%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP026^1 : 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 : n003.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 08:54:57 AM UTC 2026
% Result : Theorem 15.64s 3.65s
% Output : Refutation 15.64s
% Verified :
% SZS Type : Refutation
% Derivation depth : 25
% Number of leaves : 17
% Syntax : Number of formulae : 145 ( 116 unt; 0 typ; 2 def)
% Number of atoms : 523 ( 212 equ; 0 cnn)
% Maximal formula atoms : 4 ( 3 avg)
% Number of connectives : 752 ( 42 ~; 25 |; 0 &; 578 @)
% ( 2 <=>; 54 =>; 0 <=; 0 <~>)
% Maximal formula depth : 7 ( 2 avg)
% Number of types : 20 ( 19 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 112 ( 107 usr; 22 con; 0-5 aty)
% ( 41 !!; 10 ??; 0 @@+; 0 @@-)
% Number of variables : 149 ( 73 ^; 74 !; 2 ?; 149 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
carrie1963041556t_unit: $tType ).
thf(type_def_6,type,
set_a: $tType ).
thf(type_def_8,type,
carrie722926983_a_b_c: $tType ).
thf(type_def_9,type,
carrie1950868226xt_b_d: $tType ).
thf(type_def_10,type,
set_b: $tType ).
thf(type_def_11,type,
carrie1821755406_e_b_f: $tType ).
thf(type_def_12,type,
set_e: $tType ).
thf(type_def_13,type,
a: $tType ).
thf(type_def_14,type,
carrie980467975t_unit: $tType ).
thf(type_def_15,type,
carrie1065537299t_unit: $tType ).
thf(type_def_16,type,
carrie292259835t_unit: $tType ).
thf(type_def_17,type,
carrie377329159t_unit: $tType ).
thf(type_def_18,type,
nat: $tType ).
thf(type_def_19,type,
b: $tType ).
thf(type_def_20,type,
e: $tType ).
thf(type_def_21,type,
set_nat: $tType ).
thf(type_def_22,type,
set_nat_a: $tType ).
thf(type_def_23,type,
set_nat_b: $tType ).
thf(type_def_24,type,
set_nat_e: $tType ).
thf(func_def_0,type,
carrie377329159t_unit: $tType ).
thf(func_def_1,type,
carrie292259835t_unit: $tType ).
thf(func_def_2,type,
carrie1065537299t_unit: $tType ).
thf(func_def_3,type,
carrie980467975t_unit: $tType ).
thf(func_def_4,type,
carrie1963041556t_unit: $tType ).
thf(func_def_5,type,
carrie1821755406_e_b_f: $tType ).
thf(func_def_6,type,
carrie722926983_a_b_c: $tType ).
thf(func_def_7,type,
carrie1950868226xt_b_d: $tType ).
thf(func_def_8,type,
set_nat_e: $tType ).
thf(func_def_9,type,
set_nat_b: $tType ).
thf(func_def_10,type,
set_nat_a: $tType ).
thf(func_def_11,type,
set_nat: $tType ).
thf(func_def_12,type,
set_e: $tType ).
thf(func_def_13,type,
set_b: $tType ).
thf(func_def_14,type,
set_a: $tType ).
thf(func_def_15,type,
nat: $tType ).
thf(func_def_16,type,
e: $tType ).
thf(func_def_17,type,
b: $tType ).
thf(func_def_18,type,
a: $tType ).
thf(func_def_19,type,
carrie1074654371t_unit: carrie1963041556t_unit > set_a ).
thf(func_def_20,type,
carrie2021454486_a_b_c: carrie722926983_a_b_c > set_a ).
thf(func_def_21,type,
carrie2079586589xt_b_d: carrie1950868226xt_b_d > set_b ).
thf(func_def_22,type,
carrie730238621_e_b_f: carrie1821755406_e_b_f > set_e ).
thf(func_def_23,type,
ring_b_d: carrie1950868226xt_b_d > $o ).
thf(func_def_24,type,
zero_a1261444626_a_b_c: carrie722926983_a_b_c > a ).
thf(func_def_25,type,
ideal_b_d: carrie1950868226xt_b_d > set_b > $o ).
thf(func_def_26,type,
annihi259882159t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > set_b ).
thf(func_def_27,type,
annihilator_b_d_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > set_b ).
thf(func_def_28,type,
annihilator_b_d_e_f: carrie1950868226xt_b_d > carrie1821755406_e_b_f > set_b ).
thf(func_def_29,type,
hOM_b_1103679019t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > carrie1963041556t_unit > carrie980467975t_unit ).
thf(func_def_30,type,
hOM_b_796081438it_a_c: carrie1950868226xt_b_d > carrie1963041556t_unit > carrie722926983_a_b_c > carrie980467975t_unit ).
thf(func_def_31,type,
hOM_b_1959675421it_e_f: carrie1950868226xt_b_d > carrie1963041556t_unit > carrie1821755406_e_b_f > carrie1065537299t_unit ).
thf(func_def_32,type,
hOM_b_717364638t_unit: carrie1950868226xt_b_d > carrie722926983_a_b_c > carrie1963041556t_unit > carrie980467975t_unit ).
thf(func_def_33,type,
hOM_b_d_a_c_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > carrie722926983_a_b_c > carrie980467975t_unit ).
thf(func_def_34,type,
hOM_b_d_a_c_e_f: carrie1950868226xt_b_d > carrie722926983_a_b_c > carrie1821755406_e_b_f > carrie1065537299t_unit ).
thf(func_def_35,type,
hOM_b_1375846429t_unit: carrie1950868226xt_b_d > carrie1821755406_e_b_f > carrie1963041556t_unit > carrie292259835t_unit ).
thf(func_def_36,type,
hOM_b_d_e_f_a_c: carrie1950868226xt_b_d > carrie1821755406_e_b_f > carrie722926983_a_b_c > carrie292259835t_unit ).
thf(func_def_37,type,
hOM_b_d_e_f_e_f: carrie1950868226xt_b_d > carrie1821755406_e_b_f > carrie1821755406_e_b_f > carrie377329159t_unit ).
thf(func_def_38,type,
module589927589unit_d: carrie980467975t_unit > carrie1950868226xt_b_d > $o ).
thf(func_def_39,type,
module1648870817unit_d: carrie1065537299t_unit > carrie1950868226xt_b_d > $o ).
thf(func_def_40,type,
module1330273449unit_d: carrie292259835t_unit > carrie1950868226xt_b_d > $o ).
thf(func_def_41,type,
module241733029unit_d: carrie377329159t_unit > carrie1950868226xt_b_d > $o ).
thf(func_def_42,type,
module1821517916unit_d: carrie1963041556t_unit > carrie1950868226xt_b_d > $o ).
thf(func_def_43,type,
module_a_b_c_d: carrie722926983_a_b_c > carrie1950868226xt_b_d > $o ).
thf(func_def_44,type,
module_e_b_f_d: carrie1821755406_e_b_f > carrie1950868226xt_b_d > $o ).
thf(func_def_45,type,
free_g637607517t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > set_a > $o ).
thf(func_def_46,type,
free_g1087686480_d_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > set_a > $o ).
thf(func_def_47,type,
free_g103796815_d_e_f: carrie1950868226xt_b_d > carrie1821755406_e_b_f > set_e > $o ).
thf(func_def_48,type,
genera1692266857t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > set_a > $o ).
thf(func_def_49,type,
generator_b_d_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > set_a > $o ).
thf(func_def_50,type,
generator_b_d_e_f: carrie1950868226xt_b_d > carrie1821755406_e_b_f > set_e > $o ).
thf(func_def_51,type,
l_comb1138968323t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > nat > ( nat > b ) > ( nat > a ) > a ).
thf(func_def_52,type,
l_comb_b_d_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > nat > ( nat > b ) > ( nat > a ) > a ).
thf(func_def_53,type,
l_comb_b_d_e_f: carrie1950868226xt_b_d > carrie1821755406_e_b_f > nat > ( nat > b ) > ( nat > e ) > e ).
thf(func_def_54,type,
linear_span_b_d_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > set_b > set_a > set_a ).
thf(func_def_55,type,
mdl_a_b_c: carrie722926983_a_b_c > set_a > carrie1963041556t_unit ).
thf(func_def_56,type,
misomo493403663t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > carrie1963041556t_unit > $o ).
thf(func_def_57,type,
misomo1935876354it_a_c: carrie1950868226xt_b_d > carrie1963041556t_unit > carrie722926983_a_b_c > $o ).
thf(func_def_58,type,
misomo951986689it_e_f: carrie1950868226xt_b_d > carrie1963041556t_unit > carrie1821755406_e_b_f > $o ).
thf(func_def_59,type,
misomo1857159554t_unit: carrie1950868226xt_b_d > carrie722926983_a_b_c > carrie1963041556t_unit > $o ).
thf(func_def_60,type,
misomo1282343797_c_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > carrie722926983_a_b_c > $o ).
thf(func_def_61,type,
misomo298454132_c_e_f: carrie1950868226xt_b_d > carrie722926983_a_b_c > carrie1821755406_e_b_f > $o ).
thf(func_def_62,type,
misomo368157697t_unit: carrie1950868226xt_b_d > carrie1821755406_e_b_f > carrie1963041556t_unit > $o ).
thf(func_def_63,type,
misomo1752826100_f_a_c: carrie1950868226xt_b_d > carrie1821755406_e_b_f > carrie722926983_a_b_c > $o ).
thf(func_def_64,type,
misomo768936435_f_e_f: carrie1950868226xt_b_d > carrie1821755406_e_b_f > carrie1821755406_e_b_f > $o ).
thf(func_def_65,type,
quotie1196826005t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > set_a > set_a > set_b ).
thf(func_def_66,type,
quotie1153752712_d_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > set_a > set_a > set_b ).
thf(func_def_67,type,
quotie169863047_d_e_f: carrie1950868226xt_b_d > carrie1821755406_e_b_f > set_e > set_e > set_b ).
thf(func_def_68,type,
smodul132175289t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > set_b > set_a ).
thf(func_def_69,type,
smodul818989740_d_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > set_b > set_a ).
thf(func_def_70,type,
smodul1982583723_d_e_f: carrie1950868226xt_b_d > carrie1821755406_e_b_f > set_b > set_e ).
thf(func_def_71,type,
submod903911234t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > set_a > $o ).
thf(func_def_72,type,
submodule_b_d_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > set_a > $o ).
thf(func_def_73,type,
submodule_b_d_e_f: carrie1950868226xt_b_d > carrie1821755406_e_b_f > set_e > $o ).
thf(func_def_74,type,
algebr1039611956t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > set_a > set_a ).
thf(func_def_75,type,
algebr1152250919_d_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > set_a > set_a ).
thf(func_def_76,type,
algebr168361254_d_e_f: carrie1950868226xt_b_d > carrie1821755406_e_b_f > set_e > set_e ).
thf(func_def_77,type,
pi_nat_a: set_nat > ( nat > set_a ) > set_nat_a ).
thf(func_def_78,type,
pi_nat_b: set_nat > ( nat > set_b ) > set_nat_b ).
thf(func_def_79,type,
pi_nat_e: set_nat > ( nat > set_e ) > set_nat_e ).
thf(func_def_80,type,
minus_minus_nat: nat > nat > nat ).
thf(func_def_81,type,
minus_1788767276_nat_a: set_nat_a > set_nat_a > set_nat_a ).
thf(func_def_82,type,
minus_780847917_nat_b: set_nat_b > set_nat_b > set_nat_b ).
thf(func_def_83,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(func_def_84,type,
zero_zero_nat: nat ).
thf(func_def_85,type,
suc: nat > nat ).
thf(func_def_86,type,
bot_bot_set_nat_a: set_nat_a ).
thf(func_def_87,type,
bot_bot_set_nat_b: set_nat_b ).
thf(func_def_88,type,
bot_bot_set_nat: set_nat ).
thf(func_def_89,type,
bot_bot_set_a: set_a ).
thf(func_def_90,type,
bot_bot_set_e: set_e ).
thf(func_def_91,type,
ord_less_nat: nat > nat > $o ).
thf(func_def_92,type,
ord_less_set_a: set_a > set_a > $o ).
thf(func_def_93,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_94,type,
ord_le157368549_nat_a: set_nat_a > set_nat_a > $o ).
thf(func_def_95,type,
ord_le1296932838_nat_b: set_nat_b > set_nat_b > $o ).
thf(func_def_96,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(func_def_97,type,
ord_less_eq_set_e: set_e > set_e > $o ).
thf(func_def_98,type,
collect_nat_a: ( ( nat > a ) > $o ) > set_nat_a ).
thf(func_def_99,type,
collect_nat_b: ( ( nat > b ) > $o ) > set_nat_b ).
thf(func_def_100,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(func_def_101,type,
collect_a: ( a > $o ) > set_a ).
thf(func_def_102,type,
insert_nat_a: ( nat > a ) > set_nat_a > set_nat_a ).
thf(func_def_103,type,
insert_nat_b: ( nat > b ) > set_nat_b > set_nat_b ).
thf(func_def_104,type,
insert_a: a > set_a > set_a ).
thf(func_def_105,type,
member_nat_a: ( nat > a ) > set_nat_a > $o ).
thf(func_def_106,type,
member_nat_b: ( nat > b ) > set_nat_b > $o ).
thf(func_def_107,type,
member_nat_e: ( nat > e ) > set_nat_e > $o ).
thf(func_def_108,type,
member_nat: nat > set_nat > $o ).
thf(func_def_109,type,
member_a: a > set_a > $o ).
thf(func_def_110,type,
member_e: e > set_e > $o ).
thf(func_def_111,type,
h: set_a ).
thf(func_def_112,type,
h1: set_a ).
thf(func_def_113,type,
m: carrie722926983_a_b_c ).
thf(func_def_114,type,
n: carrie1821755406_e_b_f ).
thf(func_def_115,type,
r: carrie1950868226xt_b_d ).
thf(func_def_116,type,
g: nat > a ).
thf(func_def_117,type,
h2: a ).
thf(func_def_118,type,
m2: nat ).
thf(func_def_119,type,
t: nat > b ).
thf(func_def_120,type,
x: a ).
thf(func_def_121,type,
xa: nat ).
thf(func_def_143,type,
ph1:
!>[X0: $tType] : X0 ).
thf(func_def_144,type,
sK2: nat ).
thf(f3066,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f2052,f3055,f3065]) ).
thf(f3065,plain,
~ spl0_1,
inference(avatar_contradiction_clause,[status(thm)],[f3064]) ).
thf(f3064,plain,
( $false
| ~ spl0_1 ),
inference(trivial_inequality_removal,[status(thm)],[f3063]) ).
thf(f3063,plain,
( ( $false = $true )
| ~ spl0_1 ),
inference(forward_demodulation,[status(thm)],[f3062,f2048]) ).
thf(f2048,plain,
( ( $false
= ( ord_less_eq_nat @ xa @ m2 ) )
| ~ spl0_1 ),
inference(avatar_component_clause,[status(thm)],[f2046]) ).
thf(f2046,definition,
( spl0_1
<=> ( $false
= ( ord_less_eq_nat @ xa @ m2 ) ) ),
introduced(definition,[new_symbols(naming,[spl0_1])],[avatar_definition]) ).
thf(f3062,plain,
( $true
= ( ord_less_eq_nat @ xa @ m2 ) ),
inference(forward_demodulation,[status(thm)],[f3061,f2971]) ).
thf(f2971,plain,
( m2
= ( suc @ sK2 ) ),
inference(equality_proxy_clausification,[status(thm)],[f2967]) ).
thf(f2967,plain,
( ( ( suc @ sK2 )
= m2 )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f2965]) ).
thf(f2965,plain,
( $true
= ( ^ [Y0: nat] :
( ( suc @ Y0 )
= m2 )
@ sK2 ) ),
inference(sigma_clausification,[status(thm)],[f2948]) ).
thf(f2948,plain,
( $true
= ( ?? @ nat
@ ^ [Y0: nat] :
( ( suc @ Y0 )
= m2 ) ) ),
inference(boolean_simplification,[status(thm)],[f2947]) ).
thf(f2947,plain,
( $true
= ( $true
=> ( ?? @ nat
@ ^ [Y0: nat] :
( ( suc @ Y0 )
= m2 ) ) ) ),
inference(boolean_simplification,[status(thm)],[f2935]) ).
thf(f2935,plain,
( $true
= ( ~ $false
=> ( ?? @ nat
@ ^ [Y0: nat] :
( ( suc @ Y0 )
= m2 ) ) ) ),
inference(superposition,[status(thm)],[f2912,f1951]) ).
thf(f1951,plain,
( $false
= ( ord_less_eq_nat @ m2 @ zero_zero_nat ) ),
inference(not_proxy_clausification,[status(thm)],[f1917]) ).
thf(f1917,plain,
( $true
= ( ~ ( ord_less_eq_nat @ m2 @ zero_zero_nat ) ) ),
inference(equality_proxy_clausification,[status(thm)],[f1878]) ).
thf(f1878,plain,
( $true
= ( ( ~ ( ord_less_eq_nat @ m2 @ zero_zero_nat ) )
= $true ) ),
inference(superposition,[status(thm)],[f1873,f1574]) ).
thf(f1574,plain,
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ m2 )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1485]) ).
thf(f1485,plain,
( $true
= ( ^ [Y0: nat] : ( ord_less_eq_nat @ ( suc @ Y0 ) )
@ zero_zero_nat
@ m2 ) ),
inference(definition_unfolding,[status(thm)],[f1120,f1112]) ).
thf(f1112,plain,
( ord_less_nat
= ( ^ [Y0: nat] : ( ord_less_eq_nat @ ( suc @ Y0 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f462]) ).
thf(f462,plain,
( ord_less_nat
= ( ^ [Y0: nat] : ( ord_less_eq_nat @ ( suc @ Y0 ) ) ) ),
inference(fool_elimination,[status(thm)],[f461]) ).
thf(f461,plain,
( ord_less_nat
= ( ^ [X0: nat] : ( ord_less_eq_nat @ ( suc @ X0 ) ) ) ),
inference(rectify,[status(thm)],[f238]) ).
thf(f238,axiom,
( ord_less_nat
= ( ^ [X45: nat] : ( ord_less_eq_nat @ ( suc @ X45 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.sXLfLlVcSw/DTF2THF_9318.p',fact_237_less__eq__Suc__le) ).
thf(f1120,plain,
( ( ord_less_nat @ zero_zero_nat @ m2 )
= $true ),
inference(cnf_transformation,[status(thm)],[f1077]) ).
thf(f1077,plain,
( ( ord_less_nat @ zero_zero_nat @ m2 )
= $true ),
inference(fool_elimination,[status(thm)],[f1076]) ).
thf(f1076,plain,
ord_less_nat @ zero_zero_nat @ m2,
inference(rectify,[status(thm)],[f366]) ).
thf(f366,axiom,
ord_less_nat @ zero_zero_nat @ m2,
file('/export/starexec/sandbox/tmp/tmp.sXLfLlVcSw/DTF2THF_9318.p',conj_16) ).
thf(f1873,plain,
! [X1: nat] :
( $true
= ( ( ~ ( ord_less_eq_nat @ X1 @ zero_zero_nat ) )
= ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ X1 ) ) ),
inference(forward_demodulation,[status(thm)],[f1872,f1770]) ).
thf(f1770,plain,
! [X1: nat] :
( ( zero_zero_nat = X1 )
= ( ord_less_eq_nat @ X1 @ zero_zero_nat ) ),
inference(equality_proxy_clausification,[status(thm)],[f1769]) ).
thf(f1769,plain,
! [X1: nat] :
( $true
= ( ( zero_zero_nat = X1 )
= ( ord_less_eq_nat @ X1 @ zero_zero_nat ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1768]) ).
thf(f1768,plain,
! [X1: nat] :
( $true
= ( ^ [Y0: nat] :
( ( zero_zero_nat = Y0 )
= ( ord_less_eq_nat @ Y0 @ zero_zero_nat ) )
@ X1 ) ),
inference(pi_clausification,[status(thm)],[f1139]) ).
thf(f1139,plain,
( $true
= ( !! @ nat
@ ^ [Y0: nat] :
( ( zero_zero_nat = Y0 )
= ( ord_less_eq_nat @ Y0 @ zero_zero_nat ) ) ) ),
inference(cnf_transformation,[status(thm)],[f452]) ).
thf(f452,plain,
( $true
= ( !! @ nat
@ ^ [Y0: nat] :
( ( zero_zero_nat = Y0 )
= ( ord_less_eq_nat @ Y0 @ zero_zero_nat ) ) ) ),
inference(fool_elimination,[status(thm)],[f451]) ).
thf(f451,plain,
! [X0: nat] :
( ( zero_zero_nat = X0 )
= ( ord_less_eq_nat @ X0 @ zero_zero_nat ) ),
inference(rectify,[status(thm)],[f168]) ).
thf(f168,axiom,
! [X7: nat] :
( ( zero_zero_nat = X7 )
= ( ord_less_eq_nat @ X7 @ zero_zero_nat ) ),
file('/export/starexec/sandbox/tmp/tmp.sXLfLlVcSw/DTF2THF_9318.p',fact_167_le__0__eq) ).
thf(f1872,plain,
! [X1: nat] :
( $true
= ( ( zero_zero_nat != X1 )
= ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ X1 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1871]) ).
thf(f1871,plain,
! [X1: nat] :
( $true
= ( ^ [Y0: nat] :
( ( zero_zero_nat != Y0 )
= ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ Y0 ) )
@ X1 ) ),
inference(pi_clausification,[status(thm)],[f1566]) ).
thf(f1566,plain,
( ( !! @ nat
@ ^ [Y0: nat] :
( ( zero_zero_nat != Y0 )
= ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ Y0 ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1552]) ).
thf(f1552,plain,
( $true
= ( !! @ nat
@ ^ [Y0: nat] :
( ( zero_zero_nat != Y0 )
= ( ^ [Y1: nat] : ( ord_less_eq_nat @ ( suc @ Y1 ) )
@ zero_zero_nat
@ Y0 ) ) ) ),
inference(definition_unfolding,[status(thm)],[f1420,f1112]) ).
thf(f1420,plain,
( ( !! @ nat
@ ^ [Y0: nat] :
( ( zero_zero_nat != Y0 )
= ( ord_less_nat @ zero_zero_nat @ Y0 ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f651]) ).
thf(f651,plain,
( ( !! @ nat
@ ^ [Y0: nat] :
( ( zero_zero_nat != Y0 )
= ( ord_less_nat @ zero_zero_nat @ Y0 ) ) )
= $true ),
inference(fool_elimination,[status(thm)],[f650]) ).
thf(f650,plain,
! [X0: nat] :
( ( zero_zero_nat != X0 )
= ( ord_less_nat @ zero_zero_nat @ X0 ) ),
inference(rectify,[status(thm)],[f76]) ).
thf(f76,axiom,
! [X2: nat] :
( ( zero_zero_nat != X2 )
= ( ord_less_nat @ zero_zero_nat @ X2 ) ),
file('/export/starexec/sandbox/tmp/tmp.sXLfLlVcSw/DTF2THF_9318.p',fact_75_bot__nat__0_Onot__eq__extremum) ).
thf(f2912,plain,
! [X1: nat] :
( ( ~ ( ord_less_eq_nat @ X1 @ zero_zero_nat )
=> ( ?? @ nat
@ ^ [Y0: nat] :
( ( suc @ Y0 )
= X1 ) ) )
= $true ),
inference(forward_demodulation,[status(thm)],[f2911,f1874]) ).
thf(f1874,plain,
! [X1: nat] :
( ( ~ ( ord_less_eq_nat @ X1 @ zero_zero_nat ) )
= ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ X1 ) ),
inference(equality_proxy_clausification,[status(thm)],[f1873]) ).
thf(f2911,plain,
! [X1: nat] :
( ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ X1 )
=> ( ?? @ nat
@ ^ [Y0: nat] :
( ( suc @ Y0 )
= X1 ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f2910]) ).
thf(f2910,plain,
! [X1: nat] :
( $true
= ( ^ [Y0: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ Y0 )
=> ( ?? @ nat
@ ^ [Y1: nat] :
( ( suc @ Y1 )
= Y0 ) ) )
@ X1 ) ),
inference(pi_clausification,[status(thm)],[f1642]) ).
thf(f1642,plain,
( ( !! @ nat
@ ^ [Y0: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ Y0 )
=> ( ?? @ nat
@ ^ [Y1: nat] :
( ( suc @ Y1 )
= Y0 ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1548]) ).
thf(f1548,plain,
( ( !! @ nat
@ ^ [Y0: nat] :
( ( ^ [Y1: nat] : ( ord_less_eq_nat @ ( suc @ Y1 ) )
@ zero_zero_nat
@ Y0 )
=> ( ?? @ nat
@ ^ [Y1: nat] :
( ( suc @ Y1 )
= Y0 ) ) ) )
= $true ),
inference(definition_unfolding,[status(thm)],[f1407,f1112]) ).
thf(f1407,plain,
( ( !! @ nat
@ ^ [Y0: nat] :
( ( ord_less_nat @ zero_zero_nat @ Y0 )
=> ( ?? @ nat
@ ^ [Y1: nat] :
( ( suc @ Y1 )
= Y0 ) ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f941]) ).
thf(f941,plain,
( ( !! @ nat
@ ^ [Y0: nat] :
( ( ord_less_nat @ zero_zero_nat @ Y0 )
=> ( ?? @ nat
@ ^ [Y1: nat] :
( ( suc @ Y1 )
= Y0 ) ) ) )
= $true ),
inference(fool_elimination,[status(thm)],[f940]) ).
thf(f940,plain,
! [X0: nat] :
( ( ord_less_nat @ zero_zero_nat @ X0 )
=> ? [X1: nat] :
( ( suc @ X1 )
= X0 ) ),
inference(rectify,[status(thm)],[f248]) ).
thf(f248,axiom,
! [X7: nat] :
( ( ord_less_nat @ zero_zero_nat @ X7 )
=> ? [X40: nat] :
( ( suc @ X40 )
= X7 ) ),
file('/export/starexec/sandbox/tmp/tmp.sXLfLlVcSw/DTF2THF_9318.p',fact_247_gr0__implies__Suc) ).
thf(f3061,plain,
( $true
= ( ord_less_eq_nat @ xa @ ( suc @ sK2 ) ) ),
inference(forward_demodulation,[status(thm)],[f3060,f1717]) ).
thf(f1717,plain,
! [X1: nat] :
( ( minus_minus_nat @ X1 @ zero_zero_nat )
= X1 ),
inference(equality_proxy_clausification,[status(thm)],[f1716]) ).
thf(f1716,plain,
! [X1: nat] :
( $true
= ( ( minus_minus_nat @ X1 @ zero_zero_nat )
= X1 ) ),
inference(beta_eta_normalization,[status(thm)],[f1715]) ).
thf(f1715,plain,
! [X1: nat] :
( ( ^ [Y0: nat] :
( ( minus_minus_nat @ Y0 @ zero_zero_nat )
= Y0 )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f1264]) ).
thf(f1264,plain,
( $true
= ( !! @ nat
@ ^ [Y0: nat] :
( ( minus_minus_nat @ Y0 @ zero_zero_nat )
= Y0 ) ) ),
inference(cnf_transformation,[status(thm)],[f880]) ).
thf(f880,plain,
( $true
= ( !! @ nat
@ ^ [Y0: nat] :
( ( minus_minus_nat @ Y0 @ zero_zero_nat )
= Y0 ) ) ),
inference(fool_elimination,[status(thm)],[f879]) ).
thf(f879,plain,
! [X0: nat] :
( ( minus_minus_nat @ X0 @ zero_zero_nat )
= X0 ),
inference(rectify,[status(thm)],[f327]) ).
thf(f327,axiom,
! [X2: nat] :
( ( minus_minus_nat @ X2 @ zero_zero_nat )
= X2 ),
file('/export/starexec/sandbox/tmp/tmp.sXLfLlVcSw/DTF2THF_9318.p',fact_326_diff__zero) ).
thf(f3060,plain,
( $true
= ( ord_less_eq_nat @ xa @ ( suc @ ( minus_minus_nat @ sK2 @ zero_zero_nat ) ) ) ),
inference(forward_demodulation,[status(thm)],[f2317,f3025]) ).
thf(f3025,plain,
! [X0: nat] :
( ( minus_minus_nat @ m2 @ ( suc @ X0 ) )
= ( minus_minus_nat @ sK2 @ X0 ) ),
inference(superposition,[status(thm)],[f3002,f2971]) ).
thf(f3002,plain,
! [X2: nat,X1: nat] :
( ( minus_minus_nat @ ( suc @ X1 ) @ ( suc @ X2 ) )
= ( minus_minus_nat @ X1 @ X2 ) ),
inference(equality_proxy_clausification,[status(thm)],[f2978]) ).
thf(f2978,plain,
! [X2: nat,X1: nat] :
( $true
= ( ( minus_minus_nat @ ( suc @ X1 ) @ ( suc @ X2 ) )
= ( minus_minus_nat @ X1 @ X2 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f2977]) ).
thf(f2977,plain,
! [X2: nat,X1: nat] :
( ( ^ [Y0: nat] :
( ( minus_minus_nat @ ( suc @ X1 ) @ ( suc @ Y0 ) )
= ( minus_minus_nat @ X1 @ Y0 ) )
@ X2 )
= $true ),
inference(pi_clausification,[status(thm)],[f2970]) ).
thf(f2970,plain,
! [X1: nat] :
( ( !! @ nat
@ ^ [Y0: nat] :
( ( minus_minus_nat @ ( suc @ X1 ) @ ( suc @ Y0 ) )
= ( minus_minus_nat @ X1 @ Y0 ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f2969]) ).
thf(f2969,plain,
! [X1: nat] :
( $true
= ( ^ [Y0: nat] :
( !! @ nat
@ ^ [Y1: nat] :
( ( minus_minus_nat @ ( suc @ Y0 ) @ ( suc @ Y1 ) )
= ( minus_minus_nat @ Y0 @ Y1 ) ) )
@ X1 ) ),
inference(pi_clausification,[status(thm)],[f1113]) ).
thf(f1113,plain,
( ( !! @ nat
@ ^ [Y0: nat] :
( !! @ nat
@ ^ [Y1: nat] :
( ( minus_minus_nat @ ( suc @ Y0 ) @ ( suc @ Y1 ) )
= ( minus_minus_nat @ Y0 @ Y1 ) ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f649]) ).
thf(f649,plain,
( ( !! @ nat
@ ^ [Y0: nat] :
( !! @ nat
@ ^ [Y1: nat] :
( ( minus_minus_nat @ ( suc @ Y0 ) @ ( suc @ Y1 ) )
= ( minus_minus_nat @ Y0 @ Y1 ) ) ) )
= $true ),
inference(fool_elimination,[status(thm)],[f648]) ).
thf(f648,plain,
! [X0: nat,X1: nat] :
( ( minus_minus_nat @ ( suc @ X1 ) @ ( suc @ X0 ) )
= ( minus_minus_nat @ X1 @ X0 ) ),
inference(rectify,[status(thm)],[f81]) ).
thf(f81,axiom,
! [X7: nat,X14: nat] :
( ( minus_minus_nat @ X14 @ X7 )
= ( minus_minus_nat @ ( suc @ X14 ) @ ( suc @ X7 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.sXLfLlVcSw/DTF2THF_9318.p',fact_80_diff__Suc__Suc) ).
thf(f2317,plain,
( $true
= ( ord_less_eq_nat @ xa @ ( suc @ ( minus_minus_nat @ m2 @ ( suc @ zero_zero_nat ) ) ) ) ),
inference(boolean_simplification,[status(thm)],[f2305]) ).
thf(f2305,plain,
( ( $true
=> ( ord_less_eq_nat @ xa @ ( suc @ ( minus_minus_nat @ m2 @ ( suc @ zero_zero_nat ) ) ) ) )
= $true ),
inference(superposition,[status(thm)],[f2283,f1146]) ).
thf(f1146,plain,
( ( ord_less_eq_nat @ xa @ ( minus_minus_nat @ m2 @ ( suc @ zero_zero_nat ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f826]) ).
thf(f826,plain,
( ( ord_less_eq_nat @ xa @ ( minus_minus_nat @ m2 @ ( suc @ zero_zero_nat ) ) )
= $true ),
inference(fool_elimination,[status(thm)],[f825]) ).
thf(f825,plain,
ord_less_eq_nat @ xa @ ( minus_minus_nat @ m2 @ ( suc @ zero_zero_nat ) ),
inference(rectify,[status(thm)],[f369]) ).
thf(f369,axiom,
ord_less_eq_nat @ xa @ ( minus_minus_nat @ m2 @ ( suc @ zero_zero_nat ) ),
file('/export/starexec/sandbox/tmp/tmp.sXLfLlVcSw/DTF2THF_9318.p',conj_19) ).
thf(f2283,plain,
! [X2: nat,X1: nat] :
( $true
= ( ( ord_less_eq_nat @ X2 @ X1 )
=> ( ord_less_eq_nat @ X2 @ ( suc @ X1 ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f2282]) ).
thf(f2282,plain,
! [X2: nat,X1: nat] :
( ( ^ [Y0: nat] :
( ( ord_less_eq_nat @ Y0 @ X1 )
=> ( ord_less_eq_nat @ Y0 @ ( suc @ X1 ) ) )
@ X2 )
= $true ),
inference(pi_clausification,[status(thm)],[f2279]) ).
thf(f2279,plain,
! [X1: nat] :
( $true
= ( !! @ nat
@ ^ [Y0: nat] :
( ( ord_less_eq_nat @ Y0 @ X1 )
=> ( ord_less_eq_nat @ Y0 @ ( suc @ X1 ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f2278]) ).
thf(f2278,plain,
! [X1: nat] :
( ( ^ [Y0: nat] :
( !! @ nat
@ ^ [Y1: nat] :
( ( ord_less_eq_nat @ Y1 @ Y0 )
=> ( ord_less_eq_nat @ Y1 @ ( suc @ Y0 ) ) ) )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f1304]) ).
thf(f1304,plain,
( $true
= ( !! @ nat
@ ^ [Y0: nat] :
( !! @ nat
@ ^ [Y1: nat] :
( ( ord_less_eq_nat @ Y1 @ Y0 )
=> ( ord_less_eq_nat @ Y1 @ ( suc @ Y0 ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f1043]) ).
thf(f1043,plain,
( $true
= ( !! @ nat
@ ^ [Y0: nat] :
( !! @ nat
@ ^ [Y1: nat] :
( ( ord_less_eq_nat @ Y1 @ Y0 )
=> ( ord_less_eq_nat @ Y1 @ ( suc @ Y0 ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f1042]) ).
thf(f1042,plain,
! [X0: nat,X1: nat] :
( ( ord_less_eq_nat @ X0 @ X1 )
=> ( ord_less_eq_nat @ X0 @ ( suc @ X1 ) ) ),
inference(rectify,[status(thm)],[f163]) ).
thf(f163,axiom,
! [X14: nat,X7: nat] :
( ( ord_less_eq_nat @ X14 @ X7 )
=> ( ord_less_eq_nat @ X14 @ ( suc @ X7 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.sXLfLlVcSw/DTF2THF_9318.p',fact_162_le__SucI) ).
thf(f3055,plain,
~ spl0_2,
inference(avatar_contradiction_clause,[status(thm)],[f3054]) ).
thf(f3054,plain,
( $false
| ~ spl0_2 ),
inference(trivial_inequality_removal,[status(thm)],[f3053]) ).
thf(f3053,plain,
( ( $false = $true )
| ~ spl0_2 ),
inference(forward_demodulation,[status(thm)],[f3052,f2983]) ).
thf(f2983,plain,
( $false
= ( ord_less_eq_nat @ m2 @ sK2 ) ),
inference(superposition,[status(thm)],[f1690,f2971]) ).
thf(f1690,plain,
! [X1: nat] :
( $false
= ( ord_less_eq_nat @ ( suc @ X1 ) @ X1 ) ),
inference(not_proxy_clausification,[status(thm)],[f1689]) ).
thf(f1689,plain,
! [X1: nat] :
( $true
= ( ~ ( ord_less_eq_nat @ ( suc @ X1 ) @ X1 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1688]) ).
thf(f1688,plain,
! [X1: nat] :
( ( ^ [Y0: nat] :
~ ( ord_less_eq_nat @ ( suc @ Y0 ) @ Y0 )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f1621]) ).
thf(f1621,plain,
( $true
= ( !! @ nat
@ ^ [Y0: nat] :
~ ( ord_less_eq_nat @ ( suc @ Y0 ) @ Y0 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1562]) ).
thf(f1562,plain,
( ( !! @ nat
@ ^ [Y0: nat] :
~ ( ^ [Y1: nat] : ( ord_less_eq_nat @ ( suc @ Y1 ) )
@ Y0
@ Y0 ) )
= $true ),
inference(definition_unfolding,[status(thm)],[f1480,f1112]) ).
thf(f1480,plain,
( $true
= ( !! @ nat
@ ^ [Y0: nat] :
~ ( ord_less_nat @ Y0 @ Y0 ) ) ),
inference(cnf_transformation,[status(thm)],[f1013]) ).
thf(f1013,plain,
( $true
= ( !! @ nat
@ ^ [Y0: nat] :
~ ( ord_less_nat @ Y0 @ Y0 ) ) ),
inference(fool_elimination,[status(thm)],[f1012]) ).
thf(f1012,plain,
! [X0: nat] :
~ ( ord_less_nat @ X0 @ X0 ),
inference(rectify,[status(thm)],[f147]) ).
thf(f147,axiom,
! [X7: nat] :
~ ( ord_less_nat @ X7 @ X7 ),
file('/export/starexec/sandbox/tmp/tmp.sXLfLlVcSw/DTF2THF_9318.p',fact_146_less__not__refl) ).
thf(f3052,plain,
( ( ( ord_less_eq_nat @ m2 @ sK2 )
= $true )
| ~ spl0_2 ),
inference(forward_demodulation,[status(thm)],[f3038,f1717]) ).
thf(f3038,plain,
( ( ( ord_less_eq_nat @ m2 @ ( minus_minus_nat @ sK2 @ zero_zero_nat ) )
= $true )
| ~ spl0_2 ),
inference(backward_demodulation,[status(thm)],[f2529,f3025]) ).
thf(f2529,plain,
( ( ( ord_less_eq_nat @ m2 @ ( minus_minus_nat @ m2 @ ( suc @ zero_zero_nat ) ) )
= $true )
| ~ spl0_2 ),
inference(backward_demodulation,[status(thm)],[f1146,f2522]) ).
thf(f2522,plain,
( ( m2 = xa )
| ~ spl0_2 ),
inference(equality_proxy_clausification,[status(thm)],[f2512]) ).
thf(f2512,plain,
( ( $true
= ( xa = m2 ) )
| ~ spl0_2 ),
inference(boolean_simplification,[status(thm)],[f2511]) ).
thf(f2511,plain,
( ( ( $true
=> ( xa = m2 ) )
= $true )
| ~ spl0_2 ),
inference(boolean_simplification,[status(thm)],[f2510]) ).
thf(f2510,plain,
( ( $true
= ( ( ( g @ m2 )
= ( g @ m2 ) )
=> ( xa = m2 ) ) )
| ~ spl0_2 ),
inference(boolean_simplification,[status(thm)],[f2507]) ).
thf(f2507,plain,
( ( ( $true
=> ( ( ( g @ m2 )
= ( g @ m2 ) )
=> ( xa = m2 ) ) )
= $true )
| ~ spl0_2 ),
inference(superposition,[status(thm)],[f2051,f1756]) ).
thf(f1756,plain,
! [X0: nat] :
( ( ord_less_eq_nat @ X0 @ X0 )
= $true ),
inference(superposition,[status(thm)],[f1753,f1717]) ).
thf(f1753,plain,
! [X2: nat,X1: nat] :
( $true
= ( ord_less_eq_nat @ ( minus_minus_nat @ X2 @ X1 ) @ X2 ) ),
inference(beta_eta_normalization,[status(thm)],[f1752]) ).
thf(f1752,plain,
! [X2: nat,X1: nat] :
( $true
= ( ^ [Y0: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ Y0 @ X1 ) @ Y0 )
@ X2 ) ),
inference(pi_clausification,[status(thm)],[f1751]) ).
thf(f1751,plain,
! [X1: nat] :
( $true
= ( !! @ nat
@ ^ [Y0: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ Y0 @ X1 ) @ Y0 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1750]) ).
thf(f1750,plain,
! [X1: nat] :
( ( ^ [Y0: nat] :
( !! @ nat
@ ^ [Y1: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ Y1 @ Y0 ) @ Y1 ) )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f1467]) ).
thf(f1467,plain,
( ( !! @ nat
@ ^ [Y0: nat] :
( !! @ nat
@ ^ [Y1: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ Y1 @ Y0 ) @ Y1 ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f927]) ).
thf(f927,plain,
( ( !! @ nat
@ ^ [Y0: nat] :
( !! @ nat
@ ^ [Y1: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ Y1 @ Y0 ) @ Y1 ) ) )
= $true ),
inference(fool_elimination,[status(thm)],[f926]) ).
thf(f926,plain,
! [X0: nat,X1: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ X0 @ X1 ) @ X0 ),
inference(rectify,[status(thm)],[f215]) ).
thf(f215,axiom,
! [X14: nat,X7: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ X14 @ X7 ) @ X14 ),
file('/export/starexec/sandbox/tmp/tmp.sXLfLlVcSw/DTF2THF_9318.p',fact_214_diff__le__self) ).
thf(f2051,plain,
( ! [X0: nat] :
( $true
= ( ( ord_less_eq_nat @ X0 @ m2 )
=> ( ( ( g @ X0 )
= ( g @ m2 ) )
=> ( xa = X0 ) ) ) )
| ~ spl0_2 ),
inference(avatar_component_clause,[status(thm)],[f2050]) ).
thf(f2050,definition,
( spl0_2
<=> ! [X0: nat] :
( $true
= ( ( ord_less_eq_nat @ X0 @ m2 )
=> ( ( ( g @ X0 )
= ( g @ m2 ) )
=> ( xa = X0 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[spl0_2])],[avatar_definition]) ).
thf(f2052,plain,
( spl0_1
| spl0_2 ),
inference(avatar_split_clause,[status(thm)],[f2023,f2050,f2046]) ).
thf(f2023,plain,
! [X0: nat] :
( ( $true
= ( ( ord_less_eq_nat @ X0 @ m2 )
=> ( ( ( g @ X0 )
= ( g @ m2 ) )
=> ( xa = X0 ) ) ) )
| ( $false
= ( ord_less_eq_nat @ xa @ m2 ) ) ),
inference(superposition,[status(thm)],[f1709,f1655]) ).
thf(f1655,plain,
( ( g @ m2 )
= ( g @ xa ) ),
inference(equality_proxy_clausification,[status(thm)],[f1650]) ).
thf(f1650,plain,
( ( ( g @ m2 )
= ( g @ xa ) )
= $true ),
inference(boolean_simplification,[status(thm)],[f1648]) ).
thf(f1648,plain,
( ( $false
| ( ( g @ m2 )
= ( g @ xa ) ) )
= $true ),
inference(backward_demodulation,[status(thm)],[f1513,f1647]) ).
thf(f1647,plain,
( ( member_a @ ( g @ xa ) @ h1 )
= $false ),
inference(not_proxy_clausification,[status(thm)],[f1266]) ).
thf(f1266,plain,
( $true
= ( ~ ( member_a @ ( g @ xa ) @ h1 ) ) ),
inference(cnf_transformation,[status(thm)],[f1001]) ).
thf(f1001,plain,
( $true
= ( ~ ( member_a @ ( g @ xa ) @ h1 ) ) ),
inference(fool_elimination,[status(thm)],[f1000]) ).
thf(f1000,plain,
~ ( member_a @ ( g @ xa ) @ h1 ),
inference(rectify,[status(thm)],[f372]) ).
thf(f372,negated_conjecture,
~ ( member_a @ ( g @ xa ) @ h1 ),
inference(negated_conjecture,[status(cth)],[f371]) ).
thf(f371,conjecture,
member_a @ ( g @ xa ) @ h1,
file('/export/starexec/sandbox/tmp/tmp.sXLfLlVcSw/DTF2THF_9318.p',conj_21) ).
thf(f1513,plain,
( $true
= ( ( member_a @ ( g @ xa ) @ h1 )
| ( ( g @ m2 )
= ( g @ xa ) ) ) ),
inference(definition_unfolding,[status(thm)],[f1247,f1259]) ).
thf(f1259,plain,
( h2
= ( g @ m2 ) ),
inference(cnf_transformation,[status(thm)],[f744]) ).
thf(f744,plain,
( h2
= ( g @ m2 ) ),
inference(fool_elimination,[status(thm)],[f353]) ).
thf(f353,axiom,
( h2
= ( g @ m2 ) ),
file('/export/starexec/sandbox/tmp/tmp.sXLfLlVcSw/DTF2THF_9318.p',conj_3) ).
thf(f1247,plain,
( $true
= ( ( member_a @ ( g @ xa ) @ h1 )
| ( h2
= ( g @ xa ) ) ) ),
inference(cnf_transformation,[status(thm)],[f713]) ).
thf(f713,plain,
( $true
= ( ( member_a @ ( g @ xa ) @ h1 )
| ( h2
= ( g @ xa ) ) ) ),
inference(fool_elimination,[status(thm)],[f712]) ).
thf(f712,plain,
( ( h2
= ( g @ xa ) )
| ( member_a @ ( g @ xa ) @ h1 ) ),
inference(rectify,[status(thm)],[f370]) ).
thf(f370,axiom,
( ( h2
= ( g @ xa ) )
| ( member_a @ ( g @ xa ) @ h1 ) ),
file('/export/starexec/sandbox/tmp/tmp.sXLfLlVcSw/DTF2THF_9318.p',conj_20) ).
thf(f1709,plain,
! [X2: nat,X1: nat] :
( ( $true
= ( ( ord_less_eq_nat @ X2 @ m2 )
=> ( ( ( g @ X2 )
= ( g @ X1 ) )
=> ( X1 = X2 ) ) ) )
| ( $false
= ( ord_less_eq_nat @ X1 @ m2 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1705]) ).
thf(f1705,plain,
! [X2: nat,X1: nat] :
( ( ( ^ [Y0: nat] :
( ( ord_less_eq_nat @ Y0 @ m2 )
=> ( ( ( g @ Y0 )
= ( g @ X1 ) )
=> ( X1 = Y0 ) ) )
@ X2 )
= $true )
| ( $false
= ( ord_less_eq_nat @ X1 @ m2 ) ) ),
inference(pi_clausification,[status(thm)],[f1670]) ).
thf(f1670,plain,
! [X1: nat] :
( ( ( !! @ nat
@ ^ [Y0: nat] :
( ( ord_less_eq_nat @ Y0 @ m2 )
=> ( ( ( g @ Y0 )
= ( g @ X1 ) )
=> ( X1 = Y0 ) ) ) )
= $true )
| ( $false
= ( ord_less_eq_nat @ X1 @ m2 ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f1666]) ).
thf(f1666,plain,
! [X1: nat] :
( $true
= ( ( ord_less_eq_nat @ X1 @ m2 )
=> ( !! @ nat
@ ^ [Y0: nat] :
( ( ord_less_eq_nat @ Y0 @ m2 )
=> ( ( ( g @ Y0 )
= ( g @ X1 ) )
=> ( X1 = Y0 ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1665]) ).
thf(f1665,plain,
! [X1: nat] :
( $true
= ( ^ [Y0: nat] :
( ( ord_less_eq_nat @ Y0 @ m2 )
=> ( !! @ nat
@ ^ [Y1: nat] :
( ( ord_less_eq_nat @ Y1 @ m2 )
=> ( ( ( g @ Y1 )
= ( g @ Y0 ) )
=> ( Y0 = Y1 ) ) ) ) )
@ X1 ) ),
inference(pi_clausification,[status(thm)],[f1268]) ).
thf(f1268,plain,
( $true
= ( !! @ nat
@ ^ [Y0: nat] :
( ( ord_less_eq_nat @ Y0 @ m2 )
=> ( !! @ nat
@ ^ [Y1: nat] :
( ( ord_less_eq_nat @ Y1 @ m2 )
=> ( ( ( g @ Y1 )
= ( g @ Y0 ) )
=> ( Y0 = Y1 ) ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f603]) ).
thf(f603,plain,
( $true
= ( !! @ nat
@ ^ [Y0: nat] :
( ( ord_less_eq_nat @ Y0 @ m2 )
=> ( !! @ nat
@ ^ [Y1: nat] :
( ( ord_less_eq_nat @ Y1 @ m2 )
=> ( ( ( g @ Y1 )
= ( g @ Y0 ) )
=> ( Y0 = Y1 ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f602]) ).
thf(f602,plain,
! [X0: nat] :
( ( ord_less_eq_nat @ X0 @ m2 )
=> ! [X1: nat] :
( ( ord_less_eq_nat @ X1 @ m2 )
=> ( ( ( g @ X1 )
= ( g @ X0 ) )
=> ( X0 = X1 ) ) ) ),
inference(rectify,[status(thm)],[f351]) ).
thf(f351,axiom,
! [X56: nat] :
( ( ord_less_eq_nat @ X56 @ m2 )
=> ! [X34: nat] :
( ( ord_less_eq_nat @ X34 @ m2 )
=> ( ( ( g @ X56 )
= ( g @ X34 ) )
=> ( X34 = X56 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.sXLfLlVcSw/DTF2THF_9318.p',conj_1) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP026^1 : TPTP v9.2.1. Released v7.5.0.
% 0.11/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n003.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.35 % CPULimit : 300
% 0.16/0.35 % WCLimit : 300
% 0.16/0.35 % DateTime : Tue Feb 24 08:23:40 EST 2026
% 0.16/0.35 % CPUTime :
% 0.16/0.35 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.30/0.47 ---- Original DTF file ---
% 0.30/0.47 thf(spec,logic,$$dhol).
% 0.30/0.47 %------------------------------------------------------------------------------
% 0.30/0.47 % File : ITP026^1 : TPTP v9.2.1. Released v7.5.0.
% 0.30/0.47 % Domain : Interactive Theorem Proving
% 0.30/0.47 % Problem : Sledgehammer Algebra8 problem prob_1404__6432666_1
% 0.30/0.47 % Version : Especial.
% 0.30/0.47 % English :
% 0.30/0.47
% 0.30/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.30/0.47 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.30/0.47 % Source : [Des21]
% 0.30/0.47 % Names : Algebra8/prob_1404__6432666_1 [Des21]
% 0.30/0.47
% 0.30/0.47 % Status : Theorem
% 0.30/0.47 % Rating : 0.33 v9.1.0, 0.38 v9.0.0, 0.40 v8.2.0, 0.31 v8.1.0, 0.36 v7.5.0
% 0.30/0.47 % Syntax : Number of formulae : 493 ( 128 unt; 122 typ; 0 def)
% 0.30/0.47 % Number of atoms : 1156 ( 221 equ; 0 cnn)
% 0.30/0.47 % Maximal formula atoms : 12 ( 3 avg)
% 0.30/0.47 % Number of connectives : 4133 ( 74 ~; 13 |; 56 &;3386 @)
% 0.30/0.47 % ( 0 <=>; 604 =>; 0 <=; 0 <~>)
% 0.30/0.47 % Maximal formula depth : 21 ( 8 avg)
% 0.30/0.47 % Number of types : 20 ( 19 usr)
% 0.30/0.47 % Number of type conns : 375 ( 375 >; 0 *; 0 +; 0 <<)
% 0.30/0.47 % Number of symbols : 105 ( 103 usr; 16 con; 0-5 aty)
% 0.30/0.47 % Number of variables : 1109 ( 166 ^; 904 !; 39 ?;1109 :)
% 0.30/0.47 % SPC : TH0_THM_EQU_NAR
% 0.30/0.47
% 0.30/0.47 % Comments : This file was generated by Sledgehammer 2021-02-23 15:46:47.396
% 0.30/0.47 %------------------------------------------------------------------------------
% 0.30/0.47 % Could-be-implicit typings (19)
% 0.30/0.47 thf(ty_n_t__Algebra1__Ocarrier__Ocarrier____ext_I_062_Itf__e_Mtf__e_J_Mt__Algebra4__OaGroup__OaGroup____ext_I_062_Itf__e_Mtf__e_J_Mt__Algebra7__OModule__OModule____ext_I_062_Itf__e_Mtf__e_J_Mtf__b_Mt__Product____Type__Ounit_J_J_J,type,
% 0.30/0.47 carrie377329159t_unit: $tType ).
% 0.30/0.47
% 0.30/0.47 thf(ty_n_t__Algebra1__Ocarrier__Ocarrier____ext_I_062_Itf__e_Mtf__a_J_Mt__Algebra4__OaGroup__OaGroup____ext_I_062_Itf__e_Mtf__a_J_Mt__Algebra7__OModule__OModule____ext_I_062_Itf__e_Mtf__a_J_Mtf__b_Mt__Product____Type__Ounit_J_J_J,type,
% 0.30/0.47 carrie292259835t_unit: $tType ).
% 0.30/0.47
% 0.30/0.47 thf(ty_n_t__Algebra1__Ocarrier__Ocarrier____ext_I_062_Itf__a_Mtf__e_J_Mt__Algebra4__OaGroup__OaGroup____ext_I_062_Itf__a_Mtf__e_J_Mt__Algebra7__OModule__OModule____ext_I_062_Itf__a_Mtf__e_J_Mtf__b_Mt__Product____Type__Ounit_J_J_J,type,
% 0.30/0.47 carrie1065537299t_unit: $tType ).
% 0.30/0.47
% 0.30/0.47 thf(ty_n_t__Algebra1__Ocarrier__Ocarrier____ext_I_062_Itf__a_Mtf__a_J_Mt__Algebra4__OaGroup__OaGroup____ext_I_062_Itf__a_Mtf__a_J_Mt__Algebra7__OModule__OModule____ext_I_062_Itf__a_Mtf__a_J_Mtf__b_Mt__Product____Type__Ounit_J_J_J,type,
% 0.30/0.47 carrie980467975t_unit: $tType ).
% 0.30/0.47
% 0.30/0.47 thf(ty_n_t__Algebra1__Ocarrier__Ocarrier____ext_Itf__a_Mt__Algebra4__OaGroup__OaGroup____ext_Itf__a_Mt__Algebra7__OModule__OModule____ext_Itf__a_Mtf__b_Mt__Product____Type__Ounit_J_J_J,type,
% 0.30/0.47 carrie1963041556t_unit: $tType ).
% 0.30/0.47
% 0.30/0.47 thf(ty_n_t__Algebra1__Ocarrier__Ocarrier____ext_Itf__e_Mt__Algebra4__OaGroup__OaGroup____ext_Itf__e_Mt__Algebra7__OModule__OModule____ext_Itf__e_Mtf__b_Mtf__f_J_J_J,type,
% 0.30/0.47 carrie1821755406_e_b_f: $tType ).
% 0.30/0.47
% 0.30/0.47 thf(ty_n_t__Algebra1__Ocarrier__Ocarrier____ext_Itf__a_Mt__Algebra4__OaGroup__OaGroup____ext_Itf__a_Mt__Algebra7__OModule__OModule____ext_Itf__a_Mtf__b_Mtf__c_J_J_J,type,
% 0.30/0.47 carrie722926983_a_b_c: $tType ).
% 0.30/0.47
% 0.30/0.47 thf(ty_n_t__Algebra1__Ocarrier__Ocarrier____ext_Itf__b_Mt__Algebra4__OaGroup__OaGroup____ext_Itf__b_Mt__Algebra4__ORing__ORing____ext_Itf__b_Mtf__d_J_J_J,type,
% 0.30/0.47 carrie1950868226xt_b_d: $tType ).
% 0.30/0.47
% 0.30/0.47 thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mtf__e_J_J,type,
% 0.30/0.47 set_nat_e: $tType ).
% 0.30/0.47
% 0.30/0.47 thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mtf__b_J_J,type,
% 0.30/0.47 set_nat_b: $tType ).
% 0.30/0.47
% 0.30/0.47 thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
% 0.30/0.47 set_nat_a: $tType ).
% 0.30/0.47
% 0.30/0.47 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 0.30/0.47 set_nat: $tType ).
% 0.30/0.47
% 0.30/0.47 thf(ty_n_t__Set__Oset_Itf__e_J,type,
% 0.30/0.47 set_e: $tType ).
% 0.30/0.47
% 0.30/0.47 thf(ty_n_t__Set__Oset_Itf__b_J,type,
% 0.30/0.47 set_b: $tType ).
% 0.30/0.47
% 0.30/0.47 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 0.30/0.47 set_a: $tType ).
% 0.30/0.47
% 0.30/0.47 thf(ty_n_t__Nat__Onat,type,
% 0.30/0.47 nat: $tType ).
% 0.30/0.47
% 0.30/0.47 thf(ty_n_tf__e,type,
% 0.30/0.47 e: $tType ).
% 0.30/0.47
% 0.30/0.47 thf(ty_n_tf__b,type,
% 0.30/0.47 b: $tType ).
% 0.30/0.47
% 0.30/0.47 thf(ty_n_tf__a,type,
% 0.30/0.47 a: $tType ).
% 0.30/0.47
% 0.30/0.47 % Explicit typings (103)
% 0.30/0.47 thf(sy_c_Algebra1_Ocarrier_Ocarrier_001tf__a_001t__Algebra4__OaGroup__OaGroup____ext_Itf__a_Mt__Algebra7__OModule__OModule____ext_Itf__a_Mtf__b_Mt__Product____Type__Ounit_J_J,type,
% 0.30/0.47 carrie1074654371t_unit: carrie1963041556t_unit > set_a ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Algebra1_Ocarrier_Ocarrier_001tf__a_001t__Algebra4__OaGroup__OaGroup____ext_Itf__a_Mt__Algebra7__OModule__OModule____ext_Itf__a_Mtf__b_Mtf__c_J_J,type,
% 0.30/0.47 carrie2021454486_a_b_c: carrie722926983_a_b_c > set_a ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Algebra1_Ocarrier_Ocarrier_001tf__b_001t__Algebra4__OaGroup__OaGroup____ext_Itf__b_Mt__Algebra4__ORing__ORing____ext_Itf__b_Mtf__d_J_J,type,
% 0.30/0.47 carrie2079586589xt_b_d: carrie1950868226xt_b_d > set_b ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Algebra1_Ocarrier_Ocarrier_001tf__e_001t__Algebra4__OaGroup__OaGroup____ext_Itf__e_Mt__Algebra7__OModule__OModule____ext_Itf__e_Mtf__b_Mtf__f_J_J,type,
% 0.30/0.47 carrie730238621_e_b_f: carrie1821755406_e_b_f > set_e ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Algebra4_ORing_001tf__b_001tf__d,type,
% 0.30/0.47 ring_b_d: carrie1950868226xt_b_d > $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Algebra4_OaGroup_Ozero_001tf__a_001t__Algebra7__OModule__OModule____ext_Itf__a_Mtf__b_Mtf__c_J,type,
% 0.30/0.47 zero_a1261444626_a_b_c: carrie722926983_a_b_c > a ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Algebra4_Oideal_001tf__b_001tf__d,type,
% 0.30/0.47 ideal_b_d: carrie1950868226xt_b_d > set_b > $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Algebra7_OAnnihilator_001tf__b_001tf__d_001tf__a_001t__Product____Type__Ounit,type,
% 0.30/0.47 annihi259882159t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > set_b ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Algebra7_OAnnihilator_001tf__b_001tf__d_001tf__a_001tf__c,type,
% 0.30/0.47 annihilator_b_d_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > set_b ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Algebra7_OAnnihilator_001tf__b_001tf__d_001tf__e_001tf__f,type,
% 0.30/0.47 annihilator_b_d_e_f: carrie1950868226xt_b_d > carrie1821755406_e_b_f > set_b ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Algebra7_OHOM_001tf__b_001tf__d_001tf__a_001t__Product____Type__Ounit_001tf__a_001t__Product____Type__Ounit,type,
% 0.30/0.47 hOM_b_1103679019t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > carrie1963041556t_unit > carrie980467975t_unit ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Algebra7_OHOM_001tf__b_001tf__d_001tf__a_001t__Product____Type__Ounit_001tf__a_001tf__c,type,
% 0.30/0.47 hOM_b_796081438it_a_c: carrie1950868226xt_b_d > carrie1963041556t_unit > carrie722926983_a_b_c > carrie980467975t_unit ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Algebra7_OHOM_001tf__b_001tf__d_001tf__a_001t__Product____Type__Ounit_001tf__e_001tf__f,type,
% 0.30/0.47 hOM_b_1959675421it_e_f: carrie1950868226xt_b_d > carrie1963041556t_unit > carrie1821755406_e_b_f > carrie1065537299t_unit ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Algebra7_OHOM_001tf__b_001tf__d_001tf__a_001tf__c_001tf__a_001t__Product____Type__Ounit,type,
% 0.30/0.47 hOM_b_717364638t_unit: carrie1950868226xt_b_d > carrie722926983_a_b_c > carrie1963041556t_unit > carrie980467975t_unit ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Algebra7_OHOM_001tf__b_001tf__d_001tf__a_001tf__c_001tf__a_001tf__c,type,
% 0.30/0.47 hOM_b_d_a_c_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > carrie722926983_a_b_c > carrie980467975t_unit ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Algebra7_OHOM_001tf__b_001tf__d_001tf__a_001tf__c_001tf__e_001tf__f,type,
% 0.30/0.47 hOM_b_d_a_c_e_f: carrie1950868226xt_b_d > carrie722926983_a_b_c > carrie1821755406_e_b_f > carrie1065537299t_unit ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Algebra7_OHOM_001tf__b_001tf__d_001tf__e_001tf__f_001tf__a_001t__Product____Type__Ounit,type,
% 0.30/0.47 hOM_b_1375846429t_unit: carrie1950868226xt_b_d > carrie1821755406_e_b_f > carrie1963041556t_unit > carrie292259835t_unit ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Algebra7_OHOM_001tf__b_001tf__d_001tf__e_001tf__f_001tf__a_001tf__c,type,
% 0.30/0.47 hOM_b_d_e_f_a_c: carrie1950868226xt_b_d > carrie1821755406_e_b_f > carrie722926983_a_b_c > carrie292259835t_unit ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Algebra7_OHOM_001tf__b_001tf__d_001tf__e_001tf__f_001tf__e_001tf__f,type,
% 0.30/0.47 hOM_b_d_e_f_e_f: carrie1950868226xt_b_d > carrie1821755406_e_b_f > carrie1821755406_e_b_f > carrie377329159t_unit ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Algebra7_OModule_001_062_Itf__a_Mtf__a_J_001tf__b_001t__Product____Type__Ounit_001tf__d,type,
% 0.30/0.47 module589927589unit_d: carrie980467975t_unit > carrie1950868226xt_b_d > $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Algebra7_OModule_001_062_Itf__a_Mtf__e_J_001tf__b_001t__Product____Type__Ounit_001tf__d,type,
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% 0.30/0.48
% 0.30/0.48 % submodule_subset
% 0.30/0.48 thf(fact_4_free__generator__sub,axiom,
% 0.30/0.48 ! [H: set_a] :
% 0.30/0.48 ( ( free_g1087686480_d_a_c @ r @ m @ H )
% 0.30/0.48 => ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % free_generator_sub
% 0.30/0.48 thf(fact_5_elem__fgs,axiom,
% 0.30/0.48 ! [A: set_a,X: a] :
% 0.30/0.48 ( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) )
% 0.30/0.48 => ( ( member_a @ X @ A )
% 0.30/0.48 => ( member_a @ X @ ( algebr1152250919_d_a_c @ r @ m @ A ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % elem_fgs
% 0.30/0.48 thf(fact_6_fgs__sub__carrier,axiom,
% 0.30/0.48 ! [A: set_a] :
% 0.30/0.48 ( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) )
% 0.30/0.48 => ( ord_less_eq_set_a @ ( algebr1152250919_d_a_c @ r @ m @ A ) @ ( carrie2021454486_a_b_c @ m ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % fgs_sub_carrier
% 0.30/0.48 thf(fact_7_fgs__mono,axiom,
% 0.30/0.48 ! [H: set_a,J: set_a,K: set_a] :
% 0.30/0.48 ( ( free_g1087686480_d_a_c @ r @ m @ H )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ J @ K )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ K @ H )
% 0.30/0.48 => ( ord_less_eq_set_a @ ( algebr1152250919_d_a_c @ r @ m @ J ) @ ( algebr1152250919_d_a_c @ r @ m @ K ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % fgs_mono
% 0.30/0.48 thf(fact_8_fgs__submodule,axiom,
% 0.30/0.48 ! [A: set_a] :
% 0.30/0.48 ( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) )
% 0.30/0.48 => ( submodule_b_d_a_c @ r @ m @ ( algebr1152250919_d_a_c @ r @ m @ A ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % fgs_submodule
% 0.30/0.48 thf(fact_9_linear__comb__eq,axiom,
% 0.30/0.48 ! [H: set_a,S: nat > b,N: nat,F: nat > a,G: nat > a] :
% 0.30/0.48 ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 0.30/0.48 => ( ( member_nat_b @ S
% 0.30/0.48 @ ( pi_nat_b
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ r ) ) )
% 0.30/0.48 => ( ( member_nat_a @ F
% 0.30/0.48 @ ( pi_nat_a
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : H ) )
% 0.30/0.48 => ( ( member_nat_a @ G
% 0.30/0.48 @ ( pi_nat_a
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : H ) )
% 0.30/0.48 => ( ! [X2: nat] :
% 0.30/0.48 ( ( member_nat @ X2
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) ) )
% 0.30/0.48 => ( ( F @ X2 )
% 0.30/0.48 = ( G @ X2 ) ) )
% 0.30/0.48 => ( ( l_comb_b_d_a_c @ r @ m @ N @ S @ F )
% 0.30/0.48 = ( l_comb_b_d_a_c @ r @ m @ N @ S @ G ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % linear_comb_eq
% 0.30/0.48 thf(fact_10_linear__comb__eqTr,axiom,
% 0.30/0.48 ! [H: set_a,S: nat > b,N: nat,F: nat > a,G: nat > a] :
% 0.30/0.48 ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 0.30/0.48 => ( ( ( member_nat_b @ S
% 0.30/0.48 @ ( pi_nat_b
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ r ) ) )
% 0.30/0.48 & ( member_nat_a @ F
% 0.30/0.48 @ ( pi_nat_a
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : H ) )
% 0.30/0.48 & ( member_nat_a @ G
% 0.30/0.48 @ ( pi_nat_a
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : H ) )
% 0.30/0.48 & ! [X2: nat] :
% 0.30/0.48 ( ( member_nat @ X2
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) ) )
% 0.30/0.48 => ( ( F @ X2 )
% 0.30/0.48 = ( G @ X2 ) ) ) )
% 0.30/0.48 => ( ( l_comb_b_d_a_c @ r @ m @ N @ S @ F )
% 0.30/0.48 = ( l_comb_b_d_a_c @ r @ m @ N @ S @ G ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % linear_comb_eqTr
% 0.30/0.48 thf(fact_11_l__comb__mem,axiom,
% 0.30/0.48 ! [A2: set_b,H: set_a,S: nat > b,N: nat,M: nat > a] :
% 0.30/0.48 ( ( ideal_b_d @ r @ A2 )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 0.30/0.48 => ( ( member_nat_b @ S
% 0.30/0.48 @ ( pi_nat_b
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : A2 ) )
% 0.30/0.48 => ( ( member_nat_a @ M
% 0.30/0.48 @ ( pi_nat_a
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : H ) )
% 0.30/0.48 => ( member_a @ ( l_comb_b_d_a_c @ r @ m @ N @ S @ M ) @ ( carrie2021454486_a_b_c @ m ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % l_comb_mem
% 0.30/0.48 thf(fact_12_liear__comb__memTr,axiom,
% 0.30/0.48 ! [A2: set_b,H: set_a,N: nat] :
% 0.30/0.48 ( ( ideal_b_d @ r @ A2 )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 0.30/0.48 => ! [S2: nat > b,M2: nat > a] :
% 0.30/0.48 ( ( ( member_nat_b @ S2
% 0.30/0.48 @ ( pi_nat_b
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : A2 ) )
% 0.30/0.48 & ( member_nat_a @ M2
% 0.30/0.48 @ ( pi_nat_a
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : H ) ) )
% 0.30/0.48 => ( member_a @ ( l_comb_b_d_a_c @ r @ m @ N @ S2 @ M2 ) @ ( carrie2021454486_a_b_c @ m ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % liear_comb_memTr
% 0.30/0.48 thf(fact_13_Ann__is__ideal,axiom,
% 0.30/0.48 ideal_b_d @ r @ ( annihilator_b_d_a_c @ r @ m ) ).
