%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP103^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n020.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 08:55:34 AM UTC 2026
% Result : Theorem 2.71s 1.77s
% Output : Refutation 2.71s
% Verified :
% SZS Type : Refutation
% Derivation depth : 11
% Number of leaves : 14
% Syntax : Number of formulae : 71 ( 17 unt; 0 typ; 5 def)
% Number of atoms : 227 ( 84 equ; 0 cnn)
% Maximal formula atoms : 3 ( 3 avg)
% Number of connectives : 351 ( 60 ~; 42 |; 0 &; 225 @)
% ( 5 <=>; 19 =>; 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 : 68 ( 65 usr; 11 con; 0-3 aty)
% Number of variables : 61 ( 0 ^; 61 !; 0 ?; 61 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
nat: $tType ).
thf(type_def_7,type,
list_P1467822637od_a_a: $tType ).
thf(type_def_8,type,
list_P1891567085od_a_a: $tType ).
thf(type_def_9,type,
list_a: $tType ).
thf(type_def_10,type,
list_P319927492_a_a_a: $tType ).
thf(type_def_11,type,
list_P1458844246od_a_a: $tType ).
thf(func_def_0,type,
list_P1891567085od_a_a: $tType ).
thf(func_def_1,type,
list_P1458844246od_a_a: $tType ).
thf(func_def_2,type,
list_P319927492_a_a_a: $tType ).
thf(func_def_3,type,
list_P1467822637od_a_a: $tType ).
thf(func_def_4,type,
list_a: $tType ).
thf(func_def_5,type,
nat: $tType ).
thf(func_def_6,type,
times_times_nat: nat > nat > nat ).
thf(func_def_7,type,
zero_zero_nat: nat ).
thf(func_def_8,type,
produc1055075589od_a_a: list_P1467822637od_a_a > list_P1467822637od_a_a > list_P1891567085od_a_a ).
thf(func_def_9,type,
produc778776796_a_a_a: list_P1467822637od_a_a > list_a > list_P319927492_a_a_a ).
thf(func_def_10,type,
produc16441326od_a_a: list_a > list_P1467822637od_a_a > list_P1458844246od_a_a ).
thf(func_def_11,type,
product_a_a: list_a > list_a > list_P1467822637od_a_a ).
thf(func_def_12,type,
suc: nat > nat ).
thf(func_def_13,type,
size_s1052379225od_a_a: list_P1891567085od_a_a > nat ).
thf(func_def_14,type,
size_s736232240_a_a_a: list_P319927492_a_a_a > nat ).
thf(func_def_15,type,
size_s1875148994od_a_a: list_P1458844246od_a_a > nat ).
thf(func_def_16,type,
size_s642698393od_a_a: list_P1467822637od_a_a > nat ).
thf(func_def_17,type,
size_size_list_a: list_a > nat ).
thf(func_def_18,type,
ord_less_nat: nat > nat > $o ).
thf(func_def_19,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_20,type,
divide_divide_nat: nat > nat > nat ).
thf(func_def_21,type,
k: nat ).
thf(func_def_22,type,
n: nat ).
thf(func_def_23,type,
xs: list_a ).
thf(func_def_35,type,
sK0: nat > nat > nat ).
thf(func_def_36,type,
sK1: ( nat > nat > $o ) > nat ).
thf(func_def_37,type,
sK2: ( nat > nat > $o ) > nat ).
thf(func_def_38,type,
sK3: ( nat > nat > $o ) > nat ).
thf(func_def_39,type,
sK4: ( nat > nat > $o ) > nat ).
thf(func_def_40,type,
sK5: ( nat > $o ) > nat > nat > nat ).
thf(func_def_41,type,
sK6: nat > nat > nat ).
thf(func_def_42,type,
sK7: ( nat > $o ) > nat ).
thf(func_def_43,type,
sK8: nat > ( nat > $o ) > nat ).
thf(func_def_44,type,
sK9: nat > ( nat > $o ) > nat ).
thf(func_def_45,type,
sK10: ( nat > $o ) > nat ).
thf(func_def_46,type,
sK11: ( list_P1467822637od_a_a > $o ) > list_P1467822637od_a_a ).
thf(func_def_47,type,
sK12: nat > nat ).
thf(func_def_48,type,
sK13: ( nat > $o ) > nat ).
thf(func_def_49,type,
sK14: ( nat > $o ) > nat ).
thf(func_def_50,type,
sK15: nat > nat > nat ).
thf(func_def_51,type,
sK16: nat > nat ).
thf(func_def_52,type,
sK17: ( nat > $o ) > nat ).
thf(func_def_53,type,
sK18: nat > list_P1467822637od_a_a ).
thf(func_def_54,type,
sK19: nat > ( nat > $o ) > nat ).
thf(func_def_55,type,
sK20: nat > ( nat > $o ) > nat ).
thf(func_def_56,type,
sK21: ( nat > $o ) > nat > nat ).
thf(func_def_57,type,
sK22: ( nat > $o ) > nat > nat ).
thf(func_def_58,type,
sK23: nat > nat > nat ).
thf(func_def_59,type,
sK24: nat > nat ).
thf(func_def_60,type,
sK25: ( nat > nat > $o ) > nat ).
thf(func_def_61,type,
sK26: ( nat > nat > $o ) > nat ).
thf(func_def_62,type,
sK27: ( nat > nat > $o ) > nat ).
thf(func_def_63,type,
sK28: ( nat > nat > $o ) > nat ).
thf(func_def_64,type,
sK29: nat > ( nat > $o ) > nat ).
thf(func_def_65,type,
sK30: nat > ( nat > $o ) > nat ).
thf(func_def_66,type,
sK31: nat > nat ).
thf(func_def_67,type,
sK32: nat > nat ).
thf(func_def_68,type,
sK33: ( list_a > $o ) > list_a ).
thf(func_def_69,type,
sK34: ( nat > $o ) > nat > nat ).
thf(func_def_70,type,
sK35: ( nat > $o ) > nat > nat ).
thf(func_def_71,type,
sK36: nat > list_a ).
thf(func_def_72,type,
sK37: ( nat > nat ) > nat ).
thf(func_def_73,type,
sK38: ( nat > $o ) > nat ).
thf(func_def_74,type,
sK39: ( nat > $o ) > nat ).
thf(func_def_75,type,
sK40: ( nat > nat ) > nat ).
thf(func_def_77,type,
ph42:
!>[X0: $tType] : X0 ).
thf(f2970,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f1273,f2156,f2365,f2383,f2675,f2949]) ).
thf(f2949,plain,
~ spl41_28,
inference(avatar_contradiction_clause,[status(thm)],[f2948]) ).
thf(f2948,plain,
( $false
| ~ spl41_28 ),
inference(trivial_inequality_removal,[status(thm)],[f2911]) ).
thf(f2911,plain,
( ( n != n )
| ~ spl41_28 ),
inference(superposition,[status(thm)],[f1029,f2834]) ).
thf(f2834,plain,
( ( n
= ( suc @ n ) )
| ~ spl41_28 ),
inference(superposition,[status(thm)],[f1468,f923]) ).
thf(f923,plain,
( ( divide_divide_nat @ ( size_size_list_a @ xs ) @ k )
= ( suc @ n ) ),
inference(cnf_transformation,[status(thm)],[f184]) ).
thf(f184,axiom,
( ( divide_divide_nat @ ( size_size_list_a @ xs ) @ k )
= ( suc @ n ) ),
file('/export/starexec/sandbox/tmp/tmp.2CmgJp5TsA/DTF2THF_7532.p',conj_0) ).
thf(f1468,plain,
( ( ( divide_divide_nat @ ( size_size_list_a @ xs ) @ k )
= n )
| ~ spl41_28 ),
inference(avatar_component_clause,[status(thm)],[f1466]) ).
thf(f1466,definition,
( spl41_28
<=> ( ( divide_divide_nat @ ( size_size_list_a @ xs ) @ k )
= n ) ),
introduced(definition,[new_symbols(naming,[spl41_28])],[avatar_definition]) ).
thf(f1029,plain,
! [X0: nat] :
( ( suc @ X0 )
!= X0 ),
inference(cnf_transformation,[status(thm)],[f28]) ).
thf(f28,axiom,
! [X0: nat] :
( ( suc @ X0 )
!= X0 ),
file('/export/starexec/sandbox/tmp/tmp.2CmgJp5TsA/DTF2THF_7532.p',fact_27_n__not__Suc__n) ).
thf(f2675,plain,
( spl41_28
| spl41_17
| ~ spl41_8 ),
inference(avatar_split_clause,[status(thm)],[f2467,f1256,f1416,f1466]) ).
thf(f1416,definition,
( spl41_17
<=> ( zero_zero_nat = k ) ),
introduced(definition,[new_symbols(naming,[spl41_17])],[avatar_definition]) ).
thf(f1256,definition,
( spl41_8
<=> ( ( size_size_list_a @ xs )
= ( times_times_nat @ n @ k ) ) ),
introduced(definition,[new_symbols(naming,[spl41_8])],[avatar_definition]) ).
thf(f2467,plain,
( ( zero_zero_nat = k )
| ( ( divide_divide_nat @ ( size_size_list_a @ xs ) @ k )
= n )
| ~ spl41_8 ),
inference(superposition,[status(thm)],[f1028,f1258]) ).
thf(f1258,plain,
( ( ( size_size_list_a @ xs )
= ( times_times_nat @ n @ k ) )
| ~ spl41_8 ),
inference(avatar_component_clause,[status(thm)],[f1256]) ).
thf(f1028,plain,
! [X0: nat,X1: nat] :
( ( ( divide_divide_nat @ ( times_times_nat @ X0 @ X1 ) @ X1 )
= X0 )
| ( zero_zero_nat = X1 ) ),
inference(cnf_transformation,[status(thm)],[f556]) ).
thf(f556,plain,
! [X0: nat,X1: nat] :
( ( zero_zero_nat = X1 )
| ( ( divide_divide_nat @ ( times_times_nat @ X0 @ X1 ) @ X1 )
= X0 ) ),
inference(ennf_transformation,[status(thm)],[f503]) ).
thf(f503,plain,
! [X0: nat,X1: nat] :
( ( zero_zero_nat != X1 )
=> ( ( divide_divide_nat @ ( times_times_nat @ X0 @ X1 ) @ X1 )
= X0 ) ),
inference(rectify,[status(thm)],[f99]) ).
thf(f99,axiom,
! [X7: nat,X9: nat] :
( ( zero_zero_nat != X9 )
=> ( ( divide_divide_nat @ ( times_times_nat @ X7 @ X9 ) @ X9 )
= X7 ) ),
file('/export/starexec/sandbox/tmp/tmp.2CmgJp5TsA/DTF2THF_7532.p',fact_98_nonzero__mult__div__cancel__right) ).
thf(f2383,plain,
~ spl41_64,
inference(avatar_contradiction_clause,[status(thm)],[f2382]) ).
thf(f2382,plain,
( $false
| ~ spl41_64 ),
inference(trivial_inequality_removal,[status(thm)],[f2367]) ).
thf(f2367,plain,
( ( $true != $true )
| ~ spl41_64 ),
inference(superposition,[status(thm)],[f1139,f1808]) ).
thf(f1808,plain,
( ( ( ord_less_nat @ n @ n )
= $true )
| ~ spl41_64 ),
inference(avatar_component_clause,[status(thm)],[f1806]) ).
thf(f1806,definition,
( spl41_64
<=> ( ( ord_less_nat @ n @ n )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl41_64])],[avatar_definition]) ).
thf(f1139,plain,
! [X1: nat] :
( ( ord_less_nat @ X1 @ X1 )
!= $true ),
inference(equality_resolution,[status(thm)],[f906]) ).
thf(f906,plain,
! [X0: nat,X1: nat] :
( ( X0 != X1 )
| ( ( ord_less_nat @ X0 @ X1 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f646]) ).
thf(f646,plain,
! [X0: nat,X1: nat] :
( ( X0 != X1 )
| ( ( ord_less_nat @ X0 @ X1 )
!= $true ) ),
inference(ennf_transformation,[status(thm)],[f424]) ).
thf(f424,plain,
! [X0: nat,X1: nat] :
( ( ( ord_less_nat @ X0 @ X1 )
= $true )
=> ( X0 != X1 ) ),
inference(fool_elimination,[status(thm)],[f423]) ).
thf(f423,plain,
! [X0: nat,X1: nat] :
( ( ord_less_nat @ X0 @ X1 )
=> ( X0 != X1 ) ),
inference(rectify,[status(thm)],[f35]) ).
thf(f35,axiom,
! [X0: nat,X1: nat] :
( ( ord_less_nat @ X0 @ X1 )
=> ( X0 != X1 ) ),
file('/export/starexec/sandbox/tmp/tmp.2CmgJp5TsA/DTF2THF_7532.p',fact_34_less__not__refl2) ).
thf(f2365,plain,
( spl41_64
| ~ spl41_9 ),
inference(avatar_split_clause,[status(thm)],[f2364,f1260,f1806]) ).
thf(f1260,definition,
( spl41_9
<=> ( ( ord_less_nat @ ( size_size_list_a @ xs ) @ ( times_times_nat @ n @ k ) )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl41_9])],[avatar_definition]) ).
thf(f2364,plain,
( ( ( ord_less_nat @ n @ n )
= $true )
| ~ spl41_9 ),
inference(trivial_inequality_removal,[status(thm)],[f2363]) ).
thf(f2363,plain,
( ( ( ord_less_nat @ n @ n )
= $true )
| ( $true != $true )
| ~ spl41_9 ),
inference(superposition,[status(thm)],[f1114,f1937]) ).
thf(f1937,plain,
( ( $true
= ( ord_less_nat @ ( suc @ n ) @ n ) )
| ~ spl41_9 ),
inference(trivial_inequality_removal,[status(thm)],[f1921]) ).
thf(f1921,plain,
( ( $true
= ( ord_less_nat @ ( suc @ n ) @ n ) )
| ( $true != $true )
| ~ spl41_9 ),
inference(superposition,[status(thm)],[f1195,f1262]) ).
thf(f1262,plain,
( ( ( ord_less_nat @ ( size_size_list_a @ xs ) @ ( times_times_nat @ n @ k ) )
= $true )
| ~ spl41_9 ),
inference(avatar_component_clause,[status(thm)],[f1260]) ).
thf(f1195,plain,
! [X0: nat] :
( ( $true
!= ( ord_less_nat @ ( size_size_list_a @ xs ) @ ( times_times_nat @ X0 @ k ) ) )
| ( ( ord_less_nat @ ( suc @ n ) @ X0 )
= $true ) ),
inference(superposition,[status(thm)],[f930,f923]) ).
thf(f930,plain,
! [X2: nat,X0: nat,X1: nat] :
( ( $true
= ( ord_less_nat @ ( divide_divide_nat @ X0 @ X2 ) @ X1 ) )
| ( ( ord_less_nat @ X0 @ ( times_times_nat @ X1 @ X2 ) )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f526]) ).
thf(f526,plain,
! [X0: nat,X1: nat,X2: nat] :
( ( ( ord_less_nat @ X0 @ ( times_times_nat @ X1 @ X2 ) )
!= $true )
| ( $true
= ( ord_less_nat @ ( divide_divide_nat @ X0 @ X2 ) @ X1 ) ) ),
inference(ennf_transformation,[status(thm)],[f241]) ).
thf(f241,plain,
! [X0: nat,X1: nat,X2: nat] :
( ( ( ord_less_nat @ X0 @ ( times_times_nat @ X1 @ X2 ) )
= $true )
=> ( $true
= ( ord_less_nat @ ( divide_divide_nat @ X0 @ X2 ) @ X1 ) ) ),
inference(fool_elimination,[status(thm)],[f240]) ).
thf(f240,plain,
! [X0: nat,X1: nat,X2: nat] :
( ( ord_less_nat @ X0 @ ( times_times_nat @ X1 @ X2 ) )
=> ( ord_less_nat @ ( divide_divide_nat @ X0 @ X2 ) @ X1 ) ),
inference(rectify,[status(thm)],[f125]) ).
thf(f125,axiom,
! [X0: nat,X2: nat,X1: nat] :
( ( ord_less_nat @ X0 @ ( times_times_nat @ X2 @ X1 ) )
=> ( ord_less_nat @ ( divide_divide_nat @ X0 @ X1 ) @ X2 ) ),
file('/export/starexec/sandbox/tmp/tmp.2CmgJp5TsA/DTF2THF_7532.p',fact_124_Util__Div_Oless__mult__imp__div__less) ).
thf(f1114,plain,
! [X0: nat,X1: nat] :
( ( ( ord_less_nat @ ( suc @ X0 ) @ X1 )
!= $true )
| ( ( ord_less_nat @ X0 @ X1 )
= $true ) ),
inference(cnf_transformation,[status(thm)],[f641]) ).
thf(f641,plain,
! [X0: nat,X1: nat] :
( ( ( ord_less_nat @ ( suc @ X0 ) @ X1 )
!= $true )
| ( ( ord_less_nat @ X0 @ X1 )
= $true ) ),
inference(ennf_transformation,[status(thm)],[f306]) ).
thf(f306,plain,
! [X0: nat,X1: nat] :
( ( ( ord_less_nat @ ( suc @ X0 ) @ X1 )
= $true )
=> ( ( ord_less_nat @ X0 @ X1 )
= $true ) ),
inference(fool_elimination,[status(thm)],[f305]) ).
thf(f305,plain,
! [X0: nat,X1: nat] :
( ( ord_less_nat @ ( suc @ X0 ) @ X1 )
=> ( ord_less_nat @ X0 @ X1 ) ),
inference(rectify,[status(thm)],[f75]) ).
thf(f75,axiom,
! [X1: nat,X0: nat] :
( ( ord_less_nat @ ( suc @ X1 ) @ X0 )
=> ( ord_less_nat @ X1 @ X0 ) ),
file('/export/starexec/sandbox/tmp/tmp.2CmgJp5TsA/DTF2THF_7532.p',fact_74_Suc__lessD) ).
thf(f2156,plain,
~ spl41_17,
inference(avatar_contradiction_clause,[status(thm)],[f2155]) ).
thf(f2155,plain,
( $false
| ~ spl41_17 ),
inference(trivial_inequality_removal,[status(thm)],[f2146]) ).
thf(f2146,plain,
( ( zero_zero_nat != zero_zero_nat )
| ~ spl41_17 ),
inference(superposition,[status(thm)],[f1075,f1418]) ).
thf(f1418,plain,
( ( zero_zero_nat = k )
| ~ spl41_17 ),
inference(avatar_component_clause,[status(thm)],[f1416]) ).
thf(f1075,plain,
zero_zero_nat != k,
inference(cnf_transformation,[status(thm)],[f185]) ).
thf(f185,axiom,
zero_zero_nat != k,
file('/export/starexec/sandbox/tmp/tmp.2CmgJp5TsA/DTF2THF_7532.p',conj_1) ).
thf(f1273,plain,
( spl41_8
| spl41_9 ),
inference(avatar_split_clause,[status(thm)],[f1252,f1260,f1256]) ).
thf(f1252,plain,
( ( ( size_size_list_a @ xs )
= ( times_times_nat @ n @ k ) )
| ( ( ord_less_nat @ ( size_size_list_a @ xs ) @ ( times_times_nat @ n @ k ) )
= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f1246]) ).
thf(f1246,plain,
( ( ( ord_less_nat @ ( size_size_list_a @ xs ) @ ( times_times_nat @ n @ k ) )
= $true )
| ( ( size_size_list_a @ xs )
= ( times_times_nat @ n @ k ) )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f967,f822]) ).
thf(f822,plain,
! [X0: nat,X1: nat] :
( ( ( ord_less_nat @ X1 @ X0 )
= $true )
| ( X0 = X1 )
| ( ( ord_less_nat @ X0 @ X1 )
= $true ) ),
inference(cnf_transformation,[status(thm)],[f560]) ).
thf(f560,plain,
! [X0: nat,X1: nat] :
( ( X0 = X1 )
| ( ( ord_less_nat @ X0 @ X1 )
= $true )
| ( ( ord_less_nat @ X1 @ X0 )
= $true ) ),
inference(flattening,[status(thm)],[f559]) ).
thf(f559,plain,
! [X0: nat,X1: nat] :
( ( ( ord_less_nat @ X1 @ X0 )
= $true )
| ( ( ord_less_nat @ X0 @ X1 )
= $true )
| ( X0 = X1 ) ),
inference(ennf_transformation,[status(thm)],[f459]) ).
thf(f459,plain,
! [X0: nat,X1: nat] :
( ( X0 != X1 )
=> ( ( ( ord_less_nat @ X0 @ X1 )
!= $true )
=> ( ( ord_less_nat @ X1 @ X0 )
= $true ) ) ),
inference(flattening,[status(thm)],[f217]) ).
thf(f217,plain,
! [X0: nat,X1: nat] :
( ( X0 != X1 )
=> ( ( ( ord_less_nat @ X0 @ X1 )
!= $true )
=> ( ( ord_less_nat @ X1 @ X0 )
= $true ) ) ),
inference(fool_elimination,[status(thm)],[f216]) ).
thf(f216,plain,
! [X0: nat,X1: nat] :
( ( X0 != X1 )
=> ( ~ ( ord_less_nat @ X0 @ X1 )
=> ( ord_less_nat @ X1 @ X0 ) ) ),
inference(rectify,[status(thm)],[f30]) ).
thf(f30,axiom,
! [X10: nat,X11: nat] :
( ( X10 != X11 )
=> ( ~ ( ord_less_nat @ X10 @ X11 )
=> ( ord_less_nat @ X11 @ X10 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.2CmgJp5TsA/DTF2THF_7532.p',fact_29_linorder__neqE__nat) ).
thf(f967,plain,
( ( ord_less_nat @ ( times_times_nat @ n @ k ) @ ( size_size_list_a @ xs ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f464]) ).
thf(f464,plain,
( ( ord_less_nat @ ( times_times_nat @ n @ k ) @ ( size_size_list_a @ xs ) )
!= $true ),
inference(flattening,[status(thm)],[f258]) ).
thf(f258,plain,
( ( ord_less_nat @ ( times_times_nat @ n @ k ) @ ( size_size_list_a @ xs ) )
!= $true ),
inference(fool_elimination,[status(thm)],[f257]) ).
thf(f257,plain,
~ ( ord_less_nat @ ( times_times_nat @ n @ k ) @ ( size_size_list_a @ xs ) ),
inference(rectify,[status(thm)],[f187]) ).
thf(f187,negated_conjecture,
~ ( ord_less_nat @ ( times_times_nat @ n @ k ) @ ( size_size_list_a @ xs ) ),
inference(negated_conjecture,[status(cth)],[f186]) ).
thf(f186,conjecture,
ord_less_nat @ ( times_times_nat @ n @ k ) @ ( size_size_list_a @ xs ),
file('/export/starexec/sandbox/tmp/tmp.2CmgJp5TsA/DTF2THF_7532.p',conj_2) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP103^1 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.15/0.34 % Computer : n020.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 08:26:57 EST 2026
% 0.15/0.34 % CPUTime :
% 0.15/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.29/0.46 ---- Original DTF file ---
% 0.29/0.47 thf(spec,logic,$$dhol).
% 0.29/0.47 %------------------------------------------------------------------------------
% 0.29/0.47 % File : ITP103^1 : TPTP v9.2.1. Released v7.5.0.
