%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU664^2 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n011.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 09:36:03 AM UTC 2026
% Result : Theorem 1.52s 1.30s
% Output : Refutation 1.66s
% Verified :
% SZS Type : Refutation
% Derivation depth : 24
% Number of leaves : 12
% Syntax : Number of formulae : 78 ( 13 unt; 0 typ; 0 def)
% Number of atoms : 827 ( 300 equ; 0 cnn)
% Maximal formula atoms : 10 ( 10 avg)
% Number of connectives : 1193 ( 142 ~; 122 |; 76 &; 808 @)
% ( 6 <=>; 39 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 7 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 27 ( 24 usr; 19 con; 0-3 aty)
% Number of variables : 285 ( 0 ^; 220 !; 65 ?; 285 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_1,type,
kpair: $i > $i > $i ).
thf(func_def_2,type,
cartprod: $i > $i > $i ).
thf(func_def_5,type,
kfst: $i > $i ).
thf(func_def_6,type,
ksnd: $i > $i ).
thf(func_def_21,type,
sK10: $i > $i > $i > $i ).
thf(func_def_22,type,
sK11: $i > $i > $i > $i ).
thf(f125,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f122,f39]) ).
thf(f39,plain,
( sK0
!= ( kpair @ ( kfst @ sK0 ) @ ( ksnd @ sK0 ) ) ),
inference(cnf_transformation,[status(thm)],[f22]) ).
thf(f22,plain,
( ( cartprodsndpairEq = $true )
& ( cartprodmempair1 = $true )
& ( cartprodfstpairEq = $true )
& ( ( in @ sK0 @ ( cartprod @ sK2 @ sK1 ) )
= $true )
& ( sK0
!= ( kpair @ ( kfst @ sK0 ) @ ( ksnd @ sK0 ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2])],[f20,f21]) ).
thf(f21,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( cartprod @ X2 @ X1 ) )
= $true )
& ( ( kpair @ ( kfst @ X0 ) @ ( ksnd @ X0 ) )
!= X0 ) )
=> ( ( ( in @ sK0 @ ( cartprod @ sK2 @ sK1 ) )
= $true )
& ( sK0
!= ( kpair @ ( kfst @ sK0 ) @ ( ksnd @ sK0 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f20,plain,
( ( cartprodsndpairEq = $true )
& ( cartprodmempair1 = $true )
& ( cartprodfstpairEq = $true )
& ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( cartprod @ X2 @ X1 ) )
= $true )
& ( ( kpair @ ( kfst @ X0 ) @ ( ksnd @ X0 ) )
!= X0 ) ) ),
inference(rectify,[status(thm)],[f19]) ).
thf(f19,plain,
( ( cartprodsndpairEq = $true )
& ( cartprodmempair1 = $true )
& ( cartprodfstpairEq = $true )
& ? [X2: $i,X1: $i,X0: $i] :
( ( ( in @ X2 @ ( cartprod @ X0 @ X1 ) )
= $true )
& ( ( kpair @ ( kfst @ X2 ) @ ( ksnd @ X2 ) )
!= X2 ) ) ),
inference(flattening,[status(thm)],[f18]) ).
thf(f18,plain,
( ? [X2: $i,X1: $i,X0: $i] :
( ( ( in @ X2 @ ( cartprod @ X0 @ X1 ) )
= $true )
& ( ( kpair @ ( kfst @ X2 ) @ ( ksnd @ X2 ) )
!= X2 ) )
& ( cartprodsndpairEq = $true )
& ( cartprodfstpairEq = $true )
& ( cartprodmempair1 = $true ) ),
inference(ennf_transformation,[status(thm)],[f12]) ).
thf(f12,plain,
~ ( ( cartprodmempair1 = $true )
=> ( ( cartprodfstpairEq = $true )
=> ( ( cartprodsndpairEq = $true )
=> ! [X1: $i,X2: $i,X0: $i] :
( ( ( in @ X2 @ ( cartprod @ X0 @ X1 ) )
= $true )
=> ( ( kpair @ ( kfst @ X2 ) @ ( ksnd @ X2 ) )
= X2 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f11]) ).
thf(f11,plain,
~ ( cartprodmempair1
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X2 @ ( cartprod @ X0 @ X1 ) )
=> ( ( kpair @ ( kfst @ X2 ) @ ( ksnd @ X2 ) )
= X2 ) ) ) ) ),
inference(rectify,[status(thm)],[f5]) ).
thf(f5,negated_conjecture,
~ ( cartprodmempair1
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X2 @ ( cartprod @ X0 @ X1 ) )
=> ( ( kpair @ ( kfst @ X2 ) @ ( ksnd @ X2 ) )
= X2 ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f4]) ).
thf(f4,conjecture,
( cartprodmempair1
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X2 @ ( cartprod @ X0 @ X1 ) )
=> ( ( kpair @ ( kfst @ X2 ) @ ( ksnd @ X2 ) )
= X2 ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.3hJx7CzvFR/DTF2THF_17518.p',cartprodpairsurjEq) ).
thf(f122,plain,
( sK0
= ( kpair @ ( kfst @ sK0 ) @ ( ksnd @ sK0 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f119]) ).
thf(f119,plain,
( ( $true != $true )
| ( sK0
= ( kpair @ ( kfst @ sK0 ) @ ( ksnd @ sK0 ) ) ) ),
inference(superposition,[status(thm)],[f118,f40]) ).
thf(f40,plain,
( ( in @ sK0 @ ( cartprod @ sK2 @ sK1 ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f22]) ).
thf(f118,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
!= $true )
| ( ( kpair @ ( kfst @ X2 ) @ ( ksnd @ X2 ) )
= X2 ) ),
inference(duplicate_literal_removal,[status(thm)],[f115]) ).
thf(f115,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( ( kpair @ ( kfst @ X2 ) @ ( ksnd @ X2 ) )
= X2 )
| ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
!= $true )
| ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
!= $true ) ),
inference(superposition,[status(thm)],[f90,f94]) ).
thf(f94,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( ( ksnd @ X2 )
= ( sK11 @ X0 @ X1 @ X2 ) )
| ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f93]) ).
thf(f93,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( $true != $true )
| ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
!= $true )
| ( ( ksnd @ X2 )
= ( sK11 @ X0 @ X1 @ X2 ) ) ),
inference(duplicate_literal_removal,[status(thm)],[f92]) ).
thf(f92,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
!= $true )
| ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
!= $true )
| ( $true != $true )
| ( ( ksnd @ X2 )
= ( sK11 @ X0 @ X1 @ X2 ) ) ),
inference(superposition,[status(thm)],[f85,f73]) ).
thf(f73,plain,
! [X6: $i,X7: $i,X5: $i] :
( ( ( in @ ( sK10 @ X7 @ X6 @ X5 ) @ X6 )
= $true )
| ( ( in @ X5 @ ( cartprod @ X6 @ X7 ) )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f63]) ).
thf(f63,plain,
! [X6: $i,X7: $i,X5: $i] :
( ( ( in @ ( sK10 @ X7 @ X6 @ X5 ) @ X6 )
= $true )
| ( $true != $true )
| ( ( in @ X5 @ ( cartprod @ X6 @ X7 ) )
!= $true ) ),
inference(definition_unfolding,[status(thm)],[f50,f42]) ).
thf(f42,plain,
cartprodmempair1 = $true,
inference(cnf_transformation,[status(thm)],[f22]) ).
thf(f50,plain,
! [X6: $i,X7: $i,X5: $i] :
( ( ( in @ X5 @ ( cartprod @ X6 @ X7 ) )
!= $true )
| ( ( in @ ( sK10 @ X7 @ X6 @ X5 ) @ X6 )
= $true )
| ( cartprodmempair1 != $true ) ),
inference(cnf_transformation,[status(thm)],[f33]) ).
thf(f33,plain,
( ( ( cartprodmempair1 = $true )
| ( ( ( in @ sK7 @ ( cartprod @ sK8 @ sK9 ) )
= $true )
& ! [X3: $i] :
( ( ( in @ X3 @ sK8 )
!= $true )
| ! [X4: $i] :
( ( ( kpair @ X3 @ X4 )
!= sK7 )
| ( $true
!= ( in @ X4 @ sK9 ) ) ) ) ) )
& ( ! [X5: $i,X6: $i,X7: $i] :
( ( ( in @ X5 @ ( cartprod @ X6 @ X7 ) )
!= $true )
| ( ( ( in @ ( sK10 @ X7 @ X6 @ X5 ) @ X6 )
= $true )
& ( ( kpair @ ( sK10 @ X7 @ X6 @ X5 ) @ ( sK11 @ X7 @ X6 @ X5 ) )
= X5 )
& ( ( in @ ( sK11 @ X7 @ X6 @ X5 ) @ X7 )
= $true ) ) )
| ( cartprodmempair1 != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK7,sK8,sK9,sK10,sK11])],[f29,f32,f31,f30]) ).
thf(f30,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( cartprod @ X1 @ X2 ) )
= $true )
& ! [X3: $i] :
( ( ( in @ X3 @ X1 )
!= $true )
| ! [X4: $i] :
( ( ( kpair @ X3 @ X4 )
!= X0 )
| ( ( in @ X4 @ X2 )
!= $true ) ) ) )
=> ( ( ( in @ sK7 @ ( cartprod @ sK8 @ sK9 ) )
= $true )
& ! [X3: $i] :
( ( ( in @ X3 @ sK8 )
!= $true )
| ! [X4: $i] :
( ( ( kpair @ X3 @ X4 )
!= sK7 )
| ( $true
!= ( in @ X4 @ sK9 ) ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f31,plain,
! [X5: $i,X6: $i,X7: $i] :
( ? [X8: $i] :
( ( ( in @ X8 @ X6 )
= $true )
& ? [X9: $i] :
( ( ( kpair @ X8 @ X9 )
= X5 )
& ( ( in @ X9 @ X7 )
= $true ) ) )
=> ( ( ( in @ ( sK10 @ X7 @ X6 @ X5 ) @ X6 )
= $true )
& ? [X9: $i] :
( ( ( kpair @ ( sK10 @ X7 @ X6 @ X5 ) @ X9 )
= X5 )
& ( ( in @ X9 @ X7 )
= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f32,plain,
! [X5: $i,X6: $i,X7: $i] :
( ? [X9: $i] :
( ( ( kpair @ ( sK10 @ X7 @ X6 @ X5 ) @ X9 )
= X5 )
& ( ( in @ X9 @ X7 )
= $true ) )
=> ( ( ( kpair @ ( sK10 @ X7 @ X6 @ X5 ) @ ( sK11 @ X7 @ X6 @ X5 ) )
= X5 )
& ( ( in @ ( sK11 @ X7 @ X6 @ X5 ) @ X7 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f29,plain,
( ( ( cartprodmempair1 = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( cartprod @ X1 @ X2 ) )
= $true )
& ! [X3: $i] :
( ( ( in @ X3 @ X1 )
!= $true )
| ! [X4: $i] :
( ( ( kpair @ X3 @ X4 )
!= X0 )
| ( ( in @ X4 @ X2 )
!= $true ) ) ) ) )
& ( ! [X5: $i,X6: $i,X7: $i] :
( ( ( in @ X5 @ ( cartprod @ X6 @ X7 ) )
!= $true )
| ? [X8: $i] :
( ( ( in @ X8 @ X6 )
= $true )
& ? [X9: $i] :
( ( ( kpair @ X8 @ X9 )
= X5 )
& ( ( in @ X9 @ X7 )
= $true ) ) ) )
| ( cartprodmempair1 != $true ) ) ),
inference(rectify,[status(thm)],[f28]) ).
