%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU659^2 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n016.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 09:36:01 AM UTC 2026
% Result : Theorem 1.65s 1.34s
% Output : Refutation 1.65s
% Verified :
% SZS Type : Refutation
% Derivation depth : 17
% Number of leaves : 8
% Syntax : Number of formulae : 47 ( 10 unt; 0 typ; 0 def)
% Number of atoms : 453 ( 162 equ; 0 cnn)
% Maximal formula atoms : 10 ( 9 avg)
% Number of connectives : 625 ( 67 ~; 53 |; 50 &; 428 @)
% ( 4 <=>; 23 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 7 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 21 ( 18 usr; 15 con; 0-3 aty)
% Number of variables : 177 ( 0 ^; 124 !; 53 ?; 177 :)
% 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_15,type,
sK7: $i > $i > $i > $i ).
thf(func_def_16,type,
sK8: $i > $i > $i > $i ).
thf(f69,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f68,f39]) ).
thf(f39,plain,
( ( in @ sK12 @ sK9 )
!= $true ),
inference(cnf_transformation,[status(thm)],[f28]) ).
thf(f28,plain,
( ( $true
= ( in @ ( kpair @ sK12 @ sK10 ) @ ( cartprod @ sK9 @ sK11 ) ) )
& ( ( in @ sK12 @ sK9 )
!= $true )
& ( cartprodmempair1 = $true )
& ( setukpairinjL = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK9,sK10,sK11,sK12])],[f26,f27]) ).
thf(f27,plain,
( ? [X0: $i,X1: $i,X2: $i,X3: $i] :
( ( ( in @ ( kpair @ X3 @ X1 ) @ ( cartprod @ X0 @ X2 ) )
= $true )
& ( ( in @ X3 @ X0 )
!= $true ) )
=> ( ( $true
= ( in @ ( kpair @ sK12 @ sK10 ) @ ( cartprod @ sK9 @ sK11 ) ) )
& ( ( in @ sK12 @ sK9 )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f26,plain,
( ? [X0: $i,X1: $i,X2: $i,X3: $i] :
( ( ( in @ ( kpair @ X3 @ X1 ) @ ( cartprod @ X0 @ X2 ) )
= $true )
& ( ( in @ X3 @ X0 )
!= $true ) )
& ( cartprodmempair1 = $true )
& ( setukpairinjL = $true ) ),
inference(rectify,[status(thm)],[f14]) ).
thf(f14,plain,
( ? [X1: $i,X0: $i,X2: $i,X3: $i] :
( ( $true
= ( in @ ( kpair @ X3 @ X0 ) @ ( cartprod @ X1 @ X2 ) ) )
& ( ( in @ X3 @ X1 )
!= $true ) )
& ( cartprodmempair1 = $true )
& ( setukpairinjL = $true ) ),
inference(flattening,[status(thm)],[f13]) ).
thf(f13,plain,
( ? [X1: $i,X0: $i,X2: $i,X3: $i] :
( ( $true
= ( in @ ( kpair @ X3 @ X0 ) @ ( cartprod @ X1 @ X2 ) ) )
& ( ( in @ X3 @ X1 )
!= $true ) )
& ( setukpairinjL = $true )
& ( cartprodmempair1 = $true ) ),
inference(ennf_transformation,[status(thm)],[f11]) ).
thf(f11,plain,
~ ( ( cartprodmempair1 = $true )
=> ( ( setukpairinjL = $true )
=> ! [X0: $i,X2: $i,X3: $i,X1: $i] :
( ( $true
= ( in @ ( kpair @ X3 @ X0 ) @ ( cartprod @ X1 @ X2 ) ) )
=> ( ( in @ X3 @ X1 )
= $true ) ) ) ),
inference(fool_elimination,[status(thm)],[f10]) ).
thf(f10,plain,
~ ( cartprodmempair1
=> ( setukpairinjL
=> ! [X0: $i,X1: $i,X2: $i,X3: $i] :
( ( in @ ( kpair @ X3 @ X0 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) ) ) ),
inference(rectify,[status(thm)],[f4]) ).
