%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP101^1 : 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 08:55:33 AM UTC 2026
% Result : Theorem 2.76s 1.64s
% Output : Refutation 2.76s
% Verified :
% SZS Type : Refutation
% Derivation depth : 9
% Number of leaves : 3
% Syntax : Number of formulae : 18 ( 10 unt; 0 typ; 0 def)
% Number of atoms : 117 ( 52 equ; 0 cnn)
% Maximal formula atoms : 10 ( 6 avg)
% Number of connectives : 338 ( 24 ~; 10 |; 15 &; 265 @)
% ( 0 <=>; 24 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 4 avg)
% Number of types : 7 ( 6 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 122 ( 119 usr; 6 con; 0-4 aty)
% Number of variables : 7 ( 0 ^; 7 !; 0 ?; 7 :)
% 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(type_def_10,type,
set_nat: $tType ).
thf(type_def_11,type,
set_a: $tType ).
thf(func_def_0,type,
list_nat: $tType ).
thf(func_def_1,type,
set_nat: $tType ).
thf(func_def_2,type,
list_a: $tType ).
thf(func_def_3,type,
set_a: $tType ).
thf(func_def_4,type,
nat: $tType ).
thf(func_def_5,type,
a: $tType ).
thf(func_def_6,type,
minus_minus_nat: nat > nat > nat ).
thf(func_def_7,type,
one_one_nat: nat ).
thf(func_def_8,type,
if_nat: $o > nat > nat > nat ).
thf(func_def_9,type,
if_a: $o > a > a > a ).
thf(func_def_10,type,
list_asc_nat: list_nat > $o ).
thf(func_def_11,type,
list_desc_nat: list_nat > $o ).
thf(func_def_12,type,
list_strict_asc_nat: list_nat > $o ).
thf(func_def_13,type,
list_strict_desc_nat: list_nat > $o ).
thf(func_def_14,type,
listIn923761578nd_nat: list_nat > ( nat > nat ) > nat > nat ).
thf(func_def_15,type,
listIn1312259492pend_a: list_a > ( nat > a ) > nat > a ).
thf(func_def_16,type,
append_nat: list_nat > list_nat > list_nat ).
thf(func_def_17,type,
append_a: list_a > list_a > list_a ).
thf(func_def_18,type,
butlast_nat: list_nat > list_nat ).
thf(func_def_19,type,
butlast_a: list_a > list_a ).
thf(func_def_20,type,
set_nat2: list_nat > set_nat ).
thf(func_def_21,type,
set_a2: list_a > set_a ).
thf(func_def_22,type,
list_ex_nat: ( nat > $o ) > list_nat > $o ).
thf(func_def_23,type,
list_ex_a: ( a > $o ) > list_a > $o ).
thf(func_def_24,type,
list_update_nat: list_nat > nat > nat > list_nat ).
thf(func_def_25,type,
list_update_a: list_a > nat > a > list_a ).
thf(func_def_26,type,
nth_nat: list_nat > nat > nat ).
thf(func_def_27,type,
nth_a: list_a > nat > a ).
thf(func_def_28,type,
size_size_list_nat: list_nat > nat ).
thf(func_def_29,type,
size_size_list_a: list_a > nat ).
thf(func_def_30,type,
ord_less_nat: nat > nat > $o ).
thf(func_def_31,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_32,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(func_def_33,type,
collect_a: ( a > $o ) > set_a ).
thf(func_def_34,type,
member_nat: nat > set_nat > $o ).
thf(func_def_35,type,
member_a: a > set_a > $o ).
thf(func_def_36,type,
f: nat > a ).
thf(func_def_37,type,
g: nat > a ).
thf(func_def_38,type,
x: nat ).
thf(func_def_39,type,
xs: list_a ).
thf(func_def_40,type,
ys: list_a ).
thf(func_def_55,type,
sP0: nat > nat > nat > $o ).
thf(func_def_56,type,
sP1: nat > nat > nat > $o ).
thf(func_def_57,type,
sP2: nat > nat > nat > $o ).
thf(func_def_58,type,
sP3: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).
thf(func_def_59,type,
sP4: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).
thf(func_def_60,type,
sP5: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).
thf(func_def_61,type,
sP6: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).
thf(func_def_62,type,
sK7: ( nat > nat ) > nat ).
thf(func_def_63,type,
sK8: ( nat > nat ) > nat ).
thf(func_def_64,type,
sK9: ( nat > nat > $o ) > nat ).
thf(func_def_65,type,
sK10: ( nat > nat > $o ) > nat ).
thf(func_def_66,type,
sK11: ( nat > nat > $o ) > nat ).
thf(func_def_67,type,
sK12: ( nat > nat > $o ) > nat ).
thf(func_def_68,type,
sK13: ( nat > nat > $o ) > nat ).
thf(func_def_69,type,
sK14: ( nat > nat ) > nat ).
thf(func_def_70,type,
sK15: ( nat > nat ) > nat ).
thf(func_def_71,type,
sK16: nat > ( nat > $o ) > nat ).
thf(func_def_72,type,
sK17: nat > ( nat > $o ) > nat ).
thf(func_def_73,type,
sK18: ( nat > nat ) > nat ).
thf(func_def_74,type,
sK19: ( nat > nat ) > nat ).
thf(func_def_75,type,
sK20: list_a > list_a > nat ).
thf(func_def_76,type,
sK21: ( nat > nat ) > nat ).
thf(func_def_77,type,
sK22: ( nat > nat ) > nat ).
thf(func_def_78,type,
sK23: nat > nat ).
thf(func_def_79,type,
sK24: nat > nat ).
thf(func_def_80,type,
sK25: ( nat > nat ) > nat ).
thf(func_def_81,type,
sK26: ( nat > nat ) > nat ).
thf(func_def_82,type,
sK27: ( nat > $o ) > nat ).
thf(func_def_83,type,
sK28: ( nat > nat ) > nat ).
thf(func_def_84,type,
sK29: ( nat > nat ) > nat ).
thf(func_def_85,type,
sK30: nat > ( nat > a > $o ) > list_a ).
thf(func_def_86,type,
sK31: nat > ( nat > a > $o ) > nat ).
thf(func_def_87,type,
sK32: nat > ( nat > a > $o ) > a ).
thf(func_def_88,type,
sK33: list_a > nat > ( nat > a > $o ) > nat ).
thf(func_def_89,type,
sK34: nat > nat > ( nat > $o ) > nat ).
thf(func_def_90,type,
sK35: nat > nat > ( nat > $o ) > nat ).
thf(func_def_91,type,
sK36: nat > nat ).
thf(func_def_92,type,
sK37: nat > nat ).
thf(func_def_93,type,
sK38: nat > nat ).
thf(func_def_94,type,
sK39: nat > nat ).
thf(func_def_95,type,
sK40: ( nat > $o ) > nat ).
thf(func_def_96,type,
sK41: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_97,type,
sK42: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_98,type,
sK43: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_99,type,
sK44: ( nat > $o ) > nat > nat ).
thf(func_def_100,type,
sK45: ( nat > $o ) > nat ).
thf(func_def_101,type,
sK46: ( nat > nat ) > nat ).
thf(func_def_102,type,
sK47: ( nat > nat ) > nat ).
thf(func_def_103,type,
sK48: nat > nat ).
thf(func_def_104,type,
sK49: ( nat > nat ) > nat ).
thf(func_def_105,type,
sK50: ( nat > nat ) > nat ).
thf(func_def_106,type,
sK51: ( nat > $o ) > nat ).
thf(func_def_107,type,
sK52: ( nat > nat ) > nat ).
thf(func_def_108,type,
sK53: ( nat > nat ) > nat ).
thf(func_def_109,type,
sK54: ( nat > nat ) > nat ).
thf(func_def_110,type,
sK55: ( nat > nat ) > nat ).
thf(func_def_111,type,
sK56: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_112,type,
sK57: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_113,type,
sK58: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_114,type,
sK59: nat > nat ).
thf(func_def_115,type,
sK60: ( nat > nat ) > nat ).
thf(func_def_116,type,
sK61: ( nat > nat ) > nat ).
thf(func_def_117,type,
sK62: ( nat > nat ) > nat ).
thf(func_def_118,type,
sK63: ( nat > nat ) > nat ).
thf(func_def_119,type,
sK64: nat > nat ).
thf(func_def_120,type,
sK65: ( nat > nat > $o ) > nat ).
thf(func_def_121,type,
sK66: ( nat > nat > $o ) > nat ).
thf(func_def_122,type,
sK67: ( nat > nat > $o ) > nat ).
thf(func_def_123,type,
sK68: ( nat > nat > $o ) > nat ).
thf(func_def_124,type,
sK69: nat > list_a ).
thf(func_def_125,type,
sK70: ( list_a > $o ) > list_a ).
thf(func_def_126,type,
sK71: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_127,type,
sK72: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_128,type,
sK73: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_129,type,
sK74: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_130,type,
sK75: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_131,type,
sK76: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_132,type,
sK77: nat > nat ).
thf(func_def_133,type,
sK78: ( nat > nat ) > nat ).
thf(func_def_134,type,
sK79: ( nat > nat ) > nat ).
thf(func_def_135,type,
sK80: nat > nat ).
thf(func_def_136,type,
sK81: nat > nat ).
thf(func_def_137,type,
sK82: nat > nat ).
thf(func_def_139,type,
ph84:
!>[X0: $tType] : X0 ).
thf(f1571,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f1566,f1225]) ).
thf(f1225,plain,
( ( ord_less_nat @ x @ ( size_size_list_a @ ys ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f377]) ).
thf(f377,plain,
( ( ord_less_nat @ x @ ( size_size_list_a @ ys ) )
= $true ),
inference(fool_elimination,[status(thm)],[f376]) ).
thf(f376,plain,
ord_less_nat @ x @ ( size_size_list_a @ ys ),
inference(rectify,[status(thm)],[f266]) ).
thf(f266,axiom,
ord_less_nat @ x @ ( size_size_list_a @ ys ),
file('/export/starexec/sandbox2/tmp/tmp.wBFq0AA5QN/DTF2THF_27082.p',conj_2) ).
thf(f1566,plain,
( ( ord_less_nat @ x @ ( size_size_list_a @ ys ) )
!= $true ),
inference(trivial_inequality_removal,[status(thm)],[f1562]) ).
thf(f1562,plain,
( ( ( nth_a @ xs @ x )
!= ( nth_a @ xs @ x ) )
| ( ( ord_less_nat @ x @ ( size_size_list_a @ ys ) )
!= $true ) ),
inference(superposition,[status(thm)],[f1333,f1455]) ).
thf(f1455,plain,
! [X0: nat] :
( ( ( nth_a @ ys @ X0 )
= ( nth_a @ xs @ X0 ) )
| ( ( ord_less_nat @ X0 @ ( size_size_list_a @ ys ) )
!= $true ) ),
inference(duplicate_literal_removal,[status(thm)],[f1305]) ).
thf(f1305,plain,
! [X0: nat] :
( ( ( ord_less_nat @ X0 @ ( size_size_list_a @ ys ) )
!= $true )
| ( ( ord_less_nat @ X0 @ ( size_size_list_a @ ys ) )
!= $true )
| ( ( nth_a @ ys @ X0 )
= ( nth_a @ xs @ X0 ) ) ),
inference(cnf_transformation,[status(thm)],[f971]) ).
thf(f971,plain,
! [X0: nat] :
( ( ( ( ord_less_nat @ X0 @ ( size_size_list_a @ ys ) )
= $true )
| ( ( ( ( ord_less_nat @ X0 @ ( size_size_list_a @ ys ) )
= $true )
| ( ( g @ ( minus_minus_nat @ X0 @ ( size_size_list_a @ ys ) ) )
= ( f @ ( minus_minus_nat @ X0 @ ( size_size_list_a @ xs ) ) ) ) )
& ( ( ( g @ ( minus_minus_nat @ X0 @ ( size_size_list_a @ ys ) ) )
= ( nth_a @ xs @ X0 ) )
| ( ( ord_less_nat @ X0 @ ( size_size_list_a @ ys ) )
!= $true ) ) ) )
& ( ( ( ord_less_nat @ X0 @ ( size_size_list_a @ ys ) )
!= $true )
| ( ( ( ( nth_a @ ys @ X0 )
= ( nth_a @ xs @ X0 ) )
| ( ( ord_less_nat @ X0 @ ( size_size_list_a @ ys ) )
!= $true ) )
& ( ( ( ord_less_nat @ X0 @ ( size_size_list_a @ ys ) )
= $true )
| ( ( nth_a @ ys @ X0 )
= ( f @ ( minus_minus_nat @ X0 @ ( size_size_list_a @ xs ) ) ) ) ) ) ) ),
inference(ennf_transformation,[status(thm)],[f749]) ).
thf(f749,plain,
! [X0: nat] :
( ( ( ( ord_less_nat @ X0 @ ( size_size_list_a @ ys ) )
!= $true )
=> ( ( ( ( ord_less_nat @ X0 @ ( size_size_list_a @ ys ) )
!= $true )
=> ( ( g @ ( minus_minus_nat @ X0 @ ( size_size_list_a @ ys ) ) )
= ( f @ ( minus_minus_nat @ X0 @ ( size_size_list_a @ xs ) ) ) ) )
& ( ( ( ord_less_nat @ X0 @ ( size_size_list_a @ ys ) )
= $true )
=> ( ( g @ ( minus_minus_nat @ X0 @ ( size_size_list_a @ ys ) ) )
= ( nth_a @ xs @ X0 ) ) ) ) )
& ( ( ( ord_less_nat @ X0 @ ( size_size_list_a @ ys ) )
= $true )
=> ( ( ( ( ord_less_nat @ X0 @ ( size_size_list_a @ ys ) )
= $true )
=> ( ( nth_a @ ys @ X0 )
= ( nth_a @ xs @ X0 ) ) )
& ( ( ( ord_less_nat @ X0 @ ( size_size_list_a @ ys ) )
!= $true )
=> ( ( nth_a @ ys @ X0 )
= ( f @ ( minus_minus_nat @ X0 @ ( size_size_list_a @ xs ) ) ) ) ) ) ) ),
inference(flattening,[status(thm)],[f569]) ).
thf(f569,plain,
! [X0: nat] :
( ( ( ( ord_less_nat @ X0 @ ( size_size_list_a @ ys ) )
= $true )
=> ( ( ( ( ord_less_nat @ X0 @ ( size_size_list_a @ ys ) )
= $true )
=> ( ( nth_a @ ys @ X0 )
= ( nth_a @ xs @ X0 ) ) )
& ( ( ( ord_less_nat @ X0 @ ( size_size_list_a @ ys ) )
!= $true )
=> ( ( nth_a @ ys @ X0 )
= ( f @ ( minus_minus_nat @ X0 @ ( size_size_list_a @ xs ) ) ) ) ) ) )
& ( ( ( ord_less_nat @ X0 @ ( size_size_list_a @ ys ) )
!= $true )
=> ( ( ( ( ord_less_nat @ X0 @ ( size_size_list_a @ ys ) )
= $true )
=> ( ( g @ ( minus_minus_nat @ X0 @ ( size_size_list_a @ ys ) ) )
= ( nth_a @ xs @ X0 ) ) )
& ( ( ( ord_less_nat @ X0 @ ( size_size_list_a @ ys ) )
!= $true )
=> ( ( g @ ( minus_minus_nat @ X0 @ ( size_size_list_a @ ys ) ) )
= ( f @ ( minus_minus_nat @ X0 @ ( size_size_list_a @ xs ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f568]) ).
thf(f568,plain,
! [X0: nat] :
( ( ( ord_less_nat @ X0 @ ( size_size_list_a @ ys ) )
=> ( ( ( ord_less_nat @ X0 @ ( size_size_list_a @ ys ) )
=> ( ( nth_a @ ys @ X0 )
= ( nth_a @ xs @ X0 ) ) )
& ( ~ ( ord_less_nat @ X0 @ ( size_size_list_a @ ys ) )
=> ( ( nth_a @ ys @ X0 )
= ( f @ ( minus_minus_nat @ X0 @ ( size_size_list_a @ xs ) ) ) ) ) ) )
& ( ~ ( ord_less_nat @ X0 @ ( size_size_list_a @ ys ) )
=> ( ( ( ord_less_nat @ X0 @ ( size_size_list_a @ ys ) )
=> ( ( g @ ( minus_minus_nat @ X0 @ ( size_size_list_a @ ys ) ) )
= ( nth_a @ xs @ X0 ) ) )
& ( ~ ( ord_less_nat @ X0 @ ( size_size_list_a @ ys ) )
=> ( ( g @ ( minus_minus_nat @ X0 @ ( size_size_list_a @ ys ) ) )
= ( f @ ( minus_minus_nat @ X0 @ ( size_size_list_a @ xs ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f265]) ).
thf(f265,axiom,
! [X52: nat] :
( ( ( ord_less_nat @ X52 @ ( size_size_list_a @ ys ) )
=> ( ( ( ord_less_nat @ X52 @ ( size_size_list_a @ ys ) )
=> ( ( nth_a @ xs @ X52 )
= ( nth_a @ ys @ X52 ) ) )
& ( ~ ( ord_less_nat @ X52 @ ( size_size_list_a @ ys ) )
=> ( ( nth_a @ ys @ X52 )
= ( f @ ( minus_minus_nat @ X52 @ ( size_size_list_a @ xs ) ) ) ) ) ) )
& ( ~ ( ord_less_nat @ X52 @ ( size_size_list_a @ ys ) )
=> ( ( ( ord_less_nat @ X52 @ ( size_size_list_a @ ys ) )
=> ( ( nth_a @ xs @ X52 )
= ( g @ ( minus_minus_nat @ X52 @ ( size_size_list_a @ ys ) ) ) ) )
& ( ~ ( ord_less_nat @ X52 @ ( size_size_list_a @ ys ) )
=> ( ( f @ ( minus_minus_nat @ X52 @ ( size_size_list_a @ xs ) ) )
= ( g @ ( minus_minus_nat @ X52 @ ( size_size_list_a @ ys ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.wBFq0AA5QN/DTF2THF_27082.p',conj_1) ).
thf(f1333,plain,
( ( nth_a @ xs @ x )
!= ( nth_a @ ys @ x ) ),
inference(cnf_transformation,[status(thm)],[f748]) ).
thf(f748,plain,
( ( nth_a @ xs @ x )
!= ( nth_a @ ys @ x ) ),
inference(flattening,[status(thm)],[f268]) ).
thf(f268,negated_conjecture,
( ( nth_a @ xs @ x )
!= ( nth_a @ ys @ x ) ),
inference(negated_conjecture,[status(cth)],[f267]) ).
thf(f267,conjecture,
( ( nth_a @ xs @ x )
= ( nth_a @ ys @ x ) ),
file('/export/starexec/sandbox2/tmp/tmp.wBFq0AA5QN/DTF2THF_27082.p',conj_3) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP101^1 : TPTP v9.2.1. Released v7.5.0.
% 0.13/0.13 % 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 08:26:46 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
% 0.20/0.46 ---- Original DTF file ---
% 0.20/0.46 thf(spec,logic,$$dhol).
% 0.20/0.46 %------------------------------------------------------------------------------
% 0.20/0.46 % File : ITP101^1 : TPTP v9.2.1. Released v7.5.0.
% 0.20/0.46 % Domain : Interactive Theorem Proving
% 0.20/0.46 % Problem : Sledgehammer ListInf problem prob_63__5408460_1
% 0.20/0.46 % Version : Especial.
% 0.20/0.46 % English :
% 0.20/0.46
% 0.20/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.20/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.20/0.46 % Source : [Des21]
% 0.20/0.46 % Names : ListInf/prob_63__5408460_1 [Des21]
% 0.20/0.46
% 0.20/0.46 % Status : Theorem
% 0.20/0.46 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.08 v8.1.0, 0.09 v7.5.0
% 0.20/0.46 % Syntax : Number of formulae : 308 ( 88 unt; 41 typ; 0 def)
% 0.20/0.46 % Number of atoms : 785 ( 279 equ; 0 cnn)
% 0.20/0.46 % Maximal formula atoms : 12 ( 2 avg)
% 0.20/0.46 % Number of connectives : 2464 ( 92 ~; 26 |; 43 &;1920 @)
% 0.20/0.46 % ( 0 <=>; 383 =>; 0 <=; 0 <~>)
% 0.20/0.46 % Maximal formula depth : 18 ( 7 avg)
% 0.20/0.46 % Number of types : 7 ( 6 usr)
% 0.20/0.46 % Number of type conns : 128 ( 128 >; 0 *; 0 +; 0 <<)
% 0.20/0.46 % Number of symbols : 38 ( 35 usr; 6 con; 0-3 aty)
% 0.20/0.46 % Number of variables : 816 ( 71 ^; 704 !; 41 ?; 816 :)
% 0.20/0.46 % SPC : TH0_THM_EQU_NAR
% 0.20/0.46
% 0.20/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 15:30:16.735
% 0.20/0.46 %------------------------------------------------------------------------------
% 0.20/0.46 % Could-be-implicit typings (6)
% 0.20/0.46 thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
% 0.20/0.46 list_nat: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 0.20/0.46 set_nat: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_n_t__List__Olist_Itf__a_J,type,
% 0.20/0.46 list_a: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 0.20/0.46 set_a: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_n_t__Nat__Onat,type,
% 0.20/0.46 nat: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_n_tf__a,type,
% 0.20/0.46 a: $tType ).
% 0.20/0.46
% 0.20/0.46 % Explicit typings (35)
% 0.20/0.46 thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
% 0.20/0.46 minus_minus_nat: nat > nat > nat ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
% 0.20/0.46 one_one_nat: nat ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_If_001t__Nat__Onat,type,
% 0.20/0.46 if_nat: $o > nat > nat > nat ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_If_001tf__a,type,
% 0.20/0.46 if_a: $o > a > a > a ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_List2_Olist__asc_001t__Nat__Onat,type,
% 0.20/0.46 list_asc_nat: list_nat > $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_List2_Olist__desc_001t__Nat__Onat,type,
% 0.20/0.46 list_desc_nat: list_nat > $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_List2_Olist__strict__asc_001t__Nat__Onat,type,
% 0.20/0.46 list_strict_asc_nat: list_nat > $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_List2_Olist__strict__desc_001t__Nat__Onat,type,
% 0.20/0.46 list_strict_desc_nat: list_nat > $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_ListInf__Mirabelle__akbajwqfbr_Oi__append_001t__Nat__Onat,type,
% 0.20/0.46 listIn923761578nd_nat: list_nat > ( nat > nat ) > nat > nat ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_ListInf__Mirabelle__akbajwqfbr_Oi__append_001tf__a,type,
% 0.20/0.46 listIn1312259492pend_a: list_a > ( nat > a ) > nat > a ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_List_Oappend_001t__Nat__Onat,type,
% 0.20/0.46 append_nat: list_nat > list_nat > list_nat ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_List_Oappend_001tf__a,type,
% 0.20/0.46 append_a: list_a > list_a > list_a ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_List_Obutlast_001t__Nat__Onat,type,
% 0.20/0.46 butlast_nat: list_nat > list_nat ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_List_Obutlast_001tf__a,type,
% 0.20/0.46 butlast_a: list_a > list_a ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
% 0.20/0.46 set_nat2: list_nat > set_nat ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_List_Olist_Oset_001tf__a,type,
% 0.20/0.46 set_a2: list_a > set_a ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_List_Olist__ex_001t__Nat__Onat,type,
% 0.20/0.46 list_ex_nat: ( nat > $o ) > list_nat > $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_List_Olist__ex_001tf__a,type,
% 0.20/0.46 list_ex_a: ( a > $o ) > list_a > $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
% 0.20/0.46 list_update_nat: list_nat > nat > nat > list_nat ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_List_Olist__update_001tf__a,type,
% 0.20/0.46 list_update_a: list_a > nat > a > list_a ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_List_Onth_001t__Nat__Onat,type,
% 0.20/0.46 nth_nat: list_nat > nat > nat ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_List_Onth_001tf__a,type,
% 0.20/0.46 nth_a: list_a > nat > a ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
% 0.20/0.46 size_size_list_nat: list_nat > nat ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
% 0.20/0.46 size_size_list_a: list_a > nat ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 0.20/0.46 ord_less_nat: nat > nat > $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 0.20/0.46 ord_less_eq_nat: nat > nat > $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
% 0.20/0.46 ord_less_eq_set_a: set_a > set_a > $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Set_OCollect_001tf__a,type,
% 0.20/0.46 collect_a: ( a > $o ) > set_a ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_member_001t__Nat__Onat,type,
% 0.20/0.46 member_nat: nat > set_nat > $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_member_001tf__a,type,
% 0.20/0.46 member_a: a > set_a > $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_v_f,type,
% 0.20/0.46 f: nat > a ).
% 0.20/0.46
% 0.20/0.46 thf(sy_v_g,type,
% 0.20/0.46 g: nat > a ).
% 0.20/0.46
% 0.20/0.46 thf(sy_v_x,type,
% 0.20/0.46 x: nat ).
% 0.20/0.46
% 0.20/0.46 thf(sy_v_xs,type,
% 0.20/0.46 xs: list_a ).
% 0.20/0.46
% 0.20/0.46 thf(sy_v_ys,type,
% 0.20/0.46 ys: list_a ).
% 0.20/0.46
% 0.20/0.46 % Relevant facts (258)
% 0.20/0.46 thf(fact_0_nth__equalityI,axiom,
% 0.20/0.46 ! [Xs: list_nat,Ys: list_nat] :
% 0.20/0.46 ( ( ( size_size_list_nat @ Xs )
% 0.20/0.46 = ( size_size_list_nat @ Ys ) )
% 0.20/0.46 => ( ! [I: nat] :
% 0.20/0.46 ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
% 0.20/0.46 => ( ( nth_nat @ Xs @ I )
% 0.20/0.46 = ( nth_nat @ Ys @ I ) ) )
% 0.20/0.46 => ( Xs = Ys ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % nth_equalityI
% 0.20/0.46 thf(fact_1_nth__equalityI,axiom,
% 0.20/0.46 ! [Xs: list_a,Ys: list_a] :
% 0.20/0.46 ( ( ( size_size_list_a @ Xs )
% 0.20/0.46 = ( size_size_list_a @ Ys ) )
% 0.20/0.46 => ( ! [I: nat] :
% 0.20/0.46 ( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
% 0.20/0.46 => ( ( nth_a @ Xs @ I )
% 0.20/0.46 = ( nth_a @ Ys @ I ) ) )
% 0.20/0.46 => ( Xs = Ys ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % nth_equalityI
% 0.20/0.46 thf(fact_2_Skolem__list__nth,axiom,
% 0.20/0.46 ! [K: nat,P: nat > nat > $o] :
% 0.20/0.46 ( ( ! [I2: nat] :
% 0.20/0.46 ( ( ord_less_nat @ I2 @ K )
% 0.20/0.46 => ? [X: nat] : ( P @ I2 @ X ) ) )
% 0.20/0.46 = ( ? [Xs2: list_nat] :
% 0.20/0.46 ( ( ( size_size_list_nat @ Xs2 )
% 0.20/0.46 = K )
% 0.20/0.46 & ! [I2: nat] :
% 0.20/0.46 ( ( ord_less_nat @ I2 @ K )
% 0.20/0.46 => ( P @ I2 @ ( nth_nat @ Xs2 @ I2 ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Skolem_list_nth
% 0.20/0.46 thf(fact_3_Skolem__list__nth,axiom,
% 0.20/0.46 ! [K: nat,P: nat > a > $o] :
% 0.20/0.46 ( ( ! [I2: nat] :
% 0.20/0.46 ( ( ord_less_nat @ I2 @ K )
% 0.20/0.46 => ? [X: a] : ( P @ I2 @ X ) ) )
% 0.20/0.46 = ( ? [Xs2: list_a] :
% 0.20/0.46 ( ( ( size_size_list_a @ Xs2 )
% 0.20/0.46 = K )
% 0.20/0.46 & ! [I2: nat] :
% 0.20/0.46 ( ( ord_less_nat @ I2 @ K )
% 0.20/0.46 => ( P @ I2 @ ( nth_a @ Xs2 @ I2 ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Skolem_list_nth
% 0.20/0.46 thf(fact_4_list__eq__iff__nth__eq,axiom,
% 0.20/0.46 ( ( ^ [Y: list_nat,Z: list_nat] : ( Y = Z ) )
% 0.20/0.46 = ( ^ [Xs2: list_nat,Ys2: list_nat] :
% 0.20/0.46 ( ( ( size_size_list_nat @ Xs2 )
% 0.20/0.46 = ( size_size_list_nat @ Ys2 ) )
% 0.20/0.46 & ! [I2: nat] :
% 0.20/0.46 ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
% 0.20/0.46 => ( ( nth_nat @ Xs2 @ I2 )
% 0.20/0.46 = ( nth_nat @ Ys2 @ I2 ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % list_eq_iff_nth_eq
% 0.20/0.46 thf(fact_5_list__eq__iff__nth__eq,axiom,
% 0.20/0.46 ( ( ^ [Y: list_a,Z: list_a] : ( Y = Z ) )
% 0.20/0.46 = ( ^ [Xs2: list_a,Ys2: list_a] :
% 0.20/0.46 ( ( ( size_size_list_a @ Xs2 )
% 0.20/0.46 = ( size_size_list_a @ Ys2 ) )
% 0.20/0.46 & ! [I2: nat] :
% 0.20/0.46 ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs2 ) )
% 0.20/0.46 => ( ( nth_a @ Xs2 @ I2 )
% 0.20/0.46 = ( nth_a @ Ys2 @ I2 ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % list_eq_iff_nth_eq
% 0.20/0.46 thf(fact_6_i__append__nth1,axiom,
% 0.20/0.46 ! [N: nat,Xs: list_nat,F: nat > nat] :
% 0.20/0.46 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 0.20/0.46 => ( ( listIn923761578nd_nat @ Xs @ F @ N )
% 0.20/0.46 = ( nth_nat @ Xs @ N ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % i_append_nth1
% 0.20/0.46 thf(fact_7_i__append__nth1,axiom,
% 0.20/0.46 ! [N: nat,Xs: list_a,F: nat > a] :
% 0.20/0.46 ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
% 0.20/0.46 => ( ( listIn1312259492pend_a @ Xs @ F @ N )
% 0.20/0.46 = ( nth_a @ Xs @ N ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % i_append_nth1
% 0.20/0.46 thf(fact_8_diff__less__mono2,axiom,
% 0.20/0.46 ! [M: nat,N: nat,L: nat] :
% 0.20/0.46 ( ( ord_less_nat @ M @ N )
% 0.20/0.46 => ( ( ord_less_nat @ M @ L )
% 0.20/0.46 => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % diff_less_mono2
% 0.20/0.46 thf(fact_9_less__imp__diff__less,axiom,
% 0.20/0.46 ! [J: nat,K: nat,N: nat] :
% 0.20/0.46 ( ( ord_less_nat @ J @ K )
% 0.20/0.46 => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% 0.20/0.46
% 0.20/0.46 % less_imp_diff_less
% 0.20/0.46 thf(fact_10_less__diff__imp__less,axiom,
% 0.20/0.46 ! [I3: nat,J: nat,M: nat] :
% 0.20/0.46 ( ( ord_less_nat @ I3 @ ( minus_minus_nat @ J @ M ) )
% 0.20/0.46 => ( ord_less_nat @ I3 @ J ) ) ).
% 0.20/0.46
% 0.20/0.46 % less_diff_imp_less
% 0.20/0.46 thf(fact_11_nat__diff__left__cancel__eq1,axiom,
% 0.20/0.46 ! [K: nat,M: nat,N: nat] :
% 0.20/0.46 ( ( ( minus_minus_nat @ K @ M )
% 0.20/0.46 = ( minus_minus_nat @ K @ N ) )
% 0.20/0.46 => ( ( ord_less_nat @ M @ K )
% 0.20/0.46 => ( M = N ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % nat_diff_left_cancel_eq1
% 0.20/0.46 thf(fact_12_nat__diff__left__cancel__eq2,axiom,
% 0.20/0.46 ! [K: nat,M: nat,N: nat] :
% 0.20/0.46 ( ( ( minus_minus_nat @ K @ M )
% 0.20/0.46 = ( minus_minus_nat @ K @ N ) )
% 0.20/0.46 => ( ( ord_less_nat @ N @ K )
% 0.20/0.46 => ( M = N ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % nat_diff_left_cancel_eq2
% 0.20/0.46 thf(fact_13_nat__diff__left__cancel__less,axiom,
% 0.20/0.46 ! [K: nat,M: nat,N: nat] :
% 0.20/0.46 ( ( ord_less_nat @ ( minus_minus_nat @ K @ M ) @ ( minus_minus_nat @ K @ N ) )
% 0.20/0.46 => ( ord_less_nat @ N @ M ) ) ).
% 0.20/0.46
% 0.20/0.46 % nat_diff_left_cancel_less
% 0.20/0.46 thf(fact_14_nat__diff__right__cancel__eq1,axiom,
% 0.20/0.46 ! [M: nat,K: nat,N: nat] :
% 0.20/0.46 ( ( ( minus_minus_nat @ M @ K )
% 0.20/0.46 = ( minus_minus_nat @ N @ K ) )
% 0.20/0.46 => ( ( ord_less_nat @ K @ M )
% 0.20/0.46 => ( M = N ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % nat_diff_right_cancel_eq1
% 0.20/0.46 thf(fact_15_linorder__neqE__nat,axiom,
% 0.20/0.46 ! [X2: nat,Y2: nat] :
% 0.20/0.46 ( ( X2 != Y2 )
% 0.20/0.46 => ( ~ ( ord_less_nat @ X2 @ Y2 )
% 0.20/0.46 => ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % linorder_neqE_nat
% 0.20/0.46 thf(fact_16_infinite__descent,axiom,
% 0.20/0.46 ! [P: nat > $o,N: nat] :
% 0.20/0.46 ( ! [N2: nat] :
% 0.20/0.46 ( ~ ( P @ N2 )
% 0.20/0.46 => ? [M2: nat] :
% 0.20/0.46 ( ( ord_less_nat @ M2 @ N2 )
% 0.20/0.46 & ~ ( P @ M2 ) ) )
% 0.20/0.46 => ( P @ N ) ) ).