thf(f28,plain,
( ( ( cartprodmempair1 = $true )
| ? [X0: $i,X2: $i,X1: $i] :
( ( ( in @ X0 @ ( cartprod @ X2 @ X1 ) )
= $true )
& ! [X3: $i] :
( ( ( in @ X3 @ X2 )
!= $true )
| ! [X4: $i] :
( ( ( kpair @ X3 @ X4 )
!= X0 )
| ( ( in @ X4 @ X1 )
!= $true ) ) ) ) )
& ( ! [X0: $i,X2: $i,X1: $i] :
( ( ( in @ X0 @ ( cartprod @ X2 @ X1 ) )
!= $true )
| ? [X3: $i] :
( ( ( in @ X3 @ X2 )
= $true )
& ? [X4: $i] :
( ( ( kpair @ X3 @ X4 )
= X0 )
& ( ( in @ X4 @ X1 )
= $true ) ) ) )
| ( cartprodmempair1 != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f16]) ).
thf(f16,plain,
( ( cartprodmempair1 = $true )
<=> ! [X0: $i,X2: $i,X1: $i] :
( ( ( in @ X0 @ ( cartprod @ X2 @ X1 ) )
!= $true )
| ? [X3: $i] :
( ( ( in @ X3 @ X2 )
= $true )
& ? [X4: $i] :
( ( ( kpair @ X3 @ X4 )
= X0 )
& ( ( in @ X4 @ X1 )
= $true ) ) ) ) ),
inference(ennf_transformation,[status(thm)],[f10]) ).
thf(f10,plain,
( ! [X1: $i,X2: $i,X0: $i] :
( ( ( in @ X0 @ ( cartprod @ X2 @ X1 ) )
= $true )
=> ? [X3: $i] :
( ( ( in @ X3 @ X2 )
= $true )
& ? [X4: $i] :
( ( ( kpair @ X3 @ X4 )
= X0 )
& ( ( in @ X4 @ X1 )
= $true ) ) ) )
<=> ( cartprodmempair1 = $true ) ),
inference(fool_elimination,[status(thm)],[f9]) ).
thf(f9,plain,
! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( cartprod @ X2 @ X1 ) )
=> ? [X3: $i] :
( ? [X4: $i] :
( ( ( kpair @ X3 @ X4 )
= X0 )
& ( in @ X4 @ X1 ) )
& ( in @ X3 @ X2 ) ) )
= cartprodmempair1 ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
! [X2: $i,X1: $i,X0: $i] :
( ( ( in @ X2 @ ( cartprod @ X0 @ X1 ) )
=> ? [X3: $i] :
( ? [X4: $i] :
( ( ( kpair @ X3 @ X4 )
= X2 )
& ( in @ X4 @ X1 ) )
& ( in @ X3 @ X0 ) ) )
= cartprodmempair1 ),
file('/export/starexec/sandbox2/tmp/tmp.3hJx7CzvFR/DTF2THF_17518.p',cartprodmempair1) ).
thf(f85,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i] :
( ( ( in @ ( sK10 @ X0 @ X1 @ X2 ) @ X3 )
!= $true )
| ( ( ksnd @ X2 )
= ( sK11 @ X0 @ X1 @ X2 ) )
| ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f84]) ).
thf(f84,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i] :
( ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
!= $true )
| ( $true != $true )
| ( ( in @ ( sK10 @ X0 @ X1 @ X2 ) @ X3 )
!= $true )
| ( ( ksnd @ X2 )
= ( sK11 @ X0 @ X1 @ X2 ) ) ),
inference(duplicate_literal_removal,[status(thm)],[f83]) ).
thf(f83,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i] :
( ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
!= $true )
| ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
!= $true )
| ( $true != $true )
| ( ( in @ ( sK10 @ X0 @ X1 @ X2 ) @ X3 )
!= $true )
| ( ( ksnd @ X2 )
= ( sK11 @ X0 @ X1 @ X2 ) ) ),
inference(superposition,[status(thm)],[f76,f72]) ).
thf(f72,plain,
! [X6: $i,X7: $i,X5: $i] :
( ( ( in @ ( sK11 @ X7 @ X6 @ X5 ) @ X7 )
= $true )
| ( ( in @ X5 @ ( cartprod @ X6 @ X7 ) )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f65]) ).
thf(f65,plain,
! [X6: $i,X7: $i,X5: $i] :
( ( $true != $true )
| ( ( in @ ( sK11 @ X7 @ X6 @ X5 ) @ X7 )
= $true )
| ( ( in @ X5 @ ( cartprod @ X6 @ X7 ) )
!= $true ) ),
inference(definition_unfolding,[status(thm)],[f48,f42]) ).
thf(f48,plain,
! [X6: $i,X7: $i,X5: $i] :
( ( ( in @ X5 @ ( cartprod @ X6 @ X7 ) )
!= $true )
| ( ( in @ ( sK11 @ X7 @ X6 @ X5 ) @ X7 )
= $true )
| ( cartprodmempair1 != $true ) ),
inference(cnf_transformation,[status(thm)],[f33]) ).
thf(f76,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i,X4: $i] :
( ( ( in @ ( sK11 @ X0 @ X1 @ X2 ) @ X4 )
!= $true )
| ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
!= $true )
| ( ( in @ ( sK10 @ X0 @ X1 @ X2 ) @ X3 )
!= $true )
| ( ( ksnd @ X2 )
= ( sK11 @ X0 @ X1 @ X2 ) ) ),
inference(superposition,[status(thm)],[f70,f74]) ).
thf(f74,plain,
! [X6: $i,X7: $i,X5: $i] :
( ( ( kpair @ ( sK10 @ X7 @ X6 @ X5 ) @ ( sK11 @ X7 @ X6 @ X5 ) )
= X5 )
| ( ( in @ X5 @ ( cartprod @ X6 @ X7 ) )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f64]) ).
thf(f64,plain,
! [X6: $i,X7: $i,X5: $i] :
( ( ( kpair @ ( sK10 @ X7 @ X6 @ X5 ) @ ( sK11 @ X7 @ X6 @ X5 ) )
= X5 )
| ( $true != $true )
| ( ( in @ X5 @ ( cartprod @ X6 @ X7 ) )
!= $true ) ),
inference(definition_unfolding,[status(thm)],[f49,f42]) ).
thf(f49,plain,
! [X6: $i,X7: $i,X5: $i] :
( ( ( in @ X5 @ ( cartprod @ X6 @ X7 ) )
!= $true )
| ( ( kpair @ ( sK10 @ X7 @ X6 @ X5 ) @ ( sK11 @ X7 @ X6 @ X5 ) )
= X5 )
| ( cartprodmempair1 != $true ) ),
inference(cnf_transformation,[status(thm)],[f33]) ).
thf(f70,plain,
! [X6: $i,X7: $i,X4: $i,X5: $i] :
( ( ( ksnd @ ( kpair @ X5 @ X7 ) )
= X7 )
| ( $true
!= ( in @ X5 @ X6 ) )
| ( ( in @ X7 @ X4 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f60]) ).
thf(f60,plain,
! [X6: $i,X7: $i,X4: $i,X5: $i] :
( ( ( in @ X7 @ X4 )
!= $true )
| ( ( ksnd @ ( kpair @ X5 @ X7 ) )
= X7 )
| ( $true != $true )
| ( $true
!= ( in @ X5 @ X6 ) ) ),
inference(definition_unfolding,[status(thm)],[f44,f43]) ).
thf(f43,plain,
cartprodsndpairEq = $true,
inference(cnf_transformation,[status(thm)],[f22]) ).
thf(f44,plain,
! [X6: $i,X7: $i,X4: $i,X5: $i] :
( ( ( ksnd @ ( kpair @ X5 @ X7 ) )
= X7 )
| ( ( in @ X7 @ X4 )
!= $true )
| ( $true
!= ( in @ X5 @ X6 ) )
| ( cartprodsndpairEq != $true ) ),
inference(cnf_transformation,[status(thm)],[f27]) ).
thf(f27,plain,
( ( ( cartprodsndpairEq = $true )
| ( ( sK6
!= ( ksnd @ ( kpair @ sK4 @ sK6 ) ) )
& ( ( in @ sK6 @ sK3 )
= $true )
& ( ( in @ sK4 @ sK5 )
= $true ) ) )
& ( ! [X4: $i,X5: $i,X6: $i] :
( ! [X7: $i] :
( ( ( ksnd @ ( kpair @ X5 @ X7 ) )
= X7 )
| ( ( in @ X7 @ X4 )
!= $true ) )
| ( $true
!= ( in @ X5 @ X6 ) ) )
| ( cartprodsndpairEq != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK3,sK4,sK5,sK6])],[f24,f26,f25]) ).
