%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP099^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n015.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 08:55:32 AM UTC 2026
% Result : Theorem 2.64s 1.84s
% Output : Refutation 2.64s
% Verified :
% SZS Type : Refutation
% Derivation depth : 10
% Number of leaves : 9
% Syntax : Number of formulae : 52 ( 3 unt; 0 typ; 4 def)
% Number of atoms : 212 ( 76 equ; 0 cnn)
% Maximal formula atoms : 4 ( 4 avg)
% Number of connectives : 526 ( 46 ~; 35 |; 8 &; 412 @)
% ( 6 <=>; 19 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 4 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 10 ( 10 >; 0 *; 0 +; 0 <<)
% Number of symbols : 125 ( 122 usr; 10 con; 0-4 aty)
% Number of variables : 52 ( 0 ^; 52 !; 0 ?; 52 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
nat: $tType ).
thf(type_def_7,type,
a: $tType ).
thf(type_def_8,type,
list_nat: $tType ).
thf(type_def_9,type,
list_a: $tType ).
thf(func_def_0,type,
list_nat: $tType ).
thf(func_def_1,type,
list_a: $tType ).
thf(func_def_2,type,
nat: $tType ).
thf(func_def_3,type,
a: $tType ).
thf(func_def_4,type,
minus_minus_nat: nat > nat > nat ).
thf(func_def_5,type,
plus_plus_nat: nat > nat > nat ).
thf(func_def_6,type,
if_nat: $o > nat > nat > nat ).
thf(func_def_7,type,
if_a: $o > a > a > a ).
thf(func_def_8,type,
list_asc_nat: list_nat > $o ).
thf(func_def_9,type,
list_desc_nat: list_nat > $o ).
thf(func_def_10,type,
list_strict_asc_nat: list_nat > $o ).
thf(func_def_11,type,
list_strict_desc_nat: list_nat > $o ).
thf(func_def_12,type,
listIn923761578nd_nat: list_nat > ( nat > nat ) > nat > nat ).
thf(func_def_13,type,
listIn1312259492pend_a: list_a > ( nat > a ) > nat > a ).
thf(func_def_14,type,
listIn870223551op_nat: nat > ( nat > nat ) > nat > nat ).
thf(func_def_15,type,
listIn1417627087drop_a: nat > ( nat > a ) > nat > a ).
thf(func_def_16,type,
listIn1299132951ke_nat: nat > ( nat > nat ) > list_nat ).
thf(func_def_17,type,
listIn84854903take_a: nat > ( nat > a ) > list_a ).
thf(func_def_18,type,
append_nat: list_nat > list_nat > list_nat ).
thf(func_def_19,type,
append_a: list_a > list_a > list_a ).
thf(func_def_20,type,
drop_nat: nat > list_nat > list_nat ).
thf(func_def_21,type,
drop_a: nat > list_a > list_a ).
thf(func_def_22,type,
last_nat: list_nat > nat ).
thf(func_def_23,type,
last_a: list_a > a ).
thf(func_def_24,type,
nth_nat: list_nat > nat > nat ).
thf(func_def_25,type,
nth_a: list_a > nat > a ).
thf(func_def_26,type,
size_size_list_nat: list_nat > nat ).
thf(func_def_27,type,
size_size_list_a: list_a > nat ).
thf(func_def_28,type,
ord_less_nat: nat > nat > $o ).
thf(func_def_29,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_30,type,
f: nat > a ).
thf(func_def_31,type,
n: nat ).
thf(func_def_32,type,
xs: list_a ).
thf(func_def_47,type,
sP0: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).
thf(func_def_48,type,
sP1: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).
thf(func_def_49,type,
sP2: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).
thf(func_def_50,type,
sP3: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).
thf(func_def_51,type,
sP4: nat > nat > nat > $o ).
thf(func_def_52,type,
sP5: nat > nat > nat > $o ).
thf(func_def_53,type,
sP6: nat > nat > nat > $o ).
thf(func_def_54,type,
sK7: ( nat > nat ) > nat ).
thf(func_def_55,type,
sK8: ( nat > nat ) > nat ).
thf(func_def_56,type,
sK9: ( nat > $o ) > nat ).
thf(func_def_57,type,
sK10: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_58,type,
sK11: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_59,type,
sK12: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_60,type,
sK13: nat > nat ).
thf(func_def_61,type,
sK14: ( nat > nat ) > nat ).
thf(func_def_62,type,
sK15: ( nat > nat ) > nat ).
thf(func_def_63,type,
sK16: nat > nat ).
thf(func_def_64,type,
sK17: ( list_a > $o ) > list_a ).
thf(func_def_65,type,
sK18: nat > ( nat > $o ) > nat ).
thf(func_def_66,type,
sK19: nat > ( nat > $o ) > nat ).
thf(func_def_67,type,
sK20: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_68,type,
sK21: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_69,type,
sK22: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_70,type,
sK23: ( nat > nat ) > nat ).
thf(func_def_71,type,
sK24: ( nat > nat ) > nat ).
thf(func_def_72,type,
sK25: ( nat > nat ) > nat ).
thf(func_def_73,type,
sK26: ( nat > nat ) > nat ).
thf(func_def_74,type,
sK27: nat > nat ).
thf(func_def_75,type,
sK28: nat > ( nat > a > $o ) > a ).
thf(func_def_76,type,
sK29: list_a > nat > ( nat > a > $o ) > nat ).
thf(func_def_77,type,
sK30: nat > ( nat > a > $o ) > list_a ).
thf(func_def_78,type,
sK31: nat > ( nat > a > $o ) > nat ).
thf(func_def_79,type,
sK32: nat > nat ).
thf(func_def_80,type,
sK33: nat > nat ).
thf(func_def_81,type,
sK34: ( nat > nat > $o ) > nat ).
thf(func_def_82,type,
sK35: ( nat > nat > $o ) > nat ).
thf(func_def_83,type,
sK36: ( nat > nat > $o ) > nat ).
thf(func_def_84,type,
sK37: ( nat > nat > $o ) > nat ).
thf(func_def_85,type,
sK38: ( nat > nat ) > nat ).
thf(func_def_86,type,
sK39: ( nat > nat ) > nat ).
thf(func_def_87,type,
sK40: nat > nat ).
thf(func_def_88,type,
sK41: nat > nat ).
thf(func_def_89,type,
sK42: ( nat > nat ) > nat ).
thf(func_def_90,type,
sK43: ( nat > nat ) > nat ).
thf(func_def_91,type,
sK44: ( nat > nat ) > nat ).
thf(func_def_92,type,
sK45: ( nat > nat ) > nat ).
thf(func_def_93,type,
sK46: nat > list_a ).
thf(func_def_94,type,
sK47: nat > nat ).
thf(func_def_95,type,
sK48: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_96,type,
sK49: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_97,type,
sK50: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_98,type,
sK51: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_99,type,
sK52: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_100,type,
sK53: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_101,type,
sK54: ( nat > $o ) > nat ).
thf(func_def_102,type,
sK55: ( nat > a ) > list_a > ( nat > a ) > list_a > list_a ).
thf(func_def_103,type,
sK56: nat > nat ).
thf(func_def_104,type,
sK57: ( nat > nat > $o ) > nat ).
thf(func_def_105,type,
sK58: ( nat > nat > $o ) > nat ).
thf(func_def_106,type,
sK59: ( nat > nat > $o ) > nat ).
thf(func_def_107,type,
sK60: ( nat > nat > $o ) > nat ).
thf(func_def_108,type,
sK61: ( nat > nat > $o ) > nat ).
thf(func_def_109,type,
sK62: nat > nat ).
thf(func_def_110,type,
sK63: nat > nat ).
thf(func_def_111,type,
sK64: ( nat > nat ) > nat ).
thf(func_def_112,type,
sK65: ( nat > nat ) > nat ).
thf(func_def_113,type,
sK66: list_a > list_a > ( nat > a ) > ( nat > a ) > list_a ).
thf(func_def_114,type,
sK67: ( nat > nat ) > nat ).
thf(func_def_115,type,
sK68: ( nat > nat ) > nat ).
thf(func_def_116,type,
sK69: ( nat > $o ) > nat > nat > nat ).
thf(func_def_117,type,
sK70: nat > ( nat > $o ) > nat > nat ).
thf(func_def_118,type,
sK71: ( nat > nat ) > nat ).
thf(func_def_119,type,
sK72: ( nat > nat ) > nat ).
thf(func_def_120,type,
sK73: ( nat > nat ) > nat ).
thf(func_def_121,type,
sK74: ( nat > nat ) > nat ).
thf(func_def_122,type,
sK75: list_a > list_a > nat ).
thf(func_def_123,type,
sK76: nat > nat ).
thf(func_def_124,type,
sK77: list_a > list_a > list_a > list_a > list_a ).
thf(func_def_125,type,
sK78: ( nat > $o ) > nat ).
thf(func_def_126,type,
sK79: ( nat > $o ) > nat > nat ).
thf(func_def_127,type,
sK80: ( nat > $o ) > nat ).
thf(func_def_128,type,
sK81: nat > nat ).
thf(func_def_129,type,
sK82: ( nat > nat ) > nat ).
thf(func_def_130,type,
sK83: ( nat > nat ) > nat ).
thf(func_def_131,type,
sK84: ( nat > nat ) > nat ).
thf(func_def_132,type,
sK85: ( nat > nat ) > nat ).
thf(func_def_134,type,
ph87:
!>[X0: $tType] : X0 ).
thf(func_def_135,type,
sK88: nat ).
thf(func_def_136,type,
sK89: nat ).
thf(f1711,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f1562,f1567,f1626,f1629,f1708]) ).
thf(f1708,plain,
( spl86_1
| spl86_4 ),
inference(avatar_contradiction_clause,[status(thm)],[f1707]) ).
thf(f1707,plain,
( $false
| spl86_1
| spl86_4 ),
inference(subsumption_resolution,[status(thm)],[f1659,f1576]) ).
thf(f1576,plain,
( ( $true
!= ( ord_less_eq_nat @ ( size_size_list_a @ xs ) @ n ) )
| spl86_4 ),
inference(avatar_component_clause,[status(thm)],[f1574]) ).
thf(f1574,definition,
( spl86_4
<=> ( $true
= ( ord_less_eq_nat @ ( size_size_list_a @ xs ) @ n ) ) ),
introduced(definition,[new_symbols(naming,[spl86_4])],[avatar_definition]) ).
thf(f1659,plain,
( ( $true
= ( ord_less_eq_nat @ ( size_size_list_a @ xs ) @ n ) )
| spl86_1 ),
inference(trivial_inequality_removal,[status(thm)],[f1634]) ).
thf(f1634,plain,
( ( $true
= ( ord_less_eq_nat @ ( size_size_list_a @ xs ) @ n ) )
| ( $true != $true )
| spl86_1 ),
inference(superposition,[status(thm)],[f1557,f1171]) ).
thf(f1171,plain,
! [X0: nat,X1: nat] :
( ( $true
= ( ord_less_nat @ X0 @ X1 ) )
| ( ( ord_less_eq_nat @ X1 @ X0 )
= $true ) ),
inference(cnf_transformation,[status(thm)],[f1000]) ).
thf(f1000,plain,
! [X0: nat,X1: nat] :
( ( ( $true
!= ( ord_less_nat @ X0 @ X1 ) )
| ( ( ord_less_eq_nat @ X1 @ X0 )
!= $true ) )
& ( ( ( ord_less_eq_nat @ X1 @ X0 )
= $true )
| ( $true
= ( ord_less_nat @ X0 @ X1 ) ) ) ),
inference(nnf_transformation,[status(thm)],[f737]) ).
thf(f737,plain,
! [X0: nat,X1: nat] :
( ( $true
!= ( ord_less_nat @ X0 @ X1 ) )
<=> ( ( ord_less_eq_nat @ X1 @ X0 )
= $true ) ),
inference(flattening,[status(thm)],[f615]) ).
thf(f615,plain,
! [X0: nat,X1: nat] :
( ( $true
!= ( ord_less_nat @ X0 @ X1 ) )
<=> ( ( ord_less_eq_nat @ X1 @ X0 )
= $true ) ),
inference(fool_elimination,[status(thm)],[f614]) ).
thf(f614,plain,
! [X0: nat,X1: nat] :
( ( ~ ( ord_less_nat @ X0 @ X1 ) )
= ( ord_less_eq_nat @ X1 @ X0 ) ),
inference(rectify,[status(thm)],[f198]) ).
thf(f198,axiom,
! [X13: nat,X14: nat] :
( ( ~ ( ord_less_nat @ X13 @ X14 ) )
= ( ord_less_eq_nat @ X14 @ X13 ) ),
file('/export/starexec/sandbox2/tmp/tmp.YUokPb62dV/DTF2THF_25681.p',fact_197_not__less) ).
thf(f1557,plain,
( ( ( ord_less_nat @ n @ ( size_size_list_a @ xs ) )
!= $true )
| spl86_1 ),
inference(avatar_component_clause,[status(thm)],[f1555]) ).
thf(f1555,definition,
( spl86_1
<=> ( ( ord_less_nat @ n @ ( size_size_list_a @ xs ) )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl86_1])],[avatar_definition]) ).
thf(f1629,plain,
( ~ spl86_4
| spl86_3 ),
inference(avatar_split_clause,[status(thm)],[f1585,f1564,f1574]) ).
thf(f1564,definition,
( spl86_3
<=> ( ( listIn1417627087drop_a @ n @ ( listIn1312259492pend_a @ xs @ f ) )
= ( listIn1417627087drop_a @ ( minus_minus_nat @ n @ ( size_size_list_a @ xs ) ) @ f ) ) ),
introduced(definition,[new_symbols(naming,[spl86_3])],[avatar_definition]) ).
thf(f1585,plain,
( ( $true
!= ( ord_less_eq_nat @ ( size_size_list_a @ xs ) @ n ) )
| spl86_3 ),
inference(trivial_inequality_removal,[status(thm)],[f1583]) ).
thf(f1583,plain,
( ( $true
!= ( ord_less_eq_nat @ ( size_size_list_a @ xs ) @ n ) )
| ( ( listIn1417627087drop_a @ ( minus_minus_nat @ n @ ( size_size_list_a @ xs ) ) @ f )
!= ( listIn1417627087drop_a @ ( minus_minus_nat @ n @ ( size_size_list_a @ xs ) ) @ f ) )
| spl86_3 ),
inference(superposition,[status(thm)],[f1566,f1193]) ).
thf(f1193,plain,
! [X2: list_a,X0: nat,X1: nat > a] :
( ( ( listIn1417627087drop_a @ X0 @ ( listIn1312259492pend_a @ X2 @ X1 ) )
= ( listIn1417627087drop_a @ ( minus_minus_nat @ X0 @ ( size_size_list_a @ X2 ) ) @ X1 ) )
| ( $true
!= ( ord_less_eq_nat @ ( size_size_list_a @ X2 ) @ X0 ) ) ),
inference(cnf_transformation,[status(thm)],[f841]) ).
thf(f841,plain,
! [X0: nat,X1: nat > a,X2: list_a] :
( ( $true
!= ( ord_less_eq_nat @ ( size_size_list_a @ X2 ) @ X0 ) )
| ( ( listIn1417627087drop_a @ X0 @ ( listIn1312259492pend_a @ X2 @ X1 ) )
= ( listIn1417627087drop_a @ ( minus_minus_nat @ X0 @ ( size_size_list_a @ X2 ) ) @ X1 ) ) ),
inference(ennf_transformation,[status(thm)],[f597]) ).
thf(f597,plain,
! [X0: nat,X1: nat > a,X2: list_a] :
( ( $true
= ( ord_less_eq_nat @ ( size_size_list_a @ X2 ) @ X0 ) )
=> ( ( listIn1417627087drop_a @ X0 @ ( listIn1312259492pend_a @ X2 @ X1 ) )
= ( listIn1417627087drop_a @ ( minus_minus_nat @ X0 @ ( size_size_list_a @ X2 ) ) @ X1 ) ) ),
inference(fool_elimination,[status(thm)],[f596]) ).
thf(f596,plain,
! [X0: nat,X1: nat > a,X2: list_a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ X2 ) @ X0 )
=> ( ( listIn1417627087drop_a @ X0 @ ( listIn1312259492pend_a @ X2 @ X1 ) )
= ( listIn1417627087drop_a @ ( minus_minus_nat @ X0 @ ( size_size_list_a @ X2 ) ) @ X1 ) ) ),
inference(rectify,[status(thm)],[f36]) ).
thf(f36,axiom,
! [X4: nat,X1: nat > a,X0: list_a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ X0 ) @ X4 )
=> ( ( listIn1417627087drop_a @ X4 @ ( listIn1312259492pend_a @ X0 @ X1 ) )
= ( listIn1417627087drop_a @ ( minus_minus_nat @ X4 @ ( size_size_list_a @ X0 ) ) @ X1 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.YUokPb62dV/DTF2THF_25681.p',fact_35_i__append__i__drop__eq2) ).
thf(f1566,plain,
( ( ( listIn1417627087drop_a @ n @ ( listIn1312259492pend_a @ xs @ f ) )
!= ( listIn1417627087drop_a @ ( minus_minus_nat @ n @ ( size_size_list_a @ xs ) ) @ f ) )
| spl86_3 ),
inference(avatar_component_clause,[status(thm)],[f1564]) ).
thf(f1626,plain,
( ~ spl86_1
| spl86_2 ),
inference(avatar_split_clause,[status(thm)],[f1604,f1559,f1555]) ).
thf(f1559,definition,
( spl86_2
<=> ( ( listIn1417627087drop_a @ n @ ( listIn1312259492pend_a @ xs @ f ) )
= ( listIn1312259492pend_a @ ( drop_a @ n @ xs ) @ f ) ) ),
introduced(definition,[new_symbols(naming,[spl86_2])],[avatar_definition]) ).
thf(f1604,plain,
( ( ( ord_less_nat @ n @ ( size_size_list_a @ xs ) )
!= $true )
| spl86_2 ),
inference(trivial_inequality_removal,[status(thm)],[f1593]) ).
thf(f1593,plain,
( ( ( ord_less_nat @ n @ ( size_size_list_a @ xs ) )
!= $true )
| ( $true != $true )
| spl86_2 ),
inference(superposition,[status(thm)],[f1572,f1312]) ).
thf(f1312,plain,
! [X0: nat,X1: nat] :
( ( $true
= ( ord_less_eq_nat @ X0 @ X1 ) )
| ( $true
!= ( ord_less_nat @ X0 @ X1 ) ) ),
inference(cnf_transformation,[status(thm)],[f768]) ).
thf(f768,plain,
! [X0: nat,X1: nat] :
( ( $true
= ( ord_less_eq_nat @ X0 @ X1 ) )
| ( $true
!= ( ord_less_nat @ X0 @ X1 ) ) ),
inference(ennf_transformation,[status(thm)],[f637]) ).
thf(f637,plain,
! [X0: nat,X1: nat] :
( ( $true
= ( ord_less_nat @ X0 @ X1 ) )
=> ( $true
= ( ord_less_eq_nat @ X0 @ X1 ) ) ),
inference(fool_elimination,[status(thm)],[f636]) ).
thf(f636,plain,
! [X0: nat,X1: nat] :
( ( ord_less_nat @ X0 @ X1 )
=> ( ord_less_eq_nat @ X0 @ X1 ) ),
inference(rectify,[status(thm)],[f72]) ).
thf(f72,axiom,
! [X5: nat,X4: nat] :
( ( ord_less_nat @ X5 @ X4 )
=> ( ord_less_eq_nat @ X5 @ X4 ) ),
file('/export/starexec/sandbox2/tmp/tmp.YUokPb62dV/DTF2THF_25681.p',fact_71_less__imp__le__nat) ).
thf(f1572,plain,
( ( ( ord_less_eq_nat @ n @ ( size_size_list_a @ xs ) )
!= $true )
| spl86_2 ),
inference(trivial_inequality_removal,[status(thm)],[f1571]) ).
thf(f1571,plain,
( ( ( listIn1312259492pend_a @ ( drop_a @ n @ xs ) @ f )
!= ( listIn1312259492pend_a @ ( drop_a @ n @ xs ) @ f ) )
| ( ( ord_less_eq_nat @ n @ ( size_size_list_a @ xs ) )
!= $true )
| spl86_2 ),
inference(superposition,[status(thm)],[f1561,f1266]) ).
thf(f1266,plain,
! [X2: nat > a,X0: nat,X1: list_a] :
( ( ( listIn1417627087drop_a @ X0 @ ( listIn1312259492pend_a @ X1 @ X2 ) )
= ( listIn1312259492pend_a @ ( drop_a @ X0 @ X1 ) @ X2 ) )
| ( ( ord_less_eq_nat @ X0 @ ( size_size_list_a @ X1 ) )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f802]) ).
thf(f802,plain,
! [X0: nat,X1: list_a,X2: nat > a] :
( ( ( listIn1417627087drop_a @ X0 @ ( listIn1312259492pend_a @ X1 @ X2 ) )
= ( listIn1312259492pend_a @ ( drop_a @ X0 @ X1 ) @ X2 ) )
| ( ( ord_less_eq_nat @ X0 @ ( size_size_list_a @ X1 ) )
!= $true ) ),
inference(ennf_transformation,[status(thm)],[f437]) ).
thf(f437,plain,
! [X0: nat,X1: list_a,X2: nat > a] :
( ( ( ord_less_eq_nat @ X0 @ ( size_size_list_a @ X1 ) )
= $true )
=> ( ( listIn1417627087drop_a @ X0 @ ( listIn1312259492pend_a @ X1 @ X2 ) )
= ( listIn1312259492pend_a @ ( drop_a @ X0 @ X1 ) @ X2 ) ) ),
inference(fool_elimination,[status(thm)],[f436]) ).
thf(f436,plain,
! [X0: nat,X1: list_a,X2: nat > a] :
( ( ord_less_eq_nat @ X0 @ ( size_size_list_a @ X1 ) )
=> ( ( listIn1417627087drop_a @ X0 @ ( listIn1312259492pend_a @ X1 @ X2 ) )
= ( listIn1312259492pend_a @ ( drop_a @ X0 @ X1 ) @ X2 ) ) ),
inference(rectify,[status(thm)],[f34]) ).
thf(f34,axiom,
! [X4: nat,X0: list_a,X1: nat > a] :
( ( ord_less_eq_nat @ X4 @ ( size_size_list_a @ X0 ) )
=> ( ( listIn1417627087drop_a @ X4 @ ( listIn1312259492pend_a @ X0 @ X1 ) )
= ( listIn1312259492pend_a @ ( drop_a @ X4 @ X0 ) @ X1 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.YUokPb62dV/DTF2THF_25681.p',fact_33_i__append__i__drop__eq1) ).
thf(f1561,plain,
( ( ( listIn1417627087drop_a @ n @ ( listIn1312259492pend_a @ xs @ f ) )
!= ( listIn1312259492pend_a @ ( drop_a @ n @ xs ) @ f ) )
| spl86_2 ),
inference(avatar_component_clause,[status(thm)],[f1559]) ).
thf(f1567,plain,
( spl86_1
| ~ spl86_3 ),
inference(avatar_split_clause,[status(thm)],[f1278,f1564,f1555]) ).
thf(f1278,plain,
( ( ( ord_less_nat @ n @ ( size_size_list_a @ xs ) )
= $true )
| ( ( listIn1417627087drop_a @ n @ ( listIn1312259492pend_a @ xs @ f ) )
!= ( listIn1417627087drop_a @ ( minus_minus_nat @ n @ ( size_size_list_a @ xs ) ) @ f ) ) ),
inference(cnf_transformation,[status(thm)],[f958]) ).
thf(f958,plain,
( ( ( ( ord_less_nat @ n @ ( size_size_list_a @ xs ) )
= $true )
& ( ( listIn1417627087drop_a @ n @ ( listIn1312259492pend_a @ xs @ f ) )
!= ( listIn1312259492pend_a @ ( drop_a @ n @ xs ) @ f ) ) )
| ( ( ( ord_less_nat @ n @ ( size_size_list_a @ xs ) )
!= $true )
& ( ( listIn1417627087drop_a @ n @ ( listIn1312259492pend_a @ xs @ f ) )
!= ( listIn1417627087drop_a @ ( minus_minus_nat @ n @ ( size_size_list_a @ xs ) ) @ f ) ) ) ),
inference(ennf_transformation,[status(thm)],[f702]) ).
thf(f702,plain,
~ ( ( ( ( ord_less_nat @ n @ ( size_size_list_a @ xs ) )
= $true )
=> ( ( listIn1417627087drop_a @ n @ ( listIn1312259492pend_a @ xs @ f ) )
= ( listIn1312259492pend_a @ ( drop_a @ n @ xs ) @ f ) ) )
& ( ( ( ord_less_nat @ n @ ( size_size_list_a @ xs ) )
!= $true )
=> ( ( listIn1417627087drop_a @ n @ ( listIn1312259492pend_a @ xs @ f ) )
= ( listIn1417627087drop_a @ ( minus_minus_nat @ n @ ( size_size_list_a @ xs ) ) @ f ) ) ) ),
inference(flattening,[status(thm)],[f369]) ).
thf(f369,plain,
~ ( ( ( ( ord_less_nat @ n @ ( size_size_list_a @ xs ) )
= $true )
=> ( ( listIn1417627087drop_a @ n @ ( listIn1312259492pend_a @ xs @ f ) )
= ( listIn1312259492pend_a @ ( drop_a @ n @ xs ) @ f ) ) )
& ( ( ( ord_less_nat @ n @ ( size_size_list_a @ xs ) )
!= $true )
=> ( ( listIn1417627087drop_a @ n @ ( listIn1312259492pend_a @ xs @ f ) )
= ( listIn1417627087drop_a @ ( minus_minus_nat @ n @ ( size_size_list_a @ xs ) ) @ f ) ) ) ),
inference(fool_elimination,[status(thm)],[f368]) ).
thf(f368,plain,
~ ( ( ( ord_less_nat @ n @ ( size_size_list_a @ xs ) )
=> ( ( listIn1417627087drop_a @ n @ ( listIn1312259492pend_a @ xs @ f ) )
= ( listIn1312259492pend_a @ ( drop_a @ n @ xs ) @ f ) ) )
& ( ~ ( ord_less_nat @ n @ ( size_size_list_a @ xs ) )
=> ( ( listIn1417627087drop_a @ n @ ( listIn1312259492pend_a @ xs @ f ) )
= ( listIn1417627087drop_a @ ( minus_minus_nat @ n @ ( size_size_list_a @ xs ) ) @ f ) ) ) ),
inference(rectify,[status(thm)],[f254]) ).
thf(f254,negated_conjecture,
~ ( ( ( ord_less_nat @ n @ ( size_size_list_a @ xs ) )
=> ( ( listIn1417627087drop_a @ n @ ( listIn1312259492pend_a @ xs @ f ) )
= ( listIn1312259492pend_a @ ( drop_a @ n @ xs ) @ f ) ) )
& ( ~ ( ord_less_nat @ n @ ( size_size_list_a @ xs ) )
=> ( ( listIn1417627087drop_a @ n @ ( listIn1312259492pend_a @ xs @ f ) )
= ( listIn1417627087drop_a @ ( minus_minus_nat @ n @ ( size_size_list_a @ xs ) ) @ f ) ) ) ),
inference(negated_conjecture,[status(cth)],[f253]) ).
thf(f253,conjecture,
( ( ( ord_less_nat @ n @ ( size_size_list_a @ xs ) )
=> ( ( listIn1417627087drop_a @ n @ ( listIn1312259492pend_a @ xs @ f ) )
= ( listIn1312259492pend_a @ ( drop_a @ n @ xs ) @ f ) ) )
& ( ~ ( ord_less_nat @ n @ ( size_size_list_a @ xs ) )
=> ( ( listIn1417627087drop_a @ n @ ( listIn1312259492pend_a @ xs @ f ) )
= ( listIn1417627087drop_a @ ( minus_minus_nat @ n @ ( size_size_list_a @ xs ) ) @ f ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.YUokPb62dV/DTF2THF_25681.p',conj_0) ).
thf(f1562,plain,
( ~ spl86_1
| ~ spl86_2 ),
inference(avatar_split_clause,[status(thm)],[f1277,f1559,f1555]) ).
thf(f1277,plain,
( ( ( ord_less_nat @ n @ ( size_size_list_a @ xs ) )
!= $true )
| ( ( listIn1417627087drop_a @ n @ ( listIn1312259492pend_a @ xs @ f ) )
!= ( listIn1312259492pend_a @ ( drop_a @ n @ xs ) @ f ) ) ),
inference(cnf_transformation,[status(thm)],[f958]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP099^1 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.15/0.34 % Computer : n015.cluster.edu
% 0.15/0.34 % Model : x86_64 x86_64
% 0.15/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.34 % Memory : 8042.1875MB
% 0.15/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.34 % CPULimit : 300
% 0.15/0.34 % WCLimit : 300
% 0.15/0.34 % DateTime : Tue Feb 24 08:26:31 EST 2026
% 0.15/0.34 % CPUTime :
% 0.15/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.19/0.47 ---- Original DTF file ---
% 0.19/0.47 thf(spec,logic,$$dhol).
% 0.19/0.47 %------------------------------------------------------------------------------
% 0.19/0.47 % File : ITP099^1 : TPTP v9.2.1. Released v7.5.0.
% 0.19/0.47 % Domain : Interactive Theorem Proving
% 0.19/0.47 % Problem : Sledgehammer ListInf problem prob_412__5414926_1
% 0.19/0.47 % Version : Especial.
% 0.19/0.47 % English :
% 0.19/0.47
% 0.19/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.19/0.47 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.19/0.47 % Source : [Des21]
% 0.19/0.47 % Names : ListInf/prob_412__5414926_1 [Des21]
% 0.19/0.47
% 0.19/0.47 % Status : Theorem
% 0.19/0.47 % Rating : 0.33 v9.1.0, 0.25 v9.0.0, 0.40 v8.2.0, 0.38 v8.1.0, 0.36 v7.5.0
% 0.19/0.47 % Syntax : Number of formulae : 286 ( 89 unt; 33 typ; 0 def)
% 0.19/0.47 % Number of atoms : 736 ( 272 equ; 0 cnn)
% 0.19/0.47 % Maximal formula atoms : 12 ( 2 avg)
% 0.19/0.47 % Number of connectives : 2212 ( 82 ~; 25 |; 37 &;1727 @)
% 0.19/0.47 % ( 0 <=>; 341 =>; 0 <=; 0 <~>)
% 0.19/0.47 % Maximal formula depth : 18 ( 7 avg)
% 0.19/0.47 % Number of types : 5 ( 4 usr)
% 0.19/0.47 % Number of type conns : 134 ( 134 >; 0 *; 0 +; 0 <<)
% 0.19/0.47 % Number of symbols : 32 ( 29 usr; 4 con; 0-3 aty)
% 0.19/0.47 % Number of variables : 781 ( 70 ^; 671 !; 40 ?; 781 :)
% 0.19/0.47 % SPC : TH0_THM_EQU_NAR
% 0.19/0.47
% 0.19/0.47 % Comments : This file was generated by Sledgehammer 2021-02-23 15:32:11.229
% 0.19/0.47 %------------------------------------------------------------------------------
% 0.19/0.47 % Could-be-implicit typings (4)
% 0.19/0.47 thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
% 0.19/0.47 list_nat: $tType ).
% 0.19/0.47
% 0.19/0.47 thf(ty_n_t__List__Olist_Itf__a_J,type,
% 0.19/0.47 list_a: $tType ).
% 0.19/0.47
% 0.19/0.47 thf(ty_n_t__Nat__Onat,type,
% 0.19/0.47 nat: $tType ).
% 0.19/0.47
% 0.19/0.47 thf(ty_n_tf__a,type,
% 0.19/0.47 a: $tType ).
% 0.19/0.47
% 0.19/0.47 % Explicit typings (29)
% 0.19/0.47 thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
% 0.19/0.47 minus_minus_nat: nat > nat > nat ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
% 0.19/0.47 plus_plus_nat: nat > nat > nat ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_If_001t__Nat__Onat,type,
% 0.19/0.47 if_nat: $o > nat > nat > nat ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_If_001tf__a,type,
% 0.19/0.47 if_a: $o > a > a > a ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_List2_Olist__asc_001t__Nat__Onat,type,
% 0.19/0.47 list_asc_nat: list_nat > $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_List2_Olist__desc_001t__Nat__Onat,type,
% 0.19/0.47 list_desc_nat: list_nat > $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_List2_Olist__strict__asc_001t__Nat__Onat,type,
% 0.19/0.47 list_strict_asc_nat: list_nat > $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_List2_Olist__strict__desc_001t__Nat__Onat,type,
% 0.19/0.47 list_strict_desc_nat: list_nat > $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_ListInf__Mirabelle__akbajwqfbr_Oi__append_001t__Nat__Onat,type,
% 0.19/0.47 listIn923761578nd_nat: list_nat > ( nat > nat ) > nat > nat ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_ListInf__Mirabelle__akbajwqfbr_Oi__append_001tf__a,type,
% 0.19/0.47 listIn1312259492pend_a: list_a > ( nat > a ) > nat > a ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_ListInf__Mirabelle__akbajwqfbr_Oi__drop_001t__Nat__Onat,type,
% 0.19/0.47 listIn870223551op_nat: nat > ( nat > nat ) > nat > nat ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_ListInf__Mirabelle__akbajwqfbr_Oi__drop_001tf__a,type,
% 0.19/0.47 listIn1417627087drop_a: nat > ( nat > a ) > nat > a ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_ListInf__Mirabelle__akbajwqfbr_Oi__take_001t__Nat__Onat,type,
% 0.19/0.47 listIn1299132951ke_nat: nat > ( nat > nat ) > list_nat ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_ListInf__Mirabelle__akbajwqfbr_Oi__take_001tf__a,type,
% 0.19/0.47 listIn84854903take_a: nat > ( nat > a ) > list_a ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_List_Oappend_001t__Nat__Onat,type,
% 0.19/0.47 append_nat: list_nat > list_nat > list_nat ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_List_Oappend_001tf__a,type,
% 0.19/0.47 append_a: list_a > list_a > list_a ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_List_Odrop_001t__Nat__Onat,type,
% 0.19/0.47 drop_nat: nat > list_nat > list_nat ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_List_Odrop_001tf__a,type,
% 0.19/0.47 drop_a: nat > list_a > list_a ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_List_Olast_001t__Nat__Onat,type,
% 0.19/0.47 last_nat: list_nat > nat ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_List_Olast_001tf__a,type,
% 0.19/0.47 last_a: list_a > a ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_List_Onth_001t__Nat__Onat,type,
% 0.19/0.47 nth_nat: list_nat > nat > nat ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_List_Onth_001tf__a,type,
% 0.19/0.47 nth_a: list_a > nat > a ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
% 0.19/0.47 size_size_list_nat: list_nat > nat ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
% 0.19/0.47 size_size_list_a: list_a > nat ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 0.19/0.47 ord_less_nat: nat > nat > $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 0.19/0.47 ord_less_eq_nat: nat > nat > $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_v_f,type,
% 0.19/0.47 f: nat > a ).
% 0.19/0.47
% 0.19/0.47 thf(sy_v_n,type,
% 0.19/0.47 n: nat ).
% 0.19/0.47
% 0.19/0.47 thf(sy_v_xs,type,
% 0.19/0.47 xs: list_a ).