% 0.20/0.46
% 0.20/0.46 % infinite_descent
% 0.20/0.46 thf(fact_17_nat__less__induct,axiom,
% 0.20/0.46 ! [P: nat > $o,N: nat] :
% 0.20/0.46 ( ! [N2: nat] :
% 0.20/0.46 ( ! [M2: nat] :
% 0.20/0.46 ( ( ord_less_nat @ M2 @ N2 )
% 0.20/0.46 => ( P @ M2 ) )
% 0.20/0.46 => ( P @ N2 ) )
% 0.20/0.46 => ( P @ N ) ) ).
% 0.20/0.46
% 0.20/0.46 % nat_less_induct
% 0.20/0.46 thf(fact_18_less__irrefl__nat,axiom,
% 0.20/0.46 ! [N: nat] :
% 0.20/0.46 ~ ( ord_less_nat @ N @ N ) ).
% 0.20/0.46
% 0.20/0.46 % less_irrefl_nat
% 0.20/0.46 thf(fact_19_less__not__refl3,axiom,
% 0.20/0.46 ! [S: nat,T: nat] :
% 0.20/0.46 ( ( ord_less_nat @ S @ T )
% 0.20/0.46 => ( S != T ) ) ).
% 0.20/0.46
% 0.20/0.46 % less_not_refl3
% 0.20/0.46 thf(fact_20_less__not__refl2,axiom,
% 0.20/0.46 ! [N: nat,M: nat] :
% 0.20/0.46 ( ( ord_less_nat @ N @ M )
% 0.20/0.46 => ( M != N ) ) ).
% 0.20/0.46
% 0.20/0.46 % less_not_refl2
% 0.20/0.46 thf(fact_21_less__not__refl,axiom,
% 0.20/0.46 ! [N: nat] :
% 0.20/0.46 ~ ( ord_less_nat @ N @ N ) ).
% 0.20/0.46
% 0.20/0.46 % less_not_refl
% 0.20/0.46 thf(fact_22_nat__neq__iff,axiom,
% 0.20/0.46 ! [M: nat,N: nat] :
% 0.20/0.46 ( ( M != N )
% 0.20/0.46 = ( ( ord_less_nat @ M @ N )
% 0.20/0.46 | ( ord_less_nat @ N @ M ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % nat_neq_iff
% 0.20/0.46 thf(fact_23_size__neq__size__imp__neq,axiom,
% 0.20/0.46 ! [X2: list_nat,Y2: list_nat] :
% 0.20/0.46 ( ( ( size_size_list_nat @ X2 )
% 0.20/0.46 != ( size_size_list_nat @ Y2 ) )
% 0.20/0.46 => ( X2 != Y2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % size_neq_size_imp_neq
% 0.20/0.46 thf(fact_24_size__neq__size__imp__neq,axiom,
% 0.20/0.46 ! [X2: list_a,Y2: list_a] :
% 0.20/0.46 ( ( ( size_size_list_a @ X2 )
% 0.20/0.46 != ( size_size_list_a @ Y2 ) )
% 0.20/0.46 => ( X2 != Y2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % size_neq_size_imp_neq
% 0.20/0.46 thf(fact_25_neq__if__length__neq,axiom,
% 0.20/0.46 ! [Xs: list_nat,Ys: list_nat] :
% 0.20/0.46 ( ( ( size_size_list_nat @ Xs )
% 0.20/0.46 != ( size_size_list_nat @ Ys ) )
% 0.20/0.46 => ( Xs != Ys ) ) ).
% 0.20/0.46
% 0.20/0.46 % neq_if_length_neq
% 0.20/0.46 thf(fact_26_neq__if__length__neq,axiom,
% 0.20/0.46 ! [Xs: list_a,Ys: list_a] :
% 0.20/0.46 ( ( ( size_size_list_a @ Xs )
% 0.20/0.46 != ( size_size_list_a @ Ys ) )
% 0.20/0.46 => ( Xs != Ys ) ) ).
% 0.20/0.46
% 0.20/0.46 % neq_if_length_neq
% 0.20/0.46 thf(fact_27_Ex__list__of__length,axiom,
% 0.20/0.46 ! [N: nat] :
% 0.20/0.46 ? [Xs3: list_nat] :
% 0.20/0.46 ( ( size_size_list_nat @ Xs3 )
% 0.20/0.46 = N ) ).
% 0.20/0.46
% 0.20/0.46 % Ex_list_of_length
% 0.20/0.46 thf(fact_28_Ex__list__of__length,axiom,
% 0.20/0.46 ! [N: nat] :
% 0.20/0.46 ? [Xs3: list_a] :
% 0.20/0.46 ( ( size_size_list_a @ Xs3 )
% 0.20/0.46 = N ) ).
% 0.20/0.46
% 0.20/0.46 % Ex_list_of_length
% 0.20/0.46 thf(fact_29_eq__imp__diff__eq,axiom,
% 0.20/0.46 ! [M: nat,N: nat,K: nat] :
% 0.20/0.46 ( ( M = N )
% 0.20/0.46 => ( ( minus_minus_nat @ M @ K )
% 0.20/0.46 = ( minus_minus_nat @ N @ K ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % eq_imp_diff_eq
% 0.20/0.46 thf(fact_30_diff__commute,axiom,
% 0.20/0.46 ! [I3: nat,J: nat,K: nat] :
% 0.20/0.46 ( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J ) @ K )
% 0.20/0.46 = ( minus_minus_nat @ ( minus_minus_nat @ I3 @ K ) @ J ) ) ).
% 0.20/0.46
% 0.20/0.46 % diff_commute
% 0.20/0.46 thf(fact_31_i__append__nth,axiom,
% 0.20/0.46 ( listIn923761578nd_nat
% 0.20/0.46 = ( ^ [Xs2: list_nat,F2: nat > nat,N3: nat] : ( if_nat @ ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs2 ) ) @ ( nth_nat @ Xs2 @ N3 ) @ ( F2 @ ( minus_minus_nat @ N3 @ ( size_size_list_nat @ Xs2 ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % i_append_nth
% 0.20/0.46 thf(fact_32_i__append__nth,axiom,
% 0.20/0.46 ( listIn1312259492pend_a
% 0.20/0.46 = ( ^ [Xs2: list_a,F2: nat > a,N3: nat] : ( if_a @ ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs2 ) ) @ ( nth_a @ Xs2 @ N3 ) @ ( F2 @ ( minus_minus_nat @ N3 @ ( size_size_list_a @ Xs2 ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % i_append_nth
% 0.20/0.46 thf(fact_33_i__append__def,axiom,
% 0.20/0.46 ( listIn923761578nd_nat
% 0.20/0.46 = ( ^ [Xs2: list_nat,F2: nat > nat,N3: nat] : ( if_nat @ ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs2 ) ) @ ( nth_nat @ Xs2 @ N3 ) @ ( F2 @ ( minus_minus_nat @ N3 @ ( size_size_list_nat @ Xs2 ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % i_append_def
% 0.20/0.46 thf(fact_34_i__append__def,axiom,
% 0.20/0.46 ( listIn1312259492pend_a
% 0.20/0.46 = ( ^ [Xs2: list_a,F2: nat > a,N3: nat] : ( if_a @ ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs2 ) ) @ ( nth_a @ Xs2 @ N3 ) @ ( F2 @ ( minus_minus_nat @ N3 @ ( size_size_list_a @ Xs2 ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % i_append_def
% 0.20/0.46 thf(fact_35_length__induct,axiom,
% 0.20/0.46 ! [P: list_nat > $o,Xs: list_nat] :
% 0.20/0.46 ( ! [Xs3: list_nat] :
% 0.20/0.46 ( ! [Ys3: list_nat] :
% 0.20/0.46 ( ( ord_less_nat @ ( size_size_list_nat @ Ys3 ) @ ( size_size_list_nat @ Xs3 ) )
% 0.20/0.46 => ( P @ Ys3 ) )
% 0.20/0.46 => ( P @ Xs3 ) )
% 0.20/0.46 => ( P @ Xs ) ) ).
% 0.20/0.46
% 0.20/0.46 % length_induct
% 0.20/0.46 thf(fact_36_length__induct,axiom,
% 0.20/0.46 ! [P: list_a > $o,Xs: list_a] :
% 0.20/0.46 ( ! [Xs3: list_a] :
% 0.20/0.46 ( ! [Ys3: list_a] :
% 0.20/0.46 ( ( ord_less_nat @ ( size_size_list_a @ Ys3 ) @ ( size_size_list_a @ Xs3 ) )
% 0.20/0.46 => ( P @ Ys3 ) )
% 0.20/0.46 => ( P @ Xs3 ) )
% 0.20/0.46 => ( P @ Xs ) ) ).
% 0.20/0.46
% 0.20/0.46 % length_induct
% 0.20/0.46 thf(fact_37_nat__diff__right__cancel__less,axiom,
% 0.20/0.46 ! [N: nat,K: nat,M: nat] :
% 0.20/0.46 ( ( ord_less_nat @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M @ K ) )
% 0.20/0.46 => ( ord_less_nat @ N @ M ) ) ).
% 0.20/0.46
% 0.20/0.46 % nat_diff_right_cancel_less
% 0.20/0.46 thf(fact_38_nat__diff__right__cancel__eq2,axiom,
% 0.20/0.46 ! [M: nat,K: nat,N: nat] :
% 0.20/0.46 ( ( ( minus_minus_nat @ M @ K )
% 0.20/0.46 = ( minus_minus_nat @ N @ K ) )
% 0.20/0.46 => ( ( ord_less_nat @ K @ N )
% 0.20/0.46 => ( M = N ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % nat_diff_right_cancel_eq2
% 0.20/0.46 thf(fact_39_list__strict__desc__trans,axiom,
% 0.20/0.46 ( list_strict_desc_nat
% 0.20/0.46 = ( ^ [Xs2: list_nat] :
% 0.20/0.46 ! [J2: nat] :
% 0.20/0.46 ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs2 ) )
% 0.20/0.46 => ! [I2: nat] :
% 0.20/0.46 ( ( ord_less_nat @ I2 @ J2 )
% 0.20/0.46 => ( ord_less_nat @ ( nth_nat @ Xs2 @ J2 ) @ ( nth_nat @ Xs2 @ I2 ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % list_strict_desc_trans
% 0.20/0.46 thf(fact_40_list__strict__asc__trans,axiom,
% 0.20/0.46 ( list_strict_asc_nat
% 0.20/0.46 = ( ^ [Xs2: list_nat] :
% 0.20/0.46 ! [J2: nat] :
% 0.20/0.46 ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs2 ) )
% 0.20/0.46 => ! [I2: nat] :
% 0.20/0.46 ( ( ord_less_nat @ I2 @ J2 )
% 0.20/0.46 => ( ord_less_nat @ ( nth_nat @ Xs2 @ I2 ) @ ( nth_nat @ Xs2 @ J2 ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % list_strict_asc_trans
% 0.20/0.46 thf(fact_41_list__ex__length,axiom,
% 0.20/0.46 ( list_ex_nat
% 0.20/0.46 = ( ^ [P2: nat > $o,Xs2: list_nat] :
% 0.20/0.46 ? [N3: nat] :
% 0.20/0.46 ( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs2 ) )
% 0.20/0.46 & ( P2 @ ( nth_nat @ Xs2 @ N3 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % list_ex_length
% 0.20/0.46 thf(fact_42_list__ex__length,axiom,
% 0.20/0.46 ( list_ex_a
% 0.20/0.46 = ( ^ [P2: a > $o,Xs2: list_a] :
% 0.20/0.46 ? [N3: nat] :
% 0.20/0.46 ( ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs2 ) )
% 0.20/0.46 & ( P2 @ ( nth_a @ Xs2 @ N3 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % list_ex_length
% 0.20/0.46 thf(fact_43_i__append__nth2,axiom,
% 0.20/0.46 ! [Xs: list_nat,N: nat,F: nat > nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
% 0.20/0.46 => ( ( listIn923761578nd_nat @ Xs @ F @ N )
% 0.20/0.46 = ( F @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % i_append_nth2
% 0.20/0.46 thf(fact_44_i__append__nth2,axiom,
% 0.20/0.46 ! [Xs: list_a,N: nat,F: nat > a] :
% 0.20/0.46 ( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N )
% 0.20/0.46 => ( ( listIn1312259492pend_a @ Xs @ F @ N )
% 0.20/0.46 = ( F @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % i_append_nth2
% 0.20/0.46 thf(fact_45_nth__append,axiom,
% 0.20/0.46 ! [N: nat,Xs: list_nat,Ys: list_nat] :
% 0.20/0.46 ( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 0.20/0.46 => ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
% 0.20/0.46 = ( nth_nat @ Xs @ N ) ) )
% 0.20/0.46 & ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 0.20/0.46 => ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
% 0.20/0.46 = ( nth_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % nth_append
% 0.20/0.46 thf(fact_46_nth__append,axiom,
% 0.20/0.46 ! [N: nat,Xs: list_a,Ys: list_a] :
% 0.20/0.46 ( ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
% 0.20/0.46 => ( ( nth_a @ ( append_a @ Xs @ Ys ) @ N )
% 0.20/0.46 = ( nth_a @ Xs @ N ) ) )
% 0.20/0.46 & ( ~ ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
% 0.20/0.46 => ( ( nth_a @ ( append_a @ Xs @ Ys ) @ N )
% 0.20/0.46 = ( nth_a @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % nth_append
% 0.20/0.46 thf(fact_47_nth__list__update__eq,axiom,
% 0.20/0.46 ! [I3: nat,Xs: list_nat,X2: nat] :
% 0.20/0.46 ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
% 0.20/0.46 => ( ( nth_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) @ I3 )
% 0.20/0.46 = X2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % nth_list_update_eq
% 0.20/0.46 thf(fact_48_nth__list__update__eq,axiom,
% 0.20/0.46 ! [I3: nat,Xs: list_a,X2: a] :
% 0.20/0.46 ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
% 0.20/0.46 => ( ( nth_a @ ( list_update_a @ Xs @ I3 @ X2 ) @ I3 )
% 0.20/0.46 = X2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % nth_list_update_eq
% 0.20/0.46 thf(fact_49_nth__butlast,axiom,
% 0.20/0.46 ! [N: nat,Xs: list_nat] :
% 0.20/0.46 ( ( ord_less_nat @ N @ ( size_size_list_nat @ ( butlast_nat @ Xs ) ) )
% 0.20/0.46 => ( ( nth_nat @ ( butlast_nat @ Xs ) @ N )
% 0.20/0.46 = ( nth_nat @ Xs @ N ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % nth_butlast
% 0.20/0.46 thf(fact_50_nth__butlast,axiom,
% 0.20/0.46 ! [N: nat,Xs: list_a] :
% 0.20/0.46 ( ( ord_less_nat @ N @ ( size_size_list_a @ ( butlast_a @ Xs ) ) )
% 0.20/0.46 => ( ( nth_a @ ( butlast_a @ Xs ) @ N )
% 0.20/0.46 = ( nth_a @ Xs @ N ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % nth_butlast
% 0.20/0.46 thf(fact_51_mem__Collect__eq,axiom,
% 0.20/0.46 ! [A: a,P: a > $o] :
% 0.20/0.46 ( ( member_a @ A @ ( collect_a @ P ) )
% 0.20/0.46 = ( P @ A ) ) ).
% 0.20/0.46
% 0.20/0.46 % mem_Collect_eq
% 0.20/0.46 thf(fact_52_Collect__mem__eq,axiom,
% 0.20/0.46 ! [A2: set_a] :
% 0.20/0.46 ( ( collect_a
% 0.20/0.46 @ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
% 0.20/0.46 = A2 ) ).
% 0.20/0.46
% 0.20/0.46 % Collect_mem_eq
% 0.20/0.46 thf(fact_53_append_Oassoc,axiom,
% 0.20/0.46 ! [A: list_a,B: list_a,C: list_a] :
% 0.20/0.46 ( ( append_a @ ( append_a @ A @ B ) @ C )
% 0.20/0.46 = ( append_a @ A @ ( append_a @ B @ C ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % append.assoc
% 0.20/0.46 thf(fact_54_append__assoc,axiom,
% 0.20/0.46 ! [Xs: list_a,Ys: list_a,Zs: list_a] :
% 0.20/0.46 ( ( append_a @ ( append_a @ Xs @ Ys ) @ Zs )
% 0.20/0.46 = ( append_a @ Xs @ ( append_a @ Ys @ Zs ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % append_assoc
% 0.20/0.46 thf(fact_55_append__same__eq,axiom,
% 0.20/0.46 ! [Ys: list_a,Xs: list_a,Zs: list_a] :
% 0.20/0.46 ( ( ( append_a @ Ys @ Xs )
% 0.20/0.46 = ( append_a @ Zs @ Xs ) )
% 0.20/0.46 = ( Ys = Zs ) ) ).
% 0.20/0.46
% 0.20/0.46 % append_same_eq
% 0.20/0.46 thf(fact_56_same__append__eq,axiom,
% 0.20/0.46 ! [Xs: list_a,Ys: list_a,Zs: list_a] :
% 0.20/0.46 ( ( ( append_a @ Xs @ Ys )
% 0.20/0.46 = ( append_a @ Xs @ Zs ) )
% 0.20/0.46 = ( Ys = Zs ) ) ).
% 0.20/0.46
% 0.20/0.46 % same_append_eq
% 0.20/0.46 thf(fact_57_list__update__overwrite,axiom,
% 0.20/0.46 ! [Xs: list_a,I3: nat,X2: a,Y2: a] :
% 0.20/0.46 ( ( list_update_a @ ( list_update_a @ Xs @ I3 @ X2 ) @ I3 @ Y2 )
% 0.20/0.46 = ( list_update_a @ Xs @ I3 @ Y2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % list_update_overwrite
% 0.20/0.46 thf(fact_58_diff__diff__cancel,axiom,
% 0.20/0.46 ! [I3: nat,N: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ I3 @ N )
% 0.20/0.46 => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I3 ) )
% 0.20/0.46 = I3 ) ) ).
% 0.20/0.46
% 0.20/0.46 % diff_diff_cancel
% 0.20/0.46 thf(fact_59_append__eq__append__conv,axiom,
% 0.20/0.46 ! [Xs: list_nat,Ys: list_nat,Us: list_nat,Vs: list_nat] :
% 0.20/0.46 ( ( ( ( size_size_list_nat @ Xs )
% 0.20/0.46 = ( size_size_list_nat @ Ys ) )
% 0.20/0.46 | ( ( size_size_list_nat @ Us )
% 0.20/0.46 = ( size_size_list_nat @ Vs ) ) )
% 0.20/0.46 => ( ( ( append_nat @ Xs @ Us )
% 0.20/0.46 = ( append_nat @ Ys @ Vs ) )
% 0.20/0.46 = ( ( Xs = Ys )
% 0.20/0.46 & ( Us = Vs ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % append_eq_append_conv
% 0.20/0.46 thf(fact_60_append__eq__append__conv,axiom,
% 0.20/0.46 ! [Xs: list_a,Ys: list_a,Us: list_a,Vs: list_a] :
% 0.20/0.46 ( ( ( ( size_size_list_a @ Xs )
% 0.20/0.46 = ( size_size_list_a @ Ys ) )
% 0.20/0.46 | ( ( size_size_list_a @ Us )
% 0.20/0.46 = ( size_size_list_a @ Vs ) ) )
% 0.20/0.46 => ( ( ( append_a @ Xs @ Us )
% 0.20/0.46 = ( append_a @ Ys @ Vs ) )
% 0.20/0.46 = ( ( Xs = Ys )
% 0.20/0.46 & ( Us = Vs ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % append_eq_append_conv
% 0.20/0.46 thf(fact_61_length__list__update,axiom,
% 0.20/0.46 ! [Xs: list_nat,I3: nat,X2: nat] :
% 0.20/0.46 ( ( size_size_list_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) )
% 0.20/0.46 = ( size_size_list_nat @ Xs ) ) ).
% 0.20/0.46
% 0.20/0.46 % length_list_update
% 0.20/0.46 thf(fact_62_length__list__update,axiom,
% 0.20/0.46 ! [Xs: list_a,I3: nat,X2: a] :
% 0.20/0.46 ( ( size_size_list_a @ ( list_update_a @ Xs @ I3 @ X2 ) )
% 0.20/0.46 = ( size_size_list_a @ Xs ) ) ).
% 0.20/0.46
% 0.20/0.46 % length_list_update
% 0.20/0.46 thf(fact_63_list__update__id,axiom,
% 0.20/0.46 ! [Xs: list_nat,I3: nat] :
% 0.20/0.46 ( ( list_update_nat @ Xs @ I3 @ ( nth_nat @ Xs @ I3 ) )
% 0.20/0.46 = Xs ) ).
% 0.20/0.46
% 0.20/0.46 % list_update_id
% 0.20/0.46 thf(fact_64_list__update__id,axiom,
% 0.20/0.46 ! [Xs: list_a,I3: nat] :
% 0.20/0.46 ( ( list_update_a @ Xs @ I3 @ ( nth_a @ Xs @ I3 ) )
% 0.20/0.46 = Xs ) ).
% 0.20/0.46
% 0.20/0.46 % list_update_id
% 0.20/0.46 thf(fact_65_nth__list__update__neq,axiom,
% 0.20/0.46 ! [I3: nat,J: nat,Xs: list_nat,X2: nat] :
% 0.20/0.46 ( ( I3 != J )
% 0.20/0.46 => ( ( nth_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) @ J )
% 0.20/0.46 = ( nth_nat @ Xs @ J ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % nth_list_update_neq
% 0.20/0.46 thf(fact_66_nth__list__update__neq,axiom,
% 0.20/0.46 ! [I3: nat,J: nat,Xs: list_a,X2: a] :
% 0.20/0.46 ( ( I3 != J )
% 0.20/0.46 => ( ( nth_a @ ( list_update_a @ Xs @ I3 @ X2 ) @ J )
% 0.20/0.46 = ( nth_a @ Xs @ J ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % nth_list_update_neq
% 0.20/0.46 thf(fact_67_list__ex__append,axiom,
% 0.20/0.46 ! [P: a > $o,Xs: list_a,Ys: list_a] :
% 0.20/0.46 ( ( list_ex_a @ P @ ( append_a @ Xs @ Ys ) )
% 0.20/0.46 = ( ( list_ex_a @ P @ Xs )
% 0.20/0.46 | ( list_ex_a @ P @ Ys ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % list_ex_append
% 0.20/0.46 thf(fact_68_i__append__assoc,axiom,
% 0.20/0.46 ! [Xs: list_a,Ys: list_a,F: nat > a] :
% 0.20/0.46 ( ( listIn1312259492pend_a @ Xs @ ( listIn1312259492pend_a @ Ys @ F ) )
% 0.20/0.46 = ( listIn1312259492pend_a @ ( append_a @ Xs @ Ys ) @ F ) ) ).
% 0.20/0.46
% 0.20/0.46 % i_append_assoc
% 0.20/0.46 thf(fact_69_list__update__beyond,axiom,
% 0.20/0.46 ! [Xs: list_nat,I3: nat,X2: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I3 )
% 0.20/0.46 => ( ( list_update_nat @ Xs @ I3 @ X2 )
% 0.20/0.46 = Xs ) ) ).
% 0.20/0.46
% 0.20/0.46 % list_update_beyond
% 0.20/0.46 thf(fact_70_list__update__beyond,axiom,
% 0.20/0.46 ! [Xs: list_a,I3: nat,X2: a] :
% 0.20/0.46 ( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ I3 )
% 0.20/0.46 => ( ( list_update_a @ Xs @ I3 @ X2 )
% 0.20/0.46 = Xs ) ) ).
% 0.20/0.46
% 0.20/0.46 % list_update_beyond
% 0.20/0.46 thf(fact_71_le__refl,axiom,
% 0.20/0.46 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 0.20/0.46
% 0.20/0.46 % le_refl
% 0.20/0.46 thf(fact_72_le__trans,axiom,
% 0.20/0.46 ! [I3: nat,J: nat,K: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ I3 @ J )
% 0.20/0.46 => ( ( ord_less_eq_nat @ J @ K )
% 0.20/0.46 => ( ord_less_eq_nat @ I3 @ K ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % le_trans
% 0.20/0.46 thf(fact_73_eq__imp__le,axiom,
% 0.20/0.46 ! [M: nat,N: nat] :
% 0.20/0.46 ( ( M = N )
% 0.20/0.46 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.20/0.46
% 0.20/0.46 % eq_imp_le
% 0.20/0.46 thf(fact_74_le__antisym,axiom,
% 0.20/0.46 ! [M: nat,N: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ M @ N )
% 0.20/0.46 => ( ( ord_less_eq_nat @ N @ M )
% 0.20/0.46 => ( M = N ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % le_antisym
% 0.20/0.46 thf(fact_75_nat__le__linear,axiom,
% 0.20/0.46 ! [M: nat,N: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ M @ N )
% 0.20/0.46 | ( ord_less_eq_nat @ N @ M ) ) ).
% 0.20/0.46
% 0.20/0.46 % nat_le_linear
% 0.20/0.46 thf(fact_76_list__update__swap,axiom,
% 0.20/0.46 ! [I3: nat,I4: nat,Xs: list_a,X2: a,X4: a] :
% 0.20/0.46 ( ( I3 != I4 )
% 0.20/0.46 => ( ( list_update_a @ ( list_update_a @ Xs @ I3 @ X2 ) @ I4 @ X4 )
% 0.20/0.46 = ( list_update_a @ ( list_update_a @ Xs @ I4 @ X4 ) @ I3 @ X2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % list_update_swap
% 0.20/0.46 thf(fact_77_append__eq__appendI,axiom,
% 0.20/0.46 ! [Xs: list_a,Xs1: list_a,Zs: list_a,Ys: list_a,Us: list_a] :
% 0.20/0.46 ( ( ( append_a @ Xs @ Xs1 )
% 0.20/0.46 = Zs )
% 0.20/0.46 => ( ( Ys
% 0.20/0.46 = ( append_a @ Xs1 @ Us ) )
% 0.20/0.46 => ( ( append_a @ Xs @ Ys )
% 0.20/0.46 = ( append_a @ Zs @ Us ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % append_eq_appendI
% 0.20/0.46 thf(fact_78_Nat_Oex__has__greatest__nat,axiom,
% 0.20/0.46 ! [P: nat > $o,K: nat,B: nat] :
% 0.20/0.46 ( ( P @ K )
% 0.20/0.46 => ( ! [Y3: nat] :
% 0.20/0.46 ( ( P @ Y3 )
% 0.20/0.46 => ( ord_less_eq_nat @ Y3 @ B ) )
% 0.20/0.46 => ? [X5: nat] :
% 0.20/0.46 ( ( P @ X5 )
% 0.20/0.46 & ! [Y4: nat] :
% 0.20/0.46 ( ( P @ Y4 )
% 0.20/0.46 => ( ord_less_eq_nat @ Y4 @ X5 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Nat.ex_has_greatest_nat
% 0.20/0.46 thf(fact_79_append__eq__append__conv2,axiom,
% 0.20/0.46 ! [Xs: list_a,Ys: list_a,Zs: list_a,Ts: list_a] :
% 0.20/0.46 ( ( ( append_a @ Xs @ Ys )
% 0.20/0.46 = ( append_a @ Zs @ Ts ) )
% 0.20/0.46 = ( ? [Us2: list_a] :
% 0.20/0.46 ( ( ( Xs
% 0.20/0.46 = ( append_a @ Zs @ Us2 ) )
% 0.20/0.46 & ( ( append_a @ Us2 @ Ys )
% 0.20/0.46 = Ts ) )
% 0.20/0.46 | ( ( ( append_a @ Xs @ Us2 )
% 0.20/0.46 = Zs )
% 0.20/0.46 & ( Ys
% 0.20/0.46 = ( append_a @ Us2 @ Ts ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % append_eq_append_conv2
% 0.20/0.46 thf(fact_80_list__strict__asc__trans__le,axiom,
% 0.20/0.46 ! [Xs: list_nat] :
% 0.20/0.46 ( ( list_strict_asc_nat @ Xs )
% 0.20/0.46 => ! [J3: nat] :
% 0.20/0.46 ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
% 0.20/0.46 => ! [I5: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ I5 @ J3 )
% 0.20/0.46 => ( ord_less_eq_nat @ ( nth_nat @ Xs @ I5 ) @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % list_strict_asc_trans_le
% 0.20/0.46 thf(fact_81_list__update__append1,axiom,
% 0.20/0.46 ! [I3: nat,Xs: list_nat,Ys: list_nat,X2: nat] :
% 0.20/0.46 ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
% 0.20/0.46 => ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ I3 @ X2 )
% 0.20/0.46 = ( append_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) @ Ys ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % list_update_append1
% 0.20/0.46 thf(fact_82_list__update__append1,axiom,
% 0.20/0.46 ! [I3: nat,Xs: list_a,Ys: list_a,X2: a] :
% 0.20/0.46 ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
% 0.20/0.46 => ( ( list_update_a @ ( append_a @ Xs @ Ys ) @ I3 @ X2 )
% 0.20/0.46 = ( append_a @ ( list_update_a @ Xs @ I3 @ X2 ) @ Ys ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % list_update_append1
% 0.20/0.46 thf(fact_83_nth__append2,axiom,
% 0.20/0.46 ! [Xs: list_nat,N: nat,Ys: list_nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
% 0.20/0.46 => ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
% 0.20/0.46 = ( nth_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % nth_append2
% 0.20/0.46 thf(fact_84_nth__append2,axiom,
% 0.20/0.46 ! [Xs: list_a,N: nat,Ys: list_a] :
% 0.20/0.46 ( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N )
% 0.20/0.46 => ( ( nth_a @ ( append_a @ Xs @ Ys ) @ N )
% 0.20/0.46 = ( nth_a @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % nth_append2
% 0.20/0.46 thf(fact_85_ge__less__neq__conv,axiom,
% 0.20/0.46 ( ord_less_eq_nat
% 0.20/0.46 = ( ^ [A3: nat,N3: nat] :
% 0.20/0.46 ! [X3: nat] :
% 0.20/0.46 ( ( ord_less_nat @ X3 @ A3 )
% 0.20/0.46 => ( N3 != X3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % ge_less_neq_conv
% 0.20/0.46 thf(fact_86_less__ge__neq__conv,axiom,
% 0.20/0.46 ( ord_less_nat
% 0.20/0.46 = ( ^ [N3: nat,A3: nat] :
% 0.20/0.46 ! [X3: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ A3 @ X3 )
% 0.20/0.46 => ( N3 != X3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % less_ge_neq_conv
% 0.20/0.46 thf(fact_87_greater__le__neq__conv,axiom,
% 0.20/0.46 ( ord_less_nat
% 0.20/0.46 = ( ^ [A3: nat,N3: nat] :
% 0.20/0.46 ! [X3: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ X3 @ A3 )
% 0.20/0.46 => ( N3 != X3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % greater_le_neq_conv
% 0.20/0.46 thf(fact_88_le__greater__neq__conv,axiom,
% 0.20/0.46 ( ord_less_eq_nat
% 0.20/0.46 = ( ^ [N3: nat,A3: nat] :
% 0.20/0.46 ! [X3: nat] :
% 0.20/0.46 ( ( ord_less_nat @ A3 @ X3 )
% 0.20/0.46 => ( N3 != X3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % le_greater_neq_conv
% 0.20/0.46 thf(fact_89_nat__less__le,axiom,
% 0.20/0.46 ( ord_less_nat
% 0.20/0.46 = ( ^ [M3: nat,N3: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ M3 @ N3 )
% 0.20/0.46 & ( M3 != N3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % nat_less_le
% 0.20/0.46 thf(fact_90_less__imp__le__nat,axiom,
% 0.20/0.46 ! [M: nat,N: nat] :
% 0.20/0.46 ( ( ord_less_nat @ M @ N )
% 0.20/0.46 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.20/0.46
% 0.20/0.46 % less_imp_le_nat
% 0.20/0.46 thf(fact_91_le__eq__less__or__eq,axiom,
% 0.20/0.46 ( ord_less_eq_nat
% 0.20/0.46 = ( ^ [M3: nat,N3: nat] :
% 0.20/0.46 ( ( ord_less_nat @ M3 @ N3 )
% 0.20/0.46 | ( M3 = N3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % le_eq_less_or_eq
% 0.20/0.46 thf(fact_92_less__or__eq__imp__le,axiom,
% 0.20/0.46 ! [M: nat,N: nat] :
% 0.20/0.46 ( ( ( ord_less_nat @ M @ N )
% 0.20/0.46 | ( M = N ) )
% 0.20/0.46 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.20/0.46
% 0.20/0.46 % less_or_eq_imp_le
% 0.20/0.46 thf(fact_93_le__neq__implies__less,axiom,
% 0.20/0.46 ! [M: nat,N: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ M @ N )
% 0.20/0.46 => ( ( M != N )
% 0.20/0.46 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % le_neq_implies_less
% 0.20/0.46 thf(fact_94_less__mono__imp__le__mono,axiom,
% 0.20/0.46 ! [F: nat > nat,I3: nat,J: nat] :
% 0.20/0.46 ( ! [I: nat,J4: nat] :
% 0.20/0.46 ( ( ord_less_nat @ I @ J4 )
% 0.20/0.46 => ( ord_less_nat @ ( F @ I ) @ ( F @ J4 ) ) )
% 0.20/0.46 => ( ( ord_less_eq_nat @ I3 @ J )
% 0.20/0.46 => ( ord_less_eq_nat @ ( F @ I3 ) @ ( F @ J ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % less_mono_imp_le_mono
% 0.20/0.46 thf(fact_95_eq__diff__iff,axiom,
% 0.20/0.46 ! [K: nat,M: nat,N: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ K @ M )
% 0.20/0.46 => ( ( ord_less_eq_nat @ K @ N )
% 0.20/0.46 => ( ( ( minus_minus_nat @ M @ K )
% 0.20/0.46 = ( minus_minus_nat @ N @ K ) )
% 0.20/0.46 = ( M = N ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % eq_diff_iff
% 0.20/0.46 thf(fact_96_le__diff__iff,axiom,
% 0.20/0.46 ! [K: nat,M: nat,N: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ K @ M )
% 0.20/0.46 => ( ( ord_less_eq_nat @ K @ N )
% 0.20/0.46 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.20/0.46 = ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % le_diff_iff
% 0.20/0.46 thf(fact_97_Nat_Odiff__diff__eq,axiom,
% 0.20/0.46 ! [K: nat,M: nat,N: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ K @ M )
% 0.20/0.46 => ( ( ord_less_eq_nat @ K @ N )
% 0.20/0.46 => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.20/0.46 = ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Nat.diff_diff_eq
% 0.20/0.46 thf(fact_98_diff__le__mono,axiom,
% 0.20/0.46 ! [M: nat,N: nat,L: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ M @ N )
% 0.20/0.46 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % diff_le_mono
% 0.20/0.46 thf(fact_99_diff__le__self,axiom,
% 0.20/0.46 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% 0.20/0.46
% 0.20/0.46 % diff_le_self
% 0.20/0.46 thf(fact_100_le__diff__iff_H,axiom,
% 0.20/0.46 ! [A: nat,C: nat,B: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ A @ C )
% 0.20/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.20/0.46 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
% 0.20/0.46 = ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % le_diff_iff'
% 0.20/0.46 thf(fact_101_diff__le__mono2,axiom,
% 0.20/0.46 ! [M: nat,N: nat,L: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ M @ N )
% 0.20/0.46 => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % diff_le_mono2
% 0.20/0.46 thf(fact_102_le__diff__swap,axiom,
% 0.20/0.46 ! [I3: nat,K: nat,J: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ I3 @ K )
% 0.20/0.46 => ( ( ord_less_eq_nat @ J @ K )
% 0.20/0.46 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ K @ J ) @ I3 )
% 0.20/0.46 = ( ord_less_eq_nat @ ( minus_minus_nat @ K @ I3 ) @ J ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % le_diff_swap
% 0.20/0.46 thf(fact_103_le__diff__imp__le,axiom,
% 0.20/0.46 ! [I3: nat,J: nat,M: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ I3 @ ( minus_minus_nat @ J @ M ) )
% 0.20/0.46 => ( ord_less_eq_nat @ I3 @ J ) ) ).