% 0.30/0.48
% 0.30/0.48 % Ann_is_ideal
% 0.30/0.48 thf(fact_14_mem__fgs__l__comb,axiom,
% 0.30/0.48 ! [K: set_a,X: a] :
% 0.30/0.48 ( ( K != bot_bot_set_a )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ K @ ( carrie2021454486_a_b_c @ m ) )
% 0.30/0.48 => ( ( member_a @ X @ ( algebr1152250919_d_a_c @ r @ m @ K ) )
% 0.30/0.48 => ? [N2: nat,X2: nat > a] :
% 0.30/0.48 ( ( member_nat_a @ X2
% 0.30/0.48 @ ( pi_nat_a
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
% 0.30/0.48 @ ^ [Uu: nat] : K ) )
% 0.30/0.48 & ? [Xa: nat > b] :
% 0.30/0.48 ( ( member_nat_b @ Xa
% 0.30/0.48 @ ( pi_nat_b
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
% 0.30/0.48 @ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ r ) ) )
% 0.30/0.48 & ( X
% 0.30/0.48 = ( l_comb_b_d_a_c @ r @ m @ N2 @ Xa @ X2 ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mem_fgs_l_comb
% 0.30/0.48 thf(fact_15_Module__axioms,axiom,
% 0.30/0.48 module_a_b_c_d @ m @ r ).
% 0.30/0.48
% 0.30/0.48 % Module_axioms
% 0.30/0.48 thf(fact_16_quotient__of__submodules__is__ideal,axiom,
% 0.30/0.48 ! [P: set_a,Q: set_a] :
% 0.30/0.48 ( ( submodule_b_d_a_c @ r @ m @ P )
% 0.30/0.48 => ( ( submodule_b_d_a_c @ r @ m @ Q )
% 0.30/0.48 => ( ideal_b_d @ r @ ( quotie1153752712_d_a_c @ r @ m @ P @ Q ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % quotient_of_submodules_is_ideal
% 0.30/0.48 thf(fact_17_smodule__ideal__coeff__is__Submodule,axiom,
% 0.30/0.48 ! [A2: set_b] :
% 0.30/0.48 ( ( ideal_b_d @ r @ A2 )
% 0.30/0.48 => ( submodule_b_d_a_c @ r @ m @ ( smodul818989740_d_a_c @ r @ m @ A2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % smodule_ideal_coeff_is_Submodule
% 0.30/0.48 thf(fact_18_mem__smodule__ideal__coeff,axiom,
% 0.30/0.48 ! [A2: set_b,X: a] :
% 0.30/0.48 ( ( ideal_b_d @ r @ A2 )
% 0.30/0.48 => ( ( member_a @ X @ ( smodul818989740_d_a_c @ r @ m @ A2 ) )
% 0.30/0.48 => ? [N2: nat,X2: nat > b] :
% 0.30/0.48 ( ( member_nat_b @ X2
% 0.30/0.48 @ ( pi_nat_b
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
% 0.30/0.48 @ ^ [Uu: nat] : A2 ) )
% 0.30/0.48 & ? [Xa: nat > a] :
% 0.30/0.48 ( ( member_nat_a @ Xa
% 0.30/0.48 @ ( pi_nat_a
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
% 0.30/0.48 @ ^ [Uu: nat] : ( carrie2021454486_a_b_c @ m ) ) )
% 0.30/0.48 & ( X
% 0.30/0.48 = ( l_comb_b_d_a_c @ r @ m @ N2 @ X2 @ Xa ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mem_smodule_ideal_coeff
% 0.30/0.48 thf(fact_19_Module_Oelem__fgs,axiom,
% 0.30/0.48 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,A: set_e,X: e] :
% 0.30/0.48 ( ( module_e_b_f_d @ M3 @ R )
% 0.30/0.48 => ( ( ord_less_eq_set_e @ A @ ( carrie730238621_e_b_f @ M3 ) )
% 0.30/0.48 => ( ( member_e @ X @ A )
% 0.30/0.48 => ( member_e @ X @ ( algebr168361254_d_e_f @ R @ M3 @ A ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.elem_fgs
% 0.30/0.48 thf(fact_20_Module_Oelem__fgs,axiom,
% 0.30/0.48 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,A: set_a,X: a] :
% 0.30/0.48 ( ( module1821517916unit_d @ M3 @ R )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ A @ ( carrie1074654371t_unit @ M3 ) )
% 0.30/0.48 => ( ( member_a @ X @ A )
% 0.30/0.48 => ( member_a @ X @ ( algebr1039611956t_unit @ R @ M3 @ A ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.elem_fgs
% 0.30/0.48 thf(fact_21_Module_Oelem__fgs,axiom,
% 0.30/0.48 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,A: set_a,X: a] :
% 0.30/0.48 ( ( module_a_b_c_d @ M3 @ R )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ M3 ) )
% 0.30/0.48 => ( ( member_a @ X @ A )
% 0.30/0.48 => ( member_a @ X @ ( algebr1152250919_d_a_c @ R @ M3 @ A ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.elem_fgs
% 0.30/0.48 thf(fact_22_Module_Ofgs__mono,axiom,
% 0.30/0.48 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,H: set_e,J: set_e,K: set_e] :
% 0.30/0.48 ( ( module_e_b_f_d @ M3 @ R )
% 0.30/0.48 => ( ( free_g103796815_d_e_f @ R @ M3 @ H )
% 0.30/0.48 => ( ( ord_less_eq_set_e @ J @ K )
% 0.30/0.48 => ( ( ord_less_eq_set_e @ K @ H )
% 0.30/0.48 => ( ord_less_eq_set_e @ ( algebr168361254_d_e_f @ R @ M3 @ J ) @ ( algebr168361254_d_e_f @ R @ M3 @ K ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.fgs_mono
% 0.30/0.48 thf(fact_23_Module_Ofgs__mono,axiom,
% 0.30/0.48 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,H: set_a,J: set_a,K: set_a] :
% 0.30/0.48 ( ( module1821517916unit_d @ M3 @ R )
% 0.30/0.48 => ( ( free_g637607517t_unit @ R @ M3 @ H )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ J @ K )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ K @ H )
% 0.30/0.48 => ( ord_less_eq_set_a @ ( algebr1039611956t_unit @ R @ M3 @ J ) @ ( algebr1039611956t_unit @ R @ M3 @ K ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.fgs_mono
% 0.30/0.48 thf(fact_24_Module_Ofgs__mono,axiom,
% 0.30/0.48 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,H: set_a,J: set_a,K: set_a] :
% 0.30/0.48 ( ( module_a_b_c_d @ M3 @ R )
% 0.30/0.48 => ( ( free_g1087686480_d_a_c @ R @ M3 @ H )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ J @ K )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ K @ H )
% 0.30/0.48 => ( ord_less_eq_set_a @ ( algebr1152250919_d_a_c @ R @ M3 @ J ) @ ( algebr1152250919_d_a_c @ R @ M3 @ K ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.fgs_mono
% 0.30/0.48 thf(fact_25_Module_Ofgs__submodule,axiom,
% 0.30/0.48 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,A: set_e] :
% 0.30/0.48 ( ( module_e_b_f_d @ M3 @ R )
% 0.30/0.48 => ( ( ord_less_eq_set_e @ A @ ( carrie730238621_e_b_f @ M3 ) )
% 0.30/0.48 => ( submodule_b_d_e_f @ R @ M3 @ ( algebr168361254_d_e_f @ R @ M3 @ A ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.fgs_submodule
% 0.30/0.48 thf(fact_26_Module_Ofgs__submodule,axiom,
% 0.30/0.48 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,A: set_a] :
% 0.30/0.48 ( ( module1821517916unit_d @ M3 @ R )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ A @ ( carrie1074654371t_unit @ M3 ) )
% 0.30/0.48 => ( submod903911234t_unit @ R @ M3 @ ( algebr1039611956t_unit @ R @ M3 @ A ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.fgs_submodule
% 0.30/0.48 thf(fact_27_Module_Ofgs__submodule,axiom,
% 0.30/0.48 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,A: set_a] :
% 0.30/0.48 ( ( module_a_b_c_d @ M3 @ R )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ M3 ) )
% 0.30/0.48 => ( submodule_b_d_a_c @ R @ M3 @ ( algebr1152250919_d_a_c @ R @ M3 @ A ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.fgs_submodule
% 0.30/0.48 thf(fact_28_Module_Omem__fgs__l__comb,axiom,
% 0.30/0.48 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,K: set_e,X: e] :
% 0.30/0.48 ( ( module_e_b_f_d @ M3 @ R )
% 0.30/0.48 => ( ( K != bot_bot_set_e )
% 0.30/0.48 => ( ( ord_less_eq_set_e @ K @ ( carrie730238621_e_b_f @ M3 ) )
% 0.30/0.48 => ( ( member_e @ X @ ( algebr168361254_d_e_f @ R @ M3 @ K ) )
% 0.30/0.48 => ? [N2: nat,X2: nat > e] :
% 0.30/0.48 ( ( member_nat_e @ X2
% 0.30/0.48 @ ( pi_nat_e
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
% 0.30/0.48 @ ^ [Uu: nat] : K ) )
% 0.30/0.48 & ? [Xa: nat > b] :
% 0.30/0.48 ( ( member_nat_b @ Xa
% 0.30/0.48 @ ( pi_nat_b
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
% 0.30/0.48 @ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ R ) ) )
% 0.30/0.48 & ( X
% 0.30/0.48 = ( l_comb_b_d_e_f @ R @ M3 @ N2 @ Xa @ X2 ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.mem_fgs_l_comb
% 0.30/0.48 thf(fact_29_Module_Omem__fgs__l__comb,axiom,
% 0.30/0.48 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,K: set_a,X: a] :
% 0.30/0.48 ( ( module1821517916unit_d @ M3 @ R )
% 0.30/0.48 => ( ( K != bot_bot_set_a )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ K @ ( carrie1074654371t_unit @ M3 ) )
% 0.30/0.48 => ( ( member_a @ X @ ( algebr1039611956t_unit @ R @ M3 @ K ) )
% 0.30/0.48 => ? [N2: nat,X2: nat > a] :
% 0.30/0.48 ( ( member_nat_a @ X2
% 0.30/0.48 @ ( pi_nat_a
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
% 0.30/0.48 @ ^ [Uu: nat] : K ) )
% 0.30/0.48 & ? [Xa: nat > b] :
% 0.30/0.48 ( ( member_nat_b @ Xa
% 0.30/0.48 @ ( pi_nat_b
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
% 0.30/0.48 @ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ R ) ) )
% 0.30/0.48 & ( X
% 0.30/0.48 = ( l_comb1138968323t_unit @ R @ M3 @ N2 @ Xa @ X2 ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.mem_fgs_l_comb
% 0.30/0.48 thf(fact_30_Module_Omem__fgs__l__comb,axiom,
% 0.30/0.48 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,K: set_a,X: a] :
% 0.30/0.48 ( ( module_a_b_c_d @ M3 @ R )
% 0.30/0.48 => ( ( K != bot_bot_set_a )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ K @ ( carrie2021454486_a_b_c @ M3 ) )
% 0.30/0.48 => ( ( member_a @ X @ ( algebr1152250919_d_a_c @ R @ M3 @ K ) )
% 0.30/0.48 => ? [N2: nat,X2: nat > a] :
% 0.30/0.48 ( ( member_nat_a @ X2
% 0.30/0.48 @ ( pi_nat_a
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
% 0.30/0.48 @ ^ [Uu: nat] : K ) )
% 0.30/0.48 & ? [Xa: nat > b] :
% 0.30/0.48 ( ( member_nat_b @ Xa
% 0.30/0.48 @ ( pi_nat_b
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
% 0.30/0.48 @ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ R ) ) )
% 0.30/0.48 & ( X
% 0.30/0.48 = ( l_comb_b_d_a_c @ R @ M3 @ N2 @ Xa @ X2 ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.mem_fgs_l_comb
% 0.30/0.48 thf(fact_31_Module_Ofgs__sub__carrier,axiom,
% 0.30/0.48 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,A: set_e] :
% 0.30/0.48 ( ( module_e_b_f_d @ M3 @ R )
% 0.30/0.48 => ( ( ord_less_eq_set_e @ A @ ( carrie730238621_e_b_f @ M3 ) )
% 0.30/0.48 => ( ord_less_eq_set_e @ ( algebr168361254_d_e_f @ R @ M3 @ A ) @ ( carrie730238621_e_b_f @ M3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.fgs_sub_carrier
% 0.30/0.48 thf(fact_32_Module_Ofgs__sub__carrier,axiom,
% 0.30/0.48 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,A: set_a] :
% 0.30/0.48 ( ( module1821517916unit_d @ M3 @ R )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ A @ ( carrie1074654371t_unit @ M3 ) )
% 0.30/0.48 => ( ord_less_eq_set_a @ ( algebr1039611956t_unit @ R @ M3 @ A ) @ ( carrie1074654371t_unit @ M3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.fgs_sub_carrier
% 0.30/0.48 thf(fact_33_Module_Ofgs__sub__carrier,axiom,
% 0.30/0.48 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,A: set_a] :
% 0.30/0.48 ( ( module_a_b_c_d @ M3 @ R )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ M3 ) )
% 0.30/0.48 => ( ord_less_eq_set_a @ ( algebr1152250919_d_a_c @ R @ M3 @ A ) @ ( carrie2021454486_a_b_c @ M3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.fgs_sub_carrier
% 0.30/0.48 thf(fact_34_misom__self,axiom,
% 0.30/0.48 misomo1282343797_c_a_c @ r @ m @ m ).
% 0.30/0.48
% 0.30/0.48 % misom_self
% 0.30/0.48 thf(fact_35_HOM__is__module,axiom,
% 0.30/0.48 ! [N3: carrie1821755406_e_b_f] :
% 0.30/0.48 ( ( module_e_b_f_d @ N3 @ r )
% 0.30/0.48 => ( module1648870817unit_d @ ( hOM_b_d_a_c_e_f @ r @ m @ N3 ) @ r ) ) ).
% 0.30/0.48
% 0.30/0.48 % HOM_is_module
% 0.30/0.48 thf(fact_36_HOM__is__module,axiom,
% 0.30/0.48 ! [N3: carrie722926983_a_b_c] :
% 0.30/0.48 ( ( module_a_b_c_d @ N3 @ r )
% 0.30/0.48 => ( module589927589unit_d @ ( hOM_b_d_a_c_a_c @ r @ m @ N3 ) @ r ) ) ).
% 0.30/0.48
% 0.30/0.48 % HOM_is_module
% 0.30/0.48 thf(fact_37_HOM__is__module,axiom,
% 0.30/0.48 ! [N3: carrie1963041556t_unit] :
% 0.30/0.48 ( ( module1821517916unit_d @ N3 @ r )
% 0.30/0.48 => ( module589927589unit_d @ ( hOM_b_717364638t_unit @ r @ m @ N3 ) @ r ) ) ).
% 0.30/0.48
% 0.30/0.48 % HOM_is_module
% 0.30/0.48 thf(fact_38_Suc__pred,axiom,
% 0.30/0.48 ! [N: nat] :
% 0.30/0.48 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.48 => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 0.30/0.48 = N ) ) ).
% 0.30/0.48
% 0.30/0.48 % Suc_pred
% 0.30/0.48 thf(fact_39_generator__sub__carrier,axiom,
% 0.30/0.48 ! [H: set_a] :
% 0.30/0.48 ( ( generator_b_d_a_c @ r @ m @ H )
% 0.30/0.48 => ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % generator_sub_carrier
% 0.30/0.48 thf(fact_40_free__generator__generator,axiom,
% 0.30/0.48 ! [H: set_a] :
% 0.30/0.48 ( ( free_g1087686480_d_a_c @ r @ m @ H )
% 0.30/0.48 => ( generator_b_d_a_c @ r @ m @ H ) ) ).
% 0.30/0.48
% 0.30/0.48 % free_generator_generator
% 0.30/0.48 thf(fact_41_Module_Ol__comb__mem,axiom,
% 0.30/0.48 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,A2: set_b,H: set_e,S: nat > b,N: nat,M: nat > e] :
% 0.30/0.48 ( ( module_e_b_f_d @ M3 @ R )
% 0.30/0.48 => ( ( ideal_b_d @ R @ A2 )
% 0.30/0.48 => ( ( ord_less_eq_set_e @ H @ ( carrie730238621_e_b_f @ M3 ) )
% 0.30/0.48 => ( ( member_nat_b @ S
% 0.30/0.48 @ ( pi_nat_b
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : A2 ) )
% 0.30/0.48 => ( ( member_nat_e @ M
% 0.30/0.48 @ ( pi_nat_e
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : H ) )
% 0.30/0.48 => ( member_e @ ( l_comb_b_d_e_f @ R @ M3 @ N @ S @ M ) @ ( carrie730238621_e_b_f @ M3 ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.l_comb_mem
% 0.30/0.48 thf(fact_42_Module_Ol__comb__mem,axiom,
% 0.30/0.48 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,A2: set_b,H: set_a,S: nat > b,N: nat,M: nat > a] :
% 0.30/0.48 ( ( module_a_b_c_d @ M3 @ R )
% 0.30/0.48 => ( ( ideal_b_d @ R @ A2 )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ M3 ) )
% 0.30/0.48 => ( ( member_nat_b @ S
% 0.30/0.48 @ ( pi_nat_b
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : A2 ) )
% 0.30/0.48 => ( ( member_nat_a @ M
% 0.30/0.48 @ ( pi_nat_a
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : H ) )
% 0.30/0.48 => ( member_a @ ( l_comb_b_d_a_c @ R @ M3 @ N @ S @ M ) @ ( carrie2021454486_a_b_c @ M3 ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.l_comb_mem
% 0.30/0.48 thf(fact_43_Module_Ol__comb__mem,axiom,
% 0.30/0.48 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,A2: set_b,H: set_a,S: nat > b,N: nat,M: nat > a] :
% 0.30/0.48 ( ( module1821517916unit_d @ M3 @ R )
% 0.30/0.48 => ( ( ideal_b_d @ R @ A2 )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ H @ ( carrie1074654371t_unit @ M3 ) )
% 0.30/0.48 => ( ( member_nat_b @ S
% 0.30/0.48 @ ( pi_nat_b
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : A2 ) )
% 0.30/0.48 => ( ( member_nat_a @ M
% 0.30/0.48 @ ( pi_nat_a
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : H ) )
% 0.30/0.48 => ( member_a @ ( l_comb1138968323t_unit @ R @ M3 @ N @ S @ M ) @ ( carrie1074654371t_unit @ M3 ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.l_comb_mem
% 0.30/0.48 thf(fact_44_Module_Oliear__comb__memTr,axiom,
% 0.30/0.48 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,A2: set_b,H: set_e,N: nat] :
% 0.30/0.48 ( ( module_e_b_f_d @ M3 @ R )
% 0.30/0.48 => ( ( ideal_b_d @ R @ A2 )
% 0.30/0.48 => ( ( ord_less_eq_set_e @ H @ ( carrie730238621_e_b_f @ M3 ) )
% 0.30/0.48 => ! [S2: nat > b,M2: nat > e] :
% 0.30/0.48 ( ( ( member_nat_b @ S2
% 0.30/0.48 @ ( pi_nat_b
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : A2 ) )
% 0.30/0.48 & ( member_nat_e @ M2
% 0.30/0.48 @ ( pi_nat_e
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : H ) ) )
% 0.30/0.48 => ( member_e @ ( l_comb_b_d_e_f @ R @ M3 @ N @ S2 @ M2 ) @ ( carrie730238621_e_b_f @ M3 ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.liear_comb_memTr
% 0.30/0.48 thf(fact_45_Module_Oliear__comb__memTr,axiom,
% 0.30/0.48 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,A2: set_b,H: set_a,N: nat] :
% 0.30/0.48 ( ( module_a_b_c_d @ M3 @ R )
% 0.30/0.48 => ( ( ideal_b_d @ R @ A2 )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ M3 ) )
% 0.30/0.48 => ! [S2: nat > b,M2: nat > a] :
% 0.30/0.48 ( ( ( member_nat_b @ S2
% 0.30/0.48 @ ( pi_nat_b
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : A2 ) )
% 0.30/0.48 & ( member_nat_a @ M2
% 0.30/0.48 @ ( pi_nat_a
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : H ) ) )
% 0.30/0.48 => ( member_a @ ( l_comb_b_d_a_c @ R @ M3 @ N @ S2 @ M2 ) @ ( carrie2021454486_a_b_c @ M3 ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.liear_comb_memTr
% 0.30/0.48 thf(fact_46_Module_Oliear__comb__memTr,axiom,
% 0.30/0.48 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,A2: set_b,H: set_a,N: nat] :
% 0.30/0.48 ( ( module1821517916unit_d @ M3 @ R )
% 0.30/0.48 => ( ( ideal_b_d @ R @ A2 )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ H @ ( carrie1074654371t_unit @ M3 ) )
% 0.30/0.48 => ! [S2: nat > b,M2: nat > a] :
% 0.30/0.48 ( ( ( member_nat_b @ S2
% 0.30/0.48 @ ( pi_nat_b
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : A2 ) )
% 0.30/0.48 & ( member_nat_a @ M2
% 0.30/0.48 @ ( pi_nat_a
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : H ) ) )
% 0.30/0.48 => ( member_a @ ( l_comb1138968323t_unit @ R @ M3 @ N @ S2 @ M2 ) @ ( carrie1074654371t_unit @ M3 ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.liear_comb_memTr
% 0.30/0.48 thf(fact_47_zero__less__diff,axiom,
% 0.30/0.48 ! [N: nat,M: nat] :
% 0.30/0.48 ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
% 0.30/0.48 = ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % zero_less_diff
% 0.30/0.48 thf(fact_48_Module_Olinear__comb__eq,axiom,
% 0.30/0.48 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,H: set_e,S: nat > b,N: nat,F: nat > e,G: nat > e] :
% 0.30/0.48 ( ( module_e_b_f_d @ M3 @ R )
% 0.30/0.48 => ( ( ord_less_eq_set_e @ H @ ( carrie730238621_e_b_f @ M3 ) )
% 0.30/0.48 => ( ( member_nat_b @ S
% 0.30/0.48 @ ( pi_nat_b
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ R ) ) )
% 0.30/0.48 => ( ( member_nat_e @ F
% 0.30/0.48 @ ( pi_nat_e
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : H ) )
% 0.30/0.48 => ( ( member_nat_e @ G
% 0.30/0.48 @ ( pi_nat_e
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : H ) )
% 0.30/0.48 => ( ! [X2: nat] :
% 0.30/0.48 ( ( member_nat @ X2
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) ) )
% 0.30/0.48 => ( ( F @ X2 )
% 0.30/0.48 = ( G @ X2 ) ) )
% 0.30/0.48 => ( ( l_comb_b_d_e_f @ R @ M3 @ N @ S @ F )
% 0.30/0.48 = ( l_comb_b_d_e_f @ R @ M3 @ N @ S @ G ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.linear_comb_eq
% 0.30/0.48 thf(fact_49_Module_Olinear__comb__eq,axiom,
% 0.30/0.48 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,H: set_a,S: nat > b,N: nat,F: nat > a,G: nat > a] :
% 0.30/0.48 ( ( module_a_b_c_d @ M3 @ R )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ M3 ) )
% 0.30/0.48 => ( ( member_nat_b @ S
% 0.30/0.48 @ ( pi_nat_b
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ R ) ) )
% 0.30/0.48 => ( ( member_nat_a @ F
% 0.30/0.48 @ ( pi_nat_a
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : H ) )
% 0.30/0.48 => ( ( member_nat_a @ G
% 0.30/0.48 @ ( pi_nat_a
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : H ) )
% 0.30/0.48 => ( ! [X2: nat] :
% 0.30/0.48 ( ( member_nat @ X2
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) ) )
% 0.30/0.48 => ( ( F @ X2 )
% 0.30/0.48 = ( G @ X2 ) ) )
% 0.30/0.48 => ( ( l_comb_b_d_a_c @ R @ M3 @ N @ S @ F )
% 0.30/0.48 = ( l_comb_b_d_a_c @ R @ M3 @ N @ S @ G ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.linear_comb_eq
% 0.30/0.48 thf(fact_50_Module_Olinear__comb__eq,axiom,
% 0.30/0.48 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,H: set_a,S: nat > b,N: nat,F: nat > a,G: nat > a] :
% 0.30/0.48 ( ( module1821517916unit_d @ M3 @ R )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ H @ ( carrie1074654371t_unit @ M3 ) )
% 0.30/0.48 => ( ( member_nat_b @ S
% 0.30/0.48 @ ( pi_nat_b
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ R ) ) )
% 0.30/0.48 => ( ( member_nat_a @ F
% 0.30/0.48 @ ( pi_nat_a
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : H ) )
% 0.30/0.48 => ( ( member_nat_a @ G
% 0.30/0.48 @ ( pi_nat_a
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : H ) )
% 0.30/0.48 => ( ! [X2: nat] :
% 0.30/0.48 ( ( member_nat @ X2
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) ) )
% 0.30/0.48 => ( ( F @ X2 )
% 0.30/0.48 = ( G @ X2 ) ) )
% 0.30/0.48 => ( ( l_comb1138968323t_unit @ R @ M3 @ N @ S @ F )
% 0.30/0.48 = ( l_comb1138968323t_unit @ R @ M3 @ N @ S @ G ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.linear_comb_eq
% 0.30/0.48 thf(fact_51_Module_Olinear__comb__eqTr,axiom,
% 0.30/0.48 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,H: set_e,S: nat > b,N: nat,F: nat > e,G: nat > e] :
% 0.30/0.48 ( ( module_e_b_f_d @ M3 @ R )
% 0.30/0.48 => ( ( ord_less_eq_set_e @ H @ ( carrie730238621_e_b_f @ M3 ) )
% 0.30/0.48 => ( ( ( member_nat_b @ S
% 0.30/0.48 @ ( pi_nat_b
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ R ) ) )
% 0.30/0.48 & ( member_nat_e @ F
% 0.30/0.48 @ ( pi_nat_e
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : H ) )
% 0.30/0.48 & ( member_nat_e @ G
% 0.30/0.48 @ ( pi_nat_e
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : H ) )
% 0.30/0.48 & ! [X2: nat] :
% 0.30/0.48 ( ( member_nat @ X2
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) ) )
% 0.30/0.48 => ( ( F @ X2 )
% 0.30/0.48 = ( G @ X2 ) ) ) )
% 0.30/0.48 => ( ( l_comb_b_d_e_f @ R @ M3 @ N @ S @ F )
% 0.30/0.48 = ( l_comb_b_d_e_f @ R @ M3 @ N @ S @ G ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.linear_comb_eqTr
% 0.30/0.48 thf(fact_52_Module_Olinear__comb__eqTr,axiom,
% 0.30/0.48 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,H: set_a,S: nat > b,N: nat,F: nat > a,G: nat > a] :
% 0.30/0.48 ( ( module_a_b_c_d @ M3 @ R )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ M3 ) )
% 0.30/0.48 => ( ( ( member_nat_b @ S
% 0.30/0.48 @ ( pi_nat_b
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ R ) ) )
% 0.30/0.48 & ( member_nat_a @ F
% 0.30/0.48 @ ( pi_nat_a
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : H ) )
% 0.30/0.48 & ( member_nat_a @ G
% 0.30/0.48 @ ( pi_nat_a
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : H ) )
% 0.30/0.48 & ! [X2: nat] :
% 0.30/0.48 ( ( member_nat @ X2
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) ) )
% 0.30/0.48 => ( ( F @ X2 )
% 0.30/0.48 = ( G @ X2 ) ) ) )
% 0.30/0.48 => ( ( l_comb_b_d_a_c @ R @ M3 @ N @ S @ F )
% 0.30/0.48 = ( l_comb_b_d_a_c @ R @ M3 @ N @ S @ G ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.linear_comb_eqTr
% 0.30/0.48 thf(fact_53_Module_Olinear__comb__eqTr,axiom,
% 0.30/0.48 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,H: set_a,S: nat > b,N: nat,F: nat > a,G: nat > a] :
% 0.30/0.48 ( ( module1821517916unit_d @ M3 @ R )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ H @ ( carrie1074654371t_unit @ M3 ) )
% 0.30/0.48 => ( ( ( member_nat_b @ S
% 0.30/0.48 @ ( pi_nat_b
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ R ) ) )
% 0.30/0.48 & ( member_nat_a @ F
% 0.30/0.48 @ ( pi_nat_a
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : H ) )
% 0.30/0.48 & ( member_nat_a @ G
% 0.30/0.48 @ ( pi_nat_a
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : H ) )
% 0.30/0.48 & ! [X2: nat] :
% 0.30/0.48 ( ( member_nat @ X2
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) ) )
% 0.30/0.48 => ( ( F @ X2 )
% 0.30/0.48 = ( G @ X2 ) ) ) )
% 0.30/0.48 => ( ( l_comb1138968323t_unit @ R @ M3 @ N @ S @ F )
% 0.30/0.48 = ( l_comb1138968323t_unit @ R @ M3 @ N @ S @ G ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.linear_comb_eqTr
% 0.30/0.48 thf(fact_54_diff__is__0__eq,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ( minus_minus_nat @ M @ N )
% 0.30/0.48 = zero_zero_nat )
% 0.30/0.48 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % diff_is_0_eq
% 0.30/0.48 thf(fact_55_diff__is__0__eq_H,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.48 => ( ( minus_minus_nat @ M @ N )
% 0.30/0.48 = zero_zero_nat ) ) ).
% 0.30/0.48
% 0.30/0.48 % diff_is_0_eq'
% 0.30/0.48 thf(fact_56_old_Onat_Oinject,axiom,
% 0.30/0.48 ! [Nat: nat,Nat2: nat] :
% 0.30/0.48 ( ( ( suc @ Nat )
% 0.30/0.48 = ( suc @ Nat2 ) )
% 0.30/0.48 = ( Nat = Nat2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % old.nat.inject
% 0.30/0.48 thf(fact_57_nat_Oinject,axiom,
% 0.30/0.48 ! [X22: nat,Y2: nat] :
% 0.30/0.48 ( ( ( suc @ X22 )
% 0.30/0.48 = ( suc @ Y2 ) )
% 0.30/0.48 = ( X22 = Y2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % nat.inject
% 0.30/0.48 thf(fact_58_misom__sym,axiom,
% 0.30/0.48 ! [N3: carrie1821755406_e_b_f] :
% 0.30/0.48 ( ( module_e_b_f_d @ N3 @ r )
% 0.30/0.48 => ( ( misomo298454132_c_e_f @ r @ m @ N3 )
% 0.30/0.48 => ( misomo1752826100_f_a_c @ r @ N3 @ m ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % misom_sym
% 0.30/0.48 thf(fact_59_misom__sym,axiom,
% 0.30/0.48 ! [N3: carrie1963041556t_unit] :
% 0.30/0.48 ( ( module1821517916unit_d @ N3 @ r )
% 0.30/0.48 => ( ( misomo1857159554t_unit @ r @ m @ N3 )
% 0.30/0.48 => ( misomo1935876354it_a_c @ r @ N3 @ m ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % misom_sym
% 0.30/0.48 thf(fact_60_misom__sym,axiom,
% 0.30/0.48 ! [N3: carrie722926983_a_b_c] :
% 0.30/0.48 ( ( module_a_b_c_d @ N3 @ r )
% 0.30/0.48 => ( ( misomo1282343797_c_a_c @ r @ m @ N3 )
% 0.30/0.48 => ( misomo1282343797_c_a_c @ r @ N3 @ m ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % misom_sym
% 0.30/0.48 thf(fact_61_Suc__le__mono,axiom,
% 0.30/0.48 ! [N: nat,M: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
% 0.30/0.48 = ( ord_less_eq_nat @ N @ M ) ) ).
% 0.30/0.48
% 0.30/0.48 % Suc_le_mono
% 0.30/0.48 thf(fact_62_bot__nat__0_Oextremum,axiom,
% 0.30/0.48 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 0.30/0.48
% 0.30/0.48 % bot_nat_0.extremum
% 0.30/0.48 thf(fact_63_le0,axiom,
% 0.30/0.48 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.30/0.48
% 0.30/0.48 % le0
% 0.30/0.48 thf(fact_64_Suc__less__eq,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.30/0.48 = ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % Suc_less_eq
% 0.30/0.48 thf(fact_65_Suc__mono,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_nat @ M @ N )
% 0.30/0.48 => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Suc_mono
% 0.30/0.48 thf(fact_66_mem__Collect__eq,axiom,
% 0.30/0.48 ! [A: nat > a,P: ( nat > a ) > $o] :
% 0.30/0.48 ( ( member_nat_a @ A @ ( collect_nat_a @ P ) )
% 0.30/0.48 = ( P @ A ) ) ).
% 0.30/0.48
% 0.30/0.48 % mem_Collect_eq
% 0.30/0.48 thf(fact_67_mem__Collect__eq,axiom,
% 0.30/0.48 ! [A: nat > b,P: ( nat > b ) > $o] :
% 0.30/0.48 ( ( member_nat_b @ A @ ( collect_nat_b @ P ) )
% 0.30/0.48 = ( P @ A ) ) ).
% 0.30/0.48
% 0.30/0.48 % mem_Collect_eq
% 0.30/0.48 thf(fact_68_mem__Collect__eq,axiom,
% 0.30/0.48 ! [A: a,P: a > $o] :
% 0.30/0.48 ( ( member_a @ A @ ( collect_a @ P ) )
% 0.30/0.48 = ( P @ A ) ) ).
% 0.30/0.48
% 0.30/0.48 % mem_Collect_eq
% 0.30/0.48 thf(fact_69_mem__Collect__eq,axiom,
% 0.30/0.48 ! [A: nat,P: nat > $o] :
% 0.30/0.48 ( ( member_nat @ A @ ( collect_nat @ P ) )
% 0.30/0.48 = ( P @ A ) ) ).
% 0.30/0.48
% 0.30/0.48 % mem_Collect_eq
% 0.30/0.48 thf(fact_70_Collect__mem__eq,axiom,
% 0.30/0.48 ! [A2: set_nat_a] :
% 0.30/0.48 ( ( collect_nat_a
% 0.30/0.48 @ ^ [X3: nat > a] : ( member_nat_a @ X3 @ A2 ) )
% 0.30/0.48 = A2 ) ).
% 0.30/0.48
% 0.30/0.48 % Collect_mem_eq
% 0.30/0.48 thf(fact_71_Collect__mem__eq,axiom,
% 0.30/0.48 ! [A2: set_nat_b] :
% 0.30/0.48 ( ( collect_nat_b
% 0.30/0.48 @ ^ [X3: nat > b] : ( member_nat_b @ X3 @ A2 ) )
% 0.30/0.48 = A2 ) ).
% 0.30/0.48
% 0.30/0.48 % Collect_mem_eq
% 0.30/0.48 thf(fact_72_Collect__mem__eq,axiom,
% 0.30/0.48 ! [A2: set_a] :
% 0.30/0.48 ( ( collect_a
% 0.30/0.48 @ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
% 0.30/0.48 = A2 ) ).
% 0.30/0.48
% 0.30/0.48 % Collect_mem_eq
% 0.30/0.48 thf(fact_73_Collect__mem__eq,axiom,
% 0.30/0.48 ! [A2: set_nat] :
% 0.30/0.48 ( ( collect_nat
% 0.30/0.48 @ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
% 0.30/0.48 = A2 ) ).
% 0.30/0.48
% 0.30/0.48 % Collect_mem_eq
% 0.30/0.48 thf(fact_74_Collect__cong,axiom,
% 0.30/0.48 ! [P: nat > $o,Q: nat > $o] :
% 0.30/0.48 ( ! [X2: nat] :
% 0.30/0.48 ( ( P @ X2 )
% 0.30/0.48 = ( Q @ X2 ) )
% 0.30/0.48 => ( ( collect_nat @ P )
% 0.30/0.48 = ( collect_nat @ Q ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Collect_cong
% 0.30/0.48 thf(fact_75_bot__nat__0_Onot__eq__extremum,axiom,
% 0.30/0.48 ! [A: nat] :
% 0.30/0.48 ( ( A != zero_zero_nat )
% 0.30/0.48 = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 0.30/0.48
% 0.30/0.48 % bot_nat_0.not_eq_extremum
% 0.30/0.48 thf(fact_76_less__nat__zero__code,axiom,
% 0.30/0.48 ! [N: nat] :
% 0.30/0.48 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.30/0.48
% 0.30/0.48 % less_nat_zero_code
% 0.30/0.48 thf(fact_77_neq0__conv,axiom,
% 0.30/0.48 ! [N: nat] :
% 0.30/0.48 ( ( N != zero_zero_nat )
% 0.30/0.48 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % neq0_conv
% 0.30/0.48 thf(fact_78_diff__diff__cancel,axiom,
% 0.30/0.48 ! [I: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ I @ N )
% 0.30/0.48 => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
% 0.30/0.48 = I ) ) ).
% 0.30/0.48
% 0.30/0.48 % diff_diff_cancel
% 0.30/0.48 thf(fact_79_Suc__diff__diff,axiom,
% 0.30/0.48 ! [M: nat,N: nat,K2: nat] :
% 0.30/0.48 ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K2 ) )
% 0.30/0.48 = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % Suc_diff_diff
% 0.30/0.48 thf(fact_80_diff__Suc__Suc,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.30/0.48 = ( minus_minus_nat @ M @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % diff_Suc_Suc
% 0.30/0.48 thf(fact_81_diff__self__eq__0,axiom,
% 0.30/0.48 ! [M: nat] :
% 0.30/0.48 ( ( minus_minus_nat @ M @ M )
% 0.30/0.48 = zero_zero_nat ) ).
% 0.30/0.48
% 0.30/0.48 % diff_self_eq_0
% 0.30/0.48 thf(fact_82_diff__0__eq__0,axiom,
% 0.30/0.48 ! [N: nat] :
% 0.30/0.48 ( ( minus_minus_nat @ zero_zero_nat @ N )
% 0.30/0.48 = zero_zero_nat ) ).
% 0.30/0.48
% 0.30/0.48 % diff_0_eq_0
% 0.30/0.48 thf(fact_83_misom__trans,axiom,
% 0.30/0.48 ! [L: carrie1821755406_e_b_f,N3: carrie1821755406_e_b_f] :
% 0.30/0.48 ( ( module_e_b_f_d @ L @ r )
% 0.30/0.48 => ( ( module_e_b_f_d @ N3 @ r )
% 0.30/0.48 => ( ( misomo1752826100_f_a_c @ r @ L @ m )
% 0.30/0.48 => ( ( misomo298454132_c_e_f @ r @ m @ N3 )
% 0.30/0.48 => ( misomo768936435_f_e_f @ r @ L @ N3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % misom_trans
% 0.30/0.48 thf(fact_84_misom__trans,axiom,
% 0.30/0.48 ! [L: carrie1821755406_e_b_f,N3: carrie1963041556t_unit] :
% 0.30/0.48 ( ( module_e_b_f_d @ L @ r )
% 0.30/0.48 => ( ( module1821517916unit_d @ N3 @ r )
% 0.30/0.48 => ( ( misomo1752826100_f_a_c @ r @ L @ m )
% 0.30/0.48 => ( ( misomo1857159554t_unit @ r @ m @ N3 )
% 0.30/0.48 => ( misomo368157697t_unit @ r @ L @ N3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % misom_trans
% 0.30/0.48 thf(fact_85_misom__trans,axiom,
% 0.30/0.48 ! [L: carrie1963041556t_unit,N3: carrie1821755406_e_b_f] :
% 0.30/0.48 ( ( module1821517916unit_d @ L @ r )
% 0.30/0.48 => ( ( module_e_b_f_d @ N3 @ r )
% 0.30/0.48 => ( ( misomo1935876354it_a_c @ r @ L @ m )
% 0.30/0.48 => ( ( misomo298454132_c_e_f @ r @ m @ N3 )
% 0.30/0.48 => ( misomo951986689it_e_f @ r @ L @ N3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % misom_trans
% 0.30/0.48 thf(fact_86_misom__trans,axiom,
% 0.30/0.48 ! [L: carrie1963041556t_unit,N3: carrie1963041556t_unit] :
% 0.30/0.48 ( ( module1821517916unit_d @ L @ r )
% 0.30/0.48 => ( ( module1821517916unit_d @ N3 @ r )
% 0.30/0.48 => ( ( misomo1935876354it_a_c @ r @ L @ m )
% 0.30/0.48 => ( ( misomo1857159554t_unit @ r @ m @ N3 )
% 0.30/0.48 => ( misomo493403663t_unit @ r @ L @ N3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % misom_trans
% 0.30/0.48 thf(fact_87_misom__trans,axiom,
% 0.30/0.48 ! [L: carrie1821755406_e_b_f,N3: carrie722926983_a_b_c] :
% 0.30/0.48 ( ( module_e_b_f_d @ L @ r )
% 0.30/0.48 => ( ( module_a_b_c_d @ N3 @ r )
% 0.30/0.48 => ( ( misomo1752826100_f_a_c @ r @ L @ m )
% 0.30/0.48 => ( ( misomo1282343797_c_a_c @ r @ m @ N3 )
% 0.30/0.48 => ( misomo1752826100_f_a_c @ r @ L @ N3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % misom_trans
% 0.30/0.48 thf(fact_88_misom__trans,axiom,
% 0.30/0.48 ! [L: carrie1963041556t_unit,N3: carrie722926983_a_b_c] :
% 0.30/0.48 ( ( module1821517916unit_d @ L @ r )
% 0.30/0.48 => ( ( module_a_b_c_d @ N3 @ r )
% 0.30/0.48 => ( ( misomo1935876354it_a_c @ r @ L @ m )
% 0.30/0.48 => ( ( misomo1282343797_c_a_c @ r @ m @ N3 )
% 0.30/0.48 => ( misomo1935876354it_a_c @ r @ L @ N3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % misom_trans
% 0.30/0.48 thf(fact_89_misom__trans,axiom,
% 0.30/0.48 ! [L: carrie722926983_a_b_c,N3: carrie1821755406_e_b_f] :
% 0.30/0.48 ( ( module_a_b_c_d @ L @ r )
% 0.30/0.48 => ( ( module_e_b_f_d @ N3 @ r )
% 0.30/0.48 => ( ( misomo1282343797_c_a_c @ r @ L @ m )
% 0.30/0.48 => ( ( misomo298454132_c_e_f @ r @ m @ N3 )
% 0.30/0.48 => ( misomo298454132_c_e_f @ r @ L @ N3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % misom_trans
% 0.30/0.48 thf(fact_90_misom__trans,axiom,
% 0.30/0.48 ! [L: carrie722926983_a_b_c,N3: carrie1963041556t_unit] :
% 0.30/0.48 ( ( module_a_b_c_d @ L @ r )
% 0.30/0.48 => ( ( module1821517916unit_d @ N3 @ r )
% 0.30/0.48 => ( ( misomo1282343797_c_a_c @ r @ L @ m )
% 0.30/0.48 => ( ( misomo1857159554t_unit @ r @ m @ N3 )
% 0.30/0.48 => ( misomo1857159554t_unit @ r @ L @ N3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % misom_trans
% 0.30/0.48 thf(fact_91_misom__trans,axiom,
% 0.30/0.48 ! [L: carrie722926983_a_b_c,N3: carrie722926983_a_b_c] :
% 0.30/0.48 ( ( module_a_b_c_d @ L @ r )
% 0.30/0.48 => ( ( module_a_b_c_d @ N3 @ r )
% 0.30/0.48 => ( ( misomo1282343797_c_a_c @ r @ L @ m )
% 0.30/0.48 => ( ( misomo1282343797_c_a_c @ r @ m @ N3 )
% 0.30/0.48 => ( misomo1282343797_c_a_c @ r @ L @ N3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % misom_trans
% 0.30/0.48 thf(fact_92_zero__less__Suc,axiom,
% 0.30/0.48 ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % zero_less_Suc
% 0.30/0.48 thf(fact_93_less__Suc0,axiom,
% 0.30/0.48 ! [N: nat] :
% 0.30/0.48 ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.30/0.48 = ( N = zero_zero_nat ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_Suc0
% 0.30/0.48 thf(fact_94_Module_OHOM__is__module,axiom,
% 0.30/0.48 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,N3: carrie1821755406_e_b_f] :
% 0.30/0.48 ( ( module_e_b_f_d @ M3 @ R )
% 0.30/0.48 => ( ( module_e_b_f_d @ N3 @ R )
% 0.30/0.48 => ( module241733029unit_d @ ( hOM_b_d_e_f_e_f @ R @ M3 @ N3 ) @ R ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.HOM_is_module
% 0.30/0.48 thf(fact_95_Module_OHOM__is__module,axiom,
% 0.30/0.48 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,N3: carrie722926983_a_b_c] :
% 0.30/0.48 ( ( module_e_b_f_d @ M3 @ R )
% 0.30/0.48 => ( ( module_a_b_c_d @ N3 @ R )
% 0.30/0.48 => ( module1330273449unit_d @ ( hOM_b_d_e_f_a_c @ R @ M3 @ N3 ) @ R ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.HOM_is_module
% 0.30/0.48 thf(fact_96_Module_OHOM__is__module,axiom,
% 0.30/0.48 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,N3: carrie1963041556t_unit] :
% 0.30/0.48 ( ( module_e_b_f_d @ M3 @ R )
% 0.30/0.48 => ( ( module1821517916unit_d @ N3 @ R )
% 0.30/0.48 => ( module1330273449unit_d @ ( hOM_b_1375846429t_unit @ R @ M3 @ N3 ) @ R ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.HOM_is_module
% 0.30/0.48 thf(fact_97_Module_OHOM__is__module,axiom,
% 0.30/0.48 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,N3: carrie1821755406_e_b_f] :
% 0.30/0.48 ( ( module_a_b_c_d @ M3 @ R )
% 0.30/0.48 => ( ( module_e_b_f_d @ N3 @ R )
% 0.30/0.48 => ( module1648870817unit_d @ ( hOM_b_d_a_c_e_f @ R @ M3 @ N3 ) @ R ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.HOM_is_module
% 0.30/0.48 thf(fact_98_Module_OHOM__is__module,axiom,
% 0.30/0.48 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,N3: carrie722926983_a_b_c] :
% 0.30/0.48 ( ( module_a_b_c_d @ M3 @ R )
% 0.30/0.48 => ( ( module_a_b_c_d @ N3 @ R )
% 0.30/0.48 => ( module589927589unit_d @ ( hOM_b_d_a_c_a_c @ R @ M3 @ N3 ) @ R ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.HOM_is_module
% 0.30/0.48 thf(fact_99_Module_OHOM__is__module,axiom,
% 0.30/0.48 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,N3: carrie1963041556t_unit] :
% 0.30/0.48 ( ( module_a_b_c_d @ M3 @ R )
% 0.30/0.48 => ( ( module1821517916unit_d @ N3 @ R )
% 0.30/0.48 => ( module589927589unit_d @ ( hOM_b_717364638t_unit @ R @ M3 @ N3 ) @ R ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.HOM_is_module
% 0.30/0.48 thf(fact_100_Module_OHOM__is__module,axiom,
% 0.30/0.48 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,N3: carrie1821755406_e_b_f] :
% 0.30/0.48 ( ( module1821517916unit_d @ M3 @ R )
% 0.30/0.48 => ( ( module_e_b_f_d @ N3 @ R )
% 0.30/0.48 => ( module1648870817unit_d @ ( hOM_b_1959675421it_e_f @ R @ M3 @ N3 ) @ R ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.HOM_is_module
% 0.30/0.48 thf(fact_101_Module_OHOM__is__module,axiom,
% 0.30/0.48 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,N3: carrie722926983_a_b_c] :
% 0.30/0.48 ( ( module1821517916unit_d @ M3 @ R )
% 0.30/0.48 => ( ( module_a_b_c_d @ N3 @ R )
% 0.30/0.48 => ( module589927589unit_d @ ( hOM_b_796081438it_a_c @ R @ M3 @ N3 ) @ R ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.HOM_is_module
% 0.30/0.48 thf(fact_102_Module_OHOM__is__module,axiom,
% 0.30/0.48 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,N3: carrie1963041556t_unit] :
% 0.30/0.48 ( ( module1821517916unit_d @ M3 @ R )
% 0.30/0.48 => ( ( module1821517916unit_d @ N3 @ R )
% 0.30/0.48 => ( module589927589unit_d @ ( hOM_b_1103679019t_unit @ R @ M3 @ N3 ) @ R ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.HOM_is_module
% 0.30/0.48 thf(fact_103_Module_Omisom__trans,axiom,
% 0.30/0.48 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,L: carrie1821755406_e_b_f,N3: carrie1821755406_e_b_f] :
% 0.30/0.48 ( ( module_e_b_f_d @ M3 @ R )
% 0.30/0.48 => ( ( module_e_b_f_d @ L @ R )
% 0.30/0.48 => ( ( module_e_b_f_d @ N3 @ R )
% 0.30/0.48 => ( ( misomo768936435_f_e_f @ R @ L @ M3 )
% 0.30/0.48 => ( ( misomo768936435_f_e_f @ R @ M3 @ N3 )
% 0.30/0.48 => ( misomo768936435_f_e_f @ R @ L @ N3 ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.misom_trans
% 0.30/0.48 thf(fact_104_Module_Omisom__trans,axiom,
% 0.30/0.48 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,L: carrie1821755406_e_b_f,N3: carrie722926983_a_b_c] :
% 0.30/0.48 ( ( module_e_b_f_d @ M3 @ R )
% 0.30/0.48 => ( ( module_e_b_f_d @ L @ R )
% 0.30/0.48 => ( ( module_a_b_c_d @ N3 @ R )
% 0.30/0.48 => ( ( misomo768936435_f_e_f @ R @ L @ M3 )
% 0.30/0.48 => ( ( misomo1752826100_f_a_c @ R @ M3 @ N3 )
% 0.30/0.48 => ( misomo1752826100_f_a_c @ R @ L @ N3 ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.misom_trans
% 0.30/0.48 thf(fact_105_Module_Omisom__trans,axiom,
% 0.30/0.48 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,L: carrie1821755406_e_b_f,N3: carrie1963041556t_unit] :
% 0.30/0.48 ( ( module_e_b_f_d @ M3 @ R )
% 0.30/0.48 => ( ( module_e_b_f_d @ L @ R )
% 0.30/0.48 => ( ( module1821517916unit_d @ N3 @ R )
% 0.30/0.48 => ( ( misomo768936435_f_e_f @ R @ L @ M3 )
% 0.30/0.48 => ( ( misomo368157697t_unit @ R @ M3 @ N3 )
% 0.30/0.48 => ( misomo368157697t_unit @ R @ L @ N3 ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.misom_trans
% 0.30/0.48 thf(fact_106_Module_Omisom__trans,axiom,
% 0.30/0.48 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,L: carrie722926983_a_b_c,N3: carrie1821755406_e_b_f] :
% 0.30/0.48 ( ( module_e_b_f_d @ M3 @ R )
% 0.30/0.48 => ( ( module_a_b_c_d @ L @ R )
% 0.30/0.48 => ( ( module_e_b_f_d @ N3 @ R )
% 0.30/0.48 => ( ( misomo298454132_c_e_f @ R @ L @ M3 )
% 0.30/0.48 => ( ( misomo768936435_f_e_f @ R @ M3 @ N3 )
% 0.30/0.48 => ( misomo298454132_c_e_f @ R @ L @ N3 ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.misom_trans
% 0.30/0.48 thf(fact_107_Module_Omisom__trans,axiom,
% 0.30/0.48 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,L: carrie722926983_a_b_c,N3: carrie1963041556t_unit] :
% 0.30/0.48 ( ( module_e_b_f_d @ M3 @ R )
% 0.30/0.48 => ( ( module_a_b_c_d @ L @ R )
% 0.30/0.48 => ( ( module1821517916unit_d @ N3 @ R )
% 0.30/0.48 => ( ( misomo298454132_c_e_f @ R @ L @ M3 )
% 0.30/0.48 => ( ( misomo368157697t_unit @ R @ M3 @ N3 )
% 0.30/0.48 => ( misomo1857159554t_unit @ R @ L @ N3 ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.misom_trans
% 0.30/0.48 thf(fact_108_Module_Omisom__trans,axiom,
% 0.30/0.48 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,L: carrie1963041556t_unit,N3: carrie1821755406_e_b_f] :
% 0.30/0.48 ( ( module_e_b_f_d @ M3 @ R )
% 0.30/0.48 => ( ( module1821517916unit_d @ L @ R )
% 0.30/0.48 => ( ( module_e_b_f_d @ N3 @ R )
% 0.30/0.48 => ( ( misomo951986689it_e_f @ R @ L @ M3 )
% 0.30/0.48 => ( ( misomo768936435_f_e_f @ R @ M3 @ N3 )
% 0.30/0.48 => ( misomo951986689it_e_f @ R @ L @ N3 ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.misom_trans
% 0.30/0.48 thf(fact_109_Module_Omisom__trans,axiom,
% 0.30/0.48 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,L: carrie1963041556t_unit,N3: carrie722926983_a_b_c] :
% 0.30/0.48 ( ( module_e_b_f_d @ M3 @ R )
% 0.30/0.48 => ( ( module1821517916unit_d @ L @ R )
% 0.30/0.48 => ( ( module_a_b_c_d @ N3 @ R )
% 0.30/0.48 => ( ( misomo951986689it_e_f @ R @ L @ M3 )
% 0.30/0.48 => ( ( misomo1752826100_f_a_c @ R @ M3 @ N3 )
% 0.30/0.48 => ( misomo1935876354it_a_c @ R @ L @ N3 ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.misom_trans
% 0.30/0.48 thf(fact_110_Module_Omisom__trans,axiom,
% 0.30/0.48 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,L: carrie1963041556t_unit,N3: carrie1963041556t_unit] :
% 0.30/0.48 ( ( module_e_b_f_d @ M3 @ R )
% 0.30/0.48 => ( ( module1821517916unit_d @ L @ R )
% 0.30/0.48 => ( ( module1821517916unit_d @ N3 @ R )
% 0.30/0.48 => ( ( misomo951986689it_e_f @ R @ L @ M3 )
% 0.30/0.48 => ( ( misomo368157697t_unit @ R @ M3 @ N3 )
% 0.30/0.48 => ( misomo493403663t_unit @ R @ L @ N3 ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.misom_trans
% 0.30/0.48 thf(fact_111_Module_Omisom__trans,axiom,
% 0.30/0.48 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,L: carrie1821755406_e_b_f,N3: carrie1821755406_e_b_f] :
% 0.30/0.48 ( ( module_a_b_c_d @ M3 @ R )
% 0.30/0.48 => ( ( module_e_b_f_d @ L @ R )
% 0.30/0.48 => ( ( module_e_b_f_d @ N3 @ R )
% 0.30/0.48 => ( ( misomo1752826100_f_a_c @ R @ L @ M3 )
% 0.30/0.48 => ( ( misomo298454132_c_e_f @ R @ M3 @ N3 )
% 0.30/0.48 => ( misomo768936435_f_e_f @ R @ L @ N3 ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.misom_trans
% 0.30/0.48 thf(fact_112_Module_Omisom__trans,axiom,
% 0.30/0.48 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,L: carrie1821755406_e_b_f,N3: carrie1963041556t_unit] :
% 0.30/0.48 ( ( module_a_b_c_d @ M3 @ R )
% 0.30/0.48 => ( ( module_e_b_f_d @ L @ R )
% 0.30/0.48 => ( ( module1821517916unit_d @ N3 @ R )
% 0.30/0.48 => ( ( misomo1752826100_f_a_c @ R @ L @ M3 )
% 0.30/0.48 => ( ( misomo1857159554t_unit @ R @ M3 @ N3 )
% 0.30/0.48 => ( misomo368157697t_unit @ R @ L @ N3 ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.misom_trans
% 0.30/0.48 thf(fact_113_Module_Omisom__self,axiom,
% 0.30/0.48 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d] :
% 0.30/0.48 ( ( module_e_b_f_d @ M3 @ R )
% 0.30/0.48 => ( misomo768936435_f_e_f @ R @ M3 @ M3 ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.misom_self
% 0.30/0.48 thf(fact_114_Module_Omisom__self,axiom,
% 0.30/0.48 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d] :
% 0.30/0.48 ( ( module1821517916unit_d @ M3 @ R )
% 0.30/0.48 => ( misomo493403663t_unit @ R @ M3 @ M3 ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.misom_self
% 0.30/0.48 thf(fact_115_Module_Omisom__self,axiom,
% 0.30/0.48 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d] :
% 0.30/0.48 ( ( module_a_b_c_d @ M3 @ R )
% 0.30/0.48 => ( misomo1282343797_c_a_c @ R @ M3 @ M3 ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.misom_self
% 0.30/0.48 thf(fact_116_Module_Omisom__sym,axiom,
% 0.30/0.48 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,N3: carrie1821755406_e_b_f] :
% 0.30/0.48 ( ( module_e_b_f_d @ M3 @ R )
% 0.30/0.48 => ( ( module_e_b_f_d @ N3 @ R )
% 0.30/0.48 => ( ( misomo768936435_f_e_f @ R @ M3 @ N3 )
% 0.30/0.48 => ( misomo768936435_f_e_f @ R @ N3 @ M3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.misom_sym
% 0.30/0.48 thf(fact_117_Module_Omisom__sym,axiom,
% 0.30/0.48 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,N3: carrie722926983_a_b_c] :
% 0.30/0.48 ( ( module_e_b_f_d @ M3 @ R )
% 0.30/0.48 => ( ( module_a_b_c_d @ N3 @ R )
% 0.30/0.48 => ( ( misomo1752826100_f_a_c @ R @ M3 @ N3 )
% 0.30/0.48 => ( misomo298454132_c_e_f @ R @ N3 @ M3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.misom_sym
% 0.30/0.48 thf(fact_118_Module_Omisom__sym,axiom,
% 0.30/0.48 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,N3: carrie1963041556t_unit] :
% 0.30/0.48 ( ( module_e_b_f_d @ M3 @ R )
% 0.30/0.48 => ( ( module1821517916unit_d @ N3 @ R )
% 0.30/0.48 => ( ( misomo368157697t_unit @ R @ M3 @ N3 )
% 0.30/0.48 => ( misomo951986689it_e_f @ R @ N3 @ M3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.misom_sym
% 0.30/0.48 thf(fact_119_Module_Omisom__sym,axiom,
% 0.30/0.48 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,N3: carrie1821755406_e_b_f] :
% 0.30/0.48 ( ( module_a_b_c_d @ M3 @ R )
% 0.30/0.48 => ( ( module_e_b_f_d @ N3 @ R )
% 0.30/0.48 => ( ( misomo298454132_c_e_f @ R @ M3 @ N3 )
% 0.30/0.48 => ( misomo1752826100_f_a_c @ R @ N3 @ M3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.misom_sym
% 0.30/0.48 thf(fact_120_Module_Omisom__sym,axiom,
% 0.30/0.48 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,N3: carrie1963041556t_unit] :
% 0.30/0.48 ( ( module_a_b_c_d @ M3 @ R )
% 0.30/0.48 => ( ( module1821517916unit_d @ N3 @ R )
% 0.30/0.48 => ( ( misomo1857159554t_unit @ R @ M3 @ N3 )
% 0.30/0.48 => ( misomo1935876354it_a_c @ R @ N3 @ M3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.misom_sym
% 0.30/0.48 thf(fact_121_Module_Omisom__sym,axiom,
% 0.30/0.48 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,N3: carrie1821755406_e_b_f] :
% 0.30/0.48 ( ( module1821517916unit_d @ M3 @ R )
% 0.30/0.48 => ( ( module_e_b_f_d @ N3 @ R )
% 0.30/0.48 => ( ( misomo951986689it_e_f @ R @ M3 @ N3 )
% 0.30/0.48 => ( misomo368157697t_unit @ R @ N3 @ M3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.misom_sym
% 0.30/0.48 thf(fact_122_Module_Omisom__sym,axiom,
% 0.30/0.48 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,N3: carrie722926983_a_b_c] :
% 0.30/0.48 ( ( module1821517916unit_d @ M3 @ R )
% 0.30/0.48 => ( ( module_a_b_c_d @ N3 @ R )
% 0.30/0.48 => ( ( misomo1935876354it_a_c @ R @ M3 @ N3 )
% 0.30/0.48 => ( misomo1857159554t_unit @ R @ N3 @ M3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.misom_sym
% 0.30/0.48 thf(fact_123_Module_Omisom__sym,axiom,
% 0.30/0.48 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,N3: carrie1963041556t_unit] :
% 0.30/0.48 ( ( module1821517916unit_d @ M3 @ R )
% 0.30/0.48 => ( ( module1821517916unit_d @ N3 @ R )
% 0.30/0.48 => ( ( misomo493403663t_unit @ R @ M3 @ N3 )
% 0.30/0.48 => ( misomo493403663t_unit @ R @ N3 @ M3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.misom_sym
% 0.30/0.48 thf(fact_124_Module_Omisom__sym,axiom,
% 0.30/0.48 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,N3: carrie722926983_a_b_c] :
% 0.30/0.48 ( ( module_a_b_c_d @ M3 @ R )
% 0.30/0.48 => ( ( module_a_b_c_d @ N3 @ R )
% 0.30/0.48 => ( ( misomo1282343797_c_a_c @ R @ M3 @ N3 )
% 0.30/0.48 => ( misomo1282343797_c_a_c @ R @ N3 @ M3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.misom_sym
% 0.30/0.48 thf(fact_125_Module_Ofree__generator__generator,axiom,
% 0.30/0.48 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,H: set_e] :
% 0.30/0.48 ( ( module_e_b_f_d @ M3 @ R )
% 0.30/0.48 => ( ( free_g103796815_d_e_f @ R @ M3 @ H )
% 0.30/0.48 => ( generator_b_d_e_f @ R @ M3 @ H ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.free_generator_generator
% 0.30/0.48 thf(fact_126_Module_Ofree__generator__generator,axiom,
% 0.30/0.48 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,H: set_a] :
% 0.30/0.48 ( ( module1821517916unit_d @ M3 @ R )
% 0.30/0.48 => ( ( free_g637607517t_unit @ R @ M3 @ H )
% 0.30/0.48 => ( genera1692266857t_unit @ R @ M3 @ H ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.free_generator_generator
% 0.30/0.48 thf(fact_127_Module_Ofree__generator__generator,axiom,
% 0.30/0.48 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,H: set_a] :
% 0.30/0.48 ( ( module_a_b_c_d @ M3 @ R )
% 0.30/0.48 => ( ( free_g1087686480_d_a_c @ R @ M3 @ H )
% 0.30/0.48 => ( generator_b_d_a_c @ R @ M3 @ H ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.free_generator_generator
% 0.30/0.48 thf(fact_128_Module_Ogenerator__sub__carrier,axiom,
% 0.30/0.48 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,H: set_e] :
% 0.30/0.48 ( ( module_e_b_f_d @ M3 @ R )
% 0.30/0.48 => ( ( generator_b_d_e_f @ R @ M3 @ H )
% 0.30/0.48 => ( ord_less_eq_set_e @ H @ ( carrie730238621_e_b_f @ M3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.generator_sub_carrier
% 0.30/0.48 thf(fact_129_Module_Ogenerator__sub__carrier,axiom,
% 0.30/0.48 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,H: set_a] :
% 0.30/0.48 ( ( module_a_b_c_d @ M3 @ R )
% 0.30/0.48 => ( ( generator_b_d_a_c @ R @ M3 @ H )
% 0.30/0.48 => ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ M3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.generator_sub_carrier
% 0.30/0.48 thf(fact_130_Module_Ogenerator__sub__carrier,axiom,
% 0.30/0.48 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,H: set_a] :
% 0.30/0.48 ( ( module1821517916unit_d @ M3 @ R )
% 0.30/0.48 => ( ( genera1692266857t_unit @ R @ M3 @ H )
% 0.30/0.48 => ( ord_less_eq_set_a @ H @ ( carrie1074654371t_unit @ M3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.generator_sub_carrier
% 0.30/0.48 thf(fact_131_Module_OAnn__is__ideal,axiom,
% 0.30/0.48 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d] :
% 0.30/0.48 ( ( module_e_b_f_d @ M3 @ R )
% 0.30/0.48 => ( ideal_b_d @ R @ ( annihilator_b_d_e_f @ R @ M3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.Ann_is_ideal
% 0.30/0.48 thf(fact_132_Module_OAnn__is__ideal,axiom,
% 0.30/0.48 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d] :
% 0.30/0.48 ( ( module1821517916unit_d @ M3 @ R )
% 0.30/0.48 => ( ideal_b_d @ R @ ( annihi259882159t_unit @ R @ M3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.Ann_is_ideal
% 0.30/0.48 thf(fact_133_Module_OAnn__is__ideal,axiom,
% 0.30/0.48 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d] :
% 0.30/0.48 ( ( module_a_b_c_d @ M3 @ R )
% 0.30/0.48 => ( ideal_b_d @ R @ ( annihilator_b_d_a_c @ R @ M3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.Ann_is_ideal
% 0.30/0.48 thf(fact_134_Nat_Oex__has__greatest__nat,axiom,
% 0.30/0.48 ! [P: nat > $o,K2: nat,B: nat] :
% 0.30/0.48 ( ( P @ K2 )
% 0.30/0.48 => ( ! [Y: nat] :
% 0.30/0.48 ( ( P @ Y )
% 0.30/0.48 => ( ord_less_eq_nat @ Y @ B ) )
% 0.30/0.48 => ? [X2: nat] :
% 0.30/0.48 ( ( P @ X2 )
% 0.30/0.48 & ! [Y3: nat] :
% 0.30/0.48 ( ( P @ Y3 )
% 0.30/0.48 => ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Nat.ex_has_greatest_nat
% 0.30/0.48 thf(fact_135_nat__le__linear,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.48 | ( ord_less_eq_nat @ N @ M ) ) ).
% 0.30/0.48
% 0.30/0.48 % nat_le_linear
% 0.30/0.48 thf(fact_136_le__antisym,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.48 => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.48 => ( M = N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_antisym
% 0.30/0.48 thf(fact_137_le__trans,axiom,
% 0.30/0.48 ! [I: nat,J3: nat,K2: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ I @ J3 )
% 0.30/0.48 => ( ( ord_less_eq_nat @ J3 @ K2 )
% 0.30/0.48 => ( ord_less_eq_nat @ I @ K2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_trans
% 0.30/0.48 thf(fact_138_n__not__Suc__n,axiom,
% 0.30/0.48 ! [N: nat] :
% 0.30/0.48 ( N
% 0.30/0.48 != ( suc @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % n_not_Suc_n
% 0.30/0.48 thf(fact_139_Suc__inject,axiom,
% 0.30/0.48 ! [X: nat,Y4: nat] :
% 0.30/0.48 ( ( ( suc @ X )
% 0.30/0.48 = ( suc @ Y4 ) )
% 0.30/0.48 => ( X = Y4 ) ) ).
% 0.30/0.48
% 0.30/0.48 % Suc_inject
% 0.30/0.48 thf(fact_140_linorder__neqE__nat,axiom,
% 0.30/0.48 ! [X: nat,Y4: nat] :
% 0.30/0.48 ( ( X != Y4 )
% 0.30/0.48 => ( ~ ( ord_less_nat @ X @ Y4 )
% 0.30/0.48 => ( ord_less_nat @ Y4 @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % linorder_neqE_nat
% 0.30/0.48 thf(fact_141_infinite__descent,axiom,
% 0.30/0.48 ! [P: nat > $o,N: nat] :
% 0.30/0.48 ( ! [N2: nat] :
% 0.30/0.48 ( ~ ( P @ N2 )
% 0.30/0.48 => ? [M2: nat] :
% 0.30/0.48 ( ( ord_less_nat @ M2 @ N2 )
% 0.30/0.48 & ~ ( P @ M2 ) ) )
% 0.30/0.48 => ( P @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % infinite_descent
% 0.30/0.48 thf(fact_142_nat__less__induct,axiom,
% 0.30/0.48 ! [P: nat > $o,N: nat] :
% 0.30/0.48 ( ! [N2: nat] :
% 0.30/0.48 ( ! [M2: nat] :
% 0.30/0.48 ( ( ord_less_nat @ M2 @ N2 )
% 0.30/0.48 => ( P @ M2 ) )
% 0.30/0.48 => ( P @ N2 ) )
% 0.30/0.48 => ( P @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % nat_less_induct
% 0.30/0.48 thf(fact_143_less__irrefl__nat,axiom,
% 0.30/0.48 ! [N: nat] :
% 0.30/0.48 ~ ( ord_less_nat @ N @ N ) ).
