%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP131^1 : 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 : n023.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:48 AM UTC 2026
% Result : ContradictoryAxioms 2.52s 1.64s
% Output : Refutation 2.52s
% Verified :
% SZS Type : Refutation
% Derivation depth : 25
% Number of leaves : 14
% Syntax : Number of formulae : 142 ( 135 unt; 0 typ; 0 def)
% Number of atoms : 416 ( 251 equ; 0 cnn)
% Maximal formula atoms : 2 ( 2 avg)
% Number of connectives : 868 ( 20 ~; 5 |; 14 &; 760 @)
% ( 0 <=>; 8 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 2 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 25 ( 20 usr; 13 con; 0-2 aty)
% ( 40 !!; 21 ??; 0 @@+; 0 @@-)
% Number of variables : 238 ( 129 ^; 107 !; 2 ?; 238 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
nat: $tType ).
thf(func_def_0,type,
nat: $tType ).
thf(func_def_1,type,
minus_minus_nat: nat > nat > nat ).
thf(func_def_2,type,
plus_plus_nat: nat > nat > nat ).
thf(func_def_3,type,
times_times_nat: nat > nat > nat ).
thf(func_def_4,type,
zero_zero_nat: nat ).
thf(func_def_5,type,
number1551313001itions: ( nat > nat ) > nat > $o ).
thf(func_def_6,type,
ord_less_nat: nat > nat > $o ).
thf(func_def_7,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_8,type,
i: nat ).
thf(func_def_9,type,
k: nat ).
thf(func_def_10,type,
n: nat ).
thf(func_def_11,type,
p: nat > nat ).
thf(func_def_31,type,
ph1:
!>[X0: $tType] : X0 ).
thf(func_def_32,type,
sK2: nat ).
thf(func_def_33,type,
sK3: nat ).
thf(func_def_34,type,
sK4: nat > nat ).
thf(func_def_35,type,
sK5: nat > nat ).
thf(func_def_36,type,
sK6: nat ).
thf(func_def_37,type,
sK7: nat > nat > nat ).
thf(func_def_38,type,
sK8: nat ).
thf(func_def_39,type,
sK9: nat ).
thf(f2010,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f2009,f1211]) ).
thf(f1211,plain,
! [X0: nat] :
( ( plus_plus_nat @ X0 @ zero_zero_nat )
= X0 ),
inference(superposition,[status(thm)],[f1169,f1203]) ).
thf(f1203,plain,
! [X1: nat] :
( ( plus_plus_nat @ zero_zero_nat @ X1 )
= X1 ),
inference(equality_proxy_clausification,[status(thm)],[f1202]) ).
thf(f1202,plain,
! [X1: nat] :
( ( ( plus_plus_nat @ zero_zero_nat @ X1 )
= X1 )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1201]) ).
thf(f1201,plain,
! [X1: nat] :
( ( ^ [Y0: nat] :
( ( plus_plus_nat @ zero_zero_nat @ Y0 )
= Y0 )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f669]) ).
thf(f669,plain,
( ( !! @ nat
@ ^ [Y0: nat] :
( ( plus_plus_nat @ zero_zero_nat @ Y0 )
= Y0 ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f635]) ).
thf(f635,plain,
( ( !! @ nat
@ ^ [Y0: nat] :
( ( plus_plus_nat @ zero_zero_nat @ Y0 )
= Y0 ) )
= $true ),
inference(fool_elimination,[status(thm)],[f634]) ).
thf(f634,plain,
! [X0: nat] :
( ( plus_plus_nat @ zero_zero_nat @ X0 )
= X0 ),
inference(rectify,[status(thm)],[f35]) ).
thf(f35,axiom,
! [X7: nat] :
( ( plus_plus_nat @ zero_zero_nat @ X7 )
= X7 ),
file('/export/starexec/sandbox/tmp/tmp.GJlgnNEeHF/DTF2THF_19775.p',fact_34_add_Oleft__neutral) ).
thf(f1169,plain,
! [X2: nat,X1: nat] :
( ( plus_plus_nat @ X1 @ X2 )
= ( plus_plus_nat @ X2 @ X1 ) ),
inference(beta_eta_normalization,[status(thm)],[f1168]) ).
thf(f1168,plain,
! [X2: nat,X1: nat] :
( ( plus_plus_nat @ X1 @ X2 )
= ( ^ [Y0: nat] : ( plus_plus_nat @ Y0 @ X1 )
@ X2 ) ),
inference(argument_congruence,[status(thm)],[f1162]) ).
thf(f1162,plain,
! [X1: nat] :
( ( plus_plus_nat @ X1 )
= ( ^ [Y0: nat] : ( plus_plus_nat @ Y0 @ X1 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1161]) ).
thf(f1161,plain,
! [X1: nat] :
( ( plus_plus_nat @ X1 )
= ( ^ [Y0: nat,Y1: nat] : ( plus_plus_nat @ Y1 @ Y0 )
@ X1 ) ),
inference(argument_congruence,[status(thm)],[f807]) ).
thf(f807,plain,
( plus_plus_nat
= ( ^ [Y0: nat,Y1: nat] : ( plus_plus_nat @ Y1 @ Y0 ) ) ),
inference(cnf_transformation,[status(thm)],[f328]) ).
thf(f328,plain,
( plus_plus_nat
= ( ^ [Y0: nat,Y1: nat] : ( plus_plus_nat @ Y1 @ Y0 ) ) ),
inference(fool_elimination,[status(thm)],[f327]) ).
thf(f327,plain,
( plus_plus_nat
= ( ^ [X0: nat,X1: nat] : ( plus_plus_nat @ X1 @ X0 ) ) ),
inference(rectify,[status(thm)],[f130]) ).
thf(f130,axiom,
( plus_plus_nat
= ( ^ [X16: nat,X17: nat] : ( plus_plus_nat @ X17 @ X16 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.GJlgnNEeHF/DTF2THF_19775.p',fact_129_add_Ocommute) ).
thf(f2009,plain,
! [X0: nat] :
( ( plus_plus_nat @ X0 @ zero_zero_nat )
!= X0 ),
inference(forward_demodulation,[status(thm)],[f2000,f1581]) ).
thf(f1581,plain,
! [X0: nat] :
( zero_zero_nat
= ( sK7 @ X0 @ zero_zero_nat ) ),
inference(forward_demodulation,[status(thm)],[f1577,f1220]) ).
thf(f1220,plain,
! [X1: nat] :
( zero_zero_nat
= ( minus_minus_nat @ X1 @ X1 ) ),
inference(equality_proxy_clausification,[status(thm)],[f1219]) ).
thf(f1219,plain,
! [X1: nat] :
( ( zero_zero_nat
= ( minus_minus_nat @ X1 @ X1 ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1218]) ).
thf(f1218,plain,
! [X1: nat] :
( ( ^ [Y0: nat] :
( zero_zero_nat
= ( minus_minus_nat @ Y0 @ Y0 ) )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f699]) ).
thf(f699,plain,
( ( !! @ nat
@ ^ [Y0: nat] :
( zero_zero_nat
= ( minus_minus_nat @ Y0 @ Y0 ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f308]) ).
thf(f308,plain,
( ( !! @ nat
@ ^ [Y0: nat] :
( zero_zero_nat
= ( minus_minus_nat @ Y0 @ Y0 ) ) )
= $true ),
inference(fool_elimination,[status(thm)],[f307]) ).
thf(f307,plain,
! [X0: nat] :
( zero_zero_nat
= ( minus_minus_nat @ X0 @ X0 ) ),
inference(rectify,[status(thm)],[f36]) ).
thf(f36,axiom,
! [X7: nat] :
( zero_zero_nat
= ( minus_minus_nat @ X7 @ X7 ) ),
file('/export/starexec/sandbox/tmp/tmp.GJlgnNEeHF/DTF2THF_19775.p',fact_35_cancel__comm__monoid__add__class_Odiff__cancel) ).
thf(f1577,plain,
! [X0: nat] :
( ( sK7 @ X0 @ zero_zero_nat )
= ( minus_minus_nat @ X0 @ X0 ) ),
inference(superposition,[status(thm)],[f1338,f1443]) ).
thf(f1443,plain,
! [X0: nat] :
( ( plus_plus_nat @ X0 @ ( sK7 @ X0 @ zero_zero_nat ) )
= X0 ),
inference(superposition,[status(thm)],[f1413,f1231]) ).
thf(f1231,plain,
! [X1: nat] :
( ( minus_minus_nat @ X1 @ zero_zero_nat )
= X1 ),
inference(equality_proxy_clausification,[status(thm)],[f1230]) ).
thf(f1230,plain,
! [X1: nat] :
( ( ( minus_minus_nat @ X1 @ zero_zero_nat )
= X1 )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1229]) ).
thf(f1229,plain,
! [X1: nat] :
( ( ^ [Y0: nat] :
( ( minus_minus_nat @ Y0 @ zero_zero_nat )
= Y0 )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f802]) ).
thf(f802,plain,
( ( !! @ nat
@ ^ [Y0: nat] :
( ( minus_minus_nat @ Y0 @ zero_zero_nat )
= Y0 ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f258]) ).
thf(f258,plain,
( ( !! @ nat
@ ^ [Y0: nat] :
( ( minus_minus_nat @ Y0 @ zero_zero_nat )
= Y0 ) )
= $true ),
inference(fool_elimination,[status(thm)],[f257]) ).
thf(f257,plain,
! [X0: nat] :
( ( minus_minus_nat @ X0 @ zero_zero_nat )
= X0 ),
inference(rectify,[status(thm)],[f77]) ).
thf(f77,axiom,
! [X4: nat] :
( ( minus_minus_nat @ X4 @ zero_zero_nat )
= X4 ),
file('/export/starexec/sandbox/tmp/tmp.GJlgnNEeHF/DTF2THF_19775.p',fact_76_minus__nat_Odiff__0) ).
thf(f1413,plain,
! [X2: nat,X1: nat] :
( ( plus_plus_nat @ ( minus_minus_nat @ X1 @ X2 ) @ ( sK7 @ X1 @ X2 ) )
= X1 ),
inference(equality_proxy_clausification,[status(thm)],[f1406]) ).
thf(f1406,plain,
! [X2: nat,X1: nat] :
( $true
= ( ( plus_plus_nat @ ( minus_minus_nat @ X1 @ X2 ) @ ( sK7 @ X1 @ X2 ) )
= X1 ) ),
inference(beta_eta_normalization,[status(thm)],[f1403]) ).
thf(f1403,plain,
! [X2: nat,X1: nat] :
( ( ^ [Y0: nat] :
( ( plus_plus_nat @ ( minus_minus_nat @ X1 @ X2 ) @ Y0 )
= X1 )
@ ( sK7 @ X1 @ X2 ) )
= $true ),
inference(sigma_clausification,[status(thm)],[f1399]) ).
thf(f1399,plain,
! [X2: nat,X1: nat] :
( ( ?? @ nat
@ ^ [Y0: nat] :
( ( plus_plus_nat @ ( minus_minus_nat @ X1 @ X2 ) @ Y0 )
= X1 ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1398]) ).
thf(f1398,plain,
! [X2: nat,X1: nat] :
( ( ^ [Y0: nat] :
( ?? @ nat
@ ^ [Y1: nat] :
( ( plus_plus_nat @ ( minus_minus_nat @ X1 @ Y0 ) @ Y1 )
= X1 ) )
@ X2 )
= $true ),
inference(pi_clausification,[status(thm)],[f1395]) ).
thf(f1395,plain,
! [X1: nat] :
( ( !! @ nat
@ ^ [Y0: nat] :
( ?? @ nat
@ ^ [Y1: nat] :
( ( plus_plus_nat @ ( minus_minus_nat @ X1 @ Y0 ) @ Y1 )
= X1 ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1394]) ).
thf(f1394,plain,
! [X1: nat] :
( ( ^ [Y0: nat] :
( !! @ nat
@ ^ [Y1: nat] :
( ?? @ nat
@ ^ [Y2: nat] :
( ( plus_plus_nat @ ( minus_minus_nat @ Y0 @ Y1 ) @ Y2 )
= Y0 ) ) )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f1119]) ).
thf(f1119,plain,
( ( !! @ nat
@ ^ [Y0: nat] :
( !! @ nat
@ ^ [Y1: nat] :
( ?? @ nat
@ ^ [Y2: nat] :
( ( plus_plus_nat @ ( minus_minus_nat @ Y0 @ Y1 ) @ Y2 )
= Y0 ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f991]) ).
thf(f991,plain,
( ( !! @ nat
@ ^ [Y0: nat] :
( !! @ nat
@ ^ [Y1: nat] :
( ^ [Y2: nat,Y3: nat] :
( ?? @ nat
@ ^ [Y4: nat] :
( ( plus_plus_nat @ Y2 @ Y4 )
= Y3 ) )
@ ( minus_minus_nat @ Y0 @ Y1 )
@ Y0 ) ) )
= $true ),
inference(definition_unfolding,[status(thm)],[f838,f698]) ).
thf(f698,plain,
( ord_less_eq_nat
= ( ^ [Y0: nat,Y1: nat] :
( ?? @ nat
@ ^ [Y2: nat] :
( ( plus_plus_nat @ Y0 @ Y2 )
= Y1 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f342]) ).
thf(f342,plain,
( ord_less_eq_nat
= ( ^ [Y0: nat,Y1: nat] :
( ?? @ nat
@ ^ [Y2: nat] :
( ( plus_plus_nat @ Y0 @ Y2 )
= Y1 ) ) ) ),
inference(fool_elimination,[status(thm)],[f341]) ).
thf(f341,plain,
( ord_less_eq_nat
= ( ^ [X0: nat,X1: nat] :
? [X2: nat] :
( ( plus_plus_nat @ X0 @ X2 )
= X1 ) ) ),
inference(rectify,[status(thm)],[f161]) ).
thf(f161,axiom,
( ord_less_eq_nat
= ( ^ [X16: nat,X17: nat] :
? [X27: nat] :
( ( plus_plus_nat @ X16 @ X27 )
= X17 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.GJlgnNEeHF/DTF2THF_19775.p',fact_160_le__iff__add) ).
thf(f838,plain,
( ( !! @ nat
@ ^ [Y0: nat] :
( !! @ nat
@ ^ [Y1: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ Y0 @ Y1 ) @ Y0 ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f399]) ).
thf(f399,plain,
( ( !! @ nat
@ ^ [Y0: nat] :
( !! @ nat
@ ^ [Y1: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ Y0 @ Y1 ) @ Y0 ) ) )
= $true ),
inference(fool_elimination,[status(thm)],[f398]) ).
thf(f398,plain,
! [X0: nat,X1: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ X1 @ X0 ) @ X1 ),
inference(rectify,[status(thm)],[f83]) ).
thf(f83,axiom,
! [X2: nat,X4: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ X4 @ X2 ) @ X4 ),
file('/export/starexec/sandbox/tmp/tmp.GJlgnNEeHF/DTF2THF_19775.p',fact_82_diff__le__self) ).
thf(f1338,plain,
! [X0: nat,X1: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ X1 @ X0 ) @ X1 )
= X0 ),
inference(superposition,[status(thm)],[f1317,f1169]) ).
thf(f1317,plain,
! [X2: nat,X1: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ X1 @ X2 ) @ X2 )
= X1 ),
inference(equality_proxy_clausification,[status(thm)],[f1316]) ).
thf(f1316,plain,
! [X2: nat,X1: nat] :
( ( ( minus_minus_nat @ ( plus_plus_nat @ X1 @ X2 ) @ X2 )
= X1 )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1315]) ).
thf(f1315,plain,
! [X2: nat,X1: nat] :
( ( ^ [Y0: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ X1 @ Y0 ) @ Y0 )
= X1 )
@ X2 )
= $true ),
inference(pi_clausification,[status(thm)],[f1314]) ).
thf(f1314,plain,
! [X1: nat] :
( ( !! @ nat
@ ^ [Y0: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ X1 @ Y0 ) @ Y0 )
= X1 ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1313]) ).
thf(f1313,plain,
! [X1: nat] :
( ( ^ [Y0: nat] :
( !! @ nat
@ ^ [Y1: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ Y0 @ Y1 ) @ Y1 )
= Y0 ) )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f662]) ).
thf(f662,plain,
( ( !! @ nat
@ ^ [Y0: nat] :
( !! @ nat
@ ^ [Y1: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ Y0 @ Y1 ) @ Y1 )
= Y0 ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f310]) ).
thf(f310,plain,
( ( !! @ nat
@ ^ [Y0: nat] :
( !! @ nat
@ ^ [Y1: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ Y0 @ Y1 ) @ Y1 )
= Y0 ) ) )
= $true ),
inference(fool_elimination,[status(thm)],[f309]) ).
thf(f309,plain,
! [X0: nat,X1: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ X1 @ X0 ) @ X0 )
= X1 ),
inference(rectify,[status(thm)],[f88]) ).
thf(f88,axiom,
! [X2: nat,X4: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ X4 @ X2 ) @ X2 )
= X4 ),
file('/export/starexec/sandbox/tmp/tmp.GJlgnNEeHF/DTF2THF_19775.p',fact_87_diff__add__inverse2) ).
thf(f2000,plain,
! [X0: nat] :
( ( plus_plus_nat @ X0 @ ( sK7 @ n @ zero_zero_nat ) )
!= X0 ),
inference(backward_demodulation,[status(thm)],[f1949,f1991]) ).
thf(f1991,plain,
zero_zero_nat = sK8,
inference(backward_demodulation,[status(thm)],[f1775,f1977]) ).
thf(f1977,plain,
( zero_zero_nat
= ( minus_minus_nat @ n @ ( minus_minus_nat @ n @ sK8 ) ) ),
inference(superposition,[status(thm)],[f1269,f1963]) ).
thf(f1963,plain,
( ( minus_minus_nat @ n @ sK8 )
= ( plus_plus_nat @ n @ sK9 ) ),
inference(equality_proxy_clausification,[status(thm)],[f1958]) ).
thf(f1958,plain,
( ( ( plus_plus_nat @ n @ sK9 )
= ( minus_minus_nat @ n @ sK8 ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1956]) ).
thf(f1956,plain,
( ( ^ [Y0: nat] :
( ( plus_plus_nat @ n @ Y0 )
= ( minus_minus_nat @ n @ sK8 ) )
@ sK9 )
= $true ),
inference(sigma_clausification,[status(thm)],[f1951]) ).
thf(f1951,plain,
( ( ?? @ nat
@ ^ [Y0: nat] :
( ( plus_plus_nat @ n @ Y0 )
= ( minus_minus_nat @ n @ sK8 ) ) )
= $true ),
inference(boolean_simplification,[status(thm)],[f1950]) ).
thf(f1950,plain,
( ( ( ?? @ nat
@ ^ [Y0: nat] :
( ( plus_plus_nat @ n @ Y0 )
= ( minus_minus_nat @ n @ sK8 ) ) )
& $true )
= $true ),
inference(backward_demodulation,[status(thm)],[f1772,f1944]) ).
thf(f1944,plain,
( ( zero_zero_nat
!= ( sK7 @ n @ sK8 ) )
= $true ),
inference(binary_proxy_clausification,[status(thm)],[f1772]) ).
thf(f1772,plain,
( $true
= ( ( ?? @ nat
@ ^ [Y0: nat] :
( ( plus_plus_nat @ n @ Y0 )
= ( minus_minus_nat @ n @ sK8 ) ) )
& ( zero_zero_nat
!= ( sK7 @ n @ sK8 ) ) ) ),
inference(forward_demodulation,[status(thm)],[f1771,f1655]) ).
thf(f1655,plain,
! [X0: nat,X1: nat] :
( ( zero_zero_nat
= ( sK7 @ X0 @ X1 ) )
= ( X0
= ( minus_minus_nat @ X0 @ X1 ) ) ),
inference(superposition,[status(thm)],[f1615,f1413]) ).
thf(f1615,plain,
! [X2: nat,X1: nat] :
( ( ( plus_plus_nat @ X2 @ X1 )
= X2 )
= ( zero_zero_nat = X1 ) ),
inference(equality_proxy_clausification,[status(thm)],[f1591]) ).
thf(f1591,plain,
! [X2: nat,X1: nat] :
( ( ( zero_zero_nat = X1 )
= ( ( plus_plus_nat @ X2 @ X1 )
= X2 ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1590]) ).
thf(f1590,plain,
! [X2: nat,X1: nat] :
( ( ^ [Y0: nat] :
( ( zero_zero_nat = X1 )
= ( ( plus_plus_nat @ Y0 @ X1 )
= Y0 ) )
@ X2 )
= $true ),
inference(pi_clausification,[status(thm)],[f1587]) ).
thf(f1587,plain,
! [X1: nat] :
( ( !! @ nat
@ ^ [Y0: nat] :
( ( zero_zero_nat = X1 )
= ( ( plus_plus_nat @ Y0 @ X1 )
= Y0 ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1586]) ).
thf(f1586,plain,
! [X1: nat] :
( ( ^ [Y0: nat] :
( !! @ nat
@ ^ [Y1: nat] :
( ( zero_zero_nat = Y0 )
= ( ( plus_plus_nat @ Y1 @ Y0 )
= Y1 ) ) )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f730]) ).
thf(f730,plain,
( ( !! @ nat
@ ^ [Y0: nat] :
( !! @ nat
@ ^ [Y1: nat] :
( ( zero_zero_nat = Y0 )
= ( ( plus_plus_nat @ Y1 @ Y0 )
= Y1 ) ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f397]) ).
thf(f397,plain,
( ( !! @ nat
@ ^ [Y0: nat] :
( !! @ nat
@ ^ [Y1: nat] :
( ( zero_zero_nat = Y0 )
= ( ( plus_plus_nat @ Y1 @ Y0 )
= Y1 ) ) ) )
= $true ),
inference(fool_elimination,[status(thm)],[f396]) ).
thf(f396,plain,
! [X0: nat,X1: nat] :
( ( zero_zero_nat = X1 )
= ( ( plus_plus_nat @ X0 @ X1 )
= X0 ) ),
inference(rectify,[status(thm)],[f30]) ).
thf(f30,axiom,
! [X7: nat,X6: nat] :
( ( zero_zero_nat = X6 )
= ( ( plus_plus_nat @ X7 @ X6 )
= X7 ) ),
file('/export/starexec/sandbox/tmp/tmp.GJlgnNEeHF/DTF2THF_19775.p',fact_29_add__cancel__right__right) ).
thf(f1771,plain,
( ( ( ?? @ nat
@ ^ [Y0: nat] :
( ( plus_plus_nat @ n @ Y0 )
= ( minus_minus_nat @ n @ sK8 ) ) )
& ( n
!= ( minus_minus_nat @ n @ sK8 ) ) )
= $true ),
inference(backward_demodulation,[status(thm)],[f1160,f1768]) ).
thf(f1768,plain,
( ( minus_minus_nat @ n @ sK8 )
= ( plus_plus_nat @ ( times_times_nat @ i @ ( p @ i ) ) @ ( times_times_nat @ k @ ( p @ k ) ) ) ),
inference(superposition,[status(thm)],[f1338,f1751]) ).
thf(f1751,plain,
( n
= ( plus_plus_nat @ sK8 @ ( plus_plus_nat @ ( times_times_nat @ i @ ( p @ i ) ) @ ( times_times_nat @ k @ ( p @ k ) ) ) ) ),
inference(equality_proxy_clausification,[status(thm)],[f1748]) ).
thf(f1748,plain,
( ( ( plus_plus_nat @ sK8 @ ( plus_plus_nat @ ( times_times_nat @ i @ ( p @ i ) ) @ ( times_times_nat @ k @ ( p @ k ) ) ) )
= n )
= $true ),
inference(forward_demodulation,[status(thm)],[f1747,f1158]) ).
thf(f1158,plain,
! [X2: nat,X1: nat] :
( ( times_times_nat @ X1 @ X2 )
= ( times_times_nat @ X2 @ X1 ) ),
inference(beta_eta_normalization,[status(thm)],[f1157]) ).
thf(f1157,plain,
! [X2: nat,X1: nat] :
( ( times_times_nat @ X1 @ X2 )
= ( ^ [Y0: nat] : ( times_times_nat @ Y0 @ X1 )
@ X2 ) ),
inference(argument_congruence,[status(thm)],[f1156]) ).
thf(f1156,plain,
! [X1: nat] :
( ( times_times_nat @ X1 )
= ( ^ [Y0: nat] : ( times_times_nat @ Y0 @ X1 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1155]) ).
thf(f1155,plain,
! [X1: nat] :
( ( times_times_nat @ X1 )
= ( ^ [Y0: nat,Y1: nat] : ( times_times_nat @ Y1 @ Y0 )
@ X1 ) ),
inference(argument_congruence,[status(thm)],[f670]) ).
thf(f670,plain,
( times_times_nat
= ( ^ [Y0: nat,Y1: nat] : ( times_times_nat @ Y1 @ Y0 ) ) ),
inference(cnf_transformation,[status(thm)],[f633]) ).
thf(f633,plain,
( times_times_nat
= ( ^ [Y0: nat,Y1: nat] : ( times_times_nat @ Y1 @ Y0 ) ) ),
inference(fool_elimination,[status(thm)],[f632]) ).
thf(f632,plain,
( times_times_nat
= ( ^ [X0: nat,X1: nat] : ( times_times_nat @ X1 @ X0 ) ) ),
inference(rectify,[status(thm)],[f137]) ).
thf(f137,axiom,
( times_times_nat
= ( ^ [X16: nat,X17: nat] : ( times_times_nat @ X17 @ X16 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.GJlgnNEeHF/DTF2THF_19775.p',fact_136_mult_Ocommute) ).
thf(f1747,plain,
( ( ( plus_plus_nat @ sK8 @ ( plus_plus_nat @ ( times_times_nat @ i @ ( p @ i ) ) @ ( times_times_nat @ ( p @ k ) @ k ) ) )
= n )
= $true ),
inference(forward_demodulation,[status(thm)],[f1746,f1158]) ).
thf(f1746,plain,
( ( ( plus_plus_nat @ sK8 @ ( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) ) )
= n )
= $true ),
inference(forward_demodulation,[status(thm)],[f1745,f1169]) ).
thf(f1745,plain,
( ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) ) @ sK8 )
= n )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1744]) ).
thf(f1744,plain,
( ( ^ [Y0: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) ) @ Y0 )
= n )
@ sK8 )
= $true ),
inference(sigma_clausification,[status(thm)],[f1137]) ).
thf(f1137,plain,
( ( ?? @ nat
@ ^ [Y0: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) ) @ Y0 )
= n ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f904]) ).
thf(f904,plain,
( ( ^ [Y0: nat,Y1: nat] :
( ?? @ nat
@ ^ [Y2: nat] :
( ( plus_plus_nat @ Y0 @ Y2 )
= Y1 ) )
@ ( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) )
@ n )
= $true ),
inference(definition_unfolding,[status(thm)],[f711,f698]) ).
thf(f711,plain,
( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) ) @ n )
= $true ),
inference(cnf_transformation,[status(thm)],[f264]) ).
thf(f264,plain,
( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) ) @ n )
= $true ),
inference(fool_elimination,[status(thm)],[f263]) ).
thf(f263,plain,
ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) ) @ n,
inference(rectify,[status(thm)],[f3]) ).
thf(f3,axiom,
ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) ) @ n,
file('/export/starexec/sandbox/tmp/tmp.GJlgnNEeHF/DTF2THF_19775.p',fact_2_upper__bound) ).
thf(f1160,plain,
( ( ( ?? @ nat
@ ^ [Y0: nat] :
( ( plus_plus_nat @ n @ Y0 )
= ( plus_plus_nat @ ( times_times_nat @ i @ ( p @ i ) ) @ ( times_times_nat @ k @ ( p @ k ) ) ) ) )
& ( n
!= ( plus_plus_nat @ ( times_times_nat @ i @ ( p @ i ) ) @ ( times_times_nat @ k @ ( p @ k ) ) ) ) )
= $true ),
inference(forward_demodulation,[status(thm)],[f1159,f1158]) ).
thf(f1159,plain,
( ( ( ?? @ nat
@ ^ [Y0: nat] :
( ( plus_plus_nat @ n @ Y0 )
= ( plus_plus_nat @ ( times_times_nat @ i @ ( p @ i ) ) @ ( times_times_nat @ ( p @ k ) @ k ) ) ) )
& ( n
!= ( plus_plus_nat @ ( times_times_nat @ i @ ( p @ i ) ) @ ( times_times_nat @ ( p @ k ) @ k ) ) ) )
= $true ),
inference(backward_demodulation,[status(thm)],[f1118,f1158]) ).