thf(f25,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( ( ksnd @ ( kpair @ X1 @ X3 ) )
!= X3 )
& ( ( in @ X3 @ X0 )
= $true ) )
& ( ( in @ X1 @ X2 )
= $true ) )
=> ( ? [X3: $i] :
( ( ( ksnd @ ( kpair @ sK4 @ X3 ) )
!= X3 )
& ( ( in @ X3 @ sK3 )
= $true ) )
& ( ( in @ sK4 @ sK5 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f26,plain,
( ? [X3: $i] :
( ( ( ksnd @ ( kpair @ sK4 @ X3 ) )
!= X3 )
& ( ( in @ X3 @ sK3 )
= $true ) )
=> ( ( sK6
!= ( ksnd @ ( kpair @ sK4 @ sK6 ) ) )
& ( ( in @ sK6 @ sK3 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f24,plain,
( ( ( cartprodsndpairEq = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( ( ksnd @ ( kpair @ X1 @ X3 ) )
!= X3 )
& ( ( in @ X3 @ X0 )
= $true ) )
& ( ( in @ X1 @ X2 )
= $true ) ) )
& ( ! [X4: $i,X5: $i,X6: $i] :
( ! [X7: $i] :
( ( ( ksnd @ ( kpair @ X5 @ X7 ) )
= X7 )
| ( ( in @ X7 @ X4 )
!= $true ) )
| ( $true
!= ( in @ X5 @ X6 ) ) )
| ( cartprodsndpairEq != $true ) ) ),
inference(rectify,[status(thm)],[f23]) ).
thf(f23,plain,
( ( ( cartprodsndpairEq = $true )
| ? [X1: $i,X0: $i,X2: $i] :
( ? [X3: $i] :
( ( ( ksnd @ ( kpair @ X0 @ X3 ) )
!= X3 )
& ( ( in @ X3 @ X1 )
= $true ) )
& ( ( in @ X0 @ X2 )
= $true ) ) )
& ( ! [X1: $i,X0: $i,X2: $i] :
( ! [X3: $i] :
( ( ( ksnd @ ( kpair @ X0 @ X3 ) )
= X3 )
| ( ( in @ X3 @ X1 )
!= $true ) )
| ( ( in @ X0 @ X2 )
!= $true ) )
| ( cartprodsndpairEq != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f15]) ).
thf(f15,plain,
( ( cartprodsndpairEq = $true )
<=> ! [X1: $i,X0: $i,X2: $i] :
( ! [X3: $i] :
( ( ( ksnd @ ( kpair @ X0 @ X3 ) )
= X3 )
| ( ( in @ X3 @ X1 )
!= $true ) )
| ( ( in @ X0 @ X2 )
!= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f8]) ).
thf(f8,plain,
( ! [X2: $i,X1: $i,X0: $i] :
( ( ( in @ X0 @ X2 )
= $true )
=> ! [X3: $i] :
( ( ( in @ X3 @ X1 )
= $true )
=> ( ( ksnd @ ( kpair @ X0 @ X3 ) )
= X3 ) ) )
<=> ( cartprodsndpairEq = $true ) ),
inference(fool_elimination,[status(thm)],[f7]) ).
thf(f7,plain,
! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ X2 )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( ksnd @ ( kpair @ X0 @ X3 ) )
= X3 ) ) )
= cartprodsndpairEq ),
inference(rectify,[status(thm)],[f3]) ).
thf(f3,axiom,
! [X3: $i,X1: $i,X0: $i] :
( ( ( in @ X3 @ X0 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
= cartprodsndpairEq ),
file('/export/starexec/sandbox2/tmp/tmp.3hJx7CzvFR/DTF2THF_17518.p',cartprodsndpairEq) ).
thf(f90,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( ( kpair @ ( kfst @ X2 ) @ ( sK11 @ X0 @ X1 @ X2 ) )
= X2 )
| ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
!= $true ) ),
inference(duplicate_literal_removal,[status(thm)],[f87]) ).
thf(f87,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
!= $true )
| ( ( kpair @ ( kfst @ X2 ) @ ( sK11 @ X0 @ X1 @ X2 ) )
= X2 )
| ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
!= $true ) ),
inference(superposition,[status(thm)],[f74,f82]) ).
thf(f82,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( ( kfst @ X2 )
= ( sK10 @ X0 @ X1 @ X2 ) )
| ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f81]) ).
thf(f81,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( $true != $true )
| ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
!= $true )
| ( ( kfst @ X2 )
= ( sK10 @ X0 @ X1 @ X2 ) ) ),
inference(duplicate_literal_removal,[status(thm)],[f80]) ).
thf(f80,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( $true != $true )
| ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
!= $true )
| ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
!= $true )
| ( ( kfst @ X2 )
= ( sK10 @ X0 @ X1 @ X2 ) ) ),
inference(superposition,[status(thm)],[f79,f73]) ).
thf(f79,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i] :
( ( ( in @ ( sK10 @ X0 @ X1 @ X2 ) @ X3 )
!= $true )
| ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
!= $true )
| ( ( kfst @ X2 )
= ( sK10 @ X0 @ X1 @ X2 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f78]) ).
thf(f78,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i] :
( ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
!= $true )
| ( ( in @ ( sK10 @ X0 @ X1 @ X2 ) @ X3 )
!= $true )
| ( $true != $true )
| ( ( kfst @ X2 )
= ( sK10 @ X0 @ X1 @ X2 ) ) ),
inference(duplicate_literal_removal,[status(thm)],[f77]) ).
thf(f77,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i] :
( ( $true != $true )
| ( ( kfst @ X2 )
= ( sK10 @ X0 @ X1 @ X2 ) )
| ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
!= $true )
| ( ( in @ ( sK10 @ X0 @ X1 @ X2 ) @ X3 )
!= $true )
| ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
!= $true ) ),
inference(superposition,[status(thm)],[f75,f72]) ).
thf(f75,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i,X4: $i] :
( ( ( in @ ( sK11 @ X0 @ X1 @ X2 ) @ X3 )
!= $true )
| ( ( in @ ( sK10 @ X0 @ X1 @ X2 ) @ X4 )
!= $true )
| ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
!= $true )
| ( ( kfst @ X2 )
= ( sK10 @ X0 @ X1 @ X2 ) ) ),
inference(superposition,[status(thm)],[f71,f74]) ).
thf(f71,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i] :
( ( ( kfst @ ( kpair @ X2 @ X3 ) )
= X2 )
| ( ( in @ X3 @ X1 )
!= $true )
| ( ( in @ X2 @ X0 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f66]) ).
thf(f66,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i] :
( ( ( kfst @ ( kpair @ X2 @ X3 ) )
= X2 )
| ( ( in @ X3 @ X1 )
!= $true )
| ( $true != $true )
| ( ( in @ X2 @ X0 )
!= $true ) ),
inference(definition_unfolding,[status(thm)],[f56,f41]) ).
thf(f41,plain,
cartprodfstpairEq = $true,
inference(cnf_transformation,[status(thm)],[f22]) ).
thf(f56,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i] :
( ( ( in @ X2 @ X0 )
!= $true )
| ( ( kfst @ ( kpair @ X2 @ X3 ) )
= X2 )
| ( ( in @ X3 @ X1 )
!= $true )
| ( cartprodfstpairEq != $true ) ),
inference(cnf_transformation,[status(thm)],[f38]) ).
thf(f38,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X0 )
!= $true )
| ! [X3: $i] :
( ( ( kfst @ ( kpair @ X2 @ X3 ) )
= X2 )
| ( ( in @ X3 @ X1 )
!= $true ) ) )
| ( cartprodfstpairEq != $true ) )
& ( ( cartprodfstpairEq = $true )
| ( ( ( in @ sK14 @ sK12 )
= $true )
& ( ( kfst @ ( kpair @ sK14 @ sK15 ) )
!= sK14 )
& ( ( in @ sK15 @ sK13 )
= $true ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK12,sK13,sK14,sK15])],[f35,f37,f36]) ).
thf(f36,plain,
( ? [X4: $i,X5: $i,X6: $i] :
( ( ( in @ X6 @ X4 )
= $true )
& ? [X7: $i] :
( ( ( kfst @ ( kpair @ X6 @ X7 ) )
!= X6 )
& ( ( in @ X7 @ X5 )
= $true ) ) )
=> ( ( ( in @ sK14 @ sK12 )
= $true )
& ? [X7: $i] :
( ( sK14
!= ( kfst @ ( kpair @ sK14 @ X7 ) ) )
& ( ( in @ X7 @ sK13 )
= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f37,plain,
( ? [X7: $i] :
( ( sK14
!= ( kfst @ ( kpair @ sK14 @ X7 ) ) )
& ( ( in @ X7 @ sK13 )
= $true ) )
=> ( ( ( kfst @ ( kpair @ sK14 @ sK15 ) )
!= sK14 )
& ( ( in @ sK15 @ sK13 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f35,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X0 )
!= $true )
| ! [X3: $i] :
( ( ( kfst @ ( kpair @ X2 @ X3 ) )
= X2 )
| ( ( in @ X3 @ X1 )
!= $true ) ) )
| ( cartprodfstpairEq != $true ) )
& ( ( cartprodfstpairEq = $true )
| ? [X4: $i,X5: $i,X6: $i] :
( ( ( in @ X6 @ X4 )
= $true )
& ? [X7: $i] :
( ( ( kfst @ ( kpair @ X6 @ X7 ) )
!= X6 )
& ( ( in @ X7 @ X5 )
= $true ) ) ) ) ),
inference(rectify,[status(thm)],[f34]) ).
thf(f34,plain,
( ( ! [X2: $i,X1: $i,X0: $i] :
( ( ( in @ X0 @ X2 )
!= $true )
| ! [X3: $i] :
( ( ( kfst @ ( kpair @ X0 @ X3 ) )
= X0 )
| ( ( in @ X3 @ X1 )
!= $true ) ) )
| ( cartprodfstpairEq != $true ) )
& ( ( cartprodfstpairEq = $true )
| ? [X2: $i,X1: $i,X0: $i] :
( ( ( in @ X0 @ X2 )
= $true )
& ? [X3: $i] :
( ( ( kfst @ ( kpair @ X0 @ X3 ) )
!= X0 )
& ( ( in @ X3 @ X1 )
= $true ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f17]) ).
thf(f17,plain,
( ! [X2: $i,X1: $i,X0: $i] :
( ( ( in @ X0 @ X2 )
!= $true )
| ! [X3: $i] :
( ( ( kfst @ ( kpair @ X0 @ X3 ) )
= X0 )
| ( ( in @ X3 @ X1 )
!= $true ) ) )
<=> ( cartprodfstpairEq = $true ) ),
inference(ennf_transformation,[status(thm)],[f14]) ).
thf(f14,plain,
( ! [X2: $i,X1: $i,X0: $i] :
( ( ( in @ X0 @ X2 )
= $true )
=> ! [X3: $i] :
( ( ( in @ X3 @ X1 )
= $true )
=> ( ( kfst @ ( kpair @ X0 @ X3 ) )
= X0 ) ) )
<=> ( cartprodfstpairEq = $true ) ),
inference(fool_elimination,[status(thm)],[f13]) ).
thf(f13,plain,
! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ X2 )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( kfst @ ( kpair @ X0 @ X3 ) )
= X0 ) ) )
= cartprodfstpairEq ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
! [X3: $i,X1: $i,X0: $i] :
( ( ( in @ X3 @ X0 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
= cartprodfstpairEq ),
file('/export/starexec/sandbox2/tmp/tmp.3hJx7CzvFR/DTF2THF_17518.p',cartprodfstpairEq) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU664^2 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n011.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 12:29:47 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.30/0.45 ---- Original DTF file ---
% 0.30/0.45 thf(spec,logic,$$dhol).