% 0.20/0.46
% 0.20/0.46 % le_diff_imp_le
% 0.20/0.46 thf(fact_104_le__imp__diff__le,axiom,
% 0.20/0.46 ! [J: nat,K: nat,N: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ J @ K )
% 0.20/0.46 => ( ord_less_eq_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% 0.20/0.46
% 0.20/0.46 % le_imp_diff_le
% 0.20/0.46 thf(fact_105_eq__diff__left__iff,axiom,
% 0.20/0.46 ! [M: nat,K: nat,N: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ M @ K )
% 0.20/0.46 => ( ( ord_less_eq_nat @ N @ K )
% 0.20/0.46 => ( ( ( minus_minus_nat @ K @ M )
% 0.20/0.46 = ( minus_minus_nat @ K @ N ) )
% 0.20/0.46 = ( M = N ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % eq_diff_left_iff
% 0.20/0.46 thf(fact_106_le__diff__le__imp__le,axiom,
% 0.20/0.46 ! [I3: nat,J: nat,M: nat,N: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ I3 @ ( minus_minus_nat @ J @ M ) )
% 0.20/0.46 => ( ( ord_less_eq_nat @ N @ M )
% 0.20/0.46 => ( ord_less_eq_nat @ I3 @ ( minus_minus_nat @ J @ N ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % le_diff_le_imp_le
% 0.20/0.46 thf(fact_107_nat__diff__left__cancel__le2,axiom,
% 0.20/0.46 ! [K: nat,M: nat,N: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ ( minus_minus_nat @ K @ M ) @ ( minus_minus_nat @ K @ N ) )
% 0.20/0.46 => ( ( ord_less_eq_nat @ N @ K )
% 0.20/0.46 => ( ord_less_eq_nat @ N @ M ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % nat_diff_left_cancel_le2
% 0.20/0.46 thf(fact_108_nat__diff__right__cancel__le2,axiom,
% 0.20/0.46 ! [M: nat,K: nat,N: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.20/0.46 => ( ( ord_less_eq_nat @ K @ N )
% 0.20/0.46 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % nat_diff_right_cancel_le2
% 0.20/0.46 thf(fact_109_list__update__append,axiom,
% 0.20/0.46 ! [N: nat,Xs: list_nat,Ys: list_nat,X2: nat] :
% 0.20/0.46 ( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 0.20/0.46 => ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ N @ X2 )
% 0.20/0.46 = ( append_nat @ ( list_update_nat @ Xs @ N @ X2 ) @ Ys ) ) )
% 0.20/0.46 & ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 0.20/0.46 => ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ N @ X2 )
% 0.20/0.46 = ( append_nat @ Xs @ ( list_update_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ X2 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % list_update_append
% 0.20/0.46 thf(fact_110_list__update__append,axiom,
% 0.20/0.46 ! [N: nat,Xs: list_a,Ys: list_a,X2: a] :
% 0.20/0.46 ( ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
% 0.20/0.46 => ( ( list_update_a @ ( append_a @ Xs @ Ys ) @ N @ X2 )
% 0.20/0.46 = ( append_a @ ( list_update_a @ Xs @ N @ X2 ) @ Ys ) ) )
% 0.20/0.46 & ( ~ ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
% 0.20/0.46 => ( ( list_update_a @ ( append_a @ Xs @ Ys ) @ N @ X2 )
% 0.20/0.46 = ( append_a @ Xs @ ( list_update_a @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) @ X2 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % list_update_append
% 0.20/0.46 thf(fact_111_nth__append1,axiom,
% 0.20/0.46 ! [N: nat,Xs: list_nat,Ys: list_nat] :
% 0.20/0.46 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 0.20/0.46 => ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
% 0.20/0.46 = ( nth_nat @ Xs @ N ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % nth_append1
% 0.20/0.46 thf(fact_112_nth__append1,axiom,
% 0.20/0.46 ! [N: nat,Xs: list_a,Ys: list_a] :
% 0.20/0.46 ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
% 0.20/0.46 => ( ( nth_a @ ( append_a @ Xs @ Ys ) @ N )
% 0.20/0.46 = ( nth_a @ Xs @ N ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % nth_append1
% 0.20/0.46 thf(fact_113_less__diff__iff,axiom,
% 0.20/0.46 ! [K: nat,M: nat,N: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ K @ M )
% 0.20/0.46 => ( ( ord_less_eq_nat @ K @ N )
% 0.20/0.46 => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.20/0.46 = ( ord_less_nat @ M @ N ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % less_diff_iff
% 0.20/0.46 thf(fact_114_diff__less__mono,axiom,
% 0.20/0.46 ! [A: nat,B: nat,C: nat] :
% 0.20/0.46 ( ( ord_less_nat @ A @ B )
% 0.20/0.46 => ( ( ord_less_eq_nat @ C @ A )
% 0.20/0.46 => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % diff_less_mono
% 0.20/0.46 thf(fact_115_less__diff__le__imp__less,axiom,
% 0.20/0.46 ! [I3: nat,J: nat,M: nat,N: nat] :
% 0.20/0.46 ( ( ord_less_nat @ I3 @ ( minus_minus_nat @ J @ M ) )
% 0.20/0.46 => ( ( ord_less_eq_nat @ N @ M )
% 0.20/0.46 => ( ord_less_nat @ I3 @ ( minus_minus_nat @ J @ N ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % less_diff_le_imp_less
% 0.20/0.46 thf(fact_116_nat__diff__left__cancel__le1,axiom,
% 0.20/0.46 ! [K: nat,M: nat,N: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ ( minus_minus_nat @ K @ M ) @ ( minus_minus_nat @ K @ N ) )
% 0.20/0.46 => ( ( ord_less_nat @ M @ K )
% 0.20/0.46 => ( ord_less_eq_nat @ N @ M ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % nat_diff_left_cancel_le1
% 0.20/0.46 thf(fact_117_nat__diff__right__cancel__le1,axiom,
% 0.20/0.46 ! [M: nat,K: nat,N: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.20/0.46 => ( ( ord_less_nat @ K @ M )
% 0.20/0.46 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % nat_diff_right_cancel_le1
% 0.20/0.46 thf(fact_118_nth__list__update,axiom,
% 0.20/0.46 ! [I3: nat,Xs: list_nat,J: nat,X2: nat] :
% 0.20/0.46 ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
% 0.20/0.46 => ( ( ( I3 = J )
% 0.20/0.46 => ( ( nth_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) @ J )
% 0.20/0.46 = X2 ) )
% 0.20/0.46 & ( ( I3 != J )
% 0.20/0.46 => ( ( nth_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) @ J )
% 0.20/0.46 = ( nth_nat @ Xs @ J ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % nth_list_update
% 0.20/0.46 thf(fact_119_nth__list__update,axiom,
% 0.20/0.46 ! [I3: nat,Xs: list_a,J: nat,X2: a] :
% 0.20/0.46 ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
% 0.20/0.46 => ( ( ( I3 = J )
% 0.20/0.46 => ( ( nth_a @ ( list_update_a @ Xs @ I3 @ X2 ) @ J )
% 0.20/0.46 = X2 ) )
% 0.20/0.46 & ( ( I3 != J )
% 0.20/0.46 => ( ( nth_a @ ( list_update_a @ Xs @ I3 @ X2 ) @ J )
% 0.20/0.46 = ( nth_a @ Xs @ J ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % nth_list_update
% 0.20/0.46 thf(fact_120_list__update__same__conv,axiom,
% 0.20/0.46 ! [I3: nat,Xs: list_nat,X2: nat] :
% 0.20/0.46 ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
% 0.20/0.46 => ( ( ( list_update_nat @ Xs @ I3 @ X2 )
% 0.20/0.46 = Xs )
% 0.20/0.46 = ( ( nth_nat @ Xs @ I3 )
% 0.20/0.46 = X2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % list_update_same_conv
% 0.20/0.46 thf(fact_121_list__update__same__conv,axiom,
% 0.20/0.46 ! [I3: nat,Xs: list_a,X2: a] :
% 0.20/0.46 ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
% 0.20/0.46 => ( ( ( list_update_a @ Xs @ I3 @ X2 )
% 0.20/0.46 = Xs )
% 0.20/0.46 = ( ( nth_a @ Xs @ I3 )
% 0.20/0.46 = X2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % list_update_same_conv
% 0.20/0.46 thf(fact_122_diff__right__commute,axiom,
% 0.20/0.46 ! [A: nat,C: nat,B: nat] :
% 0.20/0.46 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
% 0.20/0.46 = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% 0.20/0.46
% 0.20/0.46 % diff_right_commute
% 0.20/0.46 thf(fact_123_list__desc__trans__le,axiom,
% 0.20/0.46 ( list_desc_nat
% 0.20/0.46 = ( ^ [Xs2: list_nat] :
% 0.20/0.46 ! [J2: nat] :
% 0.20/0.46 ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs2 ) )
% 0.20/0.46 => ! [I2: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ I2 @ J2 )
% 0.20/0.46 => ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ J2 ) @ ( nth_nat @ Xs2 @ I2 ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % list_desc_trans_le
% 0.20/0.46 thf(fact_124_list__asc__trans__le,axiom,
% 0.20/0.46 ( list_asc_nat
% 0.20/0.46 = ( ^ [Xs2: list_nat] :
% 0.20/0.46 ! [J2: nat] :
% 0.20/0.46 ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs2 ) )
% 0.20/0.46 => ! [I2: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ I2 @ J2 )
% 0.20/0.46 => ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ I2 ) @ ( nth_nat @ Xs2 @ J2 ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % list_asc_trans_le
% 0.20/0.46 thf(fact_125_list__desc__trans,axiom,
% 0.20/0.46 ( list_desc_nat
% 0.20/0.46 = ( ^ [Xs2: list_nat] :
% 0.20/0.46 ! [J2: nat] :
% 0.20/0.46 ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs2 ) )
% 0.20/0.46 => ! [I2: nat] :
% 0.20/0.46 ( ( ord_less_nat @ I2 @ J2 )
% 0.20/0.46 => ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ J2 ) @ ( nth_nat @ Xs2 @ I2 ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % list_desc_trans
% 0.20/0.46 thf(fact_126_list__asc__trans,axiom,
% 0.20/0.46 ( list_asc_nat
% 0.20/0.46 = ( ^ [Xs2: list_nat] :
% 0.20/0.46 ! [J2: nat] :
% 0.20/0.46 ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs2 ) )
% 0.20/0.46 => ! [I2: nat] :
% 0.20/0.46 ( ( ord_less_nat @ I2 @ J2 )
% 0.20/0.46 => ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ I2 ) @ ( nth_nat @ Xs2 @ J2 ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % list_asc_trans
% 0.20/0.46 thf(fact_127_order__refl,axiom,
% 0.20/0.46 ! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% 0.20/0.46
% 0.20/0.46 % order_refl
% 0.20/0.46 thf(fact_128_nat__descend__induct,axiom,
% 0.20/0.46 ! [N: nat,P: nat > $o,M: nat] :
% 0.20/0.46 ( ! [K2: nat] :
% 0.20/0.46 ( ( ord_less_nat @ N @ K2 )
% 0.20/0.46 => ( P @ K2 ) )
% 0.20/0.46 => ( ! [K2: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ K2 @ N )
% 0.20/0.46 => ( ! [I5: nat] :
% 0.20/0.46 ( ( ord_less_nat @ K2 @ I5 )
% 0.20/0.46 => ( P @ I5 ) )
% 0.20/0.46 => ( P @ K2 ) ) )
% 0.20/0.46 => ( P @ M ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % nat_descend_induct
% 0.20/0.46 thf(fact_129_minf_I8_J,axiom,
% 0.20/0.46 ! [T: nat] :
% 0.20/0.46 ? [Z2: nat] :
% 0.20/0.46 ! [X6: nat] :
% 0.20/0.46 ( ( ord_less_nat @ X6 @ Z2 )
% 0.20/0.46 => ~ ( ord_less_eq_nat @ T @ X6 ) ) ).
% 0.20/0.46
% 0.20/0.46 % minf(8)
% 0.20/0.46 thf(fact_130_order__subst1,axiom,
% 0.20/0.46 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.20/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.20/0.46 => ( ! [X5: nat,Y3: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 0.20/0.46 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 0.20/0.46 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % order_subst1
% 0.20/0.46 thf(fact_131_order__subst2,axiom,
% 0.20/0.46 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.20/0.46 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.20/0.46 => ( ! [X5: nat,Y3: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 0.20/0.46 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 0.20/0.46 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % order_subst2
% 0.20/0.46 thf(fact_132_ord__eq__le__subst,axiom,
% 0.20/0.46 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.20/0.46 ( ( A
% 0.20/0.46 = ( F @ B ) )
% 0.20/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.20/0.46 => ( ! [X5: nat,Y3: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 0.20/0.46 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 0.20/0.46 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % ord_eq_le_subst
% 0.20/0.46 thf(fact_133_ord__le__eq__subst,axiom,
% 0.20/0.46 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.20/0.46 => ( ( ( F @ B )
% 0.20/0.46 = C )
% 0.20/0.46 => ( ! [X5: nat,Y3: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 0.20/0.46 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 0.20/0.46 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % ord_le_eq_subst
% 0.20/0.46 thf(fact_134_eq__iff,axiom,
% 0.20/0.46 ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
% 0.20/0.46 = ( ^ [X3: nat,Y5: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ X3 @ Y5 )
% 0.20/0.46 & ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % eq_iff
% 0.20/0.46 thf(fact_135_antisym,axiom,
% 0.20/0.46 ! [X2: nat,Y2: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.20/0.46 => ( ( ord_less_eq_nat @ Y2 @ X2 )
% 0.20/0.46 => ( X2 = Y2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % antisym
% 0.20/0.46 thf(fact_136_linear,axiom,
% 0.20/0.46 ! [X2: nat,Y2: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.20/0.46 | ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % linear
% 0.20/0.46 thf(fact_137_eq__refl,axiom,
% 0.20/0.46 ! [X2: nat,Y2: nat] :
% 0.20/0.46 ( ( X2 = Y2 )
% 0.20/0.46 => ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % eq_refl
% 0.20/0.46 thf(fact_138_le__cases,axiom,
% 0.20/0.46 ! [X2: nat,Y2: nat] :
% 0.20/0.46 ( ~ ( ord_less_eq_nat @ X2 @ Y2 )
% 0.20/0.46 => ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % le_cases
% 0.20/0.46 thf(fact_139_order_Otrans,axiom,
% 0.20/0.46 ! [A: nat,B: nat,C: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.20/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.20/0.46 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % order.trans
% 0.20/0.46 thf(fact_140_le__cases3,axiom,
% 0.20/0.46 ! [X2: nat,Y2: nat,Z3: nat] :
% 0.20/0.46 ( ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.20/0.46 => ~ ( ord_less_eq_nat @ Y2 @ Z3 ) )
% 0.20/0.46 => ( ( ( ord_less_eq_nat @ Y2 @ X2 )
% 0.20/0.46 => ~ ( ord_less_eq_nat @ X2 @ Z3 ) )
% 0.20/0.46 => ( ( ( ord_less_eq_nat @ X2 @ Z3 )
% 0.20/0.46 => ~ ( ord_less_eq_nat @ Z3 @ Y2 ) )
% 0.20/0.46 => ( ( ( ord_less_eq_nat @ Z3 @ Y2 )
% 0.20/0.46 => ~ ( ord_less_eq_nat @ Y2 @ X2 ) )
% 0.20/0.46 => ( ( ( ord_less_eq_nat @ Y2 @ Z3 )
% 0.20/0.46 => ~ ( ord_less_eq_nat @ Z3 @ X2 ) )
% 0.20/0.46 => ~ ( ( ord_less_eq_nat @ Z3 @ X2 )
% 0.20/0.46 => ~ ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % le_cases3
% 0.20/0.46 thf(fact_141_antisym__conv,axiom,
% 0.20/0.46 ! [Y2: nat,X2: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ Y2 @ X2 )
% 0.20/0.46 => ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.20/0.46 = ( X2 = Y2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % antisym_conv
% 0.20/0.46 thf(fact_142_order__class_Oorder_Oeq__iff,axiom,
% 0.20/0.46 ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
% 0.20/0.46 = ( ^ [A3: nat,B2: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ A3 @ B2 )
% 0.20/0.46 & ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % order_class.order.eq_iff
% 0.20/0.46 thf(fact_143_ord__eq__le__trans,axiom,
% 0.20/0.46 ! [A: nat,B: nat,C: nat] :
% 0.20/0.46 ( ( A = B )
% 0.20/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.20/0.46 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % ord_eq_le_trans
% 0.20/0.46 thf(fact_144_ord__le__eq__trans,axiom,
% 0.20/0.46 ! [A: nat,B: nat,C: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.20/0.46 => ( ( B = C )
% 0.20/0.46 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % ord_le_eq_trans
% 0.20/0.46 thf(fact_145_order__class_Oorder_Oantisym,axiom,
% 0.20/0.46 ! [A: nat,B: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.20/0.46 => ( ( ord_less_eq_nat @ B @ A )
% 0.20/0.46 => ( A = B ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % order_class.order.antisym
% 0.20/0.46 thf(fact_146_order__trans,axiom,
% 0.20/0.46 ! [X2: nat,Y2: nat,Z3: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.20/0.46 => ( ( ord_less_eq_nat @ Y2 @ Z3 )
% 0.20/0.46 => ( ord_less_eq_nat @ X2 @ Z3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % order_trans
% 0.20/0.46 thf(fact_147_dual__order_Orefl,axiom,
% 0.20/0.46 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 0.20/0.46
% 0.20/0.46 % dual_order.refl
% 0.20/0.46 thf(fact_148_linorder__wlog,axiom,
% 0.20/0.46 ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.20/0.46 ( ! [A4: nat,B3: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ A4 @ B3 )
% 0.20/0.46 => ( P @ A4 @ B3 ) )
% 0.20/0.46 => ( ! [A4: nat,B3: nat] :
% 0.20/0.46 ( ( P @ B3 @ A4 )
% 0.20/0.46 => ( P @ A4 @ B3 ) )
% 0.20/0.46 => ( P @ A @ B ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % linorder_wlog
% 0.20/0.46 thf(fact_149_dual__order_Otrans,axiom,
% 0.20/0.46 ! [B: nat,A: nat,C: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ B @ A )
% 0.20/0.46 => ( ( ord_less_eq_nat @ C @ B )
% 0.20/0.46 => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % dual_order.trans
% 0.20/0.46 thf(fact_150_dual__order_Oeq__iff,axiom,
% 0.20/0.46 ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
% 0.20/0.46 = ( ^ [A3: nat,B2: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ B2 @ A3 )
% 0.20/0.46 & ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % dual_order.eq_iff
% 0.20/0.46 thf(fact_151_dual__order_Oantisym,axiom,
% 0.20/0.46 ! [B: nat,A: nat] :
% 0.20/0.46 ( ( ord_less_eq_nat @ B @ A )
% 0.20/0.46 => ( ( ord_less_eq_nat @ A @ B )
% 0.20/0.46 => ( A = B ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % dual_order.antisym
% 0.20/0.46 thf(fact_152_dual__order_Ostrict__implies__not__eq,axiom,
% 0.20/0.46 ! [B: nat,A: nat] :
% 0.20/0.46 ( ( ord_less_nat @ B @ A )
% 0.20/0.46 => ( A != B ) ) ).
% 0.20/0.46
% 0.20/0.46 % dual_order.strict_implies_not_eq
% 0.20/0.46 thf(fact_153_order_Ostrict__implies__not__eq,axiom,
% 0.20/0.46 ! [A: nat,B: nat] :
% 0.20/0.46 ( ( ord_less_nat @ A @ B )
% 0.20/0.46 => ( A != B ) ) ).
% 0.20/0.46
% 0.20/0.46 % order.strict_implies_not_eq
% 0.20/0.46 thf(fact_154_not__less__iff__gr__or__eq,axiom,
% 0.20/0.46 ! [X2: nat,Y2: nat] :
% 0.20/0.46 ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
% 0.20/0.46 = ( ( ord_less_nat @ Y2 @ X2 )
% 0.20/0.46 | ( X2 = Y2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % not_less_iff_gr_or_eq
% 0.20/0.46 thf(fact_155_dual__order_Ostrict__trans,axiom,
% 0.20/0.46 ! [B: nat,A: nat,C: nat] :
% 0.20/0.46 ( ( ord_less_nat @ B @ A )
% 0.20/0.46 => ( ( ord_less_nat @ C @ B )
% 0.20/0.46 => ( ord_less_nat @ C @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % dual_order.strict_trans
% 0.20/0.46 thf(fact_156_linorder__less__wlog,axiom,
% 0.20/0.46 ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.20/0.46 ( ! [A4: nat,B3: nat] :
% 0.20/0.46 ( ( ord_less_nat @ A4 @ B3 )
% 0.20/0.46 => ( P @ A4 @ B3 ) )
% 0.20/0.46 => ( ! [A4: nat] : ( P @ A4 @ A4 )
% 0.20/0.46 => ( ! [A4: nat,B3: nat] :
% 0.20/0.46 ( ( P @ B3 @ A4 )
% 0.20/0.46 => ( P @ A4 @ B3 ) )
% 0.20/0.46 => ( P @ A @ B ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % linorder_less_wlog
% 0.20/0.46 thf(fact_157_exists__least__iff,axiom,
% 0.20/0.46 ( ( ^ [P3: nat > $o] :
% 0.20/0.46 ? [X: nat] : ( P3 @ X ) )
% 0.20/0.46 = ( ^ [P2: nat > $o] :
% 0.20/0.46 ? [N3: nat] :
% 0.20/0.46 ( ( P2 @ N3 )
% 0.20/0.46 & ! [M3: nat] :
% 0.20/0.46 ( ( ord_less_nat @ M3 @ N3 )
% 0.20/0.46 => ~ ( P2 @ M3 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % exists_least_iff
% 0.20/0.46 thf(fact_158_less__imp__not__less,axiom,
% 0.20/0.46 ! [X2: nat,Y2: nat] :
% 0.20/0.46 ( ( ord_less_nat @ X2 @ Y2 )
% 0.20/0.46 => ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % less_imp_not_less
% 0.20/0.46 thf(fact_159_order_Ostrict__trans,axiom,
% 0.20/0.46 ! [A: nat,B: nat,C: nat] :
% 0.20/0.46 ( ( ord_less_nat @ A @ B )
% 0.20/0.46 => ( ( ord_less_nat @ B @ C )
% 0.20/0.46 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % order.strict_trans
% 0.20/0.46 thf(fact_160_dual__order_Oirrefl,axiom,
% 0.20/0.46 ! [A: nat] :
% 0.20/0.46 ~ ( ord_less_nat @ A @ A ) ).
% 0.20/0.46
% 0.20/0.46 % dual_order.irrefl
% 0.20/0.46 thf(fact_161_linorder__cases,axiom,
% 0.20/0.46 ! [X2: nat,Y2: nat] :
% 0.20/0.46 ( ~ ( ord_less_nat @ X2 @ Y2 )
% 0.20/0.46 => ( ( X2 != Y2 )
% 0.20/0.46 => ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % linorder_cases
% 0.20/0.46 thf(fact_162_less__imp__triv,axiom,
% 0.20/0.46 ! [X2: nat,Y2: nat,P: $o] :
% 0.20/0.46 ( ( ord_less_nat @ X2 @ Y2 )
% 0.20/0.46 => ( ( ord_less_nat @ Y2 @ X2 )
% 0.20/0.46 => P ) ) ).
% 0.20/0.46
% 0.20/0.46 % less_imp_triv
% 0.20/0.46 thf(fact_163_less__imp__not__eq2,axiom,
% 0.20/0.46 ! [X2: nat,Y2: nat] :
% 0.20/0.46 ( ( ord_less_nat @ X2 @ Y2 )
% 0.20/0.46 => ( Y2 != X2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % less_imp_not_eq2
% 0.20/0.46 thf(fact_164_antisym__conv3,axiom,
% 0.20/0.46 ! [Y2: nat,X2: nat] :
% 0.20/0.46 ( ~ ( ord_less_nat @ Y2 @ X2 )
% 0.20/0.46 => ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
% 0.20/0.46 = ( X2 = Y2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % antisym_conv3
% 0.20/0.46 thf(fact_165_less__induct,axiom,
% 0.20/0.46 ! [P: nat > $o,A: nat] :
% 0.20/0.46 ( ! [X5: nat] :
% 0.20/0.46 ( ! [Y4: nat] :
% 0.20/0.46 ( ( ord_less_nat @ Y4 @ X5 )
% 0.20/0.46 => ( P @ Y4 ) )
% 0.20/0.46 => ( P @ X5 ) )
% 0.20/0.46 => ( P @ A ) ) ).
% 0.20/0.46
% 0.20/0.46 % less_induct
% 0.20/0.46 thf(fact_166_less__not__sym,axiom,
% 0.20/0.46 ! [X2: nat,Y2: nat] :
% 0.20/0.46 ( ( ord_less_nat @ X2 @ Y2 )
% 0.20/0.46 => ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % less_not_sym
% 0.20/0.46 thf(fact_167_less__imp__not__eq,axiom,
% 0.20/0.46 ! [X2: nat,Y2: nat] :
% 0.20/0.46 ( ( ord_less_nat @ X2 @ Y2 )
% 0.20/0.46 => ( X2 != Y2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % less_imp_not_eq
% 0.20/0.46 thf(fact_168_dual__order_Oasym,axiom,
% 0.20/0.46 ! [B: nat,A: nat] :
% 0.20/0.46 ( ( ord_less_nat @ B @ A )
% 0.20/0.46 => ~ ( ord_less_nat @ A @ B ) ) ).
% 0.20/0.46
% 0.20/0.46 % dual_order.asym
% 0.20/0.46 thf(fact_169_ord__less__eq__trans,axiom,
% 0.20/0.46 ! [A: nat,B: nat,C: nat] :
% 0.20/0.46 ( ( ord_less_nat @ A @ B )
% 0.20/0.46 => ( ( B = C )
% 0.20/0.46 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % ord_less_eq_trans
% 0.20/0.46 thf(fact_170_ord__eq__less__trans,axiom,
% 0.20/0.46 ! [A: nat,B: nat,C: nat] :
% 0.20/0.46 ( ( A = B )
% 0.20/0.46 => ( ( ord_less_nat @ B @ C )
% 0.20/0.46 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % ord_eq_less_trans
% 0.20/0.46 thf(fact_171_less__irrefl,axiom,
% 0.20/0.46 ! [X2: nat] :
% 0.20/0.46 ~ ( ord_less_nat @ X2 @ X2 ) ).
% 0.20/0.46
% 0.20/0.46 % less_irrefl
% 0.20/0.46 thf(fact_172_less__linear,axiom,
% 0.20/0.46 ! [X2: nat,Y2: nat] :
% 0.20/0.46 ( ( ord_less_nat @ X2 @ Y2 )
% 0.20/0.46 | ( X2 = Y2 )
% 0.20/0.46 | ( ord_less_nat @ Y2 @ X2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % less_linear
% 0.20/0.46 thf(fact_173_less__trans,axiom,
% 0.20/0.46 ! [X2: nat,Y2: nat,Z3: nat] :
% 0.20/0.46 ( ( ord_less_nat @ X2 @ Y2 )
% 0.20/0.46 => ( ( ord_less_nat @ Y2 @ Z3 )
% 0.20/0.46 => ( ord_less_nat @ X2 @ Z3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % less_trans
% 0.20/0.47 thf(fact_174_less__asym_H,axiom,
% 0.20/0.47 ! [A: nat,B: nat] :
% 0.20/0.47 ( ( ord_less_nat @ A @ B )
% 0.20/0.47 => ~ ( ord_less_nat @ B @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 % less_asym'
% 0.20/0.47 thf(fact_175_less__asym,axiom,
% 0.20/0.47 ! [X2: nat,Y2: nat] :
% 0.20/0.47 ( ( ord_less_nat @ X2 @ Y2 )
% 0.20/0.47 => ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % less_asym
% 0.20/0.47 thf(fact_176_less__imp__neq,axiom,
% 0.20/0.47 ! [X2: nat,Y2: nat] :
% 0.20/0.47 ( ( ord_less_nat @ X2 @ Y2 )
% 0.20/0.47 => ( X2 != Y2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % less_imp_neq
% 0.20/0.47 thf(fact_177_order_Oasym,axiom,
% 0.20/0.47 ! [A: nat,B: nat] :
% 0.20/0.47 ( ( ord_less_nat @ A @ B )
% 0.20/0.47 => ~ ( ord_less_nat @ B @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 % order.asym
% 0.20/0.47 thf(fact_178_neq__iff,axiom,
% 0.20/0.47 ! [X2: nat,Y2: nat] :
% 0.20/0.47 ( ( X2 != Y2 )
% 0.20/0.47 = ( ( ord_less_nat @ X2 @ Y2 )
% 0.20/0.47 | ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % neq_iff
% 0.20/0.47 thf(fact_179_neqE,axiom,
% 0.20/0.47 ! [X2: nat,Y2: nat] :
% 0.20/0.47 ( ( X2 != Y2 )
% 0.20/0.47 => ( ~ ( ord_less_nat @ X2 @ Y2 )
% 0.20/0.47 => ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % neqE
% 0.20/0.47 thf(fact_180_gt__ex,axiom,
% 0.20/0.47 ! [X2: nat] :
% 0.20/0.47 ? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% 0.20/0.47
% 0.20/0.47 % gt_ex
% 0.20/0.47 thf(fact_181_order__less__subst2,axiom,
% 0.20/0.47 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.20/0.47 ( ( ord_less_nat @ A @ B )
% 0.20/0.47 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.20/0.47 => ( ! [X5: nat,Y3: nat] :
% 0.20/0.47 ( ( ord_less_nat @ X5 @ Y3 )
% 0.20/0.47 => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 0.20/0.47 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % order_less_subst2
% 0.20/0.47 thf(fact_182_order__less__subst1,axiom,
% 0.20/0.47 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.20/0.47 ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.20/0.47 => ( ( ord_less_nat @ B @ C )
% 0.20/0.47 => ( ! [X5: nat,Y3: nat] :
% 0.20/0.47 ( ( ord_less_nat @ X5 @ Y3 )
% 0.20/0.47 => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 0.20/0.47 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % order_less_subst1
% 0.20/0.47 thf(fact_183_ord__less__eq__subst,axiom,
% 0.20/0.47 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.20/0.47 ( ( ord_less_nat @ A @ B )
% 0.20/0.47 => ( ( ( F @ B )
% 0.20/0.47 = C )
% 0.20/0.47 => ( ! [X5: nat,Y3: nat] :
% 0.20/0.47 ( ( ord_less_nat @ X5 @ Y3 )
% 0.20/0.47 => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 0.20/0.47 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % ord_less_eq_subst
% 0.20/0.47 thf(fact_184_ord__eq__less__subst,axiom,
% 0.20/0.47 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.20/0.47 ( ( A
% 0.20/0.47 = ( F @ B ) )
% 0.20/0.47 => ( ( ord_less_nat @ B @ C )
% 0.20/0.47 => ( ! [X5: nat,Y3: nat] :
% 0.20/0.47 ( ( ord_less_nat @ X5 @ Y3 )
% 0.20/0.47 => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 0.20/0.47 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % ord_eq_less_subst
% 0.20/0.47 thf(fact_185_pinf_I1_J,axiom,
% 0.20/0.47 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 0.20/0.47 ( ? [Z4: nat] :
% 0.20/0.47 ! [X5: nat] :
% 0.20/0.47 ( ( ord_less_nat @ Z4 @ X5 )
% 0.20/0.47 => ( ( P @ X5 )
% 0.20/0.47 = ( P4 @ X5 ) ) )
% 0.20/0.47 => ( ? [Z4: nat] :
% 0.20/0.47 ! [X5: nat] :
% 0.20/0.47 ( ( ord_less_nat @ Z4 @ X5 )
% 0.20/0.47 => ( ( Q @ X5 )
% 0.20/0.47 = ( Q2 @ X5 ) ) )
% 0.20/0.47 => ? [Z2: nat] :
% 0.20/0.47 ! [X6: nat] :
% 0.20/0.47 ( ( ord_less_nat @ Z2 @ X6 )
% 0.20/0.47 => ( ( ( P @ X6 )
% 0.20/0.47 & ( Q @ X6 ) )
% 0.20/0.47 = ( ( P4 @ X6 )
% 0.20/0.47 & ( Q2 @ X6 ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % pinf(1)
% 0.20/0.47 thf(fact_186_pinf_I2_J,axiom,
% 0.20/0.47 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 0.20/0.47 ( ? [Z4: nat] :
% 0.20/0.47 ! [X5: nat] :
% 0.20/0.47 ( ( ord_less_nat @ Z4 @ X5 )
% 0.20/0.47 => ( ( P @ X5 )
% 0.20/0.47 = ( P4 @ X5 ) ) )
% 0.20/0.47 => ( ? [Z4: nat] :
% 0.20/0.47 ! [X5: nat] :
% 0.20/0.47 ( ( ord_less_nat @ Z4 @ X5 )
% 0.20/0.47 => ( ( Q @ X5 )
% 0.20/0.47 = ( Q2 @ X5 ) ) )
% 0.20/0.47 => ? [Z2: nat] :
% 0.20/0.47 ! [X6: nat] :
% 0.20/0.47 ( ( ord_less_nat @ Z2 @ X6 )
% 0.20/0.47 => ( ( ( P @ X6 )
% 0.20/0.47 | ( Q @ X6 ) )
% 0.20/0.47 = ( ( P4 @ X6 )
% 0.20/0.47 | ( Q2 @ X6 ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % pinf(2)
% 0.20/0.47 thf(fact_187_pinf_I3_J,axiom,
% 0.20/0.47 ! [T: nat] :
% 0.20/0.47 ? [Z2: nat] :
% 0.20/0.47 ! [X6: nat] :
% 0.20/0.47 ( ( ord_less_nat @ Z2 @ X6 )
% 0.20/0.47 => ( X6 != T ) ) ).