thf(f4,negated_conjecture,
~ ( cartprodmempair1
=> ( setukpairinjL
=> ! [X4: $i,X0: $i,X1: $i,X3: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X0 @ X1 ) )
=> ( in @ X3 @ X0 ) ) ) ),
inference(negated_conjecture,[status(cth)],[f3]) ).
thf(f3,conjecture,
( cartprodmempair1
=> ( setukpairinjL
=> ! [X4: $i,X0: $i,X1: $i,X3: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X0 @ X1 ) )
=> ( in @ X3 @ X0 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.RLP1xBcmXt/DTF2THF_18578.p',cartprodpairmemEL) ).
thf(f68,plain,
( ( in @ sK12 @ sK9 )
= $true ),
inference(equality_resolution,[status(thm)],[f66]) ).
thf(f66,plain,
! [X0: $i,X1: $i] :
( ( ( kpair @ X0 @ X1 )
!= ( kpair @ sK12 @ sK10 ) )
| ( ( in @ X0 @ sK9 )
= $true ) ),
inference(subsumption_resolution,[status(thm)],[f64,f40]) ).
thf(f40,plain,
( $true
= ( in @ ( kpair @ sK12 @ sK10 ) @ ( cartprod @ sK9 @ sK11 ) ) ),
inference(cnf_transformation,[status(thm)],[f28]) ).
thf(f64,plain,
! [X0: $i,X1: $i] :
( ( ( in @ X0 @ sK9 )
= $true )
| ( ( kpair @ X0 @ X1 )
!= ( kpair @ sK12 @ sK10 ) )
| ( $true
!= ( in @ ( kpair @ sK12 @ sK10 ) @ ( cartprod @ sK9 @ sK11 ) ) ) ),
inference(superposition,[status(thm)],[f52,f60]) ).
thf(f60,plain,
! [X0: $i,X1: $i] :
( ( ( sK7 @ sK11 @ sK9 @ ( kpair @ sK12 @ sK10 ) )
= X0 )
| ( ( kpair @ X0 @ X1 )
!= ( kpair @ sK12 @ sK10 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f56]) ).
thf(f56,plain,
! [X0: $i,X1: $i] :
( ( ( sK7 @ sK11 @ sK9 @ ( kpair @ sK12 @ sK10 ) )
= X0 )
| ( $true != $true )
| ( ( kpair @ X0 @ X1 )
!= ( kpair @ sK12 @ sK10 ) ) ),
inference(superposition,[status(thm)],[f54,f40]) ).
thf(f54,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i,X4: $i] :
( ( $true
!= ( in @ X2 @ ( cartprod @ X1 @ X0 ) ) )
| ( ( sK7 @ X0 @ X1 @ X2 )
= X3 )
| ( ( kpair @ X3 @ X4 )
!= X2 ) ),
inference(superposition,[status(thm)],[f50,f51]) ).
thf(f51,plain,
! [X6: $i,X7: $i,X5: $i] :
( ( ( kpair @ ( sK7 @ X7 @ X6 @ X5 ) @ ( sK8 @ X7 @ X6 @ X5 ) )
= X5 )
| ( $true
!= ( in @ X5 @ ( cartprod @ X6 @ X7 ) ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f48]) ).
thf(f48,plain,
! [X6: $i,X7: $i,X5: $i] :
( ( ( kpair @ ( sK7 @ X7 @ X6 @ X5 ) @ ( sK8 @ X7 @ X6 @ X5 ) )
= X5 )
| ( $true != $true )
| ( $true
!= ( in @ X5 @ ( cartprod @ X6 @ X7 ) ) ) ),
inference(definition_unfolding,[status(thm)],[f32,f38]) ).
thf(f38,plain,
cartprodmempair1 = $true,
inference(cnf_transformation,[status(thm)],[f28]) ).
thf(f32,plain,
! [X6: $i,X7: $i,X5: $i] :
( ( ( kpair @ ( sK7 @ X7 @ X6 @ X5 ) @ ( sK8 @ X7 @ X6 @ X5 ) )
= X5 )
| ( $true
!= ( in @ X5 @ ( cartprod @ X6 @ X7 ) ) )
| ( cartprodmempair1 != $true ) ),
inference(cnf_transformation,[status(thm)],[f25]) ).
