%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP133^1 : 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 : n031.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:48 AM UTC 2026
% Result : Theorem 2.60s 1.70s
% Output : Refutation 2.60s
% Verified :
% SZS Type : Refutation
% Derivation depth : 15
% Number of leaves : 17
% Syntax : Number of formulae : 97 ( 7 unt; 0 typ; 8 def)
% Number of atoms : 418 ( 229 equ; 0 cnn)
% Maximal formula atoms : 10 ( 4 avg)
% Number of connectives : 831 ( 135 ~; 129 |; 44 &; 487 @)
% ( 12 <=>; 23 =>; 0 <=; 1 <~>)
% Maximal formula depth : 11 ( 5 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 29 ( 29 >; 0 *; 0 +; 0 <<)
% Number of symbols : 119 ( 116 usr; 15 con; 0-6 aty)
% Number of variables : 158 ( 49 ^; 100 !; 9 ?; 158 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
nat: $tType ).
thf(type_def_7,type,
set_nat: $tType ).
thf(type_def_8,type,
set_set_nat: $tType ).
thf(func_def_0,type,
set_set_nat: $tType ).
thf(func_def_1,type,
set_nat: $tType ).
thf(func_def_2,type,
nat: $tType ).
thf(func_def_3,type,
one_one_nat: nat ).
thf(func_def_4,type,
times_times_nat: nat > nat > nat ).
thf(func_def_5,type,
zero_zero_nat: nat ).
thf(func_def_6,type,
groups1842438620at_nat: ( nat > nat ) > set_nat > nat ).
thf(func_def_7,type,
number1551313001itions: ( nat > nat ) > nat > $o ).
thf(func_def_8,type,
ord_less_nat: nat > nat > $o ).
thf(func_def_9,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(func_def_10,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(func_def_11,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_12,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(func_def_13,type,
ord_le1613022364et_nat: set_set_nat > set_set_nat > $o ).
thf(func_def_14,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(func_def_15,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(func_def_16,type,
set_ord_atMost_nat: nat > set_nat ).
thf(func_def_17,type,
set_or1086813439et_nat: set_nat > set_set_nat ).
thf(func_def_18,type,
member_nat: nat > set_nat > $o ).
thf(func_def_19,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(func_def_20,type,
p: nat > nat ).
thf(func_def_35,type,
sP0: nat > nat > nat > $o ).
thf(func_def_36,type,
sP1: nat > nat > nat > $o ).
thf(func_def_37,type,
sP2: nat > nat > nat > $o ).
thf(func_def_38,type,
sP3: ( nat > nat ) > ( nat > nat ) > set_nat > $o ).
thf(func_def_39,type,
sP4: ( nat > nat ) > set_nat > set_nat > $o ).
thf(func_def_40,type,
sK5: nat > ( nat > $o ) > nat ).
thf(func_def_41,type,
sK6: ( nat > $o ) > nat ).
thf(func_def_42,type,
sK7: ( nat > nat ) > nat ).
thf(func_def_43,type,
sK8: ( nat > nat ) > nat ).
thf(func_def_44,type,
sK9: ( nat > nat ) > nat ).
thf(func_def_45,type,
sK10: ( nat > nat ) > nat ).
thf(func_def_46,type,
sK11: ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_47,type,
sK12: ( set_nat > set_nat ) > set_nat ).
thf(func_def_48,type,
sK13: ( set_nat > set_nat ) > set_nat ).
thf(func_def_49,type,
sK14: ( nat > nat ) > set_nat > nat ).
thf(func_def_50,type,
sK15: ( nat > nat ) > nat ).
thf(func_def_51,type,
sK16: ( nat > nat ) > nat ).
thf(func_def_52,type,
sK17: ( nat > nat ) > nat ).
thf(func_def_53,type,
sK18: ( nat > nat ) > nat ).
thf(func_def_54,type,
sK19: ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_55,type,
sK20: ( nat > set_nat ) > nat ).
thf(func_def_56,type,
sK21: ( nat > set_nat ) > nat ).
thf(func_def_57,type,
sK22: ( nat > nat ) > nat ).
thf(func_def_58,type,
sK23: ( nat > nat ) > nat ).
thf(func_def_59,type,
sK24: ( nat > nat ) > set_nat > ( nat > nat ) > set_nat > ( nat > nat ) > nat ).
thf(func_def_60,type,
sK25: set_nat > ( nat > nat ) > set_nat > nat ).
thf(func_def_61,type,
sK26: nat > set_nat > ( nat > nat ) > set_nat > nat ).
thf(func_def_62,type,
sK27: ( nat > nat ) > nat ).
thf(func_def_63,type,
sK28: ( nat > nat ) > nat ).
thf(func_def_64,type,
sK29: ( set_nat > nat ) > set_nat ).
thf(func_def_65,type,
sK30: ( set_nat > nat ) > set_nat ).
thf(func_def_66,type,
sK31: ( set_nat > set_nat ) > set_nat ).
thf(func_def_67,type,
sK32: ( set_nat > set_nat ) > set_nat ).
thf(func_def_68,type,
sK33: set_nat > ( nat > nat ) > nat ).
thf(func_def_69,type,
sK34: ( nat > nat ) > nat ).
thf(func_def_70,type,
sK35: ( nat > nat ) > nat ).
thf(func_def_71,type,
sK36: ( set_nat > nat ) > set_nat ).
thf(func_def_72,type,
sK37: ( set_nat > nat ) > set_nat ).
thf(func_def_73,type,
sK38: ( nat > nat > $o ) > nat ).
thf(func_def_74,type,
sK39: ( nat > nat > $o ) > nat ).
thf(func_def_75,type,
sK40: ( nat > nat > $o ) > nat ).
thf(func_def_76,type,
sK41: ( nat > nat > $o ) > nat ).
thf(func_def_77,type,
sK42: ( nat > nat > $o ) > nat ).
thf(func_def_78,type,
sK43: ( nat > $o ) > nat ).
thf(func_def_79,type,
sK44: ( set_nat > set_nat ) > set_nat ).
thf(func_def_80,type,
sK45: ( set_nat > set_nat ) > set_nat ).
thf(func_def_81,type,
sK46: ( nat > nat ) > set_nat > nat ).
thf(func_def_82,type,
sK47: ( set_nat > nat ) > set_nat ).
thf(func_def_83,type,
sK48: ( set_nat > nat ) > set_nat ).
thf(func_def_84,type,
sK49: ( nat > set_nat ) > nat ).
thf(func_def_85,type,
sK50: ( nat > set_nat ) > nat ).
thf(func_def_86,type,
sK51: set_nat > ( nat > $o ) > nat ).
thf(func_def_87,type,
sK52: ( set_nat > set_nat ) > set_nat ).
thf(func_def_88,type,
sK53: ( set_nat > set_nat ) > set_nat ).
thf(func_def_89,type,
sK54: ( set_nat > nat ) > set_nat ).
thf(func_def_90,type,
sK55: ( set_nat > nat ) > set_nat ).
thf(func_def_91,type,
sK56: ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_92,type,
sK57: nat > nat ).
thf(func_def_93,type,
sK58: nat > ( nat > $o ) > nat ).
thf(func_def_94,type,
sK59: ( nat > $o ) > nat ).
thf(func_def_95,type,
sK60: set_nat > set_nat > nat ).
thf(func_def_96,type,
sK61: ( nat > nat ) > ( nat > nat ) > ( nat > nat ) > set_nat > ( nat > nat ) > set_nat > nat ).
thf(func_def_97,type,
sK62: ( nat > nat ) > set_nat > ( nat > nat ) > set_nat > nat ).
thf(func_def_98,type,
sK63: ( nat > nat ) > nat ).
thf(func_def_99,type,
sK64: ( nat > nat ) > nat ).
thf(func_def_100,type,
sK65: ( nat > set_nat ) > nat ).
thf(func_def_101,type,
sK66: ( nat > set_nat ) > nat ).
thf(func_def_102,type,
sK67: ( nat > nat ) > nat ).
thf(func_def_103,type,
sK68: ( nat > nat ) > nat ).
thf(func_def_104,type,
sK69: nat ).
thf(func_def_105,type,
sK70: ( nat > nat ) > ( nat > nat ) > set_nat > nat ).
thf(func_def_106,type,
sK71: ( nat > nat ) > nat > nat ).
thf(func_def_107,type,
sK72: ( nat > nat > $o ) > nat ).
thf(func_def_108,type,
sK73: ( nat > nat > $o ) > nat ).
thf(func_def_109,type,
sK74: ( nat > nat > $o ) > nat ).
thf(func_def_110,type,
sK75: ( nat > nat > $o ) > nat ).
thf(func_def_111,type,
sK76: ( nat > nat ) > set_nat > set_nat > nat ).
thf(func_def_112,type,
sK77: ( nat > nat ) > ( nat > nat ) > set_nat > nat ).
thf(func_def_113,type,
sK78: set_nat > set_nat > ( nat > nat ) > nat ).
thf(func_def_114,type,
sK79: set_nat > ( nat > nat ) > ( nat > nat ) > ( nat > nat ) > nat ).
thf(func_def_115,type,
sK80: set_nat > ( nat > nat ) > ( nat > nat ) > nat ).
thf(func_def_116,type,
sK81: ( nat > nat ) > ( nat > nat ) > set_nat > nat ).
thf(func_def_117,type,
sK82: ( nat > set_nat ) > nat ).
thf(func_def_118,type,
sK83: ( nat > set_nat ) > nat ).
thf(func_def_119,type,
sK84: set_nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_120,type,
sK85: ( nat > nat ) > set_nat > nat ).
thf(func_def_121,type,
sK86: ( nat > $o ) > nat ).
thf(func_def_123,type,
ph88:
!>[X0: $tType] : X0 ).
thf(func_def_124,type,
sK89: nat ).
thf(func_def_125,type,
sK90: nat ).
thf(f4129,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f1730,f1744,f1748,f2008,f2628,f3128,f3682,f3896,f4126]) ).
thf(f4126,plain,
( ~ spl87_3
| spl87_24 ),
inference(avatar_contradiction_clause,[status(thm)],[f4125]) ).
thf(f4125,plain,
( $false
| ~ spl87_3
| spl87_24 ),
inference(trivial_inequality_removal,[status(thm)],[f4117]) ).
thf(f4117,plain,
( ( zero_zero_nat != zero_zero_nat )
| ~ spl87_3
| spl87_24 ),
inference(superposition,[status(thm)],[f2281,f3182]) ).
thf(f3182,plain,
( ! [X0: set_nat] :
( zero_zero_nat
= ( groups1842438620at_nat @ p @ X0 ) )
| ~ spl87_3 ),
inference(trivial_inequality_removal,[status(thm)],[f3152]) ).
thf(f3152,plain,
( ! [X0: set_nat] :
( ( zero_zero_nat
= ( groups1842438620at_nat @ p @ X0 ) )
| ( zero_zero_nat != zero_zero_nat ) )
| ~ spl87_3 ),
inference(superposition,[status(thm)],[f1337,f3131]) ).
thf(f3131,plain,
( ! [X1: nat] :
( zero_zero_nat
= ( p @ X1 ) )
| ~ spl87_3 ),
inference(beta_eta_normalization,[status(thm)],[f3129]) ).
thf(f3129,plain,
( ! [X1: nat] :
( ( p @ X1 )
= ( ^ [Y0: nat] : zero_zero_nat
@ X1 ) )
| ~ spl87_3 ),
inference(argument_congruence,[status(thm)],[f1728]) ).
thf(f1728,plain,
( ( p
= ( ^ [Y0: nat] : zero_zero_nat ) )
| ~ spl87_3 ),
inference(avatar_component_clause,[status(thm)],[f1727]) ).
thf(f1727,definition,
( spl87_3
<=> ( p
= ( ^ [Y0: nat] : zero_zero_nat ) ) ),
introduced(definition,[new_symbols(naming,[spl87_3])],[avatar_definition]) ).
thf(f1337,plain,
! [X0: nat > nat,X1: set_nat] :
( ( zero_zero_nat
!= ( X0 @ ( sK33 @ X1 @ X0 ) ) )
| ( zero_zero_nat
= ( groups1842438620at_nat @ X0 @ X1 ) ) ),
inference(cnf_transformation,[status(thm)],[f1092]) ).
thf(f1092,plain,
! [X0: nat > nat,X1: set_nat] :
( ( zero_zero_nat
= ( groups1842438620at_nat @ X0 @ X1 ) )
| ( ( ( member_nat @ ( sK33 @ X1 @ X0 ) @ X1 )
= $true )
& ( zero_zero_nat
!= ( X0 @ ( sK33 @ X1 @ X0 ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK33])],[f985,f1091]) ).
thf(f1091,plain,
! [X0: nat > nat,X1: set_nat] :
( ? [X2: nat] :
( ( ( member_nat @ X2 @ X1 )
= $true )
& ( zero_zero_nat
!= ( X0 @ X2 ) ) )
=> ( ( ( member_nat @ ( sK33 @ X1 @ X0 ) @ X1 )
= $true )
& ( zero_zero_nat
!= ( X0 @ ( sK33 @ X1 @ X0 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f985,plain,
! [X0: nat > nat,X1: set_nat] :
( ( zero_zero_nat
= ( groups1842438620at_nat @ X0 @ X1 ) )
| ? [X2: nat] :
( ( ( member_nat @ X2 @ X1 )
= $true )
& ( zero_zero_nat
!= ( X0 @ X2 ) ) ) ),
inference(ennf_transformation,[status(thm)],[f477]) ).
thf(f477,plain,
! [X0: nat > nat,X1: set_nat] :
( ( zero_zero_nat
!= ( groups1842438620at_nat @ X0 @ X1 ) )
=> ~ ! [X2: nat] :
( ( ( member_nat @ X2 @ X1 )
= $true )
=> ( zero_zero_nat
= ( X0 @ X2 ) ) ) ),
inference(fool_elimination,[status(thm)],[f476]) ).
thf(f476,plain,
! [X0: nat > nat,X1: set_nat] :
( ( zero_zero_nat
!= ( groups1842438620at_nat @ X0 @ X1 ) )
=> ~ ! [X2: nat] :
( ( member_nat @ X2 @ X1 )
=> ( zero_zero_nat
= ( X0 @ X2 ) ) ) ),
inference(rectify,[status(thm)],[f41]) ).
thf(f41,axiom,
! [X28: nat > nat,X0: set_nat] :
( ( zero_zero_nat
!= ( groups1842438620at_nat @ X28 @ X0 ) )
=> ~ ! [X29: nat] :
( ( member_nat @ X29 @ X0 )
=> ( zero_zero_nat
= ( X28 @ X29 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.53O3If2WxG/DTF2THF_29954.p',fact_40_sum_Onot__neutral__contains__not__neutral) ).
thf(f2281,plain,
( ( zero_zero_nat
!= ( groups1842438620at_nat @ p @ ( set_ord_atMost_nat @ zero_zero_nat ) ) )
| spl87_24 ),
inference(avatar_component_clause,[status(thm)],[f2279]) ).
thf(f2279,definition,
( spl87_24
<=> ( zero_zero_nat
= ( groups1842438620at_nat @ p @ ( set_ord_atMost_nat @ zero_zero_nat ) ) ) ),
introduced(definition,[new_symbols(naming,[spl87_24])],[avatar_definition]) ).
thf(f3896,plain,
( ~ spl87_24
| spl87_1
| ~ spl87_3 ),
inference(avatar_split_clause,[status(thm)],[f3836,f1727,f1719,f2279]) ).
thf(f1719,definition,
( spl87_1
<=> ( zero_zero_nat
= ( groups1842438620at_nat
@ ^ [Y0: nat] : ( times_times_nat @ ( p @ Y0 ) @ Y0 )
@ ( set_ord_atMost_nat @ zero_zero_nat ) ) ) ),
introduced(definition,[new_symbols(naming,[spl87_1])],[avatar_definition]) ).
thf(f3836,plain,
( ( zero_zero_nat
!= ( groups1842438620at_nat @ p @ ( set_ord_atMost_nat @ zero_zero_nat ) ) )
| spl87_1
| ~ spl87_3 ),
inference(trivial_inequality_removal,[status(thm)],[f3831]) ).
thf(f3831,plain,
( ( zero_zero_nat != zero_zero_nat )
| ( zero_zero_nat
!= ( groups1842438620at_nat @ p @ ( set_ord_atMost_nat @ zero_zero_nat ) ) )
| spl87_1
| ~ spl87_3 ),
inference(superposition,[status(thm)],[f3814,f3131]) ).
thf(f3814,plain,
( ! [X0: nat > nat] :
( ( zero_zero_nat
!= ( X0
@ ( sK70
@ ^ [Y0: nat] : ( times_times_nat @ ( p @ Y0 ) @ Y0 )
@ X0
@ ( set_ord_atMost_nat @ zero_zero_nat ) ) ) )
| ( zero_zero_nat
!= ( groups1842438620at_nat @ X0 @ ( set_ord_atMost_nat @ zero_zero_nat ) ) ) )
| spl87_1
| ~ spl87_3 ),
inference(forward_demodulation,[status(thm)],[f3813,f1630]) ).
thf(f1630,plain,
! [X1: nat] :
( zero_zero_nat
= ( times_times_nat @ zero_zero_nat @ X1 ) ),
inference(equality_resolution,[status(thm)],[f1245]) ).
thf(f1245,plain,
! [X0: nat,X1: nat] :
( ( zero_zero_nat
= ( times_times_nat @ X0 @ X1 ) )
| ( zero_zero_nat != X0 ) ),
inference(cnf_transformation,[status(thm)],[f1050]) ).
thf(f1050,plain,
! [X0: nat,X1: nat] :
( ( ( zero_zero_nat
= ( times_times_nat @ X0 @ X1 ) )
| ( ( zero_zero_nat != X0 )
& ( zero_zero_nat != X1 ) ) )
& ( ( zero_zero_nat = X0 )
| ( zero_zero_nat = X1 )
| ( zero_zero_nat
!= ( times_times_nat @ X0 @ X1 ) ) ) ),
inference(flattening,[status(thm)],[f1049]) ).
thf(f1049,plain,
! [X0: nat,X1: nat] :
( ( ( zero_zero_nat
= ( times_times_nat @ X0 @ X1 ) )
| ( ( zero_zero_nat != X0 )
& ( zero_zero_nat != X1 ) ) )
& ( ( zero_zero_nat = X0 )
| ( zero_zero_nat = X1 )
| ( zero_zero_nat
!= ( times_times_nat @ X0 @ X1 ) ) ) ),
inference(nnf_transformation,[status(thm)],[f489]) ).
thf(f489,plain,
! [X0: nat,X1: nat] :
( ( zero_zero_nat
= ( times_times_nat @ X0 @ X1 ) )
<=> ( ( zero_zero_nat = X0 )
| ( zero_zero_nat = X1 ) ) ),
inference(fool_elimination,[status(thm)],[f488]) ).
thf(f488,plain,
! [X0: nat,X1: nat] :
( ( ( zero_zero_nat
= ( times_times_nat @ X0 @ X1 ) )
= ( zero_zero_nat = X0 ) )
| ( zero_zero_nat = X1 ) ),
inference(rectify,[status(thm)],[f22]) ).
thf(f22,axiom,
! [X9: nat,X3: nat] :
( ( ( zero_zero_nat
= ( times_times_nat @ X9 @ X3 ) )
= ( zero_zero_nat = X9 ) )
| ( zero_zero_nat = X3 ) ),
file('/export/starexec/sandbox/tmp/tmp.53O3If2WxG/DTF2THF_29954.p',fact_21_mult__is__0) ).
thf(f3813,plain,
( ! [X0: nat > nat] :
( ( zero_zero_nat
!= ( groups1842438620at_nat @ X0 @ ( set_ord_atMost_nat @ zero_zero_nat ) ) )
| ( ( X0
@ ( sK70
@ ^ [Y0: nat] : ( times_times_nat @ ( p @ Y0 ) @ Y0 )
@ X0
@ ( set_ord_atMost_nat @ zero_zero_nat ) ) )
!= ( times_times_nat @ zero_zero_nat
@ ( sK70
@ ^ [Y0: nat] : ( times_times_nat @ ( p @ Y0 ) @ Y0 )
@ X0
@ ( set_ord_atMost_nat @ zero_zero_nat ) ) ) ) )
| spl87_1
| ~ spl87_3 ),
inference(forward_demodulation,[status(thm)],[f3137,f3131]) ).
thf(f3137,plain,
( ! [X0: nat > nat] :
( ( ( times_times_nat
@ ( p
@ ( sK70
@ ^ [Y0: nat] : ( times_times_nat @ ( p @ Y0 ) @ Y0 )
@ X0
@ ( set_ord_atMost_nat @ zero_zero_nat ) ) )
@ ( sK70
@ ^ [Y0: nat] : ( times_times_nat @ ( p @ Y0 ) @ Y0 )
@ X0
@ ( set_ord_atMost_nat @ zero_zero_nat ) ) )
!= ( X0
@ ( sK70
@ ^ [Y0: nat] : ( times_times_nat @ ( p @ Y0 ) @ Y0 )
@ X0
@ ( set_ord_atMost_nat @ zero_zero_nat ) ) ) )
| ( zero_zero_nat
!= ( groups1842438620at_nat @ X0 @ ( set_ord_atMost_nat @ zero_zero_nat ) ) ) )
| spl87_1 ),
inference(beta_eta_normalization,[status(thm)],[f3133]) ).
thf(f3133,plain,
( ! [X0: nat > nat] :
( ( zero_zero_nat
!= ( groups1842438620at_nat @ X0 @ ( set_ord_atMost_nat @ zero_zero_nat ) ) )
| ( ( X0
@ ( sK70
@ ^ [Y0: nat] : ( times_times_nat @ ( p @ Y0 ) @ Y0 )
@ X0
@ ( set_ord_atMost_nat @ zero_zero_nat ) ) )
!= ( ^ [Y0: nat] : ( times_times_nat @ ( p @ Y0 ) @ Y0 )
@ ( sK70
@ ^ [Y0: nat] : ( times_times_nat @ ( p @ Y0 ) @ Y0 )
@ X0
@ ( set_ord_atMost_nat @ zero_zero_nat ) ) ) ) )
| spl87_1 ),
inference(superposition,[status(thm)],[f1721,f1687]) ).
thf(f1687,plain,
! [X2: nat > nat,X3: nat > nat,X1: set_nat] :
( ( ( groups1842438620at_nat @ X2 @ X1 )
= ( groups1842438620at_nat @ X3 @ X1 ) )
| ( ( X3 @ ( sK70 @ X3 @ X2 @ X1 ) )
!= ( X2 @ ( sK70 @ X3 @ X2 @ X1 ) ) ) ),
inference(equality_resolution,[status(thm)],[f1526]) ).
thf(f1526,plain,
! [X2: nat > nat,X3: nat > nat,X0: set_nat,X1: set_nat] :
( ( ( groups1842438620at_nat @ X3 @ X0 )
= ( groups1842438620at_nat @ X2 @ X1 ) )
| ( ( X2 @ ( sK70 @ X3 @ X2 @ X0 ) )
!= ( X3 @ ( sK70 @ X3 @ X2 @ X0 ) ) )
| ( X0 != X1 ) ),
inference(cnf_transformation,[status(thm)],[f1173]) ).
thf(f1173,plain,
! [X0: set_nat,X1: set_nat,X2: nat > nat,X3: nat > nat] :
( ( ( groups1842438620at_nat @ X3 @ X0 )
= ( groups1842438620at_nat @ X2 @ X1 ) )
| ( ( ( X2 @ ( sK70 @ X3 @ X2 @ X0 ) )
!= ( X3 @ ( sK70 @ X3 @ X2 @ X0 ) ) )
& ( ( member_nat @ ( sK70 @ X3 @ X2 @ X0 ) @ X0 )
= $true ) )
| ( X0 != X1 ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK70])],[f841,f1172]) ).
thf(f1172,plain,
! [X0: set_nat,X2: nat > nat,X3: nat > nat] :
( ? [X4: nat] :
( ( ( X2 @ X4 )
!= ( X3 @ X4 ) )
& ( ( member_nat @ X4 @ X0 )
= $true ) )
=> ( ( ( X2 @ ( sK70 @ X3 @ X2 @ X0 ) )
!= ( X3 @ ( sK70 @ X3 @ X2 @ X0 ) ) )
& ( ( member_nat @ ( sK70 @ X3 @ X2 @ X0 ) @ X0 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f841,plain,
! [X0: set_nat,X1: set_nat,X2: nat > nat,X3: nat > nat] :
( ( ( groups1842438620at_nat @ X3 @ X0 )
= ( groups1842438620at_nat @ X2 @ X1 ) )
| ? [X4: nat] :
( ( ( X2 @ X4 )
!= ( X3 @ X4 ) )
& ( ( member_nat @ X4 @ X0 )
= $true ) )
| ( X0 != X1 ) ),
inference(flattening,[status(thm)],[f840]) ).
thf(f840,plain,
! [X0: set_nat,X1: set_nat,X2: nat > nat,X3: nat > nat] :
( ( ( groups1842438620at_nat @ X3 @ X0 )
= ( groups1842438620at_nat @ X2 @ X1 ) )
| ? [X4: nat] :
( ( ( X2 @ X4 )
!= ( X3 @ X4 ) )
& ( ( member_nat @ X4 @ X0 )
= $true ) )
| ( X0 != X1 ) ),
inference(ennf_transformation,[status(thm)],[f623]) ).
thf(f623,plain,
! [X0: set_nat,X1: set_nat,X2: nat > nat,X3: nat > nat] :
( ( X0 = X1 )
=> ( ! [X4: nat] :
( ( ( member_nat @ X4 @ X0 )
= $true )
=> ( ( X2 @ X4 )
= ( X3 @ X4 ) ) )
=> ( ( groups1842438620at_nat @ X3 @ X0 )
= ( groups1842438620at_nat @ X2 @ X1 ) ) ) ),
inference(fool_elimination,[status(thm)],[f622]) ).
thf(f622,plain,
! [X0: set_nat,X1: set_nat,X2: nat > nat,X3: nat > nat] :
( ( X0 = X1 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ X0 )
=> ( ( X2 @ X4 )
= ( X3 @ X4 ) ) )
=> ( ( groups1842438620at_nat @ X3 @ X0 )
= ( groups1842438620at_nat @ X2 @ X1 ) ) ) ),
inference(rectify,[status(thm)],[f33]) ).
thf(f33,axiom,
! [X31: set_nat,X0: set_nat,X28: nat > nat,X27: nat > nat] :
( ( X0 = X31 )
=> ( ! [X19: nat] :
( ( member_nat @ X19 @ X31 )
=> ( ( X27 @ X19 )
= ( X28 @ X19 ) ) )
=> ( ( groups1842438620at_nat @ X28 @ X0 )
= ( groups1842438620at_nat @ X27 @ X31 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.53O3If2WxG/DTF2THF_29954.p',fact_32_sum_Ocong) ).
thf(f1721,plain,
( ( zero_zero_nat
!= ( groups1842438620at_nat
@ ^ [Y0: nat] : ( times_times_nat @ ( p @ Y0 ) @ Y0 )
@ ( set_ord_atMost_nat @ zero_zero_nat ) ) )
| spl87_1 ),
inference(avatar_component_clause,[status(thm)],[f1719]) ).
thf(f3682,plain,
~ spl87_15,
inference(avatar_contradiction_clause,[status(thm)],[f3681]) ).
thf(f3681,plain,
( $false
| ~ spl87_15 ),
inference(trivial_inequality_removal,[status(thm)],[f3602]) ).
thf(f3602,plain,
( ( one_one_nat != one_one_nat )
| ~ spl87_15 ),
inference(superposition,[status(thm)],[f1214,f1928]) ).
thf(f1928,plain,
( ( zero_zero_nat = one_one_nat )
| ~ spl87_15 ),
inference(avatar_component_clause,[status(thm)],[f1926]) ).
thf(f1926,definition,
( spl87_15
<=> ( zero_zero_nat = one_one_nat ) ),
introduced(definition,[new_symbols(naming,[spl87_15])],[avatar_definition]) ).
thf(f1214,plain,
zero_zero_nat != one_one_nat,
inference(cnf_transformation,[status(thm)],[f40]) ).
thf(f40,axiom,
zero_zero_nat != one_one_nat,
file('/export/starexec/sandbox/tmp/tmp.53O3If2WxG/DTF2THF_29954.p',fact_39_zero__neq__one) ).
thf(f3128,plain,
( ~ spl87_16
| spl87_3
| ~ spl87_7 ),
inference(avatar_split_clause,[status(thm)],[f3127,f1746,f1727,f1930]) ).
thf(f1930,definition,
( spl87_16
<=> ( zero_zero_nat
= ( p @ zero_zero_nat ) ) ),
introduced(definition,[new_symbols(naming,[spl87_16])],[avatar_definition]) ).
thf(f1746,definition,
( spl87_7
<=> ! [X1: nat] :
( ( ( ord_less_eq_nat @ X1 @ zero_zero_nat )
= $true )
| ( zero_zero_nat
= ( p @ X1 ) ) ) ),
introduced(definition,[new_symbols(naming,[spl87_7])],[avatar_definition]) ).
thf(f3127,plain,
( ( zero_zero_nat
!= ( p @ zero_zero_nat ) )
| spl87_3
| ~ spl87_7 ),
inference(superposition,[status(thm)],[f2597,f2734]) ).
thf(f2734,plain,
( ( zero_zero_nat = sK90 )
| spl87_3
| ~ spl87_7 ),
inference(trivial_inequality_removal,[status(thm)],[f2689]) ).
thf(f2689,plain,
( ( zero_zero_nat != zero_zero_nat )
| ( zero_zero_nat = sK90 )
| spl87_3
| ~ spl87_7 ),
inference(superposition,[status(thm)],[f2597,f2644]) ).
thf(f2644,plain,
( ! [X0: nat] :
( ( zero_zero_nat
= ( p @ X0 ) )
| ( zero_zero_nat = X0 ) )
| ~ spl87_7 ),
inference(trivial_inequality_removal,[status(thm)],[f2632]) ).
thf(f2632,plain,
( ! [X0: nat] :
( ( zero_zero_nat = X0 )
| ( zero_zero_nat
= ( p @ X0 ) )
| ( $true != $true ) )
| ~ spl87_7 ),
inference(superposition,[status(thm)],[f1537,f1747]) ).
thf(f1747,plain,
( ! [X1: nat] :
( ( ( ord_less_eq_nat @ X1 @ zero_zero_nat )
= $true )
| ( zero_zero_nat
= ( p @ X1 ) ) )
| ~ spl87_7 ),
inference(avatar_component_clause,[status(thm)],[f1746]) ).
thf(f1537,plain,
! [X0: nat] :
( ( ( ord_less_eq_nat @ X0 @ zero_zero_nat )
!= $true )
| ( zero_zero_nat = X0 ) ),
inference(cnf_transformation,[status(thm)],[f1176]) ).
thf(f1176,plain,
! [X0: nat] :
( ( ( ( ord_less_eq_nat @ X0 @ zero_zero_nat )
= $true )
| ( zero_zero_nat != X0 ) )
& ( ( zero_zero_nat = X0 )
| ( ( ord_less_eq_nat @ X0 @ zero_zero_nat )
!= $true ) ) ),
inference(nnf_transformation,[status(thm)],[f541]) ).
thf(f541,plain,
! [X0: nat] :
( ( ( ord_less_eq_nat @ X0 @ zero_zero_nat )
= $true )
<=> ( zero_zero_nat = X0 ) ),
inference(fool_elimination,[status(thm)],[f540]) ).
thf(f540,plain,
! [X0: nat] :
( ( ord_less_eq_nat @ X0 @ zero_zero_nat )
= ( zero_zero_nat = X0 ) ),
inference(rectify,[status(thm)],[f48]) ).
thf(f48,axiom,
! [X3: nat] :
( ( ord_less_eq_nat @ X3 @ zero_zero_nat )
= ( zero_zero_nat = X3 ) ),
file('/export/starexec/sandbox/tmp/tmp.53O3If2WxG/DTF2THF_29954.p',fact_47_le__0__eq) ).
thf(f2597,plain,
( ( zero_zero_nat
!= ( p @ sK90 ) )
| spl87_3 ),
inference(beta_eta_normalization,[status(thm)],[f2596]) ).
thf(f2596,plain,
( ( ( ^ [Y0: nat] : zero_zero_nat
@ sK90 )
!= ( p @ sK90 ) )
| spl87_3 ),
inference(negative_extensionality,[status(thm)],[f1729]) ).
thf(f1729,plain,
( ( p
!= ( ^ [Y0: nat] : zero_zero_nat ) )
| spl87_3 ),
inference(avatar_component_clause,[status(thm)],[f1727]) ).
thf(f2628,plain,
( spl87_16
| spl87_15
| ~ spl87_6 ),
inference(avatar_split_clause,[status(thm)],[f2611,f1742,f1926,f1930]) ).
thf(f1742,definition,
( spl87_6
<=> ! [X1: nat] :
( ( zero_zero_nat
= ( p @ X1 ) )
| ( ( ord_less_eq_nat @ one_one_nat @ X1 )
= $true ) ) ),
introduced(definition,[new_symbols(naming,[spl87_6])],[avatar_definition]) ).
thf(f2611,plain,
( ( zero_zero_nat
= ( p @ zero_zero_nat ) )
| ( zero_zero_nat = one_one_nat )
| ~ spl87_6 ),
inference(trivial_inequality_removal,[status(thm)],[f2607]) ).
thf(f2607,plain,
( ( zero_zero_nat
= ( p @ zero_zero_nat ) )
| ( zero_zero_nat = one_one_nat )
| ( $true != $true )
| ~ spl87_6 ),
inference(superposition,[status(thm)],[f1537,f1743]) ).
thf(f1743,plain,
( ! [X1: nat] :
( ( ( ord_less_eq_nat @ one_one_nat @ X1 )
= $true )
| ( zero_zero_nat
= ( p @ X1 ) ) )
| ~ spl87_6 ),
inference(avatar_component_clause,[status(thm)],[f1742]) ).
thf(f2008,plain,
( spl87_2
| ~ spl87_3 ),
inference(avatar_contradiction_clause,[status(thm)],[f2007]) ).
thf(f2007,plain,
( $false
| spl87_2
| ~ spl87_3 ),
inference(trivial_inequality_removal,[status(thm)],[f1962]) ).
thf(f1962,plain,
( ( zero_zero_nat != zero_zero_nat )
| spl87_2
| ~ spl87_3 ),
inference(superposition,[status(thm)],[f1725,f1939]) ).
thf(f1939,plain,
( ! [X1: nat] :
( zero_zero_nat
= ( p @ X1 ) )
| ~ spl87_3 ),
inference(beta_eta_normalization,[status(thm)],[f1937]) ).
thf(f1937,plain,
( ! [X1: nat] :
( ( p @ X1 )
= ( ^ [Y0: nat] : zero_zero_nat
@ X1 ) )
| ~ spl87_3 ),
inference(argument_congruence,[status(thm)],[f1728]) ).
thf(f1725,plain,
( ( zero_zero_nat
!= ( p @ sK69 ) )
| spl87_2 ),
inference(avatar_component_clause,[status(thm)],[f1723]) ).
thf(f1723,definition,
( spl87_2
<=> ( zero_zero_nat
= ( p @ sK69 ) ) ),
introduced(definition,[new_symbols(naming,[spl87_2])],[avatar_definition]) ).
thf(f1748,plain,
( spl87_3
| spl87_7 ),
inference(avatar_split_clause,[status(thm)],[f1521,f1746,f1727]) ).
thf(f1521,plain,
! [X1: nat] :
( ( p
= ( ^ [Y0: nat] : zero_zero_nat ) )
| ( ( ord_less_eq_nat @ X1 @ zero_zero_nat )
= $true )
| ( zero_zero_nat
= ( p @ X1 ) ) ),
inference(cnf_transformation,[status(thm)],[f1171]) ).
thf(f1171,plain,
( ( ( p
!= ( ^ [Y0: nat] : zero_zero_nat ) )
| ( zero_zero_nat
!= ( groups1842438620at_nat
@ ^ [Y0: nat] : ( times_times_nat @ ( p @ Y0 ) @ Y0 )
@ ( set_ord_atMost_nat @ zero_zero_nat ) ) )
| ( ( zero_zero_nat
!= ( p @ sK69 ) )
& ( ( ( ord_less_eq_nat @ sK69 @ zero_zero_nat )
!= $true )
| ( ( ord_less_eq_nat @ one_one_nat @ sK69 )
!= $true ) ) ) )
& ( ( p
= ( ^ [Y0: nat] : zero_zero_nat ) )
| ( ( zero_zero_nat
= ( groups1842438620at_nat
@ ^ [Y0: nat] : ( times_times_nat @ ( p @ Y0 ) @ Y0 )
@ ( set_ord_atMost_nat @ zero_zero_nat ) ) )
& ! [X1: nat] :
( ( zero_zero_nat
= ( p @ X1 ) )
| ( ( ( ord_less_eq_nat @ X1 @ zero_zero_nat )
= $true )
& ( ( ord_less_eq_nat @ one_one_nat @ X1 )
= $true ) ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK69])],[f1169,f1170]) ).
thf(f1170,plain,
( ? [X0: nat] :
( ( zero_zero_nat
!= ( p @ X0 ) )
& ( ( ( ord_less_eq_nat @ X0 @ zero_zero_nat )
!= $true )
| ( ( ord_less_eq_nat @ one_one_nat @ X0 )
!= $true ) ) )
=> ( ( zero_zero_nat
!= ( p @ sK69 ) )
& ( ( ( ord_less_eq_nat @ sK69 @ zero_zero_nat )
!= $true )
| ( ( ord_less_eq_nat @ one_one_nat @ sK69 )
!= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1169,plain,
( ( ( p
!= ( ^ [Y0: nat] : zero_zero_nat ) )
| ( zero_zero_nat
!= ( groups1842438620at_nat
@ ^ [Y0: nat] : ( times_times_nat @ ( p @ Y0 ) @ Y0 )
@ ( set_ord_atMost_nat @ zero_zero_nat ) ) )
| ? [X0: nat] :
( ( zero_zero_nat
!= ( p @ X0 ) )
& ( ( ( ord_less_eq_nat @ X0 @ zero_zero_nat )
!= $true )
| ( ( ord_less_eq_nat @ one_one_nat @ X0 )
!= $true ) ) ) )
& ( ( p
= ( ^ [Y0: nat] : zero_zero_nat ) )
| ( ( zero_zero_nat
= ( groups1842438620at_nat
@ ^ [Y0: nat] : ( times_times_nat @ ( p @ Y0 ) @ Y0 )
@ ( set_ord_atMost_nat @ zero_zero_nat ) ) )
& ! [X1: nat] :
( ( zero_zero_nat
= ( p @ X1 ) )
| ( ( ( ord_less_eq_nat @ X1 @ zero_zero_nat )
= $true )
& ( ( ord_less_eq_nat @ one_one_nat @ X1 )
= $true ) ) ) ) ) ),
inference(rectify,[status(thm)],[f1168]) ).
thf(f1168,plain,
( ( ( p
!= ( ^ [Y0: nat] : zero_zero_nat ) )
| ( zero_zero_nat
!= ( groups1842438620at_nat
@ ^ [Y0: nat] : ( times_times_nat @ ( p @ Y0 ) @ Y0 )
@ ( set_ord_atMost_nat @ zero_zero_nat ) ) )
| ? [X0: nat] :
( ( zero_zero_nat
!= ( p @ X0 ) )
& ( ( ( ord_less_eq_nat @ X0 @ zero_zero_nat )
!= $true )
| ( ( ord_less_eq_nat @ one_one_nat @ X0 )
!= $true ) ) ) )
& ( ( p
= ( ^ [Y0: nat] : zero_zero_nat ) )
| ( ( zero_zero_nat
= ( groups1842438620at_nat
@ ^ [Y0: nat] : ( times_times_nat @ ( p @ Y0 ) @ Y0 )
@ ( set_ord_atMost_nat @ zero_zero_nat ) ) )
& ! [X0: nat] :
( ( zero_zero_nat
= ( p @ X0 ) )
| ( ( ( ord_less_eq_nat @ X0 @ zero_zero_nat )
= $true )
& ( ( ord_less_eq_nat @ one_one_nat @ X0 )
= $true ) ) ) ) ) ),
inference(flattening,[status(thm)],[f1167]) ).