% 0.30/0.45 %------------------------------------------------------------------------------
% 0.30/0.45 % File : SEU664^2 : TPTP v9.2.1. Released v3.7.0.
% 0.30/0.45 % Domain : Set Theory
% 0.30/0.45 % Problem : Ordered Pairs - Properties of Pairs
% 0.30/0.45 % Version : Especial > Reduced > Especial.
% 0.30/0.45 % English : (! A:i.! B:i.! u:i.in u (cartprod A B) -> kpair (kfst u)
% 0.30/0.45 % (ksnd u) = u)
% 0.30/0.45
% 0.30/0.45 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.30/0.45 % Source : [Bro08]
% 0.30/0.45 % Names : ZFC166l [Bro08]
% 0.30/0.45
% 0.30/0.45 % Status : Theorem
% 0.30/0.45 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.31 v8.1.0, 0.18 v7.5.0, 0.14 v7.4.0, 0.22 v7.2.0, 0.12 v7.1.0, 0.38 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.29 v6.1.0, 0.43 v5.5.0, 0.33 v5.4.0, 0.40 v5.1.0, 0.60 v5.0.0, 0.40 v4.1.0, 0.33 v3.7.0
% 0.30/0.45 % Syntax : Number of formulae : 12 ( 3 unt; 8 typ; 3 def)
% 0.30/0.45 % Number of atoms : 21 ( 7 equ; 0 cnn)
% 0.30/0.45 % Maximal formula atoms : 5 ( 5 avg)
% 0.30/0.45 % Number of connectives : 43 ( 0 ~; 0 |; 2 &; 32 @)
% 0.30/0.45 % ( 0 <=>; 9 =>; 0 <=; 0 <~>)
% 0.30/0.45 % Maximal formula depth : 11 ( 4 avg)
% 0.30/0.45 % Number of types : 2 ( 0 usr)
% 0.30/0.45 % Number of type conns : 8 ( 8 >; 0 *; 0 +; 0 <<)
% 0.30/0.45 % Number of symbols : 9 ( 8 usr; 3 con; 0-2 aty)
% 0.30/0.45 % Number of variables : 16 ( 0 ^; 14 !; 2 ?; 16 :)
% 0.30/0.45 % SPC : TH0_THM_EQU_NAR
% 0.30/0.45
% 0.30/0.45 % Comments : http://mathgate.info/detsetitem.php?id=218
% 0.30/0.45 %------------------------------------------------------------------------------
% 0.30/0.45 thf(in_type,type,
% 0.30/0.45 in: $i > $i > $o ).
% 0.30/0.45
% 0.30/0.45 thf(kpair_type,type,
% 0.30/0.45 kpair: $i > $i > $i ).
% 0.30/0.45
% 0.30/0.45 thf(cartprod_type,type,
% 0.30/0.45 cartprod: $i > $i > $i ).
% 0.30/0.45
% 0.30/0.45 thf(cartprodmempair1_type,type,
% 0.30/0.45 cartprodmempair1: $o ).
% 0.30/0.45
% 0.30/0.45 thf(cartprodmempair1,definition,
% 0.30/0.45 ( cartprodmempair1
% 0.30/0.45 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/0.45 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/0.45 => ? [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 & ? [Xy: $i] :
% 0.30/0.45 ( ( in @ Xy @ B )
% 0.30/0.45 & ( Xu
% 0.30/0.45 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(kfst_type,type,
% 0.30/0.45 kfst: $i > $i ).
% 0.30/0.45
% 0.30/0.45 thf(ksnd_type,type,
% 0.30/0.45 ksnd: $i > $i ).
% 0.30/0.45
% 0.30/0.45 thf(cartprodfstpairEq_type,type,
% 0.30/0.45 cartprodfstpairEq: $o ).
% 0.30/0.45
% 0.30/0.45 thf(cartprodfstpairEq,definition,
% 0.30/0.45 ( cartprodfstpairEq
% 0.30/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 => ! [Xy: $i] :
% 0.30/0.45 ( ( in @ Xy @ B )
% 0.30/0.45 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.30/0.45 = Xx ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(cartprodsndpairEq_type,type,
% 0.30/0.45 cartprodsndpairEq: $o ).
% 0.30/0.45
% 0.30/0.45 thf(cartprodsndpairEq,definition,
% 0.30/0.45 ( cartprodsndpairEq
% 0.30/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 => ! [Xy: $i] :
% 0.30/0.45 ( ( in @ Xy @ B )
% 0.30/0.45 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.30/0.45 = Xy ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(cartprodpairsurjEq,conjecture,
% 0.30/0.45 ( cartprodmempair1
% 0.30/0.45 => ( cartprodfstpairEq
% 0.30/0.45 => ( cartprodsndpairEq
% 0.30/0.45 => ! [A: $i,B: $i,Xu: $i] :
% 0.30/0.45 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/0.45 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.30/0.45 = Xu ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 %------------------------------------------------------------------------------
% 0.30/0.45 ------------------------
% 1.52/1.26 ---- Embedded in THF ---
% 1.52/1.26 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.52/1.26 %%% Generated on Tue Feb 24 12:29:48 EST 2026
% 1.52/1.26 %%% using '$$dhol' embedding, version 1.3.0.
% 1.52/1.26 %%% Logic specification used:
% 1.52/1.26 %%% thf(spec, logic, $$dhol).
% 1.52/1.26
% 1.52/1.26 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.3hJx7CzvFR/DTF2DTF17518.p
% See solution above
% 1.52/1.26 ------------------------
% 1.52/1.27 ---- Cleaned THF ---
% 1.52/1.27 thf(in_type,type,
% 1.52/1.27 in: $i > $i > $o ).
% 1.52/1.27
% 1.52/1.27 thf(kpair_type,type,
% 1.52/1.27 kpair: $i > $i > $i ).
% 1.52/1.27
% 1.52/1.27 thf(cartprod_type,type,
% 1.52/1.27 cartprod: $i > $i > $i ).
% 1.52/1.27
% 1.52/1.27 thf(cartprodmempair1_type,type,
% 1.52/1.27 cartprodmempair1: $o ).
% 1.52/1.27
% 1.52/1.27 thf(cartprodmempair1,definition,
% 1.52/1.27 ( cartprodmempair1
% 1.52/1.27 = ( ! [A: $i,B: $i,Xu: $i] :
% 1.52/1.27 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 1.52/1.27 => ? [Xx: $i] :
% 1.52/1.27 ( ( in @ Xx @ A )
% 1.52/1.27 & ? [Xy: $i] :
% 1.52/1.27 ( ( in @ Xy @ B )
% 1.52/1.27 & ( Xu
% 1.52/1.27 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 1.52/1.27
% 1.52/1.27 thf(kfst_type,type,
% 1.52/1.27 kfst: $i > $i ).
% 1.52/1.27
% 1.52/1.27 thf(ksnd_type,type,
% 1.52/1.27 ksnd: $i > $i ).
% 1.52/1.27
% 1.52/1.27 thf(cartprodfstpairEq_type,type,
% 1.52/1.27 cartprodfstpairEq: $o ).
% 1.52/1.27
% 1.52/1.27 thf(cartprodfstpairEq,definition,
% 1.52/1.27 ( cartprodfstpairEq
% 1.52/1.27 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.52/1.27 ( ( in @ Xx @ A )
% 1.52/1.27 => ! [Xy: $i] :
% 1.52/1.27 ( ( in @ Xy @ B )
% 1.52/1.27 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 1.52/1.27 = Xx ) ) ) ) ) ).
% 1.52/1.27
% 1.52/1.27 thf(cartprodsndpairEq_type,type,
% 1.52/1.27 cartprodsndpairEq: $o ).
% 1.52/1.27
% 1.52/1.27 thf(cartprodsndpairEq,definition,
% 1.52/1.27 ( cartprodsndpairEq
% 1.52/1.27 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.52/1.27 ( ( in @ Xx @ A )
% 1.52/1.27 => ! [Xy: $i] :
% 1.52/1.27 ( ( in @ Xy @ B )
% 1.52/1.27 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 1.52/1.27 = Xy ) ) ) ) ) ).
% 1.52/1.27
% 1.52/1.27 thf(cartprodpairsurjEq,conjecture,
% 1.52/1.27 ( cartprodmempair1
% 1.52/1.27 => ( cartprodfstpairEq
% 1.52/1.27 => ( cartprodsndpairEq
% 1.52/1.27 => ! [A: $i,B: $i,Xu: $i] :
% 1.52/1.27 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 1.52/1.27 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 1.52/1.27 = Xu ) ) ) ) ) ).
% 1.52/1.27 ------------------------
% 1.52/1.29 % (17740)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_17518 for (2999ds/275Mi)
% 1.52/1.29 % (17736)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_17518 for (2999ds/4Mi)
% 1.52/1.29 % (17736)Instruction limit reached!
% 1.52/1.29 % (17736)------------------------------
% 1.52/1.29 % (17736)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.52/1.29 % (17736)Termination reason: Unknown
% 1.52/1.29 % (17736)Termination phase: Saturation
% 1.52/1.29
% 1.52/1.29 % (17736)Memory used [KB]: 5500
% 1.52/1.29 % (17736)Time elapsed: 0.003 s
% 1.52/1.29 % (17736)Instructions burned: 5 (million)
% 1.52/1.29 % (17736)------------------------------
% 1.52/1.29 % (17736)------------------------------
% 1.52/1.29 % (17740)First to succeed.