thf(f25,plain,
( ( ( cartprodmempair1 = $true )
| ( ! [X3: $i] :
( ( $true
!= ( in @ X3 @ sK5 ) )
| ! [X4: $i] :
( ( $true
!= ( in @ X4 @ sK6 ) )
| ( ( kpair @ X3 @ X4 )
!= sK4 ) ) )
& ( $true
= ( in @ sK4 @ ( cartprod @ sK5 @ sK6 ) ) ) ) )
& ( ! [X5: $i,X6: $i,X7: $i] :
( ( ( ( in @ ( sK7 @ X7 @ X6 @ X5 ) @ X6 )
= $true )
& ( ( in @ ( sK8 @ X7 @ X6 @ X5 ) @ X7 )
= $true )
& ( ( kpair @ ( sK7 @ X7 @ X6 @ X5 ) @ ( sK8 @ X7 @ X6 @ X5 ) )
= X5 ) )
| ( $true
!= ( in @ X5 @ ( cartprod @ X6 @ X7 ) ) ) )
| ( cartprodmempair1 != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK4,sK5,sK6,sK7,sK8])],[f21,f24,f23,f22]) ).
thf(f22,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ! [X3: $i] :
( ( ( in @ X3 @ X1 )
!= $true )
| ! [X4: $i] :
( ( ( in @ X4 @ X2 )
!= $true )
| ( ( kpair @ X3 @ X4 )
!= X0 ) ) )
& ( $true
= ( in @ X0 @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X3: $i] :
( ( $true
!= ( in @ X3 @ sK5 ) )
| ! [X4: $i] :
( ( $true
!= ( in @ X4 @ sK6 ) )
| ( ( kpair @ X3 @ X4 )
!= sK4 ) ) )
& ( $true
= ( in @ sK4 @ ( cartprod @ sK5 @ sK6 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f23,plain,
! [X5: $i,X6: $i,X7: $i] :
( ? [X8: $i] :
( ( $true
= ( in @ X8 @ X6 ) )
& ? [X9: $i] :
( ( ( in @ X9 @ X7 )
= $true )
& ( ( kpair @ X8 @ X9 )
= X5 ) ) )
=> ( ( ( in @ ( sK7 @ X7 @ X6 @ X5 ) @ X6 )
= $true )
& ? [X9: $i] :
( ( ( in @ X9 @ X7 )
= $true )
& ( ( kpair @ ( sK7 @ X7 @ X6 @ X5 ) @ X9 )
= X5 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f24,plain,
! [X5: $i,X6: $i,X7: $i] :
( ? [X9: $i] :
( ( ( in @ X9 @ X7 )
= $true )
& ( ( kpair @ ( sK7 @ X7 @ X6 @ X5 ) @ X9 )
= X5 ) )
=> ( ( ( in @ ( sK8 @ X7 @ X6 @ X5 ) @ X7 )
= $true )
& ( ( kpair @ ( sK7 @ X7 @ X6 @ X5 ) @ ( sK8 @ X7 @ X6 @ X5 ) )
= X5 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f21,plain,
( ( ( cartprodmempair1 = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ! [X3: $i] :
( ( ( in @ X3 @ X1 )
!= $true )
| ! [X4: $i] :
( ( ( in @ X4 @ X2 )
!= $true )
| ( ( kpair @ X3 @ X4 )
!= X0 ) ) )
& ( $true
= ( in @ X0 @ ( cartprod @ X1 @ X2 ) ) ) ) )
& ( ! [X5: $i,X6: $i,X7: $i] :
( ? [X8: $i] :
( ( $true
= ( in @ X8 @ X6 ) )
& ? [X9: $i] :
( ( ( in @ X9 @ X7 )
= $true )
& ( ( kpair @ X8 @ X9 )
= X5 ) ) )
| ( $true
!= ( in @ X5 @ ( cartprod @ X6 @ X7 ) ) ) )
| ( cartprodmempair1 != $true ) ) ),
inference(rectify,[status(thm)],[f20]) ).