thf(f1167,plain,
( ( ( p
!= ( ^ [Y0: nat] : zero_zero_nat ) )
| ( zero_zero_nat
!= ( groups1842438620at_nat
@ ^ [Y0: nat] : ( times_times_nat @ ( p @ Y0 ) @ Y0 )
@ ( set_ord_atMost_nat @ zero_zero_nat ) ) )
| ? [X0: nat] :
( ( zero_zero_nat
!= ( p @ X0 ) )
& ( ( ( ord_less_eq_nat @ X0 @ zero_zero_nat )
!= $true )
| ( ( ord_less_eq_nat @ one_one_nat @ X0 )
!= $true ) ) ) )
& ( ( p
= ( ^ [Y0: nat] : zero_zero_nat ) )
| ( ( zero_zero_nat
= ( groups1842438620at_nat
@ ^ [Y0: nat] : ( times_times_nat @ ( p @ Y0 ) @ Y0 )
@ ( set_ord_atMost_nat @ zero_zero_nat ) ) )
& ! [X0: nat] :
( ( zero_zero_nat
= ( p @ X0 ) )
| ( ( ( ord_less_eq_nat @ X0 @ zero_zero_nat )
= $true )
& ( ( ord_less_eq_nat @ one_one_nat @ X0 )
= $true ) ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f989]) ).
thf(f989,plain,
( ( ( zero_zero_nat
= ( groups1842438620at_nat
@ ^ [Y0: nat] : ( times_times_nat @ ( p @ Y0 ) @ Y0 )
@ ( set_ord_atMost_nat @ zero_zero_nat ) ) )
& ! [X0: nat] :
( ( zero_zero_nat
= ( p @ X0 ) )
| ( ( ( ord_less_eq_nat @ X0 @ zero_zero_nat )
= $true )
& ( ( ord_less_eq_nat @ one_one_nat @ X0 )
= $true ) ) ) )
<~> ( p
= ( ^ [Y0: nat] : zero_zero_nat ) ) ),
inference(ennf_transformation,[status(thm)],[f767]) ).
thf(f767,plain,
~ ( ( p
= ( ^ [Y0: nat] : zero_zero_nat ) )
<=> ( ! [X0: nat] :
( ( zero_zero_nat
!= ( p @ X0 ) )
=> ( ( ( ord_less_eq_nat @ X0 @ zero_zero_nat )
= $true )
& ( ( ord_less_eq_nat @ one_one_nat @ X0 )
= $true ) ) )
& ( zero_zero_nat
= ( groups1842438620at_nat
@ ^ [Y0: nat] : ( times_times_nat @ ( p @ Y0 ) @ Y0 )
@ ( set_ord_atMost_nat @ zero_zero_nat ) ) ) ) ),
inference(rectify,[status(thm)],[f493]) ).
thf(f493,plain,
~ ( ( ( zero_zero_nat
= ( groups1842438620at_nat
@ ^ [Y0: nat] : ( times_times_nat @ ( p @ Y0 ) @ Y0 )
@ ( set_ord_atMost_nat @ zero_zero_nat ) ) )
& ! [X1: nat] :
( ( zero_zero_nat
!= ( p @ X1 ) )
=> ( ( ( ord_less_eq_nat @ X1 @ zero_zero_nat )
= $true )
& ( ( ord_less_eq_nat @ one_one_nat @ X1 )
= $true ) ) ) )
<=> ( p
= ( ^ [Y0: nat] : zero_zero_nat ) ) ),
inference(fool_elimination,[status(thm)],[f492]) ).
thf(f492,plain,
~ ( ( zero_zero_nat
= ( groups1842438620at_nat
@ ^ [X0: nat] : ( times_times_nat @ ( p @ X0 ) @ X0 )
@ ( set_ord_atMost_nat @ zero_zero_nat ) ) )
& ! [X1: nat] :
( ( ( zero_zero_nat
!= ( p @ X1 ) )
=> ( ( ord_less_eq_nat @ X1 @ zero_zero_nat )
& ( ord_less_eq_nat @ one_one_nat @ X1 ) ) )
= ( ( ^ [X2: nat] : zero_zero_nat )
= p ) ) ),
inference(rectify,[status(thm)],[f263]) ).
thf(f263,negated_conjecture,
~ ( ( zero_zero_nat
= ( groups1842438620at_nat
@ ^ [X5: nat] : ( times_times_nat @ ( p @ X5 ) @ X5 )
@ ( set_ord_atMost_nat @ zero_zero_nat ) ) )
& ! [X5: nat] :
( ( ( zero_zero_nat
!= ( p @ X5 ) )
=> ( ( ord_less_eq_nat @ X5 @ zero_zero_nat )
& ( ord_less_eq_nat @ one_one_nat @ X5 ) ) )
= ( ( ^ [X5: nat] : zero_zero_nat )
= p ) ) ),
inference(negated_conjecture,[status(cth)],[f262]) ).
thf(f262,conjecture,
( ( zero_zero_nat
= ( groups1842438620at_nat
@ ^ [X5: nat] : ( times_times_nat @ ( p @ X5 ) @ X5 )
@ ( set_ord_atMost_nat @ zero_zero_nat ) ) )
& ! [X5: nat] :
( ( ( zero_zero_nat
!= ( p @ X5 ) )
=> ( ( ord_less_eq_nat @ X5 @ zero_zero_nat )
& ( ord_less_eq_nat @ one_one_nat @ X5 ) ) )
= ( ( ^ [X5: nat] : zero_zero_nat )
= p ) ) ),
file('/export/starexec/sandbox/tmp/tmp.53O3If2WxG/DTF2THF_29954.p',conj_0) ).
thf(f1744,plain,
( spl87_6
| spl87_3 ),
inference(avatar_split_clause,[status(thm)],[f1520,f1727,f1742]) ).
thf(f1520,plain,
! [X1: nat] :
( ( zero_zero_nat
= ( p @ X1 ) )
| ( p
= ( ^ [Y0: nat] : zero_zero_nat ) )
| ( ( ord_less_eq_nat @ one_one_nat @ X1 )
= $true ) ),
inference(cnf_transformation,[status(thm)],[f1171]) ).
thf(f1730,plain,
( ~ spl87_1
| ~ spl87_2
| ~ spl87_3 ),
inference(avatar_split_clause,[status(thm)],[f1524,f1727,f1723,f1719]) ).
thf(f1524,plain,
( ( zero_zero_nat
!= ( groups1842438620at_nat
@ ^ [Y0: nat] : ( times_times_nat @ ( p @ Y0 ) @ Y0 )
@ ( set_ord_atMost_nat @ zero_zero_nat ) ) )
| ( zero_zero_nat
!= ( p @ sK69 ) )
| ( p
!= ( ^ [Y0: nat] : zero_zero_nat ) ) ),
inference(cnf_transformation,[status(thm)],[f1171]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP133^1 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n031.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
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 08:28:33 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.21/0.45 ---- Original DTF file ---
% 0.21/0.45 thf(spec,logic,$$dhol).
% 0.21/0.45 %------------------------------------------------------------------------------
% 0.21/0.45 % File : ITP133^1 : TPTP v9.2.1. Released v7.5.0.
% 0.21/0.45 % Domain : Interactive Theorem Proving
% 0.21/0.45 % Problem : Sledgehammer Number_Partition problem prob_28__5323650_1
% 0.21/0.45 % Version : Especial.
% 0.21/0.45 % English :
% 0.21/0.45
% 0.21/0.45 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.21/0.45 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.21/0.45 % Source : [Des21]
% 0.21/0.45 % Names : Number_Partition/prob_28__5323650_1 [Des21]
% 0.21/0.45
% 0.21/0.45 % Status : Theorem
% 0.21/0.45 % Rating : 0.33 v9.1.0, 0.25 v9.0.0, 0.50 v8.2.0, 0.38 v8.1.0, 0.36 v7.5.0
% 0.21/0.45 % Syntax : Number of formulae : 283 ( 117 unt; 21 typ; 0 def)
% 0.21/0.45 % Number of atoms : 777 ( 256 equ; 0 cnn)
% 0.21/0.45 % Maximal formula atoms : 12 ( 2 avg)
% 0.21/0.45 % Number of connectives : 1961 ( 82 ~; 23 |; 46 &;1469 @)
% 0.21/0.45 % ( 0 <=>; 341 =>; 0 <=; 0 <~>)
% 0.21/0.45 % Maximal formula depth : 17 ( 6 avg)
% 0.21/0.45 % Number of types : 4 ( 3 usr)
% 0.21/0.45 % Number of type conns : 107 ( 107 >; 0 *; 0 +; 0 <<)
% 0.21/0.45 % Number of symbols : 19 ( 18 usr; 2 con; 0-2 aty)
% 0.21/0.45 % Number of variables : 719 ( 97 ^; 616 !; 6 ?; 719 :)
% 0.21/0.45 % SPC : TH0_THM_EQU_NAR
% 0.21/0.45
% 0.21/0.45 % Comments : This file was generated by Sledgehammer 2021-02-23 15:41:56.283
% 0.21/0.45 %------------------------------------------------------------------------------
% 0.21/0.45 % Could-be-implicit typings (3)
% 0.21/0.45 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 0.21/0.45 set_set_nat: $tType ).
% 0.21/0.45
% 0.21/0.45 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 0.21/0.45 set_nat: $tType ).
% 0.21/0.45
% 0.21/0.45 thf(ty_n_t__Nat__Onat,type,
% 0.21/0.45 nat: $tType ).
% 0.21/0.45
% 0.21/0.45 % Explicit typings (18)
% 0.21/0.45 thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
% 0.21/0.45 one_one_nat: nat ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
% 0.21/0.45 times_times_nat: nat > nat > nat ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 0.21/0.45 zero_zero_nat: nat ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.21/0.45 groups1842438620at_nat: ( nat > nat ) > set_nat > nat ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Number__Partition__Mirabelle__zerdlymyoj_Opartitions,type,
% 0.21/0.45 number1551313001itions: ( nat > nat ) > nat > $o ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 0.21/0.45 ord_less_nat: nat > nat > $o ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.21/0.45 ord_less_set_nat: set_nat > set_nat > $o ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
% 0.21/0.45 ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 0.21/0.45 ord_less_eq_nat: nat > nat > $o ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.21/0.45 ord_less_eq_set_nat: set_nat > set_nat > $o ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 0.21/0.45 ord_le1613022364et_nat: set_set_nat > set_set_nat > $o ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
% 0.21/0.45 collect_nat: ( nat > $o ) > set_nat ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.21/0.45 collect_set_nat: ( set_nat > $o ) > set_set_nat ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
% 0.21/0.45 set_ord_atMost_nat: nat > set_nat ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.21/0.45 set_or1086813439et_nat: set_nat > set_set_nat ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_member_001t__Nat__Onat,type,
% 0.21/0.45 member_nat: nat > set_nat > $o ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.21/0.45 member_set_nat: set_nat > set_set_nat > $o ).
% 0.21/0.45
% 0.21/0.45 thf(sy_v_p,type,
% 0.21/0.45 p: nat > nat ).
% 0.21/0.45
% 0.21/0.45 % Relevant facts (261)
% 0.21/0.45 thf(fact_0_sum_Oneutral__const,axiom,
% 0.21/0.45 ! [A: set_nat] :
% 0.21/0.45 ( ( groups1842438620at_nat
% 0.21/0.45 @ ^ [Uu: nat] : zero_zero_nat
% 0.21/0.45 @ A )
% 0.21/0.45 = zero_zero_nat ) ).
% 0.21/0.45
% 0.21/0.45 % sum.neutral_const
% 0.21/0.45 thf(fact_1_partitionsE,axiom,
% 0.21/0.45 ! [P: nat > nat,N: nat] :
% 0.21/0.45 ( ( number1551313001itions @ P @ N )
% 0.21/0.45 => ~ ( ! [I: nat] :
% 0.21/0.45 ( ( ( P @ I )
% 0.21/0.45 != zero_zero_nat )
% 0.21/0.45 => ( ( ord_less_eq_nat @ one_one_nat @ I )
% 0.21/0.45 & ( ord_less_eq_nat @ I @ N ) ) )
% 0.21/0.45 => ( ( groups1842438620at_nat
% 0.21/0.45 @ ^ [I2: nat] : ( times_times_nat @ ( P @ I2 ) @ I2 )
% 0.21/0.45 @ ( set_ord_atMost_nat @ N ) )
% 0.21/0.45 != N ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % partitionsE
% 0.21/0.45 thf(fact_2_partitionsI,axiom,
% 0.21/0.45 ! [P: nat > nat,N: nat] :
% 0.21/0.45 ( ! [I3: nat] :
% 0.21/0.45 ( ( ( P @ I3 )
% 0.21/0.45 != zero_zero_nat )
% 0.21/0.45 => ( ( ord_less_eq_nat @ one_one_nat @ I3 )
% 0.21/0.45 & ( ord_less_eq_nat @ I3 @ N ) ) )
% 0.21/0.45 => ( ( ( groups1842438620at_nat
% 0.21/0.45 @ ^ [I2: nat] : ( times_times_nat @ ( P @ I2 ) @ I2 )
% 0.21/0.45 @ ( set_ord_atMost_nat @ N ) )
% 0.21/0.45 = N )
% 0.21/0.45 => ( number1551313001itions @ P @ N ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % partitionsI
% 0.21/0.45 thf(fact_3_partitions__def,axiom,
% 0.21/0.45 ( number1551313001itions
% 0.21/0.45 = ( ^ [P2: nat > nat,N2: nat] :
% 0.21/0.45 ( ! [I2: nat] :
% 0.21/0.45 ( ( ( P2 @ I2 )
% 0.21/0.45 != zero_zero_nat )
% 0.21/0.45 => ( ( ord_less_eq_nat @ one_one_nat @ I2 )
% 0.21/0.45 & ( ord_less_eq_nat @ I2 @ N2 ) ) )
% 0.21/0.45 & ( ( groups1842438620at_nat
% 0.21/0.45 @ ^ [I2: nat] : ( times_times_nat @ ( P2 @ I2 ) @ I2 )
% 0.21/0.45 @ ( set_ord_atMost_nat @ N2 ) )
% 0.21/0.45 = N2 ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % partitions_def
% 0.21/0.45 thf(fact_4_nat__1__eq__mult__iff,axiom,
% 0.21/0.45 ! [M: nat,N: nat] :
% 0.21/0.45 ( ( one_one_nat
% 0.21/0.45 = ( times_times_nat @ M @ N ) )
% 0.21/0.45 = ( ( M = one_one_nat )
% 0.21/0.45 & ( N = one_one_nat ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % nat_1_eq_mult_iff
% 0.21/0.45 thf(fact_5_nat__mult__eq__1__iff,axiom,
% 0.21/0.45 ! [M: nat,N: nat] :
% 0.21/0.45 ( ( ( times_times_nat @ M @ N )
% 0.21/0.45 = one_one_nat )
% 0.21/0.45 = ( ( M = one_one_nat )
% 0.21/0.45 & ( N = one_one_nat ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % nat_mult_eq_1_iff
% 0.21/0.45 thf(fact_6_atMost__subset__iff,axiom,
% 0.21/0.45 ! [X: set_nat,Y: set_nat] :
% 0.21/0.45 ( ( ord_le1613022364et_nat @ ( set_or1086813439et_nat @ X ) @ ( set_or1086813439et_nat @ Y ) )
% 0.21/0.45 = ( ord_less_eq_set_nat @ X @ Y ) ) ).
% 0.21/0.45
% 0.21/0.45 % atMost_subset_iff
% 0.21/0.45 thf(fact_7_atMost__subset__iff,axiom,
% 0.21/0.45 ! [X: nat,Y: nat] :
% 0.21/0.45 ( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X ) @ ( set_ord_atMost_nat @ Y ) )
% 0.21/0.45 = ( ord_less_eq_nat @ X @ Y ) ) ).
% 0.21/0.45
% 0.21/0.45 % atMost_subset_iff
% 0.21/0.45 thf(fact_8_atMost__iff,axiom,
% 0.21/0.45 ! [I4: set_nat,K: set_nat] :
% 0.21/0.45 ( ( member_set_nat @ I4 @ ( set_or1086813439et_nat @ K ) )
% 0.21/0.45 = ( ord_less_eq_set_nat @ I4 @ K ) ) ).
% 0.21/0.45
% 0.21/0.45 % atMost_iff
% 0.21/0.45 thf(fact_9_atMost__iff,axiom,
% 0.21/0.45 ! [I4: nat,K: nat] :
% 0.21/0.45 ( ( member_nat @ I4 @ ( set_ord_atMost_nat @ K ) )
% 0.21/0.45 = ( ord_less_eq_nat @ I4 @ K ) ) ).
% 0.21/0.45
% 0.21/0.45 % atMost_iff
% 0.21/0.45 thf(fact_10_atMost__eq__iff,axiom,
% 0.21/0.45 ! [X: nat,Y: nat] :
% 0.21/0.45 ( ( ( set_ord_atMost_nat @ X )
% 0.21/0.45 = ( set_ord_atMost_nat @ Y ) )
% 0.21/0.45 = ( X = Y ) ) ).
% 0.21/0.45
% 0.21/0.45 % atMost_eq_iff
% 0.21/0.45 thf(fact_11_mult__cancel__right,axiom,
% 0.21/0.45 ! [A2: nat,C: nat,B: nat] :
% 0.21/0.45 ( ( ( times_times_nat @ A2 @ C )
% 0.21/0.45 = ( times_times_nat @ B @ C ) )
% 0.21/0.45 = ( ( C = zero_zero_nat )
% 0.21/0.45 | ( A2 = B ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % mult_cancel_right
% 0.21/0.45 thf(fact_12_mult__cancel__left,axiom,
% 0.21/0.45 ! [C: nat,A2: nat,B: nat] :
% 0.21/0.45 ( ( ( times_times_nat @ C @ A2 )
% 0.21/0.45 = ( times_times_nat @ C @ B ) )
% 0.21/0.45 = ( ( C = zero_zero_nat )
% 0.21/0.45 | ( A2 = B ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % mult_cancel_left
% 0.21/0.45 thf(fact_13_mult__eq__0__iff,axiom,
% 0.21/0.45 ! [A2: nat,B: nat] :
% 0.21/0.45 ( ( ( times_times_nat @ A2 @ B )
% 0.21/0.45 = zero_zero_nat )
% 0.21/0.45 = ( ( A2 = zero_zero_nat )
% 0.21/0.45 | ( B = zero_zero_nat ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % mult_eq_0_iff
% 0.21/0.45 thf(fact_14_mult__zero__right,axiom,
% 0.21/0.45 ! [A2: nat] :
% 0.21/0.45 ( ( times_times_nat @ A2 @ zero_zero_nat )
% 0.21/0.45 = zero_zero_nat ) ).
% 0.21/0.45
% 0.21/0.45 % mult_zero_right
% 0.21/0.45 thf(fact_15_mult__zero__left,axiom,
% 0.21/0.45 ! [A2: nat] :
% 0.21/0.45 ( ( times_times_nat @ zero_zero_nat @ A2 )
% 0.21/0.45 = zero_zero_nat ) ).
% 0.21/0.45
% 0.21/0.45 % mult_zero_left
% 0.21/0.45 thf(fact_16_bot__nat__0_Oextremum,axiom,
% 0.21/0.45 ! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% 0.21/0.45
% 0.21/0.45 % bot_nat_0.extremum
% 0.21/0.45 thf(fact_17_le0,axiom,
% 0.21/0.45 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.21/0.45
% 0.21/0.45 % le0
% 0.21/0.45 thf(fact_18_mult__cancel2,axiom,
% 0.21/0.45 ! [M: nat,K: nat,N: nat] :
% 0.21/0.45 ( ( ( times_times_nat @ M @ K )
% 0.21/0.45 = ( times_times_nat @ N @ K ) )
% 0.21/0.45 = ( ( M = N )
% 0.21/0.45 | ( K = zero_zero_nat ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % mult_cancel2
% 0.21/0.45 thf(fact_19_mult__cancel1,axiom,
% 0.21/0.45 ! [K: nat,M: nat,N: nat] :
% 0.21/0.45 ( ( ( times_times_nat @ K @ M )
% 0.21/0.45 = ( times_times_nat @ K @ N ) )
% 0.21/0.45 = ( ( M = N )
% 0.21/0.45 | ( K = zero_zero_nat ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % mult_cancel1
% 0.21/0.45 thf(fact_20_mult__0__right,axiom,
% 0.21/0.45 ! [M: nat] :
% 0.21/0.45 ( ( times_times_nat @ M @ zero_zero_nat )
% 0.21/0.45 = zero_zero_nat ) ).
% 0.21/0.45
% 0.21/0.45 % mult_0_right
% 0.21/0.45 thf(fact_21_mult__is__0,axiom,
% 0.21/0.45 ! [M: nat,N: nat] :
% 0.21/0.45 ( ( ( times_times_nat @ M @ N )
% 0.21/0.45 = zero_zero_nat )
% 0.21/0.45 = ( ( M = zero_zero_nat )
% 0.21/0.45 | ( N = zero_zero_nat ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % mult_is_0
% 0.21/0.45 thf(fact_22_bounded__Max__nat,axiom,
% 0.21/0.45 ! [P3: nat > $o,X: nat,M2: nat] :
% 0.21/0.45 ( ( P3 @ X )
% 0.21/0.45 => ( ! [X2: nat] :
% 0.21/0.45 ( ( P3 @ X2 )
% 0.21/0.45 => ( ord_less_eq_nat @ X2 @ M2 ) )
% 0.21/0.45 => ~ ! [M3: nat] :
% 0.21/0.45 ( ( P3 @ M3 )
% 0.21/0.45 => ~ ! [X3: nat] :
% 0.21/0.45 ( ( P3 @ X3 )
% 0.21/0.45 => ( ord_less_eq_nat @ X3 @ M3 ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % bounded_Max_nat
% 0.21/0.45 thf(fact_23_Nat_Oex__has__greatest__nat,axiom,
% 0.21/0.45 ! [P3: nat > $o,K: nat,B: nat] :
% 0.21/0.45 ( ( P3 @ K )
% 0.21/0.45 => ( ! [Y2: nat] :
% 0.21/0.45 ( ( P3 @ Y2 )
% 0.21/0.45 => ( ord_less_eq_nat @ Y2 @ B ) )
% 0.21/0.45 => ? [X2: nat] :
% 0.21/0.45 ( ( P3 @ X2 )
% 0.21/0.45 & ! [Y3: nat] :
% 0.21/0.45 ( ( P3 @ Y3 )
% 0.21/0.45 => ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % Nat.ex_has_greatest_nat
% 0.21/0.45 thf(fact_24_nat__le__linear,axiom,
% 0.21/0.45 ! [M: nat,N: nat] :
% 0.21/0.45 ( ( ord_less_eq_nat @ M @ N )
% 0.21/0.45 | ( ord_less_eq_nat @ N @ M ) ) ).
% 0.21/0.45
% 0.21/0.45 % nat_le_linear
% 0.21/0.45 thf(fact_25_le__antisym,axiom,
% 0.21/0.45 ! [M: nat,N: nat] :
% 0.21/0.45 ( ( ord_less_eq_nat @ M @ N )
% 0.21/0.45 => ( ( ord_less_eq_nat @ N @ M )
% 0.21/0.45 => ( M = N ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % le_antisym
% 0.21/0.45 thf(fact_26_eq__imp__le,axiom,
% 0.21/0.45 ! [M: nat,N: nat] :
% 0.21/0.45 ( ( M = N )
% 0.21/0.45 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.21/0.45
% 0.21/0.45 % eq_imp_le
% 0.21/0.45 thf(fact_27_le__trans,axiom,
% 0.21/0.45 ! [I4: nat,J: nat,K: nat] :
% 0.21/0.45 ( ( ord_less_eq_nat @ I4 @ J )
% 0.21/0.45 => ( ( ord_less_eq_nat @ J @ K )
% 0.21/0.45 => ( ord_less_eq_nat @ I4 @ K ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % le_trans
% 0.21/0.45 thf(fact_28_le__refl,axiom,
% 0.21/0.45 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 0.21/0.45
% 0.21/0.45 % le_refl
% 0.21/0.45 thf(fact_29_sum_Oreindex__bij__witness,axiom,
% 0.21/0.45 ! [S: set_nat,I4: nat > nat,J: nat > nat,T: set_nat,H: nat > nat,G: nat > nat] :
% 0.21/0.45 ( ! [A3: nat] :
% 0.21/0.45 ( ( member_nat @ A3 @ S )
% 0.21/0.45 => ( ( I4 @ ( J @ A3 ) )
% 0.21/0.45 = A3 ) )
% 0.21/0.45 => ( ! [A3: nat] :
% 0.21/0.45 ( ( member_nat @ A3 @ S )
% 0.21/0.45 => ( member_nat @ ( J @ A3 ) @ T ) )
% 0.21/0.45 => ( ! [B2: nat] :
% 0.21/0.45 ( ( member_nat @ B2 @ T )
% 0.21/0.45 => ( ( J @ ( I4 @ B2 ) )
% 0.21/0.45 = B2 ) )
% 0.21/0.45 => ( ! [B2: nat] :
% 0.21/0.45 ( ( member_nat @ B2 @ T )
% 0.21/0.45 => ( member_nat @ ( I4 @ B2 ) @ S ) )
% 0.21/0.45 => ( ! [A3: nat] :
% 0.21/0.45 ( ( member_nat @ A3 @ S )
% 0.21/0.45 => ( ( H @ ( J @ A3 ) )
% 0.21/0.45 = ( G @ A3 ) ) )
% 0.21/0.45 => ( ( groups1842438620at_nat @ G @ S )
% 0.21/0.45 = ( groups1842438620at_nat @ H @ T ) ) ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % sum.reindex_bij_witness
% 0.21/0.45 thf(fact_30_sum_Oeq__general__inverses,axiom,
% 0.21/0.45 ! [B3: set_nat,K: nat > nat,A: set_nat,H: nat > nat,Gamma: nat > nat,Phi: nat > nat] :
% 0.21/0.45 ( ! [Y2: nat] :
% 0.21/0.45 ( ( member_nat @ Y2 @ B3 )
% 0.21/0.45 => ( ( member_nat @ ( K @ Y2 ) @ A )
% 0.21/0.45 & ( ( H @ ( K @ Y2 ) )
% 0.21/0.45 = Y2 ) ) )
% 0.21/0.45 => ( ! [X2: nat] :
% 0.21/0.45 ( ( member_nat @ X2 @ A )
% 0.21/0.45 => ( ( member_nat @ ( H @ X2 ) @ B3 )
% 0.21/0.45 & ( ( K @ ( H @ X2 ) )
% 0.21/0.45 = X2 )
% 0.21/0.45 & ( ( Gamma @ ( H @ X2 ) )
% 0.21/0.45 = ( Phi @ X2 ) ) ) )
% 0.21/0.45 => ( ( groups1842438620at_nat @ Phi @ A )
% 0.21/0.45 = ( groups1842438620at_nat @ Gamma @ B3 ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % sum.eq_general_inverses
% 0.21/0.45 thf(fact_31_sum_Oeq__general,axiom,
% 0.21/0.45 ! [B3: set_nat,A: set_nat,H: nat > nat,Gamma: nat > nat,Phi: nat > nat] :
% 0.21/0.45 ( ! [Y2: nat] :
% 0.21/0.45 ( ( member_nat @ Y2 @ B3 )
% 0.21/0.45 => ? [X3: nat] :
% 0.21/0.45 ( ( member_nat @ X3 @ A )
% 0.21/0.45 & ( ( H @ X3 )
% 0.21/0.45 = Y2 )
% 0.21/0.45 & ! [Ya: nat] :
% 0.21/0.45 ( ( ( member_nat @ Ya @ A )
% 0.21/0.45 & ( ( H @ Ya )
% 0.21/0.45 = Y2 ) )
% 0.21/0.45 => ( Ya = X3 ) ) ) )
% 0.21/0.45 => ( ! [X2: nat] :
% 0.21/0.45 ( ( member_nat @ X2 @ A )
% 0.21/0.45 => ( ( member_nat @ ( H @ X2 ) @ B3 )
% 0.21/0.45 & ( ( Gamma @ ( H @ X2 ) )
% 0.21/0.45 = ( Phi @ X2 ) ) ) )
% 0.21/0.45 => ( ( groups1842438620at_nat @ Phi @ A )
% 0.21/0.45 = ( groups1842438620at_nat @ Gamma @ B3 ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % sum.eq_general
% 0.21/0.45 thf(fact_32_sum_Ocong,axiom,
% 0.21/0.45 ! [A: set_nat,B3: set_nat,G: nat > nat,H: nat > nat] :
% 0.21/0.45 ( ( A = B3 )
% 0.21/0.45 => ( ! [X2: nat] :
% 0.21/0.45 ( ( member_nat @ X2 @ B3 )
% 0.21/0.45 => ( ( G @ X2 )
% 0.21/0.45 = ( H @ X2 ) ) )
% 0.21/0.45 => ( ( groups1842438620at_nat @ G @ A )
% 0.21/0.45 = ( groups1842438620at_nat @ H @ B3 ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % sum.cong
% 0.21/0.45 thf(fact_33_sum_Oswap,axiom,
% 0.21/0.45 ! [G: nat > nat > nat,B3: set_nat,A: set_nat] :
% 0.21/0.45 ( ( groups1842438620at_nat
% 0.21/0.45 @ ^ [I2: nat] : ( groups1842438620at_nat @ ( G @ I2 ) @ B3 )
% 0.21/0.45 @ A )
% 0.21/0.45 = ( groups1842438620at_nat
% 0.21/0.45 @ ^ [J2: nat] :
% 0.21/0.45 ( groups1842438620at_nat
% 0.21/0.45 @ ^ [I2: nat] : ( G @ I2 @ J2 )
% 0.21/0.45 @ A )
% 0.21/0.45 @ B3 ) ) ).
% 0.21/0.45
% 0.21/0.45 % sum.swap
% 0.21/0.45 thf(fact_34_mult__right__cancel,axiom,
% 0.21/0.45 ! [C: nat,A2: nat,B: nat] :
% 0.21/0.45 ( ( C != zero_zero_nat )
% 0.21/0.45 => ( ( ( times_times_nat @ A2 @ C )
% 0.21/0.45 = ( times_times_nat @ B @ C ) )
% 0.21/0.45 = ( A2 = B ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % mult_right_cancel
% 0.21/0.45 thf(fact_35_mult__left__cancel,axiom,
% 0.21/0.45 ! [C: nat,A2: nat,B: nat] :
% 0.21/0.45 ( ( C != zero_zero_nat )
% 0.21/0.45 => ( ( ( times_times_nat @ C @ A2 )
% 0.21/0.45 = ( times_times_nat @ C @ B ) )
% 0.21/0.45 = ( A2 = B ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % mult_left_cancel
% 0.21/0.45 thf(fact_36_no__zero__divisors,axiom,
% 0.21/0.45 ! [A2: nat,B: nat] :
% 0.21/0.45 ( ( A2 != zero_zero_nat )
% 0.21/0.45 => ( ( B != zero_zero_nat )
% 0.21/0.45 => ( ( times_times_nat @ A2 @ B )
% 0.21/0.45 != zero_zero_nat ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % no_zero_divisors
% 0.21/0.45 thf(fact_37_divisors__zero,axiom,
% 0.21/0.45 ! [A2: nat,B: nat] :
% 0.21/0.45 ( ( ( times_times_nat @ A2 @ B )
% 0.21/0.45 = zero_zero_nat )
% 0.21/0.45 => ( ( A2 = zero_zero_nat )
% 0.21/0.45 | ( B = zero_zero_nat ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % divisors_zero
% 0.21/0.45 thf(fact_38_mult__not__zero,axiom,
% 0.21/0.45 ! [A2: nat,B: nat] :
% 0.21/0.45 ( ( ( times_times_nat @ A2 @ B )
% 0.21/0.45 != zero_zero_nat )
% 0.21/0.45 => ( ( A2 != zero_zero_nat )
% 0.21/0.45 & ( B != zero_zero_nat ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % mult_not_zero
% 0.21/0.45 thf(fact_39_zero__neq__one,axiom,
% 0.21/0.45 zero_zero_nat != one_one_nat ).
% 0.21/0.45
% 0.21/0.45 % zero_neq_one
% 0.21/0.45 thf(fact_40_sum_Onot__neutral__contains__not__neutral,axiom,
% 0.21/0.45 ! [G: nat > nat,A: set_nat] :
% 0.21/0.45 ( ( ( groups1842438620at_nat @ G @ A )
% 0.21/0.45 != zero_zero_nat )
% 0.21/0.45 => ~ ! [A3: nat] :
% 0.21/0.45 ( ( member_nat @ A3 @ A )
% 0.21/0.45 => ( ( G @ A3 )
% 0.21/0.45 = zero_zero_nat ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % sum.not_neutral_contains_not_neutral
% 0.21/0.45 thf(fact_41_sum_Oneutral,axiom,
% 0.21/0.45 ! [A: set_nat,G: nat > nat] :
% 0.21/0.45 ( ! [X2: nat] :
% 0.21/0.45 ( ( member_nat @ X2 @ A )
% 0.21/0.45 => ( ( G @ X2 )
% 0.21/0.45 = zero_zero_nat ) )
% 0.21/0.45 => ( ( groups1842438620at_nat @ G @ A )
% 0.21/0.45 = zero_zero_nat ) ) ).
% 0.21/0.45
% 0.21/0.45 % sum.neutral
% 0.21/0.45 thf(fact_42_mem__Collect__eq,axiom,
% 0.21/0.45 ! [A2: nat,P3: nat > $o] :
% 0.21/0.45 ( ( member_nat @ A2 @ ( collect_nat @ P3 ) )
% 0.21/0.45 = ( P3 @ A2 ) ) ).
% 0.21/0.45
% 0.21/0.45 % mem_Collect_eq
% 0.21/0.45 thf(fact_43_Collect__mem__eq,axiom,
% 0.21/0.45 ! [A: set_nat] :
% 0.21/0.45 ( ( collect_nat
% 0.21/0.45 @ ^ [X4: nat] : ( member_nat @ X4 @ A ) )
% 0.21/0.45 = A ) ).
% 0.21/0.45
% 0.21/0.45 % Collect_mem_eq
% 0.21/0.45 thf(fact_44_Collect__cong,axiom,
% 0.21/0.45 ! [P3: nat > $o,Q: nat > $o] :
% 0.21/0.45 ( ! [X2: nat] :
% 0.21/0.45 ( ( P3 @ X2 )
% 0.21/0.45 = ( Q @ X2 ) )
% 0.21/0.45 => ( ( collect_nat @ P3 )
% 0.21/0.45 = ( collect_nat @ Q ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % Collect_cong
% 0.21/0.45 thf(fact_45_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.21/0.45 ! [A2: nat] :
% 0.21/0.45 ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 0.21/0.45 => ( A2 = zero_zero_nat ) ) ).
% 0.21/0.45
% 0.21/0.45 % bot_nat_0.extremum_uniqueI
% 0.21/0.45 thf(fact_46_bot__nat__0_Oextremum__unique,axiom,
% 0.21/0.45 ! [A2: nat] :
% 0.21/0.45 ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 0.21/0.45 = ( A2 = zero_zero_nat ) ) ).
% 0.21/0.45
% 0.21/0.45 % bot_nat_0.extremum_unique
% 0.21/0.45 thf(fact_47_le__0__eq,axiom,
% 0.21/0.45 ! [N: nat] :
% 0.21/0.45 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.21/0.45 = ( N = zero_zero_nat ) ) ).
% 0.21/0.45
% 0.21/0.45 % le_0_eq
% 0.21/0.45 thf(fact_48_less__eq__nat_Osimps_I1_J,axiom,
% 0.21/0.45 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.21/0.45
% 0.21/0.45 % less_eq_nat.simps(1)
% 0.21/0.45 thf(fact_49_mult__0,axiom,
% 0.21/0.45 ! [N: nat] :
% 0.21/0.45 ( ( times_times_nat @ zero_zero_nat @ N )
% 0.21/0.45 = zero_zero_nat ) ).
% 0.21/0.45
% 0.21/0.45 % mult_0
% 0.21/0.45 thf(fact_50_mult__le__mono2,axiom,
% 0.21/0.45 ! [I4: nat,J: nat,K: nat] :
% 0.21/0.45 ( ( ord_less_eq_nat @ I4 @ J )
% 0.21/0.45 => ( ord_less_eq_nat @ ( times_times_nat @ K @ I4 ) @ ( times_times_nat @ K @ J ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % mult_le_mono2
% 0.21/0.45 thf(fact_51_mult__le__mono1,axiom,
% 0.21/0.45 ! [I4: nat,J: nat,K: nat] :
% 0.21/0.45 ( ( ord_less_eq_nat @ I4 @ J )
% 0.21/0.45 => ( ord_less_eq_nat @ ( times_times_nat @ I4 @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % mult_le_mono1
% 0.21/0.45 thf(fact_52_mult__le__mono,axiom,
% 0.21/0.45 ! [I4: nat,J: nat,K: nat,L: nat] :
% 0.21/0.45 ( ( ord_less_eq_nat @ I4 @ J )
% 0.21/0.45 => ( ( ord_less_eq_nat @ K @ L )
% 0.21/0.45 => ( ord_less_eq_nat @ ( times_times_nat @ I4 @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % mult_le_mono
% 0.21/0.45 thf(fact_53_le__square,axiom,
% 0.21/0.45 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% 0.21/0.45
% 0.21/0.45 % le_square
% 0.21/0.45 thf(fact_54_le__cube,axiom,
% 0.21/0.45 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % le_cube
% 0.21/0.45 thf(fact_55_nat__mult__1__right,axiom,
% 0.21/0.45 ! [N: nat] :
% 0.21/0.45 ( ( times_times_nat @ N @ one_one_nat )
% 0.21/0.45 = N ) ).
% 0.21/0.45
% 0.21/0.45 % nat_mult_1_right
% 0.21/0.45 thf(fact_56_nat__mult__1,axiom,
% 0.21/0.45 ! [N: nat] :
% 0.21/0.45 ( ( times_times_nat @ one_one_nat @ N )
% 0.21/0.45 = N ) ).
% 0.21/0.45
% 0.21/0.45 % nat_mult_1
% 0.21/0.45 thf(fact_57_lambda__zero,axiom,
% 0.21/0.45 ( ( ^ [H2: nat] : zero_zero_nat )
% 0.21/0.45 = ( times_times_nat @ zero_zero_nat ) ) ).
% 0.21/0.45
% 0.21/0.45 % lambda_zero
% 0.21/0.45 thf(fact_58_lambda__one,axiom,
% 0.21/0.45 ( ( ^ [X4: nat] : X4 )
% 0.21/0.45 = ( times_times_nat @ one_one_nat ) ) ).
% 0.21/0.45
% 0.21/0.45 % lambda_one
% 0.21/0.45 thf(fact_59_sum__mono,axiom,
% 0.21/0.45 ! [K2: set_nat,F: nat > nat,G: nat > nat] :
% 0.21/0.45 ( ! [I3: nat] :
% 0.21/0.45 ( ( member_nat @ I3 @ K2 )
% 0.21/0.45 => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.21/0.45 => ( ord_less_eq_nat @ ( groups1842438620at_nat @ F @ K2 ) @ ( groups1842438620at_nat @ G @ K2 ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % sum_mono
% 0.21/0.45 thf(fact_60_sum__distrib__right,axiom,
% 0.21/0.45 ! [F: nat > nat,A: set_nat,R: nat] :
% 0.21/0.45 ( ( times_times_nat @ ( groups1842438620at_nat @ F @ A ) @ R )
% 0.21/0.45 = ( groups1842438620at_nat
% 0.21/0.45 @ ^ [N2: nat] : ( times_times_nat @ ( F @ N2 ) @ R )
% 0.21/0.45 @ A ) ) ).