% 0.29/0.47 % Domain : Interactive Theorem Proving
% 0.29/0.47 % Problem : Sledgehammer ListSlice problem prob_142__5617098_1
% 0.29/0.47 % Version : Especial.
% 0.29/0.47 % English :
% 0.29/0.47
% 0.29/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.29/0.47 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.29/0.47 % Source : [Des21]
% 0.29/0.47 % Names : ListSlice/prob_142__5617098_1 [Des21]
% 0.29/0.47
% 0.29/0.47 % Status : Theorem
% 0.29/0.47 % Rating : 0.22 v9.1.0, 0.25 v9.0.0, 0.20 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% 0.29/0.47 % Syntax : Number of formulae : 210 ( 98 unt; 24 typ; 0 def)
% 0.29/0.47 % Number of atoms : 465 ( 229 equ; 0 cnn)
% 0.29/0.47 % Maximal formula atoms : 5 ( 2 avg)
% 0.29/0.47 % Number of connectives : 1304 ( 84 ~; 19 |; 28 &;1000 @)
% 0.29/0.47 % ( 0 <=>; 173 =>; 0 <=; 0 <~>)
% 0.29/0.47 % Maximal formula depth : 16 ( 5 avg)
% 0.29/0.47 % Number of types : 7 ( 6 usr)
% 0.29/0.47 % Number of type conns : 43 ( 43 >; 0 *; 0 +; 0 <<)
% 0.29/0.47 % Number of symbols : 19 ( 18 usr; 4 con; 0-2 aty)
% 0.29/0.47 % Number of variables : 412 ( 10 ^; 386 !; 16 ?; 412 :)
% 0.29/0.47 % SPC : TH0_THM_EQU_NAR
% 0.29/0.47
% 0.29/0.47 % Comments : This file was generated by Sledgehammer 2021-02-23 15:31:19.022
% 0.29/0.47 %------------------------------------------------------------------------------
% 0.29/0.47 % Could-be-implicit typings (6)
% 0.29/0.47 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 0.29/0.47 list_P1891567085od_a_a: $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 0.29/0.47 list_P1458844246od_a_a: $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_J,type,
% 0.29/0.47 list_P319927492_a_a_a: $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 0.29/0.47 list_P1467822637od_a_a: $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_n_t__List__Olist_Itf__a_J,type,
% 0.29/0.47 list_a: $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_n_t__Nat__Onat,type,
% 0.29/0.47 nat: $tType ).
% 0.29/0.47
% 0.29/0.47 % Explicit typings (18)
% 0.29/0.47 thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
% 0.29/0.47 times_times_nat: nat > nat > nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 0.29/0.47 zero_zero_nat: nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_List_Oproduct_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.29/0.47 produc1055075589od_a_a: list_P1467822637od_a_a > list_P1467822637od_a_a > list_P1891567085od_a_a ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_List_Oproduct_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001tf__a,type,
% 0.29/0.47 produc778776796_a_a_a: list_P1467822637od_a_a > list_a > list_P319927492_a_a_a ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_List_Oproduct_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.29/0.47 produc16441326od_a_a: list_a > list_P1467822637od_a_a > list_P1458844246od_a_a ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_List_Oproduct_001tf__a_001tf__a,type,
% 0.29/0.47 product_a_a: list_a > list_a > list_P1467822637od_a_a ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Nat_OSuc,type,
% 0.29/0.47 suc: nat > nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 0.29/0.47 size_s1052379225od_a_a: list_P1891567085od_a_a > nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_J,type,
% 0.29/0.47 size_s736232240_a_a_a: list_P319927492_a_a_a > nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 0.29/0.47 size_s1875148994od_a_a: list_P1458844246od_a_a > nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 0.29/0.47 size_s642698393od_a_a: list_P1467822637od_a_a > nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
% 0.29/0.47 size_size_list_a: list_a > nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 0.29/0.47 ord_less_nat: nat > nat > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 0.29/0.47 ord_less_eq_nat: nat > nat > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
% 0.29/0.47 divide_divide_nat: nat > nat > nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_v_k,type,
% 0.29/0.47 k: nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_v_n,type,
% 0.29/0.47 n: nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_v_xs,type,
% 0.29/0.47 xs: list_a ).
% 0.29/0.47
% 0.29/0.47 % Relevant facts (183)
% 0.29/0.47 thf(fact_0_div__mult__self__is__m,axiom,
% 0.29/0.47 ! [N: nat,M: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.47 => ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
% 0.29/0.47 = M ) ) ).
% 0.29/0.47
% 0.29/0.47 % div_mult_self_is_m
% 0.29/0.47 thf(fact_1_div__mult__self1__is__m,axiom,
% 0.29/0.47 ! [N: nat,M: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.47 => ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
% 0.29/0.47 = M ) ) ).
% 0.29/0.47
% 0.29/0.47 % div_mult_self1_is_m
% 0.29/0.47 thf(fact_2_nat__mult__div__cancel__disj,axiom,
% 0.29/0.47 ! [K: nat,M: nat,N: nat] :
% 0.29/0.47 ( ( ( K = zero_zero_nat )
% 0.29/0.47 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.29/0.47 = zero_zero_nat ) )
% 0.29/0.47 & ( ( K != zero_zero_nat )
% 0.29/0.47 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.29/0.47 = ( divide_divide_nat @ M @ N ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % nat_mult_div_cancel_disj
% 0.29/0.47 thf(fact_3_div__less,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M @ N )
% 0.29/0.47 => ( ( divide_divide_nat @ M @ N )
% 0.29/0.47 = zero_zero_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % div_less
% 0.29/0.47 thf(fact_4_div__by__Suc__0,axiom,
% 0.29/0.47 ! [M: nat] :
% 0.29/0.47 ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.29/0.47 = M ) ).
% 0.29/0.47
% 0.29/0.47 % div_by_Suc_0
% 0.29/0.47 thf(fact_5_mult__less__cancel2,axiom,
% 0.29/0.47 ! [M: nat,K: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 0.29/0.47 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.47 & ( ord_less_nat @ M @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mult_less_cancel2
% 0.29/0.47 thf(fact_6_nat__0__less__mult__iff,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
% 0.29/0.47 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.29/0.47 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % nat_0_less_mult_iff
% 0.29/0.47 thf(fact_7_nat__mult__less__cancel__disj,axiom,
% 0.29/0.47 ! [K: nat,M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.29/0.47 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.47 & ( ord_less_nat @ M @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % nat_mult_less_cancel_disj
% 0.29/0.47 thf(fact_8_mult__eq__1__iff,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ( times_times_nat @ M @ N )
% 0.29/0.47 = ( suc @ zero_zero_nat ) )
% 0.29/0.47 = ( ( M
% 0.29/0.47 = ( suc @ zero_zero_nat ) )
% 0.29/0.47 & ( N
% 0.29/0.47 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mult_eq_1_iff
% 0.29/0.47 thf(fact_9_one__eq__mult__iff,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ( suc @ zero_zero_nat )
% 0.29/0.47 = ( times_times_nat @ M @ N ) )
% 0.29/0.47 = ( ( M
% 0.29/0.47 = ( suc @ zero_zero_nat ) )
% 0.29/0.47 & ( N
% 0.29/0.47 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % one_eq_mult_iff
% 0.29/0.47 thf(fact_10_less__Suc0,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.29/0.47 = ( N = zero_zero_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_Suc0
% 0.29/0.47 thf(fact_11_zero__less__Suc,axiom,
% 0.29/0.47 ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % zero_less_Suc
% 0.29/0.47 thf(fact_12_old_Onat_Oinject,axiom,
% 0.29/0.47 ! [Nat: nat,Nat2: nat] :
% 0.29/0.47 ( ( ( suc @ Nat )
% 0.29/0.47 = ( suc @ Nat2 ) )
% 0.29/0.47 = ( Nat = Nat2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % old.nat.inject
% 0.29/0.47 thf(fact_13_nat_Oinject,axiom,
% 0.29/0.47 ! [X2: nat,Y2: nat] :
% 0.29/0.47 ( ( ( suc @ X2 )
% 0.29/0.47 = ( suc @ Y2 ) )
% 0.29/0.47 = ( X2 = Y2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % nat.inject
% 0.29/0.47 thf(fact_14_bot__nat__0_Onot__eq__extremum,axiom,
% 0.29/0.47 ! [A: nat] :
% 0.29/0.47 ( ( A != zero_zero_nat )
% 0.29/0.47 = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 % bot_nat_0.not_eq_extremum
% 0.29/0.47 thf(fact_15_less__nat__zero__code,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 % less_nat_zero_code
% 0.29/0.47 thf(fact_16_neq0__conv,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( N != zero_zero_nat )
% 0.29/0.47 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % neq0_conv
% 0.29/0.47 thf(fact_17_Suc__less__eq,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.29/0.47 = ( ord_less_nat @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % Suc_less_eq
% 0.29/0.47 thf(fact_18_Suc__mono,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M @ N )
% 0.29/0.47 => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Suc_mono
% 0.29/0.47 thf(fact_19_lessI,axiom,
% 0.29/0.47 ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % lessI
% 0.29/0.47 thf(fact_20_mult__cancel2,axiom,
% 0.29/0.47 ! [M: nat,K: nat,N: nat] :
% 0.29/0.47 ( ( ( times_times_nat @ M @ K )
% 0.29/0.47 = ( times_times_nat @ N @ K ) )
% 0.29/0.47 = ( ( M = N )
% 0.29/0.47 | ( K = zero_zero_nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mult_cancel2
% 0.29/0.47 thf(fact_21_mult__cancel1,axiom,
% 0.29/0.47 ! [K: nat,M: nat,N: nat] :
% 0.29/0.47 ( ( ( times_times_nat @ K @ M )
% 0.29/0.47 = ( times_times_nat @ K @ N ) )
% 0.29/0.47 = ( ( M = N )
% 0.29/0.47 | ( K = zero_zero_nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mult_cancel1
% 0.29/0.47 thf(fact_22_mult__0__right,axiom,
% 0.29/0.47 ! [M: nat] :
% 0.29/0.47 ( ( times_times_nat @ M @ zero_zero_nat )
% 0.29/0.47 = zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 % mult_0_right
% 0.29/0.47 thf(fact_23_mult__is__0,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ( times_times_nat @ M @ N )
% 0.29/0.47 = zero_zero_nat )
% 0.29/0.47 = ( ( M = zero_zero_nat )
% 0.29/0.47 | ( N = zero_zero_nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mult_is_0
% 0.29/0.47 thf(fact_24_div__mult__mult1__if,axiom,
% 0.29/0.47 ! [C: nat,A: nat,B: nat] :
% 0.29/0.47 ( ( ( C = zero_zero_nat )
% 0.29/0.47 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.29/0.47 = zero_zero_nat ) )
% 0.29/0.47 & ( ( C != zero_zero_nat )
% 0.29/0.47 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.29/0.47 = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % div_mult_mult1_if
% 0.29/0.47 thf(fact_25_div__mult__mult2,axiom,
% 0.29/0.47 ! [C: nat,A: nat,B: nat] :
% 0.29/0.47 ( ( C != zero_zero_nat )
% 0.29/0.47 => ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 0.29/0.47 = ( divide_divide_nat @ A @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % div_mult_mult2
% 0.29/0.47 thf(fact_26_div__mult__mult1,axiom,
% 0.29/0.47 ! [C: nat,A: nat,B: nat] :
% 0.29/0.47 ( ( C != zero_zero_nat )
% 0.29/0.47 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.29/0.47 = ( divide_divide_nat @ A @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % div_mult_mult1
% 0.29/0.47 thf(fact_27_n__not__Suc__n,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( N
% 0.29/0.47 != ( suc @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % n_not_Suc_n
% 0.29/0.47 thf(fact_28_Suc__inject,axiom,
% 0.29/0.47 ! [X: nat,Y: nat] :
% 0.29/0.47 ( ( ( suc @ X )
% 0.29/0.47 = ( suc @ Y ) )
% 0.29/0.47 => ( X = Y ) ) ).
% 0.29/0.47
% 0.29/0.47 % Suc_inject
% 0.29/0.47 thf(fact_29_linorder__neqE__nat,axiom,
% 0.29/0.47 ! [X: nat,Y: nat] :
% 0.29/0.47 ( ( X != Y )
% 0.29/0.47 => ( ~ ( ord_less_nat @ X @ Y )
% 0.29/0.47 => ( ord_less_nat @ Y @ X ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % linorder_neqE_nat
% 0.29/0.47 thf(fact_30_infinite__descent,axiom,
% 0.29/0.47 ! [P: nat > $o,N: nat] :
% 0.29/0.47 ( ! [N2: nat] :
% 0.29/0.47 ( ~ ( P @ N2 )
% 0.29/0.47 => ? [M2: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M2 @ N2 )
% 0.29/0.47 & ~ ( P @ M2 ) ) )
% 0.29/0.47 => ( P @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % infinite_descent
% 0.29/0.47 thf(fact_31_nat__less__induct,axiom,
% 0.29/0.47 ! [P: nat > $o,N: nat] :
% 0.29/0.47 ( ! [N2: nat] :
% 0.29/0.47 ( ! [M2: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M2 @ N2 )
% 0.29/0.47 => ( P @ M2 ) )
% 0.29/0.47 => ( P @ N2 ) )
% 0.29/0.47 => ( P @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % nat_less_induct
% 0.29/0.47 thf(fact_32_less__irrefl__nat,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ~ ( ord_less_nat @ N @ N ) ).
% 0.29/0.47
% 0.29/0.47 % less_irrefl_nat
% 0.29/0.47 thf(fact_33_less__not__refl3,axiom,
% 0.29/0.47 ! [S: nat,T: nat] :
% 0.29/0.47 ( ( ord_less_nat @ S @ T )
% 0.29/0.47 => ( S != T ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_not_refl3
% 0.29/0.47 thf(fact_34_less__not__refl2,axiom,
% 0.29/0.47 ! [N: nat,M: nat] :
% 0.29/0.47 ( ( ord_less_nat @ N @ M )
% 0.29/0.47 => ( M != N ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_not_refl2
% 0.29/0.47 thf(fact_35_less__not__refl,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ~ ( ord_less_nat @ N @ N ) ).
% 0.29/0.47
% 0.29/0.47 % less_not_refl
% 0.29/0.47 thf(fact_36_nat__neq__iff,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( M != N )
% 0.29/0.47 = ( ( ord_less_nat @ M @ N )
% 0.29/0.47 | ( ord_less_nat @ N @ M ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % nat_neq_iff
% 0.29/0.47 thf(fact_37_size__neq__size__imp__neq,axiom,
% 0.29/0.47 ! [X: list_P1467822637od_a_a,Y: list_P1467822637od_a_a] :
% 0.29/0.47 ( ( ( size_s642698393od_a_a @ X )
% 0.29/0.47 != ( size_s642698393od_a_a @ Y ) )
% 0.29/0.47 => ( X != Y ) ) ).
% 0.29/0.47
% 0.29/0.47 % size_neq_size_imp_neq
% 0.29/0.47 thf(fact_38_size__neq__size__imp__neq,axiom,
% 0.29/0.47 ! [X: list_a,Y: list_a] :
% 0.29/0.47 ( ( ( size_size_list_a @ X )
% 0.29/0.47 != ( size_size_list_a @ Y ) )
% 0.29/0.47 => ( X != Y ) ) ).
% 0.29/0.47
% 0.29/0.47 % size_neq_size_imp_neq
% 0.29/0.47 thf(fact_39_not0__implies__Suc,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( N != zero_zero_nat )
% 0.29/0.47 => ? [M3: nat] :
% 0.29/0.47 ( N
% 0.29/0.47 = ( suc @ M3 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % not0_implies_Suc
% 0.29/0.47 thf(fact_40_old_Onat_Oinducts,axiom,
% 0.29/0.47 ! [P: nat > $o,Nat: nat] :
% 0.29/0.47 ( ( P @ zero_zero_nat )
% 0.29/0.47 => ( ! [Nat3: nat] :
% 0.29/0.47 ( ( P @ Nat3 )
% 0.29/0.47 => ( P @ ( suc @ Nat3 ) ) )
% 0.29/0.47 => ( P @ Nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % old.nat.inducts
% 0.29/0.47 thf(fact_41_old_Onat_Oexhaust,axiom,
% 0.29/0.47 ! [Y: nat] :
% 0.29/0.47 ( ( Y != zero_zero_nat )
% 0.29/0.47 => ~ ! [Nat3: nat] :
% 0.29/0.47 ( Y
% 0.29/0.47 != ( suc @ Nat3 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % old.nat.exhaust
% 0.29/0.47 thf(fact_42_Zero__not__Suc,axiom,
% 0.29/0.47 ! [M: nat] :
% 0.29/0.47 ( zero_zero_nat
% 0.29/0.47 != ( suc @ M ) ) ).
% 0.29/0.47
% 0.29/0.47 % Zero_not_Suc
% 0.29/0.47 thf(fact_43_Zero__neq__Suc,axiom,
% 0.29/0.47 ! [M: nat] :
% 0.29/0.47 ( zero_zero_nat
% 0.29/0.47 != ( suc @ M ) ) ).
% 0.29/0.47
% 0.29/0.47 % Zero_neq_Suc
% 0.29/0.47 thf(fact_44_Suc__neq__Zero,axiom,
% 0.29/0.47 ! [M: nat] :
% 0.29/0.47 ( ( suc @ M )
% 0.29/0.47 != zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 % Suc_neq_Zero
% 0.29/0.47 thf(fact_45_zero__induct,axiom,
% 0.29/0.47 ! [P: nat > $o,K: nat] :
% 0.29/0.47 ( ( P @ K )
% 0.29/0.47 => ( ! [N2: nat] :
% 0.29/0.47 ( ( P @ ( suc @ N2 ) )
% 0.29/0.47 => ( P @ N2 ) )
% 0.29/0.47 => ( P @ zero_zero_nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % zero_induct
% 0.29/0.47 thf(fact_46_diff__induct,axiom,
% 0.29/0.47 ! [P: nat > nat > $o,M: nat,N: nat] :
% 0.29/0.47 ( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
% 0.29/0.47 => ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
% 0.29/0.47 => ( ! [X3: nat,Y3: nat] :
% 0.29/0.47 ( ( P @ X3 @ Y3 )
% 0.29/0.47 => ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
% 0.29/0.47 => ( P @ M @ N ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % diff_induct
% 0.29/0.47 thf(fact_47_nat__induct,axiom,
% 0.29/0.47 ! [P: nat > $o,N: nat] :
% 0.29/0.47 ( ( P @ zero_zero_nat )
% 0.29/0.47 => ( ! [N2: nat] :
% 0.29/0.47 ( ( P @ N2 )
% 0.29/0.47 => ( P @ ( suc @ N2 ) ) )
% 0.29/0.47 => ( P @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % nat_induct
% 0.29/0.47 thf(fact_48_nat_OdiscI,axiom,
% 0.29/0.47 ! [Nat: nat,X2: nat] :
% 0.29/0.47 ( ( Nat
% 0.29/0.47 = ( suc @ X2 ) )
% 0.29/0.47 => ( Nat != zero_zero_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % nat.discI
% 0.29/0.47 thf(fact_49_old_Onat_Odistinct_I1_J,axiom,
% 0.29/0.47 ! [Nat2: nat] :
% 0.29/0.47 ( zero_zero_nat
% 0.29/0.47 != ( suc @ Nat2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % old.nat.distinct(1)
% 0.29/0.47 thf(fact_50_old_Onat_Odistinct_I2_J,axiom,
% 0.29/0.47 ! [Nat2: nat] :
% 0.29/0.47 ( ( suc @ Nat2 )
% 0.29/0.47 != zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 % old.nat.distinct(2)
% 0.29/0.47 thf(fact_51_nat_Odistinct_I1_J,axiom,
% 0.29/0.47 ! [X2: nat] :
% 0.29/0.47 ( zero_zero_nat
% 0.29/0.47 != ( suc @ X2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % nat.distinct(1)
% 0.29/0.47 thf(fact_52_bot__nat__0_Oextremum__strict,axiom,
% 0.29/0.47 ! [A: nat] :
% 0.29/0.47 ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 % bot_nat_0.extremum_strict
% 0.29/0.47 thf(fact_53_infinite__descent0,axiom,
% 0.29/0.47 ! [P: nat > $o,N: nat] :
% 0.29/0.47 ( ( P @ zero_zero_nat )
% 0.29/0.47 => ( ! [N2: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.47 => ( ~ ( P @ N2 )
% 0.29/0.47 => ? [M2: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M2 @ N2 )
% 0.29/0.47 & ~ ( P @ M2 ) ) ) )
% 0.29/0.47 => ( P @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % infinite_descent0
% 0.29/0.47 thf(fact_54_gr__implies__not0,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M @ N )
% 0.29/0.47 => ( N != zero_zero_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % gr_implies_not0
% 0.29/0.47 thf(fact_55_less__zeroE,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 % less_zeroE
% 0.29/0.47 thf(fact_56_not__less0,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 % not_less0
% 0.29/0.47 thf(fact_57_not__gr0,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.29/0.47 = ( N = zero_zero_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % not_gr0
% 0.29/0.47 thf(fact_58_gr0I,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( N != zero_zero_nat )
% 0.29/0.47 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % gr0I
% 0.29/0.47 thf(fact_59_not__less__less__Suc__eq,axiom,
% 0.29/0.47 ! [N: nat,M: nat] :
% 0.29/0.47 ( ~ ( ord_less_nat @ N @ M )
% 0.29/0.47 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.29/0.47 = ( N = M ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % not_less_less_Suc_eq
% 0.29/0.47 thf(fact_60_strict__inc__induct,axiom,
% 0.29/0.47 ! [I: nat,J: nat,P: nat > $o] :
% 0.29/0.47 ( ( ord_less_nat @ I @ J )
% 0.29/0.47 => ( ! [I2: nat] :
% 0.29/0.47 ( ( J
% 0.29/0.47 = ( suc @ I2 ) )
% 0.29/0.47 => ( P @ I2 ) )
% 0.29/0.47 => ( ! [I2: nat] :
% 0.29/0.47 ( ( ord_less_nat @ I2 @ J )
% 0.29/0.47 => ( ( P @ ( suc @ I2 ) )
% 0.29/0.47 => ( P @ I2 ) ) )
% 0.29/0.47 => ( P @ I ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % strict_inc_induct
% 0.29/0.47 thf(fact_61_less__Suc__induct,axiom,
% 0.29/0.47 ! [I: nat,J: nat,P: nat > nat > $o] :
% 0.29/0.47 ( ( ord_less_nat @ I @ J )
% 0.29/0.47 => ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
% 0.29/0.47 => ( ! [I2: nat,J2: nat,K2: nat] :
% 0.29/0.47 ( ( ord_less_nat @ I2 @ J2 )
% 0.29/0.47 => ( ( ord_less_nat @ J2 @ K2 )
% 0.29/0.47 => ( ( P @ I2 @ J2 )
% 0.29/0.47 => ( ( P @ J2 @ K2 )
% 0.29/0.47 => ( P @ I2 @ K2 ) ) ) ) )
% 0.29/0.47 => ( P @ I @ J ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_Suc_induct
% 0.29/0.47 thf(fact_62_less__trans__Suc,axiom,
% 0.29/0.47 ! [I: nat,J: nat,K: nat] :
% 0.29/0.47 ( ( ord_less_nat @ I @ J )
% 0.29/0.47 => ( ( ord_less_nat @ J @ K )
% 0.29/0.47 => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_trans_Suc
% 0.29/0.47 thf(fact_63_Suc__less__SucD,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.29/0.47 => ( ord_less_nat @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % Suc_less_SucD
% 0.29/0.47 thf(fact_64_less__antisym,axiom,
% 0.29/0.47 ! [N: nat,M: nat] :
% 0.29/0.47 ( ~ ( ord_less_nat @ N @ M )
% 0.29/0.47 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.29/0.47 => ( M = N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_antisym
% 0.29/0.47 thf(fact_65_Suc__less__eq2,axiom,
% 0.29/0.47 ! [N: nat,M: nat] :
% 0.29/0.47 ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.29/0.47 = ( ? [M4: nat] :
% 0.29/0.47 ( ( M
% 0.29/0.47 = ( suc @ M4 ) )
% 0.29/0.47 & ( ord_less_nat @ N @ M4 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Suc_less_eq2
% 0.29/0.47 thf(fact_66_All__less__Suc,axiom,
% 0.29/0.47 ! [N: nat,P: nat > $o] :
% 0.29/0.47 ( ( ! [I3: nat] :
% 0.29/0.47 ( ( ord_less_nat @ I3 @ ( suc @ N ) )
% 0.29/0.47 => ( P @ I3 ) ) )
% 0.29/0.47 = ( ( P @ N )
% 0.29/0.47 & ! [I3: nat] :
% 0.29/0.47 ( ( ord_less_nat @ I3 @ N )
% 0.29/0.47 => ( P @ I3 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % All_less_Suc
% 0.29/0.47 thf(fact_67_not__less__eq,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ~ ( ord_less_nat @ M @ N ) )
% 0.29/0.47 = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % not_less_eq
% 0.29/0.47 thf(fact_68_less__Suc__eq,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.29/0.47 = ( ( ord_less_nat @ M @ N )
% 0.29/0.47 | ( M = N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_Suc_eq
% 0.29/0.47 thf(fact_69_Ex__less__Suc,axiom,
% 0.29/0.47 ! [N: nat,P: nat > $o] :
% 0.29/0.47 ( ( ? [I3: nat] :
% 0.29/0.47 ( ( ord_less_nat @ I3 @ ( suc @ N ) )
% 0.29/0.47 & ( P @ I3 ) ) )
% 0.29/0.47 = ( ( P @ N )
% 0.29/0.47 | ? [I3: nat] :
% 0.29/0.47 ( ( ord_less_nat @ I3 @ N )
% 0.29/0.47 & ( P @ I3 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Ex_less_Suc
% 0.29/0.47 thf(fact_70_less__SucI,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M @ N )
% 0.29/0.47 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_SucI
% 0.29/0.47 thf(fact_71_less__SucE,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.29/0.47 => ( ~ ( ord_less_nat @ M @ N )
% 0.29/0.47 => ( M = N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_SucE
% 0.29/0.47 thf(fact_72_Suc__lessI,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M @ N )
% 0.29/0.47 => ( ( ( suc @ M )
% 0.29/0.47 != N )
% 0.29/0.47 => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Suc_lessI
% 0.29/0.47 thf(fact_73_Suc__lessE,axiom,
% 0.29/0.47 ! [I: nat,K: nat] :
% 0.29/0.47 ( ( ord_less_nat @ ( suc @ I ) @ K )
% 0.29/0.47 => ~ ! [J2: nat] :
% 0.29/0.47 ( ( ord_less_nat @ I @ J2 )
% 0.29/0.47 => ( K
% 0.29/0.47 != ( suc @ J2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Suc_lessE
% 0.29/0.47 thf(fact_74_Suc__lessD,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ ( suc @ M ) @ N )
% 0.29/0.47 => ( ord_less_nat @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % Suc_lessD
% 0.29/0.47 thf(fact_75_Nat_OlessE,axiom,
% 0.29/0.47 ! [I: nat,K: nat] :
% 0.29/0.47 ( ( ord_less_nat @ I @ K )
% 0.29/0.47 => ( ( K
% 0.29/0.47 != ( suc @ I ) )
% 0.29/0.47 => ~ ! [J2: nat] :
% 0.29/0.47 ( ( ord_less_nat @ I @ J2 )
% 0.29/0.47 => ( K
% 0.29/0.47 != ( suc @ J2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Nat.lessE
% 0.29/0.47 thf(fact_76_nat__mult__eq__cancel__disj,axiom,
% 0.29/0.47 ! [K: nat,M: nat,N: nat] :
% 0.29/0.47 ( ( ( times_times_nat @ K @ M )
% 0.29/0.47 = ( times_times_nat @ K @ N ) )
% 0.29/0.47 = ( ( K = zero_zero_nat )
% 0.29/0.47 | ( M = N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % nat_mult_eq_cancel_disj
% 0.29/0.47 thf(fact_77_mult__0,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( times_times_nat @ zero_zero_nat @ N )
% 0.29/0.47 = zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 % mult_0
% 0.29/0.47 thf(fact_78_Suc__mult__cancel1,axiom,
% 0.29/0.47 ! [K: nat,M: nat,N: nat] :
% 0.29/0.47 ( ( ( times_times_nat @ ( suc @ K ) @ M )
% 0.29/0.47 = ( times_times_nat @ ( suc @ K ) @ N ) )
% 0.29/0.47 = ( M = N ) ) ).