% 0.19/0.47
% 0.19/0.47 % Relevant facts (247)
% 0.19/0.47 thf(fact_0_same__i__append__eq,axiom,
% 0.19/0.47 ! [Xs: list_a,F: nat > a,G: nat > a] :
% 0.19/0.47 ( ( ( listIn1312259492pend_a @ Xs @ F )
% 0.19/0.47 = ( listIn1312259492pend_a @ Xs @ G ) )
% 0.19/0.47 = ( F = G ) ) ).
% 0.19/0.47
% 0.19/0.47 % same_i_append_eq
% 0.19/0.47 thf(fact_1_i__append__eq__i__append__conv,axiom,
% 0.19/0.47 ! [Xs: list_nat,Ys: list_nat,F: nat > nat,G: nat > nat] :
% 0.19/0.47 ( ( ( size_size_list_nat @ Xs )
% 0.19/0.47 = ( size_size_list_nat @ Ys ) )
% 0.19/0.47 => ( ( ( listIn923761578nd_nat @ Xs @ F )
% 0.19/0.47 = ( listIn923761578nd_nat @ Ys @ G ) )
% 0.19/0.47 = ( ( Xs = Ys )
% 0.19/0.47 & ( F = G ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % i_append_eq_i_append_conv
% 0.19/0.47 thf(fact_2_i__append__eq__i__append__conv,axiom,
% 0.19/0.47 ! [Xs: list_a,Ys: list_a,F: nat > a,G: nat > a] :
% 0.19/0.47 ( ( ( size_size_list_a @ Xs )
% 0.19/0.47 = ( size_size_list_a @ Ys ) )
% 0.19/0.47 => ( ( ( listIn1312259492pend_a @ Xs @ F )
% 0.19/0.47 = ( listIn1312259492pend_a @ Ys @ G ) )
% 0.19/0.47 = ( ( Xs = Ys )
% 0.19/0.47 & ( F = G ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % i_append_eq_i_append_conv
% 0.19/0.47 thf(fact_3_length__drop,axiom,
% 0.19/0.47 ! [N: nat,Xs: list_nat] :
% 0.19/0.47 ( ( size_size_list_nat @ ( drop_nat @ N @ Xs ) )
% 0.19/0.47 = ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).
% 0.19/0.47
% 0.19/0.47 % length_drop
% 0.19/0.47 thf(fact_4_length__drop,axiom,
% 0.19/0.47 ! [N: nat,Xs: list_a] :
% 0.19/0.47 ( ( size_size_list_a @ ( drop_a @ N @ Xs ) )
% 0.19/0.47 = ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ N ) ) ).
% 0.19/0.47
% 0.19/0.47 % length_drop
% 0.19/0.47 thf(fact_5_diff__less__mono2,axiom,
% 0.19/0.47 ! [M: nat,N: nat,L: nat] :
% 0.19/0.47 ( ( ord_less_nat @ M @ N )
% 0.19/0.47 => ( ( ord_less_nat @ M @ L )
% 0.19/0.47 => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % diff_less_mono2
% 0.19/0.47 thf(fact_6_less__imp__diff__less,axiom,
% 0.19/0.47 ! [J: nat,K: nat,N: nat] :
% 0.19/0.47 ( ( ord_less_nat @ J @ K )
% 0.19/0.47 => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_imp_diff_less
% 0.19/0.47 thf(fact_7_less__diff__imp__less,axiom,
% 0.19/0.47 ! [I: nat,J: nat,M: nat] :
% 0.19/0.47 ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ M ) )
% 0.19/0.47 => ( ord_less_nat @ I @ J ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_diff_imp_less
% 0.19/0.47 thf(fact_8_nat__diff__left__cancel__eq1,axiom,
% 0.19/0.47 ! [K: nat,M: nat,N: nat] :
% 0.19/0.47 ( ( ( minus_minus_nat @ K @ M )
% 0.19/0.47 = ( minus_minus_nat @ K @ N ) )
% 0.19/0.47 => ( ( ord_less_nat @ M @ K )
% 0.19/0.47 => ( M = N ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % nat_diff_left_cancel_eq1
% 0.19/0.47 thf(fact_9_nat__diff__left__cancel__eq2,axiom,
% 0.19/0.47 ! [K: nat,M: nat,N: nat] :
% 0.19/0.47 ( ( ( minus_minus_nat @ K @ M )
% 0.19/0.47 = ( minus_minus_nat @ K @ N ) )
% 0.19/0.47 => ( ( ord_less_nat @ N @ K )
% 0.19/0.47 => ( M = N ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % nat_diff_left_cancel_eq2
% 0.19/0.47 thf(fact_10_nat__diff__left__cancel__less,axiom,
% 0.19/0.47 ! [K: nat,M: nat,N: nat] :
% 0.19/0.47 ( ( ord_less_nat @ ( minus_minus_nat @ K @ M ) @ ( minus_minus_nat @ K @ N ) )
% 0.19/0.47 => ( ord_less_nat @ N @ M ) ) ).
% 0.19/0.47
% 0.19/0.47 % nat_diff_left_cancel_less
% 0.19/0.47 thf(fact_11_nat__diff__right__cancel__eq1,axiom,
% 0.19/0.47 ! [M: nat,K: nat,N: nat] :
% 0.19/0.47 ( ( ( minus_minus_nat @ M @ K )
% 0.19/0.47 = ( minus_minus_nat @ N @ K ) )
% 0.19/0.47 => ( ( ord_less_nat @ K @ M )
% 0.19/0.47 => ( M = N ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % nat_diff_right_cancel_eq1
% 0.19/0.47 thf(fact_12_nat__diff__right__cancel__eq2,axiom,
% 0.19/0.47 ! [M: nat,K: nat,N: nat] :
% 0.19/0.47 ( ( ( minus_minus_nat @ M @ K )
% 0.19/0.47 = ( minus_minus_nat @ N @ K ) )
% 0.19/0.47 => ( ( ord_less_nat @ K @ N )
% 0.19/0.47 => ( M = N ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % nat_diff_right_cancel_eq2
% 0.19/0.47 thf(fact_13_nat__diff__right__cancel__less,axiom,
% 0.19/0.47 ! [N: nat,K: nat,M: nat] :
% 0.19/0.47 ( ( ord_less_nat @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M @ K ) )
% 0.19/0.47 => ( ord_less_nat @ N @ M ) ) ).
% 0.19/0.47
% 0.19/0.47 % nat_diff_right_cancel_less
% 0.19/0.47 thf(fact_14_length__induct,axiom,
% 0.19/0.47 ! [P: list_nat > $o,Xs: list_nat] :
% 0.19/0.47 ( ! [Xs2: list_nat] :
% 0.19/0.47 ( ! [Ys2: list_nat] :
% 0.19/0.47 ( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs2 ) )
% 0.19/0.47 => ( P @ Ys2 ) )
% 0.19/0.47 => ( P @ Xs2 ) )
% 0.19/0.47 => ( P @ Xs ) ) ).
% 0.19/0.47
% 0.19/0.47 % length_induct
% 0.19/0.47 thf(fact_15_length__induct,axiom,
% 0.19/0.47 ! [P: list_a > $o,Xs: list_a] :
% 0.19/0.47 ( ! [Xs2: list_a] :
% 0.19/0.47 ( ! [Ys2: list_a] :
% 0.19/0.47 ( ( ord_less_nat @ ( size_size_list_a @ Ys2 ) @ ( size_size_list_a @ Xs2 ) )
% 0.19/0.47 => ( P @ Ys2 ) )
% 0.19/0.47 => ( P @ Xs2 ) )
% 0.19/0.47 => ( P @ Xs ) ) ).
% 0.19/0.47
% 0.19/0.47 % length_induct
% 0.19/0.47 thf(fact_16_linorder__neqE__nat,axiom,
% 0.19/0.47 ! [X: nat,Y: nat] :
% 0.19/0.47 ( ( X != Y )
% 0.19/0.47 => ( ~ ( ord_less_nat @ X @ Y )
% 0.19/0.47 => ( ord_less_nat @ Y @ X ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % linorder_neqE_nat
% 0.19/0.47 thf(fact_17_infinite__descent,axiom,
% 0.19/0.47 ! [P: nat > $o,N: nat] :
% 0.19/0.47 ( ! [N2: nat] :
% 0.19/0.47 ( ~ ( P @ N2 )
% 0.19/0.47 => ? [M2: nat] :
% 0.19/0.47 ( ( ord_less_nat @ M2 @ N2 )
% 0.19/0.47 & ~ ( P @ M2 ) ) )
% 0.19/0.47 => ( P @ N ) ) ).
% 0.19/0.47
% 0.19/0.47 % infinite_descent
% 0.19/0.47 thf(fact_18_nat__less__induct,axiom,
% 0.19/0.47 ! [P: nat > $o,N: nat] :
% 0.19/0.47 ( ! [N2: nat] :
% 0.19/0.47 ( ! [M2: nat] :
% 0.19/0.47 ( ( ord_less_nat @ M2 @ N2 )
% 0.19/0.47 => ( P @ M2 ) )
% 0.19/0.47 => ( P @ N2 ) )
% 0.19/0.47 => ( P @ N ) ) ).
% 0.19/0.47
% 0.19/0.47 % nat_less_induct
% 0.19/0.47 thf(fact_19_less__irrefl__nat,axiom,
% 0.19/0.47 ! [N: nat] :
% 0.19/0.47 ~ ( ord_less_nat @ N @ N ) ).
% 0.19/0.47
% 0.19/0.47 % less_irrefl_nat
% 0.19/0.47 thf(fact_20_less__not__refl3,axiom,
% 0.19/0.47 ! [S: nat,T: nat] :
% 0.19/0.47 ( ( ord_less_nat @ S @ T )
% 0.19/0.47 => ( S != T ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_not_refl3
% 0.19/0.47 thf(fact_21_less__not__refl2,axiom,
% 0.19/0.47 ! [N: nat,M: nat] :
% 0.19/0.47 ( ( ord_less_nat @ N @ M )
% 0.19/0.47 => ( M != N ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_not_refl2
% 0.19/0.47 thf(fact_22_less__not__refl,axiom,
% 0.19/0.47 ! [N: nat] :
% 0.19/0.47 ~ ( ord_less_nat @ N @ N ) ).
% 0.19/0.47
% 0.19/0.47 % less_not_refl
% 0.19/0.47 thf(fact_23_nat__neq__iff,axiom,
% 0.19/0.47 ! [M: nat,N: nat] :
% 0.19/0.47 ( ( M != N )
% 0.19/0.47 = ( ( ord_less_nat @ M @ N )
% 0.19/0.47 | ( ord_less_nat @ N @ M ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % nat_neq_iff
% 0.19/0.47 thf(fact_24_size__neq__size__imp__neq,axiom,
% 0.19/0.47 ! [X: list_nat,Y: list_nat] :
% 0.19/0.47 ( ( ( size_size_list_nat @ X )
% 0.19/0.47 != ( size_size_list_nat @ Y ) )
% 0.19/0.47 => ( X != Y ) ) ).
% 0.19/0.47
% 0.19/0.47 % size_neq_size_imp_neq
% 0.19/0.47 thf(fact_25_size__neq__size__imp__neq,axiom,
% 0.19/0.47 ! [X: list_a,Y: list_a] :
% 0.19/0.47 ( ( ( size_size_list_a @ X )
% 0.19/0.47 != ( size_size_list_a @ Y ) )
% 0.19/0.47 => ( X != Y ) ) ).
% 0.19/0.47
% 0.19/0.47 % size_neq_size_imp_neq
% 0.19/0.47 thf(fact_26_neq__if__length__neq,axiom,
% 0.19/0.47 ! [Xs: list_nat,Ys: list_nat] :
% 0.19/0.47 ( ( ( size_size_list_nat @ Xs )
% 0.19/0.47 != ( size_size_list_nat @ Ys ) )
% 0.19/0.47 => ( Xs != Ys ) ) ).
% 0.19/0.47
% 0.19/0.47 % neq_if_length_neq
% 0.19/0.47 thf(fact_27_neq__if__length__neq,axiom,
% 0.19/0.47 ! [Xs: list_a,Ys: list_a] :
% 0.19/0.47 ( ( ( size_size_list_a @ Xs )
% 0.19/0.47 != ( size_size_list_a @ Ys ) )
% 0.19/0.47 => ( Xs != Ys ) ) ).
% 0.19/0.47
% 0.19/0.47 % neq_if_length_neq
% 0.19/0.47 thf(fact_28_Ex__list__of__length,axiom,
% 0.19/0.47 ! [N: nat] :
% 0.19/0.47 ? [Xs2: list_nat] :
% 0.19/0.47 ( ( size_size_list_nat @ Xs2 )
% 0.19/0.47 = N ) ).
% 0.19/0.47
% 0.19/0.47 % Ex_list_of_length
% 0.19/0.47 thf(fact_29_Ex__list__of__length,axiom,
% 0.19/0.47 ! [N: nat] :
% 0.19/0.47 ? [Xs2: list_a] :
% 0.19/0.47 ( ( size_size_list_a @ Xs2 )
% 0.19/0.47 = N ) ).
% 0.19/0.47
% 0.19/0.47 % Ex_list_of_length
% 0.19/0.47 thf(fact_30_eq__imp__diff__eq,axiom,
% 0.19/0.47 ! [M: nat,N: nat,K: nat] :
% 0.19/0.47 ( ( M = N )
% 0.19/0.47 => ( ( minus_minus_nat @ M @ K )
% 0.19/0.47 = ( minus_minus_nat @ N @ K ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % eq_imp_diff_eq
% 0.19/0.47 thf(fact_31_diff__commute,axiom,
% 0.19/0.47 ! [I: nat,J: nat,K: nat] :
% 0.19/0.47 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 0.19/0.47 = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% 0.19/0.47
% 0.19/0.47 % diff_commute
% 0.19/0.47 thf(fact_32_i__append__i__drop__eq1,axiom,
% 0.19/0.47 ! [N: nat,Xs: list_nat,F: nat > nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
% 0.19/0.47 => ( ( listIn870223551op_nat @ N @ ( listIn923761578nd_nat @ Xs @ F ) )
% 0.19/0.47 = ( listIn923761578nd_nat @ ( drop_nat @ N @ Xs ) @ F ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % i_append_i_drop_eq1
% 0.19/0.47 thf(fact_33_i__append__i__drop__eq1,axiom,
% 0.19/0.47 ! [N: nat,Xs: list_a,F: nat > a] :
% 0.19/0.47 ( ( ord_less_eq_nat @ N @ ( size_size_list_a @ Xs ) )
% 0.19/0.47 => ( ( listIn1417627087drop_a @ N @ ( listIn1312259492pend_a @ Xs @ F ) )
% 0.19/0.47 = ( listIn1312259492pend_a @ ( drop_a @ N @ Xs ) @ F ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % i_append_i_drop_eq1
% 0.19/0.47 thf(fact_34_i__append__i__drop__eq2,axiom,
% 0.19/0.47 ! [Xs: list_nat,N: nat,F: nat > nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
% 0.19/0.47 => ( ( listIn870223551op_nat @ N @ ( listIn923761578nd_nat @ Xs @ F ) )
% 0.19/0.47 = ( listIn870223551op_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ F ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % i_append_i_drop_eq2
% 0.19/0.47 thf(fact_35_i__append__i__drop__eq2,axiom,
% 0.19/0.47 ! [Xs: list_a,N: nat,F: nat > a] :
% 0.19/0.47 ( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N )
% 0.19/0.47 => ( ( listIn1417627087drop_a @ N @ ( listIn1312259492pend_a @ Xs @ F ) )
% 0.19/0.47 = ( listIn1417627087drop_a @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) @ F ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % i_append_i_drop_eq2
% 0.19/0.47 thf(fact_36_last__drop,axiom,
% 0.19/0.47 ! [N: nat,Xs: list_nat] :
% 0.19/0.47 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 0.19/0.47 => ( ( last_nat @ ( drop_nat @ N @ Xs ) )
% 0.19/0.47 = ( last_nat @ Xs ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % last_drop
% 0.19/0.47 thf(fact_37_last__drop,axiom,
% 0.19/0.47 ! [N: nat,Xs: list_a] :
% 0.19/0.47 ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
% 0.19/0.47 => ( ( last_a @ ( drop_a @ N @ Xs ) )
% 0.19/0.47 = ( last_a @ Xs ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % last_drop
% 0.19/0.47 thf(fact_38_i__append__nth2,axiom,
% 0.19/0.47 ! [Xs: list_nat,N: nat,F: nat > nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
% 0.19/0.47 => ( ( listIn923761578nd_nat @ Xs @ F @ N )
% 0.19/0.47 = ( F @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % i_append_nth2
% 0.19/0.47 thf(fact_39_i__append__nth2,axiom,
% 0.19/0.47 ! [Xs: list_a,N: nat,F: nat > a] :
% 0.19/0.47 ( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N )
% 0.19/0.47 => ( ( listIn1312259492pend_a @ Xs @ F @ N )
% 0.19/0.47 = ( F @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % i_append_nth2
% 0.19/0.47 thf(fact_40_i__append__nth,axiom,
% 0.19/0.47 ( listIn923761578nd_nat
% 0.19/0.47 = ( ^ [Xs3: list_nat,F2: nat > nat,N3: nat] : ( if_nat @ ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs3 ) ) @ ( nth_nat @ Xs3 @ N3 ) @ ( F2 @ ( minus_minus_nat @ N3 @ ( size_size_list_nat @ Xs3 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % i_append_nth
% 0.19/0.47 thf(fact_41_i__append__nth,axiom,
% 0.19/0.47 ( listIn1312259492pend_a
% 0.19/0.47 = ( ^ [Xs3: list_a,F2: nat > a,N3: nat] : ( if_a @ ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs3 ) ) @ ( nth_a @ Xs3 @ N3 ) @ ( F2 @ ( minus_minus_nat @ N3 @ ( size_size_list_a @ Xs3 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % i_append_nth
% 0.19/0.47 thf(fact_42_i__append__def,axiom,
% 0.19/0.47 ( listIn923761578nd_nat
% 0.19/0.47 = ( ^ [Xs3: list_nat,F2: nat > nat,N3: nat] : ( if_nat @ ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs3 ) ) @ ( nth_nat @ Xs3 @ N3 ) @ ( F2 @ ( minus_minus_nat @ N3 @ ( size_size_list_nat @ Xs3 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % i_append_def
% 0.19/0.47 thf(fact_43_i__append__def,axiom,
% 0.19/0.47 ( listIn1312259492pend_a
% 0.19/0.47 = ( ^ [Xs3: list_a,F2: nat > a,N3: nat] : ( if_a @ ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs3 ) ) @ ( nth_a @ Xs3 @ N3 ) @ ( F2 @ ( minus_minus_nat @ N3 @ ( size_size_list_a @ Xs3 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % i_append_def
% 0.19/0.47 thf(fact_44_drop__append,axiom,
% 0.19/0.47 ! [N: nat,Xs: list_nat,Ys: list_nat] :
% 0.19/0.47 ( ( drop_nat @ N @ ( append_nat @ Xs @ Ys ) )
% 0.19/0.47 = ( append_nat @ ( drop_nat @ N @ Xs ) @ ( drop_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % drop_append
% 0.19/0.47 thf(fact_45_drop__append,axiom,
% 0.19/0.47 ! [N: nat,Xs: list_a,Ys: list_a] :
% 0.19/0.47 ( ( drop_a @ N @ ( append_a @ Xs @ Ys ) )
% 0.19/0.47 = ( append_a @ ( drop_a @ N @ Xs ) @ ( drop_a @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) @ Ys ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % drop_append
% 0.19/0.47 thf(fact_46_append_Oassoc,axiom,
% 0.19/0.47 ! [A: list_a,B: list_a,C: list_a] :
% 0.19/0.47 ( ( append_a @ ( append_a @ A @ B ) @ C )
% 0.19/0.47 = ( append_a @ A @ ( append_a @ B @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % append.assoc
% 0.19/0.47 thf(fact_47_append__assoc,axiom,
% 0.19/0.47 ! [Xs: list_a,Ys: list_a,Zs: list_a] :
% 0.19/0.47 ( ( append_a @ ( append_a @ Xs @ Ys ) @ Zs )
% 0.19/0.47 = ( append_a @ Xs @ ( append_a @ Ys @ Zs ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % append_assoc
% 0.19/0.47 thf(fact_48_append__same__eq,axiom,
% 0.19/0.47 ! [Ys: list_a,Xs: list_a,Zs: list_a] :
% 0.19/0.47 ( ( ( append_a @ Ys @ Xs )
% 0.19/0.47 = ( append_a @ Zs @ Xs ) )
% 0.19/0.47 = ( Ys = Zs ) ) ).
% 0.19/0.47
% 0.19/0.47 % append_same_eq
% 0.19/0.47 thf(fact_49_same__append__eq,axiom,
% 0.19/0.47 ! [Xs: list_a,Ys: list_a,Zs: list_a] :
% 0.19/0.47 ( ( ( append_a @ Xs @ Ys )
% 0.19/0.47 = ( append_a @ Xs @ Zs ) )
% 0.19/0.47 = ( Ys = Zs ) ) ).
% 0.19/0.47
% 0.19/0.47 % same_append_eq
% 0.19/0.47 thf(fact_50_diff__diff__cancel,axiom,
% 0.19/0.47 ! [I: nat,N: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ I @ N )
% 0.19/0.47 => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
% 0.19/0.47 = I ) ) ).
% 0.19/0.47
% 0.19/0.47 % diff_diff_cancel
% 0.19/0.47 thf(fact_51_append__eq__append__conv,axiom,
% 0.19/0.47 ! [Xs: list_nat,Ys: list_nat,Us: list_nat,Vs: list_nat] :
% 0.19/0.47 ( ( ( ( size_size_list_nat @ Xs )
% 0.19/0.47 = ( size_size_list_nat @ Ys ) )
% 0.19/0.47 | ( ( size_size_list_nat @ Us )
% 0.19/0.47 = ( size_size_list_nat @ Vs ) ) )
% 0.19/0.47 => ( ( ( append_nat @ Xs @ Us )
% 0.19/0.47 = ( append_nat @ Ys @ Vs ) )
% 0.19/0.47 = ( ( Xs = Ys )
% 0.19/0.47 & ( Us = Vs ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % append_eq_append_conv
% 0.19/0.47 thf(fact_52_append__eq__append__conv,axiom,
% 0.19/0.47 ! [Xs: list_a,Ys: list_a,Us: list_a,Vs: list_a] :
% 0.19/0.47 ( ( ( ( size_size_list_a @ Xs )
% 0.19/0.47 = ( size_size_list_a @ Ys ) )
% 0.19/0.47 | ( ( size_size_list_a @ Us )
% 0.19/0.47 = ( size_size_list_a @ Vs ) ) )
% 0.19/0.47 => ( ( ( append_a @ Xs @ Us )
% 0.19/0.47 = ( append_a @ Ys @ Vs ) )
% 0.19/0.47 = ( ( Xs = Ys )
% 0.19/0.47 & ( Us = Vs ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % append_eq_append_conv
% 0.19/0.47 thf(fact_53_i__append__assoc,axiom,
% 0.19/0.47 ! [Xs: list_a,Ys: list_a,F: nat > a] :
% 0.19/0.47 ( ( listIn1312259492pend_a @ Xs @ ( listIn1312259492pend_a @ Ys @ F ) )
% 0.19/0.47 = ( listIn1312259492pend_a @ ( append_a @ Xs @ Ys ) @ F ) ) ).
% 0.19/0.47
% 0.19/0.47 % i_append_assoc
% 0.19/0.47 thf(fact_54_i__append__nth1,axiom,
% 0.19/0.47 ! [N: nat,Xs: list_nat,F: nat > nat] :
% 0.19/0.47 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 0.19/0.47 => ( ( listIn923761578nd_nat @ Xs @ F @ N )
% 0.19/0.47 = ( nth_nat @ Xs @ N ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % i_append_nth1
% 0.19/0.47 thf(fact_55_i__append__nth1,axiom,
% 0.19/0.47 ! [N: nat,Xs: list_a,F: nat > a] :
% 0.19/0.47 ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
% 0.19/0.47 => ( ( listIn1312259492pend_a @ Xs @ F @ N )
% 0.19/0.47 = ( nth_a @ Xs @ N ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % i_append_nth1
% 0.19/0.47 thf(fact_56_le__refl,axiom,
% 0.19/0.47 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 0.19/0.47
% 0.19/0.47 % le_refl
% 0.19/0.47 thf(fact_57_le__trans,axiom,
% 0.19/0.47 ! [I: nat,J: nat,K: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ I @ J )
% 0.19/0.47 => ( ( ord_less_eq_nat @ J @ K )
% 0.19/0.47 => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_trans
% 0.19/0.47 thf(fact_58_eq__imp__le,axiom,
% 0.19/0.47 ! [M: nat,N: nat] :
% 0.19/0.47 ( ( M = N )
% 0.19/0.47 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.19/0.47
% 0.19/0.47 % eq_imp_le
% 0.19/0.47 thf(fact_59_le__antisym,axiom,
% 0.19/0.47 ! [M: nat,N: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ M @ N )
% 0.19/0.47 => ( ( ord_less_eq_nat @ N @ M )
% 0.19/0.47 => ( M = N ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_antisym
% 0.19/0.47 thf(fact_60_nat__le__linear,axiom,
% 0.19/0.47 ! [M: nat,N: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ M @ N )
% 0.19/0.47 | ( ord_less_eq_nat @ N @ M ) ) ).