% 0.20/0.47
% 0.20/0.47 % pinf(3)
% 0.20/0.47 thf(fact_188_pinf_I4_J,axiom,
% 0.20/0.47 ! [T: nat] :
% 0.20/0.47 ? [Z2: nat] :
% 0.20/0.47 ! [X6: nat] :
% 0.20/0.47 ( ( ord_less_nat @ Z2 @ X6 )
% 0.20/0.47 => ( X6 != T ) ) ).
% 0.20/0.47
% 0.20/0.47 % pinf(4)
% 0.20/0.47 thf(fact_189_pinf_I5_J,axiom,
% 0.20/0.47 ! [T: nat] :
% 0.20/0.47 ? [Z2: nat] :
% 0.20/0.47 ! [X6: nat] :
% 0.20/0.47 ( ( ord_less_nat @ Z2 @ X6 )
% 0.20/0.47 => ~ ( ord_less_nat @ X6 @ T ) ) ).
% 0.20/0.47
% 0.20/0.47 % pinf(5)
% 0.20/0.47 thf(fact_190_pinf_I7_J,axiom,
% 0.20/0.47 ! [T: nat] :
% 0.20/0.47 ? [Z2: nat] :
% 0.20/0.47 ! [X6: nat] :
% 0.20/0.47 ( ( ord_less_nat @ Z2 @ X6 )
% 0.20/0.47 => ( ord_less_nat @ T @ X6 ) ) ).
% 0.20/0.47
% 0.20/0.47 % pinf(7)
% 0.20/0.47 thf(fact_191_minf_I1_J,axiom,
% 0.20/0.47 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 0.20/0.47 ( ? [Z4: nat] :
% 0.20/0.47 ! [X5: nat] :
% 0.20/0.47 ( ( ord_less_nat @ X5 @ Z4 )
% 0.20/0.47 => ( ( P @ X5 )
% 0.20/0.47 = ( P4 @ X5 ) ) )
% 0.20/0.47 => ( ? [Z4: nat] :
% 0.20/0.47 ! [X5: nat] :
% 0.20/0.47 ( ( ord_less_nat @ X5 @ Z4 )
% 0.20/0.47 => ( ( Q @ X5 )
% 0.20/0.47 = ( Q2 @ X5 ) ) )
% 0.20/0.47 => ? [Z2: nat] :
% 0.20/0.47 ! [X6: nat] :
% 0.20/0.47 ( ( ord_less_nat @ X6 @ Z2 )
% 0.20/0.47 => ( ( ( P @ X6 )
% 0.20/0.47 & ( Q @ X6 ) )
% 0.20/0.47 = ( ( P4 @ X6 )
% 0.20/0.47 & ( Q2 @ X6 ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % minf(1)
% 0.20/0.47 thf(fact_192_minf_I2_J,axiom,
% 0.20/0.47 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 0.20/0.47 ( ? [Z4: nat] :
% 0.20/0.47 ! [X5: nat] :
% 0.20/0.47 ( ( ord_less_nat @ X5 @ Z4 )
% 0.20/0.47 => ( ( P @ X5 )
% 0.20/0.47 = ( P4 @ X5 ) ) )
% 0.20/0.47 => ( ? [Z4: nat] :
% 0.20/0.47 ! [X5: nat] :
% 0.20/0.47 ( ( ord_less_nat @ X5 @ Z4 )
% 0.20/0.47 => ( ( Q @ X5 )
% 0.20/0.47 = ( Q2 @ X5 ) ) )
% 0.20/0.47 => ? [Z2: nat] :
% 0.20/0.47 ! [X6: nat] :
% 0.20/0.47 ( ( ord_less_nat @ X6 @ Z2 )
% 0.20/0.47 => ( ( ( P @ X6 )
% 0.20/0.47 | ( Q @ X6 ) )
% 0.20/0.47 = ( ( P4 @ X6 )
% 0.20/0.47 | ( Q2 @ X6 ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % minf(2)
% 0.20/0.47 thf(fact_193_minf_I3_J,axiom,
% 0.20/0.47 ! [T: nat] :
% 0.20/0.47 ? [Z2: nat] :
% 0.20/0.47 ! [X6: nat] :
% 0.20/0.47 ( ( ord_less_nat @ X6 @ Z2 )
% 0.20/0.47 => ( X6 != T ) ) ).
% 0.20/0.47
% 0.20/0.47 % minf(3)
% 0.20/0.47 thf(fact_194_minf_I4_J,axiom,
% 0.20/0.47 ! [T: nat] :
% 0.20/0.47 ? [Z2: nat] :
% 0.20/0.47 ! [X6: nat] :
% 0.20/0.47 ( ( ord_less_nat @ X6 @ Z2 )
% 0.20/0.47 => ( X6 != T ) ) ).
% 0.20/0.47
% 0.20/0.47 % minf(4)
% 0.20/0.47 thf(fact_195_minf_I5_J,axiom,
% 0.20/0.47 ! [T: nat] :
% 0.20/0.47 ? [Z2: nat] :
% 0.20/0.47 ! [X6: nat] :
% 0.20/0.47 ( ( ord_less_nat @ X6 @ Z2 )
% 0.20/0.47 => ( ord_less_nat @ X6 @ T ) ) ).
% 0.20/0.47
% 0.20/0.47 % minf(5)
% 0.20/0.47 thf(fact_196_minf_I7_J,axiom,
% 0.20/0.47 ! [T: nat] :
% 0.20/0.47 ? [Z2: nat] :
% 0.20/0.47 ! [X6: nat] :
% 0.20/0.47 ( ( ord_less_nat @ X6 @ Z2 )
% 0.20/0.47 => ~ ( ord_less_nat @ T @ X6 ) ) ).
% 0.20/0.47
% 0.20/0.47 % minf(7)
% 0.20/0.47 thf(fact_197_list__strict__asc__imp__list__asc,axiom,
% 0.20/0.47 ! [Xs: list_nat] :
% 0.20/0.47 ( ( list_strict_asc_nat @ Xs )
% 0.20/0.47 => ( list_asc_nat @ Xs ) ) ).
% 0.20/0.47
% 0.20/0.47 % list_strict_asc_imp_list_asc
% 0.20/0.47 thf(fact_198_list__strict__desc__imp__list__desc,axiom,
% 0.20/0.47 ! [Xs: list_nat] :
% 0.20/0.47 ( ( list_strict_desc_nat @ Xs )
% 0.20/0.47 => ( list_desc_nat @ Xs ) ) ).
% 0.20/0.47
% 0.20/0.47 % list_strict_desc_imp_list_desc
% 0.20/0.47 thf(fact_199_order_Onot__eq__order__implies__strict,axiom,
% 0.20/0.47 ! [A: nat,B: nat] :
% 0.20/0.47 ( ( A != B )
% 0.20/0.47 => ( ( ord_less_eq_nat @ A @ B )
% 0.20/0.47 => ( ord_less_nat @ A @ B ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % order.not_eq_order_implies_strict
% 0.20/0.47 thf(fact_200_dual__order_Ostrict__implies__order,axiom,
% 0.20/0.47 ! [B: nat,A: nat] :
% 0.20/0.47 ( ( ord_less_nat @ B @ A )
% 0.20/0.47 => ( ord_less_eq_nat @ B @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 % dual_order.strict_implies_order
% 0.20/0.47 thf(fact_201_dual__order_Ostrict__iff__order,axiom,
% 0.20/0.47 ( ord_less_nat
% 0.20/0.47 = ( ^ [B2: nat,A3: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ B2 @ A3 )
% 0.20/0.47 & ( A3 != B2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % dual_order.strict_iff_order
% 0.20/0.47 thf(fact_202_dual__order_Oorder__iff__strict,axiom,
% 0.20/0.47 ( ord_less_eq_nat
% 0.20/0.47 = ( ^ [B2: nat,A3: nat] :
% 0.20/0.47 ( ( ord_less_nat @ B2 @ A3 )
% 0.20/0.47 | ( A3 = B2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % dual_order.order_iff_strict
% 0.20/0.47 thf(fact_203_order_Ostrict__implies__order,axiom,
% 0.20/0.47 ! [A: nat,B: nat] :
% 0.20/0.47 ( ( ord_less_nat @ A @ B )
% 0.20/0.47 => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.20/0.47
% 0.20/0.47 % order.strict_implies_order
% 0.20/0.47 thf(fact_204_dual__order_Ostrict__trans2,axiom,
% 0.20/0.47 ! [B: nat,A: nat,C: nat] :
% 0.20/0.47 ( ( ord_less_nat @ B @ A )
% 0.20/0.47 => ( ( ord_less_eq_nat @ C @ B )
% 0.20/0.47 => ( ord_less_nat @ C @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % dual_order.strict_trans2
% 0.20/0.47 thf(fact_205_dual__order_Ostrict__trans1,axiom,
% 0.20/0.47 ! [B: nat,A: nat,C: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ B @ A )
% 0.20/0.47 => ( ( ord_less_nat @ C @ B )
% 0.20/0.47 => ( ord_less_nat @ C @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % dual_order.strict_trans1
% 0.20/0.47 thf(fact_206_order_Ostrict__iff__order,axiom,
% 0.20/0.47 ( ord_less_nat
% 0.20/0.47 = ( ^ [A3: nat,B2: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ A3 @ B2 )
% 0.20/0.47 & ( A3 != B2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % order.strict_iff_order
% 0.20/0.47 thf(fact_207_order_Oorder__iff__strict,axiom,
% 0.20/0.47 ( ord_less_eq_nat
% 0.20/0.47 = ( ^ [A3: nat,B2: nat] :
% 0.20/0.47 ( ( ord_less_nat @ A3 @ B2 )
% 0.20/0.47 | ( A3 = B2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % order.order_iff_strict
% 0.20/0.47 thf(fact_208_order_Ostrict__trans2,axiom,
% 0.20/0.47 ! [A: nat,B: nat,C: nat] :
% 0.20/0.47 ( ( ord_less_nat @ A @ B )
% 0.20/0.47 => ( ( ord_less_eq_nat @ B @ C )
% 0.20/0.47 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % order.strict_trans2
% 0.20/0.47 thf(fact_209_order_Ostrict__trans1,axiom,
% 0.20/0.47 ! [A: nat,B: nat,C: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ A @ B )
% 0.20/0.47 => ( ( ord_less_nat @ B @ C )
% 0.20/0.47 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % order.strict_trans1
% 0.20/0.47 thf(fact_210_not__le__imp__less,axiom,
% 0.20/0.47 ! [Y2: nat,X2: nat] :
% 0.20/0.47 ( ~ ( ord_less_eq_nat @ Y2 @ X2 )
% 0.20/0.47 => ( ord_less_nat @ X2 @ Y2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % not_le_imp_less
% 0.20/0.47 thf(fact_211_less__le__not__le,axiom,
% 0.20/0.47 ( ord_less_nat
% 0.20/0.47 = ( ^ [X3: nat,Y5: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ X3 @ Y5 )
% 0.20/0.47 & ~ ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % less_le_not_le
% 0.20/0.47 thf(fact_212_le__imp__less__or__eq,axiom,
% 0.20/0.47 ! [X2: nat,Y2: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.20/0.47 => ( ( ord_less_nat @ X2 @ Y2 )
% 0.20/0.47 | ( X2 = Y2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % le_imp_less_or_eq
% 0.20/0.47 thf(fact_213_le__less__linear,axiom,
% 0.20/0.47 ! [X2: nat,Y2: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.20/0.47 | ( ord_less_nat @ Y2 @ X2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % le_less_linear
% 0.20/0.47 thf(fact_214_less__le__trans,axiom,
% 0.20/0.47 ! [X2: nat,Y2: nat,Z3: nat] :
% 0.20/0.47 ( ( ord_less_nat @ X2 @ Y2 )
% 0.20/0.47 => ( ( ord_less_eq_nat @ Y2 @ Z3 )
% 0.20/0.47 => ( ord_less_nat @ X2 @ Z3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % less_le_trans
% 0.20/0.47 thf(fact_215_le__less__trans,axiom,
% 0.20/0.47 ! [X2: nat,Y2: nat,Z3: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.20/0.47 => ( ( ord_less_nat @ Y2 @ Z3 )
% 0.20/0.47 => ( ord_less_nat @ X2 @ Z3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % le_less_trans
% 0.20/0.47 thf(fact_216_less__imp__le,axiom,
% 0.20/0.47 ! [X2: nat,Y2: nat] :
% 0.20/0.47 ( ( ord_less_nat @ X2 @ Y2 )
% 0.20/0.47 => ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % less_imp_le
% 0.20/0.47 thf(fact_217_antisym__conv2,axiom,
% 0.20/0.47 ! [X2: nat,Y2: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.20/0.47 => ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
% 0.20/0.47 = ( X2 = Y2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % antisym_conv2
% 0.20/0.47 thf(fact_218_antisym__conv1,axiom,
% 0.20/0.47 ! [X2: nat,Y2: nat] :
% 0.20/0.47 ( ~ ( ord_less_nat @ X2 @ Y2 )
% 0.20/0.47 => ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.20/0.47 = ( X2 = Y2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % antisym_conv1
% 0.20/0.47 thf(fact_219_le__neq__trans,axiom,
% 0.20/0.47 ! [A: nat,B: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ A @ B )
% 0.20/0.47 => ( ( A != B )
% 0.20/0.47 => ( ord_less_nat @ A @ B ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % le_neq_trans
% 0.20/0.47 thf(fact_220_not__less,axiom,
% 0.20/0.47 ! [X2: nat,Y2: nat] :
% 0.20/0.47 ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
% 0.20/0.47 = ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % not_less
% 0.20/0.47 thf(fact_221_not__le,axiom,
% 0.20/0.47 ! [X2: nat,Y2: nat] :
% 0.20/0.47 ( ( ~ ( ord_less_eq_nat @ X2 @ Y2 ) )
% 0.20/0.47 = ( ord_less_nat @ Y2 @ X2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % not_le
% 0.20/0.47 thf(fact_222_order__less__le__subst2,axiom,
% 0.20/0.47 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.20/0.47 ( ( ord_less_nat @ A @ B )
% 0.20/0.47 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.20/0.47 => ( ! [X5: nat,Y3: nat] :
% 0.20/0.47 ( ( ord_less_nat @ X5 @ Y3 )
% 0.20/0.47 => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 0.20/0.47 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % order_less_le_subst2
% 0.20/0.47 thf(fact_223_order__less__le__subst1,axiom,
% 0.20/0.47 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.20/0.47 ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.20/0.47 => ( ( ord_less_eq_nat @ B @ C )
% 0.20/0.47 => ( ! [X5: nat,Y3: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 0.20/0.47 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 0.20/0.47 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % order_less_le_subst1
% 0.20/0.47 thf(fact_224_order__le__less__subst2,axiom,
% 0.20/0.47 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ A @ B )
% 0.20/0.47 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.20/0.47 => ( ! [X5: nat,Y3: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 0.20/0.47 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 0.20/0.47 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % order_le_less_subst2
% 0.20/0.47 thf(fact_225_order__le__less__subst1,axiom,
% 0.20/0.47 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.20/0.47 => ( ( ord_less_nat @ B @ C )
% 0.20/0.47 => ( ! [X5: nat,Y3: nat] :
% 0.20/0.47 ( ( ord_less_nat @ X5 @ Y3 )
% 0.20/0.47 => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 0.20/0.47 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % order_le_less_subst1
% 0.20/0.47 thf(fact_226_less__le,axiom,
% 0.20/0.47 ( ord_less_nat
% 0.20/0.47 = ( ^ [X3: nat,Y5: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ X3 @ Y5 )
% 0.20/0.47 & ( X3 != Y5 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % less_le
% 0.20/0.47 thf(fact_227_le__less,axiom,
% 0.20/0.47 ( ord_less_eq_nat
% 0.20/0.47 = ( ^ [X3: nat,Y5: nat] :
% 0.20/0.47 ( ( ord_less_nat @ X3 @ Y5 )
% 0.20/0.47 | ( X3 = Y5 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % le_less
% 0.20/0.47 thf(fact_228_leI,axiom,
% 0.20/0.47 ! [X2: nat,Y2: nat] :
% 0.20/0.47 ( ~ ( ord_less_nat @ X2 @ Y2 )
% 0.20/0.47 => ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % leI
% 0.20/0.47 thf(fact_229_leD,axiom,
% 0.20/0.47 ! [Y2: nat,X2: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ Y2 @ X2 )
% 0.20/0.47 => ~ ( ord_less_nat @ X2 @ Y2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % leD
% 0.20/0.47 thf(fact_230_pinf_I6_J,axiom,
% 0.20/0.47 ! [T: nat] :
% 0.20/0.47 ? [Z2: nat] :
% 0.20/0.47 ! [X6: nat] :
% 0.20/0.47 ( ( ord_less_nat @ Z2 @ X6 )
% 0.20/0.47 => ~ ( ord_less_eq_nat @ X6 @ T ) ) ).
% 0.20/0.47
% 0.20/0.47 % pinf(6)
% 0.20/0.47 thf(fact_231_pinf_I8_J,axiom,
% 0.20/0.47 ! [T: nat] :
% 0.20/0.47 ? [Z2: nat] :
% 0.20/0.47 ! [X6: nat] :
% 0.20/0.47 ( ( ord_less_nat @ Z2 @ X6 )
% 0.20/0.47 => ( ord_less_eq_nat @ T @ X6 ) ) ).
% 0.20/0.47
% 0.20/0.47 % pinf(8)
% 0.20/0.47 thf(fact_232_minf_I6_J,axiom,
% 0.20/0.47 ! [T: nat] :
% 0.20/0.47 ? [Z2: nat] :
% 0.20/0.47 ! [X6: nat] :
% 0.20/0.47 ( ( ord_less_nat @ X6 @ Z2 )
% 0.20/0.47 => ( ord_less_eq_nat @ X6 @ T ) ) ).
% 0.20/0.47
% 0.20/0.47 % minf(6)
% 0.20/0.47 thf(fact_233_complete__interval,axiom,
% 0.20/0.47 ! [A: nat,B: nat,P: nat > $o] :
% 0.20/0.47 ( ( ord_less_nat @ A @ B )
% 0.20/0.47 => ( ( P @ A )
% 0.20/0.47 => ( ~ ( P @ B )
% 0.20/0.47 => ? [C2: nat] :
% 0.20/0.47 ( ( ord_less_eq_nat @ A @ C2 )
% 0.20/0.47 & ( ord_less_eq_nat @ C2 @ B )
% 0.20/0.47 & ! [X6: nat] :
% 0.20/0.47 ( ( ( ord_less_eq_nat @ A @ X6 )
% 0.20/0.47 & ( ord_less_nat @ X6 @ C2 ) )
% 0.20/0.47 => ( P @ X6 ) )
% 0.20/0.47 & ! [D: nat] :
% 0.20/0.47 ( ! [X5: nat] :
% 0.20/0.47 ( ( ( ord_less_eq_nat @ A @ X5 )
% 0.20/0.47 & ( ord_less_nat @ X5 @ D ) )
% 0.20/0.47 => ( P @ X5 ) )
% 0.20/0.47 => ( ord_less_eq_nat @ D @ C2 ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % complete_interval
% 0.20/0.47 thf(fact_234_verit__comp__simplify1_I3_J,axiom,
% 0.20/0.47 ! [B4: nat,A5: nat] :
% 0.20/0.47 ( ( ~ ( ord_less_eq_nat @ B4 @ A5 ) )
% 0.20/0.47 = ( ord_less_nat @ A5 @ B4 ) ) ).
% 0.20/0.47
% 0.20/0.47 % verit_comp_simplify1(3)
% 0.20/0.47 thf(fact_235_set__swap,axiom,
% 0.20/0.47 ! [I3: nat,Xs: list_nat,J: nat] :
% 0.20/0.47 ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
% 0.20/0.47 => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
% 0.20/0.47 => ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I3 @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I3 ) ) )
% 0.20/0.47 = ( set_nat2 @ Xs ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % set_swap
% 0.20/0.47 thf(fact_236_set__swap,axiom,
% 0.20/0.47 ! [I3: nat,Xs: list_a,J: nat] :
% 0.20/0.47 ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
% 0.20/0.47 => ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs ) )
% 0.20/0.47 => ( ( set_a2 @ ( list_update_a @ ( list_update_a @ Xs @ I3 @ ( nth_a @ Xs @ J ) ) @ J @ ( nth_a @ Xs @ I3 ) ) )
% 0.20/0.47 = ( set_a2 @ Xs ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % set_swap
% 0.20/0.47 thf(fact_237_butlast__list__update,axiom,
% 0.20/0.47 ! [K: nat,Xs: list_nat,X2: nat] :
% 0.20/0.47 ( ( ( K
% 0.20/0.47 = ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
% 0.20/0.47 => ( ( butlast_nat @ ( list_update_nat @ Xs @ K @ X2 ) )
% 0.20/0.47 = ( butlast_nat @ Xs ) ) )
% 0.20/0.47 & ( ( K
% 0.20/0.47 != ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
% 0.20/0.47 => ( ( butlast_nat @ ( list_update_nat @ Xs @ K @ X2 ) )
% 0.20/0.47 = ( list_update_nat @ ( butlast_nat @ Xs ) @ K @ X2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % butlast_list_update
% 0.20/0.47 thf(fact_238_butlast__list__update,axiom,
% 0.20/0.47 ! [K: nat,Xs: list_a,X2: a] :
% 0.20/0.47 ( ( ( K
% 0.20/0.47 = ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) )
% 0.20/0.47 => ( ( butlast_a @ ( list_update_a @ Xs @ K @ X2 ) )
% 0.20/0.47 = ( butlast_a @ Xs ) ) )
% 0.20/0.47 & ( ( K
% 0.20/0.47 != ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) )
% 0.20/0.47 => ( ( butlast_a @ ( list_update_a @ Xs @ K @ X2 ) )
% 0.20/0.47 = ( list_update_a @ ( butlast_a @ Xs ) @ K @ X2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % butlast_list_update
% 0.20/0.47 thf(fact_239_length__butlast,axiom,
% 0.20/0.47 ! [Xs: list_nat] :
% 0.20/0.47 ( ( size_size_list_nat @ ( butlast_nat @ Xs ) )
% 0.20/0.47 = ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % length_butlast
% 0.20/0.47 thf(fact_240_length__butlast,axiom,
% 0.20/0.47 ! [Xs: list_a] :
% 0.20/0.47 ( ( size_size_list_a @ ( butlast_a @ Xs ) )
% 0.20/0.47 = ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % length_butlast
% 0.20/0.47 thf(fact_241_subset__code_I1_J,axiom,
% 0.20/0.47 ! [Xs: list_a,B5: set_a] :
% 0.20/0.47 ( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B5 )
% 0.20/0.47 = ( ! [X3: a] :
% 0.20/0.47 ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
% 0.20/0.47 => ( member_a @ X3 @ B5 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % subset_code(1)
% 0.20/0.47 thf(fact_242_set__update__subsetI,axiom,
% 0.20/0.47 ! [Xs: list_a,A2: set_a,X2: a,I3: nat] :
% 0.20/0.47 ( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ A2 )
% 0.20/0.47 => ( ( member_a @ X2 @ A2 )
% 0.20/0.47 => ( ord_less_eq_set_a @ ( set_a2 @ ( list_update_a @ Xs @ I3 @ X2 ) ) @ A2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % set_update_subsetI
% 0.20/0.47 thf(fact_243_in__set__butlastD,axiom,
% 0.20/0.47 ! [X2: a,Xs: list_a] :
% 0.20/0.47 ( ( member_a @ X2 @ ( set_a2 @ ( butlast_a @ Xs ) ) )
% 0.20/0.47 => ( member_a @ X2 @ ( set_a2 @ Xs ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % in_set_butlastD
% 0.20/0.47 thf(fact_244_one__reorient,axiom,
% 0.20/0.47 ! [X2: nat] :
% 0.20/0.47 ( ( one_one_nat = X2 )
% 0.20/0.47 = ( X2 = one_one_nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % one_reorient
% 0.20/0.47 thf(fact_245_list__ex__cong,axiom,
% 0.20/0.47 ! [Xs: list_a,Ys: list_a,F: a > $o,G: a > $o] :
% 0.20/0.47 ( ( Xs = Ys )
% 0.20/0.47 => ( ! [X5: a] :
% 0.20/0.47 ( ( member_a @ X5 @ ( set_a2 @ Ys ) )
% 0.20/0.47 => ( ( F @ X5 )
% 0.20/0.47 = ( G @ X5 ) ) )
% 0.20/0.47 => ( ( list_ex_a @ F @ Xs )
% 0.20/0.47 = ( list_ex_a @ G @ Ys ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % list_ex_cong
% 0.20/0.47 thf(fact_246_in__set__butlast__appendI,axiom,
% 0.20/0.47 ! [X2: a,Xs: list_a,Ys: list_a] :
% 0.20/0.47 ( ( ( member_a @ X2 @ ( set_a2 @ ( butlast_a @ Xs ) ) )
% 0.20/0.47 | ( member_a @ X2 @ ( set_a2 @ ( butlast_a @ Ys ) ) ) )
% 0.20/0.47 => ( member_a @ X2 @ ( set_a2 @ ( butlast_a @ ( append_a @ Xs @ Ys ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % in_set_butlast_appendI
% 0.20/0.47 thf(fact_247_nth__mem,axiom,
% 0.20/0.47 ! [N: nat,Xs: list_nat] :
% 0.20/0.47 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 0.20/0.47 => ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % nth_mem
% 0.20/0.47 thf(fact_248_nth__mem,axiom,
% 0.20/0.47 ! [N: nat,Xs: list_a] :
% 0.20/0.47 ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
% 0.20/0.47 => ( member_a @ ( nth_a @ Xs @ N ) @ ( set_a2 @ Xs ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % nth_mem
% 0.20/0.47 thf(fact_249_list__ball__nth,axiom,
% 0.20/0.47 ! [N: nat,Xs: list_nat,P: nat > $o] :
% 0.20/0.47 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 0.20/0.47 => ( ! [X5: nat] :
% 0.20/0.47 ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
% 0.20/0.47 => ( P @ X5 ) )
% 0.20/0.47 => ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % list_ball_nth
% 0.20/0.47 thf(fact_250_list__ball__nth,axiom,
% 0.20/0.47 ! [N: nat,Xs: list_a,P: a > $o] :
% 0.20/0.47 ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
% 0.20/0.47 => ( ! [X5: a] :
% 0.20/0.47 ( ( member_a @ X5 @ ( set_a2 @ Xs ) )
% 0.20/0.47 => ( P @ X5 ) )
% 0.20/0.47 => ( P @ ( nth_a @ Xs @ N ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % list_ball_nth
% 0.20/0.47 thf(fact_251_in__set__conv__nth,axiom,
% 0.20/0.47 ! [X2: nat,Xs: list_nat] :
% 0.20/0.47 ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
% 0.20/0.47 = ( ? [I2: nat] :
% 0.20/0.47 ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
% 0.20/0.47 & ( ( nth_nat @ Xs @ I2 )
% 0.20/0.47 = X2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % in_set_conv_nth
% 0.20/0.47 thf(fact_252_in__set__conv__nth,axiom,
% 0.20/0.47 ! [X2: a,Xs: list_a] :
% 0.20/0.47 ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
% 0.20/0.47 = ( ? [I2: nat] :
% 0.20/0.47 ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
% 0.20/0.47 & ( ( nth_a @ Xs @ I2 )
% 0.20/0.47 = X2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % in_set_conv_nth
% 0.20/0.47 thf(fact_253_all__nth__imp__all__set,axiom,
% 0.20/0.47 ! [Xs: list_nat,P: nat > $o,X2: nat] :
% 0.20/0.47 ( ! [I: nat] :
% 0.20/0.47 ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
% 0.20/0.47 => ( P @ ( nth_nat @ Xs @ I ) ) )
% 0.20/0.47 => ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
% 0.20/0.47 => ( P @ X2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % all_nth_imp_all_set
% 0.20/0.47 thf(fact_254_all__nth__imp__all__set,axiom,
% 0.20/0.47 ! [Xs: list_a,P: a > $o,X2: a] :
% 0.20/0.47 ( ! [I: nat] :
% 0.20/0.47 ( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
% 0.20/0.47 => ( P @ ( nth_a @ Xs @ I ) ) )
% 0.20/0.47 => ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
% 0.20/0.47 => ( P @ X2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % all_nth_imp_all_set
% 0.20/0.47 thf(fact_255_all__set__conv__all__nth,axiom,
% 0.20/0.47 ! [Xs: list_nat,P: nat > $o] :
% 0.20/0.47 ( ( ! [X3: nat] :
% 0.20/0.47 ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
% 0.20/0.47 => ( P @ X3 ) ) )
% 0.20/0.47 = ( ! [I2: nat] :
% 0.20/0.47 ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
% 0.20/0.47 => ( P @ ( nth_nat @ Xs @ I2 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % all_set_conv_all_nth
% 0.20/0.47 thf(fact_256_all__set__conv__all__nth,axiom,
% 0.20/0.47 ! [Xs: list_a,P: a > $o] :
% 0.20/0.47 ( ( ! [X3: a] :
% 0.20/0.47 ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
% 0.20/0.47 => ( P @ X3 ) ) )
% 0.20/0.47 = ( ! [I2: nat] :
% 0.20/0.47 ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
% 0.20/0.47 => ( P @ ( nth_a @ Xs @ I2 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % all_set_conv_all_nth
% 0.20/0.47 thf(fact_257_verit__la__disequality,axiom,
% 0.20/0.47 ! [A: nat,B: nat] :
% 0.20/0.47 ( ( A = B )
% 0.20/0.47 | ~ ( ord_less_eq_nat @ A @ B )
% 0.20/0.47 | ~ ( ord_less_eq_nat @ B @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 % verit_la_disequality
% 0.20/0.47
% 0.20/0.47 % Helper facts (5)
% 0.20/0.47 thf(help_If_2_1_If_001tf__a_T,axiom,
% 0.20/0.47 ! [X2: a,Y2: a] :
% 0.20/0.47 ( ( if_a @ $false @ X2 @ Y2 )
% 0.20/0.47 = Y2 ) ).
% 0.20/0.47
% 0.20/0.47 thf(help_If_1_1_If_001tf__a_T,axiom,
% 0.20/0.47 ! [X2: a,Y2: a] :
% 0.20/0.47 ( ( if_a @ $true @ X2 @ Y2 )
% 0.20/0.47 = X2 ) ).