% 0.29/0.47
% 0.29/0.47 % Suc_mult_cancel1
% 0.29/0.47 thf(fact_79_div__mult2__eq,axiom,
% 0.29/0.47 ! [M: nat,N: nat,Q: nat] :
% 0.29/0.47 ( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q ) )
% 0.29/0.47 = ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q ) ) ).
% 0.29/0.47
% 0.29/0.47 % div_mult2_eq
% 0.29/0.47 thf(fact_80_lift__Suc__mono__less__iff,axiom,
% 0.29/0.47 ! [F: nat > nat,N: nat,M: nat] :
% 0.29/0.47 ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.29/0.47 => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
% 0.29/0.47 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % lift_Suc_mono_less_iff
% 0.29/0.47 thf(fact_81_lift__Suc__mono__less,axiom,
% 0.29/0.47 ! [F: nat > nat,N: nat,N3: nat] :
% 0.29/0.47 ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.29/0.47 => ( ( ord_less_nat @ N @ N3 )
% 0.29/0.47 => ( ord_less_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % lift_Suc_mono_less
% 0.29/0.47 thf(fact_82_less__Suc__eq__0__disj,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.29/0.47 = ( ( M = zero_zero_nat )
% 0.29/0.47 | ? [J3: nat] :
% 0.29/0.47 ( ( M
% 0.29/0.47 = ( suc @ J3 ) )
% 0.29/0.47 & ( ord_less_nat @ J3 @ N ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_Suc_eq_0_disj
% 0.29/0.47 thf(fact_83_gr0__implies__Suc,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.47 => ? [M3: nat] :
% 0.29/0.47 ( N
% 0.29/0.47 = ( suc @ M3 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % gr0_implies_Suc
% 0.29/0.47 thf(fact_84_All__less__Suc2,axiom,
% 0.29/0.47 ! [N: nat,P: nat > $o] :
% 0.29/0.47 ( ( ! [I3: nat] :
% 0.29/0.47 ( ( ord_less_nat @ I3 @ ( suc @ N ) )
% 0.29/0.47 => ( P @ I3 ) ) )
% 0.29/0.47 = ( ( P @ zero_zero_nat )
% 0.29/0.47 & ! [I3: nat] :
% 0.29/0.47 ( ( ord_less_nat @ I3 @ N )
% 0.29/0.47 => ( P @ ( suc @ I3 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % All_less_Suc2
% 0.29/0.47 thf(fact_85_gr0__conv__Suc,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.47 = ( ? [M5: nat] :
% 0.29/0.47 ( N
% 0.29/0.47 = ( suc @ M5 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % gr0_conv_Suc
% 0.29/0.47 thf(fact_86_Ex__less__Suc2,axiom,
% 0.29/0.47 ! [N: nat,P: nat > $o] :
% 0.29/0.47 ( ( ? [I3: nat] :
% 0.29/0.47 ( ( ord_less_nat @ I3 @ ( suc @ N ) )
% 0.29/0.47 & ( P @ I3 ) ) )
% 0.29/0.47 = ( ( P @ zero_zero_nat )
% 0.29/0.47 | ? [I3: nat] :
% 0.29/0.47 ( ( ord_less_nat @ I3 @ N )
% 0.29/0.47 & ( P @ ( suc @ I3 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Ex_less_Suc2
% 0.29/0.47 thf(fact_87_nat__mult__less__cancel1,axiom,
% 0.29/0.47 ! [K: nat,M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.47 => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.29/0.47 = ( ord_less_nat @ M @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % nat_mult_less_cancel1
% 0.29/0.47 thf(fact_88_nat__mult__eq__cancel1,axiom,
% 0.29/0.47 ! [K: nat,M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.47 => ( ( ( times_times_nat @ K @ M )
% 0.29/0.47 = ( times_times_nat @ K @ N ) )
% 0.29/0.47 = ( M = N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % nat_mult_eq_cancel1
% 0.29/0.47 thf(fact_89_mult__less__mono2,axiom,
% 0.29/0.47 ! [I: nat,J: nat,K: nat] :
% 0.29/0.47 ( ( ord_less_nat @ I @ J )
% 0.29/0.47 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.47 => ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mult_less_mono2
% 0.29/0.47 thf(fact_90_mult__less__mono1,axiom,
% 0.29/0.47 ! [I: nat,J: nat,K: nat] :
% 0.29/0.47 ( ( ord_less_nat @ I @ J )
% 0.29/0.47 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.47 => ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mult_less_mono1
% 0.29/0.47 thf(fact_91_Suc__mult__less__cancel1,axiom,
% 0.29/0.47 ! [K: nat,M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 0.29/0.47 = ( ord_less_nat @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % Suc_mult_less_cancel1
% 0.29/0.47 thf(fact_92_Euclidean__Division_Odiv__eq__0__iff,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ( divide_divide_nat @ M @ N )
% 0.29/0.47 = zero_zero_nat )
% 0.29/0.47 = ( ( ord_less_nat @ M @ N )
% 0.29/0.47 | ( N = zero_zero_nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Euclidean_Division.div_eq_0_iff
% 0.29/0.47 thf(fact_93_Euclidean__Division_Oless__mult__imp__div__less,axiom,
% 0.29/0.47 ! [M: nat,I: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
% 0.29/0.47 => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% 0.29/0.47
% 0.29/0.47 % Euclidean_Division.less_mult_imp_div_less
% 0.29/0.47 thf(fact_94_n__less__n__mult__m,axiom,
% 0.29/0.47 ! [N: nat,M: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.47 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.29/0.47 => ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % n_less_n_mult_m
% 0.29/0.47 thf(fact_95_n__less__m__mult__n,axiom,
% 0.29/0.47 ! [N: nat,M: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.47 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.29/0.47 => ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % n_less_m_mult_n
% 0.29/0.47 thf(fact_96_one__less__mult,axiom,
% 0.29/0.47 ! [N: nat,M: nat] :
% 0.29/0.47 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.29/0.47 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.29/0.47 => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % one_less_mult
% 0.29/0.47 thf(fact_97_nat__mult__div__cancel1,axiom,
% 0.29/0.47 ! [K: nat,M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.47 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.29/0.47 = ( divide_divide_nat @ M @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % nat_mult_div_cancel1
% 0.29/0.47 thf(fact_98_nonzero__mult__div__cancel__right,axiom,
% 0.29/0.47 ! [B: nat,A: nat] :
% 0.29/0.47 ( ( B != zero_zero_nat )
% 0.29/0.47 => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
% 0.29/0.47 = A ) ) ).
% 0.29/0.47
% 0.29/0.47 % nonzero_mult_div_cancel_right
% 0.29/0.47 thf(fact_99_nonzero__mult__div__cancel__left,axiom,
% 0.29/0.47 ! [A: nat,B: nat] :
% 0.29/0.47 ( ( A != zero_zero_nat )
% 0.29/0.47 => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
% 0.29/0.47 = B ) ) ).
% 0.29/0.47
% 0.29/0.47 % nonzero_mult_div_cancel_left
% 0.29/0.47 thf(fact_100_less__div__Suc__mult,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.29/0.47 => ( ord_less_nat @ N @ ( times_times_nat @ ( suc @ ( divide_divide_nat @ N @ M ) ) @ M ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_div_Suc_mult
% 0.29/0.47 thf(fact_101_mult__zero__left,axiom,
% 0.29/0.47 ! [A: nat] :
% 0.29/0.47 ( ( times_times_nat @ zero_zero_nat @ A )
% 0.29/0.47 = zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 % mult_zero_left
% 0.29/0.47 thf(fact_102_mult__zero__right,axiom,
% 0.29/0.47 ! [A: nat] :
% 0.29/0.47 ( ( times_times_nat @ A @ zero_zero_nat )
% 0.29/0.47 = zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 % mult_zero_right
% 0.29/0.47 thf(fact_103_mult__eq__0__iff,axiom,
% 0.29/0.47 ! [A: nat,B: nat] :
% 0.29/0.47 ( ( ( times_times_nat @ A @ B )
% 0.29/0.47 = zero_zero_nat )
% 0.29/0.47 = ( ( A = zero_zero_nat )
% 0.29/0.47 | ( B = zero_zero_nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mult_eq_0_iff
% 0.29/0.47 thf(fact_104_mult__cancel__left,axiom,
% 0.29/0.47 ! [C: nat,A: nat,B: nat] :
% 0.29/0.47 ( ( ( times_times_nat @ C @ A )
% 0.29/0.47 = ( times_times_nat @ C @ B ) )
% 0.29/0.47 = ( ( C = zero_zero_nat )
% 0.29/0.47 | ( A = B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mult_cancel_left
% 0.29/0.47 thf(fact_105_mult__cancel__right,axiom,
% 0.29/0.47 ! [A: nat,C: nat,B: nat] :
% 0.29/0.47 ( ( ( times_times_nat @ A @ C )
% 0.29/0.47 = ( times_times_nat @ B @ C ) )
% 0.29/0.47 = ( ( C = zero_zero_nat )
% 0.29/0.47 | ( A = B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mult_cancel_right
% 0.29/0.47 thf(fact_106_div__0,axiom,
% 0.29/0.47 ! [A: nat] :
% 0.29/0.47 ( ( divide_divide_nat @ zero_zero_nat @ A )
% 0.29/0.47 = zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 % div_0
% 0.29/0.47 thf(fact_107_div__by__0,axiom,
% 0.29/0.47 ! [A: nat] :
% 0.29/0.47 ( ( divide_divide_nat @ A @ zero_zero_nat )
% 0.29/0.47 = zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 % div_by_0
% 0.29/0.47 thf(fact_108_mult__not__zero,axiom,
% 0.29/0.47 ! [A: nat,B: nat] :
% 0.29/0.47 ( ( ( times_times_nat @ A @ B )
% 0.29/0.47 != zero_zero_nat )
% 0.29/0.47 => ( ( A != zero_zero_nat )
% 0.29/0.47 & ( B != zero_zero_nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mult_not_zero
% 0.29/0.47 thf(fact_109_divisors__zero,axiom,
% 0.29/0.47 ! [A: nat,B: nat] :
% 0.29/0.47 ( ( ( times_times_nat @ A @ B )
% 0.29/0.47 = zero_zero_nat )
% 0.29/0.47 => ( ( A = zero_zero_nat )
% 0.29/0.47 | ( B = zero_zero_nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % divisors_zero
% 0.29/0.47 thf(fact_110_no__zero__divisors,axiom,
% 0.29/0.47 ! [A: nat,B: nat] :
% 0.29/0.47 ( ( A != zero_zero_nat )
% 0.29/0.47 => ( ( B != zero_zero_nat )
% 0.29/0.47 => ( ( times_times_nat @ A @ B )
% 0.29/0.47 != zero_zero_nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % no_zero_divisors
% 0.29/0.47 thf(fact_111_mult__left__cancel,axiom,
% 0.29/0.47 ! [C: nat,A: nat,B: nat] :
% 0.29/0.47 ( ( C != zero_zero_nat )
% 0.29/0.47 => ( ( ( times_times_nat @ C @ A )
% 0.29/0.47 = ( times_times_nat @ C @ B ) )
% 0.29/0.47 = ( A = B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mult_left_cancel
% 0.29/0.47 thf(fact_112_mult__right__cancel,axiom,
% 0.29/0.47 ! [C: nat,A: nat,B: nat] :
% 0.29/0.47 ( ( C != zero_zero_nat )
% 0.29/0.47 => ( ( ( times_times_nat @ A @ C )
% 0.29/0.47 = ( times_times_nat @ B @ C ) )
% 0.29/0.47 = ( A = B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mult_right_cancel
% 0.29/0.47 thf(fact_113_mult__neg__pos,axiom,
% 0.29/0.47 ! [A: nat,B: nat] :
% 0.29/0.47 ( ( ord_less_nat @ A @ zero_zero_nat )
% 0.29/0.47 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.29/0.47 => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mult_neg_pos
% 0.29/0.47 thf(fact_114_mult__pos__neg,axiom,
% 0.29/0.47 ! [A: nat,B: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.29/0.47 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 0.29/0.47 => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mult_pos_neg
% 0.29/0.47 thf(fact_115_mult__pos__pos,axiom,
% 0.29/0.47 ! [A: nat,B: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.29/0.47 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.29/0.47 => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mult_pos_pos
% 0.29/0.47 thf(fact_116_mult__pos__neg2,axiom,
% 0.29/0.47 ! [A: nat,B: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.29/0.47 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 0.29/0.47 => ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mult_pos_neg2
% 0.29/0.47 thf(fact_117_zero__less__mult__pos,axiom,
% 0.29/0.47 ! [A: nat,B: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
% 0.29/0.47 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.29/0.47 => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % zero_less_mult_pos
% 0.29/0.47 thf(fact_118_zero__less__mult__pos2,axiom,
% 0.29/0.47 ! [B: nat,A: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
% 0.29/0.47 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.29/0.47 => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % zero_less_mult_pos2
% 0.29/0.47 thf(fact_119_mult__strict__left__mono,axiom,
% 0.29/0.47 ! [A: nat,B: nat,C: nat] :
% 0.29/0.47 ( ( ord_less_nat @ A @ B )
% 0.29/0.47 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.29/0.47 => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mult_strict_left_mono
% 0.29/0.47 thf(fact_120_mult__strict__right__mono,axiom,
% 0.29/0.47 ! [A: nat,B: nat,C: nat] :
% 0.29/0.47 ( ( ord_less_nat @ A @ B )
% 0.29/0.47 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.29/0.47 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mult_strict_right_mono
% 0.29/0.47 thf(fact_121_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.29/0.47 ! [A: nat,B: nat,C: nat] :
% 0.29/0.47 ( ( ord_less_nat @ A @ B )
% 0.29/0.47 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.29/0.47 => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.29/0.47 thf(fact_122_div__eq__0__conv,axiom,
% 0.29/0.47 ! [N: nat,M: nat] :
% 0.29/0.47 ( ( ( divide_divide_nat @ N @ M )
% 0.29/0.47 = zero_zero_nat )
% 0.29/0.47 = ( ( M = zero_zero_nat )
% 0.29/0.47 | ( ord_less_nat @ N @ M ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % div_eq_0_conv
% 0.29/0.47 thf(fact_123_div__eq__0__conv_H,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.29/0.47 => ( ( ( divide_divide_nat @ N @ M )
% 0.29/0.47 = zero_zero_nat )
% 0.29/0.47 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % div_eq_0_conv'
% 0.29/0.47 thf(fact_124_Util__Div_Oless__mult__imp__div__less,axiom,
% 0.29/0.47 ! [N: nat,K: nat,M: nat] :
% 0.29/0.47 ( ( ord_less_nat @ N @ ( times_times_nat @ K @ M ) )
% 0.29/0.47 => ( ord_less_nat @ ( divide_divide_nat @ N @ M ) @ K ) ) ).
% 0.29/0.47
% 0.29/0.47 % Util_Div.less_mult_imp_div_less
% 0.29/0.47 thf(fact_125_div__less__conv,axiom,
% 0.29/0.47 ! [M: nat,N: nat,K: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.29/0.47 => ( ( ord_less_nat @ ( divide_divide_nat @ N @ M ) @ K )
% 0.29/0.47 = ( ord_less_nat @ N @ ( times_times_nat @ K @ M ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % div_less_conv
% 0.29/0.47 thf(fact_126_div__less__imp__less__mult,axiom,
% 0.29/0.47 ! [M: nat,N: nat,K: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.29/0.47 => ( ( ord_less_nat @ ( divide_divide_nat @ N @ M ) @ K )
% 0.29/0.47 => ( ord_less_nat @ N @ ( times_times_nat @ K @ M ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % div_less_imp_less_mult
% 0.29/0.47 thf(fact_127_bits__div__0,axiom,
% 0.29/0.47 ! [A: nat] :
% 0.29/0.47 ( ( divide_divide_nat @ zero_zero_nat @ A )
% 0.29/0.47 = zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 % bits_div_0
% 0.29/0.47 thf(fact_128_bits__div__by__0,axiom,
% 0.29/0.47 ! [A: nat] :
% 0.29/0.47 ( ( divide_divide_nat @ A @ zero_zero_nat )
% 0.29/0.47 = zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 % bits_div_by_0
% 0.29/0.47 thf(fact_129_not__gr__zero,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.29/0.47 = ( N = zero_zero_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % not_gr_zero
% 0.29/0.47 thf(fact_130_mult__cancel2__gr0,axiom,
% 0.29/0.47 ! [K: nat,M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.47 => ( ( ( times_times_nat @ M @ K )
% 0.29/0.47 = ( times_times_nat @ N @ K ) )
% 0.29/0.47 = ( M = N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mult_cancel2_gr0
% 0.29/0.47 thf(fact_131_zero__reorient,axiom,
% 0.29/0.47 ! [X: nat] :
% 0.29/0.47 ( ( zero_zero_nat = X )
% 0.29/0.47 = ( X = zero_zero_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % zero_reorient
% 0.29/0.47 thf(fact_132_mult_Oleft__commute,axiom,
% 0.29/0.47 ! [B: nat,A: nat,C: nat] :
% 0.29/0.47 ( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
% 0.29/0.47 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mult.left_commute
% 0.29/0.47 thf(fact_133_mult_Ocommute,axiom,
% 0.29/0.47 ( times_times_nat
% 0.29/0.47 = ( ^ [A2: nat,B2: nat] : ( times_times_nat @ B2 @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mult.commute
% 0.29/0.47 thf(fact_134_mult_Oassoc,axiom,
% 0.29/0.47 ! [A: nat,B: nat,C: nat] :
% 0.29/0.47 ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.29/0.47 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mult.assoc
% 0.29/0.47 thf(fact_135_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.29/0.47 ! [A: nat,B: nat,C: nat] :
% 0.29/0.47 ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.29/0.47 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ab_semigroup_mult_class.mult_ac(1)
% 0.29/0.47 thf(fact_136_zero__less__iff__neq__zero,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.47 = ( N != zero_zero_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % zero_less_iff_neq_zero
% 0.29/0.47 thf(fact_137_gr__implies__not__zero,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M @ N )
% 0.29/0.47 => ( N != zero_zero_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % gr_implies_not_zero
% 0.29/0.47 thf(fact_138_not__less__zero,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 % not_less_zero
% 0.29/0.47 thf(fact_139_gr__zeroI,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( N != zero_zero_nat )
% 0.29/0.47 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % gr_zeroI
% 0.29/0.47 thf(fact_140_gr__implies__gr0,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M @ N )
% 0.29/0.47 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % gr_implies_gr0
% 0.29/0.47 thf(fact_141_nat__lessSucSuc0__conv,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ N @ ( suc @ ( suc @ zero_zero_nat ) ) )
% 0.29/0.47 = ( ( N = zero_zero_nat )
% 0.29/0.47 | ( N
% 0.29/0.47 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % nat_lessSucSuc0_conv
% 0.29/0.47 thf(fact_142_nat__grSuc0__conv,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.29/0.47 = ( ( N != zero_zero_nat )
% 0.29/0.47 & ( N
% 0.29/0.47 != ( suc @ zero_zero_nat ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % nat_grSuc0_conv
% 0.29/0.47 thf(fact_143_length__induct,axiom,
% 0.29/0.47 ! [P: list_P1467822637od_a_a > $o,Xs: list_P1467822637od_a_a] :
% 0.29/0.47 ( ! [Xs2: list_P1467822637od_a_a] :
% 0.29/0.47 ( ! [Ys: list_P1467822637od_a_a] :
% 0.29/0.47 ( ( ord_less_nat @ ( size_s642698393od_a_a @ Ys ) @ ( size_s642698393od_a_a @ Xs2 ) )
% 0.29/0.47 => ( P @ Ys ) )
% 0.29/0.47 => ( P @ Xs2 ) )
% 0.29/0.47 => ( P @ Xs ) ) ).
