( multiset_nat > multiset_nat ) > multiset_nat > multiset_a ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Fun_Ocomp_001t__Multiset__Omultiset_It__Nat__Onat_J_001t__Nat__Onat_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
% 2.68/1.53 comp_m346803701et_nat: ( multiset_nat > nat ) > ( multiset_nat > multiset_nat ) > multiset_nat > nat ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Fun_Ocomp_001t__Multiset__Omultiset_Itf__a_J_001t__Multiset__Omultiset_It__Nat__Onat_J_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
% 2.68/1.53 comp_m1962190563et_nat: ( multiset_a > multiset_nat ) > ( multiset_nat > multiset_a ) > multiset_nat > multiset_nat ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Fun_Ocomp_001t__Multiset__Omultiset_Itf__a_J_001t__Multiset__Omultiset_Itf__a_J_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
% 2.68/1.53 comp_m540523139et_nat: ( multiset_a > multiset_a ) > ( multiset_nat > multiset_a ) > multiset_nat > multiset_a ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Fun_Ocomp_001t__Multiset__Omultiset_Itf__a_J_001t__Nat__Onat_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
% 2.68/1.53 comp_m1512105363et_nat: ( multiset_a > nat ) > ( multiset_nat > multiset_a ) > multiset_nat > nat ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Int__Oint_001t__Nat__Onat,type,
% 2.68/1.53 comp_nat_int_nat: ( nat > int ) > ( nat > nat ) > nat > int ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
% 2.68/1.53 comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001tf__a_001t__Nat__Onat,type,
% 2.68/1.53 comp_nat_a_nat: ( nat > a ) > ( nat > nat ) > nat > a ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Fun_Ocomp_001tf__a_001t__Nat__Onat_001t__Nat__Onat,type,
% 2.68/1.53 comp_a_nat_nat: ( a > nat ) > ( nat > a ) > nat > nat ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001t__Nat__Onat,type,
% 2.68/1.53 comp_a_a_nat: ( a > a ) > ( nat > a ) > nat > a ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Fun_Oinj__on_001t__Int__Oint_001t__Int__Oint,type,
% 2.68/1.53 inj_on_int_int: ( int > int ) > set_int > $o ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Fun_Oinj__on_001t__Int__Oint_001t__Nat__Onat,type,
% 2.68/1.53 inj_on_int_nat: ( int > nat ) > set_int > $o ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Int__Oint,type,
% 2.68/1.53 inj_on_nat_int: ( nat > int ) > set_nat > $o ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
% 2.68/1.53 inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Real__Oreal,type,
% 2.68/1.53 inj_on_nat_real: ( nat > real ) > set_nat > $o ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001tf__a,type,
% 2.68/1.53 inj_on_nat_a: ( nat > a ) > set_nat > $o ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
% 2.68/1.53 uminus_uminus_int: int > int ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
% 2.68/1.53 uminus_uminus_real: real > real ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
% 2.68/1.53 zero_zero_int: int ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
% 2.68/1.53 zero_z2005226153et_nat: multiset_nat ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_Itf__a_J,type,
% 2.68/1.53 zero_zero_multiset_a: multiset_a ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 2.68/1.53 zero_zero_nat: nat ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
% 2.68/1.53 zero_zero_real: real ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_If_001t__Int__Oint,type,
% 2.68/1.53 if_int: $o > int > int > int ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_If_001t__Nat__Onat,type,
% 2.68/1.53 if_nat: $o > nat > nat > nat ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_If_001t__Real__Oreal,type,
% 2.68/1.53 if_real: $o > real > real > real ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Int_Onat,type,
% 2.68/1.53 nat2: int > nat ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
% 2.68/1.53 ring_1_of_int_int: int > int ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
% 2.68/1.53 ring_1_of_int_real: int > real ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_List_Ogen__length_001tf__a,type,
% 2.68/1.53 gen_length_a: nat > list_a > nat ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
% 2.68/1.53 map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001tf__a,type,
% 2.68/1.53 map_nat_a: ( nat > a ) > list_nat > list_a ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_List_Olist_Omap_001tf__a_001t__Nat__Onat,type,
% 2.68/1.53 map_a_nat: ( a > nat ) > list_a > list_nat ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_List_Olist_Omap_001tf__a_001tf__a,type,
% 2.68/1.53 map_a_a: ( a > a ) > list_a > list_a ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_List_Olist__ex_001tf__a,type,
% 2.68/1.53 list_ex_a: ( a > $o ) > list_a > $o ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_List_Omap__tailrec_001t__Nat__Onat_001tf__a,type,
% 2.68/1.53 map_tailrec_nat_a: ( nat > a ) > list_nat > list_a ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_List_Onth_001t__Nat__Onat,type,
% 2.68/1.53 nth_nat: list_nat > nat > nat ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_List_Onth_001tf__a,type,
% 2.68/1.53 nth_a: list_a > nat > a ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_List_Oupt,type,
% 2.68/1.53 upt: nat > nat > list_nat ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Multiset_Oimage__mset_001t__Nat__Onat_001t__Int__Oint,type,
% 2.68/1.53 image_mset_nat_int: ( nat > int ) > multiset_nat > multiset_int ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Multiset_Oimage__mset_001t__Nat__Onat_001t__Nat__Onat,type,
% 2.68/1.53 image_mset_nat_nat: ( nat > nat ) > multiset_nat > multiset_nat ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Multiset_Oimage__mset_001t__Nat__Onat_001tf__a,type,
% 2.68/1.53 image_mset_nat_a: ( nat > a ) > multiset_nat > multiset_a ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Multiset_Oimage__mset_001tf__a_001t__Nat__Onat,type,
% 2.68/1.53 image_mset_a_nat: ( a > nat ) > multiset_a > multiset_nat ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Multiset_Oimage__mset_001tf__a_001tf__a,type,
% 2.68/1.53 image_mset_a_a: ( a > a ) > multiset_a > multiset_a ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Multiset_Omset_001t__Nat__Onat,type,
% 2.68/1.53 mset_nat: list_nat > multiset_nat ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Multiset_Omset_001tf__a,type,
% 2.68/1.53 mset_a: list_a > multiset_a ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Multiset_Omset__set_001t__Int__Oint,type,
% 2.68/1.53 mset_set_int: set_int > multiset_int ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Multiset_Omset__set_001t__Nat__Onat,type,
% 2.68/1.53 mset_set_nat: set_nat > multiset_nat ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Multiset_Omset__set_001tf__a,type,
% 2.68/1.53 mset_set_a: set_a > multiset_a ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Multiset_Osize__multiset_001t__Nat__Onat,type,
% 2.68/1.53 size_multiset_nat: ( nat > nat ) > multiset_nat > nat ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Multiset_Osize__multiset_001tf__a,type,
% 2.68/1.53 size_multiset_a: ( a > nat ) > multiset_a > nat ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
% 2.68/1.53 semiri2019852685at_int: nat > int ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
% 2.68/1.53 semiri1382578993at_nat: nat > nat ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
% 2.68/1.53 semiri2110766477t_real: nat > real ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
% 2.68/1.53 size_size_list_nat: list_nat > nat ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
% 2.68/1.53 size_size_list_a: list_a > nat ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
% 2.68/1.53 size_s1679505588et_nat: multiset_nat > nat ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_Itf__a_J,type,
% 2.68/1.53 size_size_multiset_a: multiset_a > nat ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
% 2.68/1.53 ord_less_int: int > int > $o ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 2.68/1.53 ord_less_nat: nat > nat > $o ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
% 2.68/1.53 ord_less_real: real > real > $o ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
% 2.68/1.53 ord_less_set_int: set_int > set_int > $o ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
% 2.68/1.53 ord_less_set_nat: set_nat > set_nat > $o ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
% 2.68/1.53 ord_less_set_real: set_real > set_real > $o ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
% 2.68/1.53 ord_less_eq_int: int > int > $o ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 2.68/1.53 ord_less_eq_nat: nat > nat > $o ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
% 2.68/1.53 ord_less_eq_real: real > real > $o ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
% 2.68/1.53 ord_less_eq_set_int: set_int > set_int > $o ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
% 2.68/1.53 ord_less_eq_set_nat: set_nat > set_nat > $o ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
% 2.68/1.53 image_int_int: ( int > int ) > set_int > set_int ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Nat__Onat,type,
% 2.68/1.53 image_int_nat: ( int > nat ) > set_int > set_nat ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
% 2.68/1.53 image_nat_int: ( nat > int ) > set_nat > set_int ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
% 2.68/1.53 image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Real__Oreal,type,
% 2.68/1.53 image_nat_real: ( nat > real ) > set_nat > set_real ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__a,type,
% 2.68/1.53 image_nat_a: ( nat > a ) > set_nat > set_a ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
% 2.68/1.53 set_or1199280219an_int: int > int > set_int ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
% 2.68/1.53 set_or562006527an_nat: nat > nat > set_nat ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Real__Oreal,type,
% 2.68/1.53 set_or2075149659n_real: real > real > set_real ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
% 2.68/1.53 set_ord_lessThan_int: int > set_int ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
% 2.68/1.53 set_ord_lessThan_nat: nat > set_nat ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
% 2.68/1.53 set_or1211449801n_real: real > set_real ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_member_001t__Int__Oint,type,
% 2.68/1.53 member_int: int > set_int > $o ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
% 2.68/1.53 member_list_a: list_a > set_list_a > $o ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_member_001t__Nat__Onat,type,
% 2.68/1.53 member_nat: nat > set_nat > $o ).
% 2.68/1.53
% 2.68/1.53 thf(sy_c_member_001t__Real__Oreal,type,
% 2.68/1.53 member_real: real > set_real > $o ).
% 2.68/1.53
% 2.68/1.53 thf(sy_v_f,type,
% 2.68/1.53 f: nat > nat ).
% 2.68/1.53
% 2.68/1.53 thf(sy_v_xs,type,
% 2.68/1.53 xs: list_a ).
% 2.68/1.53
% 2.68/1.53 thf(sy_v_ys,type,
% 2.68/1.53 ys: list_a ).
% 2.68/1.53
% 2.68/1.53 thf(fact_0__C1_C,axiom,
% 2.68/1.53 inj_on_nat_nat @ f @ ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ xs ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_1_calculation,axiom,
% 2.68/1.53 ( ( mset_a @ xs )
% 2.68/1.53 = ( image_mset_nat_a @ ( nth_a @ ys ) @ ( image_mset_nat_nat @ f @ ( mset_set_nat @ ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ xs ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_2_assms_I2_J,axiom,
% 2.68/1.53 ! [I: nat] :
% 2.68/1.53 ( ( ord_less_nat @ I @ ( size_size_list_a @ xs ) )
% 2.68/1.53 => ( ( nth_a @ xs @ I )
% 2.68/1.53 = ( nth_a @ ys @ ( f @ I ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_3__C2_C,axiom,
% 2.68/1.53 ( ( image_nat_nat @ f @ ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ xs ) ) )
% 2.68/1.53 = ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ ys ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_4_assms_I1_J,axiom,
% 2.68/1.53 bij_betw_nat_nat @ f @ ( set_ord_lessThan_nat @ ( size_size_list_a @ xs ) ) @ ( set_ord_lessThan_nat @ ( size_size_list_a @ ys ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_5__092_060open_062mset_A_Imap_A_I_I_B_J_Ays_A_092_060circ_062_Af_J_A_0910_O_O_060length_Axs_093_J_A_061_Aimage__mset_A_I_I_B_J_Ays_J_A_Iimage__mset_Af_A_Imset__set_A_1230_O_O_060length_Axs_125_J_J_092_060close_062,axiom,
% 2.68/1.53 ( ( mset_a @ ( map_nat_a @ ( comp_nat_a_nat @ ( nth_a @ ys ) @ f ) @ ( upt @ zero_zero_nat @ ( size_size_list_a @ xs ) ) ) )
% 2.68/1.53 = ( image_mset_nat_a @ ( nth_a @ ys ) @ ( image_mset_nat_nat @ f @ ( mset_set_nat @ ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ xs ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_6__092_060open_062xs_A_061_Amap_A_I_I_B_J_Ays_A_092_060circ_062_Af_J_A_0910_O_O_060length_Axs_093_092_060close_062,axiom,
% 2.68/1.53 ( xs
% 2.68/1.53 = ( map_nat_a @ ( comp_nat_a_nat @ ( nth_a @ ys ) @ f ) @ ( upt @ zero_zero_nat @ ( size_size_list_a @ xs ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_7_Ex__list__of__length,axiom,
% 2.68/1.53 ! [N: nat] :
% 2.68/1.53 ? [Xs: list_a] :
% 2.68/1.53 ( ( size_size_list_a @ Xs )
% 2.68/1.53 = N ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_8_neq__if__length__neq,axiom,
% 2.68/1.53 ! [Xs2: list_a,Ys: list_a] :
% 2.68/1.53 ( ( ( size_size_list_a @ Xs2 )
% 2.68/1.53 != ( size_size_list_a @ Ys ) )
% 2.68/1.53 => ( Xs2 != Ys ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_9_size__neq__size__imp__neq,axiom,
% 2.68/1.53 ! [X: list_a,Y: list_a] :
% 2.68/1.53 ( ( ( size_size_list_a @ X )
% 2.68/1.53 != ( size_size_list_a @ Y ) )
% 2.68/1.53 => ( X != Y ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_10_not__gr__zero,axiom,
% 2.68/1.53 ! [N: nat] :
% 2.68/1.53 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 2.68/1.53 = ( N = zero_zero_nat ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_11_bot__nat__0_Onot__eq__extremum,axiom,
% 2.68/1.53 ! [A: nat] :
% 2.68/1.53 ( ( A != zero_zero_nat )
% 2.68/1.53 = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_12_less__nat__zero__code,axiom,
% 2.68/1.53 ! [N: nat] :
% 2.68/1.53 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_13_neq0__conv,axiom,
% 2.68/1.53 ! [N: nat] :
% 2.68/1.53 ( ( N != zero_zero_nat )
% 2.68/1.53 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_14_map__map,axiom,
% 2.68/1.53 ! [F: a > a,G: nat > a,Xs2: list_nat] :
% 2.68/1.53 ( ( map_a_a @ F @ ( map_nat_a @ G @ Xs2 ) )
% 2.68/1.53 = ( map_nat_a @ ( comp_a_a_nat @ F @ G ) @ Xs2 ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_15_map__map,axiom,
% 2.68/1.53 ! [F: nat > a,G: nat > nat,Xs2: list_nat] :
% 2.68/1.53 ( ( map_nat_a @ F @ ( map_nat_nat @ G @ Xs2 ) )
% 2.68/1.53 = ( map_nat_a @ ( comp_nat_a_nat @ F @ G ) @ Xs2 ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_16_List_Omap_Ocomp,axiom,
% 2.68/1.53 ! [F: a > a,G: nat > a] :
% 2.68/1.53 ( ( comp_l1234327363st_nat @ ( map_a_a @ F ) @ ( map_nat_a @ G ) )
% 2.68/1.53 = ( map_nat_a @ ( comp_a_a_nat @ F @ G ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_17_List_Omap_Ocomp,axiom,
% 2.68/1.53 ! [F: nat > a,G: nat > nat] :
% 2.68/1.53 ( ( comp_l728830689st_nat @ ( map_nat_a @ F ) @ ( map_nat_nat @ G ) )
% 2.68/1.53 = ( map_nat_a @ ( comp_nat_a_nat @ F @ G ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_18_map__comp__map,axiom,
% 2.68/1.53 ! [F: a > a,G: nat > a] :
% 2.68/1.53 ( ( comp_l1234327363st_nat @ ( map_a_a @ F ) @ ( map_nat_a @ G ) )
% 2.68/1.53 = ( map_nat_a @ ( comp_a_a_nat @ F @ G ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_19_map__comp__map,axiom,
% 2.68/1.53 ! [F: nat > a,G: nat > nat] :
% 2.68/1.53 ( ( comp_l728830689st_nat @ ( map_nat_a @ F ) @ ( map_nat_nat @ G ) )
% 2.68/1.53 = ( map_nat_a @ ( comp_nat_a_nat @ F @ G ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_20_list_Omap__comp,axiom,
% 2.68/1.53 ! [G: a > a,F: nat > a,V: list_nat] :
% 2.68/1.53 ( ( map_a_a @ G @ ( map_nat_a @ F @ V ) )
% 2.68/1.53 = ( map_nat_a @ ( comp_a_a_nat @ G @ F ) @ V ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_21_list_Omap__comp,axiom,
% 2.68/1.53 ! [G: nat > a,F: nat > nat,V: list_nat] :
% 2.68/1.53 ( ( map_nat_a @ G @ ( map_nat_nat @ F @ V ) )
% 2.68/1.53 = ( map_nat_a @ ( comp_nat_a_nat @ G @ F ) @ V ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_22_List_Omap_Ocompositionality,axiom,
% 2.68/1.53 ! [F: a > a,G: nat > a,List: list_nat] :
% 2.68/1.53 ( ( map_a_a @ F @ ( map_nat_a @ G @ List ) )