% 0.20/0.47
% 0.20/0.47 thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
% 0.20/0.47 ! [P: $o] :
% 0.20/0.47 ( ( P = $true )
% 0.20/0.47 | ( P = $false ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 0.20/0.47 ! [X2: nat,Y2: nat] :
% 0.20/0.47 ( ( if_nat @ $false @ X2 @ Y2 )
% 0.20/0.47 = Y2 ) ).
% 0.20/0.47
% 0.20/0.47 thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 0.20/0.47 ! [X2: nat,Y2: nat] :
% 0.20/0.47 ( ( if_nat @ $true @ X2 @ Y2 )
% 0.20/0.47 = X2 ) ).
% 0.20/0.47
% 0.20/0.47 % Conjectures (4)
% 0.20/0.47 thf(conj_0,hypothesis,
% 0.20/0.47 ( ( size_size_list_a @ xs )
% 0.20/0.47 = ( size_size_list_a @ ys ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(conj_1,hypothesis,
% 0.20/0.47 ! [X6: nat] :
% 0.20/0.47 ( ( ( ord_less_nat @ X6 @ ( size_size_list_a @ ys ) )
% 0.20/0.47 => ( ( ( ord_less_nat @ X6 @ ( size_size_list_a @ ys ) )
% 0.20/0.47 => ( ( nth_a @ xs @ X6 )
% 0.20/0.47 = ( nth_a @ ys @ X6 ) ) )
% 0.20/0.47 & ( ~ ( ord_less_nat @ X6 @ ( size_size_list_a @ ys ) )
% 0.20/0.47 => ( ( f @ ( minus_minus_nat @ X6 @ ( size_size_list_a @ xs ) ) )
% 0.20/0.47 = ( nth_a @ ys @ X6 ) ) ) ) )
% 0.20/0.47 & ( ~ ( ord_less_nat @ X6 @ ( size_size_list_a @ ys ) )
% 0.20/0.47 => ( ( ( ord_less_nat @ X6 @ ( size_size_list_a @ ys ) )
% 0.20/0.47 => ( ( nth_a @ xs @ X6 )
% 0.20/0.47 = ( g @ ( minus_minus_nat @ X6 @ ( size_size_list_a @ ys ) ) ) ) )
% 0.20/0.47 & ( ~ ( ord_less_nat @ X6 @ ( size_size_list_a @ ys ) )
% 0.20/0.47 => ( ( f @ ( minus_minus_nat @ X6 @ ( size_size_list_a @ xs ) ) )
% 0.20/0.47 = ( g @ ( minus_minus_nat @ X6 @ ( size_size_list_a @ ys ) ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(conj_2,hypothesis,
% 0.20/0.47 ord_less_nat @ x @ ( size_size_list_a @ ys ) ).
% 0.20/0.47
% 0.20/0.47 thf(conj_3,conjecture,
% 0.20/0.47 ( ( nth_a @ xs @ x )
% 0.20/0.47 = ( nth_a @ ys @ x ) ) ).
% 0.20/0.47
% 0.20/0.47 %------------------------------------------------------------------------------
% 0.20/0.47 ------------------------
% 2.57/1.55 ---- Embedded in THF ---
% 2.57/1.56 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.57/1.56 %%% Generated on Tue Feb 24 08:26:47 EST 2026
% 2.57/1.56 %%% using '$$dhol' embedding, version 1.3.0.
% 2.57/1.56 %%% Logic specification used:
% 2.57/1.56 %%% thf(spec, logic, $$dhol).
% 2.57/1.56
% 2.57/1.56 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.wBFq0AA5QN/DTF2DTF27082.p
% See solution above
% 2.57/1.56 ------------------------
% 2.57/1.57 ---- Cleaned THF ---
% 2.57/1.58 thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
% 2.57/1.58 list_nat: $tType ).
% 2.57/1.58
% 2.57/1.58 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 2.57/1.58 set_nat: $tType ).
% 2.57/1.58
% 2.57/1.58 thf(ty_n_t__List__Olist_Itf__a_J,type,
% 2.57/1.58 list_a: $tType ).
% 2.57/1.58
% 2.57/1.58 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 2.57/1.58 set_a: $tType ).
% 2.57/1.58
% 2.57/1.58 thf(ty_n_t__Nat__Onat,type,
% 2.57/1.58 nat: $tType ).
% 2.57/1.58
% 2.57/1.58 thf(ty_n_tf__a,type,
% 2.57/1.58 a: $tType ).
% 2.57/1.58
% 2.57/1.58 thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
% 2.57/1.58 minus_minus_nat: nat > nat > nat ).
% 2.57/1.58
% 2.57/1.58 thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
% 2.57/1.58 one_one_nat: nat ).
% 2.57/1.58
% 2.57/1.58 thf(sy_c_If_001t__Nat__Onat,type,
% 2.57/1.58 if_nat: $o > nat > nat > nat ).
% 2.57/1.58
% 2.57/1.58 thf(sy_c_If_001tf__a,type,
% 2.57/1.58 if_a: $o > a > a > a ).
% 2.57/1.58
% 2.57/1.58 thf(sy_c_List2_Olist__asc_001t__Nat__Onat,type,
% 2.57/1.58 list_asc_nat: list_nat > $o ).
% 2.57/1.58
% 2.57/1.58 thf(sy_c_List2_Olist__desc_001t__Nat__Onat,type,
% 2.57/1.58 list_desc_nat: list_nat > $o ).
% 2.57/1.58
% 2.57/1.58 thf(sy_c_List2_Olist__strict__asc_001t__Nat__Onat,type,
% 2.57/1.58 list_strict_asc_nat: list_nat > $o ).
% 2.57/1.58
% 2.57/1.58 thf(sy_c_List2_Olist__strict__desc_001t__Nat__Onat,type,
% 2.57/1.58 list_strict_desc_nat: list_nat > $o ).
% 2.57/1.58
% 2.57/1.58 thf(sy_c_ListInf__Mirabelle__akbajwqfbr_Oi__append_001t__Nat__Onat,type,
% 2.57/1.58 listIn923761578nd_nat: list_nat > ( nat > nat ) > nat > nat ).
% 2.57/1.58
% 2.57/1.58 thf(sy_c_ListInf__Mirabelle__akbajwqfbr_Oi__append_001tf__a,type,
% 2.57/1.58 listIn1312259492pend_a: list_a > ( nat > a ) > nat > a ).
% 2.57/1.58
% 2.57/1.58 thf(sy_c_List_Oappend_001t__Nat__Onat,type,
% 2.57/1.58 append_nat: list_nat > list_nat > list_nat ).
% 2.57/1.58
% 2.57/1.58 thf(sy_c_List_Oappend_001tf__a,type,
% 2.57/1.58 append_a: list_a > list_a > list_a ).
% 2.57/1.58
% 2.57/1.58 thf(sy_c_List_Obutlast_001t__Nat__Onat,type,
% 2.57/1.58 butlast_nat: list_nat > list_nat ).
% 2.57/1.58
% 2.57/1.58 thf(sy_c_List_Obutlast_001tf__a,type,
% 2.57/1.58 butlast_a: list_a > list_a ).
% 2.57/1.58
% 2.57/1.58 thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
% 2.57/1.58 set_nat2: list_nat > set_nat ).
% 2.57/1.58
% 2.57/1.58 thf(sy_c_List_Olist_Oset_001tf__a,type,
% 2.57/1.58 set_a2: list_a > set_a ).
% 2.57/1.58
% 2.57/1.58 thf(sy_c_List_Olist__ex_001t__Nat__Onat,type,
% 2.57/1.58 list_ex_nat: ( nat > $o ) > list_nat > $o ).
% 2.57/1.58
% 2.57/1.58 thf(sy_c_List_Olist__ex_001tf__a,type,
% 2.57/1.58 list_ex_a: ( a > $o ) > list_a > $o ).
% 2.57/1.58
% 2.57/1.58 thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
% 2.57/1.58 list_update_nat: list_nat > nat > nat > list_nat ).
% 2.57/1.58
% 2.57/1.58 thf(sy_c_List_Olist__update_001tf__a,type,
% 2.57/1.58 list_update_a: list_a > nat > a > list_a ).
% 2.57/1.58
% 2.57/1.58 thf(sy_c_List_Onth_001t__Nat__Onat,type,
% 2.57/1.58 nth_nat: list_nat > nat > nat ).
% 2.57/1.58
% 2.57/1.58 thf(sy_c_List_Onth_001tf__a,type,
% 2.57/1.58 nth_a: list_a > nat > a ).
% 2.57/1.58
% 2.57/1.58 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
% 2.57/1.58 size_size_list_nat: list_nat > nat ).
% 2.57/1.58
% 2.57/1.58 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
% 2.57/1.58 size_size_list_a: list_a > nat ).
% 2.57/1.58
% 2.57/1.58 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 2.57/1.58 ord_less_nat: nat > nat > $o ).
% 2.57/1.58
% 2.57/1.58 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 2.57/1.58 ord_less_eq_nat: nat > nat > $o ).
% 2.57/1.58
% 2.57/1.58 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
% 2.57/1.58 ord_less_eq_set_a: set_a > set_a > $o ).
% 2.57/1.58
% 2.57/1.58 thf(sy_c_Set_OCollect_001tf__a,type,
% 2.57/1.58 collect_a: ( a > $o ) > set_a ).
% 2.57/1.58
% 2.57/1.58 thf(sy_c_member_001t__Nat__Onat,type,
% 2.57/1.58 member_nat: nat > set_nat > $o ).
% 2.57/1.58
% 2.57/1.58 thf(sy_c_member_001tf__a,type,
% 2.57/1.58 member_a: a > set_a > $o ).
% 2.57/1.58
% 2.57/1.58 thf(sy_v_f,type,
% 2.57/1.58 f: nat > a ).
% 2.57/1.58
% 2.57/1.58 thf(sy_v_g,type,
% 2.57/1.58 g: nat > a ).
% 2.57/1.58
% 2.57/1.58 thf(sy_v_x,type,
% 2.57/1.58 x: nat ).
% 2.57/1.58
% 2.57/1.58 thf(sy_v_xs,type,
% 2.57/1.58 xs: list_a ).
% 2.57/1.58
% 2.57/1.58 thf(sy_v_ys,type,
% 2.57/1.58 ys: list_a ).
% 2.57/1.58
% 2.57/1.58 thf(fact_0_nth__equalityI,axiom,
% 2.57/1.58 ! [Xs: list_nat,Ys: list_nat] :
% 2.57/1.58 ( ( ( size_size_list_nat @ Xs )
% 2.57/1.58 = ( size_size_list_nat @ Ys ) )
% 2.57/1.58 => ( ! [I: nat] :
% 2.57/1.58 ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
% 2.57/1.58 => ( ( nth_nat @ Xs @ I )
% 2.57/1.58 = ( nth_nat @ Ys @ I ) ) )
% 2.57/1.58 => ( Xs = Ys ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_1_nth__equalityI,axiom,
% 2.57/1.58 ! [Xs: list_a,Ys: list_a] :
% 2.57/1.58 ( ( ( size_size_list_a @ Xs )
% 2.57/1.58 = ( size_size_list_a @ Ys ) )
% 2.57/1.58 => ( ! [I: nat] :
% 2.57/1.58 ( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
% 2.57/1.58 => ( ( nth_a @ Xs @ I )
% 2.57/1.58 = ( nth_a @ Ys @ I ) ) )
% 2.57/1.58 => ( Xs = Ys ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_2_Skolem__list__nth,axiom,
% 2.57/1.58 ! [K: nat,P: nat > nat > $o] :
% 2.57/1.58 ( ( ! [I2: nat] :
% 2.57/1.58 ( ( ord_less_nat @ I2 @ K )
% 2.57/1.58 => ? [X: nat] : ( P @ I2 @ X ) ) )
% 2.57/1.58 = ( ? [Xs2: list_nat] :
% 2.57/1.58 ( ( ( size_size_list_nat @ Xs2 )
% 2.57/1.58 = K )
% 2.57/1.58 & ! [I2: nat] :
% 2.57/1.58 ( ( ord_less_nat @ I2 @ K )
% 2.57/1.58 => ( P @ I2 @ ( nth_nat @ Xs2 @ I2 ) ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_3_Skolem__list__nth,axiom,
% 2.57/1.58 ! [K: nat,P: nat > a > $o] :
% 2.57/1.58 ( ( ! [I2: nat] :
% 2.57/1.58 ( ( ord_less_nat @ I2 @ K )
% 2.57/1.58 => ? [X: a] : ( P @ I2 @ X ) ) )
% 2.57/1.58 = ( ? [Xs2: list_a] :
% 2.57/1.58 ( ( ( size_size_list_a @ Xs2 )
% 2.57/1.58 = K )
% 2.57/1.58 & ! [I2: nat] :
% 2.57/1.58 ( ( ord_less_nat @ I2 @ K )
% 2.57/1.58 => ( P @ I2 @ ( nth_a @ Xs2 @ I2 ) ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_4_list__eq__iff__nth__eq,axiom,
% 2.57/1.58 ( ( ^ [Y: list_nat,Z: list_nat] : ( Y = Z ) )
% 2.57/1.58 = ( ^ [Xs2: list_nat,Ys2: list_nat] :
% 2.57/1.58 ( ( ( size_size_list_nat @ Xs2 )
% 2.57/1.58 = ( size_size_list_nat @ Ys2 ) )
% 2.57/1.58 & ! [I2: nat] :
% 2.57/1.58 ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
% 2.57/1.58 => ( ( nth_nat @ Xs2 @ I2 )
% 2.57/1.58 = ( nth_nat @ Ys2 @ I2 ) ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_5_list__eq__iff__nth__eq,axiom,
% 2.57/1.58 ( ( ^ [Y: list_a,Z: list_a] : ( Y = Z ) )
% 2.57/1.58 = ( ^ [Xs2: list_a,Ys2: list_a] :
% 2.57/1.58 ( ( ( size_size_list_a @ Xs2 )
% 2.57/1.58 = ( size_size_list_a @ Ys2 ) )
% 2.57/1.58 & ! [I2: nat] :
% 2.57/1.58 ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs2 ) )
% 2.57/1.58 => ( ( nth_a @ Xs2 @ I2 )
% 2.57/1.58 = ( nth_a @ Ys2 @ I2 ) ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_6_i__append__nth1,axiom,
% 2.57/1.58 ! [N: nat,Xs: list_nat,F: nat > nat] :
% 2.57/1.58 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 2.57/1.58 => ( ( listIn923761578nd_nat @ Xs @ F @ N )
% 2.57/1.58 = ( nth_nat @ Xs @ N ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_7_i__append__nth1,axiom,
% 2.57/1.58 ! [N: nat,Xs: list_a,F: nat > a] :
% 2.57/1.58 ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
% 2.57/1.58 => ( ( listIn1312259492pend_a @ Xs @ F @ N )
% 2.57/1.58 = ( nth_a @ Xs @ N ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_8_diff__less__mono2,axiom,
% 2.57/1.58 ! [M: nat,N: nat,L: nat] :
% 2.57/1.58 ( ( ord_less_nat @ M @ N )
% 2.57/1.58 => ( ( ord_less_nat @ M @ L )
% 2.57/1.58 => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_9_less__imp__diff__less,axiom,
% 2.57/1.58 ! [J: nat,K: nat,N: nat] :
% 2.57/1.58 ( ( ord_less_nat @ J @ K )
% 2.57/1.58 => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_10_less__diff__imp__less,axiom,
% 2.57/1.58 ! [I3: nat,J: nat,M: nat] :
% 2.57/1.58 ( ( ord_less_nat @ I3 @ ( minus_minus_nat @ J @ M ) )
% 2.57/1.58 => ( ord_less_nat @ I3 @ J ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_11_nat__diff__left__cancel__eq1,axiom,
% 2.57/1.58 ! [K: nat,M: nat,N: nat] :
% 2.57/1.58 ( ( ( minus_minus_nat @ K @ M )
% 2.57/1.58 = ( minus_minus_nat @ K @ N ) )
% 2.57/1.58 => ( ( ord_less_nat @ M @ K )
% 2.57/1.58 => ( M = N ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_12_nat__diff__left__cancel__eq2,axiom,
% 2.57/1.58 ! [K: nat,M: nat,N: nat] :
% 2.57/1.58 ( ( ( minus_minus_nat @ K @ M )
% 2.57/1.58 = ( minus_minus_nat @ K @ N ) )
% 2.57/1.58 => ( ( ord_less_nat @ N @ K )
% 2.57/1.58 => ( M = N ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_13_nat__diff__left__cancel__less,axiom,
% 2.57/1.58 ! [K: nat,M: nat,N: nat] :
% 2.57/1.58 ( ( ord_less_nat @ ( minus_minus_nat @ K @ M ) @ ( minus_minus_nat @ K @ N ) )
% 2.57/1.58 => ( ord_less_nat @ N @ M ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_14_nat__diff__right__cancel__eq1,axiom,
% 2.57/1.58 ! [M: nat,K: nat,N: nat] :
% 2.57/1.58 ( ( ( minus_minus_nat @ M @ K )
% 2.57/1.58 = ( minus_minus_nat @ N @ K ) )
% 2.57/1.58 => ( ( ord_less_nat @ K @ M )
% 2.57/1.58 => ( M = N ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_15_linorder__neqE__nat,axiom,
% 2.57/1.58 ! [X2: nat,Y2: nat] :
% 2.57/1.58 ( ( X2 != Y2 )
% 2.57/1.58 => ( ~ ( ord_less_nat @ X2 @ Y2 )
% 2.57/1.58 => ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_16_infinite__descent,axiom,
% 2.57/1.58 ! [P: nat > $o,N: nat] :
% 2.57/1.58 ( ! [N2: nat] :
% 2.57/1.58 ( ~ ( P @ N2 )
% 2.57/1.58 => ? [M2: nat] :
% 2.57/1.58 ( ( ord_less_nat @ M2 @ N2 )
% 2.57/1.58 & ~ ( P @ M2 ) ) )
% 2.57/1.58 => ( P @ N ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_17_nat__less__induct,axiom,
% 2.57/1.58 ! [P: nat > $o,N: nat] :
% 2.57/1.58 ( ! [N2: nat] :
% 2.57/1.58 ( ! [M2: nat] :
% 2.57/1.58 ( ( ord_less_nat @ M2 @ N2 )
% 2.57/1.58 => ( P @ M2 ) )
% 2.57/1.58 => ( P @ N2 ) )
% 2.57/1.58 => ( P @ N ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_18_less__irrefl__nat,axiom,
% 2.57/1.58 ! [N: nat] :
% 2.57/1.58 ~ ( ord_less_nat @ N @ N ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_19_less__not__refl3,axiom,
% 2.57/1.58 ! [S: nat,T: nat] :
% 2.57/1.58 ( ( ord_less_nat @ S @ T )
% 2.57/1.58 => ( S != T ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_20_less__not__refl2,axiom,
% 2.57/1.58 ! [N: nat,M: nat] :
% 2.57/1.58 ( ( ord_less_nat @ N @ M )
% 2.57/1.58 => ( M != N ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_21_less__not__refl,axiom,
% 2.57/1.58 ! [N: nat] :
% 2.57/1.58 ~ ( ord_less_nat @ N @ N ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_22_nat__neq__iff,axiom,
% 2.57/1.58 ! [M: nat,N: nat] :
% 2.57/1.58 ( ( M != N )
% 2.57/1.58 = ( ( ord_less_nat @ M @ N )
% 2.57/1.58 | ( ord_less_nat @ N @ M ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_23_size__neq__size__imp__neq,axiom,
% 2.57/1.58 ! [X2: list_nat,Y2: list_nat] :
% 2.57/1.58 ( ( ( size_size_list_nat @ X2 )
% 2.57/1.58 != ( size_size_list_nat @ Y2 ) )
% 2.57/1.58 => ( X2 != Y2 ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_24_size__neq__size__imp__neq,axiom,
% 2.57/1.58 ! [X2: list_a,Y2: list_a] :
% 2.57/1.58 ( ( ( size_size_list_a @ X2 )
% 2.57/1.58 != ( size_size_list_a @ Y2 ) )
% 2.57/1.58 => ( X2 != Y2 ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_25_neq__if__length__neq,axiom,
% 2.57/1.58 ! [Xs: list_nat,Ys: list_nat] :
% 2.57/1.58 ( ( ( size_size_list_nat @ Xs )
% 2.57/1.58 != ( size_size_list_nat @ Ys ) )
% 2.57/1.58 => ( Xs != Ys ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_26_neq__if__length__neq,axiom,
% 2.57/1.58 ! [Xs: list_a,Ys: list_a] :
% 2.57/1.58 ( ( ( size_size_list_a @ Xs )
% 2.57/1.58 != ( size_size_list_a @ Ys ) )
% 2.57/1.58 => ( Xs != Ys ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_27_Ex__list__of__length,axiom,
% 2.57/1.58 ! [N: nat] :
% 2.57/1.58 ? [Xs3: list_nat] :
% 2.57/1.58 ( ( size_size_list_nat @ Xs3 )
% 2.57/1.58 = N ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_28_Ex__list__of__length,axiom,
% 2.57/1.58 ! [N: nat] :
% 2.57/1.58 ? [Xs3: list_a] :
% 2.57/1.58 ( ( size_size_list_a @ Xs3 )
% 2.57/1.58 = N ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_29_eq__imp__diff__eq,axiom,
% 2.57/1.58 ! [M: nat,N: nat,K: nat] :
% 2.57/1.58 ( ( M = N )
% 2.57/1.58 => ( ( minus_minus_nat @ M @ K )
% 2.57/1.58 = ( minus_minus_nat @ N @ K ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_30_diff__commute,axiom,
% 2.57/1.58 ! [I3: nat,J: nat,K: nat] :
% 2.57/1.58 ( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J ) @ K )
% 2.57/1.58 = ( minus_minus_nat @ ( minus_minus_nat @ I3 @ K ) @ J ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_31_i__append__nth,axiom,
% 2.57/1.58 ( listIn923761578nd_nat
% 2.57/1.58 = ( ^ [Xs2: list_nat,F2: nat > nat,N3: nat] : ( if_nat @ ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs2 ) ) @ ( nth_nat @ Xs2 @ N3 ) @ ( F2 @ ( minus_minus_nat @ N3 @ ( size_size_list_nat @ Xs2 ) ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_32_i__append__nth,axiom,
% 2.57/1.58 ( listIn1312259492pend_a
% 2.57/1.58 = ( ^ [Xs2: list_a,F2: nat > a,N3: nat] : ( if_a @ ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs2 ) ) @ ( nth_a @ Xs2 @ N3 ) @ ( F2 @ ( minus_minus_nat @ N3 @ ( size_size_list_a @ Xs2 ) ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_33_i__append__def,axiom,
% 2.57/1.58 ( listIn923761578nd_nat
% 2.57/1.58 = ( ^ [Xs2: list_nat,F2: nat > nat,N3: nat] : ( if_nat @ ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs2 ) ) @ ( nth_nat @ Xs2 @ N3 ) @ ( F2 @ ( minus_minus_nat @ N3 @ ( size_size_list_nat @ Xs2 ) ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_34_i__append__def,axiom,
% 2.57/1.58 ( listIn1312259492pend_a
% 2.57/1.58 = ( ^ [Xs2: list_a,F2: nat > a,N3: nat] : ( if_a @ ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs2 ) ) @ ( nth_a @ Xs2 @ N3 ) @ ( F2 @ ( minus_minus_nat @ N3 @ ( size_size_list_a @ Xs2 ) ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_35_length__induct,axiom,
% 2.57/1.58 ! [P: list_nat > $o,Xs: list_nat] :
% 2.57/1.58 ( ! [Xs3: list_nat] :
% 2.57/1.58 ( ! [Ys3: list_nat] :
% 2.57/1.58 ( ( ord_less_nat @ ( size_size_list_nat @ Ys3 ) @ ( size_size_list_nat @ Xs3 ) )
% 2.57/1.58 => ( P @ Ys3 ) )
% 2.57/1.58 => ( P @ Xs3 ) )
% 2.57/1.58 => ( P @ Xs ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_36_length__induct,axiom,
% 2.57/1.58 ! [P: list_a > $o,Xs: list_a] :
% 2.57/1.58 ( ! [Xs3: list_a] :
% 2.57/1.58 ( ! [Ys3: list_a] :
% 2.57/1.58 ( ( ord_less_nat @ ( size_size_list_a @ Ys3 ) @ ( size_size_list_a @ Xs3 ) )
% 2.57/1.58 => ( P @ Ys3 ) )
% 2.57/1.58 => ( P @ Xs3 ) )
% 2.57/1.58 => ( P @ Xs ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_37_nat__diff__right__cancel__less,axiom,
% 2.57/1.58 ! [N: nat,K: nat,M: nat] :
% 2.57/1.58 ( ( ord_less_nat @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M @ K ) )
% 2.57/1.58 => ( ord_less_nat @ N @ M ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_38_nat__diff__right__cancel__eq2,axiom,
% 2.57/1.58 ! [M: nat,K: nat,N: nat] :
% 2.57/1.58 ( ( ( minus_minus_nat @ M @ K )
% 2.57/1.58 = ( minus_minus_nat @ N @ K ) )
% 2.57/1.58 => ( ( ord_less_nat @ K @ N )
% 2.57/1.58 => ( M = N ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_39_list__strict__desc__trans,axiom,
% 2.57/1.58 ( list_strict_desc_nat
% 2.57/1.58 = ( ^ [Xs2: list_nat] :
% 2.57/1.58 ! [J2: nat] :
% 2.57/1.58 ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs2 ) )
% 2.57/1.58 => ! [I2: nat] :
% 2.57/1.58 ( ( ord_less_nat @ I2 @ J2 )
% 2.57/1.58 => ( ord_less_nat @ ( nth_nat @ Xs2 @ J2 ) @ ( nth_nat @ Xs2 @ I2 ) ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_40_list__strict__asc__trans,axiom,
% 2.57/1.58 ( list_strict_asc_nat
% 2.57/1.58 = ( ^ [Xs2: list_nat] :
% 2.57/1.58 ! [J2: nat] :
% 2.57/1.58 ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs2 ) )
% 2.57/1.58 => ! [I2: nat] :
% 2.57/1.58 ( ( ord_less_nat @ I2 @ J2 )
% 2.57/1.58 => ( ord_less_nat @ ( nth_nat @ Xs2 @ I2 ) @ ( nth_nat @ Xs2 @ J2 ) ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_41_list__ex__length,axiom,
% 2.57/1.58 ( list_ex_nat
% 2.57/1.58 = ( ^ [P2: nat > $o,Xs2: list_nat] :
% 2.57/1.58 ? [N3: nat] :
% 2.57/1.58 ( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs2 ) )
% 2.57/1.58 & ( P2 @ ( nth_nat @ Xs2 @ N3 ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_42_list__ex__length,axiom,
% 2.57/1.58 ( list_ex_a
% 2.57/1.58 = ( ^ [P2: a > $o,Xs2: list_a] :
% 2.57/1.58 ? [N3: nat] :
% 2.57/1.58 ( ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs2 ) )
% 2.57/1.58 & ( P2 @ ( nth_a @ Xs2 @ N3 ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_43_i__append__nth2,axiom,
% 2.57/1.58 ! [Xs: list_nat,N: nat,F: nat > nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
% 2.57/1.58 => ( ( listIn923761578nd_nat @ Xs @ F @ N )
% 2.57/1.58 = ( F @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_44_i__append__nth2,axiom,
% 2.57/1.58 ! [Xs: list_a,N: nat,F: nat > a] :
% 2.57/1.58 ( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N )
% 2.57/1.58 => ( ( listIn1312259492pend_a @ Xs @ F @ N )
% 2.57/1.58 = ( F @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_45_nth__append,axiom,
% 2.57/1.58 ! [N: nat,Xs: list_nat,Ys: list_nat] :
% 2.57/1.58 ( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 2.57/1.58 => ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
% 2.57/1.58 = ( nth_nat @ Xs @ N ) ) )
% 2.57/1.58 & ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 2.57/1.58 => ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
% 2.57/1.58 = ( nth_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_46_nth__append,axiom,
% 2.57/1.58 ! [N: nat,Xs: list_a,Ys: list_a] :
% 2.57/1.58 ( ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
% 2.57/1.58 => ( ( nth_a @ ( append_a @ Xs @ Ys ) @ N )
% 2.57/1.58 = ( nth_a @ Xs @ N ) ) )
% 2.57/1.58 & ( ~ ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
% 2.57/1.58 => ( ( nth_a @ ( append_a @ Xs @ Ys ) @ N )
% 2.57/1.58 = ( nth_a @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_47_nth__list__update__eq,axiom,
% 2.57/1.58 ! [I3: nat,Xs: list_nat,X2: nat] :
% 2.57/1.58 ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
% 2.57/1.58 => ( ( nth_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) @ I3 )
% 2.57/1.58 = X2 ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_48_nth__list__update__eq,axiom,
% 2.57/1.58 ! [I3: nat,Xs: list_a,X2: a] :
% 2.57/1.58 ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
% 2.57/1.58 => ( ( nth_a @ ( list_update_a @ Xs @ I3 @ X2 ) @ I3 )
% 2.57/1.58 = X2 ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_49_nth__butlast,axiom,
% 2.57/1.58 ! [N: nat,Xs: list_nat] :
% 2.57/1.58 ( ( ord_less_nat @ N @ ( size_size_list_nat @ ( butlast_nat @ Xs ) ) )
% 2.57/1.58 => ( ( nth_nat @ ( butlast_nat @ Xs ) @ N )
% 2.57/1.58 = ( nth_nat @ Xs @ N ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_50_nth__butlast,axiom,
% 2.57/1.58 ! [N: nat,Xs: list_a] :
% 2.57/1.58 ( ( ord_less_nat @ N @ ( size_size_list_a @ ( butlast_a @ Xs ) ) )
% 2.57/1.58 => ( ( nth_a @ ( butlast_a @ Xs ) @ N )
% 2.57/1.58 = ( nth_a @ Xs @ N ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_51_mem__Collect__eq,axiom,
% 2.57/1.58 ! [A: a,P: a > $o] :
% 2.57/1.58 ( ( member_a @ A @ ( collect_a @ P ) )
% 2.57/1.58 = ( P @ A ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_52_Collect__mem__eq,axiom,
% 2.57/1.58 ! [A2: set_a] :
% 2.57/1.58 ( ( collect_a
% 2.57/1.58 @ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
% 2.57/1.58 = A2 ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_53_append_Oassoc,axiom,
% 2.57/1.58 ! [A: list_a,B: list_a,C: list_a] :
% 2.57/1.58 ( ( append_a @ ( append_a @ A @ B ) @ C )
% 2.57/1.58 = ( append_a @ A @ ( append_a @ B @ C ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_54_append__assoc,axiom,
% 2.57/1.58 ! [Xs: list_a,Ys: list_a,Zs: list_a] :
% 2.57/1.58 ( ( append_a @ ( append_a @ Xs @ Ys ) @ Zs )
% 2.57/1.58 = ( append_a @ Xs @ ( append_a @ Ys @ Zs ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_55_append__same__eq,axiom,
% 2.57/1.58 ! [Ys: list_a,Xs: list_a,Zs: list_a] :
% 2.57/1.58 ( ( ( append_a @ Ys @ Xs )
% 2.57/1.58 = ( append_a @ Zs @ Xs ) )
% 2.57/1.58 = ( Ys = Zs ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_56_same__append__eq,axiom,
% 2.57/1.58 ! [Xs: list_a,Ys: list_a,Zs: list_a] :
% 2.57/1.58 ( ( ( append_a @ Xs @ Ys )
% 2.57/1.58 = ( append_a @ Xs @ Zs ) )
% 2.57/1.58 = ( Ys = Zs ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_57_list__update__overwrite,axiom,
% 2.57/1.58 ! [Xs: list_a,I3: nat,X2: a,Y2: a] :
% 2.57/1.58 ( ( list_update_a @ ( list_update_a @ Xs @ I3 @ X2 ) @ I3 @ Y2 )
% 2.57/1.58 = ( list_update_a @ Xs @ I3 @ Y2 ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_58_diff__diff__cancel,axiom,
% 2.57/1.58 ! [I3: nat,N: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ I3 @ N )
% 2.57/1.58 => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I3 ) )
% 2.57/1.58 = I3 ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_59_append__eq__append__conv,axiom,
% 2.57/1.58 ! [Xs: list_nat,Ys: list_nat,Us: list_nat,Vs: list_nat] :
% 2.57/1.58 ( ( ( ( size_size_list_nat @ Xs )
% 2.57/1.58 = ( size_size_list_nat @ Ys ) )
% 2.57/1.58 | ( ( size_size_list_nat @ Us )
% 2.57/1.58 = ( size_size_list_nat @ Vs ) ) )
% 2.57/1.58 => ( ( ( append_nat @ Xs @ Us )
% 2.57/1.58 = ( append_nat @ Ys @ Vs ) )
% 2.57/1.58 = ( ( Xs = Ys )
% 2.57/1.58 & ( Us = Vs ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_60_append__eq__append__conv,axiom,
% 2.57/1.58 ! [Xs: list_a,Ys: list_a,Us: list_a,Vs: list_a] :
% 2.57/1.58 ( ( ( ( size_size_list_a @ Xs )
% 2.57/1.58 = ( size_size_list_a @ Ys ) )
% 2.57/1.58 | ( ( size_size_list_a @ Us )
% 2.57/1.58 = ( size_size_list_a @ Vs ) ) )
% 2.57/1.58 => ( ( ( append_a @ Xs @ Us )
% 2.57/1.58 = ( append_a @ Ys @ Vs ) )
% 2.57/1.58 = ( ( Xs = Ys )
% 2.57/1.58 & ( Us = Vs ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_61_length__list__update,axiom,
% 2.57/1.58 ! [Xs: list_nat,I3: nat,X2: nat] :
% 2.57/1.58 ( ( size_size_list_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) )
% 2.57/1.58 = ( size_size_list_nat @ Xs ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_62_length__list__update,axiom,
% 2.57/1.58 ! [Xs: list_a,I3: nat,X2: a] :
% 2.57/1.58 ( ( size_size_list_a @ ( list_update_a @ Xs @ I3 @ X2 ) )
% 2.57/1.58 = ( size_size_list_a @ Xs ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_63_list__update__id,axiom,
% 2.57/1.58 ! [Xs: list_nat,I3: nat] :
% 2.57/1.58 ( ( list_update_nat @ Xs @ I3 @ ( nth_nat @ Xs @ I3 ) )
% 2.57/1.58 = Xs ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_64_list__update__id,axiom,
% 2.57/1.58 ! [Xs: list_a,I3: nat] :
% 2.57/1.58 ( ( list_update_a @ Xs @ I3 @ ( nth_a @ Xs @ I3 ) )
% 2.57/1.58 = Xs ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_65_nth__list__update__neq,axiom,
% 2.57/1.58 ! [I3: nat,J: nat,Xs: list_nat,X2: nat] :
% 2.57/1.58 ( ( I3 != J )
% 2.57/1.58 => ( ( nth_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) @ J )
% 2.57/1.58 = ( nth_nat @ Xs @ J ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_66_nth__list__update__neq,axiom,
% 2.57/1.58 ! [I3: nat,J: nat,Xs: list_a,X2: a] :
% 2.57/1.58 ( ( I3 != J )
% 2.57/1.58 => ( ( nth_a @ ( list_update_a @ Xs @ I3 @ X2 ) @ J )
% 2.57/1.58 = ( nth_a @ Xs @ J ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_67_list__ex__append,axiom,
% 2.57/1.58 ! [P: a > $o,Xs: list_a,Ys: list_a] :
% 2.57/1.58 ( ( list_ex_a @ P @ ( append_a @ Xs @ Ys ) )
% 2.57/1.58 = ( ( list_ex_a @ P @ Xs )
% 2.57/1.58 | ( list_ex_a @ P @ Ys ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_68_i__append__assoc,axiom,
% 2.57/1.58 ! [Xs: list_a,Ys: list_a,F: nat > a] :
% 2.57/1.58 ( ( listIn1312259492pend_a @ Xs @ ( listIn1312259492pend_a @ Ys @ F ) )
% 2.57/1.58 = ( listIn1312259492pend_a @ ( append_a @ Xs @ Ys ) @ F ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_69_list__update__beyond,axiom,
% 2.57/1.58 ! [Xs: list_nat,I3: nat,X2: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I3 )
% 2.57/1.58 => ( ( list_update_nat @ Xs @ I3 @ X2 )