% 0.29/0.47
% 0.29/0.47 % length_induct
% 0.29/0.47 thf(fact_144_length__induct,axiom,
% 0.29/0.47 ! [P: list_a > $o,Xs: list_a] :
% 0.29/0.47 ( ! [Xs2: list_a] :
% 0.29/0.47 ( ! [Ys: list_a] :
% 0.29/0.47 ( ( ord_less_nat @ ( size_size_list_a @ Ys ) @ ( size_size_list_a @ Xs2 ) )
% 0.29/0.47 => ( P @ Ys ) )
% 0.29/0.47 => ( P @ Xs2 ) )
% 0.29/0.47 => ( P @ Xs ) ) ).
% 0.29/0.47
% 0.29/0.47 % length_induct
% 0.29/0.47 thf(fact_145_Ex__list__of__length,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ? [Xs2: list_P1467822637od_a_a] :
% 0.29/0.47 ( ( size_s642698393od_a_a @ Xs2 )
% 0.29/0.47 = N ) ).
% 0.29/0.47
% 0.29/0.47 % Ex_list_of_length
% 0.29/0.47 thf(fact_146_Ex__list__of__length,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ? [Xs2: list_a] :
% 0.29/0.47 ( ( size_size_list_a @ Xs2 )
% 0.29/0.47 = N ) ).
% 0.29/0.47
% 0.29/0.47 % Ex_list_of_length
% 0.29/0.47 thf(fact_147_neq__if__length__neq,axiom,
% 0.29/0.47 ! [Xs: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a] :
% 0.29/0.47 ( ( ( size_s642698393od_a_a @ Xs )
% 0.29/0.47 != ( size_s642698393od_a_a @ Ys2 ) )
% 0.29/0.47 => ( Xs != Ys2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % neq_if_length_neq
% 0.29/0.47 thf(fact_148_neq__if__length__neq,axiom,
% 0.29/0.47 ! [Xs: list_a,Ys2: list_a] :
% 0.29/0.47 ( ( ( size_size_list_a @ Xs )
% 0.29/0.47 != ( size_size_list_a @ Ys2 ) )
% 0.29/0.47 => ( Xs != Ys2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % neq_if_length_neq
% 0.29/0.47 thf(fact_149_list__decode_Ocases,axiom,
% 0.29/0.47 ! [X: nat] :
% 0.29/0.47 ( ( X != zero_zero_nat )
% 0.29/0.47 => ~ ! [N2: nat] :
% 0.29/0.47 ( X
% 0.29/0.47 != ( suc @ N2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % list_decode.cases
% 0.29/0.47 thf(fact_150_exists__least__lemma,axiom,
% 0.29/0.47 ! [P: nat > $o] :
% 0.29/0.47 ( ~ ( P @ zero_zero_nat )
% 0.29/0.47 => ( ? [X_1: nat] : ( P @ X_1 )
% 0.29/0.47 => ? [N2: nat] :
% 0.29/0.47 ( ~ ( P @ N2 )
% 0.29/0.47 & ( P @ ( suc @ N2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % exists_least_lemma
% 0.29/0.47 thf(fact_151_less__numeral__extra_I3_J,axiom,
% 0.29/0.47 ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 % less_numeral_extra(3)
% 0.29/0.47 thf(fact_152_length__product,axiom,
% 0.29/0.47 ! [Xs: list_a,Ys2: list_P1467822637od_a_a] :
% 0.29/0.47 ( ( size_s1875148994od_a_a @ ( produc16441326od_a_a @ Xs @ Ys2 ) )
% 0.29/0.47 = ( times_times_nat @ ( size_size_list_a @ Xs ) @ ( size_s642698393od_a_a @ Ys2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % length_product
% 0.29/0.47 thf(fact_153_length__product,axiom,
% 0.29/0.47 ! [Xs: list_P1467822637od_a_a,Ys2: list_a] :
% 0.29/0.47 ( ( size_s736232240_a_a_a @ ( produc778776796_a_a_a @ Xs @ Ys2 ) )
% 0.29/0.47 = ( times_times_nat @ ( size_s642698393od_a_a @ Xs ) @ ( size_size_list_a @ Ys2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % length_product
% 0.29/0.47 thf(fact_154_length__product,axiom,
% 0.29/0.47 ! [Xs: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a] :
% 0.29/0.47 ( ( size_s1052379225od_a_a @ ( produc1055075589od_a_a @ Xs @ Ys2 ) )
% 0.29/0.47 = ( times_times_nat @ ( size_s642698393od_a_a @ Xs ) @ ( size_s642698393od_a_a @ Ys2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % length_product
% 0.29/0.47 thf(fact_155_length__product,axiom,
% 0.29/0.47 ! [Xs: list_a,Ys2: list_a] :
% 0.29/0.47 ( ( size_s642698393od_a_a @ ( product_a_a @ Xs @ Ys2 ) )
% 0.29/0.47 = ( times_times_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % length_product
% 0.29/0.47 thf(fact_156_split__div_H,axiom,
% 0.29/0.47 ! [P: nat > $o,M: nat,N: nat] :
% 0.29/0.47 ( ( P @ ( divide_divide_nat @ M @ N ) )
% 0.29/0.47 = ( ( ( N = zero_zero_nat )
% 0.29/0.47 & ( P @ zero_zero_nat ) )
% 0.29/0.47 | ? [Q2: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q2 ) @ M )
% 0.29/0.47 & ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q2 ) ) )
% 0.29/0.47 & ( P @ Q2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % split_div'
% 0.29/0.47 thf(fact_157_le__less__div__conv,axiom,
% 0.29/0.47 ! [M: nat,K: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.29/0.47 => ( ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ N )
% 0.29/0.47 & ( ord_less_nat @ N @ ( times_times_nat @ ( suc @ K ) @ M ) ) )
% 0.29/0.47 = ( ( divide_divide_nat @ N @ M )
% 0.29/0.47 = K ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_less_div_conv
% 0.29/0.47 thf(fact_158_le__zero__eq,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.29/0.47 = ( N = zero_zero_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_zero_eq
% 0.29/0.47 thf(fact_159_bot__nat__0_Oextremum,axiom,
% 0.29/0.47 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 0.29/0.47
% 0.29/0.47 % bot_nat_0.extremum
% 0.29/0.47 thf(fact_160_le0,axiom,
% 0.29/0.47 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.29/0.47
% 0.29/0.47 % le0
% 0.29/0.47 thf(fact_161_Suc__le__mono,axiom,
% 0.29/0.47 ! [N: nat,M: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
% 0.29/0.47 = ( ord_less_eq_nat @ N @ M ) ) ).
% 0.29/0.47
% 0.29/0.47 % Suc_le_mono
% 0.29/0.47 thf(fact_162_one__le__mult__iff,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
% 0.29/0.47 = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.29/0.47 & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % one_le_mult_iff
% 0.29/0.47 thf(fact_163_mult__le__cancel2,axiom,
% 0.29/0.47 ! [M: nat,K: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 0.29/0.47 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.47 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mult_le_cancel2
% 0.29/0.47 thf(fact_164_nat__mult__le__cancel__disj,axiom,
% 0.29/0.47 ! [K: nat,M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.29/0.47 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.47 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % nat_mult_le_cancel_disj
% 0.29/0.47 thf(fact_165_le__numeral__extra_I3_J,axiom,
% 0.29/0.47 ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 0.29/0.47
% 0.29/0.47 % le_numeral_extra(3)
% 0.29/0.47 thf(fact_166_zero__le,axiom,
% 0.29/0.47 ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% 0.29/0.47
% 0.29/0.47 % zero_le
% 0.29/0.47 thf(fact_167_ge__less__neq__conv,axiom,
% 0.29/0.47 ( ord_less_eq_nat
% 0.29/0.47 = ( ^ [A2: nat,N4: nat] :
% 0.29/0.47 ! [X4: nat] :
% 0.29/0.47 ( ( ord_less_nat @ X4 @ A2 )
% 0.29/0.47 => ( N4 != X4 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ge_less_neq_conv
% 0.29/0.47 thf(fact_168_less__ge__neq__conv,axiom,
% 0.29/0.47 ( ord_less_nat
% 0.29/0.47 = ( ^ [N4: nat,A2: nat] :
% 0.29/0.47 ! [X4: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ A2 @ X4 )
% 0.29/0.47 => ( N4 != X4 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_ge_neq_conv
% 0.29/0.47 thf(fact_169_greater__le__neq__conv,axiom,
% 0.29/0.47 ( ord_less_nat
% 0.29/0.47 = ( ^ [A2: nat,N4: nat] :
% 0.29/0.47 ! [X4: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ X4 @ A2 )
% 0.29/0.47 => ( N4 != X4 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % greater_le_neq_conv
% 0.29/0.47 thf(fact_170_le__greater__neq__conv,axiom,
% 0.29/0.47 ( ord_less_eq_nat
% 0.29/0.47 = ( ^ [N4: nat,A2: nat] :
% 0.29/0.47 ! [X4: nat] :
% 0.29/0.47 ( ( ord_less_nat @ A2 @ X4 )
% 0.29/0.47 => ( N4 != X4 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_greater_neq_conv
% 0.29/0.47 thf(fact_171_nat__geSucSuc0__conv,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ N )
% 0.29/0.47 = ( ( N != zero_zero_nat )
% 0.29/0.47 & ( N
% 0.29/0.47 != ( suc @ zero_zero_nat ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % nat_geSucSuc0_conv
% 0.29/0.47 thf(fact_172_nat__leSuc0__conv,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.29/0.47 = ( ( N = zero_zero_nat )
% 0.29/0.47 | ( N
% 0.29/0.47 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % nat_leSuc0_conv
% 0.29/0.47 thf(fact_173_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.29/0.47 ! [A: nat,B: nat,C: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.47 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.29/0.47 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.29/0.47 thf(fact_174_mult__nonneg__nonpos2,axiom,
% 0.29/0.47 ! [A: nat,B: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.29/0.47 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.29/0.47 => ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mult_nonneg_nonpos2
% 0.29/0.47 thf(fact_175_mult__nonpos__nonneg,axiom,
% 0.29/0.47 ! [A: nat,B: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.29/0.47 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.29/0.47 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mult_nonpos_nonneg
% 0.29/0.47 thf(fact_176_mult__nonneg__nonpos,axiom,
% 0.29/0.47 ! [A: nat,B: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.29/0.47 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.29/0.47 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mult_nonneg_nonpos
% 0.29/0.47 thf(fact_177_mult__nonneg__nonneg,axiom,
% 0.29/0.47 ! [A: nat,B: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.29/0.47 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.29/0.47 => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mult_nonneg_nonneg
% 0.29/0.47 thf(fact_178_split__mult__neg__le,axiom,
% 0.29/0.47 ! [A: nat,B: nat] :
% 0.29/0.47 ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.29/0.47 & ( ord_less_eq_nat @ B @ zero_zero_nat ) )
% 0.29/0.47 | ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.29/0.47 & ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
% 0.29/0.47 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % split_mult_neg_le
% 0.29/0.47 thf(fact_179_mult__right__mono,axiom,
% 0.29/0.47 ! [A: nat,B: nat,C: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.47 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.29/0.47 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mult_right_mono
% 0.29/0.47 thf(fact_180_mult__left__mono,axiom,
% 0.29/0.47 ! [A: nat,B: nat,C: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.47 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.29/0.47 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mult_left_mono
% 0.29/0.47 thf(fact_181_mult__mono_H,axiom,
% 0.29/0.47 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.47 => ( ( ord_less_eq_nat @ C @ D )
% 0.29/0.47 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.29/0.47 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.29/0.47 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mult_mono'
% 0.29/0.47 thf(fact_182_mult__mono,axiom,
% 0.29/0.47 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.47 => ( ( ord_less_eq_nat @ C @ D )
% 0.29/0.47 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.29/0.47 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.29/0.47 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % mult_mono
% 0.29/0.47
% 0.29/0.47 % Conjectures (3)
% 0.29/0.47 thf(conj_0,hypothesis,
% 0.29/0.47 ( ( divide_divide_nat @ ( size_size_list_a @ xs ) @ k )
% 0.29/0.47 = ( suc @ n ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(conj_1,hypothesis,
% 0.29/0.47 k != zero_zero_nat ).
% 0.29/0.47
% 0.29/0.47 thf(conj_2,conjecture,
% 0.29/0.47 ord_less_nat @ ( times_times_nat @ n @ k ) @ ( size_size_list_a @ xs ) ).
% 0.29/0.47
% 0.29/0.47 %------------------------------------------------------------------------------
% 0.29/0.47 ------------------------
% 2.66/1.51 ---- Embedded in THF ---
% 2.66/1.51 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.66/1.51 %%% Generated on Tue Feb 24 08:26:58 EST 2026
% 2.66/1.51 %%% using '$$dhol' embedding, version 1.3.0.
% 2.66/1.51 %%% Logic specification used:
% 2.66/1.51 %%% thf(spec, logic, $$dhol).
% 2.66/1.51
% 2.66/1.51 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.2CmgJp5TsA/DTF2DTF7532.p
% See solution above
% 2.66/1.51 ------------------------
% 2.66/1.54 ---- Cleaned THF ---
% 2.66/1.54 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 2.66/1.54 list_P1891567085od_a_a: $tType ).
% 2.66/1.54
% 2.66/1.54 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 2.66/1.54 list_P1458844246od_a_a: $tType ).
% 2.66/1.54
% 2.66/1.54 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_J,type,
% 2.66/1.54 list_P319927492_a_a_a: $tType ).
% 2.66/1.54
% 2.66/1.54 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 2.66/1.54 list_P1467822637od_a_a: $tType ).
% 2.66/1.54
% 2.66/1.54 thf(ty_n_t__List__Olist_Itf__a_J,type,
% 2.66/1.54 list_a: $tType ).
% 2.66/1.54
% 2.66/1.54 thf(ty_n_t__Nat__Onat,type,
% 2.66/1.54 nat: $tType ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
% 2.66/1.54 times_times_nat: nat > nat > nat ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 2.66/1.54 zero_zero_nat: nat ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_List_Oproduct_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 2.66/1.54 produc1055075589od_a_a: list_P1467822637od_a_a > list_P1467822637od_a_a > list_P1891567085od_a_a ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_List_Oproduct_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001tf__a,type,
% 2.66/1.54 produc778776796_a_a_a: list_P1467822637od_a_a > list_a > list_P319927492_a_a_a ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_List_Oproduct_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 2.66/1.54 produc16441326od_a_a: list_a > list_P1467822637od_a_a > list_P1458844246od_a_a ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_List_Oproduct_001tf__a_001tf__a,type,
% 2.66/1.54 product_a_a: list_a > list_a > list_P1467822637od_a_a ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_Nat_OSuc,type,
% 2.66/1.54 suc: nat > nat ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 2.66/1.54 size_s1052379225od_a_a: list_P1891567085od_a_a > nat ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_J,type,
% 2.66/1.54 size_s736232240_a_a_a: list_P319927492_a_a_a > nat ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 2.66/1.54 size_s1875148994od_a_a: list_P1458844246od_a_a > nat ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 2.66/1.54 size_s642698393od_a_a: list_P1467822637od_a_a > nat ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
% 2.66/1.54 size_size_list_a: list_a > nat ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 2.66/1.54 ord_less_nat: nat > nat > $o ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 2.66/1.54 ord_less_eq_nat: nat > nat > $o ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
% 2.66/1.54 divide_divide_nat: nat > nat > nat ).
% 2.66/1.54
% 2.66/1.54 thf(sy_v_k,type,
% 2.66/1.54 k: nat ).
% 2.66/1.54
% 2.66/1.54 thf(sy_v_n,type,
% 2.66/1.54 n: nat ).
% 2.66/1.54
% 2.66/1.54 thf(sy_v_xs,type,
% 2.66/1.54 xs: list_a ).