% 0.19/0.47
% 0.19/0.47 % nat_le_linear
% 0.19/0.47 thf(fact_61_append__eq__appendI,axiom,
% 0.19/0.47 ! [Xs: list_a,Xs1: list_a,Zs: list_a,Ys: list_a,Us: list_a] :
% 0.19/0.47 ( ( ( append_a @ Xs @ Xs1 )
% 0.19/0.47 = Zs )
% 0.19/0.47 => ( ( Ys
% 0.19/0.47 = ( append_a @ Xs1 @ Us ) )
% 0.19/0.47 => ( ( append_a @ Xs @ Ys )
% 0.19/0.47 = ( append_a @ Zs @ Us ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % append_eq_appendI
% 0.19/0.47 thf(fact_62_Nat_Oex__has__greatest__nat,axiom,
% 0.19/0.47 ! [P: nat > $o,K: nat,B: nat] :
% 0.19/0.47 ( ( P @ K )
% 0.19/0.47 => ( ! [Y2: nat] :
% 0.19/0.47 ( ( P @ Y2 )
% 0.19/0.47 => ( ord_less_eq_nat @ Y2 @ B ) )
% 0.19/0.47 => ? [X2: nat] :
% 0.19/0.47 ( ( P @ X2 )
% 0.19/0.47 & ! [Y3: nat] :
% 0.19/0.47 ( ( P @ Y3 )
% 0.19/0.47 => ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Nat.ex_has_greatest_nat
% 0.19/0.47 thf(fact_63_append__eq__append__conv2,axiom,
% 0.19/0.47 ! [Xs: list_a,Ys: list_a,Zs: list_a,Ts: list_a] :
% 0.19/0.47 ( ( ( append_a @ Xs @ Ys )
% 0.19/0.47 = ( append_a @ Zs @ Ts ) )
% 0.19/0.47 = ( ? [Us2: list_a] :
% 0.19/0.47 ( ( ( Xs
% 0.19/0.47 = ( append_a @ Zs @ Us2 ) )
% 0.19/0.47 & ( ( append_a @ Us2 @ Ys )
% 0.19/0.47 = Ts ) )
% 0.19/0.47 | ( ( ( append_a @ Xs @ Us2 )
% 0.19/0.47 = Zs )
% 0.19/0.47 & ( Ys
% 0.19/0.47 = ( append_a @ Us2 @ Ts ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % append_eq_append_conv2
% 0.19/0.47 thf(fact_64_i__append__eq__i__append__conv2__aux,axiom,
% 0.19/0.47 ! [Xs: list_nat,F: nat > nat,Ys: list_nat,G: nat > nat] :
% 0.19/0.47 ( ( ( listIn923761578nd_nat @ Xs @ F )
% 0.19/0.47 = ( listIn923761578nd_nat @ Ys @ G ) )
% 0.19/0.47 => ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) )
% 0.19/0.47 => ? [Zs2: list_nat] :
% 0.19/0.47 ( ( ( append_nat @ Xs @ Zs2 )
% 0.19/0.47 = Ys )
% 0.19/0.47 & ( F
% 0.19/0.47 = ( listIn923761578nd_nat @ Zs2 @ G ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % i_append_eq_i_append_conv2_aux
% 0.19/0.47 thf(fact_65_i__append__eq__i__append__conv2__aux,axiom,
% 0.19/0.47 ! [Xs: list_a,F: nat > a,Ys: list_a,G: nat > a] :
% 0.19/0.47 ( ( ( listIn1312259492pend_a @ Xs @ F )
% 0.19/0.47 = ( listIn1312259492pend_a @ Ys @ G ) )
% 0.19/0.47 => ( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) )
% 0.19/0.47 => ? [Zs2: list_a] :
% 0.19/0.47 ( ( ( append_a @ Xs @ Zs2 )
% 0.19/0.47 = Ys )
% 0.19/0.47 & ( F
% 0.19/0.47 = ( listIn1312259492pend_a @ Zs2 @ G ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % i_append_eq_i_append_conv2_aux
% 0.19/0.47 thf(fact_66_ge__less__neq__conv,axiom,
% 0.19/0.47 ( ord_less_eq_nat
% 0.19/0.47 = ( ^ [A2: nat,N3: nat] :
% 0.19/0.47 ! [X3: nat] :
% 0.19/0.47 ( ( ord_less_nat @ X3 @ A2 )
% 0.19/0.47 => ( N3 != X3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ge_less_neq_conv
% 0.19/0.47 thf(fact_67_less__ge__neq__conv,axiom,
% 0.19/0.47 ( ord_less_nat
% 0.19/0.47 = ( ^ [N3: nat,A2: nat] :
% 0.19/0.47 ! [X3: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ A2 @ X3 )
% 0.19/0.47 => ( N3 != X3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_ge_neq_conv
% 0.19/0.47 thf(fact_68_greater__le__neq__conv,axiom,
% 0.19/0.47 ( ord_less_nat
% 0.19/0.47 = ( ^ [A2: nat,N3: nat] :
% 0.19/0.47 ! [X3: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ X3 @ A2 )
% 0.19/0.47 => ( N3 != X3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % greater_le_neq_conv
% 0.19/0.47 thf(fact_69_le__greater__neq__conv,axiom,
% 0.19/0.47 ( ord_less_eq_nat
% 0.19/0.47 = ( ^ [N3: nat,A2: nat] :
% 0.19/0.47 ! [X3: nat] :
% 0.19/0.47 ( ( ord_less_nat @ A2 @ X3 )
% 0.19/0.47 => ( N3 != X3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_greater_neq_conv
% 0.19/0.47 thf(fact_70_nat__less__le,axiom,
% 0.19/0.47 ( ord_less_nat
% 0.19/0.47 = ( ^ [M3: nat,N3: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ M3 @ N3 )
% 0.19/0.47 & ( M3 != N3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % nat_less_le
% 0.19/0.47 thf(fact_71_less__imp__le__nat,axiom,
% 0.19/0.47 ! [M: nat,N: nat] :
% 0.19/0.47 ( ( ord_less_nat @ M @ N )
% 0.19/0.47 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_imp_le_nat
% 0.19/0.47 thf(fact_72_le__eq__less__or__eq,axiom,
% 0.19/0.47 ( ord_less_eq_nat
% 0.19/0.47 = ( ^ [M3: nat,N3: nat] :
% 0.19/0.47 ( ( ord_less_nat @ M3 @ N3 )
% 0.19/0.47 | ( M3 = N3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_eq_less_or_eq
% 0.19/0.47 thf(fact_73_less__or__eq__imp__le,axiom,
% 0.19/0.47 ! [M: nat,N: nat] :
% 0.19/0.47 ( ( ( ord_less_nat @ M @ N )
% 0.19/0.47 | ( M = N ) )
% 0.19/0.47 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_or_eq_imp_le
% 0.19/0.47 thf(fact_74_le__neq__implies__less,axiom,
% 0.19/0.47 ! [M: nat,N: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ M @ N )
% 0.19/0.47 => ( ( M != N )
% 0.19/0.47 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_neq_implies_less
% 0.19/0.47 thf(fact_75_less__mono__imp__le__mono,axiom,
% 0.19/0.47 ! [F: nat > nat,I: nat,J: nat] :
% 0.19/0.47 ( ! [I2: nat,J2: nat] :
% 0.19/0.47 ( ( ord_less_nat @ I2 @ J2 )
% 0.19/0.47 => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 0.19/0.47 => ( ( ord_less_eq_nat @ I @ J )
% 0.19/0.47 => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_mono_imp_le_mono
% 0.19/0.47 thf(fact_76_eq__diff__iff,axiom,
% 0.19/0.47 ! [K: nat,M: nat,N: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ K @ M )
% 0.19/0.47 => ( ( ord_less_eq_nat @ K @ N )
% 0.19/0.47 => ( ( ( minus_minus_nat @ M @ K )
% 0.19/0.47 = ( minus_minus_nat @ N @ K ) )
% 0.19/0.47 = ( M = N ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % eq_diff_iff
% 0.19/0.47 thf(fact_77_le__diff__iff,axiom,
% 0.19/0.47 ! [K: nat,M: nat,N: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ K @ M )
% 0.19/0.47 => ( ( ord_less_eq_nat @ K @ N )
% 0.19/0.47 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.19/0.47 = ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_diff_iff
% 0.19/0.47 thf(fact_78_Nat_Odiff__diff__eq,axiom,
% 0.19/0.47 ! [K: nat,M: nat,N: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ K @ M )
% 0.19/0.47 => ( ( ord_less_eq_nat @ K @ N )
% 0.19/0.47 => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.19/0.47 = ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Nat.diff_diff_eq
% 0.19/0.47 thf(fact_79_diff__le__mono,axiom,
% 0.19/0.47 ! [M: nat,N: nat,L: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ M @ N )
% 0.19/0.47 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % diff_le_mono
% 0.19/0.47 thf(fact_80_diff__le__self,axiom,
% 0.19/0.47 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% 0.19/0.47
% 0.19/0.47 % diff_le_self
% 0.19/0.47 thf(fact_81_le__diff__iff_H,axiom,
% 0.19/0.47 ! [A: nat,C: nat,B: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ A @ C )
% 0.19/0.47 => ( ( ord_less_eq_nat @ B @ C )
% 0.19/0.47 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
% 0.19/0.47 = ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_diff_iff'
% 0.19/0.47 thf(fact_82_diff__le__mono2,axiom,
% 0.19/0.47 ! [M: nat,N: nat,L: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ M @ N )
% 0.19/0.47 => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % diff_le_mono2
% 0.19/0.47 thf(fact_83_le__diff__swap,axiom,
% 0.19/0.47 ! [I: nat,K: nat,J: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ I @ K )
% 0.19/0.47 => ( ( ord_less_eq_nat @ J @ K )
% 0.19/0.47 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ K @ J ) @ I )
% 0.19/0.47 = ( ord_less_eq_nat @ ( minus_minus_nat @ K @ I ) @ J ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_diff_swap
% 0.19/0.47 thf(fact_84_le__diff__imp__le,axiom,
% 0.19/0.47 ! [I: nat,J: nat,M: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ M ) )
% 0.19/0.47 => ( ord_less_eq_nat @ I @ J ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_diff_imp_le
% 0.19/0.47 thf(fact_85_le__imp__diff__le,axiom,
% 0.19/0.47 ! [J: nat,K: nat,N: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ J @ K )
% 0.19/0.47 => ( ord_less_eq_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_imp_diff_le
% 0.19/0.47 thf(fact_86_eq__diff__left__iff,axiom,
% 0.19/0.47 ! [M: nat,K: nat,N: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ M @ K )
% 0.19/0.47 => ( ( ord_less_eq_nat @ N @ K )
% 0.19/0.47 => ( ( ( minus_minus_nat @ K @ M )
% 0.19/0.47 = ( minus_minus_nat @ K @ N ) )
% 0.19/0.47 = ( M = N ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % eq_diff_left_iff
% 0.19/0.47 thf(fact_87_le__diff__le__imp__le,axiom,
% 0.19/0.47 ! [I: nat,J: nat,M: nat,N: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ M ) )
% 0.19/0.47 => ( ( ord_less_eq_nat @ N @ M )
% 0.19/0.47 => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ N ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_diff_le_imp_le
% 0.19/0.47 thf(fact_88_nat__diff__left__cancel__le2,axiom,
% 0.19/0.47 ! [K: nat,M: nat,N: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ ( minus_minus_nat @ K @ M ) @ ( minus_minus_nat @ K @ N ) )
% 0.19/0.47 => ( ( ord_less_eq_nat @ N @ K )
% 0.19/0.47 => ( ord_less_eq_nat @ N @ M ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % nat_diff_left_cancel_le2
% 0.19/0.47 thf(fact_89_nat__diff__right__cancel__le2,axiom,
% 0.19/0.47 ! [M: nat,K: nat,N: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.19/0.47 => ( ( ord_less_eq_nat @ K @ N )
% 0.19/0.47 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % nat_diff_right_cancel_le2
% 0.19/0.47 thf(fact_90_i__append__eq__i__appendI,axiom,
% 0.19/0.47 ! [Xs: list_a,Xs4: list_a,Ys: list_a,F: nat > a,G: nat > a] :
% 0.19/0.47 ( ( ( append_a @ Xs @ Xs4 )
% 0.19/0.47 = Ys )
% 0.19/0.47 => ( ( F
% 0.19/0.47 = ( listIn1312259492pend_a @ Xs4 @ G ) )
% 0.19/0.47 => ( ( listIn1312259492pend_a @ Xs @ F )
% 0.19/0.47 = ( listIn1312259492pend_a @ Ys @ G ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % i_append_eq_i_appendI
% 0.19/0.47 thf(fact_91_i__append__eq__i__append__conv2,axiom,
% 0.19/0.47 ! [Xs: list_a,F: nat > a,Ys: list_a,G: nat > a] :
% 0.19/0.47 ( ( ( listIn1312259492pend_a @ Xs @ F )
% 0.19/0.47 = ( listIn1312259492pend_a @ Ys @ G ) )
% 0.19/0.47 = ( ? [Zs3: list_a] :
% 0.19/0.47 ( ( ( Xs
% 0.19/0.47 = ( append_a @ Ys @ Zs3 ) )
% 0.19/0.47 & ( ( listIn1312259492pend_a @ Zs3 @ F )
% 0.19/0.47 = G ) )
% 0.19/0.47 | ( ( ( append_a @ Xs @ Zs3 )
% 0.19/0.47 = Ys )
% 0.19/0.47 & ( F
% 0.19/0.47 = ( listIn1312259492pend_a @ Zs3 @ G ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % i_append_eq_i_append_conv2
% 0.19/0.47 thf(fact_92_nth__append,axiom,
% 0.19/0.47 ! [N: nat,Xs: list_nat,Ys: list_nat] :
% 0.19/0.47 ( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 0.19/0.47 => ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
% 0.19/0.47 = ( nth_nat @ Xs @ N ) ) )
% 0.19/0.47 & ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 0.19/0.47 => ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
% 0.19/0.47 = ( nth_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % nth_append
% 0.19/0.47 thf(fact_93_nth__append,axiom,
% 0.19/0.47 ! [N: nat,Xs: list_a,Ys: list_a] :
% 0.19/0.47 ( ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
% 0.19/0.47 => ( ( nth_a @ ( append_a @ Xs @ Ys ) @ N )
% 0.19/0.47 = ( nth_a @ Xs @ N ) ) )
% 0.19/0.47 & ( ~ ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
% 0.19/0.47 => ( ( nth_a @ ( append_a @ Xs @ Ys ) @ N )
% 0.19/0.47 = ( nth_a @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % nth_append
% 0.19/0.47 thf(fact_94_nth__equalityI,axiom,
% 0.19/0.47 ! [Xs: list_nat,Ys: list_nat] :
% 0.19/0.47 ( ( ( size_size_list_nat @ Xs )
% 0.19/0.47 = ( size_size_list_nat @ Ys ) )
% 0.19/0.47 => ( ! [I2: nat] :
% 0.19/0.47 ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
% 0.19/0.47 => ( ( nth_nat @ Xs @ I2 )
% 0.19/0.47 = ( nth_nat @ Ys @ I2 ) ) )
% 0.19/0.47 => ( Xs = Ys ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % nth_equalityI
% 0.19/0.47 thf(fact_95_nth__equalityI,axiom,
% 0.19/0.47 ! [Xs: list_a,Ys: list_a] :
% 0.19/0.47 ( ( ( size_size_list_a @ Xs )
% 0.19/0.47 = ( size_size_list_a @ Ys ) )
% 0.19/0.47 => ( ! [I2: nat] :
% 0.19/0.47 ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
% 0.19/0.47 => ( ( nth_a @ Xs @ I2 )
% 0.19/0.47 = ( nth_a @ Ys @ I2 ) ) )
% 0.19/0.47 => ( Xs = Ys ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % nth_equalityI
% 0.19/0.47 thf(fact_96_Skolem__list__nth,axiom,
% 0.19/0.47 ! [K: nat,P: nat > nat > $o] :
% 0.19/0.47 ( ( ! [I3: nat] :
% 0.19/0.47 ( ( ord_less_nat @ I3 @ K )
% 0.19/0.47 => ? [X4: nat] : ( P @ I3 @ X4 ) ) )
% 0.19/0.47 = ( ? [Xs3: list_nat] :
% 0.19/0.47 ( ( ( size_size_list_nat @ Xs3 )
% 0.19/0.47 = K )
% 0.19/0.47 & ! [I3: nat] :
% 0.19/0.47 ( ( ord_less_nat @ I3 @ K )
% 0.19/0.47 => ( P @ I3 @ ( nth_nat @ Xs3 @ I3 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Skolem_list_nth
% 0.19/0.47 thf(fact_97_Skolem__list__nth,axiom,
% 0.19/0.47 ! [K: nat,P: nat > a > $o] :
% 0.19/0.47 ( ( ! [I3: nat] :
% 0.19/0.47 ( ( ord_less_nat @ I3 @ K )
% 0.19/0.47 => ? [X4: a] : ( P @ I3 @ X4 ) ) )
% 0.19/0.47 = ( ? [Xs3: list_a] :
% 0.19/0.47 ( ( ( size_size_list_a @ Xs3 )
% 0.19/0.47 = K )
% 0.19/0.47 & ! [I3: nat] :
% 0.19/0.47 ( ( ord_less_nat @ I3 @ K )
% 0.19/0.47 => ( P @ I3 @ ( nth_a @ Xs3 @ I3 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Skolem_list_nth
% 0.19/0.47 thf(fact_98_list__eq__iff__nth__eq,axiom,
% 0.19/0.47 ( ( ^ [Y4: list_nat,Z: list_nat] : ( Y4 = Z ) )
% 0.19/0.47 = ( ^ [Xs3: list_nat,Ys3: list_nat] :
% 0.19/0.47 ( ( ( size_size_list_nat @ Xs3 )
% 0.19/0.47 = ( size_size_list_nat @ Ys3 ) )
% 0.19/0.47 & ! [I3: nat] :
% 0.19/0.47 ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs3 ) )
% 0.19/0.47 => ( ( nth_nat @ Xs3 @ I3 )
% 0.19/0.47 = ( nth_nat @ Ys3 @ I3 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % list_eq_iff_nth_eq
% 0.19/0.47 thf(fact_99_list__eq__iff__nth__eq,axiom,
% 0.19/0.47 ( ( ^ [Y4: list_a,Z: list_a] : ( Y4 = Z ) )
% 0.19/0.47 = ( ^ [Xs3: list_a,Ys3: list_a] :
% 0.19/0.47 ( ( ( size_size_list_a @ Xs3 )
% 0.19/0.47 = ( size_size_list_a @ Ys3 ) )
% 0.19/0.47 & ! [I3: nat] :
% 0.19/0.47 ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs3 ) )
% 0.19/0.47 => ( ( nth_a @ Xs3 @ I3 )
% 0.19/0.47 = ( nth_a @ Ys3 @ I3 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % list_eq_iff_nth_eq
% 0.19/0.47 thf(fact_100_less__diff__iff,axiom,
% 0.19/0.47 ! [K: nat,M: nat,N: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ K @ M )
% 0.19/0.47 => ( ( ord_less_eq_nat @ K @ N )
% 0.19/0.47 => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.19/0.47 = ( ord_less_nat @ M @ N ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_diff_iff
% 0.19/0.47 thf(fact_101_diff__less__mono,axiom,
% 0.19/0.47 ! [A: nat,B: nat,C: nat] :
% 0.19/0.47 ( ( ord_less_nat @ A @ B )
% 0.19/0.47 => ( ( ord_less_eq_nat @ C @ A )
% 0.19/0.47 => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % diff_less_mono
% 0.19/0.47 thf(fact_102_less__diff__le__imp__less,axiom,
% 0.19/0.47 ! [I: nat,J: nat,M: nat,N: nat] :
% 0.19/0.47 ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ M ) )
% 0.19/0.47 => ( ( ord_less_eq_nat @ N @ M )
% 0.19/0.47 => ( ord_less_nat @ I @ ( minus_minus_nat @ J @ N ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_diff_le_imp_less
% 0.19/0.47 thf(fact_103_nat__diff__left__cancel__le1,axiom,
% 0.19/0.47 ! [K: nat,M: nat,N: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ ( minus_minus_nat @ K @ M ) @ ( minus_minus_nat @ K @ N ) )
% 0.19/0.47 => ( ( ord_less_nat @ M @ K )
% 0.19/0.47 => ( ord_less_eq_nat @ N @ M ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % nat_diff_left_cancel_le1
% 0.19/0.47 thf(fact_104_nat__diff__right__cancel__le1,axiom,
% 0.19/0.47 ! [M: nat,K: nat,N: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.19/0.47 => ( ( ord_less_nat @ K @ M )
% 0.19/0.47 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % nat_diff_right_cancel_le1
% 0.19/0.47 thf(fact_105_i__drop__nth__sub,axiom,
% 0.19/0.47 ! [N: nat,X: nat,S: nat > a] :
% 0.19/0.47 ( ( ord_less_eq_nat @ N @ X )
% 0.19/0.47 => ( ( listIn1417627087drop_a @ N @ S @ ( minus_minus_nat @ X @ N ) )
% 0.19/0.47 = ( S @ X ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % i_drop_nth_sub
% 0.19/0.47 thf(fact_106_diff__right__commute,axiom,
% 0.19/0.47 ! [A: nat,C: nat,B: nat] :
% 0.19/0.47 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
% 0.19/0.47 = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% 0.19/0.47
% 0.19/0.47 % diff_right_commute
% 0.19/0.47 thf(fact_107_nth__append2,axiom,
% 0.19/0.47 ! [Xs: list_nat,N: nat,Ys: list_nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
% 0.19/0.47 => ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
% 0.19/0.47 = ( nth_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % nth_append2
% 0.19/0.47 thf(fact_108_nth__append2,axiom,
% 0.19/0.47 ! [Xs: list_a,N: nat,Ys: list_a] :
% 0.19/0.47 ( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N )
% 0.19/0.47 => ( ( nth_a @ ( append_a @ Xs @ Ys ) @ N )
% 0.19/0.47 = ( nth_a @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % nth_append2
% 0.19/0.47 thf(fact_109_nth__append1,axiom,
% 0.19/0.47 ! [N: nat,Xs: list_nat,Ys: list_nat] :
% 0.19/0.47 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 0.19/0.47 => ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
% 0.19/0.47 = ( nth_nat @ Xs @ N ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % nth_append1
% 0.19/0.47 thf(fact_110_nth__append1,axiom,
% 0.19/0.47 ! [N: nat,Xs: list_a,Ys: list_a] :
% 0.19/0.47 ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
% 0.19/0.47 => ( ( nth_a @ ( append_a @ Xs @ Ys ) @ N )
% 0.19/0.47 = ( nth_a @ Xs @ N ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % nth_append1
% 0.19/0.47 thf(fact_111_order__refl,axiom,
% 0.19/0.47 ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% 0.19/0.47
% 0.19/0.47 % order_refl
% 0.19/0.47 thf(fact_112_list__desc__trans__le,axiom,
% 0.19/0.47 ( list_desc_nat
% 0.19/0.47 = ( ^ [Xs3: list_nat] :
% 0.19/0.47 ! [J3: nat] :
% 0.19/0.47 ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs3 ) )
% 0.19/0.47 => ! [I3: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ I3 @ J3 )
% 0.19/0.47 => ( ord_less_eq_nat @ ( nth_nat @ Xs3 @ J3 ) @ ( nth_nat @ Xs3 @ I3 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % list_desc_trans_le
% 0.19/0.47 thf(fact_113_nat__descend__induct,axiom,
% 0.19/0.47 ! [N: nat,P: nat > $o,M: nat] :
% 0.19/0.47 ( ! [K2: nat] :
% 0.19/0.47 ( ( ord_less_nat @ N @ K2 )
% 0.19/0.47 => ( P @ K2 ) )
% 0.19/0.47 => ( ! [K2: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ K2 @ N )
% 0.19/0.47 => ( ! [I4: nat] :
% 0.19/0.47 ( ( ord_less_nat @ K2 @ I4 )
% 0.19/0.47 => ( P @ I4 ) )
% 0.19/0.47 => ( P @ K2 ) ) )
% 0.19/0.47 => ( P @ M ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % nat_descend_induct
% 0.19/0.47 thf(fact_114_list__asc__trans__le,axiom,
% 0.19/0.47 ( list_asc_nat
% 0.19/0.47 = ( ^ [Xs3: list_nat] :
% 0.19/0.47 ! [J3: nat] :
% 0.19/0.47 ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs3 ) )
% 0.19/0.47 => ! [I3: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ I3 @ J3 )
% 0.19/0.47 => ( ord_less_eq_nat @ ( nth_nat @ Xs3 @ I3 ) @ ( nth_nat @ Xs3 @ J3 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % list_asc_trans_le
% 0.19/0.47 thf(fact_115_list__strict__asc__trans__le,axiom,
% 0.19/0.47 ! [Xs: list_nat] :
% 0.19/0.47 ( ( list_strict_asc_nat @ Xs )
% 0.19/0.47 => ! [J4: nat] :
% 0.19/0.47 ( ( ord_less_nat @ J4 @ ( size_size_list_nat @ Xs ) )
% 0.19/0.47 => ! [I4: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ I4 @ J4 )
% 0.19/0.47 => ( ord_less_eq_nat @ ( nth_nat @ Xs @ I4 ) @ ( nth_nat @ Xs @ J4 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % list_strict_asc_trans_le
% 0.19/0.47 thf(fact_116_list__strict__asc__imp__list__asc,axiom,
% 0.19/0.47 ! [Xs: list_nat] :
% 0.19/0.47 ( ( list_strict_asc_nat @ Xs )
% 0.19/0.47 => ( list_asc_nat @ Xs ) ) ).
% 0.19/0.47
% 0.19/0.47 % list_strict_asc_imp_list_asc
% 0.19/0.47 thf(fact_117_order__subst1,axiom,
% 0.19/0.47 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.19/0.47 => ( ( ord_less_eq_nat @ B @ C )
% 0.19/0.47 => ( ! [X2: nat,Y2: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.19/0.47 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.19/0.47 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % order_subst1
% 0.19/0.47 thf(fact_118_order__subst2,axiom,
% 0.19/0.47 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.47 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.19/0.47 => ( ! [X2: nat,Y2: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.19/0.47 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.19/0.47 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % order_subst2
% 0.19/0.47 thf(fact_119_ord__eq__le__subst,axiom,
% 0.19/0.47 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.19/0.47 ( ( A
% 0.19/0.47 = ( F @ B ) )
% 0.19/0.47 => ( ( ord_less_eq_nat @ B @ C )
% 0.19/0.47 => ( ! [X2: nat,Y2: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.19/0.47 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.19/0.47 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ord_eq_le_subst
% 0.19/0.47 thf(fact_120_ord__le__eq__subst,axiom,
% 0.19/0.47 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.47 => ( ( ( F @ B )
% 0.19/0.47 = C )
% 0.19/0.47 => ( ! [X2: nat,Y2: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.19/0.47 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.19/0.47 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ord_le_eq_subst
% 0.19/0.47 thf(fact_121_eq__iff,axiom,
% 0.19/0.47 ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
% 0.19/0.47 = ( ^ [X3: nat,Y5: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ X3 @ Y5 )
% 0.19/0.47 & ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % eq_iff
% 0.19/0.47 thf(fact_122_antisym,axiom,
% 0.19/0.47 ! [X: nat,Y: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ X @ Y )
% 0.19/0.47 => ( ( ord_less_eq_nat @ Y @ X )
% 0.19/0.47 => ( X = Y ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % antisym
% 0.19/0.47 thf(fact_123_linear,axiom,
% 0.19/0.47 ! [X: nat,Y: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ X @ Y )
% 0.19/0.47 | ( ord_less_eq_nat @ Y @ X ) ) ).
% 0.19/0.47
% 0.19/0.47 % linear
% 0.19/0.47 thf(fact_124_eq__refl,axiom,
% 0.19/0.47 ! [X: nat,Y: nat] :
% 0.19/0.47 ( ( X = Y )
% 0.19/0.47 => ( ord_less_eq_nat @ X @ Y ) ) ).
% 0.19/0.47
% 0.19/0.47 % eq_refl
% 0.19/0.47 thf(fact_125_le__cases,axiom,
% 0.19/0.47 ! [X: nat,Y: nat] :
% 0.19/0.47 ( ~ ( ord_less_eq_nat @ X @ Y )
% 0.19/0.47 => ( ord_less_eq_nat @ Y @ X ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_cases
% 0.19/0.47 thf(fact_126_order_Otrans,axiom,
% 0.19/0.47 ! [A: nat,B: nat,C: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.47 => ( ( ord_less_eq_nat @ B @ C )
% 0.19/0.47 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % order.trans
% 0.19/0.47 thf(fact_127_le__cases3,axiom,
% 0.19/0.47 ! [X: nat,Y: nat,Z2: nat] :
% 0.19/0.47 ( ( ( ord_less_eq_nat @ X @ Y )
% 0.19/0.47 => ~ ( ord_less_eq_nat @ Y @ Z2 ) )
% 0.19/0.47 => ( ( ( ord_less_eq_nat @ Y @ X )
% 0.19/0.47 => ~ ( ord_less_eq_nat @ X @ Z2 ) )
% 0.19/0.47 => ( ( ( ord_less_eq_nat @ X @ Z2 )
% 0.19/0.47 => ~ ( ord_less_eq_nat @ Z2 @ Y ) )
% 0.19/0.47 => ( ( ( ord_less_eq_nat @ Z2 @ Y )
% 0.19/0.47 => ~ ( ord_less_eq_nat @ Y @ X ) )
% 0.19/0.47 => ( ( ( ord_less_eq_nat @ Y @ Z2 )
% 0.19/0.47 => ~ ( ord_less_eq_nat @ Z2 @ X ) )
% 0.19/0.47 => ~ ( ( ord_less_eq_nat @ Z2 @ X )
% 0.19/0.47 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_cases3
% 0.19/0.47 thf(fact_128_antisym__conv,axiom,
% 0.19/0.47 ! [Y: nat,X: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ Y @ X )
% 0.19/0.47 => ( ( ord_less_eq_nat @ X @ Y )
% 0.19/0.47 = ( X = Y ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % antisym_conv
% 0.19/0.47 thf(fact_129_order__class_Oorder_Oeq__iff,axiom,
% 0.19/0.47 ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
% 0.19/0.47 = ( ^ [A2: nat,B2: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ A2 @ B2 )
% 0.19/0.47 & ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % order_class.order.eq_iff
% 0.19/0.47 thf(fact_130_ord__eq__le__trans,axiom,
% 0.19/0.47 ! [A: nat,B: nat,C: nat] :
% 0.19/0.47 ( ( A = B )
% 0.19/0.47 => ( ( ord_less_eq_nat @ B @ C )
% 0.19/0.47 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ord_eq_le_trans
% 0.19/0.47 thf(fact_131_ord__le__eq__trans,axiom,
% 0.19/0.47 ! [A: nat,B: nat,C: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.47 => ( ( B = C )
% 0.19/0.47 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ord_le_eq_trans
% 0.19/0.47 thf(fact_132_order__class_Oorder_Oantisym,axiom,
% 0.19/0.47 ! [A: nat,B: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.47 => ( ( ord_less_eq_nat @ B @ A )
% 0.19/0.47 => ( A = B ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % order_class.order.antisym
% 0.19/0.47 thf(fact_133_order__trans,axiom,
% 0.19/0.47 ! [X: nat,Y: nat,Z2: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ X @ Y )
% 0.19/0.47 => ( ( ord_less_eq_nat @ Y @ Z2 )
% 0.19/0.47 => ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % order_trans
% 0.19/0.47 thf(fact_134_dual__order_Orefl,axiom,
% 0.19/0.47 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 0.19/0.47
% 0.19/0.47 % dual_order.refl
% 0.19/0.47 thf(fact_135_linorder__wlog,axiom,
% 0.19/0.47 ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.19/0.47 ( ! [A3: nat,B3: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ A3 @ B3 )
% 0.19/0.47 => ( P @ A3 @ B3 ) )
% 0.19/0.47 => ( ! [A3: nat,B3: nat] :
% 0.19/0.47 ( ( P @ B3 @ A3 )
% 0.19/0.47 => ( P @ A3 @ B3 ) )
% 0.19/0.47 => ( P @ A @ B ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % linorder_wlog
% 0.19/0.47 thf(fact_136_dual__order_Otrans,axiom,
% 0.19/0.47 ! [B: nat,A: nat,C: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ B @ A )
% 0.19/0.47 => ( ( ord_less_eq_nat @ C @ B )
% 0.19/0.47 => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % dual_order.trans
% 0.19/0.47 thf(fact_137_dual__order_Oeq__iff,axiom,
% 0.19/0.47 ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
% 0.19/0.47 = ( ^ [A2: nat,B2: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ B2 @ A2 )
% 0.19/0.47 & ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % dual_order.eq_iff
% 0.19/0.47 thf(fact_138_dual__order_Oantisym,axiom,
% 0.19/0.47 ! [B: nat,A: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ B @ A )
% 0.19/0.47 => ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.47 => ( A = B ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % dual_order.antisym
% 0.19/0.47 thf(fact_139_dual__order_Ostrict__implies__not__eq,axiom,
% 0.19/0.47 ! [B: nat,A: nat] :
% 0.19/0.47 ( ( ord_less_nat @ B @ A )
% 0.19/0.47 => ( A != B ) ) ).
% 0.19/0.47
% 0.19/0.47 % dual_order.strict_implies_not_eq
% 0.19/0.47 thf(fact_140_order_Ostrict__implies__not__eq,axiom,
% 0.19/0.47 ! [A: nat,B: nat] :
% 0.19/0.47 ( ( ord_less_nat @ A @ B )
% 0.19/0.47 => ( A != B ) ) ).
% 0.19/0.47
% 0.19/0.47 % order.strict_implies_not_eq
% 0.19/0.47 thf(fact_141_not__less__iff__gr__or__eq,axiom,
% 0.19/0.47 ! [X: nat,Y: nat] :
% 0.19/0.47 ( ( ~ ( ord_less_nat @ X @ Y ) )
% 0.19/0.47 = ( ( ord_less_nat @ Y @ X )
% 0.19/0.47 | ( X = Y ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % not_less_iff_gr_or_eq
% 0.19/0.47 thf(fact_142_dual__order_Ostrict__trans,axiom,
% 0.19/0.47 ! [B: nat,A: nat,C: nat] :
% 0.19/0.47 ( ( ord_less_nat @ B @ A )
% 0.19/0.47 => ( ( ord_less_nat @ C @ B )
% 0.19/0.47 => ( ord_less_nat @ C @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % dual_order.strict_trans
% 0.19/0.47 thf(fact_143_linorder__less__wlog,axiom,
% 0.19/0.47 ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.19/0.47 ( ! [A3: nat,B3: nat] :
% 0.19/0.47 ( ( ord_less_nat @ A3 @ B3 )
% 0.19/0.47 => ( P @ A3 @ B3 ) )
% 0.19/0.47 => ( ! [A3: nat] : ( P @ A3 @ A3 )
% 0.19/0.47 => ( ! [A3: nat,B3: nat] :
% 0.19/0.47 ( ( P @ B3 @ A3 )
% 0.19/0.47 => ( P @ A3 @ B3 ) )
% 0.19/0.47 => ( P @ A @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % linorder_less_wlog
% 0.19/0.47 thf(fact_144_exists__least__iff,axiom,
% 0.19/0.47 ( ( ^ [P2: nat > $o] :
% 0.19/0.47 ? [X4: nat] : ( P2 @ X4 ) )
% 0.19/0.47 = ( ^ [P3: nat > $o] :
% 0.19/0.47 ? [N3: nat] :
% 0.19/0.47 ( ( P3 @ N3 )
% 0.19/0.47 & ! [M3: nat] :
% 0.19/0.47 ( ( ord_less_nat @ M3 @ N3 )
% 0.19/0.47 => ~ ( P3 @ M3 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % exists_least_iff
% 0.19/0.47 thf(fact_145_less__imp__not__less,axiom,
% 0.19/0.47 ! [X: nat,Y: nat] :
% 0.19/0.47 ( ( ord_less_nat @ X @ Y )
% 0.19/0.47 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_imp_not_less
% 0.19/0.47 thf(fact_146_order_Ostrict__trans,axiom,
% 0.19/0.47 ! [A: nat,B: nat,C: nat] :
% 0.19/0.47 ( ( ord_less_nat @ A @ B )
% 0.19/0.47 => ( ( ord_less_nat @ B @ C )
% 0.19/0.47 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % order.strict_trans
% 0.19/0.47 thf(fact_147_dual__order_Oirrefl,axiom,
% 0.19/0.47 ! [A: nat] :
% 0.19/0.47 ~ ( ord_less_nat @ A @ A ) ).
% 0.19/0.47
% 0.19/0.47 % dual_order.irrefl
% 0.19/0.47 thf(fact_148_linorder__cases,axiom,
% 0.19/0.47 ! [X: nat,Y: nat] :
% 0.19/0.47 ( ~ ( ord_less_nat @ X @ Y )
% 0.19/0.47 => ( ( X != Y )
% 0.19/0.47 => ( ord_less_nat @ Y @ X ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % linorder_cases
% 0.19/0.47 thf(fact_149_less__imp__triv,axiom,
% 0.19/0.47 ! [X: nat,Y: nat,P: $o] :
% 0.19/0.47 ( ( ord_less_nat @ X @ Y )
% 0.19/0.47 => ( ( ord_less_nat @ Y @ X )
% 0.19/0.47 => P ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_imp_triv
% 0.19/0.47 thf(fact_150_less__imp__not__eq2,axiom,
% 0.19/0.47 ! [X: nat,Y: nat] :
% 0.19/0.47 ( ( ord_less_nat @ X @ Y )
% 0.19/0.47 => ( Y != X ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_imp_not_eq2
% 0.19/0.47 thf(fact_151_antisym__conv3,axiom,
% 0.19/0.47 ! [Y: nat,X: nat] :
% 0.19/0.47 ( ~ ( ord_less_nat @ Y @ X )
% 0.19/0.47 => ( ( ~ ( ord_less_nat @ X @ Y ) )
% 0.19/0.47 = ( X = Y ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % antisym_conv3
% 0.19/0.47 thf(fact_152_less__induct,axiom,
% 0.19/0.47 ! [P: nat > $o,A: nat] :
% 0.19/0.47 ( ! [X2: nat] :
% 0.19/0.47 ( ! [Y3: nat] :
% 0.19/0.47 ( ( ord_less_nat @ Y3 @ X2 )
% 0.19/0.47 => ( P @ Y3 ) )
% 0.19/0.47 => ( P @ X2 ) )
% 0.19/0.47 => ( P @ A ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_induct
% 0.19/0.47 thf(fact_153_less__not__sym,axiom,
% 0.19/0.47 ! [X: nat,Y: nat] :
% 0.19/0.47 ( ( ord_less_nat @ X @ Y )
% 0.19/0.47 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_not_sym
% 0.19/0.47 thf(fact_154_less__imp__not__eq,axiom,
% 0.19/0.47 ! [X: nat,Y: nat] :
% 0.19/0.47 ( ( ord_less_nat @ X @ Y )
% 0.19/0.47 => ( X != Y ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_imp_not_eq
% 0.19/0.47 thf(fact_155_dual__order_Oasym,axiom,
% 0.19/0.47 ! [B: nat,A: nat] :
% 0.19/0.47 ( ( ord_less_nat @ B @ A )
% 0.19/0.47 => ~ ( ord_less_nat @ A @ B ) ) ).
% 0.19/0.47
% 0.19/0.47 % dual_order.asym
% 0.19/0.47 thf(fact_156_ord__less__eq__trans,axiom,
% 0.19/0.47 ! [A: nat,B: nat,C: nat] :
% 0.19/0.47 ( ( ord_less_nat @ A @ B )
% 0.19/0.47 => ( ( B = C )
% 0.19/0.47 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ord_less_eq_trans
% 0.19/0.47 thf(fact_157_ord__eq__less__trans,axiom,
% 0.19/0.47 ! [A: nat,B: nat,C: nat] :
% 0.19/0.47 ( ( A = B )
% 0.19/0.47 => ( ( ord_less_nat @ B @ C )
% 0.19/0.47 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ord_eq_less_trans
% 0.19/0.47 thf(fact_158_less__irrefl,axiom,
% 0.19/0.47 ! [X: nat] :
% 0.19/0.47 ~ ( ord_less_nat @ X @ X ) ).
% 0.19/0.47
% 0.19/0.47 % less_irrefl
% 0.19/0.47 thf(fact_159_less__linear,axiom,
% 0.19/0.47 ! [X: nat,Y: nat] :
% 0.19/0.47 ( ( ord_less_nat @ X @ Y )
% 0.19/0.47 | ( X = Y )
% 0.19/0.47 | ( ord_less_nat @ Y @ X ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_linear
% 0.19/0.47 thf(fact_160_less__trans,axiom,
% 0.19/0.47 ! [X: nat,Y: nat,Z2: nat] :
% 0.19/0.47 ( ( ord_less_nat @ X @ Y )
% 0.19/0.47 => ( ( ord_less_nat @ Y @ Z2 )
% 0.19/0.47 => ( ord_less_nat @ X @ Z2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_trans
% 0.19/0.47 thf(fact_161_less__asym_H,axiom,
% 0.19/0.47 ! [A: nat,B: nat] :
% 0.19/0.47 ( ( ord_less_nat @ A @ B )
% 0.19/0.47 => ~ ( ord_less_nat @ B @ A ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_asym'
% 0.19/0.47 thf(fact_162_less__asym,axiom,
% 0.19/0.47 ! [X: nat,Y: nat] :
% 0.19/0.47 ( ( ord_less_nat @ X @ Y )
% 0.19/0.47 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_asym
% 0.19/0.47 thf(fact_163_less__imp__neq,axiom,
% 0.19/0.47 ! [X: nat,Y: nat] :
% 0.19/0.47 ( ( ord_less_nat @ X @ Y )
% 0.19/0.47 => ( X != Y ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_imp_neq
% 0.19/0.47 thf(fact_164_order_Oasym,axiom,
% 0.19/0.47 ! [A: nat,B: nat] :
% 0.19/0.47 ( ( ord_less_nat @ A @ B )
% 0.19/0.47 => ~ ( ord_less_nat @ B @ A ) ) ).
% 0.19/0.47
% 0.19/0.47 % order.asym
% 0.19/0.47 thf(fact_165_neq__iff,axiom,
% 0.19/0.47 ! [X: nat,Y: nat] :
% 0.19/0.47 ( ( X != Y )
% 0.19/0.47 = ( ( ord_less_nat @ X @ Y )
% 0.19/0.47 | ( ord_less_nat @ Y @ X ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % neq_iff
% 0.19/0.47 thf(fact_166_neqE,axiom,
% 0.19/0.47 ! [X: nat,Y: nat] :
% 0.19/0.47 ( ( X != Y )
% 0.19/0.47 => ( ~ ( ord_less_nat @ X @ Y )
% 0.19/0.47 => ( ord_less_nat @ Y @ X ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % neqE
% 0.19/0.47 thf(fact_167_gt__ex,axiom,
% 0.19/0.47 ! [X: nat] :
% 0.19/0.47 ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% 0.19/0.47
% 0.19/0.47 % gt_ex
% 0.19/0.47 thf(fact_168_order__less__subst2,axiom,
% 0.19/0.47 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.19/0.47 ( ( ord_less_nat @ A @ B )
% 0.19/0.47 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.19/0.47 => ( ! [X2: nat,Y2: nat] :
% 0.19/0.47 ( ( ord_less_nat @ X2 @ Y2 )
% 0.19/0.47 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.19/0.47 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % order_less_subst2
% 0.19/0.47 thf(fact_169_order__less__subst1,axiom,
% 0.19/0.47 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.19/0.47 ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.19/0.47 => ( ( ord_less_nat @ B @ C )
% 0.19/0.47 => ( ! [X2: nat,Y2: nat] :
% 0.19/0.47 ( ( ord_less_nat @ X2 @ Y2 )
% 0.19/0.47 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.19/0.47 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % order_less_subst1
% 0.19/0.47 thf(fact_170_ord__less__eq__subst,axiom,
% 0.19/0.47 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.19/0.47 ( ( ord_less_nat @ A @ B )
% 0.19/0.47 => ( ( ( F @ B )
% 0.19/0.47 = C )
% 0.19/0.47 => ( ! [X2: nat,Y2: nat] :
% 0.19/0.47 ( ( ord_less_nat @ X2 @ Y2 )
% 0.19/0.47 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.19/0.47 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ord_less_eq_subst
% 0.19/0.47 thf(fact_171_ord__eq__less__subst,axiom,
% 0.19/0.47 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.19/0.47 ( ( A
% 0.19/0.47 = ( F @ B ) )
% 0.19/0.47 => ( ( ord_less_nat @ B @ C )
% 0.19/0.47 => ( ! [X2: nat,Y2: nat] :
% 0.19/0.47 ( ( ord_less_nat @ X2 @ Y2 )
% 0.19/0.47 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.19/0.47 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ord_eq_less_subst
% 0.19/0.47 thf(fact_172_list__drop__eq__conv,axiom,
% 0.19/0.47 ( ( ^ [Y4: list_a,Z: list_a] : ( Y4 = Z ) )
% 0.19/0.47 = ( ^ [Xs3: list_a,Ys3: list_a] :
% 0.19/0.47 ! [N3: nat] :
% 0.19/0.47 ( ( drop_a @ N3 @ Xs3 )
% 0.19/0.47 = ( drop_a @ N3 @ Ys3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % list_drop_eq_conv
% 0.19/0.47 thf(fact_173_list__asc__trans,axiom,
% 0.19/0.47 ( list_asc_nat
% 0.19/0.47 = ( ^ [Xs3: list_nat] :
% 0.19/0.47 ! [J3: nat] :
% 0.19/0.47 ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs3 ) )
% 0.19/0.47 => ! [I3: nat] :
% 0.19/0.47 ( ( ord_less_nat @ I3 @ J3 )
% 0.19/0.47 => ( ord_less_eq_nat @ ( nth_nat @ Xs3 @ I3 ) @ ( nth_nat @ Xs3 @ J3 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % list_asc_trans
% 0.19/0.47 thf(fact_174_list__strict__asc__trans,axiom,
% 0.19/0.47 ( list_strict_asc_nat
% 0.19/0.47 = ( ^ [Xs3: list_nat] :
% 0.19/0.47 ! [J3: nat] :
% 0.19/0.47 ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs3 ) )
% 0.19/0.47 => ! [I3: nat] :
% 0.19/0.47 ( ( ord_less_nat @ I3 @ J3 )
% 0.19/0.47 => ( ord_less_nat @ ( nth_nat @ Xs3 @ I3 ) @ ( nth_nat @ Xs3 @ J3 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % list_strict_asc_trans
% 0.19/0.47 thf(fact_175_list__desc__trans,axiom,
% 0.19/0.47 ( list_desc_nat
% 0.19/0.47 = ( ^ [Xs3: list_nat] :
% 0.19/0.47 ! [J3: nat] :
% 0.19/0.47 ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs3 ) )
% 0.19/0.47 => ! [I3: nat] :
% 0.19/0.47 ( ( ord_less_nat @ I3 @ J3 )
% 0.19/0.47 => ( ord_less_eq_nat @ ( nth_nat @ Xs3 @ J3 ) @ ( nth_nat @ Xs3 @ I3 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % list_desc_trans
% 0.19/0.47 thf(fact_176_order_Onot__eq__order__implies__strict,axiom,
% 0.19/0.47 ! [A: nat,B: nat] :
% 0.19/0.47 ( ( A != B )
% 0.19/0.47 => ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.47 => ( ord_less_nat @ A @ B ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % order.not_eq_order_implies_strict
% 0.19/0.47 thf(fact_177_dual__order_Ostrict__implies__order,axiom,
% 0.19/0.47 ! [B: nat,A: nat] :
% 0.19/0.47 ( ( ord_less_nat @ B @ A )
% 0.19/0.47 => ( ord_less_eq_nat @ B @ A ) ) ).