% 2.57/1.58 = Xs ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_70_list__update__beyond,axiom,
% 2.57/1.58 ! [Xs: list_a,I3: nat,X2: a] :
% 2.57/1.58 ( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ I3 )
% 2.57/1.58 => ( ( list_update_a @ Xs @ I3 @ X2 )
% 2.57/1.58 = Xs ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_71_le__refl,axiom,
% 2.57/1.58 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_72_le__trans,axiom,
% 2.57/1.58 ! [I3: nat,J: nat,K: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ I3 @ J )
% 2.57/1.58 => ( ( ord_less_eq_nat @ J @ K )
% 2.57/1.58 => ( ord_less_eq_nat @ I3 @ K ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_73_eq__imp__le,axiom,
% 2.57/1.58 ! [M: nat,N: nat] :
% 2.57/1.58 ( ( M = N )
% 2.57/1.58 => ( ord_less_eq_nat @ M @ N ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_74_le__antisym,axiom,
% 2.57/1.58 ! [M: nat,N: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ M @ N )
% 2.57/1.58 => ( ( ord_less_eq_nat @ N @ M )
% 2.57/1.58 => ( M = N ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_75_nat__le__linear,axiom,
% 2.57/1.58 ! [M: nat,N: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ M @ N )
% 2.57/1.58 | ( ord_less_eq_nat @ N @ M ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_76_list__update__swap,axiom,
% 2.57/1.58 ! [I3: nat,I4: nat,Xs: list_a,X2: a,X4: a] :
% 2.57/1.58 ( ( I3 != I4 )
% 2.57/1.58 => ( ( list_update_a @ ( list_update_a @ Xs @ I3 @ X2 ) @ I4 @ X4 )
% 2.57/1.58 = ( list_update_a @ ( list_update_a @ Xs @ I4 @ X4 ) @ I3 @ X2 ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_77_append__eq__appendI,axiom,
% 2.57/1.58 ! [Xs: list_a,Xs1: list_a,Zs: list_a,Ys: list_a,Us: list_a] :
% 2.57/1.58 ( ( ( append_a @ Xs @ Xs1 )
% 2.57/1.58 = Zs )
% 2.57/1.58 => ( ( Ys
% 2.57/1.58 = ( append_a @ Xs1 @ Us ) )
% 2.57/1.58 => ( ( append_a @ Xs @ Ys )
% 2.57/1.58 = ( append_a @ Zs @ Us ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_78_Nat_Oex__has__greatest__nat,axiom,
% 2.57/1.58 ! [P: nat > $o,K: nat,B: nat] :
% 2.57/1.58 ( ( P @ K )
% 2.57/1.58 => ( ! [Y3: nat] :
% 2.57/1.58 ( ( P @ Y3 )
% 2.57/1.58 => ( ord_less_eq_nat @ Y3 @ B ) )
% 2.57/1.58 => ? [X5: nat] :
% 2.57/1.58 ( ( P @ X5 )
% 2.57/1.58 & ! [Y4: nat] :
% 2.57/1.58 ( ( P @ Y4 )
% 2.57/1.58 => ( ord_less_eq_nat @ Y4 @ X5 ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_79_append__eq__append__conv2,axiom,
% 2.57/1.58 ! [Xs: list_a,Ys: list_a,Zs: list_a,Ts: list_a] :
% 2.57/1.58 ( ( ( append_a @ Xs @ Ys )
% 2.57/1.58 = ( append_a @ Zs @ Ts ) )
% 2.57/1.58 = ( ? [Us2: list_a] :
% 2.57/1.58 ( ( ( Xs
% 2.57/1.58 = ( append_a @ Zs @ Us2 ) )
% 2.57/1.58 & ( ( append_a @ Us2 @ Ys )
% 2.57/1.58 = Ts ) )
% 2.57/1.58 | ( ( ( append_a @ Xs @ Us2 )
% 2.57/1.58 = Zs )
% 2.57/1.58 & ( Ys
% 2.57/1.58 = ( append_a @ Us2 @ Ts ) ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_80_list__strict__asc__trans__le,axiom,
% 2.57/1.58 ! [Xs: list_nat] :
% 2.57/1.58 ( ( list_strict_asc_nat @ Xs )
% 2.57/1.58 => ! [J3: nat] :
% 2.57/1.58 ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
% 2.57/1.58 => ! [I5: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ I5 @ J3 )
% 2.57/1.58 => ( ord_less_eq_nat @ ( nth_nat @ Xs @ I5 ) @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_81_list__update__append1,axiom,
% 2.57/1.58 ! [I3: nat,Xs: list_nat,Ys: list_nat,X2: nat] :
% 2.57/1.58 ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
% 2.57/1.58 => ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ I3 @ X2 )
% 2.57/1.58 = ( append_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) @ Ys ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_82_list__update__append1,axiom,
% 2.57/1.58 ! [I3: nat,Xs: list_a,Ys: list_a,X2: a] :
% 2.57/1.58 ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
% 2.57/1.58 => ( ( list_update_a @ ( append_a @ Xs @ Ys ) @ I3 @ X2 )
% 2.57/1.58 = ( append_a @ ( list_update_a @ Xs @ I3 @ X2 ) @ Ys ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_83_nth__append2,axiom,
% 2.57/1.58 ! [Xs: list_nat,N: nat,Ys: list_nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
% 2.57/1.58 => ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
% 2.57/1.58 = ( nth_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_84_nth__append2,axiom,
% 2.57/1.58 ! [Xs: list_a,N: nat,Ys: list_a] :
% 2.57/1.58 ( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N )
% 2.57/1.58 => ( ( nth_a @ ( append_a @ Xs @ Ys ) @ N )
% 2.57/1.58 = ( nth_a @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_85_ge__less__neq__conv,axiom,
% 2.57/1.58 ( ord_less_eq_nat
% 2.57/1.58 = ( ^ [A3: nat,N3: nat] :
% 2.57/1.58 ! [X3: nat] :
% 2.57/1.58 ( ( ord_less_nat @ X3 @ A3 )
% 2.57/1.58 => ( N3 != X3 ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_86_less__ge__neq__conv,axiom,
% 2.57/1.58 ( ord_less_nat
% 2.57/1.58 = ( ^ [N3: nat,A3: nat] :
% 2.57/1.58 ! [X3: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ A3 @ X3 )
% 2.57/1.58 => ( N3 != X3 ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_87_greater__le__neq__conv,axiom,
% 2.57/1.58 ( ord_less_nat
% 2.57/1.58 = ( ^ [A3: nat,N3: nat] :
% 2.57/1.58 ! [X3: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ X3 @ A3 )
% 2.57/1.58 => ( N3 != X3 ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_88_le__greater__neq__conv,axiom,
% 2.57/1.58 ( ord_less_eq_nat
% 2.57/1.58 = ( ^ [N3: nat,A3: nat] :
% 2.57/1.58 ! [X3: nat] :
% 2.57/1.58 ( ( ord_less_nat @ A3 @ X3 )
% 2.57/1.58 => ( N3 != X3 ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_89_nat__less__le,axiom,
% 2.57/1.58 ( ord_less_nat
% 2.57/1.58 = ( ^ [M3: nat,N3: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ M3 @ N3 )
% 2.57/1.58 & ( M3 != N3 ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_90_less__imp__le__nat,axiom,
% 2.57/1.58 ! [M: nat,N: nat] :
% 2.57/1.58 ( ( ord_less_nat @ M @ N )
% 2.57/1.58 => ( ord_less_eq_nat @ M @ N ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_91_le__eq__less__or__eq,axiom,
% 2.57/1.58 ( ord_less_eq_nat
% 2.57/1.58 = ( ^ [M3: nat,N3: nat] :
% 2.57/1.58 ( ( ord_less_nat @ M3 @ N3 )
% 2.57/1.58 | ( M3 = N3 ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_92_less__or__eq__imp__le,axiom,
% 2.57/1.58 ! [M: nat,N: nat] :
% 2.57/1.58 ( ( ( ord_less_nat @ M @ N )
% 2.57/1.58 | ( M = N ) )
% 2.57/1.58 => ( ord_less_eq_nat @ M @ N ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_93_le__neq__implies__less,axiom,
% 2.57/1.58 ! [M: nat,N: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ M @ N )
% 2.57/1.58 => ( ( M != N )
% 2.57/1.58 => ( ord_less_nat @ M @ N ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_94_less__mono__imp__le__mono,axiom,
% 2.57/1.58 ! [F: nat > nat,I3: nat,J: nat] :
% 2.57/1.58 ( ! [I: nat,J4: nat] :
% 2.57/1.58 ( ( ord_less_nat @ I @ J4 )
% 2.57/1.58 => ( ord_less_nat @ ( F @ I ) @ ( F @ J4 ) ) )
% 2.57/1.58 => ( ( ord_less_eq_nat @ I3 @ J )
% 2.57/1.58 => ( ord_less_eq_nat @ ( F @ I3 ) @ ( F @ J ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_95_eq__diff__iff,axiom,
% 2.57/1.58 ! [K: nat,M: nat,N: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ K @ M )
% 2.57/1.58 => ( ( ord_less_eq_nat @ K @ N )
% 2.57/1.58 => ( ( ( minus_minus_nat @ M @ K )
% 2.57/1.58 = ( minus_minus_nat @ N @ K ) )
% 2.57/1.58 = ( M = N ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_96_le__diff__iff,axiom,
% 2.57/1.58 ! [K: nat,M: nat,N: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ K @ M )
% 2.57/1.58 => ( ( ord_less_eq_nat @ K @ N )
% 2.57/1.58 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 2.57/1.58 = ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_97_Nat_Odiff__diff__eq,axiom,
% 2.57/1.58 ! [K: nat,M: nat,N: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ K @ M )
% 2.57/1.58 => ( ( ord_less_eq_nat @ K @ N )
% 2.57/1.58 => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 2.57/1.58 = ( minus_minus_nat @ M @ N ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_98_diff__le__mono,axiom,
% 2.57/1.58 ! [M: nat,N: nat,L: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ M @ N )
% 2.57/1.58 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_99_diff__le__self,axiom,
% 2.57/1.58 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_100_le__diff__iff_H,axiom,
% 2.57/1.58 ! [A: nat,C: nat,B: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ A @ C )
% 2.57/1.58 => ( ( ord_less_eq_nat @ B @ C )
% 2.57/1.58 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
% 2.57/1.58 = ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_101_diff__le__mono2,axiom,
% 2.57/1.58 ! [M: nat,N: nat,L: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ M @ N )
% 2.57/1.58 => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_102_le__diff__swap,axiom,
% 2.57/1.58 ! [I3: nat,K: nat,J: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ I3 @ K )
% 2.57/1.58 => ( ( ord_less_eq_nat @ J @ K )
% 2.57/1.58 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ K @ J ) @ I3 )
% 2.57/1.58 = ( ord_less_eq_nat @ ( minus_minus_nat @ K @ I3 ) @ J ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_103_le__diff__imp__le,axiom,
% 2.57/1.58 ! [I3: nat,J: nat,M: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ I3 @ ( minus_minus_nat @ J @ M ) )
% 2.57/1.58 => ( ord_less_eq_nat @ I3 @ J ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_104_le__imp__diff__le,axiom,
% 2.57/1.58 ! [J: nat,K: nat,N: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ J @ K )
% 2.57/1.58 => ( ord_less_eq_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_105_eq__diff__left__iff,axiom,
% 2.57/1.58 ! [M: nat,K: nat,N: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ M @ K )
% 2.57/1.58 => ( ( ord_less_eq_nat @ N @ K )
% 2.57/1.58 => ( ( ( minus_minus_nat @ K @ M )
% 2.57/1.58 = ( minus_minus_nat @ K @ N ) )
% 2.57/1.58 = ( M = N ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_106_le__diff__le__imp__le,axiom,
% 2.57/1.58 ! [I3: nat,J: nat,M: nat,N: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ I3 @ ( minus_minus_nat @ J @ M ) )
% 2.57/1.58 => ( ( ord_less_eq_nat @ N @ M )
% 2.57/1.58 => ( ord_less_eq_nat @ I3 @ ( minus_minus_nat @ J @ N ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_107_nat__diff__left__cancel__le2,axiom,
% 2.57/1.58 ! [K: nat,M: nat,N: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ ( minus_minus_nat @ K @ M ) @ ( minus_minus_nat @ K @ N ) )
% 2.57/1.58 => ( ( ord_less_eq_nat @ N @ K )
% 2.57/1.58 => ( ord_less_eq_nat @ N @ M ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_108_nat__diff__right__cancel__le2,axiom,
% 2.57/1.58 ! [M: nat,K: nat,N: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 2.57/1.58 => ( ( ord_less_eq_nat @ K @ N )
% 2.57/1.58 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_109_list__update__append,axiom,
% 2.57/1.58 ! [N: nat,Xs: list_nat,Ys: list_nat,X2: nat] :
% 2.57/1.58 ( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 2.57/1.58 => ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ N @ X2 )
% 2.57/1.58 = ( append_nat @ ( list_update_nat @ Xs @ N @ X2 ) @ Ys ) ) )
% 2.57/1.58 & ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 2.57/1.58 => ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ N @ X2 )
% 2.57/1.58 = ( append_nat @ Xs @ ( list_update_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ X2 ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_110_list__update__append,axiom,
% 2.57/1.58 ! [N: nat,Xs: list_a,Ys: list_a,X2: a] :
% 2.57/1.58 ( ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
% 2.57/1.58 => ( ( list_update_a @ ( append_a @ Xs @ Ys ) @ N @ X2 )
% 2.57/1.58 = ( append_a @ ( list_update_a @ Xs @ N @ X2 ) @ Ys ) ) )
% 2.57/1.58 & ( ~ ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
% 2.57/1.58 => ( ( list_update_a @ ( append_a @ Xs @ Ys ) @ N @ X2 )
% 2.57/1.58 = ( append_a @ Xs @ ( list_update_a @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) @ X2 ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_111_nth__append1,axiom,
% 2.57/1.58 ! [N: nat,Xs: list_nat,Ys: list_nat] :
% 2.57/1.58 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 2.57/1.58 => ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
% 2.57/1.58 = ( nth_nat @ Xs @ N ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_112_nth__append1,axiom,
% 2.57/1.58 ! [N: nat,Xs: list_a,Ys: list_a] :
% 2.57/1.58 ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
% 2.57/1.58 => ( ( nth_a @ ( append_a @ Xs @ Ys ) @ N )
% 2.57/1.58 = ( nth_a @ Xs @ N ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_113_less__diff__iff,axiom,
% 2.57/1.58 ! [K: nat,M: nat,N: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ K @ M )
% 2.57/1.58 => ( ( ord_less_eq_nat @ K @ N )
% 2.57/1.58 => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 2.57/1.58 = ( ord_less_nat @ M @ N ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_114_diff__less__mono,axiom,
% 2.57/1.58 ! [A: nat,B: nat,C: nat] :
% 2.57/1.58 ( ( ord_less_nat @ A @ B )
% 2.57/1.58 => ( ( ord_less_eq_nat @ C @ A )
% 2.57/1.58 => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_115_less__diff__le__imp__less,axiom,
% 2.57/1.58 ! [I3: nat,J: nat,M: nat,N: nat] :
% 2.57/1.58 ( ( ord_less_nat @ I3 @ ( minus_minus_nat @ J @ M ) )
% 2.57/1.58 => ( ( ord_less_eq_nat @ N @ M )
% 2.57/1.58 => ( ord_less_nat @ I3 @ ( minus_minus_nat @ J @ N ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_116_nat__diff__left__cancel__le1,axiom,
% 2.57/1.58 ! [K: nat,M: nat,N: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ ( minus_minus_nat @ K @ M ) @ ( minus_minus_nat @ K @ N ) )
% 2.57/1.58 => ( ( ord_less_nat @ M @ K )
% 2.57/1.58 => ( ord_less_eq_nat @ N @ M ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_117_nat__diff__right__cancel__le1,axiom,
% 2.57/1.58 ! [M: nat,K: nat,N: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 2.57/1.58 => ( ( ord_less_nat @ K @ M )
% 2.57/1.58 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_118_nth__list__update,axiom,
% 2.57/1.58 ! [I3: nat,Xs: list_nat,J: nat,X2: nat] :
% 2.57/1.58 ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
% 2.57/1.58 => ( ( ( I3 = J )
% 2.57/1.58 => ( ( nth_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) @ J )
% 2.57/1.58 = X2 ) )
% 2.57/1.58 & ( ( I3 != J )
% 2.57/1.58 => ( ( nth_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) @ J )
% 2.57/1.58 = ( nth_nat @ Xs @ J ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_119_nth__list__update,axiom,
% 2.57/1.58 ! [I3: nat,Xs: list_a,J: nat,X2: a] :
% 2.57/1.58 ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
% 2.57/1.58 => ( ( ( I3 = J )
% 2.57/1.58 => ( ( nth_a @ ( list_update_a @ Xs @ I3 @ X2 ) @ J )
% 2.57/1.58 = X2 ) )
% 2.57/1.58 & ( ( I3 != J )
% 2.57/1.58 => ( ( nth_a @ ( list_update_a @ Xs @ I3 @ X2 ) @ J )
% 2.57/1.58 = ( nth_a @ Xs @ J ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_120_list__update__same__conv,axiom,
% 2.57/1.58 ! [I3: nat,Xs: list_nat,X2: nat] :
% 2.57/1.58 ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
% 2.57/1.58 => ( ( ( list_update_nat @ Xs @ I3 @ X2 )
% 2.57/1.58 = Xs )
% 2.57/1.58 = ( ( nth_nat @ Xs @ I3 )
% 2.57/1.58 = X2 ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_121_list__update__same__conv,axiom,
% 2.57/1.58 ! [I3: nat,Xs: list_a,X2: a] :
% 2.57/1.58 ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
% 2.57/1.58 => ( ( ( list_update_a @ Xs @ I3 @ X2 )
% 2.57/1.58 = Xs )
% 2.57/1.58 = ( ( nth_a @ Xs @ I3 )
% 2.57/1.58 = X2 ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_122_diff__right__commute,axiom,
% 2.57/1.58 ! [A: nat,C: nat,B: nat] :
% 2.57/1.58 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
% 2.57/1.58 = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_123_list__desc__trans__le,axiom,
% 2.57/1.58 ( list_desc_nat
% 2.57/1.58 = ( ^ [Xs2: list_nat] :
% 2.57/1.58 ! [J2: nat] :
% 2.57/1.58 ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs2 ) )
% 2.57/1.58 => ! [I2: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ I2 @ J2 )
% 2.57/1.58 => ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ J2 ) @ ( nth_nat @ Xs2 @ I2 ) ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_124_list__asc__trans__le,axiom,
% 2.57/1.58 ( list_asc_nat
% 2.57/1.58 = ( ^ [Xs2: list_nat] :
% 2.57/1.58 ! [J2: nat] :
% 2.57/1.58 ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs2 ) )
% 2.57/1.58 => ! [I2: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ I2 @ J2 )
% 2.57/1.58 => ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ I2 ) @ ( nth_nat @ Xs2 @ J2 ) ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_125_list__desc__trans,axiom,
% 2.57/1.58 ( list_desc_nat
% 2.57/1.58 = ( ^ [Xs2: list_nat] :
% 2.57/1.58 ! [J2: nat] :
% 2.57/1.58 ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs2 ) )
% 2.57/1.58 => ! [I2: nat] :
% 2.57/1.58 ( ( ord_less_nat @ I2 @ J2 )
% 2.57/1.58 => ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ J2 ) @ ( nth_nat @ Xs2 @ I2 ) ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_126_list__asc__trans,axiom,
% 2.57/1.58 ( list_asc_nat
% 2.57/1.58 = ( ^ [Xs2: list_nat] :
% 2.57/1.58 ! [J2: nat] :
% 2.57/1.58 ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs2 ) )
% 2.57/1.58 => ! [I2: nat] :
% 2.57/1.58 ( ( ord_less_nat @ I2 @ J2 )
% 2.57/1.58 => ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ I2 ) @ ( nth_nat @ Xs2 @ J2 ) ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_127_order__refl,axiom,
% 2.57/1.58 ! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_128_nat__descend__induct,axiom,
% 2.57/1.58 ! [N: nat,P: nat > $o,M: nat] :
% 2.57/1.58 ( ! [K2: nat] :
% 2.57/1.58 ( ( ord_less_nat @ N @ K2 )
% 2.57/1.58 => ( P @ K2 ) )
% 2.57/1.58 => ( ! [K2: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ K2 @ N )
% 2.57/1.58 => ( ! [I5: nat] :
% 2.57/1.58 ( ( ord_less_nat @ K2 @ I5 )
% 2.57/1.58 => ( P @ I5 ) )
% 2.57/1.58 => ( P @ K2 ) ) )
% 2.57/1.58 => ( P @ M ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_129_minf_I8_J,axiom,
% 2.57/1.58 ! [T: nat] :
% 2.57/1.58 ? [Z2: nat] :
% 2.57/1.58 ! [X6: nat] :
% 2.57/1.58 ( ( ord_less_nat @ X6 @ Z2 )
% 2.57/1.58 => ~ ( ord_less_eq_nat @ T @ X6 ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_130_order__subst1,axiom,
% 2.57/1.58 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 2.57/1.58 => ( ( ord_less_eq_nat @ B @ C )
% 2.57/1.58 => ( ! [X5: nat,Y3: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 2.57/1.58 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 2.57/1.58 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_131_order__subst2,axiom,
% 2.57/1.58 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ A @ B )
% 2.57/1.58 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 2.57/1.58 => ( ! [X5: nat,Y3: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 2.57/1.58 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 2.57/1.58 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_132_ord__eq__le__subst,axiom,
% 2.57/1.58 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 2.57/1.58 ( ( A
% 2.57/1.58 = ( F @ B ) )
% 2.57/1.58 => ( ( ord_less_eq_nat @ B @ C )
% 2.57/1.58 => ( ! [X5: nat,Y3: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 2.57/1.58 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 2.57/1.58 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_133_ord__le__eq__subst,axiom,
% 2.57/1.58 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ A @ B )
% 2.57/1.58 => ( ( ( F @ B )
% 2.57/1.58 = C )
% 2.57/1.58 => ( ! [X5: nat,Y3: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 2.57/1.58 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 2.57/1.58 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_134_eq__iff,axiom,
% 2.57/1.58 ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
% 2.57/1.58 = ( ^ [X3: nat,Y5: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ X3 @ Y5 )
% 2.57/1.58 & ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_135_antisym,axiom,
% 2.57/1.58 ! [X2: nat,Y2: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.57/1.58 => ( ( ord_less_eq_nat @ Y2 @ X2 )
% 2.57/1.58 => ( X2 = Y2 ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_136_linear,axiom,
% 2.57/1.58 ! [X2: nat,Y2: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.57/1.58 | ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_137_eq__refl,axiom,
% 2.57/1.58 ! [X2: nat,Y2: nat] :
% 2.57/1.58 ( ( X2 = Y2 )
% 2.57/1.58 => ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_138_le__cases,axiom,
% 2.57/1.58 ! [X2: nat,Y2: nat] :
% 2.57/1.58 ( ~ ( ord_less_eq_nat @ X2 @ Y2 )
% 2.57/1.58 => ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_139_order_Otrans,axiom,
% 2.57/1.58 ! [A: nat,B: nat,C: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ A @ B )
% 2.57/1.58 => ( ( ord_less_eq_nat @ B @ C )
% 2.57/1.58 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_140_le__cases3,axiom,
% 2.57/1.58 ! [X2: nat,Y2: nat,Z3: nat] :
% 2.57/1.58 ( ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.57/1.58 => ~ ( ord_less_eq_nat @ Y2 @ Z3 ) )
% 2.57/1.58 => ( ( ( ord_less_eq_nat @ Y2 @ X2 )
% 2.57/1.58 => ~ ( ord_less_eq_nat @ X2 @ Z3 ) )
% 2.57/1.58 => ( ( ( ord_less_eq_nat @ X2 @ Z3 )
% 2.57/1.58 => ~ ( ord_less_eq_nat @ Z3 @ Y2 ) )
% 2.57/1.58 => ( ( ( ord_less_eq_nat @ Z3 @ Y2 )
% 2.57/1.58 => ~ ( ord_less_eq_nat @ Y2 @ X2 ) )
% 2.57/1.58 => ( ( ( ord_less_eq_nat @ Y2 @ Z3 )
% 2.57/1.58 => ~ ( ord_less_eq_nat @ Z3 @ X2 ) )
% 2.57/1.58 => ~ ( ( ord_less_eq_nat @ Z3 @ X2 )
% 2.57/1.58 => ~ ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_141_antisym__conv,axiom,
% 2.57/1.58 ! [Y2: nat,X2: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ Y2 @ X2 )
% 2.57/1.58 => ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.57/1.58 = ( X2 = Y2 ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_142_order__class_Oorder_Oeq__iff,axiom,
% 2.57/1.58 ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
% 2.57/1.58 = ( ^ [A3: nat,B2: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ A3 @ B2 )
% 2.57/1.58 & ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_143_ord__eq__le__trans,axiom,
% 2.57/1.58 ! [A: nat,B: nat,C: nat] :
% 2.57/1.58 ( ( A = B )
% 2.57/1.58 => ( ( ord_less_eq_nat @ B @ C )
% 2.57/1.58 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_144_ord__le__eq__trans,axiom,
% 2.57/1.58 ! [A: nat,B: nat,C: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ A @ B )
% 2.57/1.58 => ( ( B = C )
% 2.57/1.58 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_145_order__class_Oorder_Oantisym,axiom,
% 2.57/1.58 ! [A: nat,B: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ A @ B )
% 2.57/1.58 => ( ( ord_less_eq_nat @ B @ A )
% 2.57/1.58 => ( A = B ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_146_order__trans,axiom,
% 2.57/1.58 ! [X2: nat,Y2: nat,Z3: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.57/1.58 => ( ( ord_less_eq_nat @ Y2 @ Z3 )
% 2.57/1.58 => ( ord_less_eq_nat @ X2 @ Z3 ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_147_dual__order_Orefl,axiom,
% 2.57/1.58 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_148_linorder__wlog,axiom,
% 2.57/1.58 ! [P: nat > nat > $o,A: nat,B: nat] :
% 2.57/1.58 ( ! [A4: nat,B3: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ A4 @ B3 )
% 2.57/1.58 => ( P @ A4 @ B3 ) )
% 2.57/1.58 => ( ! [A4: nat,B3: nat] :
% 2.57/1.58 ( ( P @ B3 @ A4 )
% 2.57/1.58 => ( P @ A4 @ B3 ) )
% 2.57/1.58 => ( P @ A @ B ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_149_dual__order_Otrans,axiom,
% 2.57/1.58 ! [B: nat,A: nat,C: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ B @ A )
% 2.57/1.58 => ( ( ord_less_eq_nat @ C @ B )
% 2.57/1.58 => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_150_dual__order_Oeq__iff,axiom,
% 2.57/1.58 ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
% 2.57/1.58 = ( ^ [A3: nat,B2: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ B2 @ A3 )
% 2.57/1.58 & ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_151_dual__order_Oantisym,axiom,
% 2.57/1.58 ! [B: nat,A: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ B @ A )
% 2.57/1.58 => ( ( ord_less_eq_nat @ A @ B )
% 2.57/1.58 => ( A = B ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_152_dual__order_Ostrict__implies__not__eq,axiom,
% 2.57/1.58 ! [B: nat,A: nat] :
% 2.57/1.58 ( ( ord_less_nat @ B @ A )
% 2.57/1.58 => ( A != B ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_153_order_Ostrict__implies__not__eq,axiom,
% 2.57/1.58 ! [A: nat,B: nat] :
% 2.57/1.58 ( ( ord_less_nat @ A @ B )
% 2.57/1.58 => ( A != B ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_154_not__less__iff__gr__or__eq,axiom,
% 2.57/1.58 ! [X2: nat,Y2: nat] :
% 2.57/1.58 ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
% 2.57/1.58 = ( ( ord_less_nat @ Y2 @ X2 )
% 2.57/1.58 | ( X2 = Y2 ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_155_dual__order_Ostrict__trans,axiom,
% 2.57/1.58 ! [B: nat,A: nat,C: nat] :
% 2.57/1.58 ( ( ord_less_nat @ B @ A )
% 2.57/1.58 => ( ( ord_less_nat @ C @ B )
% 2.57/1.58 => ( ord_less_nat @ C @ A ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_156_linorder__less__wlog,axiom,
% 2.57/1.58 ! [P: nat > nat > $o,A: nat,B: nat] :
% 2.57/1.58 ( ! [A4: nat,B3: nat] :
% 2.57/1.58 ( ( ord_less_nat @ A4 @ B3 )
% 2.57/1.58 => ( P @ A4 @ B3 ) )
% 2.57/1.58 => ( ! [A4: nat] : ( P @ A4 @ A4 )
% 2.57/1.58 => ( ! [A4: nat,B3: nat] :
% 2.57/1.58 ( ( P @ B3 @ A4 )
% 2.57/1.58 => ( P @ A4 @ B3 ) )
% 2.57/1.58 => ( P @ A @ B ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_157_exists__least__iff,axiom,
% 2.57/1.58 ( ( ^ [P3: nat > $o] :
% 2.57/1.58 ? [X: nat] : ( P3 @ X ) )
% 2.57/1.58 = ( ^ [P2: nat > $o] :
% 2.57/1.58 ? [N3: nat] :
% 2.57/1.58 ( ( P2 @ N3 )
% 2.57/1.58 & ! [M3: nat] :
% 2.57/1.58 ( ( ord_less_nat @ M3 @ N3 )
% 2.57/1.58 => ~ ( P2 @ M3 ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_158_less__imp__not__less,axiom,
% 2.57/1.58 ! [X2: nat,Y2: nat] :
% 2.57/1.58 ( ( ord_less_nat @ X2 @ Y2 )
% 2.57/1.58 => ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_159_order_Ostrict__trans,axiom,
% 2.57/1.58 ! [A: nat,B: nat,C: nat] :
% 2.57/1.58 ( ( ord_less_nat @ A @ B )
% 2.57/1.58 => ( ( ord_less_nat @ B @ C )
% 2.57/1.58 => ( ord_less_nat @ A @ C ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_160_dual__order_Oirrefl,axiom,
% 2.57/1.58 ! [A: nat] :
% 2.57/1.58 ~ ( ord_less_nat @ A @ A ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_161_linorder__cases,axiom,
% 2.57/1.58 ! [X2: nat,Y2: nat] :
% 2.57/1.58 ( ~ ( ord_less_nat @ X2 @ Y2 )
% 2.57/1.58 => ( ( X2 != Y2 )
% 2.57/1.58 => ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_162_less__imp__triv,axiom,
% 2.57/1.58 ! [X2: nat,Y2: nat,P: $o] :
% 2.57/1.58 ( ( ord_less_nat @ X2 @ Y2 )
% 2.57/1.58 => ( ( ord_less_nat @ Y2 @ X2 )
% 2.57/1.58 => P ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_163_less__imp__not__eq2,axiom,