% 2.68/1.53 = ( map_nat_a @ ( comp_a_a_nat @ F @ G ) @ List ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_23_List_Omap_Ocompositionality,axiom,
% 2.68/1.53 ! [F: nat > a,G: nat > nat,List: list_nat] :
% 2.68/1.53 ( ( map_nat_a @ F @ ( map_nat_nat @ G @ List ) )
% 2.68/1.53 = ( map_nat_a @ ( comp_nat_a_nat @ F @ G ) @ List ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_24_length__map,axiom,
% 2.68/1.53 ! [F: nat > a,Xs2: list_nat] :
% 2.68/1.53 ( ( size_size_list_a @ ( map_nat_a @ F @ Xs2 ) )
% 2.68/1.53 = ( size_size_list_nat @ Xs2 ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_25_length__map,axiom,
% 2.68/1.53 ! [F: a > a,Xs2: list_a] :
% 2.68/1.53 ( ( size_size_list_a @ ( map_a_a @ F @ Xs2 ) )
% 2.68/1.53 = ( size_size_list_a @ Xs2 ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_26_nth__map,axiom,
% 2.68/1.53 ! [N: nat,Xs2: list_nat,F: nat > a] :
% 2.68/1.53 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
% 2.68/1.53 => ( ( nth_a @ ( map_nat_a @ F @ Xs2 ) @ N )
% 2.68/1.53 = ( F @ ( nth_nat @ Xs2 @ N ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_27_nth__map,axiom,
% 2.68/1.53 ! [N: nat,Xs2: list_a,F: a > a] :
% 2.68/1.53 ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs2 ) )
% 2.68/1.53 => ( ( nth_a @ ( map_a_a @ F @ Xs2 ) @ N )
% 2.68/1.53 = ( F @ ( nth_a @ Xs2 @ N ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_28__092_060open_062mset_Axs_A_061_Amset_A_Imap_A_I_I_B_J_Ays_A_092_060circ_062_Af_J_A_0910_O_O_060length_Axs_093_J_092_060close_062,axiom,
% 2.68/1.53 ( ( mset_a @ xs )
% 2.68/1.53 = ( mset_a @ ( map_nat_a @ ( comp_nat_a_nat @ ( nth_a @ ys ) @ f ) @ ( upt @ zero_zero_nat @ ( size_size_list_a @ xs ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_29_nat__neq__iff,axiom,
% 2.68/1.53 ! [M: nat,N: nat] :
% 2.68/1.53 ( ( M != N )
% 2.68/1.53 = ( ( ord_less_nat @ M @ N )
% 2.68/1.53 | ( ord_less_nat @ N @ M ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_30_less__not__refl,axiom,
% 2.68/1.53 ! [N: nat] :
% 2.68/1.53 ~ ( ord_less_nat @ N @ N ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_31_less__not__refl2,axiom,
% 2.68/1.53 ! [N: nat,M: nat] :
% 2.68/1.53 ( ( ord_less_nat @ N @ M )
% 2.68/1.53 => ( M != N ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_32_less__not__refl3,axiom,
% 2.68/1.53 ! [S: nat,T: nat] :
% 2.68/1.53 ( ( ord_less_nat @ S @ T )
% 2.68/1.53 => ( S != T ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_33_less__irrefl__nat,axiom,
% 2.68/1.53 ! [N: nat] :
% 2.68/1.53 ~ ( ord_less_nat @ N @ N ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_34_nat__less__induct,axiom,
% 2.68/1.53 ! [P: nat > $o,N: nat] :
% 2.68/1.53 ( ! [N2: nat] :
% 2.68/1.53 ( ! [M2: nat] :
% 2.68/1.53 ( ( ord_less_nat @ M2 @ N2 )
% 2.68/1.53 => ( P @ M2 ) )
% 2.68/1.53 => ( P @ N2 ) )
% 2.68/1.53 => ( P @ N ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_35_infinite__descent,axiom,
% 2.68/1.53 ! [P: nat > $o,N: nat] :
% 2.68/1.53 ( ! [N2: nat] :
% 2.68/1.53 ( ~ ( P @ N2 )
% 2.68/1.53 => ? [M2: nat] :
% 2.68/1.53 ( ( ord_less_nat @ M2 @ N2 )
% 2.68/1.53 & ~ ( P @ M2 ) ) )
% 2.68/1.53 => ( P @ N ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_36_linorder__neqE__nat,axiom,
% 2.68/1.53 ! [X: nat,Y: nat] :
% 2.68/1.53 ( ( X != Y )
% 2.68/1.53 => ( ~ ( ord_less_nat @ X @ Y )
% 2.68/1.53 => ( ord_less_nat @ Y @ X ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_37_map__eq__imp__length__eq,axiom,
% 2.68/1.53 ! [F: nat > a,Xs2: list_nat,G: nat > a,Ys: list_nat] :
% 2.68/1.53 ( ( ( map_nat_a @ F @ Xs2 )
% 2.68/1.53 = ( map_nat_a @ G @ Ys ) )
% 2.68/1.53 => ( ( size_size_list_nat @ Xs2 )
% 2.68/1.53 = ( size_size_list_nat @ Ys ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_38_map__eq__imp__length__eq,axiom,
% 2.68/1.53 ! [F: nat > a,Xs2: list_nat,G: a > a,Ys: list_a] :
% 2.68/1.53 ( ( ( map_nat_a @ F @ Xs2 )
% 2.68/1.53 = ( map_a_a @ G @ Ys ) )
% 2.68/1.53 => ( ( size_size_list_nat @ Xs2 )
% 2.68/1.53 = ( size_size_list_a @ Ys ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_39_map__eq__imp__length__eq,axiom,
% 2.68/1.53 ! [F: a > a,Xs2: list_a,G: nat > a,Ys: list_nat] :
% 2.68/1.53 ( ( ( map_a_a @ F @ Xs2 )
% 2.68/1.53 = ( map_nat_a @ G @ Ys ) )
% 2.68/1.53 => ( ( size_size_list_a @ Xs2 )
% 2.68/1.53 = ( size_size_list_nat @ Ys ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_40_zero__less__iff__neq__zero,axiom,
% 2.68/1.53 ! [N: nat] :
% 2.68/1.53 ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.68/1.53 = ( N != zero_zero_nat ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_41_gr__implies__not__zero,axiom,
% 2.68/1.53 ! [M: nat,N: nat] :
% 2.68/1.53 ( ( ord_less_nat @ M @ N )
% 2.68/1.53 => ( N != zero_zero_nat ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_42_not__less__zero,axiom,
% 2.68/1.53 ! [N: nat] :
% 2.68/1.53 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_43_gr__zeroI,axiom,
% 2.68/1.53 ! [N: nat] :
% 2.68/1.53 ( ( N != zero_zero_nat )
% 2.68/1.53 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_44_bot__nat__0_Oextremum__strict,axiom,
% 2.68/1.53 ! [A: nat] :
% 2.68/1.53 ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_45_infinite__descent0,axiom,
% 2.68/1.53 ! [P: nat > $o,N: nat] :
% 2.68/1.53 ( ( P @ zero_zero_nat )
% 2.68/1.53 => ( ! [N2: nat] :
% 2.68/1.53 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 2.68/1.53 => ( ~ ( P @ N2 )
% 2.68/1.53 => ? [M2: nat] :
% 2.68/1.53 ( ( ord_less_nat @ M2 @ N2 )
% 2.68/1.53 & ~ ( P @ M2 ) ) ) )
% 2.68/1.53 => ( P @ N ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_46_gr__implies__not0,axiom,
% 2.68/1.53 ! [M: nat,N: nat] :
% 2.68/1.53 ( ( ord_less_nat @ M @ N )
% 2.68/1.53 => ( N != zero_zero_nat ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_47_less__zeroE,axiom,
% 2.68/1.53 ! [N: nat] :
% 2.68/1.53 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_48_not__less0,axiom,
% 2.68/1.53 ! [N: nat] :
% 2.68/1.53 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_49_not__gr0,axiom,
% 2.68/1.53 ! [N: nat] :
% 2.68/1.53 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 2.68/1.53 = ( N = zero_zero_nat ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_50_gr0I,axiom,
% 2.68/1.53 ! [N: nat] :
% 2.68/1.53 ( ( N != zero_zero_nat )
% 2.68/1.53 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_51_length__induct,axiom,
% 2.68/1.53 ! [P: list_a > $o,Xs2: list_a] :
% 2.68/1.53 ( ! [Xs: list_a] :
% 2.68/1.53 ( ! [Ys2: list_a] :
% 2.68/1.53 ( ( ord_less_nat @ ( size_size_list_a @ Ys2 ) @ ( size_size_list_a @ Xs ) )
% 2.68/1.53 => ( P @ Ys2 ) )
% 2.68/1.53 => ( P @ Xs ) )
% 2.68/1.53 => ( P @ Xs2 ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_52_all__nat__less__eq,axiom,
% 2.68/1.53 ! [N: nat,P: nat > $o] :
% 2.68/1.53 ( ( ! [M3: nat] :
% 2.68/1.53 ( ( ord_less_nat @ M3 @ N )
% 2.68/1.53 => ( P @ M3 ) ) )
% 2.68/1.53 = ( ! [X2: nat] :
% 2.68/1.53 ( ( member_nat @ X2 @ ( set_or562006527an_nat @ zero_zero_nat @ N ) )
% 2.68/1.53 => ( P @ X2 ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_53_ex__nat__less__eq,axiom,
% 2.68/1.53 ! [N: nat,P: nat > $o] :
% 2.68/1.53 ( ( ? [M3: nat] :
% 2.68/1.53 ( ( ord_less_nat @ M3 @ N )
% 2.68/1.53 & ( P @ M3 ) ) )
% 2.68/1.53 = ( ? [X2: nat] :
% 2.68/1.53 ( ( member_nat @ X2 @ ( set_or562006527an_nat @ zero_zero_nat @ N ) )
% 2.68/1.53 & ( P @ X2 ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_54_list__eq__iff__nth__eq,axiom,
% 2.68/1.53 ( ( ^ [Y2: list_a,Z: list_a] : ( Y2 = Z ) )
% 2.68/1.53 = ( ^ [Xs3: list_a,Ys3: list_a] :
% 2.68/1.53 ( ( ( size_size_list_a @ Xs3 )
% 2.68/1.53 = ( size_size_list_a @ Ys3 ) )
% 2.68/1.53 & ! [I2: nat] :
% 2.68/1.53 ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs3 ) )
% 2.68/1.53 => ( ( nth_a @ Xs3 @ I2 )
% 2.68/1.53 = ( nth_a @ Ys3 @ I2 ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_55_Skolem__list__nth,axiom,
% 2.68/1.53 ! [K: nat,P: nat > a > $o] :
% 2.68/1.53 ( ( ! [I2: nat] :
% 2.68/1.53 ( ( ord_less_nat @ I2 @ K )
% 2.68/1.53 => ? [X3: a] : ( P @ I2 @ X3 ) ) )
% 2.68/1.53 = ( ? [Xs3: list_a] :
% 2.68/1.53 ( ( ( size_size_list_a @ Xs3 )
% 2.68/1.53 = K )
% 2.68/1.53 & ! [I2: nat] :
% 2.68/1.53 ( ( ord_less_nat @ I2 @ K )
% 2.68/1.53 => ( P @ I2 @ ( nth_a @ Xs3 @ I2 ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_56_nth__equalityI,axiom,
% 2.68/1.53 ! [Xs2: list_a,Ys: list_a] :
% 2.68/1.53 ( ( ( size_size_list_a @ Xs2 )
% 2.68/1.53 = ( size_size_list_a @ Ys ) )
% 2.68/1.53 => ( ! [I3: nat] :
% 2.68/1.53 ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs2 ) )
% 2.68/1.53 => ( ( nth_a @ Xs2 @ I3 )
% 2.68/1.53 = ( nth_a @ Ys @ I3 ) ) )
% 2.68/1.53 => ( Xs2 = Ys ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_57_zero__reorient,axiom,
% 2.68/1.53 ! [X: nat] :
% 2.68/1.53 ( ( zero_zero_nat = X )
% 2.68/1.53 = ( X = zero_zero_nat ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_58_zero__reorient,axiom,
% 2.68/1.53 ! [X: int] :
% 2.68/1.53 ( ( zero_zero_int = X )
% 2.68/1.53 = ( X = zero_zero_int ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_59_zero__reorient,axiom,
% 2.68/1.53 ! [X: real] :
% 2.68/1.53 ( ( zero_zero_real = X )
% 2.68/1.53 = ( X = zero_zero_real ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_60_mset__map,axiom,
% 2.68/1.53 ! [F: a > a,Xs2: list_a] :
% 2.68/1.53 ( ( mset_a @ ( map_a_a @ F @ Xs2 ) )
% 2.68/1.53 = ( image_mset_a_a @ F @ ( mset_a @ Xs2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_61_mset__map,axiom,
% 2.68/1.53 ! [F: a > nat,Xs2: list_a] :
% 2.68/1.53 ( ( mset_nat @ ( map_a_nat @ F @ Xs2 ) )
% 2.68/1.53 = ( image_mset_a_nat @ F @ ( mset_a @ Xs2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_62_mset__map,axiom,
% 2.68/1.53 ! [F: nat > a,Xs2: list_nat] :
% 2.68/1.53 ( ( mset_a @ ( map_nat_a @ F @ Xs2 ) )
% 2.68/1.53 = ( image_mset_nat_a @ F @ ( mset_nat @ Xs2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_63_mset__map,axiom,
% 2.68/1.53 ! [F: nat > nat,Xs2: list_nat] :
% 2.68/1.53 ( ( mset_nat @ ( map_nat_nat @ F @ Xs2 ) )
% 2.68/1.53 = ( image_mset_nat_nat @ F @ ( mset_nat @ Xs2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_64_size__mset,axiom,
% 2.68/1.53 ! [Xs2: list_nat] :
% 2.68/1.53 ( ( size_s1679505588et_nat @ ( mset_nat @ Xs2 ) )
% 2.68/1.53 = ( size_size_list_nat @ Xs2 ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_65_size__mset,axiom,
% 2.68/1.53 ! [Xs2: list_a] :
% 2.68/1.53 ( ( size_size_multiset_a @ ( mset_a @ Xs2 ) )
% 2.68/1.53 = ( size_size_list_a @ Xs2 ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_66_lessThan__iff,axiom,
% 2.68/1.53 ! [I: int,K: int] :
% 2.68/1.53 ( ( member_int @ I @ ( set_ord_lessThan_int @ K ) )
% 2.68/1.53 = ( ord_less_int @ I @ K ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_67_lessThan__iff,axiom,
% 2.68/1.53 ! [I: real,K: real] :
% 2.68/1.53 ( ( member_real @ I @ ( set_or1211449801n_real @ K ) )
% 2.68/1.53 = ( ord_less_real @ I @ K ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_68_lessThan__iff,axiom,
% 2.68/1.53 ! [I: nat,K: nat] :
% 2.68/1.53 ( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
% 2.68/1.53 = ( ord_less_nat @ I @ K ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_69_image__mset__mset__set,axiom,
% 2.68/1.53 ! [F: nat > int,A2: set_nat] :
% 2.68/1.53 ( ( inj_on_nat_int @ F @ A2 )
% 2.68/1.53 => ( ( image_mset_nat_int @ F @ ( mset_set_nat @ A2 ) )
% 2.68/1.53 = ( mset_set_int @ ( image_nat_int @ F @ A2 ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_70_image__mset__mset__set,axiom,
% 2.68/1.53 ! [F: nat > a,A2: set_nat] :
% 2.68/1.53 ( ( inj_on_nat_a @ F @ A2 )
% 2.68/1.53 => ( ( image_mset_nat_a @ F @ ( mset_set_nat @ A2 ) )
% 2.68/1.53 = ( mset_set_a @ ( image_nat_a @ F @ A2 ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_71_image__mset__mset__set,axiom,
% 2.68/1.53 ! [F: nat > nat,A2: set_nat] :
% 2.68/1.53 ( ( inj_on_nat_nat @ F @ A2 )
% 2.68/1.53 => ( ( image_mset_nat_nat @ F @ ( mset_set_nat @ A2 ) )
% 2.68/1.53 = ( mset_set_nat @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_72_lessThan__atLeast0,axiom,
% 2.68/1.53 ( set_ord_lessThan_nat
% 2.68/1.53 = ( set_or562006527an_nat @ zero_zero_nat ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_73_bij__betw__def,axiom,
% 2.68/1.53 ( bij_betw_nat_int
% 2.68/1.53 = ( ^ [F2: nat > int,A3: set_nat,B: set_int] :
% 2.68/1.53 ( ( inj_on_nat_int @ F2 @ A3 )
% 2.68/1.53 & ( ( image_nat_int @ F2 @ A3 )
% 2.68/1.53 = B ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_74_bij__betw__def,axiom,
% 2.68/1.53 ( bij_betw_nat_nat
% 2.68/1.53 = ( ^ [F2: nat > nat,A3: set_nat,B: set_nat] :
% 2.68/1.53 ( ( inj_on_nat_nat @ F2 @ A3 )
% 2.68/1.53 & ( ( image_nat_nat @ F2 @ A3 )
% 2.68/1.53 = B ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_75_bij__betw__imageI,axiom,
% 2.68/1.53 ! [F: nat > int,A2: set_nat,B2: set_int] :
% 2.68/1.53 ( ( inj_on_nat_int @ F @ A2 )
% 2.68/1.53 => ( ( ( image_nat_int @ F @ A2 )
% 2.68/1.53 = B2 )
% 2.68/1.53 => ( bij_betw_nat_int @ F @ A2 @ B2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_76_bij__betw__imageI,axiom,
% 2.68/1.53 ! [F: nat > nat,A2: set_nat,B2: set_nat] :
% 2.68/1.53 ( ( inj_on_nat_nat @ F @ A2 )
% 2.68/1.53 => ( ( ( image_nat_nat @ F @ A2 )
% 2.68/1.53 = B2 )
% 2.68/1.53 => ( bij_betw_nat_nat @ F @ A2 @ B2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_77_inj__on__imp__bij__betw,axiom,
% 2.68/1.53 ! [F: nat > int,A2: set_nat] :
% 2.68/1.53 ( ( inj_on_nat_int @ F @ A2 )
% 2.68/1.53 => ( bij_betw_nat_int @ F @ A2 @ ( image_nat_int @ F @ A2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_78_inj__on__imp__bij__betw,axiom,
% 2.68/1.53 ! [F: nat > nat,A2: set_nat] :
% 2.68/1.53 ( ( inj_on_nat_nat @ F @ A2 )
% 2.68/1.53 => ( bij_betw_nat_nat @ F @ A2 @ ( image_nat_nat @ F @ A2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_79_comp__inj__on,axiom,
% 2.68/1.53 ! [F: nat > int,A2: set_nat,G: int > nat] :
% 2.68/1.53 ( ( inj_on_nat_int @ F @ A2 )
% 2.68/1.53 => ( ( inj_on_int_nat @ G @ ( image_nat_int @ F @ A2 ) )
% 2.68/1.53 => ( inj_on_nat_nat @ ( comp_int_nat_nat @ G @ F ) @ A2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_80_comp__inj__on,axiom,
% 2.68/1.53 ! [F: nat > nat,A2: set_nat,G: nat > a] :
% 2.68/1.53 ( ( inj_on_nat_nat @ F @ A2 )
% 2.68/1.53 => ( ( inj_on_nat_a @ G @ ( image_nat_nat @ F @ A2 ) )
% 2.68/1.53 => ( inj_on_nat_a @ ( comp_nat_a_nat @ G @ F ) @ A2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_81_comp__inj__on,axiom,
% 2.68/1.53 ! [F: nat > nat,A2: set_nat,G: nat > nat] :
% 2.68/1.53 ( ( inj_on_nat_nat @ F @ A2 )
% 2.68/1.53 => ( ( inj_on_nat_nat @ G @ ( image_nat_nat @ F @ A2 ) )
% 2.68/1.53 => ( inj_on_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_82_inj__on__imageI,axiom,
% 2.68/1.53 ! [G: nat > a,F: nat > nat,A2: set_nat] :
% 2.68/1.53 ( ( inj_on_nat_a @ ( comp_nat_a_nat @ G @ F ) @ A2 )
% 2.68/1.53 => ( inj_on_nat_a @ G @ ( image_nat_nat @ F @ A2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_83_inj__on__imageI,axiom,
% 2.68/1.53 ! [G: int > nat,F: nat > int,A2: set_nat] :
% 2.68/1.53 ( ( inj_on_nat_nat @ ( comp_int_nat_nat @ G @ F ) @ A2 )
% 2.68/1.53 => ( inj_on_int_nat @ G @ ( image_nat_int @ F @ A2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_84_inj__on__imageI,axiom,
% 2.68/1.53 ! [G: nat > nat,F: nat > nat,A2: set_nat] :
% 2.68/1.53 ( ( inj_on_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A2 )
% 2.68/1.53 => ( inj_on_nat_nat @ G @ ( image_nat_nat @ F @ A2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_85_comp__inj__on__iff,axiom,
% 2.68/1.53 ! [F: nat > int,A2: set_nat,F3: int > nat] :
% 2.68/1.53 ( ( inj_on_nat_int @ F @ A2 )
% 2.68/1.53 => ( ( inj_on_int_nat @ F3 @ ( image_nat_int @ F @ A2 ) )
% 2.68/1.53 = ( inj_on_nat_nat @ ( comp_int_nat_nat @ F3 @ F ) @ A2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_86_comp__inj__on__iff,axiom,
% 2.68/1.53 ! [F: nat > nat,A2: set_nat,F3: nat > a] :
% 2.68/1.53 ( ( inj_on_nat_nat @ F @ A2 )
% 2.68/1.53 => ( ( inj_on_nat_a @ F3 @ ( image_nat_nat @ F @ A2 ) )
% 2.68/1.53 = ( inj_on_nat_a @ ( comp_nat_a_nat @ F3 @ F ) @ A2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_87_comp__inj__on__iff,axiom,
% 2.68/1.53 ! [F: nat > nat,A2: set_nat,F3: nat > nat] :
% 2.68/1.53 ( ( inj_on_nat_nat @ F @ A2 )
% 2.68/1.53 => ( ( inj_on_nat_nat @ F3 @ ( image_nat_nat @ F @ A2 ) )
% 2.68/1.53 = ( inj_on_nat_nat @ ( comp_nat_nat_nat @ F3 @ F ) @ A2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_88_comp__apply,axiom,
% 2.68/1.53 ( comp_nat_a_nat
% 2.68/1.53 = ( ^ [F2: nat > a,G2: nat > nat,X2: nat] : ( F2 @ ( G2 @ X2 ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_89_lessThan__eq__iff,axiom,