% 0.30/0.48
% 0.30/0.48 % less_irrefl_nat
% 0.30/0.48 thf(fact_144_less__not__refl3,axiom,
% 0.30/0.48 ! [S: nat,T: nat] :
% 0.30/0.48 ( ( ord_less_nat @ S @ T )
% 0.30/0.48 => ( S != T ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_not_refl3
% 0.30/0.48 thf(fact_145_less__not__refl2,axiom,
% 0.30/0.48 ! [N: nat,M: nat] :
% 0.30/0.48 ( ( ord_less_nat @ N @ M )
% 0.30/0.48 => ( M != N ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_not_refl2
% 0.30/0.48 thf(fact_146_less__not__refl,axiom,
% 0.30/0.48 ! [N: nat] :
% 0.30/0.48 ~ ( ord_less_nat @ N @ N ) ).
% 0.30/0.48
% 0.30/0.48 % less_not_refl
% 0.30/0.48 thf(fact_147_nat__neq__iff,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( M != N )
% 0.30/0.48 = ( ( ord_less_nat @ M @ N )
% 0.30/0.48 | ( ord_less_nat @ N @ M ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % nat_neq_iff
% 0.30/0.48 thf(fact_148_diff__commute,axiom,
% 0.30/0.48 ! [I: nat,J3: nat,K2: nat] :
% 0.30/0.48 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J3 ) @ K2 )
% 0.30/0.48 = ( minus_minus_nat @ ( minus_minus_nat @ I @ K2 ) @ J3 ) ) ).
% 0.30/0.48
% 0.30/0.48 % diff_commute
% 0.30/0.48 thf(fact_149_Module_Osmodule__ideal__coeff__is__Submodule,axiom,
% 0.30/0.48 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,A2: set_b] :
% 0.30/0.48 ( ( module_e_b_f_d @ M3 @ R )
% 0.30/0.48 => ( ( ideal_b_d @ R @ A2 )
% 0.30/0.48 => ( submodule_b_d_e_f @ R @ M3 @ ( smodul1982583723_d_e_f @ R @ M3 @ A2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.smodule_ideal_coeff_is_Submodule
% 0.30/0.48 thf(fact_150_Module_Osmodule__ideal__coeff__is__Submodule,axiom,
% 0.30/0.48 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,A2: set_b] :
% 0.30/0.48 ( ( module1821517916unit_d @ M3 @ R )
% 0.30/0.48 => ( ( ideal_b_d @ R @ A2 )
% 0.30/0.48 => ( submod903911234t_unit @ R @ M3 @ ( smodul132175289t_unit @ R @ M3 @ A2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.smodule_ideal_coeff_is_Submodule
% 0.30/0.48 thf(fact_151_Module_Osmodule__ideal__coeff__is__Submodule,axiom,
% 0.30/0.48 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,A2: set_b] :
% 0.30/0.48 ( ( module_a_b_c_d @ M3 @ R )
% 0.30/0.48 => ( ( ideal_b_d @ R @ A2 )
% 0.30/0.48 => ( submodule_b_d_a_c @ R @ M3 @ ( smodul818989740_d_a_c @ R @ M3 @ A2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.smodule_ideal_coeff_is_Submodule
% 0.30/0.48 thf(fact_152_Module_Oquotient__of__submodules__is__ideal,axiom,
% 0.30/0.48 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,P: set_e,Q: set_e] :
% 0.30/0.48 ( ( module_e_b_f_d @ M3 @ R )
% 0.30/0.48 => ( ( submodule_b_d_e_f @ R @ M3 @ P )
% 0.30/0.48 => ( ( submodule_b_d_e_f @ R @ M3 @ Q )
% 0.30/0.48 => ( ideal_b_d @ R @ ( quotie169863047_d_e_f @ R @ M3 @ P @ Q ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.quotient_of_submodules_is_ideal
% 0.30/0.48 thf(fact_153_Module_Oquotient__of__submodules__is__ideal,axiom,
% 0.30/0.48 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,P: set_a,Q: set_a] :
% 0.30/0.48 ( ( module1821517916unit_d @ M3 @ R )
% 0.30/0.48 => ( ( submod903911234t_unit @ R @ M3 @ P )
% 0.30/0.48 => ( ( submod903911234t_unit @ R @ M3 @ Q )
% 0.30/0.48 => ( ideal_b_d @ R @ ( quotie1196826005t_unit @ R @ M3 @ P @ Q ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.quotient_of_submodules_is_ideal
% 0.30/0.48 thf(fact_154_Module_Oquotient__of__submodules__is__ideal,axiom,
% 0.30/0.48 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,P: set_a,Q: set_a] :
% 0.30/0.48 ( ( module_a_b_c_d @ M3 @ R )
% 0.30/0.48 => ( ( submodule_b_d_a_c @ R @ M3 @ P )
% 0.30/0.48 => ( ( submodule_b_d_a_c @ R @ M3 @ Q )
% 0.30/0.48 => ( ideal_b_d @ R @ ( quotie1153752712_d_a_c @ R @ M3 @ P @ Q ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.quotient_of_submodules_is_ideal
% 0.30/0.48 thf(fact_155_transitive__stepwise__le,axiom,
% 0.30/0.48 ! [M: nat,N: nat,R: nat > nat > $o] :
% 0.30/0.48 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.48 => ( ! [X2: nat] : ( R @ X2 @ X2 )
% 0.30/0.48 => ( ! [X2: nat,Y: nat,Z: nat] :
% 0.30/0.48 ( ( R @ X2 @ Y )
% 0.30/0.48 => ( ( R @ Y @ Z )
% 0.30/0.48 => ( R @ X2 @ Z ) ) )
% 0.30/0.48 => ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
% 0.30/0.48 => ( R @ M @ N ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % transitive_stepwise_le
% 0.30/0.48 thf(fact_156_nat__induct__at__least,axiom,
% 0.30/0.48 ! [M: nat,N: nat,P: nat > $o] :
% 0.30/0.48 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.48 => ( ( P @ M )
% 0.30/0.48 => ( ! [N2: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.48 => ( ( P @ N2 )
% 0.30/0.48 => ( P @ ( suc @ N2 ) ) ) )
% 0.30/0.48 => ( P @ N ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % nat_induct_at_least
% 0.30/0.48 thf(fact_157_full__nat__induct,axiom,
% 0.30/0.48 ! [P: nat > $o,N: nat] :
% 0.30/0.48 ( ! [N2: nat] :
% 0.30/0.48 ( ! [M2: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
% 0.30/0.48 => ( P @ M2 ) )
% 0.30/0.48 => ( P @ N2 ) )
% 0.30/0.48 => ( P @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % full_nat_induct
% 0.30/0.48 thf(fact_158_not__less__eq__eq,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ~ ( ord_less_eq_nat @ M @ N ) )
% 0.30/0.48 = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% 0.30/0.48
% 0.30/0.48 % not_less_eq_eq
% 0.30/0.48 thf(fact_159_Suc__n__not__le__n,axiom,
% 0.30/0.48 ! [N: nat] :
% 0.30/0.48 ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% 0.30/0.48
% 0.30/0.48 % Suc_n_not_le_n
% 0.30/0.48 thf(fact_160_le__Suc__eq,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.30/0.48 = ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.48 | ( M
% 0.30/0.48 = ( suc @ N ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_Suc_eq
% 0.30/0.48 thf(fact_161_Suc__le__D,axiom,
% 0.30/0.48 ! [N: nat,M4: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ ( suc @ N ) @ M4 )
% 0.30/0.48 => ? [M5: nat] :
% 0.30/0.48 ( M4
% 0.30/0.48 = ( suc @ M5 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Suc_le_D
% 0.30/0.48 thf(fact_162_le__SucI,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.48 => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_SucI
% 0.30/0.48 thf(fact_163_le__SucE,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.30/0.48 => ( ~ ( ord_less_eq_nat @ M @ N )
% 0.30/0.48 => ( M
% 0.30/0.48 = ( suc @ N ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_SucE
% 0.30/0.48 thf(fact_164_Suc__leD,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.30/0.48 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % Suc_leD
% 0.30/0.48 thf(fact_165_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.30/0.48 ! [A: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.30/0.48 => ( A = zero_zero_nat ) ) ).
% 0.30/0.48
% 0.30/0.48 % bot_nat_0.extremum_uniqueI
% 0.30/0.48 thf(fact_166_bot__nat__0_Oextremum__unique,axiom,
% 0.30/0.48 ! [A: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.30/0.48 = ( A = zero_zero_nat ) ) ).
% 0.30/0.48
% 0.30/0.48 % bot_nat_0.extremum_unique
% 0.30/0.48 thf(fact_167_le__0__eq,axiom,
% 0.30/0.48 ! [N: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.30/0.48 = ( N = zero_zero_nat ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_0_eq
% 0.30/0.48 thf(fact_168_less__eq__nat_Osimps_I1_J,axiom,
% 0.30/0.48 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.30/0.48
% 0.30/0.48 % less_eq_nat.simps(1)
% 0.30/0.48 thf(fact_169_not0__implies__Suc,axiom,
% 0.30/0.48 ! [N: nat] :
% 0.30/0.48 ( ( N != zero_zero_nat )
% 0.30/0.48 => ? [M5: nat] :
% 0.30/0.48 ( N
% 0.30/0.48 = ( suc @ M5 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % not0_implies_Suc
% 0.30/0.48 thf(fact_170_old_Onat_Oinducts,axiom,
% 0.30/0.48 ! [P: nat > $o,Nat: nat] :
% 0.30/0.48 ( ( P @ zero_zero_nat )
% 0.30/0.48 => ( ! [Nat3: nat] :
% 0.30/0.48 ( ( P @ Nat3 )
% 0.30/0.48 => ( P @ ( suc @ Nat3 ) ) )
% 0.30/0.48 => ( P @ Nat ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % old.nat.inducts
% 0.30/0.48 thf(fact_171_old_Onat_Oexhaust,axiom,
% 0.30/0.48 ! [Y4: nat] :
% 0.30/0.48 ( ( Y4 != zero_zero_nat )
% 0.30/0.48 => ~ ! [Nat3: nat] :
% 0.30/0.48 ( Y4
% 0.30/0.48 != ( suc @ Nat3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % old.nat.exhaust
% 0.30/0.48 thf(fact_172_Zero__not__Suc,axiom,
% 0.30/0.48 ! [M: nat] :
% 0.30/0.48 ( zero_zero_nat
% 0.30/0.48 != ( suc @ M ) ) ).
% 0.30/0.48
% 0.30/0.48 % Zero_not_Suc
% 0.30/0.48 thf(fact_173_Zero__neq__Suc,axiom,
% 0.30/0.48 ! [M: nat] :
% 0.30/0.48 ( zero_zero_nat
% 0.30/0.48 != ( suc @ M ) ) ).
% 0.30/0.48
% 0.30/0.48 % Zero_neq_Suc
% 0.30/0.48 thf(fact_174_Suc__neq__Zero,axiom,
% 0.30/0.48 ! [M: nat] :
% 0.30/0.48 ( ( suc @ M )
% 0.30/0.48 != zero_zero_nat ) ).
% 0.30/0.48
% 0.30/0.48 % Suc_neq_Zero
% 0.30/0.48 thf(fact_175_zero__induct,axiom,
% 0.30/0.48 ! [P: nat > $o,K2: nat] :
% 0.30/0.48 ( ( P @ K2 )
% 0.30/0.48 => ( ! [N2: nat] :
% 0.30/0.48 ( ( P @ ( suc @ N2 ) )
% 0.30/0.48 => ( P @ N2 ) )
% 0.30/0.48 => ( P @ zero_zero_nat ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % zero_induct
% 0.30/0.48 thf(fact_176_diff__induct,axiom,
% 0.30/0.48 ! [P: nat > nat > $o,M: nat,N: nat] :
% 0.30/0.48 ( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
% 0.30/0.48 => ( ! [Y: nat] : ( P @ zero_zero_nat @ ( suc @ Y ) )
% 0.30/0.48 => ( ! [X2: nat,Y: nat] :
% 0.30/0.48 ( ( P @ X2 @ Y )
% 0.30/0.48 => ( P @ ( suc @ X2 ) @ ( suc @ Y ) ) )
% 0.30/0.48 => ( P @ M @ N ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % diff_induct
% 0.30/0.48 thf(fact_177_nat__induct,axiom,
% 0.30/0.48 ! [P: nat > $o,N: nat] :
% 0.30/0.48 ( ( P @ zero_zero_nat )
% 0.30/0.48 => ( ! [N2: nat] :
% 0.30/0.48 ( ( P @ N2 )
% 0.30/0.48 => ( P @ ( suc @ N2 ) ) )
% 0.30/0.48 => ( P @ N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % nat_induct
% 0.30/0.48 thf(fact_178_nat_OdiscI,axiom,
% 0.30/0.48 ! [Nat: nat,X22: nat] :
% 0.30/0.48 ( ( Nat
% 0.30/0.48 = ( suc @ X22 ) )
% 0.30/0.48 => ( Nat != zero_zero_nat ) ) ).
% 0.30/0.48
% 0.30/0.48 % nat.discI
% 0.30/0.48 thf(fact_179_old_Onat_Odistinct_I1_J,axiom,
% 0.30/0.48 ! [Nat2: nat] :
% 0.30/0.48 ( zero_zero_nat
% 0.30/0.48 != ( suc @ Nat2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % old.nat.distinct(1)
% 0.30/0.48 thf(fact_180_old_Onat_Odistinct_I2_J,axiom,
% 0.30/0.48 ! [Nat2: nat] :
% 0.30/0.48 ( ( suc @ Nat2 )
% 0.30/0.48 != zero_zero_nat ) ).
% 0.30/0.48
% 0.30/0.48 % old.nat.distinct(2)
% 0.30/0.48 thf(fact_181_nat_Odistinct_I1_J,axiom,
% 0.30/0.48 ! [X22: nat] :
% 0.30/0.48 ( zero_zero_nat
% 0.30/0.48 != ( suc @ X22 ) ) ).
% 0.30/0.48
% 0.30/0.48 % nat.distinct(1)
% 0.30/0.48 thf(fact_182_less__mono__imp__le__mono,axiom,
% 0.30/0.48 ! [F: nat > nat,I: nat,J3: nat] :
% 0.30/0.48 ( ! [I2: nat,J4: nat] :
% 0.30/0.48 ( ( ord_less_nat @ I2 @ J4 )
% 0.30/0.48 => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J4 ) ) )
% 0.30/0.48 => ( ( ord_less_eq_nat @ I @ J3 )
% 0.30/0.48 => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_mono_imp_le_mono
% 0.30/0.48 thf(fact_183_le__neq__implies__less,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.48 => ( ( M != N )
% 0.30/0.48 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_neq_implies_less
% 0.30/0.48 thf(fact_184_less__or__eq__imp__le,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ( ord_less_nat @ M @ N )
% 0.30/0.48 | ( M = N ) )
% 0.30/0.48 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_or_eq_imp_le
% 0.30/0.48 thf(fact_185_le__eq__less__or__eq,axiom,
% 0.30/0.48 ( ord_less_eq_nat
% 0.30/0.48 = ( ^ [M6: nat,N4: nat] :
% 0.30/0.48 ( ( ord_less_nat @ M6 @ N4 )
% 0.30/0.48 | ( M6 = N4 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_eq_less_or_eq
% 0.30/0.48 thf(fact_186_less__imp__le__nat,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_nat @ M @ N )
% 0.30/0.48 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_imp_le_nat
% 0.30/0.48 thf(fact_187_nat__less__le,axiom,
% 0.30/0.48 ( ord_less_nat
% 0.30/0.48 = ( ^ [M6: nat,N4: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ M6 @ N4 )
% 0.30/0.48 & ( M6 != N4 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % nat_less_le
% 0.30/0.48 thf(fact_188_not__less__less__Suc__eq,axiom,
% 0.30/0.48 ! [N: nat,M: nat] :
% 0.30/0.48 ( ~ ( ord_less_nat @ N @ M )
% 0.30/0.48 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.30/0.48 = ( N = M ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % not_less_less_Suc_eq
% 0.30/0.48 thf(fact_189_strict__inc__induct,axiom,
% 0.30/0.48 ! [I: nat,J3: nat,P: nat > $o] :
% 0.30/0.48 ( ( ord_less_nat @ I @ J3 )
% 0.30/0.48 => ( ! [I2: nat] :
% 0.30/0.48 ( ( J3
% 0.30/0.48 = ( suc @ I2 ) )
% 0.30/0.48 => ( P @ I2 ) )
% 0.30/0.48 => ( ! [I2: nat] :
% 0.30/0.48 ( ( ord_less_nat @ I2 @ J3 )
% 0.30/0.48 => ( ( P @ ( suc @ I2 ) )
% 0.30/0.48 => ( P @ I2 ) ) )
% 0.30/0.48 => ( P @ I ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % strict_inc_induct
% 0.30/0.48 thf(fact_190_less__Suc__induct,axiom,
% 0.30/0.48 ! [I: nat,J3: nat,P: nat > nat > $o] :
% 0.30/0.48 ( ( ord_less_nat @ I @ J3 )
% 0.30/0.48 => ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
% 0.30/0.48 => ( ! [I2: nat,J4: nat,K3: nat] :
% 0.30/0.48 ( ( ord_less_nat @ I2 @ J4 )
% 0.30/0.48 => ( ( ord_less_nat @ J4 @ K3 )
% 0.30/0.48 => ( ( P @ I2 @ J4 )
% 0.30/0.48 => ( ( P @ J4 @ K3 )
% 0.30/0.48 => ( P @ I2 @ K3 ) ) ) ) )
% 0.30/0.48 => ( P @ I @ J3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_Suc_induct
% 0.30/0.48 thf(fact_191_less__trans__Suc,axiom,
% 0.30/0.48 ! [I: nat,J3: nat,K2: nat] :
% 0.30/0.48 ( ( ord_less_nat @ I @ J3 )
% 0.30/0.48 => ( ( ord_less_nat @ J3 @ K2 )
% 0.30/0.48 => ( ord_less_nat @ ( suc @ I ) @ K2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_trans_Suc
% 0.30/0.48 thf(fact_192_Suc__less__SucD,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.30/0.48 => ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % Suc_less_SucD
% 0.30/0.48 thf(fact_193_less__antisym,axiom,
% 0.30/0.48 ! [N: nat,M: nat] :
% 0.30/0.48 ( ~ ( ord_less_nat @ N @ M )
% 0.30/0.48 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.30/0.48 => ( M = N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_antisym
% 0.30/0.48 thf(fact_194_Suc__less__eq2,axiom,
% 0.30/0.48 ! [N: nat,M: nat] :
% 0.30/0.48 ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.30/0.48 = ( ? [M7: nat] :
% 0.30/0.48 ( ( M
% 0.30/0.48 = ( suc @ M7 ) )
% 0.30/0.48 & ( ord_less_nat @ N @ M7 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Suc_less_eq2
% 0.30/0.48 thf(fact_195_All__less__Suc,axiom,
% 0.30/0.48 ! [N: nat,P: nat > $o] :
% 0.30/0.48 ( ( ! [I3: nat] :
% 0.30/0.48 ( ( ord_less_nat @ I3 @ ( suc @ N ) )
% 0.30/0.48 => ( P @ I3 ) ) )
% 0.30/0.48 = ( ( P @ N )
% 0.30/0.48 & ! [I3: nat] :
% 0.30/0.48 ( ( ord_less_nat @ I3 @ N )
% 0.30/0.48 => ( P @ I3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % All_less_Suc
% 0.30/0.48 thf(fact_196_not__less__eq,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ~ ( ord_less_nat @ M @ N ) )
% 0.30/0.48 = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % not_less_eq
% 0.30/0.48 thf(fact_197_less__Suc__eq,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.30/0.48 = ( ( ord_less_nat @ M @ N )
% 0.30/0.48 | ( M = N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_Suc_eq
% 0.30/0.48 thf(fact_198_Ex__less__Suc,axiom,
% 0.30/0.48 ! [N: nat,P: nat > $o] :
% 0.30/0.48 ( ( ? [I3: nat] :
% 0.30/0.48 ( ( ord_less_nat @ I3 @ ( suc @ N ) )
% 0.30/0.48 & ( P @ I3 ) ) )
% 0.30/0.48 = ( ( P @ N )
% 0.30/0.48 | ? [I3: nat] :
% 0.30/0.48 ( ( ord_less_nat @ I3 @ N )
% 0.30/0.48 & ( P @ I3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Ex_less_Suc
% 0.30/0.48 thf(fact_199_less__SucI,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_nat @ M @ N )
% 0.30/0.48 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_SucI
% 0.30/0.48 thf(fact_200_less__SucE,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.30/0.48 => ( ~ ( ord_less_nat @ M @ N )
% 0.30/0.48 => ( M = N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_SucE
% 0.30/0.48 thf(fact_201_Suc__lessI,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_nat @ M @ N )
% 0.30/0.48 => ( ( ( suc @ M )
% 0.30/0.48 != N )
% 0.30/0.48 => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Suc_lessI
% 0.30/0.48 thf(fact_202_Suc__lessE,axiom,
% 0.30/0.48 ! [I: nat,K2: nat] :
% 0.30/0.48 ( ( ord_less_nat @ ( suc @ I ) @ K2 )
% 0.30/0.48 => ~ ! [J4: nat] :
% 0.30/0.48 ( ( ord_less_nat @ I @ J4 )
% 0.30/0.48 => ( K2
% 0.30/0.48 != ( suc @ J4 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Suc_lessE
% 0.30/0.48 thf(fact_203_Suc__lessD,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_nat @ ( suc @ M ) @ N )
% 0.30/0.48 => ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % Suc_lessD
% 0.30/0.48 thf(fact_204_Nat_OlessE,axiom,
% 0.30/0.48 ! [I: nat,K2: nat] :
% 0.30/0.48 ( ( ord_less_nat @ I @ K2 )
% 0.30/0.48 => ( ( K2
% 0.30/0.48 != ( suc @ I ) )
% 0.30/0.48 => ~ ! [J4: nat] :
% 0.30/0.48 ( ( ord_less_nat @ I @ J4 )
% 0.30/0.48 => ( K2
% 0.30/0.48 != ( suc @ J4 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Nat.lessE
% 0.30/0.48 thf(fact_205_bot__nat__0_Oextremum__strict,axiom,
% 0.30/0.48 ! [A: nat] :
% 0.30/0.48 ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 0.30/0.48
% 0.30/0.48 % bot_nat_0.extremum_strict
% 0.30/0.48 thf(fact_206_infinite__descent0,axiom,
% 0.30/0.48 ! [P: nat > $o,N: nat] :
% 0.30/0.48 ( ( P @ zero_zero_nat )
% 0.30/0.48 => ( ! [N2: nat] :
% 0.30/0.48 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.48 => ( ~ ( P @ N2 )
% 0.30/0.48 => ? [M2: nat] :
% 0.30/0.48 ( ( ord_less_nat @ M2 @ N2 )
% 0.30/0.48 & ~ ( P @ M2 ) ) ) )
% 0.30/0.48 => ( P @ N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % infinite_descent0
% 0.30/0.48 thf(fact_207_gr__implies__not0,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_nat @ M @ N )
% 0.30/0.48 => ( N != zero_zero_nat ) ) ).
% 0.30/0.48
% 0.30/0.48 % gr_implies_not0
% 0.30/0.48 thf(fact_208_less__zeroE,axiom,
% 0.30/0.48 ! [N: nat] :
% 0.30/0.48 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.30/0.48
% 0.30/0.48 % less_zeroE
% 0.30/0.48 thf(fact_209_not__less0,axiom,
% 0.30/0.48 ! [N: nat] :
% 0.30/0.48 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.30/0.48
% 0.30/0.48 % not_less0
% 0.30/0.48 thf(fact_210_not__gr0,axiom,
% 0.30/0.48 ! [N: nat] :
% 0.30/0.48 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.30/0.48 = ( N = zero_zero_nat ) ) ).
% 0.30/0.48
% 0.30/0.48 % not_gr0
% 0.30/0.48 thf(fact_211_gr0I,axiom,
% 0.30/0.48 ! [N: nat] :
% 0.30/0.48 ( ( N != zero_zero_nat )
% 0.30/0.48 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % gr0I
% 0.30/0.48 thf(fact_212_diff__le__mono2,axiom,
% 0.30/0.48 ! [M: nat,N: nat,L2: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.48 => ( ord_less_eq_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % diff_le_mono2
% 0.30/0.48 thf(fact_213_le__diff__iff_H,axiom,
% 0.30/0.48 ! [A: nat,C: nat,B: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ A @ C )
% 0.30/0.48 => ( ( ord_less_eq_nat @ B @ C )
% 0.30/0.48 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
% 0.30/0.48 = ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_diff_iff'
% 0.30/0.48 thf(fact_214_diff__le__self,axiom,
% 0.30/0.48 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% 0.30/0.48
% 0.30/0.48 % diff_le_self
% 0.30/0.48 thf(fact_215_diff__le__mono,axiom,
% 0.30/0.48 ! [M: nat,N: nat,L2: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.48 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L2 ) @ ( minus_minus_nat @ N @ L2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % diff_le_mono
% 0.30/0.48 thf(fact_216_Nat_Odiff__diff__eq,axiom,
% 0.30/0.48 ! [K2: nat,M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ K2 @ M )
% 0.30/0.48 => ( ( ord_less_eq_nat @ K2 @ N )
% 0.30/0.48 => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
% 0.30/0.48 = ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Nat.diff_diff_eq
% 0.30/0.48 thf(fact_217_le__diff__iff,axiom,
% 0.30/0.48 ! [K2: nat,M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ K2 @ M )
% 0.30/0.48 => ( ( ord_less_eq_nat @ K2 @ N )
% 0.30/0.48 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
% 0.30/0.48 = ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_diff_iff
% 0.30/0.48 thf(fact_218_eq__diff__iff,axiom,
% 0.30/0.48 ! [K2: nat,M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ K2 @ M )
% 0.30/0.48 => ( ( ord_less_eq_nat @ K2 @ N )
% 0.30/0.48 => ( ( ( minus_minus_nat @ M @ K2 )
% 0.30/0.48 = ( minus_minus_nat @ N @ K2 ) )
% 0.30/0.48 = ( M = N ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % eq_diff_iff
% 0.30/0.48 thf(fact_219_zero__induct__lemma,axiom,
% 0.30/0.48 ! [P: nat > $o,K2: nat,I: nat] :
% 0.30/0.48 ( ( P @ K2 )
% 0.30/0.48 => ( ! [N2: nat] :
% 0.30/0.48 ( ( P @ ( suc @ N2 ) )
% 0.30/0.48 => ( P @ N2 ) )
% 0.30/0.48 => ( P @ ( minus_minus_nat @ K2 @ I ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % zero_induct_lemma
% 0.30/0.48 thf(fact_220_minus__nat_Odiff__0,axiom,
% 0.30/0.48 ! [M: nat] :
% 0.30/0.48 ( ( minus_minus_nat @ M @ zero_zero_nat )
% 0.30/0.48 = M ) ).
% 0.30/0.48
% 0.30/0.48 % minus_nat.diff_0
% 0.30/0.48 thf(fact_221_diffs0__imp__equal,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ( minus_minus_nat @ M @ N )
% 0.30/0.48 = zero_zero_nat )
% 0.30/0.48 => ( ( ( minus_minus_nat @ N @ M )
% 0.30/0.48 = zero_zero_nat )
% 0.30/0.48 => ( M = N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % diffs0_imp_equal
% 0.30/0.48 thf(fact_222_less__imp__diff__less,axiom,
% 0.30/0.48 ! [J3: nat,K2: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_nat @ J3 @ K2 )
% 0.30/0.48 => ( ord_less_nat @ ( minus_minus_nat @ J3 @ N ) @ K2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_imp_diff_less
% 0.30/0.48 thf(fact_223_diff__less__mono2,axiom,
% 0.30/0.48 ! [M: nat,N: nat,L2: nat] :
% 0.30/0.48 ( ( ord_less_nat @ M @ N )
% 0.30/0.48 => ( ( ord_less_nat @ M @ L2 )
% 0.30/0.48 => ( ord_less_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % diff_less_mono2
% 0.30/0.48 thf(fact_224_Module_Osc__Ring,axiom,
% 0.30/0.48 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d] :
% 0.30/0.48 ( ( module_e_b_f_d @ M3 @ R )
% 0.30/0.48 => ( ring_b_d @ R ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.sc_Ring
% 0.30/0.48 thf(fact_225_Module_Osc__Ring,axiom,
% 0.30/0.48 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d] :
% 0.30/0.48 ( ( module_a_b_c_d @ M3 @ R )
% 0.30/0.48 => ( ring_b_d @ R ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.sc_Ring
% 0.30/0.48 thf(fact_226_Module_Osc__Ring,axiom,
% 0.30/0.48 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d] :
% 0.30/0.48 ( ( module1821517916unit_d @ M3 @ R )
% 0.30/0.48 => ( ring_b_d @ R ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.sc_Ring
% 0.30/0.48 thf(fact_227_Module_Omem__smodule__ideal__coeff,axiom,
% 0.30/0.48 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,A2: set_b,X: e] :
% 0.30/0.48 ( ( module_e_b_f_d @ M3 @ R )
% 0.30/0.48 => ( ( ideal_b_d @ R @ A2 )
% 0.30/0.48 => ( ( member_e @ X @ ( smodul1982583723_d_e_f @ R @ M3 @ A2 ) )
% 0.30/0.48 => ? [N2: nat,X2: nat > b] :
% 0.30/0.48 ( ( member_nat_b @ X2
% 0.30/0.48 @ ( pi_nat_b
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
% 0.30/0.48 @ ^ [Uu: nat] : A2 ) )
% 0.30/0.48 & ? [Xa: nat > e] :
% 0.30/0.48 ( ( member_nat_e @ Xa
% 0.30/0.48 @ ( pi_nat_e
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
% 0.30/0.48 @ ^ [Uu: nat] : ( carrie730238621_e_b_f @ M3 ) ) )
% 0.30/0.48 & ( X
% 0.30/0.48 = ( l_comb_b_d_e_f @ R @ M3 @ N2 @ X2 @ Xa ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.mem_smodule_ideal_coeff
% 0.30/0.48 thf(fact_228_Module_Omem__smodule__ideal__coeff,axiom,
% 0.30/0.48 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,A2: set_b,X: a] :
% 0.30/0.48 ( ( module1821517916unit_d @ M3 @ R )
% 0.30/0.48 => ( ( ideal_b_d @ R @ A2 )
% 0.30/0.48 => ( ( member_a @ X @ ( smodul132175289t_unit @ R @ M3 @ A2 ) )
% 0.30/0.48 => ? [N2: nat,X2: nat > b] :
% 0.30/0.48 ( ( member_nat_b @ X2
% 0.30/0.48 @ ( pi_nat_b
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
% 0.30/0.48 @ ^ [Uu: nat] : A2 ) )
% 0.30/0.48 & ? [Xa: nat > a] :
% 0.30/0.48 ( ( member_nat_a @ Xa
% 0.30/0.48 @ ( pi_nat_a
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
% 0.30/0.48 @ ^ [Uu: nat] : ( carrie1074654371t_unit @ M3 ) ) )
% 0.30/0.48 & ( X
% 0.30/0.48 = ( l_comb1138968323t_unit @ R @ M3 @ N2 @ X2 @ Xa ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.mem_smodule_ideal_coeff
% 0.30/0.48 thf(fact_229_Module_Omem__smodule__ideal__coeff,axiom,
% 0.30/0.48 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,A2: set_b,X: a] :
% 0.30/0.48 ( ( module_a_b_c_d @ M3 @ R )
% 0.30/0.48 => ( ( ideal_b_d @ R @ A2 )
% 0.30/0.48 => ( ( member_a @ X @ ( smodul818989740_d_a_c @ R @ M3 @ A2 ) )
% 0.30/0.48 => ? [N2: nat,X2: nat > b] :
% 0.30/0.48 ( ( member_nat_b @ X2
% 0.30/0.48 @ ( pi_nat_b
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
% 0.30/0.48 @ ^ [Uu: nat] : A2 ) )
% 0.30/0.48 & ? [Xa: nat > a] :
% 0.30/0.48 ( ( member_nat_a @ Xa
% 0.30/0.48 @ ( pi_nat_a
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
% 0.30/0.48 @ ^ [Uu: nat] : ( carrie2021454486_a_b_c @ M3 ) ) )
% 0.30/0.48 & ( X
% 0.30/0.48 = ( l_comb_b_d_a_c @ R @ M3 @ N2 @ X2 @ Xa ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.mem_smodule_ideal_coeff
% 0.30/0.48 thf(fact_230_lift__Suc__antimono__le,axiom,
% 0.30/0.48 ! [F: nat > nat,N: nat,N5: nat] :
% 0.30/0.48 ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.30/0.48 => ( ( ord_less_eq_nat @ N @ N5 )
% 0.30/0.48 => ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lift_Suc_antimono_le
% 0.30/0.48 thf(fact_231_lift__Suc__antimono__le,axiom,
% 0.30/0.48 ! [F: nat > set_a,N: nat,N5: nat] :
% 0.30/0.48 ( ! [N2: nat] : ( ord_less_eq_set_a @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.30/0.48 => ( ( ord_less_eq_nat @ N @ N5 )
% 0.30/0.48 => ( ord_less_eq_set_a @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lift_Suc_antimono_le
% 0.30/0.48 thf(fact_232_lift__Suc__mono__le,axiom,
% 0.30/0.48 ! [F: nat > nat,N: nat,N5: nat] :
% 0.30/0.48 ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.30/0.48 => ( ( ord_less_eq_nat @ N @ N5 )
% 0.30/0.48 => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lift_Suc_mono_le
% 0.30/0.48 thf(fact_233_lift__Suc__mono__le,axiom,
% 0.30/0.48 ! [F: nat > set_a,N: nat,N5: nat] :
% 0.30/0.48 ( ! [N2: nat] : ( ord_less_eq_set_a @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.30/0.48 => ( ( ord_less_eq_nat @ N @ N5 )
% 0.30/0.48 => ( ord_less_eq_set_a @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lift_Suc_mono_le
% 0.30/0.48 thf(fact_234_lift__Suc__mono__less__iff,axiom,
% 0.30/0.48 ! [F: nat > nat,N: nat,M: nat] :
% 0.30/0.48 ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.30/0.48 => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
% 0.30/0.48 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lift_Suc_mono_less_iff
% 0.30/0.48 thf(fact_235_lift__Suc__mono__less,axiom,
% 0.30/0.48 ! [F: nat > nat,N: nat,N5: nat] :
% 0.30/0.48 ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.30/0.48 => ( ( ord_less_nat @ N @ N5 )
% 0.30/0.48 => ( ord_less_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lift_Suc_mono_less
% 0.30/0.48 thf(fact_236_le__imp__less__Suc,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.48 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_imp_less_Suc
% 0.30/0.48 thf(fact_237_less__eq__Suc__le,axiom,
% 0.30/0.48 ( ord_less_nat
% 0.30/0.48 = ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_eq_Suc_le
% 0.30/0.48 thf(fact_238_less__Suc__eq__le,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.30/0.48 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_Suc_eq_le
% 0.30/0.48 thf(fact_239_le__less__Suc__eq,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.48 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.30/0.48 = ( N = M ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_less_Suc_eq
% 0.30/0.48 thf(fact_240_Suc__le__lessD,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.30/0.48 => ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % Suc_le_lessD
% 0.30/0.48 thf(fact_241_inc__induct,axiom,
% 0.30/0.48 ! [I: nat,J3: nat,P: nat > $o] :
% 0.30/0.48 ( ( ord_less_eq_nat @ I @ J3 )
% 0.30/0.48 => ( ( P @ J3 )
% 0.30/0.48 => ( ! [N2: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ I @ N2 )
% 0.30/0.48 => ( ( ord_less_nat @ N2 @ J3 )
% 0.30/0.48 => ( ( P @ ( suc @ N2 ) )
% 0.30/0.48 => ( P @ N2 ) ) ) )
% 0.30/0.48 => ( P @ I ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % inc_induct
% 0.30/0.48 thf(fact_242_dec__induct,axiom,
% 0.30/0.48 ! [I: nat,J3: nat,P: nat > $o] :
% 0.30/0.48 ( ( ord_less_eq_nat @ I @ J3 )
% 0.30/0.48 => ( ( P @ I )
% 0.30/0.48 => ( ! [N2: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ I @ N2 )
% 0.30/0.48 => ( ( ord_less_nat @ N2 @ J3 )
% 0.30/0.48 => ( ( P @ N2 )
% 0.30/0.48 => ( P @ ( suc @ N2 ) ) ) ) )
% 0.30/0.48 => ( P @ J3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dec_induct
% 0.30/0.48 thf(fact_243_Suc__le__eq,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.30/0.48 = ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % Suc_le_eq
% 0.30/0.48 thf(fact_244_Suc__leI,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_nat @ M @ N )
% 0.30/0.48 => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % Suc_leI
% 0.30/0.48 thf(fact_245_ex__least__nat__le,axiom,
% 0.30/0.48 ! [P: nat > $o,N: nat] :
% 0.30/0.48 ( ( P @ N )
% 0.30/0.48 => ( ~ ( P @ zero_zero_nat )
% 0.30/0.48 => ? [K3: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ K3 @ N )
% 0.30/0.48 & ! [I4: nat] :
% 0.30/0.48 ( ( ord_less_nat @ I4 @ K3 )
% 0.30/0.48 => ~ ( P @ I4 ) )
% 0.30/0.48 & ( P @ K3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % ex_least_nat_le
% 0.30/0.48 thf(fact_246_less__Suc__eq__0__disj,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.30/0.48 = ( ( M = zero_zero_nat )
% 0.30/0.48 | ? [J2: nat] :
% 0.30/0.48 ( ( M
% 0.30/0.48 = ( suc @ J2 ) )
% 0.30/0.48 & ( ord_less_nat @ J2 @ N ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_Suc_eq_0_disj
% 0.30/0.48 thf(fact_247_gr0__implies__Suc,axiom,
% 0.30/0.48 ! [N: nat] :
% 0.30/0.48 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.48 => ? [M5: nat] :
% 0.30/0.48 ( N
% 0.30/0.48 = ( suc @ M5 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % gr0_implies_Suc
% 0.30/0.48 thf(fact_248_All__less__Suc2,axiom,
% 0.30/0.48 ! [N: nat,P: nat > $o] :
% 0.30/0.48 ( ( ! [I3: nat] :
% 0.30/0.48 ( ( ord_less_nat @ I3 @ ( suc @ N ) )
% 0.30/0.48 => ( P @ I3 ) ) )
% 0.30/0.48 = ( ( P @ zero_zero_nat )
% 0.30/0.48 & ! [I3: nat] :
% 0.30/0.48 ( ( ord_less_nat @ I3 @ N )
% 0.30/0.48 => ( P @ ( suc @ I3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % All_less_Suc2
% 0.30/0.48 thf(fact_249_gr0__conv__Suc,axiom,
% 0.30/0.48 ! [N: nat] :
% 0.30/0.48 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.48 = ( ? [M6: nat] :
% 0.30/0.48 ( N
% 0.30/0.48 = ( suc @ M6 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % gr0_conv_Suc
% 0.30/0.48 thf(fact_250_Ex__less__Suc2,axiom,
% 0.30/0.48 ! [N: nat,P: nat > $o] :
% 0.30/0.48 ( ( ? [I3: nat] :
% 0.30/0.48 ( ( ord_less_nat @ I3 @ ( suc @ N ) )
% 0.30/0.48 & ( P @ I3 ) ) )
% 0.30/0.48 = ( ( P @ zero_zero_nat )
% 0.30/0.48 | ? [I3: nat] :
% 0.30/0.48 ( ( ord_less_nat @ I3 @ N )
% 0.30/0.48 & ( P @ ( suc @ I3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Ex_less_Suc2
% 0.30/0.48 thf(fact_251_Suc__diff__le,axiom,
% 0.30/0.48 ! [N: nat,M: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.48 => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 0.30/0.48 = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Suc_diff_le
% 0.30/0.48 thf(fact_252_diff__less__mono,axiom,
% 0.30/0.48 ! [A: nat,B: nat,C: nat] :
% 0.30/0.48 ( ( ord_less_nat @ A @ B )
% 0.30/0.48 => ( ( ord_less_eq_nat @ C @ A )
% 0.30/0.48 => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % diff_less_mono
% 0.30/0.48 thf(fact_253_less__diff__iff,axiom,
% 0.30/0.48 ! [K2: nat,M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ K2 @ M )
% 0.30/0.48 => ( ( ord_less_eq_nat @ K2 @ N )
% 0.30/0.48 => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
% 0.30/0.48 = ( ord_less_nat @ M @ N ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_diff_iff
% 0.30/0.48 thf(fact_254_diff__less__Suc,axiom,
% 0.30/0.48 ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% 0.30/0.48
% 0.30/0.48 % diff_less_Suc
% 0.30/0.48 thf(fact_255_Suc__diff__Suc,axiom,
% 0.30/0.48 ! [N: nat,M: nat] :
% 0.30/0.48 ( ( ord_less_nat @ N @ M )
% 0.30/0.48 => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
% 0.30/0.48 = ( minus_minus_nat @ M @ N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Suc_diff_Suc
% 0.30/0.48 thf(fact_256_diff__less,axiom,
% 0.30/0.48 ! [N: nat,M: nat] :
% 0.30/0.48 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.48 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.48 => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % diff_less
% 0.30/0.48 thf(fact_257_Module_Osubmodule__subset1,axiom,
% 0.30/0.48 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,H: set_e,H2: e] :
% 0.30/0.48 ( ( module_e_b_f_d @ M3 @ R )
% 0.30/0.48 => ( ( submodule_b_d_e_f @ R @ M3 @ H )
% 0.30/0.48 => ( ( member_e @ H2 @ H )
% 0.30/0.48 => ( member_e @ H2 @ ( carrie730238621_e_b_f @ M3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.submodule_subset1
% 0.30/0.48 thf(fact_258_Module_Osubmodule__subset1,axiom,
% 0.30/0.48 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,H: set_a,H2: a] :
% 0.30/0.48 ( ( module_a_b_c_d @ M3 @ R )
% 0.30/0.48 => ( ( submodule_b_d_a_c @ R @ M3 @ H )
% 0.30/0.48 => ( ( member_a @ H2 @ H )
% 0.30/0.48 => ( member_a @ H2 @ ( carrie2021454486_a_b_c @ M3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.submodule_subset1
% 0.30/0.48 thf(fact_259_Module_Osubmodule__subset1,axiom,
% 0.30/0.48 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,H: set_a,H2: a] :
% 0.30/0.48 ( ( module1821517916unit_d @ M3 @ R )
% 0.30/0.48 => ( ( submod903911234t_unit @ R @ M3 @ H )
% 0.30/0.48 => ( ( member_a @ H2 @ H )
% 0.30/0.48 => ( member_a @ H2 @ ( carrie1074654371t_unit @ M3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.submodule_subset1
% 0.30/0.48 thf(fact_260_Module_Osubmodule__whole,axiom,
% 0.30/0.48 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d] :
% 0.30/0.48 ( ( module_e_b_f_d @ M3 @ R )
% 0.30/0.48 => ( submodule_b_d_e_f @ R @ M3 @ ( carrie730238621_e_b_f @ M3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.submodule_whole
% 0.30/0.48 thf(fact_261_Module_Osubmodule__whole,axiom,
% 0.30/0.48 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d] :
% 0.30/0.48 ( ( module_a_b_c_d @ M3 @ R )
% 0.30/0.48 => ( submodule_b_d_a_c @ R @ M3 @ ( carrie2021454486_a_b_c @ M3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.submodule_whole
% 0.30/0.48 thf(fact_262_Module_Osubmodule__whole,axiom,
% 0.30/0.48 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d] :
% 0.30/0.48 ( ( module1821517916unit_d @ M3 @ R )
% 0.30/0.48 => ( submod903911234t_unit @ R @ M3 @ ( carrie1074654371t_unit @ M3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.submodule_whole
% 0.30/0.48 thf(fact_263_ex__least__nat__less,axiom,
% 0.30/0.48 ! [P: nat > $o,N: nat] :
% 0.30/0.48 ( ( P @ N )
% 0.30/0.48 => ( ~ ( P @ zero_zero_nat )
% 0.30/0.48 => ? [K3: nat] :
% 0.30/0.48 ( ( ord_less_nat @ K3 @ N )
% 0.30/0.48 & ! [I4: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ I4 @ K3 )
% 0.30/0.48 => ~ ( P @ I4 ) )
% 0.30/0.48 & ( P @ ( suc @ K3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % ex_least_nat_less
% 0.30/0.48 thf(fact_264_diff__Suc__less,axiom,
% 0.30/0.48 ! [N: nat,I: nat] :
% 0.30/0.48 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.48 => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % diff_Suc_less
% 0.30/0.48 thf(fact_265_Module_Osubmodule__subset,axiom,
% 0.30/0.48 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,H: set_e] :
% 0.30/0.48 ( ( module_e_b_f_d @ M3 @ R )
% 0.30/0.48 => ( ( submodule_b_d_e_f @ R @ M3 @ H )
% 0.30/0.48 => ( ord_less_eq_set_e @ H @ ( carrie730238621_e_b_f @ M3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.submodule_subset
% 0.30/0.48 thf(fact_266_Module_Osubmodule__subset,axiom,
% 0.30/0.48 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,H: set_a] :
% 0.30/0.48 ( ( module_a_b_c_d @ M3 @ R )
% 0.30/0.48 => ( ( submodule_b_d_a_c @ R @ M3 @ H )
% 0.30/0.48 => ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ M3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.submodule_subset
% 0.30/0.48 thf(fact_267_Module_Osubmodule__subset,axiom,
% 0.30/0.48 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,H: set_a] :
% 0.30/0.48 ( ( module1821517916unit_d @ M3 @ R )
% 0.30/0.48 => ( ( submod903911234t_unit @ R @ M3 @ H )
% 0.30/0.48 => ( ord_less_eq_set_a @ H @ ( carrie1074654371t_unit @ M3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.submodule_subset
% 0.30/0.48 thf(fact_268_Module_Ofree__generator__sub,axiom,
% 0.30/0.48 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,H: set_e] :
% 0.30/0.48 ( ( module_e_b_f_d @ M3 @ R )
% 0.30/0.48 => ( ( free_g103796815_d_e_f @ R @ M3 @ H )
% 0.30/0.48 => ( ord_less_eq_set_e @ H @ ( carrie730238621_e_b_f @ M3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.free_generator_sub
% 0.30/0.48 thf(fact_269_Module_Ofree__generator__sub,axiom,
% 0.30/0.48 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,H: set_a] :
% 0.30/0.48 ( ( module_a_b_c_d @ M3 @ R )
% 0.30/0.48 => ( ( free_g1087686480_d_a_c @ R @ M3 @ H )
% 0.30/0.48 => ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ M3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.free_generator_sub
% 0.30/0.48 thf(fact_270_Module_Ofree__generator__sub,axiom,
% 0.30/0.48 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,H: set_a] :
% 0.30/0.48 ( ( module1821517916unit_d @ M3 @ R )
% 0.30/0.48 => ( ( free_g637607517t_unit @ R @ M3 @ H )
% 0.30/0.48 => ( ord_less_eq_set_a @ H @ ( carrie1074654371t_unit @ M3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Module.free_generator_sub
% 0.30/0.48 thf(fact_271_eSum__in__SubmoduleTr,axiom,
% 0.30/0.48 ! [H: set_a,K: set_a,F: nat > a,N: nat,S: nat > b] :
% 0.30/0.48 ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ K @ H )
% 0.30/0.48 => ( ( ( member_nat_a @ F
% 0.30/0.48 @ ( pi_nat_a
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : K ) )
% 0.30/0.48 & ( member_nat_b @ S
% 0.30/0.48 @ ( pi_nat_b
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ r ) ) ) )
% 0.30/0.48 => ( ( l_comb1138968323t_unit @ r @ ( mdl_a_b_c @ m @ ( algebr1152250919_d_a_c @ r @ m @ K ) ) @ N @ S @ F )
% 0.30/0.48 = ( l_comb_b_d_a_c @ r @ m @ N @ S @ F ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % eSum_in_SubmoduleTr
% 0.30/0.48 thf(fact_272_eSum__in__Submodule,axiom,
% 0.30/0.48 ! [H: set_a,K: set_a,F: nat > a,N: nat,S: nat > b] :
% 0.30/0.48 ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ K @ H )
% 0.30/0.48 => ( ( member_nat_a @ F
% 0.30/0.48 @ ( pi_nat_a
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : K ) )
% 0.30/0.48 => ( ( member_nat_b @ S
% 0.30/0.48 @ ( pi_nat_b
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ r ) ) )
% 0.30/0.48 => ( ( l_comb1138968323t_unit @ r @ ( mdl_a_b_c @ m @ ( algebr1152250919_d_a_c @ r @ m @ K ) ) @ N @ S @ F )
% 0.30/0.48 = ( l_comb_b_d_a_c @ r @ m @ N @ S @ F ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % eSum_in_Submodule
% 0.30/0.48 thf(fact_273_singleton__insert__inj__eq_H,axiom,
% 0.30/0.48 ! [A: a,A2: set_a,B: a] :
% 0.30/0.48 ( ( ( insert_a @ A @ A2 )
% 0.30/0.48 = ( insert_a @ B @ bot_bot_set_a ) )
% 0.30/0.48 = ( ( A = B )
% 0.30/0.48 & ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % singleton_insert_inj_eq'
% 0.30/0.48 thf(fact_274_singleton__insert__inj__eq,axiom,
% 0.30/0.48 ! [B: a,A: a,A2: set_a] :
% 0.30/0.48 ( ( ( insert_a @ B @ bot_bot_set_a )
% 0.30/0.48 = ( insert_a @ A @ A2 ) )
% 0.30/0.48 = ( ( A = B )
% 0.30/0.48 & ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % singleton_insert_inj_eq
% 0.30/0.48 thf(fact_275_l__span__closed2Tr,axiom,
% 0.30/0.48 ! [A2: set_b,H: set_a,S: nat > b,N: nat,F: nat > a] :
% 0.30/0.48 ( ( ideal_b_d @ r @ A2 )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 0.30/0.48 => ( ( ( member_nat_b @ S
% 0.30/0.48 @ ( pi_nat_b
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : A2 ) )
% 0.30/0.48 & ( member_nat_a @ F
% 0.30/0.48 @ ( pi_nat_a
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ H ) ) ) )
% 0.30/0.48 => ( member_a @ ( l_comb_b_d_a_c @ r @ m @ N @ S @ F ) @ ( linear_span_b_d_a_c @ r @ m @ A2 @ H ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % l_span_closed2Tr
% 0.30/0.48 thf(fact_276_l__span__closed2,axiom,
% 0.30/0.48 ! [A2: set_b,H: set_a,S: nat > b,N: nat,F: nat > a] :
% 0.30/0.48 ( ( ideal_b_d @ r @ A2 )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 0.30/0.48 => ( ( member_nat_b @ S
% 0.30/0.48 @ ( pi_nat_b
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : A2 ) )
% 0.30/0.48 => ( ( member_nat_a @ F
% 0.30/0.48 @ ( pi_nat_a
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ H ) ) )
% 0.30/0.48 => ( member_a @ ( l_comb_b_d_a_c @ r @ m @ N @ S @ F ) @ ( linear_span_b_d_a_c @ r @ m @ A2 @ H ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % l_span_closed2
% 0.30/0.48 thf(fact_277_linear__comb0__2Tr,axiom,
% 0.30/0.48 ! [A2: set_b,S: nat > b,N: nat,M: nat > a] :
% 0.30/0.48 ( ( ideal_b_d @ r @ A2 )
% 0.30/0.48 => ( ( ( member_nat_b @ S
% 0.30/0.48 @ ( pi_nat_b
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : A2 ) )
% 0.30/0.48 & ( member_nat_a @ M
% 0.30/0.48 @ ( pi_nat_a
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : ( insert_a @ ( zero_a1261444626_a_b_c @ m ) @ bot_bot_set_a ) ) ) )
% 0.30/0.48 => ( ( l_comb_b_d_a_c @ r @ m @ N @ S @ M )
% 0.30/0.48 = ( zero_a1261444626_a_b_c @ m ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % linear_comb0_2Tr
% 0.30/0.48 thf(fact_278_module__inc__zero,axiom,
% 0.30/0.48 member_a @ ( zero_a1261444626_a_b_c @ m ) @ ( carrie2021454486_a_b_c @ m ) ).
% 0.30/0.48
% 0.30/0.48 % module_inc_zero
% 0.30/0.48 thf(fact_279_psubsetI,axiom,
% 0.30/0.48 ! [A2: set_a,B2: set_a] :
% 0.30/0.48 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 0.30/0.48 => ( ( A2 != B2 )
% 0.30/0.48 => ( ord_less_set_a @ A2 @ B2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % psubsetI
% 0.30/0.48 thf(fact_280_subsetI,axiom,
% 0.30/0.48 ! [A2: set_nat_a,B2: set_nat_a] :
% 0.30/0.48 ( ! [X2: nat > a] :
% 0.30/0.48 ( ( member_nat_a @ X2 @ A2 )
% 0.30/0.48 => ( member_nat_a @ X2 @ B2 ) )
% 0.30/0.48 => ( ord_le157368549_nat_a @ A2 @ B2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % subsetI
% 0.30/0.48 thf(fact_281_subsetI,axiom,
% 0.30/0.48 ! [A2: set_nat_b,B2: set_nat_b] :
% 0.30/0.48 ( ! [X2: nat > b] :
% 0.30/0.48 ( ( member_nat_b @ X2 @ A2 )
% 0.30/0.48 => ( member_nat_b @ X2 @ B2 ) )
% 0.30/0.48 => ( ord_le1296932838_nat_b @ A2 @ B2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % subsetI
% 0.30/0.48 thf(fact_282_subsetI,axiom,
% 0.30/0.48 ! [A2: set_a,B2: set_a] :
% 0.30/0.48 ( ! [X2: a] :
% 0.30/0.48 ( ( member_a @ X2 @ A2 )
% 0.30/0.48 => ( member_a @ X2 @ B2 ) )
% 0.30/0.48 => ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % subsetI
% 0.30/0.48 thf(fact_283_subset__antisym,axiom,
% 0.30/0.48 ! [A2: set_a,B2: set_a] :
% 0.30/0.48 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ B2 @ A2 )
% 0.30/0.48 => ( A2 = B2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % subset_antisym
% 0.30/0.48 thf(fact_284_Diff__empty,axiom,
% 0.30/0.48 ! [A2: set_a] :
% 0.30/0.48 ( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
% 0.30/0.48 = A2 ) ).
% 0.30/0.48
% 0.30/0.48 % Diff_empty
% 0.30/0.48 thf(fact_285_empty__Diff,axiom,
% 0.30/0.48 ! [A2: set_a] :
% 0.30/0.48 ( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
% 0.30/0.48 = bot_bot_set_a ) ).
% 0.30/0.48
% 0.30/0.48 % empty_Diff
% 0.30/0.48 thf(fact_286_Diff__cancel,axiom,
% 0.30/0.48 ! [A2: set_a] :
% 0.30/0.48 ( ( minus_minus_set_a @ A2 @ A2 )
% 0.30/0.48 = bot_bot_set_a ) ).
% 0.30/0.48
% 0.30/0.48 % Diff_cancel
% 0.30/0.48 thf(fact_287_empty__iff,axiom,
% 0.30/0.48 ! [C: nat > a] :
% 0.30/0.48 ~ ( member_nat_a @ C @ bot_bot_set_nat_a ) ).
% 0.30/0.48
% 0.30/0.48 % empty_iff
% 0.30/0.48 thf(fact_288_empty__iff,axiom,
% 0.30/0.48 ! [C: nat > b] :
% 0.30/0.48 ~ ( member_nat_b @ C @ bot_bot_set_nat_b ) ).
% 0.30/0.48
% 0.30/0.48 % empty_iff
% 0.30/0.48 thf(fact_289_empty__iff,axiom,
% 0.30/0.48 ! [C: a] :
% 0.30/0.48 ~ ( member_a @ C @ bot_bot_set_a ) ).
% 0.30/0.48
% 0.30/0.48 % empty_iff
% 0.30/0.48 thf(fact_290_all__not__in__conv,axiom,
% 0.30/0.48 ! [A2: set_nat_a] :
% 0.30/0.48 ( ( ! [X3: nat > a] :
% 0.30/0.48 ~ ( member_nat_a @ X3 @ A2 ) )
% 0.30/0.48 = ( A2 = bot_bot_set_nat_a ) ) ).
% 0.30/0.48
% 0.30/0.48 % all_not_in_conv
% 0.30/0.48 thf(fact_291_all__not__in__conv,axiom,
% 0.30/0.48 ! [A2: set_nat_b] :
% 0.30/0.48 ( ( ! [X3: nat > b] :
% 0.30/0.48 ~ ( member_nat_b @ X3 @ A2 ) )
% 0.30/0.48 = ( A2 = bot_bot_set_nat_b ) ) ).
% 0.30/0.48
% 0.30/0.48 % all_not_in_conv
% 0.30/0.48 thf(fact_292_all__not__in__conv,axiom,
% 0.30/0.48 ! [A2: set_a] :
% 0.30/0.48 ( ( ! [X3: a] :
% 0.30/0.48 ~ ( member_a @ X3 @ A2 ) )
% 0.30/0.48 = ( A2 = bot_bot_set_a ) ) ).
% 0.30/0.48
% 0.30/0.48 % all_not_in_conv
% 0.30/0.48 thf(fact_293_Collect__empty__eq,axiom,
% 0.30/0.48 ! [P: nat > $o] :
% 0.30/0.48 ( ( ( collect_nat @ P )
% 0.30/0.48 = bot_bot_set_nat )
% 0.30/0.48 = ( ! [X3: nat] :
% 0.30/0.48 ~ ( P @ X3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Collect_empty_eq
% 0.30/0.48 thf(fact_294_Collect__empty__eq,axiom,
% 0.30/0.48 ! [P: a > $o] :
% 0.30/0.48 ( ( ( collect_a @ P )
% 0.30/0.48 = bot_bot_set_a )
% 0.30/0.48 = ( ! [X3: a] :
% 0.30/0.48 ~ ( P @ X3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Collect_empty_eq
% 0.30/0.48 thf(fact_295_empty__Collect__eq,axiom,
% 0.30/0.48 ! [P: nat > $o] :
% 0.30/0.48 ( ( bot_bot_set_nat
% 0.30/0.48 = ( collect_nat @ P ) )
% 0.30/0.48 = ( ! [X3: nat] :
% 0.30/0.48 ~ ( P @ X3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % empty_Collect_eq
% 0.30/0.48 thf(fact_296_empty__Collect__eq,axiom,
% 0.30/0.48 ! [P: a > $o] :
% 0.30/0.48 ( ( bot_bot_set_a
% 0.30/0.48 = ( collect_a @ P ) )
% 0.30/0.48 = ( ! [X3: a] :
% 0.30/0.48 ~ ( P @ X3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % empty_Collect_eq
% 0.30/0.48 thf(fact_297_Diff__insert0,axiom,
% 0.30/0.48 ! [X: nat > a,A2: set_nat_a,B2: set_nat_a] :
% 0.30/0.48 ( ~ ( member_nat_a @ X @ A2 )
% 0.30/0.48 => ( ( minus_1788767276_nat_a @ A2 @ ( insert_nat_a @ X @ B2 ) )
% 0.30/0.48 = ( minus_1788767276_nat_a @ A2 @ B2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Diff_insert0
% 0.30/0.48 thf(fact_298_Diff__insert0,axiom,
% 0.30/0.48 ! [X: nat > b,A2: set_nat_b,B2: set_nat_b] :
% 0.30/0.48 ( ~ ( member_nat_b @ X @ A2 )
% 0.30/0.48 => ( ( minus_780847917_nat_b @ A2 @ ( insert_nat_b @ X @ B2 ) )
% 0.30/0.48 = ( minus_780847917_nat_b @ A2 @ B2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Diff_insert0
% 0.30/0.48 thf(fact_299_Diff__insert0,axiom,
% 0.30/0.48 ! [X: a,A2: set_a,B2: set_a] :
% 0.30/0.48 ( ~ ( member_a @ X @ A2 )
% 0.30/0.48 => ( ( minus_minus_set_a @ A2 @ ( insert_a @ X @ B2 ) )
% 0.30/0.48 = ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Diff_insert0
% 0.30/0.48 thf(fact_300_insert__Diff1,axiom,
% 0.30/0.48 ! [X: nat > a,B2: set_nat_a,A2: set_nat_a] :
% 0.30/0.48 ( ( member_nat_a @ X @ B2 )
% 0.30/0.48 => ( ( minus_1788767276_nat_a @ ( insert_nat_a @ X @ A2 ) @ B2 )
% 0.30/0.48 = ( minus_1788767276_nat_a @ A2 @ B2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % insert_Diff1
% 0.30/0.48 thf(fact_301_insert__Diff1,axiom,
% 0.30/0.48 ! [X: nat > b,B2: set_nat_b,A2: set_nat_b] :
% 0.30/0.48 ( ( member_nat_b @ X @ B2 )
% 0.30/0.48 => ( ( minus_780847917_nat_b @ ( insert_nat_b @ X @ A2 ) @ B2 )
% 0.30/0.48 = ( minus_780847917_nat_b @ A2 @ B2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % insert_Diff1
% 0.30/0.48 thf(fact_302_insert__Diff1,axiom,
% 0.30/0.48 ! [X: a,B2: set_a,A2: set_a] :
% 0.30/0.48 ( ( member_a @ X @ B2 )
% 0.30/0.48 => ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B2 )
% 0.30/0.48 = ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % insert_Diff1
% 0.30/0.48 thf(fact_303_insertCI,axiom,
% 0.30/0.48 ! [A: nat > a,B2: set_nat_a,B: nat > a] :
% 0.30/0.48 ( ( ~ ( member_nat_a @ A @ B2 )
% 0.30/0.48 => ( A = B ) )
% 0.30/0.48 => ( member_nat_a @ A @ ( insert_nat_a @ B @ B2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % insertCI
% 0.30/0.48 thf(fact_304_insertCI,axiom,
% 0.30/0.48 ! [A: nat > b,B2: set_nat_b,B: nat > b] :
% 0.30/0.48 ( ( ~ ( member_nat_b @ A @ B2 )
% 0.30/0.48 => ( A = B ) )
% 0.30/0.48 => ( member_nat_b @ A @ ( insert_nat_b @ B @ B2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % insertCI
% 0.30/0.48 thf(fact_305_insertCI,axiom,
% 0.30/0.48 ! [A: a,B2: set_a,B: a] :
% 0.30/0.48 ( ( ~ ( member_a @ A @ B2 )
% 0.30/0.48 => ( A = B ) )
% 0.30/0.48 => ( member_a @ A @ ( insert_a @ B @ B2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % insertCI
% 0.30/0.48 thf(fact_306_insert__iff,axiom,
% 0.30/0.48 ! [A: nat > a,B: nat > a,A2: set_nat_a] :
% 0.30/0.48 ( ( member_nat_a @ A @ ( insert_nat_a @ B @ A2 ) )
% 0.30/0.48 = ( ( A = B )
% 0.30/0.48 | ( member_nat_a @ A @ A2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % insert_iff
% 0.30/0.48 thf(fact_307_insert__iff,axiom,
% 0.30/0.48 ! [A: nat > b,B: nat > b,A2: set_nat_b] :
% 0.30/0.48 ( ( member_nat_b @ A @ ( insert_nat_b @ B @ A2 ) )
% 0.30/0.48 = ( ( A = B )
% 0.30/0.48 | ( member_nat_b @ A @ A2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % insert_iff
% 0.30/0.48 thf(fact_308_insert__iff,axiom,
% 0.30/0.48 ! [A: a,B: a,A2: set_a] :
% 0.30/0.48 ( ( member_a @ A @ ( insert_a @ B @ A2 ) )
% 0.30/0.48 = ( ( A = B )
% 0.30/0.48 | ( member_a @ A @ A2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % insert_iff
% 0.30/0.48 thf(fact_309_insert__absorb2,axiom,
% 0.30/0.48 ! [X: a,A2: set_a] :
% 0.30/0.48 ( ( insert_a @ X @ ( insert_a @ X @ A2 ) )
% 0.30/0.48 = ( insert_a @ X @ A2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % insert_absorb2
% 0.30/0.48 thf(fact_310_submodule__inc__0,axiom,
% 0.30/0.48 ! [H: set_a] :
% 0.30/0.48 ( ( submodule_b_d_a_c @ r @ m @ H )
% 0.30/0.48 => ( member_a @ ( zero_a1261444626_a_b_c @ m ) @ H ) ) ).
% 0.30/0.48
% 0.30/0.48 % submodule_inc_0
% 0.30/0.48 thf(fact_311_mdl__carrier,axiom,
% 0.30/0.48 ! [H: set_a] :
% 0.30/0.48 ( ( submodule_b_d_a_c @ r @ m @ H )
% 0.30/0.48 => ( ( carrie1074654371t_unit @ ( mdl_a_b_c @ m @ H ) )
% 0.30/0.48 = H ) ) ).
% 0.30/0.48
% 0.30/0.48 % mdl_carrier
% 0.30/0.48 thf(fact_312_mdl__is__module,axiom,
% 0.30/0.48 ! [H: set_a] :
% 0.30/0.48 ( ( submodule_b_d_a_c @ r @ m @ H )
% 0.30/0.48 => ( module1821517916unit_d @ ( mdl_a_b_c @ m @ H ) @ r ) ) ).
% 0.30/0.48
% 0.30/0.48 % mdl_is_module
% 0.30/0.48 thf(fact_313_submodule__of__mdl,axiom,
% 0.30/0.48 ! [H: set_a,N3: set_a] :
% 0.30/0.48 ( ( submodule_b_d_a_c @ r @ m @ H )
% 0.30/0.48 => ( ( submodule_b_d_a_c @ r @ m @ N3 )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ H @ N3 )
% 0.30/0.48 => ( submod903911234t_unit @ r @ ( mdl_a_b_c @ m @ N3 ) @ H ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % submodule_of_mdl
% 0.30/0.48 thf(fact_314_h__in__linear__span,axiom,
% 0.30/0.48 ! [H: set_a,H2: a] :
% 0.30/0.48 ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 0.30/0.48 => ( ( member_a @ H2 @ H )
% 0.30/0.48 => ( member_a @ H2 @ ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ H ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % h_in_linear_span
% 0.30/0.48 thf(fact_315_l__span__cont__H,axiom,
% 0.30/0.48 ! [H: set_a] :
% 0.30/0.48 ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 0.30/0.48 => ( ord_less_eq_set_a @ H @ ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ H ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % l_span_cont_H
% 0.30/0.48 thf(fact_316_l__span__l__span,axiom,
% 0.30/0.48 ! [H: set_a] :
% 0.30/0.48 ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 0.30/0.48 => ( ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ H ) )
% 0.30/0.48 = ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ H ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % l_span_l_span
% 0.30/0.48 thf(fact_317_l__span__gen__mono,axiom,
% 0.30/0.48 ! [K: set_a,H: set_a,A2: set_b] :
% 0.30/0.48 ( ( ord_less_eq_set_a @ K @ H )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 0.30/0.48 => ( ( ideal_b_d @ r @ A2 )
% 0.30/0.48 => ( ord_less_eq_set_a @ ( linear_span_b_d_a_c @ r @ m @ A2 @ K ) @ ( linear_span_b_d_a_c @ r @ m @ A2 @ H ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % l_span_gen_mono
% 0.30/0.48 thf(fact_318_linear__span__sub,axiom,
% 0.30/0.48 ! [A2: set_b,H: set_a] :
% 0.30/0.48 ( ( ideal_b_d @ r @ A2 )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 0.30/0.48 => ( ord_less_eq_set_a @ ( linear_span_b_d_a_c @ r @ m @ A2 @ H ) @ ( carrie2021454486_a_b_c @ m ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % linear_span_sub
% 0.30/0.48 thf(fact_319_l__span__sub__submodule,axiom,
% 0.30/0.48 ! [A2: set_b,N3: set_a,H: set_a] :
% 0.30/0.48 ( ( ideal_b_d @ r @ A2 )
% 0.30/0.48 => ( ( submodule_b_d_a_c @ r @ m @ N3 )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ H @ N3 )
% 0.30/0.48 => ( ord_less_eq_set_a @ ( linear_span_b_d_a_c @ r @ m @ A2 @ H ) @ N3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % l_span_sub_submodule
% 0.30/0.48 thf(fact_320_fgs__generator,axiom,
% 0.30/0.48 ! [H: set_a] :
% 0.30/0.48 ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 0.30/0.48 => ( genera1692266857t_unit @ r @ ( mdl_a_b_c @ m @ ( algebr1152250919_d_a_c @ r @ m @ H ) ) @ H ) ) ).