% 2.66/1.54
% 2.66/1.54 thf(fact_0_div__mult__self__is__m,axiom,
% 2.66/1.54 ! [N: nat,M: nat] :
% 2.66/1.54 ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.66/1.54 => ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
% 2.66/1.54 = M ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_1_div__mult__self1__is__m,axiom,
% 2.66/1.54 ! [N: nat,M: nat] :
% 2.66/1.54 ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.66/1.54 => ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
% 2.66/1.54 = M ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_2_nat__mult__div__cancel__disj,axiom,
% 2.66/1.54 ! [K: nat,M: nat,N: nat] :
% 2.66/1.54 ( ( ( K = zero_zero_nat )
% 2.66/1.54 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 2.66/1.54 = zero_zero_nat ) )
% 2.66/1.54 & ( ( K != zero_zero_nat )
% 2.66/1.54 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 2.66/1.54 = ( divide_divide_nat @ M @ N ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_3_div__less,axiom,
% 2.66/1.54 ! [M: nat,N: nat] :
% 2.66/1.54 ( ( ord_less_nat @ M @ N )
% 2.66/1.54 => ( ( divide_divide_nat @ M @ N )
% 2.66/1.54 = zero_zero_nat ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_4_div__by__Suc__0,axiom,
% 2.66/1.54 ! [M: nat] :
% 2.66/1.54 ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
% 2.66/1.54 = M ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_5_mult__less__cancel2,axiom,
% 2.66/1.54 ! [M: nat,K: nat,N: nat] :
% 2.66/1.54 ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 2.66/1.54 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 2.66/1.54 & ( ord_less_nat @ M @ N ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_6_nat__0__less__mult__iff,axiom,
% 2.66/1.54 ! [M: nat,N: nat] :
% 2.66/1.54 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
% 2.66/1.54 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 2.66/1.54 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_7_nat__mult__less__cancel__disj,axiom,
% 2.66/1.54 ! [K: nat,M: nat,N: nat] :
% 2.66/1.54 ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 2.66/1.54 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 2.66/1.54 & ( ord_less_nat @ M @ N ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_8_mult__eq__1__iff,axiom,
% 2.66/1.54 ! [M: nat,N: nat] :
% 2.66/1.54 ( ( ( times_times_nat @ M @ N )
% 2.66/1.54 = ( suc @ zero_zero_nat ) )
% 2.66/1.54 = ( ( M
% 2.66/1.54 = ( suc @ zero_zero_nat ) )
% 2.66/1.54 & ( N
% 2.66/1.54 = ( suc @ zero_zero_nat ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_9_one__eq__mult__iff,axiom,
% 2.66/1.54 ! [M: nat,N: nat] :
% 2.66/1.54 ( ( ( suc @ zero_zero_nat )
% 2.66/1.54 = ( times_times_nat @ M @ N ) )
% 2.66/1.54 = ( ( M
% 2.66/1.54 = ( suc @ zero_zero_nat ) )
% 2.66/1.54 & ( N
% 2.66/1.54 = ( suc @ zero_zero_nat ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_10_less__Suc0,axiom,
% 2.66/1.54 ! [N: nat] :
% 2.66/1.54 ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
% 2.66/1.54 = ( N = zero_zero_nat ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_11_zero__less__Suc,axiom,
% 2.66/1.54 ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_12_old_Onat_Oinject,axiom,
% 2.66/1.54 ! [Nat: nat,Nat2: nat] :
% 2.66/1.54 ( ( ( suc @ Nat )
% 2.66/1.54 = ( suc @ Nat2 ) )
% 2.66/1.54 = ( Nat = Nat2 ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_13_nat_Oinject,axiom,
% 2.66/1.54 ! [X2: nat,Y2: nat] :
% 2.66/1.54 ( ( ( suc @ X2 )
% 2.66/1.54 = ( suc @ Y2 ) )
% 2.66/1.54 = ( X2 = Y2 ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_14_bot__nat__0_Onot__eq__extremum,axiom,
% 2.66/1.54 ! [A: nat] :
% 2.66/1.54 ( ( A != zero_zero_nat )
% 2.66/1.54 = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_15_less__nat__zero__code,axiom,
% 2.66/1.54 ! [N: nat] :
% 2.66/1.54 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_16_neq0__conv,axiom,
% 2.66/1.54 ! [N: nat] :
% 2.66/1.54 ( ( N != zero_zero_nat )
% 2.66/1.54 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_17_Suc__less__eq,axiom,
% 2.66/1.54 ! [M: nat,N: nat] :
% 2.66/1.54 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 2.66/1.54 = ( ord_less_nat @ M @ N ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_18_Suc__mono,axiom,
% 2.66/1.54 ! [M: nat,N: nat] :
% 2.66/1.54 ( ( ord_less_nat @ M @ N )
% 2.66/1.54 => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_19_lessI,axiom,
% 2.66/1.54 ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_20_mult__cancel2,axiom,
% 2.66/1.54 ! [M: nat,K: nat,N: nat] :
% 2.66/1.54 ( ( ( times_times_nat @ M @ K )
% 2.66/1.54 = ( times_times_nat @ N @ K ) )
% 2.66/1.54 = ( ( M = N )
% 2.66/1.54 | ( K = zero_zero_nat ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_21_mult__cancel1,axiom,
% 2.66/1.54 ! [K: nat,M: nat,N: nat] :
% 2.66/1.54 ( ( ( times_times_nat @ K @ M )
% 2.66/1.54 = ( times_times_nat @ K @ N ) )
% 2.66/1.54 = ( ( M = N )
% 2.66/1.54 | ( K = zero_zero_nat ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_22_mult__0__right,axiom,
% 2.66/1.54 ! [M: nat] :
% 2.66/1.54 ( ( times_times_nat @ M @ zero_zero_nat )
% 2.66/1.54 = zero_zero_nat ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_23_mult__is__0,axiom,
% 2.66/1.54 ! [M: nat,N: nat] :
% 2.66/1.54 ( ( ( times_times_nat @ M @ N )
% 2.66/1.54 = zero_zero_nat )
% 2.66/1.54 = ( ( M = zero_zero_nat )
% 2.66/1.54 | ( N = zero_zero_nat ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_24_div__mult__mult1__if,axiom,
% 2.66/1.54 ! [C: nat,A: nat,B: nat] :
% 2.66/1.54 ( ( ( C = zero_zero_nat )
% 2.66/1.54 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 2.66/1.54 = zero_zero_nat ) )
% 2.66/1.54 & ( ( C != zero_zero_nat )
% 2.66/1.54 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 2.66/1.54 = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_25_div__mult__mult2,axiom,
% 2.66/1.54 ! [C: nat,A: nat,B: nat] :
% 2.66/1.54 ( ( C != zero_zero_nat )
% 2.66/1.54 => ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 2.66/1.54 = ( divide_divide_nat @ A @ B ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_26_div__mult__mult1,axiom,
% 2.66/1.54 ! [C: nat,A: nat,B: nat] :
% 2.66/1.54 ( ( C != zero_zero_nat )
% 2.66/1.54 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 2.66/1.54 = ( divide_divide_nat @ A @ B ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_27_n__not__Suc__n,axiom,
% 2.66/1.54 ! [N: nat] :
% 2.66/1.54 ( N
% 2.66/1.54 != ( suc @ N ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_28_Suc__inject,axiom,
% 2.66/1.54 ! [X: nat,Y: nat] :
% 2.66/1.54 ( ( ( suc @ X )
% 2.66/1.54 = ( suc @ Y ) )
% 2.66/1.54 => ( X = Y ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_29_linorder__neqE__nat,axiom,
% 2.66/1.54 ! [X: nat,Y: nat] :
% 2.66/1.54 ( ( X != Y )
% 2.66/1.54 => ( ~ ( ord_less_nat @ X @ Y )
% 2.66/1.54 => ( ord_less_nat @ Y @ X ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_30_infinite__descent,axiom,
% 2.66/1.54 ! [P: nat > $o,N: nat] :
% 2.66/1.54 ( ! [N2: nat] :
% 2.66/1.54 ( ~ ( P @ N2 )
% 2.66/1.54 => ? [M2: nat] :
% 2.66/1.54 ( ( ord_less_nat @ M2 @ N2 )
% 2.66/1.54 & ~ ( P @ M2 ) ) )
% 2.66/1.54 => ( P @ N ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_31_nat__less__induct,axiom,
% 2.66/1.54 ! [P: nat > $o,N: nat] :
% 2.66/1.54 ( ! [N2: nat] :
% 2.66/1.54 ( ! [M2: nat] :
% 2.66/1.54 ( ( ord_less_nat @ M2 @ N2 )
% 2.66/1.54 => ( P @ M2 ) )
% 2.66/1.54 => ( P @ N2 ) )
% 2.66/1.54 => ( P @ N ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_32_less__irrefl__nat,axiom,
% 2.66/1.54 ! [N: nat] :
% 2.66/1.54 ~ ( ord_less_nat @ N @ N ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_33_less__not__refl3,axiom,
% 2.66/1.54 ! [S: nat,T: nat] :
% 2.66/1.54 ( ( ord_less_nat @ S @ T )
% 2.66/1.54 => ( S != T ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_34_less__not__refl2,axiom,
% 2.66/1.54 ! [N: nat,M: nat] :
% 2.66/1.54 ( ( ord_less_nat @ N @ M )
% 2.66/1.54 => ( M != N ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_35_less__not__refl,axiom,
% 2.66/1.54 ! [N: nat] :
% 2.66/1.54 ~ ( ord_less_nat @ N @ N ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_36_nat__neq__iff,axiom,
% 2.66/1.54 ! [M: nat,N: nat] :
% 2.66/1.54 ( ( M != N )
% 2.66/1.54 = ( ( ord_less_nat @ M @ N )
% 2.66/1.54 | ( ord_less_nat @ N @ M ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_37_size__neq__size__imp__neq,axiom,
% 2.66/1.54 ! [X: list_P1467822637od_a_a,Y: list_P1467822637od_a_a] :
% 2.66/1.54 ( ( ( size_s642698393od_a_a @ X )
% 2.66/1.54 != ( size_s642698393od_a_a @ Y ) )
% 2.66/1.54 => ( X != Y ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_38_size__neq__size__imp__neq,axiom,
% 2.66/1.54 ! [X: list_a,Y: list_a] :
% 2.66/1.54 ( ( ( size_size_list_a @ X )
% 2.66/1.54 != ( size_size_list_a @ Y ) )
% 2.66/1.54 => ( X != Y ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_39_not0__implies__Suc,axiom,
% 2.66/1.54 ! [N: nat] :
% 2.66/1.54 ( ( N != zero_zero_nat )
% 2.66/1.54 => ? [M3: nat] :
% 2.66/1.54 ( N
% 2.66/1.54 = ( suc @ M3 ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_40_old_Onat_Oinducts,axiom,
% 2.66/1.54 ! [P: nat > $o,Nat: nat] :
% 2.66/1.54 ( ( P @ zero_zero_nat )
% 2.66/1.54 => ( ! [Nat3: nat] :
% 2.66/1.54 ( ( P @ Nat3 )
% 2.66/1.54 => ( P @ ( suc @ Nat3 ) ) )
% 2.66/1.54 => ( P @ Nat ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_41_old_Onat_Oexhaust,axiom,
% 2.66/1.54 ! [Y: nat] :
% 2.66/1.54 ( ( Y != zero_zero_nat )
% 2.66/1.54 => ~ ! [Nat3: nat] :
% 2.66/1.54 ( Y
% 2.66/1.54 != ( suc @ Nat3 ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_42_Zero__not__Suc,axiom,
% 2.66/1.54 ! [M: nat] :
% 2.66/1.54 ( zero_zero_nat
% 2.66/1.54 != ( suc @ M ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_43_Zero__neq__Suc,axiom,
% 2.66/1.54 ! [M: nat] :
% 2.66/1.54 ( zero_zero_nat
% 2.66/1.54 != ( suc @ M ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_44_Suc__neq__Zero,axiom,
% 2.66/1.54 ! [M: nat] :
% 2.66/1.54 ( ( suc @ M )
% 2.66/1.54 != zero_zero_nat ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_45_zero__induct,axiom,
% 2.66/1.54 ! [P: nat > $o,K: nat] :
% 2.66/1.54 ( ( P @ K )
% 2.66/1.54 => ( ! [N2: nat] :
% 2.66/1.54 ( ( P @ ( suc @ N2 ) )
% 2.66/1.54 => ( P @ N2 ) )
% 2.66/1.54 => ( P @ zero_zero_nat ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_46_diff__induct,axiom,
% 2.66/1.54 ! [P: nat > nat > $o,M: nat,N: nat] :
% 2.66/1.54 ( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
% 2.66/1.54 => ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
% 2.66/1.54 => ( ! [X3: nat,Y3: nat] :
% 2.66/1.54 ( ( P @ X3 @ Y3 )
% 2.66/1.54 => ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
% 2.66/1.54 => ( P @ M @ N ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_47_nat__induct,axiom,
% 2.66/1.54 ! [P: nat > $o,N: nat] :
% 2.66/1.54 ( ( P @ zero_zero_nat )
% 2.66/1.54 => ( ! [N2: nat] :
% 2.66/1.54 ( ( P @ N2 )
% 2.66/1.54 => ( P @ ( suc @ N2 ) ) )
% 2.66/1.54 => ( P @ N ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_48_nat_OdiscI,axiom,
% 2.66/1.54 ! [Nat: nat,X2: nat] :
% 2.66/1.54 ( ( Nat
% 2.66/1.54 = ( suc @ X2 ) )
% 2.66/1.54 => ( Nat != zero_zero_nat ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_49_old_Onat_Odistinct_I1_J,axiom,
% 2.66/1.54 ! [Nat2: nat] :
% 2.66/1.54 ( zero_zero_nat
% 2.66/1.54 != ( suc @ Nat2 ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_50_old_Onat_Odistinct_I2_J,axiom,
% 2.66/1.54 ! [Nat2: nat] :
% 2.66/1.54 ( ( suc @ Nat2 )
% 2.66/1.54 != zero_zero_nat ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_51_nat_Odistinct_I1_J,axiom,
% 2.66/1.54 ! [X2: nat] :
% 2.66/1.54 ( zero_zero_nat
% 2.66/1.54 != ( suc @ X2 ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_52_bot__nat__0_Oextremum__strict,axiom,
% 2.66/1.54 ! [A: nat] :
% 2.66/1.54 ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_53_infinite__descent0,axiom,
% 2.66/1.54 ! [P: nat > $o,N: nat] :
% 2.66/1.54 ( ( P @ zero_zero_nat )
% 2.66/1.54 => ( ! [N2: nat] :
% 2.66/1.54 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 2.66/1.54 => ( ~ ( P @ N2 )
% 2.66/1.54 => ? [M2: nat] :
% 2.66/1.54 ( ( ord_less_nat @ M2 @ N2 )
% 2.66/1.54 & ~ ( P @ M2 ) ) ) )
% 2.66/1.54 => ( P @ N ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_54_gr__implies__not0,axiom,
% 2.66/1.54 ! [M: nat,N: nat] :
% 2.66/1.54 ( ( ord_less_nat @ M @ N )
% 2.66/1.54 => ( N != zero_zero_nat ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_55_less__zeroE,axiom,
% 2.66/1.54 ! [N: nat] :
% 2.66/1.54 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_56_not__less0,axiom,
% 2.66/1.54 ! [N: nat] :
% 2.66/1.54 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_57_not__gr0,axiom,
% 2.66/1.54 ! [N: nat] :
% 2.66/1.54 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 2.66/1.54 = ( N = zero_zero_nat ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_58_gr0I,axiom,
% 2.66/1.54 ! [N: nat] :
% 2.66/1.54 ( ( N != zero_zero_nat )
% 2.66/1.54 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_59_not__less__less__Suc__eq,axiom,
% 2.66/1.54 ! [N: nat,M: nat] :
% 2.66/1.54 ( ~ ( ord_less_nat @ N @ M )
% 2.66/1.54 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 2.66/1.54 = ( N = M ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_60_strict__inc__induct,axiom,
% 2.66/1.54 ! [I: nat,J: nat,P: nat > $o] :
% 2.66/1.54 ( ( ord_less_nat @ I @ J )
% 2.66/1.54 => ( ! [I2: nat] :
% 2.66/1.54 ( ( J
% 2.66/1.54 = ( suc @ I2 ) )
% 2.66/1.54 => ( P @ I2 ) )
% 2.66/1.54 => ( ! [I2: nat] :
% 2.66/1.54 ( ( ord_less_nat @ I2 @ J )
% 2.66/1.54 => ( ( P @ ( suc @ I2 ) )
% 2.66/1.54 => ( P @ I2 ) ) )
% 2.66/1.54 => ( P @ I ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_61_less__Suc__induct,axiom,
% 2.66/1.54 ! [I: nat,J: nat,P: nat > nat > $o] :
% 2.66/1.54 ( ( ord_less_nat @ I @ J )
% 2.66/1.54 => ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
% 2.66/1.54 => ( ! [I2: nat,J2: nat,K2: nat] :
% 2.66/1.54 ( ( ord_less_nat @ I2 @ J2 )
% 2.66/1.54 => ( ( ord_less_nat @ J2 @ K2 )
% 2.66/1.54 => ( ( P @ I2 @ J2 )
% 2.66/1.54 => ( ( P @ J2 @ K2 )
% 2.66/1.54 => ( P @ I2 @ K2 ) ) ) ) )
% 2.66/1.54 => ( P @ I @ J ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_62_less__trans__Suc,axiom,
% 2.66/1.54 ! [I: nat,J: nat,K: nat] :
% 2.66/1.54 ( ( ord_less_nat @ I @ J )
% 2.66/1.54 => ( ( ord_less_nat @ J @ K )
% 2.66/1.54 => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_63_Suc__less__SucD,axiom,
% 2.66/1.54 ! [M: nat,N: nat] :
% 2.66/1.54 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 2.66/1.54 => ( ord_less_nat @ M @ N ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_64_less__antisym,axiom,
% 2.66/1.54 ! [N: nat,M: nat] :
% 2.66/1.54 ( ~ ( ord_less_nat @ N @ M )
% 2.66/1.54 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 2.66/1.54 => ( M = N ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_65_Suc__less__eq2,axiom,
% 2.66/1.54 ! [N: nat,M: nat] :
% 2.66/1.54 ( ( ord_less_nat @ ( suc @ N ) @ M )
% 2.66/1.54 = ( ? [M4: nat] :
% 2.66/1.54 ( ( M
% 2.66/1.54 = ( suc @ M4 ) )
% 2.66/1.54 & ( ord_less_nat @ N @ M4 ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_66_All__less__Suc,axiom,
% 2.66/1.54 ! [N: nat,P: nat > $o] :
% 2.66/1.54 ( ( ! [I3: nat] :
% 2.66/1.54 ( ( ord_less_nat @ I3 @ ( suc @ N ) )
% 2.66/1.54 => ( P @ I3 ) ) )
% 2.66/1.54 = ( ( P @ N )
% 2.66/1.54 & ! [I3: nat] :
% 2.66/1.54 ( ( ord_less_nat @ I3 @ N )
% 2.66/1.54 => ( P @ I3 ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_67_not__less__eq,axiom,
% 2.66/1.54 ! [M: nat,N: nat] :
% 2.66/1.54 ( ( ~ ( ord_less_nat @ M @ N ) )
% 2.66/1.54 = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_68_less__Suc__eq,axiom,
% 2.66/1.54 ! [M: nat,N: nat] :
% 2.66/1.54 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 2.66/1.54 = ( ( ord_less_nat @ M @ N )
% 2.66/1.54 | ( M = N ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_69_Ex__less__Suc,axiom,
% 2.66/1.54 ! [N: nat,P: nat > $o] :
% 2.66/1.54 ( ( ? [I3: nat] :
% 2.66/1.54 ( ( ord_less_nat @ I3 @ ( suc @ N ) )
% 2.66/1.54 & ( P @ I3 ) ) )
% 2.66/1.54 = ( ( P @ N )
% 2.66/1.54 | ? [I3: nat] :
% 2.66/1.54 ( ( ord_less_nat @ I3 @ N )
% 2.66/1.54 & ( P @ I3 ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_70_less__SucI,axiom,
% 2.66/1.54 ! [M: nat,N: nat] :
% 2.66/1.54 ( ( ord_less_nat @ M @ N )
% 2.66/1.54 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_71_less__SucE,axiom,
% 2.66/1.54 ! [M: nat,N: nat] :
% 2.66/1.54 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 2.66/1.54 => ( ~ ( ord_less_nat @ M @ N )
% 2.66/1.54 => ( M = N ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_72_Suc__lessI,axiom,
% 2.66/1.54 ! [M: nat,N: nat] :
% 2.66/1.54 ( ( ord_less_nat @ M @ N )
% 2.66/1.54 => ( ( ( suc @ M )
% 2.66/1.54 != N )
% 2.66/1.54 => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_73_Suc__lessE,axiom,
% 2.66/1.54 ! [I: nat,K: nat] :
% 2.66/1.54 ( ( ord_less_nat @ ( suc @ I ) @ K )
% 2.66/1.54 => ~ ! [J2: nat] :
% 2.66/1.54 ( ( ord_less_nat @ I @ J2 )
% 2.66/1.54 => ( K
% 2.66/1.54 != ( suc @ J2 ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_74_Suc__lessD,axiom,
% 2.66/1.54 ! [M: nat,N: nat] :
% 2.66/1.54 ( ( ord_less_nat @ ( suc @ M ) @ N )
% 2.66/1.54 => ( ord_less_nat @ M @ N ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_75_Nat_OlessE,axiom,
% 2.66/1.54 ! [I: nat,K: nat] :
% 2.66/1.54 ( ( ord_less_nat @ I @ K )
% 2.66/1.54 => ( ( K
% 2.66/1.54 != ( suc @ I ) )
% 2.66/1.54 => ~ ! [J2: nat] :
% 2.66/1.54 ( ( ord_less_nat @ I @ J2 )
% 2.66/1.54 => ( K
% 2.66/1.54 != ( suc @ J2 ) ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_76_nat__mult__eq__cancel__disj,axiom,
% 2.66/1.54 ! [K: nat,M: nat,N: nat] :
% 2.66/1.54 ( ( ( times_times_nat @ K @ M )
% 2.66/1.54 = ( times_times_nat @ K @ N ) )
% 2.66/1.54 = ( ( K = zero_zero_nat )
% 2.66/1.54 | ( M = N ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_77_mult__0,axiom,
% 2.66/1.54 ! [N: nat] :
% 2.66/1.54 ( ( times_times_nat @ zero_zero_nat @ N )
% 2.66/1.54 = zero_zero_nat ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_78_Suc__mult__cancel1,axiom,
% 2.66/1.54 ! [K: nat,M: nat,N: nat] :
% 2.66/1.54 ( ( ( times_times_nat @ ( suc @ K ) @ M )
% 2.66/1.54 = ( times_times_nat @ ( suc @ K ) @ N ) )
% 2.66/1.54 = ( M = N ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_79_div__mult2__eq,axiom,
% 2.66/1.54 ! [M: nat,N: nat,Q: nat] :
% 2.66/1.54 ( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q ) )
% 2.66/1.54 = ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_80_lift__Suc__mono__less__iff,axiom,
% 2.66/1.54 ! [F: nat > nat,N: nat,M: nat] :
% 2.66/1.54 ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 2.66/1.54 => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
% 2.66/1.54 = ( ord_less_nat @ N @ M ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_81_lift__Suc__mono__less,axiom,
% 2.66/1.54 ! [F: nat > nat,N: nat,N3: nat] :
% 2.66/1.54 ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 2.66/1.54 => ( ( ord_less_nat @ N @ N3 )
% 2.66/1.54 => ( ord_less_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_82_less__Suc__eq__0__disj,axiom,
% 2.66/1.54 ! [M: nat,N: nat] :
% 2.66/1.54 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 2.66/1.54 = ( ( M = zero_zero_nat )
% 2.66/1.54 | ? [J3: nat] :
% 2.66/1.54 ( ( M
% 2.66/1.54 = ( suc @ J3 ) )
% 2.66/1.54 & ( ord_less_nat @ J3 @ N ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_83_gr0__implies__Suc,axiom,
% 2.66/1.54 ! [N: nat] :
% 2.66/1.54 ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.66/1.54 => ? [M3: nat] :
% 2.66/1.54 ( N
% 2.66/1.54 = ( suc @ M3 ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_84_All__less__Suc2,axiom,
% 2.66/1.54 ! [N: nat,P: nat > $o] :
% 2.66/1.54 ( ( ! [I3: nat] :
% 2.66/1.54 ( ( ord_less_nat @ I3 @ ( suc @ N ) )
% 2.66/1.54 => ( P @ I3 ) ) )
% 2.66/1.54 = ( ( P @ zero_zero_nat )
% 2.66/1.54 & ! [I3: nat] :
% 2.66/1.54 ( ( ord_less_nat @ I3 @ N )
% 2.66/1.54 => ( P @ ( suc @ I3 ) ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_85_gr0__conv__Suc,axiom,
% 2.66/1.54 ! [N: nat] :
% 2.66/1.54 ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.66/1.54 = ( ? [M5: nat] :
% 2.66/1.54 ( N
% 2.66/1.54 = ( suc @ M5 ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_86_Ex__less__Suc2,axiom,
% 2.66/1.54 ! [N: nat,P: nat > $o] :
% 2.66/1.54 ( ( ? [I3: nat] :
% 2.66/1.54 ( ( ord_less_nat @ I3 @ ( suc @ N ) )
% 2.66/1.54 & ( P @ I3 ) ) )
% 2.66/1.54 = ( ( P @ zero_zero_nat )
% 2.66/1.54 | ? [I3: nat] :
% 2.66/1.54 ( ( ord_less_nat @ I3 @ N )
% 2.66/1.54 & ( P @ ( suc @ I3 ) ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_87_nat__mult__less__cancel1,axiom,
% 2.66/1.54 ! [K: nat,M: nat,N: nat] :
% 2.66/1.54 ( ( ord_less_nat @ zero_zero_nat @ K )
% 2.66/1.54 => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 2.66/1.54 = ( ord_less_nat @ M @ N ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_88_nat__mult__eq__cancel1,axiom,
% 2.66/1.54 ! [K: nat,M: nat,N: nat] :
% 2.66/1.54 ( ( ord_less_nat @ zero_zero_nat @ K )
% 2.66/1.54 => ( ( ( times_times_nat @ K @ M )
% 2.66/1.54 = ( times_times_nat @ K @ N ) )
% 2.66/1.54 = ( M = N ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_89_mult__less__mono2,axiom,
% 2.66/1.54 ! [I: nat,J: nat,K: nat] :
% 2.66/1.54 ( ( ord_less_nat @ I @ J )
% 2.66/1.54 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 2.66/1.54 => ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_90_mult__less__mono1,axiom,
% 2.66/1.54 ! [I: nat,J: nat,K: nat] :
% 2.66/1.54 ( ( ord_less_nat @ I @ J )
% 2.66/1.54 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 2.66/1.54 => ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_91_Suc__mult__less__cancel1,axiom,
% 2.66/1.54 ! [K: nat,M: nat,N: nat] :
% 2.66/1.54 ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 2.66/1.54 = ( ord_less_nat @ M @ N ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_92_Euclidean__Division_Odiv__eq__0__iff,axiom,
% 2.66/1.54 ! [M: nat,N: nat] :
% 2.66/1.54 ( ( ( divide_divide_nat @ M @ N )
% 2.66/1.54 = zero_zero_nat )
% 2.66/1.54 = ( ( ord_less_nat @ M @ N )
% 2.66/1.54 | ( N = zero_zero_nat ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_93_Euclidean__Division_Oless__mult__imp__div__less,axiom,
% 2.66/1.54 ! [M: nat,I: nat,N: nat] :
% 2.66/1.54 ( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
% 2.66/1.54 => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_94_n__less__n__mult__m,axiom,