% 2.68/1.53 ! [X: nat,Y: nat] :
% 2.68/1.53 ( ( ( set_ord_lessThan_nat @ X )
% 2.68/1.53 = ( set_ord_lessThan_nat @ Y ) )
% 2.68/1.53 = ( X = Y ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_90_image__mset__empty,axiom,
% 2.68/1.53 ! [F: nat > a] :
% 2.68/1.53 ( ( image_mset_nat_a @ F @ zero_z2005226153et_nat )
% 2.68/1.53 = zero_zero_multiset_a ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_91_image__mset__empty,axiom,
% 2.68/1.53 ! [F: nat > nat] :
% 2.68/1.53 ( ( image_mset_nat_nat @ F @ zero_z2005226153et_nat )
% 2.68/1.53 = zero_z2005226153et_nat ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_92_image__mset__is__empty__iff,axiom,
% 2.68/1.53 ! [F: nat > a,M4: multiset_nat] :
% 2.68/1.53 ( ( ( image_mset_nat_a @ F @ M4 )
% 2.68/1.53 = zero_zero_multiset_a )
% 2.68/1.53 = ( M4 = zero_z2005226153et_nat ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_93_image__mset__is__empty__iff,axiom,
% 2.68/1.53 ! [F: nat > nat,M4: multiset_nat] :
% 2.68/1.53 ( ( ( image_mset_nat_nat @ F @ M4 )
% 2.68/1.53 = zero_z2005226153et_nat )
% 2.68/1.53 = ( M4 = zero_z2005226153et_nat ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_94_size__image__mset,axiom,
% 2.68/1.53 ! [F: nat > a,M4: multiset_nat] :
% 2.68/1.53 ( ( size_size_multiset_a @ ( image_mset_nat_a @ F @ M4 ) )
% 2.68/1.53 = ( size_s1679505588et_nat @ M4 ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_95_size__image__mset,axiom,
% 2.68/1.53 ! [F: nat > nat,M4: multiset_nat] :
% 2.68/1.53 ( ( size_s1679505588et_nat @ ( image_mset_nat_nat @ F @ M4 ) )
% 2.68/1.53 = ( size_s1679505588et_nat @ M4 ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_96_mset__upt,axiom,
% 2.68/1.53 ! [M: nat,N: nat] :
% 2.68/1.53 ( ( mset_nat @ ( upt @ M @ N ) )
% 2.68/1.53 = ( mset_set_nat @ ( set_or562006527an_nat @ M @ N ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_97_inj__on__strict__subset,axiom,
% 2.68/1.53 ! [F: nat > int,B2: set_nat,A2: set_nat] :
% 2.68/1.53 ( ( inj_on_nat_int @ F @ B2 )
% 2.68/1.53 => ( ( ord_less_set_nat @ A2 @ B2 )
% 2.68/1.53 => ( ord_less_set_int @ ( image_nat_int @ F @ A2 ) @ ( image_nat_int @ F @ B2 ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_98_inj__on__strict__subset,axiom,
% 2.68/1.53 ! [F: nat > nat,B2: set_nat,A2: set_nat] :
% 2.68/1.53 ( ( inj_on_nat_nat @ F @ B2 )
% 2.68/1.53 => ( ( ord_less_set_nat @ A2 @ B2 )
% 2.68/1.53 => ( ord_less_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B2 ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_99_comp__eq__dest__lhs,axiom,
% 2.68/1.53 ! [A: nat > a,B3: nat > nat,C: nat > a,V: nat] :
% 2.68/1.53 ( ( ( comp_nat_a_nat @ A @ B3 )
% 2.68/1.53 = C )
% 2.68/1.53 => ( ( A @ ( B3 @ V ) )
% 2.68/1.53 = ( C @ V ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_100_comp__eq__elim,axiom,
% 2.68/1.53 ! [A: nat > a,B3: nat > nat,C: nat > a,D: nat > nat] :
% 2.68/1.53 ( ( ( comp_nat_a_nat @ A @ B3 )
% 2.68/1.53 = ( comp_nat_a_nat @ C @ D ) )
% 2.68/1.53 => ! [V2: nat] :
% 2.68/1.53 ( ( A @ ( B3 @ V2 ) )
% 2.68/1.53 = ( C @ ( D @ V2 ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_101_comp__eq__dest,axiom,
% 2.68/1.53 ! [A: nat > a,B3: nat > nat,C: nat > a,D: nat > nat,V: nat] :
% 2.68/1.53 ( ( ( comp_nat_a_nat @ A @ B3 )
% 2.68/1.53 = ( comp_nat_a_nat @ C @ D ) )
% 2.68/1.53 => ( ( A @ ( B3 @ V ) )
% 2.68/1.53 = ( C @ ( D @ V ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_102_comp__assoc,axiom,
% 2.68/1.53 ! [F: a > a,G: nat > a,H: nat > nat] :
% 2.68/1.53 ( ( comp_nat_a_nat @ ( comp_a_a_nat @ F @ G ) @ H )
% 2.68/1.53 = ( comp_a_a_nat @ F @ ( comp_nat_a_nat @ G @ H ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_103_comp__assoc,axiom,
% 2.68/1.53 ! [F: nat > a,G: nat > nat,H: nat > nat] :
% 2.68/1.53 ( ( comp_nat_a_nat @ ( comp_nat_a_nat @ F @ G ) @ H )
% 2.68/1.53 = ( comp_nat_a_nat @ F @ ( comp_nat_nat_nat @ G @ H ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_104_comp__def,axiom,
% 2.68/1.53 ( comp_nat_a_nat
% 2.68/1.53 = ( ^ [F2: nat > a,G2: nat > nat,X2: nat] : ( F2 @ ( G2 @ X2 ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_105_inj__on__inverseI,axiom,
% 2.68/1.53 ! [A2: set_nat,G: nat > nat,F: nat > nat] :
% 2.68/1.53 ( ! [X4: nat] :
% 2.68/1.53 ( ( member_nat @ X4 @ A2 )
% 2.68/1.53 => ( ( G @ ( F @ X4 ) )
% 2.68/1.53 = X4 ) )
% 2.68/1.53 => ( inj_on_nat_nat @ F @ A2 ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_106_inj__on__contraD,axiom,
% 2.68/1.53 ! [F: nat > nat,A2: set_nat,X: nat,Y: nat] :
% 2.68/1.53 ( ( inj_on_nat_nat @ F @ A2 )
% 2.68/1.53 => ( ( X != Y )
% 2.68/1.53 => ( ( member_nat @ X @ A2 )
% 2.68/1.53 => ( ( member_nat @ Y @ A2 )
% 2.68/1.53 => ( ( F @ X )
% 2.68/1.53 != ( F @ Y ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_107_inj__on__eq__iff,axiom,
% 2.68/1.53 ! [F: nat > nat,A2: set_nat,X: nat,Y: nat] :
% 2.68/1.53 ( ( inj_on_nat_nat @ F @ A2 )
% 2.68/1.53 => ( ( member_nat @ X @ A2 )
% 2.68/1.53 => ( ( member_nat @ Y @ A2 )
% 2.68/1.53 => ( ( ( F @ X )
% 2.68/1.53 = ( F @ Y ) )
% 2.68/1.53 = ( X = Y ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_108_inj__on__cong,axiom,
% 2.68/1.53 ! [A2: set_nat,F: nat > nat,G: nat > nat] :
% 2.68/1.53 ( ! [A4: nat] :
% 2.68/1.53 ( ( member_nat @ A4 @ A2 )
% 2.68/1.53 => ( ( F @ A4 )
% 2.68/1.53 = ( G @ A4 ) ) )
% 2.68/1.53 => ( ( inj_on_nat_nat @ F @ A2 )
% 2.68/1.53 = ( inj_on_nat_nat @ G @ A2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_109_inj__on__def,axiom,
% 2.68/1.53 ( inj_on_nat_nat
% 2.68/1.53 = ( ^ [F2: nat > nat,A3: set_nat] :
% 2.68/1.53 ! [X2: nat] :
% 2.68/1.53 ( ( member_nat @ X2 @ A3 )
% 2.68/1.53 => ! [Y3: nat] :
% 2.68/1.53 ( ( member_nat @ Y3 @ A3 )
% 2.68/1.53 => ( ( ( F2 @ X2 )
% 2.68/1.53 = ( F2 @ Y3 ) )
% 2.68/1.53 => ( X2 = Y3 ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_110_inj__onI,axiom,
% 2.68/1.53 ! [A2: set_nat,F: nat > nat] :
% 2.68/1.53 ( ! [X4: nat,Y4: nat] :
% 2.68/1.53 ( ( member_nat @ X4 @ A2 )
% 2.68/1.53 => ( ( member_nat @ Y4 @ A2 )
% 2.68/1.53 => ( ( ( F @ X4 )
% 2.68/1.53 = ( F @ Y4 ) )
% 2.68/1.53 => ( X4 = Y4 ) ) ) )
% 2.68/1.53 => ( inj_on_nat_nat @ F @ A2 ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_111_inj__onD,axiom,
% 2.68/1.53 ! [F: nat > nat,A2: set_nat,X: nat,Y: nat] :
% 2.68/1.53 ( ( inj_on_nat_nat @ F @ A2 )
% 2.68/1.53 => ( ( ( F @ X )
% 2.68/1.53 = ( F @ Y ) )
% 2.68/1.53 => ( ( member_nat @ X @ A2 )
% 2.68/1.53 => ( ( member_nat @ Y @ A2 )
% 2.68/1.53 => ( X = Y ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_112_bij__betw__iff__bijections,axiom,
% 2.68/1.53 ( bij_betw_nat_nat
% 2.68/1.53 = ( ^ [F2: nat > nat,A3: set_nat,B: set_nat] :
% 2.68/1.53 ? [G2: nat > nat] :
% 2.68/1.53 ( ! [X2: nat] :
% 2.68/1.53 ( ( member_nat @ X2 @ A3 )
% 2.68/1.53 => ( ( member_nat @ ( F2 @ X2 ) @ B )
% 2.68/1.53 & ( ( G2 @ ( F2 @ X2 ) )
% 2.68/1.53 = X2 ) ) )
% 2.68/1.53 & ! [X2: nat] :
% 2.68/1.53 ( ( member_nat @ X2 @ B )
% 2.68/1.53 => ( ( member_nat @ ( G2 @ X2 ) @ A3 )
% 2.68/1.53 & ( ( F2 @ ( G2 @ X2 ) )
% 2.68/1.53 = X2 ) ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_113_bij__betw__apply,axiom,
% 2.68/1.53 ! [F: nat > nat,A2: set_nat,B2: set_nat,A: nat] :
% 2.68/1.53 ( ( bij_betw_nat_nat @ F @ A2 @ B2 )
% 2.68/1.53 => ( ( member_nat @ A @ A2 )
% 2.68/1.53 => ( member_nat @ ( F @ A ) @ B2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_114_bij__betw__cong,axiom,
% 2.68/1.53 ! [A2: set_nat,F: nat > nat,G: nat > nat,A5: set_nat] :
% 2.68/1.53 ( ! [A4: nat] :
% 2.68/1.53 ( ( member_nat @ A4 @ A2 )
% 2.68/1.53 => ( ( F @ A4 )
% 2.68/1.53 = ( G @ A4 ) ) )
% 2.68/1.53 => ( ( bij_betw_nat_nat @ F @ A2 @ A5 )
% 2.68/1.53 = ( bij_betw_nat_nat @ G @ A2 @ A5 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_115_bij__betw__inv,axiom,
% 2.68/1.53 ! [F: nat > nat,A2: set_nat,B2: set_nat] :
% 2.68/1.53 ( ( bij_betw_nat_nat @ F @ A2 @ B2 )
% 2.68/1.53 => ? [G3: nat > nat] : ( bij_betw_nat_nat @ G3 @ B2 @ A2 ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_116_bij__betwE,axiom,
% 2.68/1.53 ! [F: nat > nat,A2: set_nat,B2: set_nat] :
% 2.68/1.53 ( ( bij_betw_nat_nat @ F @ A2 @ B2 )
% 2.68/1.53 => ! [X5: nat] :
% 2.68/1.53 ( ( member_nat @ X5 @ A2 )
% 2.68/1.53 => ( member_nat @ ( F @ X5 ) @ B2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_117_ex__mset,axiom,
% 2.68/1.53 ! [X6: multiset_a] :
% 2.68/1.53 ? [Xs: list_a] :
% 2.68/1.53 ( ( mset_a @ Xs )
% 2.68/1.53 = X6 ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_118_ex__mset,axiom,
% 2.68/1.53 ! [X6: multiset_nat] :
% 2.68/1.53 ? [Xs: list_nat] :
% 2.68/1.53 ( ( mset_nat @ Xs )
% 2.68/1.53 = X6 ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_119_image__eq__imp__comp,axiom,
% 2.68/1.53 ! [F: nat > nat,A2: set_nat,G: nat > nat,B2: set_nat,H: nat > a] :
% 2.68/1.53 ( ( ( image_nat_nat @ F @ A2 )
% 2.68/1.53 = ( image_nat_nat @ G @ B2 ) )
% 2.68/1.53 => ( ( image_nat_a @ ( comp_nat_a_nat @ H @ F ) @ A2 )
% 2.68/1.53 = ( image_nat_a @ ( comp_nat_a_nat @ H @ G ) @ B2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_120_image__eq__imp__comp,axiom,
% 2.68/1.53 ! [F: nat > nat,A2: set_nat,G: nat > nat,B2: set_nat,H: nat > nat] :
% 2.68/1.53 ( ( ( image_nat_nat @ F @ A2 )
% 2.68/1.53 = ( image_nat_nat @ G @ B2 ) )
% 2.68/1.53 => ( ( image_nat_nat @ ( comp_nat_nat_nat @ H @ F ) @ A2 )
% 2.68/1.53 = ( image_nat_nat @ ( comp_nat_nat_nat @ H @ G ) @ B2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_121_image__eq__imp__comp,axiom,
% 2.68/1.53 ! [F: nat > nat,A2: set_nat,G: nat > nat,B2: set_nat,H: nat > int] :
% 2.68/1.53 ( ( ( image_nat_nat @ F @ A2 )
% 2.68/1.53 = ( image_nat_nat @ G @ B2 ) )
% 2.68/1.53 => ( ( image_nat_int @ ( comp_nat_int_nat @ H @ F ) @ A2 )
% 2.68/1.53 = ( image_nat_int @ ( comp_nat_int_nat @ H @ G ) @ B2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_122_image__eq__imp__comp,axiom,
% 2.68/1.53 ! [F: nat > int,A2: set_nat,G: nat > int,B2: set_nat,H: int > nat] :
% 2.68/1.53 ( ( ( image_nat_int @ F @ A2 )
% 2.68/1.53 = ( image_nat_int @ G @ B2 ) )
% 2.68/1.53 => ( ( image_nat_nat @ ( comp_int_nat_nat @ H @ F ) @ A2 )
% 2.68/1.53 = ( image_nat_nat @ ( comp_int_nat_nat @ H @ G ) @ B2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_123_image__eq__imp__comp,axiom,
% 2.68/1.53 ! [F: nat > int,A2: set_nat,G: nat > int,B2: set_nat,H: int > int] :
% 2.68/1.53 ( ( ( image_nat_int @ F @ A2 )
% 2.68/1.53 = ( image_nat_int @ G @ B2 ) )
% 2.68/1.53 => ( ( image_nat_int @ ( comp_int_int_nat @ H @ F ) @ A2 )
% 2.68/1.53 = ( image_nat_int @ ( comp_int_int_nat @ H @ G ) @ B2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_124_image__comp,axiom,
% 2.68/1.53 ! [F: nat > a,G: nat > nat,R: set_nat] :
% 2.68/1.53 ( ( image_nat_a @ F @ ( image_nat_nat @ G @ R ) )
% 2.68/1.53 = ( image_nat_a @ ( comp_nat_a_nat @ F @ G ) @ R ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_125_image__comp,axiom,
% 2.68/1.53 ! [F: int > nat,G: nat > int,R: set_nat] :
% 2.68/1.53 ( ( image_int_nat @ F @ ( image_nat_int @ G @ R ) )
% 2.68/1.53 = ( image_nat_nat @ ( comp_int_nat_nat @ F @ G ) @ R ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_126_image__comp,axiom,
% 2.68/1.53 ! [F: int > int,G: nat > int,R: set_nat] :
% 2.68/1.53 ( ( image_int_int @ F @ ( image_nat_int @ G @ R ) )
% 2.68/1.53 = ( image_nat_int @ ( comp_int_int_nat @ F @ G ) @ R ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_127_image__comp,axiom,
% 2.68/1.53 ! [F: nat > nat,G: nat > nat,R: set_nat] :
% 2.68/1.53 ( ( image_nat_nat @ F @ ( image_nat_nat @ G @ R ) )
% 2.68/1.53 = ( image_nat_nat @ ( comp_nat_nat_nat @ F @ G ) @ R ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_128_image__comp,axiom,
% 2.68/1.53 ! [F: nat > int,G: nat > nat,R: set_nat] :
% 2.68/1.53 ( ( image_nat_int @ F @ ( image_nat_nat @ G @ R ) )
% 2.68/1.53 = ( image_nat_int @ ( comp_nat_int_nat @ F @ G ) @ R ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_129_atLeastLessThan__inj_I2_J,axiom,
% 2.68/1.53 ! [A: real,B3: real,C: real,D: real] :
% 2.68/1.53 ( ( ( set_or2075149659n_real @ A @ B3 )
% 2.68/1.53 = ( set_or2075149659n_real @ C @ D ) )
% 2.68/1.53 => ( ( ord_less_real @ A @ B3 )
% 2.68/1.53 => ( ( ord_less_real @ C @ D )
% 2.68/1.53 => ( B3 = D ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_130_atLeastLessThan__inj_I2_J,axiom,
% 2.68/1.53 ! [A: nat,B3: nat,C: nat,D: nat] :
% 2.68/1.53 ( ( ( set_or562006527an_nat @ A @ B3 )
% 2.68/1.53 = ( set_or562006527an_nat @ C @ D ) )
% 2.68/1.53 => ( ( ord_less_nat @ A @ B3 )
% 2.68/1.53 => ( ( ord_less_nat @ C @ D )
% 2.68/1.53 => ( B3 = D ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_131_atLeastLessThan__inj_I2_J,axiom,
% 2.68/1.53 ! [A: int,B3: int,C: int,D: int] :
% 2.68/1.53 ( ( ( set_or1199280219an_int @ A @ B3 )
% 2.68/1.53 = ( set_or1199280219an_int @ C @ D ) )
% 2.68/1.53 => ( ( ord_less_int @ A @ B3 )
% 2.68/1.53 => ( ( ord_less_int @ C @ D )
% 2.68/1.53 => ( B3 = D ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_132_atLeastLessThan__inj_I1_J,axiom,
% 2.68/1.53 ! [A: real,B3: real,C: real,D: real] :
% 2.68/1.53 ( ( ( set_or2075149659n_real @ A @ B3 )
% 2.68/1.53 = ( set_or2075149659n_real @ C @ D ) )
% 2.68/1.53 => ( ( ord_less_real @ A @ B3 )
% 2.68/1.53 => ( ( ord_less_real @ C @ D )
% 2.68/1.53 => ( A = C ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_133_atLeastLessThan__inj_I1_J,axiom,
% 2.68/1.53 ! [A: nat,B3: nat,C: nat,D: nat] :
% 2.68/1.53 ( ( ( set_or562006527an_nat @ A @ B3 )
% 2.68/1.53 = ( set_or562006527an_nat @ C @ D ) )
% 2.68/1.53 => ( ( ord_less_nat @ A @ B3 )
% 2.68/1.53 => ( ( ord_less_nat @ C @ D )
% 2.68/1.53 => ( A = C ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_134_atLeastLessThan__inj_I1_J,axiom,
% 2.68/1.53 ! [A: int,B3: int,C: int,D: int] :
% 2.68/1.53 ( ( ( set_or1199280219an_int @ A @ B3 )
% 2.68/1.53 = ( set_or1199280219an_int @ C @ D ) )
% 2.68/1.53 => ( ( ord_less_int @ A @ B3 )
% 2.68/1.53 => ( ( ord_less_int @ C @ D )
% 2.68/1.53 => ( A = C ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_135_atLeastLessThan__eq__iff,axiom,
% 2.68/1.53 ! [A: real,B3: real,C: real,D: real] :
% 2.68/1.53 ( ( ord_less_real @ A @ B3 )
% 2.68/1.53 => ( ( ord_less_real @ C @ D )
% 2.68/1.53 => ( ( ( set_or2075149659n_real @ A @ B3 )
% 2.68/1.53 = ( set_or2075149659n_real @ C @ D ) )
% 2.68/1.53 = ( ( A = C )
% 2.68/1.53 & ( B3 = D ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_136_atLeastLessThan__eq__iff,axiom,
% 2.68/1.53 ! [A: nat,B3: nat,C: nat,D: nat] :
% 2.68/1.53 ( ( ord_less_nat @ A @ B3 )
% 2.68/1.53 => ( ( ord_less_nat @ C @ D )
% 2.68/1.53 => ( ( ( set_or562006527an_nat @ A @ B3 )
% 2.68/1.53 = ( set_or562006527an_nat @ C @ D ) )
% 2.68/1.53 = ( ( A = C )
% 2.68/1.53 & ( B3 = D ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_137_atLeastLessThan__eq__iff,axiom,
% 2.68/1.53 ! [A: int,B3: int,C: int,D: int] :
% 2.68/1.53 ( ( ord_less_int @ A @ B3 )
% 2.68/1.53 => ( ( ord_less_int @ C @ D )
% 2.68/1.53 => ( ( ( set_or1199280219an_int @ A @ B3 )
% 2.68/1.53 = ( set_or1199280219an_int @ C @ D ) )
% 2.68/1.53 = ( ( A = C )
% 2.68/1.53 & ( B3 = D ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_138_lessThan__strict__subset__iff,axiom,
% 2.68/1.53 ! [M: int,N: int] :
% 2.68/1.53 ( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N ) )
% 2.68/1.53 = ( ord_less_int @ M @ N ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_139_lessThan__strict__subset__iff,axiom,
% 2.68/1.53 ! [M: real,N: real] :
% 2.68/1.53 ( ( ord_less_set_real @ ( set_or1211449801n_real @ M ) @ ( set_or1211449801n_real @ N ) )
% 2.68/1.53 = ( ord_less_real @ M @ N ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_140_lessThan__strict__subset__iff,axiom,
% 2.68/1.53 ! [M: nat,N: nat] :
% 2.68/1.53 ( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
% 2.68/1.53 = ( ord_less_nat @ M @ N ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_141_inj__on__image__iff,axiom,
% 2.68/1.53 ! [A2: set_nat,G: nat > nat,F: nat > nat] :
% 2.68/1.53 ( ! [X4: nat] :
% 2.68/1.53 ( ( member_nat @ X4 @ A2 )
% 2.68/1.53 => ! [Xa: nat] :
% 2.68/1.53 ( ( member_nat @ Xa @ A2 )
% 2.68/1.53 => ( ( ( G @ ( F @ X4 ) )
% 2.68/1.53 = ( G @ ( F @ Xa ) ) )
% 2.68/1.53 = ( ( G @ X4 )
% 2.68/1.53 = ( G @ Xa ) ) ) ) )
% 2.68/1.53 => ( ( inj_on_nat_nat @ F @ A2 )
% 2.68/1.53 => ( ( inj_on_nat_nat @ G @ ( image_nat_nat @ F @ A2 ) )
% 2.68/1.53 = ( inj_on_nat_nat @ G @ A2 ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_142_inj__on__imageI2,axiom,
% 2.68/1.53 ! [F3: nat > a,F: nat > nat,A2: set_nat] :
% 2.68/1.53 ( ( inj_on_nat_a @ ( comp_nat_a_nat @ F3 @ F ) @ A2 )