% 0.30/0.48
% 0.30/0.48 % fgs_generator
% 0.30/0.48 thf(fact_321_submodule__0,axiom,
% 0.30/0.48 submodule_b_d_a_c @ r @ m @ ( insert_a @ ( zero_a1261444626_a_b_c @ m ) @ bot_bot_set_a ) ).
% 0.30/0.48
% 0.30/0.48 % submodule_0
% 0.30/0.48 thf(fact_322_empty__fgs,axiom,
% 0.30/0.48 ( ( algebr1152250919_d_a_c @ r @ m @ bot_bot_set_a )
% 0.30/0.48 = ( insert_a @ ( zero_a1261444626_a_b_c @ m ) @ bot_bot_set_a ) ) ).
% 0.30/0.48
% 0.30/0.48 % empty_fgs
% 0.30/0.48 thf(fact_323_le__zero__eq,axiom,
% 0.30/0.48 ! [N: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.30/0.48 = ( N = zero_zero_nat ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_zero_eq
% 0.30/0.48 thf(fact_324_not__gr__zero,axiom,
% 0.30/0.48 ! [N: nat] :
% 0.30/0.48 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.30/0.48 = ( N = zero_zero_nat ) ) ).
% 0.30/0.48
% 0.30/0.48 % not_gr_zero
% 0.30/0.48 thf(fact_325_zero__diff,axiom,
% 0.30/0.48 ! [A: nat] :
% 0.30/0.48 ( ( minus_minus_nat @ zero_zero_nat @ A )
% 0.30/0.48 = zero_zero_nat ) ).
% 0.30/0.48
% 0.30/0.48 % zero_diff
% 0.30/0.48 thf(fact_326_diff__zero,axiom,
% 0.30/0.48 ! [A: nat] :
% 0.30/0.48 ( ( minus_minus_nat @ A @ zero_zero_nat )
% 0.30/0.48 = A ) ).
% 0.30/0.48
% 0.30/0.48 % diff_zero
% 0.30/0.48 thf(fact_327_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.30/0.48 ! [A: nat] :
% 0.30/0.48 ( ( minus_minus_nat @ A @ A )
% 0.30/0.48 = zero_zero_nat ) ).
% 0.30/0.48
% 0.30/0.48 % cancel_comm_monoid_add_class.diff_cancel
% 0.30/0.48 thf(fact_328_Diff__eq__empty__iff,axiom,
% 0.30/0.48 ! [A2: set_a,B2: set_a] :
% 0.30/0.48 ( ( ( minus_minus_set_a @ A2 @ B2 )
% 0.30/0.48 = bot_bot_set_a )
% 0.30/0.48 = ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % Diff_eq_empty_iff
% 0.30/0.48 thf(fact_329_subset__empty,axiom,
% 0.30/0.48 ! [A2: set_a] :
% 0.30/0.48 ( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
% 0.30/0.48 = ( A2 = bot_bot_set_a ) ) ).
% 0.30/0.48
% 0.30/0.48 % subset_empty
% 0.30/0.48 thf(fact_330_empty__subsetI,axiom,
% 0.30/0.48 ! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% 0.30/0.48
% 0.30/0.48 % empty_subsetI
% 0.30/0.48 thf(fact_331_insert__subset,axiom,
% 0.30/0.48 ! [X: nat > a,A2: set_nat_a,B2: set_nat_a] :
% 0.30/0.48 ( ( ord_le157368549_nat_a @ ( insert_nat_a @ X @ A2 ) @ B2 )
% 0.30/0.48 = ( ( member_nat_a @ X @ B2 )
% 0.30/0.48 & ( ord_le157368549_nat_a @ A2 @ B2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % insert_subset
% 0.30/0.48 thf(fact_332_insert__subset,axiom,
% 0.30/0.48 ! [X: nat > b,A2: set_nat_b,B2: set_nat_b] :
% 0.30/0.48 ( ( ord_le1296932838_nat_b @ ( insert_nat_b @ X @ A2 ) @ B2 )
% 0.30/0.48 = ( ( member_nat_b @ X @ B2 )
% 0.30/0.48 & ( ord_le1296932838_nat_b @ A2 @ B2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % insert_subset
% 0.30/0.48 thf(fact_333_insert__subset,axiom,
% 0.30/0.48 ! [X: a,A2: set_a,B2: set_a] :
% 0.30/0.48 ( ( ord_less_eq_set_a @ ( insert_a @ X @ A2 ) @ B2 )
% 0.30/0.48 = ( ( member_a @ X @ B2 )
% 0.30/0.48 & ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % insert_subset
% 0.30/0.48 thf(fact_334_insert__Diff__single,axiom,
% 0.30/0.48 ! [A: a,A2: set_a] :
% 0.30/0.48 ( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
% 0.30/0.48 = ( insert_a @ A @ A2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % insert_Diff_single
% 0.30/0.48 thf(fact_335_singletonI,axiom,
% 0.30/0.48 ! [A: nat > a] : ( member_nat_a @ A @ ( insert_nat_a @ A @ bot_bot_set_nat_a ) ) ).
% 0.30/0.48
% 0.30/0.48 % singletonI
% 0.30/0.48 thf(fact_336_singletonI,axiom,
% 0.30/0.48 ! [A: nat > b] : ( member_nat_b @ A @ ( insert_nat_b @ A @ bot_bot_set_nat_b ) ) ).
% 0.30/0.48
% 0.30/0.48 % singletonI
% 0.30/0.48 thf(fact_337_singletonI,axiom,
% 0.30/0.48 ! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% 0.30/0.48
% 0.30/0.48 % singletonI
% 0.30/0.48 thf(fact_338_linear__span__subModule,axiom,
% 0.30/0.48 ! [A2: set_b,H: set_a] :
% 0.30/0.48 ( ( ideal_b_d @ r @ A2 )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 0.30/0.48 => ( submodule_b_d_a_c @ r @ m @ ( linear_span_b_d_a_c @ r @ m @ A2 @ H ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % linear_span_subModule
% 0.30/0.48 thf(fact_339_l__spanA__l__span,axiom,
% 0.30/0.48 ! [A2: set_b,H: set_a] :
% 0.30/0.48 ( ( ideal_b_d @ r @ A2 )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 0.30/0.48 => ( ( linear_span_b_d_a_c @ r @ m @ A2 @ ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ H ) )
% 0.30/0.48 = ( linear_span_b_d_a_c @ r @ m @ A2 @ H ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % l_spanA_l_span
% 0.30/0.48 thf(fact_340_lin__span__coeff__mono,axiom,
% 0.30/0.48 ! [A2: set_b,H: set_a] :
% 0.30/0.48 ( ( ideal_b_d @ r @ A2 )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 0.30/0.48 => ( ord_less_eq_set_a @ ( linear_span_b_d_a_c @ r @ m @ A2 @ H ) @ ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ H ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lin_span_coeff_mono
% 0.30/0.48 thf(fact_341_linear__span__inc__0,axiom,
% 0.30/0.48 ! [A2: set_b,H: set_a] :
% 0.30/0.48 ( ( ideal_b_d @ r @ A2 )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 0.30/0.48 => ( member_a @ ( zero_a1261444626_a_b_c @ m ) @ ( linear_span_b_d_a_c @ r @ m @ A2 @ H ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % linear_span_inc_0
% 0.30/0.48 thf(fact_342_generator__generator,axiom,
% 0.30/0.48 ! [H: set_a,H1: set_a] :
% 0.30/0.48 ( ( generator_b_d_a_c @ r @ m @ H )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ H1 @ ( carrie2021454486_a_b_c @ m ) )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ H @ ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ H1 ) )
% 0.30/0.48 => ( generator_b_d_a_c @ r @ m @ H1 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % generator_generator
% 0.30/0.48 thf(fact_343_l__span__zero,axiom,
% 0.30/0.48 ! [A2: set_b] :
% 0.30/0.48 ( ( ideal_b_d @ r @ A2 )
% 0.30/0.48 => ( ( linear_span_b_d_a_c @ r @ m @ A2 @ ( insert_a @ ( zero_a1261444626_a_b_c @ m ) @ bot_bot_set_a ) )
% 0.30/0.48 = ( insert_a @ ( zero_a1261444626_a_b_c @ m ) @ bot_bot_set_a ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % l_span_zero
% 0.30/0.48 thf(fact_344_l__span__closed3,axiom,
% 0.30/0.48 ! [A2: set_b,H: set_a] :
% 0.30/0.48 ( ( ideal_b_d @ r @ A2 )
% 0.30/0.48 => ( ( generator_b_d_a_c @ r @ m @ H )
% 0.30/0.48 => ( ( ( smodul818989740_d_a_c @ r @ m @ A2 )
% 0.30/0.48 = ( carrie2021454486_a_b_c @ m ) )
% 0.30/0.48 => ( ( linear_span_b_d_a_c @ r @ m @ A2 @ H )
% 0.30/0.48 = ( carrie2021454486_a_b_c @ m ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % l_span_closed3
% 0.30/0.48 thf(fact_345_singleton__conv,axiom,
% 0.30/0.48 ! [A: a] :
% 0.30/0.48 ( ( collect_a
% 0.30/0.48 @ ^ [X3: a] : ( X3 = A ) )
% 0.30/0.48 = ( insert_a @ A @ bot_bot_set_a ) ) ).
% 0.30/0.48
% 0.30/0.48 % singleton_conv
% 0.30/0.48 thf(fact_346_l__comb__mem__linear__span,axiom,
% 0.30/0.48 ! [A2: set_b,H: set_a,S: nat > b,N: nat,F: nat > a] :
% 0.30/0.48 ( ( ideal_b_d @ r @ A2 )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 0.30/0.48 => ( ( member_nat_b @ S
% 0.30/0.48 @ ( pi_nat_b
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : A2 ) )
% 0.30/0.48 => ( ( member_nat_a @ F
% 0.30/0.48 @ ( pi_nat_a
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : H ) )
% 0.30/0.48 => ( member_a @ ( l_comb_b_d_a_c @ r @ m @ N @ S @ F ) @ ( linear_span_b_d_a_c @ r @ m @ A2 @ H ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % l_comb_mem_linear_span
% 0.30/0.48 thf(fact_347_l__span__closed,axiom,
% 0.30/0.48 ! [A2: set_b,H: set_a,S: nat > b,N: nat,F: nat > a] :
% 0.30/0.48 ( ( ideal_b_d @ r @ A2 )
% 0.30/0.48 => ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 0.30/0.48 => ( ( member_nat_b @ S
% 0.30/0.48 @ ( pi_nat_b
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : A2 ) )
% 0.30/0.48 => ( ( member_nat_a @ F
% 0.30/0.48 @ ( pi_nat_a
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : ( linear_span_b_d_a_c @ r @ m @ A2 @ H ) ) )
% 0.30/0.48 => ( member_a @ ( l_comb_b_d_a_c @ r @ m @ N @ S @ F ) @ ( linear_span_b_d_a_c @ r @ m @ A2 @ H ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % l_span_closed
% 0.30/0.48 thf(fact_348_linear__comb0__2,axiom,
% 0.30/0.48 ! [A2: set_b,S: nat > b,N: nat,M: nat > a] :
% 0.30/0.48 ( ( ideal_b_d @ r @ A2 )
% 0.30/0.48 => ( ( member_nat_b @ S
% 0.30/0.48 @ ( pi_nat_b
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : A2 ) )
% 0.30/0.48 => ( ( member_nat_a @ M
% 0.30/0.48 @ ( pi_nat_a
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 0.30/0.48 @ ^ [Uu: nat] : ( insert_a @ ( zero_a1261444626_a_b_c @ m ) @ bot_bot_set_a ) ) )
% 0.30/0.48 => ( ( l_comb_b_d_a_c @ r @ m @ N @ S @ M )
% 0.30/0.48 = ( zero_a1261444626_a_b_c @ m ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % linear_comb0_2
% 0.30/0.48
% 0.30/0.48 % Conjectures (22)
% 0.30/0.48 thf(conj_0,hypothesis,
% 0.30/0.48 ( member_nat_a @ g
% 0.30/0.48 @ ( pi_nat_a
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [K4: nat] : ( ord_less_eq_nat @ K4 @ m2 ) )
% 0.30/0.48 @ ^ [Uu: nat] : ( insert_a @ h2 @ h1 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(conj_1,hypothesis,
% 0.30/0.48 ! [X4: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ X4 @ m2 )
% 0.30/0.48 => ! [Y3: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ Y3 @ m2 )
% 0.30/0.48 => ( ( ( g @ X4 )
% 0.30/0.48 = ( g @ Y3 ) )
% 0.30/0.48 => ( X4 = Y3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(conj_2,hypothesis,
% 0.30/0.48 ( member_nat_b @ t
% 0.30/0.48 @ ( pi_nat_b
% 0.30/0.48 @ ( collect_nat
% 0.30/0.48 @ ^ [K4: nat] : ( ord_less_eq_nat @ K4 @ m2 ) )
% 0.30/0.48 @ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ r ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(conj_3,hypothesis,
% 0.30/0.48 ( ( g @ m2 )
% 0.30/0.48 = h2 ) ).
% 0.30/0.48
% 0.30/0.48 thf(conj_4,hypothesis,
% 0.30/0.48 module_e_b_f_d @ n @ r ).
% 0.30/0.48
% 0.30/0.48 thf(conj_5,hypothesis,
% 0.30/0.48 free_g1087686480_d_a_c @ r @ m @ h ).
% 0.30/0.48
% 0.30/0.48 thf(conj_6,hypothesis,
% 0.30/0.48 ord_less_eq_set_a @ h1 @ h ).
% 0.30/0.48
% 0.30/0.48 thf(conj_7,hypothesis,
% 0.30/0.48 $true ).
% 0.30/0.48
% 0.30/0.48 thf(conj_8,hypothesis,
% 0.30/0.48 ideal_b_d @ r @ ( carrie2079586589xt_b_d @ r ) ).
% 0.30/0.49
% 0.30/0.49 thf(conj_9,hypothesis,
% 0.30/0.49 ord_less_eq_set_a @ h @ ( carrie2021454486_a_b_c @ m ) ).
% 0.30/0.49
% 0.30/0.49 thf(conj_10,hypothesis,
% 0.30/0.49 ord_less_eq_set_a @ h1 @ ( carrie2021454486_a_b_c @ m ) ).
% 0.30/0.49
% 0.30/0.49 thf(conj_11,hypothesis,
% 0.30/0.49 ord_less_eq_set_a @ ( algebr1152250919_d_a_c @ r @ m @ h1 ) @ ( carrie2021454486_a_b_c @ m ) ).
% 0.30/0.49
% 0.30/0.49 thf(conj_12,hypothesis,
% 0.30/0.49 member_a @ ( l_comb_b_d_a_c @ r @ m @ m2 @ t @ g ) @ ( algebr1152250919_d_a_c @ r @ m @ ( insert_a @ h2 @ h1 ) ) ).
% 0.30/0.49
% 0.30/0.49 thf(conj_13,hypothesis,
% 0.30/0.49 ( x
% 0.30/0.49 = ( l_comb_b_d_a_c @ r @ m @ m2 @ t @ g ) ) ).
% 0.30/0.49
% 0.30/0.49 thf(conj_14,hypothesis,
% 0.30/0.49 member_a @ h2 @ h ).
% 0.30/0.49
% 0.30/0.49 thf(conj_15,hypothesis,
% 0.30/0.49 ~ ( member_a @ h2 @ h1 ) ).
% 0.30/0.49
% 0.30/0.49 thf(conj_16,hypothesis,
% 0.30/0.49 ord_less_nat @ zero_zero_nat @ m2 ).
% 0.30/0.49
% 0.30/0.49 thf(conj_17,hypothesis,
% 0.30/0.49 member_a @ h2 @ ( carrie2021454486_a_b_c @ m ) ).
% 0.30/0.49
% 0.30/0.49 thf(conj_18,hypothesis,
% 0.30/0.49 submodule_b_d_a_c @ r @ m @ ( algebr1152250919_d_a_c @ r @ m @ ( insert_a @ h2 @ bot_bot_set_a ) ) ).
% 0.30/0.49
% 0.30/0.49 thf(conj_19,hypothesis,
% 0.30/0.49 ord_less_eq_nat @ xa @ ( minus_minus_nat @ m2 @ ( suc @ zero_zero_nat ) ) ).
% 0.30/0.49
% 0.30/0.49 thf(conj_20,hypothesis,
% 0.30/0.49 ( ( ( g @ xa )
% 0.30/0.49 = h2 )
% 0.30/0.49 | ( member_a @ ( g @ xa ) @ h1 ) ) ).
% 0.30/0.49
% 0.30/0.49 thf(conj_21,conjecture,
% 0.30/0.49 member_a @ ( g @ xa ) @ h1 ).
% 0.30/0.49
% 0.30/0.49 %------------------------------------------------------------------------------
% 0.30/0.49 ------------------------
% 2.96/1.63 ---- Embedded in THF ---
% 2.96/1.64 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.96/1.64 %%% Generated on Tue Feb 24 08:23:42 EST 2026
% 2.96/1.64 %%% using '$$dhol' embedding, version 1.3.0.
% 2.96/1.64 %%% Logic specification used:
% 2.96/1.64 %%% thf(spec, logic, $$dhol).
% 2.96/1.64
% 2.96/1.64 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.sXLfLlVcSw/DTF2DTF9318.p
% See solution above
% 2.96/1.64 ------------------------
% 3.01/1.70 ---- Cleaned THF ---
% 3.01/1.70 thf(ty_n_t__Algebra1__Ocarrier__Ocarrier____ext_I_062_Itf__e_Mtf__e_J_Mt__Algebra4__OaGroup__OaGroup____ext_I_062_Itf__e_Mtf__e_J_Mt__Algebra7__OModule__OModule____ext_I_062_Itf__e_Mtf__e_J_Mtf__b_Mt__Product____Type__Ounit_J_J_J,type,
% 3.01/1.70 carrie377329159t_unit: $tType ).
% 3.01/1.70
% 3.01/1.70 thf(ty_n_t__Algebra1__Ocarrier__Ocarrier____ext_I_062_Itf__e_Mtf__a_J_Mt__Algebra4__OaGroup__OaGroup____ext_I_062_Itf__e_Mtf__a_J_Mt__Algebra7__OModule__OModule____ext_I_062_Itf__e_Mtf__a_J_Mtf__b_Mt__Product____Type__Ounit_J_J_J,type,
% 3.01/1.70 carrie292259835t_unit: $tType ).
% 3.01/1.70
% 3.01/1.70 thf(ty_n_t__Algebra1__Ocarrier__Ocarrier____ext_I_062_Itf__a_Mtf__e_J_Mt__Algebra4__OaGroup__OaGroup____ext_I_062_Itf__a_Mtf__e_J_Mt__Algebra7__OModule__OModule____ext_I_062_Itf__a_Mtf__e_J_Mtf__b_Mt__Product____Type__Ounit_J_J_J,type,
% 3.01/1.70 carrie1065537299t_unit: $tType ).
% 3.01/1.70
% 3.01/1.70 thf(ty_n_t__Algebra1__Ocarrier__Ocarrier____ext_I_062_Itf__a_Mtf__a_J_Mt__Algebra4__OaGroup__OaGroup____ext_I_062_Itf__a_Mtf__a_J_Mt__Algebra7__OModule__OModule____ext_I_062_Itf__a_Mtf__a_J_Mtf__b_Mt__Product____Type__Ounit_J_J_J,type,
% 3.01/1.70 carrie980467975t_unit: $tType ).
% 3.01/1.70
% 3.01/1.70 thf(ty_n_t__Algebra1__Ocarrier__Ocarrier____ext_Itf__a_Mt__Algebra4__OaGroup__OaGroup____ext_Itf__a_Mt__Algebra7__OModule__OModule____ext_Itf__a_Mtf__b_Mt__Product____Type__Ounit_J_J_J,type,
% 3.01/1.70 carrie1963041556t_unit: $tType ).
% 3.01/1.70
% 3.01/1.70 thf(ty_n_t__Algebra1__Ocarrier__Ocarrier____ext_Itf__e_Mt__Algebra4__OaGroup__OaGroup____ext_Itf__e_Mt__Algebra7__OModule__OModule____ext_Itf__e_Mtf__b_Mtf__f_J_J_J,type,
% 3.01/1.70 carrie1821755406_e_b_f: $tType ).
% 3.01/1.70
% 3.01/1.70 thf(ty_n_t__Algebra1__Ocarrier__Ocarrier____ext_Itf__a_Mt__Algebra4__OaGroup__OaGroup____ext_Itf__a_Mt__Algebra7__OModule__OModule____ext_Itf__a_Mtf__b_Mtf__c_J_J_J,type,
% 3.01/1.70 carrie722926983_a_b_c: $tType ).
% 3.01/1.70
% 3.01/1.70 thf(ty_n_t__Algebra1__Ocarrier__Ocarrier____ext_Itf__b_Mt__Algebra4__OaGroup__OaGroup____ext_Itf__b_Mt__Algebra4__ORing__ORing____ext_Itf__b_Mtf__d_J_J_J,type,
% 3.01/1.70 carrie1950868226xt_b_d: $tType ).
% 3.01/1.70
% 3.01/1.70 thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mtf__e_J_J,type,
% 3.01/1.70 set_nat_e: $tType ).
% 3.01/1.70
% 3.01/1.70 thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mtf__b_J_J,type,
% 3.01/1.70 set_nat_b: $tType ).
% 3.01/1.70
% 3.01/1.70 thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
% 3.01/1.70 set_nat_a: $tType ).
% 3.01/1.70
% 3.01/1.70 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 3.01/1.70 set_nat: $tType ).
% 3.01/1.70
% 3.01/1.70 thf(ty_n_t__Set__Oset_Itf__e_J,type,
% 3.01/1.70 set_e: $tType ).
% 3.01/1.70
% 3.01/1.70 thf(ty_n_t__Set__Oset_Itf__b_J,type,
% 3.01/1.70 set_b: $tType ).
% 3.01/1.70
% 3.01/1.70 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 3.01/1.70 set_a: $tType ).
% 3.01/1.70
% 3.01/1.70 thf(ty_n_t__Nat__Onat,type,
% 3.01/1.70 nat: $tType ).
% 3.01/1.70
% 3.01/1.70 thf(ty_n_tf__e,type,
% 3.01/1.70 e: $tType ).
% 3.01/1.70
% 3.01/1.70 thf(ty_n_tf__b,type,
% 3.01/1.70 b: $tType ).
% 3.01/1.70
% 3.01/1.70 thf(ty_n_tf__a,type,
% 3.01/1.70 a: $tType ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra1_Ocarrier_Ocarrier_001tf__a_001t__Algebra4__OaGroup__OaGroup____ext_Itf__a_Mt__Algebra7__OModule__OModule____ext_Itf__a_Mtf__b_Mt__Product____Type__Ounit_J_J,type,
% 3.01/1.70 carrie1074654371t_unit: carrie1963041556t_unit > set_a ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra1_Ocarrier_Ocarrier_001tf__a_001t__Algebra4__OaGroup__OaGroup____ext_Itf__a_Mt__Algebra7__OModule__OModule____ext_Itf__a_Mtf__b_Mtf__c_J_J,type,
% 3.01/1.70 carrie2021454486_a_b_c: carrie722926983_a_b_c > set_a ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra1_Ocarrier_Ocarrier_001tf__b_001t__Algebra4__OaGroup__OaGroup____ext_Itf__b_Mt__Algebra4__ORing__ORing____ext_Itf__b_Mtf__d_J_J,type,
% 3.01/1.70 carrie2079586589xt_b_d: carrie1950868226xt_b_d > set_b ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra1_Ocarrier_Ocarrier_001tf__e_001t__Algebra4__OaGroup__OaGroup____ext_Itf__e_Mt__Algebra7__OModule__OModule____ext_Itf__e_Mtf__b_Mtf__f_J_J,type,
% 3.01/1.70 carrie730238621_e_b_f: carrie1821755406_e_b_f > set_e ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra4_ORing_001tf__b_001tf__d,type,
% 3.01/1.70 ring_b_d: carrie1950868226xt_b_d > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra4_OaGroup_Ozero_001tf__a_001t__Algebra7__OModule__OModule____ext_Itf__a_Mtf__b_Mtf__c_J,type,
% 3.01/1.70 zero_a1261444626_a_b_c: carrie722926983_a_b_c > a ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra4_Oideal_001tf__b_001tf__d,type,
% 3.01/1.70 ideal_b_d: carrie1950868226xt_b_d > set_b > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_OAnnihilator_001tf__b_001tf__d_001tf__a_001t__Product____Type__Ounit,type,
% 3.01/1.70 annihi259882159t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > set_b ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_OAnnihilator_001tf__b_001tf__d_001tf__a_001tf__c,type,
% 3.01/1.70 annihilator_b_d_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > set_b ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_OAnnihilator_001tf__b_001tf__d_001tf__e_001tf__f,type,
% 3.01/1.70 annihilator_b_d_e_f: carrie1950868226xt_b_d > carrie1821755406_e_b_f > set_b ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_OHOM_001tf__b_001tf__d_001tf__a_001t__Product____Type__Ounit_001tf__a_001t__Product____Type__Ounit,type,
% 3.01/1.70 hOM_b_1103679019t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > carrie1963041556t_unit > carrie980467975t_unit ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_OHOM_001tf__b_001tf__d_001tf__a_001t__Product____Type__Ounit_001tf__a_001tf__c,type,
% 3.01/1.70 hOM_b_796081438it_a_c: carrie1950868226xt_b_d > carrie1963041556t_unit > carrie722926983_a_b_c > carrie980467975t_unit ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_OHOM_001tf__b_001tf__d_001tf__a_001t__Product____Type__Ounit_001tf__e_001tf__f,type,
% 3.01/1.70 hOM_b_1959675421it_e_f: carrie1950868226xt_b_d > carrie1963041556t_unit > carrie1821755406_e_b_f > carrie1065537299t_unit ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_OHOM_001tf__b_001tf__d_001tf__a_001tf__c_001tf__a_001t__Product____Type__Ounit,type,
% 3.01/1.70 hOM_b_717364638t_unit: carrie1950868226xt_b_d > carrie722926983_a_b_c > carrie1963041556t_unit > carrie980467975t_unit ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_OHOM_001tf__b_001tf__d_001tf__a_001tf__c_001tf__a_001tf__c,type,
% 3.01/1.70 hOM_b_d_a_c_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > carrie722926983_a_b_c > carrie980467975t_unit ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_OHOM_001tf__b_001tf__d_001tf__a_001tf__c_001tf__e_001tf__f,type,
% 3.01/1.70 hOM_b_d_a_c_e_f: carrie1950868226xt_b_d > carrie722926983_a_b_c > carrie1821755406_e_b_f > carrie1065537299t_unit ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_OHOM_001tf__b_001tf__d_001tf__e_001tf__f_001tf__a_001t__Product____Type__Ounit,type,
% 3.01/1.70 hOM_b_1375846429t_unit: carrie1950868226xt_b_d > carrie1821755406_e_b_f > carrie1963041556t_unit > carrie292259835t_unit ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_OHOM_001tf__b_001tf__d_001tf__e_001tf__f_001tf__a_001tf__c,type,
% 3.01/1.70 hOM_b_d_e_f_a_c: carrie1950868226xt_b_d > carrie1821755406_e_b_f > carrie722926983_a_b_c > carrie292259835t_unit ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_OHOM_001tf__b_001tf__d_001tf__e_001tf__f_001tf__e_001tf__f,type,
% 3.01/1.70 hOM_b_d_e_f_e_f: carrie1950868226xt_b_d > carrie1821755406_e_b_f > carrie1821755406_e_b_f > carrie377329159t_unit ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_OModule_001_062_Itf__a_Mtf__a_J_001tf__b_001t__Product____Type__Ounit_001tf__d,type,
% 3.01/1.70 module589927589unit_d: carrie980467975t_unit > carrie1950868226xt_b_d > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_OModule_001_062_Itf__a_Mtf__e_J_001tf__b_001t__Product____Type__Ounit_001tf__d,type,
% 3.01/1.70 module1648870817unit_d: carrie1065537299t_unit > carrie1950868226xt_b_d > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_OModule_001_062_Itf__e_Mtf__a_J_001tf__b_001t__Product____Type__Ounit_001tf__d,type,
% 3.01/1.70 module1330273449unit_d: carrie292259835t_unit > carrie1950868226xt_b_d > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_OModule_001_062_Itf__e_Mtf__e_J_001tf__b_001t__Product____Type__Ounit_001tf__d,type,
% 3.01/1.70 module241733029unit_d: carrie377329159t_unit > carrie1950868226xt_b_d > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_OModule_001tf__a_001tf__b_001t__Product____Type__Ounit_001tf__d,type,
% 3.01/1.70 module1821517916unit_d: carrie1963041556t_unit > carrie1950868226xt_b_d > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_OModule_001tf__a_001tf__b_001tf__c_001tf__d,type,
% 3.01/1.70 module_a_b_c_d: carrie722926983_a_b_c > carrie1950868226xt_b_d > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_OModule_001tf__e_001tf__b_001tf__f_001tf__d,type,
% 3.01/1.70 module_e_b_f_d: carrie1821755406_e_b_f > carrie1950868226xt_b_d > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_Ofree__generator_001tf__b_001tf__d_001tf__a_001t__Product____Type__Ounit,type,
% 3.01/1.70 free_g637607517t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > set_a > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_Ofree__generator_001tf__b_001tf__d_001tf__a_001tf__c,type,
% 3.01/1.70 free_g1087686480_d_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > set_a > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_Ofree__generator_001tf__b_001tf__d_001tf__e_001tf__f,type,
% 3.01/1.70 free_g103796815_d_e_f: carrie1950868226xt_b_d > carrie1821755406_e_b_f > set_e > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_Ogenerator_001tf__b_001tf__d_001tf__a_001t__Product____Type__Ounit,type,
% 3.01/1.70 genera1692266857t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > set_a > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_Ogenerator_001tf__b_001tf__d_001tf__a_001tf__c,type,
% 3.01/1.70 generator_b_d_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > set_a > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_Ogenerator_001tf__b_001tf__d_001tf__e_001tf__f,type,
% 3.01/1.70 generator_b_d_e_f: carrie1950868226xt_b_d > carrie1821755406_e_b_f > set_e > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_Ol__comb_001tf__b_001tf__d_001tf__a_001t__Product____Type__Ounit,type,
% 3.01/1.70 l_comb1138968323t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > nat > ( nat > b ) > ( nat > a ) > a ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_Ol__comb_001tf__b_001tf__d_001tf__a_001tf__c,type,
% 3.01/1.70 l_comb_b_d_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > nat > ( nat > b ) > ( nat > a ) > a ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_Ol__comb_001tf__b_001tf__d_001tf__e_001tf__f,type,
% 3.01/1.70 l_comb_b_d_e_f: carrie1950868226xt_b_d > carrie1821755406_e_b_f > nat > ( nat > b ) > ( nat > e ) > e ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_Olinear__span_001tf__b_001tf__d_001tf__a_001tf__c,type,
% 3.01/1.70 linear_span_b_d_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > set_b > set_a > set_a ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_Omdl_001tf__a_001tf__b_001tf__c,type,
% 3.01/1.70 mdl_a_b_c: carrie722926983_a_b_c > set_a > carrie1963041556t_unit ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_Omisomorphic_001tf__b_001tf__d_001tf__a_001t__Product____Type__Ounit_001tf__a_001t__Product____Type__Ounit,type,
% 3.01/1.70 misomo493403663t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > carrie1963041556t_unit > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_Omisomorphic_001tf__b_001tf__d_001tf__a_001t__Product____Type__Ounit_001tf__a_001tf__c,type,
% 3.01/1.70 misomo1935876354it_a_c: carrie1950868226xt_b_d > carrie1963041556t_unit > carrie722926983_a_b_c > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_Omisomorphic_001tf__b_001tf__d_001tf__a_001t__Product____Type__Ounit_001tf__e_001tf__f,type,
% 3.01/1.70 misomo951986689it_e_f: carrie1950868226xt_b_d > carrie1963041556t_unit > carrie1821755406_e_b_f > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_Omisomorphic_001tf__b_001tf__d_001tf__a_001tf__c_001tf__a_001t__Product____Type__Ounit,type,
% 3.01/1.70 misomo1857159554t_unit: carrie1950868226xt_b_d > carrie722926983_a_b_c > carrie1963041556t_unit > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_Omisomorphic_001tf__b_001tf__d_001tf__a_001tf__c_001tf__a_001tf__c,type,
% 3.01/1.70 misomo1282343797_c_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > carrie722926983_a_b_c > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_Omisomorphic_001tf__b_001tf__d_001tf__a_001tf__c_001tf__e_001tf__f,type,
% 3.01/1.70 misomo298454132_c_e_f: carrie1950868226xt_b_d > carrie722926983_a_b_c > carrie1821755406_e_b_f > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_Omisomorphic_001tf__b_001tf__d_001tf__e_001tf__f_001tf__a_001t__Product____Type__Ounit,type,
% 3.01/1.70 misomo368157697t_unit: carrie1950868226xt_b_d > carrie1821755406_e_b_f > carrie1963041556t_unit > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_Omisomorphic_001tf__b_001tf__d_001tf__e_001tf__f_001tf__a_001tf__c,type,
% 3.01/1.70 misomo1752826100_f_a_c: carrie1950868226xt_b_d > carrie1821755406_e_b_f > carrie722926983_a_b_c > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_Omisomorphic_001tf__b_001tf__d_001tf__e_001tf__f_001tf__e_001tf__f,type,
% 3.01/1.70 misomo768936435_f_e_f: carrie1950868226xt_b_d > carrie1821755406_e_b_f > carrie1821755406_e_b_f > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_Oquotient__of__submodules_001tf__b_001tf__d_001tf__a_001t__Product____Type__Ounit,type,
% 3.01/1.70 quotie1196826005t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > set_a > set_a > set_b ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_Oquotient__of__submodules_001tf__b_001tf__d_001tf__a_001tf__c,type,
% 3.01/1.70 quotie1153752712_d_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > set_a > set_a > set_b ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_Oquotient__of__submodules_001tf__b_001tf__d_001tf__e_001tf__f,type,
% 3.01/1.70 quotie169863047_d_e_f: carrie1950868226xt_b_d > carrie1821755406_e_b_f > set_e > set_e > set_b ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_Osmodule__ideal__coeff_001tf__b_001tf__d_001tf__a_001t__Product____Type__Ounit,type,
% 3.01/1.70 smodul132175289t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > set_b > set_a ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_Osmodule__ideal__coeff_001tf__b_001tf__d_001tf__a_001tf__c,type,
% 3.01/1.70 smodul818989740_d_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > set_b > set_a ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_Osmodule__ideal__coeff_001tf__b_001tf__d_001tf__e_001tf__f,type,
% 3.01/1.70 smodul1982583723_d_e_f: carrie1950868226xt_b_d > carrie1821755406_e_b_f > set_b > set_e ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_Osubmodule_001tf__b_001tf__d_001tf__a_001t__Product____Type__Ounit,type,
% 3.01/1.70 submod903911234t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > set_a > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_Osubmodule_001tf__b_001tf__d_001tf__a_001tf__c,type,
% 3.01/1.70 submodule_b_d_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > set_a > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra7_Osubmodule_001tf__b_001tf__d_001tf__e_001tf__f,type,
% 3.01/1.70 submodule_b_d_e_f: carrie1950868226xt_b_d > carrie1821755406_e_b_f > set_e > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra8__Mirabelle__tltoludslr_Ofgs_001tf__b_001tf__d_001tf__a_001t__Product____Type__Ounit,type,
% 3.01/1.70 algebr1039611956t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > set_a > set_a ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra8__Mirabelle__tltoludslr_Ofgs_001tf__b_001tf__d_001tf__a_001tf__c,type,
% 3.01/1.70 algebr1152250919_d_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > set_a > set_a ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Algebra8__Mirabelle__tltoludslr_Ofgs_001tf__b_001tf__d_001tf__e_001tf__f,type,
% 3.01/1.70 algebr168361254_d_e_f: carrie1950868226xt_b_d > carrie1821755406_e_b_f > set_e > set_e ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_FuncSet_OPi_001t__Nat__Onat_001tf__a,type,
% 3.01/1.70 pi_nat_a: set_nat > ( nat > set_a ) > set_nat_a ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_FuncSet_OPi_001t__Nat__Onat_001tf__b,type,
% 3.01/1.70 pi_nat_b: set_nat > ( nat > set_b ) > set_nat_b ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_FuncSet_OPi_001t__Nat__Onat_001tf__e,type,
% 3.01/1.70 pi_nat_e: set_nat > ( nat > set_e ) > set_nat_e ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
% 3.01/1.70 minus_minus_nat: nat > nat > nat ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
% 3.01/1.70 minus_1788767276_nat_a: set_nat_a > set_nat_a > set_nat_a ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__b_J_J,type,
% 3.01/1.70 minus_780847917_nat_b: set_nat_b > set_nat_b > set_nat_b ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
% 3.01/1.70 minus_minus_set_a: set_a > set_a > set_a ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 3.01/1.70 zero_zero_nat: nat ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Nat_OSuc,type,
% 3.01/1.70 suc: nat > nat ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
% 3.01/1.70 bot_bot_set_nat_a: set_nat_a ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__b_J_J,type,
% 3.01/1.70 bot_bot_set_nat_b: set_nat_b ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
% 3.01/1.70 bot_bot_set_nat: set_nat ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
% 3.01/1.70 bot_bot_set_a: set_a ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__e_J,type,
% 3.01/1.70 bot_bot_set_e: set_e ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 3.01/1.70 ord_less_nat: nat > nat > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
% 3.01/1.70 ord_less_set_a: set_a > set_a > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 3.01/1.70 ord_less_eq_nat: nat > nat > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
% 3.01/1.70 ord_le157368549_nat_a: set_nat_a > set_nat_a > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__b_J_J,type,
% 3.01/1.70 ord_le1296932838_nat_b: set_nat_b > set_nat_b > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
% 3.01/1.70 ord_less_eq_set_a: set_a > set_a > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__e_J,type,
% 3.01/1.70 ord_less_eq_set_e: set_e > set_e > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mtf__a_J,type,
% 3.01/1.70 collect_nat_a: ( ( nat > a ) > $o ) > set_nat_a ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mtf__b_J,type,
% 3.01/1.70 collect_nat_b: ( ( nat > b ) > $o ) > set_nat_b ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
% 3.01/1.70 collect_nat: ( nat > $o ) > set_nat ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Set_OCollect_001tf__a,type,
% 3.01/1.70 collect_a: ( a > $o ) > set_a ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Set_Oinsert_001_062_It__Nat__Onat_Mtf__a_J,type,
% 3.01/1.70 insert_nat_a: ( nat > a ) > set_nat_a > set_nat_a ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Set_Oinsert_001_062_It__Nat__Onat_Mtf__b_J,type,
% 3.01/1.70 insert_nat_b: ( nat > b ) > set_nat_b > set_nat_b ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_Set_Oinsert_001tf__a,type,
% 3.01/1.70 insert_a: a > set_a > set_a ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_member_001_062_It__Nat__Onat_Mtf__a_J,type,
% 3.01/1.70 member_nat_a: ( nat > a ) > set_nat_a > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_member_001_062_It__Nat__Onat_Mtf__b_J,type,
% 3.01/1.70 member_nat_b: ( nat > b ) > set_nat_b > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_member_001_062_It__Nat__Onat_Mtf__e_J,type,
% 3.01/1.70 member_nat_e: ( nat > e ) > set_nat_e > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_member_001t__Nat__Onat,type,
% 3.01/1.70 member_nat: nat > set_nat > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_member_001tf__a,type,
% 3.01/1.70 member_a: a > set_a > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_c_member_001tf__e,type,
% 3.01/1.70 member_e: e > set_e > $o ).
% 3.01/1.70
% 3.01/1.70 thf(sy_v_H,type,
% 3.01/1.70 h: set_a ).
% 3.01/1.70
% 3.01/1.70 thf(sy_v_H1,type,
% 3.01/1.70 h1: set_a ).
% 3.01/1.70
% 3.01/1.70 thf(sy_v_M,type,
% 3.01/1.70 m: carrie722926983_a_b_c ).
% 3.01/1.70
% 3.01/1.70 thf(sy_v_N,type,
% 3.01/1.70 n: carrie1821755406_e_b_f ).
% 3.01/1.70
% 3.01/1.70 thf(sy_v_R,type,
% 3.01/1.70 r: carrie1950868226xt_b_d ).
% 3.01/1.70
% 3.01/1.70 thf(sy_v_g,type,
% 3.01/1.70 g: nat > a ).
% 3.01/1.70
% 3.01/1.70 thf(sy_v_h,type,
% 3.01/1.70 h2: a ).
% 3.01/1.70
% 3.01/1.70 thf(sy_v_m,type,
% 3.01/1.70 m2: nat ).
% 3.01/1.70
% 3.01/1.70 thf(sy_v_t,type,
% 3.01/1.70 t: nat > b ).
% 3.01/1.70
% 3.01/1.70 thf(sy_v_x,type,
% 3.01/1.70 x: a ).
% 3.01/1.70
% 3.01/1.70 thf(sy_v_xa,type,
% 3.01/1.70 xa: nat ).
% 3.01/1.70
% 3.01/1.70 thf(fact_0_sc__Ring,axiom,
% 3.01/1.70 ring_b_d @ r ).
% 3.01/1.70
% 3.01/1.70 thf(fact_1_submodule__subset1,axiom,
% 3.01/1.70 ! [H: set_a,H2: a] :
% 3.01/1.70 ( ( submodule_b_d_a_c @ r @ m @ H )
% 3.01/1.70 => ( ( member_a @ H2 @ H )
% 3.01/1.70 => ( member_a @ H2 @ ( carrie2021454486_a_b_c @ m ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_2_submodule__whole,axiom,
% 3.01/1.70 submodule_b_d_a_c @ r @ m @ ( carrie2021454486_a_b_c @ m ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_3_submodule__subset,axiom,
% 3.01/1.70 ! [H: set_a] :
% 3.01/1.70 ( ( submodule_b_d_a_c @ r @ m @ H )
% 3.01/1.70 => ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_4_free__generator__sub,axiom,
% 3.01/1.70 ! [H: set_a] :
% 3.01/1.70 ( ( free_g1087686480_d_a_c @ r @ m @ H )
% 3.01/1.70 => ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_5_elem__fgs,axiom,
% 3.01/1.70 ! [A: set_a,X: a] :
% 3.01/1.70 ( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) )
% 3.01/1.70 => ( ( member_a @ X @ A )
% 3.01/1.70 => ( member_a @ X @ ( algebr1152250919_d_a_c @ r @ m @ A ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_6_fgs__sub__carrier,axiom,
% 3.01/1.70 ! [A: set_a] :
% 3.01/1.70 ( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) )
% 3.01/1.70 => ( ord_less_eq_set_a @ ( algebr1152250919_d_a_c @ r @ m @ A ) @ ( carrie2021454486_a_b_c @ m ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_7_fgs__mono,axiom,
% 3.01/1.70 ! [H: set_a,J: set_a,K: set_a] :
% 3.01/1.70 ( ( free_g1087686480_d_a_c @ r @ m @ H )
% 3.01/1.70 => ( ( ord_less_eq_set_a @ J @ K )
% 3.01/1.70 => ( ( ord_less_eq_set_a @ K @ H )
% 3.01/1.70 => ( ord_less_eq_set_a @ ( algebr1152250919_d_a_c @ r @ m @ J ) @ ( algebr1152250919_d_a_c @ r @ m @ K ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_8_fgs__submodule,axiom,
% 3.01/1.70 ! [A: set_a] :
% 3.01/1.70 ( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) )
% 3.01/1.70 => ( submodule_b_d_a_c @ r @ m @ ( algebr1152250919_d_a_c @ r @ m @ A ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_9_linear__comb__eq,axiom,
% 3.01/1.70 ! [H: set_a,S: nat > b,N: nat,F: nat > a,G: nat > a] :
% 3.01/1.70 ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 3.01/1.70 => ( ( member_nat_b @ S
% 3.01/1.70 @ ( pi_nat_b
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ r ) ) )
% 3.01/1.70 => ( ( member_nat_a @ F
% 3.01/1.70 @ ( pi_nat_a
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : H ) )
% 3.01/1.70 => ( ( member_nat_a @ G
% 3.01/1.70 @ ( pi_nat_a
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : H ) )
% 3.01/1.70 => ( ! [X2: nat] :
% 3.01/1.70 ( ( member_nat @ X2
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) ) )
% 3.01/1.70 => ( ( F @ X2 )
% 3.01/1.70 = ( G @ X2 ) ) )
% 3.01/1.70 => ( ( l_comb_b_d_a_c @ r @ m @ N @ S @ F )
% 3.01/1.70 = ( l_comb_b_d_a_c @ r @ m @ N @ S @ G ) ) ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_10_linear__comb__eqTr,axiom,
% 3.01/1.70 ! [H: set_a,S: nat > b,N: nat,F: nat > a,G: nat > a] :
% 3.01/1.70 ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 3.01/1.70 => ( ( ( member_nat_b @ S
% 3.01/1.70 @ ( pi_nat_b
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ r ) ) )
% 3.01/1.70 & ( member_nat_a @ F
% 3.01/1.70 @ ( pi_nat_a
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : H ) )
% 3.01/1.70 & ( member_nat_a @ G
% 3.01/1.70 @ ( pi_nat_a
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : H ) )
% 3.01/1.70 & ! [X2: nat] :
% 3.01/1.70 ( ( member_nat @ X2
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) ) )
% 3.01/1.70 => ( ( F @ X2 )
% 3.01/1.70 = ( G @ X2 ) ) ) )
% 3.01/1.70 => ( ( l_comb_b_d_a_c @ r @ m @ N @ S @ F )
% 3.01/1.70 = ( l_comb_b_d_a_c @ r @ m @ N @ S @ G ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_11_l__comb__mem,axiom,
% 3.01/1.70 ! [A2: set_b,H: set_a,S: nat > b,N: nat,M: nat > a] :
% 3.01/1.70 ( ( ideal_b_d @ r @ A2 )
% 3.01/1.70 => ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 3.01/1.70 => ( ( member_nat_b @ S
% 3.01/1.70 @ ( pi_nat_b
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : A2 ) )
% 3.01/1.70 => ( ( member_nat_a @ M
% 3.01/1.70 @ ( pi_nat_a
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : H ) )
% 3.01/1.70 => ( member_a @ ( l_comb_b_d_a_c @ r @ m @ N @ S @ M ) @ ( carrie2021454486_a_b_c @ m ) ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_12_liear__comb__memTr,axiom,
% 3.01/1.70 ! [A2: set_b,H: set_a,N: nat] :
% 3.01/1.70 ( ( ideal_b_d @ r @ A2 )
% 3.01/1.70 => ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 3.01/1.70 => ! [S2: nat > b,M2: nat > a] :
% 3.01/1.70 ( ( ( member_nat_b @ S2
% 3.01/1.70 @ ( pi_nat_b
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : A2 ) )
% 3.01/1.70 & ( member_nat_a @ M2
% 3.01/1.70 @ ( pi_nat_a
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : H ) ) )
% 3.01/1.70 => ( member_a @ ( l_comb_b_d_a_c @ r @ m @ N @ S2 @ M2 ) @ ( carrie2021454486_a_b_c @ m ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_13_Ann__is__ideal,axiom,
% 3.01/1.70 ideal_b_d @ r @ ( annihilator_b_d_a_c @ r @ m ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_14_mem__fgs__l__comb,axiom,
% 3.01/1.70 ! [K: set_a,X: a] :
% 3.01/1.70 ( ( K != bot_bot_set_a )
% 3.01/1.70 => ( ( ord_less_eq_set_a @ K @ ( carrie2021454486_a_b_c @ m ) )
% 3.01/1.70 => ( ( member_a @ X @ ( algebr1152250919_d_a_c @ r @ m @ K ) )
% 3.01/1.70 => ? [N2: nat,X2: nat > a] :
% 3.01/1.70 ( ( member_nat_a @ X2
% 3.01/1.70 @ ( pi_nat_a
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
% 3.01/1.70 @ ^ [Uu: nat] : K ) )
% 3.01/1.70 & ? [Xa: nat > b] :
% 3.01/1.70 ( ( member_nat_b @ Xa
% 3.01/1.70 @ ( pi_nat_b
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
% 3.01/1.70 @ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ r ) ) )
% 3.01/1.70 & ( X
% 3.01/1.70 = ( l_comb_b_d_a_c @ r @ m @ N2 @ Xa @ X2 ) ) ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_15_Module__axioms,axiom,
% 3.01/1.70 module_a_b_c_d @ m @ r ).
% 3.01/1.70
% 3.01/1.70 thf(fact_16_quotient__of__submodules__is__ideal,axiom,
% 3.01/1.70 ! [P: set_a,Q: set_a] :
% 3.01/1.70 ( ( submodule_b_d_a_c @ r @ m @ P )
% 3.01/1.70 => ( ( submodule_b_d_a_c @ r @ m @ Q )
% 3.01/1.70 => ( ideal_b_d @ r @ ( quotie1153752712_d_a_c @ r @ m @ P @ Q ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_17_smodule__ideal__coeff__is__Submodule,axiom,
% 3.01/1.70 ! [A2: set_b] :
% 3.01/1.70 ( ( ideal_b_d @ r @ A2 )
% 3.01/1.70 => ( submodule_b_d_a_c @ r @ m @ ( smodul818989740_d_a_c @ r @ m @ A2 ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_18_mem__smodule__ideal__coeff,axiom,
% 3.01/1.70 ! [A2: set_b,X: a] :
% 3.01/1.70 ( ( ideal_b_d @ r @ A2 )
% 3.01/1.70 => ( ( member_a @ X @ ( smodul818989740_d_a_c @ r @ m @ A2 ) )
% 3.01/1.70 => ? [N2: nat,X2: nat > b] :
% 3.01/1.70 ( ( member_nat_b @ X2
% 3.01/1.70 @ ( pi_nat_b
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
% 3.01/1.70 @ ^ [Uu: nat] : A2 ) )
% 3.01/1.70 & ? [Xa: nat > a] :
% 3.01/1.70 ( ( member_nat_a @ Xa
% 3.01/1.70 @ ( pi_nat_a
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
% 3.01/1.70 @ ^ [Uu: nat] : ( carrie2021454486_a_b_c @ m ) ) )
% 3.01/1.70 & ( X
% 3.01/1.70 = ( l_comb_b_d_a_c @ r @ m @ N2 @ X2 @ Xa ) ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_19_Module_Oelem__fgs,axiom,
% 3.01/1.70 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,A: set_e,X: e] :
% 3.01/1.70 ( ( module_e_b_f_d @ M3 @ R )
% 3.01/1.70 => ( ( ord_less_eq_set_e @ A @ ( carrie730238621_e_b_f @ M3 ) )
% 3.01/1.70 => ( ( member_e @ X @ A )
% 3.01/1.70 => ( member_e @ X @ ( algebr168361254_d_e_f @ R @ M3 @ A ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_20_Module_Oelem__fgs,axiom,
% 3.01/1.70 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,A: set_a,X: a] :
% 3.01/1.70 ( ( module1821517916unit_d @ M3 @ R )
% 3.01/1.70 => ( ( ord_less_eq_set_a @ A @ ( carrie1074654371t_unit @ M3 ) )
% 3.01/1.70 => ( ( member_a @ X @ A )
% 3.01/1.70 => ( member_a @ X @ ( algebr1039611956t_unit @ R @ M3 @ A ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_21_Module_Oelem__fgs,axiom,
% 3.01/1.70 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,A: set_a,X: a] :
% 3.01/1.70 ( ( module_a_b_c_d @ M3 @ R )
% 3.01/1.70 => ( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ M3 ) )
% 3.01/1.70 => ( ( member_a @ X @ A )
% 3.01/1.70 => ( member_a @ X @ ( algebr1152250919_d_a_c @ R @ M3 @ A ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_22_Module_Ofgs__mono,axiom,
% 3.01/1.70 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,H: set_e,J: set_e,K: set_e] :
% 3.01/1.70 ( ( module_e_b_f_d @ M3 @ R )
% 3.01/1.70 => ( ( free_g103796815_d_e_f @ R @ M3 @ H )
% 3.01/1.70 => ( ( ord_less_eq_set_e @ J @ K )
% 3.01/1.70 => ( ( ord_less_eq_set_e @ K @ H )
% 3.01/1.70 => ( ord_less_eq_set_e @ ( algebr168361254_d_e_f @ R @ M3 @ J ) @ ( algebr168361254_d_e_f @ R @ M3 @ K ) ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_23_Module_Ofgs__mono,axiom,
% 3.01/1.70 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,H: set_a,J: set_a,K: set_a] :
% 3.01/1.70 ( ( module1821517916unit_d @ M3 @ R )
% 3.01/1.70 => ( ( free_g637607517t_unit @ R @ M3 @ H )
% 3.01/1.70 => ( ( ord_less_eq_set_a @ J @ K )
% 3.01/1.70 => ( ( ord_less_eq_set_a @ K @ H )
% 3.01/1.70 => ( ord_less_eq_set_a @ ( algebr1039611956t_unit @ R @ M3 @ J ) @ ( algebr1039611956t_unit @ R @ M3 @ K ) ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_24_Module_Ofgs__mono,axiom,
% 3.01/1.70 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,H: set_a,J: set_a,K: set_a] :
% 3.01/1.70 ( ( module_a_b_c_d @ M3 @ R )
% 3.01/1.70 => ( ( free_g1087686480_d_a_c @ R @ M3 @ H )
% 3.01/1.70 => ( ( ord_less_eq_set_a @ J @ K )
% 3.01/1.70 => ( ( ord_less_eq_set_a @ K @ H )
% 3.01/1.70 => ( ord_less_eq_set_a @ ( algebr1152250919_d_a_c @ R @ M3 @ J ) @ ( algebr1152250919_d_a_c @ R @ M3 @ K ) ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_25_Module_Ofgs__submodule,axiom,
% 3.01/1.70 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,A: set_e] :
% 3.01/1.70 ( ( module_e_b_f_d @ M3 @ R )
% 3.01/1.70 => ( ( ord_less_eq_set_e @ A @ ( carrie730238621_e_b_f @ M3 ) )
% 3.01/1.70 => ( submodule_b_d_e_f @ R @ M3 @ ( algebr168361254_d_e_f @ R @ M3 @ A ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_26_Module_Ofgs__submodule,axiom,
% 3.01/1.70 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,A: set_a] :
% 3.01/1.70 ( ( module1821517916unit_d @ M3 @ R )
% 3.01/1.70 => ( ( ord_less_eq_set_a @ A @ ( carrie1074654371t_unit @ M3 ) )
% 3.01/1.70 => ( submod903911234t_unit @ R @ M3 @ ( algebr1039611956t_unit @ R @ M3 @ A ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_27_Module_Ofgs__submodule,axiom,
% 3.01/1.70 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,A: set_a] :
% 3.01/1.70 ( ( module_a_b_c_d @ M3 @ R )
% 3.01/1.70 => ( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ M3 ) )
% 3.01/1.70 => ( submodule_b_d_a_c @ R @ M3 @ ( algebr1152250919_d_a_c @ R @ M3 @ A ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_28_Module_Omem__fgs__l__comb,axiom,
% 3.01/1.70 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,K: set_e,X: e] :
% 3.01/1.70 ( ( module_e_b_f_d @ M3 @ R )
% 3.01/1.70 => ( ( K != bot_bot_set_e )
% 3.01/1.70 => ( ( ord_less_eq_set_e @ K @ ( carrie730238621_e_b_f @ M3 ) )
% 3.01/1.70 => ( ( member_e @ X @ ( algebr168361254_d_e_f @ R @ M3 @ K ) )
% 3.01/1.70 => ? [N2: nat,X2: nat > e] :
% 3.01/1.70 ( ( member_nat_e @ X2
% 3.01/1.70 @ ( pi_nat_e
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
% 3.01/1.70 @ ^ [Uu: nat] : K ) )
% 3.01/1.70 & ? [Xa: nat > b] :
% 3.01/1.70 ( ( member_nat_b @ Xa
% 3.01/1.70 @ ( pi_nat_b
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
% 3.01/1.70 @ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ R ) ) )
% 3.01/1.70 & ( X
% 3.01/1.70 = ( l_comb_b_d_e_f @ R @ M3 @ N2 @ Xa @ X2 ) ) ) ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_29_Module_Omem__fgs__l__comb,axiom,
% 3.01/1.70 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,K: set_a,X: a] :
% 3.01/1.70 ( ( module1821517916unit_d @ M3 @ R )
% 3.01/1.70 => ( ( K != bot_bot_set_a )
% 3.01/1.70 => ( ( ord_less_eq_set_a @ K @ ( carrie1074654371t_unit @ M3 ) )
% 3.01/1.70 => ( ( member_a @ X @ ( algebr1039611956t_unit @ R @ M3 @ K ) )
% 3.01/1.70 => ? [N2: nat,X2: nat > a] :
% 3.01/1.70 ( ( member_nat_a @ X2
% 3.01/1.70 @ ( pi_nat_a
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
% 3.01/1.70 @ ^ [Uu: nat] : K ) )
% 3.01/1.70 & ? [Xa: nat > b] :
% 3.01/1.70 ( ( member_nat_b @ Xa
% 3.01/1.70 @ ( pi_nat_b
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
% 3.01/1.70 @ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ R ) ) )
% 3.01/1.70 & ( X
% 3.01/1.70 = ( l_comb1138968323t_unit @ R @ M3 @ N2 @ Xa @ X2 ) ) ) ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_30_Module_Omem__fgs__l__comb,axiom,
% 3.01/1.70 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,K: set_a,X: a] :
% 3.01/1.70 ( ( module_a_b_c_d @ M3 @ R )
% 3.01/1.70 => ( ( K != bot_bot_set_a )
% 3.01/1.70 => ( ( ord_less_eq_set_a @ K @ ( carrie2021454486_a_b_c @ M3 ) )
% 3.01/1.70 => ( ( member_a @ X @ ( algebr1152250919_d_a_c @ R @ M3 @ K ) )
% 3.01/1.70 => ? [N2: nat,X2: nat > a] :
% 3.01/1.70 ( ( member_nat_a @ X2
% 3.01/1.70 @ ( pi_nat_a
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
% 3.01/1.70 @ ^ [Uu: nat] : K ) )
% 3.01/1.70 & ? [Xa: nat > b] :
% 3.01/1.70 ( ( member_nat_b @ Xa
% 3.01/1.70 @ ( pi_nat_b
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
% 3.01/1.70 @ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ R ) ) )
% 3.01/1.70 & ( X
% 3.01/1.70 = ( l_comb_b_d_a_c @ R @ M3 @ N2 @ Xa @ X2 ) ) ) ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_31_Module_Ofgs__sub__carrier,axiom,
% 3.01/1.70 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,A: set_e] :
% 3.01/1.70 ( ( module_e_b_f_d @ M3 @ R )
% 3.01/1.70 => ( ( ord_less_eq_set_e @ A @ ( carrie730238621_e_b_f @ M3 ) )
% 3.01/1.70 => ( ord_less_eq_set_e @ ( algebr168361254_d_e_f @ R @ M3 @ A ) @ ( carrie730238621_e_b_f @ M3 ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_32_Module_Ofgs__sub__carrier,axiom,
% 3.01/1.70 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,A: set_a] :
% 3.01/1.70 ( ( module1821517916unit_d @ M3 @ R )
% 3.01/1.70 => ( ( ord_less_eq_set_a @ A @ ( carrie1074654371t_unit @ M3 ) )
% 3.01/1.70 => ( ord_less_eq_set_a @ ( algebr1039611956t_unit @ R @ M3 @ A ) @ ( carrie1074654371t_unit @ M3 ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_33_Module_Ofgs__sub__carrier,axiom,
% 3.01/1.70 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,A: set_a] :
% 3.01/1.70 ( ( module_a_b_c_d @ M3 @ R )
% 3.01/1.70 => ( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ M3 ) )
% 3.01/1.70 => ( ord_less_eq_set_a @ ( algebr1152250919_d_a_c @ R @ M3 @ A ) @ ( carrie2021454486_a_b_c @ M3 ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_34_misom__self,axiom,
% 3.01/1.70 misomo1282343797_c_a_c @ r @ m @ m ).