thf(f20,plain,
( ( ( cartprodmempair1 = $true )
| ? [X0: $i,X2: $i,X1: $i] :
( ! [X3: $i] :
( ( $true
!= ( in @ X3 @ X2 ) )
| ! [X4: $i] :
( ( ( in @ X4 @ X1 )
!= $true )
| ( ( kpair @ X3 @ X4 )
!= X0 ) ) )
& ( ( in @ X0 @ ( cartprod @ X2 @ X1 ) )
= $true ) ) )
& ( ! [X0: $i,X2: $i,X1: $i] :
( ? [X3: $i] :
( ( $true
= ( in @ X3 @ X2 ) )
& ? [X4: $i] :
( ( ( in @ X4 @ X1 )
= $true )
& ( ( kpair @ X3 @ X4 )
= X0 ) ) )
| ( ( in @ X0 @ ( cartprod @ X2 @ X1 ) )
!= $true ) )
| ( cartprodmempair1 != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f12]) ).
thf(f12,plain,
( ( cartprodmempair1 = $true )
<=> ! [X0: $i,X2: $i,X1: $i] :
( ? [X3: $i] :
( ( $true
= ( in @ X3 @ X2 ) )
& ? [X4: $i] :
( ( ( in @ X4 @ X1 )
= $true )
& ( ( kpair @ X3 @ X4 )
= X0 ) ) )
| ( ( in @ X0 @ ( cartprod @ X2 @ X1 ) )
!= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f7]) ).
thf(f7,plain,
( ( cartprodmempair1 = $true )
<=> ! [X0: $i,X2: $i,X1: $i] :
( ( ( in @ X0 @ ( cartprod @ X2 @ X1 ) )
= $true )
=> ? [X3: $i] :
( ( $true
= ( in @ X3 @ X2 ) )
& ? [X4: $i] :
( ( ( in @ X4 @ X1 )
= $true )
& ( ( kpair @ X3 @ X4 )
= X0 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f6]) ).
thf(f6,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/sandbox/tmp/tmp.RLP1xBcmXt/DTF2THF_18578.p',cartprodmempair1) ).
thf(f50,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i] :
( ( ( kpair @ X1 @ X2 )
!= ( kpair @ X0 @ X3 ) )
| ( X0 = X1 ) ),
inference(trivial_inequality_removal,[status(thm)],[f41]) ).
thf(f41,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i] :
( ( ( kpair @ X1 @ X2 )
!= ( kpair @ X0 @ X3 ) )
| ( $true != $true )
| ( X0 = X1 ) ),
inference(definition_unfolding,[status(thm)],[f31,f37]) ).
thf(f37,plain,
setukpairinjL = $true,
inference(cnf_transformation,[status(thm)],[f28]) ).
thf(f31,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i] :
( ( ( kpair @ X1 @ X2 )
!= ( kpair @ X0 @ X3 ) )
| ( X0 = X1 )
| ( setukpairinjL != $true ) ),
inference(cnf_transformation,[status(thm)],[f19]) ).
thf(f19,plain,
( ( ! [X0: $i,X1: $i,X2: $i,X3: $i] :
( ( ( kpair @ X1 @ X2 )
!= ( kpair @ X0 @ X3 ) )
| ( X0 = X1 ) )
| ( setukpairinjL != $true ) )
& ( ( setukpairinjL = $true )
| ( ( ( kpair @ sK0 @ sK3 )
= ( kpair @ sK1 @ sK2 ) )
& ( sK1 != sK0 ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2,sK3])],[f17,f18]) ).
thf(f18,plain,
( ? [X4: $i,X5: $i,X6: $i,X7: $i] :
( ( ( kpair @ X4 @ X7 )
= ( kpair @ X5 @ X6 ) )
& ( X4 != X5 ) )
=> ( ( ( kpair @ sK0 @ sK3 )
= ( kpair @ sK1 @ sK2 ) )
& ( sK1 != sK0 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f17,plain,
( ( ! [X0: $i,X1: $i,X2: $i,X3: $i] :
( ( ( kpair @ X1 @ X2 )
!= ( kpair @ X0 @ X3 ) )
| ( X0 = X1 ) )
| ( setukpairinjL != $true ) )
& ( ( setukpairinjL = $true )
| ? [X4: $i,X5: $i,X6: $i,X7: $i] :
( ( ( kpair @ X4 @ X7 )
= ( kpair @ X5 @ X6 ) )
& ( X4 != X5 ) ) ) ),
inference(rectify,[status(thm)],[f16]) ).