% 2.68/1.53 => ( inj_on_nat_nat @ F @ A2 ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_143_inj__on__imageI2,axiom,
% 2.68/1.53 ! [F3: nat > nat,F: nat > nat,A2: set_nat] :
% 2.68/1.53 ( ( inj_on_nat_nat @ ( comp_nat_nat_nat @ F3 @ F ) @ A2 )
% 2.68/1.53 => ( inj_on_nat_nat @ F @ A2 ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_144_bij__betw__imp__surj__on,axiom,
% 2.68/1.53 ! [F: nat > int,A2: set_nat,B2: set_int] :
% 2.68/1.53 ( ( bij_betw_nat_int @ F @ A2 @ B2 )
% 2.68/1.53 => ( ( image_nat_int @ F @ A2 )
% 2.68/1.53 = B2 ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_145_bij__betw__imp__surj__on,axiom,
% 2.68/1.53 ! [F: nat > nat,A2: set_nat,B2: set_nat] :
% 2.68/1.53 ( ( bij_betw_nat_nat @ F @ A2 @ B2 )
% 2.68/1.53 => ( ( image_nat_nat @ F @ A2 )
% 2.68/1.53 = B2 ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_146_bij__betw__comp__iff,axiom,
% 2.68/1.53 ! [F: nat > nat,A2: set_nat,A5: set_nat,F3: nat > a,A6: set_a] :
% 2.68/1.53 ( ( bij_betw_nat_nat @ F @ A2 @ A5 )
% 2.68/1.53 => ( ( bij_betw_nat_a @ F3 @ A5 @ A6 )
% 2.68/1.53 = ( bij_betw_nat_a @ ( comp_nat_a_nat @ F3 @ F ) @ A2 @ A6 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_147_bij__betw__comp__iff,axiom,
% 2.68/1.53 ! [F: nat > nat,A2: set_nat,A5: set_nat,F3: nat > nat,A6: set_nat] :
% 2.68/1.53 ( ( bij_betw_nat_nat @ F @ A2 @ A5 )
% 2.68/1.53 => ( ( bij_betw_nat_nat @ F3 @ A5 @ A6 )
% 2.68/1.53 = ( bij_betw_nat_nat @ ( comp_nat_nat_nat @ F3 @ F ) @ A2 @ A6 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_148_bij__betw__trans,axiom,
% 2.68/1.53 ! [F: nat > nat,A2: set_nat,B2: set_nat,G: nat > a,C2: set_a] :
% 2.68/1.53 ( ( bij_betw_nat_nat @ F @ A2 @ B2 )
% 2.68/1.53 => ( ( bij_betw_nat_a @ G @ B2 @ C2 )
% 2.68/1.53 => ( bij_betw_nat_a @ ( comp_nat_a_nat @ G @ F ) @ A2 @ C2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_149_bij__betw__trans,axiom,
% 2.68/1.53 ! [F: nat > nat,A2: set_nat,B2: set_nat,G: nat > nat,C2: set_nat] :
% 2.68/1.53 ( ( bij_betw_nat_nat @ F @ A2 @ B2 )
% 2.68/1.53 => ( ( bij_betw_nat_nat @ G @ B2 @ C2 )
% 2.68/1.53 => ( bij_betw_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A2 @ C2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_150_multiset_Omap__comp,axiom,
% 2.68/1.53 ! [G: a > a,F: nat > a,V: multiset_nat] :
% 2.68/1.53 ( ( image_mset_a_a @ G @ ( image_mset_nat_a @ F @ V ) )
% 2.68/1.53 = ( image_mset_nat_a @ ( comp_a_a_nat @ G @ F ) @ V ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_151_multiset_Omap__comp,axiom,
% 2.68/1.53 ! [G: a > nat,F: nat > a,V: multiset_nat] :
% 2.68/1.53 ( ( image_mset_a_nat @ G @ ( image_mset_nat_a @ F @ V ) )
% 2.68/1.53 = ( image_mset_nat_nat @ ( comp_a_nat_nat @ G @ F ) @ V ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_152_multiset_Omap__comp,axiom,
% 2.68/1.53 ! [G: nat > a,F: nat > nat,V: multiset_nat] :
% 2.68/1.53 ( ( image_mset_nat_a @ G @ ( image_mset_nat_nat @ F @ V ) )
% 2.68/1.53 = ( image_mset_nat_a @ ( comp_nat_a_nat @ G @ F ) @ V ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_153_multiset_Omap__comp,axiom,
% 2.68/1.53 ! [G: nat > nat,F: nat > nat,V: multiset_nat] :
% 2.68/1.53 ( ( image_mset_nat_nat @ G @ ( image_mset_nat_nat @ F @ V ) )
% 2.68/1.53 = ( image_mset_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ V ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_154_image__mset_Ocompositionality,axiom,
% 2.68/1.53 ! [F: a > a,G: nat > a,Multiset: multiset_nat] :
% 2.68/1.53 ( ( image_mset_a_a @ F @ ( image_mset_nat_a @ G @ Multiset ) )
% 2.68/1.53 = ( image_mset_nat_a @ ( comp_a_a_nat @ F @ G ) @ Multiset ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_155_image__mset_Ocompositionality,axiom,
% 2.68/1.53 ! [F: a > nat,G: nat > a,Multiset: multiset_nat] :
% 2.68/1.53 ( ( image_mset_a_nat @ F @ ( image_mset_nat_a @ G @ Multiset ) )
% 2.68/1.53 = ( image_mset_nat_nat @ ( comp_a_nat_nat @ F @ G ) @ Multiset ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_156_image__mset_Ocompositionality,axiom,
% 2.68/1.53 ! [F: nat > a,G: nat > nat,Multiset: multiset_nat] :
% 2.68/1.53 ( ( image_mset_nat_a @ F @ ( image_mset_nat_nat @ G @ Multiset ) )
% 2.68/1.53 = ( image_mset_nat_a @ ( comp_nat_a_nat @ F @ G ) @ Multiset ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_157_image__mset_Ocompositionality,axiom,
% 2.68/1.53 ! [F: nat > nat,G: nat > nat,Multiset: multiset_nat] :
% 2.68/1.53 ( ( image_mset_nat_nat @ F @ ( image_mset_nat_nat @ G @ Multiset ) )
% 2.68/1.53 = ( image_mset_nat_nat @ ( comp_nat_nat_nat @ F @ G ) @ Multiset ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_158_image__mset_Ocomp,axiom,
% 2.68/1.53 ! [F: a > a,G: nat > a] :
% 2.68/1.53 ( ( comp_m540523139et_nat @ ( image_mset_a_a @ F ) @ ( image_mset_nat_a @ G ) )
% 2.68/1.53 = ( image_mset_nat_a @ ( comp_a_a_nat @ F @ G ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_159_image__mset_Ocomp,axiom,
% 2.68/1.53 ! [F: a > nat,G: nat > a] :
% 2.68/1.53 ( ( comp_m1962190563et_nat @ ( image_mset_a_nat @ F ) @ ( image_mset_nat_a @ G ) )
% 2.68/1.53 = ( image_mset_nat_nat @ ( comp_a_nat_nat @ F @ G ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_160_image__mset_Ocomp,axiom,
% 2.68/1.53 ! [F: nat > a,G: nat > nat] :
% 2.68/1.53 ( ( comp_m1844659105et_nat @ ( image_mset_nat_a @ F ) @ ( image_mset_nat_nat @ G ) )
% 2.68/1.53 = ( image_mset_nat_a @ ( comp_nat_a_nat @ F @ G ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_161_image__mset_Ocomp,axiom,
% 2.68/1.53 ! [F: nat > nat,G: nat > nat] :
% 2.68/1.53 ( ( comp_m1386984005et_nat @ ( image_mset_nat_nat @ F ) @ ( image_mset_nat_nat @ G ) )
% 2.68/1.53 = ( image_mset_nat_nat @ ( comp_nat_nat_nat @ F @ G ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_162_bij__betw__imp__inj__on,axiom,
% 2.68/1.53 ! [F: nat > nat,A2: set_nat,B2: set_nat] :
% 2.68/1.53 ( ( bij_betw_nat_nat @ F @ A2 @ B2 )
% 2.68/1.53 => ( inj_on_nat_nat @ F @ A2 ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_163_mset__eq__length,axiom,
% 2.68/1.53 ! [Xs2: list_nat,Ys: list_nat] :
% 2.68/1.53 ( ( ( mset_nat @ Xs2 )
% 2.68/1.53 = ( mset_nat @ Ys ) )
% 2.68/1.53 => ( ( size_size_list_nat @ Xs2 )
% 2.68/1.53 = ( size_size_list_nat @ Ys ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_164_mset__eq__length,axiom,
% 2.68/1.53 ! [Xs2: list_a,Ys: list_a] :
% 2.68/1.53 ( ( ( mset_a @ Xs2 )
% 2.68/1.53 = ( mset_a @ Ys ) )
% 2.68/1.53 => ( ( size_size_list_a @ Xs2 )
% 2.68/1.53 = ( size_size_list_a @ Ys ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_165_image__eqI,axiom,
% 2.68/1.53 ! [B3: nat,F: nat > nat,X: nat,A2: set_nat] :
% 2.68/1.53 ( ( B3
% 2.68/1.53 = ( F @ X ) )
% 2.68/1.53 => ( ( member_nat @ X @ A2 )
% 2.68/1.53 => ( member_nat @ B3 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_166_image__eqI,axiom,
% 2.68/1.53 ! [B3: int,F: nat > int,X: nat,A2: set_nat] :
% 2.68/1.53 ( ( B3
% 2.68/1.53 = ( F @ X ) )
% 2.68/1.53 => ( ( member_nat @ X @ A2 )
% 2.68/1.53 => ( member_int @ B3 @ ( image_nat_int @ F @ A2 ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_167_mbs_Oless__not__eq,axiom,
% 2.68/1.53 ! [X: list_a,A2: set_list_a,Y: list_a] :
% 2.68/1.53 ( ( member_list_a @ X @ A2 )
% 2.68/1.53 => ( ( ord_less_nat @ ( size_size_list_a @ X ) @ ( size_size_list_a @ Y ) )
% 2.68/1.53 => ( X != Y ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_168_Sup_OSUP__image,axiom,
% 2.68/1.53 ! [Sup: set_a > a,G: nat > a,F: nat > nat,A2: set_nat] :
% 2.68/1.53 ( ( Sup @ ( image_nat_a @ G @ ( image_nat_nat @ F @ A2 ) ) )
% 2.68/1.53 = ( Sup @ ( image_nat_a @ ( comp_nat_a_nat @ G @ F ) @ A2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_169_Sup_OSUP__image,axiom,
% 2.68/1.53 ! [Sup: set_nat > nat,G: int > nat,F: nat > int,A2: set_nat] :
% 2.68/1.53 ( ( Sup @ ( image_int_nat @ G @ ( image_nat_int @ F @ A2 ) ) )
% 2.68/1.53 = ( Sup @ ( image_nat_nat @ ( comp_int_nat_nat @ G @ F ) @ A2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_170_Sup_OSUP__image,axiom,
% 2.68/1.53 ! [Sup: set_int > int,G: int > int,F: nat > int,A2: set_nat] :
% 2.68/1.53 ( ( Sup @ ( image_int_int @ G @ ( image_nat_int @ F @ A2 ) ) )
% 2.68/1.53 = ( Sup @ ( image_nat_int @ ( comp_int_int_nat @ G @ F ) @ A2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_171_Sup_OSUP__image,axiom,
% 2.68/1.53 ! [Sup: set_nat > nat,G: nat > nat,F: nat > nat,A2: set_nat] :
% 2.68/1.53 ( ( Sup @ ( image_nat_nat @ G @ ( image_nat_nat @ F @ A2 ) ) )
% 2.68/1.53 = ( Sup @ ( image_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_172_Sup_OSUP__image,axiom,
% 2.68/1.53 ! [Sup: set_int > int,G: nat > int,F: nat > nat,A2: set_nat] :
% 2.68/1.53 ( ( Sup @ ( image_nat_int @ G @ ( image_nat_nat @ F @ A2 ) ) )
% 2.68/1.53 = ( Sup @ ( image_nat_int @ ( comp_nat_int_nat @ G @ F ) @ A2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_173_Inf_OINF__image,axiom,
% 2.68/1.53 ! [Inf: set_a > a,G: nat > a,F: nat > nat,A2: set_nat] :
% 2.68/1.53 ( ( Inf @ ( image_nat_a @ G @ ( image_nat_nat @ F @ A2 ) ) )
% 2.68/1.53 = ( Inf @ ( image_nat_a @ ( comp_nat_a_nat @ G @ F ) @ A2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_174_Inf_OINF__image,axiom,
% 2.68/1.53 ! [Inf: set_nat > nat,G: int > nat,F: nat > int,A2: set_nat] :
% 2.68/1.53 ( ( Inf @ ( image_int_nat @ G @ ( image_nat_int @ F @ A2 ) ) )
% 2.68/1.53 = ( Inf @ ( image_nat_nat @ ( comp_int_nat_nat @ G @ F ) @ A2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_175_Inf_OINF__image,axiom,
% 2.68/1.53 ! [Inf: set_int > int,G: int > int,F: nat > int,A2: set_nat] :
% 2.68/1.53 ( ( Inf @ ( image_int_int @ G @ ( image_nat_int @ F @ A2 ) ) )
% 2.68/1.53 = ( Inf @ ( image_nat_int @ ( comp_int_int_nat @ G @ F ) @ A2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_176_Inf_OINF__image,axiom,
% 2.68/1.53 ! [Inf: set_nat > nat,G: nat > nat,F: nat > nat,A2: set_nat] :
% 2.68/1.53 ( ( Inf @ ( image_nat_nat @ G @ ( image_nat_nat @ F @ A2 ) ) )
% 2.68/1.53 = ( Inf @ ( image_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_177_Inf_OINF__image,axiom,
% 2.68/1.53 ! [Inf: set_int > int,G: nat > int,F: nat > nat,A2: set_nat] :
% 2.68/1.53 ( ( Inf @ ( image_nat_int @ G @ ( image_nat_nat @ F @ A2 ) ) )
% 2.68/1.53 = ( Inf @ ( image_nat_int @ ( comp_nat_int_nat @ G @ F ) @ A2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_178_less__numeral__extra_I3_J,axiom,
% 2.68/1.53 ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_179_less__numeral__extra_I3_J,axiom,
% 2.68/1.53 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_180_less__numeral__extra_I3_J,axiom,
% 2.68/1.53 ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_181_field__lbound__gt__zero,axiom,
% 2.68/1.53 ! [D1: real,D2: real] :
% 2.68/1.53 ( ( ord_less_real @ zero_zero_real @ D1 )
% 2.68/1.53 => ( ( ord_less_real @ zero_zero_real @ D2 )
% 2.68/1.53 => ? [E: real] :
% 2.68/1.53 ( ( ord_less_real @ zero_zero_real @ E )
% 2.68/1.53 & ( ord_less_real @ E @ D1 )
% 2.68/1.53 & ( ord_less_real @ E @ D2 ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_182_length__code,axiom,
% 2.68/1.53 ( size_size_list_a
% 2.68/1.53 = ( gen_length_a @ zero_zero_nat ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_183_imageI,axiom,
% 2.68/1.53 ! [X: nat,A2: set_nat,F: nat > nat] :
% 2.68/1.53 ( ( member_nat @ X @ A2 )
% 2.68/1.53 => ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_184_imageI,axiom,
% 2.68/1.53 ! [X: nat,A2: set_nat,F: nat > int] :
% 2.68/1.53 ( ( member_nat @ X @ A2 )
% 2.68/1.53 => ( member_int @ ( F @ X ) @ ( image_nat_int @ F @ A2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_185_image__iff,axiom,
% 2.68/1.53 ! [Z2: nat,F: nat > nat,A2: set_nat] :
% 2.68/1.53 ( ( member_nat @ Z2 @ ( image_nat_nat @ F @ A2 ) )
% 2.68/1.53 = ( ? [X2: nat] :
% 2.68/1.53 ( ( member_nat @ X2 @ A2 )
% 2.68/1.53 & ( Z2
% 2.68/1.53 = ( F @ X2 ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_186_image__iff,axiom,
% 2.68/1.53 ! [Z2: int,F: nat > int,A2: set_nat] :
% 2.68/1.53 ( ( member_int @ Z2 @ ( image_nat_int @ F @ A2 ) )
% 2.68/1.53 = ( ? [X2: nat] :
% 2.68/1.53 ( ( member_nat @ X2 @ A2 )
% 2.68/1.53 & ( Z2
% 2.68/1.53 = ( F @ X2 ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_187_bex__imageD,axiom,
% 2.68/1.53 ! [F: nat > nat,A2: set_nat,P: nat > $o] :
% 2.68/1.53 ( ? [X5: nat] :
% 2.68/1.53 ( ( member_nat @ X5 @ ( image_nat_nat @ F @ A2 ) )
% 2.68/1.53 & ( P @ X5 ) )
% 2.68/1.53 => ? [X4: nat] :
% 2.68/1.53 ( ( member_nat @ X4 @ A2 )
% 2.68/1.53 & ( P @ ( F @ X4 ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_188_bex__imageD,axiom,
% 2.68/1.53 ! [F: nat > int,A2: set_nat,P: int > $o] :
% 2.68/1.53 ( ? [X5: int] :
% 2.68/1.53 ( ( member_int @ X5 @ ( image_nat_int @ F @ A2 ) )
% 2.68/1.53 & ( P @ X5 ) )
% 2.68/1.53 => ? [X4: nat] :
% 2.68/1.53 ( ( member_nat @ X4 @ A2 )
% 2.68/1.53 & ( P @ ( F @ X4 ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_189_image__cong,axiom,
% 2.68/1.53 ! [M4: set_nat,N3: set_nat,F: nat > nat,G: nat > nat] :
% 2.68/1.53 ( ( M4 = N3 )
% 2.68/1.53 => ( ! [X4: nat] :
% 2.68/1.53 ( ( member_nat @ X4 @ N3 )
% 2.68/1.53 => ( ( F @ X4 )
% 2.68/1.53 = ( G @ X4 ) ) )
% 2.68/1.53 => ( ( image_nat_nat @ F @ M4 )
% 2.68/1.53 = ( image_nat_nat @ G @ N3 ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_190_image__cong,axiom,
% 2.68/1.53 ! [M4: set_nat,N3: set_nat,F: nat > int,G: nat > int] :
% 2.68/1.53 ( ( M4 = N3 )
% 2.68/1.53 => ( ! [X4: nat] :
% 2.68/1.53 ( ( member_nat @ X4 @ N3 )
% 2.68/1.53 => ( ( F @ X4 )
% 2.68/1.53 = ( G @ X4 ) ) )
% 2.68/1.53 => ( ( image_nat_int @ F @ M4 )
% 2.68/1.53 = ( image_nat_int @ G @ N3 ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_191_ball__imageD,axiom,
% 2.68/1.53 ! [F: nat > nat,A2: set_nat,P: nat > $o] :
% 2.68/1.53 ( ! [X4: nat] :
% 2.68/1.53 ( ( member_nat @ X4 @ ( image_nat_nat @ F @ A2 ) )
% 2.68/1.53 => ( P @ X4 ) )
% 2.68/1.53 => ! [X5: nat] :
% 2.68/1.53 ( ( member_nat @ X5 @ A2 )
% 2.68/1.53 => ( P @ ( F @ X5 ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_192_ball__imageD,axiom,
% 2.68/1.53 ! [F: nat > int,A2: set_nat,P: int > $o] :
% 2.68/1.53 ( ! [X4: int] :
% 2.68/1.53 ( ( member_int @ X4 @ ( image_nat_int @ F @ A2 ) )
% 2.68/1.53 => ( P @ X4 ) )
% 2.68/1.53 => ! [X5: nat] :
% 2.68/1.53 ( ( member_nat @ X5 @ A2 )
% 2.68/1.53 => ( P @ ( F @ X5 ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_193_rev__image__eqI,axiom,
% 2.68/1.53 ! [X: nat,A2: set_nat,B3: nat,F: nat > nat] :
% 2.68/1.53 ( ( member_nat @ X @ A2 )
% 2.68/1.53 => ( ( B3
% 2.68/1.53 = ( F @ X ) )
% 2.68/1.53 => ( member_nat @ B3 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_194_rev__image__eqI,axiom,
% 2.68/1.53 ! [X: nat,A2: set_nat,B3: int,F: nat > int] :
% 2.68/1.53 ( ( member_nat @ X @ A2 )
% 2.68/1.53 => ( ( B3
% 2.68/1.53 = ( F @ X ) )
% 2.68/1.53 => ( member_int @ B3 @ ( image_nat_int @ F @ A2 ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_195_Inf_OINF__cong,axiom,
% 2.68/1.53 ! [A2: set_nat,B2: set_nat,C2: nat > nat,D3: nat > nat,Inf: set_nat > nat] :
% 2.68/1.53 ( ( A2 = B2 )
% 2.68/1.53 => ( ! [X4: nat] :
% 2.68/1.53 ( ( member_nat @ X4 @ B2 )
% 2.68/1.53 => ( ( C2 @ X4 )
% 2.68/1.53 = ( D3 @ X4 ) ) )
% 2.68/1.53 => ( ( Inf @ ( image_nat_nat @ C2 @ A2 ) )
% 2.68/1.53 = ( Inf @ ( image_nat_nat @ D3 @ B2 ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_196_Inf_OINF__cong,axiom,
% 2.68/1.53 ! [A2: set_nat,B2: set_nat,C2: nat > int,D3: nat > int,Inf: set_int > int] :
% 2.68/1.53 ( ( A2 = B2 )
% 2.68/1.53 => ( ! [X4: nat] :
% 2.68/1.53 ( ( member_nat @ X4 @ B2 )
% 2.68/1.53 => ( ( C2 @ X4 )
% 2.68/1.53 = ( D3 @ X4 ) ) )
% 2.68/1.53 => ( ( Inf @ ( image_nat_int @ C2 @ A2 ) )
% 2.68/1.53 = ( Inf @ ( image_nat_int @ D3 @ B2 ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_197_Sup_OSUP__cong,axiom,
% 2.68/1.53 ! [A2: set_nat,B2: set_nat,C2: nat > nat,D3: nat > nat,Sup: set_nat > nat] :
% 2.68/1.53 ( ( A2 = B2 )
% 2.68/1.53 => ( ! [X4: nat] :
% 2.68/1.53 ( ( member_nat @ X4 @ B2 )
% 2.68/1.53 => ( ( C2 @ X4 )
% 2.68/1.53 = ( D3 @ X4 ) ) )
% 2.68/1.53 => ( ( Sup @ ( image_nat_nat @ C2 @ A2 ) )
% 2.68/1.53 = ( Sup @ ( image_nat_nat @ D3 @ B2 ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_198_Sup_OSUP__cong,axiom,
% 2.68/1.53 ! [A2: set_nat,B2: set_nat,C2: nat > int,D3: nat > int,Sup: set_int > int] :
% 2.68/1.53 ( ( A2 = B2 )
% 2.68/1.53 => ( ! [X4: nat] :
% 2.68/1.53 ( ( member_nat @ X4 @ B2 )
% 2.68/1.53 => ( ( C2 @ X4 )
% 2.68/1.53 = ( D3 @ X4 ) ) )
% 2.68/1.53 => ( ( Sup @ ( image_nat_int @ C2 @ A2 ) )
% 2.68/1.53 = ( Sup @ ( image_nat_int @ D3 @ B2 ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_199_map__eq__map__tailrec,axiom,
% 2.68/1.53 map_nat_a = map_tailrec_nat_a ).