% 0.19/0.47
% 0.19/0.47 % dual_order.strict_implies_order
% 0.19/0.47 thf(fact_178_dual__order_Ostrict__iff__order,axiom,
% 0.19/0.47 ( ord_less_nat
% 0.19/0.47 = ( ^ [B2: nat,A2: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ B2 @ A2 )
% 0.19/0.47 & ( A2 != B2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % dual_order.strict_iff_order
% 0.19/0.47 thf(fact_179_dual__order_Oorder__iff__strict,axiom,
% 0.19/0.47 ( ord_less_eq_nat
% 0.19/0.47 = ( ^ [B2: nat,A2: nat] :
% 0.19/0.47 ( ( ord_less_nat @ B2 @ A2 )
% 0.19/0.47 | ( A2 = B2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % dual_order.order_iff_strict
% 0.19/0.47 thf(fact_180_order_Ostrict__implies__order,axiom,
% 0.19/0.47 ! [A: nat,B: nat] :
% 0.19/0.47 ( ( ord_less_nat @ A @ B )
% 0.19/0.47 => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.19/0.47
% 0.19/0.47 % order.strict_implies_order
% 0.19/0.47 thf(fact_181_dual__order_Ostrict__trans2,axiom,
% 0.19/0.47 ! [B: nat,A: nat,C: nat] :
% 0.19/0.47 ( ( ord_less_nat @ B @ A )
% 0.19/0.47 => ( ( ord_less_eq_nat @ C @ B )
% 0.19/0.47 => ( ord_less_nat @ C @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % dual_order.strict_trans2
% 0.19/0.47 thf(fact_182_dual__order_Ostrict__trans1,axiom,
% 0.19/0.47 ! [B: nat,A: nat,C: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ B @ A )
% 0.19/0.47 => ( ( ord_less_nat @ C @ B )
% 0.19/0.47 => ( ord_less_nat @ C @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % dual_order.strict_trans1
% 0.19/0.47 thf(fact_183_order_Ostrict__iff__order,axiom,
% 0.19/0.47 ( ord_less_nat
% 0.19/0.47 = ( ^ [A2: nat,B2: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ A2 @ B2 )
% 0.19/0.47 & ( A2 != B2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % order.strict_iff_order
% 0.19/0.47 thf(fact_184_order_Oorder__iff__strict,axiom,
% 0.19/0.47 ( ord_less_eq_nat
% 0.19/0.47 = ( ^ [A2: nat,B2: nat] :
% 0.19/0.47 ( ( ord_less_nat @ A2 @ B2 )
% 0.19/0.47 | ( A2 = B2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % order.order_iff_strict
% 0.19/0.47 thf(fact_185_order_Ostrict__trans2,axiom,
% 0.19/0.47 ! [A: nat,B: nat,C: nat] :
% 0.19/0.47 ( ( ord_less_nat @ A @ B )
% 0.19/0.47 => ( ( ord_less_eq_nat @ B @ C )
% 0.19/0.47 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % order.strict_trans2
% 0.19/0.47 thf(fact_186_order_Ostrict__trans1,axiom,
% 0.19/0.47 ! [A: nat,B: nat,C: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.47 => ( ( ord_less_nat @ B @ C )
% 0.19/0.47 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % order.strict_trans1
% 0.19/0.47 thf(fact_187_not__le__imp__less,axiom,
% 0.19/0.47 ! [Y: nat,X: nat] :
% 0.19/0.47 ( ~ ( ord_less_eq_nat @ Y @ X )
% 0.19/0.47 => ( ord_less_nat @ X @ Y ) ) ).
% 0.19/0.47
% 0.19/0.47 % not_le_imp_less
% 0.19/0.47 thf(fact_188_less__le__not__le,axiom,
% 0.19/0.47 ( ord_less_nat
% 0.19/0.47 = ( ^ [X3: nat,Y5: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ X3 @ Y5 )
% 0.19/0.47 & ~ ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_le_not_le
% 0.19/0.47 thf(fact_189_le__imp__less__or__eq,axiom,
% 0.19/0.47 ! [X: nat,Y: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ X @ Y )
% 0.19/0.47 => ( ( ord_less_nat @ X @ Y )
% 0.19/0.47 | ( X = Y ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_imp_less_or_eq
% 0.19/0.47 thf(fact_190_le__less__linear,axiom,
% 0.19/0.47 ! [X: nat,Y: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ X @ Y )
% 0.19/0.47 | ( ord_less_nat @ Y @ X ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_less_linear
% 0.19/0.47 thf(fact_191_less__le__trans,axiom,
% 0.19/0.47 ! [X: nat,Y: nat,Z2: nat] :
% 0.19/0.47 ( ( ord_less_nat @ X @ Y )
% 0.19/0.47 => ( ( ord_less_eq_nat @ Y @ Z2 )
% 0.19/0.47 => ( ord_less_nat @ X @ Z2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_le_trans
% 0.19/0.47 thf(fact_192_le__less__trans,axiom,
% 0.19/0.47 ! [X: nat,Y: nat,Z2: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ X @ Y )
% 0.19/0.47 => ( ( ord_less_nat @ Y @ Z2 )
% 0.19/0.47 => ( ord_less_nat @ X @ Z2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_less_trans
% 0.19/0.47 thf(fact_193_less__imp__le,axiom,
% 0.19/0.47 ! [X: nat,Y: nat] :
% 0.19/0.47 ( ( ord_less_nat @ X @ Y )
% 0.19/0.47 => ( ord_less_eq_nat @ X @ Y ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_imp_le
% 0.19/0.47 thf(fact_194_antisym__conv2,axiom,
% 0.19/0.47 ! [X: nat,Y: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ X @ Y )
% 0.19/0.47 => ( ( ~ ( ord_less_nat @ X @ Y ) )
% 0.19/0.47 = ( X = Y ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % antisym_conv2
% 0.19/0.47 thf(fact_195_antisym__conv1,axiom,
% 0.19/0.47 ! [X: nat,Y: nat] :
% 0.19/0.47 ( ~ ( ord_less_nat @ X @ Y )
% 0.19/0.47 => ( ( ord_less_eq_nat @ X @ Y )
% 0.19/0.47 = ( X = Y ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % antisym_conv1
% 0.19/0.47 thf(fact_196_le__neq__trans,axiom,
% 0.19/0.47 ! [A: nat,B: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.47 => ( ( A != B )
% 0.19/0.47 => ( ord_less_nat @ A @ B ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_neq_trans
% 0.19/0.47 thf(fact_197_not__less,axiom,
% 0.19/0.47 ! [X: nat,Y: nat] :
% 0.19/0.47 ( ( ~ ( ord_less_nat @ X @ Y ) )
% 0.19/0.47 = ( ord_less_eq_nat @ Y @ X ) ) ).
% 0.19/0.47
% 0.19/0.47 % not_less
% 0.19/0.47 thf(fact_198_not__le,axiom,
% 0.19/0.47 ! [X: nat,Y: nat] :
% 0.19/0.47 ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
% 0.19/0.47 = ( ord_less_nat @ Y @ X ) ) ).
% 0.19/0.47
% 0.19/0.47 % not_le
% 0.19/0.47 thf(fact_199_order__less__le__subst2,axiom,
% 0.19/0.47 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.19/0.47 ( ( ord_less_nat @ A @ B )
% 0.19/0.47 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.19/0.47 => ( ! [X2: nat,Y2: nat] :
% 0.19/0.47 ( ( ord_less_nat @ X2 @ Y2 )
% 0.19/0.47 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.19/0.47 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % order_less_le_subst2
% 0.19/0.47 thf(fact_200_order__less__le__subst1,axiom,
% 0.19/0.47 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.19/0.47 ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.19/0.47 => ( ( ord_less_eq_nat @ B @ C )
% 0.19/0.47 => ( ! [X2: nat,Y2: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.19/0.47 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.19/0.47 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % order_less_le_subst1
% 0.19/0.47 thf(fact_201_order__le__less__subst2,axiom,
% 0.19/0.47 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.47 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.19/0.47 => ( ! [X2: nat,Y2: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.19/0.47 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.19/0.47 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % order_le_less_subst2
% 0.19/0.47 thf(fact_202_order__le__less__subst1,axiom,
% 0.19/0.47 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.19/0.47 => ( ( ord_less_nat @ B @ C )
% 0.19/0.47 => ( ! [X2: nat,Y2: nat] :
% 0.19/0.47 ( ( ord_less_nat @ X2 @ Y2 )
% 0.19/0.47 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.19/0.47 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % order_le_less_subst1
% 0.19/0.47 thf(fact_203_less__le,axiom,
% 0.19/0.47 ( ord_less_nat
% 0.19/0.47 = ( ^ [X3: nat,Y5: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ X3 @ Y5 )
% 0.19/0.47 & ( X3 != Y5 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % less_le
% 0.19/0.47 thf(fact_204_le__less,axiom,
% 0.19/0.47 ( ord_less_eq_nat
% 0.19/0.47 = ( ^ [X3: nat,Y5: nat] :
% 0.19/0.47 ( ( ord_less_nat @ X3 @ Y5 )
% 0.19/0.47 | ( X3 = Y5 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_less
% 0.19/0.47 thf(fact_205_leI,axiom,
% 0.19/0.47 ! [X: nat,Y: nat] :
% 0.19/0.47 ( ~ ( ord_less_nat @ X @ Y )
% 0.19/0.47 => ( ord_less_eq_nat @ Y @ X ) ) ).
% 0.19/0.47
% 0.19/0.47 % leI
% 0.19/0.47 thf(fact_206_leD,axiom,
% 0.19/0.47 ! [Y: nat,X: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ Y @ X )
% 0.19/0.47 => ~ ( ord_less_nat @ X @ Y ) ) ).
% 0.19/0.47
% 0.19/0.47 % leD
% 0.19/0.47 thf(fact_207_list__strict__desc__trans,axiom,
% 0.19/0.47 ( list_strict_desc_nat
% 0.19/0.47 = ( ^ [Xs3: list_nat] :
% 0.19/0.47 ! [J3: nat] :
% 0.19/0.47 ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs3 ) )
% 0.19/0.47 => ! [I3: nat] :
% 0.19/0.47 ( ( ord_less_nat @ I3 @ J3 )
% 0.19/0.47 => ( ord_less_nat @ ( nth_nat @ Xs3 @ J3 ) @ ( nth_nat @ Xs3 @ I3 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % list_strict_desc_trans
% 0.19/0.47 thf(fact_208_complete__interval,axiom,
% 0.19/0.47 ! [A: nat,B: nat,P: nat > $o] :
% 0.19/0.47 ( ( ord_less_nat @ A @ B )
% 0.19/0.47 => ( ( P @ A )
% 0.19/0.47 => ( ~ ( P @ B )
% 0.19/0.47 => ? [C2: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ A @ C2 )
% 0.19/0.47 & ( ord_less_eq_nat @ C2 @ B )
% 0.19/0.47 & ! [X5: nat] :
% 0.19/0.47 ( ( ( ord_less_eq_nat @ A @ X5 )
% 0.19/0.47 & ( ord_less_nat @ X5 @ C2 ) )
% 0.19/0.47 => ( P @ X5 ) )
% 0.19/0.47 & ! [D: nat] :
% 0.19/0.47 ( ! [X2: nat] :
% 0.19/0.47 ( ( ( ord_less_eq_nat @ A @ X2 )
% 0.19/0.47 & ( ord_less_nat @ X2 @ D ) )
% 0.19/0.47 => ( P @ X2 ) )
% 0.19/0.47 => ( ord_less_eq_nat @ D @ C2 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % complete_interval
% 0.19/0.47 thf(fact_209_verit__comp__simplify1_I3_J,axiom,
% 0.19/0.47 ! [B4: nat,A4: nat] :
% 0.19/0.47 ( ( ~ ( ord_less_eq_nat @ B4 @ A4 ) )
% 0.19/0.47 = ( ord_less_nat @ A4 @ B4 ) ) ).
% 0.19/0.47
% 0.19/0.47 % verit_comp_simplify1(3)
% 0.19/0.47 thf(fact_210_pinf_I6_J,axiom,
% 0.19/0.47 ! [T: nat] :
% 0.19/0.47 ? [Z3: nat] :
% 0.19/0.47 ! [X5: nat] :
% 0.19/0.47 ( ( ord_less_nat @ Z3 @ X5 )
% 0.19/0.47 => ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% 0.19/0.47
% 0.19/0.47 % pinf(6)
% 0.19/0.47 thf(fact_211_pinf_I8_J,axiom,
% 0.19/0.47 ! [T: nat] :
% 0.19/0.47 ? [Z3: nat] :
% 0.19/0.47 ! [X5: nat] :
% 0.19/0.47 ( ( ord_less_nat @ Z3 @ X5 )
% 0.19/0.47 => ( ord_less_eq_nat @ T @ X5 ) ) ).
% 0.19/0.47
% 0.19/0.47 % pinf(8)
% 0.19/0.47 thf(fact_212_verit__la__disequality,axiom,
% 0.19/0.47 ! [A: nat,B: nat] :
% 0.19/0.47 ( ( A = B )
% 0.19/0.47 | ~ ( ord_less_eq_nat @ A @ B )
% 0.19/0.47 | ~ ( ord_less_eq_nat @ B @ A ) ) ).
% 0.19/0.47
% 0.19/0.47 % verit_la_disequality
% 0.19/0.47 thf(fact_213_minf_I7_J,axiom,
% 0.19/0.47 ! [T: nat] :
% 0.19/0.47 ? [Z3: nat] :
% 0.19/0.47 ! [X5: nat] :
% 0.19/0.47 ( ( ord_less_nat @ X5 @ Z3 )
% 0.19/0.47 => ~ ( ord_less_nat @ T @ X5 ) ) ).
% 0.19/0.47
% 0.19/0.47 % minf(7)
% 0.19/0.47 thf(fact_214_minf_I5_J,axiom,
% 0.19/0.47 ! [T: nat] :
% 0.19/0.47 ? [Z3: nat] :
% 0.19/0.47 ! [X5: nat] :
% 0.19/0.47 ( ( ord_less_nat @ X5 @ Z3 )
% 0.19/0.47 => ( ord_less_nat @ X5 @ T ) ) ).
% 0.19/0.47
% 0.19/0.47 % minf(5)
% 0.19/0.47 thf(fact_215_minf_I4_J,axiom,
% 0.19/0.47 ! [T: nat] :
% 0.19/0.47 ? [Z3: nat] :
% 0.19/0.47 ! [X5: nat] :
% 0.19/0.47 ( ( ord_less_nat @ X5 @ Z3 )
% 0.19/0.47 => ( X5 != T ) ) ).
% 0.19/0.47
% 0.19/0.47 % minf(4)
% 0.19/0.47 thf(fact_216_minf_I3_J,axiom,
% 0.19/0.47 ! [T: nat] :
% 0.19/0.47 ? [Z3: nat] :
% 0.19/0.47 ! [X5: nat] :
% 0.19/0.47 ( ( ord_less_nat @ X5 @ Z3 )
% 0.19/0.47 => ( X5 != T ) ) ).
% 0.19/0.47
% 0.19/0.47 % minf(3)
% 0.19/0.47 thf(fact_217_minf_I2_J,axiom,
% 0.19/0.47 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 0.19/0.47 ( ? [Z4: nat] :
% 0.19/0.47 ! [X2: nat] :
% 0.19/0.47 ( ( ord_less_nat @ X2 @ Z4 )
% 0.19/0.47 => ( ( P @ X2 )
% 0.19/0.47 = ( P4 @ X2 ) ) )
% 0.19/0.47 => ( ? [Z4: nat] :
% 0.19/0.47 ! [X2: nat] :
% 0.19/0.47 ( ( ord_less_nat @ X2 @ Z4 )
% 0.19/0.47 => ( ( Q @ X2 )
% 0.19/0.47 = ( Q2 @ X2 ) ) )
% 0.19/0.47 => ? [Z3: nat] :
% 0.19/0.47 ! [X5: nat] :
% 0.19/0.47 ( ( ord_less_nat @ X5 @ Z3 )
% 0.19/0.47 => ( ( ( P @ X5 )
% 0.19/0.47 | ( Q @ X5 ) )
% 0.19/0.47 = ( ( P4 @ X5 )
% 0.19/0.47 | ( Q2 @ X5 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % minf(2)
% 0.19/0.47 thf(fact_218_minf_I1_J,axiom,
% 0.19/0.47 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 0.19/0.47 ( ? [Z4: nat] :
% 0.19/0.47 ! [X2: nat] :
% 0.19/0.47 ( ( ord_less_nat @ X2 @ Z4 )
% 0.19/0.47 => ( ( P @ X2 )
% 0.19/0.47 = ( P4 @ X2 ) ) )
% 0.19/0.47 => ( ? [Z4: nat] :
% 0.19/0.47 ! [X2: nat] :
% 0.19/0.47 ( ( ord_less_nat @ X2 @ Z4 )
% 0.19/0.47 => ( ( Q @ X2 )
% 0.19/0.47 = ( Q2 @ X2 ) ) )
% 0.19/0.47 => ? [Z3: nat] :
% 0.19/0.47 ! [X5: nat] :
% 0.19/0.47 ( ( ord_less_nat @ X5 @ Z3 )
% 0.19/0.47 => ( ( ( P @ X5 )
% 0.19/0.47 & ( Q @ X5 ) )
% 0.19/0.47 = ( ( P4 @ X5 )
% 0.19/0.47 & ( Q2 @ X5 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % minf(1)
% 0.19/0.47 thf(fact_219_pinf_I7_J,axiom,
% 0.19/0.47 ! [T: nat] :
% 0.19/0.47 ? [Z3: nat] :
% 0.19/0.47 ! [X5: nat] :
% 0.19/0.47 ( ( ord_less_nat @ Z3 @ X5 )
% 0.19/0.47 => ( ord_less_nat @ T @ X5 ) ) ).
% 0.19/0.47
% 0.19/0.47 % pinf(7)
% 0.19/0.47 thf(fact_220_pinf_I5_J,axiom,
% 0.19/0.47 ! [T: nat] :
% 0.19/0.47 ? [Z3: nat] :
% 0.19/0.47 ! [X5: nat] :
% 0.19/0.47 ( ( ord_less_nat @ Z3 @ X5 )
% 0.19/0.47 => ~ ( ord_less_nat @ X5 @ T ) ) ).
% 0.19/0.47
% 0.19/0.47 % pinf(5)
% 0.19/0.47 thf(fact_221_pinf_I4_J,axiom,
% 0.19/0.47 ! [T: nat] :
% 0.19/0.47 ? [Z3: nat] :
% 0.19/0.47 ! [X5: nat] :
% 0.19/0.47 ( ( ord_less_nat @ Z3 @ X5 )
% 0.19/0.47 => ( X5 != T ) ) ).
% 0.19/0.47
% 0.19/0.47 % pinf(4)
% 0.19/0.47 thf(fact_222_pinf_I3_J,axiom,
% 0.19/0.47 ! [T: nat] :
% 0.19/0.47 ? [Z3: nat] :
% 0.19/0.47 ! [X5: nat] :
% 0.19/0.47 ( ( ord_less_nat @ Z3 @ X5 )
% 0.19/0.47 => ( X5 != T ) ) ).
% 0.19/0.47
% 0.19/0.47 % pinf(3)
% 0.19/0.47 thf(fact_223_pinf_I2_J,axiom,
% 0.19/0.47 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 0.19/0.47 ( ? [Z4: nat] :
% 0.19/0.47 ! [X2: nat] :
% 0.19/0.47 ( ( ord_less_nat @ Z4 @ X2 )
% 0.19/0.47 => ( ( P @ X2 )
% 0.19/0.47 = ( P4 @ X2 ) ) )
% 0.19/0.47 => ( ? [Z4: nat] :
% 0.19/0.47 ! [X2: nat] :
% 0.19/0.47 ( ( ord_less_nat @ Z4 @ X2 )
% 0.19/0.47 => ( ( Q @ X2 )
% 0.19/0.47 = ( Q2 @ X2 ) ) )
% 0.19/0.47 => ? [Z3: nat] :
% 0.19/0.47 ! [X5: nat] :
% 0.19/0.47 ( ( ord_less_nat @ Z3 @ X5 )
% 0.19/0.47 => ( ( ( P @ X5 )
% 0.19/0.47 | ( Q @ X5 ) )
% 0.19/0.47 = ( ( P4 @ X5 )
% 0.19/0.47 | ( Q2 @ X5 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % pinf(2)
% 0.19/0.47 thf(fact_224_pinf_I1_J,axiom,
% 0.19/0.47 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 0.19/0.47 ( ? [Z4: nat] :
% 0.19/0.47 ! [X2: nat] :
% 0.19/0.47 ( ( ord_less_nat @ Z4 @ X2 )
% 0.19/0.47 => ( ( P @ X2 )
% 0.19/0.47 = ( P4 @ X2 ) ) )
% 0.19/0.47 => ( ? [Z4: nat] :
% 0.19/0.47 ! [X2: nat] :
% 0.19/0.47 ( ( ord_less_nat @ Z4 @ X2 )
% 0.19/0.47 => ( ( Q @ X2 )
% 0.19/0.47 = ( Q2 @ X2 ) ) )
% 0.19/0.47 => ? [Z3: nat] :
% 0.19/0.47 ! [X5: nat] :
% 0.19/0.47 ( ( ord_less_nat @ Z3 @ X5 )
% 0.19/0.47 => ( ( ( P @ X5 )
% 0.19/0.47 & ( Q @ X5 ) )
% 0.19/0.47 = ( ( P4 @ X5 )
% 0.19/0.47 & ( Q2 @ X5 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % pinf(1)
% 0.19/0.47 thf(fact_225_verit__comp__simplify1_I1_J,axiom,
% 0.19/0.47 ! [A: nat] :
% 0.19/0.47 ~ ( ord_less_nat @ A @ A ) ).
% 0.19/0.47
% 0.19/0.47 % verit_comp_simplify1(1)
% 0.19/0.47 thf(fact_226_list__strict__desc__imp__list__desc,axiom,
% 0.19/0.47 ! [Xs: list_nat] :
% 0.19/0.47 ( ( list_strict_desc_nat @ Xs )
% 0.19/0.47 => ( list_desc_nat @ Xs ) ) ).
% 0.19/0.47
% 0.19/0.47 % list_strict_desc_imp_list_desc
% 0.19/0.47 thf(fact_227_minf_I8_J,axiom,
% 0.19/0.47 ! [T: nat] :
% 0.19/0.47 ? [Z3: nat] :
% 0.19/0.47 ! [X5: nat] :
% 0.19/0.47 ( ( ord_less_nat @ X5 @ Z3 )
% 0.19/0.47 => ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% 0.19/0.47
% 0.19/0.47 % minf(8)
% 0.19/0.47 thf(fact_228_minf_I6_J,axiom,
% 0.19/0.47 ! [T: nat] :
% 0.19/0.47 ? [Z3: nat] :
% 0.19/0.47 ! [X5: nat] :
% 0.19/0.47 ( ( ord_less_nat @ X5 @ Z3 )
% 0.19/0.47 => ( ord_less_eq_nat @ X5 @ T ) ) ).
% 0.19/0.47
% 0.19/0.47 % minf(6)
% 0.19/0.47 thf(fact_229_i__append__i__take__eq2,axiom,
% 0.19/0.47 ! [Xs: list_nat,N: nat,F: nat > nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
% 0.19/0.47 => ( ( listIn1299132951ke_nat @ N @ ( listIn923761578nd_nat @ Xs @ F ) )
% 0.19/0.47 = ( append_nat @ Xs @ ( listIn1299132951ke_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ F ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % i_append_i_take_eq2
% 0.19/0.47 thf(fact_230_i__append__i__take__eq2,axiom,
% 0.19/0.47 ! [Xs: list_a,N: nat,F: nat > a] :
% 0.19/0.47 ( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N )
% 0.19/0.47 => ( ( listIn84854903take_a @ N @ ( listIn1312259492pend_a @ Xs @ F ) )
% 0.19/0.47 = ( append_a @ Xs @ ( listIn84854903take_a @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) @ F ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % i_append_i_take_eq2
% 0.19/0.47 thf(fact_231_nth__drop,axiom,
% 0.19/0.47 ! [N: nat,Xs: list_nat,I: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
% 0.19/0.47 => ( ( nth_nat @ ( drop_nat @ N @ Xs ) @ I )
% 0.19/0.47 = ( nth_nat @ Xs @ ( plus_plus_nat @ N @ I ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % nth_drop
% 0.19/0.47 thf(fact_232_nth__drop,axiom,
% 0.19/0.47 ! [N: nat,Xs: list_a,I: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ N @ ( size_size_list_a @ Xs ) )
% 0.19/0.47 => ( ( nth_a @ ( drop_a @ N @ Xs ) @ I )
% 0.19/0.47 = ( nth_a @ Xs @ ( plus_plus_nat @ N @ I ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % nth_drop
% 0.19/0.47 thf(fact_233_add__right__cancel,axiom,
% 0.19/0.47 ! [B: nat,A: nat,C: nat] :
% 0.19/0.47 ( ( ( plus_plus_nat @ B @ A )
% 0.19/0.47 = ( plus_plus_nat @ C @ A ) )
% 0.19/0.47 = ( B = C ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_right_cancel
% 0.19/0.47 thf(fact_234_add__left__cancel,axiom,
% 0.19/0.47 ! [A: nat,B: nat,C: nat] :
% 0.19/0.47 ( ( ( plus_plus_nat @ A @ B )
% 0.19/0.47 = ( plus_plus_nat @ A @ C ) )
% 0.19/0.47 = ( B = C ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_left_cancel
% 0.19/0.47 thf(fact_235_add__le__cancel__right,axiom,
% 0.19/0.47 ! [A: nat,C: nat,B: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.19/0.47 = ( ord_less_eq_nat @ A @ B ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_le_cancel_right
% 0.19/0.47 thf(fact_236_add__le__cancel__left,axiom,
% 0.19/0.47 ! [C: nat,A: nat,B: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.19/0.47 = ( ord_less_eq_nat @ A @ B ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_le_cancel_left
% 0.19/0.47 thf(fact_237_add__less__cancel__left,axiom,
% 0.19/0.47 ! [C: nat,A: nat,B: nat] :
% 0.19/0.47 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.19/0.47 = ( ord_less_nat @ A @ B ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_less_cancel_left
% 0.19/0.47 thf(fact_238_add__less__cancel__right,axiom,
% 0.19/0.47 ! [A: nat,C: nat,B: nat] :
% 0.19/0.47 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.19/0.47 = ( ord_less_nat @ A @ B ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_less_cancel_right
% 0.19/0.47 thf(fact_239_add__diff__cancel__left,axiom,
% 0.19/0.47 ! [C: nat,A: nat,B: nat] :
% 0.19/0.47 ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.19/0.47 = ( minus_minus_nat @ A @ B ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_diff_cancel_left
% 0.19/0.47 thf(fact_240_add__diff__cancel__left_H,axiom,
% 0.19/0.47 ! [A: nat,B: nat] :
% 0.19/0.47 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
% 0.19/0.47 = B ) ).
% 0.19/0.47
% 0.19/0.47 % add_diff_cancel_left'
% 0.19/0.47 thf(fact_241_add__diff__cancel__right,axiom,
% 0.19/0.47 ! [A: nat,C: nat,B: nat] :
% 0.19/0.47 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.19/0.47 = ( minus_minus_nat @ A @ B ) ) ).
% 0.19/0.47
% 0.19/0.47 % add_diff_cancel_right
% 0.19/0.47 thf(fact_242_add__diff__cancel__right_H,axiom,
% 0.19/0.47 ! [A: nat,B: nat] :
% 0.19/0.47 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.19/0.47 = A ) ).
% 0.19/0.47
% 0.19/0.47 % add_diff_cancel_right'
% 0.19/0.47 thf(fact_243_nat__add__left__cancel__less,axiom,
% 0.19/0.47 ! [K: nat,M: nat,N: nat] :
% 0.19/0.47 ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.19/0.47 = ( ord_less_nat @ M @ N ) ) ).
% 0.19/0.47
% 0.19/0.47 % nat_add_left_cancel_less
% 0.19/0.47 thf(fact_244_nat__add__left__cancel__le,axiom,
% 0.19/0.47 ! [K: nat,M: nat,N: nat] :
% 0.19/0.47 ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.19/0.47 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.19/0.47
% 0.19/0.47 % nat_add_left_cancel_le
% 0.19/0.47 thf(fact_245_diff__diff__left,axiom,
% 0.19/0.47 ! [I: nat,J: nat,K: nat] :
% 0.19/0.47 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 0.19/0.47 = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % diff_diff_left
% 0.19/0.47 thf(fact_246_drop__drop,axiom,
% 0.19/0.47 ! [N: nat,M: nat,Xs: list_a] :
% 0.19/0.47 ( ( drop_a @ N @ ( drop_a @ M @ Xs ) )
% 0.19/0.47 = ( drop_a @ ( plus_plus_nat @ N @ M ) @ Xs ) ) ).
% 0.19/0.47
% 0.19/0.47 % drop_drop
% 0.19/0.47
% 0.19/0.47 % Helper facts (5)
% 0.19/0.47 thf(help_If_2_1_If_001tf__a_T,axiom,
% 0.19/0.47 ! [X: a,Y: a] :
% 0.19/0.47 ( ( if_a @ $false @ X @ Y )
% 0.19/0.47 = Y ) ).
% 0.19/0.47
% 0.19/0.47 thf(help_If_1_1_If_001tf__a_T,axiom,
% 0.19/0.47 ! [X: a,Y: a] :
% 0.19/0.47 ( ( if_a @ $true @ X @ Y )
% 0.19/0.47 = X ) ).
% 0.19/0.47
% 0.19/0.47 thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
% 0.19/0.47 ! [P: $o] :
% 0.19/0.47 ( ( P = $true )
% 0.19/0.47 | ( P = $false ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 0.19/0.47 ! [X: nat,Y: nat] :
% 0.19/0.47 ( ( if_nat @ $false @ X @ Y )
% 0.19/0.47 = Y ) ).
% 0.19/0.47
% 0.19/0.47 thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 0.19/0.47 ! [X: nat,Y: nat] :
% 0.19/0.47 ( ( if_nat @ $true @ X @ Y )
% 0.19/0.47 = X ) ).
% 0.19/0.47
% 0.19/0.47 % Conjectures (1)
% 0.19/0.47 thf(conj_0,conjecture,
% 0.19/0.47 ( ( ( ord_less_nat @ n @ ( size_size_list_a @ xs ) )
% 0.19/0.47 => ( ( listIn1417627087drop_a @ n @ ( listIn1312259492pend_a @ xs @ f ) )
% 0.19/0.47 = ( listIn1312259492pend_a @ ( drop_a @ n @ xs ) @ f ) ) )
% 0.19/0.47 & ( ~ ( ord_less_nat @ n @ ( size_size_list_a @ xs ) )
% 0.19/0.47 => ( ( listIn1417627087drop_a @ n @ ( listIn1312259492pend_a @ xs @ f ) )
% 0.19/0.47 = ( listIn1417627087drop_a @ ( minus_minus_nat @ n @ ( size_size_list_a @ xs ) ) @ f ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 %------------------------------------------------------------------------------
% 0.19/0.47 ------------------------
% 2.34/1.70 ---- Embedded in THF ---
% 2.59/1.70 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.59/1.70 %%% Generated on Tue Feb 24 08:26:33 EST 2026
% 2.59/1.70 %%% using '$$dhol' embedding, version 1.3.0.
% 2.59/1.70 %%% Logic specification used:
% 2.59/1.70 %%% thf(spec, logic, $$dhol).
% 2.59/1.70
% 2.59/1.70 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.YUokPb62dV/DTF2DTF25681.p
% See solution above
% 2.59/1.70 ------------------------
% 2.59/1.73 ---- Cleaned THF ---
% 2.59/1.73 thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
% 2.59/1.73 list_nat: $tType ).
% 2.59/1.73
% 2.59/1.73 thf(ty_n_t__List__Olist_Itf__a_J,type,
% 2.59/1.73 list_a: $tType ).
% 2.59/1.73
% 2.59/1.73 thf(ty_n_t__Nat__Onat,type,
% 2.59/1.73 nat: $tType ).
% 2.59/1.73
% 2.59/1.73 thf(ty_n_tf__a,type,
% 2.59/1.73 a: $tType ).
% 2.59/1.73
% 2.59/1.73 thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
% 2.59/1.73 minus_minus_nat: nat > nat > nat ).
% 2.59/1.73
% 2.59/1.73 thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
% 2.59/1.73 plus_plus_nat: nat > nat > nat ).