% 2.66/1.54 ! [N: nat,M: nat] :
% 2.66/1.54 ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.66/1.54 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 2.66/1.54 => ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_95_n__less__m__mult__n,axiom,
% 2.66/1.54 ! [N: nat,M: nat] :
% 2.66/1.54 ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.66/1.54 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 2.66/1.54 => ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_96_one__less__mult,axiom,
% 2.66/1.54 ! [N: nat,M: nat] :
% 2.66/1.54 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 2.66/1.54 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 2.66/1.54 => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_97_nat__mult__div__cancel1,axiom,
% 2.66/1.54 ! [K: nat,M: nat,N: nat] :
% 2.66/1.54 ( ( ord_less_nat @ zero_zero_nat @ K )
% 2.66/1.54 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 2.66/1.54 = ( divide_divide_nat @ M @ N ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_98_nonzero__mult__div__cancel__right,axiom,
% 2.66/1.54 ! [B: nat,A: nat] :
% 2.66/1.54 ( ( B != zero_zero_nat )
% 2.66/1.54 => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
% 2.66/1.54 = A ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_99_nonzero__mult__div__cancel__left,axiom,
% 2.66/1.54 ! [A: nat,B: nat] :
% 2.66/1.54 ( ( A != zero_zero_nat )
% 2.66/1.54 => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
% 2.66/1.54 = B ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_100_less__div__Suc__mult,axiom,
% 2.66/1.54 ! [M: nat,N: nat] :
% 2.66/1.54 ( ( ord_less_nat @ zero_zero_nat @ M )
% 2.66/1.54 => ( ord_less_nat @ N @ ( times_times_nat @ ( suc @ ( divide_divide_nat @ N @ M ) ) @ M ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_101_mult__zero__left,axiom,
% 2.66/1.54 ! [A: nat] :
% 2.66/1.54 ( ( times_times_nat @ zero_zero_nat @ A )
% 2.66/1.54 = zero_zero_nat ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_102_mult__zero__right,axiom,
% 2.66/1.54 ! [A: nat] :
% 2.66/1.54 ( ( times_times_nat @ A @ zero_zero_nat )
% 2.66/1.54 = zero_zero_nat ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_103_mult__eq__0__iff,axiom,
% 2.66/1.54 ! [A: nat,B: nat] :
% 2.66/1.54 ( ( ( times_times_nat @ A @ B )
% 2.66/1.54 = zero_zero_nat )
% 2.66/1.54 = ( ( A = zero_zero_nat )
% 2.66/1.54 | ( B = zero_zero_nat ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_104_mult__cancel__left,axiom,
% 2.66/1.54 ! [C: nat,A: nat,B: nat] :
% 2.66/1.54 ( ( ( times_times_nat @ C @ A )
% 2.66/1.54 = ( times_times_nat @ C @ B ) )
% 2.66/1.54 = ( ( C = zero_zero_nat )
% 2.66/1.54 | ( A = B ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_105_mult__cancel__right,axiom,
% 2.66/1.54 ! [A: nat,C: nat,B: nat] :
% 2.66/1.54 ( ( ( times_times_nat @ A @ C )
% 2.66/1.54 = ( times_times_nat @ B @ C ) )
% 2.66/1.54 = ( ( C = zero_zero_nat )
% 2.66/1.54 | ( A = B ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_106_div__0,axiom,
% 2.66/1.54 ! [A: nat] :
% 2.66/1.54 ( ( divide_divide_nat @ zero_zero_nat @ A )
% 2.66/1.54 = zero_zero_nat ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_107_div__by__0,axiom,
% 2.66/1.54 ! [A: nat] :
% 2.66/1.54 ( ( divide_divide_nat @ A @ zero_zero_nat )
% 2.66/1.54 = zero_zero_nat ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_108_mult__not__zero,axiom,
% 2.66/1.54 ! [A: nat,B: nat] :
% 2.66/1.54 ( ( ( times_times_nat @ A @ B )
% 2.66/1.54 != zero_zero_nat )
% 2.66/1.54 => ( ( A != zero_zero_nat )
% 2.66/1.54 & ( B != zero_zero_nat ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_109_divisors__zero,axiom,
% 2.66/1.54 ! [A: nat,B: nat] :
% 2.66/1.54 ( ( ( times_times_nat @ A @ B )
% 2.66/1.54 = zero_zero_nat )
% 2.66/1.54 => ( ( A = zero_zero_nat )
% 2.66/1.54 | ( B = zero_zero_nat ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_110_no__zero__divisors,axiom,
% 2.66/1.54 ! [A: nat,B: nat] :
% 2.66/1.54 ( ( A != zero_zero_nat )
% 2.66/1.54 => ( ( B != zero_zero_nat )
% 2.66/1.54 => ( ( times_times_nat @ A @ B )
% 2.66/1.54 != zero_zero_nat ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_111_mult__left__cancel,axiom,
% 2.66/1.54 ! [C: nat,A: nat,B: nat] :
% 2.66/1.54 ( ( C != zero_zero_nat )
% 2.66/1.54 => ( ( ( times_times_nat @ C @ A )
% 2.66/1.54 = ( times_times_nat @ C @ B ) )
% 2.66/1.54 = ( A = B ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_112_mult__right__cancel,axiom,
% 2.66/1.54 ! [C: nat,A: nat,B: nat] :
% 2.66/1.54 ( ( C != zero_zero_nat )
% 2.66/1.54 => ( ( ( times_times_nat @ A @ C )
% 2.66/1.54 = ( times_times_nat @ B @ C ) )
% 2.66/1.54 = ( A = B ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_113_mult__neg__pos,axiom,
% 2.66/1.54 ! [A: nat,B: nat] :
% 2.66/1.54 ( ( ord_less_nat @ A @ zero_zero_nat )
% 2.66/1.54 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 2.66/1.54 => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_114_mult__pos__neg,axiom,
% 2.66/1.54 ! [A: nat,B: nat] :
% 2.66/1.54 ( ( ord_less_nat @ zero_zero_nat @ A )
% 2.66/1.54 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 2.66/1.54 => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_115_mult__pos__pos,axiom,
% 2.66/1.54 ! [A: nat,B: nat] :
% 2.66/1.54 ( ( ord_less_nat @ zero_zero_nat @ A )
% 2.66/1.54 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 2.66/1.54 => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_116_mult__pos__neg2,axiom,
% 2.66/1.54 ! [A: nat,B: nat] :
% 2.66/1.54 ( ( ord_less_nat @ zero_zero_nat @ A )
% 2.66/1.54 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 2.66/1.54 => ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_117_zero__less__mult__pos,axiom,
% 2.66/1.54 ! [A: nat,B: nat] :
% 2.66/1.54 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
% 2.66/1.54 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 2.66/1.54 => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_118_zero__less__mult__pos2,axiom,
% 2.66/1.54 ! [B: nat,A: nat] :
% 2.66/1.54 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
% 2.66/1.54 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 2.66/1.54 => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_119_mult__strict__left__mono,axiom,
% 2.66/1.54 ! [A: nat,B: nat,C: nat] :
% 2.66/1.54 ( ( ord_less_nat @ A @ B )
% 2.66/1.54 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 2.66/1.54 => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_120_mult__strict__right__mono,axiom,
% 2.66/1.54 ! [A: nat,B: nat,C: nat] :
% 2.66/1.54 ( ( ord_less_nat @ A @ B )
% 2.66/1.54 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 2.66/1.54 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_121_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 2.66/1.54 ! [A: nat,B: nat,C: nat] :
% 2.66/1.54 ( ( ord_less_nat @ A @ B )
% 2.66/1.54 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 2.66/1.54 => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_122_div__eq__0__conv,axiom,
% 2.66/1.54 ! [N: nat,M: nat] :
% 2.66/1.54 ( ( ( divide_divide_nat @ N @ M )
% 2.66/1.54 = zero_zero_nat )
% 2.66/1.54 = ( ( M = zero_zero_nat )
% 2.66/1.54 | ( ord_less_nat @ N @ M ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_123_div__eq__0__conv_H,axiom,
% 2.66/1.54 ! [M: nat,N: nat] :
% 2.66/1.54 ( ( ord_less_nat @ zero_zero_nat @ M )
% 2.66/1.54 => ( ( ( divide_divide_nat @ N @ M )
% 2.66/1.54 = zero_zero_nat )
% 2.66/1.54 = ( ord_less_nat @ N @ M ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_124_Util__Div_Oless__mult__imp__div__less,axiom,
% 2.66/1.54 ! [N: nat,K: nat,M: nat] :
% 2.66/1.54 ( ( ord_less_nat @ N @ ( times_times_nat @ K @ M ) )
% 2.66/1.54 => ( ord_less_nat @ ( divide_divide_nat @ N @ M ) @ K ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_125_div__less__conv,axiom,
% 2.66/1.54 ! [M: nat,N: nat,K: nat] :
% 2.66/1.54 ( ( ord_less_nat @ zero_zero_nat @ M )
% 2.66/1.54 => ( ( ord_less_nat @ ( divide_divide_nat @ N @ M ) @ K )
% 2.66/1.54 = ( ord_less_nat @ N @ ( times_times_nat @ K @ M ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_126_div__less__imp__less__mult,axiom,
% 2.66/1.54 ! [M: nat,N: nat,K: nat] :
% 2.66/1.54 ( ( ord_less_nat @ zero_zero_nat @ M )
% 2.66/1.54 => ( ( ord_less_nat @ ( divide_divide_nat @ N @ M ) @ K )
% 2.66/1.54 => ( ord_less_nat @ N @ ( times_times_nat @ K @ M ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_127_bits__div__0,axiom,
% 2.66/1.54 ! [A: nat] :
% 2.66/1.54 ( ( divide_divide_nat @ zero_zero_nat @ A )
% 2.66/1.54 = zero_zero_nat ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_128_bits__div__by__0,axiom,
% 2.66/1.54 ! [A: nat] :
% 2.66/1.54 ( ( divide_divide_nat @ A @ zero_zero_nat )
% 2.66/1.54 = zero_zero_nat ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_129_not__gr__zero,axiom,
% 2.66/1.54 ! [N: nat] :
% 2.66/1.54 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 2.66/1.54 = ( N = zero_zero_nat ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_130_mult__cancel2__gr0,axiom,
% 2.66/1.54 ! [K: nat,M: nat,N: nat] :
% 2.66/1.54 ( ( ord_less_nat @ zero_zero_nat @ K )
% 2.66/1.54 => ( ( ( times_times_nat @ M @ K )
% 2.66/1.54 = ( times_times_nat @ N @ K ) )
% 2.66/1.54 = ( M = N ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_131_zero__reorient,axiom,
% 2.66/1.54 ! [X: nat] :
% 2.66/1.54 ( ( zero_zero_nat = X )
% 2.66/1.54 = ( X = zero_zero_nat ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_132_mult_Oleft__commute,axiom,
% 2.66/1.54 ! [B: nat,A: nat,C: nat] :
% 2.66/1.54 ( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
% 2.66/1.54 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_133_mult_Ocommute,axiom,
% 2.66/1.54 ( times_times_nat
% 2.66/1.54 = ( ^ [A2: nat,B2: nat] : ( times_times_nat @ B2 @ A2 ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_134_mult_Oassoc,axiom,
% 2.66/1.54 ! [A: nat,B: nat,C: nat] :
% 2.66/1.54 ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 2.66/1.54 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_135_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 2.66/1.54 ! [A: nat,B: nat,C: nat] :
% 2.66/1.54 ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 2.66/1.54 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_136_zero__less__iff__neq__zero,axiom,
% 2.66/1.54 ! [N: nat] :
% 2.66/1.54 ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.66/1.54 = ( N != zero_zero_nat ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_137_gr__implies__not__zero,axiom,
% 2.66/1.54 ! [M: nat,N: nat] :
% 2.66/1.54 ( ( ord_less_nat @ M @ N )
% 2.66/1.54 => ( N != zero_zero_nat ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_138_not__less__zero,axiom,
% 2.66/1.54 ! [N: nat] :
% 2.66/1.54 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_139_gr__zeroI,axiom,
% 2.66/1.54 ! [N: nat] :
% 2.66/1.54 ( ( N != zero_zero_nat )
% 2.66/1.54 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_140_gr__implies__gr0,axiom,
% 2.66/1.54 ! [M: nat,N: nat] :
% 2.66/1.54 ( ( ord_less_nat @ M @ N )
% 2.66/1.54 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_141_nat__lessSucSuc0__conv,axiom,
% 2.66/1.54 ! [N: nat] :
% 2.66/1.54 ( ( ord_less_nat @ N @ ( suc @ ( suc @ zero_zero_nat ) ) )
% 2.66/1.54 = ( ( N = zero_zero_nat )
% 2.66/1.54 | ( N
% 2.66/1.54 = ( suc @ zero_zero_nat ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_142_nat__grSuc0__conv,axiom,
% 2.66/1.54 ! [N: nat] :
% 2.66/1.54 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 2.66/1.54 = ( ( N != zero_zero_nat )
% 2.66/1.54 & ( N
% 2.66/1.54 != ( suc @ zero_zero_nat ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_143_length__induct,axiom,
% 2.66/1.54 ! [P: list_P1467822637od_a_a > $o,Xs: list_P1467822637od_a_a] :
% 2.66/1.54 ( ! [Xs2: list_P1467822637od_a_a] :
% 2.66/1.54 ( ! [Ys: list_P1467822637od_a_a] :
% 2.66/1.54 ( ( ord_less_nat @ ( size_s642698393od_a_a @ Ys ) @ ( size_s642698393od_a_a @ Xs2 ) )
% 2.66/1.54 => ( P @ Ys ) )
% 2.66/1.54 => ( P @ Xs2 ) )
% 2.66/1.54 => ( P @ Xs ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_144_length__induct,axiom,
% 2.66/1.54 ! [P: list_a > $o,Xs: list_a] :
% 2.66/1.54 ( ! [Xs2: list_a] :
% 2.66/1.54 ( ! [Ys: list_a] :
% 2.66/1.54 ( ( ord_less_nat @ ( size_size_list_a @ Ys ) @ ( size_size_list_a @ Xs2 ) )
% 2.66/1.54 => ( P @ Ys ) )
% 2.66/1.54 => ( P @ Xs2 ) )
% 2.66/1.54 => ( P @ Xs ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_145_Ex__list__of__length,axiom,
% 2.66/1.54 ! [N: nat] :
% 2.66/1.54 ? [Xs2: list_P1467822637od_a_a] :
% 2.66/1.54 ( ( size_s642698393od_a_a @ Xs2 )
% 2.66/1.54 = N ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_146_Ex__list__of__length,axiom,
% 2.66/1.54 ! [N: nat] :
% 2.66/1.54 ? [Xs2: list_a] :
% 2.66/1.54 ( ( size_size_list_a @ Xs2 )
% 2.66/1.54 = N ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_147_neq__if__length__neq,axiom,
% 2.66/1.54 ! [Xs: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a] :
% 2.66/1.54 ( ( ( size_s642698393od_a_a @ Xs )
% 2.66/1.54 != ( size_s642698393od_a_a @ Ys2 ) )
% 2.66/1.54 => ( Xs != Ys2 ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_148_neq__if__length__neq,axiom,
% 2.66/1.54 ! [Xs: list_a,Ys2: list_a] :
% 2.66/1.54 ( ( ( size_size_list_a @ Xs )
% 2.66/1.54 != ( size_size_list_a @ Ys2 ) )
% 2.66/1.54 => ( Xs != Ys2 ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_149_list__decode_Ocases,axiom,
% 2.66/1.54 ! [X: nat] :
% 2.66/1.54 ( ( X != zero_zero_nat )
% 2.66/1.54 => ~ ! [N2: nat] :
% 2.66/1.54 ( X
% 2.66/1.54 != ( suc @ N2 ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_150_exists__least__lemma,axiom,
% 2.66/1.54 ! [P: nat > $o] :
% 2.66/1.54 ( ~ ( P @ zero_zero_nat )
% 2.66/1.54 => ( ? [X_1: nat] : ( P @ X_1 )
% 2.66/1.54 => ? [N2: nat] :
% 2.66/1.54 ( ~ ( P @ N2 )
% 2.66/1.54 & ( P @ ( suc @ N2 ) ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_151_less__numeral__extra_I3_J,axiom,
% 2.66/1.54 ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_152_length__product,axiom,
% 2.66/1.54 ! [Xs: list_a,Ys2: list_P1467822637od_a_a] :
% 2.66/1.54 ( ( size_s1875148994od_a_a @ ( produc16441326od_a_a @ Xs @ Ys2 ) )
% 2.66/1.54 = ( times_times_nat @ ( size_size_list_a @ Xs ) @ ( size_s642698393od_a_a @ Ys2 ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_153_length__product,axiom,
% 2.66/1.54 ! [Xs: list_P1467822637od_a_a,Ys2: list_a] :
% 2.66/1.54 ( ( size_s736232240_a_a_a @ ( produc778776796_a_a_a @ Xs @ Ys2 ) )
% 2.66/1.54 = ( times_times_nat @ ( size_s642698393od_a_a @ Xs ) @ ( size_size_list_a @ Ys2 ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_154_length__product,axiom,
% 2.66/1.54 ! [Xs: list_P1467822637od_a_a,Ys2: list_P1467822637od_a_a] :
% 2.66/1.54 ( ( size_s1052379225od_a_a @ ( produc1055075589od_a_a @ Xs @ Ys2 ) )
% 2.66/1.54 = ( times_times_nat @ ( size_s642698393od_a_a @ Xs ) @ ( size_s642698393od_a_a @ Ys2 ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_155_length__product,axiom,
% 2.66/1.54 ! [Xs: list_a,Ys2: list_a] :
% 2.66/1.54 ( ( size_s642698393od_a_a @ ( product_a_a @ Xs @ Ys2 ) )
% 2.66/1.54 = ( times_times_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys2 ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_156_split__div_H,axiom,
% 2.66/1.54 ! [P: nat > $o,M: nat,N: nat] :
% 2.66/1.54 ( ( P @ ( divide_divide_nat @ M @ N ) )
% 2.66/1.54 = ( ( ( N = zero_zero_nat )
% 2.66/1.54 & ( P @ zero_zero_nat ) )
% 2.66/1.54 | ? [Q2: nat] :
% 2.66/1.54 ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q2 ) @ M )
% 2.66/1.54 & ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q2 ) ) )
% 2.66/1.54 & ( P @ Q2 ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_157_le__less__div__conv,axiom,
% 2.66/1.54 ! [M: nat,K: nat,N: nat] :
% 2.66/1.54 ( ( ord_less_nat @ zero_zero_nat @ M )
% 2.66/1.54 => ( ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ N )
% 2.66/1.54 & ( ord_less_nat @ N @ ( times_times_nat @ ( suc @ K ) @ M ) ) )
% 2.66/1.54 = ( ( divide_divide_nat @ N @ M )
% 2.66/1.54 = K ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_158_le__zero__eq,axiom,
% 2.66/1.54 ! [N: nat] :
% 2.66/1.54 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 2.66/1.54 = ( N = zero_zero_nat ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_159_bot__nat__0_Oextremum,axiom,
% 2.66/1.54 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_160_le0,axiom,
% 2.66/1.54 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_161_Suc__le__mono,axiom,
% 2.66/1.54 ! [N: nat,M: nat] :
% 2.66/1.54 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
% 2.66/1.54 = ( ord_less_eq_nat @ N @ M ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_162_one__le__mult__iff,axiom,
% 2.66/1.54 ! [M: nat,N: nat] :
% 2.66/1.54 ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
% 2.66/1.54 = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
% 2.66/1.54 & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_163_mult__le__cancel2,axiom,
% 2.66/1.54 ! [M: nat,K: nat,N: nat] :
% 2.66/1.54 ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 2.66/1.54 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 2.66/1.54 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_164_nat__mult__le__cancel__disj,axiom,
% 2.66/1.54 ! [K: nat,M: nat,N: nat] :
% 2.66/1.54 ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 2.66/1.54 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 2.66/1.54 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_165_le__numeral__extra_I3_J,axiom,
% 2.66/1.54 ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 2.66/1.54
% 2.66/1.54 thf(fact_166_zero__le,axiom,
% 2.66/1.54 ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_167_ge__less__neq__conv,axiom,
% 2.66/1.54 ( ord_less_eq_nat
% 2.66/1.54 = ( ^ [A2: nat,N4: nat] :
% 2.66/1.54 ! [X4: nat] :
% 2.66/1.54 ( ( ord_less_nat @ X4 @ A2 )
% 2.66/1.54 => ( N4 != X4 ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_168_less__ge__neq__conv,axiom,
% 2.66/1.54 ( ord_less_nat
% 2.66/1.54 = ( ^ [N4: nat,A2: nat] :
% 2.66/1.54 ! [X4: nat] :
% 2.66/1.54 ( ( ord_less_eq_nat @ A2 @ X4 )
% 2.66/1.54 => ( N4 != X4 ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_169_greater__le__neq__conv,axiom,
% 2.66/1.54 ( ord_less_nat
% 2.66/1.54 = ( ^ [A2: nat,N4: nat] :
% 2.66/1.54 ! [X4: nat] :
% 2.66/1.54 ( ( ord_less_eq_nat @ X4 @ A2 )
% 2.66/1.54 => ( N4 != X4 ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_170_le__greater__neq__conv,axiom,
% 2.66/1.54 ( ord_less_eq_nat
% 2.66/1.54 = ( ^ [N4: nat,A2: nat] :
% 2.66/1.54 ! [X4: nat] :
% 2.66/1.54 ( ( ord_less_nat @ A2 @ X4 )
% 2.66/1.54 => ( N4 != X4 ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_171_nat__geSucSuc0__conv,axiom,
% 2.66/1.54 ! [N: nat] :
% 2.66/1.54 ( ( ord_less_eq_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ N )
% 2.66/1.54 = ( ( N != zero_zero_nat )
% 2.66/1.54 & ( N
% 2.66/1.54 != ( suc @ zero_zero_nat ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_172_nat__leSuc0__conv,axiom,
% 2.66/1.54 ! [N: nat] :
% 2.66/1.54 ( ( ord_less_eq_nat @ N @ ( suc @ zero_zero_nat ) )
% 2.66/1.54 = ( ( N = zero_zero_nat )
% 2.66/1.54 | ( N
% 2.66/1.54 = ( suc @ zero_zero_nat ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_173_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 2.66/1.54 ! [A: nat,B: nat,C: nat] :
% 2.66/1.54 ( ( ord_less_eq_nat @ A @ B )
% 2.66/1.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 2.66/1.54 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_174_mult__nonneg__nonpos2,axiom,
% 2.66/1.54 ! [A: nat,B: nat] :
% 2.66/1.54 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 2.66/1.54 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 2.66/1.54 => ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_175_mult__nonpos__nonneg,axiom,
% 2.66/1.54 ! [A: nat,B: nat] :
% 2.66/1.54 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 2.66/1.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 2.66/1.54 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_176_mult__nonneg__nonpos,axiom,
% 2.66/1.54 ! [A: nat,B: nat] :
% 2.66/1.54 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 2.66/1.54 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 2.66/1.54 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_177_mult__nonneg__nonneg,axiom,
% 2.66/1.54 ! [A: nat,B: nat] :
% 2.66/1.54 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 2.66/1.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 2.66/1.54 => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_178_split__mult__neg__le,axiom,
% 2.66/1.54 ! [A: nat,B: nat] :
% 2.66/1.54 ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 2.66/1.54 & ( ord_less_eq_nat @ B @ zero_zero_nat ) )
% 2.66/1.54 | ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 2.66/1.54 & ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
% 2.66/1.54 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_179_mult__right__mono,axiom,
% 2.66/1.54 ! [A: nat,B: nat,C: nat] :
% 2.66/1.54 ( ( ord_less_eq_nat @ A @ B )
% 2.66/1.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 2.66/1.54 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_180_mult__left__mono,axiom,
% 2.66/1.54 ! [A: nat,B: nat,C: nat] :
% 2.66/1.54 ( ( ord_less_eq_nat @ A @ B )
% 2.66/1.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 2.66/1.54 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_181_mult__mono_H,axiom,
% 2.66/1.54 ! [A: nat,B: nat,C: nat,D: nat] :
% 2.66/1.54 ( ( ord_less_eq_nat @ A @ B )
% 2.66/1.54 => ( ( ord_less_eq_nat @ C @ D )
% 2.66/1.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 2.66/1.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 2.66/1.54 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_182_mult__mono,axiom,
% 2.66/1.54 ! [A: nat,B: nat,C: nat,D: nat] :
% 2.66/1.54 ( ( ord_less_eq_nat @ A @ B )
% 2.66/1.54 => ( ( ord_less_eq_nat @ C @ D )
% 2.66/1.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 2.66/1.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 2.66/1.54 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(conj_0,hypothesis,
% 2.66/1.54 ( ( divide_divide_nat @ ( size_size_list_a @ xs ) @ k )
% 2.66/1.54 = ( suc @ n ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(conj_1,hypothesis,
% 2.66/1.54 k != zero_zero_nat ).
% 2.66/1.54
% 2.66/1.54 thf(conj_2,conjecture,
% 2.66/1.54 ord_less_nat @ ( times_times_nat @ n @ k ) @ ( size_size_list_a @ xs ) ).