% 2.68/1.53
% 2.68/1.53 thf(fact_200_list__ex__length,axiom,
% 2.68/1.53 ( list_ex_a
% 2.68/1.53 = ( ^ [P2: a > $o,Xs3: list_a] :
% 2.68/1.53 ? [N4: nat] :
% 2.68/1.53 ( ( ord_less_nat @ N4 @ ( size_size_list_a @ Xs3 ) )
% 2.68/1.53 & ( P2 @ ( nth_a @ Xs3 @ N4 ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_201_of__nat__0__less__iff,axiom,
% 2.68/1.53 ! [N: nat] :
% 2.68/1.53 ( ( ord_less_nat @ zero_zero_nat @ ( semiri1382578993at_nat @ N ) )
% 2.68/1.53 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_202_of__nat__0__less__iff,axiom,
% 2.68/1.53 ! [N: nat] :
% 2.68/1.53 ( ( ord_less_int @ zero_zero_int @ ( semiri2019852685at_int @ N ) )
% 2.68/1.53 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_203_of__nat__0__less__iff,axiom,
% 2.68/1.53 ! [N: nat] :
% 2.68/1.53 ( ( ord_less_real @ zero_zero_real @ ( semiri2110766477t_real @ N ) )
% 2.68/1.53 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_204_of__nat__eq__iff,axiom,
% 2.68/1.53 ! [M: nat,N: nat] :
% 2.68/1.53 ( ( ( semiri2019852685at_int @ M )
% 2.68/1.53 = ( semiri2019852685at_int @ N ) )
% 2.68/1.53 = ( M = N ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_205_of__nat__eq__iff,axiom,
% 2.68/1.53 ! [M: nat,N: nat] :
% 2.68/1.53 ( ( ( semiri2110766477t_real @ M )
% 2.68/1.53 = ( semiri2110766477t_real @ N ) )
% 2.68/1.53 = ( M = N ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_206_of__nat__0,axiom,
% 2.68/1.53 ( ( semiri1382578993at_nat @ zero_zero_nat )
% 2.68/1.53 = zero_zero_nat ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_207_of__nat__0,axiom,
% 2.68/1.53 ( ( semiri2019852685at_int @ zero_zero_nat )
% 2.68/1.53 = zero_zero_int ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_208_of__nat__0,axiom,
% 2.68/1.53 ( ( semiri2110766477t_real @ zero_zero_nat )
% 2.68/1.53 = zero_zero_real ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_209_of__nat__0__eq__iff,axiom,
% 2.68/1.53 ! [N: nat] :
% 2.68/1.53 ( ( zero_zero_nat
% 2.68/1.53 = ( semiri1382578993at_nat @ N ) )
% 2.68/1.53 = ( zero_zero_nat = N ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_210_of__nat__0__eq__iff,axiom,
% 2.68/1.53 ! [N: nat] :
% 2.68/1.53 ( ( zero_zero_int
% 2.68/1.53 = ( semiri2019852685at_int @ N ) )
% 2.68/1.53 = ( zero_zero_nat = N ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_211_of__nat__0__eq__iff,axiom,
% 2.68/1.53 ! [N: nat] :
% 2.68/1.53 ( ( zero_zero_real
% 2.68/1.53 = ( semiri2110766477t_real @ N ) )
% 2.68/1.53 = ( zero_zero_nat = N ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_212_of__nat__eq__0__iff,axiom,
% 2.68/1.53 ! [M: nat] :
% 2.68/1.53 ( ( ( semiri1382578993at_nat @ M )
% 2.68/1.53 = zero_zero_nat )
% 2.68/1.53 = ( M = zero_zero_nat ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_213_of__nat__eq__0__iff,axiom,
% 2.68/1.53 ! [M: nat] :
% 2.68/1.53 ( ( ( semiri2019852685at_int @ M )
% 2.68/1.53 = zero_zero_int )
% 2.68/1.53 = ( M = zero_zero_nat ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_214_of__nat__eq__0__iff,axiom,
% 2.68/1.53 ! [M: nat] :
% 2.68/1.53 ( ( ( semiri2110766477t_real @ M )
% 2.68/1.53 = zero_zero_real )
% 2.68/1.53 = ( M = zero_zero_nat ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_215_of__nat__less__iff,axiom,
% 2.68/1.53 ! [M: nat,N: nat] :
% 2.68/1.53 ( ( ord_less_nat @ ( semiri1382578993at_nat @ M ) @ ( semiri1382578993at_nat @ N ) )
% 2.68/1.53 = ( ord_less_nat @ M @ N ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_216_of__nat__less__iff,axiom,
% 2.68/1.53 ! [M: nat,N: nat] :
% 2.68/1.53 ( ( ord_less_int @ ( semiri2019852685at_int @ M ) @ ( semiri2019852685at_int @ N ) )
% 2.68/1.53 = ( ord_less_nat @ M @ N ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_217_of__nat__less__iff,axiom,
% 2.68/1.53 ! [M: nat,N: nat] :
% 2.68/1.53 ( ( ord_less_real @ ( semiri2110766477t_real @ M ) @ ( semiri2110766477t_real @ N ) )
% 2.68/1.53 = ( ord_less_nat @ M @ N ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_218_bij__betw__of__nat,axiom,
% 2.68/1.53 ! [N3: set_nat,A2: set_nat] :
% 2.68/1.53 ( ( bij_betw_nat_nat @ semiri1382578993at_nat @ N3 @ A2 )
% 2.68/1.53 = ( ( image_nat_nat @ semiri1382578993at_nat @ N3 )
% 2.68/1.53 = A2 ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_219_bij__betw__of__nat,axiom,
% 2.68/1.53 ! [N3: set_nat,A2: set_int] :
% 2.68/1.53 ( ( bij_betw_nat_int @ semiri2019852685at_int @ N3 @ A2 )
% 2.68/1.53 = ( ( image_nat_int @ semiri2019852685at_int @ N3 )
% 2.68/1.53 = A2 ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_220_bij__betw__of__nat,axiom,
% 2.68/1.53 ! [N3: set_nat,A2: set_real] :
% 2.68/1.53 ( ( bij_betw_nat_real @ semiri2110766477t_real @ N3 @ A2 )
% 2.68/1.53 = ( ( image_nat_real @ semiri2110766477t_real @ N3 )
% 2.68/1.53 = A2 ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_221_image__int__atLeastLessThan,axiom,
% 2.68/1.53 ! [A: nat,B3: nat] :
% 2.68/1.53 ( ( image_nat_int @ semiri2019852685at_int @ ( set_or562006527an_nat @ A @ B3 ) )
% 2.68/1.53 = ( set_or1199280219an_int @ ( semiri2019852685at_int @ A ) @ ( semiri2019852685at_int @ B3 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_222_inj__on__of__nat,axiom,
% 2.68/1.53 ! [N3: set_nat] : ( inj_on_nat_nat @ semiri1382578993at_nat @ N3 ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_223_inj__on__of__nat,axiom,
% 2.68/1.53 ! [N3: set_nat] : ( inj_on_nat_int @ semiri2019852685at_int @ N3 ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_224_inj__on__of__nat,axiom,
% 2.68/1.53 ! [N3: set_nat] : ( inj_on_nat_real @ semiri2110766477t_real @ N3 ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_225_of__nat__less__0__iff,axiom,
% 2.68/1.53 ! [M: nat] :
% 2.68/1.53 ~ ( ord_less_nat @ ( semiri1382578993at_nat @ M ) @ zero_zero_nat ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_226_of__nat__less__0__iff,axiom,
% 2.68/1.53 ! [M: nat] :
% 2.68/1.53 ~ ( ord_less_int @ ( semiri2019852685at_int @ M ) @ zero_zero_int ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_227_of__nat__less__0__iff,axiom,
% 2.68/1.53 ! [M: nat] :
% 2.68/1.53 ~ ( ord_less_real @ ( semiri2110766477t_real @ M ) @ zero_zero_real ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_228_of__nat__less__imp__less,axiom,
% 2.68/1.53 ! [M: nat,N: nat] :
% 2.68/1.53 ( ( ord_less_nat @ ( semiri1382578993at_nat @ M ) @ ( semiri1382578993at_nat @ N ) )
% 2.68/1.53 => ( ord_less_nat @ M @ N ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_229_of__nat__less__imp__less,axiom,
% 2.68/1.53 ! [M: nat,N: nat] :
% 2.68/1.53 ( ( ord_less_int @ ( semiri2019852685at_int @ M ) @ ( semiri2019852685at_int @ N ) )
% 2.68/1.53 => ( ord_less_nat @ M @ N ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_230_of__nat__less__imp__less,axiom,
% 2.68/1.53 ! [M: nat,N: nat] :
% 2.68/1.53 ( ( ord_less_real @ ( semiri2110766477t_real @ M ) @ ( semiri2110766477t_real @ N ) )
% 2.68/1.53 => ( ord_less_nat @ M @ N ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_231_less__imp__of__nat__less,axiom,
% 2.68/1.53 ! [M: nat,N: nat] :
% 2.68/1.53 ( ( ord_less_nat @ M @ N )
% 2.68/1.53 => ( ord_less_nat @ ( semiri1382578993at_nat @ M ) @ ( semiri1382578993at_nat @ N ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_232_less__imp__of__nat__less,axiom,
% 2.68/1.53 ! [M: nat,N: nat] :
% 2.68/1.53 ( ( ord_less_nat @ M @ N )
% 2.68/1.53 => ( ord_less_int @ ( semiri2019852685at_int @ M ) @ ( semiri2019852685at_int @ N ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_233_less__imp__of__nat__less,axiom,
% 2.68/1.53 ! [M: nat,N: nat] :
% 2.68/1.53 ( ( ord_less_nat @ M @ N )
% 2.68/1.53 => ( ord_less_real @ ( semiri2110766477t_real @ M ) @ ( semiri2110766477t_real @ N ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_234_pos__int__cases,axiom,
% 2.68/1.53 ! [K: int] :
% 2.68/1.53 ( ( ord_less_int @ zero_zero_int @ K )
% 2.68/1.53 => ~ ! [N2: nat] :
% 2.68/1.53 ( ( K
% 2.68/1.53 = ( semiri2019852685at_int @ N2 ) )
% 2.68/1.53 => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_235_zero__less__imp__eq__int,axiom,
% 2.68/1.53 ! [K: int] :
% 2.68/1.53 ( ( ord_less_int @ zero_zero_int @ K )
% 2.68/1.53 => ? [N2: nat] :
% 2.68/1.53 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 2.68/1.53 & ( K
% 2.68/1.53 = ( semiri2019852685at_int @ N2 ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_236_reals__Archimedean2,axiom,
% 2.68/1.53 ! [X: real] :
% 2.68/1.53 ? [N2: nat] : ( ord_less_real @ X @ ( semiri2110766477t_real @ N2 ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_237_int__int__eq,axiom,
% 2.68/1.53 ! [M: nat,N: nat] :
% 2.68/1.53 ( ( ( semiri2019852685at_int @ M )
% 2.68/1.53 = ( semiri2019852685at_int @ N ) )
% 2.68/1.53 = ( M = N ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_238_less__int__code_I1_J,axiom,
% 2.68/1.53 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_239_size__multiset__o__map,axiom,
% 2.68/1.53 ! [G: a > nat,F: nat > a] :
% 2.68/1.53 ( ( comp_m1512105363et_nat @ ( size_multiset_a @ G ) @ ( image_mset_nat_a @ F ) )
% 2.68/1.53 = ( size_multiset_nat @ ( comp_a_nat_nat @ G @ F ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_240_size__multiset__o__map,axiom,
% 2.68/1.53 ! [G: nat > nat,F: nat > nat] :
% 2.68/1.53 ( ( comp_m346803701et_nat @ ( size_multiset_nat @ G ) @ ( image_mset_nat_nat @ F ) )
% 2.68/1.53 = ( size_multiset_nat @ ( comp_nat_nat_nat @ G @ F ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_241_nat__int__comparison_I2_J,axiom,
% 2.68/1.53 ( ord_less_nat
% 2.68/1.53 = ( ^ [A7: nat,B4: nat] : ( ord_less_int @ ( semiri2019852685at_int @ A7 ) @ ( semiri2019852685at_int @ B4 ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_242_int__ops_I1_J,axiom,
% 2.68/1.53 ( ( semiri2019852685at_int @ zero_zero_nat )
% 2.68/1.53 = zero_zero_int ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_243_neg__int__cases,axiom,
% 2.68/1.53 ! [K: int] :
% 2.68/1.53 ( ( ord_less_int @ K @ zero_zero_int )
% 2.68/1.53 => ~ ! [N2: nat] :
% 2.68/1.53 ( ( K
% 2.68/1.53 = ( uminus_uminus_int @ ( semiri2019852685at_int @ N2 ) ) )
% 2.68/1.53 => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_244_add_Oinverse__inverse,axiom,
% 2.68/1.53 ! [A: int] :
% 2.68/1.53 ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
% 2.68/1.53 = A ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_245_neg__equal__iff__equal,axiom,
% 2.68/1.53 ! [A: int,B3: int] :
% 2.68/1.53 ( ( ( uminus_uminus_int @ A )
% 2.68/1.53 = ( uminus_uminus_int @ B3 ) )
% 2.68/1.53 = ( A = B3 ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_246_add_Oinverse__neutral,axiom,
% 2.68/1.53 ( ( uminus_uminus_real @ zero_zero_real )
% 2.68/1.53 = zero_zero_real ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_247_add_Oinverse__neutral,axiom,
% 2.68/1.53 ( ( uminus_uminus_int @ zero_zero_int )
% 2.68/1.53 = zero_zero_int ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_248_neg__0__equal__iff__equal,axiom,
% 2.68/1.53 ! [A: real] :
% 2.68/1.53 ( ( zero_zero_real
% 2.68/1.53 = ( uminus_uminus_real @ A ) )
% 2.68/1.53 = ( zero_zero_real = A ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_249_neg__0__equal__iff__equal,axiom,
% 2.68/1.53 ! [A: int] :
% 2.68/1.53 ( ( zero_zero_int
% 2.68/1.53 = ( uminus_uminus_int @ A ) )
% 2.68/1.53 = ( zero_zero_int = A ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_250_neg__equal__0__iff__equal,axiom,
% 2.68/1.53 ! [A: real] :
% 2.68/1.53 ( ( ( uminus_uminus_real @ A )
% 2.68/1.53 = zero_zero_real )
% 2.68/1.53 = ( A = zero_zero_real ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_251_neg__equal__0__iff__equal,axiom,
% 2.68/1.53 ! [A: int] :
% 2.68/1.53 ( ( ( uminus_uminus_int @ A )
% 2.68/1.53 = zero_zero_int )
% 2.68/1.53 = ( A = zero_zero_int ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_252_equal__neg__zero,axiom,
% 2.68/1.53 ! [A: real] :
% 2.68/1.53 ( ( A
% 2.68/1.53 = ( uminus_uminus_real @ A ) )
% 2.68/1.53 = ( A = zero_zero_real ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_253_equal__neg__zero,axiom,
% 2.68/1.53 ! [A: int] :
% 2.68/1.53 ( ( A
% 2.68/1.53 = ( uminus_uminus_int @ A ) )
% 2.68/1.53 = ( A = zero_zero_int ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_254_neg__equal__zero,axiom,
% 2.68/1.53 ! [A: real] :
% 2.68/1.53 ( ( ( uminus_uminus_real @ A )
% 2.68/1.53 = A )
% 2.68/1.53 = ( A = zero_zero_real ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_255_neg__equal__zero,axiom,
% 2.68/1.53 ! [A: int] :
% 2.68/1.53 ( ( ( uminus_uminus_int @ A )
% 2.68/1.53 = A )
% 2.68/1.53 = ( A = zero_zero_int ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_256_neg__less__iff__less,axiom,
% 2.68/1.53 ! [B3: real,A: real] :
% 2.68/1.53 ( ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A ) )
% 2.68/1.53 = ( ord_less_real @ A @ B3 ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_257_neg__less__iff__less,axiom,
% 2.68/1.53 ! [B3: int,A: int] :
% 2.68/1.53 ( ( ord_less_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A ) )
% 2.68/1.53 = ( ord_less_int @ A @ B3 ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_258_inj__uminus,axiom,
% 2.68/1.53 ! [A2: set_int] : ( inj_on_int_int @ uminus_uminus_int @ A2 ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_259_neg__less__0__iff__less,axiom,
% 2.68/1.53 ! [A: real] :
% 2.68/1.53 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 2.68/1.53 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_260_neg__less__0__iff__less,axiom,
% 2.68/1.53 ! [A: int] :
% 2.68/1.53 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 2.68/1.53 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_261_neg__0__less__iff__less,axiom,
% 2.68/1.53 ! [A: real] :
% 2.68/1.53 ( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 2.68/1.53 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_262_neg__0__less__iff__less,axiom,
% 2.68/1.53 ! [A: int] :
% 2.68/1.53 ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 2.68/1.53 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_263_neg__less__pos,axiom,
% 2.68/1.53 ! [A: real] :
% 2.68/1.53 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
% 2.68/1.53 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_264_neg__less__pos,axiom,
% 2.68/1.53 ! [A: int] :
% 2.68/1.53 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
% 2.68/1.53 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_265_less__neg__neg,axiom,
% 2.68/1.53 ! [A: real] :
% 2.68/1.53 ( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
% 2.68/1.53 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_266_less__neg__neg,axiom,
% 2.68/1.53 ! [A: int] :
% 2.68/1.53 ( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
% 2.68/1.53 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_267_negative__eq__positive,axiom,
% 2.68/1.53 ! [N: nat,M: nat] :
% 2.68/1.53 ( ( ( uminus_uminus_int @ ( semiri2019852685at_int @ N ) )
% 2.68/1.53 = ( semiri2019852685at_int @ M ) )
% 2.68/1.53 = ( ( N = zero_zero_nat )
% 2.68/1.53 & ( M = zero_zero_nat ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_268_int__cases2,axiom,
% 2.68/1.53 ! [Z2: int] :
% 2.68/1.53 ( ! [N2: nat] :
% 2.68/1.53 ( Z2
% 2.68/1.53 != ( semiri2019852685at_int @ N2 ) )
% 2.68/1.53 => ~ ! [N2: nat] :
% 2.68/1.53 ( Z2
% 2.68/1.53 != ( uminus_uminus_int @ ( semiri2019852685at_int @ N2 ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_269_minus__less__iff,axiom,
% 2.68/1.53 ! [A: real,B3: real] :
% 2.68/1.53 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B3 )
% 2.68/1.53 = ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ A ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_270_minus__less__iff,axiom,
% 2.68/1.53 ! [A: int,B3: int] :
% 2.68/1.53 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B3 )
% 2.68/1.53 = ( ord_less_int @ ( uminus_uminus_int @ B3 ) @ A ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_271_less__minus__iff,axiom,
% 2.68/1.53 ! [A: real,B3: real] :
% 2.68/1.53 ( ( ord_less_real @ A @ ( uminus_uminus_real @ B3 ) )
% 2.68/1.53 = ( ord_less_real @ B3 @ ( uminus_uminus_real @ A ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_272_less__minus__iff,axiom,
% 2.68/1.53 ! [A: int,B3: int] :
% 2.68/1.53 ( ( ord_less_int @ A @ ( uminus_uminus_int @ B3 ) )
% 2.68/1.53 = ( ord_less_int @ B3 @ ( uminus_uminus_int @ A ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_273_uminus__int__code_I1_J,axiom,
% 2.68/1.53 ( ( uminus_uminus_int @ zero_zero_int )
% 2.68/1.53 = zero_zero_int ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_274_verit__negate__coefficient_I2_J,axiom,
% 2.68/1.53 ! [A: real,B3: real] :
% 2.68/1.53 ( ( ord_less_real @ A @ B3 )
% 2.68/1.53 => ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_275_verit__negate__coefficient_I2_J,axiom,
% 2.68/1.53 ! [A: int,B3: int] :
% 2.68/1.53 ( ( ord_less_int @ A @ B3 )
% 2.68/1.53 => ( ord_less_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_276_equation__minus__iff,axiom,
% 2.68/1.53 ! [A: int,B3: int] :
% 2.68/1.53 ( ( A
% 2.68/1.53 = ( uminus_uminus_int @ B3 ) )
% 2.68/1.53 = ( B3
% 2.68/1.53 = ( uminus_uminus_int @ A ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_277_minus__equation__iff,axiom,
% 2.68/1.53 ! [A: int,B3: int] :
% 2.68/1.53 ( ( ( uminus_uminus_int @ A )
% 2.68/1.53 = B3 )
% 2.68/1.53 = ( ( uminus_uminus_int @ B3 )
% 2.68/1.53 = A ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_278_not__int__zless__negative,axiom,
% 2.68/1.53 ! [N: nat,M: nat] :
% 2.68/1.53 ~ ( ord_less_int @ ( semiri2019852685at_int @ N ) @ ( uminus_uminus_int @ ( semiri2019852685at_int @ M ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_279_int__cases4,axiom,
% 2.68/1.53 ! [M: int] :
% 2.68/1.53 ( ! [N2: nat] :
% 2.68/1.53 ( M
% 2.68/1.53 != ( semiri2019852685at_int @ N2 ) )
% 2.68/1.53 => ~ ! [N2: nat] :
% 2.68/1.53 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 2.68/1.53 => ( M
% 2.68/1.53 != ( uminus_uminus_int @ ( semiri2019852685at_int @ N2 ) ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_280_verit__comp__simplify1_I1_J,axiom,
% 2.68/1.53 ! [A: nat] :
% 2.68/1.53 ~ ( ord_less_nat @ A @ A ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_281_verit__comp__simplify1_I1_J,axiom,
% 2.68/1.53 ! [A: int] :
% 2.68/1.53 ~ ( ord_less_int @ A @ A ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_282_verit__comp__simplify1_I1_J,axiom,
% 2.68/1.53 ! [A: real] :
% 2.68/1.53 ~ ( ord_less_real @ A @ A ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_283_nat__int__comparison_I1_J,axiom,
% 2.68/1.53 ( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
% 2.68/1.53 = ( ^ [A7: nat,B4: nat] :
% 2.68/1.53 ( ( semiri2019852685at_int @ A7 )
% 2.68/1.53 = ( semiri2019852685at_int @ B4 ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_284_int__if,axiom,
% 2.68/1.53 ! [P: $o,A: nat,B3: nat] :
% 2.68/1.53 ( ( P
% 2.68/1.53 => ( ( semiri2019852685at_int @ ( if_nat @ P @ A @ B3 ) )
% 2.68/1.53 = ( semiri2019852685at_int @ A ) ) )
% 2.68/1.53 & ( ~ P
% 2.68/1.53 => ( ( semiri2019852685at_int @ ( if_nat @ P @ A @ B3 ) )
% 2.68/1.53 = ( semiri2019852685at_int @ B3 ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_285_int__cases3,axiom,
% 2.68/1.53 ! [K: int] :
% 2.68/1.53 ( ( K != zero_zero_int )
% 2.68/1.53 => ( ! [N2: nat] :
% 2.68/1.53 ( ( K
% 2.68/1.53 = ( semiri2019852685at_int @ N2 ) )
% 2.68/1.53 => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
% 2.68/1.53 => ~ ! [N2: nat] :
% 2.68/1.53 ( ( K
% 2.68/1.53 = ( uminus_uminus_int @ ( semiri2019852685at_int @ N2 ) ) )
% 2.68/1.53 => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_286_norm__frac_Oinduct,axiom,
% 2.68/1.53 ! [P: int > int > $o,A0: int,A1: int] :
% 2.68/1.53 ( ! [A4: int,B5: int] :
% 2.68/1.53 ( ( ( ord_less_int @ B5 @ zero_zero_int )
% 2.68/1.53 => ( P @ ( uminus_uminus_int @ A4 ) @ ( uminus_uminus_int @ B5 ) ) )
% 2.68/1.53 => ( P @ A4 @ B5 ) )
% 2.68/1.53 => ( P @ A0 @ A1 ) ) ).