% 2.59/1.73
% 2.59/1.73 thf(sy_c_If_001t__Nat__Onat,type,
% 2.59/1.73 if_nat: $o > nat > nat > nat ).
% 2.59/1.73
% 2.59/1.73 thf(sy_c_If_001tf__a,type,
% 2.59/1.73 if_a: $o > a > a > a ).
% 2.59/1.73
% 2.59/1.73 thf(sy_c_List2_Olist__asc_001t__Nat__Onat,type,
% 2.59/1.73 list_asc_nat: list_nat > $o ).
% 2.59/1.73
% 2.59/1.73 thf(sy_c_List2_Olist__desc_001t__Nat__Onat,type,
% 2.59/1.73 list_desc_nat: list_nat > $o ).
% 2.59/1.73
% 2.59/1.73 thf(sy_c_List2_Olist__strict__asc_001t__Nat__Onat,type,
% 2.59/1.73 list_strict_asc_nat: list_nat > $o ).
% 2.59/1.73
% 2.59/1.73 thf(sy_c_List2_Olist__strict__desc_001t__Nat__Onat,type,
% 2.59/1.73 list_strict_desc_nat: list_nat > $o ).
% 2.59/1.73
% 2.59/1.73 thf(sy_c_ListInf__Mirabelle__akbajwqfbr_Oi__append_001t__Nat__Onat,type,
% 2.59/1.73 listIn923761578nd_nat: list_nat > ( nat > nat ) > nat > nat ).
% 2.59/1.73
% 2.59/1.73 thf(sy_c_ListInf__Mirabelle__akbajwqfbr_Oi__append_001tf__a,type,
% 2.59/1.73 listIn1312259492pend_a: list_a > ( nat > a ) > nat > a ).
% 2.59/1.73
% 2.59/1.73 thf(sy_c_ListInf__Mirabelle__akbajwqfbr_Oi__drop_001t__Nat__Onat,type,
% 2.59/1.73 listIn870223551op_nat: nat > ( nat > nat ) > nat > nat ).
% 2.59/1.73
% 2.59/1.73 thf(sy_c_ListInf__Mirabelle__akbajwqfbr_Oi__drop_001tf__a,type,
% 2.59/1.73 listIn1417627087drop_a: nat > ( nat > a ) > nat > a ).
% 2.59/1.73
% 2.59/1.73 thf(sy_c_ListInf__Mirabelle__akbajwqfbr_Oi__take_001t__Nat__Onat,type,
% 2.59/1.73 listIn1299132951ke_nat: nat > ( nat > nat ) > list_nat ).
% 2.59/1.73
% 2.59/1.73 thf(sy_c_ListInf__Mirabelle__akbajwqfbr_Oi__take_001tf__a,type,
% 2.59/1.73 listIn84854903take_a: nat > ( nat > a ) > list_a ).
% 2.59/1.73
% 2.59/1.73 thf(sy_c_List_Oappend_001t__Nat__Onat,type,
% 2.59/1.73 append_nat: list_nat > list_nat > list_nat ).
% 2.59/1.73
% 2.59/1.73 thf(sy_c_List_Oappend_001tf__a,type,
% 2.59/1.73 append_a: list_a > list_a > list_a ).
% 2.59/1.73
% 2.59/1.73 thf(sy_c_List_Odrop_001t__Nat__Onat,type,
% 2.59/1.73 drop_nat: nat > list_nat > list_nat ).
% 2.59/1.73
% 2.59/1.73 thf(sy_c_List_Odrop_001tf__a,type,
% 2.59/1.73 drop_a: nat > list_a > list_a ).
% 2.59/1.73
% 2.59/1.73 thf(sy_c_List_Olast_001t__Nat__Onat,type,
% 2.59/1.73 last_nat: list_nat > nat ).
% 2.59/1.73
% 2.59/1.73 thf(sy_c_List_Olast_001tf__a,type,
% 2.59/1.73 last_a: list_a > a ).
% 2.59/1.73
% 2.59/1.73 thf(sy_c_List_Onth_001t__Nat__Onat,type,
% 2.59/1.73 nth_nat: list_nat > nat > nat ).
% 2.59/1.73
% 2.59/1.73 thf(sy_c_List_Onth_001tf__a,type,
% 2.59/1.73 nth_a: list_a > nat > a ).
% 2.59/1.73
% 2.59/1.73 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
% 2.59/1.73 size_size_list_nat: list_nat > nat ).
% 2.59/1.73
% 2.59/1.73 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
% 2.59/1.73 size_size_list_a: list_a > nat ).
% 2.59/1.73
% 2.59/1.73 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 2.59/1.73 ord_less_nat: nat > nat > $o ).
% 2.59/1.73
% 2.59/1.73 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 2.59/1.73 ord_less_eq_nat: nat > nat > $o ).
% 2.59/1.73
% 2.59/1.73 thf(sy_v_f,type,
% 2.59/1.73 f: nat > a ).
% 2.59/1.73
% 2.59/1.73 thf(sy_v_n,type,
% 2.59/1.73 n: nat ).
% 2.59/1.73
% 2.59/1.73 thf(sy_v_xs,type,
% 2.59/1.73 xs: list_a ).
% 2.59/1.73
% 2.59/1.73 thf(fact_0_same__i__append__eq,axiom,
% 2.59/1.73 ! [Xs: list_a,F: nat > a,G: nat > a] :
% 2.59/1.73 ( ( ( listIn1312259492pend_a @ Xs @ F )
% 2.59/1.73 = ( listIn1312259492pend_a @ Xs @ G ) )
% 2.59/1.73 = ( F = G ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_1_i__append__eq__i__append__conv,axiom,
% 2.59/1.73 ! [Xs: list_nat,Ys: list_nat,F: nat > nat,G: nat > nat] :
% 2.59/1.73 ( ( ( size_size_list_nat @ Xs )
% 2.59/1.73 = ( size_size_list_nat @ Ys ) )
% 2.59/1.73 => ( ( ( listIn923761578nd_nat @ Xs @ F )
% 2.59/1.73 = ( listIn923761578nd_nat @ Ys @ G ) )
% 2.59/1.73 = ( ( Xs = Ys )
% 2.59/1.73 & ( F = G ) ) ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_2_i__append__eq__i__append__conv,axiom,
% 2.59/1.73 ! [Xs: list_a,Ys: list_a,F: nat > a,G: nat > a] :
% 2.59/1.73 ( ( ( size_size_list_a @ Xs )
% 2.59/1.73 = ( size_size_list_a @ Ys ) )
% 2.59/1.73 => ( ( ( listIn1312259492pend_a @ Xs @ F )
% 2.59/1.73 = ( listIn1312259492pend_a @ Ys @ G ) )
% 2.59/1.73 = ( ( Xs = Ys )
% 2.59/1.73 & ( F = G ) ) ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_3_length__drop,axiom,
% 2.59/1.73 ! [N: nat,Xs: list_nat] :
% 2.59/1.73 ( ( size_size_list_nat @ ( drop_nat @ N @ Xs ) )
% 2.59/1.73 = ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_4_length__drop,axiom,
% 2.59/1.73 ! [N: nat,Xs: list_a] :
% 2.59/1.73 ( ( size_size_list_a @ ( drop_a @ N @ Xs ) )
% 2.59/1.73 = ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ N ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_5_diff__less__mono2,axiom,
% 2.59/1.73 ! [M: nat,N: nat,L: nat] :
% 2.59/1.73 ( ( ord_less_nat @ M @ N )
% 2.59/1.73 => ( ( ord_less_nat @ M @ L )
% 2.59/1.73 => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_6_less__imp__diff__less,axiom,
% 2.59/1.73 ! [J: nat,K: nat,N: nat] :
% 2.59/1.73 ( ( ord_less_nat @ J @ K )
% 2.59/1.73 => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_7_less__diff__imp__less,axiom,
% 2.59/1.73 ! [I: nat,J: nat,M: nat] :
% 2.59/1.73 ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ M ) )
% 2.59/1.73 => ( ord_less_nat @ I @ J ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_8_nat__diff__left__cancel__eq1,axiom,
% 2.59/1.73 ! [K: nat,M: nat,N: nat] :
% 2.59/1.73 ( ( ( minus_minus_nat @ K @ M )
% 2.59/1.73 = ( minus_minus_nat @ K @ N ) )
% 2.59/1.73 => ( ( ord_less_nat @ M @ K )
% 2.59/1.73 => ( M = N ) ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_9_nat__diff__left__cancel__eq2,axiom,
% 2.59/1.73 ! [K: nat,M: nat,N: nat] :
% 2.59/1.73 ( ( ( minus_minus_nat @ K @ M )
% 2.59/1.73 = ( minus_minus_nat @ K @ N ) )
% 2.59/1.73 => ( ( ord_less_nat @ N @ K )
% 2.59/1.73 => ( M = N ) ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_10_nat__diff__left__cancel__less,axiom,
% 2.59/1.73 ! [K: nat,M: nat,N: nat] :
% 2.59/1.73 ( ( ord_less_nat @ ( minus_minus_nat @ K @ M ) @ ( minus_minus_nat @ K @ N ) )
% 2.59/1.73 => ( ord_less_nat @ N @ M ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_11_nat__diff__right__cancel__eq1,axiom,
% 2.59/1.73 ! [M: nat,K: nat,N: nat] :
% 2.59/1.73 ( ( ( minus_minus_nat @ M @ K )
% 2.59/1.73 = ( minus_minus_nat @ N @ K ) )
% 2.59/1.73 => ( ( ord_less_nat @ K @ M )
% 2.59/1.73 => ( M = N ) ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_12_nat__diff__right__cancel__eq2,axiom,
% 2.59/1.73 ! [M: nat,K: nat,N: nat] :
% 2.59/1.73 ( ( ( minus_minus_nat @ M @ K )
% 2.59/1.73 = ( minus_minus_nat @ N @ K ) )
% 2.59/1.73 => ( ( ord_less_nat @ K @ N )
% 2.59/1.73 => ( M = N ) ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_13_nat__diff__right__cancel__less,axiom,
% 2.59/1.73 ! [N: nat,K: nat,M: nat] :
% 2.59/1.73 ( ( ord_less_nat @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M @ K ) )
% 2.59/1.73 => ( ord_less_nat @ N @ M ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_14_length__induct,axiom,
% 2.59/1.73 ! [P: list_nat > $o,Xs: list_nat] :
% 2.59/1.73 ( ! [Xs2: list_nat] :
% 2.59/1.73 ( ! [Ys2: list_nat] :
% 2.59/1.73 ( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs2 ) )
% 2.59/1.73 => ( P @ Ys2 ) )
% 2.59/1.73 => ( P @ Xs2 ) )
% 2.59/1.73 => ( P @ Xs ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_15_length__induct,axiom,
% 2.59/1.73 ! [P: list_a > $o,Xs: list_a] :
% 2.59/1.73 ( ! [Xs2: list_a] :
% 2.59/1.73 ( ! [Ys2: list_a] :
% 2.59/1.73 ( ( ord_less_nat @ ( size_size_list_a @ Ys2 ) @ ( size_size_list_a @ Xs2 ) )
% 2.59/1.73 => ( P @ Ys2 ) )
% 2.59/1.73 => ( P @ Xs2 ) )
% 2.59/1.73 => ( P @ Xs ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_16_linorder__neqE__nat,axiom,
% 2.59/1.73 ! [X: nat,Y: nat] :
% 2.59/1.73 ( ( X != Y )
% 2.59/1.73 => ( ~ ( ord_less_nat @ X @ Y )
% 2.59/1.73 => ( ord_less_nat @ Y @ X ) ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_17_infinite__descent,axiom,
% 2.59/1.73 ! [P: nat > $o,N: nat] :
% 2.59/1.73 ( ! [N2: nat] :
% 2.59/1.73 ( ~ ( P @ N2 )
% 2.59/1.73 => ? [M2: nat] :
% 2.59/1.73 ( ( ord_less_nat @ M2 @ N2 )
% 2.59/1.73 & ~ ( P @ M2 ) ) )
% 2.59/1.73 => ( P @ N ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_18_nat__less__induct,axiom,
% 2.59/1.73 ! [P: nat > $o,N: nat] :
% 2.59/1.73 ( ! [N2: nat] :
% 2.59/1.73 ( ! [M2: nat] :
% 2.59/1.73 ( ( ord_less_nat @ M2 @ N2 )
% 2.59/1.73 => ( P @ M2 ) )
% 2.59/1.73 => ( P @ N2 ) )
% 2.59/1.73 => ( P @ N ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_19_less__irrefl__nat,axiom,
% 2.59/1.73 ! [N: nat] :
% 2.59/1.73 ~ ( ord_less_nat @ N @ N ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_20_less__not__refl3,axiom,
% 2.59/1.73 ! [S: nat,T: nat] :
% 2.59/1.73 ( ( ord_less_nat @ S @ T )
% 2.59/1.73 => ( S != T ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_21_less__not__refl2,axiom,
% 2.59/1.73 ! [N: nat,M: nat] :
% 2.59/1.73 ( ( ord_less_nat @ N @ M )
% 2.59/1.73 => ( M != N ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_22_less__not__refl,axiom,
% 2.59/1.73 ! [N: nat] :
% 2.59/1.73 ~ ( ord_less_nat @ N @ N ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_23_nat__neq__iff,axiom,
% 2.59/1.73 ! [M: nat,N: nat] :
% 2.59/1.73 ( ( M != N )
% 2.59/1.73 = ( ( ord_less_nat @ M @ N )
% 2.59/1.73 | ( ord_less_nat @ N @ M ) ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_24_size__neq__size__imp__neq,axiom,
% 2.59/1.73 ! [X: list_nat,Y: list_nat] :
% 2.59/1.73 ( ( ( size_size_list_nat @ X )
% 2.59/1.73 != ( size_size_list_nat @ Y ) )
% 2.59/1.73 => ( X != Y ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_25_size__neq__size__imp__neq,axiom,
% 2.59/1.73 ! [X: list_a,Y: list_a] :
% 2.59/1.73 ( ( ( size_size_list_a @ X )
% 2.59/1.73 != ( size_size_list_a @ Y ) )
% 2.59/1.73 => ( X != Y ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_26_neq__if__length__neq,axiom,
% 2.59/1.73 ! [Xs: list_nat,Ys: list_nat] :
% 2.59/1.73 ( ( ( size_size_list_nat @ Xs )
% 2.59/1.73 != ( size_size_list_nat @ Ys ) )
% 2.59/1.73 => ( Xs != Ys ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_27_neq__if__length__neq,axiom,
% 2.59/1.73 ! [Xs: list_a,Ys: list_a] :
% 2.59/1.73 ( ( ( size_size_list_a @ Xs )
% 2.59/1.73 != ( size_size_list_a @ Ys ) )
% 2.59/1.73 => ( Xs != Ys ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_28_Ex__list__of__length,axiom,
% 2.59/1.73 ! [N: nat] :
% 2.59/1.73 ? [Xs2: list_nat] :
% 2.59/1.73 ( ( size_size_list_nat @ Xs2 )
% 2.59/1.73 = N ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_29_Ex__list__of__length,axiom,
% 2.59/1.73 ! [N: nat] :
% 2.59/1.73 ? [Xs2: list_a] :
% 2.59/1.73 ( ( size_size_list_a @ Xs2 )
% 2.59/1.73 = N ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_30_eq__imp__diff__eq,axiom,
% 2.59/1.73 ! [M: nat,N: nat,K: nat] :
% 2.59/1.73 ( ( M = N )
% 2.59/1.73 => ( ( minus_minus_nat @ M @ K )
% 2.59/1.73 = ( minus_minus_nat @ N @ K ) ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_31_diff__commute,axiom,
% 2.59/1.73 ! [I: nat,J: nat,K: nat] :
% 2.59/1.73 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 2.59/1.73 = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_32_i__append__i__drop__eq1,axiom,
% 2.59/1.73 ! [N: nat,Xs: list_nat,F: nat > nat] :
% 2.59/1.73 ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
% 2.59/1.73 => ( ( listIn870223551op_nat @ N @ ( listIn923761578nd_nat @ Xs @ F ) )
% 2.59/1.73 = ( listIn923761578nd_nat @ ( drop_nat @ N @ Xs ) @ F ) ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_33_i__append__i__drop__eq1,axiom,
% 2.59/1.73 ! [N: nat,Xs: list_a,F: nat > a] :
% 2.59/1.73 ( ( ord_less_eq_nat @ N @ ( size_size_list_a @ Xs ) )
% 2.59/1.73 => ( ( listIn1417627087drop_a @ N @ ( listIn1312259492pend_a @ Xs @ F ) )
% 2.59/1.73 = ( listIn1312259492pend_a @ ( drop_a @ N @ Xs ) @ F ) ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_34_i__append__i__drop__eq2,axiom,
% 2.59/1.73 ! [Xs: list_nat,N: nat,F: nat > nat] :
% 2.59/1.73 ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
% 2.59/1.73 => ( ( listIn870223551op_nat @ N @ ( listIn923761578nd_nat @ Xs @ F ) )
% 2.59/1.73 = ( listIn870223551op_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ F ) ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_35_i__append__i__drop__eq2,axiom,
% 2.59/1.73 ! [Xs: list_a,N: nat,F: nat > a] :
% 2.59/1.73 ( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N )
% 2.59/1.73 => ( ( listIn1417627087drop_a @ N @ ( listIn1312259492pend_a @ Xs @ F ) )
% 2.59/1.73 = ( listIn1417627087drop_a @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) @ F ) ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_36_last__drop,axiom,
% 2.59/1.73 ! [N: nat,Xs: list_nat] :
% 2.59/1.73 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 2.59/1.73 => ( ( last_nat @ ( drop_nat @ N @ Xs ) )
% 2.59/1.73 = ( last_nat @ Xs ) ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_37_last__drop,axiom,
% 2.59/1.73 ! [N: nat,Xs: list_a] :
% 2.59/1.73 ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
% 2.59/1.73 => ( ( last_a @ ( drop_a @ N @ Xs ) )
% 2.59/1.73 = ( last_a @ Xs ) ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_38_i__append__nth2,axiom,
% 2.59/1.73 ! [Xs: list_nat,N: nat,F: nat > nat] :
% 2.59/1.73 ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
% 2.59/1.73 => ( ( listIn923761578nd_nat @ Xs @ F @ N )
% 2.59/1.73 = ( F @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_39_i__append__nth2,axiom,
% 2.59/1.73 ! [Xs: list_a,N: nat,F: nat > a] :
% 2.59/1.73 ( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N )
% 2.59/1.73 => ( ( listIn1312259492pend_a @ Xs @ F @ N )
% 2.59/1.73 = ( F @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) ) ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_40_i__append__nth,axiom,
% 2.59/1.73 ( listIn923761578nd_nat
% 2.59/1.73 = ( ^ [Xs3: list_nat,F2: nat > nat,N3: nat] : ( if_nat @ ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs3 ) ) @ ( nth_nat @ Xs3 @ N3 ) @ ( F2 @ ( minus_minus_nat @ N3 @ ( size_size_list_nat @ Xs3 ) ) ) ) ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_41_i__append__nth,axiom,
% 2.59/1.73 ( listIn1312259492pend_a
% 2.59/1.73 = ( ^ [Xs3: list_a,F2: nat > a,N3: nat] : ( if_a @ ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs3 ) ) @ ( nth_a @ Xs3 @ N3 ) @ ( F2 @ ( minus_minus_nat @ N3 @ ( size_size_list_a @ Xs3 ) ) ) ) ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_42_i__append__def,axiom,
% 2.59/1.73 ( listIn923761578nd_nat
% 2.59/1.73 = ( ^ [Xs3: list_nat,F2: nat > nat,N3: nat] : ( if_nat @ ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs3 ) ) @ ( nth_nat @ Xs3 @ N3 ) @ ( F2 @ ( minus_minus_nat @ N3 @ ( size_size_list_nat @ Xs3 ) ) ) ) ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_43_i__append__def,axiom,
% 2.59/1.73 ( listIn1312259492pend_a
% 2.59/1.73 = ( ^ [Xs3: list_a,F2: nat > a,N3: nat] : ( if_a @ ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs3 ) ) @ ( nth_a @ Xs3 @ N3 ) @ ( F2 @ ( minus_minus_nat @ N3 @ ( size_size_list_a @ Xs3 ) ) ) ) ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_44_drop__append,axiom,
% 2.59/1.73 ! [N: nat,Xs: list_nat,Ys: list_nat] :
% 2.59/1.73 ( ( drop_nat @ N @ ( append_nat @ Xs @ Ys ) )
% 2.59/1.73 = ( append_nat @ ( drop_nat @ N @ Xs ) @ ( drop_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys ) ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_45_drop__append,axiom,
% 2.59/1.73 ! [N: nat,Xs: list_a,Ys: list_a] :
% 2.59/1.73 ( ( drop_a @ N @ ( append_a @ Xs @ Ys ) )
% 2.59/1.73 = ( append_a @ ( drop_a @ N @ Xs ) @ ( drop_a @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) @ Ys ) ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_46_append_Oassoc,axiom,
% 2.59/1.73 ! [A: list_a,B: list_a,C: list_a] :
% 2.59/1.73 ( ( append_a @ ( append_a @ A @ B ) @ C )
% 2.59/1.73 = ( append_a @ A @ ( append_a @ B @ C ) ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_47_append__assoc,axiom,
% 2.59/1.73 ! [Xs: list_a,Ys: list_a,Zs: list_a] :
% 2.59/1.73 ( ( append_a @ ( append_a @ Xs @ Ys ) @ Zs )
% 2.59/1.73 = ( append_a @ Xs @ ( append_a @ Ys @ Zs ) ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_48_append__same__eq,axiom,
% 2.59/1.73 ! [Ys: list_a,Xs: list_a,Zs: list_a] :
% 2.59/1.73 ( ( ( append_a @ Ys @ Xs )
% 2.59/1.73 = ( append_a @ Zs @ Xs ) )
% 2.59/1.73 = ( Ys = Zs ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_49_same__append__eq,axiom,
% 2.59/1.73 ! [Xs: list_a,Ys: list_a,Zs: list_a] :
% 2.59/1.73 ( ( ( append_a @ Xs @ Ys )
% 2.59/1.73 = ( append_a @ Xs @ Zs ) )
% 2.59/1.73 = ( Ys = Zs ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_50_diff__diff__cancel,axiom,
% 2.59/1.73 ! [I: nat,N: nat] :
% 2.59/1.73 ( ( ord_less_eq_nat @ I @ N )
% 2.59/1.73 => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
% 2.59/1.73 = I ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_51_append__eq__append__conv,axiom,
% 2.59/1.73 ! [Xs: list_nat,Ys: list_nat,Us: list_nat,Vs: list_nat] :
% 2.59/1.73 ( ( ( ( size_size_list_nat @ Xs )
% 2.59/1.73 = ( size_size_list_nat @ Ys ) )
% 2.59/1.73 | ( ( size_size_list_nat @ Us )
% 2.59/1.73 = ( size_size_list_nat @ Vs ) ) )
% 2.59/1.73 => ( ( ( append_nat @ Xs @ Us )
% 2.59/1.73 = ( append_nat @ Ys @ Vs ) )
% 2.59/1.73 = ( ( Xs = Ys )
% 2.59/1.73 & ( Us = Vs ) ) ) ) ).
% 2.59/1.73
% 2.59/1.73 thf(fact_52_append__eq__append__conv,axiom,
% 2.59/1.73 ! [Xs: list_a,Ys: list_a,Us: list_a,Vs: list_a] :
% 2.59/1.73 ( ( ( ( size_size_list_a @ Xs )
% 2.59/1.73 = ( size_size_list_a @ Ys ) )
% 2.59/1.73 | ( ( size_size_list_a @ Us )
% 2.59/1.73 = ( size_size_list_a @ Vs ) ) )
% 2.59/1.74 => ( ( ( append_a @ Xs @ Us )
% 2.59/1.74 = ( append_a @ Ys @ Vs ) )
% 2.59/1.74 = ( ( Xs = Ys )
% 2.59/1.74 & ( Us = Vs ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_53_i__append__assoc,axiom,
% 2.59/1.74 ! [Xs: list_a,Ys: list_a,F: nat > a] :
% 2.59/1.74 ( ( listIn1312259492pend_a @ Xs @ ( listIn1312259492pend_a @ Ys @ F ) )
% 2.59/1.74 = ( listIn1312259492pend_a @ ( append_a @ Xs @ Ys ) @ F ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_54_i__append__nth1,axiom,
% 2.59/1.74 ! [N: nat,Xs: list_nat,F: nat > nat] :
% 2.59/1.74 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 2.59/1.74 => ( ( listIn923761578nd_nat @ Xs @ F @ N )
% 2.59/1.74 = ( nth_nat @ Xs @ N ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_55_i__append__nth1,axiom,
% 2.59/1.74 ! [N: nat,Xs: list_a,F: nat > a] :
% 2.59/1.74 ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
% 2.59/1.74 => ( ( listIn1312259492pend_a @ Xs @ F @ N )
% 2.59/1.74 = ( nth_a @ Xs @ N ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_56_le__refl,axiom,
% 2.59/1.74 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_57_le__trans,axiom,
% 2.59/1.74 ! [I: nat,J: nat,K: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ I @ J )
% 2.59/1.74 => ( ( ord_less_eq_nat @ J @ K )
% 2.59/1.74 => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_58_eq__imp__le,axiom,
% 2.59/1.74 ! [M: nat,N: nat] :
% 2.59/1.74 ( ( M = N )
% 2.59/1.74 => ( ord_less_eq_nat @ M @ N ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_59_le__antisym,axiom,
% 2.59/1.74 ! [M: nat,N: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ M @ N )
% 2.59/1.74 => ( ( ord_less_eq_nat @ N @ M )
% 2.59/1.74 => ( M = N ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_60_nat__le__linear,axiom,
% 2.59/1.74 ! [M: nat,N: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ M @ N )
% 2.59/1.74 | ( ord_less_eq_nat @ N @ M ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_61_append__eq__appendI,axiom,
% 2.59/1.74 ! [Xs: list_a,Xs1: list_a,Zs: list_a,Ys: list_a,Us: list_a] :
% 2.59/1.74 ( ( ( append_a @ Xs @ Xs1 )
% 2.59/1.74 = Zs )
% 2.59/1.74 => ( ( Ys
% 2.59/1.74 = ( append_a @ Xs1 @ Us ) )
% 2.59/1.74 => ( ( append_a @ Xs @ Ys )
% 2.59/1.74 = ( append_a @ Zs @ Us ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_62_Nat_Oex__has__greatest__nat,axiom,
% 2.59/1.74 ! [P: nat > $o,K: nat,B: nat] :
% 2.59/1.74 ( ( P @ K )
% 2.59/1.74 => ( ! [Y2: nat] :
% 2.59/1.74 ( ( P @ Y2 )
% 2.59/1.74 => ( ord_less_eq_nat @ Y2 @ B ) )
% 2.59/1.74 => ? [X2: nat] :
% 2.59/1.74 ( ( P @ X2 )
% 2.59/1.74 & ! [Y3: nat] :
% 2.59/1.74 ( ( P @ Y3 )
% 2.59/1.74 => ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_63_append__eq__append__conv2,axiom,
% 2.59/1.74 ! [Xs: list_a,Ys: list_a,Zs: list_a,Ts: list_a] :
% 2.59/1.74 ( ( ( append_a @ Xs @ Ys )
% 2.59/1.74 = ( append_a @ Zs @ Ts ) )
% 2.59/1.74 = ( ? [Us2: list_a] :
% 2.59/1.74 ( ( ( Xs
% 2.59/1.74 = ( append_a @ Zs @ Us2 ) )
% 2.59/1.74 & ( ( append_a @ Us2 @ Ys )
% 2.59/1.74 = Ts ) )
% 2.59/1.74 | ( ( ( append_a @ Xs @ Us2 )
% 2.59/1.74 = Zs )
% 2.59/1.74 & ( Ys
% 2.59/1.74 = ( append_a @ Us2 @ Ts ) ) ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_64_i__append__eq__i__append__conv2__aux,axiom,
% 2.59/1.74 ! [Xs: list_nat,F: nat > nat,Ys: list_nat,G: nat > nat] :
% 2.59/1.74 ( ( ( listIn923761578nd_nat @ Xs @ F )
% 2.59/1.74 = ( listIn923761578nd_nat @ Ys @ G ) )
% 2.59/1.74 => ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) )
% 2.59/1.74 => ? [Zs2: list_nat] :
% 2.59/1.74 ( ( ( append_nat @ Xs @ Zs2 )
% 2.59/1.74 = Ys )
% 2.59/1.74 & ( F
% 2.59/1.74 = ( listIn923761578nd_nat @ Zs2 @ G ) ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_65_i__append__eq__i__append__conv2__aux,axiom,
% 2.59/1.74 ! [Xs: list_a,F: nat > a,Ys: list_a,G: nat > a] :
% 2.59/1.74 ( ( ( listIn1312259492pend_a @ Xs @ F )
% 2.59/1.74 = ( listIn1312259492pend_a @ Ys @ G ) )
% 2.59/1.74 => ( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) )
% 2.59/1.74 => ? [Zs2: list_a] :
% 2.59/1.74 ( ( ( append_a @ Xs @ Zs2 )
% 2.59/1.74 = Ys )
% 2.59/1.74 & ( F
% 2.59/1.74 = ( listIn1312259492pend_a @ Zs2 @ G ) ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_66_ge__less__neq__conv,axiom,
% 2.59/1.74 ( ord_less_eq_nat
% 2.59/1.74 = ( ^ [A2: nat,N3: nat] :
% 2.59/1.74 ! [X3: nat] :
% 2.59/1.74 ( ( ord_less_nat @ X3 @ A2 )
% 2.59/1.74 => ( N3 != X3 ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_67_less__ge__neq__conv,axiom,
% 2.59/1.74 ( ord_less_nat
% 2.59/1.74 = ( ^ [N3: nat,A2: nat] :
% 2.59/1.74 ! [X3: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ A2 @ X3 )
% 2.59/1.74 => ( N3 != X3 ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_68_greater__le__neq__conv,axiom,
% 2.59/1.74 ( ord_less_nat
% 2.59/1.74 = ( ^ [A2: nat,N3: nat] :
% 2.59/1.74 ! [X3: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ X3 @ A2 )
% 2.59/1.74 => ( N3 != X3 ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_69_le__greater__neq__conv,axiom,
% 2.59/1.74 ( ord_less_eq_nat
% 2.59/1.74 = ( ^ [N3: nat,A2: nat] :
% 2.59/1.74 ! [X3: nat] :
% 2.59/1.74 ( ( ord_less_nat @ A2 @ X3 )
% 2.59/1.74 => ( N3 != X3 ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_70_nat__less__le,axiom,
% 2.59/1.74 ( ord_less_nat
% 2.59/1.74 = ( ^ [M3: nat,N3: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ M3 @ N3 )
% 2.59/1.74 & ( M3 != N3 ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_71_less__imp__le__nat,axiom,
% 2.59/1.74 ! [M: nat,N: nat] :
% 2.59/1.74 ( ( ord_less_nat @ M @ N )
% 2.59/1.74 => ( ord_less_eq_nat @ M @ N ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_72_le__eq__less__or__eq,axiom,
% 2.59/1.74 ( ord_less_eq_nat
% 2.59/1.74 = ( ^ [M3: nat,N3: nat] :
% 2.59/1.74 ( ( ord_less_nat @ M3 @ N3 )
% 2.59/1.74 | ( M3 = N3 ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_73_less__or__eq__imp__le,axiom,
% 2.59/1.74 ! [M: nat,N: nat] :
% 2.59/1.74 ( ( ( ord_less_nat @ M @ N )
% 2.59/1.74 | ( M = N ) )
% 2.59/1.74 => ( ord_less_eq_nat @ M @ N ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_74_le__neq__implies__less,axiom,
% 2.59/1.74 ! [M: nat,N: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ M @ N )
% 2.59/1.74 => ( ( M != N )
% 2.59/1.74 => ( ord_less_nat @ M @ N ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_75_less__mono__imp__le__mono,axiom,
% 2.59/1.74 ! [F: nat > nat,I: nat,J: nat] :
% 2.59/1.74 ( ! [I2: nat,J2: nat] :
% 2.59/1.74 ( ( ord_less_nat @ I2 @ J2 )
% 2.59/1.74 => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 2.59/1.74 => ( ( ord_less_eq_nat @ I @ J )
% 2.59/1.74 => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_76_eq__diff__iff,axiom,
% 2.59/1.74 ! [K: nat,M: nat,N: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ K @ M )
% 2.59/1.74 => ( ( ord_less_eq_nat @ K @ N )
% 2.59/1.74 => ( ( ( minus_minus_nat @ M @ K )
% 2.59/1.74 = ( minus_minus_nat @ N @ K ) )
% 2.59/1.74 = ( M = N ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_77_le__diff__iff,axiom,
% 2.59/1.74 ! [K: nat,M: nat,N: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ K @ M )
% 2.59/1.74 => ( ( ord_less_eq_nat @ K @ N )
% 2.59/1.74 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 2.59/1.74 = ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_78_Nat_Odiff__diff__eq,axiom,
% 2.59/1.74 ! [K: nat,M: nat,N: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ K @ M )
% 2.59/1.74 => ( ( ord_less_eq_nat @ K @ N )
% 2.59/1.74 => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 2.59/1.74 = ( minus_minus_nat @ M @ N ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_79_diff__le__mono,axiom,
% 2.59/1.74 ! [M: nat,N: nat,L: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ M @ N )
% 2.59/1.74 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_80_diff__le__self,axiom,
% 2.59/1.74 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_81_le__diff__iff_H,axiom,
% 2.59/1.74 ! [A: nat,C: nat,B: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ A @ C )
% 2.59/1.74 => ( ( ord_less_eq_nat @ B @ C )
% 2.59/1.74 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
% 2.59/1.74 = ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_82_diff__le__mono2,axiom,
% 2.59/1.74 ! [M: nat,N: nat,L: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ M @ N )
% 2.59/1.74 => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_83_le__diff__swap,axiom,
% 2.59/1.74 ! [I: nat,K: nat,J: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ I @ K )
% 2.59/1.74 => ( ( ord_less_eq_nat @ J @ K )
% 2.59/1.74 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ K @ J ) @ I )
% 2.59/1.74 = ( ord_less_eq_nat @ ( minus_minus_nat @ K @ I ) @ J ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_84_le__diff__imp__le,axiom,
% 2.59/1.74 ! [I: nat,J: nat,M: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ M ) )
% 2.59/1.74 => ( ord_less_eq_nat @ I @ J ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_85_le__imp__diff__le,axiom,
% 2.59/1.74 ! [J: nat,K: nat,N: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ J @ K )
% 2.59/1.74 => ( ord_less_eq_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_86_eq__diff__left__iff,axiom,
% 2.59/1.74 ! [M: nat,K: nat,N: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ M @ K )
% 2.59/1.74 => ( ( ord_less_eq_nat @ N @ K )
% 2.59/1.74 => ( ( ( minus_minus_nat @ K @ M )
% 2.59/1.74 = ( minus_minus_nat @ K @ N ) )
% 2.59/1.74 = ( M = N ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_87_le__diff__le__imp__le,axiom,
% 2.59/1.74 ! [I: nat,J: nat,M: nat,N: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ M ) )
% 2.59/1.74 => ( ( ord_less_eq_nat @ N @ M )
% 2.59/1.74 => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ N ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_88_nat__diff__left__cancel__le2,axiom,
% 2.59/1.74 ! [K: nat,M: nat,N: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ ( minus_minus_nat @ K @ M ) @ ( minus_minus_nat @ K @ N ) )
% 2.59/1.74 => ( ( ord_less_eq_nat @ N @ K )
% 2.59/1.74 => ( ord_less_eq_nat @ N @ M ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_89_nat__diff__right__cancel__le2,axiom,
% 2.59/1.74 ! [M: nat,K: nat,N: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 2.59/1.74 => ( ( ord_less_eq_nat @ K @ N )
% 2.59/1.74 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_90_i__append__eq__i__appendI,axiom,
% 2.59/1.74 ! [Xs: list_a,Xs4: list_a,Ys: list_a,F: nat > a,G: nat > a] :
% 2.59/1.74 ( ( ( append_a @ Xs @ Xs4 )
% 2.59/1.74 = Ys )
% 2.59/1.74 => ( ( F
% 2.59/1.74 = ( listIn1312259492pend_a @ Xs4 @ G ) )
% 2.59/1.74 => ( ( listIn1312259492pend_a @ Xs @ F )
% 2.59/1.74 = ( listIn1312259492pend_a @ Ys @ G ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_91_i__append__eq__i__append__conv2,axiom,