thf(f16,plain,
( ( ! [X0: $i,X1: $i,X2: $i,X3: $i] :
( ( ( kpair @ X1 @ X2 )
!= ( kpair @ X0 @ X3 ) )
| ( X0 = X1 ) )
| ( setukpairinjL != $true ) )
& ( ( setukpairinjL = $true )
| ? [X0: $i,X1: $i,X2: $i,X3: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X0 @ X3 ) )
& ( X0 != X1 ) ) ) ),
inference(nnf_transformation,[status(thm)],[f15]) ).
thf(f15,plain,
( ! [X0: $i,X1: $i,X2: $i,X3: $i] :
( ( ( kpair @ X1 @ X2 )
!= ( kpair @ X0 @ X3 ) )
| ( X0 = X1 ) )
<=> ( setukpairinjL = $true ) ),
inference(ennf_transformation,[status(thm)],[f9]) ).
thf(f9,plain,
( ( setukpairinjL = $true )
<=> ! [X0: $i,X1: $i,X2: $i,X3: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X0 @ X3 ) )
=> ( X0 = X1 ) ) ),
inference(fool_elimination,[status(thm)],[f8]) ).
thf(f8,plain,
! [X0: $i,X1: $i,X2: $i,X3: $i] :
( ( ( ( kpair @ X1 @ X2 )
= ( kpair @ X0 @ X3 ) )
=> ( X0 = X1 ) )
= setukpairinjL ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
! [X3: $i,X5: $i,X2: $i,X4: $i] :
( ( ( ( kpair @ X3 @ X4 )
= ( kpair @ X5 @ X2 ) )
=> ( X3 = X5 ) )
= setukpairinjL ),
file('/export/starexec/sandbox/tmp/tmp.RLP1xBcmXt/DTF2THF_18578.p',setukpairinjL) ).
thf(f52,plain,
! [X6: $i,X7: $i,X5: $i] :
( ( ( in @ ( sK7 @ X7 @ X6 @ X5 ) @ X6 )
= $true )
| ( $true
!= ( in @ X5 @ ( cartprod @ X6 @ X7 ) ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f46]) ).
thf(f46,plain,
! [X6: $i,X7: $i,X5: $i] :
( ( $true
!= ( in @ X5 @ ( cartprod @ X6 @ X7 ) ) )
| ( $true != $true )
| ( ( in @ ( sK7 @ X7 @ X6 @ X5 ) @ X6 )
= $true ) ),
inference(definition_unfolding,[status(thm)],[f34,f38]) ).
thf(f34,plain,
! [X6: $i,X7: $i,X5: $i] :
( ( ( in @ ( sK7 @ X7 @ X6 @ X5 ) @ X6 )
= $true )
| ( $true
!= ( in @ X5 @ ( cartprod @ X6 @ X7 ) ) )
| ( cartprodmempair1 != $true ) ),
inference(cnf_transformation,[status(thm)],[f25]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU659^2 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n016.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 12:30:06 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.20/0.46 ---- Original DTF file ---
% 0.20/0.46 thf(spec,logic,$$dhol).
% 0.20/0.46 %------------------------------------------------------------------------------
% 0.20/0.46 % File : SEU659^2 : TPTP v9.2.1. Released v3.7.0.
% 0.20/0.46 % Domain : Set Theory
% 0.20/0.46 % Problem : Ordered Pairs - Properties of Pairs
% 0.20/0.46 % Version : Especial > Reduced > Especial.