% 1.52/1.29 % (17735)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_17518 for (2999ds/183Mi)
% 1.52/1.29 % (17737)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_17518 for (2999ds/27Mi)
% 1.52/1.29 % (17738)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_17518 for (2999ds/2Mi)
% 1.52/1.29 % (17739)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_17518 for (2999ds/2Mi)
% 1.52/1.29 % (17741)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_17518 for (2999ds/18Mi)
% 1.52/1.30 % (17738)Instruction limit reached!
% 1.52/1.30 % (17738)------------------------------
% 1.52/1.30 % (17738)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.52/1.30 % (17738)Termination reason: Unknown
% 1.52/1.30 % (17738)Termination phase: Property scanning
% 1.52/1.30
% 1.52/1.30 % (17738)Memory used [KB]: 895
% 1.52/1.30 % (17738)Time elapsed: 0.003 s
% 1.52/1.30 % (17738)Instructions burned: 2 (million)
% 1.52/1.30 % (17738)------------------------------
% 1.52/1.30 % (17738)------------------------------
% 1.52/1.30 % (17739)Instruction limit reached!
% 1.52/1.30 % (17739)------------------------------
% 1.52/1.30 % (17739)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.52/1.30 % (17739)Termination reason: Unknown
% 1.52/1.30 % (17739)Termination phase: shuffling
% 1.52/1.30
% 1.52/1.30 % (17739)Memory used [KB]: 895
% 1.52/1.30 % (17739)Time elapsed: 0.003 s
% 1.52/1.30 % (17739)Instructions burned: 2 (million)
% 1.52/1.30 % (17739)------------------------------
% 1.52/1.30 % (17739)------------------------------
% 1.52/1.30 % (17740)Refutation found. Thanks to Tanya!
% 1.52/1.30 % SZS status Theorem for DTF2THF_17518
% 1.52/1.30 % SZS output start Proof for DTF2THF_17518
% 1.52/1.30 thf(func_def_0, type, in: $i > $i > $o).
% 1.52/1.30 thf(func_def_1, type, kpair: $i > $i > $i).
% 1.52/1.30 thf(func_def_2, type, cartprod: $i > $i > $i).
% 1.52/1.30 thf(func_def_5, type, kfst: $i > $i).
% 1.52/1.30 thf(func_def_6, type, ksnd: $i > $i).
% 1.52/1.30 thf(func_def_21, type, sK10: $i > $i > $i > $i).
% 1.52/1.30 thf(func_def_22, type, sK11: $i > $i > $i > $i).
% 1.52/1.30 thf(f125,plain,(
% 1.52/1.30 $false),
% 1.52/1.30 inference(subsumption_resolution,[status(thm)],[f122,f39])).
% 1.52/1.30 thf(f39,plain,(
% 1.52/1.30 (sK0 != (kpair @ (kfst @ sK0) @ (ksnd @ sK0)))),
% 1.52/1.30 inference(cnf_transformation,[status(thm)],[f22])).
% 1.52/1.30 thf(f22,plain,(
% 1.52/1.30 (cartprodsndpairEq = $true) & (cartprodmempair1 = $true) & (cartprodfstpairEq = $true) & (((in @ sK0 @ (cartprod @ sK2 @ sK1)) = $true) & (sK0 != (kpair @ (kfst @ sK0) @ (ksnd @ sK0))))),
% 1.52/1.30 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2])],[f20,f21])).
% 1.52/1.30 thf(f21,plain,(
% 1.52/1.30 ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ (cartprod @ X2 @ X1)) = $true) & ((kpair @ (kfst @ X0) @ (ksnd @ X0)) != X0)) => (((in @ sK0 @ (cartprod @ sK2 @ sK1)) = $true) & (sK0 != (kpair @ (kfst @ sK0) @ (ksnd @ sK0))))),
% 1.66/1.30 introduced(definition,[],[choice_axiom])).
% 1.66/1.30 thf(f20,plain,(
% 1.66/1.30 (cartprodsndpairEq = $true) & (cartprodmempair1 = $true) & (cartprodfstpairEq = $true) & ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ (cartprod @ X2 @ X1)) = $true) & ((kpair @ (kfst @ X0) @ (ksnd @ X0)) != X0))),
% 1.66/1.30 inference(rectify,[status(thm)],[f19])).
% 1.66/1.30 thf(f19,plain,(
% 1.66/1.30 (cartprodsndpairEq = $true) & (cartprodmempair1 = $true) & (cartprodfstpairEq = $true) & ? [X2 : $i,X1 : $i,X0 : $i] : (((in @ X2 @ (cartprod @ X0 @ X1)) = $true) & ((kpair @ (kfst @ X2) @ (ksnd @ X2)) != X2))),
% 1.66/1.30 inference(flattening,[status(thm)],[f18])).
% 1.66/1.30 thf(f18,plain,(
% 1.66/1.30 ((? [X2 : $i,X1 : $i,X0 : $i] : (((in @ X2 @ (cartprod @ X0 @ X1)) = $true) & ((kpair @ (kfst @ X2) @ (ksnd @ X2)) != X2)) & (cartprodsndpairEq = $true)) & (cartprodfstpairEq = $true)) & (cartprodmempair1 = $true)),
% 1.66/1.30 inference(ennf_transformation,[status(thm)],[f12])).
% 1.66/1.30 thf(f12,plain,(
% 1.66/1.30 ~((cartprodmempair1 = $true) => ((cartprodfstpairEq = $true) => ((cartprodsndpairEq = $true) => ! [X1 : $i,X2 : $i,X0 : $i] : (((in @ X2 @ (cartprod @ X0 @ X1)) = $true) => ((kpair @ (kfst @ X2) @ (ksnd @ X2)) = X2)))))),
% 1.66/1.30 inference(fool_elimination,[status(thm)],[f11])).
% 1.66/1.30 thf(f11,plain,(
% 1.66/1.30 ~(cartprodmempair1 => (cartprodfstpairEq => (cartprodsndpairEq => ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X2 @ (cartprod @ X0 @ X1)) => ((kpair @ (kfst @ X2) @ (ksnd @ X2)) = X2)))))),
% 1.66/1.30 inference(rectify,[status(thm)],[f5])).
% 1.66/1.30 thf(f5,negated_conjecture,(
% 1.66/1.30 ~(cartprodmempair1 => (cartprodfstpairEq => (cartprodsndpairEq => ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X2 @ (cartprod @ X0 @ X1)) => ((kpair @ (kfst @ X2) @ (ksnd @ X2)) = X2)))))),
% 1.66/1.30 inference(negated_conjecture,[status(cth)],[f4])).
% 1.66/1.30 thf(f4,conjecture,(
% 1.66/1.30 cartprodmempair1 => (cartprodfstpairEq => (cartprodsndpairEq => ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X2 @ (cartprod @ X0 @ X1)) => ((kpair @ (kfst @ X2) @ (ksnd @ X2)) = X2))))),
% 1.66/1.30 file('/export/starexec/sandbox2/tmp/tmp.3hJx7CzvFR/DTF2THF_17518.p',cartprodpairsurjEq)).
% 1.66/1.30 thf(f122,plain,(
% 1.66/1.30 (sK0 = (kpair @ (kfst @ sK0) @ (ksnd @ sK0)))),
% 1.66/1.30 inference(trivial_inequality_removal,[status(thm)],[f119])).
% 1.66/1.30 thf(f119,plain,(
% 1.66/1.30 ($true != $true) | (sK0 = (kpair @ (kfst @ sK0) @ (ksnd @ sK0)))),
% 1.66/1.30 inference(superposition,[status(thm)],[f118,f40])).
% 1.66/1.30 thf(f40,plain,(
% 1.66/1.30 ((in @ sK0 @ (cartprod @ sK2 @ sK1)) = $true)),
% 1.66/1.30 inference(cnf_transformation,[status(thm)],[f22])).
% 1.66/1.30 thf(f118,plain,(
% 1.66/1.30 ( ! [X2 : $i,X0 : $i,X1 : $i] : (((in @ X2 @ (cartprod @ X1 @ X0)) != $true) | ((kpair @ (kfst @ X2) @ (ksnd @ X2)) = X2)) )),
% 1.66/1.30 inference(duplicate_literal_removal,[status(thm)],[f115])).
% 1.66/1.30 thf(f115,plain,(
% 1.66/1.30 ( ! [X2 : $i,X0 : $i,X1 : $i] : (((kpair @ (kfst @ X2) @ (ksnd @ X2)) = X2) | ((in @ X2 @ (cartprod @ X1 @ X0)) != $true) | ((in @ X2 @ (cartprod @ X1 @ X0)) != $true)) )),
% 1.66/1.30 inference(superposition,[status(thm)],[f90,f94])).
% 1.66/1.30 thf(f94,plain,(
% 1.66/1.30 ( ! [X2 : $i,X0 : $i,X1 : $i] : (((ksnd @ X2) = (sK11 @ X0 @ X1 @ X2)) | ((in @ X2 @ (cartprod @ X1 @ X0)) != $true)) )),
% 1.66/1.30 inference(trivial_inequality_removal,[status(thm)],[f93])).
% 1.66/1.30 thf(f93,plain,(
% 1.66/1.30 ( ! [X2 : $i,X0 : $i,X1 : $i] : (($true != $true) | ((in @ X2 @ (cartprod @ X1 @ X0)) != $true) | ((ksnd @ X2) = (sK11 @ X0 @ X1 @ X2))) )),
% 1.66/1.30 inference(duplicate_literal_removal,[status(thm)],[f92])).
% 1.66/1.30 thf(f92,plain,(
% 1.66/1.30 ( ! [X2 : $i,X0 : $i,X1 : $i] : (((in @ X2 @ (cartprod @ X1 @ X0)) != $true) | ((in @ X2 @ (cartprod @ X1 @ X0)) != $true) | ($true != $true) | ((ksnd @ X2) = (sK11 @ X0 @ X1 @ X2))) )),
% 1.66/1.30 inference(superposition,[status(thm)],[f85,f73])).