% 2.68/1.53
% 2.68/1.53 thf(fact_287_ex__inverse__of__nat__less,axiom,
% 2.68/1.53 ! [X: real] :
% 2.68/1.53 ( ( ord_less_real @ zero_zero_real @ X )
% 2.68/1.53 => ? [N2: nat] :
% 2.68/1.53 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 2.68/1.53 & ( ord_less_real @ ( inverse_inverse_real @ ( semiri2110766477t_real @ N2 ) ) @ X ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_288_zero__less__nat__eq,axiom,
% 2.68/1.54 ! [Z2: int] :
% 2.68/1.54 ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z2 ) )
% 2.68/1.54 = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_289_nat__int,axiom,
% 2.68/1.54 ! [N: nat] :
% 2.68/1.54 ( ( nat2 @ ( semiri2019852685at_int @ N ) )
% 2.68/1.54 = N ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_290_zless__nat__conj,axiom,
% 2.68/1.54 ! [W: int,Z2: int] :
% 2.68/1.54 ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
% 2.68/1.54 = ( ( ord_less_int @ zero_zero_int @ Z2 )
% 2.68/1.54 & ( ord_less_int @ W @ Z2 ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_291_nat__zminus__int,axiom,
% 2.68/1.54 ! [N: nat] :
% 2.68/1.54 ( ( nat2 @ ( uminus_uminus_int @ ( semiri2019852685at_int @ N ) ) )
% 2.68/1.54 = zero_zero_nat ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_292_nat__zero__as__int,axiom,
% 2.68/1.54 ( zero_zero_nat
% 2.68/1.54 = ( nat2 @ zero_zero_int ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_293_forall__pos__mono,axiom,
% 2.68/1.54 ! [P: real > $o,E2: real] :
% 2.68/1.54 ( ! [D4: real,E: real] :
% 2.68/1.54 ( ( ord_less_real @ D4 @ E )
% 2.68/1.54 => ( ( P @ D4 )
% 2.68/1.54 => ( P @ E ) ) )
% 2.68/1.54 => ( ! [N2: nat] :
% 2.68/1.54 ( ( N2 != zero_zero_nat )
% 2.68/1.54 => ( P @ ( inverse_inverse_real @ ( semiri2110766477t_real @ N2 ) ) ) )
% 2.68/1.54 => ( ( ord_less_real @ zero_zero_real @ E2 )
% 2.68/1.54 => ( P @ E2 ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_294_real__arch__inverse,axiom,
% 2.68/1.54 ! [E2: real] :
% 2.68/1.54 ( ( ord_less_real @ zero_zero_real @ E2 )
% 2.68/1.54 = ( ? [N4: nat] :
% 2.68/1.54 ( ( N4 != zero_zero_nat )
% 2.68/1.54 & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri2110766477t_real @ N4 ) ) )
% 2.68/1.54 & ( ord_less_real @ ( inverse_inverse_real @ ( semiri2110766477t_real @ N4 ) ) @ E2 ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_295_nat__mono__iff,axiom,
% 2.68/1.54 ! [Z2: int,W: int] :
% 2.68/1.54 ( ( ord_less_int @ zero_zero_int @ Z2 )
% 2.68/1.54 => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
% 2.68/1.54 = ( ord_less_int @ W @ Z2 ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_296_zless__nat__eq__int__zless,axiom,
% 2.68/1.54 ! [M: nat,Z2: int] :
% 2.68/1.54 ( ( ord_less_nat @ M @ ( nat2 @ Z2 ) )
% 2.68/1.54 = ( ord_less_int @ ( semiri2019852685at_int @ M ) @ Z2 ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_297_split__nat,axiom,
% 2.68/1.54 ! [P: nat > $o,I: int] :
% 2.68/1.54 ( ( P @ ( nat2 @ I ) )
% 2.68/1.54 = ( ! [N4: nat] :
% 2.68/1.54 ( ( I
% 2.68/1.54 = ( semiri2019852685at_int @ N4 ) )
% 2.68/1.54 => ( P @ N4 ) )
% 2.68/1.54 & ( ( ord_less_int @ I @ zero_zero_int )
% 2.68/1.54 => ( P @ zero_zero_nat ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_298_inverse__positive__iff__positive,axiom,
% 2.68/1.54 ! [A: real] :
% 2.68/1.54 ( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 2.68/1.54 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_299_inverse__negative__iff__negative,axiom,
% 2.68/1.54 ! [A: real] :
% 2.68/1.54 ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 2.68/1.54 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_300_inverse__zero,axiom,
% 2.68/1.54 ( ( inverse_inverse_real @ zero_zero_real )
% 2.68/1.54 = zero_zero_real ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_301_inverse__nonzero__iff__nonzero,axiom,
% 2.68/1.54 ! [A: real] :
% 2.68/1.54 ( ( ( inverse_inverse_real @ A )
% 2.68/1.54 = zero_zero_real )
% 2.68/1.54 = ( A = zero_zero_real ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_302_inverse__less__iff__less,axiom,
% 2.68/1.54 ! [A: real,B3: real] :
% 2.68/1.54 ( ( ord_less_real @ zero_zero_real @ A )
% 2.68/1.54 => ( ( ord_less_real @ zero_zero_real @ B3 )
% 2.68/1.54 => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B3 ) )
% 2.68/1.54 = ( ord_less_real @ B3 @ A ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_303_inverse__less__iff__less__neg,axiom,
% 2.68/1.54 ! [A: real,B3: real] :
% 2.68/1.54 ( ( ord_less_real @ A @ zero_zero_real )
% 2.68/1.54 => ( ( ord_less_real @ B3 @ zero_zero_real )
% 2.68/1.54 => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B3 ) )
% 2.68/1.54 = ( ord_less_real @ B3 @ A ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_304_linordered__field__no__lb,axiom,
% 2.68/1.54 ! [X5: real] :
% 2.68/1.54 ? [Y4: real] : ( ord_less_real @ Y4 @ X5 ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_305_linordered__field__no__ub,axiom,
% 2.68/1.54 ! [X5: real] :
% 2.68/1.54 ? [X_1: real] : ( ord_less_real @ X5 @ X_1 ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_306_field__class_Ofield__inverse__zero,axiom,
% 2.68/1.54 ( ( inverse_inverse_real @ zero_zero_real )
% 2.68/1.54 = zero_zero_real ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_307_inverse__zero__imp__zero,axiom,
% 2.68/1.54 ! [A: real] :
% 2.68/1.54 ( ( ( inverse_inverse_real @ A )
% 2.68/1.54 = zero_zero_real )
% 2.68/1.54 => ( A = zero_zero_real ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_308_nonzero__inverse__eq__imp__eq,axiom,
% 2.68/1.54 ! [A: real,B3: real] :
% 2.68/1.54 ( ( ( inverse_inverse_real @ A )
% 2.68/1.54 = ( inverse_inverse_real @ B3 ) )
% 2.68/1.54 => ( ( A != zero_zero_real )
% 2.68/1.54 => ( ( B3 != zero_zero_real )
% 2.68/1.54 => ( A = B3 ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_309_nonzero__inverse__inverse__eq,axiom,
% 2.68/1.54 ! [A: real] :
% 2.68/1.54 ( ( A != zero_zero_real )
% 2.68/1.54 => ( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
% 2.68/1.54 = A ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_310_nonzero__imp__inverse__nonzero,axiom,
% 2.68/1.54 ! [A: real] :
% 2.68/1.54 ( ( A != zero_zero_real )
% 2.68/1.54 => ( ( inverse_inverse_real @ A )
% 2.68/1.54 != zero_zero_real ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_311_inverse__less__imp__less,axiom,
% 2.68/1.54 ! [A: real,B3: real] :
% 2.68/1.54 ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B3 ) )
% 2.68/1.54 => ( ( ord_less_real @ zero_zero_real @ A )
% 2.68/1.54 => ( ord_less_real @ B3 @ A ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_312_less__imp__inverse__less,axiom,
% 2.68/1.54 ! [A: real,B3: real] :
% 2.68/1.54 ( ( ord_less_real @ A @ B3 )
% 2.68/1.54 => ( ( ord_less_real @ zero_zero_real @ A )
% 2.68/1.54 => ( ord_less_real @ ( inverse_inverse_real @ B3 ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_313_inverse__less__imp__less__neg,axiom,
% 2.68/1.54 ! [A: real,B3: real] :
% 2.68/1.54 ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B3 ) )
% 2.68/1.54 => ( ( ord_less_real @ B3 @ zero_zero_real )
% 2.68/1.54 => ( ord_less_real @ B3 @ A ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_314_less__imp__inverse__less__neg,axiom,
% 2.68/1.54 ! [A: real,B3: real] :
% 2.68/1.54 ( ( ord_less_real @ A @ B3 )
% 2.68/1.54 => ( ( ord_less_real @ B3 @ zero_zero_real )
% 2.68/1.54 => ( ord_less_real @ ( inverse_inverse_real @ B3 ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_315_inverse__negative__imp__negative,axiom,
% 2.68/1.54 ! [A: real] :
% 2.68/1.54 ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 2.68/1.54 => ( ( A != zero_zero_real )
% 2.68/1.54 => ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_316_inverse__positive__imp__positive,axiom,
% 2.68/1.54 ! [A: real] :
% 2.68/1.54 ( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 2.68/1.54 => ( ( A != zero_zero_real )
% 2.68/1.54 => ( ord_less_real @ zero_zero_real @ A ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_317_negative__imp__inverse__negative,axiom,
% 2.68/1.54 ! [A: real] :
% 2.68/1.54 ( ( ord_less_real @ A @ zero_zero_real )
% 2.68/1.54 => ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_318_positive__imp__inverse__positive,axiom,
% 2.68/1.54 ! [A: real] :
% 2.68/1.54 ( ( ord_less_real @ zero_zero_real @ A )
% 2.68/1.54 => ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_319_nonzero__inverse__minus__eq,axiom,
% 2.68/1.54 ! [A: real] :
% 2.68/1.54 ( ( A != zero_zero_real )
% 2.68/1.54 => ( ( inverse_inverse_real @ ( uminus_uminus_real @ A ) )
% 2.68/1.54 = ( uminus_uminus_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_320_image__atLeastZeroLessThan__int,axiom,
% 2.68/1.54 ! [U: int] :
% 2.68/1.54 ( ( ord_less_eq_int @ zero_zero_int @ U )
% 2.68/1.54 => ( ( set_or1199280219an_int @ zero_zero_int @ U )
% 2.68/1.54 = ( image_nat_int @ semiri2019852685at_int @ ( set_ord_lessThan_nat @ ( nat2 @ U ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_321_of__int__of__nat,axiom,
% 2.68/1.54 ( ring_1_of_int_int
% 2.68/1.54 = ( ^ [K2: int] : ( if_int @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_int @ ( semiri2019852685at_int @ ( nat2 @ ( uminus_uminus_int @ K2 ) ) ) ) @ ( semiri2019852685at_int @ ( nat2 @ K2 ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_322_of__int__of__nat,axiom,
% 2.68/1.54 ( ring_1_of_int_real
% 2.68/1.54 = ( ^ [K2: int] : ( if_real @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_real @ ( semiri2110766477t_real @ ( nat2 @ ( uminus_uminus_int @ K2 ) ) ) ) @ ( semiri2110766477t_real @ ( nat2 @ K2 ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_323_le__zero__eq,axiom,
% 2.68/1.54 ! [N: nat] :
% 2.68/1.54 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 2.68/1.54 = ( N = zero_zero_nat ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_324_neg__le__iff__le,axiom,
% 2.68/1.54 ! [B3: int,A: int] :
% 2.68/1.54 ( ( ord_less_eq_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A ) )
% 2.68/1.54 = ( ord_less_eq_int @ A @ B3 ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_325_of__nat__le__iff,axiom,
% 2.68/1.54 ! [M: nat,N: nat] :
% 2.68/1.54 ( ( ord_less_eq_real @ ( semiri2110766477t_real @ M ) @ ( semiri2110766477t_real @ N ) )
% 2.68/1.54 = ( ord_less_eq_nat @ M @ N ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_326_of__nat__le__iff,axiom,
% 2.68/1.54 ! [M: nat,N: nat] :
% 2.68/1.54 ( ( ord_less_eq_int @ ( semiri2019852685at_int @ M ) @ ( semiri2019852685at_int @ N ) )
% 2.68/1.54 = ( ord_less_eq_nat @ M @ N ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_327_ivl__subset,axiom,
% 2.68/1.54 ! [I: nat,J: nat,M: nat,N: nat] :
% 2.68/1.54 ( ( ord_less_eq_set_nat @ ( set_or562006527an_nat @ I @ J ) @ ( set_or562006527an_nat @ M @ N ) )
% 2.68/1.54 = ( ( ord_less_eq_nat @ J @ I )
% 2.68/1.54 | ( ( ord_less_eq_nat @ M @ I )
% 2.68/1.54 & ( ord_less_eq_nat @ J @ N ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_328_ivl__subset,axiom,
% 2.68/1.54 ! [I: int,J: int,M: int,N: int] :
% 2.68/1.54 ( ( ord_less_eq_set_int @ ( set_or1199280219an_int @ I @ J ) @ ( set_or1199280219an_int @ M @ N ) )
% 2.68/1.54 = ( ( ord_less_eq_int @ J @ I )
% 2.68/1.54 | ( ( ord_less_eq_int @ M @ I )
% 2.68/1.54 & ( ord_less_eq_int @ J @ N ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_329_lessThan__subset__iff,axiom,
% 2.68/1.54 ! [X: int,Y: int] :
% 2.68/1.54 ( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X ) @ ( set_ord_lessThan_int @ Y ) )
% 2.68/1.54 = ( ord_less_eq_int @ X @ Y ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_330_lessThan__subset__iff,axiom,
% 2.68/1.54 ! [X: nat,Y: nat] :
% 2.68/1.54 ( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y ) )
% 2.68/1.54 = ( ord_less_eq_nat @ X @ Y ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_331_neg__0__le__iff__le,axiom,
% 2.68/1.54 ! [A: real] :
% 2.68/1.54 ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 2.68/1.54 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_332_neg__0__le__iff__le,axiom,
% 2.68/1.54 ! [A: int] :
% 2.68/1.54 ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 2.68/1.54 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_333_neg__le__0__iff__le,axiom,
% 2.68/1.54 ! [A: real] :
% 2.68/1.54 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 2.68/1.54 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_334_neg__le__0__iff__le,axiom,
% 2.68/1.54 ! [A: int] :
% 2.68/1.54 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 2.68/1.54 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_335_less__eq__neg__nonpos,axiom,
% 2.68/1.54 ! [A: real] :
% 2.68/1.54 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
% 2.68/1.54 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_336_less__eq__neg__nonpos,axiom,
% 2.68/1.54 ! [A: int] :
% 2.68/1.54 ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
% 2.68/1.54 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_337_neg__less__eq__nonneg,axiom,
% 2.68/1.54 ! [A: real] :
% 2.68/1.54 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
% 2.68/1.54 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_338_neg__less__eq__nonneg,axiom,
% 2.68/1.54 ! [A: int] :
% 2.68/1.54 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
% 2.68/1.54 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_339_inverse__nonpositive__iff__nonpositive,axiom,
% 2.68/1.54 ! [A: real] :
% 2.68/1.54 ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 2.68/1.54 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_340_inverse__nonnegative__iff__nonnegative,axiom,
% 2.68/1.54 ! [A: real] :
% 2.68/1.54 ( ( ord_less_eq_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 2.68/1.54 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_341_atLeastLessThan__iff,axiom,
% 2.68/1.54 ! [I: real,L: real,U: real] :
% 2.68/1.54 ( ( member_real @ I @ ( set_or2075149659n_real @ L @ U ) )
% 2.68/1.54 = ( ( ord_less_eq_real @ L @ I )
% 2.68/1.54 & ( ord_less_real @ I @ U ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_342_atLeastLessThan__iff,axiom,
% 2.68/1.54 ! [I: nat,L: nat,U: nat] :
% 2.68/1.54 ( ( member_nat @ I @ ( set_or562006527an_nat @ L @ U ) )
% 2.68/1.54 = ( ( ord_less_eq_nat @ L @ I )
% 2.68/1.54 & ( ord_less_nat @ I @ U ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_343_atLeastLessThan__iff,axiom,
% 2.68/1.54 ! [I: int,L: int,U: int] :
% 2.68/1.54 ( ( member_int @ I @ ( set_or1199280219an_int @ L @ U ) )
% 2.68/1.54 = ( ( ord_less_eq_int @ L @ I )
% 2.68/1.54 & ( ord_less_int @ I @ U ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_344_of__int__eq__0__iff,axiom,
% 2.68/1.54 ! [Z2: int] :
% 2.68/1.54 ( ( ( ring_1_of_int_real @ Z2 )
% 2.68/1.54 = zero_zero_real )
% 2.68/1.54 = ( Z2 = zero_zero_int ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_345_negative__zle,axiom,
% 2.68/1.54 ! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri2019852685at_int @ N ) ) @ ( semiri2019852685at_int @ M ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_346_nat__le__0,axiom,
% 2.68/1.54 ! [Z2: int] :
% 2.68/1.54 ( ( ord_less_eq_int @ Z2 @ zero_zero_int )
% 2.68/1.54 => ( ( nat2 @ Z2 )
% 2.68/1.54 = zero_zero_nat ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_347_nat__0__iff,axiom,
% 2.68/1.54 ! [I: int] :
% 2.68/1.54 ( ( ( nat2 @ I )
% 2.68/1.54 = zero_zero_nat )
% 2.68/1.54 = ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(fact_348_int__nat__eq,axiom,
% 2.68/1.54 ! [Z2: int] :
% 2.68/1.54 ( ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 2.68/1.54 => ( ( semiri2019852685at_int @ ( nat2 @ Z2 ) )
% 2.68/1.54 = Z2 ) )
% 2.68/1.54 & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 2.68/1.54 => ( ( semiri2019852685at_int @ ( nat2 @ Z2 ) )
% 2.68/1.54 = zero_zero_int ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
% 2.68/1.54 ! [X: int,Y: int] :
% 2.68/1.54 ( ( if_int @ $false @ X @ Y )
% 2.68/1.54 = Y ) ).
% 2.68/1.54
% 2.68/1.54 thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
% 2.68/1.54 ! [X: int,Y: int] :
% 2.68/1.54 ( ( if_int @ $true @ X @ Y )
% 2.68/1.54 = X ) ).
% 2.68/1.54
% 2.68/1.54 thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 2.68/1.54 ! [X: nat,Y: nat] :
% 2.68/1.54 ( ( if_nat @ $false @ X @ Y )
% 2.68/1.54 = Y ) ).
% 2.68/1.54
% 2.68/1.54 thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 2.68/1.54 ! [X: nat,Y: nat] :
% 2.68/1.54 ( ( if_nat @ $true @ X @ Y )
% 2.68/1.54 = X ) ).
% 2.68/1.54
% 2.68/1.54 thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
% 2.68/1.54 ! [P: $o] :
% 2.68/1.54 ( ( P = $true )
% 2.68/1.54 | ( P = $false ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
% 2.68/1.54 ! [X: real,Y: real] :
% 2.68/1.54 ( ( if_real @ $false @ X @ Y )
% 2.68/1.54 = Y ) ).
% 2.68/1.54
% 2.68/1.54 thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
% 2.68/1.54 ! [X: real,Y: real] :
% 2.68/1.54 ( ( if_real @ $true @ X @ Y )
% 2.68/1.54 = X ) ).
% 2.68/1.54
% 2.68/1.54 thf(conj_0,conjecture,
% 2.68/1.54 ( ( image_mset_nat_a @ ( nth_a @ ys ) @ ( image_mset_nat_nat @ f @ ( mset_set_nat @ ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ xs ) ) ) ) )
% 2.68/1.54 = ( image_mset_nat_a @ ( nth_a @ ys ) @ ( mset_set_nat @ ( set_or562006527an_nat @ zero_zero_nat @ ( size_size_list_a @ ys ) ) ) ) ) ).
% 2.68/1.54 ------------------------
% 2.68/1.56 % (4225)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_3941 for (2999ds/4Mi)
% 2.68/1.57 % (4226)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_3941 for (2999ds/27Mi)
% 2.68/1.57 % (4227)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_3941 for (2999ds/2Mi)
% 2.68/1.57 % (4228)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_3941 for (2999ds/2Mi)
% 2.68/1.57 % (4225)Instruction limit reached!
% 2.68/1.57 % (4225)------------------------------
% 2.68/1.57 % (4225)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.57 % (4225)Termination reason: Unknown
% 2.68/1.57 % (4225)Termination phase: shuffling
% 2.68/1.57
% 2.68/1.57 % (4225)Memory used [KB]: 1535
% 2.68/1.57 % (4225)Time elapsed: 0.003 s
% 2.68/1.57 % (4225)Instructions burned: 6 (million)
% 2.68/1.57 % (4225)------------------------------
% 2.68/1.57 % (4225)------------------------------
% 2.68/1.57 % (4227)Instruction limit reached!