thf(f1118,plain,
( ( ( ?? @ nat
@ ^ [Y0: nat] :
( ( plus_plus_nat @ n @ Y0 )
= ( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) ) ) )
& ( n
!= ( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f879]) ).
thf(f879,plain,
( ( ^ [Y0: nat,Y1: nat] :
( ( ^ [Y2: nat,Y3: nat] :
( ?? @ nat
@ ^ [Y4: nat] :
( ( plus_plus_nat @ Y2 @ Y4 )
= Y3 ) )
@ Y0
@ Y1 )
& ( Y0 != Y1 ) )
@ n
@ ( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) ) )
= $true ),
inference(definition_unfolding,[status(thm)],[f681,f851]) ).
thf(f851,plain,
( ord_less_nat
= ( ^ [Y0: nat,Y1: nat] :
( ( ^ [Y2: nat,Y3: nat] :
( ?? @ nat
@ ^ [Y4: nat] :
( ( plus_plus_nat @ Y2 @ Y4 )
= Y3 ) )
@ Y0
@ Y1 )
& ( Y0 != Y1 ) ) ) ),
inference(definition_unfolding,[status(thm)],[f724,f698]) ).
thf(f724,plain,
( ord_less_nat
= ( ^ [Y0: nat,Y1: nat] :
( ( ord_less_eq_nat @ Y0 @ Y1 )
& ( Y0 != Y1 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f296]) ).
thf(f296,plain,
( ord_less_nat
= ( ^ [Y0: nat,Y1: nat] :
( ( ord_less_eq_nat @ Y0 @ Y1 )
& ( Y0 != Y1 ) ) ) ),
inference(fool_elimination,[status(thm)],[f295]) ).
thf(f295,plain,
( ( ^ [X0: nat,X1: nat] :
( ( X0 != X1 )
& ( ord_less_eq_nat @ X0 @ X1 ) ) )
= ord_less_nat ),
inference(rectify,[status(thm)],[f197]) ).
thf(f197,axiom,
( ( ^ [X33: nat,X34: nat] :
( ( X33 != X34 )
& ( ord_less_eq_nat @ X33 @ X34 ) ) )
= ord_less_nat ),
file('/export/starexec/sandbox/tmp/tmp.GJlgnNEeHF/DTF2THF_19775.p',fact_196_nat__less__le) ).
thf(f681,plain,
( ( ord_less_nat @ n @ ( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f601]) ).
thf(f601,plain,
( ( ord_less_nat @ n @ ( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) ) )
= $true ),
inference(fool_elimination,[status(thm)],[f600]) ).
thf(f600,plain,
ord_less_nat @ n @ ( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) ),
inference(rectify,[status(thm)],[f4]) ).
thf(f4,axiom,
ord_less_nat @ n @ ( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) ),
file('/export/starexec/sandbox/tmp/tmp.GJlgnNEeHF/DTF2THF_19775.p',fact_3_lower__bound) ).
thf(f1269,plain,
! [X2: nat,X1: nat] :
( zero_zero_nat
= ( minus_minus_nat @ X2 @ ( plus_plus_nat @ X2 @ X1 ) ) ),
inference(equality_proxy_clausification,[status(thm)],[f1268]) ).
thf(f1268,plain,
! [X2: nat,X1: nat] :
( ( zero_zero_nat
= ( minus_minus_nat @ X2 @ ( plus_plus_nat @ X2 @ X1 ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1267]) ).
thf(f1267,plain,
! [X2: nat,X1: nat] :
( ( ^ [Y0: nat] :
( zero_zero_nat
= ( minus_minus_nat @ Y0 @ ( plus_plus_nat @ Y0 @ X1 ) ) )
@ X2 )
= $true ),
inference(pi_clausification,[status(thm)],[f1265]) ).
thf(f1265,plain,
! [X1: nat] :
( $true
= ( !! @ nat
@ ^ [Y0: nat] :
( zero_zero_nat
= ( minus_minus_nat @ Y0 @ ( plus_plus_nat @ Y0 @ X1 ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1264]) ).
thf(f1264,plain,
! [X1: nat] :
( ( ^ [Y0: nat] :
( !! @ nat
@ ^ [Y1: nat] :
( zero_zero_nat
= ( minus_minus_nat @ Y1 @ ( plus_plus_nat @ Y1 @ Y0 ) ) ) )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f791]) ).
thf(f791,plain,
( ( !! @ nat
@ ^ [Y0: nat] :
( !! @ nat
@ ^ [Y1: nat] :
( zero_zero_nat
= ( minus_minus_nat @ Y1 @ ( plus_plus_nat @ Y1 @ Y0 ) ) ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f522]) ).
thf(f522,plain,
( ( !! @ nat
@ ^ [Y0: nat] :
( !! @ nat
@ ^ [Y1: nat] :
( zero_zero_nat
= ( minus_minus_nat @ Y1 @ ( plus_plus_nat @ Y1 @ Y0 ) ) ) ) )
= $true ),
inference(fool_elimination,[status(thm)],[f521]) ).
thf(f521,plain,
! [X0: nat,X1: nat] :
( zero_zero_nat
= ( minus_minus_nat @ X0 @ ( plus_plus_nat @ X0 @ X1 ) ) ),
inference(rectify,[status(thm)],[f66]) ).
thf(f66,axiom,
! [X7: nat,X6: nat] :
( zero_zero_nat
= ( minus_minus_nat @ X7 @ ( plus_plus_nat @ X7 @ X6 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.GJlgnNEeHF/DTF2THF_19775.p',fact_65_diff__add__zero) ).
thf(f1775,plain,
( ( minus_minus_nat @ n @ ( minus_minus_nat @ n @ sK8 ) )
= sK8 ),
inference(forward_demodulation,[status(thm)],[f1767,f1768]) ).
thf(f1767,plain,
( sK8
= ( minus_minus_nat @ n @ ( plus_plus_nat @ ( times_times_nat @ i @ ( p @ i ) ) @ ( times_times_nat @ k @ ( p @ k ) ) ) ) ),
inference(superposition,[status(thm)],[f1317,f1751]) ).
thf(f1949,plain,
! [X0: nat] :
( ( plus_plus_nat @ X0 @ ( sK7 @ n @ sK8 ) )
!= X0 ),
inference(trivial_inequality_removal,[status(thm)],[f1948]) ).
thf(f1948,plain,
! [X0: nat] :
( ( ( plus_plus_nat @ X0 @ ( sK7 @ n @ sK8 ) )
!= X0 )
| ( $false = $true ) ),
inference(boolean_simplification,[status(thm)],[f1947]) ).
thf(f1947,plain,
! [X0: nat] :
( ( ( plus_plus_nat @ X0 @ ( sK7 @ n @ sK8 ) )
!= X0 )
| ( ( ( ?? @ nat
@ ^ [Y0: nat] :
( ( plus_plus_nat @ n @ Y0 )
= ( minus_minus_nat @ n @ sK8 ) ) )
& $false )
= $true ) ),
inference(boolean_simplification,[status(thm)],[f1946]) ).
thf(f1946,plain,
! [X0: nat] :
( ( ( ( ?? @ nat
@ ^ [Y0: nat] :
( ( plus_plus_nat @ n @ Y0 )
= ( minus_minus_nat @ n @ sK8 ) ) )
& ~ $true )
= $true )
| ( ( plus_plus_nat @ X0 @ ( sK7 @ n @ sK8 ) )
!= X0 ) ),
inference(superposition,[status(thm)],[f1772,f1782]) ).
thf(f1782,plain,
! [X2: nat,X1: nat] :
( ( ( zero_zero_nat = X1 )
= $true )
| ( ( plus_plus_nat @ X2 @ X1 )
!= X2 ) ),
inference(equality_proxy_clausification,[status(thm)],[f1480]) ).
thf(f1480,plain,
! [X2: nat,X1: nat] :
( ( ( ( plus_plus_nat @ X2 @ X1 )
= X2 )
= $false )
| ( ( zero_zero_nat = X1 )
= $true ) ),
inference(binary_proxy_clausification,[status(thm)],[f1478]) ).
thf(f1478,plain,
! [X2: nat,X1: nat] :
( ( ( ( plus_plus_nat @ X2 @ X1 )
= X2 )
=> ( zero_zero_nat = X1 ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1477]) ).
thf(f1477,plain,
! [X2: nat,X1: nat] :
( ( ^ [Y0: nat] :
( ( ( plus_plus_nat @ Y0 @ X1 )
= Y0 )
=> ( zero_zero_nat = X1 ) )
@ X2 )
= $true ),
inference(pi_clausification,[status(thm)],[f1475]) ).
thf(f1475,plain,
! [X1: nat] :
( ( !! @ nat
@ ^ [Y0: nat] :
( ( ( plus_plus_nat @ Y0 @ X1 )
= Y0 )
=> ( zero_zero_nat = X1 ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1474]) ).
thf(f1474,plain,
! [X1: nat] :
( ( ^ [Y0: nat] :
( !! @ nat
@ ^ [Y1: nat] :
( ( ( plus_plus_nat @ Y1 @ Y0 )
= Y1 )
=> ( zero_zero_nat = Y0 ) ) )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f672]) ).
thf(f672,plain,
( $true
= ( !! @ nat
@ ^ [Y0: nat] :
( !! @ nat
@ ^ [Y1: nat] :
( ( ( plus_plus_nat @ Y1 @ Y0 )
= Y1 )
=> ( zero_zero_nat = Y0 ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f490]) ).
thf(f490,plain,
( $true
= ( !! @ nat
@ ^ [Y0: nat] :
( !! @ nat
@ ^ [Y1: nat] :
( ( ( plus_plus_nat @ Y1 @ Y0 )
= Y1 )
=> ( zero_zero_nat = Y0 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f489]) ).
thf(f489,plain,
! [X0: nat,X1: nat] :
( ( ( plus_plus_nat @ X0 @ X1 )
= X0 )
=> ( zero_zero_nat = X1 ) ),
inference(rectify,[status(thm)],[f190]) ).
thf(f190,axiom,
! [X4: nat,X2: nat] :
( ( ( plus_plus_nat @ X4 @ X2 )
= X4 )
=> ( zero_zero_nat = X2 ) ),
file('/export/starexec/sandbox/tmp/tmp.GJlgnNEeHF/DTF2THF_19775.p',fact_189_add__eq__self__zero) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP131^1 : 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
% 0.15/0.33 % Computer : n023.cluster.edu
% 0.15/0.33 % Model : x86_64 x86_64
% 0.15/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.33 % Memory : 8042.1875MB
% 0.15/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.33 % CPULimit : 300
% 0.15/0.33 % WCLimit : 300
% 0.15/0.33 % DateTime : Tue Feb 24 08:28:04 EST 2026
% 0.15/0.33 % CPUTime :
% 0.15/0.33 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.29/0.45 ---- Original DTF file ---
% 0.29/0.45 thf(spec,logic,$$dhol).
% 0.29/0.45 %------------------------------------------------------------------------------
% 0.29/0.45 % File : ITP131^1 : TPTP v9.2.1. Released v7.5.0.
% 0.29/0.45 % Domain : Interactive Theorem Proving
% 0.29/0.45 % Problem : Sledgehammer Number_Partition problem prob_126__5325690_1
% 0.29/0.45 % Version : Especial.
% 0.29/0.45 % English :
% 0.29/0.45
% 0.29/0.45 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.29/0.45 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.29/0.45 % Source : [Des21]
% 0.29/0.45 % Names : Number_Partition/prob_126__5325690_1 [Des21]
% 0.29/0.45
% 0.29/0.45 % Status : ContradictoryAxioms
% 0.29/0.45 % Rating : 0.33 v9.1.0, 0.38 v9.0.0, 0.30 v8.2.0, 0.31 v8.1.0, 0.36 v7.5.0
% 0.29/0.45 % Syntax : Number of formulae : 225 ( 119 unt; 12 typ; 0 def)
% 0.29/0.45 % Number of atoms : 490 ( 225 equ; 0 cnn)
% 0.29/0.45 % Maximal formula atoms : 4 ( 2 avg)
% 0.29/0.45 % Number of connectives : 1495 ( 37 ~; 10 |; 20 &;1299 @)
% 0.29/0.45 % ( 0 <=>; 129 =>; 0 <=; 0 <~>)
% 0.29/0.45 % Maximal formula depth : 12 ( 5 avg)
% 0.29/0.45 % Number of types : 2 ( 1 usr)
% 0.29/0.45 % Number of type conns : 23 ( 23 >; 0 *; 0 +; 0 <<)
% 0.29/0.45 % Number of symbols : 13 ( 11 usr; 5 con; 0-2 aty)
% 0.29/0.45 % Number of variables : 519 ( 13 ^; 500 !; 6 ?; 519 :)
% 0.29/0.45 % SPC : TH0_CAX_EQU_NAR
% 0.29/0.45
% 0.29/0.45 % Comments : This file was generated by Sledgehammer 2021-02-23 15:42:42.866
% 0.29/0.45 %------------------------------------------------------------------------------
% 0.29/0.45 % Could-be-implicit typings (1)
% 0.29/0.45 thf(ty_n_t__Nat__Onat,type,
% 0.29/0.45 nat: $tType ).
% 0.29/0.45
% 0.29/0.45 % Explicit typings (11)
% 0.29/0.45 thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
% 0.29/0.45 minus_minus_nat: nat > nat > nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
% 0.29/0.45 plus_plus_nat: nat > nat > nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
% 0.29/0.45 times_times_nat: nat > nat > nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 0.29/0.45 zero_zero_nat: nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Number__Partition__Mirabelle__zerdlymyoj_Opartitions,type,
% 0.29/0.45 number1551313001itions: ( nat > nat ) > nat > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 0.29/0.45 ord_less_nat: nat > nat > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 0.29/0.45 ord_less_eq_nat: nat > nat > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_i____,type,
% 0.29/0.45 i: nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_k,type,
% 0.29/0.45 k: nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_n,type,
% 0.29/0.45 n: nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_p,type,
% 0.29/0.45 p: nat > nat ).
% 0.29/0.45
% 0.29/0.45 % Relevant facts (212)
% 0.29/0.45 thf(fact_0__092_060open_062i_A_092_060le_062_An_092_060close_062,axiom,
% 0.29/0.45 ord_less_eq_nat @ i @ n ).
% 0.29/0.45
% 0.29/0.45 % \<open>i \<le> n\<close>
% 0.29/0.45 thf(fact_1_assms_I1_J,axiom,
% 0.29/0.45 number1551313001itions @ p @ n ).
% 0.29/0.45
% 0.29/0.45 % assms(1)
% 0.29/0.45 thf(fact_2_upper__bound,axiom,
% 0.29/0.45 ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) ) @ n ).
% 0.29/0.45
% 0.29/0.45 % upper_bound
% 0.29/0.45 thf(fact_3_lower__bound,axiom,
% 0.29/0.45 ord_less_nat @ n @ ( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) ) ).
% 0.29/0.45
% 0.29/0.45 % lower_bound
% 0.29/0.45 thf(fact_4__092_060open_062k_A_092_060le_062_An_092_060close_062,axiom,
% 0.29/0.45 ord_less_eq_nat @ k @ n ).
% 0.29/0.45
% 0.29/0.45 % \<open>k \<le> n\<close>
% 0.29/0.45 thf(fact_5__092_060open_062p_Ai_A_092_060noteq_062_A0_092_060close_062,axiom,
% 0.29/0.45 ( ( p @ i )
% 0.29/0.45 != zero_zero_nat ) ).
% 0.29/0.45
% 0.29/0.45 % \<open>p i \<noteq> 0\<close>
% 0.29/0.45 thf(fact_6_assms_I2_J,axiom,
% 0.29/0.45 ord_less_nat @ zero_zero_nat @ ( p @ k ) ).
% 0.29/0.45
% 0.29/0.45 % assms(2)
% 0.29/0.45 thf(fact_7__092_060open_062i_A_092_060noteq_062_Ak_092_060close_062,axiom,
% 0.29/0.45 i != k ).
% 0.29/0.45
% 0.29/0.45 % \<open>i \<noteq> k\<close>
% 0.29/0.45 thf(fact_8__092_060open_062n_A_N_Ak_A_060_Ai_092_060close_062,axiom,
% 0.29/0.45 ord_less_nat @ ( minus_minus_nat @ n @ k ) @ i ).
% 0.29/0.45
% 0.29/0.45 % \<open>n - k < i\<close>
% 0.29/0.45 thf(fact_9_partitions__zero,axiom,
% 0.29/0.45 ! [P: nat > nat] :
% 0.29/0.45 ( ( number1551313001itions @ P @ zero_zero_nat )
% 0.29/0.45 = ( P
% 0.29/0.45 = ( ^ [I: nat] : zero_zero_nat ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % partitions_zero
% 0.29/0.45 thf(fact_10_partitions__bounds,axiom,
% 0.29/0.45 ! [P: nat > nat,N: nat,I2: nat] :
% 0.29/0.45 ( ( number1551313001itions @ P @ N )
% 0.29/0.45 => ( ord_less_eq_nat @ ( P @ I2 ) @ N ) ) ).
% 0.29/0.45
% 0.29/0.45 % partitions_bounds
% 0.29/0.45 thf(fact_11_mult__le__cancel2,axiom,
% 0.29/0.45 ! [M: nat,K: nat,N: nat] :
% 0.29/0.45 ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 0.29/0.45 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.45 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % mult_le_cancel2
% 0.29/0.45 thf(fact_12_nat__mult__le__cancel__disj,axiom,
% 0.29/0.45 ! [K: nat,M: nat,N: nat] :
% 0.29/0.45 ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.29/0.45 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.45 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % nat_mult_le_cancel_disj
% 0.29/0.45 thf(fact_13_mult__less__cancel2,axiom,
% 0.29/0.45 ! [M: nat,K: nat,N: nat] :
% 0.29/0.45 ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 0.29/0.45 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.45 & ( ord_less_nat @ M @ N ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % mult_less_cancel2
% 0.29/0.45 thf(fact_14_nat__0__less__mult__iff,axiom,
% 0.29/0.45 ! [M: nat,N: nat] :
% 0.29/0.45 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
% 0.29/0.45 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.29/0.45 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % nat_0_less_mult_iff
% 0.29/0.45 thf(fact_15_nat__mult__less__cancel__disj,axiom,
% 0.29/0.45 ! [K: nat,M: nat,N: nat] :
% 0.29/0.45 ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.29/0.45 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.45 & ( ord_less_nat @ M @ N ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % nat_mult_less_cancel_disj
% 0.29/0.45 thf(fact_16_add__gr__0,axiom,
% 0.29/0.45 ! [M: nat,N: nat] :
% 0.29/0.45 ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
% 0.29/0.45 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.29/0.45 | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_gr_0
% 0.29/0.45 thf(fact_17_add__less__same__cancel1,axiom,
% 0.29/0.45 ! [B: nat,A: nat] :
% 0.29/0.45 ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 0.29/0.45 = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_less_same_cancel1
% 0.29/0.45 thf(fact_18_add__less__same__cancel2,axiom,
% 0.29/0.45 ! [A: nat,B: nat] :
% 0.29/0.45 ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.29/0.45 = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_less_same_cancel2
% 0.29/0.45 thf(fact_19_less__add__same__cancel1,axiom,
% 0.29/0.45 ! [A: nat,B: nat] :
% 0.29/0.45 ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.29/0.45 = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_add_same_cancel1
% 0.29/0.45 thf(fact_20_less__add__same__cancel2,axiom,
% 0.29/0.45 ! [A: nat,B: nat] :
% 0.29/0.45 ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 0.29/0.45 = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_add_same_cancel2
% 0.29/0.45 thf(fact_21_add__right__cancel,axiom,
% 0.29/0.45 ! [B: nat,A: nat,C: nat] :
% 0.29/0.45 ( ( ( plus_plus_nat @ B @ A )
% 0.29/0.45 = ( plus_plus_nat @ C @ A ) )
% 0.29/0.45 = ( B = C ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_right_cancel
% 0.29/0.45 thf(fact_22_add__left__cancel,axiom,
% 0.29/0.45 ! [A: nat,B: nat,C: nat] :
% 0.29/0.45 ( ( ( plus_plus_nat @ A @ B )
% 0.29/0.45 = ( plus_plus_nat @ A @ C ) )
% 0.29/0.45 = ( B = C ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_left_cancel
% 0.29/0.45 thf(fact_23_le__zero__eq,axiom,
% 0.29/0.45 ! [N: nat] :
% 0.29/0.45 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.29/0.45 = ( N = zero_zero_nat ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_zero_eq
% 0.29/0.45 thf(fact_24_not__gr__zero,axiom,
% 0.29/0.45 ! [N: nat] :
% 0.29/0.45 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.29/0.45 = ( N = zero_zero_nat ) ) ).
% 0.29/0.45
% 0.29/0.45 % not_gr_zero
% 0.29/0.45 thf(fact_25_add__le__cancel__right,axiom,
% 0.29/0.45 ! [A: nat,C: nat,B: nat] :
% 0.29/0.45 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.29/0.45 = ( ord_less_eq_nat @ A @ B ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_le_cancel_right
% 0.29/0.45 thf(fact_26_add__le__cancel__left,axiom,
% 0.29/0.45 ! [C: nat,A: nat,B: nat] :
% 0.29/0.45 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.29/0.45 = ( ord_less_eq_nat @ A @ B ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_le_cancel_left
% 0.29/0.45 thf(fact_27_zero__eq__add__iff__both__eq__0,axiom,
% 0.29/0.45 ! [X: nat,Y: nat] :
% 0.29/0.45 ( ( zero_zero_nat
% 0.29/0.45 = ( plus_plus_nat @ X @ Y ) )
% 0.29/0.45 = ( ( X = zero_zero_nat )
% 0.29/0.45 & ( Y = zero_zero_nat ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % zero_eq_add_iff_both_eq_0
% 0.29/0.45 thf(fact_28_add__eq__0__iff__both__eq__0,axiom,
% 0.29/0.45 ! [X: nat,Y: nat] :
% 0.29/0.45 ( ( ( plus_plus_nat @ X @ Y )
% 0.29/0.45 = zero_zero_nat )
% 0.29/0.45 = ( ( X = zero_zero_nat )
% 0.29/0.45 & ( Y = zero_zero_nat ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_eq_0_iff_both_eq_0
% 0.29/0.45 thf(fact_29_add__cancel__right__right,axiom,
% 0.29/0.45 ! [A: nat,B: nat] :
% 0.29/0.45 ( ( A
% 0.29/0.45 = ( plus_plus_nat @ A @ B ) )
% 0.29/0.45 = ( B = zero_zero_nat ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_cancel_right_right
% 0.29/0.45 thf(fact_30_add__cancel__right__left,axiom,
% 0.29/0.45 ! [A: nat,B: nat] :
% 0.29/0.45 ( ( A
% 0.29/0.45 = ( plus_plus_nat @ B @ A ) )
% 0.29/0.45 = ( B = zero_zero_nat ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_cancel_right_left
% 0.29/0.45 thf(fact_31_add__cancel__left__right,axiom,
% 0.29/0.45 ! [A: nat,B: nat] :
% 0.29/0.45 ( ( ( plus_plus_nat @ A @ B )
% 0.29/0.45 = A )
% 0.29/0.45 = ( B = zero_zero_nat ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_cancel_left_right
% 0.29/0.45 thf(fact_32_add__cancel__left__left,axiom,
% 0.29/0.45 ! [B: nat,A: nat] :
% 0.29/0.45 ( ( ( plus_plus_nat @ B @ A )
% 0.29/0.45 = A )
% 0.29/0.45 = ( B = zero_zero_nat ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_cancel_left_left
% 0.29/0.45 thf(fact_33_add_Oright__neutral,axiom,
% 0.29/0.45 ! [A: nat] :
% 0.29/0.45 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 0.29/0.45 = A ) ).
% 0.29/0.45
% 0.29/0.45 % add.right_neutral
% 0.29/0.45 thf(fact_34_add_Oleft__neutral,axiom,
% 0.29/0.45 ! [A: nat] :
% 0.29/0.45 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 0.29/0.45 = A ) ).
% 0.29/0.45
% 0.29/0.45 % add.left_neutral
% 0.29/0.45 thf(fact_35_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.29/0.45 ! [A: nat] :
% 0.29/0.45 ( ( minus_minus_nat @ A @ A )
% 0.29/0.45 = zero_zero_nat ) ).
% 0.29/0.45
% 0.29/0.45 % cancel_comm_monoid_add_class.diff_cancel
% 0.29/0.45 thf(fact_36_diff__zero,axiom,
% 0.29/0.45 ! [A: nat] :
% 0.29/0.45 ( ( minus_minus_nat @ A @ zero_zero_nat )
% 0.29/0.45 = A ) ).
% 0.29/0.45
% 0.29/0.45 % diff_zero
% 0.29/0.45 thf(fact_37_zero__diff,axiom,
% 0.29/0.45 ! [A: nat] :
% 0.29/0.45 ( ( minus_minus_nat @ zero_zero_nat @ A )
% 0.29/0.45 = zero_zero_nat ) ).
% 0.29/0.45
% 0.29/0.45 % zero_diff
% 0.29/0.45 thf(fact_38_add__less__cancel__right,axiom,
% 0.29/0.45 ! [A: nat,C: nat,B: nat] :
% 0.29/0.45 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.29/0.45 = ( ord_less_nat @ A @ B ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_less_cancel_right
% 0.29/0.45 thf(fact_39_add__less__cancel__left,axiom,
% 0.29/0.45 ! [C: nat,A: nat,B: nat] :
% 0.29/0.45 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.29/0.45 = ( ord_less_nat @ A @ B ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_less_cancel_left
% 0.29/0.45 thf(fact_40_add__diff__cancel__right_H,axiom,
% 0.29/0.45 ! [A: nat,B: nat] :
% 0.29/0.45 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.29/0.45 = A ) ).
% 0.29/0.45
% 0.29/0.45 % add_diff_cancel_right'
% 0.29/0.45 thf(fact_41_add__diff__cancel__right,axiom,
% 0.29/0.45 ! [A: nat,C: nat,B: nat] :
% 0.29/0.45 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.29/0.45 = ( minus_minus_nat @ A @ B ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_diff_cancel_right
% 0.29/0.45 thf(fact_42_add__diff__cancel__left_H,axiom,
% 0.29/0.45 ! [A: nat,B: nat] :
% 0.29/0.45 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
% 0.29/0.45 = B ) ).
% 0.29/0.45
% 0.29/0.45 % add_diff_cancel_left'
% 0.29/0.45 thf(fact_43_add__diff__cancel__left,axiom,
% 0.29/0.45 ! [C: nat,A: nat,B: nat] :
% 0.29/0.45 ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.29/0.45 = ( minus_minus_nat @ A @ B ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_diff_cancel_left
% 0.29/0.45 thf(fact_44_bot__nat__0_Onot__eq__extremum,axiom,
% 0.29/0.45 ! [A: nat] :
% 0.29/0.45 ( ( A != zero_zero_nat )
% 0.29/0.45 = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 0.29/0.45
% 0.29/0.45 % bot_nat_0.not_eq_extremum
% 0.29/0.45 thf(fact_45_less__nat__zero__code,axiom,
% 0.29/0.45 ! [N: nat] :
% 0.29/0.45 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.29/0.45
% 0.29/0.45 % less_nat_zero_code
% 0.29/0.45 thf(fact_46_neq0__conv,axiom,
% 0.29/0.45 ! [N: nat] :
% 0.29/0.45 ( ( N != zero_zero_nat )
% 0.29/0.45 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.29/0.45
% 0.29/0.45 % neq0_conv
% 0.29/0.45 thf(fact_47_bot__nat__0_Oextremum,axiom,
% 0.29/0.45 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 0.29/0.45
% 0.29/0.45 % bot_nat_0.extremum
% 0.29/0.45 thf(fact_48_le0,axiom,
% 0.29/0.45 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.29/0.45
% 0.29/0.45 % le0
% 0.29/0.45 thf(fact_49_Nat_Oadd__0__right,axiom,
% 0.29/0.45 ! [M: nat] :
% 0.29/0.45 ( ( plus_plus_nat @ M @ zero_zero_nat )
% 0.29/0.45 = M ) ).