% 0.21/0.45
% 0.21/0.45 % sum_distrib_right
% 0.21/0.45 thf(fact_61_sum__distrib__left,axiom,
% 0.21/0.45 ! [R: nat,F: nat > nat,A: set_nat] :
% 0.21/0.45 ( ( times_times_nat @ R @ ( groups1842438620at_nat @ F @ A ) )
% 0.21/0.45 = ( groups1842438620at_nat
% 0.21/0.45 @ ^ [N2: nat] : ( times_times_nat @ R @ ( F @ N2 ) )
% 0.21/0.45 @ A ) ) ).
% 0.21/0.45
% 0.21/0.45 % sum_distrib_left
% 0.21/0.45 thf(fact_62_sum__product,axiom,
% 0.21/0.45 ! [F: nat > nat,A: set_nat,G: nat > nat,B3: set_nat] :
% 0.21/0.45 ( ( times_times_nat @ ( groups1842438620at_nat @ F @ A ) @ ( groups1842438620at_nat @ G @ B3 ) )
% 0.21/0.45 = ( groups1842438620at_nat
% 0.21/0.45 @ ^ [I2: nat] :
% 0.21/0.45 ( groups1842438620at_nat
% 0.21/0.45 @ ^ [J2: nat] : ( times_times_nat @ ( F @ I2 ) @ ( G @ J2 ) )
% 0.21/0.45 @ B3 )
% 0.21/0.45 @ A ) ) ).
% 0.21/0.45
% 0.21/0.45 % sum_product
% 0.21/0.45 thf(fact_63_atMost__def,axiom,
% 0.21/0.45 ( set_or1086813439et_nat
% 0.21/0.45 = ( ^ [U: set_nat] :
% 0.21/0.45 ( collect_set_nat
% 0.21/0.45 @ ^ [X4: set_nat] : ( ord_less_eq_set_nat @ X4 @ U ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % atMost_def
% 0.21/0.45 thf(fact_64_atMost__def,axiom,
% 0.21/0.45 ( set_ord_atMost_nat
% 0.21/0.45 = ( ^ [U: nat] :
% 0.21/0.45 ( collect_nat
% 0.21/0.45 @ ^ [X4: nat] : ( ord_less_eq_nat @ X4 @ U ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % atMost_def
% 0.21/0.45 thf(fact_65_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.21/0.45 ! [A2: nat,B: nat,C: nat] :
% 0.21/0.45 ( ( ord_less_eq_nat @ A2 @ B )
% 0.21/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.21/0.45 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.21/0.45 thf(fact_66_mult__nonneg__nonpos2,axiom,
% 0.21/0.45 ! [A2: nat,B: nat] :
% 0.21/0.45 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.21/0.45 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.21/0.45 => ( ord_less_eq_nat @ ( times_times_nat @ B @ A2 ) @ zero_zero_nat ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % mult_nonneg_nonpos2
% 0.21/0.45 thf(fact_67_mult__nonpos__nonneg,axiom,
% 0.21/0.45 ! [A2: nat,B: nat] :
% 0.21/0.45 ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 0.21/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.21/0.45 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % mult_nonpos_nonneg
% 0.21/0.45 thf(fact_68_mult__nonneg__nonpos,axiom,
% 0.21/0.45 ! [A2: nat,B: nat] :
% 0.21/0.45 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.21/0.45 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.21/0.45 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % mult_nonneg_nonpos
% 0.21/0.45 thf(fact_69_mult__nonneg__nonneg,axiom,
% 0.21/0.45 ! [A2: nat,B: nat] :
% 0.21/0.45 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.21/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.21/0.45 => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % mult_nonneg_nonneg
% 0.21/0.45 thf(fact_70_split__mult__neg__le,axiom,
% 0.21/0.45 ! [A2: nat,B: nat] :
% 0.21/0.45 ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.21/0.45 & ( ord_less_eq_nat @ B @ zero_zero_nat ) )
% 0.21/0.45 | ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 0.21/0.45 & ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
% 0.21/0.45 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ).
% 0.21/0.45
% 0.21/0.45 % split_mult_neg_le
% 0.21/0.45 thf(fact_71_mult__right__mono,axiom,
% 0.21/0.45 ! [A2: nat,B: nat,C: nat] :
% 0.21/0.45 ( ( ord_less_eq_nat @ A2 @ B )
% 0.21/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.21/0.45 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % mult_right_mono
% 0.21/0.45 thf(fact_72_mult__left__mono,axiom,
% 0.21/0.45 ! [A2: nat,B: nat,C: nat] :
% 0.21/0.45 ( ( ord_less_eq_nat @ A2 @ B )
% 0.21/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.21/0.45 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % mult_left_mono
% 0.21/0.45 thf(fact_73_mult__mono_H,axiom,
% 0.21/0.45 ! [A2: nat,B: nat,C: nat,D: nat] :
% 0.21/0.45 ( ( ord_less_eq_nat @ A2 @ B )
% 0.21/0.45 => ( ( ord_less_eq_nat @ C @ D )
% 0.21/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.21/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.21/0.45 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % mult_mono'
% 0.21/0.45 thf(fact_74_mult__mono,axiom,
% 0.21/0.45 ! [A2: nat,B: nat,C: nat,D: nat] :
% 0.21/0.45 ( ( ord_less_eq_nat @ A2 @ B )
% 0.21/0.45 => ( ( ord_less_eq_nat @ C @ D )
% 0.21/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.21/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.21/0.45 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % mult_mono
% 0.21/0.45 thf(fact_75_not__one__le__zero,axiom,
% 0.21/0.45 ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% 0.21/0.45
% 0.21/0.45 % not_one_le_zero
% 0.21/0.45 thf(fact_76_zero__le__one,axiom,
% 0.21/0.45 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 0.21/0.45
% 0.21/0.45 % zero_le_one
% 0.21/0.45 thf(fact_77_sum__nonpos,axiom,
% 0.21/0.45 ! [A: set_nat,F: nat > nat] :
% 0.21/0.45 ( ! [X2: nat] :
% 0.21/0.45 ( ( member_nat @ X2 @ A )
% 0.21/0.45 => ( ord_less_eq_nat @ ( F @ X2 ) @ zero_zero_nat ) )
% 0.21/0.45 => ( ord_less_eq_nat @ ( groups1842438620at_nat @ F @ A ) @ zero_zero_nat ) ) ).
% 0.21/0.45
% 0.21/0.45 % sum_nonpos
% 0.21/0.45 thf(fact_78_sum__nonneg,axiom,
% 0.21/0.45 ! [A: set_nat,F: nat > nat] :
% 0.21/0.45 ( ! [X2: nat] :
% 0.21/0.45 ( ( member_nat @ X2 @ A )
% 0.21/0.45 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
% 0.21/0.45 => ( ord_less_eq_nat @ zero_zero_nat @ ( groups1842438620at_nat @ F @ A ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % sum_nonneg
% 0.21/0.45 thf(fact_79_mult__eq__self__implies__10,axiom,
% 0.21/0.45 ! [M: nat,N: nat] :
% 0.21/0.45 ( ( M
% 0.21/0.45 = ( times_times_nat @ M @ N ) )
% 0.21/0.45 => ( ( N = one_one_nat )
% 0.21/0.45 | ( M = zero_zero_nat ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % mult_eq_self_implies_10
% 0.21/0.45 thf(fact_80_mult__left__le,axiom,
% 0.21/0.45 ! [C: nat,A2: nat] :
% 0.21/0.45 ( ( ord_less_eq_nat @ C @ one_one_nat )
% 0.21/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.21/0.45 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ A2 ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % mult_left_le
% 0.21/0.45 thf(fact_81_mult__le__one,axiom,
% 0.21/0.45 ! [A2: nat,B: nat] :
% 0.21/0.45 ( ( ord_less_eq_nat @ A2 @ one_one_nat )
% 0.21/0.45 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.21/0.45 => ( ( ord_less_eq_nat @ B @ one_one_nat )
% 0.21/0.45 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ one_one_nat ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % mult_le_one
% 0.21/0.45 thf(fact_82_mult_Oright__neutral,axiom,
% 0.21/0.45 ! [A2: nat] :
% 0.21/0.45 ( ( times_times_nat @ A2 @ one_one_nat )
% 0.21/0.45 = A2 ) ).
% 0.21/0.45
% 0.21/0.45 % mult.right_neutral
% 0.21/0.45 thf(fact_83_mult_Oleft__neutral,axiom,
% 0.21/0.45 ! [A2: nat] :
% 0.21/0.45 ( ( times_times_nat @ one_one_nat @ A2 )
% 0.21/0.45 = A2 ) ).
% 0.21/0.45
% 0.21/0.45 % mult.left_neutral
% 0.21/0.45 thf(fact_84_le__zero__eq,axiom,
% 0.21/0.45 ! [N: nat] :
% 0.21/0.45 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.21/0.45 = ( N = zero_zero_nat ) ) ).
% 0.21/0.45
% 0.21/0.45 % le_zero_eq
% 0.21/0.45 thf(fact_85_subsetI,axiom,
% 0.21/0.45 ! [A: set_nat,B3: set_nat] :
% 0.21/0.45 ( ! [X2: nat] :
% 0.21/0.45 ( ( member_nat @ X2 @ A )
% 0.21/0.45 => ( member_nat @ X2 @ B3 ) )
% 0.21/0.45 => ( ord_less_eq_set_nat @ A @ B3 ) ) ).
% 0.21/0.45
% 0.21/0.45 % subsetI
% 0.21/0.45 thf(fact_86_subset__antisym,axiom,
% 0.21/0.45 ! [A: set_nat,B3: set_nat] :
% 0.21/0.45 ( ( ord_less_eq_set_nat @ A @ B3 )
% 0.21/0.45 => ( ( ord_less_eq_set_nat @ B3 @ A )
% 0.21/0.45 => ( A = B3 ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % subset_antisym
% 0.21/0.45 thf(fact_87_order__refl,axiom,
% 0.21/0.45 ! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% 0.21/0.45
% 0.21/0.45 % order_refl
% 0.21/0.45 thf(fact_88_order__refl,axiom,
% 0.21/0.45 ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% 0.21/0.45
% 0.21/0.45 % order_refl
% 0.21/0.45 thf(fact_89_nat__mult__eq__cancel__disj,axiom,
% 0.21/0.45 ! [K: nat,M: nat,N: nat] :
% 0.21/0.45 ( ( ( times_times_nat @ K @ M )
% 0.21/0.45 = ( times_times_nat @ K @ N ) )
% 0.21/0.45 = ( ( K = zero_zero_nat )
% 0.21/0.45 | ( M = N ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % nat_mult_eq_cancel_disj
% 0.21/0.45 thf(fact_90_mult_Ocomm__neutral,axiom,
% 0.21/0.45 ! [A2: nat] :
% 0.21/0.45 ( ( times_times_nat @ A2 @ one_one_nat )
% 0.21/0.45 = A2 ) ).
% 0.21/0.45
% 0.21/0.45 % mult.comm_neutral
% 0.21/0.45 thf(fact_91_comm__monoid__mult__class_Omult__1,axiom,
% 0.21/0.45 ! [A2: nat] :
% 0.21/0.45 ( ( times_times_nat @ one_one_nat @ A2 )
% 0.21/0.45 = A2 ) ).
% 0.21/0.45
% 0.21/0.45 % comm_monoid_mult_class.mult_1
% 0.21/0.45 thf(fact_92_order__subst1,axiom,
% 0.21/0.45 ! [A2: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
% 0.21/0.45 ( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
% 0.21/0.45 => ( ( ord_less_eq_set_nat @ B @ C )
% 0.21/0.45 => ( ! [X2: set_nat,Y2: set_nat] :
% 0.21/0.45 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 0.21/0.45 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.21/0.45 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % order_subst1
% 0.21/0.45 thf(fact_93_order__subst1,axiom,
% 0.21/0.45 ! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
% 0.21/0.45 ( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
% 0.21/0.45 => ( ( ord_less_eq_nat @ B @ C )
% 0.21/0.45 => ( ! [X2: nat,Y2: nat] :
% 0.21/0.45 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.21/0.45 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.21/0.45 => ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % order_subst1
% 0.21/0.45 thf(fact_94_order__subst1,axiom,
% 0.21/0.45 ! [A2: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
% 0.21/0.45 ( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
% 0.21/0.45 => ( ( ord_less_eq_set_nat @ B @ C )
% 0.21/0.45 => ( ! [X2: set_nat,Y2: set_nat] :
% 0.21/0.45 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 0.21/0.45 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.21/0.45 => ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % order_subst1
% 0.21/0.45 thf(fact_95_order__subst1,axiom,
% 0.21/0.45 ! [A2: nat,F: nat > nat,B: nat,C: nat] :
% 0.21/0.45 ( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
% 0.21/0.45 => ( ( ord_less_eq_nat @ B @ C )
% 0.21/0.45 => ( ! [X2: nat,Y2: nat] :
% 0.21/0.45 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.21/0.45 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.21/0.45 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % order_subst1
% 0.21/0.45 thf(fact_96_order__subst2,axiom,
% 0.21/0.45 ! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
% 0.21/0.45 ( ( ord_less_eq_nat @ A2 @ B )
% 0.21/0.45 => ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
% 0.21/0.45 => ( ! [X2: nat,Y2: nat] :
% 0.21/0.45 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.21/0.45 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.21/0.45 => ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % order_subst2
% 0.21/0.45 thf(fact_97_order__subst2,axiom,
% 0.21/0.45 ! [A2: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
% 0.21/0.45 ( ( ord_less_eq_set_nat @ A2 @ B )
% 0.21/0.45 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.21/0.45 => ( ! [X2: set_nat,Y2: set_nat] :
% 0.21/0.45 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 0.21/0.45 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.21/0.45 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % order_subst2
% 0.21/0.45 thf(fact_98_order__subst2,axiom,
% 0.21/0.45 ! [A2: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
% 0.21/0.45 ( ( ord_less_eq_set_nat @ A2 @ B )
% 0.21/0.45 => ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
% 0.21/0.45 => ( ! [X2: set_nat,Y2: set_nat] :
% 0.21/0.45 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 0.21/0.45 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.21/0.45 => ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % order_subst2
% 0.21/0.45 thf(fact_99_order__subst2,axiom,
% 0.21/0.45 ! [A2: nat,B: nat,F: nat > nat,C: nat] :
% 0.21/0.45 ( ( ord_less_eq_nat @ A2 @ B )
% 0.21/0.45 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.21/0.45 => ( ! [X2: nat,Y2: nat] :
% 0.21/0.45 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.21/0.45 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.21/0.45 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % order_subst2
% 0.21/0.45 thf(fact_100_ord__eq__le__subst,axiom,
% 0.21/0.45 ! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
% 0.21/0.45 ( ( A2
% 0.21/0.45 = ( F @ B ) )
% 0.21/0.45 => ( ( ord_less_eq_nat @ B @ C )
% 0.21/0.45 => ( ! [X2: nat,Y2: nat] :
% 0.21/0.45 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.21/0.45 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.21/0.45 => ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % ord_eq_le_subst
% 0.21/0.45 thf(fact_101_ord__eq__le__subst,axiom,
% 0.21/0.45 ! [A2: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
% 0.21/0.45 ( ( A2
% 0.21/0.45 = ( F @ B ) )
% 0.21/0.45 => ( ( ord_less_eq_set_nat @ B @ C )
% 0.21/0.45 => ( ! [X2: set_nat,Y2: set_nat] :
% 0.21/0.46 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 0.21/0.46 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.21/0.46 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % ord_eq_le_subst
% 0.21/0.46 thf(fact_102_ord__eq__le__subst,axiom,
% 0.21/0.46 ! [A2: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
% 0.21/0.46 ( ( A2
% 0.21/0.46 = ( F @ B ) )
% 0.21/0.46 => ( ( ord_less_eq_set_nat @ B @ C )
% 0.21/0.46 => ( ! [X2: set_nat,Y2: set_nat] :
% 0.21/0.46 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 0.21/0.46 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.21/0.46 => ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % ord_eq_le_subst
% 0.21/0.46 thf(fact_103_ord__eq__le__subst,axiom,
% 0.21/0.46 ! [A2: nat,F: nat > nat,B: nat,C: nat] :
% 0.21/0.46 ( ( A2
% 0.21/0.46 = ( F @ B ) )
% 0.21/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.21/0.46 => ( ! [X2: nat,Y2: nat] :
% 0.21/0.46 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.21/0.46 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.21/0.46 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % ord_eq_le_subst
% 0.21/0.46 thf(fact_104_ord__le__eq__subst,axiom,
% 0.21/0.46 ! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
% 0.21/0.46 ( ( ord_less_eq_nat @ A2 @ B )
% 0.21/0.46 => ( ( ( F @ B )
% 0.21/0.46 = C )
% 0.21/0.46 => ( ! [X2: nat,Y2: nat] :
% 0.21/0.46 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.21/0.46 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.21/0.46 => ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % ord_le_eq_subst
% 0.21/0.46 thf(fact_105_ord__le__eq__subst,axiom,
% 0.21/0.46 ! [A2: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
% 0.21/0.46 ( ( ord_less_eq_set_nat @ A2 @ B )
% 0.21/0.46 => ( ( ( F @ B )
% 0.21/0.46 = C )
% 0.21/0.46 => ( ! [X2: set_nat,Y2: set_nat] :
% 0.21/0.46 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 0.21/0.46 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.21/0.46 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % ord_le_eq_subst
% 0.21/0.46 thf(fact_106_ord__le__eq__subst,axiom,
% 0.21/0.46 ! [A2: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
% 0.21/0.46 ( ( ord_less_eq_set_nat @ A2 @ B )
% 0.21/0.46 => ( ( ( F @ B )
% 0.21/0.46 = C )
% 0.21/0.46 => ( ! [X2: set_nat,Y2: set_nat] :
% 0.21/0.46 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 0.21/0.46 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.21/0.46 => ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % ord_le_eq_subst
% 0.21/0.46 thf(fact_107_ord__le__eq__subst,axiom,
% 0.21/0.46 ! [A2: nat,B: nat,F: nat > nat,C: nat] :
% 0.21/0.46 ( ( ord_less_eq_nat @ A2 @ B )
% 0.21/0.46 => ( ( ( F @ B )
% 0.21/0.46 = C )
% 0.21/0.46 => ( ! [X2: nat,Y2: nat] :
% 0.21/0.46 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.21/0.46 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.21/0.46 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % ord_le_eq_subst
% 0.21/0.46 thf(fact_108_eq__iff,axiom,
% 0.21/0.46 ( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
% 0.21/0.46 = ( ^ [X4: set_nat,Y5: set_nat] :
% 0.21/0.46 ( ( ord_less_eq_set_nat @ X4 @ Y5 )
% 0.21/0.46 & ( ord_less_eq_set_nat @ Y5 @ X4 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % eq_iff
% 0.21/0.46 thf(fact_109_eq__iff,axiom,
% 0.21/0.46 ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
% 0.21/0.46 = ( ^ [X4: nat,Y5: nat] :
% 0.21/0.46 ( ( ord_less_eq_nat @ X4 @ Y5 )
% 0.21/0.46 & ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % eq_iff
% 0.21/0.46 thf(fact_110_antisym,axiom,
% 0.21/0.46 ! [X: set_nat,Y: set_nat] :
% 0.21/0.46 ( ( ord_less_eq_set_nat @ X @ Y )
% 0.21/0.46 => ( ( ord_less_eq_set_nat @ Y @ X )
% 0.21/0.46 => ( X = Y ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % antisym
% 0.21/0.46 thf(fact_111_antisym,axiom,
% 0.21/0.46 ! [X: nat,Y: nat] :
% 0.21/0.46 ( ( ord_less_eq_nat @ X @ Y )
% 0.21/0.46 => ( ( ord_less_eq_nat @ Y @ X )
% 0.21/0.46 => ( X = Y ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % antisym
% 0.21/0.46 thf(fact_112_linear,axiom,
% 0.21/0.46 ! [X: nat,Y: nat] :
% 0.21/0.46 ( ( ord_less_eq_nat @ X @ Y )
% 0.21/0.46 | ( ord_less_eq_nat @ Y @ X ) ) ).
% 0.21/0.46
% 0.21/0.46 % linear
% 0.21/0.46 thf(fact_113_eq__refl,axiom,
% 0.21/0.46 ! [X: set_nat,Y: set_nat] :
% 0.21/0.46 ( ( X = Y )
% 0.21/0.46 => ( ord_less_eq_set_nat @ X @ Y ) ) ).
% 0.21/0.46
% 0.21/0.46 % eq_refl
% 0.21/0.46 thf(fact_114_eq__refl,axiom,
% 0.21/0.46 ! [X: nat,Y: nat] :
% 0.21/0.46 ( ( X = Y )
% 0.21/0.46 => ( ord_less_eq_nat @ X @ Y ) ) ).
% 0.21/0.46
% 0.21/0.46 % eq_refl
% 0.21/0.46 thf(fact_115_le__cases,axiom,
% 0.21/0.46 ! [X: nat,Y: nat] :
% 0.21/0.46 ( ~ ( ord_less_eq_nat @ X @ Y )
% 0.21/0.46 => ( ord_less_eq_nat @ Y @ X ) ) ).
% 0.21/0.46
% 0.21/0.46 % le_cases
% 0.21/0.46 thf(fact_116_order_Otrans,axiom,
% 0.21/0.46 ! [A2: set_nat,B: set_nat,C: set_nat] :
% 0.21/0.46 ( ( ord_less_eq_set_nat @ A2 @ B )
% 0.21/0.46 => ( ( ord_less_eq_set_nat @ B @ C )
% 0.21/0.46 => ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % order.trans
% 0.21/0.46 thf(fact_117_order_Otrans,axiom,
% 0.21/0.46 ! [A2: nat,B: nat,C: nat] :
% 0.21/0.46 ( ( ord_less_eq_nat @ A2 @ B )
% 0.21/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.21/0.46 => ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % order.trans
% 0.21/0.46 thf(fact_118_le__cases3,axiom,
% 0.21/0.46 ! [X: nat,Y: nat,Z2: nat] :
% 0.21/0.46 ( ( ( ord_less_eq_nat @ X @ Y )
% 0.21/0.46 => ~ ( ord_less_eq_nat @ Y @ Z2 ) )
% 0.21/0.46 => ( ( ( ord_less_eq_nat @ Y @ X )
% 0.21/0.46 => ~ ( ord_less_eq_nat @ X @ Z2 ) )
% 0.21/0.46 => ( ( ( ord_less_eq_nat @ X @ Z2 )
% 0.21/0.46 => ~ ( ord_less_eq_nat @ Z2 @ Y ) )
% 0.21/0.46 => ( ( ( ord_less_eq_nat @ Z2 @ Y )
% 0.21/0.46 => ~ ( ord_less_eq_nat @ Y @ X ) )
% 0.21/0.46 => ( ( ( ord_less_eq_nat @ Y @ Z2 )
% 0.21/0.46 => ~ ( ord_less_eq_nat @ Z2 @ X ) )
% 0.21/0.46 => ~ ( ( ord_less_eq_nat @ Z2 @ X )
% 0.21/0.46 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % le_cases3
% 0.21/0.46 thf(fact_119_antisym__conv,axiom,
% 0.21/0.46 ! [Y: set_nat,X: set_nat] :
% 0.21/0.46 ( ( ord_less_eq_set_nat @ Y @ X )
% 0.21/0.46 => ( ( ord_less_eq_set_nat @ X @ Y )
% 0.21/0.46 = ( X = Y ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % antisym_conv
% 0.21/0.46 thf(fact_120_antisym__conv,axiom,
% 0.21/0.46 ! [Y: nat,X: nat] :
% 0.21/0.46 ( ( ord_less_eq_nat @ Y @ X )
% 0.21/0.46 => ( ( ord_less_eq_nat @ X @ Y )
% 0.21/0.46 = ( X = Y ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % antisym_conv
% 0.21/0.46 thf(fact_121_order__class_Oorder_Oeq__iff,axiom,
% 0.21/0.46 ( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
% 0.21/0.46 = ( ^ [A4: set_nat,B4: set_nat] :
% 0.21/0.46 ( ( ord_less_eq_set_nat @ A4 @ B4 )
% 0.21/0.46 & ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % order_class.order.eq_iff
% 0.21/0.46 thf(fact_122_order__class_Oorder_Oeq__iff,axiom,
% 0.21/0.46 ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
% 0.21/0.46 = ( ^ [A4: nat,B4: nat] :
% 0.21/0.46 ( ( ord_less_eq_nat @ A4 @ B4 )
% 0.21/0.46 & ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % order_class.order.eq_iff
% 0.21/0.46 thf(fact_123_ord__eq__le__trans,axiom,
% 0.21/0.46 ! [A2: set_nat,B: set_nat,C: set_nat] :
% 0.21/0.46 ( ( A2 = B )
% 0.21/0.46 => ( ( ord_less_eq_set_nat @ B @ C )
% 0.21/0.46 => ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % ord_eq_le_trans
% 0.21/0.46 thf(fact_124_ord__eq__le__trans,axiom,
% 0.21/0.46 ! [A2: nat,B: nat,C: nat] :
% 0.21/0.46 ( ( A2 = B )
% 0.21/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.21/0.46 => ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % ord_eq_le_trans
% 0.21/0.46 thf(fact_125_ord__le__eq__trans,axiom,
% 0.21/0.46 ! [A2: set_nat,B: set_nat,C: set_nat] :
% 0.21/0.46 ( ( ord_less_eq_set_nat @ A2 @ B )
% 0.21/0.46 => ( ( B = C )
% 0.21/0.46 => ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % ord_le_eq_trans
% 0.21/0.46 thf(fact_126_ord__le__eq__trans,axiom,
% 0.21/0.46 ! [A2: nat,B: nat,C: nat] :
% 0.21/0.46 ( ( ord_less_eq_nat @ A2 @ B )
% 0.21/0.46 => ( ( B = C )
% 0.21/0.46 => ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % ord_le_eq_trans
% 0.21/0.46 thf(fact_127_order__class_Oorder_Oantisym,axiom,
% 0.21/0.46 ! [A2: set_nat,B: set_nat] :
% 0.21/0.46 ( ( ord_less_eq_set_nat @ A2 @ B )
% 0.21/0.46 => ( ( ord_less_eq_set_nat @ B @ A2 )
% 0.21/0.46 => ( A2 = B ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % order_class.order.antisym
% 0.21/0.46 thf(fact_128_order__class_Oorder_Oantisym,axiom,
% 0.21/0.46 ! [A2: nat,B: nat] :
% 0.21/0.46 ( ( ord_less_eq_nat @ A2 @ B )
% 0.21/0.46 => ( ( ord_less_eq_nat @ B @ A2 )
% 0.21/0.46 => ( A2 = B ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % order_class.order.antisym
% 0.21/0.46 thf(fact_129_order__trans,axiom,
% 0.21/0.46 ! [X: set_nat,Y: set_nat,Z2: set_nat] :
% 0.21/0.46 ( ( ord_less_eq_set_nat @ X @ Y )
% 0.21/0.46 => ( ( ord_less_eq_set_nat @ Y @ Z2 )
% 0.21/0.46 => ( ord_less_eq_set_nat @ X @ Z2 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % order_trans
% 0.21/0.46 thf(fact_130_order__trans,axiom,
% 0.21/0.46 ! [X: nat,Y: nat,Z2: nat] :
% 0.21/0.46 ( ( ord_less_eq_nat @ X @ Y )
% 0.21/0.46 => ( ( ord_less_eq_nat @ Y @ Z2 )
% 0.21/0.46 => ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % order_trans
% 0.21/0.46 thf(fact_131_dual__order_Orefl,axiom,
% 0.21/0.46 ! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% 0.21/0.46
% 0.21/0.46 % dual_order.refl
% 0.21/0.46 thf(fact_132_dual__order_Orefl,axiom,
% 0.21/0.46 ! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% 0.21/0.46
% 0.21/0.46 % dual_order.refl
% 0.21/0.46 thf(fact_133_linorder__wlog,axiom,
% 0.21/0.46 ! [P3: nat > nat > $o,A2: nat,B: nat] :
% 0.21/0.46 ( ! [A3: nat,B2: nat] :
% 0.21/0.46 ( ( ord_less_eq_nat @ A3 @ B2 )
% 0.21/0.46 => ( P3 @ A3 @ B2 ) )
% 0.21/0.46 => ( ! [A3: nat,B2: nat] :
% 0.21/0.46 ( ( P3 @ B2 @ A3 )
% 0.21/0.46 => ( P3 @ A3 @ B2 ) )
% 0.21/0.46 => ( P3 @ A2 @ B ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % linorder_wlog
% 0.21/0.46 thf(fact_134_dual__order_Otrans,axiom,
% 0.21/0.46 ! [B: set_nat,A2: set_nat,C: set_nat] :
% 0.21/0.46 ( ( ord_less_eq_set_nat @ B @ A2 )
% 0.21/0.46 => ( ( ord_less_eq_set_nat @ C @ B )
% 0.21/0.46 => ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % dual_order.trans
% 0.21/0.46 thf(fact_135_dual__order_Otrans,axiom,
% 0.21/0.46 ! [B: nat,A2: nat,C: nat] :
% 0.21/0.46 ( ( ord_less_eq_nat @ B @ A2 )
% 0.21/0.46 => ( ( ord_less_eq_nat @ C @ B )
% 0.21/0.46 => ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % dual_order.trans
% 0.21/0.46 thf(fact_136_dual__order_Oeq__iff,axiom,
% 0.21/0.46 ( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
% 0.21/0.46 = ( ^ [A4: set_nat,B4: set_nat] :
% 0.21/0.46 ( ( ord_less_eq_set_nat @ B4 @ A4 )
% 0.21/0.46 & ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % dual_order.eq_iff
% 0.21/0.46 thf(fact_137_dual__order_Oeq__iff,axiom,
% 0.21/0.46 ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
% 0.21/0.46 = ( ^ [A4: nat,B4: nat] :
% 0.21/0.46 ( ( ord_less_eq_nat @ B4 @ A4 )
% 0.21/0.46 & ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % dual_order.eq_iff
% 0.21/0.46 thf(fact_138_dual__order_Oantisym,axiom,
% 0.21/0.46 ! [B: set_nat,A2: set_nat] :
% 0.21/0.46 ( ( ord_less_eq_set_nat @ B @ A2 )
% 0.21/0.46 => ( ( ord_less_eq_set_nat @ A2 @ B )
% 0.21/0.46 => ( A2 = B ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % dual_order.antisym
% 0.21/0.46 thf(fact_139_dual__order_Oantisym,axiom,
% 0.21/0.46 ! [B: nat,A2: nat] :
% 0.21/0.46 ( ( ord_less_eq_nat @ B @ A2 )
% 0.21/0.46 => ( ( ord_less_eq_nat @ A2 @ B )
% 0.21/0.46 => ( A2 = B ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % dual_order.antisym
% 0.21/0.46 thf(fact_140_zero__reorient,axiom,
% 0.21/0.46 ! [X: nat] :
% 0.21/0.46 ( ( zero_zero_nat = X )
% 0.21/0.46 = ( X = zero_zero_nat ) ) ).
% 0.21/0.46
% 0.21/0.46 % zero_reorient
% 0.21/0.46 thf(fact_141_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.21/0.46 ! [A2: nat,B: nat,C: nat] :
% 0.21/0.46 ( ( times_times_nat @ ( times_times_nat @ A2 @ B ) @ C )
% 0.21/0.46 = ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % ab_semigroup_mult_class.mult_ac(1)
% 0.21/0.46 thf(fact_142_mult_Oassoc,axiom,
% 0.21/0.46 ! [A2: nat,B: nat,C: nat] :
% 0.21/0.46 ( ( times_times_nat @ ( times_times_nat @ A2 @ B ) @ C )
% 0.21/0.46 = ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % mult.assoc
% 0.21/0.46 thf(fact_143_mult_Ocommute,axiom,
% 0.21/0.46 ( times_times_nat
% 0.21/0.46 = ( ^ [A4: nat,B4: nat] : ( times_times_nat @ B4 @ A4 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % mult.commute
% 0.21/0.46 thf(fact_144_mult_Oleft__commute,axiom,
% 0.21/0.46 ! [B: nat,A2: nat,C: nat] :
% 0.21/0.46 ( ( times_times_nat @ B @ ( times_times_nat @ A2 @ C ) )
% 0.21/0.46 = ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % mult.left_commute
% 0.21/0.46 thf(fact_145_one__reorient,axiom,
% 0.21/0.46 ! [X: nat] :
% 0.21/0.46 ( ( one_one_nat = X )
% 0.21/0.46 = ( X = one_one_nat ) ) ).
% 0.21/0.46
% 0.21/0.46 % one_reorient
% 0.21/0.46 thf(fact_146_Collect__mono__iff,axiom,
% 0.21/0.46 ! [P3: nat > $o,Q: nat > $o] :
% 0.21/0.46 ( ( ord_less_eq_set_nat @ ( collect_nat @ P3 ) @ ( collect_nat @ Q ) )
% 0.21/0.46 = ( ! [X4: nat] :
% 0.21/0.46 ( ( P3 @ X4 )
% 0.21/0.46 => ( Q @ X4 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % Collect_mono_iff
% 0.21/0.46 thf(fact_147_set__eq__subset,axiom,
% 0.21/0.46 ( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
% 0.21/0.46 = ( ^ [A5: set_nat,B5: set_nat] :
% 0.21/0.46 ( ( ord_less_eq_set_nat @ A5 @ B5 )
% 0.21/0.46 & ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % set_eq_subset
% 0.21/0.46 thf(fact_148_subset__trans,axiom,
% 0.21/0.46 ! [A: set_nat,B3: set_nat,C2: set_nat] :
% 0.21/0.46 ( ( ord_less_eq_set_nat @ A @ B3 )
% 0.21/0.46 => ( ( ord_less_eq_set_nat @ B3 @ C2 )
% 0.21/0.46 => ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % subset_trans
% 0.21/0.46 thf(fact_149_Collect__mono,axiom,
% 0.21/0.46 ! [P3: nat > $o,Q: nat > $o] :
% 0.21/0.46 ( ! [X2: nat] :
% 0.21/0.46 ( ( P3 @ X2 )
% 0.21/0.46 => ( Q @ X2 ) )
% 0.21/0.46 => ( ord_less_eq_set_nat @ ( collect_nat @ P3 ) @ ( collect_nat @ Q ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % Collect_mono
% 0.21/0.46 thf(fact_150_subset__refl,axiom,
% 0.21/0.46 ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% 0.21/0.46
% 0.21/0.46 % subset_refl
% 0.21/0.46 thf(fact_151_subset__iff,axiom,
% 0.21/0.46 ( ord_less_eq_set_nat
% 0.21/0.46 = ( ^ [A5: set_nat,B5: set_nat] :
% 0.21/0.46 ! [T2: nat] :
% 0.21/0.46 ( ( member_nat @ T2 @ A5 )
% 0.21/0.46 => ( member_nat @ T2 @ B5 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % subset_iff
% 0.21/0.46 thf(fact_152_equalityD2,axiom,
% 0.21/0.46 ! [A: set_nat,B3: set_nat] :
% 0.21/0.46 ( ( A = B3 )
% 0.21/0.46 => ( ord_less_eq_set_nat @ B3 @ A ) ) ).
% 0.21/0.46
% 0.21/0.46 % equalityD2
% 0.21/0.46 thf(fact_153_equalityD1,axiom,
% 0.21/0.46 ! [A: set_nat,B3: set_nat] :
% 0.21/0.46 ( ( A = B3 )
% 0.21/0.46 => ( ord_less_eq_set_nat @ A @ B3 ) ) ).
% 0.21/0.46
% 0.21/0.46 % equalityD1
% 0.21/0.46 thf(fact_154_subset__eq,axiom,
% 0.21/0.46 ( ord_less_eq_set_nat
% 0.21/0.46 = ( ^ [A5: set_nat,B5: set_nat] :
% 0.21/0.46 ! [X4: nat] :
% 0.21/0.46 ( ( member_nat @ X4 @ A5 )
% 0.21/0.46 => ( member_nat @ X4 @ B5 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % subset_eq
% 0.21/0.46 thf(fact_155_equalityE,axiom,
% 0.21/0.46 ! [A: set_nat,B3: set_nat] :
% 0.21/0.46 ( ( A = B3 )
% 0.21/0.46 => ~ ( ( ord_less_eq_set_nat @ A @ B3 )
% 0.21/0.46 => ~ ( ord_less_eq_set_nat @ B3 @ A ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % equalityE
% 0.21/0.46 thf(fact_156_subsetD,axiom,
% 0.21/0.46 ! [A: set_nat,B3: set_nat,C: nat] :
% 0.21/0.46 ( ( ord_less_eq_set_nat @ A @ B3 )
% 0.21/0.46 => ( ( member_nat @ C @ A )
% 0.21/0.46 => ( member_nat @ C @ B3 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % subsetD
% 0.21/0.46 thf(fact_157_in__mono,axiom,
% 0.21/0.46 ! [A: set_nat,B3: set_nat,X: nat] :
% 0.21/0.46 ( ( ord_less_eq_set_nat @ A @ B3 )
% 0.21/0.46 => ( ( member_nat @ X @ A )
% 0.21/0.46 => ( member_nat @ X @ B3 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % in_mono
% 0.21/0.46 thf(fact_158_less__eq__set__def,axiom,
% 0.21/0.46 ( ord_less_eq_set_nat
% 0.21/0.46 = ( ^ [A5: set_nat,B5: set_nat] :
% 0.21/0.46 ( ord_less_eq_nat_o
% 0.21/0.46 @ ^ [X4: nat] : ( member_nat @ X4 @ A5 )
% 0.21/0.46 @ ^ [X4: nat] : ( member_nat @ X4 @ B5 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % less_eq_set_def
% 0.21/0.46 thf(fact_159_Collect__subset,axiom,
% 0.21/0.46 ! [A: set_nat,P3: nat > $o] :
% 0.21/0.46 ( ord_less_eq_set_nat
% 0.21/0.46 @ ( collect_nat
% 0.21/0.46 @ ^ [X4: nat] :
% 0.21/0.46 ( ( member_nat @ X4 @ A )
% 0.21/0.46 & ( P3 @ X4 ) ) )
% 0.21/0.46 @ A ) ).
% 0.21/0.46
% 0.21/0.46 % Collect_subset
% 0.21/0.46 thf(fact_160_zero__le,axiom,
% 0.21/0.46 ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% 0.21/0.46
% 0.21/0.46 % zero_le
% 0.21/0.46 thf(fact_161_le__numeral__extra_I4_J,axiom,
% 0.21/0.46 ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% 0.21/0.46
% 0.21/0.46 % le_numeral_extra(4)
% 0.21/0.46 thf(fact_162_le__numeral__extra_I3_J,axiom,
% 0.21/0.46 ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 0.21/0.46
% 0.21/0.46 % le_numeral_extra(3)
% 0.21/0.46 thf(fact_163_subset__Collect__iff,axiom,
% 0.21/0.46 ! [B3: set_nat,A: set_nat,P3: nat > $o] :
% 0.21/0.46 ( ( ord_less_eq_set_nat @ B3 @ A )
% 0.21/0.46 => ( ( ord_less_eq_set_nat @ B3
% 0.21/0.46 @ ( collect_nat
% 0.21/0.46 @ ^ [X4: nat] :
% 0.21/0.46 ( ( member_nat @ X4 @ A )
% 0.21/0.46 & ( P3 @ X4 ) ) ) )
% 0.21/0.46 = ( ! [X4: nat] :
% 0.21/0.46 ( ( member_nat @ X4 @ B3 )
% 0.21/0.46 => ( P3 @ X4 ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % subset_Collect_iff
% 0.21/0.46 thf(fact_164_subset__CollectI,axiom,
% 0.21/0.46 ! [B3: set_nat,A: set_nat,Q: nat > $o,P3: nat > $o] :
% 0.21/0.46 ( ( ord_less_eq_set_nat @ B3 @ A )
% 0.21/0.46 => ( ! [X2: nat] :
% 0.21/0.46 ( ( member_nat @ X2 @ B3 )
% 0.21/0.46 => ( ( Q @ X2 )
% 0.21/0.46 => ( P3 @ X2 ) ) )
% 0.21/0.46 => ( ord_less_eq_set_nat
% 0.21/0.46 @ ( collect_nat
% 0.21/0.46 @ ^ [X4: nat] :
% 0.21/0.46 ( ( member_nat @ X4 @ B3 )
% 0.21/0.46 & ( Q @ X4 ) ) )
% 0.21/0.46 @ ( collect_nat
% 0.21/0.46 @ ^ [X4: nat] :
% 0.21/0.46 ( ( member_nat @ X4 @ A )
% 0.21/0.46 & ( P3 @ X4 ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % subset_CollectI
% 0.21/0.46 thf(fact_165_Collect__restrict,axiom,
% 0.21/0.46 ! [X5: set_nat,P3: nat > $o] :
% 0.21/0.46 ( ord_less_eq_set_nat
% 0.21/0.46 @ ( collect_nat
% 0.21/0.46 @ ^ [X4: nat] :
% 0.21/0.46 ( ( member_nat @ X4 @ X5 )
% 0.21/0.46 & ( P3 @ X4 ) ) )
% 0.21/0.46 @ X5 ) ).