% 2.57/1.58 ! [X2: nat,Y2: nat] :
% 2.57/1.58 ( ( ord_less_nat @ X2 @ Y2 )
% 2.57/1.58 => ( Y2 != X2 ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_164_antisym__conv3,axiom,
% 2.57/1.58 ! [Y2: nat,X2: nat] :
% 2.57/1.58 ( ~ ( ord_less_nat @ Y2 @ X2 )
% 2.57/1.58 => ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
% 2.57/1.58 = ( X2 = Y2 ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_165_less__induct,axiom,
% 2.57/1.58 ! [P: nat > $o,A: nat] :
% 2.57/1.58 ( ! [X5: nat] :
% 2.57/1.58 ( ! [Y4: nat] :
% 2.57/1.58 ( ( ord_less_nat @ Y4 @ X5 )
% 2.57/1.58 => ( P @ Y4 ) )
% 2.57/1.58 => ( P @ X5 ) )
% 2.57/1.58 => ( P @ A ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_166_less__not__sym,axiom,
% 2.57/1.58 ! [X2: nat,Y2: nat] :
% 2.57/1.58 ( ( ord_less_nat @ X2 @ Y2 )
% 2.57/1.58 => ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_167_less__imp__not__eq,axiom,
% 2.57/1.58 ! [X2: nat,Y2: nat] :
% 2.57/1.58 ( ( ord_less_nat @ X2 @ Y2 )
% 2.57/1.58 => ( X2 != Y2 ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_168_dual__order_Oasym,axiom,
% 2.57/1.58 ! [B: nat,A: nat] :
% 2.57/1.58 ( ( ord_less_nat @ B @ A )
% 2.57/1.58 => ~ ( ord_less_nat @ A @ B ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_169_ord__less__eq__trans,axiom,
% 2.57/1.58 ! [A: nat,B: nat,C: nat] :
% 2.57/1.58 ( ( ord_less_nat @ A @ B )
% 2.57/1.58 => ( ( B = C )
% 2.57/1.58 => ( ord_less_nat @ A @ C ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_170_ord__eq__less__trans,axiom,
% 2.57/1.58 ! [A: nat,B: nat,C: nat] :
% 2.57/1.58 ( ( A = B )
% 2.57/1.58 => ( ( ord_less_nat @ B @ C )
% 2.57/1.58 => ( ord_less_nat @ A @ C ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_171_less__irrefl,axiom,
% 2.57/1.58 ! [X2: nat] :
% 2.57/1.58 ~ ( ord_less_nat @ X2 @ X2 ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_172_less__linear,axiom,
% 2.57/1.58 ! [X2: nat,Y2: nat] :
% 2.57/1.58 ( ( ord_less_nat @ X2 @ Y2 )
% 2.57/1.58 | ( X2 = Y2 )
% 2.57/1.58 | ( ord_less_nat @ Y2 @ X2 ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_173_less__trans,axiom,
% 2.57/1.58 ! [X2: nat,Y2: nat,Z3: nat] :
% 2.57/1.58 ( ( ord_less_nat @ X2 @ Y2 )
% 2.57/1.58 => ( ( ord_less_nat @ Y2 @ Z3 )
% 2.57/1.58 => ( ord_less_nat @ X2 @ Z3 ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_174_less__asym_H,axiom,
% 2.57/1.58 ! [A: nat,B: nat] :
% 2.57/1.58 ( ( ord_less_nat @ A @ B )
% 2.57/1.58 => ~ ( ord_less_nat @ B @ A ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_175_less__asym,axiom,
% 2.57/1.58 ! [X2: nat,Y2: nat] :
% 2.57/1.58 ( ( ord_less_nat @ X2 @ Y2 )
% 2.57/1.58 => ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_176_less__imp__neq,axiom,
% 2.57/1.58 ! [X2: nat,Y2: nat] :
% 2.57/1.58 ( ( ord_less_nat @ X2 @ Y2 )
% 2.57/1.58 => ( X2 != Y2 ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_177_order_Oasym,axiom,
% 2.57/1.58 ! [A: nat,B: nat] :
% 2.57/1.58 ( ( ord_less_nat @ A @ B )
% 2.57/1.58 => ~ ( ord_less_nat @ B @ A ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_178_neq__iff,axiom,
% 2.57/1.58 ! [X2: nat,Y2: nat] :
% 2.57/1.58 ( ( X2 != Y2 )
% 2.57/1.58 = ( ( ord_less_nat @ X2 @ Y2 )
% 2.57/1.58 | ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_179_neqE,axiom,
% 2.57/1.58 ! [X2: nat,Y2: nat] :
% 2.57/1.58 ( ( X2 != Y2 )
% 2.57/1.58 => ( ~ ( ord_less_nat @ X2 @ Y2 )
% 2.57/1.58 => ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_180_gt__ex,axiom,
% 2.57/1.58 ! [X2: nat] :
% 2.57/1.58 ? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_181_order__less__subst2,axiom,
% 2.57/1.58 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 2.57/1.58 ( ( ord_less_nat @ A @ B )
% 2.57/1.58 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 2.57/1.58 => ( ! [X5: nat,Y3: nat] :
% 2.57/1.58 ( ( ord_less_nat @ X5 @ Y3 )
% 2.57/1.58 => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 2.57/1.58 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_182_order__less__subst1,axiom,
% 2.57/1.58 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 2.57/1.58 ( ( ord_less_nat @ A @ ( F @ B ) )
% 2.57/1.58 => ( ( ord_less_nat @ B @ C )
% 2.57/1.58 => ( ! [X5: nat,Y3: nat] :
% 2.57/1.58 ( ( ord_less_nat @ X5 @ Y3 )
% 2.57/1.58 => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 2.57/1.58 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_183_ord__less__eq__subst,axiom,
% 2.57/1.58 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 2.57/1.58 ( ( ord_less_nat @ A @ B )
% 2.57/1.58 => ( ( ( F @ B )
% 2.57/1.58 = C )
% 2.57/1.58 => ( ! [X5: nat,Y3: nat] :
% 2.57/1.58 ( ( ord_less_nat @ X5 @ Y3 )
% 2.57/1.58 => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 2.57/1.58 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_184_ord__eq__less__subst,axiom,
% 2.57/1.58 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 2.57/1.58 ( ( A
% 2.57/1.58 = ( F @ B ) )
% 2.57/1.58 => ( ( ord_less_nat @ B @ C )
% 2.57/1.58 => ( ! [X5: nat,Y3: nat] :
% 2.57/1.58 ( ( ord_less_nat @ X5 @ Y3 )
% 2.57/1.58 => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 2.57/1.58 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_185_pinf_I1_J,axiom,
% 2.57/1.58 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 2.57/1.58 ( ? [Z4: nat] :
% 2.57/1.58 ! [X5: nat] :
% 2.57/1.58 ( ( ord_less_nat @ Z4 @ X5 )
% 2.57/1.58 => ( ( P @ X5 )
% 2.57/1.58 = ( P4 @ X5 ) ) )
% 2.57/1.58 => ( ? [Z4: nat] :
% 2.57/1.58 ! [X5: nat] :
% 2.57/1.58 ( ( ord_less_nat @ Z4 @ X5 )
% 2.57/1.58 => ( ( Q @ X5 )
% 2.57/1.58 = ( Q2 @ X5 ) ) )
% 2.57/1.58 => ? [Z2: nat] :
% 2.57/1.58 ! [X6: nat] :
% 2.57/1.58 ( ( ord_less_nat @ Z2 @ X6 )
% 2.57/1.58 => ( ( ( P @ X6 )
% 2.57/1.58 & ( Q @ X6 ) )
% 2.57/1.58 = ( ( P4 @ X6 )
% 2.57/1.58 & ( Q2 @ X6 ) ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_186_pinf_I2_J,axiom,
% 2.57/1.58 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 2.57/1.58 ( ? [Z4: nat] :
% 2.57/1.58 ! [X5: nat] :
% 2.57/1.58 ( ( ord_less_nat @ Z4 @ X5 )
% 2.57/1.58 => ( ( P @ X5 )
% 2.57/1.58 = ( P4 @ X5 ) ) )
% 2.57/1.58 => ( ? [Z4: nat] :
% 2.57/1.58 ! [X5: nat] :
% 2.57/1.58 ( ( ord_less_nat @ Z4 @ X5 )
% 2.57/1.58 => ( ( Q @ X5 )
% 2.57/1.58 = ( Q2 @ X5 ) ) )
% 2.57/1.58 => ? [Z2: nat] :
% 2.57/1.58 ! [X6: nat] :
% 2.57/1.58 ( ( ord_less_nat @ Z2 @ X6 )
% 2.57/1.58 => ( ( ( P @ X6 )
% 2.57/1.58 | ( Q @ X6 ) )
% 2.57/1.58 = ( ( P4 @ X6 )
% 2.57/1.58 | ( Q2 @ X6 ) ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_187_pinf_I3_J,axiom,
% 2.57/1.58 ! [T: nat] :
% 2.57/1.58 ? [Z2: nat] :
% 2.57/1.58 ! [X6: nat] :
% 2.57/1.58 ( ( ord_less_nat @ Z2 @ X6 )
% 2.57/1.58 => ( X6 != T ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_188_pinf_I4_J,axiom,
% 2.57/1.58 ! [T: nat] :
% 2.57/1.58 ? [Z2: nat] :
% 2.57/1.58 ! [X6: nat] :
% 2.57/1.58 ( ( ord_less_nat @ Z2 @ X6 )
% 2.57/1.58 => ( X6 != T ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_189_pinf_I5_J,axiom,
% 2.57/1.58 ! [T: nat] :
% 2.57/1.58 ? [Z2: nat] :
% 2.57/1.58 ! [X6: nat] :
% 2.57/1.58 ( ( ord_less_nat @ Z2 @ X6 )
% 2.57/1.58 => ~ ( ord_less_nat @ X6 @ T ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_190_pinf_I7_J,axiom,
% 2.57/1.58 ! [T: nat] :
% 2.57/1.58 ? [Z2: nat] :
% 2.57/1.58 ! [X6: nat] :
% 2.57/1.58 ( ( ord_less_nat @ Z2 @ X6 )
% 2.57/1.58 => ( ord_less_nat @ T @ X6 ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_191_minf_I1_J,axiom,
% 2.57/1.58 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 2.57/1.58 ( ? [Z4: nat] :
% 2.57/1.58 ! [X5: nat] :
% 2.57/1.58 ( ( ord_less_nat @ X5 @ Z4 )
% 2.57/1.58 => ( ( P @ X5 )
% 2.57/1.58 = ( P4 @ X5 ) ) )
% 2.57/1.58 => ( ? [Z4: nat] :
% 2.57/1.58 ! [X5: nat] :
% 2.57/1.58 ( ( ord_less_nat @ X5 @ Z4 )
% 2.57/1.58 => ( ( Q @ X5 )
% 2.57/1.58 = ( Q2 @ X5 ) ) )
% 2.57/1.58 => ? [Z2: nat] :
% 2.57/1.58 ! [X6: nat] :
% 2.57/1.58 ( ( ord_less_nat @ X6 @ Z2 )
% 2.57/1.58 => ( ( ( P @ X6 )
% 2.57/1.58 & ( Q @ X6 ) )
% 2.57/1.58 = ( ( P4 @ X6 )
% 2.57/1.58 & ( Q2 @ X6 ) ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_192_minf_I2_J,axiom,
% 2.57/1.58 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 2.57/1.58 ( ? [Z4: nat] :
% 2.57/1.58 ! [X5: nat] :
% 2.57/1.58 ( ( ord_less_nat @ X5 @ Z4 )
% 2.57/1.58 => ( ( P @ X5 )
% 2.57/1.58 = ( P4 @ X5 ) ) )
% 2.57/1.58 => ( ? [Z4: nat] :
% 2.57/1.58 ! [X5: nat] :
% 2.57/1.58 ( ( ord_less_nat @ X5 @ Z4 )
% 2.57/1.58 => ( ( Q @ X5 )
% 2.57/1.58 = ( Q2 @ X5 ) ) )
% 2.57/1.58 => ? [Z2: nat] :
% 2.57/1.58 ! [X6: nat] :
% 2.57/1.58 ( ( ord_less_nat @ X6 @ Z2 )
% 2.57/1.58 => ( ( ( P @ X6 )
% 2.57/1.58 | ( Q @ X6 ) )
% 2.57/1.58 = ( ( P4 @ X6 )
% 2.57/1.58 | ( Q2 @ X6 ) ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_193_minf_I3_J,axiom,
% 2.57/1.58 ! [T: nat] :
% 2.57/1.58 ? [Z2: nat] :
% 2.57/1.58 ! [X6: nat] :
% 2.57/1.58 ( ( ord_less_nat @ X6 @ Z2 )
% 2.57/1.58 => ( X6 != T ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_194_minf_I4_J,axiom,
% 2.57/1.58 ! [T: nat] :
% 2.57/1.58 ? [Z2: nat] :
% 2.57/1.58 ! [X6: nat] :
% 2.57/1.58 ( ( ord_less_nat @ X6 @ Z2 )
% 2.57/1.58 => ( X6 != T ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_195_minf_I5_J,axiom,
% 2.57/1.58 ! [T: nat] :
% 2.57/1.58 ? [Z2: nat] :
% 2.57/1.58 ! [X6: nat] :
% 2.57/1.58 ( ( ord_less_nat @ X6 @ Z2 )
% 2.57/1.58 => ( ord_less_nat @ X6 @ T ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_196_minf_I7_J,axiom,
% 2.57/1.58 ! [T: nat] :
% 2.57/1.58 ? [Z2: nat] :
% 2.57/1.58 ! [X6: nat] :
% 2.57/1.58 ( ( ord_less_nat @ X6 @ Z2 )
% 2.57/1.58 => ~ ( ord_less_nat @ T @ X6 ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_197_list__strict__asc__imp__list__asc,axiom,
% 2.57/1.58 ! [Xs: list_nat] :
% 2.57/1.58 ( ( list_strict_asc_nat @ Xs )
% 2.57/1.58 => ( list_asc_nat @ Xs ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_198_list__strict__desc__imp__list__desc,axiom,
% 2.57/1.58 ! [Xs: list_nat] :
% 2.57/1.58 ( ( list_strict_desc_nat @ Xs )
% 2.57/1.58 => ( list_desc_nat @ Xs ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_199_order_Onot__eq__order__implies__strict,axiom,
% 2.57/1.58 ! [A: nat,B: nat] :
% 2.57/1.58 ( ( A != B )
% 2.57/1.58 => ( ( ord_less_eq_nat @ A @ B )
% 2.57/1.58 => ( ord_less_nat @ A @ B ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_200_dual__order_Ostrict__implies__order,axiom,
% 2.57/1.58 ! [B: nat,A: nat] :
% 2.57/1.58 ( ( ord_less_nat @ B @ A )
% 2.57/1.58 => ( ord_less_eq_nat @ B @ A ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_201_dual__order_Ostrict__iff__order,axiom,
% 2.57/1.58 ( ord_less_nat
% 2.57/1.58 = ( ^ [B2: nat,A3: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ B2 @ A3 )
% 2.57/1.58 & ( A3 != B2 ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_202_dual__order_Oorder__iff__strict,axiom,
% 2.57/1.58 ( ord_less_eq_nat
% 2.57/1.58 = ( ^ [B2: nat,A3: nat] :
% 2.57/1.58 ( ( ord_less_nat @ B2 @ A3 )
% 2.57/1.58 | ( A3 = B2 ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_203_order_Ostrict__implies__order,axiom,
% 2.57/1.58 ! [A: nat,B: nat] :
% 2.57/1.58 ( ( ord_less_nat @ A @ B )
% 2.57/1.58 => ( ord_less_eq_nat @ A @ B ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_204_dual__order_Ostrict__trans2,axiom,
% 2.57/1.58 ! [B: nat,A: nat,C: nat] :
% 2.57/1.58 ( ( ord_less_nat @ B @ A )
% 2.57/1.58 => ( ( ord_less_eq_nat @ C @ B )
% 2.57/1.58 => ( ord_less_nat @ C @ A ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_205_dual__order_Ostrict__trans1,axiom,
% 2.57/1.58 ! [B: nat,A: nat,C: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ B @ A )
% 2.57/1.58 => ( ( ord_less_nat @ C @ B )
% 2.57/1.58 => ( ord_less_nat @ C @ A ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_206_order_Ostrict__iff__order,axiom,
% 2.57/1.58 ( ord_less_nat
% 2.57/1.58 = ( ^ [A3: nat,B2: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ A3 @ B2 )
% 2.57/1.58 & ( A3 != B2 ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_207_order_Oorder__iff__strict,axiom,
% 2.57/1.58 ( ord_less_eq_nat
% 2.57/1.58 = ( ^ [A3: nat,B2: nat] :
% 2.57/1.58 ( ( ord_less_nat @ A3 @ B2 )
% 2.57/1.58 | ( A3 = B2 ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_208_order_Ostrict__trans2,axiom,
% 2.57/1.58 ! [A: nat,B: nat,C: nat] :
% 2.57/1.58 ( ( ord_less_nat @ A @ B )
% 2.57/1.58 => ( ( ord_less_eq_nat @ B @ C )
% 2.57/1.58 => ( ord_less_nat @ A @ C ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_209_order_Ostrict__trans1,axiom,
% 2.57/1.58 ! [A: nat,B: nat,C: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ A @ B )
% 2.57/1.58 => ( ( ord_less_nat @ B @ C )
% 2.57/1.58 => ( ord_less_nat @ A @ C ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_210_not__le__imp__less,axiom,
% 2.57/1.58 ! [Y2: nat,X2: nat] :
% 2.57/1.58 ( ~ ( ord_less_eq_nat @ Y2 @ X2 )
% 2.57/1.58 => ( ord_less_nat @ X2 @ Y2 ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_211_less__le__not__le,axiom,
% 2.57/1.58 ( ord_less_nat
% 2.57/1.58 = ( ^ [X3: nat,Y5: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ X3 @ Y5 )
% 2.57/1.58 & ~ ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_212_le__imp__less__or__eq,axiom,
% 2.57/1.58 ! [X2: nat,Y2: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.57/1.58 => ( ( ord_less_nat @ X2 @ Y2 )
% 2.57/1.58 | ( X2 = Y2 ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_213_le__less__linear,axiom,
% 2.57/1.58 ! [X2: nat,Y2: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.57/1.58 | ( ord_less_nat @ Y2 @ X2 ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_214_less__le__trans,axiom,
% 2.57/1.58 ! [X2: nat,Y2: nat,Z3: nat] :
% 2.57/1.58 ( ( ord_less_nat @ X2 @ Y2 )
% 2.57/1.58 => ( ( ord_less_eq_nat @ Y2 @ Z3 )
% 2.57/1.58 => ( ord_less_nat @ X2 @ Z3 ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_215_le__less__trans,axiom,
% 2.57/1.58 ! [X2: nat,Y2: nat,Z3: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.57/1.58 => ( ( ord_less_nat @ Y2 @ Z3 )
% 2.57/1.58 => ( ord_less_nat @ X2 @ Z3 ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_216_less__imp__le,axiom,
% 2.57/1.58 ! [X2: nat,Y2: nat] :
% 2.57/1.58 ( ( ord_less_nat @ X2 @ Y2 )
% 2.57/1.58 => ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_217_antisym__conv2,axiom,
% 2.57/1.58 ! [X2: nat,Y2: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.57/1.58 => ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
% 2.57/1.58 = ( X2 = Y2 ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_218_antisym__conv1,axiom,
% 2.57/1.58 ! [X2: nat,Y2: nat] :
% 2.57/1.58 ( ~ ( ord_less_nat @ X2 @ Y2 )
% 2.57/1.58 => ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.57/1.58 = ( X2 = Y2 ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_219_le__neq__trans,axiom,
% 2.57/1.58 ! [A: nat,B: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ A @ B )
% 2.57/1.58 => ( ( A != B )
% 2.57/1.58 => ( ord_less_nat @ A @ B ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_220_not__less,axiom,
% 2.57/1.58 ! [X2: nat,Y2: nat] :
% 2.57/1.58 ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
% 2.57/1.58 = ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_221_not__le,axiom,
% 2.57/1.58 ! [X2: nat,Y2: nat] :
% 2.57/1.58 ( ( ~ ( ord_less_eq_nat @ X2 @ Y2 ) )
% 2.57/1.58 = ( ord_less_nat @ Y2 @ X2 ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_222_order__less__le__subst2,axiom,
% 2.57/1.58 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 2.57/1.58 ( ( ord_less_nat @ A @ B )
% 2.57/1.58 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 2.57/1.58 => ( ! [X5: nat,Y3: nat] :
% 2.57/1.58 ( ( ord_less_nat @ X5 @ Y3 )
% 2.57/1.58 => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 2.57/1.58 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_223_order__less__le__subst1,axiom,
% 2.57/1.58 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 2.57/1.58 ( ( ord_less_nat @ A @ ( F @ B ) )
% 2.57/1.58 => ( ( ord_less_eq_nat @ B @ C )
% 2.57/1.58 => ( ! [X5: nat,Y3: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 2.57/1.58 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 2.57/1.58 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_224_order__le__less__subst2,axiom,
% 2.57/1.58 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ A @ B )
% 2.57/1.58 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 2.57/1.58 => ( ! [X5: nat,Y3: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 2.57/1.58 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 2.57/1.58 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_225_order__le__less__subst1,axiom,
% 2.57/1.58 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 2.57/1.58 => ( ( ord_less_nat @ B @ C )
% 2.57/1.58 => ( ! [X5: nat,Y3: nat] :
% 2.57/1.58 ( ( ord_less_nat @ X5 @ Y3 )
% 2.57/1.58 => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
% 2.57/1.58 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_226_less__le,axiom,
% 2.57/1.58 ( ord_less_nat
% 2.57/1.58 = ( ^ [X3: nat,Y5: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ X3 @ Y5 )
% 2.57/1.58 & ( X3 != Y5 ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_227_le__less,axiom,
% 2.57/1.58 ( ord_less_eq_nat
% 2.57/1.58 = ( ^ [X3: nat,Y5: nat] :
% 2.57/1.58 ( ( ord_less_nat @ X3 @ Y5 )
% 2.57/1.58 | ( X3 = Y5 ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_228_leI,axiom,
% 2.57/1.58 ! [X2: nat,Y2: nat] :
% 2.57/1.58 ( ~ ( ord_less_nat @ X2 @ Y2 )
% 2.57/1.58 => ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_229_leD,axiom,
% 2.57/1.58 ! [Y2: nat,X2: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ Y2 @ X2 )
% 2.57/1.58 => ~ ( ord_less_nat @ X2 @ Y2 ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_230_pinf_I6_J,axiom,
% 2.57/1.58 ! [T: nat] :
% 2.57/1.58 ? [Z2: nat] :
% 2.57/1.58 ! [X6: nat] :
% 2.57/1.58 ( ( ord_less_nat @ Z2 @ X6 )
% 2.57/1.58 => ~ ( ord_less_eq_nat @ X6 @ T ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_231_pinf_I8_J,axiom,
% 2.57/1.58 ! [T: nat] :
% 2.57/1.58 ? [Z2: nat] :
% 2.57/1.58 ! [X6: nat] :
% 2.57/1.58 ( ( ord_less_nat @ Z2 @ X6 )
% 2.57/1.58 => ( ord_less_eq_nat @ T @ X6 ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_232_minf_I6_J,axiom,
% 2.57/1.58 ! [T: nat] :
% 2.57/1.58 ? [Z2: nat] :
% 2.57/1.58 ! [X6: nat] :
% 2.57/1.58 ( ( ord_less_nat @ X6 @ Z2 )
% 2.57/1.58 => ( ord_less_eq_nat @ X6 @ T ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_233_complete__interval,axiom,
% 2.57/1.58 ! [A: nat,B: nat,P: nat > $o] :
% 2.57/1.58 ( ( ord_less_nat @ A @ B )
% 2.57/1.58 => ( ( P @ A )
% 2.57/1.58 => ( ~ ( P @ B )
% 2.57/1.58 => ? [C2: nat] :
% 2.57/1.58 ( ( ord_less_eq_nat @ A @ C2 )
% 2.57/1.58 & ( ord_less_eq_nat @ C2 @ B )
% 2.57/1.58 & ! [X6: nat] :
% 2.57/1.58 ( ( ( ord_less_eq_nat @ A @ X6 )
% 2.57/1.58 & ( ord_less_nat @ X6 @ C2 ) )
% 2.57/1.58 => ( P @ X6 ) )
% 2.57/1.58 & ! [D: nat] :
% 2.57/1.58 ( ! [X5: nat] :
% 2.57/1.58 ( ( ( ord_less_eq_nat @ A @ X5 )
% 2.57/1.58 & ( ord_less_nat @ X5 @ D ) )
% 2.57/1.58 => ( P @ X5 ) )
% 2.57/1.58 => ( ord_less_eq_nat @ D @ C2 ) ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_234_verit__comp__simplify1_I3_J,axiom,
% 2.57/1.58 ! [B4: nat,A5: nat] :
% 2.57/1.58 ( ( ~ ( ord_less_eq_nat @ B4 @ A5 ) )
% 2.57/1.58 = ( ord_less_nat @ A5 @ B4 ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_235_set__swap,axiom,
% 2.57/1.58 ! [I3: nat,Xs: list_nat,J: nat] :
% 2.57/1.58 ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
% 2.57/1.58 => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
% 2.57/1.58 => ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I3 @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I3 ) ) )
% 2.57/1.58 = ( set_nat2 @ Xs ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_236_set__swap,axiom,
% 2.57/1.58 ! [I3: nat,Xs: list_a,J: nat] :
% 2.57/1.58 ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
% 2.57/1.58 => ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs ) )
% 2.57/1.58 => ( ( set_a2 @ ( list_update_a @ ( list_update_a @ Xs @ I3 @ ( nth_a @ Xs @ J ) ) @ J @ ( nth_a @ Xs @ I3 ) ) )
% 2.57/1.58 = ( set_a2 @ Xs ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_237_butlast__list__update,axiom,
% 2.57/1.58 ! [K: nat,Xs: list_nat,X2: nat] :
% 2.57/1.58 ( ( ( K
% 2.57/1.58 = ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
% 2.57/1.58 => ( ( butlast_nat @ ( list_update_nat @ Xs @ K @ X2 ) )
% 2.57/1.58 = ( butlast_nat @ Xs ) ) )
% 2.57/1.58 & ( ( K
% 2.57/1.58 != ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
% 2.57/1.58 => ( ( butlast_nat @ ( list_update_nat @ Xs @ K @ X2 ) )
% 2.57/1.58 = ( list_update_nat @ ( butlast_nat @ Xs ) @ K @ X2 ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_238_butlast__list__update,axiom,
% 2.57/1.58 ! [K: nat,Xs: list_a,X2: a] :
% 2.57/1.58 ( ( ( K
% 2.57/1.58 = ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) )
% 2.57/1.58 => ( ( butlast_a @ ( list_update_a @ Xs @ K @ X2 ) )
% 2.57/1.58 = ( butlast_a @ Xs ) ) )
% 2.57/1.58 & ( ( K
% 2.57/1.58 != ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) )
% 2.57/1.58 => ( ( butlast_a @ ( list_update_a @ Xs @ K @ X2 ) )
% 2.57/1.58 = ( list_update_a @ ( butlast_a @ Xs ) @ K @ X2 ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_239_length__butlast,axiom,
% 2.57/1.58 ! [Xs: list_nat] :
% 2.57/1.58 ( ( size_size_list_nat @ ( butlast_nat @ Xs ) )
% 2.57/1.58 = ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_240_length__butlast,axiom,
% 2.57/1.58 ! [Xs: list_a] :
% 2.57/1.58 ( ( size_size_list_a @ ( butlast_a @ Xs ) )
% 2.57/1.58 = ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_241_subset__code_I1_J,axiom,
% 2.57/1.58 ! [Xs: list_a,B5: set_a] :
% 2.57/1.58 ( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B5 )
% 2.57/1.58 = ( ! [X3: a] :
% 2.57/1.58 ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
% 2.57/1.58 => ( member_a @ X3 @ B5 ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_242_set__update__subsetI,axiom,
% 2.57/1.58 ! [Xs: list_a,A2: set_a,X2: a,I3: nat] :
% 2.57/1.58 ( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ A2 )
% 2.57/1.58 => ( ( member_a @ X2 @ A2 )
% 2.57/1.58 => ( ord_less_eq_set_a @ ( set_a2 @ ( list_update_a @ Xs @ I3 @ X2 ) ) @ A2 ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_243_in__set__butlastD,axiom,
% 2.57/1.58 ! [X2: a,Xs: list_a] :
% 2.57/1.58 ( ( member_a @ X2 @ ( set_a2 @ ( butlast_a @ Xs ) ) )
% 2.57/1.58 => ( member_a @ X2 @ ( set_a2 @ Xs ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_244_one__reorient,axiom,
% 2.57/1.58 ! [X2: nat] :
% 2.57/1.58 ( ( one_one_nat = X2 )
% 2.57/1.58 = ( X2 = one_one_nat ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_245_list__ex__cong,axiom,
% 2.57/1.58 ! [Xs: list_a,Ys: list_a,F: a > $o,G: a > $o] :
% 2.57/1.58 ( ( Xs = Ys )
% 2.57/1.58 => ( ! [X5: a] :
% 2.57/1.58 ( ( member_a @ X5 @ ( set_a2 @ Ys ) )
% 2.57/1.58 => ( ( F @ X5 )
% 2.57/1.58 = ( G @ X5 ) ) )
% 2.57/1.58 => ( ( list_ex_a @ F @ Xs )
% 2.57/1.58 = ( list_ex_a @ G @ Ys ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_246_in__set__butlast__appendI,axiom,
% 2.57/1.58 ! [X2: a,Xs: list_a,Ys: list_a] :
% 2.57/1.58 ( ( ( member_a @ X2 @ ( set_a2 @ ( butlast_a @ Xs ) ) )
% 2.57/1.58 | ( member_a @ X2 @ ( set_a2 @ ( butlast_a @ Ys ) ) ) )
% 2.57/1.58 => ( member_a @ X2 @ ( set_a2 @ ( butlast_a @ ( append_a @ Xs @ Ys ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_247_nth__mem,axiom,
% 2.57/1.58 ! [N: nat,Xs: list_nat] :
% 2.57/1.58 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 2.57/1.58 => ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_248_nth__mem,axiom,
% 2.57/1.58 ! [N: nat,Xs: list_a] :
% 2.57/1.58 ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
% 2.57/1.58 => ( member_a @ ( nth_a @ Xs @ N ) @ ( set_a2 @ Xs ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_249_list__ball__nth,axiom,
% 2.57/1.58 ! [N: nat,Xs: list_nat,P: nat > $o] :
% 2.57/1.58 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 2.57/1.58 => ( ! [X5: nat] :
% 2.57/1.58 ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
% 2.57/1.58 => ( P @ X5 ) )
% 2.57/1.58 => ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_250_list__ball__nth,axiom,
% 2.57/1.58 ! [N: nat,Xs: list_a,P: a > $o] :
% 2.57/1.58 ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
% 2.57/1.58 => ( ! [X5: a] :
% 2.57/1.58 ( ( member_a @ X5 @ ( set_a2 @ Xs ) )
% 2.57/1.58 => ( P @ X5 ) )
% 2.57/1.58 => ( P @ ( nth_a @ Xs @ N ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_251_in__set__conv__nth,axiom,
% 2.57/1.58 ! [X2: nat,Xs: list_nat] :
% 2.57/1.58 ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
% 2.57/1.58 = ( ? [I2: nat] :
% 2.57/1.58 ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
% 2.57/1.58 & ( ( nth_nat @ Xs @ I2 )
% 2.57/1.58 = X2 ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_252_in__set__conv__nth,axiom,
% 2.57/1.58 ! [X2: a,Xs: list_a] :
% 2.57/1.58 ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
% 2.57/1.58 = ( ? [I2: nat] :
% 2.57/1.58 ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
% 2.57/1.58 & ( ( nth_a @ Xs @ I2 )
% 2.57/1.58 = X2 ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_253_all__nth__imp__all__set,axiom,
% 2.57/1.58 ! [Xs: list_nat,P: nat > $o,X2: nat] :
% 2.57/1.58 ( ! [I: nat] :
% 2.57/1.58 ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
% 2.57/1.58 => ( P @ ( nth_nat @ Xs @ I ) ) )
% 2.57/1.58 => ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
% 2.57/1.58 => ( P @ X2 ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_254_all__nth__imp__all__set,axiom,
% 2.57/1.58 ! [Xs: list_a,P: a > $o,X2: a] :
% 2.57/1.58 ( ! [I: nat] :
% 2.57/1.58 ( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
% 2.57/1.58 => ( P @ ( nth_a @ Xs @ I ) ) )
% 2.57/1.58 => ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
% 2.57/1.58 => ( P @ X2 ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_255_all__set__conv__all__nth,axiom,
% 2.57/1.58 ! [Xs: list_nat,P: nat > $o] :
% 2.57/1.58 ( ( ! [X3: nat] :
% 2.57/1.58 ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
% 2.57/1.58 => ( P @ X3 ) ) )
% 2.57/1.58 = ( ! [I2: nat] :
% 2.57/1.58 ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
% 2.57/1.58 => ( P @ ( nth_nat @ Xs @ I2 ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_256_all__set__conv__all__nth,axiom,
% 2.57/1.58 ! [Xs: list_a,P: a > $o] :
% 2.57/1.58 ( ( ! [X3: a] :
% 2.57/1.58 ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
% 2.57/1.58 => ( P @ X3 ) ) )
% 2.57/1.58 = ( ! [I2: nat] :
% 2.57/1.58 ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
% 2.57/1.58 => ( P @ ( nth_a @ Xs @ I2 ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(fact_257_verit__la__disequality,axiom,
% 2.57/1.58 ! [A: nat,B: nat] :
% 2.57/1.58 ( ( A = B )
% 2.57/1.58 | ~ ( ord_less_eq_nat @ A @ B )
% 2.57/1.58 | ~ ( ord_less_eq_nat @ B @ A ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(help_If_2_1_If_001tf__a_T,axiom,
% 2.57/1.58 ! [X2: a,Y2: a] :
% 2.57/1.58 ( ( if_a @ $false @ X2 @ Y2 )
% 2.57/1.58 = Y2 ) ).