% 2.66/1.54 ------------------------
% 2.66/1.57 % (7695)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_7532 for (2999ds/2Mi)
% 2.66/1.57 % (7693)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_7532 for (2999ds/27Mi)
% 2.66/1.57 % (7697)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_7532 for (2999ds/18Mi)
% 2.66/1.57 % (7691)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_7532 for (2999ds/183Mi)
% 2.66/1.57 % (7696)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_7532 for (2999ds/275Mi)
% 2.66/1.57 % (7692)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_7532 for (2999ds/4Mi)
% 2.66/1.57 % (7694)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_7532 for (2999ds/2Mi)
% 2.66/1.57 % (7694)Instruction limit reached!
% 2.66/1.57 % (7694)------------------------------
% 2.66/1.57 % (7694)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.57 % (7694)Termination reason: Unknown
% 2.66/1.57 % (7694)Termination phase: shuffling
% 2.66/1.57
% 2.66/1.57 % (7694)Memory used [KB]: 1151
% 2.66/1.57 % (7694)Time elapsed: 0.003 s
% 2.66/1.57 % (7694)Instructions burned: 2 (million)
% 2.66/1.57 % (7694)------------------------------
% 2.66/1.57 % (7694)------------------------------
% 2.66/1.57 % (7695)Instruction limit reached!
% 2.66/1.57 % (7695)------------------------------
% 2.66/1.57 % (7695)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.57 % (7695)Termination reason: Unknown
% 2.66/1.57 % (7695)Termination phase: shuffling
% 2.66/1.57
% 2.66/1.57 % (7695)Memory used [KB]: 1151
% 2.66/1.57 % (7695)Time elapsed: 0.003 s
% 2.66/1.57 % (7695)Instructions burned: 2 (million)
% 2.66/1.57 % (7695)------------------------------
% 2.66/1.57 % (7695)------------------------------
% 2.66/1.57 % (7692)Instruction limit reached!
% 2.66/1.57 % (7692)------------------------------
% 2.66/1.57 % (7692)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.57 % (7692)Termination reason: Unknown
% 2.66/1.57 % (7692)Termination phase: shuffling
% 2.66/1.57
% 2.66/1.57 % (7692)Memory used [KB]: 1151
% 2.66/1.57 % (7692)Time elapsed: 0.004 s
% 2.66/1.57 % (7692)Instructions burned: 4 (million)
% 2.66/1.57 % (7692)------------------------------
% 2.66/1.57 % (7692)------------------------------
% 2.66/1.58 % (7697)Instruction limit reached!
% 2.66/1.58 % (7697)------------------------------
% 2.66/1.58 % (7697)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.58 % (7697)Termination reason: Unknown
% 2.66/1.58 % (7697)Termination phase: Preprocessing 2
% 2.66/1.58
% 2.66/1.58 % (7697)Memory used [KB]: 1407
% 2.66/1.58 % (7697)Time elapsed: 0.011 s
% 2.66/1.58 % (7697)Instructions burned: 18 (million)
% 2.66/1.58 % (7697)------------------------------
% 2.66/1.58 % (7697)------------------------------
% 2.66/1.58 % (7693)Instruction limit reached!
% 2.66/1.58 % (7693)------------------------------
% 2.66/1.58 % (7693)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.58 % (7693)Termination reason: Unknown
% 2.66/1.58 % (7693)Termination phase: Property scanning
% 2.66/1.58
% 2.66/1.58 % (7693)Memory used [KB]: 1535
% 2.66/1.58 % (7693)Time elapsed: 0.016 s
% 2.66/1.58 % (7693)Instructions burned: 28 (million)
% 2.66/1.58 % (7693)------------------------------
% 2.66/1.58 % (7693)------------------------------
% 2.66/1.59 % (7698)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_7532 for (2999ds/3Mi)
% 2.66/1.59 % (7699)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_7532 for (2999ds/37Mi)
% 2.66/1.59 % (7700)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_7532 for (2999ds/15Mi)
% 2.66/1.59 % (7698)Instruction limit reached!
% 2.66/1.59 % (7698)------------------------------
% 2.66/1.59 % (7698)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.59 % (7698)Termination reason: Unknown
% 2.66/1.59 % (7698)Termination phase: shuffling
% 2.66/1.59
% 2.66/1.59 % (7698)Memory used [KB]: 1151
% 2.66/1.59 % (7698)Time elapsed: 0.003 s
% 2.66/1.59 % (7698)Instructions burned: 3 (million)
% 2.66/1.59 % (7698)------------------------------
% 2.66/1.59 % (7698)------------------------------
% 2.66/1.59 % (7700)Instruction limit reached!
% 2.66/1.59 % (7700)------------------------------
% 2.66/1.59 % (7700)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.60 % (7700)Termination reason: Unknown
% 2.66/1.60 % (7700)Termination phase: Property scanning
% 2.66/1.60
% 2.66/1.60 % (7700)Memory used [KB]: 1279
% 2.66/1.60 % (7700)Time elapsed: 0.009 s
% 2.66/1.60 % (7700)Instructions burned: 15 (million)
% 2.66/1.60 % (7700)------------------------------
% 2.66/1.60 % (7700)------------------------------
% 2.66/1.60 % (7701)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_7532 for (2999ds/3Mi)
% 2.71/1.60 % (7701)Instruction limit reached!
% 2.71/1.60 % (7701)------------------------------
% 2.71/1.60 % (7701)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.60 % (7701)Termination reason: Unknown
% 2.71/1.60 % (7701)Termination phase: shuffling
% 2.71/1.60
% 2.71/1.60 % (7701)Memory used [KB]: 1151
% 2.71/1.60 % (7701)Time elapsed: 0.003 s
% 2.71/1.60 % (7701)Instructions burned: 3 (million)
% 2.71/1.60 % (7701)------------------------------
% 2.71/1.60 % (7701)------------------------------
% 2.71/1.60 % (7702)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_7532 for (2999ds/1041Mi)
% 2.71/1.60 % (7699)Instruction limit reached!
% 2.71/1.60 % (7699)------------------------------
% 2.71/1.60 % (7699)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.60 % (7699)Termination reason: Unknown
% 2.71/1.60 % (7699)Termination phase: Property scanning
% 2.71/1.60
% 2.71/1.60 % (7699)Memory used [KB]: 1663
% 2.71/1.60 % (7699)Time elapsed: 0.019 s
% 2.71/1.60 % (7699)Instructions burned: 38 (million)
% 2.71/1.60 % (7699)------------------------------
% 2.71/1.60 % (7699)------------------------------
% 2.71/1.61 % (7703)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_7532 for (2999ds/7Mi)
% 2.71/1.61 % (7703)Instruction limit reached!
% 2.71/1.61 % (7703)------------------------------
% 2.71/1.61 % (7703)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.61 % (7703)Termination reason: Unknown
% 2.71/1.61 % (7703)Termination phase: shuffling
% 2.71/1.61
% 2.71/1.61 % (7703)Memory used [KB]: 1279
% 2.71/1.61 % (7703)Time elapsed: 0.005 s
% 2.71/1.61 % (7703)Instructions burned: 7 (million)
% 2.71/1.61 % (7703)------------------------------
% 2.71/1.61 % (7703)------------------------------
% 2.71/1.61 % (7704)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_7532 for (2999ds/16Mi)
% 2.71/1.61 % (7705)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_7532 for (2999ds/3Mi)
% 2.71/1.62 % (7705)Instruction limit reached!
% 2.71/1.62 % (7705)------------------------------
% 2.71/1.62 % (7705)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.62 % (7705)Termination reason: Unknown
% 2.71/1.62 % (7705)Termination phase: shuffling
% 2.71/1.62
% 2.71/1.62 % (7705)Memory used [KB]: 1151
% 2.71/1.62 % (7705)Time elapsed: 0.003 s
% 2.71/1.62 % (7705)Instructions burned: 3 (million)
% 2.71/1.62 % (7705)------------------------------
% 2.71/1.62 % (7705)------------------------------
% 2.71/1.62 % (7704)Instruction limit reached!
% 2.71/1.62 % (7704)------------------------------
% 2.71/1.62 % (7704)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.62 % (7704)Termination reason: Unknown
% 2.71/1.62 % (7704)Termination phase: Property scanning
% 2.71/1.62
% 2.71/1.62 % (7704)Memory used [KB]: 1279
% 2.71/1.62 % (7704)Time elapsed: 0.010 s
% 2.71/1.62 % (7704)Instructions burned: 17 (million)
% 2.71/1.62 % (7704)------------------------------
% 2.71/1.62 % (7704)------------------------------
% 2.71/1.62 % (7706)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_7532 for (2999ds/3Mi)
% 2.71/1.62 % (7706)Instruction limit reached!
% 2.71/1.62 % (7706)------------------------------
% 2.71/1.62 % (7706)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.62 % (7706)Termination reason: Unknown
% 2.71/1.62 % (7706)Termination phase: shuffling
% 2.71/1.62
% 2.71/1.62 % (7706)Memory used [KB]: 1279
% 2.71/1.62 % (7706)Time elapsed: 0.004 s
% 2.71/1.62 % (7706)Instructions burned: 4 (million)
% 2.71/1.62 % (7706)------------------------------
% 2.71/1.62 % (7706)------------------------------
% 2.71/1.63 % (7707)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_7532 for (2999ds/7Mi)
% 2.71/1.63 % (7707)Instruction limit reached!
% 2.71/1.63 % (7707)------------------------------
% 2.71/1.63 % (7707)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.63 % (7707)Termination reason: Unknown
% 2.71/1.63 % (7707)Termination phase: shuffling
% 2.71/1.63
% 2.71/1.63 % (7707)Memory used [KB]: 1279
% 2.71/1.63 % (7707)Time elapsed: 0.006 s
% 2.71/1.63 % (7707)Instructions burned: 8 (million)
% 2.71/1.63 % (7707)------------------------------
% 2.71/1.63 % (7707)------------------------------
% 2.71/1.63 % (7708)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_7532 for (2999ds/3Mi)
% 2.71/1.63 % (7708)Instruction limit reached!
% 2.71/1.63 % (7708)------------------------------
% 2.71/1.63 % (7708)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.63 % (7708)Termination reason: Unknown
% 2.71/1.63 % (7708)Termination phase: shuffling
% 2.71/1.63
% 2.71/1.63 % (7708)Memory used [KB]: 1151
% 2.71/1.63 % (7708)Time elapsed: 0.003 s
% 2.71/1.63 % (7708)Instructions burned: 3 (million)
% 2.71/1.63 % (7708)------------------------------
% 2.71/1.63 % (7708)------------------------------
% 2.71/1.64 % (7709)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_7532 for (2999ds/4Mi)
% 2.71/1.64 % (7709)Instruction limit reached!
% 2.71/1.64 % (7709)------------------------------
% 2.71/1.64 % (7709)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.64 % (7709)Termination reason: Unknown
% 2.71/1.64 % (7709)Termination phase: shuffling
% 2.71/1.64
% 2.71/1.64 % (7709)Memory used [KB]: 1279
% 2.71/1.64 % (7709)Time elapsed: 0.004 s
% 2.71/1.64 % (7709)Instructions burned: 5 (million)
% 2.71/1.64 % (7709)------------------------------
% 2.71/1.64 % (7709)------------------------------
% 2.71/1.64 % (7710)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_7532 for (2999ds/18Mi)
% 2.71/1.65 % (7711)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_7532 for (2999ds/710Mi)
% 2.71/1.65 % (7710)Instruction limit reached!
% 2.71/1.65 % (7710)------------------------------
% 2.71/1.65 % (7710)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.65 % (7710)Termination reason: Unknown
% 2.71/1.65 % (7710)Termination phase: Preprocessing 1
% 2.71/1.65
% 2.71/1.65 % (7710)Memory used [KB]: 1279
% 2.71/1.65 % (7710)Time elapsed: 0.011 s
% 2.71/1.65 % (7710)Instructions burned: 19 (million)
% 2.71/1.65 % (7710)------------------------------
% 2.71/1.65 % (7710)------------------------------
% 2.71/1.65 % (7712)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_7532 for (2999ds/6Mi)
% 2.71/1.65 % (7712)Instruction limit reached!
% 2.71/1.65 % (7712)------------------------------
% 2.71/1.65 % (7712)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.65 % (7712)Termination reason: Unknown
% 2.71/1.65 % (7712)Termination phase: shuffling
% 2.71/1.65
% 2.71/1.65 % (7712)Memory used [KB]: 1279
% 2.71/1.65 % (7712)Time elapsed: 0.005 s
% 2.71/1.65 % (7712)Instructions burned: 6 (million)
% 2.71/1.65 % (7712)------------------------------
% 2.71/1.65 % (7712)------------------------------
% 2.71/1.66 % (7713)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_7532 for (2999ds/902Mi)
% 2.71/1.66 % (7691)Instruction limit reached!
% 2.71/1.66 % (7691)------------------------------
% 2.71/1.66 % (7691)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.66 % (7691)Termination reason: Unknown
% 2.71/1.66 % (7691)Termination phase: Saturation
% 2.71/1.66
% 2.71/1.66 % (7691)Memory used [KB]: 7164
% 2.71/1.66 % (7691)Time elapsed: 0.094 s
% 2.71/1.66 % (7691)Instructions burned: 183 (million)
% 2.71/1.66 % (7691)------------------------------
% 2.71/1.66 % (7691)------------------------------
% 2.71/1.67 % (7714)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_7532 for (2998ds/21Mi)
% 2.71/1.67 % (7715)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_7532 for (2998ds/5Mi)
% 2.71/1.67 % (7715)Instruction limit reached!
% 2.71/1.67 % (7715)------------------------------
% 2.71/1.67 % (7715)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.67 % (7715)Termination reason: Unknown
% 2.71/1.67 % (7715)Termination phase: shuffling
% 2.71/1.67
% 2.71/1.67 % (7715)Memory used [KB]: 1279
% 2.71/1.67 % (7715)Time elapsed: 0.004 s
% 2.71/1.67 % (7715)Instructions burned: 5 (million)
% 2.71/1.67 % (7715)------------------------------
% 2.71/1.67 % (7715)------------------------------
% 2.71/1.68 % (7716)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_7532 for (2998ds/6Mi)
% 2.71/1.68 % (7714)Instruction limit reached!
% 2.71/1.68 % (7714)------------------------------
% 2.71/1.68 % (7714)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.68 % (7714)Termination reason: Unknown
% 2.71/1.68 % (7714)Termination phase: SInE selection
% 2.71/1.68
% 2.71/1.68 % (7714)Memory used [KB]: 1407
% 2.71/1.68 % (7714)Time elapsed: 0.012 s
% 2.71/1.68 % (7714)Instructions burned: 21 (million)
% 2.71/1.68 % (7714)------------------------------
% 2.71/1.68 % (7714)------------------------------
% 2.71/1.68 % (7716)Instruction limit reached!
% 2.71/1.68 % (7716)------------------------------
% 2.71/1.68 % (7716)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.68 % (7716)Termination reason: Unknown
% 2.71/1.68 % (7716)Termination phase: shuffling
% 2.71/1.68
% 2.71/1.68 % (7716)Memory used [KB]: 1279
% 2.71/1.68 % (7716)Time elapsed: 0.005 s
% 2.71/1.68 % (7716)Instructions burned: 7 (million)
% 2.71/1.68 % (7716)------------------------------
% 2.71/1.68 % (7716)------------------------------
% 2.71/1.69 % (7717)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_7532 for (2998ds/377Mi)
% 2.71/1.70 % (7718)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_7532 for (2998ds/779Mi)
% 2.71/1.70 % (7719)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_7532 for (2998ds/19Mi)
% 2.71/1.71 % (7719)Instruction limit reached!
% 2.71/1.71 % (7719)------------------------------
% 2.71/1.71 % (7719)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.71 % (7719)Termination reason: Unknown
% 2.71/1.71 % (7719)Termination phase: shuffling
% 2.71/1.71
% 2.71/1.71 % (7719)Memory used [KB]: 1535
% 2.71/1.71 % (7719)Time elapsed: 0.011 s
% 2.71/1.71 % (7719)Instructions burned: 19 (million)
% 2.71/1.71 % (7719)------------------------------
% 2.71/1.71 % (7719)------------------------------
% 2.71/1.72 % (7696)Instruction limit reached!
% 2.71/1.72 % (7696)------------------------------
% 2.71/1.72 % (7696)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.72 % (7696)Termination reason: Unknown
% 2.71/1.72 % (7696)Termination phase: Saturation
% 2.71/1.72
% 2.71/1.72 % (7696)Memory used [KB]: 6908
% 2.71/1.72 % (7696)Time elapsed: 0.152 s
% 2.71/1.72 % (7696)Instructions burned: 275 (million)
% 2.71/1.72 % (7696)------------------------------
% 2.71/1.72 % (7696)------------------------------
% 2.71/1.73 % (7720)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_7532 for (2998ds/879Mi)
% 2.71/1.74 % (7721)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_7532 for (2998ds/17Mi)
% 2.71/1.75 % (7721)Instruction limit reached!
% 2.71/1.75 % (7721)------------------------------
% 2.71/1.75 % (7721)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.75 % (7721)Termination reason: Unknown
% 2.71/1.75 % (7721)Termination phase: Property scanning
% 2.71/1.75
% 2.71/1.75 % (7721)Memory used [KB]: 1279
% 2.71/1.75 % (7721)Time elapsed: 0.010 s
% 2.71/1.75 % (7721)Instructions burned: 17 (million)
% 2.71/1.75 % (7721)------------------------------
% 2.71/1.75 % (7721)------------------------------
% 2.71/1.76 % (7722)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_7532 for (2997ds/3Mi)
% 2.71/1.77 % (7722)Instruction limit reached!
% 2.71/1.77 % (7722)------------------------------
% 2.71/1.77 % (7722)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.77 % (7722)Termination reason: Unknown
% 2.71/1.77 % (7722)Termination phase: shuffling
% 2.71/1.77
% 2.71/1.77 % (7722)Memory used [KB]: 1151
% 2.71/1.77 % (7722)Time elapsed: 0.004 s
% 2.71/1.77 % (7722)Instructions burned: 3 (million)
% 2.71/1.77 % (7722)------------------------------
% 2.71/1.77 % (7722)------------------------------
% 2.71/1.77 % (7713)First to succeed.
% 2.71/1.77 % (7713)Refutation found. Thanks to Tanya!
% 2.71/1.77 % SZS status Theorem for DTF2THF_7532
% 2.71/1.77 % SZS output start Proof for DTF2THF_7532
% 2.71/1.77 thf(type_def_5, type, nat: $tType).
% 2.71/1.77 thf(type_def_7, type, list_P1467822637od_a_a: $tType).
% 2.71/1.77 thf(type_def_8, type, list_P1891567085od_a_a: $tType).
% 2.71/1.77 thf(type_def_9, type, list_a: $tType).
% 2.71/1.77 thf(type_def_10, type, list_P319927492_a_a_a: $tType).
% 2.71/1.77 thf(type_def_11, type, list_P1458844246od_a_a: $tType).
% 2.71/1.77 thf(func_def_0, type, list_P1891567085od_a_a: $tType).
% 2.71/1.77 thf(func_def_1, type, list_P1458844246od_a_a: $tType).
% 2.71/1.77 thf(func_def_2, type, list_P319927492_a_a_a: $tType).
% 2.71/1.77 thf(func_def_3, type, list_P1467822637od_a_a: $tType).
% 2.71/1.77 thf(func_def_4, type, list_a: $tType).
% 2.71/1.77 thf(func_def_5, type, nat: $tType).
% 2.71/1.77 thf(func_def_6, type, times_times_nat: nat > nat > nat).
% 2.71/1.77 thf(func_def_7, type, zero_zero_nat: nat).
% 2.71/1.77 thf(func_def_8, type, produc1055075589od_a_a: list_P1467822637od_a_a > list_P1467822637od_a_a > list_P1891567085od_a_a).
% 2.71/1.77 thf(func_def_9, type, produc778776796_a_a_a: list_P1467822637od_a_a > list_a > list_P319927492_a_a_a).
% 2.71/1.77 thf(func_def_10, type, produc16441326od_a_a: list_a > list_P1467822637od_a_a > list_P1458844246od_a_a).
% 2.71/1.77 thf(func_def_11, type, product_a_a: list_a > list_a > list_P1467822637od_a_a).
% 2.71/1.77 thf(func_def_12, type, suc: nat > nat).
% 2.71/1.77 thf(func_def_13, type, size_s1052379225od_a_a: list_P1891567085od_a_a > nat).
% 2.71/1.77 thf(func_def_14, type, size_s736232240_a_a_a: list_P319927492_a_a_a > nat).
% 2.71/1.77 thf(func_def_15, type, size_s1875148994od_a_a: list_P1458844246od_a_a > nat).
% 2.71/1.77 thf(func_def_16, type, size_s642698393od_a_a: list_P1467822637od_a_a > nat).
% 2.71/1.77 thf(func_def_17, type, size_size_list_a: list_a > nat).
% 2.71/1.77 thf(func_def_18, type, ord_less_nat: nat > nat > $o).
% 2.71/1.77 thf(func_def_19, type, ord_less_eq_nat: nat > nat > $o).
% 2.71/1.77 thf(func_def_20, type, divide_divide_nat: nat > nat > nat).
% 2.71/1.77 thf(func_def_21, type, k: nat).
% 2.71/1.77 thf(func_def_22, type, n: nat).
% 2.71/1.77 thf(func_def_23, type, xs: list_a).
% 2.71/1.77 thf(func_def_35, type, sK0: nat > nat > nat).
% 2.71/1.77 thf(func_def_36, type, sK1: (nat > nat > $o) > nat).
% 2.71/1.77 thf(func_def_37, type, sK2: (nat > nat > $o) > nat).
% 2.71/1.77 thf(func_def_38, type, sK3: (nat > nat > $o) > nat).
% 2.71/1.77 thf(func_def_39, type, sK4: (nat > nat > $o) > nat).
% 2.71/1.77 thf(func_def_40, type, sK5: (nat > $o) > nat > nat > nat).
% 2.71/1.77 thf(func_def_41, type, sK6: nat > nat > nat).
% 2.71/1.77 thf(func_def_42, type, sK7: (nat > $o) > nat).
% 2.71/1.77 thf(func_def_43, type, sK8: nat > (nat > $o) > nat).
% 2.71/1.77 thf(func_def_44, type, sK9: nat > (nat > $o) > nat).
% 2.71/1.77 thf(func_def_45, type, sK10: (nat > $o) > nat).
% 2.71/1.77 thf(func_def_46, type, sK11: (list_P1467822637od_a_a > $o) > list_P1467822637od_a_a).
% 2.71/1.77 thf(func_def_47, type, sK12: nat > nat).
% 2.71/1.77 thf(func_def_48, type, sK13: (nat > $o) > nat).
% 2.71/1.77 thf(func_def_49, type, sK14: (nat > $o) > nat).
% 2.71/1.77 thf(func_def_50, type, sK15: nat > nat > nat).
% 2.71/1.77 thf(func_def_51, type, sK16: nat > nat).
% 2.71/1.77 thf(func_def_52, type, sK17: (nat > $o) > nat).
% 2.71/1.77 thf(func_def_53, type, sK18: nat > list_P1467822637od_a_a).
% 2.71/1.77 thf(func_def_54, type, sK19: nat > (nat > $o) > nat).
% 2.71/1.77 thf(func_def_55, type, sK20: nat > (nat > $o) > nat).