% 0.21/0.46
% 0.21/0.46 % Collect_restrict
% 0.21/0.46 thf(fact_166_prop__restrict,axiom,
% 0.21/0.46 ! [X: nat,Z3: set_nat,X5: set_nat,P3: nat > $o] :
% 0.21/0.46 ( ( member_nat @ X @ Z3 )
% 0.21/0.46 => ( ( ord_less_eq_set_nat @ Z3
% 0.21/0.46 @ ( collect_nat
% 0.21/0.46 @ ^ [X4: nat] :
% 0.21/0.46 ( ( member_nat @ X4 @ X5 )
% 0.21/0.46 & ( P3 @ X4 ) ) ) )
% 0.21/0.46 => ( P3 @ X ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % prop_restrict
% 0.21/0.46 thf(fact_167_pred__subset__eq,axiom,
% 0.21/0.46 ! [R2: set_nat,S: set_nat] :
% 0.21/0.46 ( ( ord_less_eq_nat_o
% 0.21/0.46 @ ^ [X4: nat] : ( member_nat @ X4 @ R2 )
% 0.21/0.46 @ ^ [X4: nat] : ( member_nat @ X4 @ S ) )
% 0.21/0.46 = ( ord_less_eq_set_nat @ R2 @ S ) ) ).
% 0.21/0.46
% 0.21/0.46 % pred_subset_eq
% 0.21/0.46 thf(fact_168_conj__subset__def,axiom,
% 0.21/0.46 ! [A: set_nat,P3: nat > $o,Q: nat > $o] :
% 0.21/0.46 ( ( ord_less_eq_set_nat @ A
% 0.21/0.46 @ ( collect_nat
% 0.21/0.46 @ ^ [X4: nat] :
% 0.21/0.46 ( ( P3 @ X4 )
% 0.21/0.46 & ( Q @ X4 ) ) ) )
% 0.21/0.46 = ( ( ord_less_eq_set_nat @ A @ ( collect_nat @ P3 ) )
% 0.21/0.46 & ( ord_less_eq_set_nat @ A @ ( collect_nat @ Q ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % conj_subset_def
% 0.21/0.46 thf(fact_169_neq0__conv,axiom,
% 0.21/0.46 ! [N: nat] :
% 0.21/0.46 ( ( N != zero_zero_nat )
% 0.21/0.46 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.21/0.46
% 0.21/0.46 % neq0_conv
% 0.21/0.46 thf(fact_170_less__nat__zero__code,axiom,
% 0.21/0.46 ! [N: nat] :
% 0.21/0.46 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.21/0.46
% 0.21/0.46 % less_nat_zero_code
% 0.21/0.46 thf(fact_171_bot__nat__0_Onot__eq__extremum,axiom,
% 0.21/0.46 ! [A2: nat] :
% 0.21/0.46 ( ( A2 != zero_zero_nat )
% 0.21/0.46 = ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% 0.21/0.46
% 0.21/0.46 % bot_nat_0.not_eq_extremum
% 0.21/0.46 thf(fact_172_psubsetI,axiom,
% 0.21/0.46 ! [A: set_nat,B3: set_nat] :
% 0.21/0.46 ( ( ord_less_eq_set_nat @ A @ B3 )
% 0.21/0.46 => ( ( A != B3 )
% 0.21/0.46 => ( ord_less_set_nat @ A @ B3 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % psubsetI
% 0.21/0.46 thf(fact_173_not__gr__zero,axiom,
% 0.21/0.46 ! [N: nat] :
% 0.21/0.46 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.21/0.46 = ( N = zero_zero_nat ) ) ).
% 0.21/0.46
% 0.21/0.46 % not_gr_zero
% 0.21/0.46 thf(fact_174_nat__0__less__mult__iff,axiom,
% 0.21/0.46 ! [M: nat,N: nat] :
% 0.21/0.46 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
% 0.21/0.46 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.21/0.46 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % nat_0_less_mult_iff
% 0.21/0.46 thf(fact_175_mult__less__cancel2,axiom,
% 0.21/0.46 ! [M: nat,K: nat,N: nat] :
% 0.21/0.46 ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 0.21/0.46 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.21/0.46 & ( ord_less_nat @ M @ N ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % mult_less_cancel2
% 0.21/0.46 thf(fact_176_nat__mult__less__cancel__disj,axiom,
% 0.21/0.46 ! [K: nat,M: nat,N: nat] :
% 0.21/0.46 ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.21/0.46 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.21/0.46 & ( ord_less_nat @ M @ N ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % nat_mult_less_cancel_disj
% 0.21/0.46 thf(fact_177_less__one,axiom,
% 0.21/0.46 ! [N: nat] :
% 0.21/0.46 ( ( ord_less_nat @ N @ one_one_nat )
% 0.21/0.46 = ( N = zero_zero_nat ) ) ).
% 0.21/0.46
% 0.21/0.46 % less_one
% 0.21/0.46 thf(fact_178_mult__le__cancel2,axiom,
% 0.21/0.46 ! [M: nat,K: nat,N: nat] :
% 0.21/0.46 ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 0.21/0.46 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.21/0.46 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % mult_le_cancel2
% 0.21/0.46 thf(fact_179_nat__mult__le__cancel__disj,axiom,
% 0.21/0.46 ! [K: nat,M: nat,N: nat] :
% 0.21/0.46 ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.21/0.46 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.21/0.46 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % nat_mult_le_cancel_disj
% 0.21/0.46 thf(fact_180_dual__order_Ostrict__implies__not__eq,axiom,
% 0.21/0.46 ! [B: nat,A2: nat] :
% 0.21/0.46 ( ( ord_less_nat @ B @ A2 )
% 0.21/0.46 => ( A2 != B ) ) ).
% 0.21/0.46
% 0.21/0.46 % dual_order.strict_implies_not_eq
% 0.21/0.46 thf(fact_181_order_Ostrict__implies__not__eq,axiom,
% 0.21/0.46 ! [A2: nat,B: nat] :
% 0.21/0.46 ( ( ord_less_nat @ A2 @ B )
% 0.21/0.46 => ( A2 != B ) ) ).
% 0.21/0.46
% 0.21/0.46 % order.strict_implies_not_eq
% 0.21/0.46 thf(fact_182_not__less__iff__gr__or__eq,axiom,
% 0.21/0.46 ! [X: nat,Y: nat] :
% 0.21/0.46 ( ( ~ ( ord_less_nat @ X @ Y ) )
% 0.21/0.46 = ( ( ord_less_nat @ Y @ X )
% 0.21/0.46 | ( X = Y ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % not_less_iff_gr_or_eq
% 0.21/0.46 thf(fact_183_dual__order_Ostrict__trans,axiom,
% 0.21/0.46 ! [B: nat,A2: nat,C: nat] :
% 0.21/0.46 ( ( ord_less_nat @ B @ A2 )
% 0.21/0.46 => ( ( ord_less_nat @ C @ B )
% 0.21/0.46 => ( ord_less_nat @ C @ A2 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % dual_order.strict_trans
% 0.21/0.46 thf(fact_184_linorder__less__wlog,axiom,
% 0.21/0.46 ! [P3: nat > nat > $o,A2: nat,B: nat] :
% 0.21/0.46 ( ! [A3: nat,B2: nat] :
% 0.21/0.46 ( ( ord_less_nat @ A3 @ B2 )
% 0.21/0.46 => ( P3 @ A3 @ B2 ) )
% 0.21/0.46 => ( ! [A3: nat] : ( P3 @ A3 @ A3 )
% 0.21/0.46 => ( ! [A3: nat,B2: nat] :
% 0.21/0.46 ( ( P3 @ B2 @ A3 )
% 0.21/0.46 => ( P3 @ A3 @ B2 ) )
% 0.21/0.46 => ( P3 @ A2 @ B ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % linorder_less_wlog
% 0.21/0.46 thf(fact_185_exists__least__iff,axiom,
% 0.21/0.46 ( ( ^ [P4: nat > $o] :
% 0.21/0.46 ? [X6: nat] : ( P4 @ X6 ) )
% 0.21/0.46 = ( ^ [P5: nat > $o] :
% 0.21/0.46 ? [N2: nat] :
% 0.21/0.46 ( ( P5 @ N2 )
% 0.21/0.46 & ! [M4: nat] :
% 0.21/0.46 ( ( ord_less_nat @ M4 @ N2 )
% 0.21/0.46 => ~ ( P5 @ M4 ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % exists_least_iff
% 0.21/0.46 thf(fact_186_less__imp__not__less,axiom,
% 0.21/0.46 ! [X: nat,Y: nat] :
% 0.21/0.46 ( ( ord_less_nat @ X @ Y )
% 0.21/0.46 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 0.21/0.46
% 0.21/0.46 % less_imp_not_less
% 0.21/0.46 thf(fact_187_order_Ostrict__trans,axiom,
% 0.21/0.46 ! [A2: nat,B: nat,C: nat] :
% 0.21/0.46 ( ( ord_less_nat @ A2 @ B )
% 0.21/0.46 => ( ( ord_less_nat @ B @ C )
% 0.21/0.46 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % order.strict_trans
% 0.21/0.46 thf(fact_188_dual__order_Oirrefl,axiom,
% 0.21/0.46 ! [A2: nat] :
% 0.21/0.46 ~ ( ord_less_nat @ A2 @ A2 ) ).
% 0.21/0.46
% 0.21/0.46 % dual_order.irrefl
% 0.21/0.46 thf(fact_189_linorder__cases,axiom,
% 0.21/0.46 ! [X: nat,Y: nat] :
% 0.21/0.46 ( ~ ( ord_less_nat @ X @ Y )
% 0.21/0.46 => ( ( X != Y )
% 0.21/0.46 => ( ord_less_nat @ Y @ X ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % linorder_cases
% 0.21/0.46 thf(fact_190_less__imp__triv,axiom,
% 0.21/0.46 ! [X: nat,Y: nat,P3: $o] :
% 0.21/0.46 ( ( ord_less_nat @ X @ Y )
% 0.21/0.46 => ( ( ord_less_nat @ Y @ X )
% 0.21/0.46 => P3 ) ) ).
% 0.21/0.46
% 0.21/0.46 % less_imp_triv
% 0.21/0.46 thf(fact_191_less__imp__not__eq2,axiom,
% 0.21/0.46 ! [X: nat,Y: nat] :
% 0.21/0.46 ( ( ord_less_nat @ X @ Y )
% 0.21/0.46 => ( Y != X ) ) ).
% 0.21/0.46
% 0.21/0.46 % less_imp_not_eq2
% 0.21/0.46 thf(fact_192_antisym__conv3,axiom,
% 0.21/0.46 ! [Y: nat,X: nat] :
% 0.21/0.46 ( ~ ( ord_less_nat @ Y @ X )
% 0.21/0.46 => ( ( ~ ( ord_less_nat @ X @ Y ) )
% 0.21/0.46 = ( X = Y ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % antisym_conv3
% 0.21/0.46 thf(fact_193_less__induct,axiom,
% 0.21/0.46 ! [P3: nat > $o,A2: nat] :
% 0.21/0.46 ( ! [X2: nat] :
% 0.21/0.46 ( ! [Y3: nat] :
% 0.21/0.46 ( ( ord_less_nat @ Y3 @ X2 )
% 0.21/0.46 => ( P3 @ Y3 ) )
% 0.21/0.46 => ( P3 @ X2 ) )
% 0.21/0.46 => ( P3 @ A2 ) ) ).
% 0.21/0.46
% 0.21/0.46 % less_induct
% 0.21/0.46 thf(fact_194_less__not__sym,axiom,
% 0.21/0.46 ! [X: nat,Y: nat] :
% 0.21/0.46 ( ( ord_less_nat @ X @ Y )
% 0.21/0.46 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 0.21/0.46
% 0.21/0.46 % less_not_sym
% 0.21/0.46 thf(fact_195_less__imp__not__eq,axiom,
% 0.21/0.46 ! [X: nat,Y: nat] :
% 0.21/0.46 ( ( ord_less_nat @ X @ Y )
% 0.21/0.46 => ( X != Y ) ) ).
% 0.21/0.46
% 0.21/0.46 % less_imp_not_eq
% 0.21/0.46 thf(fact_196_dual__order_Oasym,axiom,
% 0.21/0.46 ! [B: nat,A2: nat] :
% 0.21/0.46 ( ( ord_less_nat @ B @ A2 )
% 0.21/0.46 => ~ ( ord_less_nat @ A2 @ B ) ) ).
% 0.21/0.46
% 0.21/0.46 % dual_order.asym
% 0.21/0.46 thf(fact_197_ord__less__eq__trans,axiom,
% 0.21/0.46 ! [A2: nat,B: nat,C: nat] :
% 0.21/0.46 ( ( ord_less_nat @ A2 @ B )
% 0.21/0.46 => ( ( B = C )
% 0.21/0.46 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % ord_less_eq_trans
% 0.21/0.46 thf(fact_198_ord__eq__less__trans,axiom,
% 0.21/0.46 ! [A2: nat,B: nat,C: nat] :
% 0.21/0.46 ( ( A2 = B )
% 0.21/0.46 => ( ( ord_less_nat @ B @ C )
% 0.21/0.46 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % ord_eq_less_trans
% 0.21/0.46 thf(fact_199_less__irrefl,axiom,
% 0.21/0.46 ! [X: nat] :
% 0.21/0.46 ~ ( ord_less_nat @ X @ X ) ).
% 0.21/0.46
% 0.21/0.46 % less_irrefl
% 0.21/0.46 thf(fact_200_less__linear,axiom,
% 0.21/0.46 ! [X: nat,Y: nat] :
% 0.21/0.46 ( ( ord_less_nat @ X @ Y )
% 0.21/0.46 | ( X = Y )
% 0.21/0.46 | ( ord_less_nat @ Y @ X ) ) ).
% 0.21/0.46
% 0.21/0.46 % less_linear
% 0.21/0.46 thf(fact_201_less__trans,axiom,
% 0.21/0.46 ! [X: nat,Y: nat,Z2: nat] :
% 0.21/0.46 ( ( ord_less_nat @ X @ Y )
% 0.21/0.46 => ( ( ord_less_nat @ Y @ Z2 )
% 0.21/0.46 => ( ord_less_nat @ X @ Z2 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % less_trans
% 0.21/0.46 thf(fact_202_less__asym_H,axiom,
% 0.21/0.46 ! [A2: nat,B: nat] :
% 0.21/0.46 ( ( ord_less_nat @ A2 @ B )
% 0.21/0.46 => ~ ( ord_less_nat @ B @ A2 ) ) ).
% 0.21/0.46
% 0.21/0.46 % less_asym'
% 0.21/0.46 thf(fact_203_less__asym,axiom,
% 0.21/0.46 ! [X: nat,Y: nat] :
% 0.21/0.46 ( ( ord_less_nat @ X @ Y )
% 0.21/0.46 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 0.21/0.46
% 0.21/0.46 % less_asym
% 0.21/0.46 thf(fact_204_less__imp__neq,axiom,
% 0.21/0.46 ! [X: nat,Y: nat] :
% 0.21/0.46 ( ( ord_less_nat @ X @ Y )
% 0.21/0.46 => ( X != Y ) ) ).
% 0.21/0.46
% 0.21/0.46 % less_imp_neq
% 0.21/0.46 thf(fact_205_order_Oasym,axiom,
% 0.21/0.46 ! [A2: nat,B: nat] :
% 0.21/0.46 ( ( ord_less_nat @ A2 @ B )
% 0.21/0.46 => ~ ( ord_less_nat @ B @ A2 ) ) ).
% 0.21/0.46
% 0.21/0.46 % order.asym
% 0.21/0.46 thf(fact_206_neq__iff,axiom,
% 0.21/0.46 ! [X: nat,Y: nat] :
% 0.21/0.46 ( ( X != Y )
% 0.21/0.46 = ( ( ord_less_nat @ X @ Y )
% 0.21/0.46 | ( ord_less_nat @ Y @ X ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % neq_iff
% 0.21/0.46 thf(fact_207_neqE,axiom,
% 0.21/0.46 ! [X: nat,Y: nat] :
% 0.21/0.46 ( ( X != Y )
% 0.21/0.46 => ( ~ ( ord_less_nat @ X @ Y )
% 0.21/0.46 => ( ord_less_nat @ Y @ X ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % neqE
% 0.21/0.46 thf(fact_208_gt__ex,axiom,
% 0.21/0.46 ! [X: nat] :
% 0.21/0.46 ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% 0.21/0.46
% 0.21/0.46 % gt_ex
% 0.21/0.46 thf(fact_209_order__less__subst2,axiom,
% 0.21/0.46 ! [A2: nat,B: nat,F: nat > nat,C: nat] :
% 0.21/0.46 ( ( ord_less_nat @ A2 @ B )
% 0.21/0.46 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.21/0.46 => ( ! [X2: nat,Y2: nat] :
% 0.21/0.46 ( ( ord_less_nat @ X2 @ Y2 )
% 0.21/0.46 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.21/0.46 => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % order_less_subst2
% 0.21/0.46 thf(fact_210_order__less__subst1,axiom,
% 0.21/0.46 ! [A2: nat,F: nat > nat,B: nat,C: nat] :
% 0.21/0.46 ( ( ord_less_nat @ A2 @ ( F @ B ) )
% 0.21/0.46 => ( ( ord_less_nat @ B @ C )
% 0.21/0.46 => ( ! [X2: nat,Y2: nat] :
% 0.21/0.46 ( ( ord_less_nat @ X2 @ Y2 )
% 0.21/0.46 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.21/0.46 => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % order_less_subst1
% 0.21/0.46 thf(fact_211_ord__less__eq__subst,axiom,
% 0.21/0.46 ! [A2: nat,B: nat,F: nat > nat,C: nat] :
% 0.21/0.46 ( ( ord_less_nat @ A2 @ B )
% 0.21/0.46 => ( ( ( F @ B )
% 0.21/0.46 = C )
% 0.21/0.46 => ( ! [X2: nat,Y2: nat] :
% 0.21/0.46 ( ( ord_less_nat @ X2 @ Y2 )
% 0.21/0.46 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.21/0.46 => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % ord_less_eq_subst
% 0.21/0.46 thf(fact_212_ord__eq__less__subst,axiom,
% 0.21/0.46 ! [A2: nat,F: nat > nat,B: nat,C: nat] :
% 0.21/0.46 ( ( A2
% 0.21/0.46 = ( F @ B ) )
% 0.21/0.46 => ( ( ord_less_nat @ B @ C )
% 0.21/0.46 => ( ! [X2: nat,Y2: nat] :
% 0.21/0.46 ( ( ord_less_nat @ X2 @ Y2 )
% 0.21/0.46 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.21/0.46 => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % ord_eq_less_subst
% 0.21/0.46 thf(fact_213_gr__zeroI,axiom,
% 0.21/0.46 ! [N: nat] :
% 0.21/0.46 ( ( N != zero_zero_nat )
% 0.21/0.46 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.21/0.46
% 0.21/0.46 % gr_zeroI
% 0.21/0.46 thf(fact_214_not__less__zero,axiom,
% 0.21/0.46 ! [N: nat] :
% 0.21/0.46 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.21/0.46
% 0.21/0.46 % not_less_zero
% 0.21/0.46 thf(fact_215_gr__implies__not__zero,axiom,
% 0.21/0.46 ! [M: nat,N: nat] :
% 0.21/0.46 ( ( ord_less_nat @ M @ N )
% 0.21/0.46 => ( N != zero_zero_nat ) ) ).
% 0.21/0.46
% 0.21/0.46 % gr_implies_not_zero
% 0.21/0.46 thf(fact_216_zero__less__iff__neq__zero,axiom,
% 0.21/0.46 ! [N: nat] :
% 0.21/0.46 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.21/0.46 = ( N != zero_zero_nat ) ) ).
% 0.21/0.46
% 0.21/0.46 % zero_less_iff_neq_zero
% 0.21/0.46 thf(fact_217_nat__less__le,axiom,
% 0.21/0.46 ( ord_less_nat
% 0.21/0.46 = ( ^ [M4: nat,N2: nat] :
% 0.21/0.46 ( ( ord_less_eq_nat @ M4 @ N2 )
% 0.21/0.46 & ( M4 != N2 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % nat_less_le
% 0.21/0.46 thf(fact_218_less__imp__le__nat,axiom,
% 0.21/0.46 ! [M: nat,N: nat] :
% 0.21/0.46 ( ( ord_less_nat @ M @ N )
% 0.21/0.46 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.21/0.46
% 0.21/0.46 % less_imp_le_nat
% 0.21/0.46 thf(fact_219_le__eq__less__or__eq,axiom,
% 0.21/0.46 ( ord_less_eq_nat
% 0.21/0.46 = ( ^ [M4: nat,N2: nat] :
% 0.21/0.46 ( ( ord_less_nat @ M4 @ N2 )
% 0.21/0.46 | ( M4 = N2 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % le_eq_less_or_eq
% 0.21/0.46 thf(fact_220_less__or__eq__imp__le,axiom,
% 0.21/0.46 ! [M: nat,N: nat] :
% 0.21/0.46 ( ( ( ord_less_nat @ M @ N )
% 0.21/0.46 | ( M = N ) )
% 0.21/0.46 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.21/0.46
% 0.21/0.46 % less_or_eq_imp_le
% 0.21/0.46 thf(fact_221_le__neq__implies__less,axiom,
% 0.21/0.46 ! [M: nat,N: nat] :
% 0.21/0.46 ( ( ord_less_eq_nat @ M @ N )
% 0.21/0.46 => ( ( M != N )
% 0.21/0.46 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % le_neq_implies_less
% 0.21/0.46 thf(fact_222_less__mono__imp__le__mono,axiom,
% 0.21/0.46 ! [F: nat > nat,I4: nat,J: nat] :
% 0.21/0.46 ( ! [I3: nat,J3: nat] :
% 0.21/0.46 ( ( ord_less_nat @ I3 @ J3 )
% 0.21/0.46 => ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
% 0.21/0.46 => ( ( ord_less_eq_nat @ I4 @ J )
% 0.21/0.46 => ( ord_less_eq_nat @ ( F @ I4 ) @ ( F @ J ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % less_mono_imp_le_mono
% 0.21/0.46 thf(fact_223_gr0I,axiom,
% 0.21/0.46 ! [N: nat] :
% 0.21/0.46 ( ( N != zero_zero_nat )
% 0.21/0.46 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.21/0.46
% 0.21/0.46 % gr0I
% 0.21/0.46 thf(fact_224_not__gr0,axiom,
% 0.21/0.46 ! [N: nat] :
% 0.21/0.46 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.21/0.46 = ( N = zero_zero_nat ) ) ).
% 0.21/0.46
% 0.21/0.46 % not_gr0
% 0.21/0.46 thf(fact_225_not__less0,axiom,
% 0.21/0.46 ! [N: nat] :
% 0.21/0.46 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.21/0.46
% 0.21/0.46 % not_less0
% 0.21/0.46 thf(fact_226_less__zeroE,axiom,
% 0.21/0.46 ! [N: nat] :
% 0.21/0.46 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.21/0.46
% 0.21/0.46 % less_zeroE
% 0.21/0.46 thf(fact_227_gr__implies__not0,axiom,
% 0.21/0.46 ! [M: nat,N: nat] :
% 0.21/0.46 ( ( ord_less_nat @ M @ N )
% 0.21/0.46 => ( N != zero_zero_nat ) ) ).
% 0.21/0.46
% 0.21/0.46 % gr_implies_not0
% 0.21/0.46 thf(fact_228_infinite__descent0,axiom,
% 0.21/0.46 ! [P3: nat > $o,N: nat] :
% 0.21/0.46 ( ( P3 @ zero_zero_nat )
% 0.21/0.46 => ( ! [N3: nat] :
% 0.21/0.46 ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 0.21/0.46 => ( ~ ( P3 @ N3 )
% 0.21/0.46 => ? [M5: nat] :
% 0.21/0.46 ( ( ord_less_nat @ M5 @ N3 )
% 0.21/0.46 & ~ ( P3 @ M5 ) ) ) )
% 0.21/0.46 => ( P3 @ N ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % infinite_descent0
% 0.21/0.46 thf(fact_229_bot__nat__0_Oextremum__strict,axiom,
% 0.21/0.46 ! [A2: nat] :
% 0.21/0.46 ~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% 0.21/0.46
% 0.21/0.46 % bot_nat_0.extremum_strict
% 0.21/0.46 thf(fact_230_psubsetE,axiom,
% 0.21/0.46 ! [A: set_nat,B3: set_nat] :
% 0.21/0.46 ( ( ord_less_set_nat @ A @ B3 )
% 0.21/0.46 => ~ ( ( ord_less_eq_set_nat @ A @ B3 )
% 0.21/0.46 => ( ord_less_eq_set_nat @ B3 @ A ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % psubsetE
% 0.21/0.46 thf(fact_231_psubset__eq,axiom,
% 0.21/0.46 ( ord_less_set_nat
% 0.21/0.46 = ( ^ [A5: set_nat,B5: set_nat] :
% 0.21/0.46 ( ( ord_less_eq_set_nat @ A5 @ B5 )
% 0.21/0.46 & ( A5 != B5 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % psubset_eq
% 0.21/0.46 thf(fact_232_psubset__imp__subset,axiom,
% 0.21/0.46 ! [A: set_nat,B3: set_nat] :
% 0.21/0.46 ( ( ord_less_set_nat @ A @ B3 )
% 0.21/0.46 => ( ord_less_eq_set_nat @ A @ B3 ) ) ).
% 0.21/0.46
% 0.21/0.46 % psubset_imp_subset
% 0.21/0.46 thf(fact_233_psubset__subset__trans,axiom,
% 0.21/0.46 ! [A: set_nat,B3: set_nat,C2: set_nat] :
% 0.21/0.46 ( ( ord_less_set_nat @ A @ B3 )
% 0.21/0.46 => ( ( ord_less_eq_set_nat @ B3 @ C2 )
% 0.21/0.46 => ( ord_less_set_nat @ A @ C2 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % psubset_subset_trans
% 0.21/0.46 thf(fact_234_subset__not__subset__eq,axiom,
% 0.21/0.46 ( ord_less_set_nat
% 0.21/0.46 = ( ^ [A5: set_nat,B5: set_nat] :
% 0.21/0.46 ( ( ord_less_eq_set_nat @ A5 @ B5 )
% 0.21/0.46 & ~ ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % subset_not_subset_eq
% 0.21/0.46 thf(fact_235_subset__psubset__trans,axiom,
% 0.21/0.46 ! [A: set_nat,B3: set_nat,C2: set_nat] :
% 0.21/0.46 ( ( ord_less_eq_set_nat @ A @ B3 )
% 0.21/0.46 => ( ( ord_less_set_nat @ B3 @ C2 )
% 0.21/0.46 => ( ord_less_set_nat @ A @ C2 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % subset_psubset_trans
% 0.21/0.46 thf(fact_236_subset__iff__psubset__eq,axiom,
% 0.21/0.46 ( ord_less_eq_set_nat
% 0.21/0.46 = ( ^ [A5: set_nat,B5: set_nat] :
% 0.21/0.46 ( ( ord_less_set_nat @ A5 @ B5 )
% 0.21/0.46 | ( A5 = B5 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % subset_iff_psubset_eq
% 0.21/0.46 thf(fact_237_less__numeral__extra_I4_J,axiom,
% 0.21/0.46 ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% 0.21/0.46
% 0.21/0.46 % less_numeral_extra(4)
% 0.21/0.46 thf(fact_238_less__numeral__extra_I3_J,axiom,
% 0.21/0.46 ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 0.21/0.46
% 0.21/0.46 % less_numeral_extra(3)
% 0.21/0.46 thf(fact_239_order_Onot__eq__order__implies__strict,axiom,
% 0.21/0.46 ! [A2: set_nat,B: set_nat] :
% 0.21/0.46 ( ( A2 != B )
% 0.21/0.46 => ( ( ord_less_eq_set_nat @ A2 @ B )
% 0.21/0.46 => ( ord_less_set_nat @ A2 @ B ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % order.not_eq_order_implies_strict
% 0.21/0.46 thf(fact_240_order_Onot__eq__order__implies__strict,axiom,
% 0.21/0.46 ! [A2: nat,B: nat] :
% 0.21/0.46 ( ( A2 != B )
% 0.21/0.46 => ( ( ord_less_eq_nat @ A2 @ B )
% 0.21/0.46 => ( ord_less_nat @ A2 @ B ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % order.not_eq_order_implies_strict
% 0.21/0.46 thf(fact_241_dual__order_Ostrict__implies__order,axiom,
% 0.21/0.46 ! [B: set_nat,A2: set_nat] :
% 0.21/0.46 ( ( ord_less_set_nat @ B @ A2 )
% 0.21/0.46 => ( ord_less_eq_set_nat @ B @ A2 ) ) ).
% 0.21/0.46
% 0.21/0.46 % dual_order.strict_implies_order
% 0.21/0.46 thf(fact_242_dual__order_Ostrict__implies__order,axiom,
% 0.21/0.46 ! [B: nat,A2: nat] :
% 0.21/0.46 ( ( ord_less_nat @ B @ A2 )
% 0.21/0.46 => ( ord_less_eq_nat @ B @ A2 ) ) ).
% 0.21/0.46
% 0.21/0.46 % dual_order.strict_implies_order
% 0.21/0.46 thf(fact_243_dual__order_Ostrict__iff__order,axiom,
% 0.21/0.46 ( ord_less_set_nat
% 0.21/0.46 = ( ^ [B4: set_nat,A4: set_nat] :
% 0.21/0.46 ( ( ord_less_eq_set_nat @ B4 @ A4 )
% 0.21/0.46 & ( A4 != B4 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % dual_order.strict_iff_order
% 0.21/0.46 thf(fact_244_dual__order_Ostrict__iff__order,axiom,
% 0.21/0.46 ( ord_less_nat
% 0.21/0.46 = ( ^ [B4: nat,A4: nat] :
% 0.21/0.46 ( ( ord_less_eq_nat @ B4 @ A4 )
% 0.21/0.46 & ( A4 != B4 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % dual_order.strict_iff_order
% 0.21/0.46 thf(fact_245_dual__order_Oorder__iff__strict,axiom,
% 0.21/0.46 ( ord_less_eq_set_nat
% 0.21/0.46 = ( ^ [B4: set_nat,A4: set_nat] :
% 0.21/0.46 ( ( ord_less_set_nat @ B4 @ A4 )
% 0.21/0.46 | ( A4 = B4 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % dual_order.order_iff_strict
% 0.21/0.46 thf(fact_246_dual__order_Oorder__iff__strict,axiom,
% 0.21/0.46 ( ord_less_eq_nat
% 0.21/0.46 = ( ^ [B4: nat,A4: nat] :
% 0.21/0.46 ( ( ord_less_nat @ B4 @ A4 )
% 0.21/0.46 | ( A4 = B4 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % dual_order.order_iff_strict
% 0.21/0.46 thf(fact_247_order_Ostrict__implies__order,axiom,
% 0.21/0.46 ! [A2: set_nat,B: set_nat] :
% 0.21/0.46 ( ( ord_less_set_nat @ A2 @ B )
% 0.21/0.46 => ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% 0.21/0.46
% 0.21/0.46 % order.strict_implies_order
% 0.21/0.46 thf(fact_248_order_Ostrict__implies__order,axiom,
% 0.21/0.46 ! [A2: nat,B: nat] :
% 0.21/0.46 ( ( ord_less_nat @ A2 @ B )
% 0.21/0.46 => ( ord_less_eq_nat @ A2 @ B ) ) ).
% 0.21/0.46
% 0.21/0.46 % order.strict_implies_order
% 0.21/0.46 thf(fact_249_dual__order_Ostrict__trans2,axiom,
% 0.21/0.46 ! [B: set_nat,A2: set_nat,C: set_nat] :
% 0.21/0.46 ( ( ord_less_set_nat @ B @ A2 )
% 0.21/0.46 => ( ( ord_less_eq_set_nat @ C @ B )
% 0.21/0.46 => ( ord_less_set_nat @ C @ A2 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % dual_order.strict_trans2
% 0.21/0.46 thf(fact_250_dual__order_Ostrict__trans2,axiom,
% 0.21/0.46 ! [B: nat,A2: nat,C: nat] :
% 0.21/0.46 ( ( ord_less_nat @ B @ A2 )
% 0.21/0.46 => ( ( ord_less_eq_nat @ C @ B )
% 0.21/0.46 => ( ord_less_nat @ C @ A2 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % dual_order.strict_trans2
% 0.21/0.46 thf(fact_251_dual__order_Ostrict__trans1,axiom,
% 0.21/0.46 ! [B: set_nat,A2: set_nat,C: set_nat] :
% 0.21/0.46 ( ( ord_less_eq_set_nat @ B @ A2 )
% 0.21/0.46 => ( ( ord_less_set_nat @ C @ B )
% 0.21/0.46 => ( ord_less_set_nat @ C @ A2 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % dual_order.strict_trans1
% 0.21/0.46 thf(fact_252_dual__order_Ostrict__trans1,axiom,
% 0.21/0.46 ! [B: nat,A2: nat,C: nat] :
% 0.21/0.46 ( ( ord_less_eq_nat @ B @ A2 )
% 0.21/0.46 => ( ( ord_less_nat @ C @ B )
% 0.21/0.46 => ( ord_less_nat @ C @ A2 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % dual_order.strict_trans1
% 0.21/0.46 thf(fact_253_order_Ostrict__iff__order,axiom,
% 0.21/0.46 ( ord_less_set_nat
% 0.21/0.46 = ( ^ [A4: set_nat,B4: set_nat] :
% 0.21/0.46 ( ( ord_less_eq_set_nat @ A4 @ B4 )
% 0.21/0.46 & ( A4 != B4 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % order.strict_iff_order
% 0.21/0.46 thf(fact_254_order_Ostrict__iff__order,axiom,
% 0.21/0.46 ( ord_less_nat
% 0.21/0.46 = ( ^ [A4: nat,B4: nat] :
% 0.21/0.46 ( ( ord_less_eq_nat @ A4 @ B4 )
% 0.21/0.46 & ( A4 != B4 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % order.strict_iff_order
% 0.21/0.46 thf(fact_255_order_Oorder__iff__strict,axiom,
% 0.21/0.46 ( ord_less_eq_set_nat
% 0.21/0.46 = ( ^ [A4: set_nat,B4: set_nat] :
% 0.21/0.46 ( ( ord_less_set_nat @ A4 @ B4 )
% 0.21/0.46 | ( A4 = B4 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % order.order_iff_strict
% 0.21/0.46 thf(fact_256_order_Oorder__iff__strict,axiom,
% 0.21/0.46 ( ord_less_eq_nat
% 0.21/0.46 = ( ^ [A4: nat,B4: nat] :
% 0.21/0.46 ( ( ord_less_nat @ A4 @ B4 )
% 0.21/0.46 | ( A4 = B4 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % order.order_iff_strict
% 0.21/0.46 thf(fact_257_order_Ostrict__trans2,axiom,
% 0.21/0.46 ! [A2: set_nat,B: set_nat,C: set_nat] :
% 0.21/0.46 ( ( ord_less_set_nat @ A2 @ B )
% 0.21/0.46 => ( ( ord_less_eq_set_nat @ B @ C )
% 0.21/0.46 => ( ord_less_set_nat @ A2 @ C ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % order.strict_trans2
% 0.21/0.46 thf(fact_258_order_Ostrict__trans2,axiom,
% 0.21/0.46 ! [A2: nat,B: nat,C: nat] :
% 0.21/0.46 ( ( ord_less_nat @ A2 @ B )
% 0.21/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.21/0.46 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % order.strict_trans2
% 0.21/0.46 thf(fact_259_order_Ostrict__trans1,axiom,
% 0.21/0.46 ! [A2: set_nat,B: set_nat,C: set_nat] :
% 0.21/0.46 ( ( ord_less_eq_set_nat @ A2 @ B )
% 0.21/0.46 => ( ( ord_less_set_nat @ B @ C )
% 0.21/0.46 => ( ord_less_set_nat @ A2 @ C ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % order.strict_trans1
% 0.21/0.46 thf(fact_260_order_Ostrict__trans1,axiom,
% 0.21/0.46 ! [A2: nat,B: nat,C: nat] :
% 0.21/0.46 ( ( ord_less_eq_nat @ A2 @ B )
% 0.21/0.46 => ( ( ord_less_nat @ B @ C )
% 0.21/0.46 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % order.strict_trans1
% 0.21/0.46
% 0.21/0.46 % Conjectures (1)
% 0.21/0.46 thf(conj_0,conjecture,
% 0.21/0.46 ( ( ! [I2: nat] :
% 0.21/0.46 ( ( ( p @ I2 )
% 0.21/0.46 != zero_zero_nat )
% 0.21/0.46 => ( ( ord_less_eq_nat @ one_one_nat @ I2 )
% 0.21/0.46 & ( ord_less_eq_nat @ I2 @ zero_zero_nat ) ) )
% 0.21/0.46 & ( ( groups1842438620at_nat
% 0.21/0.46 @ ^ [I2: nat] : ( times_times_nat @ ( p @ I2 ) @ I2 )
% 0.21/0.46 @ ( set_ord_atMost_nat @ zero_zero_nat ) )
% 0.21/0.46 = zero_zero_nat ) )
% 0.21/0.46 = ( p
% 0.21/0.46 = ( ^ [I2: nat] : zero_zero_nat ) ) ) ).
% 0.21/0.46
% 0.21/0.46 %------------------------------------------------------------------------------
% 0.21/0.46 ------------------------
% 2.54/1.42 ---- Embedded in THF ---
% 2.54/1.42 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.54/1.42 %%% Generated on Tue Feb 24 08:28:34 EST 2026
% 2.54/1.42 %%% using '$$dhol' embedding, version 1.3.0.
% 2.54/1.42 %%% Logic specification used:
% 2.54/1.42 %%% thf(spec, logic, $$dhol).
% 2.54/1.42
% 2.54/1.42 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.53O3If2WxG/DTF2DTF29954.p
% See solution above
% 2.54/1.42 ------------------------
% 2.54/1.44 ---- Cleaned THF ---
% 2.54/1.44 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 2.54/1.44 set_set_nat: $tType ).
% 2.54/1.44
% 2.54/1.44 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 2.54/1.44 set_nat: $tType ).
% 2.54/1.44
% 2.54/1.44 thf(ty_n_t__Nat__Onat,type,
% 2.54/1.44 nat: $tType ).
% 2.54/1.44
% 2.54/1.44 thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
% 2.54/1.44 one_one_nat: nat ).