% 0.29/0.45
% 0.29/0.45 % Nat.add_0_right
% 0.29/0.45 thf(fact_50_add__is__0,axiom,
% 0.29/0.45 ! [M: nat,N: nat] :
% 0.29/0.45 ( ( ( plus_plus_nat @ M @ N )
% 0.29/0.45 = zero_zero_nat )
% 0.29/0.45 = ( ( M = zero_zero_nat )
% 0.29/0.45 & ( N = zero_zero_nat ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_is_0
% 0.29/0.45 thf(fact_51_nat__add__left__cancel__less,axiom,
% 0.29/0.45 ! [K: nat,M: nat,N: nat] :
% 0.29/0.45 ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.29/0.45 = ( ord_less_nat @ M @ N ) ) ).
% 0.29/0.45
% 0.29/0.45 % nat_add_left_cancel_less
% 0.29/0.45 thf(fact_52_diff__self__eq__0,axiom,
% 0.29/0.45 ! [M: nat] :
% 0.29/0.45 ( ( minus_minus_nat @ M @ M )
% 0.29/0.45 = zero_zero_nat ) ).
% 0.29/0.45
% 0.29/0.45 % diff_self_eq_0
% 0.29/0.45 thf(fact_53_diff__0__eq__0,axiom,
% 0.29/0.45 ! [N: nat] :
% 0.29/0.45 ( ( minus_minus_nat @ zero_zero_nat @ N )
% 0.29/0.45 = zero_zero_nat ) ).
% 0.29/0.45
% 0.29/0.45 % diff_0_eq_0
% 0.29/0.45 thf(fact_54_mult__cancel2,axiom,
% 0.29/0.45 ! [M: nat,K: nat,N: nat] :
% 0.29/0.45 ( ( ( times_times_nat @ M @ K )
% 0.29/0.45 = ( times_times_nat @ N @ K ) )
% 0.29/0.45 = ( ( M = N )
% 0.29/0.45 | ( K = zero_zero_nat ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % mult_cancel2
% 0.29/0.45 thf(fact_55_mult__cancel1,axiom,
% 0.29/0.45 ! [K: nat,M: nat,N: nat] :
% 0.29/0.45 ( ( ( times_times_nat @ K @ M )
% 0.29/0.45 = ( times_times_nat @ K @ N ) )
% 0.29/0.45 = ( ( M = N )
% 0.29/0.45 | ( K = zero_zero_nat ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % mult_cancel1
% 0.29/0.45 thf(fact_56_mult__0__right,axiom,
% 0.29/0.45 ! [M: nat] :
% 0.29/0.45 ( ( times_times_nat @ M @ zero_zero_nat )
% 0.29/0.45 = zero_zero_nat ) ).
% 0.29/0.45
% 0.29/0.45 % mult_0_right
% 0.29/0.45 thf(fact_57_mult__is__0,axiom,
% 0.29/0.45 ! [M: nat,N: nat] :
% 0.29/0.45 ( ( ( times_times_nat @ M @ N )
% 0.29/0.45 = zero_zero_nat )
% 0.29/0.45 = ( ( M = zero_zero_nat )
% 0.29/0.45 | ( N = zero_zero_nat ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % mult_is_0
% 0.29/0.45 thf(fact_58_nat__add__left__cancel__le,axiom,
% 0.29/0.45 ! [K: nat,M: nat,N: nat] :
% 0.29/0.45 ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.29/0.45 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.29/0.45
% 0.29/0.45 % nat_add_left_cancel_le
% 0.29/0.45 thf(fact_59_diff__diff__cancel,axiom,
% 0.29/0.45 ! [I2: nat,N: nat] :
% 0.29/0.45 ( ( ord_less_eq_nat @ I2 @ N )
% 0.29/0.45 => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I2 ) )
% 0.29/0.45 = I2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % diff_diff_cancel
% 0.29/0.45 thf(fact_60_diff__diff__left,axiom,
% 0.29/0.45 ! [I2: nat,J: nat,K: nat] :
% 0.29/0.45 ( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
% 0.29/0.45 = ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % diff_diff_left
% 0.29/0.45 thf(fact_61_le__add__same__cancel2,axiom,
% 0.29/0.45 ! [A: nat,B: nat] :
% 0.29/0.45 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 0.29/0.45 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_add_same_cancel2
% 0.29/0.45 thf(fact_62_le__add__same__cancel1,axiom,
% 0.29/0.45 ! [A: nat,B: nat] :
% 0.29/0.45 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.29/0.45 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_add_same_cancel1
% 0.29/0.45 thf(fact_63_add__le__same__cancel2,axiom,
% 0.29/0.45 ! [A: nat,B: nat] :
% 0.29/0.45 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.29/0.45 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_le_same_cancel2
% 0.29/0.45 thf(fact_64_add__le__same__cancel1,axiom,
% 0.29/0.45 ! [B: nat,A: nat] :
% 0.29/0.45 ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 0.29/0.45 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_le_same_cancel1
% 0.29/0.45 thf(fact_65_diff__add__zero,axiom,
% 0.29/0.45 ! [A: nat,B: nat] :
% 0.29/0.45 ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.29/0.45 = zero_zero_nat ) ).
% 0.29/0.45
% 0.29/0.45 % diff_add_zero
% 0.29/0.45 thf(fact_66_zero__less__diff,axiom,
% 0.29/0.45 ! [N: nat,M: nat] :
% 0.29/0.45 ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
% 0.29/0.45 = ( ord_less_nat @ M @ N ) ) ).
% 0.29/0.45
% 0.29/0.45 % zero_less_diff
% 0.29/0.45 thf(fact_67_diff__is__0__eq_H,axiom,
% 0.29/0.45 ! [M: nat,N: nat] :
% 0.29/0.45 ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.45 => ( ( minus_minus_nat @ M @ N )
% 0.29/0.45 = zero_zero_nat ) ) ).
% 0.29/0.45
% 0.29/0.45 % diff_is_0_eq'
% 0.29/0.45 thf(fact_68_diff__is__0__eq,axiom,
% 0.29/0.45 ! [M: nat,N: nat] :
% 0.29/0.45 ( ( ( minus_minus_nat @ M @ N )
% 0.29/0.45 = zero_zero_nat )
% 0.29/0.45 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.29/0.45
% 0.29/0.45 % diff_is_0_eq
% 0.29/0.45 thf(fact_69_Nat_Odiff__diff__right,axiom,
% 0.29/0.45 ! [K: nat,J: nat,I2: nat] :
% 0.29/0.45 ( ( ord_less_eq_nat @ K @ J )
% 0.29/0.45 => ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
% 0.29/0.45 = ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Nat.diff_diff_right
% 0.29/0.46 thf(fact_70_Nat_Oadd__diff__assoc2,axiom,
% 0.29/0.46 ! [K: nat,J: nat,I2: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ K @ J )
% 0.29/0.46 => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
% 0.29/0.46 = ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Nat.add_diff_assoc2
% 0.29/0.46 thf(fact_71_Nat_Oadd__diff__assoc,axiom,
% 0.29/0.46 ! [K: nat,J: nat,I2: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ K @ J )
% 0.29/0.46 => ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
% 0.29/0.46 = ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Nat.add_diff_assoc
% 0.29/0.46 thf(fact_72_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
% 0.29/0.46 ! [A: nat,C: nat,B: nat] :
% 0.29/0.46 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
% 0.29/0.46 = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% 0.29/0.46
% 0.29/0.46 % cancel_ab_semigroup_add_class.diff_right_commute
% 0.29/0.46 thf(fact_73_diff__commute,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,K: nat] :
% 0.29/0.46 ( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
% 0.29/0.46 = ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_commute
% 0.29/0.46 thf(fact_74_add__implies__diff,axiom,
% 0.29/0.46 ! [C: nat,B: nat,A: nat] :
% 0.29/0.46 ( ( ( plus_plus_nat @ C @ B )
% 0.29/0.46 = A )
% 0.29/0.46 => ( C
% 0.29/0.46 = ( minus_minus_nat @ A @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_implies_diff
% 0.29/0.46 thf(fact_75_diff__diff__add,axiom,
% 0.29/0.46 ! [A: nat,B: nat,C: nat] :
% 0.29/0.46 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
% 0.29/0.46 = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_diff_add
% 0.29/0.46 thf(fact_76_minus__nat_Odiff__0,axiom,
% 0.29/0.46 ! [M: nat] :
% 0.29/0.46 ( ( minus_minus_nat @ M @ zero_zero_nat )
% 0.29/0.46 = M ) ).
% 0.29/0.46
% 0.29/0.46 % minus_nat.diff_0
% 0.29/0.46 thf(fact_77_diffs0__imp__equal,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ( minus_minus_nat @ M @ N )
% 0.29/0.46 = zero_zero_nat )
% 0.29/0.46 => ( ( ( minus_minus_nat @ N @ M )
% 0.29/0.46 = zero_zero_nat )
% 0.29/0.46 => ( M = N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diffs0_imp_equal
% 0.29/0.46 thf(fact_78_less__imp__diff__less,axiom,
% 0.29/0.46 ! [J: nat,K: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_nat @ J @ K )
% 0.29/0.46 => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_imp_diff_less
% 0.29/0.46 thf(fact_79_diff__less__mono2,axiom,
% 0.29/0.46 ! [M: nat,N: nat,L: nat] :
% 0.29/0.46 ( ( ord_less_nat @ M @ N )
% 0.29/0.46 => ( ( ord_less_nat @ M @ L )
% 0.29/0.46 => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_less_mono2
% 0.29/0.46 thf(fact_80_diff__le__mono2,axiom,
% 0.29/0.46 ! [M: nat,N: nat,L: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.46 => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_le_mono2
% 0.29/0.46 thf(fact_81_le__diff__iff_H,axiom,
% 0.29/0.46 ! [A: nat,C: nat,B: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A @ C )
% 0.29/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.29/0.46 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
% 0.29/0.46 = ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_diff_iff'
% 0.29/0.46 thf(fact_82_diff__le__self,axiom,
% 0.29/0.46 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% 0.29/0.46
% 0.29/0.46 % diff_le_self
% 0.29/0.46 thf(fact_83_diff__le__mono,axiom,
% 0.29/0.46 ! [M: nat,N: nat,L: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.46 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_le_mono
% 0.29/0.46 thf(fact_84_Nat_Odiff__diff__eq,axiom,
% 0.29/0.46 ! [K: nat,M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ K @ M )
% 0.29/0.46 => ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.46 => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.29/0.46 = ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Nat.diff_diff_eq
% 0.29/0.46 thf(fact_85_le__diff__iff,axiom,
% 0.29/0.46 ! [K: nat,M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ K @ M )
% 0.29/0.46 => ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.46 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.29/0.46 = ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_diff_iff
% 0.29/0.46 thf(fact_86_eq__diff__iff,axiom,
% 0.29/0.46 ! [K: nat,M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ K @ M )
% 0.29/0.46 => ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.46 => ( ( ( minus_minus_nat @ M @ K )
% 0.29/0.46 = ( minus_minus_nat @ N @ K ) )
% 0.29/0.46 = ( M = N ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_diff_iff
% 0.29/0.46 thf(fact_87_diff__add__inverse2,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
% 0.29/0.46 = M ) ).
% 0.29/0.46
% 0.29/0.46 % diff_add_inverse2
% 0.29/0.46 thf(fact_88_diff__add__inverse,axiom,
% 0.29/0.46 ! [N: nat,M: nat] :
% 0.29/0.46 ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
% 0.29/0.46 = M ) ).
% 0.29/0.46
% 0.29/0.46 % diff_add_inverse
% 0.29/0.46 thf(fact_89_diff__cancel2,axiom,
% 0.29/0.46 ! [M: nat,K: nat,N: nat] :
% 0.29/0.46 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
% 0.29/0.46 = ( minus_minus_nat @ M @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_cancel2
% 0.29/0.46 thf(fact_90_Nat_Odiff__cancel,axiom,
% 0.29/0.46 ! [K: nat,M: nat,N: nat] :
% 0.29/0.46 ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.29/0.46 = ( minus_minus_nat @ M @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % Nat.diff_cancel
% 0.29/0.46 thf(fact_91_diff__mult__distrib2,axiom,
% 0.29/0.46 ! [K: nat,M: nat,N: nat] :
% 0.29/0.46 ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 0.29/0.46 = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_mult_distrib2
% 0.29/0.46 thf(fact_92_diff__mult__distrib,axiom,
% 0.29/0.46 ! [M: nat,N: nat,K: nat] :
% 0.29/0.46 ( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
% 0.29/0.46 = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_mult_distrib
% 0.29/0.46 thf(fact_93_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 0.29/0.46 ! [A: nat,B: nat,C: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.46 => ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.46 => ( ( ( minus_minus_nat @ B @ A )
% 0.29/0.46 = C )
% 0.29/0.46 = ( B
% 0.29/0.46 = ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
% 0.29/0.46 thf(fact_94_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 0.29/0.46 ! [A: nat,B: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.46 => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
% 0.29/0.46 = B ) ) ).
% 0.29/0.46
% 0.29/0.46 % ordered_cancel_comm_monoid_diff_class.add_diff_inverse
% 0.29/0.46 thf(fact_95_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 0.29/0.46 ! [A: nat,B: nat,C: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.46 => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.29/0.46 = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ordered_cancel_comm_monoid_diff_class.diff_diff_right
% 0.29/0.46 thf(fact_96_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 0.29/0.46 ! [A: nat,B: nat,C: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.46 => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
% 0.29/0.46 = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
% 0.29/0.46 thf(fact_97_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 0.29/0.46 ! [A: nat,B: nat,C: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.46 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
% 0.29/0.46 = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
% 0.29/0.46 thf(fact_98_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 0.29/0.46 ! [A: nat,B: nat,C: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.46 => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
% 0.29/0.46 = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ordered_cancel_comm_monoid_diff_class.diff_add_assoc
% 0.29/0.46 thf(fact_99_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 0.29/0.46 ! [A: nat,B: nat,C: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.46 => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.29/0.46 = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ordered_cancel_comm_monoid_diff_class.add_diff_assoc
% 0.29/0.46 thf(fact_100_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 0.29/0.46 ! [A: nat,B: nat,C: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.46 => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.29/0.46 = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ordered_cancel_comm_monoid_diff_class.le_diff_conv2
% 0.29/0.46 thf(fact_101_le__add__diff,axiom,
% 0.29/0.46 ! [A: nat,B: nat,C: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.46 => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_add_diff
% 0.29/0.46 thf(fact_102_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
% 0.29/0.46 ! [A: nat,B: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.46 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
% 0.29/0.46 = B ) ) ).
% 0.29/0.46
% 0.29/0.46 % ordered_cancel_comm_monoid_diff_class.diff_add
% 0.29/0.46 thf(fact_103_diff__less,axiom,
% 0.29/0.46 ! [N: nat,M: nat] :
% 0.29/0.46 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.46 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.29/0.46 => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_less
% 0.29/0.46 thf(fact_104_diff__add__0,axiom,
% 0.29/0.46 ! [N: nat,M: nat] :
% 0.29/0.46 ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
% 0.29/0.46 = zero_zero_nat ) ).
% 0.29/0.46
% 0.29/0.46 % diff_add_0
% 0.29/0.46 thf(fact_105_diff__less__mono,axiom,
% 0.29/0.46 ! [A: nat,B: nat,C: nat] :
% 0.29/0.46 ( ( ord_less_nat @ A @ B )
% 0.29/0.46 => ( ( ord_less_eq_nat @ C @ A )
% 0.29/0.46 => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_less_mono
% 0.29/0.46 thf(fact_106_less__diff__iff,axiom,
% 0.29/0.46 ! [K: nat,M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ K @ M )
% 0.29/0.46 => ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.46 => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.29/0.46 = ( ord_less_nat @ M @ N ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_diff_iff
% 0.29/0.46 thf(fact_107_add__diff__inverse__nat,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ~ ( ord_less_nat @ M @ N )
% 0.29/0.46 => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
% 0.29/0.46 = M ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_diff_inverse_nat
% 0.29/0.46 thf(fact_108_less__diff__conv,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,K: nat] :
% 0.29/0.46 ( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
% 0.29/0.46 = ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_diff_conv
% 0.29/0.46 thf(fact_109_Nat_Ole__imp__diff__is__add,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,K: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ I2 @ J )
% 0.29/0.46 => ( ( ( minus_minus_nat @ J @ I2 )
% 0.29/0.46 = K )
% 0.29/0.46 = ( J
% 0.29/0.46 = ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Nat.le_imp_diff_is_add
% 0.29/0.46 thf(fact_110_Nat_Odiff__add__assoc2,axiom,
% 0.29/0.46 ! [K: nat,J: nat,I2: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ K @ J )
% 0.29/0.46 => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K )
% 0.29/0.46 = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Nat.diff_add_assoc2
% 0.29/0.46 thf(fact_111_Nat_Odiff__add__assoc,axiom,
% 0.29/0.46 ! [K: nat,J: nat,I2: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ K @ J )
% 0.29/0.46 => ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
% 0.29/0.46 = ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Nat.diff_add_assoc
% 0.29/0.46 thf(fact_112_Nat_Ole__diff__conv2,axiom,
% 0.29/0.46 ! [K: nat,J: nat,I2: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ K @ J )
% 0.29/0.46 => ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
% 0.29/0.46 = ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Nat.le_diff_conv2
% 0.29/0.46 thf(fact_113_le__diff__conv,axiom,
% 0.29/0.46 ! [J: nat,K: nat,I2: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
% 0.29/0.46 = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_diff_conv
% 0.29/0.46 thf(fact_114_nat__diff__split__asm,axiom,
% 0.29/0.46 ! [P2: nat > $o,A: nat,B: nat] :
% 0.29/0.46 ( ( P2 @ ( minus_minus_nat @ A @ B ) )
% 0.29/0.46 = ( ~ ( ( ( ord_less_nat @ A @ B )
% 0.29/0.46 & ~ ( P2 @ zero_zero_nat ) )
% 0.29/0.46 | ? [D: nat] :
% 0.29/0.46 ( ( A
% 0.29/0.46 = ( plus_plus_nat @ B @ D ) )
% 0.29/0.46 & ~ ( P2 @ D ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_diff_split_asm
% 0.29/0.46 thf(fact_115_nat__diff__split,axiom,
% 0.29/0.46 ! [P2: nat > $o,A: nat,B: nat] :
% 0.29/0.46 ( ( P2 @ ( minus_minus_nat @ A @ B ) )
% 0.29/0.46 = ( ( ( ord_less_nat @ A @ B )
% 0.29/0.46 => ( P2 @ zero_zero_nat ) )
% 0.29/0.46 & ! [D: nat] :
% 0.29/0.46 ( ( A
% 0.29/0.46 = ( plus_plus_nat @ B @ D ) )
% 0.29/0.46 => ( P2 @ D ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_diff_split
% 0.29/0.46 thf(fact_116_less__diff__conv2,axiom,
% 0.29/0.46 ! [K: nat,J: nat,I2: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ K @ J )
% 0.29/0.46 => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
% 0.29/0.46 = ( ord_less_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_diff_conv2
% 0.29/0.46 thf(fact_117_nat__diff__add__eq2,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ I2 @ J )
% 0.29/0.46 => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.29/0.46 = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_diff_add_eq2
% 0.29/0.46 thf(fact_118_nat__diff__add__eq1,axiom,
% 0.29/0.46 ! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ J @ I2 )
% 0.29/0.46 => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.29/0.46 = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_diff_add_eq1
% 0.29/0.46 thf(fact_119_nat__le__add__iff2,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ I2 @ J )
% 0.29/0.46 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.29/0.46 = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_le_add_iff2
% 0.29/0.46 thf(fact_120_nat__le__add__iff1,axiom,
% 0.29/0.46 ! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ J @ I2 )
% 0.29/0.46 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.29/0.46 = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_le_add_iff1
% 0.29/0.46 thf(fact_121_nat__eq__add__iff2,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ I2 @ J )
% 0.29/0.46 => ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M )
% 0.29/0.46 = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.29/0.46 = ( M
% 0.29/0.46 = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_eq_add_iff2
% 0.29/0.46 thf(fact_122_nat__eq__add__iff1,axiom,
% 0.29/0.46 ! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ J @ I2 )
% 0.29/0.46 => ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M )
% 0.29/0.46 = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.29/0.46 = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M )
% 0.29/0.46 = N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_eq_add_iff1
% 0.29/0.46 thf(fact_123_nat__less__add__iff2,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ I2 @ J )
% 0.29/0.46 => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.29/0.46 = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_less_add_iff2
% 0.29/0.46 thf(fact_124_nat__less__add__iff1,axiom,
% 0.29/0.46 ! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ J @ I2 )
% 0.29/0.46 => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.29/0.46 = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_less_add_iff1
% 0.29/0.46 thf(fact_125_zero__reorient,axiom,
% 0.29/0.46 ! [X: nat] :
% 0.29/0.46 ( ( zero_zero_nat = X )
% 0.29/0.46 = ( X = zero_zero_nat ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_reorient
% 0.29/0.46 thf(fact_126_add__right__imp__eq,axiom,
% 0.29/0.46 ! [B: nat,A: nat,C: nat] :
% 0.29/0.46 ( ( ( plus_plus_nat @ B @ A )
% 0.29/0.46 = ( plus_plus_nat @ C @ A ) )
% 0.29/0.46 => ( B = C ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_right_imp_eq
% 0.29/0.46 thf(fact_127_add__left__imp__eq,axiom,
% 0.29/0.46 ! [A: nat,B: nat,C: nat] :
% 0.29/0.46 ( ( ( plus_plus_nat @ A @ B )
% 0.29/0.46 = ( plus_plus_nat @ A @ C ) )
% 0.29/0.46 => ( B = C ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_left_imp_eq
% 0.29/0.46 thf(fact_128_add_Oleft__commute,axiom,
% 0.29/0.46 ! [B: nat,A: nat,C: nat] :
% 0.29/0.46 ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
% 0.29/0.46 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add.left_commute
% 0.29/0.46 thf(fact_129_add_Ocommute,axiom,
% 0.29/0.46 ( plus_plus_nat
% 0.29/0.46 = ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add.commute
% 0.29/0.46 thf(fact_130_add_Oassoc,axiom,
% 0.29/0.46 ! [A: nat,B: nat,C: nat] :
% 0.29/0.46 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.29/0.46 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add.assoc
% 0.29/0.46 thf(fact_131_group__cancel_Oadd2,axiom,
% 0.29/0.46 ! [B3: nat,K: nat,B: nat,A: nat] :
% 0.29/0.46 ( ( B3
% 0.29/0.46 = ( plus_plus_nat @ K @ B ) )
% 0.29/0.46 => ( ( plus_plus_nat @ A @ B3 )
% 0.29/0.46 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % group_cancel.add2
% 0.29/0.46 thf(fact_132_group__cancel_Oadd1,axiom,
% 0.29/0.46 ! [A3: nat,K: nat,A: nat,B: nat] :
% 0.29/0.46 ( ( A3
% 0.29/0.46 = ( plus_plus_nat @ K @ A ) )
% 0.29/0.46 => ( ( plus_plus_nat @ A3 @ B )
% 0.29/0.46 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % group_cancel.add1
% 0.29/0.46 thf(fact_133_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.29/0.46 ( ( ( I2 = J )
% 0.29/0.46 & ( K = L ) )
% 0.29/0.46 => ( ( plus_plus_nat @ I2 @ K )
% 0.29/0.46 = ( plus_plus_nat @ J @ L ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_mono_thms_linordered_semiring(4)
% 0.29/0.46 thf(fact_134_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.29/0.46 ! [A: nat,B: nat,C: nat] :
% 0.29/0.46 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.29/0.46 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ab_semigroup_add_class.add_ac(1)
% 0.29/0.46 thf(fact_135_mult_Oleft__commute,axiom,
% 0.29/0.46 ! [B: nat,A: nat,C: nat] :
% 0.29/0.46 ( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
% 0.29/0.46 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % mult.left_commute
% 0.29/0.46 thf(fact_136_mult_Ocommute,axiom,
% 0.29/0.46 ( times_times_nat
% 0.29/0.46 = ( ^ [A2: nat,B2: nat] : ( times_times_nat @ B2 @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % mult.commute
% 0.29/0.46 thf(fact_137_mult_Oassoc,axiom,
% 0.29/0.46 ! [A: nat,B: nat,C: nat] :
% 0.29/0.46 ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.29/0.46 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % mult.assoc
% 0.29/0.46 thf(fact_138_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.29/0.46 ! [A: nat,B: nat,C: nat] :
% 0.29/0.46 ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.29/0.46 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ab_semigroup_mult_class.mult_ac(1)
% 0.29/0.46 thf(fact_139_linorder__neqE__nat,axiom,
% 0.29/0.46 ! [X: nat,Y: nat] :
% 0.29/0.46 ( ( X != Y )
% 0.29/0.46 => ( ~ ( ord_less_nat @ X @ Y )
% 0.29/0.46 => ( ord_less_nat @ Y @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % linorder_neqE_nat
% 0.29/0.46 thf(fact_140_infinite__descent,axiom,
% 0.29/0.46 ! [P2: nat > $o,N: nat] :
% 0.29/0.46 ( ! [N2: nat] :
% 0.29/0.46 ( ~ ( P2 @ N2 )
% 0.29/0.46 => ? [M2: nat] :
% 0.29/0.46 ( ( ord_less_nat @ M2 @ N2 )
% 0.29/0.46 & ~ ( P2 @ M2 ) ) )
% 0.29/0.46 => ( P2 @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % infinite_descent
% 0.29/0.46 thf(fact_141_nat__less__induct,axiom,
% 0.29/0.46 ! [P2: nat > $o,N: nat] :
% 0.29/0.46 ( ! [N2: nat] :
% 0.29/0.46 ( ! [M2: nat] :
% 0.29/0.46 ( ( ord_less_nat @ M2 @ N2 )
% 0.29/0.46 => ( P2 @ M2 ) )
% 0.29/0.46 => ( P2 @ N2 ) )
% 0.29/0.46 => ( P2 @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_less_induct
% 0.29/0.46 thf(fact_142_less__irrefl__nat,axiom,
% 0.29/0.46 ! [N: nat] :
% 0.29/0.46 ~ ( ord_less_nat @ N @ N ) ).
% 0.29/0.46
% 0.29/0.46 % less_irrefl_nat
% 0.29/0.46 thf(fact_143_less__not__refl3,axiom,
% 0.29/0.46 ! [S: nat,T: nat] :
% 0.29/0.46 ( ( ord_less_nat @ S @ T )
% 0.29/0.46 => ( S != T ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_not_refl3
% 0.29/0.46 thf(fact_144_less__not__refl2,axiom,
% 0.29/0.46 ! [N: nat,M: nat] :
% 0.29/0.46 ( ( ord_less_nat @ N @ M )
% 0.29/0.46 => ( M != N ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_not_refl2
% 0.29/0.46 thf(fact_145_less__not__refl,axiom,
% 0.29/0.46 ! [N: nat] :
% 0.29/0.46 ~ ( ord_less_nat @ N @ N ) ).
% 0.29/0.46
% 0.29/0.46 % less_not_refl
% 0.29/0.46 thf(fact_146_nat__neq__iff,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( M != N )
% 0.29/0.46 = ( ( ord_less_nat @ M @ N )
% 0.29/0.46 | ( ord_less_nat @ N @ M ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_neq_iff
% 0.29/0.46 thf(fact_147_Nat_Oex__has__greatest__nat,axiom,
% 0.29/0.46 ! [P2: nat > $o,K: nat,B: nat] :
% 0.29/0.46 ( ( P2 @ K )
% 0.29/0.46 => ( ! [Y2: nat] :
% 0.29/0.46 ( ( P2 @ Y2 )
% 0.29/0.46 => ( ord_less_eq_nat @ Y2 @ B ) )
% 0.29/0.46 => ? [X2: nat] :
% 0.29/0.46 ( ( P2 @ X2 )
% 0.29/0.46 & ! [Y3: nat] :
% 0.29/0.46 ( ( P2 @ Y3 )
% 0.29/0.46 => ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Nat.ex_has_greatest_nat
% 0.29/0.46 thf(fact_148_nat__le__linear,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.46 | ( ord_less_eq_nat @ N @ M ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_le_linear
% 0.29/0.46 thf(fact_149_le__antisym,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.46 => ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.46 => ( M = N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_antisym
% 0.29/0.46 thf(fact_150_eq__imp__le,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( M = N )
% 0.29/0.46 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_imp_le
% 0.29/0.46 thf(fact_151_le__trans,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,K: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ I2 @ J )
% 0.29/0.46 => ( ( ord_less_eq_nat @ J @ K )
% 0.29/0.46 => ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_trans
% 0.29/0.46 thf(fact_152_le__refl,axiom,
% 0.29/0.46 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 0.29/0.46
% 0.29/0.46 % le_refl
% 0.29/0.46 thf(fact_153_zero__le,axiom,
% 0.29/0.46 ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% 0.29/0.46
% 0.29/0.46 % zero_le
% 0.29/0.46 thf(fact_154_zero__less__iff__neq__zero,axiom,
% 0.29/0.46 ! [N: nat] :
% 0.29/0.46 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.46 = ( N != zero_zero_nat ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_less_iff_neq_zero
% 0.29/0.46 thf(fact_155_gr__implies__not__zero,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_nat @ M @ N )
% 0.29/0.46 => ( N != zero_zero_nat ) ) ).