% 2.59/1.74 ! [Xs: list_a,F: nat > a,Ys: list_a,G: nat > a] :
% 2.59/1.74 ( ( ( listIn1312259492pend_a @ Xs @ F )
% 2.59/1.74 = ( listIn1312259492pend_a @ Ys @ G ) )
% 2.59/1.74 = ( ? [Zs3: list_a] :
% 2.59/1.74 ( ( ( Xs
% 2.59/1.74 = ( append_a @ Ys @ Zs3 ) )
% 2.59/1.74 & ( ( listIn1312259492pend_a @ Zs3 @ F )
% 2.59/1.74 = G ) )
% 2.59/1.74 | ( ( ( append_a @ Xs @ Zs3 )
% 2.59/1.74 = Ys )
% 2.59/1.74 & ( F
% 2.59/1.74 = ( listIn1312259492pend_a @ Zs3 @ G ) ) ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_92_nth__append,axiom,
% 2.59/1.74 ! [N: nat,Xs: list_nat,Ys: list_nat] :
% 2.59/1.74 ( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 2.59/1.74 => ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
% 2.59/1.74 = ( nth_nat @ Xs @ N ) ) )
% 2.59/1.74 & ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 2.59/1.74 => ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
% 2.59/1.74 = ( nth_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_93_nth__append,axiom,
% 2.59/1.74 ! [N: nat,Xs: list_a,Ys: list_a] :
% 2.59/1.74 ( ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
% 2.59/1.74 => ( ( nth_a @ ( append_a @ Xs @ Ys ) @ N )
% 2.59/1.74 = ( nth_a @ Xs @ N ) ) )
% 2.59/1.74 & ( ~ ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
% 2.59/1.74 => ( ( nth_a @ ( append_a @ Xs @ Ys ) @ N )
% 2.59/1.74 = ( nth_a @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_94_nth__equalityI,axiom,
% 2.59/1.74 ! [Xs: list_nat,Ys: list_nat] :
% 2.59/1.74 ( ( ( size_size_list_nat @ Xs )
% 2.59/1.74 = ( size_size_list_nat @ Ys ) )
% 2.59/1.74 => ( ! [I2: nat] :
% 2.59/1.74 ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
% 2.59/1.74 => ( ( nth_nat @ Xs @ I2 )
% 2.59/1.74 = ( nth_nat @ Ys @ I2 ) ) )
% 2.59/1.74 => ( Xs = Ys ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_95_nth__equalityI,axiom,
% 2.59/1.74 ! [Xs: list_a,Ys: list_a] :
% 2.59/1.74 ( ( ( size_size_list_a @ Xs )
% 2.59/1.74 = ( size_size_list_a @ Ys ) )
% 2.59/1.74 => ( ! [I2: nat] :
% 2.59/1.74 ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
% 2.59/1.74 => ( ( nth_a @ Xs @ I2 )
% 2.59/1.74 = ( nth_a @ Ys @ I2 ) ) )
% 2.59/1.74 => ( Xs = Ys ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_96_Skolem__list__nth,axiom,
% 2.59/1.74 ! [K: nat,P: nat > nat > $o] :
% 2.59/1.74 ( ( ! [I3: nat] :
% 2.59/1.74 ( ( ord_less_nat @ I3 @ K )
% 2.59/1.74 => ? [X4: nat] : ( P @ I3 @ X4 ) ) )
% 2.59/1.74 = ( ? [Xs3: list_nat] :
% 2.59/1.74 ( ( ( size_size_list_nat @ Xs3 )
% 2.59/1.74 = K )
% 2.59/1.74 & ! [I3: nat] :
% 2.59/1.74 ( ( ord_less_nat @ I3 @ K )
% 2.59/1.74 => ( P @ I3 @ ( nth_nat @ Xs3 @ I3 ) ) ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_97_Skolem__list__nth,axiom,
% 2.59/1.74 ! [K: nat,P: nat > a > $o] :
% 2.59/1.74 ( ( ! [I3: nat] :
% 2.59/1.74 ( ( ord_less_nat @ I3 @ K )
% 2.59/1.74 => ? [X4: a] : ( P @ I3 @ X4 ) ) )
% 2.59/1.74 = ( ? [Xs3: list_a] :
% 2.59/1.74 ( ( ( size_size_list_a @ Xs3 )
% 2.59/1.74 = K )
% 2.59/1.74 & ! [I3: nat] :
% 2.59/1.74 ( ( ord_less_nat @ I3 @ K )
% 2.59/1.74 => ( P @ I3 @ ( nth_a @ Xs3 @ I3 ) ) ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_98_list__eq__iff__nth__eq,axiom,
% 2.59/1.74 ( ( ^ [Y4: list_nat,Z: list_nat] : ( Y4 = Z ) )
% 2.59/1.74 = ( ^ [Xs3: list_nat,Ys3: list_nat] :
% 2.59/1.74 ( ( ( size_size_list_nat @ Xs3 )
% 2.59/1.74 = ( size_size_list_nat @ Ys3 ) )
% 2.59/1.74 & ! [I3: nat] :
% 2.59/1.74 ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs3 ) )
% 2.59/1.74 => ( ( nth_nat @ Xs3 @ I3 )
% 2.59/1.74 = ( nth_nat @ Ys3 @ I3 ) ) ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_99_list__eq__iff__nth__eq,axiom,
% 2.59/1.74 ( ( ^ [Y4: list_a,Z: list_a] : ( Y4 = Z ) )
% 2.59/1.74 = ( ^ [Xs3: list_a,Ys3: list_a] :
% 2.59/1.74 ( ( ( size_size_list_a @ Xs3 )
% 2.59/1.74 = ( size_size_list_a @ Ys3 ) )
% 2.59/1.74 & ! [I3: nat] :
% 2.59/1.74 ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs3 ) )
% 2.59/1.74 => ( ( nth_a @ Xs3 @ I3 )
% 2.59/1.74 = ( nth_a @ Ys3 @ I3 ) ) ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_100_less__diff__iff,axiom,
% 2.59/1.74 ! [K: nat,M: nat,N: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ K @ M )
% 2.59/1.74 => ( ( ord_less_eq_nat @ K @ N )
% 2.59/1.74 => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 2.59/1.74 = ( ord_less_nat @ M @ N ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_101_diff__less__mono,axiom,
% 2.59/1.74 ! [A: nat,B: nat,C: nat] :
% 2.59/1.74 ( ( ord_less_nat @ A @ B )
% 2.59/1.74 => ( ( ord_less_eq_nat @ C @ A )
% 2.59/1.74 => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_102_less__diff__le__imp__less,axiom,
% 2.59/1.74 ! [I: nat,J: nat,M: nat,N: nat] :
% 2.59/1.74 ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ M ) )
% 2.59/1.74 => ( ( ord_less_eq_nat @ N @ M )
% 2.59/1.74 => ( ord_less_nat @ I @ ( minus_minus_nat @ J @ N ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_103_nat__diff__left__cancel__le1,axiom,
% 2.59/1.74 ! [K: nat,M: nat,N: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ ( minus_minus_nat @ K @ M ) @ ( minus_minus_nat @ K @ N ) )
% 2.59/1.74 => ( ( ord_less_nat @ M @ K )
% 2.59/1.74 => ( ord_less_eq_nat @ N @ M ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_104_nat__diff__right__cancel__le1,axiom,
% 2.59/1.74 ! [M: nat,K: nat,N: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 2.59/1.74 => ( ( ord_less_nat @ K @ M )
% 2.59/1.74 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_105_i__drop__nth__sub,axiom,
% 2.59/1.74 ! [N: nat,X: nat,S: nat > a] :
% 2.59/1.74 ( ( ord_less_eq_nat @ N @ X )
% 2.59/1.74 => ( ( listIn1417627087drop_a @ N @ S @ ( minus_minus_nat @ X @ N ) )
% 2.59/1.74 = ( S @ X ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_106_diff__right__commute,axiom,
% 2.59/1.74 ! [A: nat,C: nat,B: nat] :
% 2.59/1.74 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
% 2.59/1.74 = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_107_nth__append2,axiom,
% 2.59/1.74 ! [Xs: list_nat,N: nat,Ys: list_nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
% 2.59/1.74 => ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
% 2.59/1.74 = ( nth_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_108_nth__append2,axiom,
% 2.59/1.74 ! [Xs: list_a,N: nat,Ys: list_a] :
% 2.59/1.74 ( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N )
% 2.59/1.74 => ( ( nth_a @ ( append_a @ Xs @ Ys ) @ N )
% 2.59/1.74 = ( nth_a @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_109_nth__append1,axiom,
% 2.59/1.74 ! [N: nat,Xs: list_nat,Ys: list_nat] :
% 2.59/1.74 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 2.59/1.74 => ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
% 2.59/1.74 = ( nth_nat @ Xs @ N ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_110_nth__append1,axiom,
% 2.59/1.74 ! [N: nat,Xs: list_a,Ys: list_a] :
% 2.59/1.74 ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
% 2.59/1.74 => ( ( nth_a @ ( append_a @ Xs @ Ys ) @ N )
% 2.59/1.74 = ( nth_a @ Xs @ N ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_111_order__refl,axiom,
% 2.59/1.74 ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_112_list__desc__trans__le,axiom,
% 2.59/1.74 ( list_desc_nat
% 2.59/1.74 = ( ^ [Xs3: list_nat] :
% 2.59/1.74 ! [J3: nat] :
% 2.59/1.74 ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs3 ) )
% 2.59/1.74 => ! [I3: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ I3 @ J3 )
% 2.59/1.74 => ( ord_less_eq_nat @ ( nth_nat @ Xs3 @ J3 ) @ ( nth_nat @ Xs3 @ I3 ) ) ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_113_nat__descend__induct,axiom,
% 2.59/1.74 ! [N: nat,P: nat > $o,M: nat] :
% 2.59/1.74 ( ! [K2: nat] :
% 2.59/1.74 ( ( ord_less_nat @ N @ K2 )
% 2.59/1.74 => ( P @ K2 ) )
% 2.59/1.74 => ( ! [K2: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ K2 @ N )
% 2.59/1.74 => ( ! [I4: nat] :
% 2.59/1.74 ( ( ord_less_nat @ K2 @ I4 )
% 2.59/1.74 => ( P @ I4 ) )
% 2.59/1.74 => ( P @ K2 ) ) )
% 2.59/1.74 => ( P @ M ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_114_list__asc__trans__le,axiom,
% 2.59/1.74 ( list_asc_nat
% 2.59/1.74 = ( ^ [Xs3: list_nat] :
% 2.59/1.74 ! [J3: nat] :
% 2.59/1.74 ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs3 ) )
% 2.59/1.74 => ! [I3: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ I3 @ J3 )
% 2.59/1.74 => ( ord_less_eq_nat @ ( nth_nat @ Xs3 @ I3 ) @ ( nth_nat @ Xs3 @ J3 ) ) ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_115_list__strict__asc__trans__le,axiom,
% 2.59/1.74 ! [Xs: list_nat] :
% 2.59/1.74 ( ( list_strict_asc_nat @ Xs )
% 2.59/1.74 => ! [J4: nat] :
% 2.59/1.74 ( ( ord_less_nat @ J4 @ ( size_size_list_nat @ Xs ) )
% 2.59/1.74 => ! [I4: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ I4 @ J4 )
% 2.59/1.74 => ( ord_less_eq_nat @ ( nth_nat @ Xs @ I4 ) @ ( nth_nat @ Xs @ J4 ) ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_116_list__strict__asc__imp__list__asc,axiom,
% 2.59/1.74 ! [Xs: list_nat] :
% 2.59/1.74 ( ( list_strict_asc_nat @ Xs )
% 2.59/1.74 => ( list_asc_nat @ Xs ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_117_order__subst1,axiom,
% 2.59/1.74 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 2.59/1.74 => ( ( ord_less_eq_nat @ B @ C )
% 2.59/1.74 => ( ! [X2: nat,Y2: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.59/1.74 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.59/1.74 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_118_order__subst2,axiom,
% 2.59/1.74 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ A @ B )
% 2.59/1.74 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 2.59/1.74 => ( ! [X2: nat,Y2: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.59/1.74 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.59/1.74 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_119_ord__eq__le__subst,axiom,
% 2.59/1.74 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 2.59/1.74 ( ( A
% 2.59/1.74 = ( F @ B ) )
% 2.59/1.74 => ( ( ord_less_eq_nat @ B @ C )
% 2.59/1.74 => ( ! [X2: nat,Y2: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.59/1.74 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.59/1.74 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_120_ord__le__eq__subst,axiom,
% 2.59/1.74 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ A @ B )
% 2.59/1.74 => ( ( ( F @ B )
% 2.59/1.74 = C )
% 2.59/1.74 => ( ! [X2: nat,Y2: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.59/1.74 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.59/1.74 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_121_eq__iff,axiom,
% 2.59/1.74 ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
% 2.59/1.74 = ( ^ [X3: nat,Y5: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ X3 @ Y5 )
% 2.59/1.74 & ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_122_antisym,axiom,
% 2.59/1.74 ! [X: nat,Y: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ X @ Y )
% 2.59/1.74 => ( ( ord_less_eq_nat @ Y @ X )
% 2.59/1.74 => ( X = Y ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_123_linear,axiom,
% 2.59/1.74 ! [X: nat,Y: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ X @ Y )
% 2.59/1.74 | ( ord_less_eq_nat @ Y @ X ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_124_eq__refl,axiom,
% 2.59/1.74 ! [X: nat,Y: nat] :
% 2.59/1.74 ( ( X = Y )
% 2.59/1.74 => ( ord_less_eq_nat @ X @ Y ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_125_le__cases,axiom,
% 2.59/1.74 ! [X: nat,Y: nat] :
% 2.59/1.74 ( ~ ( ord_less_eq_nat @ X @ Y )
% 2.59/1.74 => ( ord_less_eq_nat @ Y @ X ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_126_order_Otrans,axiom,
% 2.59/1.74 ! [A: nat,B: nat,C: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ A @ B )
% 2.59/1.74 => ( ( ord_less_eq_nat @ B @ C )
% 2.59/1.74 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_127_le__cases3,axiom,
% 2.59/1.74 ! [X: nat,Y: nat,Z2: nat] :
% 2.59/1.74 ( ( ( ord_less_eq_nat @ X @ Y )
% 2.59/1.74 => ~ ( ord_less_eq_nat @ Y @ Z2 ) )
% 2.59/1.74 => ( ( ( ord_less_eq_nat @ Y @ X )
% 2.59/1.74 => ~ ( ord_less_eq_nat @ X @ Z2 ) )
% 2.59/1.74 => ( ( ( ord_less_eq_nat @ X @ Z2 )
% 2.59/1.74 => ~ ( ord_less_eq_nat @ Z2 @ Y ) )
% 2.59/1.74 => ( ( ( ord_less_eq_nat @ Z2 @ Y )
% 2.59/1.74 => ~ ( ord_less_eq_nat @ Y @ X ) )
% 2.59/1.74 => ( ( ( ord_less_eq_nat @ Y @ Z2 )
% 2.59/1.74 => ~ ( ord_less_eq_nat @ Z2 @ X ) )
% 2.59/1.74 => ~ ( ( ord_less_eq_nat @ Z2 @ X )
% 2.59/1.74 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_128_antisym__conv,axiom,
% 2.59/1.74 ! [Y: nat,X: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ Y @ X )
% 2.59/1.74 => ( ( ord_less_eq_nat @ X @ Y )
% 2.59/1.74 = ( X = Y ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_129_order__class_Oorder_Oeq__iff,axiom,
% 2.59/1.74 ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
% 2.59/1.74 = ( ^ [A2: nat,B2: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ A2 @ B2 )
% 2.59/1.74 & ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_130_ord__eq__le__trans,axiom,
% 2.59/1.74 ! [A: nat,B: nat,C: nat] :
% 2.59/1.74 ( ( A = B )
% 2.59/1.74 => ( ( ord_less_eq_nat @ B @ C )
% 2.59/1.74 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_131_ord__le__eq__trans,axiom,
% 2.59/1.74 ! [A: nat,B: nat,C: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ A @ B )
% 2.59/1.74 => ( ( B = C )
% 2.59/1.74 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_132_order__class_Oorder_Oantisym,axiom,
% 2.59/1.74 ! [A: nat,B: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ A @ B )
% 2.59/1.74 => ( ( ord_less_eq_nat @ B @ A )
% 2.59/1.74 => ( A = B ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_133_order__trans,axiom,
% 2.59/1.74 ! [X: nat,Y: nat,Z2: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ X @ Y )
% 2.59/1.74 => ( ( ord_less_eq_nat @ Y @ Z2 )
% 2.59/1.74 => ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_134_dual__order_Orefl,axiom,
% 2.59/1.74 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_135_linorder__wlog,axiom,
% 2.59/1.74 ! [P: nat > nat > $o,A: nat,B: nat] :
% 2.59/1.74 ( ! [A3: nat,B3: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ A3 @ B3 )
% 2.59/1.74 => ( P @ A3 @ B3 ) )
% 2.59/1.74 => ( ! [A3: nat,B3: nat] :
% 2.59/1.74 ( ( P @ B3 @ A3 )
% 2.59/1.74 => ( P @ A3 @ B3 ) )
% 2.59/1.74 => ( P @ A @ B ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_136_dual__order_Otrans,axiom,
% 2.59/1.74 ! [B: nat,A: nat,C: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ B @ A )
% 2.59/1.74 => ( ( ord_less_eq_nat @ C @ B )
% 2.59/1.74 => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_137_dual__order_Oeq__iff,axiom,
% 2.59/1.74 ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
% 2.59/1.74 = ( ^ [A2: nat,B2: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ B2 @ A2 )
% 2.59/1.74 & ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_138_dual__order_Oantisym,axiom,
% 2.59/1.74 ! [B: nat,A: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ B @ A )
% 2.59/1.74 => ( ( ord_less_eq_nat @ A @ B )
% 2.59/1.74 => ( A = B ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_139_dual__order_Ostrict__implies__not__eq,axiom,
% 2.59/1.74 ! [B: nat,A: nat] :
% 2.59/1.74 ( ( ord_less_nat @ B @ A )
% 2.59/1.74 => ( A != B ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_140_order_Ostrict__implies__not__eq,axiom,
% 2.59/1.74 ! [A: nat,B: nat] :
% 2.59/1.74 ( ( ord_less_nat @ A @ B )
% 2.59/1.74 => ( A != B ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_141_not__less__iff__gr__or__eq,axiom,
% 2.59/1.74 ! [X: nat,Y: nat] :
% 2.59/1.74 ( ( ~ ( ord_less_nat @ X @ Y ) )
% 2.59/1.74 = ( ( ord_less_nat @ Y @ X )
% 2.59/1.74 | ( X = Y ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_142_dual__order_Ostrict__trans,axiom,
% 2.59/1.74 ! [B: nat,A: nat,C: nat] :
% 2.59/1.74 ( ( ord_less_nat @ B @ A )
% 2.59/1.74 => ( ( ord_less_nat @ C @ B )
% 2.59/1.74 => ( ord_less_nat @ C @ A ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_143_linorder__less__wlog,axiom,
% 2.59/1.74 ! [P: nat > nat > $o,A: nat,B: nat] :
% 2.59/1.74 ( ! [A3: nat,B3: nat] :
% 2.59/1.74 ( ( ord_less_nat @ A3 @ B3 )
% 2.59/1.74 => ( P @ A3 @ B3 ) )
% 2.59/1.74 => ( ! [A3: nat] : ( P @ A3 @ A3 )
% 2.59/1.74 => ( ! [A3: nat,B3: nat] :
% 2.59/1.74 ( ( P @ B3 @ A3 )
% 2.59/1.74 => ( P @ A3 @ B3 ) )
% 2.59/1.74 => ( P @ A @ B ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_144_exists__least__iff,axiom,
% 2.59/1.74 ( ( ^ [P2: nat > $o] :
% 2.59/1.74 ? [X4: nat] : ( P2 @ X4 ) )
% 2.59/1.74 = ( ^ [P3: nat > $o] :
% 2.59/1.74 ? [N3: nat] :
% 2.59/1.74 ( ( P3 @ N3 )
% 2.59/1.74 & ! [M3: nat] :
% 2.59/1.74 ( ( ord_less_nat @ M3 @ N3 )
% 2.59/1.74 => ~ ( P3 @ M3 ) ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_145_less__imp__not__less,axiom,
% 2.59/1.74 ! [X: nat,Y: nat] :
% 2.59/1.74 ( ( ord_less_nat @ X @ Y )
% 2.59/1.74 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_146_order_Ostrict__trans,axiom,
% 2.59/1.74 ! [A: nat,B: nat,C: nat] :
% 2.59/1.74 ( ( ord_less_nat @ A @ B )
% 2.59/1.74 => ( ( ord_less_nat @ B @ C )
% 2.59/1.74 => ( ord_less_nat @ A @ C ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_147_dual__order_Oirrefl,axiom,
% 2.59/1.74 ! [A: nat] :
% 2.59/1.74 ~ ( ord_less_nat @ A @ A ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_148_linorder__cases,axiom,
% 2.59/1.74 ! [X: nat,Y: nat] :
% 2.59/1.74 ( ~ ( ord_less_nat @ X @ Y )
% 2.59/1.74 => ( ( X != Y )
% 2.59/1.74 => ( ord_less_nat @ Y @ X ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_149_less__imp__triv,axiom,
% 2.59/1.74 ! [X: nat,Y: nat,P: $o] :
% 2.59/1.74 ( ( ord_less_nat @ X @ Y )
% 2.59/1.74 => ( ( ord_less_nat @ Y @ X )
% 2.59/1.74 => P ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_150_less__imp__not__eq2,axiom,
% 2.59/1.74 ! [X: nat,Y: nat] :
% 2.59/1.74 ( ( ord_less_nat @ X @ Y )
% 2.59/1.74 => ( Y != X ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_151_antisym__conv3,axiom,
% 2.59/1.74 ! [Y: nat,X: nat] :
% 2.59/1.74 ( ~ ( ord_less_nat @ Y @ X )
% 2.59/1.74 => ( ( ~ ( ord_less_nat @ X @ Y ) )
% 2.59/1.74 = ( X = Y ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_152_less__induct,axiom,
% 2.59/1.74 ! [P: nat > $o,A: nat] :
% 2.59/1.74 ( ! [X2: nat] :
% 2.59/1.74 ( ! [Y3: nat] :
% 2.59/1.74 ( ( ord_less_nat @ Y3 @ X2 )
% 2.59/1.74 => ( P @ Y3 ) )
% 2.59/1.74 => ( P @ X2 ) )
% 2.59/1.74 => ( P @ A ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_153_less__not__sym,axiom,
% 2.59/1.74 ! [X: nat,Y: nat] :
% 2.59/1.74 ( ( ord_less_nat @ X @ Y )
% 2.59/1.74 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_154_less__imp__not__eq,axiom,
% 2.59/1.74 ! [X: nat,Y: nat] :
% 2.59/1.74 ( ( ord_less_nat @ X @ Y )
% 2.59/1.74 => ( X != Y ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_155_dual__order_Oasym,axiom,
% 2.59/1.74 ! [B: nat,A: nat] :
% 2.59/1.74 ( ( ord_less_nat @ B @ A )
% 2.59/1.74 => ~ ( ord_less_nat @ A @ B ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_156_ord__less__eq__trans,axiom,
% 2.59/1.74 ! [A: nat,B: nat,C: nat] :
% 2.59/1.74 ( ( ord_less_nat @ A @ B )
% 2.59/1.74 => ( ( B = C )
% 2.59/1.74 => ( ord_less_nat @ A @ C ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_157_ord__eq__less__trans,axiom,
% 2.59/1.74 ! [A: nat,B: nat,C: nat] :
% 2.59/1.74 ( ( A = B )
% 2.59/1.74 => ( ( ord_less_nat @ B @ C )
% 2.59/1.74 => ( ord_less_nat @ A @ C ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_158_less__irrefl,axiom,
% 2.59/1.74 ! [X: nat] :
% 2.59/1.74 ~ ( ord_less_nat @ X @ X ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_159_less__linear,axiom,
% 2.59/1.74 ! [X: nat,Y: nat] :
% 2.59/1.74 ( ( ord_less_nat @ X @ Y )
% 2.59/1.74 | ( X = Y )
% 2.59/1.74 | ( ord_less_nat @ Y @ X ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_160_less__trans,axiom,
% 2.59/1.74 ! [X: nat,Y: nat,Z2: nat] :
% 2.59/1.74 ( ( ord_less_nat @ X @ Y )
% 2.59/1.74 => ( ( ord_less_nat @ Y @ Z2 )
% 2.59/1.74 => ( ord_less_nat @ X @ Z2 ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_161_less__asym_H,axiom,
% 2.59/1.74 ! [A: nat,B: nat] :
% 2.59/1.74 ( ( ord_less_nat @ A @ B )
% 2.59/1.74 => ~ ( ord_less_nat @ B @ A ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_162_less__asym,axiom,
% 2.59/1.74 ! [X: nat,Y: nat] :
% 2.59/1.74 ( ( ord_less_nat @ X @ Y )
% 2.59/1.74 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_163_less__imp__neq,axiom,
% 2.59/1.74 ! [X: nat,Y: nat] :
% 2.59/1.74 ( ( ord_less_nat @ X @ Y )
% 2.59/1.74 => ( X != Y ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_164_order_Oasym,axiom,
% 2.59/1.74 ! [A: nat,B: nat] :
% 2.59/1.74 ( ( ord_less_nat @ A @ B )
% 2.59/1.74 => ~ ( ord_less_nat @ B @ A ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_165_neq__iff,axiom,
% 2.59/1.74 ! [X: nat,Y: nat] :
% 2.59/1.74 ( ( X != Y )
% 2.59/1.74 = ( ( ord_less_nat @ X @ Y )
% 2.59/1.74 | ( ord_less_nat @ Y @ X ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_166_neqE,axiom,
% 2.59/1.74 ! [X: nat,Y: nat] :
% 2.59/1.74 ( ( X != Y )
% 2.59/1.74 => ( ~ ( ord_less_nat @ X @ Y )
% 2.59/1.74 => ( ord_less_nat @ Y @ X ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_167_gt__ex,axiom,
% 2.59/1.74 ! [X: nat] :
% 2.59/1.74 ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_168_order__less__subst2,axiom,
% 2.59/1.74 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 2.59/1.74 ( ( ord_less_nat @ A @ B )
% 2.59/1.74 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 2.59/1.74 => ( ! [X2: nat,Y2: nat] :
% 2.59/1.74 ( ( ord_less_nat @ X2 @ Y2 )
% 2.59/1.74 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.59/1.74 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_169_order__less__subst1,axiom,
% 2.59/1.74 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 2.59/1.74 ( ( ord_less_nat @ A @ ( F @ B ) )
% 2.59/1.74 => ( ( ord_less_nat @ B @ C )
% 2.59/1.74 => ( ! [X2: nat,Y2: nat] :
% 2.59/1.74 ( ( ord_less_nat @ X2 @ Y2 )
% 2.59/1.74 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.59/1.74 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_170_ord__less__eq__subst,axiom,
% 2.59/1.74 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 2.59/1.74 ( ( ord_less_nat @ A @ B )
% 2.59/1.74 => ( ( ( F @ B )
% 2.59/1.74 = C )
% 2.59/1.74 => ( ! [X2: nat,Y2: nat] :
% 2.59/1.74 ( ( ord_less_nat @ X2 @ Y2 )
% 2.59/1.74 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.59/1.74 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_171_ord__eq__less__subst,axiom,
% 2.59/1.74 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 2.59/1.74 ( ( A
% 2.59/1.74 = ( F @ B ) )
% 2.59/1.74 => ( ( ord_less_nat @ B @ C )
% 2.59/1.74 => ( ! [X2: nat,Y2: nat] :
% 2.59/1.74 ( ( ord_less_nat @ X2 @ Y2 )
% 2.59/1.74 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.59/1.74 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_172_list__drop__eq__conv,axiom,
% 2.59/1.74 ( ( ^ [Y4: list_a,Z: list_a] : ( Y4 = Z ) )
% 2.59/1.74 = ( ^ [Xs3: list_a,Ys3: list_a] :
% 2.59/1.74 ! [N3: nat] :
% 2.59/1.74 ( ( drop_a @ N3 @ Xs3 )
% 2.59/1.74 = ( drop_a @ N3 @ Ys3 ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_173_list__asc__trans,axiom,
% 2.59/1.74 ( list_asc_nat
% 2.59/1.74 = ( ^ [Xs3: list_nat] :
% 2.59/1.74 ! [J3: nat] :
% 2.59/1.74 ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs3 ) )
% 2.59/1.74 => ! [I3: nat] :
% 2.59/1.74 ( ( ord_less_nat @ I3 @ J3 )
% 2.59/1.74 => ( ord_less_eq_nat @ ( nth_nat @ Xs3 @ I3 ) @ ( nth_nat @ Xs3 @ J3 ) ) ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_174_list__strict__asc__trans,axiom,
% 2.59/1.74 ( list_strict_asc_nat
% 2.59/1.74 = ( ^ [Xs3: list_nat] :
% 2.59/1.74 ! [J3: nat] :
% 2.59/1.74 ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs3 ) )
% 2.59/1.74 => ! [I3: nat] :
% 2.59/1.74 ( ( ord_less_nat @ I3 @ J3 )
% 2.59/1.74 => ( ord_less_nat @ ( nth_nat @ Xs3 @ I3 ) @ ( nth_nat @ Xs3 @ J3 ) ) ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_175_list__desc__trans,axiom,
% 2.59/1.74 ( list_desc_nat
% 2.59/1.74 = ( ^ [Xs3: list_nat] :
% 2.59/1.74 ! [J3: nat] :
% 2.59/1.74 ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs3 ) )
% 2.59/1.74 => ! [I3: nat] :
% 2.59/1.74 ( ( ord_less_nat @ I3 @ J3 )
% 2.59/1.74 => ( ord_less_eq_nat @ ( nth_nat @ Xs3 @ J3 ) @ ( nth_nat @ Xs3 @ I3 ) ) ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_176_order_Onot__eq__order__implies__strict,axiom,
% 2.59/1.74 ! [A: nat,B: nat] :
% 2.59/1.74 ( ( A != B )
% 2.59/1.74 => ( ( ord_less_eq_nat @ A @ B )
% 2.59/1.74 => ( ord_less_nat @ A @ B ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_177_dual__order_Ostrict__implies__order,axiom,
% 2.59/1.74 ! [B: nat,A: nat] :
% 2.59/1.74 ( ( ord_less_nat @ B @ A )
% 2.59/1.74 => ( ord_less_eq_nat @ B @ A ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_178_dual__order_Ostrict__iff__order,axiom,
% 2.59/1.74 ( ord_less_nat
% 2.59/1.74 = ( ^ [B2: nat,A2: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ B2 @ A2 )
% 2.59/1.74 & ( A2 != B2 ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_179_dual__order_Oorder__iff__strict,axiom,
% 2.59/1.74 ( ord_less_eq_nat
% 2.59/1.74 = ( ^ [B2: nat,A2: nat] :
% 2.59/1.74 ( ( ord_less_nat @ B2 @ A2 )
% 2.59/1.74 | ( A2 = B2 ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_180_order_Ostrict__implies__order,axiom,
% 2.59/1.74 ! [A: nat,B: nat] :
% 2.59/1.74 ( ( ord_less_nat @ A @ B )
% 2.59/1.74 => ( ord_less_eq_nat @ A @ B ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_181_dual__order_Ostrict__trans2,axiom,
% 2.59/1.74 ! [B: nat,A: nat,C: nat] :
% 2.59/1.74 ( ( ord_less_nat @ B @ A )
% 2.59/1.74 => ( ( ord_less_eq_nat @ C @ B )
% 2.59/1.74 => ( ord_less_nat @ C @ A ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_182_dual__order_Ostrict__trans1,axiom,
% 2.59/1.74 ! [B: nat,A: nat,C: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ B @ A )
% 2.59/1.74 => ( ( ord_less_nat @ C @ B )
% 2.59/1.74 => ( ord_less_nat @ C @ A ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_183_order_Ostrict__iff__order,axiom,
% 2.59/1.74 ( ord_less_nat
% 2.59/1.74 = ( ^ [A2: nat,B2: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ A2 @ B2 )
% 2.59/1.74 & ( A2 != B2 ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_184_order_Oorder__iff__strict,axiom,
% 2.59/1.74 ( ord_less_eq_nat
% 2.59/1.74 = ( ^ [A2: nat,B2: nat] :
% 2.59/1.74 ( ( ord_less_nat @ A2 @ B2 )
% 2.59/1.74 | ( A2 = B2 ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_185_order_Ostrict__trans2,axiom,
% 2.59/1.74 ! [A: nat,B: nat,C: nat] :
% 2.59/1.74 ( ( ord_less_nat @ A @ B )
% 2.59/1.74 => ( ( ord_less_eq_nat @ B @ C )
% 2.59/1.74 => ( ord_less_nat @ A @ C ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_186_order_Ostrict__trans1,axiom,
% 2.59/1.74 ! [A: nat,B: nat,C: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ A @ B )
% 2.59/1.74 => ( ( ord_less_nat @ B @ C )
% 2.59/1.74 => ( ord_less_nat @ A @ C ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_187_not__le__imp__less,axiom,
% 2.59/1.74 ! [Y: nat,X: nat] :
% 2.59/1.74 ( ~ ( ord_less_eq_nat @ Y @ X )
% 2.59/1.74 => ( ord_less_nat @ X @ Y ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_188_less__le__not__le,axiom,
% 2.59/1.74 ( ord_less_nat
% 2.59/1.74 = ( ^ [X3: nat,Y5: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ X3 @ Y5 )
% 2.59/1.74 & ~ ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_189_le__imp__less__or__eq,axiom,
% 2.59/1.74 ! [X: nat,Y: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ X @ Y )
% 2.59/1.74 => ( ( ord_less_nat @ X @ Y )
% 2.59/1.74 | ( X = Y ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_190_le__less__linear,axiom,
% 2.59/1.74 ! [X: nat,Y: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ X @ Y )
% 2.59/1.74 | ( ord_less_nat @ Y @ X ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_191_less__le__trans,axiom,
% 2.59/1.74 ! [X: nat,Y: nat,Z2: nat] :
% 2.59/1.74 ( ( ord_less_nat @ X @ Y )
% 2.59/1.74 => ( ( ord_less_eq_nat @ Y @ Z2 )
% 2.59/1.74 => ( ord_less_nat @ X @ Z2 ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_192_le__less__trans,axiom,
% 2.59/1.74 ! [X: nat,Y: nat,Z2: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ X @ Y )
% 2.59/1.74 => ( ( ord_less_nat @ Y @ Z2 )
% 2.59/1.74 => ( ord_less_nat @ X @ Z2 ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_193_less__imp__le,axiom,
% 2.59/1.74 ! [X: nat,Y: nat] :
% 2.59/1.74 ( ( ord_less_nat @ X @ Y )
% 2.59/1.74 => ( ord_less_eq_nat @ X @ Y ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_194_antisym__conv2,axiom,