% 2.54/1.44
% 2.54/1.44 thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
% 2.54/1.44 times_times_nat: nat > nat > nat ).
% 2.54/1.44
% 2.54/1.44 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 2.54/1.44 zero_zero_nat: nat ).
% 2.54/1.44
% 2.54/1.44 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
% 2.54/1.44 groups1842438620at_nat: ( nat > nat ) > set_nat > nat ).
% 2.54/1.44
% 2.54/1.44 thf(sy_c_Number__Partition__Mirabelle__zerdlymyoj_Opartitions,type,
% 2.54/1.44 number1551313001itions: ( nat > nat ) > nat > $o ).
% 2.54/1.44
% 2.54/1.44 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 2.54/1.44 ord_less_nat: nat > nat > $o ).
% 2.54/1.44
% 2.54/1.44 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
% 2.54/1.44 ord_less_set_nat: set_nat > set_nat > $o ).
% 2.54/1.44
% 2.54/1.44 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
% 2.54/1.44 ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
% 2.54/1.44
% 2.54/1.44 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 2.54/1.44 ord_less_eq_nat: nat > nat > $o ).
% 2.54/1.44
% 2.54/1.44 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
% 2.54/1.44 ord_less_eq_set_nat: set_nat > set_nat > $o ).
% 2.54/1.44
% 2.54/1.44 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 2.54/1.44 ord_le1613022364et_nat: set_set_nat > set_set_nat > $o ).
% 2.54/1.44
% 2.54/1.44 thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
% 2.54/1.44 collect_nat: ( nat > $o ) > set_nat ).
% 2.54/1.44
% 2.54/1.44 thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
% 2.54/1.44 collect_set_nat: ( set_nat > $o ) > set_set_nat ).
% 2.54/1.44
% 2.54/1.44 thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
% 2.54/1.44 set_ord_atMost_nat: nat > set_nat ).
% 2.54/1.44
% 2.54/1.44 thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
% 2.54/1.44 set_or1086813439et_nat: set_nat > set_set_nat ).
% 2.54/1.44
% 2.54/1.44 thf(sy_c_member_001t__Nat__Onat,type,
% 2.54/1.44 member_nat: nat > set_nat > $o ).
% 2.54/1.44
% 2.54/1.44 thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
% 2.54/1.44 member_set_nat: set_nat > set_set_nat > $o ).
% 2.54/1.44
% 2.54/1.44 thf(sy_v_p,type,
% 2.54/1.44 p: nat > nat ).
% 2.54/1.44
% 2.54/1.44 thf(fact_0_sum_Oneutral__const,axiom,
% 2.54/1.44 ! [A: set_nat] :
% 2.54/1.44 ( ( groups1842438620at_nat
% 2.54/1.44 @ ^ [Uu: nat] : zero_zero_nat
% 2.54/1.44 @ A )
% 2.54/1.44 = zero_zero_nat ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_1_partitionsE,axiom,
% 2.54/1.44 ! [P: nat > nat,N: nat] :
% 2.54/1.44 ( ( number1551313001itions @ P @ N )
% 2.54/1.44 => ~ ( ! [I: nat] :
% 2.54/1.44 ( ( ( P @ I )
% 2.54/1.44 != zero_zero_nat )
% 2.54/1.44 => ( ( ord_less_eq_nat @ one_one_nat @ I )
% 2.54/1.44 & ( ord_less_eq_nat @ I @ N ) ) )
% 2.54/1.44 => ( ( groups1842438620at_nat
% 2.54/1.44 @ ^ [I2: nat] : ( times_times_nat @ ( P @ I2 ) @ I2 )
% 2.54/1.44 @ ( set_ord_atMost_nat @ N ) )
% 2.54/1.44 != N ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_2_partitionsI,axiom,
% 2.54/1.44 ! [P: nat > nat,N: nat] :
% 2.54/1.44 ( ! [I3: nat] :
% 2.54/1.44 ( ( ( P @ I3 )
% 2.54/1.44 != zero_zero_nat )
% 2.54/1.44 => ( ( ord_less_eq_nat @ one_one_nat @ I3 )
% 2.54/1.44 & ( ord_less_eq_nat @ I3 @ N ) ) )
% 2.54/1.44 => ( ( ( groups1842438620at_nat
% 2.54/1.44 @ ^ [I2: nat] : ( times_times_nat @ ( P @ I2 ) @ I2 )
% 2.54/1.44 @ ( set_ord_atMost_nat @ N ) )
% 2.54/1.44 = N )
% 2.54/1.44 => ( number1551313001itions @ P @ N ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_3_partitions__def,axiom,
% 2.54/1.44 ( number1551313001itions
% 2.54/1.44 = ( ^ [P2: nat > nat,N2: nat] :
% 2.54/1.44 ( ! [I2: nat] :
% 2.54/1.44 ( ( ( P2 @ I2 )
% 2.54/1.44 != zero_zero_nat )
% 2.54/1.44 => ( ( ord_less_eq_nat @ one_one_nat @ I2 )
% 2.54/1.44 & ( ord_less_eq_nat @ I2 @ N2 ) ) )
% 2.54/1.44 & ( ( groups1842438620at_nat
% 2.54/1.44 @ ^ [I2: nat] : ( times_times_nat @ ( P2 @ I2 ) @ I2 )
% 2.54/1.44 @ ( set_ord_atMost_nat @ N2 ) )
% 2.54/1.44 = N2 ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_4_nat__1__eq__mult__iff,axiom,
% 2.54/1.44 ! [M: nat,N: nat] :
% 2.54/1.44 ( ( one_one_nat
% 2.54/1.44 = ( times_times_nat @ M @ N ) )
% 2.54/1.44 = ( ( M = one_one_nat )
% 2.54/1.44 & ( N = one_one_nat ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_5_nat__mult__eq__1__iff,axiom,
% 2.54/1.44 ! [M: nat,N: nat] :
% 2.54/1.44 ( ( ( times_times_nat @ M @ N )
% 2.54/1.44 = one_one_nat )
% 2.54/1.44 = ( ( M = one_one_nat )
% 2.54/1.44 & ( N = one_one_nat ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_6_atMost__subset__iff,axiom,
% 2.54/1.44 ! [X: set_nat,Y: set_nat] :
% 2.54/1.44 ( ( ord_le1613022364et_nat @ ( set_or1086813439et_nat @ X ) @ ( set_or1086813439et_nat @ Y ) )
% 2.54/1.44 = ( ord_less_eq_set_nat @ X @ Y ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_7_atMost__subset__iff,axiom,
% 2.54/1.44 ! [X: nat,Y: nat] :
% 2.54/1.44 ( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X ) @ ( set_ord_atMost_nat @ Y ) )
% 2.54/1.44 = ( ord_less_eq_nat @ X @ Y ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_8_atMost__iff,axiom,
% 2.54/1.44 ! [I4: set_nat,K: set_nat] :
% 2.54/1.44 ( ( member_set_nat @ I4 @ ( set_or1086813439et_nat @ K ) )
% 2.54/1.44 = ( ord_less_eq_set_nat @ I4 @ K ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_9_atMost__iff,axiom,
% 2.54/1.44 ! [I4: nat,K: nat] :
% 2.54/1.44 ( ( member_nat @ I4 @ ( set_ord_atMost_nat @ K ) )
% 2.54/1.44 = ( ord_less_eq_nat @ I4 @ K ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_10_atMost__eq__iff,axiom,
% 2.54/1.44 ! [X: nat,Y: nat] :
% 2.54/1.44 ( ( ( set_ord_atMost_nat @ X )
% 2.54/1.44 = ( set_ord_atMost_nat @ Y ) )
% 2.54/1.44 = ( X = Y ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_11_mult__cancel__right,axiom,
% 2.54/1.44 ! [A2: nat,C: nat,B: nat] :
% 2.54/1.44 ( ( ( times_times_nat @ A2 @ C )
% 2.54/1.44 = ( times_times_nat @ B @ C ) )
% 2.54/1.44 = ( ( C = zero_zero_nat )
% 2.54/1.44 | ( A2 = B ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_12_mult__cancel__left,axiom,
% 2.54/1.44 ! [C: nat,A2: nat,B: nat] :
% 2.54/1.44 ( ( ( times_times_nat @ C @ A2 )
% 2.54/1.44 = ( times_times_nat @ C @ B ) )
% 2.54/1.44 = ( ( C = zero_zero_nat )
% 2.54/1.44 | ( A2 = B ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_13_mult__eq__0__iff,axiom,
% 2.54/1.44 ! [A2: nat,B: nat] :
% 2.54/1.44 ( ( ( times_times_nat @ A2 @ B )
% 2.54/1.44 = zero_zero_nat )
% 2.54/1.44 = ( ( A2 = zero_zero_nat )
% 2.54/1.44 | ( B = zero_zero_nat ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_14_mult__zero__right,axiom,
% 2.54/1.44 ! [A2: nat] :
% 2.54/1.44 ( ( times_times_nat @ A2 @ zero_zero_nat )
% 2.54/1.44 = zero_zero_nat ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_15_mult__zero__left,axiom,
% 2.54/1.44 ! [A2: nat] :
% 2.54/1.44 ( ( times_times_nat @ zero_zero_nat @ A2 )
% 2.54/1.44 = zero_zero_nat ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_16_bot__nat__0_Oextremum,axiom,
% 2.54/1.44 ! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_17_le0,axiom,
% 2.54/1.44 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_18_mult__cancel2,axiom,
% 2.54/1.44 ! [M: nat,K: nat,N: nat] :
% 2.54/1.44 ( ( ( times_times_nat @ M @ K )
% 2.54/1.44 = ( times_times_nat @ N @ K ) )
% 2.54/1.44 = ( ( M = N )
% 2.54/1.44 | ( K = zero_zero_nat ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_19_mult__cancel1,axiom,
% 2.54/1.44 ! [K: nat,M: nat,N: nat] :
% 2.54/1.44 ( ( ( times_times_nat @ K @ M )
% 2.54/1.44 = ( times_times_nat @ K @ N ) )
% 2.54/1.44 = ( ( M = N )
% 2.54/1.44 | ( K = zero_zero_nat ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_20_mult__0__right,axiom,
% 2.54/1.44 ! [M: nat] :
% 2.54/1.44 ( ( times_times_nat @ M @ zero_zero_nat )
% 2.54/1.44 = zero_zero_nat ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_21_mult__is__0,axiom,
% 2.54/1.44 ! [M: nat,N: nat] :
% 2.54/1.44 ( ( ( times_times_nat @ M @ N )
% 2.54/1.44 = zero_zero_nat )
% 2.54/1.44 = ( ( M = zero_zero_nat )
% 2.54/1.44 | ( N = zero_zero_nat ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_22_bounded__Max__nat,axiom,
% 2.54/1.44 ! [P3: nat > $o,X: nat,M2: nat] :
% 2.54/1.44 ( ( P3 @ X )
% 2.54/1.44 => ( ! [X2: nat] :
% 2.54/1.44 ( ( P3 @ X2 )
% 2.54/1.44 => ( ord_less_eq_nat @ X2 @ M2 ) )
% 2.54/1.44 => ~ ! [M3: nat] :
% 2.54/1.44 ( ( P3 @ M3 )
% 2.54/1.44 => ~ ! [X3: nat] :
% 2.54/1.44 ( ( P3 @ X3 )
% 2.54/1.44 => ( ord_less_eq_nat @ X3 @ M3 ) ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_23_Nat_Oex__has__greatest__nat,axiom,
% 2.54/1.44 ! [P3: nat > $o,K: nat,B: nat] :
% 2.54/1.44 ( ( P3 @ K )
% 2.54/1.44 => ( ! [Y2: nat] :
% 2.54/1.44 ( ( P3 @ Y2 )
% 2.54/1.44 => ( ord_less_eq_nat @ Y2 @ B ) )
% 2.54/1.44 => ? [X2: nat] :
% 2.54/1.44 ( ( P3 @ X2 )
% 2.54/1.44 & ! [Y3: nat] :
% 2.54/1.44 ( ( P3 @ Y3 )
% 2.54/1.44 => ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_24_nat__le__linear,axiom,
% 2.54/1.44 ! [M: nat,N: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ M @ N )
% 2.54/1.44 | ( ord_less_eq_nat @ N @ M ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_25_le__antisym,axiom,
% 2.54/1.44 ! [M: nat,N: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ M @ N )
% 2.54/1.44 => ( ( ord_less_eq_nat @ N @ M )
% 2.54/1.44 => ( M = N ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_26_eq__imp__le,axiom,
% 2.54/1.44 ! [M: nat,N: nat] :
% 2.54/1.44 ( ( M = N )
% 2.54/1.44 => ( ord_less_eq_nat @ M @ N ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_27_le__trans,axiom,
% 2.54/1.44 ! [I4: nat,J: nat,K: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ I4 @ J )
% 2.54/1.44 => ( ( ord_less_eq_nat @ J @ K )
% 2.54/1.44 => ( ord_less_eq_nat @ I4 @ K ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_28_le__refl,axiom,
% 2.54/1.44 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_29_sum_Oreindex__bij__witness,axiom,
% 2.54/1.44 ! [S: set_nat,I4: nat > nat,J: nat > nat,T: set_nat,H: nat > nat,G: nat > nat] :
% 2.54/1.44 ( ! [A3: nat] :
% 2.54/1.44 ( ( member_nat @ A3 @ S )
% 2.54/1.44 => ( ( I4 @ ( J @ A3 ) )
% 2.54/1.44 = A3 ) )
% 2.54/1.44 => ( ! [A3: nat] :
% 2.54/1.44 ( ( member_nat @ A3 @ S )
% 2.54/1.44 => ( member_nat @ ( J @ A3 ) @ T ) )
% 2.54/1.44 => ( ! [B2: nat] :
% 2.54/1.44 ( ( member_nat @ B2 @ T )
% 2.54/1.44 => ( ( J @ ( I4 @ B2 ) )
% 2.54/1.44 = B2 ) )
% 2.54/1.44 => ( ! [B2: nat] :
% 2.54/1.44 ( ( member_nat @ B2 @ T )
% 2.54/1.44 => ( member_nat @ ( I4 @ B2 ) @ S ) )
% 2.54/1.44 => ( ! [A3: nat] :
% 2.54/1.44 ( ( member_nat @ A3 @ S )
% 2.54/1.44 => ( ( H @ ( J @ A3 ) )
% 2.54/1.44 = ( G @ A3 ) ) )
% 2.54/1.44 => ( ( groups1842438620at_nat @ G @ S )
% 2.54/1.44 = ( groups1842438620at_nat @ H @ T ) ) ) ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_30_sum_Oeq__general__inverses,axiom,
% 2.54/1.44 ! [B3: set_nat,K: nat > nat,A: set_nat,H: nat > nat,Gamma: nat > nat,Phi: nat > nat] :
% 2.54/1.44 ( ! [Y2: nat] :
% 2.54/1.44 ( ( member_nat @ Y2 @ B3 )
% 2.54/1.44 => ( ( member_nat @ ( K @ Y2 ) @ A )
% 2.54/1.44 & ( ( H @ ( K @ Y2 ) )
% 2.54/1.44 = Y2 ) ) )
% 2.54/1.44 => ( ! [X2: nat] :
% 2.54/1.44 ( ( member_nat @ X2 @ A )
% 2.54/1.44 => ( ( member_nat @ ( H @ X2 ) @ B3 )
% 2.54/1.44 & ( ( K @ ( H @ X2 ) )
% 2.54/1.44 = X2 )
% 2.54/1.44 & ( ( Gamma @ ( H @ X2 ) )
% 2.54/1.44 = ( Phi @ X2 ) ) ) )
% 2.54/1.44 => ( ( groups1842438620at_nat @ Phi @ A )
% 2.54/1.44 = ( groups1842438620at_nat @ Gamma @ B3 ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_31_sum_Oeq__general,axiom,
% 2.54/1.44 ! [B3: set_nat,A: set_nat,H: nat > nat,Gamma: nat > nat,Phi: nat > nat] :
% 2.54/1.44 ( ! [Y2: nat] :
% 2.54/1.44 ( ( member_nat @ Y2 @ B3 )
% 2.54/1.44 => ? [X3: nat] :
% 2.54/1.44 ( ( member_nat @ X3 @ A )
% 2.54/1.44 & ( ( H @ X3 )
% 2.54/1.44 = Y2 )
% 2.54/1.44 & ! [Ya: nat] :
% 2.54/1.44 ( ( ( member_nat @ Ya @ A )
% 2.54/1.44 & ( ( H @ Ya )
% 2.54/1.44 = Y2 ) )
% 2.54/1.44 => ( Ya = X3 ) ) ) )
% 2.54/1.44 => ( ! [X2: nat] :
% 2.54/1.44 ( ( member_nat @ X2 @ A )
% 2.54/1.44 => ( ( member_nat @ ( H @ X2 ) @ B3 )
% 2.54/1.44 & ( ( Gamma @ ( H @ X2 ) )
% 2.54/1.44 = ( Phi @ X2 ) ) ) )
% 2.54/1.44 => ( ( groups1842438620at_nat @ Phi @ A )
% 2.54/1.44 = ( groups1842438620at_nat @ Gamma @ B3 ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_32_sum_Ocong,axiom,
% 2.54/1.44 ! [A: set_nat,B3: set_nat,G: nat > nat,H: nat > nat] :
% 2.54/1.44 ( ( A = B3 )
% 2.54/1.44 => ( ! [X2: nat] :
% 2.54/1.44 ( ( member_nat @ X2 @ B3 )
% 2.54/1.44 => ( ( G @ X2 )
% 2.54/1.44 = ( H @ X2 ) ) )
% 2.54/1.44 => ( ( groups1842438620at_nat @ G @ A )
% 2.54/1.44 = ( groups1842438620at_nat @ H @ B3 ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_33_sum_Oswap,axiom,
% 2.54/1.44 ! [G: nat > nat > nat,B3: set_nat,A: set_nat] :
% 2.54/1.44 ( ( groups1842438620at_nat
% 2.54/1.44 @ ^ [I2: nat] : ( groups1842438620at_nat @ ( G @ I2 ) @ B3 )
% 2.54/1.44 @ A )
% 2.54/1.44 = ( groups1842438620at_nat
% 2.54/1.44 @ ^ [J2: nat] :
% 2.54/1.44 ( groups1842438620at_nat
% 2.54/1.44 @ ^ [I2: nat] : ( G @ I2 @ J2 )
% 2.54/1.44 @ A )
% 2.54/1.44 @ B3 ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_34_mult__right__cancel,axiom,
% 2.54/1.44 ! [C: nat,A2: nat,B: nat] :
% 2.54/1.44 ( ( C != zero_zero_nat )
% 2.54/1.44 => ( ( ( times_times_nat @ A2 @ C )
% 2.54/1.44 = ( times_times_nat @ B @ C ) )
% 2.54/1.44 = ( A2 = B ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_35_mult__left__cancel,axiom,
% 2.54/1.44 ! [C: nat,A2: nat,B: nat] :
% 2.54/1.44 ( ( C != zero_zero_nat )
% 2.54/1.44 => ( ( ( times_times_nat @ C @ A2 )
% 2.54/1.44 = ( times_times_nat @ C @ B ) )
% 2.54/1.44 = ( A2 = B ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_36_no__zero__divisors,axiom,
% 2.54/1.44 ! [A2: nat,B: nat] :
% 2.54/1.44 ( ( A2 != zero_zero_nat )
% 2.54/1.44 => ( ( B != zero_zero_nat )
% 2.54/1.44 => ( ( times_times_nat @ A2 @ B )
% 2.54/1.44 != zero_zero_nat ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_37_divisors__zero,axiom,
% 2.54/1.44 ! [A2: nat,B: nat] :
% 2.54/1.44 ( ( ( times_times_nat @ A2 @ B )
% 2.54/1.44 = zero_zero_nat )
% 2.54/1.44 => ( ( A2 = zero_zero_nat )
% 2.54/1.44 | ( B = zero_zero_nat ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_38_mult__not__zero,axiom,
% 2.54/1.44 ! [A2: nat,B: nat] :
% 2.54/1.44 ( ( ( times_times_nat @ A2 @ B )
% 2.54/1.44 != zero_zero_nat )
% 2.54/1.44 => ( ( A2 != zero_zero_nat )
% 2.54/1.44 & ( B != zero_zero_nat ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_39_zero__neq__one,axiom,
% 2.54/1.44 zero_zero_nat != one_one_nat ).
% 2.54/1.44
% 2.54/1.44 thf(fact_40_sum_Onot__neutral__contains__not__neutral,axiom,
% 2.54/1.44 ! [G: nat > nat,A: set_nat] :
% 2.54/1.44 ( ( ( groups1842438620at_nat @ G @ A )
% 2.54/1.44 != zero_zero_nat )
% 2.54/1.44 => ~ ! [A3: nat] :
% 2.54/1.44 ( ( member_nat @ A3 @ A )
% 2.54/1.44 => ( ( G @ A3 )
% 2.54/1.44 = zero_zero_nat ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_41_sum_Oneutral,axiom,
% 2.54/1.44 ! [A: set_nat,G: nat > nat] :
% 2.54/1.44 ( ! [X2: nat] :
% 2.54/1.44 ( ( member_nat @ X2 @ A )
% 2.54/1.44 => ( ( G @ X2 )
% 2.54/1.44 = zero_zero_nat ) )
% 2.54/1.44 => ( ( groups1842438620at_nat @ G @ A )
% 2.54/1.44 = zero_zero_nat ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_42_mem__Collect__eq,axiom,
% 2.54/1.44 ! [A2: nat,P3: nat > $o] :
% 2.54/1.44 ( ( member_nat @ A2 @ ( collect_nat @ P3 ) )
% 2.54/1.44 = ( P3 @ A2 ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_43_Collect__mem__eq,axiom,
% 2.54/1.44 ! [A: set_nat] :
% 2.54/1.44 ( ( collect_nat
% 2.54/1.44 @ ^ [X4: nat] : ( member_nat @ X4 @ A ) )
% 2.54/1.44 = A ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_44_Collect__cong,axiom,
% 2.54/1.44 ! [P3: nat > $o,Q: nat > $o] :
% 2.54/1.44 ( ! [X2: nat] :
% 2.54/1.44 ( ( P3 @ X2 )
% 2.54/1.44 = ( Q @ X2 ) )
% 2.54/1.44 => ( ( collect_nat @ P3 )
% 2.54/1.44 = ( collect_nat @ Q ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_45_bot__nat__0_Oextremum__uniqueI,axiom,
% 2.54/1.44 ! [A2: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 2.54/1.44 => ( A2 = zero_zero_nat ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_46_bot__nat__0_Oextremum__unique,axiom,
% 2.54/1.44 ! [A2: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 2.54/1.44 = ( A2 = zero_zero_nat ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_47_le__0__eq,axiom,
% 2.54/1.44 ! [N: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 2.54/1.44 = ( N = zero_zero_nat ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_48_less__eq__nat_Osimps_I1_J,axiom,
% 2.54/1.44 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_49_mult__0,axiom,
% 2.54/1.44 ! [N: nat] :
% 2.54/1.44 ( ( times_times_nat @ zero_zero_nat @ N )
% 2.54/1.44 = zero_zero_nat ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_50_mult__le__mono2,axiom,
% 2.54/1.44 ! [I4: nat,J: nat,K: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ I4 @ J )
% 2.54/1.44 => ( ord_less_eq_nat @ ( times_times_nat @ K @ I4 ) @ ( times_times_nat @ K @ J ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_51_mult__le__mono1,axiom,
% 2.54/1.44 ! [I4: nat,J: nat,K: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ I4 @ J )
% 2.54/1.44 => ( ord_less_eq_nat @ ( times_times_nat @ I4 @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_52_mult__le__mono,axiom,
% 2.54/1.44 ! [I4: nat,J: nat,K: nat,L: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ I4 @ J )
% 2.54/1.44 => ( ( ord_less_eq_nat @ K @ L )
% 2.54/1.44 => ( ord_less_eq_nat @ ( times_times_nat @ I4 @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_53_le__square,axiom,
% 2.54/1.44 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_54_le__cube,axiom,
% 2.54/1.44 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_55_nat__mult__1__right,axiom,
% 2.54/1.44 ! [N: nat] :
% 2.54/1.44 ( ( times_times_nat @ N @ one_one_nat )
% 2.54/1.44 = N ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_56_nat__mult__1,axiom,
% 2.54/1.44 ! [N: nat] :
% 2.54/1.44 ( ( times_times_nat @ one_one_nat @ N )
% 2.54/1.44 = N ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_57_lambda__zero,axiom,
% 2.54/1.44 ( ( ^ [H2: nat] : zero_zero_nat )
% 2.54/1.44 = ( times_times_nat @ zero_zero_nat ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_58_lambda__one,axiom,
% 2.54/1.44 ( ( ^ [X4: nat] : X4 )
% 2.54/1.44 = ( times_times_nat @ one_one_nat ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_59_sum__mono,axiom,
% 2.54/1.44 ! [K2: set_nat,F: nat > nat,G: nat > nat] :
% 2.54/1.44 ( ! [I3: nat] :
% 2.54/1.44 ( ( member_nat @ I3 @ K2 )
% 2.54/1.44 => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 2.54/1.44 => ( ord_less_eq_nat @ ( groups1842438620at_nat @ F @ K2 ) @ ( groups1842438620at_nat @ G @ K2 ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_60_sum__distrib__right,axiom,
% 2.54/1.44 ! [F: nat > nat,A: set_nat,R: nat] :
% 2.54/1.44 ( ( times_times_nat @ ( groups1842438620at_nat @ F @ A ) @ R )
% 2.54/1.44 = ( groups1842438620at_nat
% 2.54/1.44 @ ^ [N2: nat] : ( times_times_nat @ ( F @ N2 ) @ R )
% 2.54/1.44 @ A ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_61_sum__distrib__left,axiom,
% 2.54/1.44 ! [R: nat,F: nat > nat,A: set_nat] :
% 2.54/1.44 ( ( times_times_nat @ R @ ( groups1842438620at_nat @ F @ A ) )
% 2.54/1.44 = ( groups1842438620at_nat
% 2.54/1.44 @ ^ [N2: nat] : ( times_times_nat @ R @ ( F @ N2 ) )
% 2.54/1.44 @ A ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_62_sum__product,axiom,
% 2.54/1.44 ! [F: nat > nat,A: set_nat,G: nat > nat,B3: set_nat] :
% 2.54/1.44 ( ( times_times_nat @ ( groups1842438620at_nat @ F @ A ) @ ( groups1842438620at_nat @ G @ B3 ) )
% 2.54/1.44 = ( groups1842438620at_nat
% 2.54/1.44 @ ^ [I2: nat] :
% 2.54/1.44 ( groups1842438620at_nat
% 2.54/1.44 @ ^ [J2: nat] : ( times_times_nat @ ( F @ I2 ) @ ( G @ J2 ) )
% 2.54/1.44 @ B3 )
% 2.54/1.44 @ A ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_63_atMost__def,axiom,
% 2.54/1.44 ( set_or1086813439et_nat
% 2.54/1.44 = ( ^ [U: set_nat] :
% 2.54/1.44 ( collect_set_nat
% 2.54/1.44 @ ^ [X4: set_nat] : ( ord_less_eq_set_nat @ X4 @ U ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_64_atMost__def,axiom,
% 2.54/1.44 ( set_ord_atMost_nat
% 2.54/1.44 = ( ^ [U: nat] :
% 2.54/1.44 ( collect_nat
% 2.54/1.44 @ ^ [X4: nat] : ( ord_less_eq_nat @ X4 @ U ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_65_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 2.54/1.44 ! [A2: nat,B: nat,C: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ A2 @ B )
% 2.54/1.44 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 2.54/1.44 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_66_mult__nonneg__nonpos2,axiom,
% 2.54/1.44 ! [A2: nat,B: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 2.54/1.44 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 2.54/1.44 => ( ord_less_eq_nat @ ( times_times_nat @ B @ A2 ) @ zero_zero_nat ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_67_mult__nonpos__nonneg,axiom,
% 2.54/1.44 ! [A2: nat,B: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 2.54/1.44 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 2.54/1.44 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_68_mult__nonneg__nonpos,axiom,
% 2.54/1.44 ! [A2: nat,B: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 2.54/1.44 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 2.54/1.44 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_69_mult__nonneg__nonneg,axiom,
% 2.54/1.44 ! [A2: nat,B: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 2.54/1.44 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 2.54/1.44 => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_70_split__mult__neg__le,axiom,
% 2.54/1.44 ! [A2: nat,B: nat] :
% 2.54/1.44 ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 2.54/1.44 & ( ord_less_eq_nat @ B @ zero_zero_nat ) )
% 2.54/1.44 | ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 2.54/1.44 & ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
% 2.54/1.44 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_71_mult__right__mono,axiom,
% 2.54/1.44 ! [A2: nat,B: nat,C: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ A2 @ B )
% 2.54/1.44 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 2.54/1.44 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_72_mult__left__mono,axiom,
% 2.54/1.44 ! [A2: nat,B: nat,C: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ A2 @ B )
% 2.54/1.44 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 2.54/1.44 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_73_mult__mono_H,axiom,
% 2.54/1.44 ! [A2: nat,B: nat,C: nat,D: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ A2 @ B )
% 2.54/1.44 => ( ( ord_less_eq_nat @ C @ D )
% 2.54/1.44 => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 2.54/1.44 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 2.54/1.44 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_74_mult__mono,axiom,
% 2.54/1.44 ! [A2: nat,B: nat,C: nat,D: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ A2 @ B )
% 2.54/1.44 => ( ( ord_less_eq_nat @ C @ D )
% 2.54/1.44 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 2.54/1.44 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 2.54/1.44 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_75_not__one__le__zero,axiom,
% 2.54/1.44 ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_76_zero__le__one,axiom,
% 2.54/1.44 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 2.54/1.44
% 2.54/1.44 thf(fact_77_sum__nonpos,axiom,
% 2.54/1.44 ! [A: set_nat,F: nat > nat] :
% 2.54/1.44 ( ! [X2: nat] :
% 2.54/1.44 ( ( member_nat @ X2 @ A )
% 2.54/1.44 => ( ord_less_eq_nat @ ( F @ X2 ) @ zero_zero_nat ) )
% 2.54/1.44 => ( ord_less_eq_nat @ ( groups1842438620at_nat @ F @ A ) @ zero_zero_nat ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_78_sum__nonneg,axiom,
% 2.54/1.44 ! [A: set_nat,F: nat > nat] :
% 2.54/1.44 ( ! [X2: nat] :
% 2.54/1.44 ( ( member_nat @ X2 @ A )
% 2.54/1.44 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
% 2.54/1.44 => ( ord_less_eq_nat @ zero_zero_nat @ ( groups1842438620at_nat @ F @ A ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_79_mult__eq__self__implies__10,axiom,
% 2.54/1.44 ! [M: nat,N: nat] :
% 2.54/1.44 ( ( M
% 2.54/1.44 = ( times_times_nat @ M @ N ) )
% 2.54/1.44 => ( ( N = one_one_nat )
% 2.54/1.44 | ( M = zero_zero_nat ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_80_mult__left__le,axiom,
% 2.54/1.44 ! [C: nat,A2: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ C @ one_one_nat )
% 2.54/1.44 => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 2.54/1.44 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ A2 ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_81_mult__le__one,axiom,
% 2.54/1.44 ! [A2: nat,B: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ A2 @ one_one_nat )
% 2.54/1.44 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 2.54/1.44 => ( ( ord_less_eq_nat @ B @ one_one_nat )
% 2.54/1.44 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ one_one_nat ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_82_mult_Oright__neutral,axiom,
% 2.54/1.44 ! [A2: nat] :
% 2.54/1.44 ( ( times_times_nat @ A2 @ one_one_nat )
% 2.54/1.44 = A2 ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_83_mult_Oleft__neutral,axiom,
% 2.54/1.44 ! [A2: nat] :
% 2.54/1.44 ( ( times_times_nat @ one_one_nat @ A2 )
% 2.54/1.44 = A2 ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_84_le__zero__eq,axiom,
% 2.54/1.44 ! [N: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 2.54/1.44 = ( N = zero_zero_nat ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_85_subsetI,axiom,
% 2.54/1.44 ! [A: set_nat,B3: set_nat] :
% 2.54/1.44 ( ! [X2: nat] :
% 2.54/1.44 ( ( member_nat @ X2 @ A )
% 2.54/1.44 => ( member_nat @ X2 @ B3 ) )
% 2.54/1.44 => ( ord_less_eq_set_nat @ A @ B3 ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_86_subset__antisym,axiom,
% 2.54/1.44 ! [A: set_nat,B3: set_nat] :
% 2.54/1.44 ( ( ord_less_eq_set_nat @ A @ B3 )
% 2.54/1.44 => ( ( ord_less_eq_set_nat @ B3 @ A )
% 2.54/1.44 => ( A = B3 ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_87_order__refl,axiom,
% 2.54/1.44 ! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_88_order__refl,axiom,
% 2.54/1.44 ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_89_nat__mult__eq__cancel__disj,axiom,
% 2.54/1.44 ! [K: nat,M: nat,N: nat] :
% 2.54/1.44 ( ( ( times_times_nat @ K @ M )
% 2.54/1.44 = ( times_times_nat @ K @ N ) )
% 2.54/1.44 = ( ( K = zero_zero_nat )
% 2.54/1.44 | ( M = N ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_90_mult_Ocomm__neutral,axiom,
% 2.54/1.44 ! [A2: nat] :
% 2.54/1.44 ( ( times_times_nat @ A2 @ one_one_nat )
% 2.54/1.44 = A2 ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_91_comm__monoid__mult__class_Omult__1,axiom,
% 2.54/1.44 ! [A2: nat] :
% 2.54/1.44 ( ( times_times_nat @ one_one_nat @ A2 )
% 2.54/1.44 = A2 ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_92_order__subst1,axiom,
% 2.54/1.44 ! [A2: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
% 2.54/1.44 => ( ( ord_less_eq_set_nat @ B @ C )
% 2.54/1.44 => ( ! [X2: set_nat,Y2: set_nat] :
% 2.54/1.44 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 2.54/1.44 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.54/1.44 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_93_order__subst1,axiom,
% 2.54/1.44 ! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
% 2.54/1.44 ( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
% 2.54/1.44 => ( ( ord_less_eq_nat @ B @ C )
% 2.54/1.44 => ( ! [X2: nat,Y2: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.54/1.44 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.54/1.44 => ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_94_order__subst1,axiom,
% 2.54/1.44 ! [A2: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
% 2.54/1.44 ( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
% 2.54/1.44 => ( ( ord_less_eq_set_nat @ B @ C )
% 2.54/1.44 => ( ! [X2: set_nat,Y2: set_nat] :
% 2.54/1.44 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 2.54/1.44 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.54/1.44 => ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_95_order__subst1,axiom,
% 2.54/1.44 ! [A2: nat,F: nat > nat,B: nat,C: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
% 2.54/1.44 => ( ( ord_less_eq_nat @ B @ C )
% 2.54/1.44 => ( ! [X2: nat,Y2: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.54/1.44 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.54/1.44 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_96_order__subst2,axiom,
% 2.54/1.44 ! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ A2 @ B )
% 2.54/1.44 => ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
% 2.54/1.44 => ( ! [X2: nat,Y2: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.54/1.44 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.54/1.44 => ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_97_order__subst2,axiom,
% 2.54/1.44 ! [A2: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
% 2.54/1.44 ( ( ord_less_eq_set_nat @ A2 @ B )
% 2.54/1.44 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 2.54/1.44 => ( ! [X2: set_nat,Y2: set_nat] :
% 2.54/1.44 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 2.54/1.44 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.54/1.44 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_98_order__subst2,axiom,
% 2.54/1.44 ! [A2: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
% 2.54/1.44 ( ( ord_less_eq_set_nat @ A2 @ B )
% 2.54/1.44 => ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
% 2.54/1.44 => ( ! [X2: set_nat,Y2: set_nat] :
% 2.54/1.44 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 2.54/1.44 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.54/1.44 => ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_99_order__subst2,axiom,
% 2.54/1.44 ! [A2: nat,B: nat,F: nat > nat,C: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ A2 @ B )
% 2.54/1.44 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 2.54/1.44 => ( ! [X2: nat,Y2: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.54/1.44 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.54/1.44 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_100_ord__eq__le__subst,axiom,
% 2.54/1.44 ! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
% 2.54/1.44 ( ( A2
% 2.54/1.44 = ( F @ B ) )
% 2.54/1.44 => ( ( ord_less_eq_nat @ B @ C )
% 2.54/1.44 => ( ! [X2: nat,Y2: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.54/1.44 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.54/1.44 => ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_101_ord__eq__le__subst,axiom,
% 2.54/1.44 ! [A2: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
% 2.54/1.44 ( ( A2
% 2.54/1.44 = ( F @ B ) )
% 2.54/1.44 => ( ( ord_less_eq_set_nat @ B @ C )
% 2.54/1.44 => ( ! [X2: set_nat,Y2: set_nat] :
% 2.54/1.44 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 2.54/1.44 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.54/1.44 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_102_ord__eq__le__subst,axiom,
% 2.54/1.44 ! [A2: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
% 2.54/1.44 ( ( A2
% 2.54/1.44 = ( F @ B ) )
% 2.54/1.44 => ( ( ord_less_eq_set_nat @ B @ C )
% 2.54/1.44 => ( ! [X2: set_nat,Y2: set_nat] :
% 2.54/1.44 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 2.54/1.44 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.54/1.44 => ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_103_ord__eq__le__subst,axiom,
% 2.54/1.44 ! [A2: nat,F: nat > nat,B: nat,C: nat] :
% 2.54/1.44 ( ( A2
% 2.54/1.44 = ( F @ B ) )
% 2.54/1.44 => ( ( ord_less_eq_nat @ B @ C )
% 2.54/1.44 => ( ! [X2: nat,Y2: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.54/1.44 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.54/1.44 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_104_ord__le__eq__subst,axiom,
% 2.54/1.44 ! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ A2 @ B )
% 2.54/1.44 => ( ( ( F @ B )
% 2.54/1.44 = C )
% 2.54/1.44 => ( ! [X2: nat,Y2: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.54/1.44 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.54/1.44 => ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_105_ord__le__eq__subst,axiom,
% 2.54/1.44 ! [A2: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
% 2.54/1.44 ( ( ord_less_eq_set_nat @ A2 @ B )
% 2.54/1.44 => ( ( ( F @ B )
% 2.54/1.44 = C )
% 2.54/1.44 => ( ! [X2: set_nat,Y2: set_nat] :
% 2.54/1.44 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 2.54/1.44 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.54/1.44 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_106_ord__le__eq__subst,axiom,
% 2.54/1.44 ! [A2: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
% 2.54/1.44 ( ( ord_less_eq_set_nat @ A2 @ B )
% 2.54/1.44 => ( ( ( F @ B )
% 2.54/1.44 = C )
% 2.54/1.44 => ( ! [X2: set_nat,Y2: set_nat] :
% 2.54/1.44 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 2.54/1.44 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.54/1.44 => ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_107_ord__le__eq__subst,axiom,
% 2.54/1.44 ! [A2: nat,B: nat,F: nat > nat,C: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ A2 @ B )
% 2.54/1.44 => ( ( ( F @ B )
% 2.54/1.44 = C )
% 2.54/1.44 => ( ! [X2: nat,Y2: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.54/1.44 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.54/1.44 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_108_eq__iff,axiom,
% 2.54/1.44 ( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
% 2.54/1.44 = ( ^ [X4: set_nat,Y5: set_nat] :
% 2.54/1.44 ( ( ord_less_eq_set_nat @ X4 @ Y5 )
% 2.54/1.44 & ( ord_less_eq_set_nat @ Y5 @ X4 ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_109_eq__iff,axiom,
% 2.54/1.44 ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
% 2.54/1.44 = ( ^ [X4: nat,Y5: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ X4 @ Y5 )
% 2.54/1.44 & ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_110_antisym,axiom,
% 2.54/1.44 ! [X: set_nat,Y: set_nat] :
% 2.54/1.44 ( ( ord_less_eq_set_nat @ X @ Y )
% 2.54/1.44 => ( ( ord_less_eq_set_nat @ Y @ X )
% 2.54/1.44 => ( X = Y ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_111_antisym,axiom,
% 2.54/1.44 ! [X: nat,Y: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ X @ Y )
% 2.54/1.44 => ( ( ord_less_eq_nat @ Y @ X )
% 2.54/1.44 => ( X = Y ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_112_linear,axiom,
% 2.54/1.44 ! [X: nat,Y: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ X @ Y )
% 2.54/1.44 | ( ord_less_eq_nat @ Y @ X ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_113_eq__refl,axiom,
% 2.54/1.44 ! [X: set_nat,Y: set_nat] :
% 2.54/1.44 ( ( X = Y )
% 2.54/1.44 => ( ord_less_eq_set_nat @ X @ Y ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_114_eq__refl,axiom,
% 2.54/1.44 ! [X: nat,Y: nat] :
% 2.54/1.44 ( ( X = Y )