% 0.29/0.46
% 0.29/0.46 % gr_implies_not_zero
% 0.29/0.46 thf(fact_156_not__less__zero,axiom,
% 0.29/0.46 ! [N: nat] :
% 0.29/0.46 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.29/0.46
% 0.29/0.46 % not_less_zero
% 0.29/0.46 thf(fact_157_gr__zeroI,axiom,
% 0.29/0.46 ! [N: nat] :
% 0.29/0.46 ( ( N != zero_zero_nat )
% 0.29/0.46 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % gr_zeroI
% 0.29/0.46 thf(fact_158_add__le__imp__le__right,axiom,
% 0.29/0.46 ! [A: nat,C: nat,B: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.29/0.46 => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_le_imp_le_right
% 0.29/0.46 thf(fact_159_add__le__imp__le__left,axiom,
% 0.29/0.46 ! [C: nat,A: nat,B: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.29/0.46 => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_le_imp_le_left
% 0.29/0.46 thf(fact_160_le__iff__add,axiom,
% 0.29/0.46 ( ord_less_eq_nat
% 0.29/0.46 = ( ^ [A2: nat,B2: nat] :
% 0.29/0.46 ? [C2: nat] :
% 0.29/0.46 ( B2
% 0.29/0.46 = ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_iff_add
% 0.29/0.46 thf(fact_161_add__right__mono,axiom,
% 0.29/0.46 ! [A: nat,B: nat,C: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.46 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_right_mono
% 0.29/0.46 thf(fact_162_less__eqE,axiom,
% 0.29/0.46 ! [A: nat,B: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.46 => ~ ! [C3: nat] :
% 0.29/0.46 ( B
% 0.29/0.46 != ( plus_plus_nat @ A @ C3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_eqE
% 0.29/0.46 thf(fact_163_add__left__mono,axiom,
% 0.29/0.46 ! [A: nat,B: nat,C: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.46 => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_left_mono
% 0.29/0.46 thf(fact_164_add__mono,axiom,
% 0.29/0.46 ! [A: nat,B: nat,C: nat,D2: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.46 => ( ( ord_less_eq_nat @ C @ D2 )
% 0.29/0.46 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_mono
% 0.29/0.46 thf(fact_165_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.29/0.46 ( ( ( ord_less_eq_nat @ I2 @ J )
% 0.29/0.46 & ( ord_less_eq_nat @ K @ L ) )
% 0.29/0.46 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_mono_thms_linordered_semiring(1)
% 0.29/0.46 thf(fact_166_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.29/0.46 ( ( ( I2 = J )
% 0.29/0.46 & ( ord_less_eq_nat @ K @ L ) )
% 0.29/0.46 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_mono_thms_linordered_semiring(2)
% 0.29/0.46 thf(fact_167_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.29/0.46 ( ( ( ord_less_eq_nat @ I2 @ J )
% 0.29/0.46 & ( K = L ) )
% 0.29/0.46 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_mono_thms_linordered_semiring(3)
% 0.29/0.46 thf(fact_168_add_Ocomm__neutral,axiom,
% 0.29/0.46 ! [A: nat] :
% 0.29/0.46 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 0.29/0.46 = A ) ).
% 0.29/0.46
% 0.29/0.46 % add.comm_neutral
% 0.29/0.46 thf(fact_169_comm__monoid__add__class_Oadd__0,axiom,
% 0.29/0.46 ! [A: nat] :
% 0.29/0.46 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 0.29/0.46 = A ) ).
% 0.29/0.46
% 0.29/0.46 % comm_monoid_add_class.add_0
% 0.29/0.46 thf(fact_170_add__less__imp__less__right,axiom,
% 0.29/0.46 ! [A: nat,C: nat,B: nat] :
% 0.29/0.46 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.29/0.46 => ( ord_less_nat @ A @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_less_imp_less_right
% 0.29/0.46 thf(fact_171_add__less__imp__less__left,axiom,
% 0.29/0.46 ! [C: nat,A: nat,B: nat] :
% 0.29/0.46 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.29/0.46 => ( ord_less_nat @ A @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_less_imp_less_left
% 0.29/0.46 thf(fact_172_add__strict__right__mono,axiom,
% 0.29/0.46 ! [A: nat,B: nat,C: nat] :
% 0.29/0.46 ( ( ord_less_nat @ A @ B )
% 0.29/0.46 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_strict_right_mono
% 0.29/0.46 thf(fact_173_add__strict__left__mono,axiom,
% 0.29/0.46 ! [A: nat,B: nat,C: nat] :
% 0.29/0.46 ( ( ord_less_nat @ A @ B )
% 0.29/0.46 => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_strict_left_mono
% 0.29/0.46 thf(fact_174_add__strict__mono,axiom,
% 0.29/0.46 ! [A: nat,B: nat,C: nat,D2: nat] :
% 0.29/0.46 ( ( ord_less_nat @ A @ B )
% 0.29/0.46 => ( ( ord_less_nat @ C @ D2 )
% 0.29/0.46 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_strict_mono
% 0.29/0.46 thf(fact_175_add__mono__thms__linordered__field_I1_J,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.29/0.46 ( ( ( ord_less_nat @ I2 @ J )
% 0.29/0.46 & ( K = L ) )
% 0.29/0.46 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_mono_thms_linordered_field(1)
% 0.29/0.46 thf(fact_176_add__mono__thms__linordered__field_I2_J,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.29/0.46 ( ( ( I2 = J )
% 0.29/0.46 & ( ord_less_nat @ K @ L ) )
% 0.29/0.46 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_mono_thms_linordered_field(2)
% 0.29/0.46 thf(fact_177_add__mono__thms__linordered__field_I5_J,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.29/0.46 ( ( ( ord_less_nat @ I2 @ J )
% 0.29/0.46 & ( ord_less_nat @ K @ L ) )
% 0.29/0.46 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_mono_thms_linordered_field(5)
% 0.29/0.46 thf(fact_178_bot__nat__0_Oextremum__strict,axiom,
% 0.29/0.46 ! [A: nat] :
% 0.29/0.46 ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 0.29/0.46
% 0.29/0.46 % bot_nat_0.extremum_strict
% 0.29/0.46 thf(fact_179_infinite__descent0,axiom,
% 0.29/0.46 ! [P2: nat > $o,N: nat] :
% 0.29/0.46 ( ( P2 @ zero_zero_nat )
% 0.29/0.46 => ( ! [N2: nat] :
% 0.29/0.46 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.46 => ( ~ ( P2 @ N2 )
% 0.29/0.46 => ? [M2: nat] :
% 0.29/0.46 ( ( ord_less_nat @ M2 @ N2 )
% 0.29/0.46 & ~ ( P2 @ M2 ) ) ) )
% 0.29/0.46 => ( P2 @ N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % infinite_descent0
% 0.29/0.46 thf(fact_180_gr__implies__not0,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_nat @ M @ N )
% 0.29/0.46 => ( N != zero_zero_nat ) ) ).
% 0.29/0.46
% 0.29/0.46 % gr_implies_not0
% 0.29/0.46 thf(fact_181_less__zeroE,axiom,
% 0.29/0.46 ! [N: nat] :
% 0.29/0.46 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.29/0.46
% 0.29/0.46 % less_zeroE
% 0.29/0.46 thf(fact_182_not__less0,axiom,
% 0.29/0.46 ! [N: nat] :
% 0.29/0.46 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.29/0.46
% 0.29/0.46 % not_less0
% 0.29/0.46 thf(fact_183_not__gr0,axiom,
% 0.29/0.46 ! [N: nat] :
% 0.29/0.46 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.29/0.46 = ( N = zero_zero_nat ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_gr0
% 0.29/0.46 thf(fact_184_gr0I,axiom,
% 0.29/0.46 ! [N: nat] :
% 0.29/0.46 ( ( N != zero_zero_nat )
% 0.29/0.46 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % gr0I
% 0.29/0.46 thf(fact_185_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.29/0.46 ! [A: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.29/0.46 => ( A = zero_zero_nat ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot_nat_0.extremum_uniqueI
% 0.29/0.46 thf(fact_186_bot__nat__0_Oextremum__unique,axiom,
% 0.29/0.46 ! [A: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.29/0.46 = ( A = zero_zero_nat ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot_nat_0.extremum_unique
% 0.29/0.46 thf(fact_187_le__0__eq,axiom,
% 0.29/0.46 ! [N: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.29/0.46 = ( N = zero_zero_nat ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_0_eq
% 0.29/0.46 thf(fact_188_less__eq__nat_Osimps_I1_J,axiom,
% 0.29/0.46 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.29/0.46
% 0.29/0.46 % less_eq_nat.simps(1)
% 0.29/0.46 thf(fact_189_add__eq__self__zero,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ( plus_plus_nat @ M @ N )
% 0.29/0.46 = M )
% 0.29/0.46 => ( N = zero_zero_nat ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_eq_self_zero
% 0.29/0.46 thf(fact_190_plus__nat_Oadd__0,axiom,
% 0.29/0.46 ! [N: nat] :
% 0.29/0.46 ( ( plus_plus_nat @ zero_zero_nat @ N )
% 0.29/0.46 = N ) ).
% 0.29/0.46
% 0.29/0.46 % plus_nat.add_0
% 0.29/0.46 thf(fact_191_less__mono__imp__le__mono,axiom,
% 0.29/0.46 ! [F: nat > nat,I2: nat,J: nat] :
% 0.29/0.46 ( ! [I3: nat,J2: nat] :
% 0.29/0.46 ( ( ord_less_nat @ I3 @ J2 )
% 0.29/0.46 => ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
% 0.29/0.46 => ( ( ord_less_eq_nat @ I2 @ J )
% 0.29/0.46 => ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_mono_imp_le_mono
% 0.29/0.46 thf(fact_192_le__neq__implies__less,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.46 => ( ( M != N )
% 0.29/0.46 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_neq_implies_less
% 0.29/0.46 thf(fact_193_less__or__eq__imp__le,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ( ord_less_nat @ M @ N )
% 0.29/0.46 | ( M = N ) )
% 0.29/0.46 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_or_eq_imp_le
% 0.29/0.46 thf(fact_194_le__eq__less__or__eq,axiom,
% 0.29/0.46 ( ord_less_eq_nat
% 0.29/0.46 = ( ^ [M3: nat,N3: nat] :
% 0.29/0.46 ( ( ord_less_nat @ M3 @ N3 )
% 0.29/0.46 | ( M3 = N3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_eq_less_or_eq
% 0.29/0.46 thf(fact_195_less__imp__le__nat,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_nat @ M @ N )
% 0.29/0.46 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_imp_le_nat
% 0.29/0.46 thf(fact_196_nat__less__le,axiom,
% 0.29/0.46 ( ord_less_nat
% 0.29/0.46 = ( ^ [M3: nat,N3: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ M3 @ N3 )
% 0.29/0.46 & ( M3 != N3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_less_le
% 0.29/0.46 thf(fact_197_less__add__eq__less,axiom,
% 0.29/0.46 ! [K: nat,L: nat,M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_nat @ K @ L )
% 0.29/0.46 => ( ( ( plus_plus_nat @ M @ L )
% 0.29/0.46 = ( plus_plus_nat @ K @ N ) )
% 0.29/0.46 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_add_eq_less
% 0.29/0.46 thf(fact_198_trans__less__add2,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,M: nat] :
% 0.29/0.46 ( ( ord_less_nat @ I2 @ J )
% 0.29/0.46 => ( ord_less_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % trans_less_add2
% 0.29/0.46 thf(fact_199_trans__less__add1,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,M: nat] :
% 0.29/0.46 ( ( ord_less_nat @ I2 @ J )
% 0.29/0.46 => ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % trans_less_add1
% 0.29/0.46 thf(fact_200_add__less__mono1,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,K: nat] :
% 0.29/0.46 ( ( ord_less_nat @ I2 @ J )
% 0.29/0.46 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_less_mono1
% 0.29/0.46 thf(fact_201_not__add__less2,axiom,
% 0.29/0.46 ! [J: nat,I2: nat] :
% 0.29/0.46 ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).
% 0.29/0.46
% 0.29/0.46 % not_add_less2
% 0.29/0.46 thf(fact_202_not__add__less1,axiom,
% 0.29/0.46 ! [I2: nat,J: nat] :
% 0.29/0.46 ~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).
% 0.29/0.46
% 0.29/0.46 % not_add_less1
% 0.29/0.46 thf(fact_203_add__less__mono,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.29/0.46 ( ( ord_less_nat @ I2 @ J )
% 0.29/0.46 => ( ( ord_less_nat @ K @ L )
% 0.29/0.46 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_less_mono
% 0.29/0.46 thf(fact_204_add__lessD1,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,K: nat] :
% 0.29/0.46 ( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
% 0.29/0.46 => ( ord_less_nat @ I2 @ K ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_lessD1
% 0.29/0.46 thf(fact_205_nat__mult__eq__cancel__disj,axiom,
% 0.29/0.46 ! [K: nat,M: nat,N: nat] :
% 0.29/0.46 ( ( ( times_times_nat @ K @ M )
% 0.29/0.46 = ( times_times_nat @ K @ N ) )
% 0.29/0.46 = ( ( K = zero_zero_nat )
% 0.29/0.46 | ( M = N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_mult_eq_cancel_disj
% 0.29/0.46 thf(fact_206_mult__0,axiom,
% 0.29/0.46 ! [N: nat] :
% 0.29/0.46 ( ( times_times_nat @ zero_zero_nat @ N )
% 0.29/0.46 = zero_zero_nat ) ).
% 0.29/0.46
% 0.29/0.46 % mult_0
% 0.29/0.46 thf(fact_207_nat__le__iff__add,axiom,
% 0.29/0.46 ( ord_less_eq_nat
% 0.29/0.46 = ( ^ [M3: nat,N3: nat] :
% 0.29/0.46 ? [K2: nat] :
% 0.29/0.46 ( N3
% 0.29/0.46 = ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_le_iff_add
% 0.29/0.46 thf(fact_208_trans__le__add2,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,M: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ I2 @ J )
% 0.29/0.46 => ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % trans_le_add2
% 0.29/0.46 thf(fact_209_trans__le__add1,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,M: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ I2 @ J )
% 0.29/0.46 => ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % trans_le_add1
% 0.29/0.46 thf(fact_210_add__le__mono1,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,K: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ I2 @ J )
% 0.29/0.46 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_le_mono1
% 0.29/0.46 thf(fact_211_add__le__mono,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ I2 @ J )
% 0.29/0.46 => ( ( ord_less_eq_nat @ K @ L )
% 0.29/0.46 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_le_mono
% 0.29/0.46
% 0.29/0.46 % Conjectures (1)
% 0.29/0.46 thf(conj_0,conjecture,
% 0.29/0.46 $false ).
% 0.29/0.46
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 ------------------------
% 2.40/1.44 ---- Embedded in THF ---
% 2.40/1.44 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.40/1.44 %%% Generated on Tue Feb 24 08:28:05 EST 2026
% 2.40/1.44 %%% using '$$dhol' embedding, version 1.3.0.
% 2.40/1.44 %%% Logic specification used:
% 2.40/1.44 %%% thf(spec, logic, $$dhol).
% 2.40/1.44
% 2.40/1.44 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.GJlgnNEeHF/DTF2DTF19775.p
% See solution above
% 2.40/1.45 ------------------------
% 2.40/1.47 ---- Cleaned THF ---
% 2.40/1.47 thf(ty_n_t__Nat__Onat,type,
% 2.40/1.47 nat: $tType ).
% 2.40/1.47
% 2.40/1.47 thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
% 2.40/1.47 minus_minus_nat: nat > nat > nat ).
% 2.40/1.47
% 2.40/1.47 thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
% 2.40/1.47 plus_plus_nat: nat > nat > nat ).
% 2.40/1.47
% 2.40/1.47 thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
% 2.40/1.47 times_times_nat: nat > nat > nat ).
% 2.40/1.47
% 2.40/1.47 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 2.40/1.47 zero_zero_nat: nat ).
% 2.40/1.47
% 2.40/1.47 thf(sy_c_Number__Partition__Mirabelle__zerdlymyoj_Opartitions,type,
% 2.40/1.47 number1551313001itions: ( nat > nat ) > nat > $o ).
% 2.40/1.47
% 2.40/1.47 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 2.40/1.47 ord_less_nat: nat > nat > $o ).
% 2.40/1.47
% 2.40/1.47 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 2.40/1.47 ord_less_eq_nat: nat > nat > $o ).
% 2.40/1.47
% 2.40/1.47 thf(sy_v_i____,type,
% 2.40/1.47 i: nat ).
% 2.40/1.47
% 2.40/1.47 thf(sy_v_k,type,
% 2.40/1.47 k: nat ).
% 2.40/1.47
% 2.40/1.47 thf(sy_v_n,type,
% 2.40/1.47 n: nat ).
% 2.40/1.47
% 2.40/1.47 thf(sy_v_p,type,
% 2.40/1.47 p: nat > nat ).
% 2.40/1.47
% 2.40/1.47 thf(fact_0__092_060open_062i_A_092_060le_062_An_092_060close_062,axiom,
% 2.40/1.47 ord_less_eq_nat @ i @ n ).
% 2.40/1.47
% 2.40/1.47 thf(fact_1_assms_I1_J,axiom,
% 2.40/1.47 number1551313001itions @ p @ n ).
% 2.40/1.47
% 2.40/1.47 thf(fact_2_upper__bound,axiom,
% 2.40/1.47 ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) ) @ n ).
% 2.40/1.47
% 2.40/1.47 thf(fact_3_lower__bound,axiom,
% 2.40/1.47 ord_less_nat @ n @ ( plus_plus_nat @ ( times_times_nat @ ( p @ i ) @ i ) @ ( times_times_nat @ ( p @ k ) @ k ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_4__092_060open_062k_A_092_060le_062_An_092_060close_062,axiom,
% 2.40/1.47 ord_less_eq_nat @ k @ n ).
% 2.40/1.47
% 2.40/1.47 thf(fact_5__092_060open_062p_Ai_A_092_060noteq_062_A0_092_060close_062,axiom,
% 2.40/1.47 ( ( p @ i )
% 2.40/1.47 != zero_zero_nat ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_6_assms_I2_J,axiom,
% 2.40/1.47 ord_less_nat @ zero_zero_nat @ ( p @ k ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_7__092_060open_062i_A_092_060noteq_062_Ak_092_060close_062,axiom,
% 2.40/1.47 i != k ).
% 2.40/1.47
% 2.40/1.47 thf(fact_8__092_060open_062n_A_N_Ak_A_060_Ai_092_060close_062,axiom,
% 2.40/1.47 ord_less_nat @ ( minus_minus_nat @ n @ k ) @ i ).
% 2.40/1.47
% 2.40/1.47 thf(fact_9_partitions__zero,axiom,
% 2.40/1.47 ! [P: nat > nat] :
% 2.40/1.47 ( ( number1551313001itions @ P @ zero_zero_nat )
% 2.40/1.47 = ( P
% 2.40/1.47 = ( ^ [I: nat] : zero_zero_nat ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_10_partitions__bounds,axiom,
% 2.40/1.47 ! [P: nat > nat,N: nat,I2: nat] :
% 2.40/1.47 ( ( number1551313001itions @ P @ N )
% 2.40/1.47 => ( ord_less_eq_nat @ ( P @ I2 ) @ N ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_11_mult__le__cancel2,axiom,
% 2.40/1.47 ! [M: nat,K: nat,N: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 2.40/1.47 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 2.40/1.47 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_12_nat__mult__le__cancel__disj,axiom,
% 2.40/1.47 ! [K: nat,M: nat,N: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 2.40/1.47 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 2.40/1.47 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_13_mult__less__cancel2,axiom,
% 2.40/1.47 ! [M: nat,K: nat,N: nat] :
% 2.40/1.47 ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 2.40/1.47 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 2.40/1.47 & ( ord_less_nat @ M @ N ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_14_nat__0__less__mult__iff,axiom,
% 2.40/1.47 ! [M: nat,N: nat] :
% 2.40/1.47 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
% 2.40/1.47 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 2.40/1.47 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_15_nat__mult__less__cancel__disj,axiom,
% 2.40/1.47 ! [K: nat,M: nat,N: nat] :
% 2.40/1.47 ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 2.40/1.47 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 2.40/1.47 & ( ord_less_nat @ M @ N ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_16_add__gr__0,axiom,
% 2.40/1.47 ! [M: nat,N: nat] :
% 2.40/1.47 ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
% 2.40/1.47 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 2.40/1.47 | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_17_add__less__same__cancel1,axiom,
% 2.40/1.47 ! [B: nat,A: nat] :
% 2.40/1.47 ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 2.40/1.47 = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_18_add__less__same__cancel2,axiom,
% 2.40/1.47 ! [A: nat,B: nat] :
% 2.40/1.47 ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 2.40/1.47 = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_19_less__add__same__cancel1,axiom,
% 2.40/1.47 ! [A: nat,B: nat] :
% 2.40/1.47 ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 2.40/1.47 = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_20_less__add__same__cancel2,axiom,
% 2.40/1.47 ! [A: nat,B: nat] :
% 2.40/1.47 ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 2.40/1.47 = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_21_add__right__cancel,axiom,
% 2.40/1.47 ! [B: nat,A: nat,C: nat] :
% 2.40/1.47 ( ( ( plus_plus_nat @ B @ A )
% 2.40/1.47 = ( plus_plus_nat @ C @ A ) )
% 2.40/1.47 = ( B = C ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_22_add__left__cancel,axiom,
% 2.40/1.47 ! [A: nat,B: nat,C: nat] :
% 2.40/1.47 ( ( ( plus_plus_nat @ A @ B )
% 2.40/1.47 = ( plus_plus_nat @ A @ C ) )
% 2.40/1.47 = ( B = C ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_23_le__zero__eq,axiom,
% 2.40/1.47 ! [N: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 2.40/1.47 = ( N = zero_zero_nat ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_24_not__gr__zero,axiom,
% 2.40/1.47 ! [N: nat] :
% 2.40/1.47 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 2.40/1.47 = ( N = zero_zero_nat ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_25_add__le__cancel__right,axiom,
% 2.40/1.47 ! [A: nat,C: nat,B: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 2.40/1.47 = ( ord_less_eq_nat @ A @ B ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_26_add__le__cancel__left,axiom,
% 2.40/1.47 ! [C: nat,A: nat,B: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 2.40/1.47 = ( ord_less_eq_nat @ A @ B ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_27_zero__eq__add__iff__both__eq__0,axiom,
% 2.40/1.47 ! [X: nat,Y: nat] :
% 2.40/1.47 ( ( zero_zero_nat
% 2.40/1.47 = ( plus_plus_nat @ X @ Y ) )
% 2.40/1.47 = ( ( X = zero_zero_nat )
% 2.40/1.47 & ( Y = zero_zero_nat ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_28_add__eq__0__iff__both__eq__0,axiom,
% 2.40/1.47 ! [X: nat,Y: nat] :
% 2.40/1.47 ( ( ( plus_plus_nat @ X @ Y )
% 2.40/1.47 = zero_zero_nat )
% 2.40/1.47 = ( ( X = zero_zero_nat )
% 2.40/1.47 & ( Y = zero_zero_nat ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_29_add__cancel__right__right,axiom,
% 2.40/1.47 ! [A: nat,B: nat] :
% 2.40/1.47 ( ( A
% 2.40/1.47 = ( plus_plus_nat @ A @ B ) )
% 2.40/1.47 = ( B = zero_zero_nat ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_30_add__cancel__right__left,axiom,
% 2.40/1.47 ! [A: nat,B: nat] :
% 2.40/1.47 ( ( A
% 2.40/1.47 = ( plus_plus_nat @ B @ A ) )
% 2.40/1.47 = ( B = zero_zero_nat ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_31_add__cancel__left__right,axiom,
% 2.40/1.47 ! [A: nat,B: nat] :
% 2.40/1.47 ( ( ( plus_plus_nat @ A @ B )
% 2.40/1.47 = A )
% 2.40/1.47 = ( B = zero_zero_nat ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_32_add__cancel__left__left,axiom,
% 2.40/1.47 ! [B: nat,A: nat] :
% 2.40/1.47 ( ( ( plus_plus_nat @ B @ A )
% 2.40/1.47 = A )
% 2.40/1.47 = ( B = zero_zero_nat ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_33_add_Oright__neutral,axiom,
% 2.40/1.47 ! [A: nat] :
% 2.40/1.47 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 2.40/1.47 = A ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_34_add_Oleft__neutral,axiom,
% 2.40/1.47 ! [A: nat] :
% 2.40/1.47 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 2.40/1.47 = A ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_35_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 2.40/1.47 ! [A: nat] :
% 2.40/1.47 ( ( minus_minus_nat @ A @ A )
% 2.40/1.47 = zero_zero_nat ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_36_diff__zero,axiom,
% 2.40/1.47 ! [A: nat] :
% 2.40/1.47 ( ( minus_minus_nat @ A @ zero_zero_nat )
% 2.40/1.47 = A ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_37_zero__diff,axiom,
% 2.40/1.47 ! [A: nat] :
% 2.40/1.47 ( ( minus_minus_nat @ zero_zero_nat @ A )
% 2.40/1.47 = zero_zero_nat ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_38_add__less__cancel__right,axiom,
% 2.40/1.47 ! [A: nat,C: nat,B: nat] :
% 2.40/1.47 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 2.40/1.47 = ( ord_less_nat @ A @ B ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_39_add__less__cancel__left,axiom,
% 2.40/1.47 ! [C: nat,A: nat,B: nat] :
% 2.40/1.47 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 2.40/1.47 = ( ord_less_nat @ A @ B ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_40_add__diff__cancel__right_H,axiom,
% 2.40/1.47 ! [A: nat,B: nat] :
% 2.40/1.47 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 2.40/1.47 = A ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_41_add__diff__cancel__right,axiom,
% 2.40/1.47 ! [A: nat,C: nat,B: nat] :
% 2.40/1.47 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 2.40/1.47 = ( minus_minus_nat @ A @ B ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_42_add__diff__cancel__left_H,axiom,
% 2.40/1.47 ! [A: nat,B: nat] :
% 2.40/1.47 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
% 2.40/1.47 = B ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_43_add__diff__cancel__left,axiom,
% 2.40/1.47 ! [C: nat,A: nat,B: nat] :
% 2.40/1.47 ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 2.40/1.47 = ( minus_minus_nat @ A @ B ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_44_bot__nat__0_Onot__eq__extremum,axiom,
% 2.40/1.47 ! [A: nat] :
% 2.40/1.47 ( ( A != zero_zero_nat )
% 2.40/1.47 = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_45_less__nat__zero__code,axiom,
% 2.40/1.47 ! [N: nat] :
% 2.40/1.47 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_46_neq0__conv,axiom,
% 2.40/1.47 ! [N: nat] :
% 2.40/1.47 ( ( N != zero_zero_nat )
% 2.40/1.47 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_47_bot__nat__0_Oextremum,axiom,
% 2.40/1.47 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_48_le0,axiom,
% 2.40/1.47 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_49_Nat_Oadd__0__right,axiom,
% 2.40/1.47 ! [M: nat] :
% 2.40/1.47 ( ( plus_plus_nat @ M @ zero_zero_nat )
% 2.40/1.47 = M ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_50_add__is__0,axiom,
% 2.40/1.47 ! [M: nat,N: nat] :
% 2.40/1.47 ( ( ( plus_plus_nat @ M @ N )
% 2.40/1.47 = zero_zero_nat )
% 2.40/1.47 = ( ( M = zero_zero_nat )
% 2.40/1.47 & ( N = zero_zero_nat ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_51_nat__add__left__cancel__less,axiom,
% 2.40/1.47 ! [K: nat,M: nat,N: nat] :
% 2.40/1.47 ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 2.40/1.47 = ( ord_less_nat @ M @ N ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_52_diff__self__eq__0,axiom,
% 2.40/1.47 ! [M: nat] :
% 2.40/1.47 ( ( minus_minus_nat @ M @ M )
% 2.40/1.47 = zero_zero_nat ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_53_diff__0__eq__0,axiom,
% 2.40/1.47 ! [N: nat] :
% 2.40/1.47 ( ( minus_minus_nat @ zero_zero_nat @ N )
% 2.40/1.47 = zero_zero_nat ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_54_mult__cancel2,axiom,
% 2.40/1.47 ! [M: nat,K: nat,N: nat] :
% 2.40/1.47 ( ( ( times_times_nat @ M @ K )
% 2.40/1.47 = ( times_times_nat @ N @ K ) )
% 2.40/1.47 = ( ( M = N )
% 2.40/1.47 | ( K = zero_zero_nat ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_55_mult__cancel1,axiom,
% 2.40/1.47 ! [K: nat,M: nat,N: nat] :
% 2.40/1.47 ( ( ( times_times_nat @ K @ M )
% 2.40/1.47 = ( times_times_nat @ K @ N ) )
% 2.40/1.47 = ( ( M = N )