% 2.57/1.58
% 2.57/1.58 thf(help_If_1_1_If_001tf__a_T,axiom,
% 2.57/1.58 ! [X2: a,Y2: a] :
% 2.57/1.58 ( ( if_a @ $true @ X2 @ Y2 )
% 2.57/1.58 = X2 ) ).
% 2.57/1.58
% 2.57/1.58 thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
% 2.57/1.58 ! [P: $o] :
% 2.57/1.58 ( ( P = $true )
% 2.57/1.58 | ( P = $false ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 2.57/1.58 ! [X2: nat,Y2: nat] :
% 2.57/1.58 ( ( if_nat @ $false @ X2 @ Y2 )
% 2.57/1.58 = Y2 ) ).
% 2.57/1.58
% 2.57/1.58 thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 2.57/1.58 ! [X2: nat,Y2: nat] :
% 2.57/1.58 ( ( if_nat @ $true @ X2 @ Y2 )
% 2.57/1.58 = X2 ) ).
% 2.57/1.58
% 2.57/1.58 thf(conj_0,hypothesis,
% 2.57/1.58 ( ( size_size_list_a @ xs )
% 2.57/1.58 = ( size_size_list_a @ ys ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(conj_1,hypothesis,
% 2.57/1.58 ! [X6: nat] :
% 2.57/1.58 ( ( ( ord_less_nat @ X6 @ ( size_size_list_a @ ys ) )
% 2.57/1.58 => ( ( ( ord_less_nat @ X6 @ ( size_size_list_a @ ys ) )
% 2.57/1.58 => ( ( nth_a @ xs @ X6 )
% 2.57/1.58 = ( nth_a @ ys @ X6 ) ) )
% 2.57/1.58 & ( ~ ( ord_less_nat @ X6 @ ( size_size_list_a @ ys ) )
% 2.57/1.58 => ( ( f @ ( minus_minus_nat @ X6 @ ( size_size_list_a @ xs ) ) )
% 2.57/1.58 = ( nth_a @ ys @ X6 ) ) ) ) )
% 2.57/1.58 & ( ~ ( ord_less_nat @ X6 @ ( size_size_list_a @ ys ) )
% 2.57/1.58 => ( ( ( ord_less_nat @ X6 @ ( size_size_list_a @ ys ) )
% 2.57/1.58 => ( ( nth_a @ xs @ X6 )
% 2.57/1.58 = ( g @ ( minus_minus_nat @ X6 @ ( size_size_list_a @ ys ) ) ) ) )
% 2.57/1.58 & ( ~ ( ord_less_nat @ X6 @ ( size_size_list_a @ ys ) )
% 2.57/1.58 => ( ( f @ ( minus_minus_nat @ X6 @ ( size_size_list_a @ xs ) ) )
% 2.57/1.58 = ( g @ ( minus_minus_nat @ X6 @ ( size_size_list_a @ ys ) ) ) ) ) ) ) ) ).
% 2.57/1.58
% 2.57/1.58 thf(conj_2,hypothesis,
% 2.57/1.58 ord_less_nat @ x @ ( size_size_list_a @ ys ) ).
% 2.57/1.58
% 2.57/1.58 thf(conj_3,conjecture,
% 2.57/1.58 ( ( nth_a @ xs @ x )
% 2.57/1.58 = ( nth_a @ ys @ x ) ) ).
% 2.57/1.58 ------------------------
% 2.76/1.60 % (27254)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_27082 for (2999ds/4Mi)
% 2.76/1.60 % (27255)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_27082 for (2999ds/27Mi)
% 2.76/1.60 % (27258)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_27082 for (2999ds/275Mi)
% 2.76/1.60 % (27256)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_27082 for (2999ds/2Mi)
% 2.76/1.60 % (27254)Instruction limit reached!
% 2.76/1.60 % (27254)------------------------------
% 2.76/1.60 % (27254)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.76/1.60 % (27254)Termination reason: Unknown
% 2.76/1.60 % (27254)Termination phase: shuffling
% 2.76/1.60
% 2.76/1.60 % (27254)Memory used [KB]: 1407
% 2.76/1.60 % (27254)Time elapsed: 0.003 s
% 2.76/1.60 % (27254)Instructions burned: 5 (million)
% 2.76/1.60 % (27254)------------------------------
% 2.76/1.60 % (27254)------------------------------
% 2.76/1.60 % (27256)Instruction limit reached!
% 2.76/1.60 % (27256)------------------------------
% 2.76/1.60 % (27256)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.76/1.60 % (27256)Termination reason: Unknown
% 2.76/1.60 % (27256)Termination phase: shuffling
% 2.76/1.60
% 2.76/1.60 % (27256)Memory used [KB]: 1279
% 2.76/1.60 % (27256)Time elapsed: 0.002 s
% 2.76/1.60 % (27256)Instructions burned: 3 (million)
% 2.76/1.60 % (27256)------------------------------
% 2.76/1.60 % (27256)------------------------------
% 2.76/1.61 % (27253)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_27082 for (2999ds/183Mi)
% 2.76/1.61 % (27259)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_27082 for (2999ds/18Mi)
% 2.76/1.61 % (27257)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_27082 for (2999ds/2Mi)
% 2.76/1.61 % (27257)Instruction limit reached!
% 2.76/1.61 % (27257)------------------------------
% 2.76/1.61 % (27257)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.76/1.61 % (27257)Termination reason: Unknown
% 2.76/1.61 % (27257)Termination phase: shuffling
% 2.76/1.61
% 2.76/1.61 % (27257)Memory used [KB]: 1279
% 2.76/1.61 % (27257)Time elapsed: 0.003 s
% 2.76/1.61 % (27257)Instructions burned: 2 (million)
% 2.76/1.61 % (27257)------------------------------
% 2.76/1.61 % (27257)------------------------------
% 2.76/1.61 % (27255)Instruction limit reached!
% 2.76/1.61 % (27255)------------------------------
% 2.76/1.61 % (27255)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.76/1.61 % (27255)Termination reason: Unknown
% 2.76/1.61 % (27255)Termination phase: Property scanning
% 2.76/1.61
% 2.76/1.61 % (27255)Memory used [KB]: 1663
% 2.76/1.61 % (27255)Time elapsed: 0.010 s
% 2.76/1.61 % (27255)Instructions burned: 27 (million)
% 2.76/1.61 % (27255)------------------------------
% 2.76/1.61 % (27255)------------------------------
% 2.76/1.61 % (27260)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_27082 for (2999ds/3Mi)
% 2.76/1.61 % (27261)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_27082 for (2999ds/37Mi)
% 2.76/1.61 % (27260)Instruction limit reached!
% 2.76/1.61 % (27260)------------------------------
% 2.76/1.61 % (27260)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.76/1.61 % (27260)Termination reason: Unknown
% 2.76/1.61 % (27260)Termination phase: shuffling
% 2.76/1.61
% 2.76/1.61 % (27260)Memory used [KB]: 1279
% 2.76/1.61 % (27260)Time elapsed: 0.002 s
% 2.76/1.61 % (27260)Instructions burned: 3 (million)
% 2.76/1.61 % (27260)------------------------------
% 2.76/1.61 % (27260)------------------------------
% 2.76/1.62 % (27259)Instruction limit reached!
% 2.76/1.62 % (27259)------------------------------
% 2.76/1.62 % (27259)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.76/1.62 % (27259)Termination reason: Unknown
% 2.76/1.62 % (27259)Termination phase: shuffling
% 2.76/1.62
% 2.76/1.62 % (27259)Memory used [KB]: 1663
% 2.76/1.62 % (27259)Time elapsed: 0.014 s
% 2.76/1.62 % (27259)Instructions burned: 19 (million)
% 2.76/1.62 % (27259)------------------------------
% 2.76/1.62 % (27259)------------------------------
% 2.76/1.62 % (27263)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_27082 for (2999ds/3Mi)
% 2.76/1.62 % (27263)Instruction limit reached!
% 2.76/1.62 % (27263)------------------------------
% 2.76/1.62 % (27263)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.76/1.62 % (27263)Termination reason: Unknown
% 2.76/1.62 % (27263)Termination phase: shuffling
% 2.76/1.62
% 2.76/1.62 % (27263)Memory used [KB]: 1279
% 2.76/1.62 % (27263)Time elapsed: 0.002 s
% 2.76/1.62 % (27263)Instructions burned: 3 (million)
% 2.76/1.62 % (27263)------------------------------
% 2.76/1.62 % (27263)------------------------------
% 2.76/1.62 % (27264)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_27082 for (2999ds/1041Mi)
% 2.76/1.62 % (27261)Instruction limit reached!
% 2.76/1.62 % (27261)------------------------------
% 2.76/1.62 % (27261)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.76/1.62 % (27261)Termination reason: Unknown
% 2.76/1.62 % (27261)Termination phase: shuffling
% 2.76/1.62
% 2.76/1.62 % (27261)Memory used [KB]: 2046
% 2.76/1.62 % (27261)Time elapsed: 0.013 s
% 2.76/1.62 % (27261)Instructions burned: 39 (million)
% 2.76/1.62 % (27261)------------------------------
% 2.76/1.62 % (27261)------------------------------
% 2.76/1.63 % (27266)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_27082 for (2999ds/16Mi)
% 2.76/1.63 % (27266)Instruction limit reached!
% 2.76/1.63 % (27266)------------------------------
% 2.76/1.63 % (27266)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.76/1.63 % (27266)Termination reason: Unknown
% 2.76/1.63 % (27266)Termination phase: shuffling
% 2.76/1.63
% 2.76/1.63 % (27266)Memory used [KB]: 1535
% 2.76/1.63 % (27266)Time elapsed: 0.006 s
% 2.76/1.63 % (27266)Instructions burned: 16 (million)
% 2.76/1.63 % (27266)------------------------------
% 2.76/1.63 % (27266)------------------------------
% 2.76/1.63 % (27267)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_27082 for (2999ds/3Mi)
% 2.76/1.63 % (27267)Instruction limit reached!
% 2.76/1.63 % (27267)------------------------------
% 2.76/1.63 % (27267)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.76/1.63 % (27267)Termination reason: Unknown
% 2.76/1.63 % (27267)Termination phase: shuffling
% 2.76/1.63
% 2.76/1.63 % (27267)Memory used [KB]: 1407
% 2.76/1.63 % (27267)Time elapsed: 0.003 s
% 2.76/1.63 % (27267)Instructions burned: 3 (million)
% 2.76/1.63 % (27267)------------------------------
% 2.76/1.63 % (27267)------------------------------
% 2.76/1.63 % (27265)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_27082 for (2999ds/7Mi)
% 2.76/1.64 % (27262)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_27082 for (2999ds/15Mi)
% 2.76/1.64 % (27258)First to succeed.
% 2.76/1.64 % (27265)Instruction limit reached!
% 2.76/1.64 % (27265)------------------------------
% 2.76/1.64 % (27265)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.76/1.64 % (27265)Termination reason: Unknown
% 2.76/1.64 % (27265)Termination phase: shuffling
% 2.76/1.64
% 2.76/1.64 % (27265)Memory used [KB]: 1407
% 2.76/1.64 % (27265)Time elapsed: 0.004 s
% 2.76/1.64 % (27265)Instructions burned: 7 (million)
% 2.76/1.64 % (27265)------------------------------
% 2.76/1.64 % (27265)------------------------------
% 2.76/1.64 % (27258)Refutation found. Thanks to Tanya!
% 2.76/1.64 % SZS status Theorem for DTF2THF_27082
% 2.76/1.64 % SZS output start Proof for DTF2THF_27082
% 2.76/1.64 thf(type_def_5, type, nat: $tType).
% 2.76/1.64 thf(type_def_7, type, a: $tType).
% 2.76/1.64 thf(type_def_8, type, list_nat: $tType).
% 2.76/1.64 thf(type_def_9, type, list_a: $tType).
% 2.76/1.64 thf(type_def_10, type, set_nat: $tType).
% 2.76/1.64 thf(type_def_11, type, set_a: $tType).
% 2.76/1.64 thf(func_def_0, type, list_nat: $tType).
% 2.76/1.64 thf(func_def_1, type, set_nat: $tType).
% 2.76/1.64 thf(func_def_2, type, list_a: $tType).
% 2.76/1.64 thf(func_def_3, type, set_a: $tType).
% 2.76/1.64 thf(func_def_4, type, nat: $tType).
% 2.76/1.64 thf(func_def_5, type, a: $tType).
% 2.76/1.64 thf(func_def_6, type, minus_minus_nat: nat > nat > nat).
% 2.76/1.64 thf(func_def_7, type, one_one_nat: nat).
% 2.76/1.64 thf(func_def_8, type, if_nat: $o > nat > nat > nat).
% 2.76/1.64 thf(func_def_9, type, if_a: $o > a > a > a).
% 2.76/1.64 thf(func_def_10, type, list_asc_nat: list_nat > $o).
% 2.76/1.64 thf(func_def_11, type, list_desc_nat: list_nat > $o).
% 2.76/1.64 thf(func_def_12, type, list_strict_asc_nat: list_nat > $o).
% 2.76/1.64 thf(func_def_13, type, list_strict_desc_nat: list_nat > $o).
% 2.76/1.64 thf(func_def_14, type, listIn923761578nd_nat: list_nat > (nat > nat) > nat > nat).
% 2.76/1.64 thf(func_def_15, type, listIn1312259492pend_a: list_a > (nat > a) > nat > a).
% 2.76/1.64 thf(func_def_16, type, append_nat: list_nat > list_nat > list_nat).
% 2.76/1.64 thf(func_def_17, type, append_a: list_a > list_a > list_a).
% 2.76/1.64 thf(func_def_18, type, butlast_nat: list_nat > list_nat).
% 2.76/1.64 thf(func_def_19, type, butlast_a: list_a > list_a).
% 2.76/1.64 thf(func_def_20, type, set_nat2: list_nat > set_nat).
% 2.76/1.64 thf(func_def_21, type, set_a2: list_a > set_a).
% 2.76/1.64 thf(func_def_22, type, list_ex_nat: (nat > $o) > list_nat > $o).
% 2.76/1.64 thf(func_def_23, type, list_ex_a: (a > $o) > list_a > $o).
% 2.76/1.64 thf(func_def_24, type, list_update_nat: list_nat > nat > nat > list_nat).
% 2.76/1.64 thf(func_def_25, type, list_update_a: list_a > nat > a > list_a).
% 2.76/1.64 thf(func_def_26, type, nth_nat: list_nat > nat > nat).
% 2.76/1.64 thf(func_def_27, type, nth_a: list_a > nat > a).
% 2.76/1.64 thf(func_def_28, type, size_size_list_nat: list_nat > nat).
% 2.76/1.64 thf(func_def_29, type, size_size_list_a: list_a > nat).
% 2.76/1.64 thf(func_def_30, type, ord_less_nat: nat > nat > $o).
% 2.76/1.64 thf(func_def_31, type, ord_less_eq_nat: nat > nat > $o).
% 2.76/1.64 thf(func_def_32, type, ord_less_eq_set_a: set_a > set_a > $o).
% 2.76/1.64 thf(func_def_33, type, collect_a: (a > $o) > set_a).
% 2.76/1.64 thf(func_def_34, type, member_nat: nat > set_nat > $o).
% 2.76/1.64 thf(func_def_35, type, member_a: a > set_a > $o).
% 2.76/1.64 thf(func_def_36, type, f: nat > a).
% 2.76/1.64 thf(func_def_37, type, g: nat > a).
% 2.76/1.64 thf(func_def_38, type, x: nat).
% 2.76/1.64 thf(func_def_39, type, xs: list_a).
% 2.76/1.64 thf(func_def_40, type, ys: list_a).
% 2.76/1.64 thf(func_def_55, type, sP0: nat > nat > nat > $o).
% 2.76/1.64 thf(func_def_56, type, sP1: nat > nat > nat > $o).
% 2.76/1.64 thf(func_def_57, type, sP2: nat > nat > nat > $o).
% 2.76/1.64 thf(func_def_58, type, sP3: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > $o).
% 2.76/1.64 thf(func_def_59, type, sP4: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > $o).
% 2.76/1.64 thf(func_def_60, type, sP5: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > $o).
% 2.76/1.64 thf(func_def_61, type, sP6: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > $o).
% 2.76/1.64 thf(func_def_62, type, sK7: (nat > nat) > nat).
% 2.76/1.64 thf(func_def_63, type, sK8: (nat > nat) > nat).
% 2.76/1.64 thf(func_def_64, type, sK9: (nat > nat > $o) > nat).
% 2.76/1.64 thf(func_def_65, type, sK10: (nat > nat > $o) > nat).
% 2.76/1.64 thf(func_def_66, type, sK11: (nat > nat > $o) > nat).
% 2.76/1.64 thf(func_def_67, type, sK12: (nat > nat > $o) > nat).
% 2.76/1.64 thf(func_def_68, type, sK13: (nat > nat > $o) > nat).
% 2.76/1.64 thf(func_def_69, type, sK14: (nat > nat) > nat).
% 2.76/1.64 thf(func_def_70, type, sK15: (nat > nat) > nat).
% 2.76/1.64 thf(func_def_71, type, sK16: nat > (nat > $o) > nat).
% 2.76/1.64 thf(func_def_72, type, sK17: nat > (nat > $o) > nat).
% 2.76/1.64 thf(func_def_73, type, sK18: (nat > nat) > nat).
% 2.76/1.64 thf(func_def_74, type, sK19: (nat > nat) > nat).
% 2.76/1.64 thf(func_def_75, type, sK20: list_a > list_a > nat).
% 2.76/1.64 thf(func_def_76, type, sK21: (nat > nat) > nat).
% 2.76/1.64 thf(func_def_77, type, sK22: (nat > nat) > nat).
% 2.76/1.64 thf(func_def_78, type, sK23: nat > nat).
% 2.76/1.64 thf(func_def_79, type, sK24: nat > nat).
% 2.76/1.64 thf(func_def_80, type, sK25: (nat > nat) > nat).
% 2.76/1.64 thf(func_def_81, type, sK26: (nat > nat) > nat).
% 2.76/1.64 thf(func_def_82, type, sK27: (nat > $o) > nat).
% 2.76/1.64 thf(func_def_83, type, sK28: (nat > nat) > nat).
% 2.76/1.64 thf(func_def_84, type, sK29: (nat > nat) > nat).
% 2.76/1.64 thf(func_def_85, type, sK30: nat > (nat > a > $o) > list_a).
% 2.76/1.64 thf(func_def_86, type, sK31: nat > (nat > a > $o) > nat).
% 2.76/1.64 thf(func_def_87, type, sK32: nat > (nat > a > $o) > a).
% 2.76/1.64 thf(func_def_88, type, sK33: list_a > nat > (nat > a > $o) > nat).
% 2.76/1.64 thf(func_def_89, type, sK34: nat > nat > (nat > $o) > nat).
% 2.76/1.64 thf(func_def_90, type, sK35: nat > nat > (nat > $o) > nat).
% 2.76/1.64 thf(func_def_91, type, sK36: nat > nat).
% 2.76/1.64 thf(func_def_92, type, sK37: nat > nat).
% 2.76/1.64 thf(func_def_93, type, sK38: nat > nat).
% 2.76/1.64 thf(func_def_94, type, sK39: nat > nat).
% 2.76/1.64 thf(func_def_95, type, sK40: (nat > $o) > nat).
% 2.76/1.64 thf(func_def_96, type, sK41: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > nat).
% 2.76/1.64 thf(func_def_97, type, sK42: nat > (nat > $o) > (nat > $o) > nat).
% 2.76/1.64 thf(func_def_98, type, sK43: nat > (nat > $o) > (nat > $o) > nat).
% 2.76/1.64 thf(func_def_99, type, sK44: (nat > $o) > nat > nat).
% 2.76/1.64 thf(func_def_100, type, sK45: (nat > $o) > nat).
% 2.76/1.64 thf(func_def_101, type, sK46: (nat > nat) > nat).
% 2.76/1.64 thf(func_def_102, type, sK47: (nat > nat) > nat).
% 2.76/1.64 thf(func_def_103, type, sK48: nat > nat).
% 2.76/1.64 thf(func_def_104, type, sK49: (nat > nat) > nat).
% 2.76/1.64 thf(func_def_105, type, sK50: (nat > nat) > nat).
% 2.76/1.64 thf(func_def_106, type, sK51: (nat > $o) > nat).
% 2.76/1.64 thf(func_def_107, type, sK52: (nat > nat) > nat).
% 2.76/1.64 thf(func_def_108, type, sK53: (nat > nat) > nat).
% 2.76/1.64 thf(func_def_109, type, sK54: (nat > nat) > nat).
% 2.76/1.64 thf(func_def_110, type, sK55: (nat > nat) > nat).
% 2.76/1.64 thf(func_def_111, type, sK56: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > nat).
% 2.76/1.64 thf(func_def_112, type, sK57: nat > (nat > $o) > (nat > $o) > nat).
% 2.76/1.64 thf(func_def_113, type, sK58: nat > (nat > $o) > (nat > $o) > nat).
% 2.76/1.64 thf(func_def_114, type, sK59: nat > nat).
% 2.76/1.64 thf(func_def_115, type, sK60: (nat > nat) > nat).
% 2.76/1.64 thf(func_def_116, type, sK61: (nat > nat) > nat).
% 2.76/1.64 thf(func_def_117, type, sK62: (nat > nat) > nat).
% 2.76/1.64 thf(func_def_118, type, sK63: (nat > nat) > nat).
% 2.76/1.64 thf(func_def_119, type, sK64: nat > nat).
% 2.76/1.64 thf(func_def_120, type, sK65: (nat > nat > $o) > nat).
% 2.76/1.64 thf(func_def_121, type, sK66: (nat > nat > $o) > nat).
% 2.76/1.64 thf(func_def_122, type, sK67: (nat > nat > $o) > nat).
% 2.76/1.64 thf(func_def_123, type, sK68: (nat > nat > $o) > nat).
% 2.76/1.64 thf(func_def_124, type, sK69: nat > list_a).
% 2.76/1.64 thf(func_def_125, type, sK70: (list_a > $o) > list_a).
% 2.76/1.64 thf(func_def_126, type, sK71: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > nat).
% 2.76/1.64 thf(func_def_127, type, sK72: nat > (nat > $o) > (nat > $o) > nat).
% 2.76/1.64 thf(func_def_128, type, sK73: nat > (nat > $o) > (nat > $o) > nat).
% 2.76/1.64 thf(func_def_129, type, sK74: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > nat).
% 2.76/1.64 thf(func_def_130, type, sK75: nat > (nat > $o) > (nat > $o) > nat).
% 2.76/1.64 thf(func_def_131, type, sK76: nat > (nat > $o) > (nat > $o) > nat).
% 2.76/1.64 thf(func_def_132, type, sK77: nat > nat).
% 2.76/1.64 thf(func_def_133, type, sK78: (nat > nat) > nat).
% 2.76/1.64 thf(func_def_134, type, sK79: (nat > nat) > nat).
% 2.76/1.64 thf(func_def_135, type, sK80: nat > nat).
% 2.76/1.64 thf(func_def_136, type, sK81: nat > nat).
% 2.76/1.64 thf(func_def_137, type, sK82: nat > nat).
% 2.76/1.64 thf(func_def_139, type, ph84: !>[X0: $tType]:(X0)).
% 2.76/1.64 thf(f1571,plain,(
% 2.76/1.64 $false),
% 2.76/1.64 inference(subsumption_resolution,[status(thm)],[f1566,f1225])).
% 2.76/1.64 thf(f1225,plain,(
% 2.76/1.64 ((ord_less_nat @ x @ (size_size_list_a @ ys)) = $true)),
% 2.76/1.64 inference(cnf_transformation,[status(thm)],[f377])).
% 2.76/1.64 thf(f377,plain,(
% 2.76/1.64 ((ord_less_nat @ x @ (size_size_list_a @ ys)) = $true)),
% 2.76/1.64 inference(fool_elimination,[status(thm)],[f376])).
% 2.76/1.64 thf(f376,plain,(
% 2.76/1.64 (ord_less_nat @ x @ (size_size_list_a @ ys))),
% 2.76/1.64 inference(rectify,[status(thm)],[f266])).
% 2.76/1.64 thf(f266,axiom,(
% 2.76/1.64 (ord_less_nat @ x @ (size_size_list_a @ ys))),
% 2.76/1.64 file('/export/starexec/sandbox2/tmp/tmp.wBFq0AA5QN/DTF2THF_27082.p',conj_2)).
% 2.76/1.64 thf(f1566,plain,(
% 2.76/1.64 ((ord_less_nat @ x @ (size_size_list_a @ ys)) != $true)),
% 2.76/1.64 inference(trivial_inequality_removal,[status(thm)],[f1562])).
% 2.76/1.64 thf(f1562,plain,(
% 2.76/1.64 ((nth_a @ xs @ x) != (nth_a @ xs @ x)) | ((ord_less_nat @ x @ (size_size_list_a @ ys)) != $true)),
% 2.76/1.64 inference(superposition,[status(thm)],[f1333,f1455])).
% 2.76/1.64 thf(f1455,plain,(
% 2.76/1.64 ( ! [X0 : nat] : (((nth_a @ ys @ X0) = (nth_a @ xs @ X0)) | ((ord_less_nat @ X0 @ (size_size_list_a @ ys)) != $true)) )),
% 2.76/1.64 inference(duplicate_literal_removal,[status(thm)],[f1305])).
% 2.76/1.64 thf(f1305,plain,(
% 2.76/1.64 ( ! [X0 : nat] : (((ord_less_nat @ X0 @ (size_size_list_a @ ys)) != $true) | ((ord_less_nat @ X0 @ (size_size_list_a @ ys)) != $true) | ((nth_a @ ys @ X0) = (nth_a @ xs @ X0))) )),
% 2.76/1.64 inference(cnf_transformation,[status(thm)],[f971])).
% 2.76/1.64 thf(f971,plain,(
% 2.76/1.64 ! [X0 : nat] : ((((ord_less_nat @ X0 @ (size_size_list_a @ ys)) = $true) | ((((ord_less_nat @ X0 @ (size_size_list_a @ ys)) = $true) | ((g @ (minus_minus_nat @ X0 @ (size_size_list_a @ ys))) = (f @ (minus_minus_nat @ X0 @ (size_size_list_a @ xs))))) & (((g @ (minus_minus_nat @ X0 @ (size_size_list_a @ ys))) = (nth_a @ xs @ X0)) | ((ord_less_nat @ X0 @ (size_size_list_a @ ys)) != $true)))) & (((ord_less_nat @ X0 @ (size_size_list_a @ ys)) != $true) | ((((nth_a @ ys @ X0) = (nth_a @ xs @ X0)) | ((ord_less_nat @ X0 @ (size_size_list_a @ ys)) != $true)) & (((ord_less_nat @ X0 @ (size_size_list_a @ ys)) = $true) | ((nth_a @ ys @ X0) = (f @ (minus_minus_nat @ X0 @ (size_size_list_a @ xs))))))))),
% 2.76/1.64 inference(ennf_transformation,[status(thm)],[f749])).
% 2.76/1.64 thf(f749,plain,(
% 2.76/1.64 ! [X0 : nat] : ((((ord_less_nat @ X0 @ (size_size_list_a @ ys)) != $true) => ((((ord_less_nat @ X0 @ (size_size_list_a @ ys)) != $true) => ((g @ (minus_minus_nat @ X0 @ (size_size_list_a @ ys))) = (f @ (minus_minus_nat @ X0 @ (size_size_list_a @ xs))))) & (((ord_less_nat @ X0 @ (size_size_list_a @ ys)) = $true) => ((g @ (minus_minus_nat @ X0 @ (size_size_list_a @ ys))) = (nth_a @ xs @ X0))))) & (((ord_less_nat @ X0 @ (size_size_list_a @ ys)) = $true) => ((((ord_less_nat @ X0 @ (size_size_list_a @ ys)) = $true) => ((nth_a @ ys @ X0) = (nth_a @ xs @ X0))) & (((ord_less_nat @ X0 @ (size_size_list_a @ ys)) != $true) => ((nth_a @ ys @ X0) = (f @ (minus_minus_nat @ X0 @ (size_size_list_a @ xs))))))))),
% 2.76/1.64 inference(flattening,[status(thm)],[f569])).
% 2.76/1.64 thf(f569,plain,(
% 2.76/1.64 ! [X0 : nat] : ((((ord_less_nat @ X0 @ (size_size_list_a @ ys)) = $true) => ((((ord_less_nat @ X0 @ (size_size_list_a @ ys)) = $true) => ((nth_a @ ys @ X0) = (nth_a @ xs @ X0))) & (~((ord_less_nat @ X0 @ (size_size_list_a @ ys)) = $true) => ((nth_a @ ys @ X0) = (f @ (minus_minus_nat @ X0 @ (size_size_list_a @ xs))))))) & (~((ord_less_nat @ X0 @ (size_size_list_a @ ys)) = $true) => ((((ord_less_nat @ X0 @ (size_size_list_a @ ys)) = $true) => ((g @ (minus_minus_nat @ X0 @ (size_size_list_a @ ys))) = (nth_a @ xs @ X0))) & (~((ord_less_nat @ X0 @ (size_size_list_a @ ys)) = $true) => ((g @ (minus_minus_nat @ X0 @ (size_size_list_a @ ys))) = (f @ (minus_minus_nat @ X0 @ (size_size_list_a @ xs))))))))),
% 2.76/1.64 inference(fool_elimination,[status(thm)],[f568])).
% 2.76/1.64 thf(f568,plain,(
% 2.76/1.64 ! [X0 : nat] : (((ord_less_nat @ X0 @ (size_size_list_a @ ys)) => (((ord_less_nat @ X0 @ (size_size_list_a @ ys)) => ((nth_a @ ys @ X0) = (nth_a @ xs @ X0))) & (~(ord_less_nat @ X0 @ (size_size_list_a @ ys)) => ((nth_a @ ys @ X0) = (f @ (minus_minus_nat @ X0 @ (size_size_list_a @ xs))))))) & (~(ord_less_nat @ X0 @ (size_size_list_a @ ys)) => (((ord_less_nat @ X0 @ (size_size_list_a @ ys)) => ((g @ (minus_minus_nat @ X0 @ (size_size_list_a @ ys))) = (nth_a @ xs @ X0))) & (~(ord_less_nat @ X0 @ (size_size_list_a @ ys)) => ((g @ (minus_minus_nat @ X0 @ (size_size_list_a @ ys))) = (f @ (minus_minus_nat @ X0 @ (size_size_list_a @ xs))))))))),
% 2.76/1.64 inference(rectify,[status(thm)],[f265])).
% 2.76/1.64 thf(f265,axiom,(
% 2.76/1.64 ! [X52 : nat] : (((ord_less_nat @ X52 @ (size_size_list_a @ ys)) => (((ord_less_nat @ X52 @ (size_size_list_a @ ys)) => ((nth_a @ xs @ X52) = (nth_a @ ys @ X52))) & (~(ord_less_nat @ X52 @ (size_size_list_a @ ys)) => ((nth_a @ ys @ X52) = (f @ (minus_minus_nat @ X52 @ (size_size_list_a @ xs))))))) & (~(ord_less_nat @ X52 @ (size_size_list_a @ ys)) => (((ord_less_nat @ X52 @ (size_size_list_a @ ys)) => ((nth_a @ xs @ X52) = (g @ (minus_minus_nat @ X52 @ (size_size_list_a @ ys))))) & (~(ord_less_nat @ X52 @ (size_size_list_a @ ys)) => ((f @ (minus_minus_nat @ X52 @ (size_size_list_a @ xs))) = (g @ (minus_minus_nat @ X52 @ (size_size_list_a @ ys))))))))),
% 2.76/1.64 file('/export/starexec/sandbox2/tmp/tmp.wBFq0AA5QN/DTF2THF_27082.p',conj_1)).
% 2.76/1.64 thf(f1333,plain,(
% 2.76/1.64 ((nth_a @ xs @ x) != (nth_a @ ys @ x))),
% 2.76/1.64 inference(cnf_transformation,[status(thm)],[f748])).
% 2.76/1.64 thf(f748,plain,(
% 2.76/1.64 ((nth_a @ xs @ x) != (nth_a @ ys @ x))),
% 2.76/1.64 inference(flattening,[status(thm)],[f268])).
% 2.76/1.64 thf(f268,negated_conjecture,(
% 2.76/1.64 ~((nth_a @ xs @ x) = (nth_a @ ys @ x))),
% 2.76/1.64 inference(negated_conjecture,[status(cth)],[f267])).
% 2.76/1.64 thf(f267,conjecture,(
% 2.76/1.64 ((nth_a @ xs @ x) = (nth_a @ ys @ x))),
% 2.76/1.64 file('/export/starexec/sandbox2/tmp/tmp.wBFq0AA5QN/DTF2THF_27082.p',conj_3)).
% 2.76/1.64 % SZS output end Proof for DTF2THF_27082
% 2.76/1.64 % (27258)------------------------------
% 2.76/1.64 % (27258)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.76/1.64 % (27258)Termination reason: Refutation
% 2.76/1.64
% 2.76/1.64 % (27258)Memory used [KB]: 6908
% 2.76/1.64 % (27258)Time elapsed: 0.038 s
% 2.76/1.64 % (27258)Instructions burned: 113 (million)
% 2.76/1.64 % (27258)------------------------------
% 2.76/1.64 % (27258)------------------------------
% 2.76/1.64 % (27252)Success in time 0.056 s
%------------------------------------------------------------------------------