% 3.01/1.70
% 3.01/1.70 thf(fact_35_HOM__is__module,axiom,
% 3.01/1.70 ! [N3: carrie1821755406_e_b_f] :
% 3.01/1.70 ( ( module_e_b_f_d @ N3 @ r )
% 3.01/1.70 => ( module1648870817unit_d @ ( hOM_b_d_a_c_e_f @ r @ m @ N3 ) @ r ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_36_HOM__is__module,axiom,
% 3.01/1.70 ! [N3: carrie722926983_a_b_c] :
% 3.01/1.70 ( ( module_a_b_c_d @ N3 @ r )
% 3.01/1.70 => ( module589927589unit_d @ ( hOM_b_d_a_c_a_c @ r @ m @ N3 ) @ r ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_37_HOM__is__module,axiom,
% 3.01/1.70 ! [N3: carrie1963041556t_unit] :
% 3.01/1.70 ( ( module1821517916unit_d @ N3 @ r )
% 3.01/1.70 => ( module589927589unit_d @ ( hOM_b_717364638t_unit @ r @ m @ N3 ) @ r ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_38_Suc__pred,axiom,
% 3.01/1.70 ! [N: nat] :
% 3.01/1.70 ( ( ord_less_nat @ zero_zero_nat @ N )
% 3.01/1.70 => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 3.01/1.70 = N ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_39_generator__sub__carrier,axiom,
% 3.01/1.70 ! [H: set_a] :
% 3.01/1.70 ( ( generator_b_d_a_c @ r @ m @ H )
% 3.01/1.70 => ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_40_free__generator__generator,axiom,
% 3.01/1.70 ! [H: set_a] :
% 3.01/1.70 ( ( free_g1087686480_d_a_c @ r @ m @ H )
% 3.01/1.70 => ( generator_b_d_a_c @ r @ m @ H ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_41_Module_Ol__comb__mem,axiom,
% 3.01/1.70 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,A2: set_b,H: set_e,S: nat > b,N: nat,M: nat > e] :
% 3.01/1.70 ( ( module_e_b_f_d @ M3 @ R )
% 3.01/1.70 => ( ( ideal_b_d @ R @ A2 )
% 3.01/1.70 => ( ( ord_less_eq_set_e @ H @ ( carrie730238621_e_b_f @ M3 ) )
% 3.01/1.70 => ( ( member_nat_b @ S
% 3.01/1.70 @ ( pi_nat_b
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : A2 ) )
% 3.01/1.70 => ( ( member_nat_e @ M
% 3.01/1.70 @ ( pi_nat_e
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : H ) )
% 3.01/1.70 => ( member_e @ ( l_comb_b_d_e_f @ R @ M3 @ N @ S @ M ) @ ( carrie730238621_e_b_f @ M3 ) ) ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_42_Module_Ol__comb__mem,axiom,
% 3.01/1.70 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,A2: set_b,H: set_a,S: nat > b,N: nat,M: nat > a] :
% 3.01/1.70 ( ( module_a_b_c_d @ M3 @ R )
% 3.01/1.70 => ( ( ideal_b_d @ R @ A2 )
% 3.01/1.70 => ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ M3 ) )
% 3.01/1.70 => ( ( member_nat_b @ S
% 3.01/1.70 @ ( pi_nat_b
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : A2 ) )
% 3.01/1.70 => ( ( member_nat_a @ M
% 3.01/1.70 @ ( pi_nat_a
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : H ) )
% 3.01/1.70 => ( member_a @ ( l_comb_b_d_a_c @ R @ M3 @ N @ S @ M ) @ ( carrie2021454486_a_b_c @ M3 ) ) ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_43_Module_Ol__comb__mem,axiom,
% 3.01/1.70 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,A2: set_b,H: set_a,S: nat > b,N: nat,M: nat > a] :
% 3.01/1.70 ( ( module1821517916unit_d @ M3 @ R )
% 3.01/1.70 => ( ( ideal_b_d @ R @ A2 )
% 3.01/1.70 => ( ( ord_less_eq_set_a @ H @ ( carrie1074654371t_unit @ M3 ) )
% 3.01/1.70 => ( ( member_nat_b @ S
% 3.01/1.70 @ ( pi_nat_b
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : A2 ) )
% 3.01/1.70 => ( ( member_nat_a @ M
% 3.01/1.70 @ ( pi_nat_a
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : H ) )
% 3.01/1.70 => ( member_a @ ( l_comb1138968323t_unit @ R @ M3 @ N @ S @ M ) @ ( carrie1074654371t_unit @ M3 ) ) ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_44_Module_Oliear__comb__memTr,axiom,
% 3.01/1.70 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,A2: set_b,H: set_e,N: nat] :
% 3.01/1.70 ( ( module_e_b_f_d @ M3 @ R )
% 3.01/1.70 => ( ( ideal_b_d @ R @ A2 )
% 3.01/1.70 => ( ( ord_less_eq_set_e @ H @ ( carrie730238621_e_b_f @ M3 ) )
% 3.01/1.70 => ! [S2: nat > b,M2: nat > e] :
% 3.01/1.70 ( ( ( member_nat_b @ S2
% 3.01/1.70 @ ( pi_nat_b
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : A2 ) )
% 3.01/1.70 & ( member_nat_e @ M2
% 3.01/1.70 @ ( pi_nat_e
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : H ) ) )
% 3.01/1.70 => ( member_e @ ( l_comb_b_d_e_f @ R @ M3 @ N @ S2 @ M2 ) @ ( carrie730238621_e_b_f @ M3 ) ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_45_Module_Oliear__comb__memTr,axiom,
% 3.01/1.70 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,A2: set_b,H: set_a,N: nat] :
% 3.01/1.70 ( ( module_a_b_c_d @ M3 @ R )
% 3.01/1.70 => ( ( ideal_b_d @ R @ A2 )
% 3.01/1.70 => ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ M3 ) )
% 3.01/1.70 => ! [S2: nat > b,M2: nat > a] :
% 3.01/1.70 ( ( ( member_nat_b @ S2
% 3.01/1.70 @ ( pi_nat_b
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : A2 ) )
% 3.01/1.70 & ( member_nat_a @ M2
% 3.01/1.70 @ ( pi_nat_a
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : H ) ) )
% 3.01/1.70 => ( member_a @ ( l_comb_b_d_a_c @ R @ M3 @ N @ S2 @ M2 ) @ ( carrie2021454486_a_b_c @ M3 ) ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_46_Module_Oliear__comb__memTr,axiom,
% 3.01/1.70 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,A2: set_b,H: set_a,N: nat] :
% 3.01/1.70 ( ( module1821517916unit_d @ M3 @ R )
% 3.01/1.70 => ( ( ideal_b_d @ R @ A2 )
% 3.01/1.70 => ( ( ord_less_eq_set_a @ H @ ( carrie1074654371t_unit @ M3 ) )
% 3.01/1.70 => ! [S2: nat > b,M2: nat > a] :
% 3.01/1.70 ( ( ( member_nat_b @ S2
% 3.01/1.70 @ ( pi_nat_b
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : A2 ) )
% 3.01/1.70 & ( member_nat_a @ M2
% 3.01/1.70 @ ( pi_nat_a
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : H ) ) )
% 3.01/1.70 => ( member_a @ ( l_comb1138968323t_unit @ R @ M3 @ N @ S2 @ M2 ) @ ( carrie1074654371t_unit @ M3 ) ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_47_zero__less__diff,axiom,
% 3.01/1.70 ! [N: nat,M: nat] :
% 3.01/1.70 ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
% 3.01/1.70 = ( ord_less_nat @ M @ N ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_48_Module_Olinear__comb__eq,axiom,
% 3.01/1.70 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,H: set_e,S: nat > b,N: nat,F: nat > e,G: nat > e] :
% 3.01/1.70 ( ( module_e_b_f_d @ M3 @ R )
% 3.01/1.70 => ( ( ord_less_eq_set_e @ H @ ( carrie730238621_e_b_f @ M3 ) )
% 3.01/1.70 => ( ( member_nat_b @ S
% 3.01/1.70 @ ( pi_nat_b
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ R ) ) )
% 3.01/1.70 => ( ( member_nat_e @ F
% 3.01/1.70 @ ( pi_nat_e
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : H ) )
% 3.01/1.70 => ( ( member_nat_e @ G
% 3.01/1.70 @ ( pi_nat_e
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : H ) )
% 3.01/1.70 => ( ! [X2: nat] :
% 3.01/1.70 ( ( member_nat @ X2
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) ) )
% 3.01/1.70 => ( ( F @ X2 )
% 3.01/1.70 = ( G @ X2 ) ) )
% 3.01/1.70 => ( ( l_comb_b_d_e_f @ R @ M3 @ N @ S @ F )
% 3.01/1.70 = ( l_comb_b_d_e_f @ R @ M3 @ N @ S @ G ) ) ) ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_49_Module_Olinear__comb__eq,axiom,
% 3.01/1.70 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,H: set_a,S: nat > b,N: nat,F: nat > a,G: nat > a] :
% 3.01/1.70 ( ( module_a_b_c_d @ M3 @ R )
% 3.01/1.70 => ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ M3 ) )
% 3.01/1.70 => ( ( member_nat_b @ S
% 3.01/1.70 @ ( pi_nat_b
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ R ) ) )
% 3.01/1.70 => ( ( member_nat_a @ F
% 3.01/1.70 @ ( pi_nat_a
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : H ) )
% 3.01/1.70 => ( ( member_nat_a @ G
% 3.01/1.70 @ ( pi_nat_a
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : H ) )
% 3.01/1.70 => ( ! [X2: nat] :
% 3.01/1.70 ( ( member_nat @ X2
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) ) )
% 3.01/1.70 => ( ( F @ X2 )
% 3.01/1.70 = ( G @ X2 ) ) )
% 3.01/1.70 => ( ( l_comb_b_d_a_c @ R @ M3 @ N @ S @ F )
% 3.01/1.70 = ( l_comb_b_d_a_c @ R @ M3 @ N @ S @ G ) ) ) ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_50_Module_Olinear__comb__eq,axiom,
% 3.01/1.70 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,H: set_a,S: nat > b,N: nat,F: nat > a,G: nat > a] :
% 3.01/1.70 ( ( module1821517916unit_d @ M3 @ R )
% 3.01/1.70 => ( ( ord_less_eq_set_a @ H @ ( carrie1074654371t_unit @ M3 ) )
% 3.01/1.70 => ( ( member_nat_b @ S
% 3.01/1.70 @ ( pi_nat_b
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ R ) ) )
% 3.01/1.70 => ( ( member_nat_a @ F
% 3.01/1.70 @ ( pi_nat_a
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : H ) )
% 3.01/1.70 => ( ( member_nat_a @ G
% 3.01/1.70 @ ( pi_nat_a
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : H ) )
% 3.01/1.70 => ( ! [X2: nat] :
% 3.01/1.70 ( ( member_nat @ X2
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) ) )
% 3.01/1.70 => ( ( F @ X2 )
% 3.01/1.70 = ( G @ X2 ) ) )
% 3.01/1.70 => ( ( l_comb1138968323t_unit @ R @ M3 @ N @ S @ F )
% 3.01/1.70 = ( l_comb1138968323t_unit @ R @ M3 @ N @ S @ G ) ) ) ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_51_Module_Olinear__comb__eqTr,axiom,
% 3.01/1.70 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,H: set_e,S: nat > b,N: nat,F: nat > e,G: nat > e] :
% 3.01/1.70 ( ( module_e_b_f_d @ M3 @ R )
% 3.01/1.70 => ( ( ord_less_eq_set_e @ H @ ( carrie730238621_e_b_f @ M3 ) )
% 3.01/1.70 => ( ( ( member_nat_b @ S
% 3.01/1.70 @ ( pi_nat_b
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ R ) ) )
% 3.01/1.70 & ( member_nat_e @ F
% 3.01/1.70 @ ( pi_nat_e
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : H ) )
% 3.01/1.70 & ( member_nat_e @ G
% 3.01/1.70 @ ( pi_nat_e
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : H ) )
% 3.01/1.70 & ! [X2: nat] :
% 3.01/1.70 ( ( member_nat @ X2
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) ) )
% 3.01/1.70 => ( ( F @ X2 )
% 3.01/1.70 = ( G @ X2 ) ) ) )
% 3.01/1.70 => ( ( l_comb_b_d_e_f @ R @ M3 @ N @ S @ F )
% 3.01/1.70 = ( l_comb_b_d_e_f @ R @ M3 @ N @ S @ G ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_52_Module_Olinear__comb__eqTr,axiom,
% 3.01/1.70 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,H: set_a,S: nat > b,N: nat,F: nat > a,G: nat > a] :
% 3.01/1.70 ( ( module_a_b_c_d @ M3 @ R )
% 3.01/1.70 => ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ M3 ) )
% 3.01/1.70 => ( ( ( member_nat_b @ S
% 3.01/1.70 @ ( pi_nat_b
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ R ) ) )
% 3.01/1.70 & ( member_nat_a @ F
% 3.01/1.70 @ ( pi_nat_a
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : H ) )
% 3.01/1.70 & ( member_nat_a @ G
% 3.01/1.70 @ ( pi_nat_a
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : H ) )
% 3.01/1.70 & ! [X2: nat] :
% 3.01/1.70 ( ( member_nat @ X2
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) ) )
% 3.01/1.70 => ( ( F @ X2 )
% 3.01/1.70 = ( G @ X2 ) ) ) )
% 3.01/1.70 => ( ( l_comb_b_d_a_c @ R @ M3 @ N @ S @ F )
% 3.01/1.70 = ( l_comb_b_d_a_c @ R @ M3 @ N @ S @ G ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_53_Module_Olinear__comb__eqTr,axiom,
% 3.01/1.70 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,H: set_a,S: nat > b,N: nat,F: nat > a,G: nat > a] :
% 3.01/1.70 ( ( module1821517916unit_d @ M3 @ R )
% 3.01/1.70 => ( ( ord_less_eq_set_a @ H @ ( carrie1074654371t_unit @ M3 ) )
% 3.01/1.70 => ( ( ( member_nat_b @ S
% 3.01/1.70 @ ( pi_nat_b
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ R ) ) )
% 3.01/1.70 & ( member_nat_a @ F
% 3.01/1.70 @ ( pi_nat_a
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : H ) )
% 3.01/1.70 & ( member_nat_a @ G
% 3.01/1.70 @ ( pi_nat_a
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.70 @ ^ [Uu: nat] : H ) )
% 3.01/1.70 & ! [X2: nat] :
% 3.01/1.70 ( ( member_nat @ X2
% 3.01/1.70 @ ( collect_nat
% 3.01/1.70 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) ) )
% 3.01/1.70 => ( ( F @ X2 )
% 3.01/1.70 = ( G @ X2 ) ) ) )
% 3.01/1.70 => ( ( l_comb1138968323t_unit @ R @ M3 @ N @ S @ F )
% 3.01/1.70 = ( l_comb1138968323t_unit @ R @ M3 @ N @ S @ G ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_54_diff__is__0__eq,axiom,
% 3.01/1.70 ! [M: nat,N: nat] :
% 3.01/1.70 ( ( ( minus_minus_nat @ M @ N )
% 3.01/1.70 = zero_zero_nat )
% 3.01/1.70 = ( ord_less_eq_nat @ M @ N ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_55_diff__is__0__eq_H,axiom,
% 3.01/1.70 ! [M: nat,N: nat] :
% 3.01/1.70 ( ( ord_less_eq_nat @ M @ N )
% 3.01/1.70 => ( ( minus_minus_nat @ M @ N )
% 3.01/1.70 = zero_zero_nat ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_56_old_Onat_Oinject,axiom,
% 3.01/1.70 ! [Nat: nat,Nat2: nat] :
% 3.01/1.70 ( ( ( suc @ Nat )
% 3.01/1.70 = ( suc @ Nat2 ) )
% 3.01/1.70 = ( Nat = Nat2 ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_57_nat_Oinject,axiom,
% 3.01/1.70 ! [X22: nat,Y2: nat] :
% 3.01/1.70 ( ( ( suc @ X22 )
% 3.01/1.70 = ( suc @ Y2 ) )
% 3.01/1.70 = ( X22 = Y2 ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_58_misom__sym,axiom,
% 3.01/1.70 ! [N3: carrie1821755406_e_b_f] :
% 3.01/1.70 ( ( module_e_b_f_d @ N3 @ r )
% 3.01/1.70 => ( ( misomo298454132_c_e_f @ r @ m @ N3 )
% 3.01/1.70 => ( misomo1752826100_f_a_c @ r @ N3 @ m ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_59_misom__sym,axiom,
% 3.01/1.70 ! [N3: carrie1963041556t_unit] :
% 3.01/1.70 ( ( module1821517916unit_d @ N3 @ r )
% 3.01/1.70 => ( ( misomo1857159554t_unit @ r @ m @ N3 )
% 3.01/1.70 => ( misomo1935876354it_a_c @ r @ N3 @ m ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_60_misom__sym,axiom,
% 3.01/1.70 ! [N3: carrie722926983_a_b_c] :
% 3.01/1.70 ( ( module_a_b_c_d @ N3 @ r )
% 3.01/1.70 => ( ( misomo1282343797_c_a_c @ r @ m @ N3 )
% 3.01/1.70 => ( misomo1282343797_c_a_c @ r @ N3 @ m ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_61_Suc__le__mono,axiom,
% 3.01/1.70 ! [N: nat,M: nat] :
% 3.01/1.70 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
% 3.01/1.70 = ( ord_less_eq_nat @ N @ M ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_62_bot__nat__0_Oextremum,axiom,
% 3.01/1.70 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_63_le0,axiom,
% 3.01/1.70 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_64_Suc__less__eq,axiom,
% 3.01/1.70 ! [M: nat,N: nat] :
% 3.01/1.70 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 3.01/1.70 = ( ord_less_nat @ M @ N ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_65_Suc__mono,axiom,
% 3.01/1.70 ! [M: nat,N: nat] :
% 3.01/1.70 ( ( ord_less_nat @ M @ N )
% 3.01/1.70 => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_66_mem__Collect__eq,axiom,
% 3.01/1.70 ! [A: nat > a,P: ( nat > a ) > $o] :
% 3.01/1.70 ( ( member_nat_a @ A @ ( collect_nat_a @ P ) )
% 3.01/1.70 = ( P @ A ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_67_mem__Collect__eq,axiom,
% 3.01/1.70 ! [A: nat > b,P: ( nat > b ) > $o] :
% 3.01/1.70 ( ( member_nat_b @ A @ ( collect_nat_b @ P ) )
% 3.01/1.70 = ( P @ A ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_68_mem__Collect__eq,axiom,
% 3.01/1.70 ! [A: a,P: a > $o] :
% 3.01/1.70 ( ( member_a @ A @ ( collect_a @ P ) )
% 3.01/1.70 = ( P @ A ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_69_mem__Collect__eq,axiom,
% 3.01/1.70 ! [A: nat,P: nat > $o] :
% 3.01/1.70 ( ( member_nat @ A @ ( collect_nat @ P ) )
% 3.01/1.70 = ( P @ A ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_70_Collect__mem__eq,axiom,
% 3.01/1.70 ! [A2: set_nat_a] :
% 3.01/1.70 ( ( collect_nat_a
% 3.01/1.70 @ ^ [X3: nat > a] : ( member_nat_a @ X3 @ A2 ) )
% 3.01/1.70 = A2 ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_71_Collect__mem__eq,axiom,
% 3.01/1.70 ! [A2: set_nat_b] :
% 3.01/1.70 ( ( collect_nat_b
% 3.01/1.70 @ ^ [X3: nat > b] : ( member_nat_b @ X3 @ A2 ) )
% 3.01/1.70 = A2 ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_72_Collect__mem__eq,axiom,
% 3.01/1.70 ! [A2: set_a] :
% 3.01/1.70 ( ( collect_a
% 3.01/1.70 @ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
% 3.01/1.70 = A2 ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_73_Collect__mem__eq,axiom,
% 3.01/1.70 ! [A2: set_nat] :
% 3.01/1.70 ( ( collect_nat
% 3.01/1.70 @ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
% 3.01/1.70 = A2 ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_74_Collect__cong,axiom,
% 3.01/1.70 ! [P: nat > $o,Q: nat > $o] :
% 3.01/1.70 ( ! [X2: nat] :
% 3.01/1.70 ( ( P @ X2 )
% 3.01/1.70 = ( Q @ X2 ) )
% 3.01/1.70 => ( ( collect_nat @ P )
% 3.01/1.70 = ( collect_nat @ Q ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_75_bot__nat__0_Onot__eq__extremum,axiom,
% 3.01/1.70 ! [A: nat] :
% 3.01/1.70 ( ( A != zero_zero_nat )
% 3.01/1.70 = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_76_less__nat__zero__code,axiom,
% 3.01/1.70 ! [N: nat] :
% 3.01/1.70 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_77_neq0__conv,axiom,
% 3.01/1.70 ! [N: nat] :
% 3.01/1.70 ( ( N != zero_zero_nat )
% 3.01/1.70 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_78_diff__diff__cancel,axiom,
% 3.01/1.70 ! [I: nat,N: nat] :
% 3.01/1.70 ( ( ord_less_eq_nat @ I @ N )
% 3.01/1.70 => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
% 3.01/1.70 = I ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_79_Suc__diff__diff,axiom,
% 3.01/1.70 ! [M: nat,N: nat,K2: nat] :
% 3.01/1.70 ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K2 ) )
% 3.01/1.70 = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K2 ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_80_diff__Suc__Suc,axiom,
% 3.01/1.70 ! [M: nat,N: nat] :
% 3.01/1.70 ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
% 3.01/1.70 = ( minus_minus_nat @ M @ N ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_81_diff__self__eq__0,axiom,
% 3.01/1.70 ! [M: nat] :
% 3.01/1.70 ( ( minus_minus_nat @ M @ M )
% 3.01/1.70 = zero_zero_nat ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_82_diff__0__eq__0,axiom,
% 3.01/1.70 ! [N: nat] :
% 3.01/1.70 ( ( minus_minus_nat @ zero_zero_nat @ N )
% 3.01/1.70 = zero_zero_nat ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_83_misom__trans,axiom,
% 3.01/1.70 ! [L: carrie1821755406_e_b_f,N3: carrie1821755406_e_b_f] :
% 3.01/1.70 ( ( module_e_b_f_d @ L @ r )
% 3.01/1.70 => ( ( module_e_b_f_d @ N3 @ r )
% 3.01/1.70 => ( ( misomo1752826100_f_a_c @ r @ L @ m )
% 3.01/1.70 => ( ( misomo298454132_c_e_f @ r @ m @ N3 )
% 3.01/1.70 => ( misomo768936435_f_e_f @ r @ L @ N3 ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_84_misom__trans,axiom,
% 3.01/1.70 ! [L: carrie1821755406_e_b_f,N3: carrie1963041556t_unit] :
% 3.01/1.70 ( ( module_e_b_f_d @ L @ r )
% 3.01/1.70 => ( ( module1821517916unit_d @ N3 @ r )
% 3.01/1.70 => ( ( misomo1752826100_f_a_c @ r @ L @ m )
% 3.01/1.70 => ( ( misomo1857159554t_unit @ r @ m @ N3 )
% 3.01/1.70 => ( misomo368157697t_unit @ r @ L @ N3 ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_85_misom__trans,axiom,
% 3.01/1.70 ! [L: carrie1963041556t_unit,N3: carrie1821755406_e_b_f] :
% 3.01/1.70 ( ( module1821517916unit_d @ L @ r )
% 3.01/1.70 => ( ( module_e_b_f_d @ N3 @ r )
% 3.01/1.70 => ( ( misomo1935876354it_a_c @ r @ L @ m )
% 3.01/1.70 => ( ( misomo298454132_c_e_f @ r @ m @ N3 )
% 3.01/1.70 => ( misomo951986689it_e_f @ r @ L @ N3 ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_86_misom__trans,axiom,
% 3.01/1.70 ! [L: carrie1963041556t_unit,N3: carrie1963041556t_unit] :
% 3.01/1.70 ( ( module1821517916unit_d @ L @ r )
% 3.01/1.70 => ( ( module1821517916unit_d @ N3 @ r )
% 3.01/1.70 => ( ( misomo1935876354it_a_c @ r @ L @ m )
% 3.01/1.70 => ( ( misomo1857159554t_unit @ r @ m @ N3 )
% 3.01/1.70 => ( misomo493403663t_unit @ r @ L @ N3 ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_87_misom__trans,axiom,
% 3.01/1.70 ! [L: carrie1821755406_e_b_f,N3: carrie722926983_a_b_c] :
% 3.01/1.70 ( ( module_e_b_f_d @ L @ r )
% 3.01/1.70 => ( ( module_a_b_c_d @ N3 @ r )
% 3.01/1.70 => ( ( misomo1752826100_f_a_c @ r @ L @ m )
% 3.01/1.70 => ( ( misomo1282343797_c_a_c @ r @ m @ N3 )
% 3.01/1.70 => ( misomo1752826100_f_a_c @ r @ L @ N3 ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_88_misom__trans,axiom,
% 3.01/1.70 ! [L: carrie1963041556t_unit,N3: carrie722926983_a_b_c] :
% 3.01/1.70 ( ( module1821517916unit_d @ L @ r )
% 3.01/1.70 => ( ( module_a_b_c_d @ N3 @ r )
% 3.01/1.70 => ( ( misomo1935876354it_a_c @ r @ L @ m )
% 3.01/1.70 => ( ( misomo1282343797_c_a_c @ r @ m @ N3 )
% 3.01/1.70 => ( misomo1935876354it_a_c @ r @ L @ N3 ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_89_misom__trans,axiom,
% 3.01/1.70 ! [L: carrie722926983_a_b_c,N3: carrie1821755406_e_b_f] :
% 3.01/1.70 ( ( module_a_b_c_d @ L @ r )
% 3.01/1.70 => ( ( module_e_b_f_d @ N3 @ r )
% 3.01/1.70 => ( ( misomo1282343797_c_a_c @ r @ L @ m )
% 3.01/1.70 => ( ( misomo298454132_c_e_f @ r @ m @ N3 )
% 3.01/1.70 => ( misomo298454132_c_e_f @ r @ L @ N3 ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_90_misom__trans,axiom,
% 3.01/1.70 ! [L: carrie722926983_a_b_c,N3: carrie1963041556t_unit] :
% 3.01/1.70 ( ( module_a_b_c_d @ L @ r )
% 3.01/1.70 => ( ( module1821517916unit_d @ N3 @ r )
% 3.01/1.70 => ( ( misomo1282343797_c_a_c @ r @ L @ m )
% 3.01/1.70 => ( ( misomo1857159554t_unit @ r @ m @ N3 )
% 3.01/1.70 => ( misomo1857159554t_unit @ r @ L @ N3 ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_91_misom__trans,axiom,
% 3.01/1.70 ! [L: carrie722926983_a_b_c,N3: carrie722926983_a_b_c] :
% 3.01/1.70 ( ( module_a_b_c_d @ L @ r )
% 3.01/1.70 => ( ( module_a_b_c_d @ N3 @ r )
% 3.01/1.70 => ( ( misomo1282343797_c_a_c @ r @ L @ m )
% 3.01/1.70 => ( ( misomo1282343797_c_a_c @ r @ m @ N3 )
% 3.01/1.70 => ( misomo1282343797_c_a_c @ r @ L @ N3 ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_92_zero__less__Suc,axiom,
% 3.01/1.70 ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_93_less__Suc0,axiom,
% 3.01/1.70 ! [N: nat] :
% 3.01/1.70 ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
% 3.01/1.70 = ( N = zero_zero_nat ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_94_Module_OHOM__is__module,axiom,
% 3.01/1.70 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,N3: carrie1821755406_e_b_f] :
% 3.01/1.70 ( ( module_e_b_f_d @ M3 @ R )
% 3.01/1.70 => ( ( module_e_b_f_d @ N3 @ R )
% 3.01/1.70 => ( module241733029unit_d @ ( hOM_b_d_e_f_e_f @ R @ M3 @ N3 ) @ R ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_95_Module_OHOM__is__module,axiom,
% 3.01/1.70 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,N3: carrie722926983_a_b_c] :
% 3.01/1.70 ( ( module_e_b_f_d @ M3 @ R )
% 3.01/1.70 => ( ( module_a_b_c_d @ N3 @ R )
% 3.01/1.70 => ( module1330273449unit_d @ ( hOM_b_d_e_f_a_c @ R @ M3 @ N3 ) @ R ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_96_Module_OHOM__is__module,axiom,
% 3.01/1.70 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,N3: carrie1963041556t_unit] :
% 3.01/1.70 ( ( module_e_b_f_d @ M3 @ R )
% 3.01/1.70 => ( ( module1821517916unit_d @ N3 @ R )
% 3.01/1.70 => ( module1330273449unit_d @ ( hOM_b_1375846429t_unit @ R @ M3 @ N3 ) @ R ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_97_Module_OHOM__is__module,axiom,
% 3.01/1.70 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,N3: carrie1821755406_e_b_f] :
% 3.01/1.70 ( ( module_a_b_c_d @ M3 @ R )
% 3.01/1.70 => ( ( module_e_b_f_d @ N3 @ R )
% 3.01/1.70 => ( module1648870817unit_d @ ( hOM_b_d_a_c_e_f @ R @ M3 @ N3 ) @ R ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_98_Module_OHOM__is__module,axiom,
% 3.01/1.70 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,N3: carrie722926983_a_b_c] :
% 3.01/1.70 ( ( module_a_b_c_d @ M3 @ R )
% 3.01/1.70 => ( ( module_a_b_c_d @ N3 @ R )
% 3.01/1.70 => ( module589927589unit_d @ ( hOM_b_d_a_c_a_c @ R @ M3 @ N3 ) @ R ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_99_Module_OHOM__is__module,axiom,
% 3.01/1.70 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,N3: carrie1963041556t_unit] :
% 3.01/1.70 ( ( module_a_b_c_d @ M3 @ R )
% 3.01/1.70 => ( ( module1821517916unit_d @ N3 @ R )
% 3.01/1.70 => ( module589927589unit_d @ ( hOM_b_717364638t_unit @ R @ M3 @ N3 ) @ R ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_100_Module_OHOM__is__module,axiom,
% 3.01/1.70 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,N3: carrie1821755406_e_b_f] :
% 3.01/1.70 ( ( module1821517916unit_d @ M3 @ R )
% 3.01/1.70 => ( ( module_e_b_f_d @ N3 @ R )
% 3.01/1.70 => ( module1648870817unit_d @ ( hOM_b_1959675421it_e_f @ R @ M3 @ N3 ) @ R ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_101_Module_OHOM__is__module,axiom,
% 3.01/1.70 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,N3: carrie722926983_a_b_c] :
% 3.01/1.70 ( ( module1821517916unit_d @ M3 @ R )
% 3.01/1.70 => ( ( module_a_b_c_d @ N3 @ R )
% 3.01/1.70 => ( module589927589unit_d @ ( hOM_b_796081438it_a_c @ R @ M3 @ N3 ) @ R ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_102_Module_OHOM__is__module,axiom,
% 3.01/1.70 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,N3: carrie1963041556t_unit] :
% 3.01/1.70 ( ( module1821517916unit_d @ M3 @ R )
% 3.01/1.70 => ( ( module1821517916unit_d @ N3 @ R )
% 3.01/1.70 => ( module589927589unit_d @ ( hOM_b_1103679019t_unit @ R @ M3 @ N3 ) @ R ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_103_Module_Omisom__trans,axiom,
% 3.01/1.70 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,L: carrie1821755406_e_b_f,N3: carrie1821755406_e_b_f] :
% 3.01/1.70 ( ( module_e_b_f_d @ M3 @ R )
% 3.01/1.70 => ( ( module_e_b_f_d @ L @ R )
% 3.01/1.70 => ( ( module_e_b_f_d @ N3 @ R )
% 3.01/1.70 => ( ( misomo768936435_f_e_f @ R @ L @ M3 )
% 3.01/1.70 => ( ( misomo768936435_f_e_f @ R @ M3 @ N3 )
% 3.01/1.70 => ( misomo768936435_f_e_f @ R @ L @ N3 ) ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_104_Module_Omisom__trans,axiom,
% 3.01/1.70 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,L: carrie1821755406_e_b_f,N3: carrie722926983_a_b_c] :
% 3.01/1.70 ( ( module_e_b_f_d @ M3 @ R )
% 3.01/1.70 => ( ( module_e_b_f_d @ L @ R )
% 3.01/1.70 => ( ( module_a_b_c_d @ N3 @ R )
% 3.01/1.70 => ( ( misomo768936435_f_e_f @ R @ L @ M3 )
% 3.01/1.70 => ( ( misomo1752826100_f_a_c @ R @ M3 @ N3 )
% 3.01/1.70 => ( misomo1752826100_f_a_c @ R @ L @ N3 ) ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_105_Module_Omisom__trans,axiom,
% 3.01/1.70 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,L: carrie1821755406_e_b_f,N3: carrie1963041556t_unit] :
% 3.01/1.70 ( ( module_e_b_f_d @ M3 @ R )
% 3.01/1.70 => ( ( module_e_b_f_d @ L @ R )
% 3.01/1.70 => ( ( module1821517916unit_d @ N3 @ R )
% 3.01/1.70 => ( ( misomo768936435_f_e_f @ R @ L @ M3 )
% 3.01/1.70 => ( ( misomo368157697t_unit @ R @ M3 @ N3 )
% 3.01/1.70 => ( misomo368157697t_unit @ R @ L @ N3 ) ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_106_Module_Omisom__trans,axiom,
% 3.01/1.70 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,L: carrie722926983_a_b_c,N3: carrie1821755406_e_b_f] :
% 3.01/1.70 ( ( module_e_b_f_d @ M3 @ R )
% 3.01/1.70 => ( ( module_a_b_c_d @ L @ R )
% 3.01/1.70 => ( ( module_e_b_f_d @ N3 @ R )
% 3.01/1.70 => ( ( misomo298454132_c_e_f @ R @ L @ M3 )
% 3.01/1.70 => ( ( misomo768936435_f_e_f @ R @ M3 @ N3 )
% 3.01/1.70 => ( misomo298454132_c_e_f @ R @ L @ N3 ) ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_107_Module_Omisom__trans,axiom,
% 3.01/1.70 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,L: carrie722926983_a_b_c,N3: carrie1963041556t_unit] :
% 3.01/1.70 ( ( module_e_b_f_d @ M3 @ R )
% 3.01/1.70 => ( ( module_a_b_c_d @ L @ R )
% 3.01/1.70 => ( ( module1821517916unit_d @ N3 @ R )
% 3.01/1.70 => ( ( misomo298454132_c_e_f @ R @ L @ M3 )
% 3.01/1.70 => ( ( misomo368157697t_unit @ R @ M3 @ N3 )
% 3.01/1.70 => ( misomo1857159554t_unit @ R @ L @ N3 ) ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_108_Module_Omisom__trans,axiom,
% 3.01/1.70 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,L: carrie1963041556t_unit,N3: carrie1821755406_e_b_f] :
% 3.01/1.70 ( ( module_e_b_f_d @ M3 @ R )
% 3.01/1.70 => ( ( module1821517916unit_d @ L @ R )
% 3.01/1.70 => ( ( module_e_b_f_d @ N3 @ R )
% 3.01/1.70 => ( ( misomo951986689it_e_f @ R @ L @ M3 )
% 3.01/1.70 => ( ( misomo768936435_f_e_f @ R @ M3 @ N3 )
% 3.01/1.70 => ( misomo951986689it_e_f @ R @ L @ N3 ) ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_109_Module_Omisom__trans,axiom,
% 3.01/1.70 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,L: carrie1963041556t_unit,N3: carrie722926983_a_b_c] :
% 3.01/1.70 ( ( module_e_b_f_d @ M3 @ R )
% 3.01/1.70 => ( ( module1821517916unit_d @ L @ R )
% 3.01/1.70 => ( ( module_a_b_c_d @ N3 @ R )
% 3.01/1.70 => ( ( misomo951986689it_e_f @ R @ L @ M3 )
% 3.01/1.70 => ( ( misomo1752826100_f_a_c @ R @ M3 @ N3 )
% 3.01/1.70 => ( misomo1935876354it_a_c @ R @ L @ N3 ) ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_110_Module_Omisom__trans,axiom,
% 3.01/1.70 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,L: carrie1963041556t_unit,N3: carrie1963041556t_unit] :
% 3.01/1.70 ( ( module_e_b_f_d @ M3 @ R )
% 3.01/1.70 => ( ( module1821517916unit_d @ L @ R )
% 3.01/1.70 => ( ( module1821517916unit_d @ N3 @ R )
% 3.01/1.70 => ( ( misomo951986689it_e_f @ R @ L @ M3 )
% 3.01/1.70 => ( ( misomo368157697t_unit @ R @ M3 @ N3 )
% 3.01/1.70 => ( misomo493403663t_unit @ R @ L @ N3 ) ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_111_Module_Omisom__trans,axiom,
% 3.01/1.70 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,L: carrie1821755406_e_b_f,N3: carrie1821755406_e_b_f] :
% 3.01/1.70 ( ( module_a_b_c_d @ M3 @ R )
% 3.01/1.70 => ( ( module_e_b_f_d @ L @ R )
% 3.01/1.70 => ( ( module_e_b_f_d @ N3 @ R )
% 3.01/1.70 => ( ( misomo1752826100_f_a_c @ R @ L @ M3 )
% 3.01/1.70 => ( ( misomo298454132_c_e_f @ R @ M3 @ N3 )
% 3.01/1.70 => ( misomo768936435_f_e_f @ R @ L @ N3 ) ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_112_Module_Omisom__trans,axiom,
% 3.01/1.70 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,L: carrie1821755406_e_b_f,N3: carrie1963041556t_unit] :
% 3.01/1.70 ( ( module_a_b_c_d @ M3 @ R )
% 3.01/1.70 => ( ( module_e_b_f_d @ L @ R )
% 3.01/1.70 => ( ( module1821517916unit_d @ N3 @ R )
% 3.01/1.70 => ( ( misomo1752826100_f_a_c @ R @ L @ M3 )
% 3.01/1.70 => ( ( misomo1857159554t_unit @ R @ M3 @ N3 )
% 3.01/1.70 => ( misomo368157697t_unit @ R @ L @ N3 ) ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_113_Module_Omisom__self,axiom,
% 3.01/1.70 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d] :
% 3.01/1.70 ( ( module_e_b_f_d @ M3 @ R )
% 3.01/1.70 => ( misomo768936435_f_e_f @ R @ M3 @ M3 ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_114_Module_Omisom__self,axiom,
% 3.01/1.70 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d] :
% 3.01/1.70 ( ( module1821517916unit_d @ M3 @ R )
% 3.01/1.70 => ( misomo493403663t_unit @ R @ M3 @ M3 ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_115_Module_Omisom__self,axiom,
% 3.01/1.70 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d] :
% 3.01/1.70 ( ( module_a_b_c_d @ M3 @ R )
% 3.01/1.70 => ( misomo1282343797_c_a_c @ R @ M3 @ M3 ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_116_Module_Omisom__sym,axiom,
% 3.01/1.70 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,N3: carrie1821755406_e_b_f] :
% 3.01/1.70 ( ( module_e_b_f_d @ M3 @ R )
% 3.01/1.70 => ( ( module_e_b_f_d @ N3 @ R )
% 3.01/1.70 => ( ( misomo768936435_f_e_f @ R @ M3 @ N3 )
% 3.01/1.70 => ( misomo768936435_f_e_f @ R @ N3 @ M3 ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_117_Module_Omisom__sym,axiom,
% 3.01/1.70 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,N3: carrie722926983_a_b_c] :
% 3.01/1.70 ( ( module_e_b_f_d @ M3 @ R )
% 3.01/1.70 => ( ( module_a_b_c_d @ N3 @ R )
% 3.01/1.70 => ( ( misomo1752826100_f_a_c @ R @ M3 @ N3 )
% 3.01/1.70 => ( misomo298454132_c_e_f @ R @ N3 @ M3 ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_118_Module_Omisom__sym,axiom,
% 3.01/1.70 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,N3: carrie1963041556t_unit] :
% 3.01/1.70 ( ( module_e_b_f_d @ M3 @ R )
% 3.01/1.70 => ( ( module1821517916unit_d @ N3 @ R )
% 3.01/1.70 => ( ( misomo368157697t_unit @ R @ M3 @ N3 )
% 3.01/1.70 => ( misomo951986689it_e_f @ R @ N3 @ M3 ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_119_Module_Omisom__sym,axiom,
% 3.01/1.70 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,N3: carrie1821755406_e_b_f] :
% 3.01/1.70 ( ( module_a_b_c_d @ M3 @ R )
% 3.01/1.70 => ( ( module_e_b_f_d @ N3 @ R )
% 3.01/1.70 => ( ( misomo298454132_c_e_f @ R @ M3 @ N3 )
% 3.01/1.70 => ( misomo1752826100_f_a_c @ R @ N3 @ M3 ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_120_Module_Omisom__sym,axiom,
% 3.01/1.70 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,N3: carrie1963041556t_unit] :
% 3.01/1.70 ( ( module_a_b_c_d @ M3 @ R )
% 3.01/1.70 => ( ( module1821517916unit_d @ N3 @ R )
% 3.01/1.70 => ( ( misomo1857159554t_unit @ R @ M3 @ N3 )
% 3.01/1.70 => ( misomo1935876354it_a_c @ R @ N3 @ M3 ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_121_Module_Omisom__sym,axiom,
% 3.01/1.70 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,N3: carrie1821755406_e_b_f] :
% 3.01/1.70 ( ( module1821517916unit_d @ M3 @ R )
% 3.01/1.70 => ( ( module_e_b_f_d @ N3 @ R )
% 3.01/1.70 => ( ( misomo951986689it_e_f @ R @ M3 @ N3 )
% 3.01/1.70 => ( misomo368157697t_unit @ R @ N3 @ M3 ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_122_Module_Omisom__sym,axiom,
% 3.01/1.70 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,N3: carrie722926983_a_b_c] :
% 3.01/1.70 ( ( module1821517916unit_d @ M3 @ R )
% 3.01/1.70 => ( ( module_a_b_c_d @ N3 @ R )
% 3.01/1.70 => ( ( misomo1935876354it_a_c @ R @ M3 @ N3 )
% 3.01/1.70 => ( misomo1857159554t_unit @ R @ N3 @ M3 ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_123_Module_Omisom__sym,axiom,
% 3.01/1.70 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,N3: carrie1963041556t_unit] :
% 3.01/1.70 ( ( module1821517916unit_d @ M3 @ R )
% 3.01/1.70 => ( ( module1821517916unit_d @ N3 @ R )
% 3.01/1.70 => ( ( misomo493403663t_unit @ R @ M3 @ N3 )
% 3.01/1.70 => ( misomo493403663t_unit @ R @ N3 @ M3 ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_124_Module_Omisom__sym,axiom,
% 3.01/1.70 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,N3: carrie722926983_a_b_c] :
% 3.01/1.70 ( ( module_a_b_c_d @ M3 @ R )
% 3.01/1.70 => ( ( module_a_b_c_d @ N3 @ R )
% 3.01/1.70 => ( ( misomo1282343797_c_a_c @ R @ M3 @ N3 )
% 3.01/1.70 => ( misomo1282343797_c_a_c @ R @ N3 @ M3 ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_125_Module_Ofree__generator__generator,axiom,
% 3.01/1.70 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,H: set_e] :
% 3.01/1.70 ( ( module_e_b_f_d @ M3 @ R )
% 3.01/1.70 => ( ( free_g103796815_d_e_f @ R @ M3 @ H )
% 3.01/1.70 => ( generator_b_d_e_f @ R @ M3 @ H ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_126_Module_Ofree__generator__generator,axiom,
% 3.01/1.70 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,H: set_a] :
% 3.01/1.70 ( ( module1821517916unit_d @ M3 @ R )
% 3.01/1.70 => ( ( free_g637607517t_unit @ R @ M3 @ H )
% 3.01/1.70 => ( genera1692266857t_unit @ R @ M3 @ H ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_127_Module_Ofree__generator__generator,axiom,
% 3.01/1.70 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,H: set_a] :
% 3.01/1.70 ( ( module_a_b_c_d @ M3 @ R )
% 3.01/1.70 => ( ( free_g1087686480_d_a_c @ R @ M3 @ H )
% 3.01/1.70 => ( generator_b_d_a_c @ R @ M3 @ H ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_128_Module_Ogenerator__sub__carrier,axiom,
% 3.01/1.70 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,H: set_e] :
% 3.01/1.70 ( ( module_e_b_f_d @ M3 @ R )
% 3.01/1.70 => ( ( generator_b_d_e_f @ R @ M3 @ H )
% 3.01/1.70 => ( ord_less_eq_set_e @ H @ ( carrie730238621_e_b_f @ M3 ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_129_Module_Ogenerator__sub__carrier,axiom,
% 3.01/1.70 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,H: set_a] :
% 3.01/1.70 ( ( module_a_b_c_d @ M3 @ R )
% 3.01/1.70 => ( ( generator_b_d_a_c @ R @ M3 @ H )
% 3.01/1.70 => ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ M3 ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_130_Module_Ogenerator__sub__carrier,axiom,
% 3.01/1.70 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,H: set_a] :
% 3.01/1.70 ( ( module1821517916unit_d @ M3 @ R )
% 3.01/1.70 => ( ( genera1692266857t_unit @ R @ M3 @ H )
% 3.01/1.70 => ( ord_less_eq_set_a @ H @ ( carrie1074654371t_unit @ M3 ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_131_Module_OAnn__is__ideal,axiom,
% 3.01/1.70 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d] :
% 3.01/1.70 ( ( module_e_b_f_d @ M3 @ R )
% 3.01/1.70 => ( ideal_b_d @ R @ ( annihilator_b_d_e_f @ R @ M3 ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_132_Module_OAnn__is__ideal,axiom,
% 3.01/1.70 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d] :
% 3.01/1.70 ( ( module1821517916unit_d @ M3 @ R )
% 3.01/1.70 => ( ideal_b_d @ R @ ( annihi259882159t_unit @ R @ M3 ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_133_Module_OAnn__is__ideal,axiom,
% 3.01/1.70 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d] :
% 3.01/1.70 ( ( module_a_b_c_d @ M3 @ R )
% 3.01/1.70 => ( ideal_b_d @ R @ ( annihilator_b_d_a_c @ R @ M3 ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_134_Nat_Oex__has__greatest__nat,axiom,
% 3.01/1.70 ! [P: nat > $o,K2: nat,B: nat] :
% 3.01/1.70 ( ( P @ K2 )
% 3.01/1.70 => ( ! [Y: nat] :
% 3.01/1.70 ( ( P @ Y )
% 3.01/1.70 => ( ord_less_eq_nat @ Y @ B ) )
% 3.01/1.70 => ? [X2: nat] :
% 3.01/1.70 ( ( P @ X2 )
% 3.01/1.70 & ! [Y3: nat] :
% 3.01/1.70 ( ( P @ Y3 )
% 3.01/1.70 => ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_135_nat__le__linear,axiom,
% 3.01/1.70 ! [M: nat,N: nat] :
% 3.01/1.70 ( ( ord_less_eq_nat @ M @ N )
% 3.01/1.70 | ( ord_less_eq_nat @ N @ M ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_136_le__antisym,axiom,
% 3.01/1.70 ! [M: nat,N: nat] :
% 3.01/1.70 ( ( ord_less_eq_nat @ M @ N )
% 3.01/1.70 => ( ( ord_less_eq_nat @ N @ M )
% 3.01/1.70 => ( M = N ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_137_le__trans,axiom,
% 3.01/1.70 ! [I: nat,J3: nat,K2: nat] :
% 3.01/1.70 ( ( ord_less_eq_nat @ I @ J3 )
% 3.01/1.70 => ( ( ord_less_eq_nat @ J3 @ K2 )
% 3.01/1.70 => ( ord_less_eq_nat @ I @ K2 ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_138_n__not__Suc__n,axiom,
% 3.01/1.70 ! [N: nat] :
% 3.01/1.70 ( N
% 3.01/1.70 != ( suc @ N ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_139_Suc__inject,axiom,
% 3.01/1.70 ! [X: nat,Y4: nat] :
% 3.01/1.70 ( ( ( suc @ X )
% 3.01/1.70 = ( suc @ Y4 ) )
% 3.01/1.70 => ( X = Y4 ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_140_linorder__neqE__nat,axiom,
% 3.01/1.70 ! [X: nat,Y4: nat] :
% 3.01/1.70 ( ( X != Y4 )
% 3.01/1.70 => ( ~ ( ord_less_nat @ X @ Y4 )
% 3.01/1.70 => ( ord_less_nat @ Y4 @ X ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_141_infinite__descent,axiom,
% 3.01/1.70 ! [P: nat > $o,N: nat] :
% 3.01/1.70 ( ! [N2: nat] :
% 3.01/1.70 ( ~ ( P @ N2 )
% 3.01/1.70 => ? [M2: nat] :
% 3.01/1.70 ( ( ord_less_nat @ M2 @ N2 )
% 3.01/1.70 & ~ ( P @ M2 ) ) )
% 3.01/1.70 => ( P @ N ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_142_nat__less__induct,axiom,
% 3.01/1.70 ! [P: nat > $o,N: nat] :
% 3.01/1.70 ( ! [N2: nat] :
% 3.01/1.70 ( ! [M2: nat] :
% 3.01/1.70 ( ( ord_less_nat @ M2 @ N2 )
% 3.01/1.70 => ( P @ M2 ) )
% 3.01/1.70 => ( P @ N2 ) )
% 3.01/1.70 => ( P @ N ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_143_less__irrefl__nat,axiom,
% 3.01/1.70 ! [N: nat] :
% 3.01/1.70 ~ ( ord_less_nat @ N @ N ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_144_less__not__refl3,axiom,
% 3.01/1.70 ! [S: nat,T: nat] :
% 3.01/1.70 ( ( ord_less_nat @ S @ T )
% 3.01/1.70 => ( S != T ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_145_less__not__refl2,axiom,
% 3.01/1.70 ! [N: nat,M: nat] :
% 3.01/1.70 ( ( ord_less_nat @ N @ M )
% 3.01/1.70 => ( M != N ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_146_less__not__refl,axiom,
% 3.01/1.70 ! [N: nat] :
% 3.01/1.70 ~ ( ord_less_nat @ N @ N ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_147_nat__neq__iff,axiom,
% 3.01/1.70 ! [M: nat,N: nat] :
% 3.01/1.70 ( ( M != N )
% 3.01/1.70 = ( ( ord_less_nat @ M @ N )
% 3.01/1.70 | ( ord_less_nat @ N @ M ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_148_diff__commute,axiom,
% 3.01/1.70 ! [I: nat,J3: nat,K2: nat] :
% 3.01/1.70 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J3 ) @ K2 )
% 3.01/1.70 = ( minus_minus_nat @ ( minus_minus_nat @ I @ K2 ) @ J3 ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_149_Module_Osmodule__ideal__coeff__is__Submodule,axiom,
% 3.01/1.70 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,A2: set_b] :
% 3.01/1.70 ( ( module_e_b_f_d @ M3 @ R )
% 3.01/1.70 => ( ( ideal_b_d @ R @ A2 )
% 3.01/1.70 => ( submodule_b_d_e_f @ R @ M3 @ ( smodul1982583723_d_e_f @ R @ M3 @ A2 ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_150_Module_Osmodule__ideal__coeff__is__Submodule,axiom,
% 3.01/1.70 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,A2: set_b] :
% 3.01/1.70 ( ( module1821517916unit_d @ M3 @ R )
% 3.01/1.70 => ( ( ideal_b_d @ R @ A2 )
% 3.01/1.70 => ( submod903911234t_unit @ R @ M3 @ ( smodul132175289t_unit @ R @ M3 @ A2 ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_151_Module_Osmodule__ideal__coeff__is__Submodule,axiom,
% 3.01/1.70 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,A2: set_b] :
% 3.01/1.70 ( ( module_a_b_c_d @ M3 @ R )
% 3.01/1.70 => ( ( ideal_b_d @ R @ A2 )
% 3.01/1.70 => ( submodule_b_d_a_c @ R @ M3 @ ( smodul818989740_d_a_c @ R @ M3 @ A2 ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_152_Module_Oquotient__of__submodules__is__ideal,axiom,
% 3.01/1.70 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,P: set_e,Q: set_e] :
% 3.01/1.70 ( ( module_e_b_f_d @ M3 @ R )
% 3.01/1.70 => ( ( submodule_b_d_e_f @ R @ M3 @ P )
% 3.01/1.70 => ( ( submodule_b_d_e_f @ R @ M3 @ Q )
% 3.01/1.70 => ( ideal_b_d @ R @ ( quotie169863047_d_e_f @ R @ M3 @ P @ Q ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_153_Module_Oquotient__of__submodules__is__ideal,axiom,
% 3.01/1.70 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,P: set_a,Q: set_a] :
% 3.01/1.70 ( ( module1821517916unit_d @ M3 @ R )
% 3.01/1.70 => ( ( submod903911234t_unit @ R @ M3 @ P )
% 3.01/1.70 => ( ( submod903911234t_unit @ R @ M3 @ Q )
% 3.01/1.70 => ( ideal_b_d @ R @ ( quotie1196826005t_unit @ R @ M3 @ P @ Q ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_154_Module_Oquotient__of__submodules__is__ideal,axiom,
% 3.01/1.70 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,P: set_a,Q: set_a] :
% 3.01/1.70 ( ( module_a_b_c_d @ M3 @ R )
% 3.01/1.70 => ( ( submodule_b_d_a_c @ R @ M3 @ P )
% 3.01/1.70 => ( ( submodule_b_d_a_c @ R @ M3 @ Q )
% 3.01/1.70 => ( ideal_b_d @ R @ ( quotie1153752712_d_a_c @ R @ M3 @ P @ Q ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_155_transitive__stepwise__le,axiom,
% 3.01/1.70 ! [M: nat,N: nat,R: nat > nat > $o] :
% 3.01/1.70 ( ( ord_less_eq_nat @ M @ N )
% 3.01/1.70 => ( ! [X2: nat] : ( R @ X2 @ X2 )
% 3.01/1.70 => ( ! [X2: nat,Y: nat,Z: nat] :
% 3.01/1.70 ( ( R @ X2 @ Y )
% 3.01/1.70 => ( ( R @ Y @ Z )
% 3.01/1.70 => ( R @ X2 @ Z ) ) )
% 3.01/1.70 => ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
% 3.01/1.70 => ( R @ M @ N ) ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_156_nat__induct__at__least,axiom,
% 3.01/1.70 ! [M: nat,N: nat,P: nat > $o] :
% 3.01/1.70 ( ( ord_less_eq_nat @ M @ N )
% 3.01/1.70 => ( ( P @ M )
% 3.01/1.70 => ( ! [N2: nat] :
% 3.01/1.70 ( ( ord_less_eq_nat @ M @ N2 )
% 3.01/1.70 => ( ( P @ N2 )
% 3.01/1.70 => ( P @ ( suc @ N2 ) ) ) )
% 3.01/1.70 => ( P @ N ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_157_full__nat__induct,axiom,
% 3.01/1.70 ! [P: nat > $o,N: nat] :
% 3.01/1.70 ( ! [N2: nat] :
% 3.01/1.70 ( ! [M2: nat] :
% 3.01/1.70 ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
% 3.01/1.70 => ( P @ M2 ) )
% 3.01/1.70 => ( P @ N2 ) )
% 3.01/1.70 => ( P @ N ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_158_not__less__eq__eq,axiom,
% 3.01/1.70 ! [M: nat,N: nat] :
% 3.01/1.70 ( ( ~ ( ord_less_eq_nat @ M @ N ) )
% 3.01/1.70 = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_159_Suc__n__not__le__n,axiom,
% 3.01/1.70 ! [N: nat] :
% 3.01/1.70 ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_160_le__Suc__eq,axiom,
% 3.01/1.70 ! [M: nat,N: nat] :
% 3.01/1.70 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 3.01/1.70 = ( ( ord_less_eq_nat @ M @ N )
% 3.01/1.70 | ( M
% 3.01/1.70 = ( suc @ N ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_161_Suc__le__D,axiom,
% 3.01/1.70 ! [N: nat,M4: nat] :
% 3.01/1.70 ( ( ord_less_eq_nat @ ( suc @ N ) @ M4 )
% 3.01/1.70 => ? [M5: nat] :
% 3.01/1.70 ( M4
% 3.01/1.70 = ( suc @ M5 ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_162_le__SucI,axiom,
% 3.01/1.70 ! [M: nat,N: nat] :
% 3.01/1.70 ( ( ord_less_eq_nat @ M @ N )
% 3.01/1.70 => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_163_le__SucE,axiom,
% 3.01/1.70 ! [M: nat,N: nat] :
% 3.01/1.70 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 3.01/1.70 => ( ~ ( ord_less_eq_nat @ M @ N )
% 3.01/1.70 => ( M
% 3.01/1.70 = ( suc @ N ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_164_Suc__leD,axiom,
% 3.01/1.70 ! [M: nat,N: nat] :
% 3.01/1.70 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 3.01/1.70 => ( ord_less_eq_nat @ M @ N ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_165_bot__nat__0_Oextremum__uniqueI,axiom,
% 3.01/1.70 ! [A: nat] :
% 3.01/1.70 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 3.01/1.70 => ( A = zero_zero_nat ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_166_bot__nat__0_Oextremum__unique,axiom,
% 3.01/1.70 ! [A: nat] :
% 3.01/1.70 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 3.01/1.70 = ( A = zero_zero_nat ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_167_le__0__eq,axiom,
% 3.01/1.70 ! [N: nat] :
% 3.01/1.70 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 3.01/1.70 = ( N = zero_zero_nat ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_168_less__eq__nat_Osimps_I1_J,axiom,
% 3.01/1.70 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_169_not0__implies__Suc,axiom,
% 3.01/1.70 ! [N: nat] :
% 3.01/1.70 ( ( N != zero_zero_nat )
% 3.01/1.70 => ? [M5: nat] :
% 3.01/1.70 ( N
% 3.01/1.70 = ( suc @ M5 ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_170_old_Onat_Oinducts,axiom,
% 3.01/1.70 ! [P: nat > $o,Nat: nat] :
% 3.01/1.70 ( ( P @ zero_zero_nat )
% 3.01/1.70 => ( ! [Nat3: nat] :
% 3.01/1.70 ( ( P @ Nat3 )
% 3.01/1.70 => ( P @ ( suc @ Nat3 ) ) )
% 3.01/1.70 => ( P @ Nat ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_171_old_Onat_Oexhaust,axiom,
% 3.01/1.70 ! [Y4: nat] :
% 3.01/1.70 ( ( Y4 != zero_zero_nat )
% 3.01/1.70 => ~ ! [Nat3: nat] :
% 3.01/1.70 ( Y4
% 3.01/1.70 != ( suc @ Nat3 ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_172_Zero__not__Suc,axiom,
% 3.01/1.70 ! [M: nat] :
% 3.01/1.70 ( zero_zero_nat
% 3.01/1.70 != ( suc @ M ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_173_Zero__neq__Suc,axiom,
% 3.01/1.70 ! [M: nat] :
% 3.01/1.70 ( zero_zero_nat
% 3.01/1.70 != ( suc @ M ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_174_Suc__neq__Zero,axiom,
% 3.01/1.70 ! [M: nat] :
% 3.01/1.70 ( ( suc @ M )
% 3.01/1.70 != zero_zero_nat ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_175_zero__induct,axiom,
% 3.01/1.70 ! [P: nat > $o,K2: nat] :
% 3.01/1.70 ( ( P @ K2 )
% 3.01/1.70 => ( ! [N2: nat] :
% 3.01/1.70 ( ( P @ ( suc @ N2 ) )
% 3.01/1.70 => ( P @ N2 ) )
% 3.01/1.70 => ( P @ zero_zero_nat ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_176_diff__induct,axiom,
% 3.01/1.70 ! [P: nat > nat > $o,M: nat,N: nat] :
% 3.01/1.70 ( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
% 3.01/1.70 => ( ! [Y: nat] : ( P @ zero_zero_nat @ ( suc @ Y ) )
% 3.01/1.70 => ( ! [X2: nat,Y: nat] :
% 3.01/1.70 ( ( P @ X2 @ Y )
% 3.01/1.70 => ( P @ ( suc @ X2 ) @ ( suc @ Y ) ) )
% 3.01/1.70 => ( P @ M @ N ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_177_nat__induct,axiom,
% 3.01/1.70 ! [P: nat > $o,N: nat] :
% 3.01/1.70 ( ( P @ zero_zero_nat )
% 3.01/1.70 => ( ! [N2: nat] :
% 3.01/1.70 ( ( P @ N2 )
% 3.01/1.70 => ( P @ ( suc @ N2 ) ) )
% 3.01/1.70 => ( P @ N ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_178_nat_OdiscI,axiom,
% 3.01/1.70 ! [Nat: nat,X22: nat] :
% 3.01/1.70 ( ( Nat
% 3.01/1.70 = ( suc @ X22 ) )
% 3.01/1.70 => ( Nat != zero_zero_nat ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_179_old_Onat_Odistinct_I1_J,axiom,
% 3.01/1.70 ! [Nat2: nat] :
% 3.01/1.70 ( zero_zero_nat
% 3.01/1.70 != ( suc @ Nat2 ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_180_old_Onat_Odistinct_I2_J,axiom,
% 3.01/1.70 ! [Nat2: nat] :
% 3.01/1.70 ( ( suc @ Nat2 )
% 3.01/1.70 != zero_zero_nat ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_181_nat_Odistinct_I1_J,axiom,
% 3.01/1.70 ! [X22: nat] :
% 3.01/1.70 ( zero_zero_nat
% 3.01/1.70 != ( suc @ X22 ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_182_less__mono__imp__le__mono,axiom,
% 3.01/1.70 ! [F: nat > nat,I: nat,J3: nat] :
% 3.01/1.70 ( ! [I2: nat,J4: nat] :
% 3.01/1.70 ( ( ord_less_nat @ I2 @ J4 )
% 3.01/1.70 => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J4 ) ) )
% 3.01/1.70 => ( ( ord_less_eq_nat @ I @ J3 )
% 3.01/1.70 => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J3 ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_183_le__neq__implies__less,axiom,
% 3.01/1.70 ! [M: nat,N: nat] :
% 3.01/1.70 ( ( ord_less_eq_nat @ M @ N )
% 3.01/1.70 => ( ( M != N )
% 3.01/1.70 => ( ord_less_nat @ M @ N ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_184_less__or__eq__imp__le,axiom,
% 3.01/1.70 ! [M: nat,N: nat] :
% 3.01/1.70 ( ( ( ord_less_nat @ M @ N )
% 3.01/1.70 | ( M = N ) )
% 3.01/1.70 => ( ord_less_eq_nat @ M @ N ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_185_le__eq__less__or__eq,axiom,
% 3.01/1.70 ( ord_less_eq_nat
% 3.01/1.70 = ( ^ [M6: nat,N4: nat] :
% 3.01/1.70 ( ( ord_less_nat @ M6 @ N4 )
% 3.01/1.70 | ( M6 = N4 ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_186_less__imp__le__nat,axiom,
% 3.01/1.70 ! [M: nat,N: nat] :
% 3.01/1.70 ( ( ord_less_nat @ M @ N )
% 3.01/1.70 => ( ord_less_eq_nat @ M @ N ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_187_nat__less__le,axiom,
% 3.01/1.70 ( ord_less_nat
% 3.01/1.70 = ( ^ [M6: nat,N4: nat] :
% 3.01/1.70 ( ( ord_less_eq_nat @ M6 @ N4 )
% 3.01/1.70 & ( M6 != N4 ) ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_188_not__less__less__Suc__eq,axiom,
% 3.01/1.70 ! [N: nat,M: nat] :
% 3.01/1.70 ( ~ ( ord_less_nat @ N @ M )
% 3.01/1.70 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 3.01/1.70 = ( N = M ) ) ) ).