% 0.20/0.46 % English : (! A:i.! B:i.! x:i.! y:i.in (kpair x y) (cartprod A B) -> in x A)
% 0.20/0.46
% 0.20/0.46 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.20/0.46 % Source : [Bro08]
% 0.20/0.46 % Names : ZFC161l [Bro08]
% 0.20/0.46
% 0.20/0.46 % Status : Theorem
% 0.20/0.46 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.10 v8.2.0, 0.23 v8.1.0, 0.09 v7.5.0, 0.00 v7.4.0, 0.11 v7.2.0, 0.00 v7.1.0, 0.12 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.00 v6.1.0, 0.14 v5.5.0, 0.17 v5.4.0, 0.20 v5.3.0, 0.40 v5.2.0, 0.20 v4.1.0, 0.00 v4.0.0, 0.33 v3.7.0
% 0.20/0.46 % Syntax : Number of formulae : 8 ( 2 unt; 5 typ; 2 def)
% 0.20/0.46 % Number of atoms : 14 ( 5 equ; 0 cnn)
% 0.20/0.46 % Maximal formula atoms : 4 ( 4 avg)
% 0.20/0.46 % Number of connectives : 29 ( 0 ~; 0 |; 2 &; 22 @)
% 0.20/0.46 % ( 0 <=>; 5 =>; 0 <=; 0 <~>)
% 0.20/0.46 % Maximal formula depth : 12 ( 5 avg)
% 0.20/0.46 % Number of types : 2 ( 0 usr)
% 0.20/0.46 % Number of type conns : 6 ( 6 >; 0 *; 0 +; 0 <<)
% 0.20/0.46 % Number of symbols : 6 ( 5 usr; 2 con; 0-2 aty)
% 0.20/0.46 % Number of variables : 13 ( 0 ^; 11 !; 2 ?; 13 :)
% 0.20/0.46 % SPC : TH0_THM_EQU_NAR
% 0.20/0.46
% 0.20/0.46 % Comments : http://mathgate.info/detsetitem.php?id=337
% 0.20/0.46 %------------------------------------------------------------------------------
% 0.20/0.46 thf(in_type,type,
% 0.20/0.46 in: $i > $i > $o ).
% 0.20/0.46
% 0.20/0.46 thf(kpair_type,type,
% 0.20/0.46 kpair: $i > $i > $i ).
% 0.20/0.46
% 0.20/0.46 thf(cartprod_type,type,
% 0.20/0.46 cartprod: $i > $i > $i ).
% 0.20/0.46
% 0.20/0.46 thf(cartprodmempair1_type,type,
% 0.20/0.46 cartprodmempair1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(cartprodmempair1,definition,
% 0.20/0.46 ( cartprodmempair1
% 0.20/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.20/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.20/0.46 => ? [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 & ? [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ B )
% 0.20/0.46 & ( Xu
% 0.20/0.46 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setukpairinjL_type,type,
% 0.20/0.46 setukpairinjL: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setukpairinjL,definition,
% 0.20/0.46 ( setukpairinjL
% 0.20/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.20/0.46 ( ( ( kpair @ Xx @ Xy )
% 0.20/0.46 = ( kpair @ Xz @ Xu ) )
% 0.20/0.46 => ( Xx = Xz ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(cartprodpairmemEL,conjecture,
% 0.20/0.46 ( cartprodmempair1
% 0.20/0.46 => ( setukpairinjL
% 0.20/0.46 => ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.20/0.46 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.20/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 %------------------------------------------------------------------------------
% 0.20/0.46 ------------------------
% 1.46/1.30 ---- Embedded in THF ---
% 1.46/1.30 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.46/1.30 %%% Generated on Tue Feb 24 12:30:07 EST 2026
% 1.46/1.30 %%% using '$$dhol' embedding, version 1.3.0.
% 1.46/1.30 %%% Logic specification used:
% 1.46/1.30 %%% thf(spec, logic, $$dhol).
% 1.46/1.30
% 1.46/1.30 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.RLP1xBcmXt/DTF2DTF18578.p
% See solution above
% 1.65/1.30 ------------------------
% 1.65/1.31 ---- Cleaned THF ---
% 1.65/1.31 thf(in_type,type,
% 1.65/1.31 in: $i > $i > $o ).
% 1.65/1.31
% 1.65/1.31 thf(kpair_type,type,
% 1.65/1.31 kpair: $i > $i > $i ).
% 1.65/1.31
% 1.65/1.31 thf(cartprod_type,type,
% 1.65/1.31 cartprod: $i > $i > $i ).
% 1.65/1.31
% 1.65/1.31 thf(cartprodmempair1_type,type,
% 1.65/1.31 cartprodmempair1: $o ).