% 1.66/1.30 thf(f73,plain,(
% 1.66/1.30 ( ! [X6 : $i,X7 : $i,X5 : $i] : (((in @ (sK10 @ X7 @ X6 @ X5) @ X6) = $true) | ((in @ X5 @ (cartprod @ X6 @ X7)) != $true)) )),
% 1.66/1.30 inference(trivial_inequality_removal,[status(thm)],[f63])).
% 1.66/1.30 thf(f63,plain,(
% 1.66/1.30 ( ! [X6 : $i,X7 : $i,X5 : $i] : (((in @ (sK10 @ X7 @ X6 @ X5) @ X6) = $true) | ($true != $true) | ((in @ X5 @ (cartprod @ X6 @ X7)) != $true)) )),
% 1.66/1.30 inference(definition_unfolding,[status(thm)],[f50,f42])).
% 1.66/1.30 thf(f42,plain,(
% 1.66/1.30 (cartprodmempair1 = $true)),
% 1.66/1.30 inference(cnf_transformation,[status(thm)],[f22])).
% 1.66/1.30 thf(f50,plain,(
% 1.66/1.30 ( ! [X6 : $i,X7 : $i,X5 : $i] : (((in @ X5 @ (cartprod @ X6 @ X7)) != $true) | ((in @ (sK10 @ X7 @ X6 @ X5) @ X6) = $true) | (cartprodmempair1 != $true)) )),
% 1.66/1.30 inference(cnf_transformation,[status(thm)],[f33])).
% 1.66/1.30 thf(f33,plain,(
% 1.66/1.30 ((cartprodmempair1 = $true) | (((in @ sK7 @ (cartprod @ sK8 @ sK9)) = $true) & ! [X3 : $i] : (((in @ X3 @ sK8) != $true) | ! [X4 : $i] : (((kpair @ X3 @ X4) != sK7) | ($true != (in @ X4 @ sK9)))))) & (! [X5 : $i,X6 : $i,X7 : $i] : (((in @ X5 @ (cartprod @ X6 @ X7)) != $true) | (((in @ (sK10 @ X7 @ X6 @ X5) @ X6) = $true) & (((kpair @ (sK10 @ X7 @ X6 @ X5) @ (sK11 @ X7 @ X6 @ X5)) = X5) & ((in @ (sK11 @ X7 @ X6 @ X5) @ X7) = $true)))) | (cartprodmempair1 != $true))),
% 1.66/1.30 inference(skolemisation,[status(esa),new_symbols(skolem,[sK7,sK8,sK9,sK10,sK11])],[f29,f32,f31,f30])).
% 1.66/1.30 thf(f30,plain,(
% 1.66/1.30 ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ (cartprod @ X1 @ X2)) = $true) & ! [X3 : $i] : (((in @ X3 @ X1) != $true) | ! [X4 : $i] : (((kpair @ X3 @ X4) != X0) | ((in @ X4 @ X2) != $true)))) => (((in @ sK7 @ (cartprod @ sK8 @ sK9)) = $true) & ! [X3 : $i] : (((in @ X3 @ sK8) != $true) | ! [X4 : $i] : (((kpair @ X3 @ X4) != sK7) | ($true != (in @ X4 @ sK9)))))),
% 1.66/1.30 introduced(definition,[],[choice_axiom])).
% 1.66/1.30 thf(f31,plain,(
% 1.66/1.30 ! [X5 : $i,X6 : $i,X7 : $i] : (? [X8 : $i] : (((in @ X8 @ X6) = $true) & ? [X9 : $i] : (((kpair @ X8 @ X9) = X5) & ((in @ X9 @ X7) = $true))) => (((in @ (sK10 @ X7 @ X6 @ X5) @ X6) = $true) & ? [X9 : $i] : (((kpair @ (sK10 @ X7 @ X6 @ X5) @ X9) = X5) & ((in @ X9 @ X7) = $true))))),
% 1.66/1.30 introduced(definition,[],[choice_axiom])).
% 1.66/1.30 thf(f32,plain,(
% 1.66/1.30 ! [X5 : $i,X6 : $i,X7 : $i] : (? [X9 : $i] : (((kpair @ (sK10 @ X7 @ X6 @ X5) @ X9) = X5) & ((in @ X9 @ X7) = $true)) => (((kpair @ (sK10 @ X7 @ X6 @ X5) @ (sK11 @ X7 @ X6 @ X5)) = X5) & ((in @ (sK11 @ X7 @ X6 @ X5) @ X7) = $true)))),
% 1.66/1.30 introduced(definition,[],[choice_axiom])).
% 1.66/1.30 thf(f29,plain,(
% 1.66/1.30 ((cartprodmempair1 = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ (cartprod @ X1 @ X2)) = $true) & ! [X3 : $i] : (((in @ X3 @ X1) != $true) | ! [X4 : $i] : (((kpair @ X3 @ X4) != X0) | ((in @ X4 @ X2) != $true))))) & (! [X5 : $i,X6 : $i,X7 : $i] : (((in @ X5 @ (cartprod @ X6 @ X7)) != $true) | ? [X8 : $i] : (((in @ X8 @ X6) = $true) & ? [X9 : $i] : (((kpair @ X8 @ X9) = X5) & ((in @ X9 @ X7) = $true)))) | (cartprodmempair1 != $true))),
% 1.66/1.30 inference(rectify,[status(thm)],[f28])).
% 1.66/1.30 thf(f28,plain,(
% 1.66/1.30 ((cartprodmempair1 = $true) | ? [X0 : $i,X2 : $i,X1 : $i] : (((in @ X0 @ (cartprod @ X2 @ X1)) = $true) & ! [X3 : $i] : (((in @ X3 @ X2) != $true) | ! [X4 : $i] : (((kpair @ X3 @ X4) != X0) | ((in @ X4 @ X1) != $true))))) & (! [X0 : $i,X2 : $i,X1 : $i] : (((in @ X0 @ (cartprod @ X2 @ X1)) != $true) | ? [X3 : $i] : (((in @ X3 @ X2) = $true) & ? [X4 : $i] : (((kpair @ X3 @ X4) = X0) & ((in @ X4 @ X1) = $true)))) | (cartprodmempair1 != $true))),
% 1.66/1.30 inference(nnf_transformation,[status(thm)],[f16])).
% 1.66/1.30 thf(f16,plain,(
% 1.66/1.30 (cartprodmempair1 = $true) <=> ! [X0 : $i,X2 : $i,X1 : $i] : (((in @ X0 @ (cartprod @ X2 @ X1)) != $true) | ? [X3 : $i] : (((in @ X3 @ X2) = $true) & ? [X4 : $i] : (((kpair @ X3 @ X4) = X0) & ((in @ X4 @ X1) = $true))))),
% 1.66/1.30 inference(ennf_transformation,[status(thm)],[f10])).
% 1.66/1.30 thf(f10,plain,(
% 1.66/1.30 ! [X1 : $i,X2 : $i,X0 : $i] : (((in @ X0 @ (cartprod @ X2 @ X1)) = $true) => ? [X3 : $i] : (((in @ X3 @ X2) = $true) & ? [X4 : $i] : (((kpair @ X3 @ X4) = X0) & ((in @ X4 @ X1) = $true)))) <=> (cartprodmempair1 = $true)),
% 1.66/1.30 inference(fool_elimination,[status(thm)],[f9])).
% 1.66/1.30 thf(f9,plain,(
% 1.66/1.30 (! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X0 @ (cartprod @ X2 @ X1)) => ? [X3 : $i] : (? [X4 : $i] : (((kpair @ X3 @ X4) = X0) & (in @ X4 @ X1)) & (in @ X3 @ X2))) = cartprodmempair1)),
% 1.66/1.30 inference(rectify,[status(thm)],[f1])).
% 1.66/1.30 thf(f1,axiom,(
% 1.66/1.30 (! [X2 : $i,X1 : $i,X0 : $i] : ((in @ X2 @ (cartprod @ X0 @ X1)) => ? [X3 : $i] : (? [X4 : $i] : (((kpair @ X3 @ X4) = X2) & (in @ X4 @ X1)) & (in @ X3 @ X0))) = cartprodmempair1)),
% 1.66/1.30 file('/export/starexec/sandbox2/tmp/tmp.3hJx7CzvFR/DTF2THF_17518.p',cartprodmempair1)).
% 1.66/1.30 thf(f85,plain,(
% 1.66/1.30 ( ! [X2 : $i,X3 : $i,X0 : $i,X1 : $i] : (((in @ (sK10 @ X0 @ X1 @ X2) @ X3) != $true) | ((ksnd @ X2) = (sK11 @ X0 @ X1 @ X2)) | ((in @ X2 @ (cartprod @ X1 @ X0)) != $true)) )),
% 1.66/1.30 inference(trivial_inequality_removal,[status(thm)],[f84])).
% 1.66/1.30 thf(f84,plain,(
% 1.66/1.30 ( ! [X2 : $i,X3 : $i,X0 : $i,X1 : $i] : (((in @ X2 @ (cartprod @ X1 @ X0)) != $true) | ($true != $true) | ((in @ (sK10 @ X0 @ X1 @ X2) @ X3) != $true) | ((ksnd @ X2) = (sK11 @ X0 @ X1 @ X2))) )),
% 1.66/1.30 inference(duplicate_literal_removal,[status(thm)],[f83])).
% 1.66/1.30 thf(f83,plain,(
% 1.66/1.30 ( ! [X2 : $i,X3 : $i,X0 : $i,X1 : $i] : (((in @ X2 @ (cartprod @ X1 @ X0)) != $true) | ((in @ X2 @ (cartprod @ X1 @ X0)) != $true) | ($true != $true) | ((in @ (sK10 @ X0 @ X1 @ X2) @ X3) != $true) | ((ksnd @ X2) = (sK11 @ X0 @ X1 @ X2))) )),
% 1.66/1.30 inference(superposition,[status(thm)],[f76,f72])).
% 1.66/1.30 thf(f72,plain,(
% 1.66/1.30 ( ! [X6 : $i,X7 : $i,X5 : $i] : (((in @ (sK11 @ X7 @ X6 @ X5) @ X7) = $true) | ((in @ X5 @ (cartprod @ X6 @ X7)) != $true)) )),
% 1.66/1.30 inference(trivial_inequality_removal,[status(thm)],[f65])).