% 3.01/1.70
% 3.01/1.70 thf(fact_189_strict__inc__induct,axiom,
% 3.01/1.70 ! [I: nat,J3: nat,P: nat > $o] :
% 3.01/1.70 ( ( ord_less_nat @ I @ J3 )
% 3.01/1.70 => ( ! [I2: nat] :
% 3.01/1.71 ( ( J3
% 3.01/1.71 = ( suc @ I2 ) )
% 3.01/1.71 => ( P @ I2 ) )
% 3.01/1.71 => ( ! [I2: nat] :
% 3.01/1.71 ( ( ord_less_nat @ I2 @ J3 )
% 3.01/1.71 => ( ( P @ ( suc @ I2 ) )
% 3.01/1.71 => ( P @ I2 ) ) )
% 3.01/1.71 => ( P @ I ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_190_less__Suc__induct,axiom,
% 3.01/1.71 ! [I: nat,J3: nat,P: nat > nat > $o] :
% 3.01/1.71 ( ( ord_less_nat @ I @ J3 )
% 3.01/1.71 => ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
% 3.01/1.71 => ( ! [I2: nat,J4: nat,K3: nat] :
% 3.01/1.71 ( ( ord_less_nat @ I2 @ J4 )
% 3.01/1.71 => ( ( ord_less_nat @ J4 @ K3 )
% 3.01/1.71 => ( ( P @ I2 @ J4 )
% 3.01/1.71 => ( ( P @ J4 @ K3 )
% 3.01/1.71 => ( P @ I2 @ K3 ) ) ) ) )
% 3.01/1.71 => ( P @ I @ J3 ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_191_less__trans__Suc,axiom,
% 3.01/1.71 ! [I: nat,J3: nat,K2: nat] :
% 3.01/1.71 ( ( ord_less_nat @ I @ J3 )
% 3.01/1.71 => ( ( ord_less_nat @ J3 @ K2 )
% 3.01/1.71 => ( ord_less_nat @ ( suc @ I ) @ K2 ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_192_Suc__less__SucD,axiom,
% 3.01/1.71 ! [M: nat,N: nat] :
% 3.01/1.71 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 3.01/1.71 => ( ord_less_nat @ M @ N ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_193_less__antisym,axiom,
% 3.01/1.71 ! [N: nat,M: nat] :
% 3.01/1.71 ( ~ ( ord_less_nat @ N @ M )
% 3.01/1.71 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 3.01/1.71 => ( M = N ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_194_Suc__less__eq2,axiom,
% 3.01/1.71 ! [N: nat,M: nat] :
% 3.01/1.71 ( ( ord_less_nat @ ( suc @ N ) @ M )
% 3.01/1.71 = ( ? [M7: nat] :
% 3.01/1.71 ( ( M
% 3.01/1.71 = ( suc @ M7 ) )
% 3.01/1.71 & ( ord_less_nat @ N @ M7 ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_195_All__less__Suc,axiom,
% 3.01/1.71 ! [N: nat,P: nat > $o] :
% 3.01/1.71 ( ( ! [I3: nat] :
% 3.01/1.71 ( ( ord_less_nat @ I3 @ ( suc @ N ) )
% 3.01/1.71 => ( P @ I3 ) ) )
% 3.01/1.71 = ( ( P @ N )
% 3.01/1.71 & ! [I3: nat] :
% 3.01/1.71 ( ( ord_less_nat @ I3 @ N )
% 3.01/1.71 => ( P @ I3 ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_196_not__less__eq,axiom,
% 3.01/1.71 ! [M: nat,N: nat] :
% 3.01/1.71 ( ( ~ ( ord_less_nat @ M @ N ) )
% 3.01/1.71 = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_197_less__Suc__eq,axiom,
% 3.01/1.71 ! [M: nat,N: nat] :
% 3.01/1.71 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 3.01/1.71 = ( ( ord_less_nat @ M @ N )
% 3.01/1.71 | ( M = N ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_198_Ex__less__Suc,axiom,
% 3.01/1.71 ! [N: nat,P: nat > $o] :
% 3.01/1.71 ( ( ? [I3: nat] :
% 3.01/1.71 ( ( ord_less_nat @ I3 @ ( suc @ N ) )
% 3.01/1.71 & ( P @ I3 ) ) )
% 3.01/1.71 = ( ( P @ N )
% 3.01/1.71 | ? [I3: nat] :
% 3.01/1.71 ( ( ord_less_nat @ I3 @ N )
% 3.01/1.71 & ( P @ I3 ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_199_less__SucI,axiom,
% 3.01/1.71 ! [M: nat,N: nat] :
% 3.01/1.71 ( ( ord_less_nat @ M @ N )
% 3.01/1.71 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_200_less__SucE,axiom,
% 3.01/1.71 ! [M: nat,N: nat] :
% 3.01/1.71 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 3.01/1.71 => ( ~ ( ord_less_nat @ M @ N )
% 3.01/1.71 => ( M = N ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_201_Suc__lessI,axiom,
% 3.01/1.71 ! [M: nat,N: nat] :
% 3.01/1.71 ( ( ord_less_nat @ M @ N )
% 3.01/1.71 => ( ( ( suc @ M )
% 3.01/1.71 != N )
% 3.01/1.71 => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_202_Suc__lessE,axiom,
% 3.01/1.71 ! [I: nat,K2: nat] :
% 3.01/1.71 ( ( ord_less_nat @ ( suc @ I ) @ K2 )
% 3.01/1.71 => ~ ! [J4: nat] :
% 3.01/1.71 ( ( ord_less_nat @ I @ J4 )
% 3.01/1.71 => ( K2
% 3.01/1.71 != ( suc @ J4 ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_203_Suc__lessD,axiom,
% 3.01/1.71 ! [M: nat,N: nat] :
% 3.01/1.71 ( ( ord_less_nat @ ( suc @ M ) @ N )
% 3.01/1.71 => ( ord_less_nat @ M @ N ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_204_Nat_OlessE,axiom,
% 3.01/1.71 ! [I: nat,K2: nat] :
% 3.01/1.71 ( ( ord_less_nat @ I @ K2 )
% 3.01/1.71 => ( ( K2
% 3.01/1.71 != ( suc @ I ) )
% 3.01/1.71 => ~ ! [J4: nat] :
% 3.01/1.71 ( ( ord_less_nat @ I @ J4 )
% 3.01/1.71 => ( K2
% 3.01/1.71 != ( suc @ J4 ) ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_205_bot__nat__0_Oextremum__strict,axiom,
% 3.01/1.71 ! [A: nat] :
% 3.01/1.71 ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_206_infinite__descent0,axiom,
% 3.01/1.71 ! [P: nat > $o,N: nat] :
% 3.01/1.71 ( ( P @ zero_zero_nat )
% 3.01/1.71 => ( ! [N2: nat] :
% 3.01/1.71 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 3.01/1.71 => ( ~ ( P @ N2 )
% 3.01/1.71 => ? [M2: nat] :
% 3.01/1.71 ( ( ord_less_nat @ M2 @ N2 )
% 3.01/1.71 & ~ ( P @ M2 ) ) ) )
% 3.01/1.71 => ( P @ N ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_207_gr__implies__not0,axiom,
% 3.01/1.71 ! [M: nat,N: nat] :
% 3.01/1.71 ( ( ord_less_nat @ M @ N )
% 3.01/1.71 => ( N != zero_zero_nat ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_208_less__zeroE,axiom,
% 3.01/1.71 ! [N: nat] :
% 3.01/1.71 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_209_not__less0,axiom,
% 3.01/1.71 ! [N: nat] :
% 3.01/1.71 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_210_not__gr0,axiom,
% 3.01/1.71 ! [N: nat] :
% 3.01/1.71 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 3.01/1.71 = ( N = zero_zero_nat ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_211_gr0I,axiom,
% 3.01/1.71 ! [N: nat] :
% 3.01/1.71 ( ( N != zero_zero_nat )
% 3.01/1.71 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_212_diff__le__mono2,axiom,
% 3.01/1.71 ! [M: nat,N: nat,L2: nat] :
% 3.01/1.71 ( ( ord_less_eq_nat @ M @ N )
% 3.01/1.71 => ( ord_less_eq_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_213_le__diff__iff_H,axiom,
% 3.01/1.71 ! [A: nat,C: nat,B: nat] :
% 3.01/1.71 ( ( ord_less_eq_nat @ A @ C )
% 3.01/1.71 => ( ( ord_less_eq_nat @ B @ C )
% 3.01/1.71 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
% 3.01/1.71 = ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_214_diff__le__self,axiom,
% 3.01/1.71 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_215_diff__le__mono,axiom,
% 3.01/1.71 ! [M: nat,N: nat,L2: nat] :
% 3.01/1.71 ( ( ord_less_eq_nat @ M @ N )
% 3.01/1.71 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L2 ) @ ( minus_minus_nat @ N @ L2 ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_216_Nat_Odiff__diff__eq,axiom,
% 3.01/1.71 ! [K2: nat,M: nat,N: nat] :
% 3.01/1.71 ( ( ord_less_eq_nat @ K2 @ M )
% 3.01/1.71 => ( ( ord_less_eq_nat @ K2 @ N )
% 3.01/1.71 => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
% 3.01/1.71 = ( minus_minus_nat @ M @ N ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_217_le__diff__iff,axiom,
% 3.01/1.71 ! [K2: nat,M: nat,N: nat] :
% 3.01/1.71 ( ( ord_less_eq_nat @ K2 @ M )
% 3.01/1.71 => ( ( ord_less_eq_nat @ K2 @ N )
% 3.01/1.71 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
% 3.01/1.71 = ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_218_eq__diff__iff,axiom,
% 3.01/1.71 ! [K2: nat,M: nat,N: nat] :
% 3.01/1.71 ( ( ord_less_eq_nat @ K2 @ M )
% 3.01/1.71 => ( ( ord_less_eq_nat @ K2 @ N )
% 3.01/1.71 => ( ( ( minus_minus_nat @ M @ K2 )
% 3.01/1.71 = ( minus_minus_nat @ N @ K2 ) )
% 3.01/1.71 = ( M = N ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_219_zero__induct__lemma,axiom,
% 3.01/1.71 ! [P: nat > $o,K2: nat,I: nat] :
% 3.01/1.71 ( ( P @ K2 )
% 3.01/1.71 => ( ! [N2: nat] :
% 3.01/1.71 ( ( P @ ( suc @ N2 ) )
% 3.01/1.71 => ( P @ N2 ) )
% 3.01/1.71 => ( P @ ( minus_minus_nat @ K2 @ I ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_220_minus__nat_Odiff__0,axiom,
% 3.01/1.71 ! [M: nat] :
% 3.01/1.71 ( ( minus_minus_nat @ M @ zero_zero_nat )
% 3.01/1.71 = M ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_221_diffs0__imp__equal,axiom,
% 3.01/1.71 ! [M: nat,N: nat] :
% 3.01/1.71 ( ( ( minus_minus_nat @ M @ N )
% 3.01/1.71 = zero_zero_nat )
% 3.01/1.71 => ( ( ( minus_minus_nat @ N @ M )
% 3.01/1.71 = zero_zero_nat )
% 3.01/1.71 => ( M = N ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_222_less__imp__diff__less,axiom,
% 3.01/1.71 ! [J3: nat,K2: nat,N: nat] :
% 3.01/1.71 ( ( ord_less_nat @ J3 @ K2 )
% 3.01/1.71 => ( ord_less_nat @ ( minus_minus_nat @ J3 @ N ) @ K2 ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_223_diff__less__mono2,axiom,
% 3.01/1.71 ! [M: nat,N: nat,L2: nat] :
% 3.01/1.71 ( ( ord_less_nat @ M @ N )
% 3.01/1.71 => ( ( ord_less_nat @ M @ L2 )
% 3.01/1.71 => ( ord_less_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_224_Module_Osc__Ring,axiom,
% 3.01/1.71 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d] :
% 3.01/1.71 ( ( module_e_b_f_d @ M3 @ R )
% 3.01/1.71 => ( ring_b_d @ R ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_225_Module_Osc__Ring,axiom,
% 3.01/1.71 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d] :
% 3.01/1.71 ( ( module_a_b_c_d @ M3 @ R )
% 3.01/1.71 => ( ring_b_d @ R ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_226_Module_Osc__Ring,axiom,
% 3.01/1.71 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d] :
% 3.01/1.71 ( ( module1821517916unit_d @ M3 @ R )
% 3.01/1.71 => ( ring_b_d @ R ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_227_Module_Omem__smodule__ideal__coeff,axiom,
% 3.01/1.71 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,A2: set_b,X: e] :
% 3.01/1.71 ( ( module_e_b_f_d @ M3 @ R )
% 3.01/1.71 => ( ( ideal_b_d @ R @ A2 )
% 3.01/1.71 => ( ( member_e @ X @ ( smodul1982583723_d_e_f @ R @ M3 @ A2 ) )
% 3.01/1.71 => ? [N2: nat,X2: nat > b] :
% 3.01/1.71 ( ( member_nat_b @ X2
% 3.01/1.71 @ ( pi_nat_b
% 3.01/1.71 @ ( collect_nat
% 3.01/1.71 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
% 3.01/1.71 @ ^ [Uu: nat] : A2 ) )
% 3.01/1.71 & ? [Xa: nat > e] :
% 3.01/1.71 ( ( member_nat_e @ Xa
% 3.01/1.71 @ ( pi_nat_e
% 3.01/1.71 @ ( collect_nat
% 3.01/1.71 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
% 3.01/1.71 @ ^ [Uu: nat] : ( carrie730238621_e_b_f @ M3 ) ) )
% 3.01/1.71 & ( X
% 3.01/1.71 = ( l_comb_b_d_e_f @ R @ M3 @ N2 @ X2 @ Xa ) ) ) ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_228_Module_Omem__smodule__ideal__coeff,axiom,
% 3.01/1.71 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,A2: set_b,X: a] :
% 3.01/1.71 ( ( module1821517916unit_d @ M3 @ R )
% 3.01/1.71 => ( ( ideal_b_d @ R @ A2 )
% 3.01/1.71 => ( ( member_a @ X @ ( smodul132175289t_unit @ R @ M3 @ A2 ) )
% 3.01/1.71 => ? [N2: nat,X2: nat > b] :
% 3.01/1.71 ( ( member_nat_b @ X2
% 3.01/1.71 @ ( pi_nat_b
% 3.01/1.71 @ ( collect_nat
% 3.01/1.71 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
% 3.01/1.71 @ ^ [Uu: nat] : A2 ) )
% 3.01/1.71 & ? [Xa: nat > a] :
% 3.01/1.71 ( ( member_nat_a @ Xa
% 3.01/1.71 @ ( pi_nat_a
% 3.01/1.71 @ ( collect_nat
% 3.01/1.71 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
% 3.01/1.71 @ ^ [Uu: nat] : ( carrie1074654371t_unit @ M3 ) ) )
% 3.01/1.71 & ( X
% 3.01/1.71 = ( l_comb1138968323t_unit @ R @ M3 @ N2 @ X2 @ Xa ) ) ) ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_229_Module_Omem__smodule__ideal__coeff,axiom,
% 3.01/1.71 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,A2: set_b,X: a] :
% 3.01/1.71 ( ( module_a_b_c_d @ M3 @ R )
% 3.01/1.71 => ( ( ideal_b_d @ R @ A2 )
% 3.01/1.71 => ( ( member_a @ X @ ( smodul818989740_d_a_c @ R @ M3 @ A2 ) )
% 3.01/1.71 => ? [N2: nat,X2: nat > b] :
% 3.01/1.71 ( ( member_nat_b @ X2
% 3.01/1.71 @ ( pi_nat_b
% 3.01/1.71 @ ( collect_nat
% 3.01/1.71 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
% 3.01/1.71 @ ^ [Uu: nat] : A2 ) )
% 3.01/1.71 & ? [Xa: nat > a] :
% 3.01/1.71 ( ( member_nat_a @ Xa
% 3.01/1.71 @ ( pi_nat_a
% 3.01/1.71 @ ( collect_nat
% 3.01/1.71 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
% 3.01/1.71 @ ^ [Uu: nat] : ( carrie2021454486_a_b_c @ M3 ) ) )
% 3.01/1.71 & ( X
% 3.01/1.71 = ( l_comb_b_d_a_c @ R @ M3 @ N2 @ X2 @ Xa ) ) ) ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_230_lift__Suc__antimono__le,axiom,
% 3.01/1.71 ! [F: nat > nat,N: nat,N5: nat] :
% 3.01/1.71 ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 3.01/1.71 => ( ( ord_less_eq_nat @ N @ N5 )
% 3.01/1.71 => ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_231_lift__Suc__antimono__le,axiom,
% 3.01/1.71 ! [F: nat > set_a,N: nat,N5: nat] :
% 3.01/1.71 ( ! [N2: nat] : ( ord_less_eq_set_a @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 3.01/1.71 => ( ( ord_less_eq_nat @ N @ N5 )
% 3.01/1.71 => ( ord_less_eq_set_a @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_232_lift__Suc__mono__le,axiom,
% 3.01/1.71 ! [F: nat > nat,N: nat,N5: nat] :
% 3.01/1.71 ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 3.01/1.71 => ( ( ord_less_eq_nat @ N @ N5 )
% 3.01/1.71 => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_233_lift__Suc__mono__le,axiom,
% 3.01/1.71 ! [F: nat > set_a,N: nat,N5: nat] :
% 3.01/1.71 ( ! [N2: nat] : ( ord_less_eq_set_a @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 3.01/1.71 => ( ( ord_less_eq_nat @ N @ N5 )
% 3.01/1.71 => ( ord_less_eq_set_a @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_234_lift__Suc__mono__less__iff,axiom,
% 3.01/1.71 ! [F: nat > nat,N: nat,M: nat] :
% 3.01/1.71 ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 3.01/1.71 => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
% 3.01/1.71 = ( ord_less_nat @ N @ M ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_235_lift__Suc__mono__less,axiom,
% 3.01/1.71 ! [F: nat > nat,N: nat,N5: nat] :
% 3.01/1.71 ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 3.01/1.71 => ( ( ord_less_nat @ N @ N5 )
% 3.01/1.71 => ( ord_less_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_236_le__imp__less__Suc,axiom,
% 3.01/1.71 ! [M: nat,N: nat] :
% 3.01/1.71 ( ( ord_less_eq_nat @ M @ N )
% 3.01/1.71 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_237_less__eq__Suc__le,axiom,
% 3.01/1.71 ( ord_less_nat
% 3.01/1.71 = ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_238_less__Suc__eq__le,axiom,
% 3.01/1.71 ! [M: nat,N: nat] :
% 3.01/1.71 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 3.01/1.71 = ( ord_less_eq_nat @ M @ N ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_239_le__less__Suc__eq,axiom,
% 3.01/1.71 ! [M: nat,N: nat] :
% 3.01/1.71 ( ( ord_less_eq_nat @ M @ N )
% 3.01/1.71 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 3.01/1.71 = ( N = M ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_240_Suc__le__lessD,axiom,
% 3.01/1.71 ! [M: nat,N: nat] :
% 3.01/1.71 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 3.01/1.71 => ( ord_less_nat @ M @ N ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_241_inc__induct,axiom,
% 3.01/1.71 ! [I: nat,J3: nat,P: nat > $o] :
% 3.01/1.71 ( ( ord_less_eq_nat @ I @ J3 )
% 3.01/1.71 => ( ( P @ J3 )
% 3.01/1.71 => ( ! [N2: nat] :
% 3.01/1.71 ( ( ord_less_eq_nat @ I @ N2 )
% 3.01/1.71 => ( ( ord_less_nat @ N2 @ J3 )
% 3.01/1.71 => ( ( P @ ( suc @ N2 ) )
% 3.01/1.71 => ( P @ N2 ) ) ) )
% 3.01/1.71 => ( P @ I ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_242_dec__induct,axiom,
% 3.01/1.71 ! [I: nat,J3: nat,P: nat > $o] :
% 3.01/1.71 ( ( ord_less_eq_nat @ I @ J3 )
% 3.01/1.71 => ( ( P @ I )
% 3.01/1.71 => ( ! [N2: nat] :
% 3.01/1.71 ( ( ord_less_eq_nat @ I @ N2 )
% 3.01/1.71 => ( ( ord_less_nat @ N2 @ J3 )
% 3.01/1.71 => ( ( P @ N2 )
% 3.01/1.71 => ( P @ ( suc @ N2 ) ) ) ) )
% 3.01/1.71 => ( P @ J3 ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_243_Suc__le__eq,axiom,
% 3.01/1.71 ! [M: nat,N: nat] :
% 3.01/1.71 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 3.01/1.71 = ( ord_less_nat @ M @ N ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_244_Suc__leI,axiom,
% 3.01/1.71 ! [M: nat,N: nat] :
% 3.01/1.71 ( ( ord_less_nat @ M @ N )
% 3.01/1.71 => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_245_ex__least__nat__le,axiom,
% 3.01/1.71 ! [P: nat > $o,N: nat] :
% 3.01/1.71 ( ( P @ N )
% 3.01/1.71 => ( ~ ( P @ zero_zero_nat )
% 3.01/1.71 => ? [K3: nat] :
% 3.01/1.71 ( ( ord_less_eq_nat @ K3 @ N )
% 3.01/1.71 & ! [I4: nat] :
% 3.01/1.71 ( ( ord_less_nat @ I4 @ K3 )
% 3.01/1.71 => ~ ( P @ I4 ) )
% 3.01/1.71 & ( P @ K3 ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_246_less__Suc__eq__0__disj,axiom,
% 3.01/1.71 ! [M: nat,N: nat] :
% 3.01/1.71 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 3.01/1.71 = ( ( M = zero_zero_nat )
% 3.01/1.71 | ? [J2: nat] :
% 3.01/1.71 ( ( M
% 3.01/1.71 = ( suc @ J2 ) )
% 3.01/1.71 & ( ord_less_nat @ J2 @ N ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_247_gr0__implies__Suc,axiom,
% 3.01/1.71 ! [N: nat] :
% 3.01/1.71 ( ( ord_less_nat @ zero_zero_nat @ N )
% 3.01/1.71 => ? [M5: nat] :
% 3.01/1.71 ( N
% 3.01/1.71 = ( suc @ M5 ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_248_All__less__Suc2,axiom,
% 3.01/1.71 ! [N: nat,P: nat > $o] :
% 3.01/1.71 ( ( ! [I3: nat] :
% 3.01/1.71 ( ( ord_less_nat @ I3 @ ( suc @ N ) )
% 3.01/1.71 => ( P @ I3 ) ) )
% 3.01/1.71 = ( ( P @ zero_zero_nat )
% 3.01/1.71 & ! [I3: nat] :
% 3.01/1.71 ( ( ord_less_nat @ I3 @ N )
% 3.01/1.71 => ( P @ ( suc @ I3 ) ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_249_gr0__conv__Suc,axiom,
% 3.01/1.71 ! [N: nat] :
% 3.01/1.71 ( ( ord_less_nat @ zero_zero_nat @ N )
% 3.01/1.71 = ( ? [M6: nat] :
% 3.01/1.71 ( N
% 3.01/1.71 = ( suc @ M6 ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_250_Ex__less__Suc2,axiom,
% 3.01/1.71 ! [N: nat,P: nat > $o] :
% 3.01/1.71 ( ( ? [I3: nat] :
% 3.01/1.71 ( ( ord_less_nat @ I3 @ ( suc @ N ) )
% 3.01/1.71 & ( P @ I3 ) ) )
% 3.01/1.71 = ( ( P @ zero_zero_nat )
% 3.01/1.71 | ? [I3: nat] :
% 3.01/1.71 ( ( ord_less_nat @ I3 @ N )
% 3.01/1.71 & ( P @ ( suc @ I3 ) ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_251_Suc__diff__le,axiom,
% 3.01/1.71 ! [N: nat,M: nat] :
% 3.01/1.71 ( ( ord_less_eq_nat @ N @ M )
% 3.01/1.71 => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 3.01/1.71 = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_252_diff__less__mono,axiom,
% 3.01/1.71 ! [A: nat,B: nat,C: nat] :
% 3.01/1.71 ( ( ord_less_nat @ A @ B )
% 3.01/1.71 => ( ( ord_less_eq_nat @ C @ A )
% 3.01/1.71 => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_253_less__diff__iff,axiom,
% 3.01/1.71 ! [K2: nat,M: nat,N: nat] :
% 3.01/1.71 ( ( ord_less_eq_nat @ K2 @ M )
% 3.01/1.71 => ( ( ord_less_eq_nat @ K2 @ N )
% 3.01/1.71 => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
% 3.01/1.71 = ( ord_less_nat @ M @ N ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_254_diff__less__Suc,axiom,
% 3.01/1.71 ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_255_Suc__diff__Suc,axiom,
% 3.01/1.71 ! [N: nat,M: nat] :
% 3.01/1.71 ( ( ord_less_nat @ N @ M )
% 3.01/1.71 => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
% 3.01/1.71 = ( minus_minus_nat @ M @ N ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_256_diff__less,axiom,
% 3.01/1.71 ! [N: nat,M: nat] :
% 3.01/1.71 ( ( ord_less_nat @ zero_zero_nat @ N )
% 3.01/1.71 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 3.01/1.71 => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_257_Module_Osubmodule__subset1,axiom,
% 3.01/1.71 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,H: set_e,H2: e] :
% 3.01/1.71 ( ( module_e_b_f_d @ M3 @ R )
% 3.01/1.71 => ( ( submodule_b_d_e_f @ R @ M3 @ H )
% 3.01/1.71 => ( ( member_e @ H2 @ H )
% 3.01/1.71 => ( member_e @ H2 @ ( carrie730238621_e_b_f @ M3 ) ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_258_Module_Osubmodule__subset1,axiom,
% 3.01/1.71 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,H: set_a,H2: a] :
% 3.01/1.71 ( ( module_a_b_c_d @ M3 @ R )
% 3.01/1.71 => ( ( submodule_b_d_a_c @ R @ M3 @ H )
% 3.01/1.71 => ( ( member_a @ H2 @ H )
% 3.01/1.71 => ( member_a @ H2 @ ( carrie2021454486_a_b_c @ M3 ) ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_259_Module_Osubmodule__subset1,axiom,
% 3.01/1.71 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,H: set_a,H2: a] :
% 3.01/1.71 ( ( module1821517916unit_d @ M3 @ R )
% 3.01/1.71 => ( ( submod903911234t_unit @ R @ M3 @ H )
% 3.01/1.71 => ( ( member_a @ H2 @ H )
% 3.01/1.71 => ( member_a @ H2 @ ( carrie1074654371t_unit @ M3 ) ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_260_Module_Osubmodule__whole,axiom,
% 3.01/1.71 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d] :
% 3.01/1.71 ( ( module_e_b_f_d @ M3 @ R )
% 3.01/1.71 => ( submodule_b_d_e_f @ R @ M3 @ ( carrie730238621_e_b_f @ M3 ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_261_Module_Osubmodule__whole,axiom,
% 3.01/1.71 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d] :
% 3.01/1.71 ( ( module_a_b_c_d @ M3 @ R )
% 3.01/1.71 => ( submodule_b_d_a_c @ R @ M3 @ ( carrie2021454486_a_b_c @ M3 ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_262_Module_Osubmodule__whole,axiom,
% 3.01/1.71 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d] :
% 3.01/1.71 ( ( module1821517916unit_d @ M3 @ R )
% 3.01/1.71 => ( submod903911234t_unit @ R @ M3 @ ( carrie1074654371t_unit @ M3 ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_263_ex__least__nat__less,axiom,
% 3.01/1.71 ! [P: nat > $o,N: nat] :
% 3.01/1.71 ( ( P @ N )
% 3.01/1.71 => ( ~ ( P @ zero_zero_nat )
% 3.01/1.71 => ? [K3: nat] :
% 3.01/1.71 ( ( ord_less_nat @ K3 @ N )
% 3.01/1.71 & ! [I4: nat] :
% 3.01/1.71 ( ( ord_less_eq_nat @ I4 @ K3 )
% 3.01/1.71 => ~ ( P @ I4 ) )
% 3.01/1.71 & ( P @ ( suc @ K3 ) ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_264_diff__Suc__less,axiom,
% 3.01/1.71 ! [N: nat,I: nat] :
% 3.01/1.71 ( ( ord_less_nat @ zero_zero_nat @ N )
% 3.01/1.71 => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_265_Module_Osubmodule__subset,axiom,
% 3.01/1.71 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,H: set_e] :
% 3.01/1.71 ( ( module_e_b_f_d @ M3 @ R )
% 3.01/1.71 => ( ( submodule_b_d_e_f @ R @ M3 @ H )
% 3.01/1.71 => ( ord_less_eq_set_e @ H @ ( carrie730238621_e_b_f @ M3 ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_266_Module_Osubmodule__subset,axiom,
% 3.01/1.71 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,H: set_a] :
% 3.01/1.71 ( ( module_a_b_c_d @ M3 @ R )
% 3.01/1.71 => ( ( submodule_b_d_a_c @ R @ M3 @ H )
% 3.01/1.71 => ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ M3 ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_267_Module_Osubmodule__subset,axiom,
% 3.01/1.71 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,H: set_a] :
% 3.01/1.71 ( ( module1821517916unit_d @ M3 @ R )
% 3.01/1.71 => ( ( submod903911234t_unit @ R @ M3 @ H )
% 3.01/1.71 => ( ord_less_eq_set_a @ H @ ( carrie1074654371t_unit @ M3 ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_268_Module_Ofree__generator__sub,axiom,
% 3.01/1.71 ! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,H: set_e] :
% 3.01/1.71 ( ( module_e_b_f_d @ M3 @ R )
% 3.01/1.71 => ( ( free_g103796815_d_e_f @ R @ M3 @ H )
% 3.01/1.71 => ( ord_less_eq_set_e @ H @ ( carrie730238621_e_b_f @ M3 ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_269_Module_Ofree__generator__sub,axiom,
% 3.01/1.71 ! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,H: set_a] :
% 3.01/1.71 ( ( module_a_b_c_d @ M3 @ R )
% 3.01/1.71 => ( ( free_g1087686480_d_a_c @ R @ M3 @ H )
% 3.01/1.71 => ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ M3 ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_270_Module_Ofree__generator__sub,axiom,
% 3.01/1.71 ! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,H: set_a] :
% 3.01/1.71 ( ( module1821517916unit_d @ M3 @ R )
% 3.01/1.71 => ( ( free_g637607517t_unit @ R @ M3 @ H )
% 3.01/1.71 => ( ord_less_eq_set_a @ H @ ( carrie1074654371t_unit @ M3 ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_271_eSum__in__SubmoduleTr,axiom,
% 3.01/1.71 ! [H: set_a,K: set_a,F: nat > a,N: nat,S: nat > b] :
% 3.01/1.71 ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 3.01/1.71 => ( ( ord_less_eq_set_a @ K @ H )
% 3.01/1.71 => ( ( ( member_nat_a @ F
% 3.01/1.71 @ ( pi_nat_a
% 3.01/1.71 @ ( collect_nat
% 3.01/1.71 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.71 @ ^ [Uu: nat] : K ) )
% 3.01/1.71 & ( member_nat_b @ S
% 3.01/1.71 @ ( pi_nat_b
% 3.01/1.71 @ ( collect_nat
% 3.01/1.71 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.71 @ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ r ) ) ) )
% 3.01/1.71 => ( ( l_comb1138968323t_unit @ r @ ( mdl_a_b_c @ m @ ( algebr1152250919_d_a_c @ r @ m @ K ) ) @ N @ S @ F )
% 3.01/1.71 = ( l_comb_b_d_a_c @ r @ m @ N @ S @ F ) ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_272_eSum__in__Submodule,axiom,
% 3.01/1.71 ! [H: set_a,K: set_a,F: nat > a,N: nat,S: nat > b] :
% 3.01/1.71 ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 3.01/1.71 => ( ( ord_less_eq_set_a @ K @ H )
% 3.01/1.71 => ( ( member_nat_a @ F
% 3.01/1.71 @ ( pi_nat_a
% 3.01/1.71 @ ( collect_nat
% 3.01/1.71 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.71 @ ^ [Uu: nat] : K ) )
% 3.01/1.71 => ( ( member_nat_b @ S
% 3.01/1.71 @ ( pi_nat_b
% 3.01/1.71 @ ( collect_nat
% 3.01/1.71 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.71 @ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ r ) ) )
% 3.01/1.71 => ( ( l_comb1138968323t_unit @ r @ ( mdl_a_b_c @ m @ ( algebr1152250919_d_a_c @ r @ m @ K ) ) @ N @ S @ F )
% 3.01/1.71 = ( l_comb_b_d_a_c @ r @ m @ N @ S @ F ) ) ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_273_singleton__insert__inj__eq_H,axiom,
% 3.01/1.71 ! [A: a,A2: set_a,B: a] :
% 3.01/1.71 ( ( ( insert_a @ A @ A2 )
% 3.01/1.71 = ( insert_a @ B @ bot_bot_set_a ) )
% 3.01/1.71 = ( ( A = B )
% 3.01/1.71 & ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_274_singleton__insert__inj__eq,axiom,
% 3.01/1.71 ! [B: a,A: a,A2: set_a] :
% 3.01/1.71 ( ( ( insert_a @ B @ bot_bot_set_a )
% 3.01/1.71 = ( insert_a @ A @ A2 ) )
% 3.01/1.71 = ( ( A = B )
% 3.01/1.71 & ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_275_l__span__closed2Tr,axiom,
% 3.01/1.71 ! [A2: set_b,H: set_a,S: nat > b,N: nat,F: nat > a] :
% 3.01/1.71 ( ( ideal_b_d @ r @ A2 )
% 3.01/1.71 => ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 3.01/1.71 => ( ( ( member_nat_b @ S
% 3.01/1.71 @ ( pi_nat_b
% 3.01/1.71 @ ( collect_nat
% 3.01/1.71 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.71 @ ^ [Uu: nat] : A2 ) )
% 3.01/1.71 & ( member_nat_a @ F
% 3.01/1.71 @ ( pi_nat_a
% 3.01/1.71 @ ( collect_nat
% 3.01/1.71 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.71 @ ^ [Uu: nat] : ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ H ) ) ) )
% 3.01/1.71 => ( member_a @ ( l_comb_b_d_a_c @ r @ m @ N @ S @ F ) @ ( linear_span_b_d_a_c @ r @ m @ A2 @ H ) ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_276_l__span__closed2,axiom,
% 3.01/1.71 ! [A2: set_b,H: set_a,S: nat > b,N: nat,F: nat > a] :
% 3.01/1.71 ( ( ideal_b_d @ r @ A2 )
% 3.01/1.71 => ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 3.01/1.71 => ( ( member_nat_b @ S
% 3.01/1.71 @ ( pi_nat_b
% 3.01/1.71 @ ( collect_nat
% 3.01/1.71 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.71 @ ^ [Uu: nat] : A2 ) )
% 3.01/1.71 => ( ( member_nat_a @ F
% 3.01/1.71 @ ( pi_nat_a
% 3.01/1.71 @ ( collect_nat
% 3.01/1.71 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.71 @ ^ [Uu: nat] : ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ H ) ) )
% 3.01/1.71 => ( member_a @ ( l_comb_b_d_a_c @ r @ m @ N @ S @ F ) @ ( linear_span_b_d_a_c @ r @ m @ A2 @ H ) ) ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_277_linear__comb0__2Tr,axiom,
% 3.01/1.71 ! [A2: set_b,S: nat > b,N: nat,M: nat > a] :
% 3.01/1.71 ( ( ideal_b_d @ r @ A2 )
% 3.01/1.71 => ( ( ( member_nat_b @ S
% 3.01/1.71 @ ( pi_nat_b
% 3.01/1.71 @ ( collect_nat
% 3.01/1.71 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.71 @ ^ [Uu: nat] : A2 ) )
% 3.01/1.71 & ( member_nat_a @ M
% 3.01/1.71 @ ( pi_nat_a
% 3.01/1.71 @ ( collect_nat
% 3.01/1.71 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.71 @ ^ [Uu: nat] : ( insert_a @ ( zero_a1261444626_a_b_c @ m ) @ bot_bot_set_a ) ) ) )
% 3.01/1.71 => ( ( l_comb_b_d_a_c @ r @ m @ N @ S @ M )
% 3.01/1.71 = ( zero_a1261444626_a_b_c @ m ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_278_module__inc__zero,axiom,
% 3.01/1.71 member_a @ ( zero_a1261444626_a_b_c @ m ) @ ( carrie2021454486_a_b_c @ m ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_279_psubsetI,axiom,
% 3.01/1.71 ! [A2: set_a,B2: set_a] :
% 3.01/1.71 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 3.01/1.71 => ( ( A2 != B2 )
% 3.01/1.71 => ( ord_less_set_a @ A2 @ B2 ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_280_subsetI,axiom,
% 3.01/1.71 ! [A2: set_nat_a,B2: set_nat_a] :
% 3.01/1.71 ( ! [X2: nat > a] :
% 3.01/1.71 ( ( member_nat_a @ X2 @ A2 )
% 3.01/1.71 => ( member_nat_a @ X2 @ B2 ) )
% 3.01/1.71 => ( ord_le157368549_nat_a @ A2 @ B2 ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_281_subsetI,axiom,
% 3.01/1.71 ! [A2: set_nat_b,B2: set_nat_b] :
% 3.01/1.71 ( ! [X2: nat > b] :
% 3.01/1.71 ( ( member_nat_b @ X2 @ A2 )
% 3.01/1.71 => ( member_nat_b @ X2 @ B2 ) )
% 3.01/1.71 => ( ord_le1296932838_nat_b @ A2 @ B2 ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_282_subsetI,axiom,
% 3.01/1.71 ! [A2: set_a,B2: set_a] :
% 3.01/1.71 ( ! [X2: a] :
% 3.01/1.71 ( ( member_a @ X2 @ A2 )
% 3.01/1.71 => ( member_a @ X2 @ B2 ) )
% 3.01/1.71 => ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_283_subset__antisym,axiom,
% 3.01/1.71 ! [A2: set_a,B2: set_a] :
% 3.01/1.71 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 3.01/1.71 => ( ( ord_less_eq_set_a @ B2 @ A2 )
% 3.01/1.71 => ( A2 = B2 ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_284_Diff__empty,axiom,
% 3.01/1.71 ! [A2: set_a] :
% 3.01/1.71 ( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
% 3.01/1.71 = A2 ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_285_empty__Diff,axiom,
% 3.01/1.71 ! [A2: set_a] :
% 3.01/1.71 ( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
% 3.01/1.71 = bot_bot_set_a ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_286_Diff__cancel,axiom,
% 3.01/1.71 ! [A2: set_a] :
% 3.01/1.71 ( ( minus_minus_set_a @ A2 @ A2 )
% 3.01/1.71 = bot_bot_set_a ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_287_empty__iff,axiom,
% 3.01/1.71 ! [C: nat > a] :
% 3.01/1.71 ~ ( member_nat_a @ C @ bot_bot_set_nat_a ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_288_empty__iff,axiom,
% 3.01/1.71 ! [C: nat > b] :
% 3.01/1.71 ~ ( member_nat_b @ C @ bot_bot_set_nat_b ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_289_empty__iff,axiom,
% 3.01/1.71 ! [C: a] :
% 3.01/1.71 ~ ( member_a @ C @ bot_bot_set_a ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_290_all__not__in__conv,axiom,
% 3.01/1.71 ! [A2: set_nat_a] :
% 3.01/1.71 ( ( ! [X3: nat > a] :
% 3.01/1.71 ~ ( member_nat_a @ X3 @ A2 ) )
% 3.01/1.71 = ( A2 = bot_bot_set_nat_a ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_291_all__not__in__conv,axiom,
% 3.01/1.71 ! [A2: set_nat_b] :
% 3.01/1.71 ( ( ! [X3: nat > b] :
% 3.01/1.71 ~ ( member_nat_b @ X3 @ A2 ) )
% 3.01/1.71 = ( A2 = bot_bot_set_nat_b ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_292_all__not__in__conv,axiom,
% 3.01/1.71 ! [A2: set_a] :
% 3.01/1.71 ( ( ! [X3: a] :
% 3.01/1.71 ~ ( member_a @ X3 @ A2 ) )
% 3.01/1.71 = ( A2 = bot_bot_set_a ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_293_Collect__empty__eq,axiom,
% 3.01/1.71 ! [P: nat > $o] :
% 3.01/1.71 ( ( ( collect_nat @ P )
% 3.01/1.71 = bot_bot_set_nat )
% 3.01/1.71 = ( ! [X3: nat] :
% 3.01/1.71 ~ ( P @ X3 ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_294_Collect__empty__eq,axiom,
% 3.01/1.71 ! [P: a > $o] :
% 3.01/1.71 ( ( ( collect_a @ P )
% 3.01/1.71 = bot_bot_set_a )
% 3.01/1.71 = ( ! [X3: a] :
% 3.01/1.71 ~ ( P @ X3 ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_295_empty__Collect__eq,axiom,
% 3.01/1.71 ! [P: nat > $o] :
% 3.01/1.71 ( ( bot_bot_set_nat
% 3.01/1.71 = ( collect_nat @ P ) )
% 3.01/1.71 = ( ! [X3: nat] :
% 3.01/1.71 ~ ( P @ X3 ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_296_empty__Collect__eq,axiom,
% 3.01/1.71 ! [P: a > $o] :
% 3.01/1.71 ( ( bot_bot_set_a
% 3.01/1.71 = ( collect_a @ P ) )
% 3.01/1.71 = ( ! [X3: a] :
% 3.01/1.71 ~ ( P @ X3 ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_297_Diff__insert0,axiom,
% 3.01/1.71 ! [X: nat > a,A2: set_nat_a,B2: set_nat_a] :
% 3.01/1.71 ( ~ ( member_nat_a @ X @ A2 )
% 3.01/1.71 => ( ( minus_1788767276_nat_a @ A2 @ ( insert_nat_a @ X @ B2 ) )
% 3.01/1.71 = ( minus_1788767276_nat_a @ A2 @ B2 ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_298_Diff__insert0,axiom,
% 3.01/1.71 ! [X: nat > b,A2: set_nat_b,B2: set_nat_b] :
% 3.01/1.71 ( ~ ( member_nat_b @ X @ A2 )
% 3.01/1.71 => ( ( minus_780847917_nat_b @ A2 @ ( insert_nat_b @ X @ B2 ) )
% 3.01/1.71 = ( minus_780847917_nat_b @ A2 @ B2 ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_299_Diff__insert0,axiom,
% 3.01/1.71 ! [X: a,A2: set_a,B2: set_a] :
% 3.01/1.71 ( ~ ( member_a @ X @ A2 )
% 3.01/1.71 => ( ( minus_minus_set_a @ A2 @ ( insert_a @ X @ B2 ) )
% 3.01/1.71 = ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_300_insert__Diff1,axiom,
% 3.01/1.71 ! [X: nat > a,B2: set_nat_a,A2: set_nat_a] :
% 3.01/1.71 ( ( member_nat_a @ X @ B2 )
% 3.01/1.71 => ( ( minus_1788767276_nat_a @ ( insert_nat_a @ X @ A2 ) @ B2 )
% 3.01/1.71 = ( minus_1788767276_nat_a @ A2 @ B2 ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_301_insert__Diff1,axiom,
% 3.01/1.71 ! [X: nat > b,B2: set_nat_b,A2: set_nat_b] :
% 3.01/1.71 ( ( member_nat_b @ X @ B2 )
% 3.01/1.71 => ( ( minus_780847917_nat_b @ ( insert_nat_b @ X @ A2 ) @ B2 )
% 3.01/1.71 = ( minus_780847917_nat_b @ A2 @ B2 ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_302_insert__Diff1,axiom,
% 3.01/1.71 ! [X: a,B2: set_a,A2: set_a] :
% 3.01/1.71 ( ( member_a @ X @ B2 )
% 3.01/1.71 => ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B2 )
% 3.01/1.71 = ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_303_insertCI,axiom,
% 3.01/1.71 ! [A: nat > a,B2: set_nat_a,B: nat > a] :
% 3.01/1.71 ( ( ~ ( member_nat_a @ A @ B2 )
% 3.01/1.71 => ( A = B ) )
% 3.01/1.71 => ( member_nat_a @ A @ ( insert_nat_a @ B @ B2 ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_304_insertCI,axiom,
% 3.01/1.71 ! [A: nat > b,B2: set_nat_b,B: nat > b] :
% 3.01/1.71 ( ( ~ ( member_nat_b @ A @ B2 )
% 3.01/1.71 => ( A = B ) )
% 3.01/1.71 => ( member_nat_b @ A @ ( insert_nat_b @ B @ B2 ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_305_insertCI,axiom,
% 3.01/1.71 ! [A: a,B2: set_a,B: a] :
% 3.01/1.71 ( ( ~ ( member_a @ A @ B2 )
% 3.01/1.71 => ( A = B ) )
% 3.01/1.71 => ( member_a @ A @ ( insert_a @ B @ B2 ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_306_insert__iff,axiom,
% 3.01/1.71 ! [A: nat > a,B: nat > a,A2: set_nat_a] :
% 3.01/1.71 ( ( member_nat_a @ A @ ( insert_nat_a @ B @ A2 ) )
% 3.01/1.71 = ( ( A = B )
% 3.01/1.71 | ( member_nat_a @ A @ A2 ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_307_insert__iff,axiom,
% 3.01/1.71 ! [A: nat > b,B: nat > b,A2: set_nat_b] :
% 3.01/1.71 ( ( member_nat_b @ A @ ( insert_nat_b @ B @ A2 ) )
% 3.01/1.71 = ( ( A = B )
% 3.01/1.71 | ( member_nat_b @ A @ A2 ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_308_insert__iff,axiom,
% 3.01/1.71 ! [A: a,B: a,A2: set_a] :
% 3.01/1.71 ( ( member_a @ A @ ( insert_a @ B @ A2 ) )
% 3.01/1.71 = ( ( A = B )
% 3.01/1.71 | ( member_a @ A @ A2 ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_309_insert__absorb2,axiom,
% 3.01/1.71 ! [X: a,A2: set_a] :
% 3.01/1.71 ( ( insert_a @ X @ ( insert_a @ X @ A2 ) )
% 3.01/1.71 = ( insert_a @ X @ A2 ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_310_submodule__inc__0,axiom,
% 3.01/1.71 ! [H: set_a] :
% 3.01/1.71 ( ( submodule_b_d_a_c @ r @ m @ H )
% 3.01/1.71 => ( member_a @ ( zero_a1261444626_a_b_c @ m ) @ H ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_311_mdl__carrier,axiom,
% 3.01/1.71 ! [H: set_a] :
% 3.01/1.71 ( ( submodule_b_d_a_c @ r @ m @ H )
% 3.01/1.71 => ( ( carrie1074654371t_unit @ ( mdl_a_b_c @ m @ H ) )
% 3.01/1.71 = H ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_312_mdl__is__module,axiom,
% 3.01/1.71 ! [H: set_a] :
% 3.01/1.71 ( ( submodule_b_d_a_c @ r @ m @ H )
% 3.01/1.71 => ( module1821517916unit_d @ ( mdl_a_b_c @ m @ H ) @ r ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_313_submodule__of__mdl,axiom,
% 3.01/1.71 ! [H: set_a,N3: set_a] :
% 3.01/1.71 ( ( submodule_b_d_a_c @ r @ m @ H )
% 3.01/1.71 => ( ( submodule_b_d_a_c @ r @ m @ N3 )
% 3.01/1.71 => ( ( ord_less_eq_set_a @ H @ N3 )
% 3.01/1.71 => ( submod903911234t_unit @ r @ ( mdl_a_b_c @ m @ N3 ) @ H ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_314_h__in__linear__span,axiom,
% 3.01/1.71 ! [H: set_a,H2: a] :
% 3.01/1.71 ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 3.01/1.71 => ( ( member_a @ H2 @ H )
% 3.01/1.71 => ( member_a @ H2 @ ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ H ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_315_l__span__cont__H,axiom,
% 3.01/1.71 ! [H: set_a] :
% 3.01/1.71 ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 3.01/1.71 => ( ord_less_eq_set_a @ H @ ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ H ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_316_l__span__l__span,axiom,
% 3.01/1.71 ! [H: set_a] :
% 3.01/1.71 ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 3.01/1.71 => ( ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ H ) )
% 3.01/1.71 = ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ H ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_317_l__span__gen__mono,axiom,
% 3.01/1.71 ! [K: set_a,H: set_a,A2: set_b] :
% 3.01/1.71 ( ( ord_less_eq_set_a @ K @ H )
% 3.01/1.71 => ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 3.01/1.71 => ( ( ideal_b_d @ r @ A2 )
% 3.01/1.71 => ( ord_less_eq_set_a @ ( linear_span_b_d_a_c @ r @ m @ A2 @ K ) @ ( linear_span_b_d_a_c @ r @ m @ A2 @ H ) ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_318_linear__span__sub,axiom,
% 3.01/1.71 ! [A2: set_b,H: set_a] :
% 3.01/1.71 ( ( ideal_b_d @ r @ A2 )
% 3.01/1.71 => ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 3.01/1.71 => ( ord_less_eq_set_a @ ( linear_span_b_d_a_c @ r @ m @ A2 @ H ) @ ( carrie2021454486_a_b_c @ m ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_319_l__span__sub__submodule,axiom,
% 3.01/1.71 ! [A2: set_b,N3: set_a,H: set_a] :
% 3.01/1.71 ( ( ideal_b_d @ r @ A2 )
% 3.01/1.71 => ( ( submodule_b_d_a_c @ r @ m @ N3 )
% 3.01/1.71 => ( ( ord_less_eq_set_a @ H @ N3 )
% 3.01/1.71 => ( ord_less_eq_set_a @ ( linear_span_b_d_a_c @ r @ m @ A2 @ H ) @ N3 ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_320_fgs__generator,axiom,
% 3.01/1.71 ! [H: set_a] :
% 3.01/1.71 ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 3.01/1.71 => ( genera1692266857t_unit @ r @ ( mdl_a_b_c @ m @ ( algebr1152250919_d_a_c @ r @ m @ H ) ) @ H ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_321_submodule__0,axiom,
% 3.01/1.71 submodule_b_d_a_c @ r @ m @ ( insert_a @ ( zero_a1261444626_a_b_c @ m ) @ bot_bot_set_a ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_322_empty__fgs,axiom,
% 3.01/1.71 ( ( algebr1152250919_d_a_c @ r @ m @ bot_bot_set_a )
% 3.01/1.71 = ( insert_a @ ( zero_a1261444626_a_b_c @ m ) @ bot_bot_set_a ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_323_le__zero__eq,axiom,
% 3.01/1.71 ! [N: nat] :
% 3.01/1.71 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 3.01/1.71 = ( N = zero_zero_nat ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_324_not__gr__zero,axiom,
% 3.01/1.71 ! [N: nat] :
% 3.01/1.71 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 3.01/1.71 = ( N = zero_zero_nat ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_325_zero__diff,axiom,
% 3.01/1.71 ! [A: nat] :
% 3.01/1.71 ( ( minus_minus_nat @ zero_zero_nat @ A )
% 3.01/1.71 = zero_zero_nat ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_326_diff__zero,axiom,
% 3.01/1.71 ! [A: nat] :
% 3.01/1.71 ( ( minus_minus_nat @ A @ zero_zero_nat )
% 3.01/1.71 = A ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_327_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 3.01/1.71 ! [A: nat] :
% 3.01/1.71 ( ( minus_minus_nat @ A @ A )
% 3.01/1.71 = zero_zero_nat ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_328_Diff__eq__empty__iff,axiom,
% 3.01/1.71 ! [A2: set_a,B2: set_a] :
% 3.01/1.71 ( ( ( minus_minus_set_a @ A2 @ B2 )
% 3.01/1.71 = bot_bot_set_a )
% 3.01/1.71 = ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_329_subset__empty,axiom,
% 3.01/1.71 ! [A2: set_a] :
% 3.01/1.71 ( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
% 3.01/1.71 = ( A2 = bot_bot_set_a ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_330_empty__subsetI,axiom,
% 3.01/1.71 ! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_331_insert__subset,axiom,
% 3.01/1.71 ! [X: nat > a,A2: set_nat_a,B2: set_nat_a] :
% 3.01/1.71 ( ( ord_le157368549_nat_a @ ( insert_nat_a @ X @ A2 ) @ B2 )
% 3.01/1.71 = ( ( member_nat_a @ X @ B2 )
% 3.01/1.71 & ( ord_le157368549_nat_a @ A2 @ B2 ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_332_insert__subset,axiom,
% 3.01/1.71 ! [X: nat > b,A2: set_nat_b,B2: set_nat_b] :
% 3.01/1.71 ( ( ord_le1296932838_nat_b @ ( insert_nat_b @ X @ A2 ) @ B2 )
% 3.01/1.71 = ( ( member_nat_b @ X @ B2 )
% 3.01/1.71 & ( ord_le1296932838_nat_b @ A2 @ B2 ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_333_insert__subset,axiom,
% 3.01/1.71 ! [X: a,A2: set_a,B2: set_a] :
% 3.01/1.71 ( ( ord_less_eq_set_a @ ( insert_a @ X @ A2 ) @ B2 )
% 3.01/1.71 = ( ( member_a @ X @ B2 )
% 3.01/1.71 & ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_334_insert__Diff__single,axiom,
% 3.01/1.71 ! [A: a,A2: set_a] :
% 3.01/1.71 ( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
% 3.01/1.71 = ( insert_a @ A @ A2 ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_335_singletonI,axiom,
% 3.01/1.71 ! [A: nat > a] : ( member_nat_a @ A @ ( insert_nat_a @ A @ bot_bot_set_nat_a ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_336_singletonI,axiom,
% 3.01/1.71 ! [A: nat > b] : ( member_nat_b @ A @ ( insert_nat_b @ A @ bot_bot_set_nat_b ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_337_singletonI,axiom,
% 3.01/1.71 ! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_338_linear__span__subModule,axiom,
% 3.01/1.71 ! [A2: set_b,H: set_a] :
% 3.01/1.71 ( ( ideal_b_d @ r @ A2 )
% 3.01/1.71 => ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 3.01/1.71 => ( submodule_b_d_a_c @ r @ m @ ( linear_span_b_d_a_c @ r @ m @ A2 @ H ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_339_l__spanA__l__span,axiom,
% 3.01/1.71 ! [A2: set_b,H: set_a] :
% 3.01/1.71 ( ( ideal_b_d @ r @ A2 )
% 3.01/1.71 => ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 3.01/1.71 => ( ( linear_span_b_d_a_c @ r @ m @ A2 @ ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ H ) )
% 3.01/1.71 = ( linear_span_b_d_a_c @ r @ m @ A2 @ H ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_340_lin__span__coeff__mono,axiom,
% 3.01/1.71 ! [A2: set_b,H: set_a] :
% 3.01/1.71 ( ( ideal_b_d @ r @ A2 )
% 3.01/1.71 => ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 3.01/1.71 => ( ord_less_eq_set_a @ ( linear_span_b_d_a_c @ r @ m @ A2 @ H ) @ ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ H ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_341_linear__span__inc__0,axiom,
% 3.01/1.71 ! [A2: set_b,H: set_a] :
% 3.01/1.71 ( ( ideal_b_d @ r @ A2 )
% 3.01/1.71 => ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 3.01/1.71 => ( member_a @ ( zero_a1261444626_a_b_c @ m ) @ ( linear_span_b_d_a_c @ r @ m @ A2 @ H ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_342_generator__generator,axiom,
% 3.01/1.71 ! [H: set_a,H1: set_a] :
% 3.01/1.71 ( ( generator_b_d_a_c @ r @ m @ H )
% 3.01/1.71 => ( ( ord_less_eq_set_a @ H1 @ ( carrie2021454486_a_b_c @ m ) )
% 3.01/1.71 => ( ( ord_less_eq_set_a @ H @ ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ H1 ) )
% 3.01/1.71 => ( generator_b_d_a_c @ r @ m @ H1 ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_343_l__span__zero,axiom,
% 3.01/1.71 ! [A2: set_b] :
% 3.01/1.71 ( ( ideal_b_d @ r @ A2 )
% 3.01/1.71 => ( ( linear_span_b_d_a_c @ r @ m @ A2 @ ( insert_a @ ( zero_a1261444626_a_b_c @ m ) @ bot_bot_set_a ) )
% 3.01/1.71 = ( insert_a @ ( zero_a1261444626_a_b_c @ m ) @ bot_bot_set_a ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_344_l__span__closed3,axiom,
% 3.01/1.71 ! [A2: set_b,H: set_a] :
% 3.01/1.71 ( ( ideal_b_d @ r @ A2 )
% 3.01/1.71 => ( ( generator_b_d_a_c @ r @ m @ H )
% 3.01/1.71 => ( ( ( smodul818989740_d_a_c @ r @ m @ A2 )
% 3.01/1.71 = ( carrie2021454486_a_b_c @ m ) )
% 3.01/1.71 => ( ( linear_span_b_d_a_c @ r @ m @ A2 @ H )
% 3.01/1.71 = ( carrie2021454486_a_b_c @ m ) ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_345_singleton__conv,axiom,
% 3.01/1.71 ! [A: a] :
% 3.01/1.71 ( ( collect_a
% 3.01/1.71 @ ^ [X3: a] : ( X3 = A ) )
% 3.01/1.71 = ( insert_a @ A @ bot_bot_set_a ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_346_l__comb__mem__linear__span,axiom,
% 3.01/1.71 ! [A2: set_b,H: set_a,S: nat > b,N: nat,F: nat > a] :
% 3.01/1.71 ( ( ideal_b_d @ r @ A2 )
% 3.01/1.71 => ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 3.01/1.71 => ( ( member_nat_b @ S
% 3.01/1.71 @ ( pi_nat_b
% 3.01/1.71 @ ( collect_nat
% 3.01/1.71 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.71 @ ^ [Uu: nat] : A2 ) )
% 3.01/1.71 => ( ( member_nat_a @ F
% 3.01/1.71 @ ( pi_nat_a
% 3.01/1.71 @ ( collect_nat
% 3.01/1.71 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.71 @ ^ [Uu: nat] : H ) )
% 3.01/1.71 => ( member_a @ ( l_comb_b_d_a_c @ r @ m @ N @ S @ F ) @ ( linear_span_b_d_a_c @ r @ m @ A2 @ H ) ) ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_347_l__span__closed,axiom,
% 3.01/1.71 ! [A2: set_b,H: set_a,S: nat > b,N: nat,F: nat > a] :
% 3.01/1.71 ( ( ideal_b_d @ r @ A2 )
% 3.01/1.71 => ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
% 3.01/1.71 => ( ( member_nat_b @ S
% 3.01/1.71 @ ( pi_nat_b
% 3.01/1.71 @ ( collect_nat
% 3.01/1.71 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.71 @ ^ [Uu: nat] : A2 ) )
% 3.01/1.71 => ( ( member_nat_a @ F
% 3.01/1.71 @ ( pi_nat_a
% 3.01/1.71 @ ( collect_nat
% 3.01/1.71 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.71 @ ^ [Uu: nat] : ( linear_span_b_d_a_c @ r @ m @ A2 @ H ) ) )
% 3.01/1.71 => ( member_a @ ( l_comb_b_d_a_c @ r @ m @ N @ S @ F ) @ ( linear_span_b_d_a_c @ r @ m @ A2 @ H ) ) ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(fact_348_linear__comb0__2,axiom,
% 3.01/1.71 ! [A2: set_b,S: nat > b,N: nat,M: nat > a] :
% 3.01/1.71 ( ( ideal_b_d @ r @ A2 )
% 3.01/1.71 => ( ( member_nat_b @ S
% 3.01/1.71 @ ( pi_nat_b
% 3.01/1.71 @ ( collect_nat
% 3.01/1.71 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.71 @ ^ [Uu: nat] : A2 ) )
% 3.01/1.71 => ( ( member_nat_a @ M
% 3.01/1.71 @ ( pi_nat_a
% 3.01/1.71 @ ( collect_nat
% 3.01/1.71 @ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
% 3.01/1.71 @ ^ [Uu: nat] : ( insert_a @ ( zero_a1261444626_a_b_c @ m ) @ bot_bot_set_a ) ) )
% 3.01/1.71 => ( ( l_comb_b_d_a_c @ r @ m @ N @ S @ M )
% 3.01/1.71 = ( zero_a1261444626_a_b_c @ m ) ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(conj_0,hypothesis,
% 3.01/1.71 ( member_nat_a @ g
% 3.01/1.71 @ ( pi_nat_a
% 3.01/1.71 @ ( collect_nat
% 3.01/1.71 @ ^ [K4: nat] : ( ord_less_eq_nat @ K4 @ m2 ) )
% 3.01/1.71 @ ^ [Uu: nat] : ( insert_a @ h2 @ h1 ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(conj_1,hypothesis,
% 3.01/1.71 ! [X4: nat] :
% 3.01/1.71 ( ( ord_less_eq_nat @ X4 @ m2 )
% 3.01/1.71 => ! [Y3: nat] :
% 3.01/1.71 ( ( ord_less_eq_nat @ Y3 @ m2 )
% 3.01/1.71 => ( ( ( g @ X4 )
% 3.01/1.71 = ( g @ Y3 ) )
% 3.01/1.71 => ( X4 = Y3 ) ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(conj_2,hypothesis,
% 3.01/1.71 ( member_nat_b @ t
% 3.01/1.71 @ ( pi_nat_b
% 3.01/1.71 @ ( collect_nat
% 3.01/1.71 @ ^ [K4: nat] : ( ord_less_eq_nat @ K4 @ m2 ) )
% 3.01/1.71 @ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ r ) ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(conj_3,hypothesis,
% 3.01/1.71 ( ( g @ m2 )
% 3.01/1.71 = h2 ) ).
% 3.01/1.71
% 3.01/1.71 thf(conj_4,hypothesis,
% 3.01/1.71 module_e_b_f_d @ n @ r ).
% 3.01/1.71
% 3.01/1.71 thf(conj_5,hypothesis,
% 3.01/1.71 free_g1087686480_d_a_c @ r @ m @ h ).
% 3.01/1.71
% 3.01/1.71 thf(conj_6,hypothesis,
% 3.01/1.71 ord_less_eq_set_a @ h1 @ h ).
% 3.01/1.71
% 3.01/1.71 thf(conj_7,hypothesis,
% 3.01/1.71 $true ).
% 3.01/1.71
% 3.01/1.71 thf(conj_8,hypothesis,
% 3.01/1.71 ideal_b_d @ r @ ( carrie2079586589xt_b_d @ r ) ).
% 3.01/1.71
% 3.01/1.71 thf(conj_9,hypothesis,
% 3.01/1.71 ord_less_eq_set_a @ h @ ( carrie2021454486_a_b_c @ m ) ).
% 3.01/1.71
% 3.01/1.71 thf(conj_10,hypothesis,
% 3.01/1.71 ord_less_eq_set_a @ h1 @ ( carrie2021454486_a_b_c @ m ) ).
% 3.01/1.71
% 3.01/1.71 thf(conj_11,hypothesis,
% 3.01/1.71 ord_less_eq_set_a @ ( algebr1152250919_d_a_c @ r @ m @ h1 ) @ ( carrie2021454486_a_b_c @ m ) ).
% 3.01/1.71
% 3.01/1.71 thf(conj_12,hypothesis,
% 3.01/1.71 member_a @ ( l_comb_b_d_a_c @ r @ m @ m2 @ t @ g ) @ ( algebr1152250919_d_a_c @ r @ m @ ( insert_a @ h2 @ h1 ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(conj_13,hypothesis,
% 3.01/1.71 ( x
% 3.01/1.71 = ( l_comb_b_d_a_c @ r @ m @ m2 @ t @ g ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(conj_14,hypothesis,
% 3.01/1.71 member_a @ h2 @ h ).
% 3.01/1.71
% 3.01/1.71 thf(conj_15,hypothesis,
% 3.01/1.71 ~ ( member_a @ h2 @ h1 ) ).
% 3.01/1.71
% 3.01/1.71 thf(conj_16,hypothesis,
% 3.01/1.71 ord_less_nat @ zero_zero_nat @ m2 ).
% 3.01/1.71
% 3.01/1.71 thf(conj_17,hypothesis,
% 3.01/1.71 member_a @ h2 @ ( carrie2021454486_a_b_c @ m ) ).
% 3.01/1.71
% 3.01/1.71 thf(conj_18,hypothesis,
% 3.01/1.71 submodule_b_d_a_c @ r @ m @ ( algebr1152250919_d_a_c @ r @ m @ ( insert_a @ h2 @ bot_bot_set_a ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(conj_19,hypothesis,
% 3.01/1.71 ord_less_eq_nat @ xa @ ( minus_minus_nat @ m2 @ ( suc @ zero_zero_nat ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(conj_20,hypothesis,
% 3.01/1.71 ( ( ( g @ xa )
% 3.01/1.71 = h2 )
% 3.01/1.71 | ( member_a @ ( g @ xa ) @ h1 ) ) ).
% 3.01/1.71
% 3.01/1.71 thf(conj_21,conjecture,
% 3.01/1.71 member_a @ ( g @ xa ) @ h1 ).
% 3.01/1.71 ------------------------
% 3.01/1.75 % (9656)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_9318 for (2999ds/27Mi)
% 3.01/1.75 % (9657)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_9318 for (2999ds/2Mi)
% 3.01/1.75 % (9660)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_9318 for (2999ds/18Mi)
% 3.01/1.75 % (9657)Instruction limit reached!
% 3.01/1.75 % (9657)------------------------------
% 3.01/1.75 % (9657)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.01/1.75 % (9657)Termination reason: Unknown
% 3.01/1.75 % (9657)Termination phase: shuffling
% 3.01/1.75
% 3.01/1.75 % (9657)Memory used [KB]: 1535
% 3.01/1.75 % (9657)Time elapsed: 0.002 s
% 3.01/1.75 % (9657)Instructions burned: 2 (million)
% 3.01/1.75 % (9657)------------------------------
% 3.01/1.75 % (9657)------------------------------
% 3.01/1.75 % (9654)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_9318 for (2999ds/183Mi)
% 3.01/1.75 % (9655)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_9318 for (2999ds/4Mi)
% 3.01/1.75 % (9660)Instruction limit reached!
% 3.01/1.75 % (9660)------------------------------
% 3.01/1.75 % (9660)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.01/1.75 % (9660)Termination reason: Unknown
% 3.01/1.75 % (9660)Termination phase: shuffling
% 3.01/1.75
% 3.01/1.75 % (9660)Memory used [KB]: 1918
% 3.01/1.75 % (9660)Time elapsed: 0.008 s
% 3.01/1.75 % (9660)Instructions burned: 20 (million)
% 3.01/1.75 % (9660)------------------------------
% 3.01/1.75 % (9660)------------------------------
% 3.01/1.75 % (9658)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_9318 for (2999ds/2Mi)
% 3.01/1.76 % (9656)Instruction limit reached!
% 3.01/1.76 % (9656)------------------------------
% 3.01/1.76 % (9656)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.01/1.76 % (9656)Termination reason: Unknown
% 3.01/1.76 % (9656)Termination phase: shuffling
% 3.01/1.76
% 3.01/1.76 % (9656)Memory used [KB]: 1918
% 3.01/1.76 % (9656)Time elapsed: 0.011 s
% 3.01/1.76 % (9656)Instructions burned: 29 (million)
% 3.01/1.76 % (9656)------------------------------
% 3.01/1.76 % (9656)------------------------------
% 3.01/1.76 % (9658)Instruction limit reached!
% 3.01/1.76 % (9658)------------------------------
% 3.01/1.76 % (9658)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.01/1.76 % (9658)Termination reason: Unknown
% 3.01/1.76 % (9658)Termination phase: shuffling
% 3.01/1.76
% 3.01/1.76 % (9658)Memory used [KB]: 1535
% 3.01/1.76 % (9658)Time elapsed: 0.004 s
% 3.01/1.76 % (9658)Instructions burned: 3 (million)
% 3.01/1.76 % (9658)------------------------------
% 3.01/1.76 % (9658)------------------------------
% 3.01/1.76 % (9659)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_9318 for (2999ds/275Mi)
% 3.01/1.76 % (9655)Instruction limit reached!
% 3.01/1.76 % (9655)------------------------------
% 3.01/1.76 % (9655)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.01/1.76 % (9655)Termination reason: Unknown
% 3.01/1.76 % (9655)Termination phase: shuffling
% 3.01/1.76
% 3.01/1.76 % (9655)Memory used [KB]: 1535
% 3.01/1.76 % (9655)Time elapsed: 0.005 s
% 3.01/1.76 % (9655)Instructions burned: 5 (million)
% 3.01/1.76 % (9655)------------------------------
% 3.01/1.76 % (9655)------------------------------
% 3.01/1.76 % (9663)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_9318 for (2999ds/3Mi)
% 3.01/1.76 % (9663)Instruction limit reached!
% 3.01/1.76 % (9663)------------------------------
% 3.01/1.76 % (9663)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.01/1.76 % (9663)Termination reason: Unknown
% 3.01/1.76 % (9663)Termination phase: shuffling
% 3.01/1.76
% 3.01/1.76 % (9663)Memory used [KB]: 1535
% 3.01/1.76 % (9663)Time elapsed: 0.003 s
% 3.01/1.76 % (9663)Instructions burned: 3 (million)
% 3.01/1.76 % (9663)------------------------------
% 3.01/1.76 % (9663)------------------------------
% 3.01/1.77 % (9665)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_9318 for (2999ds/37Mi)
% 3.01/1.77 % (9670)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_9318 for (2999ds/7Mi)
% 3.01/1.77 % (9670)Instruction limit reached!
% 3.01/1.77 % (9670)------------------------------
% 3.01/1.77 % (9670)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.01/1.77 % (9670)Termination reason: Unknown
% 3.01/1.77 % (9670)Termination phase: shuffling
% 3.01/1.77
% 3.01/1.77 % (9670)Memory used [KB]: 1663
% 3.01/1.77 % (9670)Time elapsed: 0.004 s
% 3.01/1.77 % (9670)Instructions burned: 9 (million)
% 3.01/1.77 % (9670)------------------------------
% 3.01/1.77 % (9670)------------------------------
% 3.01/1.77 % (9667)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_9318 for (2999ds/15Mi)
% 3.01/1.77 % (9668)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_9318 for (2999ds/3Mi)
% 3.01/1.77 % (9669)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_9318 for (2999ds/1041Mi)
% 3.01/1.78 % (9668)Instruction limit reached!
% 3.01/1.78 % (9668)------------------------------
% 3.01/1.78 % (9668)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.01/1.78 % (9668)Termination reason: Unknown
% 3.01/1.78 % (9668)Termination phase: shuffling
% 3.01/1.78
% 3.01/1.78 % (9668)Memory used [KB]: 1535
% 3.01/1.78 % (9668)Time elapsed: 0.004 s
% 3.01/1.78 % (9668)Instructions burned: 3 (million)
% 3.01/1.78 % (9668)------------------------------
% 3.01/1.78 % (9668)------------------------------
% 3.01/1.78 % (9665)Instruction limit reached!
% 3.01/1.78 % (9665)------------------------------
% 3.01/1.78 % (9665)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.01/1.78 % (9665)Termination reason: Unknown
% 3.01/1.78 % (9665)Termination phase: shuffling
% 3.01/1.78
% 3.01/1.78 % (9665)Memory used [KB]: 2302
% 3.01/1.78 % (9665)Time elapsed: 0.012 s
% 3.01/1.78 % (9665)Instructions burned: 37 (million)
% 3.01/1.78 % (9665)------------------------------
% 3.01/1.78 % (9665)------------------------------
% 3.01/1.78 % (9674)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_9318 for (2999ds/16Mi)
% 3.01/1.78 % (9667)Instruction limit reached!
% 3.01/1.78 % (9667)------------------------------
% 3.01/1.78 % (9667)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.01/1.78 % (9667)Termination reason: Unknown
% 3.01/1.78 % (9667)Termination phase: shuffling
% 3.01/1.78
% 3.01/1.78 % (9667)Memory used [KB]: 1791
% 3.01/1.78 % (9667)Time elapsed: 0.012 s
% 3.01/1.78 % (9667)Instructions burned: 16 (million)
% 3.01/1.78 % (9667)------------------------------
% 3.01/1.78 % (9667)------------------------------
% 3.01/1.79 % (9676)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_9318 for (2999ds/3Mi)
% 3.01/1.79 % (9674)Instruction limit reached!
% 3.01/1.79 % (9674)------------------------------
% 3.01/1.79 % (9674)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.01/1.79 % (9674)Termination reason: Unknown
% 3.01/1.79 % (9674)Termination phase: shuffling
% 3.01/1.79
% 3.01/1.79 % (9674)Memory used [KB]: 1791
% 3.01/1.79 % (9674)Time elapsed: 0.007 s
% 3.01/1.79 % (9674)Instructions burned: 18 (million)
% 3.01/1.79 % (9674)------------------------------
% 3.01/1.79 % (9674)------------------------------
% 3.01/1.79 % (9676)Instruction limit reached!
% 3.01/1.79 % (9676)------------------------------
% 3.01/1.79 % (9676)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.01/1.79 % (9676)Termination reason: Unknown
% 3.01/1.79 % (9676)Termination phase: shuffling
% 3.01/1.79
% 3.01/1.79 % (9676)Memory used [KB]: 1535
% 3.01/1.79 % (9676)Time elapsed: 0.003 s
% 3.01/1.79 % (9676)Instructions burned: 3 (million)
% 3.01/1.79 % (9676)------------------------------
% 3.01/1.79 % (9676)------------------------------
% 3.01/1.79 % (9675)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_9318 for (2999ds/3Mi)
% 3.01/1.80 % (9675)Instruction limit reached!
% 3.01/1.80 % (9675)------------------------------
% 3.01/1.80 % (9675)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.01/1.80 % (9675)Termination reason: Unknown
% 3.01/1.80 % (9675)Termination phase: shuffling
% 3.01/1.80
% 3.01/1.80 % (9675)Memory used [KB]: 1535
% 3.01/1.80 % (9675)Time elapsed: 0.004 s
% 3.01/1.80 % (9675)Instructions burned: 3 (million)
% 3.01/1.80 % (9675)------------------------------
% 3.01/1.80 % (9675)------------------------------
% 3.07/1.80 % (9679)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_9318 for (2999ds/3Mi)
% 3.07/1.80 % (9679)Instruction limit reached!
% 3.07/1.80 % (9679)------------------------------
% 3.07/1.80 % (9679)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.80 % (9679)Termination reason: Unknown
% 3.07/1.80 % (9679)Termination phase: shuffling
% 3.07/1.80
% 3.07/1.80 % (9679)Memory used [KB]: 1535
% 3.07/1.80 % (9679)Time elapsed: 0.002 s
% 3.07/1.80 % (9679)Instructions burned: 3 (million)
% 3.07/1.80 % (9679)------------------------------
% 3.07/1.80 % (9679)------------------------------
% 3.07/1.80 % (9680)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_9318 for (2999ds/4Mi)
% 3.07/1.80 % (9680)Instruction limit reached!
% 3.07/1.80 % (9680)------------------------------
% 3.07/1.80 % (9680)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.80 % (9680)Termination reason: Unknown
% 3.07/1.80 % (9680)Termination phase: shuffling
% 3.07/1.80
% 3.07/1.80 % (9680)Memory used [KB]: 1535
% 3.07/1.80 % (9680)Time elapsed: 0.003 s
% 3.07/1.80 % (9680)Instructions burned: 5 (million)
% 3.07/1.80 % (9680)------------------------------
% 3.07/1.80 % (9680)------------------------------
% 3.07/1.80 % (9678)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_9318 for (2999ds/7Mi)
% 3.07/1.81 % (9678)Instruction limit reached!
% 3.07/1.81 % (9678)------------------------------
% 3.07/1.81 % (9678)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.81 % (9678)Termination reason: Unknown
% 3.07/1.81 % (9678)Termination phase: shuffling
% 3.07/1.81
% 3.07/1.81 % (9678)Memory used [KB]: 1663
% 3.07/1.81 % (9678)Time elapsed: 0.006 s
% 3.07/1.81 % (9678)Instructions burned: 7 (million)
% 3.07/1.81 % (9678)------------------------------
% 3.07/1.81 % (9678)------------------------------
% 3.07/1.81 % (9686)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_9318 for (2999ds/710Mi)
% 3.07/1.81 % (9688)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_9318 for (2999ds/6Mi)
% 3.07/1.81 % (9688)Instruction limit reached!
% 3.07/1.81 % (9688)------------------------------
% 3.07/1.81 % (9688)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.81 % (9684)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_9318 for (2999ds/18Mi)
% 3.07/1.81 % (9688)Termination reason: Unknown
% 3.07/1.81 % (9688)Termination phase: shuffling
% 3.07/1.81
% 3.07/1.81 % (9688)Memory used [KB]: 1663
% 3.07/1.81 % (9688)Time elapsed: 0.004 s
% 3.07/1.81 % (9688)Instructions burned: 8 (million)
% 3.07/1.81 % (9688)------------------------------
% 3.07/1.81 % (9688)------------------------------
% 3.07/1.82 % (9684)Instruction limit reached!
% 3.07/1.82 % (9684)------------------------------
% 3.07/1.82 % (9684)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.82 % (9684)Termination reason: Unknown
% 3.07/1.82 % (9684)Termination phase: shuffling
% 3.07/1.82
% 3.07/1.82 % (9684)Memory used [KB]: 1791
% 3.07/1.82 % (9684)Time elapsed: 0.011 s
% 3.07/1.82 % (9684)Instructions burned: 20 (million)
% 3.07/1.82 % (9684)------------------------------
% 3.07/1.82 % (9684)------------------------------
% 3.07/1.82 % (9692)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_9318 for (2999ds/21Mi)
% 3.07/1.82 % (9691)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_9318 for (2999ds/902Mi)
% 3.07/1.83 % (9692)Instruction limit reached!
% 3.07/1.83 % (9692)------------------------------
% 3.07/1.83 % (9692)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.83 % (9692)Termination reason: Unknown
% 3.07/1.83 % (9692)Termination phase: shuffling
% 3.07/1.83
% 3.07/1.83 % (9692)Memory used [KB]: 1918
% 3.07/1.83 % (9692)Time elapsed: 0.008 s
% 3.07/1.83 % (9692)Instructions burned: 21 (million)
% 3.07/1.83 % (9692)------------------------------
% 3.07/1.83 % (9692)------------------------------
% 3.07/1.83 % (9693)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_9318 for (2998ds/5Mi)
% 3.07/1.83 % (9693)Instruction limit reached!
% 3.07/1.83 % (9693)------------------------------
% 3.07/1.83 % (9693)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.83 % (9693)Termination reason: Unknown
% 3.07/1.83 % (9693)Termination phase: shuffling
% 3.07/1.83
% 3.07/1.83 % (9693)Memory used [KB]: 1663
% 3.07/1.83 % (9693)Time elapsed: 0.004 s
% 3.07/1.83 % (9693)Instructions burned: 7 (million)
% 3.07/1.83 % (9693)------------------------------
% 3.07/1.83 % (9693)------------------------------
% 3.07/1.84 % (9654)Instruction limit reached!
% 3.07/1.84 % (9654)------------------------------
% 3.07/1.84 % (9654)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.84 % (9654)Termination reason: Unknown
% 3.07/1.84 % (9654)Termination phase: Saturation
% 3.07/1.84
% 3.07/1.84 % (9654)Memory used [KB]: 7419
% 3.07/1.84 % (9654)Time elapsed: 0.085 s
% 3.07/1.84 % (9654)Instructions burned: 184 (million)
% 3.07/1.84 % (9654)------------------------------
% 3.07/1.84 % (9654)------------------------------
% 3.07/1.84 % (9697)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_9318 for (2998ds/6Mi)
% 3.07/1.84 % (9697)Instruction limit reached!
% 3.07/1.84 % (9697)------------------------------
% 3.07/1.84 % (9697)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.84 % (9697)Termination reason: Unknown
% 3.07/1.84 % (9697)Termination phase: shuffling
% 3.07/1.84
% 3.07/1.84 % (9697)Memory used [KB]: 1663
% 3.07/1.84 % (9697)Time elapsed: 0.003 s
% 3.07/1.84 % (9697)Instructions burned: 7 (million)
% 3.07/1.84 % (9697)------------------------------
% 3.07/1.84 % (9697)------------------------------
% 3.07/1.84 % (9700)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_9318 for (2998ds/377Mi)
% 3.07/1.85 % (9702)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_9318 for (2998ds/779Mi)
% 3.07/1.85 % (9703)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_9318 for (2998ds/19Mi)
% 3.07/1.86 % (9703)Instruction limit reached!
% 3.07/1.86 % (9703)------------------------------
% 3.07/1.86 % (9703)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.86 % (9703)Termination reason: Unknown
% 3.07/1.86 % (9703)Termination phase: shuffling
% 3.07/1.86
% 3.07/1.86 % (9703)Memory used [KB]: 2046
% 3.07/1.86 % (9703)Time elapsed: 0.007 s
% 3.07/1.86 % (9703)Instructions burned: 21 (million)
% 3.07/1.86 % (9703)------------------------------
% 3.07/1.86 % (9703)------------------------------
% 3.07/1.87 % (9707)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_9318 for (2998ds/879Mi)
% 3.07/1.87 % (9659)Instruction limit reached!