% 2.71/1.77 thf(func_def_56, type, sK21: (nat > $o) > nat > nat).
% 2.71/1.77 thf(func_def_57, type, sK22: (nat > $o) > nat > nat).
% 2.71/1.77 thf(func_def_58, type, sK23: nat > nat > nat).
% 2.71/1.77 thf(func_def_59, type, sK24: nat > nat).
% 2.71/1.77 thf(func_def_60, type, sK25: (nat > nat > $o) > nat).
% 2.71/1.77 thf(func_def_61, type, sK26: (nat > nat > $o) > nat).
% 2.71/1.77 thf(func_def_62, type, sK27: (nat > nat > $o) > nat).
% 2.71/1.77 thf(func_def_63, type, sK28: (nat > nat > $o) > nat).
% 2.71/1.77 thf(func_def_64, type, sK29: nat > (nat > $o) > nat).
% 2.71/1.77 thf(func_def_65, type, sK30: nat > (nat > $o) > nat).
% 2.71/1.77 thf(func_def_66, type, sK31: nat > nat).
% 2.71/1.77 thf(func_def_67, type, sK32: nat > nat).
% 2.71/1.77 thf(func_def_68, type, sK33: (list_a > $o) > list_a).
% 2.71/1.77 thf(func_def_69, type, sK34: (nat > $o) > nat > nat).
% 2.71/1.77 thf(func_def_70, type, sK35: (nat > $o) > nat > nat).
% 2.71/1.77 thf(func_def_71, type, sK36: nat > list_a).
% 2.71/1.77 thf(func_def_72, type, sK37: (nat > nat) > nat).
% 2.71/1.77 thf(func_def_73, type, sK38: (nat > $o) > nat).
% 2.71/1.77 thf(func_def_74, type, sK39: (nat > $o) > nat).
% 2.71/1.77 thf(func_def_75, type, sK40: (nat > nat) > nat).
% 2.71/1.77 thf(func_def_77, type, ph42: !>[X0: $tType]:(X0)).
% 2.71/1.77 thf(f2970,plain,(
% 2.71/1.77 $false),
% 2.71/1.77 inference(avatar_sat_refutation,[status(thm)],[f1273,f2156,f2365,f2383,f2675,f2949])).
% 2.71/1.77 thf(f2949,plain,(
% 2.71/1.77 ~spl41_28),
% 2.71/1.77 inference(avatar_contradiction_clause,[status(thm)],[f2948])).
% 2.71/1.77 thf(f2948,plain,(
% 2.71/1.77 $false | ~spl41_28),
% 2.71/1.77 inference(trivial_inequality_removal,[status(thm)],[f2911])).
% 2.71/1.77 thf(f2911,plain,(
% 2.71/1.77 (n != n) | ~spl41_28),
% 2.71/1.77 inference(superposition,[status(thm)],[f1029,f2834])).
% 2.71/1.77 thf(f2834,plain,(
% 2.71/1.77 (n = (suc @ n)) | ~spl41_28),
% 2.71/1.77 inference(superposition,[status(thm)],[f1468,f923])).
% 2.71/1.77 thf(f923,plain,(
% 2.71/1.77 ((divide_divide_nat @ (size_size_list_a @ xs) @ k) = (suc @ n))),
% 2.71/1.77 inference(cnf_transformation,[status(thm)],[f184])).
% 2.71/1.77 thf(f184,axiom,(
% 2.71/1.77 ((divide_divide_nat @ (size_size_list_a @ xs) @ k) = (suc @ n))),
% 2.71/1.77 file('/export/starexec/sandbox/tmp/tmp.2CmgJp5TsA/DTF2THF_7532.p',conj_0)).
% 2.71/1.77 thf(f1468,plain,(
% 2.71/1.77 ((divide_divide_nat @ (size_size_list_a @ xs) @ k) = n) | ~spl41_28),
% 2.71/1.77 inference(avatar_component_clause,[status(thm)],[f1466])).
% 2.71/1.77 thf(f1466,definition,(
% 2.71/1.77 spl41_28 <=> ((divide_divide_nat @ (size_size_list_a @ xs) @ k) = n)),
% 2.71/1.77 introduced(definition,[new_symbols(naming,[spl41_28])],[avatar_definition])).
% 2.71/1.77 thf(f1029,plain,(
% 2.71/1.77 ( ! [X0 : nat] : (((suc @ X0) != X0)) )),
% 2.71/1.77 inference(cnf_transformation,[status(thm)],[f28])).
% 2.71/1.77 thf(f28,axiom,(
% 2.71/1.77 ! [X0 : nat] : ((suc @ X0) != X0)),
% 2.71/1.77 file('/export/starexec/sandbox/tmp/tmp.2CmgJp5TsA/DTF2THF_7532.p',fact_27_n__not__Suc__n)).
% 2.71/1.77 thf(f2675,plain,(
% 2.71/1.77 spl41_28 | spl41_17 | ~spl41_8),
% 2.71/1.77 inference(avatar_split_clause,[status(thm)],[f2467,f1256,f1416,f1466])).
% 2.71/1.77 thf(f1416,definition,(
% 2.71/1.77 spl41_17 <=> (zero_zero_nat = k)),
% 2.71/1.77 introduced(definition,[new_symbols(naming,[spl41_17])],[avatar_definition])).
% 2.71/1.77 thf(f1256,definition,(
% 2.71/1.77 spl41_8 <=> ((size_size_list_a @ xs) = (times_times_nat @ n @ k))),
% 2.71/1.77 introduced(definition,[new_symbols(naming,[spl41_8])],[avatar_definition])).
% 2.71/1.77 thf(f2467,plain,(
% 2.71/1.77 (zero_zero_nat = k) | ((divide_divide_nat @ (size_size_list_a @ xs) @ k) = n) | ~spl41_8),
% 2.71/1.77 inference(superposition,[status(thm)],[f1028,f1258])).
% 2.71/1.77 thf(f1258,plain,(
% 2.71/1.77 ((size_size_list_a @ xs) = (times_times_nat @ n @ k)) | ~spl41_8),
% 2.71/1.77 inference(avatar_component_clause,[status(thm)],[f1256])).
% 2.71/1.77 thf(f1028,plain,(
% 2.71/1.77 ( ! [X0 : nat,X1 : nat] : (((divide_divide_nat @ (times_times_nat @ X0 @ X1) @ X1) = X0) | (zero_zero_nat = X1)) )),
% 2.71/1.77 inference(cnf_transformation,[status(thm)],[f556])).
% 2.71/1.77 thf(f556,plain,(
% 2.71/1.77 ! [X0 : nat,X1 : nat] : ((zero_zero_nat = X1) | ((divide_divide_nat @ (times_times_nat @ X0 @ X1) @ X1) = X0))),
% 2.71/1.77 inference(ennf_transformation,[status(thm)],[f503])).
% 2.71/1.77 thf(f503,plain,(
% 2.71/1.77 ! [X0 : nat,X1 : nat] : ((zero_zero_nat != X1) => ((divide_divide_nat @ (times_times_nat @ X0 @ X1) @ X1) = X0))),
% 2.71/1.77 inference(rectify,[status(thm)],[f99])).
% 2.71/1.77 thf(f99,axiom,(
% 2.71/1.77 ! [X7 : nat,X9 : nat] : ((zero_zero_nat != X9) => ((divide_divide_nat @ (times_times_nat @ X7 @ X9) @ X9) = X7))),
% 2.71/1.77 file('/export/starexec/sandbox/tmp/tmp.2CmgJp5TsA/DTF2THF_7532.p',fact_98_nonzero__mult__div__cancel__right)).
% 2.71/1.77 thf(f2383,plain,(
% 2.71/1.77 ~spl41_64),
% 2.71/1.77 inference(avatar_contradiction_clause,[status(thm)],[f2382])).
% 2.71/1.77 thf(f2382,plain,(
% 2.71/1.77 $false | ~spl41_64),
% 2.71/1.77 inference(trivial_inequality_removal,[status(thm)],[f2367])).
% 2.71/1.77 thf(f2367,plain,(
% 2.71/1.77 ($true != $true) | ~spl41_64),
% 2.71/1.77 inference(superposition,[status(thm)],[f1139,f1808])).
% 2.71/1.77 thf(f1808,plain,(
% 2.71/1.77 ((ord_less_nat @ n @ n) = $true) | ~spl41_64),
% 2.71/1.77 inference(avatar_component_clause,[status(thm)],[f1806])).
% 2.71/1.77 thf(f1806,definition,(
% 2.71/1.77 spl41_64 <=> ((ord_less_nat @ n @ n) = $true)),
% 2.71/1.77 introduced(definition,[new_symbols(naming,[spl41_64])],[avatar_definition])).
% 2.71/1.77 thf(f1139,plain,(
% 2.71/1.77 ( ! [X1 : nat] : (((ord_less_nat @ X1 @ X1) != $true)) )),
% 2.71/1.77 inference(equality_resolution,[status(thm)],[f906])).
% 2.71/1.77 thf(f906,plain,(
% 2.71/1.77 ( ! [X0 : nat,X1 : nat] : ((X0 != X1) | ((ord_less_nat @ X0 @ X1) != $true)) )),
% 2.71/1.77 inference(cnf_transformation,[status(thm)],[f646])).
% 2.71/1.77 thf(f646,plain,(
% 2.71/1.77 ! [X0 : nat,X1 : nat] : ((X0 != X1) | ((ord_less_nat @ X0 @ X1) != $true))),
% 2.71/1.77 inference(ennf_transformation,[status(thm)],[f424])).
% 2.71/1.77 thf(f424,plain,(
% 2.71/1.77 ! [X0 : nat,X1 : nat] : (((ord_less_nat @ X0 @ X1) = $true) => (X0 != X1))),
% 2.71/1.77 inference(fool_elimination,[status(thm)],[f423])).
% 2.71/1.77 thf(f423,plain,(
% 2.71/1.77 ! [X0 : nat,X1 : nat] : ((ord_less_nat @ X0 @ X1) => (X0 != X1))),
% 2.71/1.77 inference(rectify,[status(thm)],[f35])).
% 2.71/1.77 thf(f35,axiom,(
% 2.71/1.77 ! [X0 : nat,X1 : nat] : ((ord_less_nat @ X0 @ X1) => (X0 != X1))),
% 2.71/1.77 file('/export/starexec/sandbox/tmp/tmp.2CmgJp5TsA/DTF2THF_7532.p',fact_34_less__not__refl2)).
% 2.71/1.77 thf(f2365,plain,(
% 2.71/1.77 spl41_64 | ~spl41_9),
% 2.71/1.77 inference(avatar_split_clause,[status(thm)],[f2364,f1260,f1806])).
% 2.71/1.77 thf(f1260,definition,(
% 2.71/1.77 spl41_9 <=> ((ord_less_nat @ (size_size_list_a @ xs) @ (times_times_nat @ n @ k)) = $true)),
% 2.71/1.77 introduced(definition,[new_symbols(naming,[spl41_9])],[avatar_definition])).
% 2.71/1.77 thf(f2364,plain,(
% 2.71/1.77 ((ord_less_nat @ n @ n) = $true) | ~spl41_9),
% 2.71/1.77 inference(trivial_inequality_removal,[status(thm)],[f2363])).
% 2.71/1.77 thf(f2363,plain,(
% 2.71/1.77 ((ord_less_nat @ n @ n) = $true) | ($true != $true) | ~spl41_9),
% 2.71/1.77 inference(superposition,[status(thm)],[f1114,f1937])).
% 2.71/1.77 thf(f1937,plain,(
% 2.71/1.77 ($true = (ord_less_nat @ (suc @ n) @ n)) | ~spl41_9),
% 2.71/1.77 inference(trivial_inequality_removal,[status(thm)],[f1921])).
% 2.71/1.77 thf(f1921,plain,(
% 2.71/1.77 ($true = (ord_less_nat @ (suc @ n) @ n)) | ($true != $true) | ~spl41_9),
% 2.71/1.77 inference(superposition,[status(thm)],[f1195,f1262])).
% 2.71/1.77 thf(f1262,plain,(
% 2.71/1.77 ((ord_less_nat @ (size_size_list_a @ xs) @ (times_times_nat @ n @ k)) = $true) | ~spl41_9),
% 2.71/1.77 inference(avatar_component_clause,[status(thm)],[f1260])).
% 2.71/1.77 thf(f1195,plain,(
% 2.71/1.77 ( ! [X0 : nat] : (($true != (ord_less_nat @ (size_size_list_a @ xs) @ (times_times_nat @ X0 @ k))) | ((ord_less_nat @ (suc @ n) @ X0) = $true)) )),
% 2.71/1.77 inference(superposition,[status(thm)],[f930,f923])).
% 2.71/1.77 thf(f930,plain,(
% 2.71/1.77 ( ! [X2 : nat,X0 : nat,X1 : nat] : (($true = (ord_less_nat @ (divide_divide_nat @ X0 @ X2) @ X1)) | ((ord_less_nat @ X0 @ (times_times_nat @ X1 @ X2)) != $true)) )),
% 2.71/1.77 inference(cnf_transformation,[status(thm)],[f526])).
% 2.71/1.77 thf(f526,plain,(
% 2.71/1.77 ! [X0 : nat,X1 : nat,X2 : nat] : (((ord_less_nat @ X0 @ (times_times_nat @ X1 @ X2)) != $true) | ($true = (ord_less_nat @ (divide_divide_nat @ X0 @ X2) @ X1)))),
% 2.71/1.77 inference(ennf_transformation,[status(thm)],[f241])).
% 2.71/1.77 thf(f241,plain,(
% 2.71/1.77 ! [X0 : nat,X1 : nat,X2 : nat] : (((ord_less_nat @ X0 @ (times_times_nat @ X1 @ X2)) = $true) => ($true = (ord_less_nat @ (divide_divide_nat @ X0 @ X2) @ X1)))),
% 2.71/1.77 inference(fool_elimination,[status(thm)],[f240])).
% 2.71/1.77 thf(f240,plain,(
% 2.71/1.77 ! [X0 : nat,X1 : nat,X2 : nat] : ((ord_less_nat @ X0 @ (times_times_nat @ X1 @ X2)) => (ord_less_nat @ (divide_divide_nat @ X0 @ X2) @ X1))),
% 2.71/1.77 inference(rectify,[status(thm)],[f125])).
% 2.71/1.77 thf(f125,axiom,(
% 2.71/1.77 ! [X0 : nat,X2 : nat,X1 : nat] : ((ord_less_nat @ X0 @ (times_times_nat @ X2 @ X1)) => (ord_less_nat @ (divide_divide_nat @ X0 @ X1) @ X2))),
% 2.71/1.77 file('/export/starexec/sandbox/tmp/tmp.2CmgJp5TsA/DTF2THF_7532.p',fact_124_Util__Div_Oless__mult__imp__div__less)).
% 2.71/1.77 thf(f1114,plain,(
% 2.71/1.77 ( ! [X0 : nat,X1 : nat] : (((ord_less_nat @ (suc @ X0) @ X1) != $true) | ((ord_less_nat @ X0 @ X1) = $true)) )),
% 2.71/1.77 inference(cnf_transformation,[status(thm)],[f641])).
% 2.71/1.77 thf(f641,plain,(
% 2.71/1.77 ! [X0 : nat,X1 : nat] : (((ord_less_nat @ (suc @ X0) @ X1) != $true) | ((ord_less_nat @ X0 @ X1) = $true))),
% 2.71/1.77 inference(ennf_transformation,[status(thm)],[f306])).
% 2.71/1.77 thf(f306,plain,(
% 2.71/1.77 ! [X0 : nat,X1 : nat] : (((ord_less_nat @ (suc @ X0) @ X1) = $true) => ((ord_less_nat @ X0 @ X1) = $true))),
% 2.71/1.77 inference(fool_elimination,[status(thm)],[f305])).
% 2.71/1.77 thf(f305,plain,(
% 2.71/1.77 ! [X0 : nat,X1 : nat] : ((ord_less_nat @ (suc @ X0) @ X1) => (ord_less_nat @ X0 @ X1))),
% 2.71/1.77 inference(rectify,[status(thm)],[f75])).
% 2.71/1.77 thf(f75,axiom,(
% 2.71/1.77 ! [X1 : nat,X0 : nat] : ((ord_less_nat @ (suc @ X1) @ X0) => (ord_less_nat @ X1 @ X0))),
% 2.71/1.77 file('/export/starexec/sandbox/tmp/tmp.2CmgJp5TsA/DTF2THF_7532.p',fact_74_Suc__lessD)).
% 2.71/1.77 thf(f2156,plain,(
% 2.71/1.77 ~spl41_17),
% 2.71/1.77 inference(avatar_contradiction_clause,[status(thm)],[f2155])).
% 2.71/1.77 thf(f2155,plain,(
% 2.71/1.77 $false | ~spl41_17),
% 2.71/1.77 inference(trivial_inequality_removal,[status(thm)],[f2146])).
% 2.71/1.77 thf(f2146,plain,(
% 2.71/1.77 (zero_zero_nat != zero_zero_nat) | ~spl41_17),
% 2.71/1.77 inference(superposition,[status(thm)],[f1075,f1418])).
% 2.71/1.77 thf(f1418,plain,(
% 2.71/1.77 (zero_zero_nat = k) | ~spl41_17),
% 2.71/1.77 inference(avatar_component_clause,[status(thm)],[f1416])).
% 2.71/1.77 thf(f1075,plain,(
% 2.71/1.77 (zero_zero_nat != k)),
% 2.71/1.77 inference(cnf_transformation,[status(thm)],[f185])).
% 2.71/1.77 thf(f185,axiom,(
% 2.71/1.77 (zero_zero_nat != k)),
% 2.71/1.77 file('/export/starexec/sandbox/tmp/tmp.2CmgJp5TsA/DTF2THF_7532.p',conj_1)).
% 2.71/1.77 thf(f1273,plain,(
% 2.71/1.77 spl41_8 | spl41_9),
% 2.71/1.77 inference(avatar_split_clause,[status(thm)],[f1252,f1260,f1256])).
% 2.71/1.77 thf(f1252,plain,(
% 2.71/1.77 ((size_size_list_a @ xs) = (times_times_nat @ n @ k)) | ((ord_less_nat @ (size_size_list_a @ xs) @ (times_times_nat @ n @ k)) = $true)),
% 2.71/1.77 inference(trivial_inequality_removal,[status(thm)],[f1246])).
% 2.71/1.77 thf(f1246,plain,(
% 2.71/1.77 ((ord_less_nat @ (size_size_list_a @ xs) @ (times_times_nat @ n @ k)) = $true) | ((size_size_list_a @ xs) = (times_times_nat @ n @ k)) | ($true != $true)),
% 2.71/1.77 inference(superposition,[status(thm)],[f967,f822])).
% 2.71/1.77 thf(f822,plain,(
% 2.71/1.77 ( ! [X0 : nat,X1 : nat] : (((ord_less_nat @ X1 @ X0) = $true) | (X0 = X1) | ((ord_less_nat @ X0 @ X1) = $true)) )),
% 2.71/1.77 inference(cnf_transformation,[status(thm)],[f560])).
% 2.71/1.77 thf(f560,plain,(
% 2.71/1.77 ! [X0 : nat,X1 : nat] : ((X0 = X1) | ((ord_less_nat @ X0 @ X1) = $true) | ((ord_less_nat @ X1 @ X0) = $true))),
% 2.71/1.77 inference(flattening,[status(thm)],[f559])).
% 2.71/1.77 thf(f559,plain,(
% 2.71/1.77 ! [X0 : nat,X1 : nat] : ((((ord_less_nat @ X1 @ X0) = $true) | ((ord_less_nat @ X0 @ X1) = $true)) | (X0 = X1))),
% 2.71/1.77 inference(ennf_transformation,[status(thm)],[f459])).
% 2.71/1.77 thf(f459,plain,(
% 2.71/1.77 ! [X0 : nat,X1 : nat] : ((X0 != X1) => (((ord_less_nat @ X0 @ X1) != $true) => ((ord_less_nat @ X1 @ X0) = $true)))),
% 2.71/1.77 inference(flattening,[status(thm)],[f217])).
% 2.71/1.77 thf(f217,plain,(
% 2.71/1.77 ! [X0 : nat,X1 : nat] : ((X0 != X1) => (~((ord_less_nat @ X0 @ X1) = $true) => ((ord_less_nat @ X1 @ X0) = $true)))),
% 2.71/1.77 inference(fool_elimination,[status(thm)],[f216])).
% 2.71/1.77 thf(f216,plain,(
% 2.71/1.77 ! [X0 : nat,X1 : nat] : ((X0 != X1) => (~(ord_less_nat @ X0 @ X1) => (ord_less_nat @ X1 @ X0)))),
% 2.71/1.77 inference(rectify,[status(thm)],[f30])).
% 2.71/1.77 thf(f30,axiom,(
% 2.71/1.77 ! [X10 : nat,X11 : nat] : ((X10 != X11) => (~(ord_less_nat @ X10 @ X11) => (ord_less_nat @ X11 @ X10)))),
% 2.71/1.77 file('/export/starexec/sandbox/tmp/tmp.2CmgJp5TsA/DTF2THF_7532.p',fact_29_linorder__neqE__nat)).
% 2.71/1.77 thf(f967,plain,(
% 2.71/1.77 ((ord_less_nat @ (times_times_nat @ n @ k) @ (size_size_list_a @ xs)) != $true)),
% 2.71/1.77 inference(cnf_transformation,[status(thm)],[f464])).
% 2.71/1.77 thf(f464,plain,(
% 2.71/1.77 ((ord_less_nat @ (times_times_nat @ n @ k) @ (size_size_list_a @ xs)) != $true)),
% 2.71/1.77 inference(flattening,[status(thm)],[f258])).
% 2.71/1.77 thf(f258,plain,(
% 2.71/1.77 ~((ord_less_nat @ (times_times_nat @ n @ k) @ (size_size_list_a @ xs)) = $true)),
% 2.71/1.77 inference(fool_elimination,[status(thm)],[f257])).
% 2.71/1.77 thf(f257,plain,(
% 2.71/1.77 ~(ord_less_nat @ (times_times_nat @ n @ k) @ (size_size_list_a @ xs))),
% 2.71/1.77 inference(rectify,[status(thm)],[f187])).
% 2.71/1.77 thf(f187,negated_conjecture,(
% 2.71/1.77 ~(ord_less_nat @ (times_times_nat @ n @ k) @ (size_size_list_a @ xs))),
% 2.71/1.77 inference(negated_conjecture,[status(cth)],[f186])).
% 2.71/1.77 thf(f186,conjecture,(
% 2.71/1.77 (ord_less_nat @ (times_times_nat @ n @ k) @ (size_size_list_a @ xs))),
% 2.71/1.77 file('/export/starexec/sandbox/tmp/tmp.2CmgJp5TsA/DTF2THF_7532.p',conj_2)).
% 2.71/1.77 % SZS output end Proof for DTF2THF_7532
% 2.71/1.77 % (7713)------------------------------
% 2.71/1.77 % (7713)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.77 % (7713)Termination reason: Refutation
% 2.71/1.77
% 2.71/1.77 % (7713)Memory used [KB]: 7291
% 2.71/1.77 % (7713)Time elapsed: 0.114 s
% 2.71/1.77 % (7713)Instructions burned: 206 (million)
% 2.71/1.77 % (7713)------------------------------
% 2.71/1.77 % (7713)------------------------------
% 2.71/1.77 % (7690)Success in time 0.234 s
%------------------------------------------------------------------------------