% 2.59/1.74 ! [X: nat,Y: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ X @ Y )
% 2.59/1.74 => ( ( ~ ( ord_less_nat @ X @ Y ) )
% 2.59/1.74 = ( X = Y ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_195_antisym__conv1,axiom,
% 2.59/1.74 ! [X: nat,Y: nat] :
% 2.59/1.74 ( ~ ( ord_less_nat @ X @ Y )
% 2.59/1.74 => ( ( ord_less_eq_nat @ X @ Y )
% 2.59/1.74 = ( X = Y ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_196_le__neq__trans,axiom,
% 2.59/1.74 ! [A: nat,B: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ A @ B )
% 2.59/1.74 => ( ( A != B )
% 2.59/1.74 => ( ord_less_nat @ A @ B ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_197_not__less,axiom,
% 2.59/1.74 ! [X: nat,Y: nat] :
% 2.59/1.74 ( ( ~ ( ord_less_nat @ X @ Y ) )
% 2.59/1.74 = ( ord_less_eq_nat @ Y @ X ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_198_not__le,axiom,
% 2.59/1.74 ! [X: nat,Y: nat] :
% 2.59/1.74 ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
% 2.59/1.74 = ( ord_less_nat @ Y @ X ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_199_order__less__le__subst2,axiom,
% 2.59/1.74 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 2.59/1.74 ( ( ord_less_nat @ A @ B )
% 2.59/1.74 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 2.59/1.74 => ( ! [X2: nat,Y2: nat] :
% 2.59/1.74 ( ( ord_less_nat @ X2 @ Y2 )
% 2.59/1.74 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.59/1.74 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_200_order__less__le__subst1,axiom,
% 2.59/1.74 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 2.59/1.74 ( ( ord_less_nat @ A @ ( F @ B ) )
% 2.59/1.74 => ( ( ord_less_eq_nat @ B @ C )
% 2.59/1.74 => ( ! [X2: nat,Y2: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.59/1.74 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.59/1.74 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_201_order__le__less__subst2,axiom,
% 2.59/1.74 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ A @ B )
% 2.59/1.74 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 2.59/1.74 => ( ! [X2: nat,Y2: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.59/1.74 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.59/1.74 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_202_order__le__less__subst1,axiom,
% 2.59/1.74 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 2.59/1.74 => ( ( ord_less_nat @ B @ C )
% 2.59/1.74 => ( ! [X2: nat,Y2: nat] :
% 2.59/1.74 ( ( ord_less_nat @ X2 @ Y2 )
% 2.59/1.74 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.59/1.74 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_203_less__le,axiom,
% 2.59/1.74 ( ord_less_nat
% 2.59/1.74 = ( ^ [X3: nat,Y5: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ X3 @ Y5 )
% 2.59/1.74 & ( X3 != Y5 ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_204_le__less,axiom,
% 2.59/1.74 ( ord_less_eq_nat
% 2.59/1.74 = ( ^ [X3: nat,Y5: nat] :
% 2.59/1.74 ( ( ord_less_nat @ X3 @ Y5 )
% 2.59/1.74 | ( X3 = Y5 ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_205_leI,axiom,
% 2.59/1.74 ! [X: nat,Y: nat] :
% 2.59/1.74 ( ~ ( ord_less_nat @ X @ Y )
% 2.59/1.74 => ( ord_less_eq_nat @ Y @ X ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_206_leD,axiom,
% 2.59/1.74 ! [Y: nat,X: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ Y @ X )
% 2.59/1.74 => ~ ( ord_less_nat @ X @ Y ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_207_list__strict__desc__trans,axiom,
% 2.59/1.74 ( list_strict_desc_nat
% 2.59/1.74 = ( ^ [Xs3: list_nat] :
% 2.59/1.74 ! [J3: nat] :
% 2.59/1.74 ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs3 ) )
% 2.59/1.74 => ! [I3: nat] :
% 2.59/1.74 ( ( ord_less_nat @ I3 @ J3 )
% 2.59/1.74 => ( ord_less_nat @ ( nth_nat @ Xs3 @ J3 ) @ ( nth_nat @ Xs3 @ I3 ) ) ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_208_complete__interval,axiom,
% 2.59/1.74 ! [A: nat,B: nat,P: nat > $o] :
% 2.59/1.74 ( ( ord_less_nat @ A @ B )
% 2.59/1.74 => ( ( P @ A )
% 2.59/1.74 => ( ~ ( P @ B )
% 2.59/1.74 => ? [C2: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ A @ C2 )
% 2.59/1.74 & ( ord_less_eq_nat @ C2 @ B )
% 2.59/1.74 & ! [X5: nat] :
% 2.59/1.74 ( ( ( ord_less_eq_nat @ A @ X5 )
% 2.59/1.74 & ( ord_less_nat @ X5 @ C2 ) )
% 2.59/1.74 => ( P @ X5 ) )
% 2.59/1.74 & ! [D: nat] :
% 2.59/1.74 ( ! [X2: nat] :
% 2.59/1.74 ( ( ( ord_less_eq_nat @ A @ X2 )
% 2.59/1.74 & ( ord_less_nat @ X2 @ D ) )
% 2.59/1.74 => ( P @ X2 ) )
% 2.59/1.74 => ( ord_less_eq_nat @ D @ C2 ) ) ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_209_verit__comp__simplify1_I3_J,axiom,
% 2.59/1.74 ! [B4: nat,A4: nat] :
% 2.59/1.74 ( ( ~ ( ord_less_eq_nat @ B4 @ A4 ) )
% 2.59/1.74 = ( ord_less_nat @ A4 @ B4 ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_210_pinf_I6_J,axiom,
% 2.59/1.74 ! [T: nat] :
% 2.59/1.74 ? [Z3: nat] :
% 2.59/1.74 ! [X5: nat] :
% 2.59/1.74 ( ( ord_less_nat @ Z3 @ X5 )
% 2.59/1.74 => ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_211_pinf_I8_J,axiom,
% 2.59/1.74 ! [T: nat] :
% 2.59/1.74 ? [Z3: nat] :
% 2.59/1.74 ! [X5: nat] :
% 2.59/1.74 ( ( ord_less_nat @ Z3 @ X5 )
% 2.59/1.74 => ( ord_less_eq_nat @ T @ X5 ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_212_verit__la__disequality,axiom,
% 2.59/1.74 ! [A: nat,B: nat] :
% 2.59/1.74 ( ( A = B )
% 2.59/1.74 | ~ ( ord_less_eq_nat @ A @ B )
% 2.59/1.74 | ~ ( ord_less_eq_nat @ B @ A ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_213_minf_I7_J,axiom,
% 2.59/1.74 ! [T: nat] :
% 2.59/1.74 ? [Z3: nat] :
% 2.59/1.74 ! [X5: nat] :
% 2.59/1.74 ( ( ord_less_nat @ X5 @ Z3 )
% 2.59/1.74 => ~ ( ord_less_nat @ T @ X5 ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_214_minf_I5_J,axiom,
% 2.59/1.74 ! [T: nat] :
% 2.59/1.74 ? [Z3: nat] :
% 2.59/1.74 ! [X5: nat] :
% 2.59/1.74 ( ( ord_less_nat @ X5 @ Z3 )
% 2.59/1.74 => ( ord_less_nat @ X5 @ T ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_215_minf_I4_J,axiom,
% 2.59/1.74 ! [T: nat] :
% 2.59/1.74 ? [Z3: nat] :
% 2.59/1.74 ! [X5: nat] :
% 2.59/1.74 ( ( ord_less_nat @ X5 @ Z3 )
% 2.59/1.74 => ( X5 != T ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_216_minf_I3_J,axiom,
% 2.59/1.74 ! [T: nat] :
% 2.59/1.74 ? [Z3: nat] :
% 2.59/1.74 ! [X5: nat] :
% 2.59/1.74 ( ( ord_less_nat @ X5 @ Z3 )
% 2.59/1.74 => ( X5 != T ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_217_minf_I2_J,axiom,
% 2.59/1.74 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 2.59/1.74 ( ? [Z4: nat] :
% 2.59/1.74 ! [X2: nat] :
% 2.59/1.74 ( ( ord_less_nat @ X2 @ Z4 )
% 2.59/1.74 => ( ( P @ X2 )
% 2.59/1.74 = ( P4 @ X2 ) ) )
% 2.59/1.74 => ( ? [Z4: nat] :
% 2.59/1.74 ! [X2: nat] :
% 2.59/1.74 ( ( ord_less_nat @ X2 @ Z4 )
% 2.59/1.74 => ( ( Q @ X2 )
% 2.59/1.74 = ( Q2 @ X2 ) ) )
% 2.59/1.74 => ? [Z3: nat] :
% 2.59/1.74 ! [X5: nat] :
% 2.59/1.74 ( ( ord_less_nat @ X5 @ Z3 )
% 2.59/1.74 => ( ( ( P @ X5 )
% 2.59/1.74 | ( Q @ X5 ) )
% 2.59/1.74 = ( ( P4 @ X5 )
% 2.59/1.74 | ( Q2 @ X5 ) ) ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_218_minf_I1_J,axiom,
% 2.59/1.74 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 2.59/1.74 ( ? [Z4: nat] :
% 2.59/1.74 ! [X2: nat] :
% 2.59/1.74 ( ( ord_less_nat @ X2 @ Z4 )
% 2.59/1.74 => ( ( P @ X2 )
% 2.59/1.74 = ( P4 @ X2 ) ) )
% 2.59/1.74 => ( ? [Z4: nat] :
% 2.59/1.74 ! [X2: nat] :
% 2.59/1.74 ( ( ord_less_nat @ X2 @ Z4 )
% 2.59/1.74 => ( ( Q @ X2 )
% 2.59/1.74 = ( Q2 @ X2 ) ) )
% 2.59/1.74 => ? [Z3: nat] :
% 2.59/1.74 ! [X5: nat] :
% 2.59/1.74 ( ( ord_less_nat @ X5 @ Z3 )
% 2.59/1.74 => ( ( ( P @ X5 )
% 2.59/1.74 & ( Q @ X5 ) )
% 2.59/1.74 = ( ( P4 @ X5 )
% 2.59/1.74 & ( Q2 @ X5 ) ) ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_219_pinf_I7_J,axiom,
% 2.59/1.74 ! [T: nat] :
% 2.59/1.74 ? [Z3: nat] :
% 2.59/1.74 ! [X5: nat] :
% 2.59/1.74 ( ( ord_less_nat @ Z3 @ X5 )
% 2.59/1.74 => ( ord_less_nat @ T @ X5 ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_220_pinf_I5_J,axiom,
% 2.59/1.74 ! [T: nat] :
% 2.59/1.74 ? [Z3: nat] :
% 2.59/1.74 ! [X5: nat] :
% 2.59/1.74 ( ( ord_less_nat @ Z3 @ X5 )
% 2.59/1.74 => ~ ( ord_less_nat @ X5 @ T ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_221_pinf_I4_J,axiom,
% 2.59/1.74 ! [T: nat] :
% 2.59/1.74 ? [Z3: nat] :
% 2.59/1.74 ! [X5: nat] :
% 2.59/1.74 ( ( ord_less_nat @ Z3 @ X5 )
% 2.59/1.74 => ( X5 != T ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_222_pinf_I3_J,axiom,
% 2.59/1.74 ! [T: nat] :
% 2.59/1.74 ? [Z3: nat] :
% 2.59/1.74 ! [X5: nat] :
% 2.59/1.74 ( ( ord_less_nat @ Z3 @ X5 )
% 2.59/1.74 => ( X5 != T ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_223_pinf_I2_J,axiom,
% 2.59/1.74 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 2.59/1.74 ( ? [Z4: nat] :
% 2.59/1.74 ! [X2: nat] :
% 2.59/1.74 ( ( ord_less_nat @ Z4 @ X2 )
% 2.59/1.74 => ( ( P @ X2 )
% 2.59/1.74 = ( P4 @ X2 ) ) )
% 2.59/1.74 => ( ? [Z4: nat] :
% 2.59/1.74 ! [X2: nat] :
% 2.59/1.74 ( ( ord_less_nat @ Z4 @ X2 )
% 2.59/1.74 => ( ( Q @ X2 )
% 2.59/1.74 = ( Q2 @ X2 ) ) )
% 2.59/1.74 => ? [Z3: nat] :
% 2.59/1.74 ! [X5: nat] :
% 2.59/1.74 ( ( ord_less_nat @ Z3 @ X5 )
% 2.59/1.74 => ( ( ( P @ X5 )
% 2.59/1.74 | ( Q @ X5 ) )
% 2.59/1.74 = ( ( P4 @ X5 )
% 2.59/1.74 | ( Q2 @ X5 ) ) ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_224_pinf_I1_J,axiom,
% 2.59/1.74 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 2.59/1.74 ( ? [Z4: nat] :
% 2.59/1.74 ! [X2: nat] :
% 2.59/1.74 ( ( ord_less_nat @ Z4 @ X2 )
% 2.59/1.74 => ( ( P @ X2 )
% 2.59/1.74 = ( P4 @ X2 ) ) )
% 2.59/1.74 => ( ? [Z4: nat] :
% 2.59/1.74 ! [X2: nat] :
% 2.59/1.74 ( ( ord_less_nat @ Z4 @ X2 )
% 2.59/1.74 => ( ( Q @ X2 )
% 2.59/1.74 = ( Q2 @ X2 ) ) )
% 2.59/1.74 => ? [Z3: nat] :
% 2.59/1.74 ! [X5: nat] :
% 2.59/1.74 ( ( ord_less_nat @ Z3 @ X5 )
% 2.59/1.74 => ( ( ( P @ X5 )
% 2.59/1.74 & ( Q @ X5 ) )
% 2.59/1.74 = ( ( P4 @ X5 )
% 2.59/1.74 & ( Q2 @ X5 ) ) ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_225_verit__comp__simplify1_I1_J,axiom,
% 2.59/1.74 ! [A: nat] :
% 2.59/1.74 ~ ( ord_less_nat @ A @ A ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_226_list__strict__desc__imp__list__desc,axiom,
% 2.59/1.74 ! [Xs: list_nat] :
% 2.59/1.74 ( ( list_strict_desc_nat @ Xs )
% 2.59/1.74 => ( list_desc_nat @ Xs ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_227_minf_I8_J,axiom,
% 2.59/1.74 ! [T: nat] :
% 2.59/1.74 ? [Z3: nat] :
% 2.59/1.74 ! [X5: nat] :
% 2.59/1.74 ( ( ord_less_nat @ X5 @ Z3 )
% 2.59/1.74 => ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_228_minf_I6_J,axiom,
% 2.59/1.74 ! [T: nat] :
% 2.59/1.74 ? [Z3: nat] :
% 2.59/1.74 ! [X5: nat] :
% 2.59/1.74 ( ( ord_less_nat @ X5 @ Z3 )
% 2.59/1.74 => ( ord_less_eq_nat @ X5 @ T ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_229_i__append__i__take__eq2,axiom,
% 2.59/1.74 ! [Xs: list_nat,N: nat,F: nat > nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
% 2.59/1.74 => ( ( listIn1299132951ke_nat @ N @ ( listIn923761578nd_nat @ Xs @ F ) )
% 2.59/1.74 = ( append_nat @ Xs @ ( listIn1299132951ke_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ F ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_230_i__append__i__take__eq2,axiom,
% 2.59/1.74 ! [Xs: list_a,N: nat,F: nat > a] :
% 2.59/1.74 ( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N )
% 2.59/1.74 => ( ( listIn84854903take_a @ N @ ( listIn1312259492pend_a @ Xs @ F ) )
% 2.59/1.74 = ( append_a @ Xs @ ( listIn84854903take_a @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) @ F ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_231_nth__drop,axiom,
% 2.59/1.74 ! [N: nat,Xs: list_nat,I: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
% 2.59/1.74 => ( ( nth_nat @ ( drop_nat @ N @ Xs ) @ I )
% 2.59/1.74 = ( nth_nat @ Xs @ ( plus_plus_nat @ N @ I ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_232_nth__drop,axiom,
% 2.59/1.74 ! [N: nat,Xs: list_a,I: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ N @ ( size_size_list_a @ Xs ) )
% 2.59/1.74 => ( ( nth_a @ ( drop_a @ N @ Xs ) @ I )
% 2.59/1.74 = ( nth_a @ Xs @ ( plus_plus_nat @ N @ I ) ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_233_add__right__cancel,axiom,
% 2.59/1.74 ! [B: nat,A: nat,C: nat] :
% 2.59/1.74 ( ( ( plus_plus_nat @ B @ A )
% 2.59/1.74 = ( plus_plus_nat @ C @ A ) )
% 2.59/1.74 = ( B = C ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_234_add__left__cancel,axiom,
% 2.59/1.74 ! [A: nat,B: nat,C: nat] :
% 2.59/1.74 ( ( ( plus_plus_nat @ A @ B )
% 2.59/1.74 = ( plus_plus_nat @ A @ C ) )
% 2.59/1.74 = ( B = C ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_235_add__le__cancel__right,axiom,
% 2.59/1.74 ! [A: nat,C: nat,B: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 2.59/1.74 = ( ord_less_eq_nat @ A @ B ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_236_add__le__cancel__left,axiom,
% 2.59/1.74 ! [C: nat,A: nat,B: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 2.59/1.74 = ( ord_less_eq_nat @ A @ B ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_237_add__less__cancel__left,axiom,
% 2.59/1.74 ! [C: nat,A: nat,B: nat] :
% 2.59/1.74 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 2.59/1.74 = ( ord_less_nat @ A @ B ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_238_add__less__cancel__right,axiom,
% 2.59/1.74 ! [A: nat,C: nat,B: nat] :
% 2.59/1.74 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 2.59/1.74 = ( ord_less_nat @ A @ B ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_239_add__diff__cancel__left,axiom,
% 2.59/1.74 ! [C: nat,A: nat,B: nat] :
% 2.59/1.74 ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 2.59/1.74 = ( minus_minus_nat @ A @ B ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_240_add__diff__cancel__left_H,axiom,
% 2.59/1.74 ! [A: nat,B: nat] :
% 2.59/1.74 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
% 2.59/1.74 = B ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_241_add__diff__cancel__right,axiom,
% 2.59/1.74 ! [A: nat,C: nat,B: nat] :
% 2.59/1.74 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 2.59/1.74 = ( minus_minus_nat @ A @ B ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_242_add__diff__cancel__right_H,axiom,
% 2.59/1.74 ! [A: nat,B: nat] :
% 2.59/1.74 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 2.59/1.74 = A ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_243_nat__add__left__cancel__less,axiom,
% 2.59/1.74 ! [K: nat,M: nat,N: nat] :
% 2.59/1.74 ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 2.59/1.74 = ( ord_less_nat @ M @ N ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_244_nat__add__left__cancel__le,axiom,
% 2.59/1.74 ! [K: nat,M: nat,N: nat] :
% 2.59/1.74 ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 2.59/1.74 = ( ord_less_eq_nat @ M @ N ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_245_diff__diff__left,axiom,
% 2.59/1.74 ! [I: nat,J: nat,K: nat] :
% 2.59/1.74 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 2.59/1.74 = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(fact_246_drop__drop,axiom,
% 2.59/1.74 ! [N: nat,M: nat,Xs: list_a] :
% 2.59/1.74 ( ( drop_a @ N @ ( drop_a @ M @ Xs ) )
% 2.59/1.74 = ( drop_a @ ( plus_plus_nat @ N @ M ) @ Xs ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(help_If_2_1_If_001tf__a_T,axiom,
% 2.59/1.74 ! [X: a,Y: a] :
% 2.59/1.74 ( ( if_a @ $false @ X @ Y )
% 2.59/1.74 = Y ) ).
% 2.59/1.74
% 2.59/1.74 thf(help_If_1_1_If_001tf__a_T,axiom,
% 2.59/1.74 ! [X: a,Y: a] :
% 2.59/1.74 ( ( if_a @ $true @ X @ Y )
% 2.59/1.74 = X ) ).
% 2.59/1.74
% 2.59/1.74 thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
% 2.59/1.74 ! [P: $o] :
% 2.59/1.74 ( ( P = $true )
% 2.59/1.74 | ( P = $false ) ) ).
% 2.59/1.74
% 2.59/1.74 thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 2.59/1.74 ! [X: nat,Y: nat] :
% 2.59/1.74 ( ( if_nat @ $false @ X @ Y )
% 2.59/1.74 = Y ) ).
% 2.59/1.74
% 2.59/1.74 thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 2.59/1.74 ! [X: nat,Y: nat] :
% 2.59/1.74 ( ( if_nat @ $true @ X @ Y )
% 2.59/1.74 = X ) ).
% 2.59/1.74
% 2.59/1.74 thf(conj_0,conjecture,
% 2.59/1.74 ( ( ( ord_less_nat @ n @ ( size_size_list_a @ xs ) )
% 2.59/1.74 => ( ( listIn1417627087drop_a @ n @ ( listIn1312259492pend_a @ xs @ f ) )
% 2.59/1.74 = ( listIn1312259492pend_a @ ( drop_a @ n @ xs ) @ f ) ) )
% 2.59/1.74 & ( ~ ( ord_less_nat @ n @ ( size_size_list_a @ xs ) )
% 2.59/1.74 => ( ( listIn1417627087drop_a @ n @ ( listIn1312259492pend_a @ xs @ f ) )
% 2.59/1.74 = ( listIn1417627087drop_a @ ( minus_minus_nat @ n @ ( size_size_list_a @ xs ) ) @ f ) ) ) ) ).
% 2.59/1.74 ------------------------
% 2.59/1.76 % (25867)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_25681 for (2999ds/4Mi)
% 2.59/1.76 % (25868)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_25681 for (2999ds/27Mi)
% 2.59/1.76 % (25872)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_25681 for (2999ds/18Mi)
% 2.59/1.76 % (25867)Instruction limit reached!
% 2.59/1.76 % (25867)------------------------------
% 2.59/1.76 % (25867)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.59/1.76 % (25867)Termination reason: Unknown
% 2.59/1.76 % (25867)Termination phase: shuffling
% 2.59/1.76
% 2.59/1.76 % (25867)Memory used [KB]: 1407
% 2.59/1.76 % (25867)Time elapsed: 0.003 s
% 2.59/1.76 % (25867)Instructions burned: 5 (million)
% 2.59/1.76 % (25867)------------------------------
% 2.59/1.76 % (25867)------------------------------
% 2.59/1.77 % (25872)Instruction limit reached!
% 2.59/1.77 % (25872)------------------------------
% 2.59/1.77 % (25872)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.59/1.77 % (25872)Termination reason: Unknown
% 2.59/1.77 % (25872)Termination phase: shuffling
% 2.59/1.77
% 2.59/1.77 % (25872)Memory used [KB]: 1663
% 2.59/1.77 % (25872)Time elapsed: 0.007 s
% 2.59/1.77 % (25872)Instructions burned: 18 (million)
% 2.59/1.77 % (25872)------------------------------
% 2.59/1.77 % (25872)------------------------------
% 2.59/1.77 % (25870)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_25681 for (2999ds/2Mi)
% 2.59/1.77 % (25866)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_25681 for (2999ds/183Mi)
% 2.59/1.77 % (25869)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_25681 for (2999ds/2Mi)
% 2.59/1.77 % (25871)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_25681 for (2999ds/275Mi)
% 2.59/1.77 % (25868)Instruction limit reached!
% 2.59/1.77 % (25868)------------------------------
% 2.59/1.77 % (25868)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.59/1.77 % (25868)Termination reason: Unknown
% 2.59/1.77 % (25868)Termination phase: Property scanning
% 2.59/1.77
% 2.59/1.77 % (25868)Memory used [KB]: 1663
% 2.59/1.77 % (25868)Time elapsed: 0.010 s
% 2.59/1.77 % (25868)Instructions burned: 28 (million)
% 2.59/1.77 % (25868)------------------------------
% 2.59/1.77 % (25868)------------------------------
% 2.59/1.77 % (25870)Instruction limit reached!
% 2.59/1.77 % (25870)------------------------------
% 2.59/1.77 % (25870)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.59/1.77 % (25870)Termination reason: Unknown
% 2.59/1.77 % (25870)Termination phase: shuffling
% 2.59/1.77
% 2.59/1.77 % (25870)Memory used [KB]: 1279
% 2.59/1.77 % (25870)Time elapsed: 0.003 s
% 2.59/1.77 % (25870)Instructions burned: 3 (million)
% 2.59/1.77 % (25870)------------------------------
% 2.59/1.77 % (25870)------------------------------
% 2.59/1.77 % (25869)Instruction limit reached!
% 2.59/1.77 % (25869)------------------------------
% 2.59/1.77 % (25869)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.59/1.77 % (25869)Termination reason: Unknown
% 2.59/1.77 % (25869)Termination phase: shuffling
% 2.59/1.77
% 2.59/1.77 % (25869)Memory used [KB]: 1279
% 2.59/1.77 % (25869)Time elapsed: 0.004 s
% 2.59/1.77 % (25869)Instructions burned: 3 (million)
% 2.59/1.77 % (25869)------------------------------
% 2.59/1.77 % (25869)------------------------------
% 2.59/1.77 % (25873)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_25681 for (2999ds/3Mi)
% 2.59/1.77 % (25873)Instruction limit reached!
% 2.59/1.77 % (25873)------------------------------
% 2.59/1.77 % (25873)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.59/1.77 % (25873)Termination reason: Unknown
% 2.59/1.77 % (25873)Termination phase: shuffling
% 2.59/1.77
% 2.59/1.77 % (25873)Memory used [KB]: 1279
% 2.59/1.77 % (25873)Time elapsed: 0.002 s
% 2.59/1.77 % (25873)Instructions burned: 3 (million)
% 2.59/1.77 % (25873)------------------------------
% 2.59/1.77 % (25873)------------------------------
% 2.59/1.78 % (25876)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_25681 for (2999ds/37Mi)
% 2.59/1.78 % (25880)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_25681 for (2999ds/15Mi)
% 2.59/1.78 % (25887)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_25681 for (2999ds/7Mi)
% 2.59/1.79 % (25880)Instruction limit reached!
% 2.59/1.79 % (25880)------------------------------
% 2.59/1.79 % (25880)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.59/1.79 % (25880)Termination reason: Unknown
% 2.59/1.79 % (25880)Termination phase: shuffling
% 2.59/1.79
% 2.59/1.79 % (25880)Memory used [KB]: 1535
% 2.59/1.79 % (25880)Time elapsed: 0.006 s
% 2.59/1.79 % (25880)Instructions burned: 15 (million)
% 2.59/1.79 % (25880)------------------------------
% 2.59/1.79 % (25880)------------------------------
% 2.59/1.79 % (25887)Instruction limit reached!
% 2.59/1.79 % (25887)------------------------------
% 2.59/1.79 % (25887)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.59/1.79 % (25887)Termination reason: Unknown
% 2.59/1.79 % (25887)Termination phase: shuffling
% 2.59/1.79
% 2.59/1.79 % (25887)Memory used [KB]: 1407
% 2.59/1.79 % (25887)Time elapsed: 0.004 s
% 2.59/1.79 % (25887)Instructions burned: 10 (million)
% 2.59/1.79 % (25887)------------------------------
% 2.59/1.79 % (25887)------------------------------
% 2.59/1.79 % (25876)Instruction limit reached!
% 2.59/1.79 % (25876)------------------------------
% 2.59/1.79 % (25876)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.59/1.79 % (25876)Termination reason: Unknown
% 2.59/1.79 % (25876)Termination phase: shuffling
% 2.59/1.79
% 2.59/1.79 % (25876)Memory used [KB]: 2046
% 2.59/1.79 % (25876)Time elapsed: 0.011 s
% 2.59/1.79 % (25876)Instructions burned: 37 (million)
% 2.59/1.79 % (25876)------------------------------
% 2.59/1.79 % (25876)------------------------------
% 2.59/1.79 % (25881)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_25681 for (2999ds/3Mi)
% 2.59/1.79 % (25884)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_25681 for (2999ds/1041Mi)
% 2.59/1.79 % (25881)Instruction limit reached!
% 2.59/1.79 % (25881)------------------------------
% 2.59/1.79 % (25881)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.59/1.79 % (25881)Termination reason: Unknown
% 2.59/1.79 % (25881)Termination phase: shuffling
% 2.59/1.79
% 2.59/1.79 % (25881)Memory used [KB]: 1279
% 2.59/1.79 % (25881)Time elapsed: 0.003 s
% 2.59/1.79 % (25881)Instructions burned: 3 (million)
% 2.59/1.79 % (25881)------------------------------
% 2.59/1.79 % (25881)------------------------------
% 2.59/1.80 % (25895)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_25681 for (2999ds/16Mi)
% 2.64/1.80 % (25896)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_25681 for (2999ds/3Mi)
% 2.64/1.80 % (25897)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_25681 for (2999ds/3Mi)
% 2.64/1.80 % (25896)Instruction limit reached!
% 2.64/1.80 % (25896)------------------------------
% 2.64/1.80 % (25896)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.80 % (25896)Termination reason: Unknown
% 2.64/1.80 % (25896)Termination phase: shuffling
% 2.64/1.80
% 2.64/1.80 % (25896)Memory used [KB]: 1407
% 2.64/1.80 % (25896)Time elapsed: 0.003 s
% 2.64/1.80 % (25896)Instructions burned: 5 (million)
% 2.64/1.80 % (25896)------------------------------
% 2.64/1.80 % (25896)------------------------------
% 2.64/1.80 % (25897)Instruction limit reached!
% 2.64/1.80 % (25897)------------------------------
% 2.64/1.80 % (25897)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.80 % (25897)Termination reason: Unknown
% 2.64/1.80 % (25897)Termination phase: shuffling
% 2.64/1.80
% 2.64/1.80 % (25897)Memory used [KB]: 1407
% 2.64/1.80 % (25897)Time elapsed: 0.003 s
% 2.64/1.80 % (25897)Instructions burned: 6 (million)
% 2.64/1.80 % (25897)------------------------------
% 2.64/1.80 % (25897)------------------------------
% 2.64/1.80 % (25895)Instruction limit reached!
% 2.64/1.80 % (25895)------------------------------
% 2.64/1.80 % (25895)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.80 % (25895)Termination reason: Unknown
% 2.64/1.80 % (25895)Termination phase: shuffling
% 2.64/1.80
% 2.64/1.80 % (25895)Memory used [KB]: 1663
% 2.64/1.80 % (25895)Time elapsed: 0.007 s
% 2.64/1.80 % (25895)Instructions burned: 19 (million)
% 2.64/1.80 % (25895)------------------------------
% 2.64/1.80 % (25895)------------------------------
% 2.64/1.80 % (25900)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_25681 for (2999ds/7Mi)
% 2.64/1.81 % (25900)Instruction limit reached!
% 2.64/1.81 % (25900)------------------------------
% 2.64/1.81 % (25900)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.81 % (25900)Termination reason: Unknown
% 2.64/1.81 % (25900)Termination phase: shuffling
% 2.64/1.81
% 2.64/1.81 % (25900)Memory used [KB]: 1407
% 2.64/1.81 % (25900)Time elapsed: 0.004 s
% 2.64/1.81 % (25900)Instructions burned: 8 (million)
% 2.64/1.81 % (25900)------------------------------
% 2.64/1.81 % (25900)------------------------------
% 2.64/1.81 % (25906)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_25681 for (2999ds/3Mi)
% 2.64/1.81 % (25906)Instruction limit reached!
% 2.64/1.81 % (25906)------------------------------
% 2.64/1.81 % (25906)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.81 % (25906)Termination reason: Unknown
% 2.64/1.81 % (25906)Termination phase: shuffling
% 2.64/1.81
% 2.64/1.81 % (25906)Memory used [KB]: 1279
% 2.64/1.81 % (25906)Time elapsed: 0.002 s
% 2.64/1.81 % (25906)Instructions burned: 3 (million)
% 2.64/1.81 % (25906)------------------------------
% 2.64/1.81 % (25906)------------------------------
% 2.64/1.81 % (25908)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_25681 for (2999ds/4Mi)
% 2.64/1.81 % (25909)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_25681 for (2999ds/18Mi)
% 2.64/1.81 % (25908)Instruction limit reached!
% 2.64/1.81 % (25908)------------------------------
% 2.64/1.81 % (25908)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.81 % (25908)Termination reason: Unknown
% 2.64/1.81 % (25908)Termination phase: shuffling
% 2.64/1.81
% 2.64/1.81 % (25908)Memory used [KB]: 1407
% 2.64/1.81 % (25908)Time elapsed: 0.003 s
% 2.64/1.81 % (25908)Instructions burned: 6 (million)
% 2.64/1.81 % (25908)------------------------------
% 2.64/1.81 % (25908)------------------------------
% 2.64/1.82 % (25910)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_25681 for (2999ds/710Mi)
% 2.64/1.82 % (25909)Instruction limit reached!
% 2.64/1.82 % (25909)------------------------------
% 2.64/1.82 % (25909)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.82 % (25909)Termination reason: Unknown
% 2.64/1.82 % (25909)Termination phase: shuffling
% 2.64/1.82
% 2.64/1.82 % (25909)Memory used [KB]: 1663
% 2.64/1.82 % (25909)Time elapsed: 0.007 s
% 2.64/1.82 % (25909)Instructions burned: 19 (million)
% 2.64/1.82 % (25909)------------------------------
% 2.64/1.82 % (25909)------------------------------
% 2.64/1.82 % (25911)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_25681 for (2999ds/6Mi)
% 2.64/1.82 % (25911)Instruction limit reached!
% 2.64/1.82 % (25911)------------------------------
% 2.64/1.82 % (25911)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.82 % (25911)Termination reason: Unknown
% 2.64/1.82 % (25911)Termination phase: shuffling
% 2.64/1.82
% 2.64/1.82 % (25911)Memory used [KB]: 1407
% 2.64/1.82 % (25911)Time elapsed: 0.003 s
% 2.64/1.82 % (25911)Instructions burned: 7 (million)
% 2.64/1.82 % (25911)------------------------------
% 2.64/1.82 % (25911)------------------------------
% 2.64/1.82 % (25912)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_25681 for (2999ds/902Mi)
% 2.64/1.83 % (25914)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_25681 for (2999ds/21Mi)
% 2.64/1.83 % (25919)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_25681 for (2999ds/5Mi)
% 2.64/1.83 % (25919)Instruction limit reached!