% 3.07/1.87 % (9659)------------------------------
% 3.07/1.87 % (9659)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.87 % (9659)Termination reason: Unknown
% 3.07/1.87 % (9659)Termination phase: Saturation
% 3.07/1.87
% 3.07/1.87 % (9659)Memory used [KB]: 7675
% 3.07/1.87 % (9659)Time elapsed: 0.119 s
% 3.07/1.87 % (9659)Instructions burned: 275 (million)
% 3.07/1.87 % (9659)------------------------------
% 3.07/1.87 % (9659)------------------------------
% 3.07/1.88 % (9714)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_9318 for (2998ds/17Mi)
% 3.07/1.89 % (9714)Instruction limit reached!
% 3.07/1.89 % (9714)------------------------------
% 3.07/1.89 % (9714)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.89 % (9714)Termination reason: Unknown
% 3.07/1.89 % (9714)Termination phase: shuffling
% 3.07/1.89
% 3.07/1.89 % (9714)Memory used [KB]: 1918
% 3.07/1.89 % (9714)Time elapsed: 0.008 s
% 3.07/1.89 % (9714)Instructions burned: 17 (million)
% 3.07/1.89 % (9714)------------------------------
% 3.07/1.89 % (9714)------------------------------
% 3.07/1.90 % (9723)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_9318 for (2998ds/3Mi)
% 3.07/1.90 % (9723)Instruction limit reached!
% 3.07/1.90 % (9723)------------------------------
% 3.07/1.90 % (9723)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.90 % (9723)Termination reason: Unknown
% 3.07/1.90 % (9723)Termination phase: shuffling
% 3.07/1.90
% 3.07/1.90 % (9723)Memory used [KB]: 1535
% 3.07/1.90 % (9723)Time elapsed: 0.003 s
% 3.07/1.90 % (9723)Instructions burned: 3 (million)
% 3.07/1.90 % (9723)------------------------------
% 3.07/1.90 % (9723)------------------------------
% 3.07/1.91 % (9730)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_9318 for (2998ds/30Mi)
% 3.07/1.92 % (9730)Instruction limit reached!
% 3.07/1.92 % (9730)------------------------------
% 3.07/1.92 % (9730)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.92 % (9730)Termination reason: Unknown
% 3.07/1.92 % (9730)Termination phase: shuffling
% 3.07/1.92
% 3.07/1.92 % (9730)Memory used [KB]: 2046
% 3.07/1.92 % (9730)Time elapsed: 0.013 s
% 3.07/1.92 % (9730)Instructions burned: 32 (million)
% 3.07/1.92 % (9730)------------------------------
% 3.07/1.92 % (9730)------------------------------
% 3.07/1.93 % (9743)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_9318 for (2998ds/127Mi)
% 3.07/1.96 % (9700)Instruction limit reached!
% 3.07/1.96 % (9700)------------------------------
% 3.07/1.96 % (9700)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.96 % (9700)Termination reason: Unknown
% 3.07/1.96 % (9700)Termination phase: Saturation
% 3.07/1.96
% 3.07/1.96 % (9700)Memory used [KB]: 8699
% 3.07/1.96 % (9700)Time elapsed: 0.120 s
% 3.07/1.96 % (9700)Instructions burned: 377 (million)
% 3.07/1.96 % (9700)------------------------------
% 3.07/1.96 % (9700)------------------------------
% 3.07/1.97 % (9751)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_9318 for (2997ds/100Mi)
% 3.07/1.97 % (9743)Instruction limit reached!
% 3.07/1.97 % (9743)------------------------------
% 3.07/1.97 % (9743)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.97 % (9743)Termination reason: Unknown
% 3.07/1.97 % (9743)Termination phase: Property scanning
% 3.07/1.97
% 3.07/1.97 % (9743)Memory used [KB]: 2814
% 3.07/1.97 % (9743)Time elapsed: 0.042 s
% 3.07/1.97 % (9743)Instructions burned: 130 (million)
% 3.07/1.97 % (9743)------------------------------
% 3.07/1.97 % (9743)------------------------------
% 3.07/1.98 % (9752)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_9318 for (2997ds/3Mi)
% 3.07/1.98 % (9752)Instruction limit reached!
% 3.07/1.98 % (9752)------------------------------
% 3.07/1.98 % (9752)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/1.98 % (9752)Termination reason: Unknown
% 3.07/1.98 % (9752)Termination phase: shuffling
% 3.07/1.98
% 3.07/1.98 % (9752)Memory used [KB]: 1535
% 3.07/1.98 % (9752)Time elapsed: 0.003 s
% 3.07/1.98 % (9752)Instructions burned: 4 (million)
% 3.07/1.98 % (9752)------------------------------
% 3.07/1.98 % (9752)------------------------------
% 3.07/1.99 % (9753)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_9318 for (2997ds/20Mi)
% 3.07/2.00 % (9751)Instruction limit reached!
% 3.07/2.00 % (9751)------------------------------
% 3.07/2.00 % (9751)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/2.00 % (9751)Termination reason: Unknown
% 3.07/2.00 % (9751)Termination phase: shuffling
% 3.07/2.00
% 3.07/2.00 % (9751)Memory used [KB]: 2814
% 3.07/2.00 % (9751)Time elapsed: 0.028 s
% 3.07/2.00 % (9751)Instructions burned: 103 (million)
% 3.07/2.00 % (9751)------------------------------
% 3.07/2.00 % (9751)------------------------------
% 3.07/2.00 % (9753)Instruction limit reached!
% 3.07/2.00 % (9753)------------------------------
% 3.07/2.00 % (9753)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.07/2.00 % (9753)Termination reason: Unknown
% 3.07/2.00 % (9753)Termination phase: shuffling
% 3.07/2.00
% 3.07/2.00 % (9753)Memory used [KB]: 1918
% 3.07/2.00 % (9753)Time elapsed: 0.009 s
% 3.07/2.00 % (9753)Instructions burned: 21 (million)
% 3.07/2.00 % (9753)------------------------------
% 3.07/2.00 % (9753)------------------------------
% 3.07/2.01 % (9754)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_9318 for (2997ds/86Mi)
% 4.94/2.01 % (9755)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_9318 for (2997ds/107Mi)
% 4.94/2.03 % (9686)Instruction limit reached!
% 4.94/2.03 % (9686)------------------------------
% 4.94/2.03 % (9686)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.94/2.03 % (9686)Termination reason: Unknown
% 4.94/2.03 % (9686)Termination phase: Saturation
% 4.94/2.03
% 4.94/2.03 % (9686)Memory used [KB]: 8571
% 4.94/2.03 % (9686)Time elapsed: 0.224 s
% 4.94/2.03 % (9686)Instructions burned: 713 (million)
% 4.94/2.03 % (9686)------------------------------
% 4.94/2.03 % (9686)------------------------------
% 4.94/2.04 % (9754)Instruction limit reached!
% 4.94/2.04 % (9754)------------------------------
% 4.94/2.04 % (9754)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.94/2.04 % (9754)Termination reason: Unknown
% 4.94/2.04 % (9754)Termination phase: Property scanning
% 4.94/2.04
% 4.94/2.04 % (9754)Memory used [KB]: 2686
% 4.94/2.04 % (9754)Time elapsed: 0.027 s
% 4.94/2.04 % (9754)Instructions burned: 86 (million)
% 4.94/2.04 % (9754)------------------------------
% 4.94/2.04 % (9754)------------------------------
% 4.94/2.04 % (9756)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_9318 for (2996ds/39Mi)
% 4.94/2.04 % (9757)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_9318 for (2996ds/448Mi)
% 4.94/2.05 % (9755)Instruction limit reached!
% 4.94/2.05 % (9755)------------------------------
% 4.94/2.05 % (9755)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.94/2.05 % (9755)Termination reason: Unknown
% 4.94/2.05 % (9755)Termination phase: Property scanning
% 4.94/2.05
% 4.94/2.05 % (9755)Memory used [KB]: 2686
% 4.94/2.05 % (9755)Time elapsed: 0.036 s
% 4.94/2.05 % (9755)Instructions burned: 109 (million)
% 4.94/2.05 % (9755)------------------------------
% 4.94/2.05 % (9755)------------------------------
% 4.94/2.05 % (9756)Instruction limit reached!
% 4.94/2.05 % (9756)------------------------------
% 4.94/2.05 % (9756)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.94/2.05 % (9756)Termination reason: Unknown
% 4.94/2.05 % (9756)Termination phase: shuffling
% 4.94/2.05
% 4.94/2.05 % (9756)Memory used [KB]: 2302
% 4.94/2.05 % (9756)Time elapsed: 0.012 s
% 4.94/2.05 % (9756)Instructions burned: 39 (million)
% 4.94/2.05 % (9756)------------------------------
% 4.94/2.05 % (9756)------------------------------
% 4.94/2.06 % (9758)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_9318 for (2996ds/46Mi)
% 4.94/2.06 % (9759)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_9318 for (2996ds/98Mi)
% 4.94/2.07 % (9758)Instruction limit reached!
% 4.94/2.07 % (9758)------------------------------
% 4.94/2.07 % (9758)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.94/2.07 % (9758)Termination reason: Unknown
% 4.94/2.07 % (9758)Termination phase: Property scanning
% 4.94/2.07
% 4.94/2.07 % (9758)Memory used [KB]: 2174
% 4.94/2.07 % (9758)Time elapsed: 0.018 s
% 4.94/2.07 % (9758)Instructions burned: 48 (million)
% 4.94/2.07 % (9758)------------------------------
% 4.94/2.07 % (9758)------------------------------
% 4.94/2.08 % (9702)Instruction limit reached!
% 4.94/2.08 % (9702)------------------------------
% 4.94/2.08 % (9702)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.94/2.08 % (9702)Termination reason: Unknown
% 4.94/2.08 % (9702)Termination phase: Saturation
% 4.94/2.08
% 4.94/2.08 % (9702)Memory used [KB]: 9083
% 4.94/2.08 % (9702)Time elapsed: 0.237 s
% 4.94/2.08 % (9702)Instructions burned: 781 (million)
% 4.94/2.08 % (9702)------------------------------
% 4.94/2.08 % (9702)------------------------------
% 4.94/2.08 % (9760)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_9318 for (2996ds/507Mi)
% 4.94/2.09 % (9759)Instruction limit reached!
% 4.94/2.09 % (9759)------------------------------
% 4.94/2.09 % (9759)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.94/2.09 % (9759)Termination reason: Unknown
% 4.94/2.09 % (9759)Termination phase: Property scanning
% 4.94/2.09
% 4.94/2.09 % (9759)Memory used [KB]: 2814
% 4.94/2.09 % (9759)Time elapsed: 0.027 s
% 4.94/2.09 % (9759)Instructions burned: 99 (million)
% 4.94/2.09 % (9759)------------------------------
% 4.94/2.09 % (9759)------------------------------
% 4.94/2.09 % (9761)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_9318 for (2996ds/149Mi)
% 4.94/2.10 % (9762)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_9318 for (2996ds/39Mi)
% 4.94/2.11 % (9762)Instruction limit reached!
% 4.94/2.11 % (9762)------------------------------
% 4.94/2.11 % (9762)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.94/2.11 % (9762)Termination reason: Unknown
% 4.94/2.11 % (9762)Termination phase: Property scanning
% 4.94/2.11
% 4.94/2.11 % (9762)Memory used [KB]: 2174
% 4.94/2.11 % (9762)Time elapsed: 0.015 s
% 4.94/2.11 % (9762)Instructions burned: 42 (million)
% 4.94/2.11 % (9762)------------------------------
% 4.94/2.11 % (9762)------------------------------
% 5.54/2.12 % (9763)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_9318 for (2996ds/6Mi)
% 5.54/2.12 % (9763)Instruction limit reached!
% 5.54/2.12 % (9763)------------------------------
% 5.54/2.12 % (9763)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.54/2.12 % (9763)Termination reason: Unknown
% 5.54/2.12 % (9763)Termination phase: shuffling
% 5.54/2.12
% 5.54/2.12 % (9763)Memory used [KB]: 1663
% 5.54/2.12 % (9763)Time elapsed: 0.003 s
% 5.54/2.12 % (9763)Instructions burned: 7 (million)
% 5.54/2.12 % (9763)------------------------------
% 5.54/2.12 % (9763)------------------------------
% 5.54/2.13 % (9669)Instruction limit reached!
% 5.54/2.13 % (9669)------------------------------
% 5.54/2.13 % (9669)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.54/2.13 % (9669)Termination reason: Unknown
% 5.54/2.13 % (9669)Termination phase: Saturation
% 5.54/2.13
% 5.54/2.13 % (9669)Memory used [KB]: 10234
% 5.54/2.13 % (9669)Time elapsed: 0.356 s
% 5.54/2.13 % (9669)Instructions burned: 1041 (million)
% 5.54/2.13 % (9669)------------------------------
% 5.54/2.13 % (9669)------------------------------
% 5.54/2.13 % (9764)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=1087:si=on:rtra=on_0 on DTF2THF_9318 for (2995ds/1087Mi)
% 5.54/2.13 % (9691)Instruction limit reached!
% 5.54/2.13 % (9691)------------------------------
% 5.54/2.13 % (9691)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.54/2.13 % (9691)Termination reason: Unknown
% 5.54/2.13 % (9691)Termination phase: Saturation
% 5.54/2.13
% 5.54/2.13 % (9691)Memory used [KB]: 8571
% 5.54/2.13 % (9691)Time elapsed: 0.310 s
% 5.54/2.13 % (9691)Instructions burned: 904 (million)
% 5.54/2.13 % (9691)------------------------------
% 5.54/2.13 % (9691)------------------------------
% 5.54/2.14 % (9761)Instruction limit reached!
% 5.54/2.14 % (9761)------------------------------
% 5.54/2.14 % (9761)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.54/2.14 % (9761)Termination reason: Unknown
% 5.54/2.14 % (9761)Termination phase: Saturation
% 5.54/2.14
% 5.54/2.14 % (9761)Memory used [KB]: 7419
% 5.54/2.14 % (9761)Time elapsed: 0.044 s
% 5.54/2.14 % (9761)Instructions burned: 152 (million)
% 5.54/2.14 % (9761)------------------------------
% 5.54/2.14 % (9761)------------------------------
% 5.54/2.14 % (9765)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_9318 for (2995ds/47Mi)
% 5.54/2.14 % (9766)lrs+1010_1:128_av=off:cbe=off:cnfonf=lazy_simp:fsr=off:ntd=on:ss=axioms:st=1.5:i=44:si=on:rtra=on_0 on DTF2THF_9318 for (2995ds/44Mi)
% 5.54/2.14 % (9767)dis+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:tnu=1:i=432:si=on:rtra=on_0 on DTF2THF_9318 for (2995ds/432Mi)
% 5.54/2.15 % (9766)Instruction limit reached!
% 5.54/2.15 % (9766)------------------------------
% 5.54/2.15 % (9766)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.54/2.15 % (9766)Termination reason: Unknown
% 5.54/2.15 % (9766)Termination phase: shuffling
% 5.54/2.15
% 5.54/2.15 % (9766)Memory used [KB]: 2430
% 5.54/2.15 % (9766)Time elapsed: 0.015 s
% 5.54/2.15 % (9766)Instructions burned: 44 (million)
% 5.54/2.15 % (9766)------------------------------
% 5.54/2.15 % (9766)------------------------------
% 5.54/2.16 % (9765)Instruction limit reached!
% 5.54/2.16 % (9765)------------------------------
% 5.54/2.16 % (9765)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.54/2.16 % (9765)Termination reason: Unknown
% 5.54/2.16 % (9765)Termination phase: Property scanning
% 5.54/2.16
% 5.54/2.16 % (9765)Memory used [KB]: 2174
% 5.54/2.16 % (9765)Time elapsed: 0.018 s
% 5.54/2.16 % (9765)Instructions burned: 47 (million)
% 5.54/2.16 % (9765)------------------------------
% 5.54/2.16 % (9765)------------------------------
% 5.54/2.16 % (9768)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_9318 for (2995ds/62Mi)
% 5.54/2.17 % (9769)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_9318 for (2995ds/31Mi)
% 5.54/2.17 % (9757)Instruction limit reached!
% 5.54/2.17 % (9757)------------------------------
% 5.54/2.17 % (9757)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.54/2.17 % (9757)Termination reason: Unknown
% 5.54/2.17 % (9757)Termination phase: Saturation
% 5.54/2.17
% 5.54/2.17 % (9757)Memory used [KB]: 9083
% 5.54/2.17 % (9757)Time elapsed: 0.152 s
% 5.54/2.17 % (9757)Instructions burned: 449 (million)
% 5.54/2.17 % (9757)------------------------------
% 5.54/2.17 % (9757)------------------------------
% 5.54/2.18 % (9769)Instruction limit reached!
% 5.54/2.18 % (9769)------------------------------
% 5.54/2.18 % (9769)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.54/2.18 % (9769)Termination reason: Unknown
% 5.54/2.18 % (9769)Termination phase: shuffling
% 5.54/2.18
% 5.54/2.18 % (9769)Memory used [KB]: 2046
% 5.54/2.18 % (9769)Time elapsed: 0.013 s
% 5.54/2.18 % (9769)Instructions burned: 32 (million)
% 5.54/2.18 % (9769)------------------------------
% 5.54/2.18 % (9769)------------------------------
% 5.54/2.18 % (9770)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_9318 for (2995ds/96Mi)
% 5.54/2.18 % (9768)Instruction limit reached!
% 5.54/2.18 % (9768)------------------------------
% 5.54/2.18 % (9768)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.54/2.18 % (9768)Termination reason: Unknown
% 5.54/2.18 % (9768)Termination phase: Property scanning
% 5.54/2.18
% 5.54/2.18 % (9768)Memory used [KB]: 2814
% 5.54/2.18 % (9768)Time elapsed: 0.021 s
% 5.54/2.18 % (9768)Instructions burned: 65 (million)
% 5.54/2.18 % (9768)------------------------------
% 5.54/2.18 % (9768)------------------------------
% 5.54/2.19 % (9771)lrs+1002_1:4_au=on:cbe=off:cnfonf=lazy_pi_sigma_gen:hud=15:inj=on:nwc=4.0:s2a=on:sac=on:sp=reverse_arity:i=453:si=on:rtra=on_0 on DTF2THF_9318 for (2995ds/453Mi)
% 5.54/2.19 % (9707)Instruction limit reached!
% 5.54/2.19 % (9707)------------------------------
% 5.54/2.19 % (9707)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.54/2.19 % (9707)Termination reason: Unknown
% 5.54/2.19 % (9707)Termination phase: Saturation
% 5.54/2.19
% 5.54/2.19 % (9707)Memory used [KB]: 8955
% 5.54/2.19 % (9707)Time elapsed: 0.326 s
% 5.54/2.19 % (9707)Instructions burned: 879 (million)
% 5.54/2.19 % (9707)------------------------------
% 5.54/2.19 % (9707)------------------------------
% 5.54/2.19 % (9772)lrs-1004_1:1_au=on:bsd=on:cbe=off:cnfonf=off:fe=axiom:fsd=on:i=71:si=on:rtra=on_0 on DTF2THF_9318 for (2995ds/71Mi)
% 5.98/2.20 % (9773)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_9318 for (2995ds/83Mi)
% 5.98/2.21 % (9770)Instruction limit reached!
% 5.98/2.21 % (9770)------------------------------
% 5.98/2.21 % (9770)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.98/2.21 % (9770)Termination reason: Unknown
% 5.98/2.21 % (9770)Termination phase: Property scanning
% 5.98/2.21
% 5.98/2.21 % (9770)Memory used [KB]: 2686
% 5.98/2.21 % (9770)Time elapsed: 0.030 s
% 5.98/2.21 % (9770)Instructions burned: 99 (million)
% 5.98/2.21 % (9770)------------------------------
% 5.98/2.21 % (9770)------------------------------
% 5.98/2.22 % (9772)Instruction limit reached!
% 5.98/2.22 % (9772)------------------------------
% 5.98/2.22 % (9772)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.98/2.22 % (9772)Termination reason: Unknown
% 5.98/2.22 % (9772)Termination phase: Preprocessing 3
% 5.98/2.22
% 5.98/2.22 % (9772)Memory used [KB]: 2430
% 5.98/2.22 % (9772)Time elapsed: 0.025 s
% 5.98/2.22 % (9772)Instructions burned: 71 (million)
% 5.98/2.22 % (9772)------------------------------
% 5.98/2.22 % (9772)------------------------------
% 5.98/2.22 % (9774)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_9318 for (2995ds/11Mi)
% 5.98/2.22 % (9774)Instruction limit reached!
% 5.98/2.22 % (9774)------------------------------
% 5.98/2.22 % (9774)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.98/2.22 % (9774)Termination reason: Unknown
% 5.98/2.22 % (9774)Termination phase: shuffling
% 5.98/2.22
% 5.98/2.22 % (9774)Memory used [KB]: 1791
% 5.98/2.22 % (9774)Time elapsed: 0.005 s
% 5.98/2.22 % (9774)Instructions burned: 14 (million)
% 5.98/2.22 % (9774)------------------------------
% 5.98/2.22 % (9774)------------------------------
% 5.98/2.23 % (9773)Instruction limit reached!
% 5.98/2.23 % (9773)------------------------------
% 5.98/2.23 % (9773)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.98/2.23 % (9773)Termination reason: Unknown
% 5.98/2.23 % (9773)Termination phase: Preprocessing 3
% 5.98/2.23
% 5.98/2.23 % (9773)Memory used [KB]: 2302
% 5.98/2.23 % (9773)Time elapsed: 0.026 s
% 5.98/2.23 % (9773)Instructions burned: 83 (million)
% 5.98/2.23 % (9773)------------------------------
% 5.98/2.23 % (9773)------------------------------
% 5.98/2.23 % (9775)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_gen_be_off:fsr=off:spb=non_intro:i=41:si=on:rtra=on_0 on DTF2THF_9318 for (2994ds/41Mi)
% 5.98/2.23 % (9776)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_9318 for (2994ds/6Mi)
% 5.98/2.23 % (9777)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_9318 for (2994ds/150Mi)
% 5.98/2.24 % (9776)Instruction limit reached!
% 5.98/2.24 % (9776)------------------------------
% 5.98/2.24 % (9776)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.98/2.24 % (9776)Termination reason: Unknown
% 5.98/2.24 % (9776)Termination phase: shuffling
% 5.98/2.24
% 5.98/2.24 % (9776)Memory used [KB]: 1535
% 5.98/2.24 % (9776)Time elapsed: 0.003 s
% 5.98/2.24 % (9776)Instructions burned: 6 (million)
% 5.98/2.24 % (9776)------------------------------
% 5.98/2.24 % (9776)------------------------------
% 5.98/2.24 % (9760)Instruction limit reached!
% 5.98/2.24 % (9760)------------------------------
% 5.98/2.24 % (9760)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.98/2.24 % (9760)Termination reason: Unknown
% 5.98/2.24 % (9760)Termination phase: Saturation
% 5.98/2.24
% 5.98/2.24 % (9760)Memory used [KB]: 8955
% 5.98/2.24 % (9760)Time elapsed: 0.158 s
% 5.98/2.24 % (9760)Instructions burned: 507 (million)
% 5.98/2.24 % (9760)------------------------------
% 5.98/2.24 % (9760)------------------------------
% 5.98/2.24 % (9775)Instruction limit reached!
% 5.98/2.24 % (9775)------------------------------
% 5.98/2.24 % (9775)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.98/2.24 % (9775)Termination reason: Unknown
% 5.98/2.24 % (9775)Termination phase: shuffling
% 5.98/2.24
% 5.98/2.24 % (9775)Memory used [KB]: 2302
% 5.98/2.24 % (9775)Time elapsed: 0.014 s
% 5.98/2.24 % (9775)Instructions burned: 42 (million)
% 5.98/2.24 % (9775)------------------------------
% 5.98/2.24 % (9775)------------------------------
% 5.98/2.25 % (9778)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_9318 for (2994ds/343Mi)
% 5.98/2.25 % (9780)dis+10_1:1_au=on:c=on:e2e=on:fsr=off:sos=all:sp=const_frequency:ss=axioms:i=253:si=on:rtra=on_0 on DTF2THF_9318 for (2994ds/253Mi)
% 5.98/2.25 % (9779)ott+1010_1:4_cbe=off:chr=on:hfsq=on:hfsqr=1,16:hud=3:sac=on:sp=unary_frequency:ss=axioms:st=5.0:tnu=2:i=109:si=on:rtra=on_0 on DTF2THF_9318 for (2994ds/109Mi)
% 5.98/2.27 % (9767)Instruction limit reached!
% 5.98/2.27 % (9767)------------------------------
% 5.98/2.27 % (9767)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.98/2.27 % (9767)Termination reason: Unknown
% 5.98/2.27 % (9767)Termination phase: Saturation
% 5.98/2.27
% 5.98/2.27 % (9767)Memory used [KB]: 3837
% 5.98/2.27 % (9767)Time elapsed: 0.126 s
% 5.98/2.27 % (9767)Instructions burned: 432 (million)
% 5.98/2.27 % (9767)------------------------------
% 5.98/2.27 % (9767)------------------------------
% 5.98/2.28 % (9777)Instruction limit reached!
% 5.98/2.28 % (9777)------------------------------
% 5.98/2.28 % (9777)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.98/2.28 % (9777)Termination reason: Unknown
% 5.98/2.28 % (9777)Termination phase: Saturation
% 5.98/2.28
% 5.98/2.28 % (9777)Memory used [KB]: 7419
% 5.98/2.28 % (9777)Time elapsed: 0.043 s
% 5.98/2.28 % (9777)Instructions burned: 150 (million)
% 5.98/2.28 % (9777)------------------------------
% 5.98/2.28 % (9777)------------------------------
% 5.98/2.28 % (9781)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_9318 for (2994ds/140Mi)
% 5.98/2.28 % (9779)Instruction limit reached!
% 5.98/2.28 % (9779)------------------------------
% 5.98/2.28 % (9779)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.98/2.28 % (9779)Termination reason: Unknown
% 5.98/2.28 % (9779)Termination phase: Property scanning
% 5.98/2.28
% 5.98/2.28 % (9779)Memory used [KB]: 2686
% 5.98/2.28 % (9779)Time elapsed: 0.034 s
% 5.98/2.28 % (9779)Instructions burned: 110 (million)
% 5.98/2.28 % (9779)------------------------------
% 5.98/2.28 % (9779)------------------------------
% 5.98/2.29 % (9782)dis+1010_16:1_acc=model:au=on:avsq=on:cond=fast:plsq=on:plsqc=2:plsqr=9267,262144:i=614:si=on:rtra=on_0 on DTF2THF_9318 for (2994ds/614Mi)
% 5.98/2.29 % (9783)lrs+1002_1:1_e2e=on:fde=unused:fe=axiom:prag=on:sd=1:sos=on:ss=axioms:st=2.0:i=202:si=on:rtra=on_0 on DTF2THF_9318 for (2994ds/202Mi)
% 6.53/2.32 % (9771)Instruction limit reached!
% 6.53/2.32 % (9771)------------------------------
% 6.53/2.32 % (9771)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.53/2.32 % (9771)Termination reason: Unknown
% 6.53/2.32 % (9771)Termination phase: Saturation
% 6.53/2.32
% 6.53/2.32 % (9771)Memory used [KB]: 8187
% 6.53/2.32 % (9771)Time elapsed: 0.130 s
% 6.53/2.32 % (9771)Instructions burned: 453 (million)
% 6.53/2.32 % (9771)------------------------------
% 6.53/2.32 % (9771)------------------------------
% 6.53/2.32 % (9781)Instruction limit reached!
% 6.53/2.32 % (9781)------------------------------
% 6.53/2.32 % (9781)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.53/2.32 % (9781)Termination reason: Unknown
% 6.53/2.32 % (9781)Termination phase: Saturation
% 6.53/2.32
% 6.53/2.32 % (9781)Memory used [KB]: 7291
% 6.53/2.32 % (9781)Time elapsed: 0.044 s
% 6.53/2.32 % (9781)Instructions burned: 140 (million)
% 6.53/2.32 % (9781)------------------------------
% 6.53/2.32 % (9781)------------------------------
% 6.53/2.33 % (9784)dis+1002_1:1_au=on:cs=on:ins=3:kws=inv_frequency:nwc=10.0:ss=axioms:st=3.0:i=339:si=on:rtra=on_0 on DTF2THF_9318 for (2994ds/339Mi)
% 6.53/2.33 % (9780)Instruction limit reached!
% 6.53/2.33 % (9780)------------------------------
% 6.53/2.33 % (9780)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.53/2.33 % (9780)Termination reason: Unknown
% 6.53/2.33 % (9780)Termination phase: Saturation
% 6.53/2.33
% 6.53/2.33 % (9780)Memory used [KB]: 8059
% 6.53/2.33 % (9780)Time elapsed: 0.079 s
% 6.53/2.33 % (9780)Instructions burned: 255 (million)
% 6.53/2.33 % (9780)------------------------------
% 6.53/2.33 % (9780)------------------------------
% 6.53/2.33 % (9785)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=283:si=on:rtra=on_0 on DTF2THF_9318 for (2993ds/283Mi)
% 6.53/2.34 % (9786)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_9318 for (2993ds/134Mi)
% 6.53/2.35 % (9783)Instruction limit reached!
% 6.53/2.35 % (9783)------------------------------
% 6.53/2.35 % (9783)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.53/2.35 % (9783)Termination reason: Unknown
% 6.53/2.35 % (9783)Termination phase: Saturation
% 6.53/2.35
% 6.53/2.35 % (9783)Memory used [KB]: 7931
% 6.53/2.35 % (9783)Time elapsed: 0.062 s
% 6.53/2.35 % (9783)Instructions burned: 202 (million)
% 6.53/2.35 % (9783)------------------------------
% 6.53/2.35 % (9783)------------------------------
% 6.53/2.35 % (9778)Instruction limit reached!
% 6.53/2.35 % (9778)------------------------------
% 6.53/2.35 % (9778)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.53/2.35 % (9778)Termination reason: Unknown
% 6.53/2.35 % (9778)Termination phase: Saturation
% 6.53/2.35
% 6.53/2.35 % (9778)Memory used [KB]: 7803
% 6.53/2.35 % (9778)Time elapsed: 0.109 s
% 6.53/2.35 % (9778)Instructions burned: 344 (million)
% 6.53/2.35 % (9778)------------------------------
% 6.53/2.35 % (9778)------------------------------
% 6.53/2.36 % (9787)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_9318 for (2993ds/298Mi)
% 6.53/2.36 % (9788)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_9318 for (2993ds/192Mi)
% 6.53/2.37 % (9786)Instruction limit reached!
% 6.53/2.37 % (9786)------------------------------
% 6.53/2.37 % (9786)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.53/2.37 % (9786)Termination reason: Unknown
% 6.53/2.37 % (9786)Termination phase: Property scanning
% 6.53/2.37
% 6.53/2.37 % (9786)Memory used [KB]: 2814
% 6.53/2.37 % (9786)Time elapsed: 0.038 s
% 6.53/2.37 % (9786)Instructions burned: 134 (million)
% 6.53/2.37 % (9786)------------------------------
% 6.53/2.37 % (9786)------------------------------
% 6.53/2.38 % (9789)lrs+10_1:1_bd=off:hud=1:nm=0:prag=on:sd=2:sgt=32:sos=all:ss=axioms:i=258:si=on:rtra=on_0 on DTF2THF_9318 for (2993ds/258Mi)
% 7.06/2.41 % (9785)Instruction limit reached!
% 7.06/2.41 % (9785)------------------------------
% 7.06/2.41 % (9785)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.06/2.41 % (9785)Termination reason: Unknown
% 7.06/2.41 % (9785)Termination phase: Saturation
% 7.06/2.41
% 7.06/2.41 % (9785)Memory used [KB]: 7675
% 7.06/2.41 % (9785)Time elapsed: 0.078 s
% 7.06/2.41 % (9785)Instructions burned: 283 (million)
% 7.06/2.41 % (9785)------------------------------
% 7.06/2.41 % (9785)------------------------------
% 7.06/2.42 % (9788)Instruction limit reached!
% 7.06/2.42 % (9788)------------------------------
% 7.06/2.42 % (9788)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.06/2.42 % (9788)Termination reason: Unknown
% 7.06/2.42 % (9788)Termination phase: Saturation
% 7.06/2.42
% 7.06/2.42 % (9788)Memory used [KB]: 3709
% 7.06/2.42 % (9788)Time elapsed: 0.057 s
% 7.06/2.42 % (9788)Instructions burned: 192 (million)
% 7.06/2.42 % (9788)------------------------------
% 7.06/2.42 % (9788)------------------------------
% 7.06/2.42 % (9790)lrs+1002_1:1024_au=on:cond=fast:ntd=on:piset=and:plsq=on:plsqc=1:plsqr=183491,1048576:sac=on:tgt=ground:i=90:si=on:rtra=on_0 on DTF2THF_9318 for (2993ds/90Mi)
% 7.06/2.43 % (9791)dis+1004_16:1_avsq=on:avsqr=1,16:cnfonf=lazy_gen:cs=on:fe=off:tgt=ground:tnu=1:i=231:si=on:rtra=on_0 on DTF2THF_9318 for (2993ds/231Mi)
% 7.06/2.43 % (9784)Instruction limit reached!
% 7.06/2.43 % (9784)------------------------------
% 7.06/2.43 % (9784)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.06/2.43 % (9784)Termination reason: Unknown
% 7.06/2.43 % (9784)Termination phase: Saturation
% 7.06/2.43
% 7.06/2.43 % (9784)Memory used [KB]: 7803
% 7.06/2.43 % (9784)Time elapsed: 0.105 s
% 7.06/2.43 % (9784)Instructions burned: 341 (million)
% 7.06/2.43 % (9784)------------------------------
% 7.06/2.43 % (9784)------------------------------
% 7.06/2.44 % (9792)lrs+1002_1:1_au=on:s2a=on:sgt=32:slsq=on:slsqc=1:slsql=off:ss=axioms:i=458:si=on:rtra=on_0 on DTF2THF_9318 for (2992ds/458Mi)
% 7.06/2.44 % (9787)Instruction limit reached!
% 7.06/2.44 % (9787)------------------------------
% 7.06/2.44 % (9787)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.06/2.44 % (9787)Termination reason: Unknown
% 7.06/2.44 % (9787)Termination phase: Saturation
% 7.06/2.44
% 7.06/2.44 % (9787)Memory used [KB]: 7931
% 7.06/2.44 % (9787)Time elapsed: 0.080 s
% 7.06/2.44 % (9787)Instructions burned: 300 (million)
% 7.06/2.44 % (9787)------------------------------
% 7.06/2.44 % (9787)------------------------------
% 7.06/2.45 % (9790)Instruction limit reached!
% 7.06/2.45 % (9790)------------------------------
% 7.06/2.45 % (9790)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.06/2.45 % (9790)Termination reason: Unknown
% 7.06/2.45 % (9790)Termination phase: Property scanning
% 7.06/2.45
% 7.06/2.45 % (9790)Memory used [KB]: 2814
% 7.06/2.45 % (9790)Time elapsed: 0.029 s
% 7.06/2.45 % (9790)Instructions burned: 92 (million)
% 7.06/2.45 % (9790)------------------------------
% 7.06/2.45 % (9790)------------------------------
% 7.06/2.45 % (9793)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_9318 for (2992ds/31Mi)
% 7.06/2.46 % (9794)lrs+10_1:1_cnfonf=lazy_gen:kws=inv_precedence:nwc=10.0:s2a=on:sp=weighted_frequency:ss=axioms:i=297:si=on:rtra=on_0 on DTF2THF_9318 for (2992ds/297Mi)
% 7.06/2.46 % (9789)Instruction limit reached!
% 7.06/2.46 % (9789)------------------------------
% 7.06/2.46 % (9789)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.06/2.46 % (9789)Termination reason: Unknown
% 7.06/2.46 % (9789)Termination phase: Saturation
% 7.06/2.46
% 7.06/2.46 % (9789)Memory used [KB]: 8187
% 7.06/2.46 % (9789)Time elapsed: 0.078 s
% 7.06/2.46 % (9789)Instructions burned: 258 (million)
% 7.06/2.46 % (9789)------------------------------
% 7.06/2.46 % (9789)------------------------------
% 7.06/2.46 % (9793)Instruction limit reached!
% 7.06/2.46 % (9793)------------------------------
% 7.06/2.46 % (9793)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.06/2.46 % (9793)Termination reason: Unknown
% 7.06/2.46 % (9793)Termination phase: shuffling
% 7.06/2.46
% 7.06/2.46 % (9793)Memory used [KB]: 2046
% 7.06/2.46 % (9793)Time elapsed: 0.011 s
% 7.06/2.46 % (9793)Instructions burned: 31 (million)
% 7.06/2.46 % (9793)------------------------------
% 7.06/2.46 % (9793)------------------------------
% 7.06/2.47 % (9796)lrs+10_1:1024_avsq=on:avsqc=4:avsqr=584411,1048576:bd=off:bet=on:br=off:chr=on:fsd=on:fsdmm=1:hud=10:nm=0:ntd=on:nwc=9.0:piset=or:plsq=on:plsqc=2:plsql=on:plsqr=852219,1048576:rawr=on:sp=const_frequency:spb=units:i=182:si=on:rtra=on_0 on DTF2THF_9318 for (2992ds/182Mi)
% 7.06/2.47 % (9782)Instruction limit reached!
% 7.06/2.47 % (9782)------------------------------
% 7.06/2.47 % (9782)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.06/2.47 % (9782)Termination reason: Unknown
% 7.06/2.47 % (9782)Termination phase: Saturation
% 7.06/2.47
% 7.06/2.47 % (9782)Memory used [KB]: 9210
% 7.06/2.47 % (9782)Time elapsed: 0.188 s
% 7.06/2.47 % (9782)Instructions burned: 616 (million)
% 7.06/2.47 % (9782)------------------------------
% 7.06/2.47 % (9782)------------------------------
% 7.06/2.48 % (9764)Instruction limit reached!
% 7.06/2.48 % (9764)------------------------------
% 7.06/2.48 % (9764)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.06/2.48 % (9764)Termination reason: Unknown
% 7.06/2.48 % (9764)Termination phase: Saturation
% 7.06/2.48
% 7.06/2.48 % (9764)Memory used [KB]: 8955
% 7.06/2.48 % (9764)Time elapsed: 0.346 s
% 7.06/2.48 % (9764)Instructions burned: 1088 (million)
% 7.06/2.48 % (9764)------------------------------
% 7.06/2.48 % (9764)------------------------------
% 7.06/2.48 % (9795)dis+2_1:4_amm=off:au=on:cbe=off:cnfonf=lazy_gen:hfaw=0:hfsq=on:hfsql=off:plsq=on:plsqc=1:plsqr=35,4:sac=on:i=987:si=on:rtra=on_0 on DTF2THF_9318 for (2992ds/987Mi)
% 7.06/2.48 % (9797)dis+10_1:1_au=on:e2e=on:fde=none:nwc=10.0:s2a=on:s2at=2.0:i=182:si=on:rtra=on_0 on DTF2THF_9318 for (2992ds/182Mi)
% 7.06/2.48 % (9798)dis+10_1:8_cha=on:cnfonf=lazy_pi_sigma_gen:fe=axiom:hud=5:plsq=on:plsqr=32,1:prag=on:sos=all:i=154:si=on:rtra=on_0 on DTF2THF_9318 for (2992ds/154Mi)
% 7.06/2.49 % (9791)Instruction limit reached!
% 7.06/2.49 % (9791)------------------------------
% 7.06/2.49 % (9791)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.06/2.49 % (9791)Termination reason: Unknown
% 7.06/2.49 % (9791)Termination phase: Saturation
% 7.06/2.49
% 7.06/2.49 % (9791)Memory used [KB]: 7547
% 7.06/2.49 % (9791)Time elapsed: 0.062 s
% 7.06/2.49 % (9791)Instructions burned: 232 (million)
% 7.06/2.49 % (9791)------------------------------
% 7.06/2.49 % (9791)------------------------------
% 8.18/2.50 % (9799)dis+1003_1:64_atotf=0.4:au=on:fsr=off:hflw=5:hfsq=on:hfsqc=20:hfsql=off:plsq=on:plsqc=1:plsqr=32,1:i=79:si=on:rtra=on_0 on DTF2THF_9318 for (2992ds/79Mi)
% 8.18/2.52 % (9796)Instruction limit reached!
% 8.18/2.52 % (9796)------------------------------
% 8.18/2.52 % (9796)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.18/2.52 % (9796)Termination reason: Unknown
% 8.18/2.52 % (9796)Termination phase: Saturation
% 8.18/2.52
% 8.18/2.52 % (9796)Memory used [KB]: 7547
% 8.18/2.52 % (9796)Time elapsed: 0.053 s
% 8.18/2.52 % (9796)Instructions burned: 183 (million)
% 8.18/2.52 % (9796)------------------------------
% 8.18/2.52 % (9796)------------------------------
% 8.18/2.52 % (9799)Instruction limit reached!
% 8.18/2.52 % (9799)------------------------------
% 8.18/2.52 % (9799)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.18/2.52 % (9799)Termination reason: Unknown
% 8.18/2.52 % (9799)Termination phase: Preprocessing 3
% 8.18/2.52
% 8.18/2.52 % (9799)Memory used [KB]: 2686
% 8.18/2.52 % (9799)Time elapsed: 0.025 s
% 8.18/2.52 % (9799)Instructions burned: 81 (million)
% 8.18/2.52 % (9799)------------------------------
% 8.18/2.52 % (9799)------------------------------
% 8.18/2.52 % (9798)Instruction limit reached!
% 8.18/2.52 % (9798)------------------------------
% 8.18/2.52 % (9798)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.18/2.52 % (9798)Termination reason: Unknown
% 8.18/2.52 % (9798)Termination phase: Property scanning
% 8.18/2.52
% 8.18/2.52 % (9798)Memory used [KB]: 2942
% 8.18/2.52 % (9798)Time elapsed: 0.041 s
% 8.18/2.52 % (9798)Instructions burned: 157 (million)
% 8.18/2.52 % (9798)------------------------------
% 8.18/2.52 % (9798)------------------------------
% 8.18/2.53 % (9800)dis+1010_5:2_anc=all_dependent:awrs=converge:bd=off:bet=on:c=on:cha=on:cnfonf=lazy_not_gen_be_off:fe=axiom:hud=5:kws=precedence:piset=and:prag=on:s2a=on:sac=on:slsq=on:slsqc=1:sp=reverse_arity:ss=axioms:st=1.5:i=216:si=on:rtra=on_0 on DTF2THF_9318 for (2992ds/216Mi)
% 8.18/2.53 % (9801)lrs+1002_1:1_fe=off:hud=10:ins=2:sac=on:sp=const_frequency:i=114:si=on:rtra=on_0 on DTF2THF_9318 for (2992ds/114Mi)
% 8.18/2.53 % (9802)dis+1002_1:1_acc=on:au=on:bd=off:e2e=on:sgt=8:ss=axioms:i=922:si=on:rtra=on_0 on DTF2THF_9318 for (2992ds/922Mi)
% 8.18/2.54 % (9797)Instruction limit reached!
% 8.18/2.54 % (9797)------------------------------
% 8.18/2.54 % (9797)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.18/2.54 % (9797)Termination reason: Unknown
% 8.18/2.54 % (9797)Termination phase: Saturation
% 8.18/2.54
% 8.18/2.54 % (9797)Memory used [KB]: 7547
% 8.18/2.54 % (9797)Time elapsed: 0.054 s
% 8.18/2.54 % (9797)Instructions burned: 182 (million)
% 8.18/2.54 % (9797)------------------------------
% 8.18/2.54 % (9797)------------------------------
% 8.18/2.54 % (9794)Instruction limit reached!
% 8.18/2.54 % (9794)------------------------------
% 8.18/2.54 % (9794)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.18/2.54 % (9794)Termination reason: Unknown
% 8.18/2.54 % (9794)Termination phase: Saturation
% 8.18/2.54
% 8.18/2.54 % (9794)Memory used [KB]: 7803
% 8.18/2.54 % (9794)Time elapsed: 0.084 s
% 8.18/2.54 % (9794)Instructions burned: 299 (million)
% 8.18/2.54 % (9794)------------------------------
% 8.18/2.54 % (9794)------------------------------
% 8.18/2.55 % (9803)lrs+10_1:128_au=on:bd=off:cnfonf=lazy_not_gen_be_off:nwc=10.0:piset=and:i=182:si=on:rtra=on_0 on DTF2THF_9318 for (2991ds/182Mi)
% 8.18/2.55 % (9804)ott+1010_1:1024_au=on:av=off:c=on:cnfonf=conj_eager:piset=equals:s2a=on:sp=occurrence:spb=units:ss=axioms:i=164:si=on:rtra=on_0 on DTF2THF_9318 for (2991ds/164Mi)
% 8.18/2.57 % (9801)Instruction limit reached!
% 8.18/2.57 % (9801)------------------------------
% 8.18/2.57 % (9801)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.18/2.57 % (9801)Termination reason: Unknown
% 8.18/2.57 % (9801)Termination phase: Property scanning
% 8.18/2.57
% 8.18/2.57 % (9801)Memory used [KB]: 2814
% 8.18/2.57 % (9801)Time elapsed: 0.035 s
% 8.18/2.57 % (9801)Instructions burned: 114 (million)
% 8.18/2.57 % (9801)------------------------------
% 8.18/2.57 % (9801)------------------------------
% 8.18/2.58 % (9805)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=50:si=on:rtra=on_0 on DTF2THF_9318 for (2991ds/50Mi)
% 8.18/2.58 % (9792)Instruction limit reached!
% 8.18/2.58 % (9792)------------------------------
% 8.18/2.58 % (9792)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.18/2.58 % (9792)Termination reason: Unknown
% 8.18/2.58 % (9792)Termination phase: Saturation
% 8.18/2.58
% 8.18/2.58 % (9792)Memory used [KB]: 8571
% 8.18/2.58 % (9792)Time elapsed: 0.142 s
% 8.18/2.58 % (9792)Instructions burned: 460 (million)
% 8.18/2.58 % (9792)------------------------------
% 8.18/2.58 % (9792)------------------------------
% 8.18/2.59 % (9806)dis-30_1:1_acc=on:bd=off:fsr=off:plsq=on:plsqr=32,1:i=368:si=on:rtra=on_0 on DTF2THF_9318 for (2991ds/368Mi)
% 8.18/2.59 % (9800)Instruction limit reached!
% 8.18/2.59 % (9800)------------------------------
% 8.18/2.59 % (9800)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.18/2.59 % (9800)Termination reason: Unknown
% 8.18/2.59 % (9800)Termination phase: Saturation
% 8.18/2.59
% 8.18/2.59 % (9800)Memory used [KB]: 7547
% 8.18/2.59 % (9800)Time elapsed: 0.059 s
% 8.18/2.59 % (9800)Instructions burned: 218 (million)
% 8.18/2.59 % (9800)------------------------------
% 8.18/2.59 % (9800)------------------------------
% 8.18/2.59 % (9805)Instruction limit reached!
% 8.18/2.59 % (9805)------------------------------
% 8.18/2.59 % (9805)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.18/2.59 % (9805)Termination reason: Unknown
% 8.18/2.59 % (9805)Termination phase: shuffling
% 8.18/2.59
% 8.18/2.59 % (9805)Memory used [KB]: 2430
% 8.18/2.59 % (9805)Time elapsed: 0.015 s
% 8.18/2.59 % (9805)Instructions burned: 52 (million)
% 8.18/2.59 % (9805)------------------------------
% 8.18/2.59 % (9805)------------------------------
% 8.18/2.59 % (9804)Instruction limit reached!
% 8.18/2.59 % (9804)------------------------------
% 8.18/2.59 % (9804)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.18/2.59 % (9804)Termination reason: Unknown
% 8.18/2.59 % (9804)Termination phase: Property scanning
% 8.18/2.59
% 8.18/2.59 % (9804)Memory used [KB]: 2814
% 8.18/2.59 % (9804)Time elapsed: 0.045 s
% 8.18/2.59 % (9804)Instructions burned: 165 (million)
% 8.18/2.59 % (9804)------------------------------
% 8.18/2.59 % (9804)------------------------------
% 8.18/2.59 % (9803)Instruction limit reached!
% 8.18/2.59 % (9803)------------------------------
% 8.18/2.59 % (9803)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.18/2.59 % (9803)Termination reason: Unknown
% 8.18/2.59 % (9803)Termination phase: Property scanning
% 8.18/2.59
% 8.18/2.59 % (9803)Memory used [KB]: 2942
% 8.18/2.59 % (9803)Time elapsed: 0.049 s
% 8.18/2.59 % (9803)Instructions burned: 185 (million)
% 8.18/2.59 % (9803)------------------------------
% 8.18/2.59 % (9803)------------------------------
% 8.88/2.60 % (9807)dis+2_1:1_bd=off:e2e=on:hud=1:nm=3:piset=or:prag=on:s2a=on:sp=const_max:i=1021:si=on:rtra=on_0 on DTF2THF_9318 for (2991ds/1021Mi)
% 8.88/2.60 % (9808)dis+10_1:1_cnfonf=lazy_gen:fe=off:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=267:si=on:rtra=on_0 on DTF2THF_9318 for (2991ds/267Mi)
% 8.88/2.60 % (9809)dis+21_1:5_au=on:ntd=on:nwc=5.0:s2a=on:ss=axioms:st=5.0:i=204:si=on:rtra=on_0 on DTF2THF_9318 for (2991ds/204Mi)
% 8.88/2.60 % (9810)dis+2_1:1024_atotf=0.0802881:au=on:bs=unit_only:cnfonf=lazy_not_gen_be_off:fde=none:nwc=10.0:prag=on:i=26:si=on:rtra=on_0 on DTF2THF_9318 for (2991ds/26Mi)
% 8.88/2.61 % (9810)Instruction limit reached!
% 8.88/2.61 % (9810)------------------------------
% 8.88/2.61 % (9810)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.88/2.61 % (9810)Termination reason: Unknown
% 8.88/2.61 % (9810)Termination phase: shuffling
% 8.88/2.61
% 8.88/2.61 % (9810)Memory used [KB]: 2174
% 8.88/2.61 % (9810)Time elapsed: 0.009 s
% 8.88/2.61 % (9810)Instructions burned: 29 (million)
% 8.88/2.61 % (9810)------------------------------
% 8.88/2.61 % (9810)------------------------------
% 8.88/2.62 % (9811)dis+1002_1:1_bd=preordered:c=on:cbe=off:cnfonf=lazy_not_gen_be_off:hud=10:ixr=off:prag=on:i=53:si=on:rtra=on_0 on DTF2THF_9318 for (2990ds/53Mi)
% 8.88/2.64 % (9811)Instruction limit reached!
% 8.88/2.64 % (9811)------------------------------
% 8.88/2.64 % (9811)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.88/2.64 % (9811)Termination reason: Unknown
% 8.88/2.64 % (9811)Termination phase: shuffling
% 8.88/2.64
% 8.88/2.64 % (9811)Memory used [KB]: 2558
% 8.88/2.64 % (9811)Time elapsed: 0.016 s
% 8.88/2.64 % (9811)Instructions burned: 56 (million)
% 8.88/2.64 % (9811)------------------------------
% 8.88/2.64 % (9811)------------------------------
% 8.88/2.64 % (9812)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=4436:si=on:rtra=on_0 on DTF2THF_9318 for (2990ds/4436Mi)
% 8.88/2.66 % (9809)Instruction limit reached!
% 8.88/2.66 % (9809)------------------------------
% 8.88/2.66 % (9809)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.88/2.66 % (9809)Termination reason: Unknown
% 8.88/2.66 % (9809)Termination phase: Saturation
% 8.88/2.66
% 8.88/2.66 % (9809)Memory used [KB]: 7547
% 8.88/2.66 % (9809)Time elapsed: 0.061 s
% 8.88/2.66 % (9809)Instructions burned: 204 (million)
% 8.88/2.66 % (9809)------------------------------
% 8.88/2.66 % (9809)------------------------------
% 8.88/2.67 % (9813)dis+1010_1:1_acc=model:avsq=on:ixr=off:prag=on:s2a=on:sac=on:i=1539:si=on:rtra=on_0 on DTF2THF_9318 for (2990ds/1539Mi)
% 8.88/2.67 % (9808)Instruction limit reached!
% 8.88/2.67 % (9808)------------------------------
% 8.88/2.67 % (9808)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.88/2.67 % (9808)Termination reason: Unknown
% 8.88/2.67 % (9808)Termination phase: Saturation
% 8.88/2.67
% 8.88/2.67 % (9808)Memory used [KB]: 7675
% 8.88/2.67 % (9808)Time elapsed: 0.075 s
% 8.88/2.67 % (9808)Instructions burned: 267 (million)
% 8.88/2.67 % (9808)------------------------------
% 8.88/2.67 % (9808)------------------------------
% 8.88/2.68 % (9814)lrs+2_1:1_bd=off:e2e=on:fde=none:kws=inv_precedence:plsq=on:plsqr=32,1:sos=on:i=408:si=on:rtra=on_0 on DTF2THF_9318 for (2990ds/408Mi)
% 9.06/2.70 % (9806)Instruction limit reached!
% 9.06/2.70 % (9806)------------------------------
% 9.06/2.70 % (9806)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.06/2.70 % (9806)Termination reason: Unknown
% 9.06/2.70 % (9806)Termination phase: Saturation
% 9.06/2.70
% 9.06/2.70 % (9806)Memory used [KB]: 7931
% 9.06/2.70 % (9806)Time elapsed: 0.108 s
% 9.06/2.70 % (9806)Instructions burned: 370 (million)
% 9.06/2.70 % (9806)------------------------------
% 9.06/2.70 % (9806)------------------------------
% 9.06/2.71 % (9815)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=5205:si=on:rtra=on_0 on DTF2THF_9318 for (2990ds/5205Mi)
% 9.20/2.80 % (9814)Instruction limit reached!
% 9.20/2.80 % (9814)------------------------------
% 9.20/2.80 % (9814)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.20/2.80 % (9814)Termination reason: Unknown
% 9.20/2.80 % (9814)Termination phase: Saturation
% 9.20/2.80
% 9.20/2.80 % (9814)Memory used [KB]: 8827
% 9.20/2.80 % (9814)Time elapsed: 0.121 s
% 9.20/2.80 % (9814)Instructions burned: 409 (million)
% 9.20/2.80 % (9814)------------------------------
% 9.20/2.80 % (9814)------------------------------
% 9.20/2.81 % (9816)dis+1010_1:4_au=on:av=off:cbe=off:nwc=10.0:s2a=on:s2at=3.0:sd=2:ss=axioms:st=5.0:i=1913:si=on:rtra=on_0 on DTF2THF_9318 for (2989ds/1913Mi)
% 9.20/2.82 % (9795)Instruction limit reached!
% 9.20/2.82 % (9795)------------------------------
% 9.20/2.82 % (9795)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.20/2.82 % (9795)Termination reason: Unknown
% 9.20/2.82 % (9795)Termination phase: Saturation
% 9.20/2.82
% 9.20/2.82 % (9795)Memory used [KB]: 9850
% 9.20/2.82 % (9795)Time elapsed: 0.368 s
% 9.20/2.82 % (9795)Instructions burned: 987 (million)
% 9.20/2.82 % (9795)------------------------------
% 9.20/2.82 % (9795)------------------------------
% 9.20/2.83 % (9802)Instruction limit reached!
% 9.20/2.83 % (9802)------------------------------
% 9.20/2.83 % (9802)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.20/2.83 % (9802)Termination reason: Unknown
% 9.20/2.83 % (9802)Termination phase: Saturation
% 9.20/2.83
% 9.20/2.83 % (9802)Memory used [KB]: 8955
% 9.20/2.83 % (9802)Time elapsed: 0.319 s
% 9.20/2.83 % (9802)Instructions burned: 922 (million)
% 9.20/2.83 % (9802)------------------------------
% 9.20/2.83 % (9802)------------------------------
% 9.20/2.83 % (9817)lrs+2_1:1_bd=off:cbe=off:piset=equals:sd=1:sgt=20:sos=all:ss=axioms:i=4350:si=on:rtra=on_0 on DTF2THF_9318 for (2988ds/4350Mi)
% 9.20/2.84 % (9818)ott+1010_1:4_cbe=off:chr=on:hfsq=on:hfsqr=1,16:hud=3:sac=on:sp=unary_frequency:ss=axioms:st=5.0:tnu=2:i=68:si=on:rtra=on_0 on DTF2THF_9318 for (2988ds/68Mi)
% 9.20/2.86 % (9818)Instruction limit reached!
% 9.20/2.86 % (9818)------------------------------
% 9.20/2.86 % (9818)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.20/2.86 % (9818)Termination reason: Unknown
% 9.20/2.86 % (9818)Termination phase: SInE selection
% 9.20/2.86
% 9.20/2.86 % (9818)Memory used [KB]: 2174
% 9.20/2.86 % (9818)Time elapsed: 0.021 s
% 9.20/2.86 % (9818)Instructions burned: 68 (million)
% 9.20/2.86 % (9818)------------------------------
% 9.20/2.86 % (9818)------------------------------
% 9.20/2.87 % (9819)dis+10_1:1_au=on:nwc=5.0:s2a=on:s2at=2.0:sac=on:sd=7:ss=axioms:st=2.0:i=358:si=on:rtra=on_0 on DTF2THF_9318 for (2988ds/358Mi)
% 10.01/2.91 % (9807)Instruction limit reached!
% 10.01/2.91 % (9807)------------------------------
% 10.01/2.91 % (9807)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.01/2.91 % (9807)Termination reason: Unknown
% 10.01/2.91 % (9807)Termination phase: Saturation
% 10.01/2.91
% 10.01/2.91 % (9807)Memory used [KB]: 8699
% 10.01/2.91 % (9807)Time elapsed: 0.315 s
% 10.01/2.91 % (9807)Instructions burned: 1023 (million)
% 10.01/2.91 % (9807)------------------------------
% 10.01/2.91 % (9807)------------------------------
% 10.01/2.92 % (9820)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_9318 for (2988ds/31Mi)
% 10.01/2.93 % (9820)Instruction limit reached!
% 10.01/2.93 % (9820)------------------------------
% 10.01/2.93 % (9820)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.01/2.93 % (9820)Termination reason: Unknown
% 10.01/2.93 % (9820)Termination phase: shuffling
% 10.01/2.93
% 10.01/2.93 % (9820)Memory used [KB]: 2046
% 10.01/2.93 % (9820)Time elapsed: 0.011 s
% 10.01/2.93 % (9820)Instructions burned: 31 (million)
% 10.01/2.93 % (9820)------------------------------
% 10.01/2.93 % (9820)------------------------------
% 10.01/2.94 % (9821)lrs+10_1:1_cnfonf=lazy_simp:sgt=5:sos=on:ss=axioms:i=279:si=on:rtra=on_0 on DTF2THF_9318 for (2987ds/279Mi)
% 10.01/2.98 % (9819)Instruction limit reached!
% 10.01/2.98 % (9819)------------------------------
% 10.01/2.98 % (9819)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.01/2.98 % (9819)Termination reason: Unknown
% 10.01/2.98 % (9819)Termination phase: Saturation
% 10.01/2.98
% 10.01/2.98 % (9819)Memory used [KB]: 7675
% 10.01/2.98 % (9819)Time elapsed: 0.114 s
% 10.01/2.98 % (9819)Instructions burned: 359 (million)
% 10.01/2.98 % (9819)------------------------------
% 10.01/2.98 % (9819)------------------------------
% 10.01/2.99 % (9822)lrs+10_8:1_au=on:fde=none:slsq=on:i=258:si=on:rtra=on_0 on DTF2THF_9318 for (2987ds/258Mi)
% 10.01/2.99 % (9821)Refutation not found, incomplete strategy
% 10.01/2.99 % (9821)------------------------------
% 10.01/2.99 % (9821)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.01/2.99 % (9821)Termination reason: Refutation not found, incomplete strategy
% 10.01/2.99
% 10.01/2.99
% 10.01/2.99 % (9821)Memory used [KB]: 7675
% 10.01/2.99 % (9821)Time elapsed: 0.051 s
% 10.01/2.99 % (9821)Instructions burned: 189 (million)
% 10.01/2.99 % (9821)------------------------------
% 10.01/2.99 % (9821)------------------------------
% 10.01/3.00 % (9823)dis+2_1:1024_atotf=0.0802881:au=on:bs=unit_only:cnfonf=lazy_not_gen_be_off:fde=none:nwc=10.0:prag=on:i=26:si=on:rtra=on_0 on DTF2THF_9318 for (2987ds/26Mi)
% 10.01/3.01 % (9823)Instruction limit reached!
% 10.01/3.01 % (9823)------------------------------
% 10.01/3.01 % (9823)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.01/3.01 % (9823)Termination reason: Unknown
% 10.01/3.01 % (9823)Termination phase: shuffling
% 10.01/3.01
% 10.01/3.01 % (9823)Memory used [KB]: 2046
% 10.01/3.01 % (9823)Time elapsed: 0.008 s
% 10.01/3.01 % (9823)Instructions burned: 26 (million)
% 10.01/3.01 % (9823)------------------------------
% 10.01/3.01 % (9823)------------------------------
% 11.95/3.02 % (9824)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_be_gen:ntd=on:pe=on:i=995:si=on:rtra=on_0 on DTF2THF_9318 for (2987ds/995Mi)
% 11.95/3.06 % (9822)Instruction limit reached!
% 11.95/3.06 % (9822)------------------------------
% 11.95/3.06 % (9822)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.95/3.06 % (9822)Termination reason: Unknown
% 11.95/3.06 % (9822)Termination phase: Saturation
% 11.95/3.06
% 11.95/3.06 % (9822)Memory used [KB]: 7803
% 11.95/3.06 % (9822)Time elapsed: 0.076 s
% 11.95/3.06 % (9822)Instructions burned: 258 (million)
% 11.95/3.06 % (9822)------------------------------
% 11.95/3.06 % (9822)------------------------------
% 11.95/3.07 % (9825)dis+10_1:8_amm=sco:cbe=off:hflw=5:hfsq=on:hfsqc=14:hfsql=off:hfsqr=4,1:hud=5:ile=on:lma=on:nm=2:ntd=on:prag=on:sac=on:sp=frequency:i=373:si=on:rtra=on_0 on DTF2THF_9318 for (2986ds/373Mi)
% 12.53/3.15 % (9813)Instruction limit reached!
% 12.53/3.15 % (9813)------------------------------
% 12.53/3.15 % (9813)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.53/3.15 % (9813)Termination reason: Unknown
% 12.53/3.15 % (9813)Termination phase: Saturation
% 12.53/3.15
% 12.53/3.15 % (9813)Memory used [KB]: 10746
% 12.53/3.15 % (9813)Time elapsed: 0.481 s
% 12.53/3.15 % (9813)Instructions burned: 1540 (million)
% 12.53/3.15 % (9813)------------------------------
% 12.53/3.15 % (9813)------------------------------
% 12.53/3.16 % (9826)lrs+1010_1:128_au=on:cnfonf=lazy_pi_sigma_gen:hflw=4:hfsq=on:hfsqc=20:hfsql=off:i=253:si=on:rtra=on_0 on DTF2THF_9318 for (2985ds/253Mi)
% 12.53/3.19 % (9825)Instruction limit reached!
% 12.53/3.19 % (9825)------------------------------
% 12.53/3.19 % (9825)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.53/3.19 % (9825)Termination reason: Unknown
% 12.53/3.19 % (9825)Termination phase: Saturation
% 12.53/3.19
% 12.53/3.19 % (9825)Memory used [KB]: 8187
% 12.53/3.19 % (9825)Time elapsed: 0.141 s
% 12.53/3.19 % (9825)Instructions burned: 373 (million)
% 12.53/3.19 % (9825)------------------------------
% 12.53/3.19 % (9825)------------------------------
% 13.05/3.20 % (9827)lrs+1002_1:14_au=on:bd=off:e2e=on:fde=unused:sac=on:sos=on:i=832:si=on:rtra=on_0 on DTF2THF_9318 for (2985ds/832Mi)
% 13.05/3.23 % (9826)Instruction limit reached!
% 13.05/3.23 % (9826)------------------------------
% 13.05/3.23 % (9826)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.05/3.23 % (9826)Termination reason: Unknown
% 13.05/3.23 % (9826)Termination phase: Saturation
% 13.05/3.23
% 13.05/3.23 % (9826)Memory used [KB]: 7931
% 13.05/3.23 % (9826)Time elapsed: 0.068 s
% 13.05/3.23 % (9826)Instructions burned: 254 (million)
% 13.05/3.23 % (9826)------------------------------
% 13.05/3.23 % (9826)------------------------------
% 13.05/3.24 % (9816)Instruction limit reached!
% 13.05/3.24 % (9816)------------------------------
% 13.05/3.24 % (9816)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.05/3.24 % (9816)Termination reason: Unknown
% 13.05/3.24 % (9816)Termination phase: Saturation
% 13.05/3.24
% 13.05/3.24 % (9816)Memory used [KB]: 3198
% 13.05/3.24 % (9816)Time elapsed: 0.425 s
% 13.05/3.24 % (9816)Instructions burned: 1914 (million)
% 13.05/3.24 % (9816)------------------------------
% 13.05/3.24 % (9816)------------------------------
% 13.05/3.24 % (9828)lrs+1010_1:1_chr=on:cnfonf=lazy_not_be_gen:gs=on:hud=5:plsq=on:plsqc=1:plsqr=5,1:prag=on:rawr=on:i=668:si=on:rtra=on_0 on DTF2THF_9318 for (2984ds/668Mi)
% 13.05/3.25 % (9829)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=40:si=on:rtra=on_0 on DTF2THF_9318 for (2984ds/40Mi)
% 13.05/3.26 % (9829)Instruction limit reached!
% 13.05/3.26 % (9829)------------------------------
% 13.05/3.26 % (9829)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.05/3.26 % (9829)Termination reason: Unknown
% 13.05/3.26 % (9829)Termination phase: shuffling
% 13.05/3.26
% 13.05/3.26 % (9829)Memory used [KB]: 2302
% 13.05/3.26 % (9829)Time elapsed: 0.012 s
% 13.05/3.26 % (9829)Instructions burned: 40 (million)
% 13.05/3.26 % (9829)------------------------------
% 13.05/3.26 % (9829)------------------------------
% 13.05/3.27 % (9830)lrs+1010_1:1_cnfonf=lazy_pi_sigma_gen:fd=off:ntd=on:sd=1:sgt=32:sos=on:ss=axioms:i=294:si=on:rtra=on_0 on DTF2THF_9318 for (2984ds/294Mi)
% 13.50/3.32 % (9824)Instruction limit reached!
% 13.50/3.32 % (9824)------------------------------
% 13.50/3.32 % (9824)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.50/3.32 % (9824)Termination reason: Unknown
% 13.50/3.32 % (9824)Termination phase: Saturation
% 13.50/3.32
% 13.50/3.32 % (9824)Memory used [KB]: 11513
% 13.50/3.32 % (9824)Time elapsed: 0.299 s
% 13.50/3.32 % (9824)Instructions burned: 996 (million)
% 13.50/3.32 % (9824)------------------------------
% 13.50/3.32 % (9824)------------------------------
% 13.50/3.33 % (9831)dis+1010_1:1_avsq=on:fe=off:fsr=off:hud=10:plsq=on:plsqr=32,1:prag=on:i=187:si=on:rtra=on_0 on DTF2THF_9318 for (2983ds/187Mi)
% 13.50/3.35 % (9830)Instruction limit reached!
% 13.50/3.35 % (9830)------------------------------
% 13.50/3.35 % (9830)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.50/3.35 % (9830)Termination reason: Unknown
% 13.50/3.35 % (9830)Termination phase: Saturation
% 13.50/3.35
% 13.50/3.35 % (9830)Memory used [KB]: 7675
% 13.50/3.35 % (9830)Time elapsed: 0.080 s
% 13.50/3.35 % (9830)Instructions burned: 296 (million)
% 13.50/3.35 % (9830)------------------------------
% 13.50/3.35 % (9830)------------------------------
% 13.50/3.36 % (9832)lrs+1010_2:3_au=on:fsr=off:sd=4:ss=axioms:st=1.5:i=371:si=on:rtra=on_0 on DTF2THF_9318 for (2983ds/371Mi)
% 13.50/3.38 % (9831)Instruction limit reached!