% 2.40/1.47 | ( K = zero_zero_nat ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_56_mult__0__right,axiom,
% 2.40/1.47 ! [M: nat] :
% 2.40/1.47 ( ( times_times_nat @ M @ zero_zero_nat )
% 2.40/1.47 = zero_zero_nat ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_57_mult__is__0,axiom,
% 2.40/1.47 ! [M: nat,N: nat] :
% 2.40/1.47 ( ( ( times_times_nat @ M @ N )
% 2.40/1.47 = zero_zero_nat )
% 2.40/1.47 = ( ( M = zero_zero_nat )
% 2.40/1.47 | ( N = zero_zero_nat ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_58_nat__add__left__cancel__le,axiom,
% 2.40/1.47 ! [K: nat,M: nat,N: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 2.40/1.47 = ( ord_less_eq_nat @ M @ N ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_59_diff__diff__cancel,axiom,
% 2.40/1.47 ! [I2: nat,N: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ I2 @ N )
% 2.40/1.47 => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I2 ) )
% 2.40/1.47 = I2 ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_60_diff__diff__left,axiom,
% 2.40/1.47 ! [I2: nat,J: nat,K: nat] :
% 2.40/1.47 ( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
% 2.40/1.47 = ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_61_le__add__same__cancel2,axiom,
% 2.40/1.47 ! [A: nat,B: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 2.40/1.47 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_62_le__add__same__cancel1,axiom,
% 2.40/1.47 ! [A: nat,B: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 2.40/1.47 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_63_add__le__same__cancel2,axiom,
% 2.40/1.47 ! [A: nat,B: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 2.40/1.47 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_64_add__le__same__cancel1,axiom,
% 2.40/1.47 ! [B: nat,A: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 2.40/1.47 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_65_diff__add__zero,axiom,
% 2.40/1.47 ! [A: nat,B: nat] :
% 2.40/1.47 ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 2.40/1.47 = zero_zero_nat ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_66_zero__less__diff,axiom,
% 2.40/1.47 ! [N: nat,M: nat] :
% 2.40/1.47 ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
% 2.40/1.47 = ( ord_less_nat @ M @ N ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_67_diff__is__0__eq_H,axiom,
% 2.40/1.47 ! [M: nat,N: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ M @ N )
% 2.40/1.47 => ( ( minus_minus_nat @ M @ N )
% 2.40/1.47 = zero_zero_nat ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_68_diff__is__0__eq,axiom,
% 2.40/1.47 ! [M: nat,N: nat] :
% 2.40/1.47 ( ( ( minus_minus_nat @ M @ N )
% 2.40/1.47 = zero_zero_nat )
% 2.40/1.47 = ( ord_less_eq_nat @ M @ N ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_69_Nat_Odiff__diff__right,axiom,
% 2.40/1.47 ! [K: nat,J: nat,I2: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ K @ J )
% 2.40/1.47 => ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
% 2.40/1.47 = ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_70_Nat_Oadd__diff__assoc2,axiom,
% 2.40/1.47 ! [K: nat,J: nat,I2: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ K @ J )
% 2.40/1.47 => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
% 2.40/1.47 = ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_71_Nat_Oadd__diff__assoc,axiom,
% 2.40/1.47 ! [K: nat,J: nat,I2: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ K @ J )
% 2.40/1.47 => ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
% 2.40/1.47 = ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_72_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
% 2.40/1.47 ! [A: nat,C: nat,B: nat] :
% 2.40/1.47 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
% 2.40/1.47 = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_73_diff__commute,axiom,
% 2.40/1.47 ! [I2: nat,J: nat,K: nat] :
% 2.40/1.47 ( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
% 2.40/1.47 = ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_74_add__implies__diff,axiom,
% 2.40/1.47 ! [C: nat,B: nat,A: nat] :
% 2.40/1.47 ( ( ( plus_plus_nat @ C @ B )
% 2.40/1.47 = A )
% 2.40/1.47 => ( C
% 2.40/1.47 = ( minus_minus_nat @ A @ B ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_75_diff__diff__add,axiom,
% 2.40/1.47 ! [A: nat,B: nat,C: nat] :
% 2.40/1.47 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
% 2.40/1.47 = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_76_minus__nat_Odiff__0,axiom,
% 2.40/1.47 ! [M: nat] :
% 2.40/1.47 ( ( minus_minus_nat @ M @ zero_zero_nat )
% 2.40/1.47 = M ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_77_diffs0__imp__equal,axiom,
% 2.40/1.47 ! [M: nat,N: nat] :
% 2.40/1.47 ( ( ( minus_minus_nat @ M @ N )
% 2.40/1.47 = zero_zero_nat )
% 2.40/1.47 => ( ( ( minus_minus_nat @ N @ M )
% 2.40/1.47 = zero_zero_nat )
% 2.40/1.47 => ( M = N ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_78_less__imp__diff__less,axiom,
% 2.40/1.47 ! [J: nat,K: nat,N: nat] :
% 2.40/1.47 ( ( ord_less_nat @ J @ K )
% 2.40/1.47 => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_79_diff__less__mono2,axiom,
% 2.40/1.47 ! [M: nat,N: nat,L: nat] :
% 2.40/1.47 ( ( ord_less_nat @ M @ N )
% 2.40/1.47 => ( ( ord_less_nat @ M @ L )
% 2.40/1.47 => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_80_diff__le__mono2,axiom,
% 2.40/1.47 ! [M: nat,N: nat,L: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ M @ N )
% 2.40/1.47 => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_81_le__diff__iff_H,axiom,
% 2.40/1.47 ! [A: nat,C: nat,B: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ A @ C )
% 2.40/1.47 => ( ( ord_less_eq_nat @ B @ C )
% 2.40/1.47 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
% 2.40/1.47 = ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_82_diff__le__self,axiom,
% 2.40/1.47 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_83_diff__le__mono,axiom,
% 2.40/1.47 ! [M: nat,N: nat,L: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ M @ N )
% 2.40/1.47 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_84_Nat_Odiff__diff__eq,axiom,
% 2.40/1.47 ! [K: nat,M: nat,N: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ K @ M )
% 2.40/1.47 => ( ( ord_less_eq_nat @ K @ N )
% 2.40/1.47 => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 2.40/1.47 = ( minus_minus_nat @ M @ N ) ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_85_le__diff__iff,axiom,
% 2.40/1.47 ! [K: nat,M: nat,N: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ K @ M )
% 2.40/1.47 => ( ( ord_less_eq_nat @ K @ N )
% 2.40/1.47 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 2.40/1.47 = ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_86_eq__diff__iff,axiom,
% 2.40/1.47 ! [K: nat,M: nat,N: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ K @ M )
% 2.40/1.47 => ( ( ord_less_eq_nat @ K @ N )
% 2.40/1.47 => ( ( ( minus_minus_nat @ M @ K )
% 2.40/1.47 = ( minus_minus_nat @ N @ K ) )
% 2.40/1.47 = ( M = N ) ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_87_diff__add__inverse2,axiom,
% 2.40/1.47 ! [M: nat,N: nat] :
% 2.40/1.47 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
% 2.40/1.47 = M ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_88_diff__add__inverse,axiom,
% 2.40/1.47 ! [N: nat,M: nat] :
% 2.40/1.47 ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
% 2.40/1.47 = M ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_89_diff__cancel2,axiom,
% 2.40/1.47 ! [M: nat,K: nat,N: nat] :
% 2.40/1.47 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
% 2.40/1.47 = ( minus_minus_nat @ M @ N ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_90_Nat_Odiff__cancel,axiom,
% 2.40/1.47 ! [K: nat,M: nat,N: nat] :
% 2.40/1.47 ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 2.40/1.47 = ( minus_minus_nat @ M @ N ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_91_diff__mult__distrib2,axiom,
% 2.40/1.47 ! [K: nat,M: nat,N: nat] :
% 2.40/1.47 ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 2.40/1.47 = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_92_diff__mult__distrib,axiom,
% 2.40/1.47 ! [M: nat,N: nat,K: nat] :
% 2.40/1.47 ( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
% 2.40/1.47 = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_93_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 2.40/1.47 ! [A: nat,B: nat,C: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ A @ B )
% 2.40/1.47 => ( ( ord_less_eq_nat @ A @ B )
% 2.40/1.47 => ( ( ( minus_minus_nat @ B @ A )
% 2.40/1.47 = C )
% 2.40/1.47 = ( B
% 2.40/1.47 = ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_94_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 2.40/1.47 ! [A: nat,B: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ A @ B )
% 2.40/1.47 => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
% 2.40/1.47 = B ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_95_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 2.40/1.47 ! [A: nat,B: nat,C: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ A @ B )
% 2.40/1.47 => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 2.40/1.47 = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_96_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 2.40/1.47 ! [A: nat,B: nat,C: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ A @ B )
% 2.40/1.47 => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
% 2.40/1.47 = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_97_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 2.40/1.47 ! [A: nat,B: nat,C: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ A @ B )
% 2.40/1.47 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
% 2.40/1.47 = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_98_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 2.40/1.47 ! [A: nat,B: nat,C: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ A @ B )
% 2.40/1.47 => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
% 2.40/1.47 = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_99_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 2.40/1.47 ! [A: nat,B: nat,C: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ A @ B )
% 2.40/1.47 => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 2.40/1.47 = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_100_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 2.40/1.47 ! [A: nat,B: nat,C: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ A @ B )
% 2.40/1.47 => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 2.40/1.47 = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_101_le__add__diff,axiom,
% 2.40/1.47 ! [A: nat,B: nat,C: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ A @ B )
% 2.40/1.47 => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_102_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
% 2.40/1.47 ! [A: nat,B: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ A @ B )
% 2.40/1.47 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
% 2.40/1.47 = B ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_103_diff__less,axiom,
% 2.40/1.47 ! [N: nat,M: nat] :
% 2.40/1.47 ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.40/1.47 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 2.40/1.47 => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_104_diff__add__0,axiom,
% 2.40/1.47 ! [N: nat,M: nat] :
% 2.40/1.47 ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
% 2.40/1.47 = zero_zero_nat ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_105_diff__less__mono,axiom,
% 2.40/1.47 ! [A: nat,B: nat,C: nat] :
% 2.40/1.47 ( ( ord_less_nat @ A @ B )
% 2.40/1.47 => ( ( ord_less_eq_nat @ C @ A )
% 2.40/1.47 => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_106_less__diff__iff,axiom,
% 2.40/1.47 ! [K: nat,M: nat,N: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ K @ M )
% 2.40/1.47 => ( ( ord_less_eq_nat @ K @ N )
% 2.40/1.47 => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 2.40/1.47 = ( ord_less_nat @ M @ N ) ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_107_add__diff__inverse__nat,axiom,
% 2.40/1.47 ! [M: nat,N: nat] :
% 2.40/1.47 ( ~ ( ord_less_nat @ M @ N )
% 2.40/1.47 => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
% 2.40/1.47 = M ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_108_less__diff__conv,axiom,
% 2.40/1.47 ! [I2: nat,J: nat,K: nat] :
% 2.40/1.47 ( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
% 2.40/1.47 = ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_109_Nat_Ole__imp__diff__is__add,axiom,
% 2.40/1.47 ! [I2: nat,J: nat,K: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ I2 @ J )
% 2.40/1.47 => ( ( ( minus_minus_nat @ J @ I2 )
% 2.40/1.47 = K )
% 2.40/1.47 = ( J
% 2.40/1.47 = ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_110_Nat_Odiff__add__assoc2,axiom,
% 2.40/1.47 ! [K: nat,J: nat,I2: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ K @ J )
% 2.40/1.47 => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K )
% 2.40/1.47 = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_111_Nat_Odiff__add__assoc,axiom,
% 2.40/1.47 ! [K: nat,J: nat,I2: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ K @ J )
% 2.40/1.47 => ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
% 2.40/1.47 = ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_112_Nat_Ole__diff__conv2,axiom,
% 2.40/1.47 ! [K: nat,J: nat,I2: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ K @ J )
% 2.40/1.47 => ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
% 2.40/1.47 = ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_113_le__diff__conv,axiom,
% 2.40/1.47 ! [J: nat,K: nat,I2: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
% 2.40/1.47 = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_114_nat__diff__split__asm,axiom,
% 2.40/1.47 ! [P2: nat > $o,A: nat,B: nat] :
% 2.40/1.47 ( ( P2 @ ( minus_minus_nat @ A @ B ) )
% 2.40/1.47 = ( ~ ( ( ( ord_less_nat @ A @ B )
% 2.40/1.47 & ~ ( P2 @ zero_zero_nat ) )
% 2.40/1.47 | ? [D: nat] :
% 2.40/1.47 ( ( A
% 2.40/1.47 = ( plus_plus_nat @ B @ D ) )
% 2.40/1.47 & ~ ( P2 @ D ) ) ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_115_nat__diff__split,axiom,
% 2.40/1.47 ! [P2: nat > $o,A: nat,B: nat] :
% 2.40/1.47 ( ( P2 @ ( minus_minus_nat @ A @ B ) )
% 2.40/1.47 = ( ( ( ord_less_nat @ A @ B )
% 2.40/1.47 => ( P2 @ zero_zero_nat ) )
% 2.40/1.47 & ! [D: nat] :
% 2.40/1.47 ( ( A
% 2.40/1.47 = ( plus_plus_nat @ B @ D ) )
% 2.40/1.47 => ( P2 @ D ) ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_116_less__diff__conv2,axiom,
% 2.40/1.47 ! [K: nat,J: nat,I2: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ K @ J )
% 2.40/1.47 => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
% 2.40/1.47 = ( ord_less_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_117_nat__diff__add__eq2,axiom,
% 2.40/1.47 ! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ I2 @ J )
% 2.40/1.47 => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 2.40/1.47 = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_118_nat__diff__add__eq1,axiom,
% 2.40/1.47 ! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ J @ I2 )
% 2.40/1.47 => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 2.40/1.47 = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M ) @ N ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_119_nat__le__add__iff2,axiom,
% 2.40/1.47 ! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ I2 @ J )
% 2.40/1.47 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 2.40/1.47 = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_120_nat__le__add__iff1,axiom,
% 2.40/1.47 ! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ J @ I2 )
% 2.40/1.47 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 2.40/1.47 = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M ) @ N ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_121_nat__eq__add__iff2,axiom,
% 2.40/1.47 ! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ I2 @ J )
% 2.40/1.47 => ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M )
% 2.40/1.47 = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 2.40/1.47 = ( M
% 2.40/1.47 = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_122_nat__eq__add__iff1,axiom,
% 2.40/1.47 ! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ J @ I2 )
% 2.40/1.47 => ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M )
% 2.40/1.47 = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 2.40/1.47 = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M )
% 2.40/1.47 = N ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_123_nat__less__add__iff2,axiom,
% 2.40/1.47 ! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ I2 @ J )
% 2.40/1.47 => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 2.40/1.47 = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_124_nat__less__add__iff1,axiom,
% 2.40/1.47 ! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
% 2.40/1.47 ( ( ord_less_eq_nat @ J @ I2 )
% 2.40/1.47 => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 2.40/1.47 = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M ) @ N ) ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_125_zero__reorient,axiom,
% 2.40/1.47 ! [X: nat] :
% 2.40/1.47 ( ( zero_zero_nat = X )
% 2.40/1.47 = ( X = zero_zero_nat ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_126_add__right__imp__eq,axiom,
% 2.40/1.47 ! [B: nat,A: nat,C: nat] :
% 2.40/1.47 ( ( ( plus_plus_nat @ B @ A )
% 2.40/1.47 = ( plus_plus_nat @ C @ A ) )
% 2.40/1.47 => ( B = C ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_127_add__left__imp__eq,axiom,
% 2.40/1.47 ! [A: nat,B: nat,C: nat] :
% 2.40/1.47 ( ( ( plus_plus_nat @ A @ B )
% 2.40/1.47 = ( plus_plus_nat @ A @ C ) )
% 2.40/1.47 => ( B = C ) ) ).
% 2.40/1.47
% 2.40/1.47 thf(fact_128_add_Oleft__commute,axiom,
% 2.40/1.47 ! [B: nat,A: nat,C: nat] :
% 2.40/1.48 ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
% 2.40/1.48 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_129_add_Ocommute,axiom,
% 2.40/1.48 ( plus_plus_nat
% 2.40/1.48 = ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_130_add_Oassoc,axiom,
% 2.40/1.48 ! [A: nat,B: nat,C: nat] :
% 2.40/1.48 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 2.40/1.48 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_131_group__cancel_Oadd2,axiom,
% 2.40/1.48 ! [B3: nat,K: nat,B: nat,A: nat] :
% 2.40/1.48 ( ( B3
% 2.40/1.48 = ( plus_plus_nat @ K @ B ) )
% 2.40/1.48 => ( ( plus_plus_nat @ A @ B3 )
% 2.40/1.48 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_132_group__cancel_Oadd1,axiom,
% 2.40/1.48 ! [A3: nat,K: nat,A: nat,B: nat] :
% 2.40/1.48 ( ( A3
% 2.40/1.48 = ( plus_plus_nat @ K @ A ) )
% 2.40/1.48 => ( ( plus_plus_nat @ A3 @ B )
% 2.40/1.48 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_133_add__mono__thms__linordered__semiring_I4_J,axiom,
% 2.40/1.48 ! [I2: nat,J: nat,K: nat,L: nat] :
% 2.40/1.48 ( ( ( I2 = J )
% 2.40/1.48 & ( K = L ) )
% 2.40/1.48 => ( ( plus_plus_nat @ I2 @ K )
% 2.40/1.48 = ( plus_plus_nat @ J @ L ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_134_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 2.40/1.48 ! [A: nat,B: nat,C: nat] :
% 2.40/1.48 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 2.40/1.48 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_135_mult_Oleft__commute,axiom,
% 2.40/1.48 ! [B: nat,A: nat,C: nat] :
% 2.40/1.48 ( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
% 2.40/1.48 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_136_mult_Ocommute,axiom,
% 2.40/1.48 ( times_times_nat
% 2.40/1.48 = ( ^ [A2: nat,B2: nat] : ( times_times_nat @ B2 @ A2 ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_137_mult_Oassoc,axiom,
% 2.40/1.48 ! [A: nat,B: nat,C: nat] :
% 2.40/1.48 ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 2.40/1.48 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_138_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 2.40/1.48 ! [A: nat,B: nat,C: nat] :
% 2.40/1.48 ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 2.40/1.48 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_139_linorder__neqE__nat,axiom,
% 2.40/1.48 ! [X: nat,Y: nat] :
% 2.40/1.48 ( ( X != Y )
% 2.40/1.48 => ( ~ ( ord_less_nat @ X @ Y )
% 2.40/1.48 => ( ord_less_nat @ Y @ X ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_140_infinite__descent,axiom,
% 2.40/1.48 ! [P2: nat > $o,N: nat] :
% 2.40/1.48 ( ! [N2: nat] :
% 2.40/1.48 ( ~ ( P2 @ N2 )
% 2.40/1.48 => ? [M2: nat] :
% 2.40/1.48 ( ( ord_less_nat @ M2 @ N2 )
% 2.40/1.48 & ~ ( P2 @ M2 ) ) )
% 2.40/1.48 => ( P2 @ N ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_141_nat__less__induct,axiom,
% 2.40/1.48 ! [P2: nat > $o,N: nat] :
% 2.40/1.48 ( ! [N2: nat] :
% 2.40/1.48 ( ! [M2: nat] :
% 2.40/1.48 ( ( ord_less_nat @ M2 @ N2 )
% 2.40/1.48 => ( P2 @ M2 ) )
% 2.40/1.48 => ( P2 @ N2 ) )
% 2.40/1.48 => ( P2 @ N ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_142_less__irrefl__nat,axiom,
% 2.40/1.48 ! [N: nat] :
% 2.40/1.48 ~ ( ord_less_nat @ N @ N ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_143_less__not__refl3,axiom,
% 2.40/1.48 ! [S: nat,T: nat] :
% 2.40/1.48 ( ( ord_less_nat @ S @ T )
% 2.40/1.48 => ( S != T ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_144_less__not__refl2,axiom,
% 2.40/1.48 ! [N: nat,M: nat] :
% 2.40/1.48 ( ( ord_less_nat @ N @ M )
% 2.40/1.48 => ( M != N ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_145_less__not__refl,axiom,
% 2.40/1.48 ! [N: nat] :
% 2.40/1.48 ~ ( ord_less_nat @ N @ N ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_146_nat__neq__iff,axiom,
% 2.40/1.48 ! [M: nat,N: nat] :
% 2.40/1.48 ( ( M != N )
% 2.40/1.48 = ( ( ord_less_nat @ M @ N )
% 2.40/1.48 | ( ord_less_nat @ N @ M ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_147_Nat_Oex__has__greatest__nat,axiom,
% 2.40/1.48 ! [P2: nat > $o,K: nat,B: nat] :
% 2.40/1.48 ( ( P2 @ K )
% 2.40/1.48 => ( ! [Y2: nat] :
% 2.40/1.48 ( ( P2 @ Y2 )
% 2.40/1.48 => ( ord_less_eq_nat @ Y2 @ B ) )
% 2.40/1.48 => ? [X2: nat] :
% 2.40/1.48 ( ( P2 @ X2 )
% 2.40/1.48 & ! [Y3: nat] :
% 2.40/1.48 ( ( P2 @ Y3 )
% 2.40/1.48 => ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_148_nat__le__linear,axiom,
% 2.40/1.48 ! [M: nat,N: nat] :
% 2.40/1.48 ( ( ord_less_eq_nat @ M @ N )
% 2.40/1.48 | ( ord_less_eq_nat @ N @ M ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_149_le__antisym,axiom,
% 2.40/1.48 ! [M: nat,N: nat] :
% 2.40/1.48 ( ( ord_less_eq_nat @ M @ N )
% 2.40/1.48 => ( ( ord_less_eq_nat @ N @ M )
% 2.40/1.48 => ( M = N ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_150_eq__imp__le,axiom,
% 2.40/1.48 ! [M: nat,N: nat] :
% 2.40/1.48 ( ( M = N )
% 2.40/1.48 => ( ord_less_eq_nat @ M @ N ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_151_le__trans,axiom,
% 2.40/1.48 ! [I2: nat,J: nat,K: nat] :
% 2.40/1.48 ( ( ord_less_eq_nat @ I2 @ J )
% 2.40/1.48 => ( ( ord_less_eq_nat @ J @ K )
% 2.40/1.48 => ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_152_le__refl,axiom,
% 2.40/1.48 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_153_zero__le,axiom,
% 2.40/1.48 ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_154_zero__less__iff__neq__zero,axiom,
% 2.40/1.48 ! [N: nat] :
% 2.40/1.48 ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.40/1.48 = ( N != zero_zero_nat ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_155_gr__implies__not__zero,axiom,
% 2.40/1.48 ! [M: nat,N: nat] :
% 2.40/1.48 ( ( ord_less_nat @ M @ N )
% 2.40/1.48 => ( N != zero_zero_nat ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_156_not__less__zero,axiom,
% 2.40/1.48 ! [N: nat] :
% 2.40/1.48 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_157_gr__zeroI,axiom,
% 2.40/1.48 ! [N: nat] :
% 2.40/1.48 ( ( N != zero_zero_nat )
% 2.40/1.48 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_158_add__le__imp__le__right,axiom,
% 2.40/1.48 ! [A: nat,C: nat,B: nat] :
% 2.40/1.48 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 2.40/1.48 => ( ord_less_eq_nat @ A @ B ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_159_add__le__imp__le__left,axiom,
% 2.40/1.48 ! [C: nat,A: nat,B: nat] :
% 2.40/1.48 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 2.40/1.48 => ( ord_less_eq_nat @ A @ B ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_160_le__iff__add,axiom,
% 2.40/1.48 ( ord_less_eq_nat
% 2.40/1.48 = ( ^ [A2: nat,B2: nat] :
% 2.40/1.48 ? [C2: nat] :
% 2.40/1.48 ( B2
% 2.40/1.48 = ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_161_add__right__mono,axiom,
% 2.40/1.48 ! [A: nat,B: nat,C: nat] :
% 2.40/1.48 ( ( ord_less_eq_nat @ A @ B )
% 2.40/1.48 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_162_less__eqE,axiom,
% 2.40/1.48 ! [A: nat,B: nat] :
% 2.40/1.48 ( ( ord_less_eq_nat @ A @ B )
% 2.40/1.48 => ~ ! [C3: nat] :
% 2.40/1.48 ( B
% 2.40/1.48 != ( plus_plus_nat @ A @ C3 ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_163_add__left__mono,axiom,
% 2.40/1.48 ! [A: nat,B: nat,C: nat] :
% 2.40/1.48 ( ( ord_less_eq_nat @ A @ B )
% 2.40/1.48 => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_164_add__mono,axiom,
% 2.40/1.48 ! [A: nat,B: nat,C: nat,D2: nat] :
% 2.40/1.48 ( ( ord_less_eq_nat @ A @ B )
% 2.40/1.48 => ( ( ord_less_eq_nat @ C @ D2 )
% 2.40/1.48 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_165_add__mono__thms__linordered__semiring_I1_J,axiom,
% 2.40/1.48 ! [I2: nat,J: nat,K: nat,L: nat] :
% 2.40/1.48 ( ( ( ord_less_eq_nat @ I2 @ J )
% 2.40/1.48 & ( ord_less_eq_nat @ K @ L ) )
% 2.40/1.48 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_166_add__mono__thms__linordered__semiring_I2_J,axiom,
% 2.40/1.48 ! [I2: nat,J: nat,K: nat,L: nat] :
% 2.40/1.48 ( ( ( I2 = J )
% 2.40/1.48 & ( ord_less_eq_nat @ K @ L ) )
% 2.40/1.48 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_167_add__mono__thms__linordered__semiring_I3_J,axiom,
% 2.40/1.48 ! [I2: nat,J: nat,K: nat,L: nat] :
% 2.40/1.48 ( ( ( ord_less_eq_nat @ I2 @ J )
% 2.40/1.48 & ( K = L ) )
% 2.40/1.48 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_168_add_Ocomm__neutral,axiom,
% 2.40/1.48 ! [A: nat] :
% 2.40/1.48 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 2.40/1.48 = A ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_169_comm__monoid__add__class_Oadd__0,axiom,