% 1.65/1.31
% 1.65/1.31 thf(cartprodmempair1,definition,
% 1.65/1.31 ( cartprodmempair1
% 1.65/1.31 = ( ! [A: $i,B: $i,Xu: $i] :
% 1.65/1.31 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 1.65/1.31 => ? [Xx: $i] :
% 1.65/1.31 ( ( in @ Xx @ A )
% 1.65/1.31 & ? [Xy: $i] :
% 1.65/1.31 ( ( in @ Xy @ B )
% 1.65/1.31 & ( Xu
% 1.65/1.31 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 1.65/1.31
% 1.65/1.31 thf(setukpairinjL_type,type,
% 1.65/1.31 setukpairinjL: $o ).
% 1.65/1.31
% 1.65/1.31 thf(setukpairinjL,definition,
% 1.65/1.31 ( setukpairinjL
% 1.65/1.31 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 1.65/1.31 ( ( ( kpair @ Xx @ Xy )
% 1.65/1.31 = ( kpair @ Xz @ Xu ) )
% 1.65/1.31 => ( Xx = Xz ) ) ) ) ).
% 1.65/1.31
% 1.65/1.31 thf(cartprodpairmemEL,conjecture,
% 1.65/1.31 ( cartprodmempair1
% 1.65/1.31 => ( setukpairinjL
% 1.65/1.31 => ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 1.65/1.31 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 1.65/1.31 => ( in @ Xx @ A ) ) ) ) ).
% 1.65/1.31 ------------------------
% 1.65/1.32 % (18771)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_18578 for (2999ds/2Mi)
% 1.65/1.32 % (18771)Instruction limit reached!
% 1.65/1.32 % (18771)------------------------------
% 1.65/1.32 % (18771)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.65/1.32 % (18771)Termination reason: Unknown
% 1.65/1.32 % (18771)Termination phase: Preprocessing 3
% 1.65/1.32
% 1.65/1.32 % (18771)Memory used [KB]: 895
% 1.65/1.32 % (18771)Time elapsed: 0.002 s
% 1.65/1.32 % (18771)Instructions burned: 2 (million)
% 1.65/1.32 % (18771)------------------------------
% 1.65/1.32 % (18771)------------------------------
% 1.65/1.33 % (18769)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_18578 for (2999ds/4Mi)
% 1.65/1.33 % (18768)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_18578 for (2999ds/183Mi)
% 1.65/1.33 % (18772)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_18578 for (2999ds/2Mi)
% 1.65/1.33 % (18769)Instruction limit reached!
% 1.65/1.33 % (18769)------------------------------
% 1.65/1.33 % (18769)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.65/1.33 % (18769)Termination reason: Unknown
% 1.65/1.33 % (18769)Termination phase: Saturation
% 1.65/1.33
% 1.65/1.33 % (18769)Memory used [KB]: 5500
% 1.65/1.33 % (18769)Time elapsed: 0.004 s
% 1.65/1.33 % (18769)Instructions burned: 4 (million)
% 1.65/1.33 % (18769)------------------------------
% 1.65/1.33 % (18769)------------------------------
% 1.65/1.33 % (18772)Instruction limit reached!
% 1.65/1.33 % (18772)------------------------------
% 1.65/1.33 % (18772)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.65/1.33 % (18772)Termination reason: Unknown
% 1.65/1.33 % (18772)Termination phase: Saturation
% 1.65/1.33
% 1.65/1.33 % (18772)Memory used [KB]: 1023
% 1.65/1.33 % (18772)Time elapsed: 0.004 s
% 1.65/1.33 % (18772)Instructions burned: 3 (million)
% 1.65/1.33 % (18772)------------------------------
% 1.65/1.33 % (18772)------------------------------
% 1.65/1.33 % (18773)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_18578 for (2999ds/275Mi)
% 1.65/1.34 % (18773)First to succeed.
% 1.65/1.34 % (18774)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_18578 for (2999ds/18Mi)
% 1.65/1.34 % (18773)Refutation found. Thanks to Tanya!
% 1.65/1.34 % SZS status Theorem for DTF2THF_18578
% 1.65/1.34 % SZS output start Proof for DTF2THF_18578
% 1.65/1.34 thf(func_def_0, type, in: $i > $i > $o).
% 1.65/1.34 thf(func_def_1, type, kpair: $i > $i > $i).
% 1.65/1.34 thf(func_def_2, type, cartprod: $i > $i > $i).