% 1.66/1.30 thf(f65,plain,(
% 1.66/1.30 ( ! [X6 : $i,X7 : $i,X5 : $i] : (($true != $true) | ((in @ (sK11 @ X7 @ X6 @ X5) @ X7) = $true) | ((in @ X5 @ (cartprod @ X6 @ X7)) != $true)) )),
% 1.66/1.30 inference(definition_unfolding,[status(thm)],[f48,f42])).
% 1.66/1.30 thf(f48,plain,(
% 1.66/1.30 ( ! [X6 : $i,X7 : $i,X5 : $i] : (((in @ X5 @ (cartprod @ X6 @ X7)) != $true) | ((in @ (sK11 @ X7 @ X6 @ X5) @ X7) = $true) | (cartprodmempair1 != $true)) )),
% 1.66/1.30 inference(cnf_transformation,[status(thm)],[f33])).
% 1.66/1.30 thf(f76,plain,(
% 1.66/1.30 ( ! [X2 : $i,X3 : $i,X0 : $i,X1 : $i,X4 : $i] : (((in @ (sK11 @ X0 @ X1 @ X2) @ X4) != $true) | ((in @ X2 @ (cartprod @ X1 @ X0)) != $true) | ((in @ (sK10 @ X0 @ X1 @ X2) @ X3) != $true) | ((ksnd @ X2) = (sK11 @ X0 @ X1 @ X2))) )),
% 1.66/1.30 inference(superposition,[status(thm)],[f70,f74])).
% 1.66/1.30 thf(f74,plain,(
% 1.66/1.30 ( ! [X6 : $i,X7 : $i,X5 : $i] : (((kpair @ (sK10 @ X7 @ X6 @ X5) @ (sK11 @ X7 @ X6 @ X5)) = X5) | ((in @ X5 @ (cartprod @ X6 @ X7)) != $true)) )),
% 1.66/1.30 inference(trivial_inequality_removal,[status(thm)],[f64])).
% 1.66/1.30 thf(f64,plain,(
% 1.66/1.30 ( ! [X6 : $i,X7 : $i,X5 : $i] : (((kpair @ (sK10 @ X7 @ X6 @ X5) @ (sK11 @ X7 @ X6 @ X5)) = X5) | ($true != $true) | ((in @ X5 @ (cartprod @ X6 @ X7)) != $true)) )),
% 1.66/1.30 inference(definition_unfolding,[status(thm)],[f49,f42])).
% 1.66/1.30 thf(f49,plain,(
% 1.66/1.30 ( ! [X6 : $i,X7 : $i,X5 : $i] : (((in @ X5 @ (cartprod @ X6 @ X7)) != $true) | ((kpair @ (sK10 @ X7 @ X6 @ X5) @ (sK11 @ X7 @ X6 @ X5)) = X5) | (cartprodmempair1 != $true)) )),
% 1.66/1.30 inference(cnf_transformation,[status(thm)],[f33])).
% 1.66/1.30 thf(f70,plain,(
% 1.66/1.30 ( ! [X6 : $i,X7 : $i,X4 : $i,X5 : $i] : (((ksnd @ (kpair @ X5 @ X7)) = X7) | ($true != (in @ X5 @ X6)) | ((in @ X7 @ X4) != $true)) )),
% 1.66/1.30 inference(trivial_inequality_removal,[status(thm)],[f60])).
% 1.66/1.30 thf(f60,plain,(
% 1.66/1.30 ( ! [X6 : $i,X7 : $i,X4 : $i,X5 : $i] : (((in @ X7 @ X4) != $true) | ((ksnd @ (kpair @ X5 @ X7)) = X7) | ($true != $true) | ($true != (in @ X5 @ X6))) )),
% 1.66/1.30 inference(definition_unfolding,[status(thm)],[f44,f43])).
% 1.66/1.30 thf(f43,plain,(
% 1.66/1.30 (cartprodsndpairEq = $true)),
% 1.66/1.30 inference(cnf_transformation,[status(thm)],[f22])).
% 1.66/1.30 thf(f44,plain,(
% 1.66/1.30 ( ! [X6 : $i,X7 : $i,X4 : $i,X5 : $i] : (((ksnd @ (kpair @ X5 @ X7)) = X7) | ((in @ X7 @ X4) != $true) | ($true != (in @ X5 @ X6)) | (cartprodsndpairEq != $true)) )),
% 1.66/1.30 inference(cnf_transformation,[status(thm)],[f27])).
% 1.66/1.30 thf(f27,plain,(
% 1.66/1.30 ((cartprodsndpairEq = $true) | (((sK6 != (ksnd @ (kpair @ sK4 @ sK6))) & ((in @ sK6 @ sK3) = $true)) & ((in @ sK4 @ sK5) = $true))) & (! [X4 : $i,X5 : $i,X6 : $i] : (! [X7 : $i] : (((ksnd @ (kpair @ X5 @ X7)) = X7) | ((in @ X7 @ X4) != $true)) | ($true != (in @ X5 @ X6))) | (cartprodsndpairEq != $true))),
% 1.66/1.30 inference(skolemisation,[status(esa),new_symbols(skolem,[sK3,sK4,sK5,sK6])],[f24,f26,f25])).
% 1.66/1.30 thf(f25,plain,(
% 1.66/1.30 ? [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : (((ksnd @ (kpair @ X1 @ X3)) != X3) & ((in @ X3 @ X0) = $true)) & ((in @ X1 @ X2) = $true)) => (? [X3 : $i] : (((ksnd @ (kpair @ sK4 @ X3)) != X3) & ((in @ X3 @ sK3) = $true)) & ((in @ sK4 @ sK5) = $true))),
% 1.66/1.30 introduced(definition,[],[choice_axiom])).
% 1.66/1.30 thf(f26,plain,(
% 1.66/1.30 ? [X3 : $i] : (((ksnd @ (kpair @ sK4 @ X3)) != X3) & ((in @ X3 @ sK3) = $true)) => ((sK6 != (ksnd @ (kpair @ sK4 @ sK6))) & ((in @ sK6 @ sK3) = $true))),
% 1.66/1.30 introduced(definition,[],[choice_axiom])).
% 1.66/1.30 thf(f24,plain,(
% 1.66/1.30 ((cartprodsndpairEq = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : (((ksnd @ (kpair @ X1 @ X3)) != X3) & ((in @ X3 @ X0) = $true)) & ((in @ X1 @ X2) = $true))) & (! [X4 : $i,X5 : $i,X6 : $i] : (! [X7 : $i] : (((ksnd @ (kpair @ X5 @ X7)) = X7) | ((in @ X7 @ X4) != $true)) | ($true != (in @ X5 @ X6))) | (cartprodsndpairEq != $true))),
% 1.66/1.30 inference(rectify,[status(thm)],[f23])).
% 1.66/1.30 thf(f23,plain,(
% 1.66/1.30 ((cartprodsndpairEq = $true) | ? [X1 : $i,X0 : $i,X2 : $i] : (? [X3 : $i] : (((ksnd @ (kpair @ X0 @ X3)) != X3) & ((in @ X3 @ X1) = $true)) & ((in @ X0 @ X2) = $true))) & (! [X1 : $i,X0 : $i,X2 : $i] : (! [X3 : $i] : (((ksnd @ (kpair @ X0 @ X3)) = X3) | ((in @ X3 @ X1) != $true)) | ((in @ X0 @ X2) != $true)) | (cartprodsndpairEq != $true))),
% 1.66/1.30 inference(nnf_transformation,[status(thm)],[f15])).
% 1.66/1.30 thf(f15,plain,(
% 1.66/1.30 (cartprodsndpairEq = $true) <=> ! [X1 : $i,X0 : $i,X2 : $i] : (! [X3 : $i] : (((ksnd @ (kpair @ X0 @ X3)) = X3) | ((in @ X3 @ X1) != $true)) | ((in @ X0 @ X2) != $true))),
% 1.66/1.30 inference(ennf_transformation,[status(thm)],[f8])).
% 1.66/1.30 thf(f8,plain,(
% 1.66/1.30 ! [X2 : $i,X1 : $i,X0 : $i] : (((in @ X0 @ X2) = $true) => ! [X3 : $i] : (((in @ X3 @ X1) = $true) => ((ksnd @ (kpair @ X0 @ X3)) = X3))) <=> (cartprodsndpairEq = $true)),
% 1.66/1.30 inference(fool_elimination,[status(thm)],[f7])).
% 1.66/1.30 thf(f7,plain,(
% 1.66/1.30 (! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X0 @ X2) => ! [X3 : $i] : ((in @ X3 @ X1) => ((ksnd @ (kpair @ X0 @ X3)) = X3))) = cartprodsndpairEq)),
% 1.66/1.30 inference(rectify,[status(thm)],[f3])).
% 1.66/1.30 thf(f3,axiom,(
% 1.66/1.30 (! [X3 : $i,X1 : $i,X0 : $i] : ((in @ X3 @ X0) => ! [X4 : $i] : ((in @ X4 @ X1) => ((ksnd @ (kpair @ X3 @ X4)) = X4))) = cartprodsndpairEq)),
% 1.66/1.30 file('/export/starexec/sandbox2/tmp/tmp.3hJx7CzvFR/DTF2THF_17518.p',cartprodsndpairEq)).
% 1.66/1.30 thf(f90,plain,(
% 1.66/1.30 ( ! [X2 : $i,X0 : $i,X1 : $i] : (((kpair @ (kfst @ X2) @ (sK11 @ X0 @ X1 @ X2)) = X2) | ((in @ X2 @ (cartprod @ X1 @ X0)) != $true)) )),
% 1.66/1.30 inference(duplicate_literal_removal,[status(thm)],[f87])).
% 1.66/1.30 thf(f87,plain,(
% 1.66/1.30 ( ! [X2 : $i,X0 : $i,X1 : $i] : (((in @ X2 @ (cartprod @ X1 @ X0)) != $true) | ((kpair @ (kfst @ X2) @ (sK11 @ X0 @ X1 @ X2)) = X2) | ((in @ X2 @ (cartprod @ X1 @ X0)) != $true)) )),
% 1.66/1.30 inference(superposition,[status(thm)],[f74,f82])).