% 2.40/1.48 ! [A: nat] :
% 2.40/1.48 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 2.40/1.48 = A ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_170_add__less__imp__less__right,axiom,
% 2.40/1.48 ! [A: nat,C: nat,B: nat] :
% 2.40/1.48 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 2.40/1.48 => ( ord_less_nat @ A @ B ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_171_add__less__imp__less__left,axiom,
% 2.40/1.48 ! [C: nat,A: nat,B: nat] :
% 2.40/1.48 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 2.40/1.48 => ( ord_less_nat @ A @ B ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_172_add__strict__right__mono,axiom,
% 2.40/1.48 ! [A: nat,B: nat,C: nat] :
% 2.40/1.48 ( ( ord_less_nat @ A @ B )
% 2.40/1.48 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_173_add__strict__left__mono,axiom,
% 2.40/1.48 ! [A: nat,B: nat,C: nat] :
% 2.40/1.48 ( ( ord_less_nat @ A @ B )
% 2.40/1.48 => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_174_add__strict__mono,axiom,
% 2.40/1.48 ! [A: nat,B: nat,C: nat,D2: nat] :
% 2.40/1.48 ( ( ord_less_nat @ A @ B )
% 2.40/1.48 => ( ( ord_less_nat @ C @ D2 )
% 2.40/1.48 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_175_add__mono__thms__linordered__field_I1_J,axiom,
% 2.40/1.48 ! [I2: nat,J: nat,K: nat,L: nat] :
% 2.40/1.48 ( ( ( ord_less_nat @ I2 @ J )
% 2.40/1.48 & ( K = L ) )
% 2.40/1.48 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_176_add__mono__thms__linordered__field_I2_J,axiom,
% 2.40/1.48 ! [I2: nat,J: nat,K: nat,L: nat] :
% 2.40/1.48 ( ( ( I2 = J )
% 2.40/1.48 & ( ord_less_nat @ K @ L ) )
% 2.40/1.48 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_177_add__mono__thms__linordered__field_I5_J,axiom,
% 2.40/1.48 ! [I2: nat,J: nat,K: nat,L: nat] :
% 2.40/1.48 ( ( ( ord_less_nat @ I2 @ J )
% 2.40/1.48 & ( ord_less_nat @ K @ L ) )
% 2.40/1.48 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_178_bot__nat__0_Oextremum__strict,axiom,
% 2.40/1.48 ! [A: nat] :
% 2.40/1.48 ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_179_infinite__descent0,axiom,
% 2.40/1.48 ! [P2: nat > $o,N: nat] :
% 2.40/1.48 ( ( P2 @ zero_zero_nat )
% 2.40/1.48 => ( ! [N2: nat] :
% 2.40/1.48 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 2.40/1.48 => ( ~ ( P2 @ N2 )
% 2.40/1.48 => ? [M2: nat] :
% 2.40/1.48 ( ( ord_less_nat @ M2 @ N2 )
% 2.40/1.48 & ~ ( P2 @ M2 ) ) ) )
% 2.40/1.48 => ( P2 @ N ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_180_gr__implies__not0,axiom,
% 2.40/1.48 ! [M: nat,N: nat] :
% 2.40/1.48 ( ( ord_less_nat @ M @ N )
% 2.40/1.48 => ( N != zero_zero_nat ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_181_less__zeroE,axiom,
% 2.40/1.48 ! [N: nat] :
% 2.40/1.48 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_182_not__less0,axiom,
% 2.40/1.48 ! [N: nat] :
% 2.40/1.48 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_183_not__gr0,axiom,
% 2.40/1.48 ! [N: nat] :
% 2.40/1.48 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 2.40/1.48 = ( N = zero_zero_nat ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_184_gr0I,axiom,
% 2.40/1.48 ! [N: nat] :
% 2.40/1.48 ( ( N != zero_zero_nat )
% 2.40/1.48 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_185_bot__nat__0_Oextremum__uniqueI,axiom,
% 2.40/1.48 ! [A: nat] :
% 2.40/1.48 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 2.40/1.48 => ( A = zero_zero_nat ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_186_bot__nat__0_Oextremum__unique,axiom,
% 2.40/1.48 ! [A: nat] :
% 2.40/1.48 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 2.40/1.48 = ( A = zero_zero_nat ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_187_le__0__eq,axiom,
% 2.40/1.48 ! [N: nat] :
% 2.40/1.48 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 2.40/1.48 = ( N = zero_zero_nat ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_188_less__eq__nat_Osimps_I1_J,axiom,
% 2.40/1.48 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_189_add__eq__self__zero,axiom,
% 2.40/1.48 ! [M: nat,N: nat] :
% 2.40/1.48 ( ( ( plus_plus_nat @ M @ N )
% 2.40/1.48 = M )
% 2.40/1.48 => ( N = zero_zero_nat ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_190_plus__nat_Oadd__0,axiom,
% 2.40/1.48 ! [N: nat] :
% 2.40/1.48 ( ( plus_plus_nat @ zero_zero_nat @ N )
% 2.40/1.48 = N ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_191_less__mono__imp__le__mono,axiom,
% 2.40/1.48 ! [F: nat > nat,I2: nat,J: nat] :
% 2.40/1.48 ( ! [I3: nat,J2: nat] :
% 2.40/1.48 ( ( ord_less_nat @ I3 @ J2 )
% 2.40/1.48 => ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
% 2.40/1.48 => ( ( ord_less_eq_nat @ I2 @ J )
% 2.40/1.48 => ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_192_le__neq__implies__less,axiom,
% 2.40/1.48 ! [M: nat,N: nat] :
% 2.40/1.48 ( ( ord_less_eq_nat @ M @ N )
% 2.40/1.48 => ( ( M != N )
% 2.40/1.48 => ( ord_less_nat @ M @ N ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_193_less__or__eq__imp__le,axiom,
% 2.40/1.48 ! [M: nat,N: nat] :
% 2.40/1.48 ( ( ( ord_less_nat @ M @ N )
% 2.40/1.48 | ( M = N ) )
% 2.40/1.48 => ( ord_less_eq_nat @ M @ N ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_194_le__eq__less__or__eq,axiom,
% 2.40/1.48 ( ord_less_eq_nat
% 2.40/1.48 = ( ^ [M3: nat,N3: nat] :
% 2.40/1.48 ( ( ord_less_nat @ M3 @ N3 )
% 2.40/1.48 | ( M3 = N3 ) ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_195_less__imp__le__nat,axiom,
% 2.40/1.48 ! [M: nat,N: nat] :
% 2.40/1.48 ( ( ord_less_nat @ M @ N )
% 2.40/1.48 => ( ord_less_eq_nat @ M @ N ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_196_nat__less__le,axiom,
% 2.40/1.48 ( ord_less_nat
% 2.40/1.48 = ( ^ [M3: nat,N3: nat] :
% 2.40/1.48 ( ( ord_less_eq_nat @ M3 @ N3 )
% 2.40/1.48 & ( M3 != N3 ) ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_197_less__add__eq__less,axiom,
% 2.40/1.48 ! [K: nat,L: nat,M: nat,N: nat] :
% 2.40/1.48 ( ( ord_less_nat @ K @ L )
% 2.40/1.48 => ( ( ( plus_plus_nat @ M @ L )
% 2.40/1.48 = ( plus_plus_nat @ K @ N ) )
% 2.40/1.48 => ( ord_less_nat @ M @ N ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_198_trans__less__add2,axiom,
% 2.40/1.48 ! [I2: nat,J: nat,M: nat] :
% 2.40/1.48 ( ( ord_less_nat @ I2 @ J )
% 2.40/1.48 => ( ord_less_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_199_trans__less__add1,axiom,
% 2.40/1.48 ! [I2: nat,J: nat,M: nat] :
% 2.40/1.48 ( ( ord_less_nat @ I2 @ J )
% 2.40/1.48 => ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_200_add__less__mono1,axiom,
% 2.40/1.48 ! [I2: nat,J: nat,K: nat] :
% 2.40/1.48 ( ( ord_less_nat @ I2 @ J )
% 2.40/1.48 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_201_not__add__less2,axiom,
% 2.40/1.48 ! [J: nat,I2: nat] :
% 2.40/1.48 ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_202_not__add__less1,axiom,
% 2.40/1.48 ! [I2: nat,J: nat] :
% 2.40/1.48 ~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_203_add__less__mono,axiom,
% 2.40/1.48 ! [I2: nat,J: nat,K: nat,L: nat] :
% 2.40/1.48 ( ( ord_less_nat @ I2 @ J )
% 2.40/1.48 => ( ( ord_less_nat @ K @ L )
% 2.40/1.48 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_204_add__lessD1,axiom,
% 2.40/1.48 ! [I2: nat,J: nat,K: nat] :
% 2.40/1.48 ( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
% 2.40/1.48 => ( ord_less_nat @ I2 @ K ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_205_nat__mult__eq__cancel__disj,axiom,
% 2.40/1.48 ! [K: nat,M: nat,N: nat] :
% 2.40/1.48 ( ( ( times_times_nat @ K @ M )
% 2.40/1.48 = ( times_times_nat @ K @ N ) )
% 2.40/1.48 = ( ( K = zero_zero_nat )
% 2.40/1.48 | ( M = N ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_206_mult__0,axiom,
% 2.40/1.48 ! [N: nat] :
% 2.40/1.48 ( ( times_times_nat @ zero_zero_nat @ N )
% 2.40/1.48 = zero_zero_nat ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_207_nat__le__iff__add,axiom,
% 2.40/1.48 ( ord_less_eq_nat
% 2.40/1.48 = ( ^ [M3: nat,N3: nat] :
% 2.40/1.48 ? [K2: nat] :
% 2.40/1.48 ( N3
% 2.40/1.48 = ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_208_trans__le__add2,axiom,
% 2.40/1.48 ! [I2: nat,J: nat,M: nat] :
% 2.40/1.48 ( ( ord_less_eq_nat @ I2 @ J )
% 2.40/1.48 => ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_209_trans__le__add1,axiom,
% 2.40/1.48 ! [I2: nat,J: nat,M: nat] :
% 2.40/1.48 ( ( ord_less_eq_nat @ I2 @ J )
% 2.40/1.48 => ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_210_add__le__mono1,axiom,
% 2.40/1.48 ! [I2: nat,J: nat,K: nat] :
% 2.40/1.48 ( ( ord_less_eq_nat @ I2 @ J )
% 2.40/1.48 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(fact_211_add__le__mono,axiom,
% 2.40/1.48 ! [I2: nat,J: nat,K: nat,L: nat] :
% 2.40/1.48 ( ( ord_less_eq_nat @ I2 @ J )
% 2.40/1.48 => ( ( ord_less_eq_nat @ K @ L )
% 2.40/1.48 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 2.40/1.48
% 2.40/1.48 thf(conj_0,conjecture,
% 2.40/1.48 $false ).
% 2.40/1.48 ------------------------
% 2.40/1.49 % (20049)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_19775 for (2999ds/27Mi)
% 2.52/1.50 % (20047)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_19775 for (2999ds/183Mi)
% 2.52/1.50 % (20048)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_19775 for (2999ds/4Mi)
% 2.52/1.50 % (20050)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_19775 for (2999ds/2Mi)
% 2.52/1.50 % (20052)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_19775 for (2999ds/275Mi)
% 2.52/1.50 % (20053)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_19775 for (2999ds/18Mi)
% 2.52/1.50 % (20051)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_19775 for (2999ds/2Mi)
% 2.52/1.50 % (20049)Instruction limit reached!
% 2.52/1.50 % (20049)------------------------------
% 2.52/1.50 % (20049)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.50 % (20049)Termination reason: Unknown
% 2.52/1.50 % (20049)Termination phase: Property scanning
% 2.52/1.50
% 2.52/1.50 % (20049)Memory used [KB]: 1535
% 2.52/1.50 % (20049)Time elapsed: 0.010 s
% 2.52/1.50 % (20049)Instructions burned: 27 (million)
% 2.52/1.50 % (20049)------------------------------
% 2.52/1.50 % (20049)------------------------------
% 2.52/1.50 % (20050)Instruction limit reached!
% 2.52/1.50 % (20050)------------------------------
% 2.52/1.50 % (20050)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.50 % (20050)Termination reason: Unknown
% 2.52/1.50 % (20050)Termination phase: shuffling
% 2.52/1.50
% 2.52/1.50 % (20050)Memory used [KB]: 1151
% 2.52/1.50 % (20050)Time elapsed: 0.004 s
% 2.52/1.50 % (20050)Instructions burned: 3 (million)
% 2.52/1.50 % (20050)------------------------------
% 2.52/1.50 % (20050)------------------------------
% 2.52/1.50 % (20051)Instruction limit reached!
% 2.52/1.50 % (20051)------------------------------
% 2.52/1.50 % (20051)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.50 % (20051)Termination reason: Unknown
% 2.52/1.50 % (20051)Termination phase: shuffling
% 2.52/1.50
% 2.52/1.50 % (20051)Memory used [KB]: 1151
% 2.52/1.50 % (20051)Time elapsed: 0.003 s
% 2.52/1.50 % (20051)Instructions burned: 2 (million)
% 2.52/1.50 % (20051)------------------------------
% 2.52/1.50 % (20051)------------------------------
% 2.52/1.50 % (20048)Instruction limit reached!
% 2.52/1.50 % (20048)------------------------------
% 2.52/1.50 % (20048)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.50 % (20048)Termination reason: Unknown
% 2.52/1.50 % (20048)Termination phase: shuffling
% 2.52/1.50
% 2.52/1.50 % (20048)Memory used [KB]: 1279
% 2.52/1.50 % (20048)Time elapsed: 0.005 s
% 2.52/1.50 % (20048)Instructions burned: 5 (million)
% 2.52/1.50 % (20048)------------------------------
% 2.52/1.50 % (20048)------------------------------
% 2.52/1.51 % (20053)Instruction limit reached!
% 2.52/1.51 % (20053)------------------------------
% 2.52/1.51 % (20053)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.51 % (20053)Termination reason: Unknown
% 2.52/1.51 % (20053)Termination phase: Property scanning
% 2.52/1.51
% 2.52/1.51 % (20053)Memory used [KB]: 1279
% 2.52/1.51 % (20053)Time elapsed: 0.013 s
% 2.52/1.51 % (20053)Instructions burned: 19 (million)
% 2.52/1.51 % (20053)------------------------------
% 2.52/1.51 % (20053)------------------------------
% 2.52/1.51 % (20054)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_19775 for (2999ds/3Mi)
% 2.52/1.51 % (20052)Refutation not found, incomplete strategy
% 2.52/1.51 % (20052)------------------------------
% 2.52/1.51 % (20052)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.51 % (20052)Termination reason: Refutation not found, incomplete strategy
% 2.52/1.51
% 2.52/1.51
% 2.52/1.51 % (20052)Memory used [KB]: 5884
% 2.52/1.51 % (20052)Time elapsed: 0.014 s
% 2.52/1.51 % (20052)Instructions burned: 21 (million)
% 2.52/1.51 % (20052)------------------------------
% 2.52/1.51 % (20052)------------------------------
% 2.52/1.51 % (20054)Instruction limit reached!
% 2.52/1.51 % (20054)------------------------------
% 2.52/1.51 % (20054)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.51 % (20054)Termination reason: Unknown
% 2.52/1.51 % (20054)Termination phase: shuffling
% 2.52/1.51
% 2.52/1.51 % (20054)Memory used [KB]: 1151
% 2.52/1.51 % (20054)Time elapsed: 0.003 s
% 2.52/1.51 % (20054)Instructions burned: 4 (million)
% 2.52/1.51 % (20054)------------------------------
% 2.52/1.51 % (20054)------------------------------
% 2.52/1.52 % (20055)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_19775 for (2999ds/37Mi)
% 2.52/1.52 % (20056)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_19775 for (2999ds/15Mi)
% 2.52/1.52 % (20057)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_19775 for (2999ds/3Mi)
% 2.52/1.52 % (20057)Instruction limit reached!
% 2.52/1.52 % (20057)------------------------------
% 2.52/1.52 % (20057)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.52 % (20057)Termination reason: Unknown
% 2.52/1.52 % (20057)Termination phase: shuffling
% 2.52/1.52
% 2.52/1.52 % (20057)Memory used [KB]: 1151
% 2.52/1.52 % (20057)Time elapsed: 0.004 s
% 2.52/1.52 % (20057)Instructions burned: 4 (million)
% 2.52/1.52 % (20057)------------------------------
% 2.52/1.52 % (20057)------------------------------
% 2.52/1.52 % (20059)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_19775 for (2999ds/7Mi)
% 2.52/1.52 % (20059)Instruction limit reached!
% 2.52/1.52 % (20059)------------------------------
% 2.52/1.52 % (20059)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.52 % (20059)Termination reason: Unknown
% 2.52/1.52 % (20059)Termination phase: shuffling
% 2.52/1.52
% 2.52/1.52 % (20059)Memory used [KB]: 1279
% 2.52/1.52 % (20059)Time elapsed: 0.004 s
% 2.52/1.52 % (20059)Instructions burned: 7 (million)
% 2.52/1.52 % (20059)------------------------------
% 2.52/1.52 % (20059)------------------------------
% 2.52/1.53 % (20058)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_19775 for (2999ds/1041Mi)
% 2.52/1.53 % (20056)Instruction limit reached!
% 2.52/1.53 % (20056)------------------------------
% 2.52/1.53 % (20056)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.53 % (20056)Termination reason: Unknown
% 2.52/1.53 % (20056)Termination phase: Property scanning
% 2.52/1.53
% 2.52/1.53 % (20056)Memory used [KB]: 1279
% 2.52/1.53 % (20056)Time elapsed: 0.011 s
% 2.52/1.53 % (20056)Instructions burned: 18 (million)
% 2.52/1.53 % (20056)------------------------------
% 2.52/1.53 % (20056)------------------------------
% 2.52/1.53 % (20060)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_19775 for (2999ds/16Mi)
% 2.52/1.53 % (20062)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_19775 for (2999ds/3Mi)
% 2.52/1.54 % (20061)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_19775 for (2999ds/3Mi)
% 2.52/1.54 % (20062)Instruction limit reached!
% 2.52/1.54 % (20062)------------------------------
% 2.52/1.54 % (20062)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.54 % (20062)Termination reason: Unknown
% 2.52/1.54 % (20062)Termination phase: shuffling
% 2.52/1.54
% 2.52/1.54 % (20062)Memory used [KB]: 1151
% 2.52/1.54 % (20062)Time elapsed: 0.003 s
% 2.52/1.54 % (20062)Instructions burned: 4 (million)
% 2.52/1.54 % (20062)------------------------------
% 2.52/1.54 % (20062)------------------------------
% 2.52/1.54 % (20055)Instruction limit reached!
% 2.52/1.54 % (20055)------------------------------
% 2.52/1.54 % (20055)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.54 % (20055)Termination reason: Unknown
% 2.52/1.54 % (20055)Termination phase: Property scanning
% 2.52/1.54
% 2.52/1.54 % (20055)Memory used [KB]: 1663
% 2.52/1.54 % (20055)Time elapsed: 0.020 s
% 2.52/1.54 % (20055)Instructions burned: 37 (million)
% 2.52/1.54 % (20055)------------------------------
% 2.52/1.54 % (20055)------------------------------
% 2.52/1.54 % (20061)Instruction limit reached!
% 2.52/1.54 % (20061)------------------------------
% 2.52/1.54 % (20061)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.54 % (20061)Termination reason: Unknown
% 2.52/1.54 % (20061)Termination phase: shuffling
% 2.52/1.54
% 2.52/1.54 % (20061)Memory used [KB]: 1151
% 2.52/1.54 % (20061)Time elapsed: 0.003 s
% 2.52/1.54 % (20061)Instructions burned: 3 (million)
% 2.52/1.54 % (20061)------------------------------
% 2.52/1.54 % (20061)------------------------------
% 2.52/1.54 % (20063)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_19775 for (2999ds/7Mi)
% 2.52/1.54 % (20060)Instruction limit reached!
% 2.52/1.54 % (20060)------------------------------
% 2.52/1.54 % (20060)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.54 % (20060)Termination reason: Unknown
% 2.52/1.54 % (20060)Termination phase: Property scanning
% 2.52/1.54
% 2.52/1.54 % (20060)Memory used [KB]: 1407
% 2.52/1.54 % (20060)Time elapsed: 0.011 s
% 2.52/1.54 % (20060)Instructions burned: 16 (million)
% 2.52/1.54 % (20060)------------------------------
% 2.52/1.54 % (20060)------------------------------
% 2.52/1.54 % (20063)Instruction limit reached!
% 2.52/1.54 % (20063)------------------------------
% 2.52/1.54 % (20063)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.54 % (20063)Termination reason: Unknown
% 2.52/1.54 % (20063)Termination phase: shuffling
% 2.52/1.54
% 2.52/1.54 % (20063)Memory used [KB]: 1279
% 2.52/1.54 % (20063)Time elapsed: 0.004 s
% 2.52/1.54 % (20063)Instructions burned: 7 (million)
% 2.52/1.54 % (20063)------------------------------
% 2.52/1.54 % (20063)------------------------------
% 2.52/1.55 % (20064)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_19775 for (2999ds/3Mi)
% 2.52/1.55 % (20066)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_19775 for (2999ds/18Mi)
% 2.52/1.55 % (20064)Instruction limit reached!
% 2.52/1.55 % (20064)------------------------------
% 2.52/1.55 % (20064)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.55 % (20064)Termination reason: Unknown
% 2.52/1.55 % (20064)Termination phase: shuffling
% 2.52/1.55
% 2.52/1.55 % (20064)Memory used [KB]: 1279
% 2.52/1.55 % (20064)Time elapsed: 0.003 s
% 2.52/1.55 % (20064)Instructions burned: 4 (million)
% 2.52/1.55 % (20064)------------------------------
% 2.52/1.55 % (20064)------------------------------
% 2.52/1.55 % (20068)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_19775 for (2999ds/6Mi)
% 2.52/1.55 % (20066)Instruction limit reached!
% 2.52/1.55 % (20066)------------------------------
% 2.52/1.55 % (20066)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.55 % (20066)Termination reason: Unknown
% 2.52/1.55 % (20066)Termination phase: Property scanning
% 2.52/1.55
% 2.52/1.55 % (20066)Memory used [KB]: 1407
% 2.52/1.55 % (20066)Time elapsed: 0.007 s
% 2.52/1.55 % (20066)Instructions burned: 21 (million)
% 2.52/1.55 % (20066)------------------------------
% 2.52/1.55 % (20066)------------------------------
% 2.52/1.55 % (20068)Instruction limit reached!
% 2.52/1.55 % (20068)------------------------------
% 2.52/1.55 % (20068)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.55 % (20068)Termination reason: Unknown
% 2.52/1.55 % (20068)Termination phase: shuffling
% 2.52/1.55
% 2.52/1.55 % (20068)Memory used [KB]: 1279
% 2.52/1.55 % (20068)Time elapsed: 0.004 s
% 2.52/1.55 % (20068)Instructions burned: 7 (million)
% 2.52/1.55 % (20068)------------------------------
% 2.52/1.55 % (20068)------------------------------
% 2.52/1.55 % (20065)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_19775 for (2999ds/4Mi)
% 2.52/1.56 % (20065)Instruction limit reached!
% 2.52/1.56 % (20065)------------------------------
% 2.52/1.56 % (20065)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.56 % (20065)Termination reason: Unknown
% 2.52/1.56 % (20065)Termination phase: shuffling
% 2.52/1.56
% 2.52/1.56 % (20065)Memory used [KB]: 1151
% 2.52/1.56 % (20065)Time elapsed: 0.004 s
% 2.52/1.56 % (20065)Instructions burned: 4 (million)
% 2.52/1.56 % (20065)------------------------------
% 2.52/1.56 % (20065)------------------------------
% 2.52/1.56 % (20067)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_19775 for (2999ds/710Mi)
% 2.52/1.56 % (20069)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_19775 for (2999ds/902Mi)
% 2.52/1.56 % (20070)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_19775 for (2999ds/21Mi)
% 2.52/1.56 % (20071)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_19775 for (2999ds/5Mi)
% 2.52/1.57 % (20071)Instruction limit reached!
% 2.52/1.57 % (20071)------------------------------
% 2.52/1.57 % (20071)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.57 % (20071)Termination reason: Unknown
% 2.52/1.57 % (20071)Termination phase: shuffling
% 2.52/1.57
% 2.52/1.57 % (20071)Memory used [KB]: 1279
% 2.52/1.57 % (20071)Time elapsed: 0.004 s
% 2.52/1.57 % (20071)Instructions burned: 8 (million)
% 2.52/1.57 % (20071)------------------------------
% 2.52/1.57 % (20071)------------------------------
% 2.52/1.57 % (20067)Refutation not found, incomplete strategy
% 2.52/1.57 % (20067)------------------------------
% 2.52/1.57 % (20067)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.57 % (20067)Termination reason: Refutation not found, incomplete strategy
% 2.52/1.57
% 2.52/1.57
% 2.52/1.57 % (20067)Memory used [KB]: 5884
% 2.52/1.57 % (20067)Time elapsed: 0.014 s
% 2.52/1.57 % (20067)Instructions burned: 22 (million)
% 2.52/1.57 % (20067)------------------------------
% 2.52/1.57 % (20067)------------------------------
% 2.52/1.57 % (20047)Instruction limit reached!
% 2.52/1.57 % (20047)------------------------------
% 2.52/1.57 % (20047)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.57 % (20047)Termination reason: Unknown
% 2.52/1.57 % (20047)Termination phase: Saturation
% 2.52/1.57
% 2.52/1.57 % (20047)Memory used [KB]: 7036
% 2.52/1.57 % (20047)Time elapsed: 0.072 s
% 2.52/1.57 % (20047)Instructions burned: 183 (million)
% 2.52/1.57 % (20047)------------------------------
% 2.52/1.57 % (20047)------------------------------
% 2.52/1.57 % (20070)Instruction limit reached!
% 2.52/1.57 % (20070)------------------------------
% 2.52/1.57 % (20070)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.57 % (20070)Termination reason: Unknown
% 2.52/1.57 % (20070)Termination phase: Property scanning
% 2.52/1.57
% 2.52/1.57 % (20070)Memory used [KB]: 1407
% 2.52/1.57 % (20070)Time elapsed: 0.008 s
% 2.52/1.57 % (20070)Instructions burned: 24 (million)
% 2.52/1.57 % (20070)------------------------------
% 2.52/1.57 % (20070)------------------------------
% 2.52/1.57 % (20072)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_19775 for (2999ds/6Mi)
% 2.52/1.58 % (20069)Refutation not found, incomplete strategy
% 2.52/1.58 % (20069)------------------------------
% 2.52/1.58 % (20069)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.58 % (20069)Termination reason: Refutation not found, incomplete strategy
% 2.52/1.58
% 2.52/1.58
% 2.52/1.58 % (20069)Memory used [KB]: 6140
% 2.52/1.58 % (20069)Time elapsed: 0.019 s
% 2.52/1.58 % (20069)Instructions burned: 56 (million)
% 2.52/1.58 % (20069)------------------------------
% 2.52/1.58 % (20069)------------------------------
% 2.52/1.58 % (20073)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_19775 for (2999ds/377Mi)
% 2.52/1.58 % (20072)Instruction limit reached!
% 2.52/1.58 % (20072)------------------------------
% 2.52/1.58 % (20072)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.58 % (20072)Termination reason: Unknown
% 2.52/1.58 % (20072)Termination phase: shuffling
% 2.52/1.58
% 2.52/1.58 % (20072)Memory used [KB]: 1279
% 2.52/1.58 % (20072)Time elapsed: 0.005 s
% 2.52/1.58 % (20072)Instructions burned: 6 (million)
% 2.52/1.58 % (20072)------------------------------
% 2.52/1.58 % (20072)------------------------------
% 2.52/1.58 % (20076)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_19775 for (2999ds/879Mi)
% 2.52/1.58 % (20074)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_19775 for (2999ds/779Mi)
% 2.52/1.59 % (20077)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_19775 for (2999ds/17Mi)
% 2.52/1.59 % (20076)Refutation not found, incomplete strategy
% 2.52/1.59 % (20076)------------------------------
% 2.52/1.59 % (20076)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.59 % (20076)Termination reason: Refutation not found, incomplete strategy
% 2.52/1.59
% 2.52/1.59
% 2.52/1.59 % (20076)Memory used [KB]: 5884
% 2.52/1.59 % (20076)Time elapsed: 0.008 s
% 2.52/1.59 % (20076)Instructions burned: 25 (million)
% 2.52/1.59 % (20076)------------------------------
% 2.52/1.59 % (20076)------------------------------
% 2.52/1.59 % (20075)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_19775 for (2999ds/19Mi)
% 2.52/1.59 % (20077)Instruction limit reached!
% 2.52/1.59 % (20077)------------------------------
% 2.52/1.59 % (20077)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.59 % (20077)Termination reason: Unknown
% 2.52/1.59 % (20077)Termination phase: Property scanning
% 2.52/1.59
% 2.52/1.59 % (20077)Memory used [KB]: 1407
% 2.52/1.59 % (20077)Time elapsed: 0.007 s
% 2.52/1.59 % (20077)Instructions burned: 17 (million)
% 2.52/1.59 % (20077)------------------------------
% 2.52/1.59 % (20077)------------------------------
% 2.52/1.59 % (20078)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_19775 for (2999ds/3Mi)
% 2.52/1.60 % (20078)Instruction limit reached!