% 13.50/3.38 % (9831)------------------------------
% 13.50/3.38 % (9831)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.50/3.38 % (9831)Termination reason: Unknown
% 13.50/3.38 % (9831)Termination phase: Saturation
% 13.50/3.38
% 13.50/3.38 % (9831)Memory used [KB]: 7547
% 13.50/3.38 % (9831)Time elapsed: 0.056 s
% 13.50/3.38 % (9831)Instructions burned: 189 (million)
% 13.50/3.38 % (9831)------------------------------
% 13.50/3.38 % (9831)------------------------------
% 13.50/3.39 % (9833)dis+1003_1:1_au=on:bd=preordered:fde=none:ixr=off:nwc=10.0:i=1827:si=on:rtra=on_0 on DTF2THF_9318 for (2983ds/1827Mi)
% 14.26/3.43 % (9828)Instruction limit reached!
% 14.26/3.43 % (9828)------------------------------
% 14.26/3.43 % (9828)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.26/3.43 % (9828)Termination reason: Unknown
% 14.26/3.43 % (9828)Termination phase: Saturation
% 14.26/3.43
% 14.26/3.43 % (9828)Memory used [KB]: 14583
% 14.26/3.43 % (9828)Time elapsed: 0.195 s
% 14.26/3.43 % (9828)Instructions burned: 670 (million)
% 14.26/3.43 % (9828)------------------------------
% 14.26/3.43 % (9828)------------------------------
% 14.26/3.44 % (9834)lrs+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:tnu=1:i=70:si=on:rtra=on_0 on DTF2THF_9318 for (2982ds/70Mi)
% 14.26/3.45 % (9827)Instruction limit reached!
% 14.26/3.45 % (9827)------------------------------
% 14.26/3.45 % (9827)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.26/3.45 % (9827)Termination reason: Unknown
% 14.26/3.45 % (9827)Termination phase: Saturation
% 14.26/3.45
% 14.26/3.45 % (9827)Memory used [KB]: 9083
% 14.26/3.45 % (9827)Time elapsed: 0.279 s
% 14.26/3.45 % (9827)Instructions burned: 834 (million)
% 14.26/3.45 % (9827)------------------------------
% 14.26/3.45 % (9827)------------------------------
% 14.26/3.46 % (9834)Instruction limit reached!
% 14.26/3.46 % (9834)------------------------------
% 14.26/3.46 % (9834)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.26/3.46 % (9834)Termination reason: Unknown
% 14.26/3.46 % (9834)Termination phase: Property scanning
% 14.26/3.46
% 14.26/3.46 % (9834)Memory used [KB]: 2814
% 14.26/3.46 % (9834)Time elapsed: 0.020 s
% 14.26/3.46 % (9834)Instructions burned: 70 (million)
% 14.26/3.46 % (9834)------------------------------
% 14.26/3.46 % (9834)------------------------------
% 14.26/3.46 % (9835)lrs+10_1:20_bd=off:cnfonf=lazy_not_be_gen:fe=axiom:hud=10:ins=1:nm=20:prag=on:sos=on:i=316:si=on:rtra=on_0 on DTF2THF_9318 for (2982ds/316Mi)
% 14.26/3.47 % (9832)Refutation not found, non-redundant clauses discarded% (9832)------------------------------
% 14.26/3.47 % (9832)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.26/3.47 % (9832)Termination reason: Refutation not found, non-redundant clauses discarded
% 14.26/3.47
% 14.26/3.47 % (9832)Memory used [KB]: 7803
% 14.26/3.47 % (9832)Time elapsed: 0.114 s
% 14.26/3.47 % (9832)Instructions burned: 364 (million)
% 14.26/3.47 % (9832)------------------------------
% 14.26/3.47 % (9832)------------------------------
% 14.26/3.47 % (9836)ott+1010_1:1_cnfonf=lazy_gen:fe=off:plsq=on:plsqc=1:plsqr=3548551,65536:s2a=on:i=932:si=on:rtra=on_0 on DTF2THF_9318 for (2982ds/932Mi)
% 14.26/3.48 % (9837)lrs+1010_1:128_au=on:bd=preordered:cnfonf=lazy_pi_sigma_gen:plsq=on:plsqc=1:plsqr=32,1:sos=all:sp=occurrence:i=4714:si=on:rtra=on_0 on DTF2THF_9318 for (2982ds/4714Mi)
% 15.18/3.53 % (9835)Refutation not found, incomplete strategy
% 15.18/3.53 % (9835)------------------------------
% 15.18/3.53 % (9835)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.18/3.53 % (9835)Termination reason: Refutation not found, incomplete strategy
% 15.18/3.53
% 15.18/3.53
% 15.18/3.53 % (9835)Memory used [KB]: 7803
% 15.18/3.53 % (9835)Time elapsed: 0.069 s
% 15.18/3.53 % (9835)Instructions burned: 258 (million)
% 15.18/3.53 % (9835)------------------------------
% 15.18/3.53 % (9835)------------------------------
% 15.18/3.54 % (9838)dis+10_1:1_au=on:e2e=on:fde=none:nwc=10.0:s2a=on:s2at=2.0:i=8250:si=on:rtra=on_0 on DTF2THF_9318 for (2981ds/8250Mi)
% 15.18/3.55 % (9837)Refutation not found, incomplete strategy
% 15.18/3.55 % (9837)------------------------------
% 15.18/3.55 % (9837)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.18/3.55 % (9837)Termination reason: Refutation not found, incomplete strategy
% 15.18/3.55
% 15.18/3.55
% 15.18/3.55 % (9837)Memory used [KB]: 8059
% 15.18/3.55 % (9837)Time elapsed: 0.073 s
% 15.18/3.55 % (9837)Instructions burned: 267 (million)
% 15.18/3.55 % (9837)------------------------------
% 15.18/3.55 % (9837)------------------------------
% 15.18/3.56 % (9839)lrs+1002_1:1_apa=on:cnfonf=off:fd=off:flr=on:hud=1:lcm=reverse:prag=on:sd=1:sos=on:ss=axioms:i=770:si=on:rtra=on_0 on DTF2THF_9318 for (2981ds/770Mi)
% 15.64/3.64 % (9836)First to succeed.
% 15.64/3.65 % (9836)Refutation found. Thanks to Tanya!
% 15.64/3.65 % SZS status Theorem for DTF2THF_9318
% 15.64/3.65 % SZS output start Proof for DTF2THF_9318
% 15.64/3.65 thf(type_def_5, type, carrie1963041556t_unit: $tType).
% 15.64/3.65 thf(type_def_6, type, set_a: $tType).
% 15.64/3.65 thf(type_def_8, type, carrie722926983_a_b_c: $tType).
% 15.64/3.65 thf(type_def_9, type, carrie1950868226xt_b_d: $tType).
% 15.64/3.65 thf(type_def_10, type, set_b: $tType).
% 15.64/3.65 thf(type_def_11, type, carrie1821755406_e_b_f: $tType).
% 15.64/3.65 thf(type_def_12, type, set_e: $tType).
% 15.64/3.65 thf(type_def_13, type, a: $tType).
% 15.64/3.65 thf(type_def_14, type, carrie980467975t_unit: $tType).
% 15.64/3.65 thf(type_def_15, type, carrie1065537299t_unit: $tType).
% 15.64/3.65 thf(type_def_16, type, carrie292259835t_unit: $tType).
% 15.64/3.65 thf(type_def_17, type, carrie377329159t_unit: $tType).
% 15.64/3.65 thf(type_def_18, type, nat: $tType).
% 15.64/3.65 thf(type_def_19, type, b: $tType).
% 15.64/3.65 thf(type_def_20, type, e: $tType).
% 15.64/3.65 thf(type_def_21, type, set_nat: $tType).
% 15.64/3.65 thf(type_def_22, type, set_nat_a: $tType).
% 15.64/3.65 thf(type_def_23, type, set_nat_b: $tType).
% 15.64/3.65 thf(type_def_24, type, set_nat_e: $tType).
% 15.64/3.65 thf(func_def_0, type, carrie377329159t_unit: $tType).
% 15.64/3.65 thf(func_def_1, type, carrie292259835t_unit: $tType).
% 15.64/3.65 thf(func_def_2, type, carrie1065537299t_unit: $tType).
% 15.64/3.65 thf(func_def_3, type, carrie980467975t_unit: $tType).
% 15.64/3.65 thf(func_def_4, type, carrie1963041556t_unit: $tType).
% 15.64/3.65 thf(func_def_5, type, carrie1821755406_e_b_f: $tType).
% 15.64/3.65 thf(func_def_6, type, carrie722926983_a_b_c: $tType).
% 15.64/3.65 thf(func_def_7, type, carrie1950868226xt_b_d: $tType).
% 15.64/3.65 thf(func_def_8, type, set_nat_e: $tType).
% 15.64/3.65 thf(func_def_9, type, set_nat_b: $tType).
% 15.64/3.65 thf(func_def_10, type, set_nat_a: $tType).
% 15.64/3.65 thf(func_def_11, type, set_nat: $tType).
% 15.64/3.65 thf(func_def_12, type, set_e: $tType).
% 15.64/3.65 thf(func_def_13, type, set_b: $tType).
% 15.64/3.65 thf(func_def_14, type, set_a: $tType).
% 15.64/3.65 thf(func_def_15, type, nat: $tType).
% 15.64/3.65 thf(func_def_16, type, e: $tType).
% 15.64/3.65 thf(func_def_17, type, b: $tType).
% 15.64/3.65 thf(func_def_18, type, a: $tType).
% 15.64/3.65 thf(func_def_19, type, carrie1074654371t_unit: carrie1963041556t_unit > set_a).
% 15.64/3.65 thf(func_def_20, type, carrie2021454486_a_b_c: carrie722926983_a_b_c > set_a).
% 15.64/3.65 thf(func_def_21, type, carrie2079586589xt_b_d: carrie1950868226xt_b_d > set_b).
% 15.64/3.65 thf(func_def_22, type, carrie730238621_e_b_f: carrie1821755406_e_b_f > set_e).
% 15.64/3.65 thf(func_def_23, type, ring_b_d: carrie1950868226xt_b_d > $o).
% 15.64/3.65 thf(func_def_24, type, zero_a1261444626_a_b_c: carrie722926983_a_b_c > a).
% 15.64/3.65 thf(func_def_25, type, ideal_b_d: carrie1950868226xt_b_d > set_b > $o).
% 15.64/3.65 thf(func_def_26, type, annihi259882159t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > set_b).
% 15.64/3.65 thf(func_def_27, type, annihilator_b_d_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > set_b).
% 15.64/3.65 thf(func_def_28, type, annihilator_b_d_e_f: carrie1950868226xt_b_d > carrie1821755406_e_b_f > set_b).
% 15.64/3.65 thf(func_def_29, type, hOM_b_1103679019t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > carrie1963041556t_unit > carrie980467975t_unit).
% 15.64/3.65 thf(func_def_30, type, hOM_b_796081438it_a_c: carrie1950868226xt_b_d > carrie1963041556t_unit > carrie722926983_a_b_c > carrie980467975t_unit).
% 15.64/3.65 thf(func_def_31, type, hOM_b_1959675421it_e_f: carrie1950868226xt_b_d > carrie1963041556t_unit > carrie1821755406_e_b_f > carrie1065537299t_unit).
% 15.64/3.65 thf(func_def_32, type, hOM_b_717364638t_unit: carrie1950868226xt_b_d > carrie722926983_a_b_c > carrie1963041556t_unit > carrie980467975t_unit).
% 15.64/3.65 thf(func_def_33, type, hOM_b_d_a_c_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > carrie722926983_a_b_c > carrie980467975t_unit).
% 15.64/3.65 thf(func_def_34, type, hOM_b_d_a_c_e_f: carrie1950868226xt_b_d > carrie722926983_a_b_c > carrie1821755406_e_b_f > carrie1065537299t_unit).
% 15.64/3.65 thf(func_def_35, type, hOM_b_1375846429t_unit: carrie1950868226xt_b_d > carrie1821755406_e_b_f > carrie1963041556t_unit > carrie292259835t_unit).
% 15.64/3.65 thf(func_def_36, type, hOM_b_d_e_f_a_c: carrie1950868226xt_b_d > carrie1821755406_e_b_f > carrie722926983_a_b_c > carrie292259835t_unit).
% 15.64/3.65 thf(func_def_37, type, hOM_b_d_e_f_e_f: carrie1950868226xt_b_d > carrie1821755406_e_b_f > carrie1821755406_e_b_f > carrie377329159t_unit).
% 15.64/3.65 thf(func_def_38, type, module589927589unit_d: carrie980467975t_unit > carrie1950868226xt_b_d > $o).
% 15.64/3.65 thf(func_def_39, type, module1648870817unit_d: carrie1065537299t_unit > carrie1950868226xt_b_d > $o).
% 15.64/3.65 thf(func_def_40, type, module1330273449unit_d: carrie292259835t_unit > carrie1950868226xt_b_d > $o).
% 15.64/3.65 thf(func_def_41, type, module241733029unit_d: carrie377329159t_unit > carrie1950868226xt_b_d > $o).
% 15.64/3.65 thf(func_def_42, type, module1821517916unit_d: carrie1963041556t_unit > carrie1950868226xt_b_d > $o).
% 15.64/3.65 thf(func_def_43, type, module_a_b_c_d: carrie722926983_a_b_c > carrie1950868226xt_b_d > $o).
% 15.64/3.65 thf(func_def_44, type, module_e_b_f_d: carrie1821755406_e_b_f > carrie1950868226xt_b_d > $o).
% 15.64/3.65 thf(func_def_45, type, free_g637607517t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > set_a > $o).
% 15.64/3.65 thf(func_def_46, type, free_g1087686480_d_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > set_a > $o).
% 15.64/3.65 thf(func_def_47, type, free_g103796815_d_e_f: carrie1950868226xt_b_d > carrie1821755406_e_b_f > set_e > $o).
% 15.64/3.65 thf(func_def_48, type, genera1692266857t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > set_a > $o).
% 15.64/3.65 thf(func_def_49, type, generator_b_d_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > set_a > $o).
% 15.64/3.65 thf(func_def_50, type, generator_b_d_e_f: carrie1950868226xt_b_d > carrie1821755406_e_b_f > set_e > $o).
% 15.64/3.65 thf(func_def_51, type, l_comb1138968323t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > nat > (nat > b) > (nat > a) > a).
% 15.64/3.65 thf(func_def_52, type, l_comb_b_d_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > nat > (nat > b) > (nat > a) > a).
% 15.64/3.65 thf(func_def_53, type, l_comb_b_d_e_f: carrie1950868226xt_b_d > carrie1821755406_e_b_f > nat > (nat > b) > (nat > e) > e).
% 15.64/3.65 thf(func_def_54, type, linear_span_b_d_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > set_b > set_a > set_a).
% 15.64/3.65 thf(func_def_55, type, mdl_a_b_c: carrie722926983_a_b_c > set_a > carrie1963041556t_unit).
% 15.64/3.65 thf(func_def_56, type, misomo493403663t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > carrie1963041556t_unit > $o).
% 15.64/3.65 thf(func_def_57, type, misomo1935876354it_a_c: carrie1950868226xt_b_d > carrie1963041556t_unit > carrie722926983_a_b_c > $o).
% 15.64/3.65 thf(func_def_58, type, misomo951986689it_e_f: carrie1950868226xt_b_d > carrie1963041556t_unit > carrie1821755406_e_b_f > $o).
% 15.64/3.65 thf(func_def_59, type, misomo1857159554t_unit: carrie1950868226xt_b_d > carrie722926983_a_b_c > carrie1963041556t_unit > $o).
% 15.64/3.65 thf(func_def_60, type, misomo1282343797_c_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > carrie722926983_a_b_c > $o).
% 15.64/3.65 thf(func_def_61, type, misomo298454132_c_e_f: carrie1950868226xt_b_d > carrie722926983_a_b_c > carrie1821755406_e_b_f > $o).
% 15.64/3.65 thf(func_def_62, type, misomo368157697t_unit: carrie1950868226xt_b_d > carrie1821755406_e_b_f > carrie1963041556t_unit > $o).
% 15.64/3.65 thf(func_def_63, type, misomo1752826100_f_a_c: carrie1950868226xt_b_d > carrie1821755406_e_b_f > carrie722926983_a_b_c > $o).
% 15.64/3.65 thf(func_def_64, type, misomo768936435_f_e_f: carrie1950868226xt_b_d > carrie1821755406_e_b_f > carrie1821755406_e_b_f > $o).
% 15.64/3.65 thf(func_def_65, type, quotie1196826005t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > set_a > set_a > set_b).
% 15.64/3.65 thf(func_def_66, type, quotie1153752712_d_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > set_a > set_a > set_b).
% 15.64/3.65 thf(func_def_67, type, quotie169863047_d_e_f: carrie1950868226xt_b_d > carrie1821755406_e_b_f > set_e > set_e > set_b).
% 15.64/3.65 thf(func_def_68, type, smodul132175289t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > set_b > set_a).
% 15.64/3.65 thf(func_def_69, type, smodul818989740_d_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > set_b > set_a).
% 15.64/3.65 thf(func_def_70, type, smodul1982583723_d_e_f: carrie1950868226xt_b_d > carrie1821755406_e_b_f > set_b > set_e).
% 15.64/3.65 thf(func_def_71, type, submod903911234t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > set_a > $o).
% 15.64/3.65 thf(func_def_72, type, submodule_b_d_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > set_a > $o).
% 15.64/3.65 thf(func_def_73, type, submodule_b_d_e_f: carrie1950868226xt_b_d > carrie1821755406_e_b_f > set_e > $o).
% 15.64/3.65 thf(func_def_74, type, algebr1039611956t_unit: carrie1950868226xt_b_d > carrie1963041556t_unit > set_a > set_a).
% 15.64/3.65 thf(func_def_75, type, algebr1152250919_d_a_c: carrie1950868226xt_b_d > carrie722926983_a_b_c > set_a > set_a).
% 15.64/3.65 thf(func_def_76, type, algebr168361254_d_e_f: carrie1950868226xt_b_d > carrie1821755406_e_b_f > set_e > set_e).
% 15.64/3.65 thf(func_def_77, type, pi_nat_a: set_nat > (nat > set_a) > set_nat_a).
% 15.64/3.65 thf(func_def_78, type, pi_nat_b: set_nat > (nat > set_b) > set_nat_b).
% 15.64/3.65 thf(func_def_79, type, pi_nat_e: set_nat > (nat > set_e) > set_nat_e).
% 15.64/3.65 thf(func_def_80, type, minus_minus_nat: nat > nat > nat).
% 15.64/3.65 thf(func_def_81, type, minus_1788767276_nat_a: set_nat_a > set_nat_a > set_nat_a).
% 15.64/3.65 thf(func_def_82, type, minus_780847917_nat_b: set_nat_b > set_nat_b > set_nat_b).
% 15.64/3.65 thf(func_def_83, type, minus_minus_set_a: set_a > set_a > set_a).
% 15.64/3.65 thf(func_def_84, type, zero_zero_nat: nat).
% 15.64/3.65 thf(func_def_85, type, suc: nat > nat).
% 15.64/3.65 thf(func_def_86, type, bot_bot_set_nat_a: set_nat_a).
% 15.64/3.65 thf(func_def_87, type, bot_bot_set_nat_b: set_nat_b).
% 15.64/3.65 thf(func_def_88, type, bot_bot_set_nat: set_nat).
% 15.64/3.65 thf(func_def_89, type, bot_bot_set_a: set_a).
% 15.64/3.65 thf(func_def_90, type, bot_bot_set_e: set_e).
% 15.64/3.65 thf(func_def_91, type, ord_less_nat: nat > nat > $o).
% 15.64/3.65 thf(func_def_92, type, ord_less_set_a: set_a > set_a > $o).
% 15.64/3.65 thf(func_def_93, type, ord_less_eq_nat: nat > nat > $o).
% 15.64/3.65 thf(func_def_94, type, ord_le157368549_nat_a: set_nat_a > set_nat_a > $o).
% 15.64/3.65 thf(func_def_95, type, ord_le1296932838_nat_b: set_nat_b > set_nat_b > $o).
% 15.64/3.65 thf(func_def_96, type, ord_less_eq_set_a: set_a > set_a > $o).
% 15.64/3.65 thf(func_def_97, type, ord_less_eq_set_e: set_e > set_e > $o).
% 15.64/3.65 thf(func_def_98, type, collect_nat_a: ((nat > a) > $o) > set_nat_a).
% 15.64/3.65 thf(func_def_99, type, collect_nat_b: ((nat > b) > $o) > set_nat_b).
% 15.64/3.65 thf(func_def_100, type, collect_nat: (nat > $o) > set_nat).
% 15.64/3.65 thf(func_def_101, type, collect_a: (a > $o) > set_a).
% 15.64/3.65 thf(func_def_102, type, insert_nat_a: (nat > a) > set_nat_a > set_nat_a).
% 15.64/3.65 thf(func_def_103, type, insert_nat_b: (nat > b) > set_nat_b > set_nat_b).
% 15.64/3.65 thf(func_def_104, type, insert_a: a > set_a > set_a).
% 15.64/3.65 thf(func_def_105, type, member_nat_a: (nat > a) > set_nat_a > $o).
% 15.64/3.65 thf(func_def_106, type, member_nat_b: (nat > b) > set_nat_b > $o).
% 15.64/3.65 thf(func_def_107, type, member_nat_e: (nat > e) > set_nat_e > $o).
% 15.64/3.65 thf(func_def_108, type, member_nat: nat > set_nat > $o).
% 15.64/3.65 thf(func_def_109, type, member_a: a > set_a > $o).
% 15.64/3.65 thf(func_def_110, type, member_e: e > set_e > $o).
% 15.64/3.65 thf(func_def_111, type, h: set_a).
% 15.64/3.65 thf(func_def_112, type, h1: set_a).
% 15.64/3.65 thf(func_def_113, type, m: carrie722926983_a_b_c).
% 15.64/3.65 thf(func_def_114, type, n: carrie1821755406_e_b_f).
% 15.64/3.65 thf(func_def_115, type, r: carrie1950868226xt_b_d).
% 15.64/3.65 thf(func_def_116, type, g: nat > a).
% 15.64/3.65 thf(func_def_117, type, h2: a).
% 15.64/3.65 thf(func_def_118, type, m2: nat).
% 15.64/3.65 thf(func_def_119, type, t: nat > b).
% 15.64/3.65 thf(func_def_120, type, x: a).
% 15.64/3.65 thf(func_def_121, type, xa: nat).
% 15.64/3.65 thf(func_def_143, type, ph1: !>[X0: $tType]:(X0)).
% 15.64/3.65 thf(func_def_144, type, sK2: nat).
% 15.64/3.65 thf(f3066,plain,(
% 15.64/3.65 $false),
% 15.64/3.65 inference(avatar_sat_refutation,[status(thm)],[f2052,f3055,f3065])).
% 15.64/3.65 thf(f3065,plain,(
% 15.64/3.65 ~spl0_1),
% 15.64/3.65 inference(avatar_contradiction_clause,[status(thm)],[f3064])).
% 15.64/3.65 thf(f3064,plain,(
% 15.64/3.65 $false | ~spl0_1),
% 15.64/3.65 inference(trivial_inequality_removal,[status(thm)],[f3063])).
% 15.64/3.65 thf(f3063,plain,(
% 15.64/3.65 ($false = $true) | ~spl0_1),
% 15.64/3.65 inference(forward_demodulation,[status(thm)],[f3062,f2048])).
% 15.64/3.65 thf(f2048,plain,(
% 15.64/3.65 ($false = (ord_less_eq_nat @ xa @ m2)) | ~spl0_1),
% 15.64/3.65 inference(avatar_component_clause,[status(thm)],[f2046])).
% 15.64/3.65 thf(f2046,definition,(
% 15.64/3.65 spl0_1 <=> ($false = (ord_less_eq_nat @ xa @ m2))),
% 15.64/3.65 introduced(definition,[new_symbols(naming,[spl0_1])],[avatar_definition])).
% 15.64/3.65 thf(f3062,plain,(
% 15.64/3.65 ($true = (ord_less_eq_nat @ xa @ m2))),
% 15.64/3.65 inference(forward_demodulation,[status(thm)],[f3061,f2971])).
% 15.64/3.65 thf(f2971,plain,(
% 15.64/3.65 (m2 = (suc @ sK2))),
% 15.64/3.65 inference(equality_proxy_clausification,[status(thm)],[f2967])).
% 15.64/3.65 thf(f2967,plain,(
% 15.64/3.65 (((suc @ sK2) = m2) = $true)),
% 15.64/3.65 inference(beta_eta_normalization,[status(thm)],[f2965])).
% 15.64/3.65 thf(f2965,plain,(
% 15.64/3.65 ($true = ((^[Y0 : nat]: ((suc @ Y0) = m2)) @ sK2))),
% 15.64/3.65 inference(sigma_clausification,[status(thm)],[f2948])).
% 15.64/3.65 thf(f2948,plain,(
% 15.64/3.65 ($true = (?? @ nat @ (^[Y0 : nat]: ((suc @ Y0) = m2))))),
% 15.64/3.65 inference(boolean_simplification,[status(thm)],[f2947])).
% 15.64/3.65 thf(f2947,plain,(
% 15.64/3.65 ($true = ($true => (?? @ nat @ (^[Y0 : nat]: ((suc @ Y0) = m2)))))),
% 15.64/3.65 inference(boolean_simplification,[status(thm)],[f2935])).
% 15.64/3.65 thf(f2935,plain,(
% 15.64/3.65 ($true = ((~ $false) => (?? @ nat @ (^[Y0 : nat]: ((suc @ Y0) = m2)))))),
% 15.64/3.65 inference(superposition,[status(thm)],[f2912,f1951])).
% 15.64/3.65 thf(f1951,plain,(
% 15.64/3.65 ($false = (ord_less_eq_nat @ m2 @ zero_zero_nat))),
% 15.64/3.65 inference(not_proxy_clausification,[status(thm)],[f1917])).
% 15.64/3.65 thf(f1917,plain,(
% 15.64/3.65 ($true = (~ (ord_less_eq_nat @ m2 @ zero_zero_nat)))),
% 15.64/3.65 inference(equality_proxy_clausification,[status(thm)],[f1878])).
% 15.64/3.65 thf(f1878,plain,(
% 15.64/3.65 ($true = ((~ (ord_less_eq_nat @ m2 @ zero_zero_nat)) = $true))),
% 15.64/3.65 inference(superposition,[status(thm)],[f1873,f1574])).
% 15.64/3.65 thf(f1574,plain,(
% 15.64/3.65 ((ord_less_eq_nat @ (suc @ zero_zero_nat) @ m2) = $true)),
% 15.64/3.65 inference(beta_eta_normalization,[status(thm)],[f1485])).
% 15.64/3.65 thf(f1485,plain,(
% 15.64/3.65 ($true = ((^[Y0 : nat]: (ord_less_eq_nat @ (suc @ Y0))) @ zero_zero_nat @ m2))),
% 15.64/3.65 inference(definition_unfolding,[status(thm)],[f1120,f1112])).
% 15.64/3.65 thf(f1112,plain,(
% 15.64/3.65 (ord_less_nat = (^[Y0 : nat]: (ord_less_eq_nat @ (suc @ Y0))))),
% 15.64/3.65 inference(cnf_transformation,[status(thm)],[f462])).
% 15.64/3.65 thf(f462,plain,(
% 15.64/3.65 (ord_less_nat = (^[Y0 : nat]: (ord_less_eq_nat @ (suc @ Y0))))),
% 15.64/3.65 inference(fool_elimination,[status(thm)],[f461])).
% 15.64/3.65 thf(f461,plain,(
% 15.64/3.65 (ord_less_nat = (^[X0 : nat] : (ord_less_eq_nat @ (suc @ X0))))),
% 15.64/3.65 inference(rectify,[status(thm)],[f238])).
% 15.64/3.65 thf(f238,axiom,(
% 15.64/3.65 (ord_less_nat = (^[X45 : nat] : (ord_less_eq_nat @ (suc @ X45))))),
% 15.64/3.65 file('/export/starexec/sandbox/tmp/tmp.sXLfLlVcSw/DTF2THF_9318.p',fact_237_less__eq__Suc__le)).
% 15.64/3.65 thf(f1120,plain,(
% 15.64/3.65 ((ord_less_nat @ zero_zero_nat @ m2) = $true)),
% 15.64/3.65 inference(cnf_transformation,[status(thm)],[f1077])).
% 15.64/3.65 thf(f1077,plain,(
% 15.64/3.65 ((ord_less_nat @ zero_zero_nat @ m2) = $true)),
% 15.64/3.65 inference(fool_elimination,[status(thm)],[f1076])).
% 15.64/3.65 thf(f1076,plain,(
% 15.64/3.65 (ord_less_nat @ zero_zero_nat @ m2)),
% 15.64/3.65 inference(rectify,[status(thm)],[f366])).
% 15.64/3.65 thf(f366,axiom,(
% 15.64/3.65 (ord_less_nat @ zero_zero_nat @ m2)),
% 15.64/3.65 file('/export/starexec/sandbox/tmp/tmp.sXLfLlVcSw/DTF2THF_9318.p',conj_16)).
% 15.64/3.65 thf(f1873,plain,(
% 15.64/3.65 ( ! [X1 : nat] : (($true = ((~ (ord_less_eq_nat @ X1 @ zero_zero_nat)) = (ord_less_eq_nat @ (suc @ zero_zero_nat) @ X1)))) )),
% 15.64/3.65 inference(forward_demodulation,[status(thm)],[f1872,f1770])).
% 15.64/3.65 thf(f1770,plain,(
% 15.64/3.65 ( ! [X1 : nat] : (((zero_zero_nat = X1) = (ord_less_eq_nat @ X1 @ zero_zero_nat))) )),
% 15.64/3.65 inference(equality_proxy_clausification,[status(thm)],[f1769])).
% 15.64/3.65 thf(f1769,plain,(
% 15.64/3.65 ( ! [X1 : nat] : (($true = ((zero_zero_nat = X1) = (ord_less_eq_nat @ X1 @ zero_zero_nat)))) )),
% 15.64/3.65 inference(beta_eta_normalization,[status(thm)],[f1768])).
% 15.64/3.65 thf(f1768,plain,(
% 15.64/3.65 ( ! [X1 : nat] : (($true = ((^[Y0 : nat]: ((zero_zero_nat = Y0) = (ord_less_eq_nat @ Y0 @ zero_zero_nat))) @ X1))) )),
% 15.64/3.65 inference(pi_clausification,[status(thm)],[f1139])).
% 15.64/3.65 thf(f1139,plain,(
% 15.64/3.65 ($true = (!! @ nat @ (^[Y0 : nat]: ((zero_zero_nat = Y0) = (ord_less_eq_nat @ Y0 @ zero_zero_nat)))))),
% 15.64/3.65 inference(cnf_transformation,[status(thm)],[f452])).
% 15.64/3.65 thf(f452,plain,(
% 15.64/3.65 ($true = (!! @ nat @ (^[Y0 : nat]: ((zero_zero_nat = Y0) = (ord_less_eq_nat @ Y0 @ zero_zero_nat)))))),
% 15.64/3.65 inference(fool_elimination,[status(thm)],[f451])).
% 15.64/3.65 thf(f451,plain,(
% 15.64/3.65 ! [X0 : nat] : ((zero_zero_nat = X0) = (ord_less_eq_nat @ X0 @ zero_zero_nat))),
% 15.64/3.65 inference(rectify,[status(thm)],[f168])).
% 15.64/3.65 thf(f168,axiom,(
% 15.64/3.65 ! [X7 : nat] : ((zero_zero_nat = X7) = (ord_less_eq_nat @ X7 @ zero_zero_nat))),
% 15.64/3.65 file('/export/starexec/sandbox/tmp/tmp.sXLfLlVcSw/DTF2THF_9318.p',fact_167_le__0__eq)).
% 15.64/3.65 thf(f1872,plain,(
% 15.64/3.65 ( ! [X1 : nat] : (($true = ((~ (zero_zero_nat = X1)) = (ord_less_eq_nat @ (suc @ zero_zero_nat) @ X1)))) )),
% 15.64/3.65 inference(beta_eta_normalization,[status(thm)],[f1871])).
% 15.64/3.65 thf(f1871,plain,(
% 15.64/3.65 ( ! [X1 : nat] : (($true = ((^[Y0 : nat]: ((~ (zero_zero_nat = Y0)) = (ord_less_eq_nat @ (suc @ zero_zero_nat) @ Y0))) @ X1))) )),
% 15.64/3.65 inference(pi_clausification,[status(thm)],[f1566])).
% 15.64/3.65 thf(f1566,plain,(
% 15.64/3.65 ((!! @ nat @ (^[Y0 : nat]: ((~ (zero_zero_nat = Y0)) = (ord_less_eq_nat @ (suc @ zero_zero_nat) @ Y0)))) = $true)),
% 15.64/3.65 inference(beta_eta_normalization,[status(thm)],[f1552])).
% 15.64/3.65 thf(f1552,plain,(
% 15.64/3.65 ($true = (!! @ nat @ (^[Y0 : nat]: ((~ (zero_zero_nat = Y0)) = ((^[Y1 : nat]: (ord_less_eq_nat @ (suc @ Y1))) @ zero_zero_nat @ Y0)))))),
% 15.64/3.65 inference(definition_unfolding,[status(thm)],[f1420,f1112])).
% 15.64/3.65 thf(f1420,plain,(
% 15.64/3.65 ((!! @ nat @ (^[Y0 : nat]: ((~ (zero_zero_nat = Y0)) = (ord_less_nat @ zero_zero_nat @ Y0)))) = $true)),
% 15.64/3.65 inference(cnf_transformation,[status(thm)],[f651])).
% 15.64/3.65 thf(f651,plain,(
% 15.64/3.65 ((!! @ nat @ (^[Y0 : nat]: ((~ (zero_zero_nat = Y0)) = (ord_less_nat @ zero_zero_nat @ Y0)))) = $true)),
% 15.64/3.65 inference(fool_elimination,[status(thm)],[f650])).
% 15.64/3.65 thf(f650,plain,(
% 15.64/3.65 ! [X0 : nat] : ((zero_zero_nat != X0) = (ord_less_nat @ zero_zero_nat @ X0))),
% 15.64/3.65 inference(rectify,[status(thm)],[f76])).
% 15.64/3.65 thf(f76,axiom,(
% 15.64/3.65 ! [X2 : nat] : ((zero_zero_nat != X2) = (ord_less_nat @ zero_zero_nat @ X2))),
% 15.64/3.65 file('/export/starexec/sandbox/tmp/tmp.sXLfLlVcSw/DTF2THF_9318.p',fact_75_bot__nat__0_Onot__eq__extremum)).
% 15.64/3.65 thf(f2912,plain,(
% 15.64/3.65 ( ! [X1 : nat] : ((((~ (ord_less_eq_nat @ X1 @ zero_zero_nat)) => (?? @ nat @ (^[Y0 : nat]: ((suc @ Y0) = X1)))) = $true)) )),
% 15.64/3.65 inference(forward_demodulation,[status(thm)],[f2911,f1874])).
% 15.64/3.65 thf(f1874,plain,(
% 15.64/3.65 ( ! [X1 : nat] : (((~ (ord_less_eq_nat @ X1 @ zero_zero_nat)) = (ord_less_eq_nat @ (suc @ zero_zero_nat) @ X1))) )),
% 15.64/3.65 inference(equality_proxy_clausification,[status(thm)],[f1873])).
% 15.64/3.65 thf(f2911,plain,(
% 15.64/3.65 ( ! [X1 : nat] : ((((ord_less_eq_nat @ (suc @ zero_zero_nat) @ X1) => (?? @ nat @ (^[Y0 : nat]: ((suc @ Y0) = X1)))) = $true)) )),
% 15.64/3.65 inference(beta_eta_normalization,[status(thm)],[f2910])).
% 15.64/3.65 thf(f2910,plain,(
% 15.64/3.65 ( ! [X1 : nat] : (($true = ((^[Y0 : nat]: ((ord_less_eq_nat @ (suc @ zero_zero_nat) @ Y0) => (?? @ nat @ (^[Y1 : nat]: ((suc @ Y1) = Y0))))) @ X1))) )),
% 15.64/3.65 inference(pi_clausification,[status(thm)],[f1642])).
% 15.64/3.65 thf(f1642,plain,(
% 15.64/3.65 ((!! @ nat @ (^[Y0 : nat]: ((ord_less_eq_nat @ (suc @ zero_zero_nat) @ Y0) => (?? @ nat @ (^[Y1 : nat]: ((suc @ Y1) = Y0)))))) = $true)),
% 15.64/3.65 inference(beta_eta_normalization,[status(thm)],[f1548])).
% 15.64/3.65 thf(f1548,plain,(
% 15.64/3.65 ((!! @ nat @ (^[Y0 : nat]: (((^[Y1 : nat]: (ord_less_eq_nat @ (suc @ Y1))) @ zero_zero_nat @ Y0) => (?? @ nat @ (^[Y1 : nat]: ((suc @ Y1) = Y0)))))) = $true)),
% 15.64/3.65 inference(definition_unfolding,[status(thm)],[f1407,f1112])).
% 15.64/3.65 thf(f1407,plain,(
% 15.64/3.65 ((!! @ nat @ (^[Y0 : nat]: ((ord_less_nat @ zero_zero_nat @ Y0) => (?? @ nat @ (^[Y1 : nat]: ((suc @ Y1) = Y0)))))) = $true)),
% 15.64/3.65 inference(cnf_transformation,[status(thm)],[f941])).
% 15.64/3.65 thf(f941,plain,(
% 15.64/3.65 ((!! @ nat @ (^[Y0 : nat]: ((ord_less_nat @ zero_zero_nat @ Y0) => (?? @ nat @ (^[Y1 : nat]: ((suc @ Y1) = Y0)))))) = $true)),
% 15.64/3.65 inference(fool_elimination,[status(thm)],[f940])).
% 15.64/3.65 thf(f940,plain,(
% 15.64/3.65 ! [X0 : nat] : ((ord_less_nat @ zero_zero_nat @ X0) => ? [X1 : nat] : ((suc @ X1) = X0))),
% 15.64/3.65 inference(rectify,[status(thm)],[f248])).
% 15.64/3.65 thf(f248,axiom,(
% 15.64/3.65 ! [X7 : nat] : ((ord_less_nat @ zero_zero_nat @ X7) => ? [X40 : nat] : ((suc @ X40) = X7))),
% 15.64/3.65 file('/export/starexec/sandbox/tmp/tmp.sXLfLlVcSw/DTF2THF_9318.p',fact_247_gr0__implies__Suc)).
% 15.64/3.65 thf(f3061,plain,(
% 15.64/3.65 ($true = (ord_less_eq_nat @ xa @ (suc @ sK2)))),
% 15.64/3.65 inference(forward_demodulation,[status(thm)],[f3060,f1717])).
% 15.64/3.65 thf(f1717,plain,(
% 15.64/3.65 ( ! [X1 : nat] : (((minus_minus_nat @ X1 @ zero_zero_nat) = X1)) )),
% 15.64/3.65 inference(equality_proxy_clausification,[status(thm)],[f1716])).
% 15.64/3.65 thf(f1716,plain,(
% 15.64/3.65 ( ! [X1 : nat] : (($true = ((minus_minus_nat @ X1 @ zero_zero_nat) = X1))) )),
% 15.64/3.65 inference(beta_eta_normalization,[status(thm)],[f1715])).
% 15.64/3.65 thf(f1715,plain,(
% 15.64/3.65 ( ! [X1 : nat] : ((((^[Y0 : nat]: ((minus_minus_nat @ Y0 @ zero_zero_nat) = Y0)) @ X1) = $true)) )),
% 15.64/3.65 inference(pi_clausification,[status(thm)],[f1264])).
% 15.64/3.65 thf(f1264,plain,(
% 15.64/3.65 ($true = (!! @ nat @ (^[Y0 : nat]: ((minus_minus_nat @ Y0 @ zero_zero_nat) = Y0))))),
% 15.64/3.65 inference(cnf_transformation,[status(thm)],[f880])).
% 15.64/3.65 thf(f880,plain,(
% 15.64/3.65 ($true = (!! @ nat @ (^[Y0 : nat]: ((minus_minus_nat @ Y0 @ zero_zero_nat) = Y0))))),
% 15.64/3.65 inference(fool_elimination,[status(thm)],[f879])).
% 15.64/3.65 thf(f879,plain,(
% 15.64/3.65 ! [X0 : nat] : ((minus_minus_nat @ X0 @ zero_zero_nat) = X0)),
% 15.64/3.65 inference(rectify,[status(thm)],[f327])).
% 15.64/3.65 thf(f327,axiom,(
% 15.64/3.65 ! [X2 : nat] : ((minus_minus_nat @ X2 @ zero_zero_nat) = X2)),
% 15.64/3.65 file('/export/starexec/sandbox/tmp/tmp.sXLfLlVcSw/DTF2THF_9318.p',fact_326_diff__zero)).
% 15.64/3.65 thf(f3060,plain,(
% 15.64/3.65 ($true = (ord_less_eq_nat @ xa @ (suc @ (minus_minus_nat @ sK2 @ zero_zero_nat))))),
% 15.64/3.65 inference(forward_demodulation,[status(thm)],[f2317,f3025])).
% 15.64/3.65 thf(f3025,plain,(
% 15.64/3.65 ( ! [X0 : nat] : (((minus_minus_nat @ m2 @ (suc @ X0)) = (minus_minus_nat @ sK2 @ X0))) )),
% 15.64/3.65 inference(superposition,[status(thm)],[f3002,f2971])).
% 15.64/3.65 thf(f3002,plain,(
% 15.64/3.65 ( ! [X2 : nat,X1 : nat] : (((minus_minus_nat @ (suc @ X1) @ (suc @ X2)) = (minus_minus_nat @ X1 @ X2))) )),
% 15.64/3.65 inference(equality_proxy_clausification,[status(thm)],[f2978])).
% 15.64/3.65 thf(f2978,plain,(
% 15.64/3.65 ( ! [X2 : nat,X1 : nat] : (($true = ((minus_minus_nat @ (suc @ X1) @ (suc @ X2)) = (minus_minus_nat @ X1 @ X2)))) )),
% 15.64/3.65 inference(beta_eta_normalization,[status(thm)],[f2977])).
% 15.64/3.65 thf(f2977,plain,(
% 15.64/3.65 ( ! [X2 : nat,X1 : nat] : ((((^[Y0 : nat]: ((minus_minus_nat @ (suc @ X1) @ (suc @ Y0)) = (minus_minus_nat @ X1 @ Y0))) @ X2) = $true)) )),
% 15.64/3.65 inference(pi_clausification,[status(thm)],[f2970])).
% 15.64/3.65 thf(f2970,plain,(
% 15.64/3.65 ( ! [X1 : nat] : (((!! @ nat @ (^[Y0 : nat]: ((minus_minus_nat @ (suc @ X1) @ (suc @ Y0)) = (minus_minus_nat @ X1 @ Y0)))) = $true)) )),
% 15.64/3.65 inference(beta_eta_normalization,[status(thm)],[f2969])).
% 15.64/3.65 thf(f2969,plain,(
% 15.64/3.65 ( ! [X1 : nat] : (($true = ((^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: ((minus_minus_nat @ (suc @ Y0) @ (suc @ Y1)) = (minus_minus_nat @ Y0 @ Y1))))) @ X1))) )),
% 15.64/3.65 inference(pi_clausification,[status(thm)],[f1113])).
% 15.64/3.65 thf(f1113,plain,(
% 15.64/3.65 ((!! @ nat @ (^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: ((minus_minus_nat @ (suc @ Y0) @ (suc @ Y1)) = (minus_minus_nat @ Y0 @ Y1)))))) = $true)),
% 15.64/3.65 inference(cnf_transformation,[status(thm)],[f649])).
% 15.64/3.65 thf(f649,plain,(
% 15.64/3.65 ((!! @ nat @ (^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: ((minus_minus_nat @ (suc @ Y0) @ (suc @ Y1)) = (minus_minus_nat @ Y0 @ Y1)))))) = $true)),
% 15.64/3.65 inference(fool_elimination,[status(thm)],[f648])).
% 15.64/3.65 thf(f648,plain,(
% 15.64/3.65 ! [X0 : nat,X1 : nat] : ((minus_minus_nat @ (suc @ X1) @ (suc @ X0)) = (minus_minus_nat @ X1 @ X0))),
% 15.64/3.65 inference(rectify,[status(thm)],[f81])).
% 15.64/3.65 thf(f81,axiom,(
% 15.64/3.65 ! [X7 : nat,X14 : nat] : ((minus_minus_nat @ X14 @ X7) = (minus_minus_nat @ (suc @ X14) @ (suc @ X7)))),
% 15.64/3.65 file('/export/starexec/sandbox/tmp/tmp.sXLfLlVcSw/DTF2THF_9318.p',fact_80_diff__Suc__Suc)).
% 15.64/3.65 thf(f2317,plain,(
% 15.64/3.65 ($true = (ord_less_eq_nat @ xa @ (suc @ (minus_minus_nat @ m2 @ (suc @ zero_zero_nat)))))),
% 15.64/3.65 inference(boolean_simplification,[status(thm)],[f2305])).
% 15.64/3.65 thf(f2305,plain,(
% 15.64/3.65 (($true => (ord_less_eq_nat @ xa @ (suc @ (minus_minus_nat @ m2 @ (suc @ zero_zero_nat))))) = $true)),
% 15.64/3.65 inference(superposition,[status(thm)],[f2283,f1146])).
% 15.64/3.65 thf(f1146,plain,(
% 15.64/3.65 ((ord_less_eq_nat @ xa @ (minus_minus_nat @ m2 @ (suc @ zero_zero_nat))) = $true)),
% 15.64/3.65 inference(cnf_transformation,[status(thm)],[f826])).
% 15.64/3.65 thf(f826,plain,(
% 15.64/3.65 ((ord_less_eq_nat @ xa @ (minus_minus_nat @ m2 @ (suc @ zero_zero_nat))) = $true)),
% 15.64/3.65 inference(fool_elimination,[status(thm)],[f825])).
% 15.64/3.65 thf(f825,plain,(
% 15.64/3.65 (ord_less_eq_nat @ xa @ (minus_minus_nat @ m2 @ (suc @ zero_zero_nat)))),
% 15.64/3.65 inference(rectify,[status(thm)],[f369])).
% 15.64/3.65 thf(f369,axiom,(
% 15.64/3.65 (ord_less_eq_nat @ xa @ (minus_minus_nat @ m2 @ (suc @ zero_zero_nat)))),
% 15.64/3.65 file('/export/starexec/sandbox/tmp/tmp.sXLfLlVcSw/DTF2THF_9318.p',conj_19)).
% 15.64/3.65 thf(f2283,plain,(
% 15.64/3.65 ( ! [X2 : nat,X1 : nat] : (($true = ((ord_less_eq_nat @ X2 @ X1) => (ord_less_eq_nat @ X2 @ (suc @ X1))))) )),
% 15.64/3.65 inference(beta_eta_normalization,[status(thm)],[f2282])).
% 15.64/3.65 thf(f2282,plain,(
% 15.64/3.65 ( ! [X2 : nat,X1 : nat] : ((((^[Y0 : nat]: ((ord_less_eq_nat @ Y0 @ X1) => (ord_less_eq_nat @ Y0 @ (suc @ X1)))) @ X2) = $true)) )),
% 15.64/3.65 inference(pi_clausification,[status(thm)],[f2279])).
% 15.64/3.65 thf(f2279,plain,(
% 15.64/3.65 ( ! [X1 : nat] : (($true = (!! @ nat @ (^[Y0 : nat]: ((ord_less_eq_nat @ Y0 @ X1) => (ord_less_eq_nat @ Y0 @ (suc @ X1))))))) )),
% 15.64/3.65 inference(beta_eta_normalization,[status(thm)],[f2278])).
% 15.64/3.65 thf(f2278,plain,(
% 15.64/3.65 ( ! [X1 : nat] : ((((^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: ((ord_less_eq_nat @ Y1 @ Y0) => (ord_less_eq_nat @ Y1 @ (suc @ Y0)))))) @ X1) = $true)) )),
% 15.64/3.65 inference(pi_clausification,[status(thm)],[f1304])).
% 15.64/3.65 thf(f1304,plain,(
% 15.64/3.65 ($true = (!! @ nat @ (^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: ((ord_less_eq_nat @ Y1 @ Y0) => (ord_less_eq_nat @ Y1 @ (suc @ Y0))))))))),
% 15.64/3.65 inference(cnf_transformation,[status(thm)],[f1043])).
% 15.64/3.65 thf(f1043,plain,(
% 15.64/3.65 ($true = (!! @ nat @ (^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: ((ord_less_eq_nat @ Y1 @ Y0) => (ord_less_eq_nat @ Y1 @ (suc @ Y0))))))))),
% 15.64/3.65 inference(fool_elimination,[status(thm)],[f1042])).
% 15.64/3.65 thf(f1042,plain,(
% 15.64/3.65 ! [X0 : nat,X1 : nat] : ((ord_less_eq_nat @ X0 @ X1) => (ord_less_eq_nat @ X0 @ (suc @ X1)))),
% 15.64/3.65 inference(rectify,[status(thm)],[f163])).
% 15.64/3.65 thf(f163,axiom,(
% 15.64/3.65 ! [X14 : nat,X7 : nat] : ((ord_less_eq_nat @ X14 @ X7) => (ord_less_eq_nat @ X14 @ (suc @ X7)))),
% 15.64/3.65 file('/export/starexec/sandbox/tmp/tmp.sXLfLlVcSw/DTF2THF_9318.p',fact_162_le__SucI)).
% 15.64/3.65 thf(f3055,plain,(
% 15.64/3.65 ~spl0_2),
% 15.64/3.65 inference(avatar_contradiction_clause,[status(thm)],[f3054])).
% 15.64/3.65 thf(f3054,plain,(
% 15.64/3.65 $false | ~spl0_2),
% 15.64/3.65 inference(trivial_inequality_removal,[status(thm)],[f3053])).
% 15.64/3.65 thf(f3053,plain,(
% 15.64/3.65 ($false = $true) | ~spl0_2),
% 15.64/3.65 inference(forward_demodulation,[status(thm)],[f3052,f2983])).
% 15.64/3.65 thf(f2983,plain,(
% 15.64/3.65 ($false = (ord_less_eq_nat @ m2 @ sK2))),
% 15.64/3.65 inference(superposition,[status(thm)],[f1690,f2971])).
% 15.64/3.65 thf(f1690,plain,(
% 15.64/3.65 ( ! [X1 : nat] : (($false = (ord_less_eq_nat @ (suc @ X1) @ X1))) )),
% 15.64/3.65 inference(not_proxy_clausification,[status(thm)],[f1689])).
% 15.64/3.65 thf(f1689,plain,(
% 15.64/3.65 ( ! [X1 : nat] : (($true = (~ (ord_less_eq_nat @ (suc @ X1) @ X1)))) )),
% 15.64/3.65 inference(beta_eta_normalization,[status(thm)],[f1688])).
% 15.64/3.65 thf(f1688,plain,(
% 15.64/3.65 ( ! [X1 : nat] : ((((^[Y0 : nat]: (~ (ord_less_eq_nat @ (suc @ Y0) @ Y0))) @ X1) = $true)) )),
% 15.64/3.65 inference(pi_clausification,[status(thm)],[f1621])).
% 15.64/3.65 thf(f1621,plain,(
% 15.64/3.65 ($true = (!! @ nat @ (^[Y0 : nat]: (~ (ord_less_eq_nat @ (suc @ Y0) @ Y0)))))),
% 15.64/3.65 inference(beta_eta_normalization,[status(thm)],[f1562])).
% 15.64/3.65 thf(f1562,plain,(
% 15.64/3.65 ((!! @ nat @ (^[Y0 : nat]: (~ ((^[Y1 : nat]: (ord_less_eq_nat @ (suc @ Y1))) @ Y0 @ Y0)))) = $true)),
% 15.64/3.65 inference(definition_unfolding,[status(thm)],[f1480,f1112])).
% 15.64/3.65 thf(f1480,plain,(
% 15.64/3.65 ($true = (!! @ nat @ (^[Y0 : nat]: (~ (ord_less_nat @ Y0 @ Y0)))))),
% 15.64/3.65 inference(cnf_transformation,[status(thm)],[f1013])).
% 15.64/3.65 thf(f1013,plain,(
% 15.64/3.65 ($true = (!! @ nat @ (^[Y0 : nat]: (~ (ord_less_nat @ Y0 @ Y0)))))),
% 15.64/3.65 inference(fool_elimination,[status(thm)],[f1012])).
% 15.64/3.65 thf(f1012,plain,(
% 15.64/3.65 ! [X0 : nat] : ~(ord_less_nat @ X0 @ X0)),
% 15.64/3.65 inference(rectify,[status(thm)],[f147])).
% 15.64/3.65 thf(f147,axiom,(
% 15.64/3.65 ! [X7 : nat] : ~(ord_less_nat @ X7 @ X7)),
% 15.64/3.65 file('/export/starexec/sandbox/tmp/tmp.sXLfLlVcSw/DTF2THF_9318.p',fact_146_less__not__refl)).
% 15.64/3.65 thf(f3052,plain,(
% 15.64/3.65 ((ord_less_eq_nat @ m2 @ sK2) = $true) | ~spl0_2),
% 15.64/3.65 inference(forward_demodulation,[status(thm)],[f3038,f1717])).
% 15.64/3.65 thf(f3038,plain,(
% 15.64/3.65 ((ord_less_eq_nat @ m2 @ (minus_minus_nat @ sK2 @ zero_zero_nat)) = $true) | ~spl0_2),
% 15.64/3.65 inference(backward_demodulation,[status(thm)],[f2529,f3025])).
% 15.64/3.65 thf(f2529,plain,(
% 15.64/3.65 ((ord_less_eq_nat @ m2 @ (minus_minus_nat @ m2 @ (suc @ zero_zero_nat))) = $true) | ~spl0_2),
% 15.64/3.65 inference(backward_demodulation,[status(thm)],[f1146,f2522])).
% 15.64/3.65 thf(f2522,plain,(
% 15.64/3.65 (m2 = xa) | ~spl0_2),
% 15.64/3.65 inference(equality_proxy_clausification,[status(thm)],[f2512])).
% 15.64/3.65 thf(f2512,plain,(
% 15.64/3.65 ($true = (xa = m2)) | ~spl0_2),
% 15.64/3.65 inference(boolean_simplification,[status(thm)],[f2511])).
% 15.64/3.65 thf(f2511,plain,(
% 15.64/3.65 (($true => (xa = m2)) = $true) | ~spl0_2),
% 15.64/3.65 inference(boolean_simplification,[status(thm)],[f2510])).
% 15.64/3.65 thf(f2510,plain,(
% 15.64/3.65 ($true = (((g @ m2) = (g @ m2)) => (xa = m2))) | ~spl0_2),
% 15.64/3.65 inference(boolean_simplification,[status(thm)],[f2507])).
% 15.64/3.65 thf(f2507,plain,(
% 15.64/3.65 (($true => (((g @ m2) = (g @ m2)) => (xa = m2))) = $true) | ~spl0_2),
% 15.64/3.65 inference(superposition,[status(thm)],[f2051,f1756])).
% 15.64/3.65 thf(f1756,plain,(
% 15.64/3.65 ( ! [X0 : nat] : (((ord_less_eq_nat @ X0 @ X0) = $true)) )),
% 15.64/3.65 inference(superposition,[status(thm)],[f1753,f1717])).
% 15.64/3.65 thf(f1753,plain,(
% 15.64/3.65 ( ! [X2 : nat,X1 : nat] : (($true = (ord_less_eq_nat @ (minus_minus_nat @ X2 @ X1) @ X2))) )),
% 15.64/3.65 inference(beta_eta_normalization,[status(thm)],[f1752])).
% 15.64/3.65 thf(f1752,plain,(
% 15.64/3.65 ( ! [X2 : nat,X1 : nat] : (($true = ((^[Y0 : nat]: (ord_less_eq_nat @ (minus_minus_nat @ Y0 @ X1) @ Y0)) @ X2))) )),
% 15.64/3.65 inference(pi_clausification,[status(thm)],[f1751])).
% 15.64/3.65 thf(f1751,plain,(
% 15.64/3.65 ( ! [X1 : nat] : (($true = (!! @ nat @ (^[Y0 : nat]: (ord_less_eq_nat @ (minus_minus_nat @ Y0 @ X1) @ Y0))))) )),
% 15.64/3.65 inference(beta_eta_normalization,[status(thm)],[f1750])).
% 15.64/3.65 thf(f1750,plain,(
% 15.64/3.65 ( ! [X1 : nat] : ((((^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: (ord_less_eq_nat @ (minus_minus_nat @ Y1 @ Y0) @ Y1)))) @ X1) = $true)) )),
% 15.64/3.65 inference(pi_clausification,[status(thm)],[f1467])).
% 15.64/3.65 thf(f1467,plain,(
% 15.64/3.65 ((!! @ nat @ (^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: (ord_less_eq_nat @ (minus_minus_nat @ Y1 @ Y0) @ Y1))))) = $true)),
% 15.64/3.65 inference(cnf_transformation,[status(thm)],[f927])).
% 15.64/3.65 thf(f927,plain,(
% 15.64/3.65 ((!! @ nat @ (^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: (ord_less_eq_nat @ (minus_minus_nat @ Y1 @ Y0) @ Y1))))) = $true)),
% 15.64/3.65 inference(fool_elimination,[status(thm)],[f926])).
% 15.64/3.65 thf(f926,plain,(
% 15.64/3.65 ! [X0 : nat,X1 : nat] : (ord_less_eq_nat @ (minus_minus_nat @ X0 @ X1) @ X0)),
% 15.64/3.65 inference(rectify,[status(thm)],[f215])).
% 15.64/3.65 thf(f215,axiom,(
% 15.64/3.65 ! [X14 : nat,X7 : nat] : (ord_less_eq_nat @ (minus_minus_nat @ X14 @ X7) @ X14)),
% 15.64/3.65 file('/export/starexec/sandbox/tmp/tmp.sXLfLlVcSw/DTF2THF_9318.p',fact_214_diff__le__self)).
% 15.64/3.65 thf(f2051,plain,(
% 15.64/3.65 ( ! [X0 : nat] : (($true = ((ord_less_eq_nat @ X0 @ m2) => (((g @ X0) = (g @ m2)) => (xa = X0))))) ) | ~spl0_2),
% 15.64/3.65 inference(avatar_component_clause,[status(thm)],[f2050])).
% 15.64/3.65 thf(f2050,definition,(
% 15.64/3.65 spl0_2 <=> ! [X0 : nat] : ($true = ((ord_less_eq_nat @ X0 @ m2) => (((g @ X0) = (g @ m2)) => (xa = X0))))),
% 15.64/3.65 introduced(definition,[new_symbols(naming,[spl0_2])],[avatar_definition])).
% 15.64/3.65 thf(f2052,plain,(
% 15.64/3.65 spl0_1 | spl0_2),
% 15.64/3.65 inference(avatar_split_clause,[status(thm)],[f2023,f2050,f2046])).
% 15.64/3.65 thf(f2023,plain,(
% 15.64/3.65 ( ! [X0 : nat] : (($true = ((ord_less_eq_nat @ X0 @ m2) => (((g @ X0) = (g @ m2)) => (xa = X0)))) | ($false = (ord_less_eq_nat @ xa @ m2))) )),
% 15.64/3.65 inference(superposition,[status(thm)],[f1709,f1655])).
% 15.64/3.65 thf(f1655,plain,(
% 15.64/3.65 ((g @ m2) = (g @ xa))),
% 15.64/3.65 inference(equality_proxy_clausification,[status(thm)],[f1650])).
% 15.64/3.65 thf(f1650,plain,(
% 15.64/3.65 (((g @ m2) = (g @ xa)) = $true)),
% 15.64/3.65 inference(boolean_simplification,[status(thm)],[f1648])).
% 15.64/3.65 thf(f1648,plain,(
% 15.64/3.65 (($false | ((g @ m2) = (g @ xa))) = $true)),
% 15.64/3.65 inference(backward_demodulation,[status(thm)],[f1513,f1647])).
% 15.64/3.65 thf(f1647,plain,(
% 15.64/3.65 ((member_a @ (g @ xa) @ h1) = $false)),
% 15.64/3.65 inference(not_proxy_clausification,[status(thm)],[f1266])).
% 15.64/3.65 thf(f1266,plain,(
% 15.64/3.65 ($true = (~ (member_a @ (g @ xa) @ h1)))),
% 15.64/3.65 inference(cnf_transformation,[status(thm)],[f1001])).
% 15.64/3.65 thf(f1001,plain,(
% 15.64/3.65 ($true = (~ (member_a @ (g @ xa) @ h1)))),
% 15.64/3.65 inference(fool_elimination,[status(thm)],[f1000])).
% 15.64/3.65 thf(f1000,plain,(
% 15.64/3.65 ~(member_a @ (g @ xa) @ h1)),
% 15.64/3.65 inference(rectify,[status(thm)],[f372])).
% 15.64/3.65 thf(f372,negated_conjecture,(
% 15.64/3.65 ~(member_a @ (g @ xa) @ h1)),
% 15.64/3.65 inference(negated_conjecture,[status(cth)],[f371])).
% 15.64/3.65 thf(f371,conjecture,(
% 15.64/3.65 (member_a @ (g @ xa) @ h1)),
% 15.64/3.65 file('/export/starexec/sandbox/tmp/tmp.sXLfLlVcSw/DTF2THF_9318.p',conj_21)).
% 15.64/3.65 thf(f1513,plain,(
% 15.64/3.65 ($true = ((member_a @ (g @ xa) @ h1) | ((g @ m2) = (g @ xa))))),
% 15.64/3.65 inference(definition_unfolding,[status(thm)],[f1247,f1259])).
% 15.64/3.65 thf(f1259,plain,(
% 15.64/3.65 (h2 = (g @ m2))),
% 15.64/3.65 inference(cnf_transformation,[status(thm)],[f744])).
% 15.64/3.65 thf(f744,plain,(
% 15.64/3.65 (h2 = (g @ m2))),
% 15.64/3.65 inference(fool_elimination,[status(thm)],[f353])).
% 15.64/3.65 thf(f353,axiom,(
% 15.64/3.65 (h2 = (g @ m2))),
% 15.64/3.65 file('/export/starexec/sandbox/tmp/tmp.sXLfLlVcSw/DTF2THF_9318.p',conj_3)).
% 15.64/3.65 thf(f1247,plain,(
% 15.64/3.65 ($true = ((member_a @ (g @ xa) @ h1) | (h2 = (g @ xa))))),
% 15.64/3.65 inference(cnf_transformation,[status(thm)],[f713])).
% 15.64/3.65 thf(f713,plain,(
% 15.64/3.65 ($true = ((member_a @ (g @ xa) @ h1) | (h2 = (g @ xa))))),
% 15.64/3.65 inference(fool_elimination,[status(thm)],[f712])).
% 15.64/3.65 thf(f712,plain,(
% 15.64/3.65 (h2 = (g @ xa)) | (member_a @ (g @ xa) @ h1)),
% 15.64/3.65 inference(rectify,[status(thm)],[f370])).
% 15.64/3.65 thf(f370,axiom,(
% 15.64/3.65 (h2 = (g @ xa)) | (member_a @ (g @ xa) @ h1)),
% 15.64/3.65 file('/export/starexec/sandbox/tmp/tmp.sXLfLlVcSw/DTF2THF_9318.p',conj_20)).
% 15.64/3.65 thf(f1709,plain,(
% 15.64/3.65 ( ! [X2 : nat,X1 : nat] : (($true = ((ord_less_eq_nat @ X2 @ m2) => (((g @ X2) = (g @ X1)) => (X1 = X2)))) | ($false = (ord_less_eq_nat @ X1 @ m2))) )),
% 15.64/3.65 inference(beta_eta_normalization,[status(thm)],[f1705])).
% 15.64/3.65 thf(f1705,plain,(
% 15.64/3.65 ( ! [X2 : nat,X1 : nat] : ((((^[Y0 : nat]: ((ord_less_eq_nat @ Y0 @ m2) => (((g @ Y0) = (g @ X1)) => (X1 = Y0)))) @ X2) = $true) | ($false = (ord_less_eq_nat @ X1 @ m2))) )),
% 15.64/3.65 inference(pi_clausification,[status(thm)],[f1670])).
% 15.64/3.65 thf(f1670,plain,(
% 15.64/3.65 ( ! [X1 : nat] : (((!! @ nat @ (^[Y0 : nat]: ((ord_less_eq_nat @ Y0 @ m2) => (((g @ Y0) = (g @ X1)) => (X1 = Y0))))) = $true) | ($false = (ord_less_eq_nat @ X1 @ m2))) )),
% 15.64/3.65 inference(binary_proxy_clausification,[status(thm)],[f1666])).
% 15.64/3.65 thf(f1666,plain,(
% 15.64/3.65 ( ! [X1 : nat] : (($true = ((ord_less_eq_nat @ X1 @ m2) => (!! @ nat @ (^[Y0 : nat]: ((ord_less_eq_nat @ Y0 @ m2) => (((g @ Y0) = (g @ X1)) => (X1 = Y0)))))))) )),
% 15.64/3.65 inference(beta_eta_normalization,[status(thm)],[f1665])).
% 15.64/3.65 thf(f1665,plain,(
% 15.64/3.65 ( ! [X1 : nat] : (($true = ((^[Y0 : nat]: ((ord_less_eq_nat @ Y0 @ m2) => (!! @ nat @ (^[Y1 : nat]: ((ord_less_eq_nat @ Y1 @ m2) => (((g @ Y1) = (g @ Y0)) => (Y0 = Y1))))))) @ X1))) )),
% 15.64/3.65 inference(pi_clausification,[status(thm)],[f1268])).
% 15.64/3.65 thf(f1268,plain,(
% 15.64/3.65 ($true = (!! @ nat @ (^[Y0 : nat]: ((ord_less_eq_nat @ Y0 @ m2) => (!! @ nat @ (^[Y1 : nat]: ((ord_less_eq_nat @ Y1 @ m2) => (((g @ Y1) = (g @ Y0)) => (Y0 = Y1)))))))))),
% 15.64/3.65 inference(cnf_transformation,[status(thm)],[f603])).
% 15.64/3.65 thf(f603,plain,(
% 15.64/3.65 ($true = (!! @ nat @ (^[Y0 : nat]: ((ord_less_eq_nat @ Y0 @ m2) => (!! @ nat @ (^[Y1 : nat]: ((ord_less_eq_nat @ Y1 @ m2) => (((g @ Y1) = (g @ Y0)) => (Y0 = Y1)))))))))),
% 15.64/3.65 inference(fool_elimination,[status(thm)],[f602])).
% 15.64/3.65 thf(f602,plain,(
% 15.64/3.65 ! [X0 : nat] : ((ord_less_eq_nat @ X0 @ m2) => ! [X1 : nat] : ((ord_less_eq_nat @ X1 @ m2) => (((g @ X1) = (g @ X0)) => (X0 = X1))))),
% 15.64/3.65 inference(rectify,[status(thm)],[f351])).
% 15.64/3.65 thf(f351,axiom,(
% 15.64/3.65 ! [X56 : nat] : ((ord_less_eq_nat @ X56 @ m2) => ! [X34 : nat] : ((ord_less_eq_nat @ X34 @ m2) => (((g @ X56) = (g @ X34)) => (X34 = X56))))),
% 15.64/3.65 file('/export/starexec/sandbox/tmp/tmp.sXLfLlVcSw/DTF2THF_9318.p',conj_1)).
% 15.64/3.65 % SZS output end Proof for DTF2THF_9318
% 15.64/3.65 % (9836)------------------------------
% 15.64/3.65 % (9836)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.64/3.65 % (9836)Termination reason: Refutation
% 15.64/3.65
% 15.64/3.65 % (9836)Memory used [KB]: 8571
% 15.64/3.65 % (9836)Time elapsed: 0.175 s
% 15.64/3.65 % (9836)Instructions burned: 597 (million)
% 15.64/3.65 % (9836)------------------------------
% 15.64/3.65 % (9836)------------------------------
% 15.64/3.65 % (9640)Success in time 1.931 s
%------------------------------------------------------------------------------