% 2.54/1.44 => ( ord_less_eq_nat @ X @ Y ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_115_le__cases,axiom,
% 2.54/1.44 ! [X: nat,Y: nat] :
% 2.54/1.44 ( ~ ( ord_less_eq_nat @ X @ Y )
% 2.54/1.44 => ( ord_less_eq_nat @ Y @ X ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_116_order_Otrans,axiom,
% 2.54/1.44 ! [A2: set_nat,B: set_nat,C: set_nat] :
% 2.54/1.44 ( ( ord_less_eq_set_nat @ A2 @ B )
% 2.54/1.44 => ( ( ord_less_eq_set_nat @ B @ C )
% 2.54/1.44 => ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_117_order_Otrans,axiom,
% 2.54/1.44 ! [A2: nat,B: nat,C: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ A2 @ B )
% 2.54/1.44 => ( ( ord_less_eq_nat @ B @ C )
% 2.54/1.44 => ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_118_le__cases3,axiom,
% 2.54/1.44 ! [X: nat,Y: nat,Z2: nat] :
% 2.54/1.44 ( ( ( ord_less_eq_nat @ X @ Y )
% 2.54/1.44 => ~ ( ord_less_eq_nat @ Y @ Z2 ) )
% 2.54/1.44 => ( ( ( ord_less_eq_nat @ Y @ X )
% 2.54/1.44 => ~ ( ord_less_eq_nat @ X @ Z2 ) )
% 2.54/1.44 => ( ( ( ord_less_eq_nat @ X @ Z2 )
% 2.54/1.44 => ~ ( ord_less_eq_nat @ Z2 @ Y ) )
% 2.54/1.44 => ( ( ( ord_less_eq_nat @ Z2 @ Y )
% 2.54/1.44 => ~ ( ord_less_eq_nat @ Y @ X ) )
% 2.54/1.44 => ( ( ( ord_less_eq_nat @ Y @ Z2 )
% 2.54/1.44 => ~ ( ord_less_eq_nat @ Z2 @ X ) )
% 2.54/1.44 => ~ ( ( ord_less_eq_nat @ Z2 @ X )
% 2.54/1.44 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_119_antisym__conv,axiom,
% 2.54/1.44 ! [Y: set_nat,X: set_nat] :
% 2.54/1.44 ( ( ord_less_eq_set_nat @ Y @ X )
% 2.54/1.44 => ( ( ord_less_eq_set_nat @ X @ Y )
% 2.54/1.44 = ( X = Y ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_120_antisym__conv,axiom,
% 2.54/1.44 ! [Y: nat,X: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ Y @ X )
% 2.54/1.44 => ( ( ord_less_eq_nat @ X @ Y )
% 2.54/1.44 = ( X = Y ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_121_order__class_Oorder_Oeq__iff,axiom,
% 2.54/1.44 ( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
% 2.54/1.44 = ( ^ [A4: set_nat,B4: set_nat] :
% 2.54/1.44 ( ( ord_less_eq_set_nat @ A4 @ B4 )
% 2.54/1.44 & ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_122_order__class_Oorder_Oeq__iff,axiom,
% 2.54/1.44 ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
% 2.54/1.44 = ( ^ [A4: nat,B4: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ A4 @ B4 )
% 2.54/1.44 & ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_123_ord__eq__le__trans,axiom,
% 2.54/1.44 ! [A2: set_nat,B: set_nat,C: set_nat] :
% 2.54/1.44 ( ( A2 = B )
% 2.54/1.44 => ( ( ord_less_eq_set_nat @ B @ C )
% 2.54/1.44 => ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_124_ord__eq__le__trans,axiom,
% 2.54/1.44 ! [A2: nat,B: nat,C: nat] :
% 2.54/1.44 ( ( A2 = B )
% 2.54/1.44 => ( ( ord_less_eq_nat @ B @ C )
% 2.54/1.44 => ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_125_ord__le__eq__trans,axiom,
% 2.54/1.44 ! [A2: set_nat,B: set_nat,C: set_nat] :
% 2.54/1.44 ( ( ord_less_eq_set_nat @ A2 @ B )
% 2.54/1.44 => ( ( B = C )
% 2.54/1.44 => ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_126_ord__le__eq__trans,axiom,
% 2.54/1.44 ! [A2: nat,B: nat,C: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ A2 @ B )
% 2.54/1.44 => ( ( B = C )
% 2.54/1.44 => ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_127_order__class_Oorder_Oantisym,axiom,
% 2.54/1.44 ! [A2: set_nat,B: set_nat] :
% 2.54/1.44 ( ( ord_less_eq_set_nat @ A2 @ B )
% 2.54/1.44 => ( ( ord_less_eq_set_nat @ B @ A2 )
% 2.54/1.44 => ( A2 = B ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_128_order__class_Oorder_Oantisym,axiom,
% 2.54/1.44 ! [A2: nat,B: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ A2 @ B )
% 2.54/1.44 => ( ( ord_less_eq_nat @ B @ A2 )
% 2.54/1.44 => ( A2 = B ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_129_order__trans,axiom,
% 2.54/1.44 ! [X: set_nat,Y: set_nat,Z2: set_nat] :
% 2.54/1.44 ( ( ord_less_eq_set_nat @ X @ Y )
% 2.54/1.44 => ( ( ord_less_eq_set_nat @ Y @ Z2 )
% 2.54/1.44 => ( ord_less_eq_set_nat @ X @ Z2 ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_130_order__trans,axiom,
% 2.54/1.44 ! [X: nat,Y: nat,Z2: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ X @ Y )
% 2.54/1.44 => ( ( ord_less_eq_nat @ Y @ Z2 )
% 2.54/1.44 => ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_131_dual__order_Orefl,axiom,
% 2.54/1.44 ! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_132_dual__order_Orefl,axiom,
% 2.54/1.44 ! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_133_linorder__wlog,axiom,
% 2.54/1.44 ! [P3: nat > nat > $o,A2: nat,B: nat] :
% 2.54/1.44 ( ! [A3: nat,B2: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ A3 @ B2 )
% 2.54/1.44 => ( P3 @ A3 @ B2 ) )
% 2.54/1.44 => ( ! [A3: nat,B2: nat] :
% 2.54/1.44 ( ( P3 @ B2 @ A3 )
% 2.54/1.44 => ( P3 @ A3 @ B2 ) )
% 2.54/1.44 => ( P3 @ A2 @ B ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_134_dual__order_Otrans,axiom,
% 2.54/1.44 ! [B: set_nat,A2: set_nat,C: set_nat] :
% 2.54/1.44 ( ( ord_less_eq_set_nat @ B @ A2 )
% 2.54/1.44 => ( ( ord_less_eq_set_nat @ C @ B )
% 2.54/1.44 => ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_135_dual__order_Otrans,axiom,
% 2.54/1.44 ! [B: nat,A2: nat,C: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ B @ A2 )
% 2.54/1.44 => ( ( ord_less_eq_nat @ C @ B )
% 2.54/1.44 => ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_136_dual__order_Oeq__iff,axiom,
% 2.54/1.44 ( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
% 2.54/1.44 = ( ^ [A4: set_nat,B4: set_nat] :
% 2.54/1.44 ( ( ord_less_eq_set_nat @ B4 @ A4 )
% 2.54/1.44 & ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_137_dual__order_Oeq__iff,axiom,
% 2.54/1.44 ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
% 2.54/1.44 = ( ^ [A4: nat,B4: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ B4 @ A4 )
% 2.54/1.44 & ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_138_dual__order_Oantisym,axiom,
% 2.54/1.44 ! [B: set_nat,A2: set_nat] :
% 2.54/1.44 ( ( ord_less_eq_set_nat @ B @ A2 )
% 2.54/1.44 => ( ( ord_less_eq_set_nat @ A2 @ B )
% 2.54/1.44 => ( A2 = B ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_139_dual__order_Oantisym,axiom,
% 2.54/1.44 ! [B: nat,A2: nat] :
% 2.54/1.44 ( ( ord_less_eq_nat @ B @ A2 )
% 2.54/1.44 => ( ( ord_less_eq_nat @ A2 @ B )
% 2.54/1.44 => ( A2 = B ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_140_zero__reorient,axiom,
% 2.54/1.44 ! [X: nat] :
% 2.54/1.44 ( ( zero_zero_nat = X )
% 2.54/1.44 = ( X = zero_zero_nat ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_141_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 2.54/1.44 ! [A2: nat,B: nat,C: nat] :
% 2.54/1.44 ( ( times_times_nat @ ( times_times_nat @ A2 @ B ) @ C )
% 2.54/1.44 = ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_142_mult_Oassoc,axiom,
% 2.54/1.44 ! [A2: nat,B: nat,C: nat] :
% 2.54/1.44 ( ( times_times_nat @ ( times_times_nat @ A2 @ B ) @ C )
% 2.54/1.44 = ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_143_mult_Ocommute,axiom,
% 2.54/1.44 ( times_times_nat
% 2.54/1.44 = ( ^ [A4: nat,B4: nat] : ( times_times_nat @ B4 @ A4 ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_144_mult_Oleft__commute,axiom,
% 2.54/1.44 ! [B: nat,A2: nat,C: nat] :
% 2.54/1.44 ( ( times_times_nat @ B @ ( times_times_nat @ A2 @ C ) )
% 2.54/1.44 = ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_145_one__reorient,axiom,
% 2.54/1.44 ! [X: nat] :
% 2.54/1.44 ( ( one_one_nat = X )
% 2.54/1.44 = ( X = one_one_nat ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_146_Collect__mono__iff,axiom,
% 2.54/1.44 ! [P3: nat > $o,Q: nat > $o] :
% 2.54/1.44 ( ( ord_less_eq_set_nat @ ( collect_nat @ P3 ) @ ( collect_nat @ Q ) )
% 2.54/1.44 = ( ! [X4: nat] :
% 2.54/1.44 ( ( P3 @ X4 )
% 2.54/1.44 => ( Q @ X4 ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_147_set__eq__subset,axiom,
% 2.54/1.44 ( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
% 2.54/1.44 = ( ^ [A5: set_nat,B5: set_nat] :
% 2.54/1.44 ( ( ord_less_eq_set_nat @ A5 @ B5 )
% 2.54/1.44 & ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_148_subset__trans,axiom,
% 2.54/1.44 ! [A: set_nat,B3: set_nat,C2: set_nat] :
% 2.54/1.44 ( ( ord_less_eq_set_nat @ A @ B3 )
% 2.54/1.44 => ( ( ord_less_eq_set_nat @ B3 @ C2 )
% 2.54/1.44 => ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_149_Collect__mono,axiom,
% 2.54/1.44 ! [P3: nat > $o,Q: nat > $o] :
% 2.54/1.44 ( ! [X2: nat] :
% 2.54/1.44 ( ( P3 @ X2 )
% 2.54/1.44 => ( Q @ X2 ) )
% 2.54/1.44 => ( ord_less_eq_set_nat @ ( collect_nat @ P3 ) @ ( collect_nat @ Q ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_150_subset__refl,axiom,
% 2.54/1.44 ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_151_subset__iff,axiom,
% 2.54/1.44 ( ord_less_eq_set_nat
% 2.54/1.44 = ( ^ [A5: set_nat,B5: set_nat] :
% 2.54/1.44 ! [T2: nat] :
% 2.54/1.44 ( ( member_nat @ T2 @ A5 )
% 2.54/1.44 => ( member_nat @ T2 @ B5 ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_152_equalityD2,axiom,
% 2.54/1.44 ! [A: set_nat,B3: set_nat] :
% 2.54/1.44 ( ( A = B3 )
% 2.54/1.44 => ( ord_less_eq_set_nat @ B3 @ A ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_153_equalityD1,axiom,
% 2.54/1.44 ! [A: set_nat,B3: set_nat] :
% 2.54/1.44 ( ( A = B3 )
% 2.54/1.44 => ( ord_less_eq_set_nat @ A @ B3 ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_154_subset__eq,axiom,
% 2.54/1.44 ( ord_less_eq_set_nat
% 2.54/1.44 = ( ^ [A5: set_nat,B5: set_nat] :
% 2.54/1.44 ! [X4: nat] :
% 2.54/1.44 ( ( member_nat @ X4 @ A5 )
% 2.54/1.44 => ( member_nat @ X4 @ B5 ) ) ) ) ).
% 2.54/1.44
% 2.54/1.44 thf(fact_155_equalityE,axiom,
% 2.54/1.44 ! [A: set_nat,B3: set_nat] :
% 2.54/1.44 ( ( A = B3 )
% 2.54/1.44 => ~ ( ( ord_less_eq_set_nat @ A @ B3 )
% 2.54/1.45 => ~ ( ord_less_eq_set_nat @ B3 @ A ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_156_subsetD,axiom,
% 2.54/1.45 ! [A: set_nat,B3: set_nat,C: nat] :
% 2.54/1.45 ( ( ord_less_eq_set_nat @ A @ B3 )
% 2.54/1.45 => ( ( member_nat @ C @ A )
% 2.54/1.45 => ( member_nat @ C @ B3 ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_157_in__mono,axiom,
% 2.54/1.45 ! [A: set_nat,B3: set_nat,X: nat] :
% 2.54/1.45 ( ( ord_less_eq_set_nat @ A @ B3 )
% 2.54/1.45 => ( ( member_nat @ X @ A )
% 2.54/1.45 => ( member_nat @ X @ B3 ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_158_less__eq__set__def,axiom,
% 2.54/1.45 ( ord_less_eq_set_nat
% 2.54/1.45 = ( ^ [A5: set_nat,B5: set_nat] :
% 2.54/1.45 ( ord_less_eq_nat_o
% 2.54/1.45 @ ^ [X4: nat] : ( member_nat @ X4 @ A5 )
% 2.54/1.45 @ ^ [X4: nat] : ( member_nat @ X4 @ B5 ) ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_159_Collect__subset,axiom,
% 2.54/1.45 ! [A: set_nat,P3: nat > $o] :
% 2.54/1.45 ( ord_less_eq_set_nat
% 2.54/1.45 @ ( collect_nat
% 2.54/1.45 @ ^ [X4: nat] :
% 2.54/1.45 ( ( member_nat @ X4 @ A )
% 2.54/1.45 & ( P3 @ X4 ) ) )
% 2.54/1.45 @ A ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_160_zero__le,axiom,
% 2.54/1.45 ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_161_le__numeral__extra_I4_J,axiom,
% 2.54/1.45 ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% 2.54/1.45
% 2.54/1.45 thf(fact_162_le__numeral__extra_I3_J,axiom,
% 2.54/1.45 ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 2.54/1.45
% 2.54/1.45 thf(fact_163_subset__Collect__iff,axiom,
% 2.54/1.45 ! [B3: set_nat,A: set_nat,P3: nat > $o] :
% 2.54/1.45 ( ( ord_less_eq_set_nat @ B3 @ A )
% 2.54/1.45 => ( ( ord_less_eq_set_nat @ B3
% 2.54/1.45 @ ( collect_nat
% 2.54/1.45 @ ^ [X4: nat] :
% 2.54/1.45 ( ( member_nat @ X4 @ A )
% 2.54/1.45 & ( P3 @ X4 ) ) ) )
% 2.54/1.45 = ( ! [X4: nat] :
% 2.54/1.45 ( ( member_nat @ X4 @ B3 )
% 2.54/1.45 => ( P3 @ X4 ) ) ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_164_subset__CollectI,axiom,
% 2.54/1.45 ! [B3: set_nat,A: set_nat,Q: nat > $o,P3: nat > $o] :
% 2.54/1.45 ( ( ord_less_eq_set_nat @ B3 @ A )
% 2.54/1.45 => ( ! [X2: nat] :
% 2.54/1.45 ( ( member_nat @ X2 @ B3 )
% 2.54/1.45 => ( ( Q @ X2 )
% 2.54/1.45 => ( P3 @ X2 ) ) )
% 2.54/1.45 => ( ord_less_eq_set_nat
% 2.54/1.45 @ ( collect_nat
% 2.54/1.45 @ ^ [X4: nat] :
% 2.54/1.45 ( ( member_nat @ X4 @ B3 )
% 2.54/1.45 & ( Q @ X4 ) ) )
% 2.54/1.45 @ ( collect_nat
% 2.54/1.45 @ ^ [X4: nat] :
% 2.54/1.45 ( ( member_nat @ X4 @ A )
% 2.54/1.45 & ( P3 @ X4 ) ) ) ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_165_Collect__restrict,axiom,
% 2.54/1.45 ! [X5: set_nat,P3: nat > $o] :
% 2.54/1.45 ( ord_less_eq_set_nat
% 2.54/1.45 @ ( collect_nat
% 2.54/1.45 @ ^ [X4: nat] :
% 2.54/1.45 ( ( member_nat @ X4 @ X5 )
% 2.54/1.45 & ( P3 @ X4 ) ) )
% 2.54/1.45 @ X5 ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_166_prop__restrict,axiom,
% 2.54/1.45 ! [X: nat,Z3: set_nat,X5: set_nat,P3: nat > $o] :
% 2.54/1.45 ( ( member_nat @ X @ Z3 )
% 2.54/1.45 => ( ( ord_less_eq_set_nat @ Z3
% 2.54/1.45 @ ( collect_nat
% 2.54/1.45 @ ^ [X4: nat] :
% 2.54/1.45 ( ( member_nat @ X4 @ X5 )
% 2.54/1.45 & ( P3 @ X4 ) ) ) )
% 2.54/1.45 => ( P3 @ X ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_167_pred__subset__eq,axiom,
% 2.54/1.45 ! [R2: set_nat,S: set_nat] :
% 2.54/1.45 ( ( ord_less_eq_nat_o
% 2.54/1.45 @ ^ [X4: nat] : ( member_nat @ X4 @ R2 )
% 2.54/1.45 @ ^ [X4: nat] : ( member_nat @ X4 @ S ) )
% 2.54/1.45 = ( ord_less_eq_set_nat @ R2 @ S ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_168_conj__subset__def,axiom,
% 2.54/1.45 ! [A: set_nat,P3: nat > $o,Q: nat > $o] :
% 2.54/1.45 ( ( ord_less_eq_set_nat @ A
% 2.54/1.45 @ ( collect_nat
% 2.54/1.45 @ ^ [X4: nat] :
% 2.54/1.45 ( ( P3 @ X4 )
% 2.54/1.45 & ( Q @ X4 ) ) ) )
% 2.54/1.45 = ( ( ord_less_eq_set_nat @ A @ ( collect_nat @ P3 ) )
% 2.54/1.45 & ( ord_less_eq_set_nat @ A @ ( collect_nat @ Q ) ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_169_neq0__conv,axiom,
% 2.54/1.45 ! [N: nat] :
% 2.54/1.45 ( ( N != zero_zero_nat )
% 2.54/1.45 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_170_less__nat__zero__code,axiom,
% 2.54/1.45 ! [N: nat] :
% 2.54/1.45 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_171_bot__nat__0_Onot__eq__extremum,axiom,
% 2.54/1.45 ! [A2: nat] :
% 2.54/1.45 ( ( A2 != zero_zero_nat )
% 2.54/1.45 = ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_172_psubsetI,axiom,
% 2.54/1.45 ! [A: set_nat,B3: set_nat] :
% 2.54/1.45 ( ( ord_less_eq_set_nat @ A @ B3 )
% 2.54/1.45 => ( ( A != B3 )
% 2.54/1.45 => ( ord_less_set_nat @ A @ B3 ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_173_not__gr__zero,axiom,
% 2.54/1.45 ! [N: nat] :
% 2.54/1.45 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 2.54/1.45 = ( N = zero_zero_nat ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_174_nat__0__less__mult__iff,axiom,
% 2.54/1.45 ! [M: nat,N: nat] :
% 2.54/1.45 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
% 2.54/1.45 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 2.54/1.45 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_175_mult__less__cancel2,axiom,
% 2.54/1.45 ! [M: nat,K: nat,N: nat] :
% 2.54/1.45 ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 2.54/1.45 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 2.54/1.45 & ( ord_less_nat @ M @ N ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_176_nat__mult__less__cancel__disj,axiom,
% 2.54/1.45 ! [K: nat,M: nat,N: nat] :
% 2.54/1.45 ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 2.54/1.45 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 2.54/1.45 & ( ord_less_nat @ M @ N ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_177_less__one,axiom,
% 2.54/1.45 ! [N: nat] :
% 2.54/1.45 ( ( ord_less_nat @ N @ one_one_nat )
% 2.54/1.45 = ( N = zero_zero_nat ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_178_mult__le__cancel2,axiom,
% 2.54/1.45 ! [M: nat,K: nat,N: nat] :
% 2.54/1.45 ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 2.54/1.45 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 2.54/1.45 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_179_nat__mult__le__cancel__disj,axiom,
% 2.54/1.45 ! [K: nat,M: nat,N: nat] :
% 2.54/1.45 ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 2.54/1.45 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 2.54/1.45 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_180_dual__order_Ostrict__implies__not__eq,axiom,
% 2.54/1.45 ! [B: nat,A2: nat] :
% 2.54/1.45 ( ( ord_less_nat @ B @ A2 )
% 2.54/1.45 => ( A2 != B ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_181_order_Ostrict__implies__not__eq,axiom,
% 2.54/1.45 ! [A2: nat,B: nat] :
% 2.54/1.45 ( ( ord_less_nat @ A2 @ B )
% 2.54/1.45 => ( A2 != B ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_182_not__less__iff__gr__or__eq,axiom,
% 2.54/1.45 ! [X: nat,Y: nat] :
% 2.54/1.45 ( ( ~ ( ord_less_nat @ X @ Y ) )
% 2.54/1.45 = ( ( ord_less_nat @ Y @ X )
% 2.54/1.45 | ( X = Y ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_183_dual__order_Ostrict__trans,axiom,
% 2.54/1.45 ! [B: nat,A2: nat,C: nat] :
% 2.54/1.45 ( ( ord_less_nat @ B @ A2 )
% 2.54/1.45 => ( ( ord_less_nat @ C @ B )
% 2.54/1.45 => ( ord_less_nat @ C @ A2 ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_184_linorder__less__wlog,axiom,
% 2.54/1.45 ! [P3: nat > nat > $o,A2: nat,B: nat] :
% 2.54/1.45 ( ! [A3: nat,B2: nat] :
% 2.54/1.45 ( ( ord_less_nat @ A3 @ B2 )
% 2.54/1.45 => ( P3 @ A3 @ B2 ) )
% 2.54/1.45 => ( ! [A3: nat] : ( P3 @ A3 @ A3 )
% 2.54/1.45 => ( ! [A3: nat,B2: nat] :
% 2.54/1.45 ( ( P3 @ B2 @ A3 )
% 2.54/1.45 => ( P3 @ A3 @ B2 ) )
% 2.54/1.45 => ( P3 @ A2 @ B ) ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_185_exists__least__iff,axiom,
% 2.54/1.45 ( ( ^ [P4: nat > $o] :
% 2.54/1.45 ? [X6: nat] : ( P4 @ X6 ) )
% 2.54/1.45 = ( ^ [P5: nat > $o] :
% 2.54/1.45 ? [N2: nat] :
% 2.54/1.45 ( ( P5 @ N2 )
% 2.54/1.45 & ! [M4: nat] :
% 2.54/1.45 ( ( ord_less_nat @ M4 @ N2 )
% 2.54/1.45 => ~ ( P5 @ M4 ) ) ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_186_less__imp__not__less,axiom,
% 2.54/1.45 ! [X: nat,Y: nat] :
% 2.54/1.45 ( ( ord_less_nat @ X @ Y )
% 2.54/1.45 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_187_order_Ostrict__trans,axiom,
% 2.54/1.45 ! [A2: nat,B: nat,C: nat] :
% 2.54/1.45 ( ( ord_less_nat @ A2 @ B )
% 2.54/1.45 => ( ( ord_less_nat @ B @ C )
% 2.54/1.45 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_188_dual__order_Oirrefl,axiom,
% 2.54/1.45 ! [A2: nat] :
% 2.54/1.45 ~ ( ord_less_nat @ A2 @ A2 ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_189_linorder__cases,axiom,
% 2.54/1.45 ! [X: nat,Y: nat] :
% 2.54/1.45 ( ~ ( ord_less_nat @ X @ Y )
% 2.54/1.45 => ( ( X != Y )
% 2.54/1.45 => ( ord_less_nat @ Y @ X ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_190_less__imp__triv,axiom,
% 2.54/1.45 ! [X: nat,Y: nat,P3: $o] :
% 2.54/1.45 ( ( ord_less_nat @ X @ Y )
% 2.54/1.45 => ( ( ord_less_nat @ Y @ X )
% 2.54/1.45 => P3 ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_191_less__imp__not__eq2,axiom,
% 2.54/1.45 ! [X: nat,Y: nat] :
% 2.54/1.45 ( ( ord_less_nat @ X @ Y )
% 2.54/1.45 => ( Y != X ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_192_antisym__conv3,axiom,
% 2.54/1.45 ! [Y: nat,X: nat] :
% 2.54/1.45 ( ~ ( ord_less_nat @ Y @ X )
% 2.54/1.45 => ( ( ~ ( ord_less_nat @ X @ Y ) )
% 2.54/1.45 = ( X = Y ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_193_less__induct,axiom,
% 2.54/1.45 ! [P3: nat > $o,A2: nat] :
% 2.54/1.45 ( ! [X2: nat] :
% 2.54/1.45 ( ! [Y3: nat] :
% 2.54/1.45 ( ( ord_less_nat @ Y3 @ X2 )
% 2.54/1.45 => ( P3 @ Y3 ) )
% 2.54/1.45 => ( P3 @ X2 ) )
% 2.54/1.45 => ( P3 @ A2 ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_194_less__not__sym,axiom,
% 2.54/1.45 ! [X: nat,Y: nat] :
% 2.54/1.45 ( ( ord_less_nat @ X @ Y )
% 2.54/1.45 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_195_less__imp__not__eq,axiom,
% 2.54/1.45 ! [X: nat,Y: nat] :
% 2.54/1.45 ( ( ord_less_nat @ X @ Y )
% 2.54/1.45 => ( X != Y ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_196_dual__order_Oasym,axiom,
% 2.54/1.45 ! [B: nat,A2: nat] :
% 2.54/1.45 ( ( ord_less_nat @ B @ A2 )
% 2.54/1.45 => ~ ( ord_less_nat @ A2 @ B ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_197_ord__less__eq__trans,axiom,
% 2.54/1.45 ! [A2: nat,B: nat,C: nat] :
% 2.54/1.45 ( ( ord_less_nat @ A2 @ B )
% 2.54/1.45 => ( ( B = C )
% 2.54/1.45 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_198_ord__eq__less__trans,axiom,
% 2.54/1.45 ! [A2: nat,B: nat,C: nat] :
% 2.54/1.45 ( ( A2 = B )
% 2.54/1.45 => ( ( ord_less_nat @ B @ C )
% 2.54/1.45 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_199_less__irrefl,axiom,
% 2.54/1.45 ! [X: nat] :
% 2.54/1.45 ~ ( ord_less_nat @ X @ X ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_200_less__linear,axiom,
% 2.54/1.45 ! [X: nat,Y: nat] :
% 2.54/1.45 ( ( ord_less_nat @ X @ Y )
% 2.54/1.45 | ( X = Y )
% 2.54/1.45 | ( ord_less_nat @ Y @ X ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_201_less__trans,axiom,
% 2.54/1.45 ! [X: nat,Y: nat,Z2: nat] :
% 2.54/1.45 ( ( ord_less_nat @ X @ Y )
% 2.54/1.45 => ( ( ord_less_nat @ Y @ Z2 )
% 2.54/1.45 => ( ord_less_nat @ X @ Z2 ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_202_less__asym_H,axiom,
% 2.54/1.45 ! [A2: nat,B: nat] :
% 2.54/1.45 ( ( ord_less_nat @ A2 @ B )
% 2.54/1.45 => ~ ( ord_less_nat @ B @ A2 ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_203_less__asym,axiom,
% 2.54/1.45 ! [X: nat,Y: nat] :
% 2.54/1.45 ( ( ord_less_nat @ X @ Y )
% 2.54/1.45 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_204_less__imp__neq,axiom,
% 2.54/1.45 ! [X: nat,Y: nat] :
% 2.54/1.45 ( ( ord_less_nat @ X @ Y )
% 2.54/1.45 => ( X != Y ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_205_order_Oasym,axiom,
% 2.54/1.45 ! [A2: nat,B: nat] :
% 2.54/1.45 ( ( ord_less_nat @ A2 @ B )
% 2.54/1.45 => ~ ( ord_less_nat @ B @ A2 ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_206_neq__iff,axiom,
% 2.54/1.45 ! [X: nat,Y: nat] :
% 2.54/1.45 ( ( X != Y )
% 2.54/1.45 = ( ( ord_less_nat @ X @ Y )
% 2.54/1.45 | ( ord_less_nat @ Y @ X ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_207_neqE,axiom,
% 2.54/1.45 ! [X: nat,Y: nat] :
% 2.54/1.45 ( ( X != Y )
% 2.54/1.45 => ( ~ ( ord_less_nat @ X @ Y )
% 2.54/1.45 => ( ord_less_nat @ Y @ X ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_208_gt__ex,axiom,
% 2.54/1.45 ! [X: nat] :
% 2.54/1.45 ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_209_order__less__subst2,axiom,
% 2.54/1.45 ! [A2: nat,B: nat,F: nat > nat,C: nat] :
% 2.54/1.45 ( ( ord_less_nat @ A2 @ B )
% 2.54/1.45 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 2.54/1.45 => ( ! [X2: nat,Y2: nat] :
% 2.54/1.45 ( ( ord_less_nat @ X2 @ Y2 )
% 2.54/1.45 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.54/1.45 => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_210_order__less__subst1,axiom,
% 2.54/1.45 ! [A2: nat,F: nat > nat,B: nat,C: nat] :
% 2.54/1.45 ( ( ord_less_nat @ A2 @ ( F @ B ) )
% 2.54/1.45 => ( ( ord_less_nat @ B @ C )
% 2.54/1.45 => ( ! [X2: nat,Y2: nat] :
% 2.54/1.45 ( ( ord_less_nat @ X2 @ Y2 )
% 2.54/1.45 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.54/1.45 => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_211_ord__less__eq__subst,axiom,
% 2.54/1.45 ! [A2: nat,B: nat,F: nat > nat,C: nat] :
% 2.54/1.45 ( ( ord_less_nat @ A2 @ B )
% 2.54/1.45 => ( ( ( F @ B )
% 2.54/1.45 = C )
% 2.54/1.45 => ( ! [X2: nat,Y2: nat] :
% 2.54/1.45 ( ( ord_less_nat @ X2 @ Y2 )
% 2.54/1.45 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.54/1.45 => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_212_ord__eq__less__subst,axiom,
% 2.54/1.45 ! [A2: nat,F: nat > nat,B: nat,C: nat] :
% 2.54/1.45 ( ( A2
% 2.54/1.45 = ( F @ B ) )
% 2.54/1.45 => ( ( ord_less_nat @ B @ C )
% 2.54/1.45 => ( ! [X2: nat,Y2: nat] :
% 2.54/1.45 ( ( ord_less_nat @ X2 @ Y2 )
% 2.54/1.45 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.54/1.45 => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_213_gr__zeroI,axiom,
% 2.54/1.45 ! [N: nat] :
% 2.54/1.45 ( ( N != zero_zero_nat )
% 2.54/1.45 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_214_not__less__zero,axiom,
% 2.54/1.45 ! [N: nat] :
% 2.54/1.45 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_215_gr__implies__not__zero,axiom,
% 2.54/1.45 ! [M: nat,N: nat] :
% 2.54/1.45 ( ( ord_less_nat @ M @ N )
% 2.54/1.45 => ( N != zero_zero_nat ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_216_zero__less__iff__neq__zero,axiom,
% 2.54/1.45 ! [N: nat] :
% 2.54/1.45 ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.54/1.45 = ( N != zero_zero_nat ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_217_nat__less__le,axiom,
% 2.54/1.45 ( ord_less_nat
% 2.54/1.45 = ( ^ [M4: nat,N2: nat] :
% 2.54/1.45 ( ( ord_less_eq_nat @ M4 @ N2 )
% 2.54/1.45 & ( M4 != N2 ) ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_218_less__imp__le__nat,axiom,
% 2.54/1.45 ! [M: nat,N: nat] :
% 2.54/1.45 ( ( ord_less_nat @ M @ N )
% 2.54/1.45 => ( ord_less_eq_nat @ M @ N ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_219_le__eq__less__or__eq,axiom,
% 2.54/1.45 ( ord_less_eq_nat
% 2.54/1.45 = ( ^ [M4: nat,N2: nat] :
% 2.54/1.45 ( ( ord_less_nat @ M4 @ N2 )
% 2.54/1.45 | ( M4 = N2 ) ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_220_less__or__eq__imp__le,axiom,
% 2.54/1.45 ! [M: nat,N: nat] :
% 2.54/1.45 ( ( ( ord_less_nat @ M @ N )
% 2.54/1.45 | ( M = N ) )
% 2.54/1.45 => ( ord_less_eq_nat @ M @ N ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_221_le__neq__implies__less,axiom,
% 2.54/1.45 ! [M: nat,N: nat] :
% 2.54/1.45 ( ( ord_less_eq_nat @ M @ N )
% 2.54/1.45 => ( ( M != N )
% 2.54/1.45 => ( ord_less_nat @ M @ N ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_222_less__mono__imp__le__mono,axiom,
% 2.54/1.45 ! [F: nat > nat,I4: nat,J: nat] :
% 2.54/1.45 ( ! [I3: nat,J3: nat] :
% 2.54/1.45 ( ( ord_less_nat @ I3 @ J3 )
% 2.54/1.45 => ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
% 2.54/1.45 => ( ( ord_less_eq_nat @ I4 @ J )
% 2.54/1.45 => ( ord_less_eq_nat @ ( F @ I4 ) @ ( F @ J ) ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_223_gr0I,axiom,
% 2.54/1.45 ! [N: nat] :
% 2.54/1.45 ( ( N != zero_zero_nat )
% 2.54/1.45 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_224_not__gr0,axiom,
% 2.54/1.45 ! [N: nat] :
% 2.54/1.45 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 2.54/1.45 = ( N = zero_zero_nat ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_225_not__less0,axiom,
% 2.54/1.45 ! [N: nat] :
% 2.54/1.45 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_226_less__zeroE,axiom,
% 2.54/1.45 ! [N: nat] :
% 2.54/1.45 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_227_gr__implies__not0,axiom,
% 2.54/1.45 ! [M: nat,N: nat] :
% 2.54/1.45 ( ( ord_less_nat @ M @ N )
% 2.54/1.45 => ( N != zero_zero_nat ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_228_infinite__descent0,axiom,
% 2.54/1.45 ! [P3: nat > $o,N: nat] :
% 2.54/1.45 ( ( P3 @ zero_zero_nat )
% 2.54/1.45 => ( ! [N3: nat] :
% 2.54/1.45 ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 2.54/1.45 => ( ~ ( P3 @ N3 )
% 2.54/1.45 => ? [M5: nat] :
% 2.54/1.45 ( ( ord_less_nat @ M5 @ N3 )
% 2.54/1.45 & ~ ( P3 @ M5 ) ) ) )
% 2.54/1.45 => ( P3 @ N ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_229_bot__nat__0_Oextremum__strict,axiom,
% 2.54/1.45 ! [A2: nat] :
% 2.54/1.45 ~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_230_psubsetE,axiom,
% 2.54/1.45 ! [A: set_nat,B3: set_nat] :
% 2.54/1.45 ( ( ord_less_set_nat @ A @ B3 )
% 2.54/1.45 => ~ ( ( ord_less_eq_set_nat @ A @ B3 )
% 2.54/1.45 => ( ord_less_eq_set_nat @ B3 @ A ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_231_psubset__eq,axiom,
% 2.54/1.45 ( ord_less_set_nat
% 2.54/1.45 = ( ^ [A5: set_nat,B5: set_nat] :
% 2.54/1.45 ( ( ord_less_eq_set_nat @ A5 @ B5 )
% 2.54/1.45 & ( A5 != B5 ) ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_232_psubset__imp__subset,axiom,
% 2.54/1.45 ! [A: set_nat,B3: set_nat] :
% 2.54/1.45 ( ( ord_less_set_nat @ A @ B3 )
% 2.54/1.45 => ( ord_less_eq_set_nat @ A @ B3 ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_233_psubset__subset__trans,axiom,
% 2.54/1.45 ! [A: set_nat,B3: set_nat,C2: set_nat] :
% 2.54/1.45 ( ( ord_less_set_nat @ A @ B3 )
% 2.54/1.45 => ( ( ord_less_eq_set_nat @ B3 @ C2 )
% 2.54/1.45 => ( ord_less_set_nat @ A @ C2 ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_234_subset__not__subset__eq,axiom,
% 2.54/1.45 ( ord_less_set_nat
% 2.54/1.45 = ( ^ [A5: set_nat,B5: set_nat] :
% 2.54/1.45 ( ( ord_less_eq_set_nat @ A5 @ B5 )
% 2.54/1.45 & ~ ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_235_subset__psubset__trans,axiom,
% 2.54/1.45 ! [A: set_nat,B3: set_nat,C2: set_nat] :
% 2.54/1.45 ( ( ord_less_eq_set_nat @ A @ B3 )
% 2.54/1.45 => ( ( ord_less_set_nat @ B3 @ C2 )
% 2.54/1.45 => ( ord_less_set_nat @ A @ C2 ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_236_subset__iff__psubset__eq,axiom,
% 2.54/1.45 ( ord_less_eq_set_nat
% 2.54/1.45 = ( ^ [A5: set_nat,B5: set_nat] :
% 2.54/1.45 ( ( ord_less_set_nat @ A5 @ B5 )
% 2.54/1.45 | ( A5 = B5 ) ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_237_less__numeral__extra_I4_J,axiom,
% 2.54/1.45 ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_238_less__numeral__extra_I3_J,axiom,
% 2.54/1.45 ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_239_order_Onot__eq__order__implies__strict,axiom,
% 2.54/1.45 ! [A2: set_nat,B: set_nat] :
% 2.54/1.45 ( ( A2 != B )
% 2.54/1.45 => ( ( ord_less_eq_set_nat @ A2 @ B )
% 2.54/1.45 => ( ord_less_set_nat @ A2 @ B ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_240_order_Onot__eq__order__implies__strict,axiom,
% 2.54/1.45 ! [A2: nat,B: nat] :
% 2.54/1.45 ( ( A2 != B )
% 2.54/1.45 => ( ( ord_less_eq_nat @ A2 @ B )
% 2.54/1.45 => ( ord_less_nat @ A2 @ B ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_241_dual__order_Ostrict__implies__order,axiom,
% 2.54/1.45 ! [B: set_nat,A2: set_nat] :
% 2.54/1.45 ( ( ord_less_set_nat @ B @ A2 )
% 2.54/1.45 => ( ord_less_eq_set_nat @ B @ A2 ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_242_dual__order_Ostrict__implies__order,axiom,
% 2.54/1.45 ! [B: nat,A2: nat] :
% 2.54/1.45 ( ( ord_less_nat @ B @ A2 )
% 2.54/1.45 => ( ord_less_eq_nat @ B @ A2 ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_243_dual__order_Ostrict__iff__order,axiom,
% 2.54/1.45 ( ord_less_set_nat
% 2.54/1.45 = ( ^ [B4: set_nat,A4: set_nat] :
% 2.54/1.45 ( ( ord_less_eq_set_nat @ B4 @ A4 )
% 2.54/1.45 & ( A4 != B4 ) ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_244_dual__order_Ostrict__iff__order,axiom,
% 2.54/1.45 ( ord_less_nat
% 2.54/1.45 = ( ^ [B4: nat,A4: nat] :
% 2.54/1.45 ( ( ord_less_eq_nat @ B4 @ A4 )
% 2.54/1.45 & ( A4 != B4 ) ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_245_dual__order_Oorder__iff__strict,axiom,
% 2.54/1.45 ( ord_less_eq_set_nat
% 2.54/1.45 = ( ^ [B4: set_nat,A4: set_nat] :
% 2.54/1.45 ( ( ord_less_set_nat @ B4 @ A4 )
% 2.54/1.45 | ( A4 = B4 ) ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_246_dual__order_Oorder__iff__strict,axiom,
% 2.54/1.45 ( ord_less_eq_nat
% 2.54/1.45 = ( ^ [B4: nat,A4: nat] :
% 2.54/1.45 ( ( ord_less_nat @ B4 @ A4 )
% 2.54/1.45 | ( A4 = B4 ) ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_247_order_Ostrict__implies__order,axiom,
% 2.54/1.45 ! [A2: set_nat,B: set_nat] :
% 2.54/1.45 ( ( ord_less_set_nat @ A2 @ B )
% 2.54/1.45 => ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_248_order_Ostrict__implies__order,axiom,
% 2.54/1.45 ! [A2: nat,B: nat] :
% 2.54/1.45 ( ( ord_less_nat @ A2 @ B )
% 2.54/1.45 => ( ord_less_eq_nat @ A2 @ B ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_249_dual__order_Ostrict__trans2,axiom,
% 2.54/1.45 ! [B: set_nat,A2: set_nat,C: set_nat] :
% 2.54/1.45 ( ( ord_less_set_nat @ B @ A2 )
% 2.54/1.45 => ( ( ord_less_eq_set_nat @ C @ B )
% 2.54/1.45 => ( ord_less_set_nat @ C @ A2 ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_250_dual__order_Ostrict__trans2,axiom,
% 2.54/1.45 ! [B: nat,A2: nat,C: nat] :
% 2.54/1.45 ( ( ord_less_nat @ B @ A2 )
% 2.54/1.45 => ( ( ord_less_eq_nat @ C @ B )
% 2.54/1.45 => ( ord_less_nat @ C @ A2 ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_251_dual__order_Ostrict__trans1,axiom,
% 2.54/1.45 ! [B: set_nat,A2: set_nat,C: set_nat] :
% 2.54/1.45 ( ( ord_less_eq_set_nat @ B @ A2 )
% 2.54/1.45 => ( ( ord_less_set_nat @ C @ B )
% 2.54/1.45 => ( ord_less_set_nat @ C @ A2 ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_252_dual__order_Ostrict__trans1,axiom,
% 2.54/1.45 ! [B: nat,A2: nat,C: nat] :
% 2.54/1.45 ( ( ord_less_eq_nat @ B @ A2 )
% 2.54/1.45 => ( ( ord_less_nat @ C @ B )
% 2.54/1.45 => ( ord_less_nat @ C @ A2 ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_253_order_Ostrict__iff__order,axiom,
% 2.54/1.45 ( ord_less_set_nat
% 2.54/1.45 = ( ^ [A4: set_nat,B4: set_nat] :
% 2.54/1.45 ( ( ord_less_eq_set_nat @ A4 @ B4 )
% 2.54/1.45 & ( A4 != B4 ) ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_254_order_Ostrict__iff__order,axiom,
% 2.54/1.45 ( ord_less_nat
% 2.54/1.45 = ( ^ [A4: nat,B4: nat] :
% 2.54/1.45 ( ( ord_less_eq_nat @ A4 @ B4 )
% 2.54/1.45 & ( A4 != B4 ) ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_255_order_Oorder__iff__strict,axiom,
% 2.54/1.45 ( ord_less_eq_set_nat
% 2.54/1.45 = ( ^ [A4: set_nat,B4: set_nat] :
% 2.54/1.45 ( ( ord_less_set_nat @ A4 @ B4 )
% 2.54/1.45 | ( A4 = B4 ) ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_256_order_Oorder__iff__strict,axiom,
% 2.54/1.45 ( ord_less_eq_nat
% 2.54/1.45 = ( ^ [A4: nat,B4: nat] :
% 2.54/1.45 ( ( ord_less_nat @ A4 @ B4 )
% 2.54/1.45 | ( A4 = B4 ) ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_257_order_Ostrict__trans2,axiom,
% 2.54/1.45 ! [A2: set_nat,B: set_nat,C: set_nat] :
% 2.54/1.45 ( ( ord_less_set_nat @ A2 @ B )
% 2.54/1.45 => ( ( ord_less_eq_set_nat @ B @ C )
% 2.54/1.45 => ( ord_less_set_nat @ A2 @ C ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_258_order_Ostrict__trans2,axiom,
% 2.54/1.45 ! [A2: nat,B: nat,C: nat] :
% 2.54/1.45 ( ( ord_less_nat @ A2 @ B )
% 2.54/1.45 => ( ( ord_less_eq_nat @ B @ C )
% 2.54/1.45 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_259_order_Ostrict__trans1,axiom,
% 2.54/1.45 ! [A2: set_nat,B: set_nat,C: set_nat] :
% 2.54/1.45 ( ( ord_less_eq_set_nat @ A2 @ B )
% 2.54/1.45 => ( ( ord_less_set_nat @ B @ C )
% 2.54/1.45 => ( ord_less_set_nat @ A2 @ C ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(fact_260_order_Ostrict__trans1,axiom,
% 2.54/1.45 ! [A2: nat,B: nat,C: nat] :
% 2.54/1.45 ( ( ord_less_eq_nat @ A2 @ B )
% 2.54/1.45 => ( ( ord_less_nat @ B @ C )
% 2.54/1.45 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 2.54/1.45
% 2.54/1.45 thf(conj_0,conjecture,
% 2.54/1.45 ( ( ! [I2: nat] :
% 2.54/1.45 ( ( ( p @ I2 )
% 2.54/1.45 != zero_zero_nat )
% 2.54/1.45 => ( ( ord_less_eq_nat @ one_one_nat @ I2 )
% 2.54/1.45 & ( ord_less_eq_nat @ I2 @ zero_zero_nat ) ) )
% 2.54/1.45 & ( ( groups1842438620at_nat
% 2.54/1.45 @ ^ [I2: nat] : ( times_times_nat @ ( p @ I2 ) @ I2 )
% 2.54/1.45 @ ( set_ord_atMost_nat @ zero_zero_nat ) )
% 2.54/1.45 = zero_zero_nat ) )
% 2.54/1.45 = ( p
% 2.54/1.45 = ( ^ [I2: nat] : zero_zero_nat ) ) ) ).