% 2.52/1.60 % (20078)------------------------------
% 2.52/1.60 % (20078)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.60 % (20078)Termination reason: Unknown
% 2.52/1.60 % (20078)Termination phase: shuffling
% 2.52/1.60
% 2.52/1.60 % (20078)Memory used [KB]: 1151
% 2.52/1.60 % (20078)Time elapsed: 0.004 s
% 2.52/1.60 % (20078)Instructions burned: 3 (million)
% 2.52/1.60 % (20078)------------------------------
% 2.52/1.60 % (20078)------------------------------
% 2.52/1.60 % (20079)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_19775 for (2998ds/30Mi)
% 2.52/1.60 % (20075)Instruction limit reached!
% 2.52/1.60 % (20075)------------------------------
% 2.52/1.60 % (20075)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.60 % (20075)Termination reason: Unknown
% 2.52/1.60 % (20075)Termination phase: shuffling
% 2.52/1.60
% 2.52/1.60 % (20075)Memory used [KB]: 1535
% 2.52/1.60 % (20075)Time elapsed: 0.012 s
% 2.52/1.60 % (20075)Instructions burned: 20 (million)
% 2.52/1.60 % (20075)------------------------------
% 2.52/1.60 % (20075)------------------------------
% 2.52/1.60 % (20080)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_19775 for (2998ds/127Mi)
% 2.52/1.60 % (20079)Instruction limit reached!
% 2.52/1.60 % (20079)------------------------------
% 2.52/1.60 % (20079)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.60 % (20079)Termination reason: Unknown
% 2.52/1.60 % (20079)Termination phase: Property scanning
% 2.52/1.60
% 2.52/1.60 % (20079)Memory used [KB]: 1535
% 2.52/1.60 % (20079)Time elapsed: 0.010 s
% 2.52/1.60 % (20079)Instructions burned: 31 (million)
% 2.52/1.60 % (20079)------------------------------
% 2.52/1.60 % (20079)------------------------------
% 2.52/1.61 % (20081)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_19775 for (2998ds/100Mi)
% 2.52/1.61 % (20082)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_19775 for (2998ds/3Mi)
% 2.52/1.61 % (20082)Instruction limit reached!
% 2.52/1.61 % (20082)------------------------------
% 2.52/1.61 % (20082)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.61 % (20082)Termination reason: Unknown
% 2.52/1.61 % (20082)Termination phase: shuffling
% 2.52/1.61
% 2.52/1.61 % (20082)Memory used [KB]: 1151
% 2.52/1.61 % (20082)Time elapsed: 0.003 s
% 2.52/1.61 % (20082)Instructions burned: 3 (million)
% 2.52/1.61 % (20082)------------------------------
% 2.52/1.61 % (20082)------------------------------
% 2.52/1.61 % (20083)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_19775 for (2998ds/20Mi)
% 2.52/1.62 % (20083)Instruction limit reached!
% 2.52/1.62 % (20083)------------------------------
% 2.52/1.62 % (20083)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.62 % (20083)Termination reason: Unknown
% 2.52/1.62 % (20083)Termination phase: SInE selection
% 2.52/1.62
% 2.52/1.62 % (20083)Memory used [KB]: 1279
% 2.52/1.62 % (20083)Time elapsed: 0.007 s
% 2.52/1.62 % (20083)Instructions burned: 20 (million)
% 2.52/1.62 % (20083)------------------------------
% 2.52/1.62 % (20083)------------------------------
% 2.52/1.62 % (20084)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_19775 for (2998ds/86Mi)
% 2.52/1.63 % (20085)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_19775 for (2998ds/107Mi)
% 2.52/1.64 % (20085)Refutation not found, incomplete strategy
% 2.52/1.64 % (20085)------------------------------
% 2.52/1.64 % (20085)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.64 % (20085)Termination reason: Refutation not found, incomplete strategy
% 2.52/1.64
% 2.52/1.64
% 2.52/1.64 % (20085)Memory used [KB]: 5884
% 2.52/1.64 % (20085)Time elapsed: 0.008 s
% 2.52/1.64 % (20085)Instructions burned: 25 (million)
% 2.52/1.64 % (20085)------------------------------
% 2.52/1.64 % (20085)------------------------------
% 2.52/1.64 % (20058)First to succeed.
% 2.52/1.64 % (20081)Instruction limit reached!
% 2.52/1.64 % (20081)------------------------------
% 2.52/1.64 % (20081)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.64 % (20081)Termination reason: Unknown
% 2.52/1.64 % (20081)Termination phase: Property scanning
% 2.52/1.64
% 2.52/1.64 % (20081)Memory used [KB]: 1918
% 2.52/1.64 % (20081)Time elapsed: 0.029 s
% 2.52/1.64 % (20081)Instructions burned: 100 (million)
% 2.52/1.64 % (20081)------------------------------
% 2.52/1.64 % (20081)------------------------------
% 2.52/1.64 % (20058)Refutation found. Thanks to Tanya!
% 2.52/1.64 % SZS status ContradictoryAxioms for DTF2THF_19775
% 2.52/1.64 % SZS output start Proof for DTF2THF_19775
% 2.52/1.64 thf(type_def_5, type, nat: $tType).
% 2.52/1.64 thf(func_def_0, type, nat: $tType).
% 2.52/1.64 thf(func_def_1, type, minus_minus_nat: nat > nat > nat).
% 2.52/1.64 thf(func_def_2, type, plus_plus_nat: nat > nat > nat).
% 2.52/1.64 thf(func_def_3, type, times_times_nat: nat > nat > nat).
% 2.52/1.64 thf(func_def_4, type, zero_zero_nat: nat).
% 2.52/1.64 thf(func_def_5, type, number1551313001itions: (nat > nat) > nat > $o).
% 2.52/1.64 thf(func_def_6, type, ord_less_nat: nat > nat > $o).
% 2.52/1.64 thf(func_def_7, type, ord_less_eq_nat: nat > nat > $o).
% 2.52/1.64 thf(func_def_8, type, i: nat).
% 2.52/1.64 thf(func_def_9, type, k: nat).
% 2.52/1.64 thf(func_def_10, type, n: nat).
% 2.52/1.64 thf(func_def_11, type, p: nat > nat).
% 2.52/1.64 thf(func_def_31, type, ph1: !>[X0: $tType]:(X0)).
% 2.52/1.64 thf(func_def_32, type, sK2: nat).
% 2.52/1.64 thf(func_def_33, type, sK3: nat).
% 2.52/1.64 thf(func_def_34, type, sK4: nat > nat).
% 2.52/1.64 thf(func_def_35, type, sK5: nat > nat).
% 2.52/1.64 thf(func_def_36, type, sK6: nat).
% 2.52/1.64 thf(func_def_37, type, sK7: nat > nat > nat).
% 2.52/1.64 thf(func_def_38, type, sK8: nat).
% 2.52/1.64 thf(func_def_39, type, sK9: nat).
% 2.52/1.64 thf(f2010,plain,(
% 2.52/1.64 $false),
% 2.52/1.64 inference(subsumption_resolution,[status(thm)],[f2009,f1211])).
% 2.52/1.64 thf(f1211,plain,(
% 2.52/1.64 ( ! [X0 : nat] : (((plus_plus_nat @ X0 @ zero_zero_nat) = X0)) )),
% 2.52/1.64 inference(superposition,[status(thm)],[f1169,f1203])).
% 2.52/1.64 thf(f1203,plain,(
% 2.52/1.64 ( ! [X1 : nat] : (((plus_plus_nat @ zero_zero_nat @ X1) = X1)) )),
% 2.52/1.64 inference(equality_proxy_clausification,[status(thm)],[f1202])).
% 2.52/1.64 thf(f1202,plain,(
% 2.52/1.64 ( ! [X1 : nat] : ((((plus_plus_nat @ zero_zero_nat @ X1) = X1) = $true)) )),
% 2.52/1.64 inference(beta_eta_normalization,[status(thm)],[f1201])).
% 2.52/1.64 thf(f1201,plain,(
% 2.52/1.64 ( ! [X1 : nat] : ((((^[Y0 : nat]: ((plus_plus_nat @ zero_zero_nat @ Y0) = Y0)) @ X1) = $true)) )),
% 2.52/1.64 inference(pi_clausification,[status(thm)],[f669])).
% 2.52/1.64 thf(f669,plain,(
% 2.52/1.64 ((!! @ nat @ (^[Y0 : nat]: ((plus_plus_nat @ zero_zero_nat @ Y0) = Y0))) = $true)),
% 2.52/1.64 inference(cnf_transformation,[status(thm)],[f635])).
% 2.52/1.64 thf(f635,plain,(
% 2.52/1.64 ((!! @ nat @ (^[Y0 : nat]: ((plus_plus_nat @ zero_zero_nat @ Y0) = Y0))) = $true)),
% 2.52/1.64 inference(fool_elimination,[status(thm)],[f634])).
% 2.52/1.64 thf(f634,plain,(
% 2.52/1.64 ! [X0 : nat] : ((plus_plus_nat @ zero_zero_nat @ X0) = X0)),
% 2.52/1.64 inference(rectify,[status(thm)],[f35])).
% 2.52/1.64 thf(f35,axiom,(
% 2.52/1.64 ! [X7 : nat] : ((plus_plus_nat @ zero_zero_nat @ X7) = X7)),
% 2.52/1.64 file('/export/starexec/sandbox/tmp/tmp.GJlgnNEeHF/DTF2THF_19775.p',fact_34_add_Oleft__neutral)).
% 2.52/1.64 thf(f1169,plain,(
% 2.52/1.64 ( ! [X2 : nat,X1 : nat] : (((plus_plus_nat @ X1 @ X2) = (plus_plus_nat @ X2 @ X1))) )),
% 2.52/1.64 inference(beta_eta_normalization,[status(thm)],[f1168])).
% 2.52/1.64 thf(f1168,plain,(
% 2.52/1.64 ( ! [X2 : nat,X1 : nat] : (((plus_plus_nat @ X1 @ X2) = ((^[Y0 : nat]: (plus_plus_nat @ Y0 @ X1)) @ X2))) )),
% 2.52/1.64 inference(argument_congruence,[status(thm)],[f1162])).
% 2.52/1.64 thf(f1162,plain,(
% 2.52/1.64 ( ! [X1 : nat] : (((plus_plus_nat @ X1) = (^[Y0 : nat]: (plus_plus_nat @ Y0 @ X1)))) )),
% 2.52/1.64 inference(beta_eta_normalization,[status(thm)],[f1161])).
% 2.52/1.64 thf(f1161,plain,(
% 2.52/1.64 ( ! [X1 : nat] : (((plus_plus_nat @ X1) = ((^[Y0 : nat]: ((^[Y1 : nat]: (plus_plus_nat @ Y1 @ Y0)))) @ X1))) )),
% 2.52/1.64 inference(argument_congruence,[status(thm)],[f807])).
% 2.52/1.64 thf(f807,plain,(
% 2.52/1.64 (plus_plus_nat = (^[Y0 : nat]: ((^[Y1 : nat]: (plus_plus_nat @ Y1 @ Y0)))))),
% 2.52/1.64 inference(cnf_transformation,[status(thm)],[f328])).
% 2.52/1.64 thf(f328,plain,(
% 2.52/1.64 (plus_plus_nat = (^[Y0 : nat]: ((^[Y1 : nat]: (plus_plus_nat @ Y1 @ Y0)))))),
% 2.52/1.64 inference(fool_elimination,[status(thm)],[f327])).
% 2.52/1.64 thf(f327,plain,(
% 2.52/1.64 (plus_plus_nat = (^[X0 : nat, X1 : nat] : (plus_plus_nat @ X1 @ X0)))),
% 2.52/1.64 inference(rectify,[status(thm)],[f130])).
% 2.52/1.64 thf(f130,axiom,(
% 2.52/1.64 (plus_plus_nat = (^[X16 : nat, X17 : nat] : (plus_plus_nat @ X17 @ X16)))),
% 2.52/1.64 file('/export/starexec/sandbox/tmp/tmp.GJlgnNEeHF/DTF2THF_19775.p',fact_129_add_Ocommute)).
% 2.52/1.64 thf(f2009,plain,(
% 2.52/1.64 ( ! [X0 : nat] : (((plus_plus_nat @ X0 @ zero_zero_nat) != X0)) )),
% 2.52/1.64 inference(forward_demodulation,[status(thm)],[f2000,f1581])).
% 2.52/1.64 thf(f1581,plain,(
% 2.52/1.64 ( ! [X0 : nat] : ((zero_zero_nat = (sK7 @ X0 @ zero_zero_nat))) )),
% 2.52/1.64 inference(forward_demodulation,[status(thm)],[f1577,f1220])).
% 2.52/1.64 thf(f1220,plain,(
% 2.52/1.64 ( ! [X1 : nat] : ((zero_zero_nat = (minus_minus_nat @ X1 @ X1))) )),
% 2.52/1.64 inference(equality_proxy_clausification,[status(thm)],[f1219])).
% 2.52/1.64 thf(f1219,plain,(
% 2.52/1.64 ( ! [X1 : nat] : (((zero_zero_nat = (minus_minus_nat @ X1 @ X1)) = $true)) )),
% 2.52/1.64 inference(beta_eta_normalization,[status(thm)],[f1218])).
% 2.52/1.64 thf(f1218,plain,(
% 2.52/1.64 ( ! [X1 : nat] : ((((^[Y0 : nat]: (zero_zero_nat = (minus_minus_nat @ Y0 @ Y0))) @ X1) = $true)) )),
% 2.52/1.64 inference(pi_clausification,[status(thm)],[f699])).
% 2.52/1.64 thf(f699,plain,(
% 2.52/1.64 ((!! @ nat @ (^[Y0 : nat]: (zero_zero_nat = (minus_minus_nat @ Y0 @ Y0)))) = $true)),
% 2.52/1.64 inference(cnf_transformation,[status(thm)],[f308])).
% 2.52/1.64 thf(f308,plain,(
% 2.52/1.64 ((!! @ nat @ (^[Y0 : nat]: (zero_zero_nat = (minus_minus_nat @ Y0 @ Y0)))) = $true)),
% 2.52/1.64 inference(fool_elimination,[status(thm)],[f307])).
% 2.52/1.64 thf(f307,plain,(
% 2.52/1.64 ! [X0 : nat] : (zero_zero_nat = (minus_minus_nat @ X0 @ X0))),
% 2.52/1.64 inference(rectify,[status(thm)],[f36])).
% 2.52/1.64 thf(f36,axiom,(
% 2.52/1.64 ! [X7 : nat] : (zero_zero_nat = (minus_minus_nat @ X7 @ X7))),
% 2.52/1.64 file('/export/starexec/sandbox/tmp/tmp.GJlgnNEeHF/DTF2THF_19775.p',fact_35_cancel__comm__monoid__add__class_Odiff__cancel)).
% 2.52/1.64 thf(f1577,plain,(
% 2.52/1.64 ( ! [X0 : nat] : (((sK7 @ X0 @ zero_zero_nat) = (minus_minus_nat @ X0 @ X0))) )),
% 2.52/1.64 inference(superposition,[status(thm)],[f1338,f1443])).
% 2.52/1.64 thf(f1443,plain,(
% 2.52/1.64 ( ! [X0 : nat] : (((plus_plus_nat @ X0 @ (sK7 @ X0 @ zero_zero_nat)) = X0)) )),
% 2.52/1.64 inference(superposition,[status(thm)],[f1413,f1231])).
% 2.52/1.64 thf(f1231,plain,(
% 2.52/1.64 ( ! [X1 : nat] : (((minus_minus_nat @ X1 @ zero_zero_nat) = X1)) )),
% 2.52/1.64 inference(equality_proxy_clausification,[status(thm)],[f1230])).
% 2.52/1.64 thf(f1230,plain,(
% 2.52/1.64 ( ! [X1 : nat] : ((((minus_minus_nat @ X1 @ zero_zero_nat) = X1) = $true)) )),
% 2.52/1.64 inference(beta_eta_normalization,[status(thm)],[f1229])).
% 2.52/1.64 thf(f1229,plain,(
% 2.52/1.64 ( ! [X1 : nat] : ((((^[Y0 : nat]: ((minus_minus_nat @ Y0 @ zero_zero_nat) = Y0)) @ X1) = $true)) )),
% 2.52/1.64 inference(pi_clausification,[status(thm)],[f802])).
% 2.52/1.64 thf(f802,plain,(
% 2.52/1.64 ((!! @ nat @ (^[Y0 : nat]: ((minus_minus_nat @ Y0 @ zero_zero_nat) = Y0))) = $true)),
% 2.52/1.64 inference(cnf_transformation,[status(thm)],[f258])).
% 2.52/1.64 thf(f258,plain,(
% 2.52/1.64 ((!! @ nat @ (^[Y0 : nat]: ((minus_minus_nat @ Y0 @ zero_zero_nat) = Y0))) = $true)),
% 2.52/1.64 inference(fool_elimination,[status(thm)],[f257])).
% 2.52/1.64 thf(f257,plain,(
% 2.52/1.64 ! [X0 : nat] : ((minus_minus_nat @ X0 @ zero_zero_nat) = X0)),
% 2.52/1.64 inference(rectify,[status(thm)],[f77])).
% 2.52/1.64 thf(f77,axiom,(
% 2.52/1.64 ! [X4 : nat] : ((minus_minus_nat @ X4 @ zero_zero_nat) = X4)),
% 2.52/1.64 file('/export/starexec/sandbox/tmp/tmp.GJlgnNEeHF/DTF2THF_19775.p',fact_76_minus__nat_Odiff__0)).
% 2.52/1.64 thf(f1413,plain,(
% 2.52/1.64 ( ! [X2 : nat,X1 : nat] : (((plus_plus_nat @ (minus_minus_nat @ X1 @ X2) @ (sK7 @ X1 @ X2)) = X1)) )),
% 2.52/1.64 inference(equality_proxy_clausification,[status(thm)],[f1406])).
% 2.52/1.64 thf(f1406,plain,(
% 2.52/1.64 ( ! [X2 : nat,X1 : nat] : (($true = ((plus_plus_nat @ (minus_minus_nat @ X1 @ X2) @ (sK7 @ X1 @ X2)) = X1))) )),
% 2.52/1.64 inference(beta_eta_normalization,[status(thm)],[f1403])).
% 2.52/1.64 thf(f1403,plain,(
% 2.52/1.64 ( ! [X2 : nat,X1 : nat] : ((((^[Y0 : nat]: ((plus_plus_nat @ (minus_minus_nat @ X1 @ X2) @ Y0) = X1)) @ (sK7 @ X1 @ X2)) = $true)) )),
% 2.52/1.64 inference(sigma_clausification,[status(thm)],[f1399])).
% 2.52/1.64 thf(f1399,plain,(
% 2.52/1.64 ( ! [X2 : nat,X1 : nat] : (((?? @ nat @ (^[Y0 : nat]: ((plus_plus_nat @ (minus_minus_nat @ X1 @ X2) @ Y0) = X1))) = $true)) )),
% 2.52/1.64 inference(beta_eta_normalization,[status(thm)],[f1398])).
% 2.52/1.64 thf(f1398,plain,(
% 2.52/1.64 ( ! [X2 : nat,X1 : nat] : ((((^[Y0 : nat]: (?? @ nat @ (^[Y1 : nat]: ((plus_plus_nat @ (minus_minus_nat @ X1 @ Y0) @ Y1) = X1)))) @ X2) = $true)) )),
% 2.52/1.64 inference(pi_clausification,[status(thm)],[f1395])).
% 2.52/1.64 thf(f1395,plain,(
% 2.52/1.64 ( ! [X1 : nat] : (((!! @ nat @ (^[Y0 : nat]: (?? @ nat @ (^[Y1 : nat]: ((plus_plus_nat @ (minus_minus_nat @ X1 @ Y0) @ Y1) = X1))))) = $true)) )),
% 2.52/1.64 inference(beta_eta_normalization,[status(thm)],[f1394])).
% 2.52/1.64 thf(f1394,plain,(
% 2.52/1.64 ( ! [X1 : nat] : ((((^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: (?? @ nat @ (^[Y2 : nat]: ((plus_plus_nat @ (minus_minus_nat @ Y0 @ Y1) @ Y2) = Y0)))))) @ X1) = $true)) )),
% 2.52/1.64 inference(pi_clausification,[status(thm)],[f1119])).
% 2.52/1.64 thf(f1119,plain,(
% 2.52/1.64 ((!! @ nat @ (^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: (?? @ nat @ (^[Y2 : nat]: ((plus_plus_nat @ (minus_minus_nat @ Y0 @ Y1) @ Y2) = Y0))))))) = $true)),
% 2.52/1.64 inference(beta_eta_normalization,[status(thm)],[f991])).
% 2.52/1.64 thf(f991,plain,(
% 2.52/1.64 ((!! @ nat @ (^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: ((^[Y2 : nat]: ((^[Y3 : nat]: (?? @ nat @ (^[Y4 : nat]: ((plus_plus_nat @ Y2 @ Y4) = Y3)))))) @ (minus_minus_nat @ Y0 @ Y1) @ Y0))))) = $true)),
% 2.52/1.64 inference(definition_unfolding,[status(thm)],[f838,f698])).
% 2.52/1.64 thf(f698,plain,(
% 2.52/1.64 (ord_less_eq_nat = (^[Y0 : nat]: ((^[Y1 : nat]: (?? @ nat @ (^[Y2 : nat]: ((plus_plus_nat @ Y0 @ Y2) = Y1)))))))),
% 2.52/1.64 inference(cnf_transformation,[status(thm)],[f342])).
% 2.52/1.64 thf(f342,plain,(
% 2.52/1.64 (ord_less_eq_nat = (^[Y0 : nat]: ((^[Y1 : nat]: (?? @ nat @ (^[Y2 : nat]: ((plus_plus_nat @ Y0 @ Y2) = Y1)))))))),
% 2.52/1.64 inference(fool_elimination,[status(thm)],[f341])).
% 2.52/1.64 thf(f341,plain,(
% 2.52/1.64 (ord_less_eq_nat = (^[X0 : nat, X1 : nat] : (? [X2 : nat] : ((plus_plus_nat @ X0 @ X2) = X1))))),
% 2.52/1.64 inference(rectify,[status(thm)],[f161])).
% 2.52/1.64 thf(f161,axiom,(
% 2.52/1.64 (ord_less_eq_nat = (^[X16 : nat, X17 : nat] : (? [X27 : nat] : ((plus_plus_nat @ X16 @ X27) = X17))))),
% 2.52/1.64 file('/export/starexec/sandbox/tmp/tmp.GJlgnNEeHF/DTF2THF_19775.p',fact_160_le__iff__add)).
% 2.52/1.64 thf(f838,plain,(
% 2.52/1.64 ((!! @ nat @ (^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: (ord_less_eq_nat @ (minus_minus_nat @ Y0 @ Y1) @ Y0))))) = $true)),
% 2.52/1.64 inference(cnf_transformation,[status(thm)],[f399])).
% 2.52/1.64 thf(f399,plain,(
% 2.52/1.64 ((!! @ nat @ (^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: (ord_less_eq_nat @ (minus_minus_nat @ Y0 @ Y1) @ Y0))))) = $true)),
% 2.52/1.64 inference(fool_elimination,[status(thm)],[f398])).
% 2.52/1.64 thf(f398,plain,(
% 2.52/1.64 ! [X0 : nat,X1 : nat] : (ord_less_eq_nat @ (minus_minus_nat @ X1 @ X0) @ X1)),
% 2.52/1.64 inference(rectify,[status(thm)],[f83])).
% 2.52/1.64 thf(f83,axiom,(
% 2.52/1.64 ! [X2 : nat,X4 : nat] : (ord_less_eq_nat @ (minus_minus_nat @ X4 @ X2) @ X4)),
% 2.52/1.64 file('/export/starexec/sandbox/tmp/tmp.GJlgnNEeHF/DTF2THF_19775.p',fact_82_diff__le__self)).
% 2.52/1.64 thf(f1338,plain,(
% 2.52/1.64 ( ! [X0 : nat,X1 : nat] : (((minus_minus_nat @ (plus_plus_nat @ X1 @ X0) @ X1) = X0)) )),
% 2.52/1.64 inference(superposition,[status(thm)],[f1317,f1169])).
% 2.52/1.64 thf(f1317,plain,(
% 2.52/1.64 ( ! [X2 : nat,X1 : nat] : (((minus_minus_nat @ (plus_plus_nat @ X1 @ X2) @ X2) = X1)) )),
% 2.52/1.64 inference(equality_proxy_clausification,[status(thm)],[f1316])).
% 2.52/1.64 thf(f1316,plain,(
% 2.52/1.64 ( ! [X2 : nat,X1 : nat] : ((((minus_minus_nat @ (plus_plus_nat @ X1 @ X2) @ X2) = X1) = $true)) )),
% 2.52/1.64 inference(beta_eta_normalization,[status(thm)],[f1315])).
% 2.52/1.64 thf(f1315,plain,(
% 2.52/1.64 ( ! [X2 : nat,X1 : nat] : ((((^[Y0 : nat]: ((minus_minus_nat @ (plus_plus_nat @ X1 @ Y0) @ Y0) = X1)) @ X2) = $true)) )),
% 2.52/1.64 inference(pi_clausification,[status(thm)],[f1314])).
% 2.52/1.64 thf(f1314,plain,(
% 2.52/1.64 ( ! [X1 : nat] : (((!! @ nat @ (^[Y0 : nat]: ((minus_minus_nat @ (plus_plus_nat @ X1 @ Y0) @ Y0) = X1))) = $true)) )),
% 2.52/1.64 inference(beta_eta_normalization,[status(thm)],[f1313])).
% 2.52/1.64 thf(f1313,plain,(
% 2.52/1.64 ( ! [X1 : nat] : ((((^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: ((minus_minus_nat @ (plus_plus_nat @ Y0 @ Y1) @ Y1) = Y0)))) @ X1) = $true)) )),
% 2.52/1.64 inference(pi_clausification,[status(thm)],[f662])).
% 2.52/1.64 thf(f662,plain,(
% 2.52/1.64 ((!! @ nat @ (^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: ((minus_minus_nat @ (plus_plus_nat @ Y0 @ Y1) @ Y1) = Y0))))) = $true)),
% 2.52/1.64 inference(cnf_transformation,[status(thm)],[f310])).
% 2.52/1.64 thf(f310,plain,(
% 2.52/1.64 ((!! @ nat @ (^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: ((minus_minus_nat @ (plus_plus_nat @ Y0 @ Y1) @ Y1) = Y0))))) = $true)),
% 2.52/1.64 inference(fool_elimination,[status(thm)],[f309])).
% 2.52/1.64 thf(f309,plain,(
% 2.52/1.64 ! [X0 : nat,X1 : nat] : ((minus_minus_nat @ (plus_plus_nat @ X1 @ X0) @ X0) = X1)),
% 2.52/1.64 inference(rectify,[status(thm)],[f88])).
% 2.52/1.64 thf(f88,axiom,(
% 2.52/1.64 ! [X2 : nat,X4 : nat] : ((minus_minus_nat @ (plus_plus_nat @ X4 @ X2) @ X2) = X4)),
% 2.52/1.64 file('/export/starexec/sandbox/tmp/tmp.GJlgnNEeHF/DTF2THF_19775.p',fact_87_diff__add__inverse2)).
% 2.52/1.64 thf(f2000,plain,(
% 2.52/1.64 ( ! [X0 : nat] : (((plus_plus_nat @ X0 @ (sK7 @ n @ zero_zero_nat)) != X0)) )),
% 2.52/1.64 inference(backward_demodulation,[status(thm)],[f1949,f1991])).
% 2.52/1.64 thf(f1991,plain,(
% 2.52/1.64 (zero_zero_nat = sK8)),
% 2.52/1.64 inference(backward_demodulation,[status(thm)],[f1775,f1977])).
% 2.52/1.64 thf(f1977,plain,(
% 2.52/1.64 (zero_zero_nat = (minus_minus_nat @ n @ (minus_minus_nat @ n @ sK8)))),
% 2.52/1.64 inference(superposition,[status(thm)],[f1269,f1963])).