% 2.68/1.57 % (4227)------------------------------
% 2.68/1.57 % (4227)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.57 % (4227)Termination reason: Unknown
% 2.68/1.57 % (4227)Termination phase: shuffling
% 2.68/1.57
% 2.68/1.57 % (4227)Memory used [KB]: 1407
% 2.68/1.57 % (4227)Time elapsed: 0.002 s
% 2.68/1.57 % (4227)Instructions burned: 3 (million)
% 2.68/1.57 % (4227)------------------------------
% 2.68/1.57 % (4227)------------------------------
% 2.68/1.57 % (4228)Instruction limit reached!
% 2.68/1.57 % (4228)------------------------------
% 2.68/1.57 % (4228)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.57 % (4228)Termination reason: Unknown
% 2.68/1.57 % (4228)Termination phase: shuffling
% 2.68/1.57
% 2.68/1.57 % (4228)Memory used [KB]: 1407
% 2.68/1.57 % (4228)Time elapsed: 0.002 s
% 2.68/1.57 % (4228)Instructions burned: 3 (million)
% 2.68/1.57 % (4228)------------------------------
% 2.68/1.57 % (4228)------------------------------
% 2.68/1.57 % (4224)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_3941 for (2999ds/183Mi)
% 2.68/1.57 % (4229)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_3941 for (2999ds/275Mi)
% 2.68/1.57 % (4230)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_3941 for (2999ds/18Mi)
% 2.68/1.57 % (4226)Instruction limit reached!
% 2.68/1.57 % (4226)------------------------------
% 2.68/1.57 % (4226)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.57 % (4226)Termination reason: Unknown
% 2.68/1.57 % (4226)Termination phase: Property scanning
% 2.68/1.57
% 2.68/1.57 % (4226)Memory used [KB]: 1791
% 2.68/1.57 % (4226)Time elapsed: 0.010 s
% 2.68/1.57 % (4226)Instructions burned: 29 (million)
% 2.68/1.57 % (4226)------------------------------
% 2.68/1.57 % (4226)------------------------------
% 2.68/1.58 % (4231)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_3941 for (2999ds/3Mi)
% 2.68/1.58 % (4232)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_3941 for (2999ds/37Mi)
% 2.68/1.58 % (4233)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_3941 for (2999ds/15Mi)
% 2.68/1.58 % (4231)Instruction limit reached!
% 2.68/1.58 % (4231)------------------------------
% 2.68/1.58 % (4231)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.58 % (4231)Termination reason: Unknown
% 2.68/1.58 % (4231)Termination phase: shuffling
% 2.68/1.58
% 2.68/1.58 % (4231)Memory used [KB]: 1535
% 2.68/1.58 % (4231)Time elapsed: 0.003 s
% 2.68/1.58 % (4231)Instructions burned: 6 (million)
% 2.68/1.58 % (4231)------------------------------
% 2.68/1.58 % (4231)------------------------------
% 2.68/1.58 % (4234)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_3941 for (2999ds/3Mi)
% 2.68/1.58 % (4233)Instruction limit reached!
% 2.68/1.58 % (4233)------------------------------
% 2.68/1.58 % (4233)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.58 % (4233)Termination reason: Unknown
% 2.68/1.58 % (4233)Termination phase: Property scanning
% 2.68/1.58
% 2.68/1.58 % (4233)Memory used [KB]: 1791
% 2.68/1.58 % (4233)Time elapsed: 0.008 s
% 2.68/1.58 % (4233)Instructions burned: 17 (million)
% 2.68/1.58 % (4233)------------------------------
% 2.68/1.58 % (4233)------------------------------
% 2.68/1.58 % (4234)Instruction limit reached!
% 2.68/1.58 % (4234)------------------------------
% 2.68/1.58 % (4234)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.58 % (4234)Termination reason: Unknown
% 2.68/1.58 % (4234)Termination phase: shuffling
% 2.68/1.58
% 2.68/1.58 % (4234)Memory used [KB]: 1407
% 2.68/1.58 % (4234)Time elapsed: 0.002 s
% 2.68/1.58 % (4234)Instructions burned: 3 (million)
% 2.68/1.58 % (4234)------------------------------
% 2.68/1.58 % (4234)------------------------------
% 2.68/1.59 % (4230)Instruction limit reached!
% 2.68/1.59 % (4230)------------------------------
% 2.68/1.59 % (4230)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.59 % (4230)Termination reason: Unknown
% 2.68/1.59 % (4230)Termination phase: Property scanning
% 2.68/1.59
% 2.68/1.59 % (4230)Memory used [KB]: 1791
% 2.68/1.59 % (4230)Time elapsed: 0.014 s
% 2.68/1.59 % (4230)Instructions burned: 19 (million)
% 2.68/1.59 % (4230)------------------------------
% 2.68/1.59 % (4230)------------------------------
% 2.68/1.59 % (4235)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_3941 for (2999ds/1041Mi)
% 2.68/1.59 % (4232)Instruction limit reached!
% 2.68/1.59 % (4232)------------------------------
% 2.68/1.59 % (4232)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.59 % (4232)Termination reason: Unknown
% 2.68/1.59 % (4232)Termination phase: shuffling
% 2.68/1.59
% 2.68/1.59 % (4232)Memory used [KB]: 2174
% 2.68/1.59 % (4232)Time elapsed: 0.013 s
% 2.68/1.59 % (4232)Instructions burned: 38 (million)
% 2.68/1.59 % (4232)------------------------------
% 2.68/1.59 % (4232)------------------------------
% 2.68/1.59 % (4236)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_3941 for (2999ds/7Mi)
% 2.68/1.59 % (4237)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_3941 for (2999ds/16Mi)
% 2.68/1.60 % (4236)Instruction limit reached!
% 2.68/1.60 % (4236)------------------------------
% 2.68/1.60 % (4236)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.60 % (4236)Termination reason: Unknown
% 2.68/1.60 % (4236)Termination phase: shuffling
% 2.68/1.60
% 2.68/1.60 % (4236)Memory used [KB]: 1535
% 2.68/1.60 % (4236)Time elapsed: 0.004 s
% 2.68/1.60 % (4236)Instructions burned: 7 (million)
% 2.68/1.60 % (4236)------------------------------
% 2.68/1.60 % (4236)------------------------------
% 2.72/1.60 % (4239)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_3941 for (2999ds/3Mi)
% 2.72/1.60 % (4237)Instruction limit reached!
% 2.72/1.60 % (4237)------------------------------
% 2.72/1.60 % (4237)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.60 % (4237)Termination reason: Unknown
% 2.72/1.60 % (4237)Termination phase: Property scanning
% 2.72/1.60
% 2.72/1.60 % (4237)Memory used [KB]: 1791
% 2.72/1.60 % (4237)Time elapsed: 0.007 s
% 2.72/1.60 % (4237)Instructions burned: 18 (million)
% 2.72/1.60 % (4237)------------------------------
% 2.72/1.60 % (4237)------------------------------
% 2.72/1.60 % (4239)Instruction limit reached!
% 2.72/1.60 % (4239)------------------------------
% 2.72/1.60 % (4239)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.60 % (4239)Termination reason: Unknown
% 2.72/1.60 % (4239)Termination phase: shuffling
% 2.72/1.60
% 2.72/1.60 % (4239)Memory used [KB]: 1407
% 2.72/1.60 % (4239)Time elapsed: 0.003 s
% 2.72/1.60 % (4239)Instructions burned: 4 (million)
% 2.72/1.60 % (4239)------------------------------
% 2.72/1.60 % (4239)------------------------------
% 2.72/1.60 % (4238)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_3941 for (2999ds/3Mi)
% 2.72/1.60 % (4238)Instruction limit reached!
% 2.72/1.60 % (4238)------------------------------
% 2.72/1.60 % (4238)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.60 % (4238)Termination reason: Unknown
% 2.72/1.60 % (4238)Termination phase: shuffling
% 2.72/1.60
% 2.72/1.60 % (4238)Memory used [KB]: 1407
% 2.72/1.60 % (4238)Time elapsed: 0.004 s
% 2.72/1.60 % (4238)Instructions burned: 3 (million)
% 2.72/1.60 % (4238)------------------------------
% 2.72/1.60 % (4238)------------------------------
% 2.72/1.61 % (4240)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_3941 for (2999ds/7Mi)
% 2.72/1.61 % (4240)Instruction limit reached!
% 2.72/1.61 % (4240)------------------------------
% 2.72/1.61 % (4240)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.61 % (4240)Termination reason: Unknown
% 2.72/1.61 % (4240)Termination phase: shuffling
% 2.72/1.61
% 2.72/1.61 % (4240)Memory used [KB]: 1535
% 2.72/1.61 % (4240)Time elapsed: 0.004 s
% 2.72/1.61 % (4240)Instructions burned: 7 (million)
% 2.72/1.61 % (4240)------------------------------
% 2.72/1.61 % (4240)------------------------------
% 2.72/1.61 % (4241)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_3941 for (2999ds/3Mi)
% 2.72/1.61 % (4242)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_3941 for (2999ds/4Mi)
% 2.72/1.61 % (4241)Instruction limit reached!
% 2.72/1.61 % (4241)------------------------------
% 2.72/1.61 % (4241)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.61 % (4241)Termination reason: Unknown
% 2.72/1.61 % (4241)Termination phase: shuffling
% 2.72/1.61
% 2.72/1.61 % (4241)Memory used [KB]: 1407
% 2.72/1.61 % (4241)Time elapsed: 0.002 s
% 2.72/1.61 % (4241)Instructions burned: 3 (million)
% 2.72/1.61 % (4241)------------------------------
% 2.72/1.61 % (4241)------------------------------
% 2.72/1.61 % (4242)Instruction limit reached!
% 2.72/1.61 % (4242)------------------------------
% 2.72/1.61 % (4242)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.61 % (4242)Termination reason: Unknown
% 2.72/1.61 % (4242)Termination phase: shuffling
% 2.72/1.61
% 2.72/1.61 % (4242)Memory used [KB]: 1407
% 2.72/1.61 % (4242)Time elapsed: 0.003 s
% 2.72/1.61 % (4242)Instructions burned: 4 (million)
% 2.72/1.61 % (4242)------------------------------
% 2.72/1.61 % (4242)------------------------------
% 2.72/1.62 % (4244)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_3941 for (2999ds/710Mi)
% 2.72/1.62 % (4245)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_3941 for (2999ds/6Mi)
% 2.72/1.62 % (4246)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_3941 for (2999ds/902Mi)
% 2.72/1.62 % (4245)Instruction limit reached!
% 2.72/1.62 % (4245)------------------------------
% 2.72/1.62 % (4245)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.62 % (4245)Termination reason: Unknown
% 2.72/1.62 % (4245)Termination phase: shuffling
% 2.72/1.62
% 2.72/1.62 % (4245)Memory used [KB]: 1535
% 2.72/1.62 % (4245)Time elapsed: 0.004 s
% 2.72/1.62 % (4245)Instructions burned: 8 (million)
% 2.72/1.62 % (4245)------------------------------
% 2.72/1.62 % (4245)------------------------------
% 2.72/1.62 % (4243)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_3941 for (2999ds/18Mi)
% 2.72/1.63 % (4247)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_3941 for (2999ds/21Mi)
% 2.72/1.63 % (4243)Instruction limit reached!
% 2.72/1.63 % (4243)------------------------------
% 2.72/1.63 % (4243)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.63 % (4243)Termination reason: Unknown
% 2.72/1.63 % (4243)Termination phase: Property scanning
% 2.72/1.63
% 2.72/1.63 % (4243)Memory used [KB]: 1791
% 2.72/1.63 % (4243)Time elapsed: 0.013 s
% 2.72/1.63 % (4243)Instructions burned: 18 (million)
% 2.72/1.63 % (4243)------------------------------
% 2.72/1.63 % (4243)------------------------------
% 2.72/1.64 % (4247)Instruction limit reached!
% 2.72/1.64 % (4247)------------------------------
% 2.72/1.64 % (4247)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.64 % (4247)Termination reason: Unknown
% 2.72/1.64 % (4247)Termination phase: shuffling
% 2.72/1.64
% 2.72/1.64 % (4247)Memory used [KB]: 1791
% 2.72/1.64 % (4247)Time elapsed: 0.008 s
% 2.72/1.64 % (4247)Instructions burned: 21 (million)
% 2.72/1.64 % (4247)------------------------------
% 2.72/1.64 % (4247)------------------------------
% 2.72/1.65 % (4249)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_3941 for (2999ds/6Mi)
% 2.72/1.65 % (4249)Instruction limit reached!
% 2.72/1.65 % (4249)------------------------------
% 2.72/1.65 % (4249)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.65 % (4249)Termination reason: Unknown
% 2.72/1.65 % (4249)Termination phase: shuffling
% 2.72/1.65
% 2.72/1.65 % (4249)Memory used [KB]: 1535
% 2.72/1.65 % (4249)Time elapsed: 0.003 s
% 2.72/1.65 % (4249)Instructions burned: 7 (million)
% 2.72/1.65 % (4249)------------------------------
% 2.72/1.65 % (4249)------------------------------
% 2.72/1.65 % (4248)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_3941 for (2999ds/5Mi)
% 2.72/1.65 % (4248)Instruction limit reached!
% 2.72/1.65 % (4248)------------------------------
% 2.72/1.65 % (4248)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.65 % (4248)Termination reason: Unknown
% 2.72/1.65 % (4248)Termination phase: shuffling
% 2.72/1.65
% 2.72/1.65 % (4248)Memory used [KB]: 1535
% 2.72/1.65 % (4248)Time elapsed: 0.005 s
% 2.72/1.65 % (4248)Instructions burned: 6 (million)
% 2.72/1.65 % (4248)------------------------------
% 2.72/1.65 % (4248)------------------------------
% 2.72/1.66 % (4250)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_3941 for (2998ds/377Mi)
% 2.72/1.67 % (4224)Instruction limit reached!
% 2.72/1.67 % (4224)------------------------------
% 2.72/1.67 % (4224)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.67 % (4224)Termination reason: Unknown
% 2.72/1.67 % (4224)Termination phase: Saturation
% 2.72/1.67
% 2.72/1.67 % (4224)Memory used [KB]: 7164
% 2.72/1.67 % (4224)Time elapsed: 0.097 s
% 2.72/1.67 % (4224)Instructions burned: 183 (million)
% 2.72/1.67 % (4224)------------------------------
% 2.72/1.67 % (4224)------------------------------
% 2.72/1.67 % (4251)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_3941 for (2998ds/779Mi)
% 2.72/1.68 % (4229)Instruction limit reached!
% 2.72/1.68 % (4229)------------------------------
% 2.72/1.68 % (4229)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.68 % (4229)Termination reason: Unknown
% 2.72/1.68 % (4229)Termination phase: Saturation
% 2.72/1.68
% 2.72/1.68 % (4229)Memory used [KB]: 6652
% 2.72/1.68 % (4229)Time elapsed: 0.112 s
% 2.72/1.68 % (4229)Instructions burned: 278 (million)
% 2.72/1.68 % (4229)------------------------------
% 2.72/1.68 % (4229)------------------------------
% 2.72/1.69 % (4252)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_3941 for (2998ds/19Mi)
% 2.72/1.69 % (4253)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_3941 for (2998ds/879Mi)
% 2.72/1.70 % (4252)Instruction limit reached!
% 2.72/1.70 % (4252)------------------------------
% 2.72/1.70 % (4252)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.70 % (4252)Termination reason: Unknown
% 2.72/1.70 % (4252)Termination phase: shuffling
% 2.72/1.70
% 2.72/1.70 % (4252)Memory used [KB]: 1791
% 2.72/1.70 % (4252)Time elapsed: 0.012 s
% 2.72/1.70 % (4252)Instructions burned: 20 (million)
% 2.72/1.70 % (4252)------------------------------
% 2.72/1.70 % (4252)------------------------------
% 2.72/1.71 % (4254)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_3941 for (2998ds/17Mi)
% 2.72/1.72 % (4254)Instruction limit reached!
% 2.72/1.72 % (4254)------------------------------
% 2.72/1.72 % (4254)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.72 % (4254)Termination reason: Unknown
% 2.72/1.72 % (4254)Termination phase: Property scanning
% 2.72/1.72
% 2.72/1.72 % (4254)Memory used [KB]: 1791
% 2.72/1.72 % (4254)Time elapsed: 0.013 s
% 2.72/1.72 % (4254)Instructions burned: 18 (million)
% 2.72/1.72 % (4254)------------------------------
% 2.72/1.72 % (4254)------------------------------
% 2.72/1.74 % (4255)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_3941 for (2998ds/3Mi)
% 2.72/1.74 % (4244)First to succeed.
% 2.72/1.74 % (4255)Instruction limit reached!
% 2.72/1.74 % (4255)------------------------------
% 2.72/1.74 % (4255)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.74 % (4255)Termination reason: Unknown
% 2.72/1.74 % (4255)Termination phase: shuffling
% 2.72/1.74
% 2.72/1.74 % (4255)Memory used [KB]: 1407
% 2.72/1.74 % (4255)Time elapsed: 0.004 s
% 2.72/1.74 % (4255)Instructions burned: 3 (million)
% 2.72/1.74 % (4255)------------------------------
% 2.72/1.74 % (4255)------------------------------
% 2.72/1.74 % (4244)Refutation found. Thanks to Tanya!
% 2.72/1.74 % SZS status Theorem for DTF2THF_3941
% 2.72/1.74 % SZS output start Proof for DTF2THF_3941
% 2.72/1.74 thf(type_def_5, type, real: $tType).
% 2.72/1.74 thf(type_def_7, type, nat: $tType).
% 2.72/1.74 thf(type_def_8, type, int: $tType).
% 2.72/1.74 thf(type_def_9, type, set_nat: $tType).
% 2.72/1.74 thf(type_def_10, type, set_int: $tType).
% 2.72/1.74 thf(type_def_11, type, set_real: $tType).
% 2.72/1.74 thf(type_def_12, type, a: $tType).
% 2.72/1.74 thf(type_def_13, type, set_a: $tType).
% 2.72/1.74 thf(type_def_14, type, list_nat: $tType).
% 2.72/1.74 thf(type_def_15, type, list_a: $tType).
% 2.72/1.74 thf(type_def_16, type, multiset_nat: $tType).
% 2.72/1.74 thf(type_def_17, type, multiset_a: $tType).
% 2.72/1.74 thf(type_def_18, type, multiset_int: $tType).
% 2.72/1.74 thf(type_def_19, type, set_list_a: $tType).
% 2.72/1.74 thf(func_def_0, type, set_list_a: $tType).
% 2.72/1.74 thf(func_def_1, type, multiset_nat: $tType).
% 2.72/1.74 thf(func_def_2, type, multiset_int: $tType).
% 2.72/1.74 thf(func_def_3, type, multiset_a: $tType).
% 2.72/1.74 thf(func_def_4, type, set_real: $tType).
% 2.72/1.74 thf(func_def_5, type, list_nat: $tType).
% 2.72/1.74 thf(func_def_6, type, set_nat: $tType).
% 2.72/1.74 thf(func_def_7, type, set_int: $tType).
% 2.72/1.74 thf(func_def_8, type, list_a: $tType).
% 2.72/1.74 thf(func_def_9, type, set_a: $tType).
% 2.72/1.74 thf(func_def_10, type, real: $tType).
% 2.72/1.74 thf(func_def_11, type, nat: $tType).
% 2.72/1.74 thf(func_def_12, type, int: $tType).
% 2.72/1.74 thf(func_def_13, type, a: $tType).
% 2.72/1.74 thf(func_def_14, type, inverse_inverse_real: real > real).
% 2.72/1.74 thf(func_def_15, type, bij_betw_nat_int: (nat > int) > set_nat > set_int > $o).
% 2.72/1.74 thf(func_def_16, type, bij_betw_nat_nat: (nat > nat) > set_nat > set_nat > $o).
% 2.72/1.74 thf(func_def_17, type, bij_betw_nat_real: (nat > real) > set_nat > set_real > $o).
% 2.72/1.74 thf(func_def_18, type, bij_betw_nat_a: (nat > a) > set_nat > set_a > $o).
% 2.72/1.74 thf(func_def_19, type, comp_int_int_nat: (int > int) > (nat > int) > nat > int).
% 2.72/1.74 thf(func_def_20, type, comp_int_nat_nat: (int > nat) > (nat > int) > nat > nat).
% 2.72/1.74 thf(func_def_21, type, comp_l728830689st_nat: (list_nat > list_a) > (list_nat > list_nat) > list_nat > list_a).
% 2.72/1.74 thf(func_def_22, type, comp_l1234327363st_nat: (list_a > list_a) > (list_nat > list_a) > list_nat > list_a).
% 2.72/1.74 thf(func_def_23, type, comp_m1386984005et_nat: (multiset_nat > multiset_nat) > (multiset_nat > multiset_nat) > multiset_nat > multiset_nat).
% 2.72/1.74 thf(func_def_24, type, comp_m1844659105et_nat: (multiset_nat > multiset_a) > (multiset_nat > multiset_nat) > multiset_nat > multiset_a).
% 2.72/1.74 thf(func_def_25, type, comp_m346803701et_nat: (multiset_nat > nat) > (multiset_nat > multiset_nat) > multiset_nat > nat).
% 2.72/1.74 thf(func_def_26, type, comp_m1962190563et_nat: (multiset_a > multiset_nat) > (multiset_nat > multiset_a) > multiset_nat > multiset_nat).
% 2.72/1.74 thf(func_def_27, type, comp_m540523139et_nat: (multiset_a > multiset_a) > (multiset_nat > multiset_a) > multiset_nat > multiset_a).
% 2.72/1.74 thf(func_def_28, type, comp_m1512105363et_nat: (multiset_a > nat) > (multiset_nat > multiset_a) > multiset_nat > nat).
% 2.72/1.74 thf(func_def_29, type, comp_nat_int_nat: (nat > int) > (nat > nat) > nat > int).
% 2.72/1.74 thf(func_def_30, type, comp_nat_nat_nat: (nat > nat) > (nat > nat) > nat > nat).