% 2.54/1.45 ------------------------
% 2.54/1.46 % (30151)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_29954 for (2999ds/2Mi)
% 2.54/1.46 % (30151)Instruction limit reached!
% 2.54/1.46 % (30151)------------------------------
% 2.54/1.46 % (30151)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.54/1.46 % (30151)Termination reason: Unknown
% 2.54/1.46 % (30151)Termination phase: shuffling
% 2.54/1.46
% 2.54/1.46 % (30151)Memory used [KB]: 1279
% 2.54/1.46 % (30151)Time elapsed: 0.003 s
% 2.54/1.46 % (30151)Instructions burned: 3 (million)
% 2.54/1.46 % (30151)------------------------------
% 2.54/1.46 % (30151)------------------------------
% 2.54/1.46 % (30149)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_29954 for (2999ds/27Mi)
% 2.54/1.47 % (30153)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_29954 for (2999ds/18Mi)
% 2.54/1.47 % (30148)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_29954 for (2999ds/4Mi)
% 2.54/1.47 % (30150)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_29954 for (2999ds/2Mi)
% 2.54/1.47 % (30147)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_29954 for (2999ds/183Mi)
% 2.54/1.47 % (30148)Instruction limit reached!
% 2.54/1.47 % (30148)------------------------------
% 2.54/1.47 % (30148)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.54/1.47 % (30148)Termination reason: Unknown
% 2.54/1.47 % (30148)Termination phase: shuffling
% 2.54/1.47
% 2.54/1.47 % (30148)Memory used [KB]: 1279
% 2.54/1.47 % (30148)Time elapsed: 0.004 s
% 2.54/1.47 % (30148)Instructions burned: 4 (million)
% 2.54/1.47 % (30148)------------------------------
% 2.54/1.47 % (30148)------------------------------
% 2.54/1.47 % (30150)Instruction limit reached!
% 2.54/1.47 % (30150)------------------------------
% 2.54/1.47 % (30150)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.54/1.47 % (30150)Termination reason: Unknown
% 2.54/1.47 % (30150)Termination phase: shuffling
% 2.54/1.47
% 2.54/1.47 % (30150)Memory used [KB]: 1279
% 2.54/1.47 % (30150)Time elapsed: 0.004 s
% 2.54/1.47 % (30150)Instructions burned: 3 (million)
% 2.54/1.47 % (30150)------------------------------
% 2.54/1.47 % (30150)------------------------------
% 2.54/1.47 % (30153)Instruction limit reached!
% 2.54/1.47 % (30153)------------------------------
% 2.54/1.47 % (30153)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.54/1.47 % (30153)Termination reason: Unknown
% 2.54/1.47 % (30153)Termination phase: Property scanning
% 2.54/1.47
% 2.54/1.47 % (30153)Memory used [KB]: 1663
% 2.54/1.47 % (30153)Time elapsed: 0.007 s
% 2.54/1.47 % (30153)Instructions burned: 18 (million)
% 2.54/1.47 % (30153)------------------------------
% 2.54/1.47 % (30153)------------------------------
% 2.54/1.47 % (30149)Instruction limit reached!
% 2.54/1.47 % (30149)------------------------------
% 2.54/1.47 % (30149)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.54/1.47 % (30149)Termination reason: Unknown
% 2.54/1.47 % (30149)Termination phase: Property scanning
% 2.54/1.47
% 2.54/1.47 % (30149)Memory used [KB]: 1663
% 2.54/1.47 % (30149)Time elapsed: 0.012 s
% 2.54/1.47 % (30149)Instructions burned: 28 (million)
% 2.54/1.47 % (30149)------------------------------
% 2.54/1.47 % (30149)------------------------------
% 2.54/1.48 % (30152)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_29954 for (2999ds/275Mi)
% 2.54/1.48 % (30156)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_29954 for (2999ds/15Mi)
% 2.54/1.49 % (30157)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_29954 for (2999ds/3Mi)
% 2.54/1.49 % (30157)Instruction limit reached!
% 2.54/1.49 % (30157)------------------------------
% 2.54/1.49 % (30157)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.54/1.49 % (30157)Termination reason: Unknown
% 2.54/1.49 % (30157)Termination phase: shuffling
% 2.54/1.49
% 2.54/1.49 % (30157)Memory used [KB]: 1279
% 2.54/1.49 % (30157)Time elapsed: 0.002 s
% 2.54/1.49 % (30157)Instructions burned: 3 (million)
% 2.54/1.49 % (30157)------------------------------
% 2.54/1.49 % (30157)------------------------------
% 2.54/1.49 % (30156)Instruction limit reached!
% 2.54/1.49 % (30156)------------------------------
% 2.54/1.49 % (30156)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.54/1.49 % (30156)Termination reason: Unknown
% 2.54/1.49 % (30156)Termination phase: shuffling
% 2.54/1.49
% 2.54/1.49 % (30156)Memory used [KB]: 1535
% 2.54/1.49 % (30156)Time elapsed: 0.006 s
% 2.54/1.49 % (30156)Instructions burned: 15 (million)
% 2.54/1.49 % (30156)------------------------------
% 2.54/1.49 % (30156)------------------------------
% 2.54/1.49 % (30154)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_29954 for (2999ds/3Mi)
% 2.54/1.49 % (30154)Instruction limit reached!
% 2.54/1.49 % (30154)------------------------------
% 2.54/1.49 % (30154)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.54/1.49 % (30154)Termination reason: Unknown
% 2.54/1.49 % (30154)Termination phase: shuffling
% 2.54/1.49
% 2.54/1.49 % (30154)Memory used [KB]: 1279
% 2.54/1.49 % (30154)Time elapsed: 0.003 s
% 2.54/1.49 % (30154)Instructions burned: 3 (million)
% 2.54/1.49 % (30154)------------------------------
% 2.54/1.49 % (30154)------------------------------
% 2.54/1.49 % (30155)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_29954 for (2999ds/37Mi)
% 2.54/1.49 % (30158)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_29954 for (2999ds/1041Mi)
% 2.60/1.50 % (30159)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_29954 for (2999ds/7Mi)
% 2.60/1.50 % (30160)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_29954 for (2999ds/16Mi)
% 2.60/1.50 % (30159)Instruction limit reached!
% 2.60/1.50 % (30159)------------------------------
% 2.60/1.50 % (30159)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.60/1.50 % (30159)Termination reason: Unknown
% 2.60/1.50 % (30159)Termination phase: shuffling
% 2.60/1.50
% 2.60/1.50 % (30159)Memory used [KB]: 1407
% 2.60/1.50 % (30159)Time elapsed: 0.004 s
% 2.60/1.50 % (30159)Instructions burned: 9 (million)
% 2.60/1.50 % (30159)------------------------------
% 2.60/1.50 % (30159)------------------------------
% 2.60/1.50 % (30155)Instruction limit reached!
% 2.60/1.50 % (30155)------------------------------
% 2.60/1.50 % (30155)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.60/1.50 % (30155)Termination reason: Unknown
% 2.60/1.50 % (30155)Termination phase: Property scanning
% 2.60/1.50
% 2.60/1.50 % (30155)Memory used [KB]: 2046
% 2.60/1.50 % (30155)Time elapsed: 0.015 s
% 2.60/1.50 % (30155)Instructions burned: 37 (million)
% 2.60/1.50 % (30155)------------------------------
% 2.60/1.50 % (30155)------------------------------
% 2.60/1.50 % (30160)Instruction limit reached!
% 2.60/1.50 % (30160)------------------------------
% 2.60/1.50 % (30160)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.60/1.50 % (30160)Termination reason: Unknown
% 2.60/1.50 % (30160)Termination phase: shuffling
% 2.60/1.50
% 2.60/1.50 % (30160)Memory used [KB]: 1663
% 2.60/1.50 % (30160)Time elapsed: 0.008 s
% 2.60/1.50 % (30160)Instructions burned: 18 (million)
% 2.60/1.50 % (30160)------------------------------
% 2.60/1.50 % (30160)------------------------------
% 2.60/1.51 % (30162)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_29954 for (2999ds/3Mi)
% 2.60/1.51 % (30161)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_29954 for (2999ds/3Mi)
% 2.60/1.51 % (30162)Instruction limit reached!
% 2.60/1.51 % (30162)------------------------------
% 2.60/1.51 % (30162)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.60/1.51 % (30162)Termination reason: Unknown
% 2.60/1.51 % (30162)Termination phase: shuffling
% 2.60/1.51
% 2.60/1.51 % (30162)Memory used [KB]: 1279
% 2.60/1.51 % (30162)Time elapsed: 0.002 s
% 2.60/1.51 % (30162)Instructions burned: 3 (million)
% 2.60/1.51 % (30162)------------------------------
% 2.60/1.51 % (30162)------------------------------
% 2.60/1.51 % (30161)Instruction limit reached!
% 2.60/1.51 % (30161)------------------------------
% 2.60/1.51 % (30161)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.60/1.51 % (30161)Termination reason: Unknown
% 2.60/1.51 % (30161)Termination phase: shuffling
% 2.60/1.51
% 2.60/1.51 % (30161)Memory used [KB]: 1279
% 2.60/1.51 % (30161)Time elapsed: 0.004 s
% 2.60/1.51 % (30161)Instructions burned: 4 (million)
% 2.60/1.51 % (30161)------------------------------
% 2.60/1.51 % (30161)------------------------------
% 2.60/1.51 % (30163)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_29954 for (2999ds/7Mi)
% 2.60/1.51 % (30164)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_29954 for (2999ds/3Mi)
% 2.60/1.52 % (30164)Instruction limit reached!
% 2.60/1.52 % (30164)------------------------------
% 2.60/1.52 % (30164)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.60/1.52 % (30164)Termination reason: Unknown
% 2.60/1.52 % (30164)Termination phase: shuffling
% 2.60/1.52
% 2.60/1.52 % (30164)Memory used [KB]: 1279
% 2.60/1.52 % (30164)Time elapsed: 0.003 s
% 2.60/1.52 % (30164)Instructions burned: 4 (million)
% 2.60/1.52 % (30164)------------------------------
% 2.60/1.52 % (30164)------------------------------
% 2.60/1.52 % (30163)Instruction limit reached!
% 2.60/1.52 % (30163)------------------------------
% 2.60/1.52 % (30163)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.60/1.52 % (30163)Termination reason: Unknown
% 2.60/1.52 % (30163)Termination phase: shuffling
% 2.60/1.52
% 2.60/1.52 % (30163)Memory used [KB]: 1407
% 2.60/1.52 % (30163)Time elapsed: 0.005 s
% 2.60/1.52 % (30163)Instructions burned: 9 (million)
% 2.60/1.52 % (30163)------------------------------
% 2.60/1.52 % (30163)------------------------------
% 2.60/1.52 % (30165)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_29954 for (2999ds/4Mi)
% 2.60/1.52 % (30165)Instruction limit reached!
% 2.60/1.52 % (30165)------------------------------
% 2.60/1.52 % (30165)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.60/1.52 % (30165)Termination reason: Unknown
% 2.60/1.52 % (30165)Termination phase: shuffling
% 2.60/1.52
% 2.60/1.52 % (30165)Memory used [KB]: 1407
% 2.60/1.52 % (30165)Time elapsed: 0.003 s
% 2.60/1.52 % (30165)Instructions burned: 6 (million)
% 2.60/1.52 % (30165)------------------------------
% 2.60/1.52 % (30165)------------------------------
% 2.60/1.53 % (30167)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_29954 for (2999ds/710Mi)
% 2.60/1.53 % (30168)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_29954 for (2999ds/6Mi)
% 2.60/1.53 % (30166)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_29954 for (2999ds/18Mi)
% 2.60/1.53 % (30168)Instruction limit reached!
% 2.60/1.53 % (30168)------------------------------
% 2.60/1.53 % (30168)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.60/1.53 % (30168)Termination reason: Unknown
% 2.60/1.53 % (30168)Termination phase: shuffling
% 2.60/1.53
% 2.60/1.53 % (30168)Memory used [KB]: 1407
% 2.60/1.53 % (30168)Time elapsed: 0.004 s
% 2.60/1.53 % (30168)Instructions burned: 6 (million)
% 2.60/1.53 % (30168)------------------------------
% 2.60/1.53 % (30168)------------------------------
% 2.60/1.53 % (30169)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_29954 for (2999ds/902Mi)
% 2.60/1.54 % (30166)Instruction limit reached!
% 2.60/1.54 % (30166)------------------------------
% 2.60/1.54 % (30166)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.60/1.54 % (30166)Termination reason: Unknown
% 2.60/1.54 % (30166)Termination phase: Property scanning
% 2.60/1.54
% 2.60/1.54 % (30166)Memory used [KB]: 1663
% 2.60/1.54 % (30166)Time elapsed: 0.011 s
% 2.60/1.54 % (30166)Instructions burned: 19 (million)
% 2.60/1.54 % (30166)------------------------------
% 2.60/1.54 % (30166)------------------------------
% 2.60/1.54 % (30170)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_29954 for (2999ds/21Mi)
% 2.60/1.54 % (30147)Instruction limit reached!
% 2.60/1.54 % (30147)------------------------------
% 2.60/1.54 % (30147)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.60/1.54 % (30147)Termination reason: Unknown
% 2.60/1.54 % (30147)Termination phase: Saturation
% 2.60/1.54
% 2.60/1.54 % (30147)Memory used [KB]: 7164
% 2.60/1.54 % (30147)Time elapsed: 0.076 s
% 2.60/1.54 % (30147)Instructions burned: 185 (million)
% 2.60/1.54 % (30147)------------------------------
% 2.60/1.54 % (30147)------------------------------
% 2.60/1.55 % (30170)Instruction limit reached!
% 2.60/1.55 % (30170)------------------------------
% 2.60/1.55 % (30170)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.60/1.55 % (30170)Termination reason: Unknown
% 2.60/1.55 % (30170)Termination phase: Property scanning
% 2.60/1.55
% 2.60/1.55 % (30170)Memory used [KB]: 1663
% 2.60/1.55 % (30170)Time elapsed: 0.010 s
% 2.60/1.55 % (30170)Instructions burned: 22 (million)
% 2.60/1.55 % (30170)------------------------------
% 2.60/1.55 % (30170)------------------------------
% 2.60/1.55 % (30171)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_29954 for (2999ds/5Mi)
% 2.60/1.55 % (30171)Instruction limit reached!
% 2.60/1.55 % (30171)------------------------------
% 2.60/1.55 % (30171)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.60/1.55 % (30171)Termination reason: Unknown
% 2.60/1.55 % (30171)Termination phase: shuffling
% 2.60/1.55
% 2.60/1.55 % (30171)Memory used [KB]: 1407
% 2.60/1.55 % (30171)Time elapsed: 0.003 s
% 2.60/1.55 % (30171)Instructions burned: 6 (million)
% 2.60/1.55 % (30171)------------------------------
% 2.60/1.55 % (30171)------------------------------
% 2.60/1.56 % (30172)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_29954 for (2999ds/6Mi)
% 2.60/1.56 % (30174)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_29954 for (2998ds/779Mi)
% 2.60/1.56 % (30172)Instruction limit reached!
% 2.60/1.56 % (30172)------------------------------
% 2.60/1.56 % (30172)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.60/1.56 % (30172)Termination reason: Unknown
% 2.60/1.56 % (30172)Termination phase: shuffling
% 2.60/1.56
% 2.60/1.56 % (30172)Memory used [KB]: 1407
% 2.60/1.56 % (30172)Time elapsed: 0.005 s
% 2.60/1.56 % (30172)Instructions burned: 6 (million)
% 2.60/1.56 % (30172)------------------------------
% 2.60/1.56 % (30172)------------------------------
% 2.60/1.57 % (30173)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_29954 for (2999ds/377Mi)
% 2.60/1.58 % (30175)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_29954 for (2998ds/19Mi)
% 2.60/1.59 % (30175)Instruction limit reached!
% 2.60/1.59 % (30175)------------------------------
% 2.60/1.59 % (30175)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.60/1.59 % (30175)Termination reason: Unknown
% 2.60/1.59 % (30175)Termination phase: shuffling
% 2.60/1.59
% 2.60/1.59 % (30175)Memory used [KB]: 1663
% 2.60/1.59 % (30175)Time elapsed: 0.011 s
% 2.60/1.59 % (30175)Instructions burned: 19 (million)
% 2.60/1.59 % (30175)------------------------------
% 2.60/1.59 % (30175)------------------------------
% 2.60/1.59 % (30152)Instruction limit reached!
% 2.60/1.59 % (30152)------------------------------
% 2.60/1.59 % (30152)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.60/1.59 % (30152)Termination reason: Unknown
% 2.60/1.59 % (30152)Termination phase: Saturation
% 2.60/1.59
% 2.60/1.59 % (30152)Memory used [KB]: 7164
% 2.60/1.59 % (30152)Time elapsed: 0.119 s
% 2.60/1.59 % (30152)Instructions burned: 275 (million)
% 2.60/1.59 % (30152)------------------------------
% 2.60/1.59 % (30152)------------------------------
% 2.60/1.60 % (30177)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_29954 for (2998ds/17Mi)
% 2.60/1.61 % (30176)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_29954 for (2998ds/879Mi)
% 2.60/1.61 % (30177)Instruction limit reached!
% 2.60/1.61 % (30177)------------------------------
% 2.60/1.61 % (30177)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.60/1.61 % (30177)Termination reason: Unknown
% 2.60/1.61 % (30177)Termination phase: shuffling
% 2.60/1.61
% 2.60/1.61 % (30177)Memory used [KB]: 1535
% 2.60/1.61 % (30177)Time elapsed: 0.007 s
% 2.60/1.61 % (30177)Instructions burned: 17 (million)
% 2.60/1.61 % (30177)------------------------------
% 2.60/1.61 % (30177)------------------------------
% 2.60/1.62 % (30178)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_29954 for (2998ds/3Mi)
% 2.60/1.62 % (30178)Instruction limit reached!
% 2.60/1.62 % (30178)------------------------------
% 2.60/1.62 % (30178)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.60/1.62 % (30178)Termination reason: Unknown
% 2.60/1.62 % (30178)Termination phase: shuffling
% 2.60/1.62
% 2.60/1.62 % (30178)Memory used [KB]: 1407
% 2.60/1.62 % (30178)Time elapsed: 0.003 s
% 2.60/1.62 % (30178)Instructions burned: 6 (million)
% 2.60/1.62 % (30178)------------------------------
% 2.60/1.62 % (30178)------------------------------
% 2.60/1.63 % (30179)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_29954 for (2998ds/30Mi)
% 2.60/1.64 % (30179)Instruction limit reached!
% 2.60/1.64 % (30179)------------------------------
% 2.60/1.64 % (30179)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.60/1.64 % (30179)Termination reason: Unknown
% 2.60/1.64 % (30179)Termination phase: Preprocessing 2
% 2.60/1.64
% 2.60/1.64 % (30179)Memory used [KB]: 1663
% 2.60/1.64 % (30179)Time elapsed: 0.011 s
% 2.60/1.64 % (30179)Instructions burned: 32 (million)
% 2.60/1.64 % (30179)------------------------------
% 2.60/1.64 % (30179)------------------------------
% 2.60/1.65 % (30180)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_29954 for (2998ds/127Mi)
% 2.60/1.68 % (30180)Instruction limit reached!
% 2.60/1.68 % (30180)------------------------------
% 2.60/1.68 % (30180)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.60/1.68 % (30180)Termination reason: Unknown
% 2.60/1.68 % (30180)Termination phase: Saturation
% 2.60/1.68
% 2.60/1.68 % (30180)Memory used [KB]: 6780
% 2.60/1.68 % (30180)Time elapsed: 0.036 s
% 2.60/1.68 % (30180)Instructions burned: 129 (million)
% 2.60/1.68 % (30180)------------------------------
% 2.60/1.68 % (30180)------------------------------
% 2.60/1.70 % (30181)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_29954 for (2997ds/100Mi)
% 2.60/1.70 % (30169)First to succeed.
% 2.60/1.70 % (30169)Refutation found. Thanks to Tanya!
% 2.60/1.70 % SZS status Theorem for DTF2THF_29954
% 2.60/1.70 % SZS output start Proof for DTF2THF_29954
% 2.60/1.70 thf(type_def_5, type, nat: $tType).
% 2.60/1.70 thf(type_def_7, type, set_nat: $tType).
% 2.60/1.70 thf(type_def_8, type, set_set_nat: $tType).
% 2.60/1.70 thf(func_def_0, type, set_set_nat: $tType).
% 2.60/1.70 thf(func_def_1, type, set_nat: $tType).
% 2.60/1.70 thf(func_def_2, type, nat: $tType).
% 2.60/1.70 thf(func_def_3, type, one_one_nat: nat).
% 2.60/1.70 thf(func_def_4, type, times_times_nat: nat > nat > nat).
% 2.60/1.70 thf(func_def_5, type, zero_zero_nat: nat).
% 2.60/1.70 thf(func_def_6, type, groups1842438620at_nat: (nat > nat) > set_nat > nat).
% 2.60/1.70 thf(func_def_7, type, number1551313001itions: (nat > nat) > nat > $o).
% 2.60/1.70 thf(func_def_8, type, ord_less_nat: nat > nat > $o).
% 2.60/1.70 thf(func_def_9, type, ord_less_set_nat: set_nat > set_nat > $o).
% 2.60/1.70 thf(func_def_10, type, ord_less_eq_nat_o: (nat > $o) > (nat > $o) > $o).
% 2.60/1.70 thf(func_def_11, type, ord_less_eq_nat: nat > nat > $o).
% 2.60/1.70 thf(func_def_12, type, ord_less_eq_set_nat: set_nat > set_nat > $o).
% 2.60/1.70 thf(func_def_13, type, ord_le1613022364et_nat: set_set_nat > set_set_nat > $o).
% 2.60/1.70 thf(func_def_14, type, collect_nat: (nat > $o) > set_nat).
% 2.60/1.70 thf(func_def_15, type, collect_set_nat: (set_nat > $o) > set_set_nat).
% 2.60/1.70 thf(func_def_16, type, set_ord_atMost_nat: nat > set_nat).
% 2.60/1.70 thf(func_def_17, type, set_or1086813439et_nat: set_nat > set_set_nat).
% 2.60/1.70 thf(func_def_18, type, member_nat: nat > set_nat > $o).
% 2.60/1.70 thf(func_def_19, type, member_set_nat: set_nat > set_set_nat > $o).
% 2.60/1.70 thf(func_def_20, type, p: nat > nat).
% 2.60/1.70 thf(func_def_35, type, sP0: nat > nat > nat > $o).
% 2.60/1.70 thf(func_def_36, type, sP1: nat > nat > nat > $o).
% 2.60/1.70 thf(func_def_37, type, sP2: nat > nat > nat > $o).
% 2.60/1.70 thf(func_def_38, type, sP3: (nat > nat) > (nat > nat) > set_nat > $o).
% 2.60/1.70 thf(func_def_39, type, sP4: (nat > nat) > set_nat > set_nat > $o).
% 2.60/1.70 thf(func_def_40, type, sK5: nat > (nat > $o) > nat).
% 2.60/1.70 thf(func_def_41, type, sK6: (nat > $o) > nat).
% 2.60/1.70 thf(func_def_42, type, sK7: (nat > nat) > nat).
% 2.60/1.70 thf(func_def_43, type, sK8: (nat > nat) > nat).
% 2.60/1.70 thf(func_def_44, type, sK9: (nat > nat) > nat).
% 2.60/1.70 thf(func_def_45, type, sK10: (nat > nat) > nat).
% 2.60/1.70 thf(func_def_46, type, sK11: (nat > $o) > (nat > $o) > nat).
% 2.60/1.70 thf(func_def_47, type, sK12: (set_nat > set_nat) > set_nat).
% 2.60/1.70 thf(func_def_48, type, sK13: (set_nat > set_nat) > set_nat).
% 2.60/1.70 thf(func_def_49, type, sK14: (nat > nat) > set_nat > nat).
% 2.60/1.70 thf(func_def_50, type, sK15: (nat > nat) > nat).
% 2.60/1.70 thf(func_def_51, type, sK16: (nat > nat) > nat).
% 2.60/1.70 thf(func_def_52, type, sK17: (nat > nat) > nat).
% 2.60/1.70 thf(func_def_53, type, sK18: (nat > nat) > nat).
% 2.60/1.70 thf(func_def_54, type, sK19: (nat > $o) > (nat > $o) > nat).
% 2.60/1.70 thf(func_def_55, type, sK20: (nat > set_nat) > nat).
% 2.60/1.70 thf(func_def_56, type, sK21: (nat > set_nat) > nat).
% 2.60/1.70 thf(func_def_57, type, sK22: (nat > nat) > nat).
% 2.60/1.70 thf(func_def_58, type, sK23: (nat > nat) > nat).
% 2.60/1.70 thf(func_def_59, type, sK24: (nat > nat) > set_nat > (nat > nat) > set_nat > (nat > nat) > nat).
% 2.60/1.70 thf(func_def_60, type, sK25: set_nat > (nat > nat) > set_nat > nat).
% 2.60/1.70 thf(func_def_61, type, sK26: nat > set_nat > (nat > nat) > set_nat > nat).
% 2.60/1.70 thf(func_def_62, type, sK27: (nat > nat) > nat).
% 2.60/1.70 thf(func_def_63, type, sK28: (nat > nat) > nat).
% 2.60/1.70 thf(func_def_64, type, sK29: (set_nat > nat) > set_nat).
% 2.60/1.70 thf(func_def_65, type, sK30: (set_nat > nat) > set_nat).
% 2.60/1.70 thf(func_def_66, type, sK31: (set_nat > set_nat) > set_nat).
% 2.60/1.70 thf(func_def_67, type, sK32: (set_nat > set_nat) > set_nat).
% 2.60/1.70 thf(func_def_68, type, sK33: set_nat > (nat > nat) > nat).
% 2.60/1.70 thf(func_def_69, type, sK34: (nat > nat) > nat).
% 2.60/1.70 thf(func_def_70, type, sK35: (nat > nat) > nat).
% 2.60/1.70 thf(func_def_71, type, sK36: (set_nat > nat) > set_nat).
% 2.60/1.70 thf(func_def_72, type, sK37: (set_nat > nat) > set_nat).
% 2.60/1.70 thf(func_def_73, type, sK38: (nat > nat > $o) > nat).
% 2.60/1.70 thf(func_def_74, type, sK39: (nat > nat > $o) > nat).
% 2.60/1.70 thf(func_def_75, type, sK40: (nat > nat > $o) > nat).
% 2.60/1.70 thf(func_def_76, type, sK41: (nat > nat > $o) > nat).
% 2.60/1.70 thf(func_def_77, type, sK42: (nat > nat > $o) > nat).
% 2.60/1.70 thf(func_def_78, type, sK43: (nat > $o) > nat).
% 2.60/1.70 thf(func_def_79, type, sK44: (set_nat > set_nat) > set_nat).
% 2.60/1.70 thf(func_def_80, type, sK45: (set_nat > set_nat) > set_nat).
% 2.60/1.70 thf(func_def_81, type, sK46: (nat > nat) > set_nat > nat).
% 2.60/1.70 thf(func_def_82, type, sK47: (set_nat > nat) > set_nat).
% 2.60/1.70 thf(func_def_83, type, sK48: (set_nat > nat) > set_nat).
% 2.60/1.70 thf(func_def_84, type, sK49: (nat > set_nat) > nat).
% 2.60/1.70 thf(func_def_85, type, sK50: (nat > set_nat) > nat).
% 2.60/1.70 thf(func_def_86, type, sK51: set_nat > (nat > $o) > nat).
% 2.60/1.70 thf(func_def_87, type, sK52: (set_nat > set_nat) > set_nat).
% 2.60/1.70 thf(func_def_88, type, sK53: (set_nat > set_nat) > set_nat).
% 2.60/1.70 thf(func_def_89, type, sK54: (set_nat > nat) > set_nat).
% 2.60/1.70 thf(func_def_90, type, sK55: (set_nat > nat) > set_nat).
% 2.60/1.70 thf(func_def_91, type, sK56: (nat > $o) > (nat > $o) > nat).
% 2.60/1.70 thf(func_def_92, type, sK57: nat > nat).
% 2.60/1.70 thf(func_def_93, type, sK58: nat > (nat > $o) > nat).
% 2.60/1.70 thf(func_def_94, type, sK59: (nat > $o) > nat).
% 2.60/1.70 thf(func_def_95, type, sK60: set_nat > set_nat > nat).
% 2.60/1.70 thf(func_def_96, type, sK61: (nat > nat) > (nat > nat) > (nat > nat) > set_nat > (nat > nat) > set_nat > nat).
% 2.60/1.70 thf(func_def_97, type, sK62: (nat > nat) > set_nat > (nat > nat) > set_nat > nat).
% 2.60/1.70 thf(func_def_98, type, sK63: (nat > nat) > nat).
% 2.60/1.70 thf(func_def_99, type, sK64: (nat > nat) > nat).
% 2.60/1.70 thf(func_def_100, type, sK65: (nat > set_nat) > nat).
% 2.60/1.70 thf(func_def_101, type, sK66: (nat > set_nat) > nat).
% 2.60/1.70 thf(func_def_102, type, sK67: (nat > nat) > nat).
% 2.60/1.70 thf(func_def_103, type, sK68: (nat > nat) > nat).
% 2.60/1.70 thf(func_def_104, type, sK69: nat).
% 2.60/1.70 thf(func_def_105, type, sK70: (nat > nat) > (nat > nat) > set_nat > nat).
% 2.60/1.70 thf(func_def_106, type, sK71: (nat > nat) > nat > nat).
% 2.60/1.70 thf(func_def_107, type, sK72: (nat > nat > $o) > nat).
% 2.60/1.70 thf(func_def_108, type, sK73: (nat > nat > $o) > nat).
% 2.60/1.70 thf(func_def_109, type, sK74: (nat > nat > $o) > nat).
% 2.60/1.70 thf(func_def_110, type, sK75: (nat > nat > $o) > nat).
% 2.60/1.70 thf(func_def_111, type, sK76: (nat > nat) > set_nat > set_nat > nat).
% 2.60/1.70 thf(func_def_112, type, sK77: (nat > nat) > (nat > nat) > set_nat > nat).
% 2.60/1.70 thf(func_def_113, type, sK78: set_nat > set_nat > (nat > nat) > nat).
% 2.60/1.70 thf(func_def_114, type, sK79: set_nat > (nat > nat) > (nat > nat) > (nat > nat) > nat).
% 2.60/1.70 thf(func_def_115, type, sK80: set_nat > (nat > nat) > (nat > nat) > nat).
% 2.60/1.70 thf(func_def_116, type, sK81: (nat > nat) > (nat > nat) > set_nat > nat).
% 2.60/1.70 thf(func_def_117, type, sK82: (nat > set_nat) > nat).
% 2.60/1.70 thf(func_def_118, type, sK83: (nat > set_nat) > nat).
% 2.60/1.70 thf(func_def_119, type, sK84: set_nat > (nat > $o) > (nat > $o) > nat).
% 2.60/1.70 thf(func_def_120, type, sK85: (nat > nat) > set_nat > nat).
% 2.60/1.70 thf(func_def_121, type, sK86: (nat > $o) > nat).
% 2.60/1.70 thf(func_def_123, type, ph88: !>[X0: $tType]:(X0)).
% 2.60/1.70 thf(func_def_124, type, sK89: nat).
% 2.60/1.70 thf(func_def_125, type, sK90: nat).
% 2.60/1.70 thf(f4129,plain,(
% 2.60/1.70 $false),
% 2.60/1.70 inference(avatar_sat_refutation,[status(thm)],[f1730,f1744,f1748,f2008,f2628,f3128,f3682,f3896,f4126])).
% 2.60/1.70 thf(f4126,plain,(
% 2.60/1.70 ~spl87_3 | spl87_24),
% 2.60/1.70 inference(avatar_contradiction_clause,[status(thm)],[f4125])).
% 2.60/1.70 thf(f4125,plain,(
% 2.60/1.70 $false | (~spl87_3 | spl87_24)),
% 2.60/1.70 inference(trivial_inequality_removal,[status(thm)],[f4117])).
% 2.60/1.70 thf(f4117,plain,(
% 2.60/1.70 (zero_zero_nat != zero_zero_nat) | (~spl87_3 | spl87_24)),
% 2.60/1.70 inference(superposition,[status(thm)],[f2281,f3182])).
% 2.60/1.70 thf(f3182,plain,(
% 2.60/1.70 ( ! [X0 : set_nat] : ((zero_zero_nat = (groups1842438620at_nat @ p @ X0))) ) | ~spl87_3),
% 2.60/1.70 inference(trivial_inequality_removal,[status(thm)],[f3152])).