% 2.52/1.64 thf(f1963,plain,(
% 2.52/1.64 ((minus_minus_nat @ n @ sK8) = (plus_plus_nat @ n @ sK9))),
% 2.52/1.64 inference(equality_proxy_clausification,[status(thm)],[f1958])).
% 2.52/1.64 thf(f1958,plain,(
% 2.52/1.64 (((plus_plus_nat @ n @ sK9) = (minus_minus_nat @ n @ sK8)) = $true)),
% 2.52/1.64 inference(beta_eta_normalization,[status(thm)],[f1956])).
% 2.52/1.64 thf(f1956,plain,(
% 2.52/1.64 (((^[Y0 : nat]: ((plus_plus_nat @ n @ Y0) = (minus_minus_nat @ n @ sK8))) @ sK9) = $true)),
% 2.52/1.64 inference(sigma_clausification,[status(thm)],[f1951])).
% 2.52/1.64 thf(f1951,plain,(
% 2.52/1.64 ((?? @ nat @ (^[Y0 : nat]: ((plus_plus_nat @ n @ Y0) = (minus_minus_nat @ n @ sK8)))) = $true)),
% 2.52/1.64 inference(boolean_simplification,[status(thm)],[f1950])).
% 2.52/1.64 thf(f1950,plain,(
% 2.52/1.64 (((?? @ nat @ (^[Y0 : nat]: ((plus_plus_nat @ n @ Y0) = (minus_minus_nat @ n @ sK8)))) & $true) = $true)),
% 2.52/1.64 inference(backward_demodulation,[status(thm)],[f1772,f1944])).
% 2.52/1.64 thf(f1944,plain,(
% 2.52/1.64 ((~ (zero_zero_nat = (sK7 @ n @ sK8))) = $true)),
% 2.52/1.64 inference(binary_proxy_clausification,[status(thm)],[f1772])).
% 2.52/1.64 thf(f1772,plain,(
% 2.52/1.64 ($true = ((?? @ nat @ (^[Y0 : nat]: ((plus_plus_nat @ n @ Y0) = (minus_minus_nat @ n @ sK8)))) & (~ (zero_zero_nat = (sK7 @ n @ sK8)))))),
% 2.52/1.64 inference(forward_demodulation,[status(thm)],[f1771,f1655])).
% 2.52/1.64 thf(f1655,plain,(
% 2.52/1.64 ( ! [X0 : nat,X1 : nat] : (((zero_zero_nat = (sK7 @ X0 @ X1)) = (X0 = (minus_minus_nat @ X0 @ X1)))) )),
% 2.52/1.64 inference(superposition,[status(thm)],[f1615,f1413])).
% 2.52/1.64 thf(f1615,plain,(
% 2.52/1.64 ( ! [X2 : nat,X1 : nat] : ((((plus_plus_nat @ X2 @ X1) = X2) = (zero_zero_nat = X1))) )),
% 2.52/1.64 inference(equality_proxy_clausification,[status(thm)],[f1591])).
% 2.52/1.64 thf(f1591,plain,(
% 2.52/1.64 ( ! [X2 : nat,X1 : nat] : ((((zero_zero_nat = X1) = ((plus_plus_nat @ X2 @ X1) = X2)) = $true)) )),
% 2.52/1.64 inference(beta_eta_normalization,[status(thm)],[f1590])).
% 2.52/1.64 thf(f1590,plain,(
% 2.52/1.64 ( ! [X2 : nat,X1 : nat] : ((((^[Y0 : nat]: ((zero_zero_nat = X1) = ((plus_plus_nat @ Y0 @ X1) = Y0))) @ X2) = $true)) )),
% 2.52/1.64 inference(pi_clausification,[status(thm)],[f1587])).
% 2.52/1.64 thf(f1587,plain,(
% 2.52/1.64 ( ! [X1 : nat] : (((!! @ nat @ (^[Y0 : nat]: ((zero_zero_nat = X1) = ((plus_plus_nat @ Y0 @ X1) = Y0)))) = $true)) )),
% 2.52/1.64 inference(beta_eta_normalization,[status(thm)],[f1586])).
% 2.52/1.64 thf(f1586,plain,(
% 2.52/1.64 ( ! [X1 : nat] : ((((^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: ((zero_zero_nat = Y0) = ((plus_plus_nat @ Y1 @ Y0) = Y1))))) @ X1) = $true)) )),
% 2.52/1.64 inference(pi_clausification,[status(thm)],[f730])).
% 2.52/1.64 thf(f730,plain,(
% 2.52/1.64 ((!! @ nat @ (^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: ((zero_zero_nat = Y0) = ((plus_plus_nat @ Y1 @ Y0) = Y1)))))) = $true)),
% 2.52/1.64 inference(cnf_transformation,[status(thm)],[f397])).
% 2.52/1.64 thf(f397,plain,(
% 2.52/1.64 ((!! @ nat @ (^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: ((zero_zero_nat = Y0) = ((plus_plus_nat @ Y1 @ Y0) = Y1)))))) = $true)),
% 2.52/1.64 inference(fool_elimination,[status(thm)],[f396])).
% 2.52/1.64 thf(f396,plain,(
% 2.52/1.64 ! [X0 : nat,X1 : nat] : ((zero_zero_nat = X1) = ((plus_plus_nat @ X0 @ X1) = X0))),
% 2.52/1.64 inference(rectify,[status(thm)],[f30])).
% 2.52/1.64 thf(f30,axiom,(
% 2.52/1.64 ! [X7 : nat,X6 : nat] : ((zero_zero_nat = X6) = ((plus_plus_nat @ X7 @ X6) = X7))),
% 2.52/1.64 file('/export/starexec/sandbox/tmp/tmp.GJlgnNEeHF/DTF2THF_19775.p',fact_29_add__cancel__right__right)).
% 2.52/1.64 thf(f1771,plain,(
% 2.52/1.64 (((?? @ nat @ (^[Y0 : nat]: ((plus_plus_nat @ n @ Y0) = (minus_minus_nat @ n @ sK8)))) & (~ (n = (minus_minus_nat @ n @ sK8)))) = $true)),
% 2.52/1.64 inference(backward_demodulation,[status(thm)],[f1160,f1768])).
% 2.52/1.64 thf(f1768,plain,(
% 2.52/1.64 ((minus_minus_nat @ n @ sK8) = (plus_plus_nat @ (times_times_nat @ i @ (p @ i)) @ (times_times_nat @ k @ (p @ k))))),
% 2.52/1.64 inference(superposition,[status(thm)],[f1338,f1751])).
% 2.52/1.64 thf(f1751,plain,(
% 2.52/1.64 (n = (plus_plus_nat @ sK8 @ (plus_plus_nat @ (times_times_nat @ i @ (p @ i)) @ (times_times_nat @ k @ (p @ k)))))),
% 2.52/1.64 inference(equality_proxy_clausification,[status(thm)],[f1748])).
% 2.52/1.64 thf(f1748,plain,(
% 2.52/1.64 (((plus_plus_nat @ sK8 @ (plus_plus_nat @ (times_times_nat @ i @ (p @ i)) @ (times_times_nat @ k @ (p @ k)))) = n) = $true)),
% 2.52/1.64 inference(forward_demodulation,[status(thm)],[f1747,f1158])).
% 2.52/1.64 thf(f1158,plain,(
% 2.52/1.64 ( ! [X2 : nat,X1 : nat] : (((times_times_nat @ X1 @ X2) = (times_times_nat @ X2 @ X1))) )),
% 2.52/1.64 inference(beta_eta_normalization,[status(thm)],[f1157])).
% 2.52/1.64 thf(f1157,plain,(
% 2.52/1.64 ( ! [X2 : nat,X1 : nat] : (((times_times_nat @ X1 @ X2) = ((^[Y0 : nat]: (times_times_nat @ Y0 @ X1)) @ X2))) )),
% 2.52/1.64 inference(argument_congruence,[status(thm)],[f1156])).
% 2.52/1.64 thf(f1156,plain,(
% 2.52/1.64 ( ! [X1 : nat] : (((times_times_nat @ X1) = (^[Y0 : nat]: (times_times_nat @ Y0 @ X1)))) )),
% 2.52/1.64 inference(beta_eta_normalization,[status(thm)],[f1155])).
% 2.52/1.64 thf(f1155,plain,(
% 2.52/1.64 ( ! [X1 : nat] : (((times_times_nat @ X1) = ((^[Y0 : nat]: ((^[Y1 : nat]: (times_times_nat @ Y1 @ Y0)))) @ X1))) )),
% 2.52/1.64 inference(argument_congruence,[status(thm)],[f670])).
% 2.52/1.64 thf(f670,plain,(
% 2.52/1.64 (times_times_nat = (^[Y0 : nat]: ((^[Y1 : nat]: (times_times_nat @ Y1 @ Y0)))))),
% 2.52/1.64 inference(cnf_transformation,[status(thm)],[f633])).
% 2.52/1.64 thf(f633,plain,(
% 2.52/1.64 (times_times_nat = (^[Y0 : nat]: ((^[Y1 : nat]: (times_times_nat @ Y1 @ Y0)))))),
% 2.52/1.64 inference(fool_elimination,[status(thm)],[f632])).
% 2.52/1.64 thf(f632,plain,(
% 2.52/1.64 (times_times_nat = (^[X0 : nat, X1 : nat] : (times_times_nat @ X1 @ X0)))),
% 2.52/1.64 inference(rectify,[status(thm)],[f137])).
% 2.52/1.64 thf(f137,axiom,(
% 2.52/1.64 (times_times_nat = (^[X16 : nat, X17 : nat] : (times_times_nat @ X17 @ X16)))),
% 2.52/1.64 file('/export/starexec/sandbox/tmp/tmp.GJlgnNEeHF/DTF2THF_19775.p',fact_136_mult_Ocommute)).
% 2.52/1.64 thf(f1747,plain,(
% 2.52/1.64 (((plus_plus_nat @ sK8 @ (plus_plus_nat @ (times_times_nat @ i @ (p @ i)) @ (times_times_nat @ (p @ k) @ k))) = n) = $true)),
% 2.52/1.64 inference(forward_demodulation,[status(thm)],[f1746,f1158])).
% 2.52/1.64 thf(f1746,plain,(
% 2.52/1.64 (((plus_plus_nat @ sK8 @ (plus_plus_nat @ (times_times_nat @ (p @ i) @ i) @ (times_times_nat @ (p @ k) @ k))) = n) = $true)),
% 2.52/1.64 inference(forward_demodulation,[status(thm)],[f1745,f1169])).
% 2.52/1.64 thf(f1745,plain,(
% 2.52/1.64 (((plus_plus_nat @ (plus_plus_nat @ (times_times_nat @ (p @ i) @ i) @ (times_times_nat @ (p @ k) @ k)) @ sK8) = n) = $true)),
% 2.52/1.64 inference(beta_eta_normalization,[status(thm)],[f1744])).
% 2.52/1.64 thf(f1744,plain,(
% 2.52/1.64 (((^[Y0 : nat]: ((plus_plus_nat @ (plus_plus_nat @ (times_times_nat @ (p @ i) @ i) @ (times_times_nat @ (p @ k) @ k)) @ Y0) = n)) @ sK8) = $true)),
% 2.52/1.64 inference(sigma_clausification,[status(thm)],[f1137])).
% 2.52/1.64 thf(f1137,plain,(
% 2.52/1.64 ((?? @ nat @ (^[Y0 : nat]: ((plus_plus_nat @ (plus_plus_nat @ (times_times_nat @ (p @ i) @ i) @ (times_times_nat @ (p @ k) @ k)) @ Y0) = n))) = $true)),
% 2.52/1.64 inference(beta_eta_normalization,[status(thm)],[f904])).
% 2.52/1.64 thf(f904,plain,(
% 2.52/1.64 (((^[Y0 : nat]: ((^[Y1 : nat]: (?? @ nat @ (^[Y2 : nat]: ((plus_plus_nat @ Y0 @ Y2) = Y1)))))) @ (plus_plus_nat @ (times_times_nat @ (p @ i) @ i) @ (times_times_nat @ (p @ k) @ k)) @ n) = $true)),
% 2.52/1.64 inference(definition_unfolding,[status(thm)],[f711,f698])).
% 2.52/1.64 thf(f711,plain,(
% 2.52/1.64 ((ord_less_eq_nat @ (plus_plus_nat @ (times_times_nat @ (p @ i) @ i) @ (times_times_nat @ (p @ k) @ k)) @ n) = $true)),
% 2.52/1.64 inference(cnf_transformation,[status(thm)],[f264])).
% 2.52/1.64 thf(f264,plain,(
% 2.52/1.64 ((ord_less_eq_nat @ (plus_plus_nat @ (times_times_nat @ (p @ i) @ i) @ (times_times_nat @ (p @ k) @ k)) @ n) = $true)),
% 2.52/1.64 inference(fool_elimination,[status(thm)],[f263])).
% 2.52/1.64 thf(f263,plain,(
% 2.52/1.64 (ord_less_eq_nat @ (plus_plus_nat @ (times_times_nat @ (p @ i) @ i) @ (times_times_nat @ (p @ k) @ k)) @ n)),
% 2.52/1.64 inference(rectify,[status(thm)],[f3])).
% 2.52/1.64 thf(f3,axiom,(
% 2.52/1.64 (ord_less_eq_nat @ (plus_plus_nat @ (times_times_nat @ (p @ i) @ i) @ (times_times_nat @ (p @ k) @ k)) @ n)),
% 2.52/1.64 file('/export/starexec/sandbox/tmp/tmp.GJlgnNEeHF/DTF2THF_19775.p',fact_2_upper__bound)).
% 2.52/1.64 thf(f1160,plain,(
% 2.52/1.64 (((?? @ nat @ (^[Y0 : nat]: ((plus_plus_nat @ n @ Y0) = (plus_plus_nat @ (times_times_nat @ i @ (p @ i)) @ (times_times_nat @ k @ (p @ k)))))) & (~ (n = (plus_plus_nat @ (times_times_nat @ i @ (p @ i)) @ (times_times_nat @ k @ (p @ k)))))) = $true)),
% 2.52/1.64 inference(forward_demodulation,[status(thm)],[f1159,f1158])).
% 2.52/1.64 thf(f1159,plain,(
% 2.52/1.64 (((?? @ nat @ (^[Y0 : nat]: ((plus_plus_nat @ n @ Y0) = (plus_plus_nat @ (times_times_nat @ i @ (p @ i)) @ (times_times_nat @ (p @ k) @ k))))) & (~ (n = (plus_plus_nat @ (times_times_nat @ i @ (p @ i)) @ (times_times_nat @ (p @ k) @ k))))) = $true)),
% 2.52/1.64 inference(backward_demodulation,[status(thm)],[f1118,f1158])).
% 2.52/1.64 thf(f1118,plain,(
% 2.52/1.64 (((?? @ nat @ (^[Y0 : nat]: ((plus_plus_nat @ n @ Y0) = (plus_plus_nat @ (times_times_nat @ (p @ i) @ i) @ (times_times_nat @ (p @ k) @ k))))) & (~ (n = (plus_plus_nat @ (times_times_nat @ (p @ i) @ i) @ (times_times_nat @ (p @ k) @ k))))) = $true)),
% 2.52/1.64 inference(beta_eta_normalization,[status(thm)],[f879])).
% 2.52/1.64 thf(f879,plain,(
% 2.52/1.64 (((^[Y0 : nat]: ((^[Y1 : nat]: (((^[Y2 : nat]: ((^[Y3 : nat]: (?? @ nat @ (^[Y4 : nat]: ((plus_plus_nat @ Y2 @ Y4) = Y3)))))) @ Y0 @ Y1) & (~ (Y0 = Y1)))))) @ n @ (plus_plus_nat @ (times_times_nat @ (p @ i) @ i) @ (times_times_nat @ (p @ k) @ k))) = $true)),
% 2.52/1.64 inference(definition_unfolding,[status(thm)],[f681,f851])).
% 2.52/1.64 thf(f851,plain,(
% 2.52/1.64 (ord_less_nat = (^[Y0 : nat]: ((^[Y1 : nat]: (((^[Y2 : nat]: ((^[Y3 : nat]: (?? @ nat @ (^[Y4 : nat]: ((plus_plus_nat @ Y2 @ Y4) = Y3)))))) @ Y0 @ Y1) & (~ (Y0 = Y1)))))))),
% 2.52/1.64 inference(definition_unfolding,[status(thm)],[f724,f698])).
% 2.52/1.64 thf(f724,plain,(
% 2.52/1.64 (ord_less_nat = (^[Y0 : nat]: ((^[Y1 : nat]: ((ord_less_eq_nat @ Y0 @ Y1) & (~ (Y0 = Y1)))))))),
% 2.52/1.64 inference(cnf_transformation,[status(thm)],[f296])).
% 2.52/1.64 thf(f296,plain,(
% 2.52/1.64 (ord_less_nat = (^[Y0 : nat]: ((^[Y1 : nat]: ((ord_less_eq_nat @ Y0 @ Y1) & (~ (Y0 = Y1)))))))),
% 2.52/1.64 inference(fool_elimination,[status(thm)],[f295])).
% 2.52/1.64 thf(f295,plain,(
% 2.52/1.64 ((^[X0 : nat, X1 : nat] : ((X0 != X1) & (ord_less_eq_nat @ X0 @ X1))) = ord_less_nat)),
% 2.52/1.64 inference(rectify,[status(thm)],[f197])).
% 2.52/1.64 thf(f197,axiom,(
% 2.52/1.64 ((^[X33 : nat, X34 : nat] : ((X33 != X34) & (ord_less_eq_nat @ X33 @ X34))) = ord_less_nat)),
% 2.52/1.64 file('/export/starexec/sandbox/tmp/tmp.GJlgnNEeHF/DTF2THF_19775.p',fact_196_nat__less__le)).
% 2.52/1.64 thf(f681,plain,(
% 2.52/1.64 ((ord_less_nat @ n @ (plus_plus_nat @ (times_times_nat @ (p @ i) @ i) @ (times_times_nat @ (p @ k) @ k))) = $true)),
% 2.52/1.64 inference(cnf_transformation,[status(thm)],[f601])).
% 2.52/1.64 thf(f601,plain,(
% 2.52/1.64 ((ord_less_nat @ n @ (plus_plus_nat @ (times_times_nat @ (p @ i) @ i) @ (times_times_nat @ (p @ k) @ k))) = $true)),
% 2.52/1.64 inference(fool_elimination,[status(thm)],[f600])).
% 2.52/1.64 thf(f600,plain,(
% 2.52/1.64 (ord_less_nat @ n @ (plus_plus_nat @ (times_times_nat @ (p @ i) @ i) @ (times_times_nat @ (p @ k) @ k)))),
% 2.52/1.64 inference(rectify,[status(thm)],[f4])).
% 2.52/1.64 thf(f4,axiom,(
% 2.52/1.64 (ord_less_nat @ n @ (plus_plus_nat @ (times_times_nat @ (p @ i) @ i) @ (times_times_nat @ (p @ k) @ k)))),
% 2.52/1.64 file('/export/starexec/sandbox/tmp/tmp.GJlgnNEeHF/DTF2THF_19775.p',fact_3_lower__bound)).
% 2.52/1.64 thf(f1269,plain,(
% 2.52/1.64 ( ! [X2 : nat,X1 : nat] : ((zero_zero_nat = (minus_minus_nat @ X2 @ (plus_plus_nat @ X2 @ X1)))) )),
% 2.52/1.64 inference(equality_proxy_clausification,[status(thm)],[f1268])).
% 2.52/1.64 thf(f1268,plain,(
% 2.52/1.64 ( ! [X2 : nat,X1 : nat] : (((zero_zero_nat = (minus_minus_nat @ X2 @ (plus_plus_nat @ X2 @ X1))) = $true)) )),
% 2.52/1.64 inference(beta_eta_normalization,[status(thm)],[f1267])).
% 2.52/1.64 thf(f1267,plain,(
% 2.52/1.64 ( ! [X2 : nat,X1 : nat] : ((((^[Y0 : nat]: (zero_zero_nat = (minus_minus_nat @ Y0 @ (plus_plus_nat @ Y0 @ X1)))) @ X2) = $true)) )),
% 2.52/1.64 inference(pi_clausification,[status(thm)],[f1265])).
% 2.52/1.64 thf(f1265,plain,(
% 2.52/1.64 ( ! [X1 : nat] : (($true = (!! @ nat @ (^[Y0 : nat]: (zero_zero_nat = (minus_minus_nat @ Y0 @ (plus_plus_nat @ Y0 @ X1))))))) )),
% 2.52/1.64 inference(beta_eta_normalization,[status(thm)],[f1264])).
% 2.52/1.64 thf(f1264,plain,(
% 2.52/1.64 ( ! [X1 : nat] : ((((^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: (zero_zero_nat = (minus_minus_nat @ Y1 @ (plus_plus_nat @ Y1 @ Y0)))))) @ X1) = $true)) )),
% 2.52/1.64 inference(pi_clausification,[status(thm)],[f791])).
% 2.52/1.64 thf(f791,plain,(
% 2.52/1.64 ((!! @ nat @ (^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: (zero_zero_nat = (minus_minus_nat @ Y1 @ (plus_plus_nat @ Y1 @ Y0))))))) = $true)),
% 2.52/1.64 inference(cnf_transformation,[status(thm)],[f522])).
% 2.52/1.64 thf(f522,plain,(
% 2.52/1.64 ((!! @ nat @ (^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: (zero_zero_nat = (minus_minus_nat @ Y1 @ (plus_plus_nat @ Y1 @ Y0))))))) = $true)),
% 2.52/1.64 inference(fool_elimination,[status(thm)],[f521])).
% 2.52/1.64 thf(f521,plain,(
% 2.52/1.64 ! [X0 : nat,X1 : nat] : (zero_zero_nat = (minus_minus_nat @ X0 @ (plus_plus_nat @ X0 @ X1)))),
% 2.52/1.64 inference(rectify,[status(thm)],[f66])).
% 2.52/1.64 thf(f66,axiom,(
% 2.52/1.64 ! [X7 : nat,X6 : nat] : (zero_zero_nat = (minus_minus_nat @ X7 @ (plus_plus_nat @ X7 @ X6)))),
% 2.52/1.64 file('/export/starexec/sandbox/tmp/tmp.GJlgnNEeHF/DTF2THF_19775.p',fact_65_diff__add__zero)).
% 2.52/1.64 thf(f1775,plain,(
% 2.52/1.64 ((minus_minus_nat @ n @ (minus_minus_nat @ n @ sK8)) = sK8)),
% 2.52/1.64 inference(forward_demodulation,[status(thm)],[f1767,f1768])).
% 2.52/1.64 thf(f1767,plain,(
% 2.52/1.64 (sK8 = (minus_minus_nat @ n @ (plus_plus_nat @ (times_times_nat @ i @ (p @ i)) @ (times_times_nat @ k @ (p @ k)))))),
% 2.52/1.64 inference(superposition,[status(thm)],[f1317,f1751])).
% 2.52/1.64 thf(f1949,plain,(
% 2.52/1.64 ( ! [X0 : nat] : (((plus_plus_nat @ X0 @ (sK7 @ n @ sK8)) != X0)) )),
% 2.52/1.64 inference(trivial_inequality_removal,[status(thm)],[f1948])).
% 2.52/1.64 thf(f1948,plain,(
% 2.52/1.64 ( ! [X0 : nat] : (((plus_plus_nat @ X0 @ (sK7 @ n @ sK8)) != X0) | ($false = $true)) )),
% 2.52/1.64 inference(boolean_simplification,[status(thm)],[f1947])).
% 2.52/1.64 thf(f1947,plain,(
% 2.52/1.64 ( ! [X0 : nat] : (((plus_plus_nat @ X0 @ (sK7 @ n @ sK8)) != X0) | (((?? @ nat @ (^[Y0 : nat]: ((plus_plus_nat @ n @ Y0) = (minus_minus_nat @ n @ sK8)))) & $false) = $true)) )),
% 2.52/1.64 inference(boolean_simplification,[status(thm)],[f1946])).
% 2.52/1.64 thf(f1946,plain,(
% 2.52/1.64 ( ! [X0 : nat] : ((((?? @ nat @ (^[Y0 : nat]: ((plus_plus_nat @ n @ Y0) = (minus_minus_nat @ n @ sK8)))) & (~ $true)) = $true) | ((plus_plus_nat @ X0 @ (sK7 @ n @ sK8)) != X0)) )),
% 2.52/1.64 inference(superposition,[status(thm)],[f1772,f1782])).
% 2.52/1.64 thf(f1782,plain,(
% 2.52/1.64 ( ! [X2 : nat,X1 : nat] : (((zero_zero_nat = X1) = $true) | ((plus_plus_nat @ X2 @ X1) != X2)) )),
% 2.52/1.64 inference(equality_proxy_clausification,[status(thm)],[f1480])).
% 2.52/1.64 thf(f1480,plain,(
% 2.52/1.64 ( ! [X2 : nat,X1 : nat] : ((((plus_plus_nat @ X2 @ X1) = X2) = $false) | ((zero_zero_nat = X1) = $true)) )),
% 2.52/1.64 inference(binary_proxy_clausification,[status(thm)],[f1478])).
% 2.52/1.64 thf(f1478,plain,(
% 2.52/1.64 ( ! [X2 : nat,X1 : nat] : (((((plus_plus_nat @ X2 @ X1) = X2) => (zero_zero_nat = X1)) = $true)) )),
% 2.52/1.64 inference(beta_eta_normalization,[status(thm)],[f1477])).
% 2.52/1.64 thf(f1477,plain,(
% 2.52/1.64 ( ! [X2 : nat,X1 : nat] : ((((^[Y0 : nat]: (((plus_plus_nat @ Y0 @ X1) = Y0) => (zero_zero_nat = X1))) @ X2) = $true)) )),
% 2.52/1.64 inference(pi_clausification,[status(thm)],[f1475])).
% 2.52/1.64 thf(f1475,plain,(
% 2.52/1.64 ( ! [X1 : nat] : (((!! @ nat @ (^[Y0 : nat]: (((plus_plus_nat @ Y0 @ X1) = Y0) => (zero_zero_nat = X1)))) = $true)) )),
% 2.52/1.64 inference(beta_eta_normalization,[status(thm)],[f1474])).
% 2.52/1.64 thf(f1474,plain,(
% 2.52/1.64 ( ! [X1 : nat] : ((((^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: (((plus_plus_nat @ Y1 @ Y0) = Y1) => (zero_zero_nat = Y0))))) @ X1) = $true)) )),
% 2.52/1.64 inference(pi_clausification,[status(thm)],[f672])).
% 2.52/1.64 thf(f672,plain,(
% 2.52/1.64 ($true = (!! @ nat @ (^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: (((plus_plus_nat @ Y1 @ Y0) = Y1) => (zero_zero_nat = Y0)))))))),
% 2.52/1.64 inference(cnf_transformation,[status(thm)],[f490])).
% 2.52/1.64 thf(f490,plain,(
% 2.52/1.64 ($true = (!! @ nat @ (^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: (((plus_plus_nat @ Y1 @ Y0) = Y1) => (zero_zero_nat = Y0)))))))),
% 2.52/1.64 inference(fool_elimination,[status(thm)],[f489])).
% 2.52/1.64 thf(f489,plain,(
% 2.52/1.64 ! [X0 : nat,X1 : nat] : (((plus_plus_nat @ X0 @ X1) = X0) => (zero_zero_nat = X1))),
% 2.52/1.64 inference(rectify,[status(thm)],[f190])).
% 2.52/1.64 thf(f190,axiom,(
% 2.52/1.64 ! [X4 : nat,X2 : nat] : (((plus_plus_nat @ X4 @ X2) = X4) => (zero_zero_nat = X2))),
% 2.52/1.64 file('/export/starexec/sandbox/tmp/tmp.GJlgnNEeHF/DTF2THF_19775.p',fact_189_add__eq__self__zero)).
% 2.52/1.64 % SZS output end Proof for DTF2THF_19775
% 2.52/1.64 % (20058)------------------------------
% 2.52/1.64 % (20058)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.64 % (20058)Termination reason: Refutation
% 2.52/1.64
% 2.52/1.64 % (20058)Memory used [KB]: 7419
% 2.52/1.64 % (20058)Time elapsed: 0.113 s
% 2.52/1.64 % (20058)Instructions burned: 336 (million)
% 2.52/1.64 % (20058)------------------------------
% 2.52/1.64 % (20058)------------------------------
% 2.52/1.64 % (20046)Success in time 0.169 s
%------------------------------------------------------------------------------