% 1.66/1.30 thf(f82,plain,(
% 1.66/1.30 ( ! [X2 : $i,X0 : $i,X1 : $i] : (((kfst @ X2) = (sK10 @ X0 @ X1 @ X2)) | ((in @ X2 @ (cartprod @ X1 @ X0)) != $true)) )),
% 1.66/1.30 inference(trivial_inequality_removal,[status(thm)],[f81])).
% 1.66/1.30 thf(f81,plain,(
% 1.66/1.30 ( ! [X2 : $i,X0 : $i,X1 : $i] : (($true != $true) | ((in @ X2 @ (cartprod @ X1 @ X0)) != $true) | ((kfst @ X2) = (sK10 @ X0 @ X1 @ X2))) )),
% 1.66/1.30 inference(duplicate_literal_removal,[status(thm)],[f80])).
% 1.66/1.30 thf(f80,plain,(
% 1.66/1.30 ( ! [X2 : $i,X0 : $i,X1 : $i] : (($true != $true) | ((in @ X2 @ (cartprod @ X1 @ X0)) != $true) | ((in @ X2 @ (cartprod @ X1 @ X0)) != $true) | ((kfst @ X2) = (sK10 @ X0 @ X1 @ X2))) )),
% 1.66/1.30 inference(superposition,[status(thm)],[f79,f73])).
% 1.66/1.30 thf(f79,plain,(
% 1.66/1.30 ( ! [X2 : $i,X3 : $i,X0 : $i,X1 : $i] : (((in @ (sK10 @ X0 @ X1 @ X2) @ X3) != $true) | ((in @ X2 @ (cartprod @ X1 @ X0)) != $true) | ((kfst @ X2) = (sK10 @ X0 @ X1 @ X2))) )),
% 1.66/1.30 inference(trivial_inequality_removal,[status(thm)],[f78])).
% 1.66/1.30 thf(f78,plain,(
% 1.66/1.30 ( ! [X2 : $i,X3 : $i,X0 : $i,X1 : $i] : (((in @ X2 @ (cartprod @ X1 @ X0)) != $true) | ((in @ (sK10 @ X0 @ X1 @ X2) @ X3) != $true) | ($true != $true) | ((kfst @ X2) = (sK10 @ X0 @ X1 @ X2))) )),
% 1.66/1.30 inference(duplicate_literal_removal,[status(thm)],[f77])).
% 1.66/1.30 thf(f77,plain,(
% 1.66/1.30 ( ! [X2 : $i,X3 : $i,X0 : $i,X1 : $i] : (($true != $true) | ((kfst @ X2) = (sK10 @ X0 @ X1 @ X2)) | ((in @ X2 @ (cartprod @ X1 @ X0)) != $true) | ((in @ (sK10 @ X0 @ X1 @ X2) @ X3) != $true) | ((in @ X2 @ (cartprod @ X1 @ X0)) != $true)) )),
% 1.66/1.30 inference(superposition,[status(thm)],[f75,f72])).
% 1.66/1.30 thf(f75,plain,(
% 1.66/1.30 ( ! [X2 : $i,X3 : $i,X0 : $i,X1 : $i,X4 : $i] : (((in @ (sK11 @ X0 @ X1 @ X2) @ X3) != $true) | ((in @ (sK10 @ X0 @ X1 @ X2) @ X4) != $true) | ((in @ X2 @ (cartprod @ X1 @ X0)) != $true) | ((kfst @ X2) = (sK10 @ X0 @ X1 @ X2))) )),
% 1.66/1.30 inference(superposition,[status(thm)],[f71,f74])).
% 1.66/1.30 thf(f71,plain,(
% 1.66/1.30 ( ! [X2 : $i,X3 : $i,X0 : $i,X1 : $i] : (((kfst @ (kpair @ X2 @ X3)) = X2) | ((in @ X3 @ X1) != $true) | ((in @ X2 @ X0) != $true)) )),
% 1.66/1.30 inference(trivial_inequality_removal,[status(thm)],[f66])).
% 1.66/1.30 thf(f66,plain,(
% 1.66/1.30 ( ! [X2 : $i,X3 : $i,X0 : $i,X1 : $i] : (((kfst @ (kpair @ X2 @ X3)) = X2) | ((in @ X3 @ X1) != $true) | ($true != $true) | ((in @ X2 @ X0) != $true)) )),
% 1.66/1.30 inference(definition_unfolding,[status(thm)],[f56,f41])).
% 1.66/1.30 thf(f41,plain,(
% 1.66/1.30 (cartprodfstpairEq = $true)),
% 1.66/1.30 inference(cnf_transformation,[status(thm)],[f22])).
% 1.66/1.30 thf(f56,plain,(
% 1.66/1.30 ( ! [X2 : $i,X3 : $i,X0 : $i,X1 : $i] : (((in @ X2 @ X0) != $true) | ((kfst @ (kpair @ X2 @ X3)) = X2) | ((in @ X3 @ X1) != $true) | (cartprodfstpairEq != $true)) )),
% 1.66/1.30 inference(cnf_transformation,[status(thm)],[f38])).
% 1.66/1.30 thf(f38,plain,(
% 1.66/1.30 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ X0) != $true) | ! [X3 : $i] : (((kfst @ (kpair @ X2 @ X3)) = X2) | ((in @ X3 @ X1) != $true))) | (cartprodfstpairEq != $true)) & ((cartprodfstpairEq = $true) | (((in @ sK14 @ sK12) = $true) & (((kfst @ (kpair @ sK14 @ sK15)) != sK14) & ((in @ sK15 @ sK13) = $true))))),
% 1.66/1.30 inference(skolemisation,[status(esa),new_symbols(skolem,[sK12,sK13,sK14,sK15])],[f35,f37,f36])).
% 1.66/1.30 thf(f36,plain,(
% 1.66/1.30 ? [X4 : $i,X5 : $i,X6 : $i] : (((in @ X6 @ X4) = $true) & ? [X7 : $i] : (((kfst @ (kpair @ X6 @ X7)) != X6) & ((in @ X7 @ X5) = $true))) => (((in @ sK14 @ sK12) = $true) & ? [X7 : $i] : ((sK14 != (kfst @ (kpair @ sK14 @ X7))) & ((in @ X7 @ sK13) = $true)))),
% 1.66/1.30 introduced(definition,[],[choice_axiom])).
% 1.66/1.30 thf(f37,plain,(
% 1.66/1.30 ? [X7 : $i] : ((sK14 != (kfst @ (kpair @ sK14 @ X7))) & ((in @ X7 @ sK13) = $true)) => (((kfst @ (kpair @ sK14 @ sK15)) != sK14) & ((in @ sK15 @ sK13) = $true))),
% 1.66/1.30 introduced(definition,[],[choice_axiom])).
% 1.66/1.30 thf(f35,plain,(
% 1.66/1.30 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ X0) != $true) | ! [X3 : $i] : (((kfst @ (kpair @ X2 @ X3)) = X2) | ((in @ X3 @ X1) != $true))) | (cartprodfstpairEq != $true)) & ((cartprodfstpairEq = $true) | ? [X4 : $i,X5 : $i,X6 : $i] : (((in @ X6 @ X4) = $true) & ? [X7 : $i] : (((kfst @ (kpair @ X6 @ X7)) != X6) & ((in @ X7 @ X5) = $true))))),
% 1.66/1.30 inference(rectify,[status(thm)],[f34])).
% 1.66/1.30 thf(f34,plain,(
% 1.66/1.30 (! [X2 : $i,X1 : $i,X0 : $i] : (((in @ X0 @ X2) != $true) | ! [X3 : $i] : (((kfst @ (kpair @ X0 @ X3)) = X0) | ((in @ X3 @ X1) != $true))) | (cartprodfstpairEq != $true)) & ((cartprodfstpairEq = $true) | ? [X2 : $i,X1 : $i,X0 : $i] : (((in @ X0 @ X2) = $true) & ? [X3 : $i] : (((kfst @ (kpair @ X0 @ X3)) != X0) & ((in @ X3 @ X1) = $true))))),
% 1.66/1.30 inference(nnf_transformation,[status(thm)],[f17])).
% 1.66/1.30 thf(f17,plain,(
% 1.66/1.30 ! [X2 : $i,X1 : $i,X0 : $i] : (((in @ X0 @ X2) != $true) | ! [X3 : $i] : (((kfst @ (kpair @ X0 @ X3)) = X0) | ((in @ X3 @ X1) != $true))) <=> (cartprodfstpairEq = $true)),
% 1.66/1.30 inference(ennf_transformation,[status(thm)],[f14])).
% 1.66/1.30 thf(f14,plain,(
% 1.66/1.30 ! [X2 : $i,X1 : $i,X0 : $i] : (((in @ X0 @ X2) = $true) => ! [X3 : $i] : (((in @ X3 @ X1) = $true) => ((kfst @ (kpair @ X0 @ X3)) = X0))) <=> (cartprodfstpairEq = $true)),
% 1.66/1.30 inference(fool_elimination,[status(thm)],[f13])).
% 1.66/1.30 thf(f13,plain,(
% 1.66/1.30 (! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X0 @ X2) => ! [X3 : $i] : ((in @ X3 @ X1) => ((kfst @ (kpair @ X0 @ X3)) = X0))) = cartprodfstpairEq)),
% 1.66/1.30 inference(rectify,[status(thm)],[f2])).
% 1.66/1.30 thf(f2,axiom,(
% 1.66/1.30 (! [X3 : $i,X1 : $i,X0 : $i] : ((in @ X3 @ X0) => ! [X4 : $i] : ((in @ X4 @ X1) => ((kfst @ (kpair @ X3 @ X4)) = X3))) = cartprodfstpairEq)),
% 1.66/1.30 file('/export/starexec/sandbox2/tmp/tmp.3hJx7CzvFR/DTF2THF_17518.p',cartprodfstpairEq)).
% 1.66/1.30 % SZS output end Proof for DTF2THF_17518
% 1.66/1.30 % (17740)------------------------------
% 1.66/1.30 % (17740)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.66/1.30 % (17740)Termination reason: Refutation
% 1.66/1.30
% 1.66/1.30 % (17740)Memory used [KB]: 5628
% 1.66/1.30 % (17740)Time elapsed: 0.010 s
% 1.66/1.30 % (17740)Instructions burned: 18 (million)
% 1.66/1.30 % (17740)------------------------------
% 1.66/1.30 % (17740)------------------------------
% 1.66/1.30 % (17734)Success in time 0.019 s
%------------------------------------------------------------------------------