% 2.60/1.70 thf(f3152,plain,(
% 2.60/1.70 ( ! [X0 : set_nat] : ((zero_zero_nat = (groups1842438620at_nat @ p @ X0)) | (zero_zero_nat != zero_zero_nat)) ) | ~spl87_3),
% 2.60/1.70 inference(superposition,[status(thm)],[f1337,f3131])).
% 2.60/1.70 thf(f3131,plain,(
% 2.60/1.70 ( ! [X1 : nat] : ((zero_zero_nat = (p @ X1))) ) | ~spl87_3),
% 2.60/1.70 inference(beta_eta_normalization,[status(thm)],[f3129])).
% 2.60/1.70 thf(f3129,plain,(
% 2.60/1.70 ( ! [X1 : nat] : (((p @ X1) = ((^[Y0 : nat]: (zero_zero_nat)) @ X1))) ) | ~spl87_3),
% 2.60/1.70 inference(argument_congruence,[status(thm)],[f1728])).
% 2.60/1.70 thf(f1728,plain,(
% 2.60/1.70 (p = (^[Y0 : nat]: (zero_zero_nat))) | ~spl87_3),
% 2.60/1.70 inference(avatar_component_clause,[status(thm)],[f1727])).
% 2.60/1.70 thf(f1727,definition,(
% 2.60/1.70 spl87_3 <=> (p = (^[Y0 : nat]: (zero_zero_nat)))),
% 2.60/1.70 introduced(definition,[new_symbols(naming,[spl87_3])],[avatar_definition])).
% 2.60/1.70 thf(f1337,plain,(
% 2.60/1.70 ( ! [X0 : nat > nat,X1 : set_nat] : ((zero_zero_nat != (X0 @ (sK33 @ X1 @ X0))) | (zero_zero_nat = (groups1842438620at_nat @ X0 @ X1))) )),
% 2.60/1.70 inference(cnf_transformation,[status(thm)],[f1092])).
% 2.60/1.70 thf(f1092,plain,(
% 2.60/1.70 ! [X0 : nat > nat,X1 : set_nat] : ((zero_zero_nat = (groups1842438620at_nat @ X0 @ X1)) | (((member_nat @ (sK33 @ X1 @ X0) @ X1) = $true) & (zero_zero_nat != (X0 @ (sK33 @ X1 @ X0)))))),
% 2.60/1.70 inference(skolemisation,[status(esa),new_symbols(skolem,[sK33])],[f985,f1091])).
% 2.60/1.70 thf(f1091,plain,(
% 2.60/1.70 ! [X0 : nat > nat,X1 : set_nat] : (? [X2 : nat] : (((member_nat @ X2 @ X1) = $true) & (zero_zero_nat != (X0 @ X2))) => (((member_nat @ (sK33 @ X1 @ X0) @ X1) = $true) & (zero_zero_nat != (X0 @ (sK33 @ X1 @ X0)))))),
% 2.60/1.70 introduced(definition,[],[choice_axiom])).
% 2.60/1.70 thf(f985,plain,(
% 2.60/1.70 ! [X0 : nat > nat,X1 : set_nat] : ((zero_zero_nat = (groups1842438620at_nat @ X0 @ X1)) | ? [X2 : nat] : (((member_nat @ X2 @ X1) = $true) & (zero_zero_nat != (X0 @ X2))))),
% 2.60/1.70 inference(ennf_transformation,[status(thm)],[f477])).
% 2.60/1.70 thf(f477,plain,(
% 2.60/1.70 ! [X0 : nat > nat,X1 : set_nat] : ((zero_zero_nat != (groups1842438620at_nat @ X0 @ X1)) => ~! [X2 : nat] : (((member_nat @ X2 @ X1) = $true) => (zero_zero_nat = (X0 @ X2))))),
% 2.60/1.70 inference(fool_elimination,[status(thm)],[f476])).
% 2.60/1.70 thf(f476,plain,(
% 2.60/1.70 ! [X0 : nat > nat,X1 : set_nat] : ((zero_zero_nat != (groups1842438620at_nat @ X0 @ X1)) => ~! [X2 : nat] : ((member_nat @ X2 @ X1) => (zero_zero_nat = (X0 @ X2))))),
% 2.60/1.70 inference(rectify,[status(thm)],[f41])).
% 2.60/1.70 thf(f41,axiom,(
% 2.60/1.70 ! [X28 : nat > nat,X0 : set_nat] : ((zero_zero_nat != (groups1842438620at_nat @ X28 @ X0)) => ~! [X29 : nat] : ((member_nat @ X29 @ X0) => (zero_zero_nat = (X28 @ X29))))),
% 2.60/1.70 file('/export/starexec/sandbox/tmp/tmp.53O3If2WxG/DTF2THF_29954.p',fact_40_sum_Onot__neutral__contains__not__neutral)).
% 2.60/1.70 thf(f2281,plain,(
% 2.60/1.70 (zero_zero_nat != (groups1842438620at_nat @ p @ (set_ord_atMost_nat @ zero_zero_nat))) | spl87_24),
% 2.60/1.70 inference(avatar_component_clause,[status(thm)],[f2279])).
% 2.60/1.70 thf(f2279,definition,(
% 2.60/1.70 spl87_24 <=> (zero_zero_nat = (groups1842438620at_nat @ p @ (set_ord_atMost_nat @ zero_zero_nat)))),
% 2.60/1.70 introduced(definition,[new_symbols(naming,[spl87_24])],[avatar_definition])).
% 2.60/1.70 thf(f3896,plain,(
% 2.60/1.70 ~spl87_24 | spl87_1 | ~spl87_3),
% 2.60/1.70 inference(avatar_split_clause,[status(thm)],[f3836,f1727,f1719,f2279])).
% 2.60/1.70 thf(f1719,definition,(
% 2.60/1.70 spl87_1 <=> (zero_zero_nat = (groups1842438620at_nat @ (^[Y0 : nat]: (times_times_nat @ (p @ Y0) @ Y0)) @ (set_ord_atMost_nat @ zero_zero_nat)))),
% 2.60/1.70 introduced(definition,[new_symbols(naming,[spl87_1])],[avatar_definition])).
% 2.60/1.70 thf(f3836,plain,(
% 2.60/1.70 (zero_zero_nat != (groups1842438620at_nat @ p @ (set_ord_atMost_nat @ zero_zero_nat))) | (spl87_1 | ~spl87_3)),
% 2.60/1.70 inference(trivial_inequality_removal,[status(thm)],[f3831])).
% 2.60/1.70 thf(f3831,plain,(
% 2.60/1.70 (zero_zero_nat != zero_zero_nat) | (zero_zero_nat != (groups1842438620at_nat @ p @ (set_ord_atMost_nat @ zero_zero_nat))) | (spl87_1 | ~spl87_3)),
% 2.60/1.70 inference(superposition,[status(thm)],[f3814,f3131])).
% 2.60/1.70 thf(f3814,plain,(
% 2.60/1.70 ( ! [X0 : nat > nat] : ((zero_zero_nat != (X0 @ (sK70 @ (^[Y0 : nat]: (times_times_nat @ (p @ Y0) @ Y0)) @ X0 @ (set_ord_atMost_nat @ zero_zero_nat)))) | (zero_zero_nat != (groups1842438620at_nat @ X0 @ (set_ord_atMost_nat @ zero_zero_nat)))) ) | (spl87_1 | ~spl87_3)),
% 2.60/1.70 inference(forward_demodulation,[status(thm)],[f3813,f1630])).
% 2.60/1.70 thf(f1630,plain,(
% 2.60/1.70 ( ! [X1 : nat] : ((zero_zero_nat = (times_times_nat @ zero_zero_nat @ X1))) )),
% 2.60/1.70 inference(equality_resolution,[status(thm)],[f1245])).
% 2.60/1.70 thf(f1245,plain,(
% 2.60/1.70 ( ! [X0 : nat,X1 : nat] : ((zero_zero_nat = (times_times_nat @ X0 @ X1)) | (zero_zero_nat != X0)) )),
% 2.60/1.70 inference(cnf_transformation,[status(thm)],[f1050])).
% 2.60/1.70 thf(f1050,plain,(
% 2.60/1.70 ! [X0 : nat,X1 : nat] : (((zero_zero_nat = (times_times_nat @ X0 @ X1)) | ((zero_zero_nat != X0) & (zero_zero_nat != X1))) & ((zero_zero_nat = X0) | (zero_zero_nat = X1) | (zero_zero_nat != (times_times_nat @ X0 @ X1))))),
% 2.60/1.70 inference(flattening,[status(thm)],[f1049])).
% 2.60/1.70 thf(f1049,plain,(
% 2.60/1.70 ! [X0 : nat,X1 : nat] : (((zero_zero_nat = (times_times_nat @ X0 @ X1)) | ((zero_zero_nat != X0) & (zero_zero_nat != X1))) & (((zero_zero_nat = X0) | (zero_zero_nat = X1)) | (zero_zero_nat != (times_times_nat @ X0 @ X1))))),
% 2.60/1.70 inference(nnf_transformation,[status(thm)],[f489])).
% 2.60/1.70 thf(f489,plain,(
% 2.60/1.70 ! [X0 : nat,X1 : nat] : ((zero_zero_nat = (times_times_nat @ X0 @ X1)) <=> ((zero_zero_nat = X0) | (zero_zero_nat = X1)))),
% 2.60/1.70 inference(fool_elimination,[status(thm)],[f488])).
% 2.60/1.70 thf(f488,plain,(
% 2.60/1.70 ! [X0 : nat,X1 : nat] : ((zero_zero_nat = (times_times_nat @ X0 @ X1)) = (zero_zero_nat = X0) | (zero_zero_nat = X1))),
% 2.60/1.70 inference(rectify,[status(thm)],[f22])).
% 2.60/1.70 thf(f22,axiom,(
% 2.60/1.70 ! [X9 : nat,X3 : nat] : ((zero_zero_nat = (times_times_nat @ X9 @ X3)) = (zero_zero_nat = X9) | (zero_zero_nat = X3))),
% 2.60/1.70 file('/export/starexec/sandbox/tmp/tmp.53O3If2WxG/DTF2THF_29954.p',fact_21_mult__is__0)).
% 2.60/1.70 thf(f3813,plain,(
% 2.60/1.70 ( ! [X0 : nat > nat] : ((zero_zero_nat != (groups1842438620at_nat @ X0 @ (set_ord_atMost_nat @ zero_zero_nat))) | ((X0 @ (sK70 @ (^[Y0 : nat]: (times_times_nat @ (p @ Y0) @ Y0)) @ X0 @ (set_ord_atMost_nat @ zero_zero_nat))) != (times_times_nat @ zero_zero_nat @ (sK70 @ (^[Y0 : nat]: (times_times_nat @ (p @ Y0) @ Y0)) @ X0 @ (set_ord_atMost_nat @ zero_zero_nat))))) ) | (spl87_1 | ~spl87_3)),
% 2.60/1.70 inference(forward_demodulation,[status(thm)],[f3137,f3131])).
% 2.60/1.70 thf(f3137,plain,(
% 2.60/1.70 ( ! [X0 : nat > nat] : (((times_times_nat @ (p @ (sK70 @ (^[Y0 : nat]: (times_times_nat @ (p @ Y0) @ Y0)) @ X0 @ (set_ord_atMost_nat @ zero_zero_nat))) @ (sK70 @ (^[Y0 : nat]: (times_times_nat @ (p @ Y0) @ Y0)) @ X0 @ (set_ord_atMost_nat @ zero_zero_nat))) != (X0 @ (sK70 @ (^[Y0 : nat]: (times_times_nat @ (p @ Y0) @ Y0)) @ X0 @ (set_ord_atMost_nat @ zero_zero_nat)))) | (zero_zero_nat != (groups1842438620at_nat @ X0 @ (set_ord_atMost_nat @ zero_zero_nat)))) ) | spl87_1),
% 2.60/1.70 inference(beta_eta_normalization,[status(thm)],[f3133])).
% 2.60/1.70 thf(f3133,plain,(
% 2.60/1.70 ( ! [X0 : nat > nat] : ((zero_zero_nat != (groups1842438620at_nat @ X0 @ (set_ord_atMost_nat @ zero_zero_nat))) | ((X0 @ (sK70 @ (^[Y0 : nat]: (times_times_nat @ (p @ Y0) @ Y0)) @ X0 @ (set_ord_atMost_nat @ zero_zero_nat))) != ((^[Y0 : nat]: (times_times_nat @ (p @ Y0) @ Y0)) @ (sK70 @ (^[Y0 : nat]: (times_times_nat @ (p @ Y0) @ Y0)) @ X0 @ (set_ord_atMost_nat @ zero_zero_nat))))) ) | spl87_1),
% 2.60/1.70 inference(superposition,[status(thm)],[f1721,f1687])).
% 2.60/1.70 thf(f1687,plain,(
% 2.60/1.70 ( ! [X2 : nat > nat,X3 : nat > nat,X1 : set_nat] : (((groups1842438620at_nat @ X2 @ X1) = (groups1842438620at_nat @ X3 @ X1)) | ((X3 @ (sK70 @ X3 @ X2 @ X1)) != (X2 @ (sK70 @ X3 @ X2 @ X1)))) )),
% 2.60/1.70 inference(equality_resolution,[status(thm)],[f1526])).
% 2.60/1.70 thf(f1526,plain,(
% 2.60/1.70 ( ! [X2 : nat > nat,X3 : nat > nat,X0 : set_nat,X1 : set_nat] : (((groups1842438620at_nat @ X3 @ X0) = (groups1842438620at_nat @ X2 @ X1)) | ((X2 @ (sK70 @ X3 @ X2 @ X0)) != (X3 @ (sK70 @ X3 @ X2 @ X0))) | (X0 != X1)) )),
% 2.60/1.70 inference(cnf_transformation,[status(thm)],[f1173])).
% 2.60/1.70 thf(f1173,plain,(
% 2.60/1.70 ! [X0 : set_nat,X1 : set_nat,X2 : nat > nat,X3 : nat > nat] : (((groups1842438620at_nat @ X3 @ X0) = (groups1842438620at_nat @ X2 @ X1)) | (((X2 @ (sK70 @ X3 @ X2 @ X0)) != (X3 @ (sK70 @ X3 @ X2 @ X0))) & ((member_nat @ (sK70 @ X3 @ X2 @ X0) @ X0) = $true)) | (X0 != X1))),
% 2.60/1.70 inference(skolemisation,[status(esa),new_symbols(skolem,[sK70])],[f841,f1172])).
% 2.60/1.70 thf(f1172,plain,(
% 2.60/1.70 ! [X0 : set_nat,X2 : nat > nat,X3 : nat > nat] : (? [X4 : nat] : (((X2 @ X4) != (X3 @ X4)) & ((member_nat @ X4 @ X0) = $true)) => (((X2 @ (sK70 @ X3 @ X2 @ X0)) != (X3 @ (sK70 @ X3 @ X2 @ X0))) & ((member_nat @ (sK70 @ X3 @ X2 @ X0) @ X0) = $true)))),
% 2.60/1.70 introduced(definition,[],[choice_axiom])).
% 2.60/1.70 thf(f841,plain,(
% 2.60/1.70 ! [X0 : set_nat,X1 : set_nat,X2 : nat > nat,X3 : nat > nat] : (((groups1842438620at_nat @ X3 @ X0) = (groups1842438620at_nat @ X2 @ X1)) | ? [X4 : nat] : (((X2 @ X4) != (X3 @ X4)) & ((member_nat @ X4 @ X0) = $true)) | (X0 != X1))),
% 2.60/1.70 inference(flattening,[status(thm)],[f840])).
% 2.60/1.70 thf(f840,plain,(
% 2.60/1.70 ! [X0 : set_nat,X1 : set_nat,X2 : nat > nat,X3 : nat > nat] : ((((groups1842438620at_nat @ X3 @ X0) = (groups1842438620at_nat @ X2 @ X1)) | ? [X4 : nat] : (((X2 @ X4) != (X3 @ X4)) & ((member_nat @ X4 @ X0) = $true))) | (X0 != X1))),
% 2.60/1.70 inference(ennf_transformation,[status(thm)],[f623])).
% 2.60/1.70 thf(f623,plain,(
% 2.60/1.70 ! [X0 : set_nat,X1 : set_nat,X2 : nat > nat,X3 : nat > nat] : ((X0 = X1) => (! [X4 : nat] : (((member_nat @ X4 @ X0) = $true) => ((X2 @ X4) = (X3 @ X4))) => ((groups1842438620at_nat @ X3 @ X0) = (groups1842438620at_nat @ X2 @ X1))))),
% 2.60/1.70 inference(fool_elimination,[status(thm)],[f622])).
% 2.60/1.70 thf(f622,plain,(
% 2.60/1.70 ! [X0 : set_nat,X1 : set_nat,X2 : nat > nat,X3 : nat > nat] : ((X0 = X1) => (! [X4 : nat] : ((member_nat @ X4 @ X0) => ((X2 @ X4) = (X3 @ X4))) => ((groups1842438620at_nat @ X3 @ X0) = (groups1842438620at_nat @ X2 @ X1))))),
% 2.60/1.70 inference(rectify,[status(thm)],[f33])).
% 2.60/1.70 thf(f33,axiom,(
% 2.60/1.70 ! [X31 : set_nat,X0 : set_nat,X28 : nat > nat,X27 : nat > nat] : ((X0 = X31) => (! [X19 : nat] : ((member_nat @ X19 @ X31) => ((X27 @ X19) = (X28 @ X19))) => ((groups1842438620at_nat @ X28 @ X0) = (groups1842438620at_nat @ X27 @ X31))))),
% 2.60/1.70 file('/export/starexec/sandbox/tmp/tmp.53O3If2WxG/DTF2THF_29954.p',fact_32_sum_Ocong)).
% 2.60/1.70 thf(f1721,plain,(
% 2.60/1.70 (zero_zero_nat != (groups1842438620at_nat @ (^[Y0 : nat]: (times_times_nat @ (p @ Y0) @ Y0)) @ (set_ord_atMost_nat @ zero_zero_nat))) | spl87_1),
% 2.60/1.70 inference(avatar_component_clause,[status(thm)],[f1719])).
% 2.60/1.70 thf(f3682,plain,(
% 2.60/1.70 ~spl87_15),
% 2.60/1.70 inference(avatar_contradiction_clause,[status(thm)],[f3681])).
% 2.60/1.70 thf(f3681,plain,(
% 2.60/1.70 $false | ~spl87_15),
% 2.60/1.70 inference(trivial_inequality_removal,[status(thm)],[f3602])).
% 2.60/1.70 thf(f3602,plain,(
% 2.60/1.70 (one_one_nat != one_one_nat) | ~spl87_15),
% 2.60/1.70 inference(superposition,[status(thm)],[f1214,f1928])).
% 2.60/1.70 thf(f1928,plain,(
% 2.60/1.70 (zero_zero_nat = one_one_nat) | ~spl87_15),
% 2.60/1.70 inference(avatar_component_clause,[status(thm)],[f1926])).
% 2.60/1.70 thf(f1926,definition,(
% 2.60/1.70 spl87_15 <=> (zero_zero_nat = one_one_nat)),
% 2.60/1.70 introduced(definition,[new_symbols(naming,[spl87_15])],[avatar_definition])).
% 2.60/1.70 thf(f1214,plain,(
% 2.60/1.70 (zero_zero_nat != one_one_nat)),
% 2.60/1.70 inference(cnf_transformation,[status(thm)],[f40])).
% 2.60/1.70 thf(f40,axiom,(
% 2.60/1.70 (zero_zero_nat != one_one_nat)),
% 2.60/1.70 file('/export/starexec/sandbox/tmp/tmp.53O3If2WxG/DTF2THF_29954.p',fact_39_zero__neq__one)).
% 2.60/1.70 thf(f3128,plain,(
% 2.60/1.70 ~spl87_16 | spl87_3 | ~spl87_7),
% 2.60/1.70 inference(avatar_split_clause,[status(thm)],[f3127,f1746,f1727,f1930])).
% 2.60/1.70 thf(f1930,definition,(
% 2.60/1.70 spl87_16 <=> (zero_zero_nat = (p @ zero_zero_nat))),
% 2.60/1.70 introduced(definition,[new_symbols(naming,[spl87_16])],[avatar_definition])).
% 2.60/1.70 thf(f1746,definition,(
% 2.60/1.70 spl87_7 <=> ! [X1 : nat] : (((ord_less_eq_nat @ X1 @ zero_zero_nat) = $true) | (zero_zero_nat = (p @ X1)))),
% 2.60/1.70 introduced(definition,[new_symbols(naming,[spl87_7])],[avatar_definition])).
% 2.60/1.70 thf(f3127,plain,(
% 2.60/1.70 (zero_zero_nat != (p @ zero_zero_nat)) | (spl87_3 | ~spl87_7)),
% 2.60/1.70 inference(superposition,[status(thm)],[f2597,f2734])).
% 2.60/1.70 thf(f2734,plain,(
% 2.60/1.70 (zero_zero_nat = sK90) | (spl87_3 | ~spl87_7)),
% 2.60/1.70 inference(trivial_inequality_removal,[status(thm)],[f2689])).
% 2.60/1.70 thf(f2689,plain,(
% 2.60/1.70 (zero_zero_nat != zero_zero_nat) | (zero_zero_nat = sK90) | (spl87_3 | ~spl87_7)),
% 2.60/1.70 inference(superposition,[status(thm)],[f2597,f2644])).
% 2.60/1.70 thf(f2644,plain,(
% 2.60/1.70 ( ! [X0 : nat] : ((zero_zero_nat = (p @ X0)) | (zero_zero_nat = X0)) ) | ~spl87_7),
% 2.60/1.70 inference(trivial_inequality_removal,[status(thm)],[f2632])).
% 2.60/1.70 thf(f2632,plain,(
% 2.60/1.70 ( ! [X0 : nat] : ((zero_zero_nat = X0) | (zero_zero_nat = (p @ X0)) | ($true != $true)) ) | ~spl87_7),
% 2.60/1.70 inference(superposition,[status(thm)],[f1537,f1747])).
% 2.60/1.70 thf(f1747,plain,(
% 2.60/1.70 ( ! [X1 : nat] : (((ord_less_eq_nat @ X1 @ zero_zero_nat) = $true) | (zero_zero_nat = (p @ X1))) ) | ~spl87_7),
% 2.60/1.70 inference(avatar_component_clause,[status(thm)],[f1746])).
% 2.60/1.70 thf(f1537,plain,(
% 2.60/1.70 ( ! [X0 : nat] : (((ord_less_eq_nat @ X0 @ zero_zero_nat) != $true) | (zero_zero_nat = X0)) )),
% 2.60/1.70 inference(cnf_transformation,[status(thm)],[f1176])).
% 2.60/1.70 thf(f1176,plain,(
% 2.60/1.70 ! [X0 : nat] : ((((ord_less_eq_nat @ X0 @ zero_zero_nat) = $true) | (zero_zero_nat != X0)) & ((zero_zero_nat = X0) | ((ord_less_eq_nat @ X0 @ zero_zero_nat) != $true)))),
% 2.60/1.70 inference(nnf_transformation,[status(thm)],[f541])).
% 2.60/1.70 thf(f541,plain,(
% 2.60/1.70 ! [X0 : nat] : (((ord_less_eq_nat @ X0 @ zero_zero_nat) = $true) <=> (zero_zero_nat = X0))),
% 2.60/1.70 inference(fool_elimination,[status(thm)],[f540])).
% 2.60/1.70 thf(f540,plain,(
% 2.60/1.70 ! [X0 : nat] : ((ord_less_eq_nat @ X0 @ zero_zero_nat) = (zero_zero_nat = X0))),
% 2.60/1.70 inference(rectify,[status(thm)],[f48])).
% 2.60/1.70 thf(f48,axiom,(
% 2.60/1.70 ! [X3 : nat] : ((ord_less_eq_nat @ X3 @ zero_zero_nat) = (zero_zero_nat = X3))),
% 2.60/1.70 file('/export/starexec/sandbox/tmp/tmp.53O3If2WxG/DTF2THF_29954.p',fact_47_le__0__eq)).
% 2.60/1.70 thf(f2597,plain,(
% 2.60/1.70 (zero_zero_nat != (p @ sK90)) | spl87_3),
% 2.60/1.70 inference(beta_eta_normalization,[status(thm)],[f2596])).
% 2.60/1.70 thf(f2596,plain,(
% 2.60/1.70 (((^[Y0 : nat]: (zero_zero_nat)) @ sK90) != (p @ sK90)) | spl87_3),
% 2.60/1.70 inference(negative_extensionality,[status(thm)],[f1729])).
% 2.60/1.70 thf(f1729,plain,(
% 2.60/1.70 (p != (^[Y0 : nat]: (zero_zero_nat))) | spl87_3),
% 2.60/1.70 inference(avatar_component_clause,[status(thm)],[f1727])).
% 2.60/1.70 thf(f2628,plain,(
% 2.60/1.70 spl87_16 | spl87_15 | ~spl87_6),
% 2.60/1.70 inference(avatar_split_clause,[status(thm)],[f2611,f1742,f1926,f1930])).
% 2.60/1.70 thf(f1742,definition,(
% 2.60/1.70 spl87_6 <=> ! [X1 : nat] : ((zero_zero_nat = (p @ X1)) | ((ord_less_eq_nat @ one_one_nat @ X1) = $true))),
% 2.60/1.70 introduced(definition,[new_symbols(naming,[spl87_6])],[avatar_definition])).
% 2.60/1.70 thf(f2611,plain,(
% 2.60/1.70 (zero_zero_nat = (p @ zero_zero_nat)) | (zero_zero_nat = one_one_nat) | ~spl87_6),
% 2.60/1.70 inference(trivial_inequality_removal,[status(thm)],[f2607])).
% 2.60/1.70 thf(f2607,plain,(
% 2.60/1.70 (zero_zero_nat = (p @ zero_zero_nat)) | (zero_zero_nat = one_one_nat) | ($true != $true) | ~spl87_6),
% 2.60/1.70 inference(superposition,[status(thm)],[f1537,f1743])).
% 2.60/1.70 thf(f1743,plain,(
% 2.60/1.70 ( ! [X1 : nat] : (((ord_less_eq_nat @ one_one_nat @ X1) = $true) | (zero_zero_nat = (p @ X1))) ) | ~spl87_6),
% 2.60/1.70 inference(avatar_component_clause,[status(thm)],[f1742])).
% 2.60/1.70 thf(f2008,plain,(
% 2.60/1.70 spl87_2 | ~spl87_3),
% 2.60/1.70 inference(avatar_contradiction_clause,[status(thm)],[f2007])).
% 2.60/1.70 thf(f2007,plain,(
% 2.60/1.70 $false | (spl87_2 | ~spl87_3)),
% 2.60/1.70 inference(trivial_inequality_removal,[status(thm)],[f1962])).
% 2.60/1.70 thf(f1962,plain,(
% 2.60/1.70 (zero_zero_nat != zero_zero_nat) | (spl87_2 | ~spl87_3)),
% 2.60/1.70 inference(superposition,[status(thm)],[f1725,f1939])).
% 2.60/1.70 thf(f1939,plain,(
% 2.60/1.70 ( ! [X1 : nat] : ((zero_zero_nat = (p @ X1))) ) | ~spl87_3),
% 2.60/1.70 inference(beta_eta_normalization,[status(thm)],[f1937])).
% 2.60/1.70 thf(f1937,plain,(
% 2.60/1.70 ( ! [X1 : nat] : (((p @ X1) = ((^[Y0 : nat]: (zero_zero_nat)) @ X1))) ) | ~spl87_3),
% 2.60/1.70 inference(argument_congruence,[status(thm)],[f1728])).
% 2.60/1.70 thf(f1725,plain,(
% 2.60/1.70 (zero_zero_nat != (p @ sK69)) | spl87_2),
% 2.60/1.70 inference(avatar_component_clause,[status(thm)],[f1723])).
% 2.60/1.70 thf(f1723,definition,(
% 2.60/1.70 spl87_2 <=> (zero_zero_nat = (p @ sK69))),
% 2.60/1.70 introduced(definition,[new_symbols(naming,[spl87_2])],[avatar_definition])).
% 2.60/1.70 thf(f1748,plain,(
% 2.60/1.70 spl87_3 | spl87_7),
% 2.60/1.70 inference(avatar_split_clause,[status(thm)],[f1521,f1746,f1727])).
% 2.60/1.70 thf(f1521,plain,(
% 2.60/1.70 ( ! [X1 : nat] : ((p = (^[Y0 : nat]: (zero_zero_nat))) | ((ord_less_eq_nat @ X1 @ zero_zero_nat) = $true) | (zero_zero_nat = (p @ X1))) )),
% 2.60/1.70 inference(cnf_transformation,[status(thm)],[f1171])).
% 2.60/1.70 thf(f1171,plain,(
% 2.60/1.70 ((p != (^[Y0 : nat]: (zero_zero_nat))) | (zero_zero_nat != (groups1842438620at_nat @ (^[Y0 : nat]: (times_times_nat @ (p @ Y0) @ Y0)) @ (set_ord_atMost_nat @ zero_zero_nat))) | ((zero_zero_nat != (p @ sK69)) & (((ord_less_eq_nat @ sK69 @ zero_zero_nat) != $true) | ((ord_less_eq_nat @ one_one_nat @ sK69) != $true)))) & ((p = (^[Y0 : nat]: (zero_zero_nat))) | ((zero_zero_nat = (groups1842438620at_nat @ (^[Y0 : nat]: (times_times_nat @ (p @ Y0) @ Y0)) @ (set_ord_atMost_nat @ zero_zero_nat))) & ! [X1 : nat] : ((zero_zero_nat = (p @ X1)) | (((ord_less_eq_nat @ X1 @ zero_zero_nat) = $true) & ((ord_less_eq_nat @ one_one_nat @ X1) = $true)))))),
% 2.60/1.70 inference(skolemisation,[status(esa),new_symbols(skolem,[sK69])],[f1169,f1170])).
% 2.60/1.70 thf(f1170,plain,(
% 2.60/1.70 ? [X0 : nat] : ((zero_zero_nat != (p @ X0)) & (((ord_less_eq_nat @ X0 @ zero_zero_nat) != $true) | ((ord_less_eq_nat @ one_one_nat @ X0) != $true))) => ((zero_zero_nat != (p @ sK69)) & (((ord_less_eq_nat @ sK69 @ zero_zero_nat) != $true) | ((ord_less_eq_nat @ one_one_nat @ sK69) != $true)))),
% 2.60/1.70 introduced(definition,[],[choice_axiom])).
% 2.60/1.70 thf(f1169,plain,(
% 2.60/1.70 ((p != (^[Y0 : nat]: (zero_zero_nat))) | (zero_zero_nat != (groups1842438620at_nat @ (^[Y0 : nat]: (times_times_nat @ (p @ Y0) @ Y0)) @ (set_ord_atMost_nat @ zero_zero_nat))) | ? [X0 : nat] : ((zero_zero_nat != (p @ X0)) & (((ord_less_eq_nat @ X0 @ zero_zero_nat) != $true) | ((ord_less_eq_nat @ one_one_nat @ X0) != $true)))) & ((p = (^[Y0 : nat]: (zero_zero_nat))) | ((zero_zero_nat = (groups1842438620at_nat @ (^[Y0 : nat]: (times_times_nat @ (p @ Y0) @ Y0)) @ (set_ord_atMost_nat @ zero_zero_nat))) & ! [X1 : nat] : ((zero_zero_nat = (p @ X1)) | (((ord_less_eq_nat @ X1 @ zero_zero_nat) = $true) & ((ord_less_eq_nat @ one_one_nat @ X1) = $true)))))),
% 2.60/1.70 inference(rectify,[status(thm)],[f1168])).
% 2.60/1.70 thf(f1168,plain,(
% 2.60/1.70 ((p != (^[Y0 : nat]: (zero_zero_nat))) | (zero_zero_nat != (groups1842438620at_nat @ (^[Y0 : nat]: (times_times_nat @ (p @ Y0) @ Y0)) @ (set_ord_atMost_nat @ zero_zero_nat))) | ? [X0 : nat] : ((zero_zero_nat != (p @ X0)) & (((ord_less_eq_nat @ X0 @ zero_zero_nat) != $true) | ((ord_less_eq_nat @ one_one_nat @ X0) != $true)))) & ((p = (^[Y0 : nat]: (zero_zero_nat))) | ((zero_zero_nat = (groups1842438620at_nat @ (^[Y0 : nat]: (times_times_nat @ (p @ Y0) @ Y0)) @ (set_ord_atMost_nat @ zero_zero_nat))) & ! [X0 : nat] : ((zero_zero_nat = (p @ X0)) | (((ord_less_eq_nat @ X0 @ zero_zero_nat) = $true) & ((ord_less_eq_nat @ one_one_nat @ X0) = $true)))))),
% 2.60/1.70 inference(flattening,[status(thm)],[f1167])).
% 2.60/1.70 thf(f1167,plain,(
% 2.60/1.70 ((p != (^[Y0 : nat]: (zero_zero_nat))) | ((zero_zero_nat != (groups1842438620at_nat @ (^[Y0 : nat]: (times_times_nat @ (p @ Y0) @ Y0)) @ (set_ord_atMost_nat @ zero_zero_nat))) | ? [X0 : nat] : ((zero_zero_nat != (p @ X0)) & (((ord_less_eq_nat @ X0 @ zero_zero_nat) != $true) | ((ord_less_eq_nat @ one_one_nat @ X0) != $true))))) & ((p = (^[Y0 : nat]: (zero_zero_nat))) | ((zero_zero_nat = (groups1842438620at_nat @ (^[Y0 : nat]: (times_times_nat @ (p @ Y0) @ Y0)) @ (set_ord_atMost_nat @ zero_zero_nat))) & ! [X0 : nat] : ((zero_zero_nat = (p @ X0)) | (((ord_less_eq_nat @ X0 @ zero_zero_nat) = $true) & ((ord_less_eq_nat @ one_one_nat @ X0) = $true)))))),
% 2.60/1.70 inference(nnf_transformation,[status(thm)],[f989])).
% 2.60/1.70 thf(f989,plain,(
% 2.60/1.70 ((zero_zero_nat = (groups1842438620at_nat @ (^[Y0 : nat]: (times_times_nat @ (p @ Y0) @ Y0)) @ (set_ord_atMost_nat @ zero_zero_nat))) & ! [X0 : nat] : ((zero_zero_nat = (p @ X0)) | (((ord_less_eq_nat @ X0 @ zero_zero_nat) = $true) & ((ord_less_eq_nat @ one_one_nat @ X0) = $true)))) <~> (p = (^[Y0 : nat]: (zero_zero_nat)))),
% 2.60/1.70 inference(ennf_transformation,[status(thm)],[f767])).
% 2.60/1.70 thf(f767,plain,(
% 2.60/1.70 ~((p = (^[Y0 : nat]: (zero_zero_nat))) <=> (! [X0 : nat] : ((zero_zero_nat != (p @ X0)) => (((ord_less_eq_nat @ X0 @ zero_zero_nat) = $true) & ((ord_less_eq_nat @ one_one_nat @ X0) = $true))) & (zero_zero_nat = (groups1842438620at_nat @ (^[Y0 : nat]: (times_times_nat @ (p @ Y0) @ Y0)) @ (set_ord_atMost_nat @ zero_zero_nat)))))),
% 2.60/1.70 inference(rectify,[status(thm)],[f493])).
% 2.60/1.70 thf(f493,plain,(
% 2.60/1.70 ~(((zero_zero_nat = (groups1842438620at_nat @ (^[Y0 : nat]: (times_times_nat @ (p @ Y0) @ Y0)) @ (set_ord_atMost_nat @ zero_zero_nat))) & ! [X1 : nat] : ((zero_zero_nat != (p @ X1)) => (((ord_less_eq_nat @ X1 @ zero_zero_nat) = $true) & ((ord_less_eq_nat @ one_one_nat @ X1) = $true)))) <=> (p = (^[Y0 : nat]: (zero_zero_nat))))),
% 2.60/1.70 inference(fool_elimination,[status(thm)],[f492])).
% 2.60/1.70 thf(f492,plain,(
% 2.60/1.70 ~((zero_zero_nat = (groups1842438620at_nat @ (^[X0 : nat] : (times_times_nat @ (p @ X0) @ X0)) @ (set_ord_atMost_nat @ zero_zero_nat))) & ! [X1 : nat] : ((zero_zero_nat != (p @ X1)) => ((ord_less_eq_nat @ X1 @ zero_zero_nat) & (ord_less_eq_nat @ one_one_nat @ X1))) = ((^[X2 : nat] : (zero_zero_nat)) = p))),
% 2.60/1.70 inference(rectify,[status(thm)],[f263])).
% 2.60/1.70 thf(f263,negated_conjecture,(
% 2.60/1.70 ~((zero_zero_nat = (groups1842438620at_nat @ (^[X5 : nat] : (times_times_nat @ (p @ X5) @ X5)) @ (set_ord_atMost_nat @ zero_zero_nat))) & ! [X5 : nat] : ((zero_zero_nat != (p @ X5)) => ((ord_less_eq_nat @ X5 @ zero_zero_nat) & (ord_less_eq_nat @ one_one_nat @ X5))) = ((^[X5 : nat] : (zero_zero_nat)) = p))),
% 2.60/1.70 inference(negated_conjecture,[status(cth)],[f262])).
% 2.60/1.70 thf(f262,conjecture,(
% 2.60/1.70 ((zero_zero_nat = (groups1842438620at_nat @ (^[X5 : nat] : (times_times_nat @ (p @ X5) @ X5)) @ (set_ord_atMost_nat @ zero_zero_nat))) & ! [X5 : nat] : ((zero_zero_nat != (p @ X5)) => ((ord_less_eq_nat @ X5 @ zero_zero_nat) & (ord_less_eq_nat @ one_one_nat @ X5))) = ((^[X5 : nat] : (zero_zero_nat)) = p))),
% 2.60/1.70 file('/export/starexec/sandbox/tmp/tmp.53O3If2WxG/DTF2THF_29954.p',conj_0)).
% 2.60/1.70 thf(f1744,plain,(
% 2.60/1.70 spl87_6 | spl87_3),
% 2.60/1.70 inference(avatar_split_clause,[status(thm)],[f1520,f1727,f1742])).
% 2.60/1.70 thf(f1520,plain,(
% 2.60/1.70 ( ! [X1 : nat] : ((zero_zero_nat = (p @ X1)) | (p = (^[Y0 : nat]: (zero_zero_nat))) | ((ord_less_eq_nat @ one_one_nat @ X1) = $true)) )),
% 2.60/1.70 inference(cnf_transformation,[status(thm)],[f1171])).
% 2.60/1.70 thf(f1730,plain,(
% 2.60/1.70 ~spl87_1 | ~spl87_2 | ~spl87_3),
% 2.60/1.70 inference(avatar_split_clause,[status(thm)],[f1524,f1727,f1723,f1719])).
% 2.60/1.70 thf(f1524,plain,(
% 2.60/1.70 (zero_zero_nat != (groups1842438620at_nat @ (^[Y0 : nat]: (times_times_nat @ (p @ Y0) @ Y0)) @ (set_ord_atMost_nat @ zero_zero_nat))) | (zero_zero_nat != (p @ sK69)) | (p != (^[Y0 : nat]: (zero_zero_nat)))),
% 2.60/1.70 inference(cnf_transformation,[status(thm)],[f1171])).
% 2.60/1.70 % SZS output end Proof for DTF2THF_29954
% 2.60/1.70 % (30169)------------------------------
% 2.60/1.70 % (30169)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.60/1.70 % (30169)Termination reason: Refutation
% 2.60/1.70
% 2.60/1.70 % (30169)Memory used [KB]: 7803
% 2.60/1.70 % (30169)Time elapsed: 0.173 s
% 2.60/1.70 % (30169)Instructions burned: 441 (million)
% 2.60/1.70 % (30169)------------------------------
% 2.60/1.70 % (30169)------------------------------
% 2.60/1.70 % (30146)Success in time 0.249 s
%------------------------------------------------------------------------------