% 2.72/1.74 thf(func_def_31, type, comp_nat_a_nat: (nat > a) > (nat > nat) > nat > a).
% 2.72/1.74 thf(func_def_32, type, comp_a_nat_nat: (a > nat) > (nat > a) > nat > nat).
% 2.72/1.74 thf(func_def_33, type, comp_a_a_nat: (a > a) > (nat > a) > nat > a).
% 2.72/1.74 thf(func_def_34, type, inj_on_int_int: (int > int) > set_int > $o).
% 2.72/1.74 thf(func_def_35, type, inj_on_int_nat: (int > nat) > set_int > $o).
% 2.72/1.74 thf(func_def_36, type, inj_on_nat_int: (nat > int) > set_nat > $o).
% 2.72/1.74 thf(func_def_37, type, inj_on_nat_nat: (nat > nat) > set_nat > $o).
% 2.72/1.74 thf(func_def_38, type, inj_on_nat_real: (nat > real) > set_nat > $o).
% 2.72/1.74 thf(func_def_39, type, inj_on_nat_a: (nat > a) > set_nat > $o).
% 2.72/1.74 thf(func_def_40, type, uminus_uminus_int: int > int).
% 2.72/1.74 thf(func_def_41, type, uminus_uminus_real: real > real).
% 2.72/1.74 thf(func_def_42, type, zero_zero_int: int).
% 2.72/1.74 thf(func_def_43, type, zero_z2005226153et_nat: multiset_nat).
% 2.72/1.74 thf(func_def_44, type, zero_zero_multiset_a: multiset_a).
% 2.72/1.74 thf(func_def_45, type, zero_zero_nat: nat).
% 2.72/1.74 thf(func_def_46, type, zero_zero_real: real).
% 2.72/1.74 thf(func_def_47, type, if_int: $o > int > int > int).
% 2.72/1.74 thf(func_def_48, type, if_nat: $o > nat > nat > nat).
% 2.72/1.74 thf(func_def_49, type, if_real: $o > real > real > real).
% 2.72/1.74 thf(func_def_50, type, nat2: int > nat).
% 2.72/1.74 thf(func_def_51, type, ring_1_of_int_int: int > int).
% 2.72/1.74 thf(func_def_52, type, ring_1_of_int_real: int > real).
% 2.72/1.74 thf(func_def_53, type, gen_length_a: nat > list_a > nat).
% 2.72/1.74 thf(func_def_54, type, map_nat_nat: (nat > nat) > list_nat > list_nat).
% 2.72/1.74 thf(func_def_55, type, map_nat_a: (nat > a) > list_nat > list_a).
% 2.72/1.74 thf(func_def_56, type, map_a_nat: (a > nat) > list_a > list_nat).
% 2.72/1.74 thf(func_def_57, type, map_a_a: (a > a) > list_a > list_a).
% 2.72/1.74 thf(func_def_58, type, list_ex_a: (a > $o) > list_a > $o).
% 2.72/1.74 thf(func_def_59, type, map_tailrec_nat_a: (nat > a) > list_nat > list_a).
% 2.72/1.74 thf(func_def_60, type, nth_nat: list_nat > nat > nat).
% 2.72/1.74 thf(func_def_61, type, nth_a: list_a > nat > a).
% 2.72/1.74 thf(func_def_62, type, upt: nat > nat > list_nat).
% 2.72/1.74 thf(func_def_63, type, image_mset_nat_int: (nat > int) > multiset_nat > multiset_int).
% 2.72/1.74 thf(func_def_64, type, image_mset_nat_nat: (nat > nat) > multiset_nat > multiset_nat).
% 2.72/1.74 thf(func_def_65, type, image_mset_nat_a: (nat > a) > multiset_nat > multiset_a).
% 2.72/1.74 thf(func_def_66, type, image_mset_a_nat: (a > nat) > multiset_a > multiset_nat).
% 2.72/1.74 thf(func_def_67, type, image_mset_a_a: (a > a) > multiset_a > multiset_a).
% 2.72/1.74 thf(func_def_68, type, mset_nat: list_nat > multiset_nat).
% 2.72/1.74 thf(func_def_69, type, mset_a: list_a > multiset_a).
% 2.72/1.74 thf(func_def_70, type, mset_set_int: set_int > multiset_int).
% 2.72/1.74 thf(func_def_71, type, mset_set_nat: set_nat > multiset_nat).
% 2.72/1.74 thf(func_def_72, type, mset_set_a: set_a > multiset_a).
% 2.72/1.74 thf(func_def_73, type, size_multiset_nat: (nat > nat) > multiset_nat > nat).
% 2.72/1.74 thf(func_def_74, type, size_multiset_a: (a > nat) > multiset_a > nat).
% 2.72/1.74 thf(func_def_75, type, semiri2019852685at_int: nat > int).
% 2.72/1.74 thf(func_def_76, type, semiri1382578993at_nat: nat > nat).
% 2.72/1.74 thf(func_def_77, type, semiri2110766477t_real: nat > real).
% 2.72/1.74 thf(func_def_78, type, size_size_list_nat: list_nat > nat).
% 2.72/1.74 thf(func_def_79, type, size_size_list_a: list_a > nat).
% 2.72/1.74 thf(func_def_80, type, size_s1679505588et_nat: multiset_nat > nat).
% 2.72/1.74 thf(func_def_81, type, size_size_multiset_a: multiset_a > nat).
% 2.72/1.74 thf(func_def_82, type, ord_less_int: int > int > $o).
% 2.72/1.74 thf(func_def_83, type, ord_less_nat: nat > nat > $o).
% 2.72/1.74 thf(func_def_84, type, ord_less_real: real > real > $o).
% 2.72/1.74 thf(func_def_85, type, ord_less_set_int: set_int > set_int > $o).
% 2.72/1.74 thf(func_def_86, type, ord_less_set_nat: set_nat > set_nat > $o).
% 2.72/1.74 thf(func_def_87, type, ord_less_set_real: set_real > set_real > $o).
% 2.72/1.74 thf(func_def_88, type, ord_less_eq_int: int > int > $o).
% 2.72/1.74 thf(func_def_89, type, ord_less_eq_nat: nat > nat > $o).
% 2.72/1.74 thf(func_def_90, type, ord_less_eq_real: real > real > $o).
% 2.72/1.74 thf(func_def_91, type, ord_less_eq_set_int: set_int > set_int > $o).
% 2.72/1.74 thf(func_def_92, type, ord_less_eq_set_nat: set_nat > set_nat > $o).
% 2.72/1.74 thf(func_def_93, type, image_int_int: (int > int) > set_int > set_int).
% 2.72/1.74 thf(func_def_94, type, image_int_nat: (int > nat) > set_int > set_nat).
% 2.72/1.74 thf(func_def_95, type, image_nat_int: (nat > int) > set_nat > set_int).
% 2.72/1.74 thf(func_def_96, type, image_nat_nat: (nat > nat) > set_nat > set_nat).
% 2.72/1.74 thf(func_def_97, type, image_nat_real: (nat > real) > set_nat > set_real).
% 2.72/1.74 thf(func_def_98, type, image_nat_a: (nat > a) > set_nat > set_a).
% 2.72/1.74 thf(func_def_99, type, set_or1199280219an_int: int > int > set_int).
% 2.72/1.74 thf(func_def_100, type, set_or562006527an_nat: nat > nat > set_nat).
% 2.72/1.74 thf(func_def_101, type, set_or2075149659n_real: real > real > set_real).
% 2.72/1.74 thf(func_def_102, type, set_ord_lessThan_int: int > set_int).
% 2.72/1.74 thf(func_def_103, type, set_ord_lessThan_nat: nat > set_nat).
% 2.72/1.74 thf(func_def_104, type, set_or1211449801n_real: real > set_real).
% 2.72/1.74 thf(func_def_105, type, member_int: int > set_int > $o).
% 2.72/1.74 thf(func_def_106, type, member_list_a: list_a > set_list_a > $o).
% 2.72/1.74 thf(func_def_107, type, member_nat: nat > set_nat > $o).
% 2.72/1.74 thf(func_def_108, type, member_real: real > set_real > $o).
% 2.72/1.74 thf(func_def_109, type, f: nat > nat).
% 2.72/1.74 thf(func_def_110, type, xs: list_a).
% 2.72/1.74 thf(func_def_111, type, ys: list_a).
% 2.72/1.74 thf(func_def_126, type, sK0: list_a > list_a > nat).
% 2.72/1.74 thf(func_def_127, type, sK1: nat > list_a).
% 2.72/1.74 thf(func_def_128, type, sK2: (nat > a > $o) > nat > list_a).
% 2.72/1.74 thf(func_def_129, type, sK3: (nat > a > $o) > nat > nat).
% 2.72/1.74 thf(func_def_130, type, sK4: nat > (nat > a > $o) > a).
% 2.72/1.74 thf(func_def_131, type, sK5: list_a > (nat > a > $o) > nat > nat).
% 2.72/1.74 thf(func_def_132, type, sK6: nat > (nat > $o) > nat).
% 2.72/1.74 thf(func_def_133, type, sK7: nat > (nat > $o) > nat).
% 2.72/1.74 thf(func_def_134, type, sK8: (list_a > $o) > list_a).
% 2.72/1.74 thf(func_def_135, type, sK9: nat > (nat > $o) > nat).
% 2.72/1.74 thf(func_def_136, type, sK10: nat > (nat > $o) > nat).
% 2.72/1.74 thf(func_def_137, type, sK11: (nat > $o) > nat).
% 2.72/1.74 thf(func_def_140, type, ph13: !>[X0: $tType]:(X0)).
% 2.72/1.74 thf(f2147,plain,(
% 2.72/1.74 $false),
% 2.72/1.74 inference(subsumption_resolution,[status(thm)],[f2146,f936])).
% 2.72/1.74 thf(f936,plain,(
% 2.72/1.74 ((mset_a @ xs) != (image_mset_nat_a @ (nth_a @ ys) @ (mset_set_nat @ (set_or562006527an_nat @ zero_zero_nat @ (size_size_list_a @ ys)))))),
% 2.72/1.74 inference(backward_demodulation,[status(thm)],[f910,f935])).
% 2.72/1.74 thf(f935,plain,(
% 2.72/1.74 ((mset_a @ xs) = (image_mset_nat_a @ (nth_a @ ys) @ (image_mset_nat_nat @ f @ (mset_set_nat @ (set_or562006527an_nat @ zero_zero_nat @ (size_size_list_a @ xs))))))),
% 2.72/1.75 inference(forward_demodulation,[status(thm)],[f886,f853])).
% 2.72/1.75 thf(f853,plain,(
% 2.72/1.75 (xs = (map_nat_a @ (comp_nat_a_nat @ (nth_a @ ys) @ f) @ (upt @ zero_zero_nat @ (size_size_list_a @ xs))))),
% 2.72/1.75 inference(cnf_transformation,[status(thm)],[f7])).
% 2.72/1.75 thf(f7,axiom,(
% 2.72/1.75 (xs = (map_nat_a @ (comp_nat_a_nat @ (nth_a @ ys) @ f) @ (upt @ zero_zero_nat @ (size_size_list_a @ xs))))),
% 2.72/1.75 file('/export/starexec/sandbox/tmp/tmp.VwIcuknk1l/DTF2THF_3941.p',fact_6__092_060open_062xs_A_061_Amap_A_I_I_B_J_Ays_A_092_060circ_062_Af_J_A_0910_O_O_060length_Axs_093_092_060close_062)).
% 2.72/1.75 thf(f886,plain,(
% 2.72/1.75 ((image_mset_nat_a @ (nth_a @ ys) @ (image_mset_nat_nat @ f @ (mset_set_nat @ (set_or562006527an_nat @ zero_zero_nat @ (size_size_list_a @ xs))))) = (mset_a @ (map_nat_a @ (comp_nat_a_nat @ (nth_a @ ys) @ f) @ (upt @ zero_zero_nat @ (size_size_list_a @ xs)))))),
% 2.72/1.75 inference(cnf_transformation,[status(thm)],[f6])).
% 2.72/1.75 thf(f6,axiom,(
% 2.72/1.75 ((image_mset_nat_a @ (nth_a @ ys) @ (image_mset_nat_nat @ f @ (mset_set_nat @ (set_or562006527an_nat @ zero_zero_nat @ (size_size_list_a @ xs))))) = (mset_a @ (map_nat_a @ (comp_nat_a_nat @ (nth_a @ ys) @ f) @ (upt @ zero_zero_nat @ (size_size_list_a @ xs)))))),
% 2.72/1.75 file('/export/starexec/sandbox/tmp/tmp.VwIcuknk1l/DTF2THF_3941.p',fact_5__092_060open_062mset_A_Imap_A_I_I_B_J_Ays_A_092_060circ_062_Af_J_A_0910_O_O_060length_Axs_093_J_A_061_Aimage__mset_A_I_I_B_J_Ays_J_A_Iimage__mset_Af_A_Imset__set_A_1230_O_O_060length_Axs_125_J_J_092_060close_062)).
% 2.72/1.75 thf(f910,plain,(
% 2.72/1.75 ((image_mset_nat_a @ (nth_a @ ys) @ (image_mset_nat_nat @ f @ (mset_set_nat @ (set_or562006527an_nat @ zero_zero_nat @ (size_size_list_a @ xs))))) != (image_mset_nat_a @ (nth_a @ ys) @ (mset_set_nat @ (set_or562006527an_nat @ zero_zero_nat @ (size_size_list_a @ ys)))))),
% 2.72/1.75 inference(cnf_transformation,[status(thm)],[f781])).
% 2.72/1.75 thf(f781,plain,(
% 2.72/1.75 ((image_mset_nat_a @ (nth_a @ ys) @ (image_mset_nat_nat @ f @ (mset_set_nat @ (set_or562006527an_nat @ zero_zero_nat @ (size_size_list_a @ xs))))) != (image_mset_nat_a @ (nth_a @ ys) @ (mset_set_nat @ (set_or562006527an_nat @ zero_zero_nat @ (size_size_list_a @ ys)))))),
% 2.72/1.75 inference(flattening,[status(thm)],[f358])).
% 2.72/1.75 thf(f358,negated_conjecture,(
% 2.72/1.75 ~((image_mset_nat_a @ (nth_a @ ys) @ (image_mset_nat_nat @ f @ (mset_set_nat @ (set_or562006527an_nat @ zero_zero_nat @ (size_size_list_a @ xs))))) = (image_mset_nat_a @ (nth_a @ ys) @ (mset_set_nat @ (set_or562006527an_nat @ zero_zero_nat @ (size_size_list_a @ ys)))))),
% 2.72/1.75 inference(negated_conjecture,[status(cth)],[f357])).
% 2.72/1.75 thf(f357,conjecture,(
% 2.72/1.75 ((image_mset_nat_a @ (nth_a @ ys) @ (image_mset_nat_nat @ f @ (mset_set_nat @ (set_or562006527an_nat @ zero_zero_nat @ (size_size_list_a @ xs))))) = (image_mset_nat_a @ (nth_a @ ys) @ (mset_set_nat @ (set_or562006527an_nat @ zero_zero_nat @ (size_size_list_a @ ys)))))),
% 2.72/1.75 file('/export/starexec/sandbox/tmp/tmp.VwIcuknk1l/DTF2THF_3941.p',conj_0)).
% 2.72/1.75 thf(f2146,plain,(
% 2.72/1.75 ((mset_a @ xs) = (image_mset_nat_a @ (nth_a @ ys) @ (mset_set_nat @ (set_or562006527an_nat @ zero_zero_nat @ (size_size_list_a @ ys)))))),
% 2.72/1.75 inference(backward_demodulation,[status(thm)],[f935,f2145])).
% 2.72/1.75 thf(f2145,plain,(
% 2.72/1.75 ((image_mset_nat_nat @ f @ (mset_set_nat @ (set_or562006527an_nat @ zero_zero_nat @ (size_size_list_a @ xs)))) = (mset_set_nat @ (set_or562006527an_nat @ zero_zero_nat @ (size_size_list_a @ ys))))),
% 2.72/1.75 inference(forward_demodulation,[status(thm)],[f2134,f854])).
% 2.72/1.75 thf(f854,plain,(
% 2.72/1.75 ((image_nat_nat @ f @ (set_or562006527an_nat @ zero_zero_nat @ (size_size_list_a @ xs))) = (set_or562006527an_nat @ zero_zero_nat @ (size_size_list_a @ ys)))),
% 2.72/1.75 inference(cnf_transformation,[status(thm)],[f4])).
% 2.72/1.75 thf(f4,axiom,(
% 2.72/1.75 ((image_nat_nat @ f @ (set_or562006527an_nat @ zero_zero_nat @ (size_size_list_a @ xs))) = (set_or562006527an_nat @ zero_zero_nat @ (size_size_list_a @ ys)))),
% 2.72/1.75 file('/export/starexec/sandbox/tmp/tmp.VwIcuknk1l/DTF2THF_3941.p',fact_3__C2_C)).
% 2.72/1.75 thf(f2134,plain,(
% 2.72/1.75 ((image_mset_nat_nat @ f @ (mset_set_nat @ (set_or562006527an_nat @ zero_zero_nat @ (size_size_list_a @ xs)))) = (mset_set_nat @ (image_nat_nat @ f @ (set_or562006527an_nat @ zero_zero_nat @ (size_size_list_a @ xs)))))),
% 2.72/1.75 inference(trivial_inequality_removal,[status(thm)],[f2126])).
% 2.72/1.75 thf(f2126,plain,(
% 2.72/1.75 ($true != $true) | ((image_mset_nat_nat @ f @ (mset_set_nat @ (set_or562006527an_nat @ zero_zero_nat @ (size_size_list_a @ xs)))) = (mset_set_nat @ (image_nat_nat @ f @ (set_or562006527an_nat @ zero_zero_nat @ (size_size_list_a @ xs)))))),
% 2.72/1.75 inference(superposition,[status(thm)],[f869,f866])).
% 2.72/1.75 thf(f866,plain,(
% 2.72/1.75 ((inj_on_nat_nat @ f @ (set_or562006527an_nat @ zero_zero_nat @ (size_size_list_a @ xs))) = $true)),
% 2.72/1.75 inference(cnf_transformation,[status(thm)],[f756])).
% 2.72/1.75 thf(f756,plain,(
% 2.72/1.75 ((inj_on_nat_nat @ f @ (set_or562006527an_nat @ zero_zero_nat @ (size_size_list_a @ xs))) = $true)),
% 2.72/1.75 inference(fool_elimination,[status(thm)],[f755])).
% 2.72/1.75 thf(f755,plain,(
% 2.72/1.75 (inj_on_nat_nat @ f @ (set_or562006527an_nat @ zero_zero_nat @ (size_size_list_a @ xs)))),
% 2.72/1.75 inference(rectify,[status(thm)],[f1])).
% 2.72/1.75 thf(f1,axiom,(
% 2.72/1.75 (inj_on_nat_nat @ f @ (set_or562006527an_nat @ zero_zero_nat @ (size_size_list_a @ xs)))),
% 2.72/1.75 file('/export/starexec/sandbox/tmp/tmp.VwIcuknk1l/DTF2THF_3941.p',fact_0__C1_C)).
% 2.72/1.75 thf(f869,plain,(
% 2.72/1.75 ( ! [X0 : nat > nat,X1 : set_nat] : (((inj_on_nat_nat @ X0 @ X1) != $true) | ((mset_set_nat @ (image_nat_nat @ X0 @ X1)) = (image_mset_nat_nat @ X0 @ (mset_set_nat @ X1)))) )),
% 2.72/1.75 inference(cnf_transformation,[status(thm)],[f818])).
% 2.72/1.75 thf(f818,plain,(
% 2.72/1.75 ! [X0 : nat > nat,X1 : set_nat] : (((mset_set_nat @ (image_nat_nat @ X0 @ X1)) = (image_mset_nat_nat @ X0 @ (mset_set_nat @ X1))) | ((inj_on_nat_nat @ X0 @ X1) != $true))),
% 2.72/1.75 inference(ennf_transformation,[status(thm)],[f562])).
% 2.72/1.75 thf(f562,plain,(
% 2.72/1.75 ! [X0 : nat > nat,X1 : set_nat] : (((inj_on_nat_nat @ X0 @ X1) = $true) => ((mset_set_nat @ (image_nat_nat @ X0 @ X1)) = (image_mset_nat_nat @ X0 @ (mset_set_nat @ X1))))),
% 2.72/1.75 inference(fool_elimination,[status(thm)],[f561])).
% 2.72/1.75 thf(f561,plain,(
% 2.72/1.75 ! [X0 : nat > nat,X1 : set_nat] : ((inj_on_nat_nat @ X0 @ X1) => ((mset_set_nat @ (image_nat_nat @ X0 @ X1)) = (image_mset_nat_nat @ X0 @ (mset_set_nat @ X1))))),
% 2.72/1.75 inference(rectify,[status(thm)],[f72])).
% 2.72/1.75 thf(f72,axiom,(
% 2.72/1.75 ! [X8 : nat > nat,X29 : set_nat] : ((inj_on_nat_nat @ X8 @ X29) => ((image_mset_nat_nat @ X8 @ (mset_set_nat @ X29)) = (mset_set_nat @ (image_nat_nat @ X8 @ X29))))),
% 2.72/1.75 file('/export/starexec/sandbox/tmp/tmp.VwIcuknk1l/DTF2THF_3941.p',fact_71_image__mset__mset__set)).
% 2.72/1.75 % SZS output end Proof for DTF2THF_3941
% 2.72/1.75 % (4244)------------------------------
% 2.72/1.75 % (4244)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.75 % (4244)Termination reason: Refutation
% 2.72/1.75
% 2.72/1.75 % (4244)Memory used [KB]: 7547
% 2.72/1.75 % (4244)Time elapsed: 0.127 s
% 2.72/1.75 % (4244)Instructions burned: 366 (million)
% 2.72/1.75 % (4244)------------------------------
% 2.72/1.75 % (4244)------------------------------
% 2.72/1.75 % (4223)Success in time 0.197 s
%------------------------------------------------------------------------------