% 2.64/1.83 % (25919)------------------------------
% 2.64/1.83 % (25919)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.83 % (25919)Termination reason: Unknown
% 2.64/1.83 % (25919)Termination phase: shuffling
% 2.64/1.83
% 2.64/1.83 % (25919)Memory used [KB]: 1407
% 2.64/1.83 % (25919)Time elapsed: 0.003 s
% 2.64/1.83 % (25919)Instructions burned: 6 (million)
% 2.64/1.83 % (25919)------------------------------
% 2.64/1.83 % (25919)------------------------------
% 2.64/1.83 % (25914)Instruction limit reached!
% 2.64/1.83 % (25914)------------------------------
% 2.64/1.83 % (25914)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.83 % (25914)Termination reason: Unknown
% 2.64/1.83 % (25914)Termination phase: shuffling
% 2.64/1.83
% 2.64/1.83 % (25914)Memory used [KB]: 1663
% 2.64/1.83 % (25914)Time elapsed: 0.008 s
% 2.64/1.83 % (25914)Instructions burned: 21 (million)
% 2.64/1.83 % (25914)------------------------------
% 2.64/1.83 % (25914)------------------------------
% 2.64/1.84 % (25871)First to succeed.
% 2.64/1.84 % (25930)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_25681 for (2999ds/6Mi)
% 2.64/1.84 % (25931)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_25681 for (2999ds/377Mi)
% 2.64/1.84 % (25930)Instruction limit reached!
% 2.64/1.84 % (25930)------------------------------
% 2.64/1.84 % (25930)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.84 % (25930)Termination reason: Unknown
% 2.64/1.84 % (25930)Termination phase: shuffling
% 2.64/1.84
% 2.64/1.84 % (25930)Memory used [KB]: 1407
% 2.64/1.84 % (25930)Time elapsed: 0.003 s
% 2.64/1.84 % (25930)Instructions burned: 8 (million)
% 2.64/1.84 % (25930)------------------------------
% 2.64/1.84 % (25930)------------------------------
% 2.64/1.84 % (25871)Refutation found. Thanks to Tanya!
% 2.64/1.84 % SZS status Theorem for DTF2THF_25681
% 2.64/1.84 % SZS output start Proof for DTF2THF_25681
% 2.64/1.84 thf(type_def_5, type, nat: $tType).
% 2.64/1.84 thf(type_def_7, type, a: $tType).
% 2.64/1.84 thf(type_def_8, type, list_nat: $tType).
% 2.64/1.84 thf(type_def_9, type, list_a: $tType).
% 2.64/1.84 thf(func_def_0, type, list_nat: $tType).
% 2.64/1.84 thf(func_def_1, type, list_a: $tType).
% 2.64/1.84 thf(func_def_2, type, nat: $tType).
% 2.64/1.84 thf(func_def_3, type, a: $tType).
% 2.64/1.84 thf(func_def_4, type, minus_minus_nat: nat > nat > nat).
% 2.64/1.84 thf(func_def_5, type, plus_plus_nat: nat > nat > nat).
% 2.64/1.84 thf(func_def_6, type, if_nat: $o > nat > nat > nat).
% 2.64/1.84 thf(func_def_7, type, if_a: $o > a > a > a).
% 2.64/1.84 thf(func_def_8, type, list_asc_nat: list_nat > $o).
% 2.64/1.84 thf(func_def_9, type, list_desc_nat: list_nat > $o).
% 2.64/1.84 thf(func_def_10, type, list_strict_asc_nat: list_nat > $o).
% 2.64/1.84 thf(func_def_11, type, list_strict_desc_nat: list_nat > $o).
% 2.64/1.84 thf(func_def_12, type, listIn923761578nd_nat: list_nat > (nat > nat) > nat > nat).
% 2.64/1.84 thf(func_def_13, type, listIn1312259492pend_a: list_a > (nat > a) > nat > a).
% 2.64/1.84 thf(func_def_14, type, listIn870223551op_nat: nat > (nat > nat) > nat > nat).
% 2.64/1.84 thf(func_def_15, type, listIn1417627087drop_a: nat > (nat > a) > nat > a).
% 2.64/1.84 thf(func_def_16, type, listIn1299132951ke_nat: nat > (nat > nat) > list_nat).
% 2.64/1.84 thf(func_def_17, type, listIn84854903take_a: nat > (nat > a) > list_a).
% 2.64/1.84 thf(func_def_18, type, append_nat: list_nat > list_nat > list_nat).
% 2.64/1.84 thf(func_def_19, type, append_a: list_a > list_a > list_a).
% 2.64/1.84 thf(func_def_20, type, drop_nat: nat > list_nat > list_nat).
% 2.64/1.84 thf(func_def_21, type, drop_a: nat > list_a > list_a).
% 2.64/1.84 thf(func_def_22, type, last_nat: list_nat > nat).
% 2.64/1.84 thf(func_def_23, type, last_a: list_a > a).
% 2.64/1.84 thf(func_def_24, type, nth_nat: list_nat > nat > nat).
% 2.64/1.84 thf(func_def_25, type, nth_a: list_a > nat > a).
% 2.64/1.84 thf(func_def_26, type, size_size_list_nat: list_nat > nat).
% 2.64/1.84 thf(func_def_27, type, size_size_list_a: list_a > nat).
% 2.64/1.84 thf(func_def_28, type, ord_less_nat: nat > nat > $o).
% 2.64/1.84 thf(func_def_29, type, ord_less_eq_nat: nat > nat > $o).
% 2.64/1.84 thf(func_def_30, type, f: nat > a).
% 2.64/1.84 thf(func_def_31, type, n: nat).
% 2.64/1.84 thf(func_def_32, type, xs: list_a).
% 2.64/1.84 thf(func_def_47, type, sP0: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > $o).
% 2.64/1.84 thf(func_def_48, type, sP1: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > $o).
% 2.64/1.84 thf(func_def_49, type, sP2: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > $o).
% 2.64/1.84 thf(func_def_50, type, sP3: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > $o).
% 2.64/1.84 thf(func_def_51, type, sP4: nat > nat > nat > $o).
% 2.64/1.84 thf(func_def_52, type, sP5: nat > nat > nat > $o).
% 2.64/1.84 thf(func_def_53, type, sP6: nat > nat > nat > $o).
% 2.64/1.84 thf(func_def_54, type, sK7: (nat > nat) > nat).
% 2.64/1.84 thf(func_def_55, type, sK8: (nat > nat) > nat).
% 2.64/1.84 thf(func_def_56, type, sK9: (nat > $o) > nat).
% 2.64/1.84 thf(func_def_57, type, sK10: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > nat).
% 2.64/1.84 thf(func_def_58, type, sK11: nat > (nat > $o) > (nat > $o) > nat).
% 2.64/1.84 thf(func_def_59, type, sK12: nat > (nat > $o) > (nat > $o) > nat).
% 2.64/1.84 thf(func_def_60, type, sK13: nat > nat).
% 2.64/1.84 thf(func_def_61, type, sK14: (nat > nat) > nat).
% 2.64/1.84 thf(func_def_62, type, sK15: (nat > nat) > nat).
% 2.64/1.84 thf(func_def_63, type, sK16: nat > nat).
% 2.64/1.84 thf(func_def_64, type, sK17: (list_a > $o) > list_a).
% 2.64/1.84 thf(func_def_65, type, sK18: nat > (nat > $o) > nat).
% 2.64/1.84 thf(func_def_66, type, sK19: nat > (nat > $o) > nat).
% 2.64/1.84 thf(func_def_67, type, sK20: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > nat).
% 2.64/1.84 thf(func_def_68, type, sK21: nat > (nat > $o) > (nat > $o) > nat).
% 2.64/1.84 thf(func_def_69, type, sK22: nat > (nat > $o) > (nat > $o) > nat).
% 2.64/1.84 thf(func_def_70, type, sK23: (nat > nat) > nat).
% 2.64/1.84 thf(func_def_71, type, sK24: (nat > nat) > nat).
% 2.64/1.84 thf(func_def_72, type, sK25: (nat > nat) > nat).
% 2.64/1.84 thf(func_def_73, type, sK26: (nat > nat) > nat).
% 2.64/1.84 thf(func_def_74, type, sK27: nat > nat).
% 2.64/1.84 thf(func_def_75, type, sK28: nat > (nat > a > $o) > a).
% 2.64/1.84 thf(func_def_76, type, sK29: list_a > nat > (nat > a > $o) > nat).
% 2.64/1.84 thf(func_def_77, type, sK30: nat > (nat > a > $o) > list_a).
% 2.64/1.84 thf(func_def_78, type, sK31: nat > (nat > a > $o) > nat).
% 2.64/1.84 thf(func_def_79, type, sK32: nat > nat).
% 2.64/1.84 thf(func_def_80, type, sK33: nat > nat).
% 2.64/1.84 thf(func_def_81, type, sK34: (nat > nat > $o) > nat).
% 2.64/1.84 thf(func_def_82, type, sK35: (nat > nat > $o) > nat).
% 2.64/1.84 thf(func_def_83, type, sK36: (nat > nat > $o) > nat).
% 2.64/1.84 thf(func_def_84, type, sK37: (nat > nat > $o) > nat).
% 2.64/1.84 thf(func_def_85, type, sK38: (nat > nat) > nat).
% 2.64/1.84 thf(func_def_86, type, sK39: (nat > nat) > nat).
% 2.64/1.84 thf(func_def_87, type, sK40: nat > nat).
% 2.64/1.84 thf(func_def_88, type, sK41: nat > nat).
% 2.64/1.84 thf(func_def_89, type, sK42: (nat > nat) > nat).
% 2.64/1.84 thf(func_def_90, type, sK43: (nat > nat) > nat).
% 2.64/1.84 thf(func_def_91, type, sK44: (nat > nat) > nat).
% 2.64/1.84 thf(func_def_92, type, sK45: (nat > nat) > nat).
% 2.64/1.84 thf(func_def_93, type, sK46: nat > list_a).
% 2.64/1.84 thf(func_def_94, type, sK47: nat > nat).
% 2.64/1.84 thf(func_def_95, type, sK48: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > nat).
% 2.64/1.84 thf(func_def_96, type, sK49: nat > (nat > $o) > (nat > $o) > nat).
% 2.64/1.84 thf(func_def_97, type, sK50: nat > (nat > $o) > (nat > $o) > nat).
% 2.64/1.84 thf(func_def_98, type, sK51: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > nat).
% 2.64/1.84 thf(func_def_99, type, sK52: nat > (nat > $o) > (nat > $o) > nat).
% 2.64/1.84 thf(func_def_100, type, sK53: nat > (nat > $o) > (nat > $o) > nat).
% 2.64/1.84 thf(func_def_101, type, sK54: (nat > $o) > nat).
% 2.64/1.84 thf(func_def_102, type, sK55: (nat > a) > list_a > (nat > a) > list_a > list_a).
% 2.64/1.84 thf(func_def_103, type, sK56: nat > nat).
% 2.64/1.84 thf(func_def_104, type, sK57: (nat > nat > $o) > nat).
% 2.64/1.84 thf(func_def_105, type, sK58: (nat > nat > $o) > nat).
% 2.64/1.84 thf(func_def_106, type, sK59: (nat > nat > $o) > nat).
% 2.64/1.84 thf(func_def_107, type, sK60: (nat > nat > $o) > nat).
% 2.64/1.84 thf(func_def_108, type, sK61: (nat > nat > $o) > nat).
% 2.64/1.84 thf(func_def_109, type, sK62: nat > nat).
% 2.64/1.84 thf(func_def_110, type, sK63: nat > nat).
% 2.64/1.84 thf(func_def_111, type, sK64: (nat > nat) > nat).
% 2.64/1.84 thf(func_def_112, type, sK65: (nat > nat) > nat).
% 2.64/1.84 thf(func_def_113, type, sK66: list_a > list_a > (nat > a) > (nat > a) > list_a).
% 2.64/1.84 thf(func_def_114, type, sK67: (nat > nat) > nat).
% 2.64/1.84 thf(func_def_115, type, sK68: (nat > nat) > nat).
% 2.64/1.84 thf(func_def_116, type, sK69: (nat > $o) > nat > nat > nat).
% 2.64/1.84 thf(func_def_117, type, sK70: nat > (nat > $o) > nat > nat).
% 2.64/1.84 thf(func_def_118, type, sK71: (nat > nat) > nat).
% 2.64/1.84 thf(func_def_119, type, sK72: (nat > nat) > nat).
% 2.64/1.84 thf(func_def_120, type, sK73: (nat > nat) > nat).
% 2.64/1.84 thf(func_def_121, type, sK74: (nat > nat) > nat).
% 2.64/1.84 thf(func_def_122, type, sK75: list_a > list_a > nat).
% 2.64/1.84 thf(func_def_123, type, sK76: nat > nat).
% 2.64/1.84 thf(func_def_124, type, sK77: list_a > list_a > list_a > list_a > list_a).
% 2.64/1.84 thf(func_def_125, type, sK78: (nat > $o) > nat).
% 2.64/1.84 thf(func_def_126, type, sK79: (nat > $o) > nat > nat).
% 2.64/1.84 thf(func_def_127, type, sK80: (nat > $o) > nat).
% 2.64/1.84 thf(func_def_128, type, sK81: nat > nat).
% 2.64/1.84 thf(func_def_129, type, sK82: (nat > nat) > nat).
% 2.64/1.84 thf(func_def_130, type, sK83: (nat > nat) > nat).
% 2.64/1.84 thf(func_def_131, type, sK84: (nat > nat) > nat).
% 2.64/1.84 thf(func_def_132, type, sK85: (nat > nat) > nat).
% 2.64/1.84 thf(func_def_134, type, ph87: !>[X0: $tType]:(X0)).
% 2.64/1.84 thf(func_def_135, type, sK88: nat).
% 2.64/1.84 thf(func_def_136, type, sK89: nat).
% 2.64/1.84 thf(f1711,plain,(
% 2.64/1.84 $false),
% 2.64/1.84 inference(avatar_sat_refutation,[status(thm)],[f1562,f1567,f1626,f1629,f1708])).
% 2.64/1.84 thf(f1708,plain,(
% 2.64/1.84 spl86_1 | spl86_4),
% 2.64/1.84 inference(avatar_contradiction_clause,[status(thm)],[f1707])).
% 2.64/1.84 thf(f1707,plain,(
% 2.64/1.84 $false | (spl86_1 | spl86_4)),
% 2.64/1.84 inference(subsumption_resolution,[status(thm)],[f1659,f1576])).
% 2.64/1.84 thf(f1576,plain,(
% 2.64/1.84 ($true != (ord_less_eq_nat @ (size_size_list_a @ xs) @ n)) | spl86_4),
% 2.64/1.84 inference(avatar_component_clause,[status(thm)],[f1574])).
% 2.64/1.84 thf(f1574,definition,(
% 2.64/1.84 spl86_4 <=> ($true = (ord_less_eq_nat @ (size_size_list_a @ xs) @ n))),
% 2.64/1.84 introduced(definition,[new_symbols(naming,[spl86_4])],[avatar_definition])).
% 2.64/1.84 thf(f1659,plain,(
% 2.64/1.84 ($true = (ord_less_eq_nat @ (size_size_list_a @ xs) @ n)) | spl86_1),
% 2.64/1.84 inference(trivial_inequality_removal,[status(thm)],[f1634])).
% 2.64/1.84 thf(f1634,plain,(
% 2.64/1.84 ($true = (ord_less_eq_nat @ (size_size_list_a @ xs) @ n)) | ($true != $true) | spl86_1),
% 2.64/1.84 inference(superposition,[status(thm)],[f1557,f1171])).
% 2.64/1.84 thf(f1171,plain,(
% 2.64/1.84 ( ! [X0 : nat,X1 : nat] : (($true = (ord_less_nat @ X0 @ X1)) | ((ord_less_eq_nat @ X1 @ X0) = $true)) )),
% 2.64/1.84 inference(cnf_transformation,[status(thm)],[f1000])).
% 2.64/1.84 thf(f1000,plain,(
% 2.64/1.84 ! [X0 : nat,X1 : nat] : ((($true != (ord_less_nat @ X0 @ X1)) | ((ord_less_eq_nat @ X1 @ X0) != $true)) & (((ord_less_eq_nat @ X1 @ X0) = $true) | ($true = (ord_less_nat @ X0 @ X1))))),
% 2.64/1.84 inference(nnf_transformation,[status(thm)],[f737])).
% 2.64/1.84 thf(f737,plain,(
% 2.64/1.84 ! [X0 : nat,X1 : nat] : (($true != (ord_less_nat @ X0 @ X1)) <=> ((ord_less_eq_nat @ X1 @ X0) = $true))),
% 2.64/1.84 inference(flattening,[status(thm)],[f615])).
% 2.64/1.84 thf(f615,plain,(
% 2.64/1.84 ! [X0 : nat,X1 : nat] : (~($true = (ord_less_nat @ X0 @ X1)) <=> ((ord_less_eq_nat @ X1 @ X0) = $true))),
% 2.64/1.84 inference(fool_elimination,[status(thm)],[f614])).
% 2.64/1.84 thf(f614,plain,(
% 2.64/1.84 ! [X0 : nat,X1 : nat] : (~(ord_less_nat @ X0 @ X1) = (ord_less_eq_nat @ X1 @ X0))),
% 2.64/1.84 inference(rectify,[status(thm)],[f198])).
% 2.64/1.84 thf(f198,axiom,(
% 2.64/1.84 ! [X13 : nat,X14 : nat] : (~(ord_less_nat @ X13 @ X14) = (ord_less_eq_nat @ X14 @ X13))),
% 2.64/1.84 file('/export/starexec/sandbox2/tmp/tmp.YUokPb62dV/DTF2THF_25681.p',fact_197_not__less)).
% 2.64/1.84 thf(f1557,plain,(
% 2.64/1.84 ((ord_less_nat @ n @ (size_size_list_a @ xs)) != $true) | spl86_1),
% 2.64/1.84 inference(avatar_component_clause,[status(thm)],[f1555])).
% 2.64/1.84 thf(f1555,definition,(
% 2.64/1.84 spl86_1 <=> ((ord_less_nat @ n @ (size_size_list_a @ xs)) = $true)),
% 2.64/1.84 introduced(definition,[new_symbols(naming,[spl86_1])],[avatar_definition])).
% 2.64/1.84 thf(f1629,plain,(
% 2.64/1.84 ~spl86_4 | spl86_3),
% 2.64/1.84 inference(avatar_split_clause,[status(thm)],[f1585,f1564,f1574])).
% 2.64/1.84 thf(f1564,definition,(
% 2.64/1.84 spl86_3 <=> ((listIn1417627087drop_a @ n @ (listIn1312259492pend_a @ xs @ f)) = (listIn1417627087drop_a @ (minus_minus_nat @ n @ (size_size_list_a @ xs)) @ f))),
% 2.64/1.84 introduced(definition,[new_symbols(naming,[spl86_3])],[avatar_definition])).
% 2.64/1.84 thf(f1585,plain,(
% 2.64/1.84 ($true != (ord_less_eq_nat @ (size_size_list_a @ xs) @ n)) | spl86_3),
% 2.64/1.84 inference(trivial_inequality_removal,[status(thm)],[f1583])).
% 2.64/1.84 thf(f1583,plain,(
% 2.64/1.84 ($true != (ord_less_eq_nat @ (size_size_list_a @ xs) @ n)) | ((listIn1417627087drop_a @ (minus_minus_nat @ n @ (size_size_list_a @ xs)) @ f) != (listIn1417627087drop_a @ (minus_minus_nat @ n @ (size_size_list_a @ xs)) @ f)) | spl86_3),
% 2.64/1.84 inference(superposition,[status(thm)],[f1566,f1193])).
% 2.64/1.84 thf(f1193,plain,(
% 2.64/1.84 ( ! [X2 : list_a,X0 : nat,X1 : nat > a] : (((listIn1417627087drop_a @ X0 @ (listIn1312259492pend_a @ X2 @ X1)) = (listIn1417627087drop_a @ (minus_minus_nat @ X0 @ (size_size_list_a @ X2)) @ X1)) | ($true != (ord_less_eq_nat @ (size_size_list_a @ X2) @ X0))) )),
% 2.64/1.84 inference(cnf_transformation,[status(thm)],[f841])).
% 2.64/1.84 thf(f841,plain,(
% 2.64/1.84 ! [X0 : nat,X1 : nat > a,X2 : list_a] : (($true != (ord_less_eq_nat @ (size_size_list_a @ X2) @ X0)) | ((listIn1417627087drop_a @ X0 @ (listIn1312259492pend_a @ X2 @ X1)) = (listIn1417627087drop_a @ (minus_minus_nat @ X0 @ (size_size_list_a @ X2)) @ X1)))),
% 2.64/1.84 inference(ennf_transformation,[status(thm)],[f597])).
% 2.64/1.84 thf(f597,plain,(
% 2.64/1.84 ! [X0 : nat,X1 : nat > a,X2 : list_a] : (($true = (ord_less_eq_nat @ (size_size_list_a @ X2) @ X0)) => ((listIn1417627087drop_a @ X0 @ (listIn1312259492pend_a @ X2 @ X1)) = (listIn1417627087drop_a @ (minus_minus_nat @ X0 @ (size_size_list_a @ X2)) @ X1)))),
% 2.64/1.84 inference(fool_elimination,[status(thm)],[f596])).
% 2.64/1.84 thf(f596,plain,(
% 2.64/1.84 ! [X0 : nat,X1 : nat > a,X2 : list_a] : ((ord_less_eq_nat @ (size_size_list_a @ X2) @ X0) => ((listIn1417627087drop_a @ X0 @ (listIn1312259492pend_a @ X2 @ X1)) = (listIn1417627087drop_a @ (minus_minus_nat @ X0 @ (size_size_list_a @ X2)) @ X1)))),
% 2.64/1.84 inference(rectify,[status(thm)],[f36])).
% 2.64/1.84 thf(f36,axiom,(
% 2.64/1.84 ! [X4 : nat,X1 : nat > a,X0 : list_a] : ((ord_less_eq_nat @ (size_size_list_a @ X0) @ X4) => ((listIn1417627087drop_a @ X4 @ (listIn1312259492pend_a @ X0 @ X1)) = (listIn1417627087drop_a @ (minus_minus_nat @ X4 @ (size_size_list_a @ X0)) @ X1)))),
% 2.64/1.84 file('/export/starexec/sandbox2/tmp/tmp.YUokPb62dV/DTF2THF_25681.p',fact_35_i__append__i__drop__eq2)).
% 2.64/1.84 thf(f1566,plain,(
% 2.64/1.84 ((listIn1417627087drop_a @ n @ (listIn1312259492pend_a @ xs @ f)) != (listIn1417627087drop_a @ (minus_minus_nat @ n @ (size_size_list_a @ xs)) @ f)) | spl86_3),
% 2.64/1.84 inference(avatar_component_clause,[status(thm)],[f1564])).
% 2.64/1.84 thf(f1626,plain,(
% 2.64/1.84 ~spl86_1 | spl86_2),
% 2.64/1.84 inference(avatar_split_clause,[status(thm)],[f1604,f1559,f1555])).
% 2.64/1.84 thf(f1559,definition,(
% 2.64/1.84 spl86_2 <=> ((listIn1417627087drop_a @ n @ (listIn1312259492pend_a @ xs @ f)) = (listIn1312259492pend_a @ (drop_a @ n @ xs) @ f))),
% 2.64/1.84 introduced(definition,[new_symbols(naming,[spl86_2])],[avatar_definition])).
% 2.64/1.84 thf(f1604,plain,(
% 2.64/1.84 ((ord_less_nat @ n @ (size_size_list_a @ xs)) != $true) | spl86_2),
% 2.64/1.84 inference(trivial_inequality_removal,[status(thm)],[f1593])).
% 2.64/1.84 thf(f1593,plain,(
% 2.64/1.84 ((ord_less_nat @ n @ (size_size_list_a @ xs)) != $true) | ($true != $true) | spl86_2),
% 2.64/1.84 inference(superposition,[status(thm)],[f1572,f1312])).
% 2.64/1.84 thf(f1312,plain,(
% 2.64/1.84 ( ! [X0 : nat,X1 : nat] : (($true = (ord_less_eq_nat @ X0 @ X1)) | ($true != (ord_less_nat @ X0 @ X1))) )),
% 2.64/1.84 inference(cnf_transformation,[status(thm)],[f768])).
% 2.64/1.84 thf(f768,plain,(
% 2.64/1.84 ! [X0 : nat,X1 : nat] : (($true = (ord_less_eq_nat @ X0 @ X1)) | ($true != (ord_less_nat @ X0 @ X1)))),
% 2.64/1.84 inference(ennf_transformation,[status(thm)],[f637])).
% 2.64/1.84 thf(f637,plain,(
% 2.64/1.84 ! [X0 : nat,X1 : nat] : (($true = (ord_less_nat @ X0 @ X1)) => ($true = (ord_less_eq_nat @ X0 @ X1)))),
% 2.64/1.84 inference(fool_elimination,[status(thm)],[f636])).
% 2.64/1.84 thf(f636,plain,(
% 2.64/1.84 ! [X0 : nat,X1 : nat] : ((ord_less_nat @ X0 @ X1) => (ord_less_eq_nat @ X0 @ X1))),
% 2.64/1.84 inference(rectify,[status(thm)],[f72])).
% 2.64/1.84 thf(f72,axiom,(
% 2.64/1.84 ! [X5 : nat,X4 : nat] : ((ord_less_nat @ X5 @ X4) => (ord_less_eq_nat @ X5 @ X4))),
% 2.64/1.84 file('/export/starexec/sandbox2/tmp/tmp.YUokPb62dV/DTF2THF_25681.p',fact_71_less__imp__le__nat)).
% 2.64/1.84 thf(f1572,plain,(
% 2.64/1.84 ((ord_less_eq_nat @ n @ (size_size_list_a @ xs)) != $true) | spl86_2),
% 2.64/1.84 inference(trivial_inequality_removal,[status(thm)],[f1571])).
% 2.64/1.84 thf(f1571,plain,(
% 2.64/1.84 ((listIn1312259492pend_a @ (drop_a @ n @ xs) @ f) != (listIn1312259492pend_a @ (drop_a @ n @ xs) @ f)) | ((ord_less_eq_nat @ n @ (size_size_list_a @ xs)) != $true) | spl86_2),
% 2.64/1.84 inference(superposition,[status(thm)],[f1561,f1266])).
% 2.64/1.84 thf(f1266,plain,(
% 2.64/1.84 ( ! [X2 : nat > a,X0 : nat,X1 : list_a] : (((listIn1417627087drop_a @ X0 @ (listIn1312259492pend_a @ X1 @ X2)) = (listIn1312259492pend_a @ (drop_a @ X0 @ X1) @ X2)) | ((ord_less_eq_nat @ X0 @ (size_size_list_a @ X1)) != $true)) )),
% 2.64/1.84 inference(cnf_transformation,[status(thm)],[f802])).
% 2.64/1.84 thf(f802,plain,(
% 2.64/1.84 ! [X0 : nat,X1 : list_a,X2 : nat > a] : (((listIn1417627087drop_a @ X0 @ (listIn1312259492pend_a @ X1 @ X2)) = (listIn1312259492pend_a @ (drop_a @ X0 @ X1) @ X2)) | ((ord_less_eq_nat @ X0 @ (size_size_list_a @ X1)) != $true))),
% 2.64/1.84 inference(ennf_transformation,[status(thm)],[f437])).
% 2.64/1.84 thf(f437,plain,(
% 2.64/1.84 ! [X0 : nat,X1 : list_a,X2 : nat > a] : (((ord_less_eq_nat @ X0 @ (size_size_list_a @ X1)) = $true) => ((listIn1417627087drop_a @ X0 @ (listIn1312259492pend_a @ X1 @ X2)) = (listIn1312259492pend_a @ (drop_a @ X0 @ X1) @ X2)))),
% 2.64/1.84 inference(fool_elimination,[status(thm)],[f436])).
% 2.64/1.84 thf(f436,plain,(
% 2.64/1.84 ! [X0 : nat,X1 : list_a,X2 : nat > a] : ((ord_less_eq_nat @ X0 @ (size_size_list_a @ X1)) => ((listIn1417627087drop_a @ X0 @ (listIn1312259492pend_a @ X1 @ X2)) = (listIn1312259492pend_a @ (drop_a @ X0 @ X1) @ X2)))),
% 2.64/1.84 inference(rectify,[status(thm)],[f34])).
% 2.64/1.84 thf(f34,axiom,(
% 2.64/1.84 ! [X4 : nat,X0 : list_a,X1 : nat > a] : ((ord_less_eq_nat @ X4 @ (size_size_list_a @ X0)) => ((listIn1417627087drop_a @ X4 @ (listIn1312259492pend_a @ X0 @ X1)) = (listIn1312259492pend_a @ (drop_a @ X4 @ X0) @ X1)))),
% 2.64/1.84 file('/export/starexec/sandbox2/tmp/tmp.YUokPb62dV/DTF2THF_25681.p',fact_33_i__append__i__drop__eq1)).
% 2.64/1.84 thf(f1561,plain,(
% 2.64/1.84 ((listIn1417627087drop_a @ n @ (listIn1312259492pend_a @ xs @ f)) != (listIn1312259492pend_a @ (drop_a @ n @ xs) @ f)) | spl86_2),
% 2.64/1.84 inference(avatar_component_clause,[status(thm)],[f1559])).
% 2.64/1.84 thf(f1567,plain,(
% 2.64/1.84 spl86_1 | ~spl86_3),
% 2.64/1.84 inference(avatar_split_clause,[status(thm)],[f1278,f1564,f1555])).
% 2.64/1.84 thf(f1278,plain,(
% 2.64/1.84 ((ord_less_nat @ n @ (size_size_list_a @ xs)) = $true) | ((listIn1417627087drop_a @ n @ (listIn1312259492pend_a @ xs @ f)) != (listIn1417627087drop_a @ (minus_minus_nat @ n @ (size_size_list_a @ xs)) @ f))),
% 2.64/1.84 inference(cnf_transformation,[status(thm)],[f958])).
% 2.64/1.84 thf(f958,plain,(
% 2.64/1.84 (((ord_less_nat @ n @ (size_size_list_a @ xs)) = $true) & ((listIn1417627087drop_a @ n @ (listIn1312259492pend_a @ xs @ f)) != (listIn1312259492pend_a @ (drop_a @ n @ xs) @ f))) | (((ord_less_nat @ n @ (size_size_list_a @ xs)) != $true) & ((listIn1417627087drop_a @ n @ (listIn1312259492pend_a @ xs @ f)) != (listIn1417627087drop_a @ (minus_minus_nat @ n @ (size_size_list_a @ xs)) @ f)))),
% 2.64/1.84 inference(ennf_transformation,[status(thm)],[f702])).
% 2.64/1.84 thf(f702,plain,(
% 2.64/1.84 ~((((ord_less_nat @ n @ (size_size_list_a @ xs)) = $true) => ((listIn1417627087drop_a @ n @ (listIn1312259492pend_a @ xs @ f)) = (listIn1312259492pend_a @ (drop_a @ n @ xs) @ f))) & (((ord_less_nat @ n @ (size_size_list_a @ xs)) != $true) => ((listIn1417627087drop_a @ n @ (listIn1312259492pend_a @ xs @ f)) = (listIn1417627087drop_a @ (minus_minus_nat @ n @ (size_size_list_a @ xs)) @ f))))),
% 2.64/1.84 inference(flattening,[status(thm)],[f369])).
% 2.64/1.84 thf(f369,plain,(
% 2.64/1.84 ~((((ord_less_nat @ n @ (size_size_list_a @ xs)) = $true) => ((listIn1417627087drop_a @ n @ (listIn1312259492pend_a @ xs @ f)) = (listIn1312259492pend_a @ (drop_a @ n @ xs) @ f))) & (~((ord_less_nat @ n @ (size_size_list_a @ xs)) = $true) => ((listIn1417627087drop_a @ n @ (listIn1312259492pend_a @ xs @ f)) = (listIn1417627087drop_a @ (minus_minus_nat @ n @ (size_size_list_a @ xs)) @ f))))),
% 2.64/1.84 inference(fool_elimination,[status(thm)],[f368])).
% 2.64/1.84 thf(f368,plain,(
% 2.64/1.84 ~(((ord_less_nat @ n @ (size_size_list_a @ xs)) => ((listIn1417627087drop_a @ n @ (listIn1312259492pend_a @ xs @ f)) = (listIn1312259492pend_a @ (drop_a @ n @ xs) @ f))) & (~(ord_less_nat @ n @ (size_size_list_a @ xs)) => ((listIn1417627087drop_a @ n @ (listIn1312259492pend_a @ xs @ f)) = (listIn1417627087drop_a @ (minus_minus_nat @ n @ (size_size_list_a @ xs)) @ f))))),
% 2.64/1.84 inference(rectify,[status(thm)],[f254])).
% 2.64/1.84 thf(f254,negated_conjecture,(
% 2.64/1.84 ~(((ord_less_nat @ n @ (size_size_list_a @ xs)) => ((listIn1417627087drop_a @ n @ (listIn1312259492pend_a @ xs @ f)) = (listIn1312259492pend_a @ (drop_a @ n @ xs) @ f))) & (~(ord_less_nat @ n @ (size_size_list_a @ xs)) => ((listIn1417627087drop_a @ n @ (listIn1312259492pend_a @ xs @ f)) = (listIn1417627087drop_a @ (minus_minus_nat @ n @ (size_size_list_a @ xs)) @ f))))),
% 2.64/1.84 inference(negated_conjecture,[status(cth)],[f253])).
% 2.64/1.84 thf(f253,conjecture,(
% 2.64/1.84 ((ord_less_nat @ n @ (size_size_list_a @ xs)) => ((listIn1417627087drop_a @ n @ (listIn1312259492pend_a @ xs @ f)) = (listIn1312259492pend_a @ (drop_a @ n @ xs) @ f))) & (~(ord_less_nat @ n @ (size_size_list_a @ xs)) => ((listIn1417627087drop_a @ n @ (listIn1312259492pend_a @ xs @ f)) = (listIn1417627087drop_a @ (minus_minus_nat @ n @ (size_size_list_a @ xs)) @ f)))),
% 2.64/1.84 file('/export/starexec/sandbox2/tmp/tmp.YUokPb62dV/DTF2THF_25681.p',conj_0)).
% 2.64/1.84 thf(f1562,plain,(
% 2.64/1.84 ~spl86_1 | ~spl86_2),
% 2.64/1.84 inference(avatar_split_clause,[status(thm)],[f1277,f1559,f1555])).
% 2.64/1.84 thf(f1277,plain,(
% 2.64/1.84 ((ord_less_nat @ n @ (size_size_list_a @ xs)) != $true) | ((listIn1417627087drop_a @ n @ (listIn1312259492pend_a @ xs @ f)) != (listIn1312259492pend_a @ (drop_a @ n @ xs) @ f))),
% 2.64/1.84 inference(cnf_transformation,[status(thm)],[f958])).
% 2.64/1.84 % SZS output end Proof for DTF2THF_25681
% 2.64/1.84 % (25871)------------------------------
% 2.64/1.84 % (25871)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.84 % (25871)Termination reason: Refutation
% 2.64/1.84
% 2.64/1.84 % (25871)Memory used [KB]: 7036
% 2.64/1.84 % (25871)Time elapsed: 0.073 s
% 2.64/1.84 % (25871)Instructions burned: 127 (million)
% 2.64/1.84 % (25871)------------------------------
% 2.64/1.84 % (25871)------------------------------
% 2.64/1.84 % (25856)Success in time 0.093 s
%------------------------------------------------------------------------------