% 1.65/1.34 thf(func_def_15, type, sK7: $i > $i > $i > $i).
% 1.65/1.34 thf(func_def_16, type, sK8: $i > $i > $i > $i).
% 1.65/1.34 thf(f69,plain,(
% 1.65/1.34 $false),
% 1.65/1.34 inference(subsumption_resolution,[status(thm)],[f68,f39])).
% 1.65/1.34 thf(f39,plain,(
% 1.65/1.34 ((in @ sK12 @ sK9) != $true)),
% 1.65/1.34 inference(cnf_transformation,[status(thm)],[f28])).
% 1.65/1.34 thf(f28,plain,(
% 1.65/1.34 (($true = (in @ (kpair @ sK12 @ sK10) @ (cartprod @ sK9 @ sK11))) & ((in @ sK12 @ sK9) != $true)) & (cartprodmempair1 = $true) & (setukpairinjL = $true)),
% 1.65/1.34 inference(skolemisation,[status(esa),new_symbols(skolem,[sK9,sK10,sK11,sK12])],[f26,f27])).
% 1.65/1.34 thf(f27,plain,(
% 1.65/1.34 ? [X0 : $i,X1 : $i,X2 : $i,X3 : $i] : (((in @ (kpair @ X3 @ X1) @ (cartprod @ X0 @ X2)) = $true) & ((in @ X3 @ X0) != $true)) => (($true = (in @ (kpair @ sK12 @ sK10) @ (cartprod @ sK9 @ sK11))) & ((in @ sK12 @ sK9) != $true))),
% 1.65/1.34 introduced(definition,[],[choice_axiom])).
% 1.65/1.34 thf(f26,plain,(
% 1.65/1.34 ? [X0 : $i,X1 : $i,X2 : $i,X3 : $i] : (((in @ (kpair @ X3 @ X1) @ (cartprod @ X0 @ X2)) = $true) & ((in @ X3 @ X0) != $true)) & (cartprodmempair1 = $true) & (setukpairinjL = $true)),
% 1.65/1.34 inference(rectify,[status(thm)],[f14])).
% 1.65/1.34 thf(f14,plain,(
% 1.65/1.34 ? [X1 : $i,X0 : $i,X2 : $i,X3 : $i] : (($true = (in @ (kpair @ X3 @ X0) @ (cartprod @ X1 @ X2))) & ((in @ X3 @ X1) != $true)) & (cartprodmempair1 = $true) & (setukpairinjL = $true)),
% 1.65/1.34 inference(flattening,[status(thm)],[f13])).
% 1.65/1.34 thf(f13,plain,(
% 1.65/1.34 (? [X1 : $i,X0 : $i,X2 : $i,X3 : $i] : (($true = (in @ (kpair @ X3 @ X0) @ (cartprod @ X1 @ X2))) & ((in @ X3 @ X1) != $true)) & (setukpairinjL = $true)) & (cartprodmempair1 = $true)),
% 1.65/1.34 inference(ennf_transformation,[status(thm)],[f11])).
% 1.65/1.34 thf(f11,plain,(
% 1.65/1.34 ~((cartprodmempair1 = $true) => ((setukpairinjL = $true) => ! [X0 : $i,X2 : $i,X3 : $i,X1 : $i] : (($true = (in @ (kpair @ X3 @ X0) @ (cartprod @ X1 @ X2))) => ((in @ X3 @ X1) = $true))))),
% 1.65/1.34 inference(fool_elimination,[status(thm)],[f10])).
% 1.65/1.34 thf(f10,plain,(
% 1.65/1.34 ~(cartprodmempair1 => (setukpairinjL => ! [X0 : $i,X1 : $i,X2 : $i,X3 : $i] : ((in @ (kpair @ X3 @ X0) @ (cartprod @ X1 @ X2)) => (in @ X3 @ X1))))),
% 1.65/1.34 inference(rectify,[status(thm)],[f4])).
% 1.65/1.34 thf(f4,negated_conjecture,(
% 1.65/1.34 ~(cartprodmempair1 => (setukpairinjL => ! [X4 : $i,X0 : $i,X1 : $i,X3 : $i] : ((in @ (kpair @ X3 @ X4) @ (cartprod @ X0 @ X1)) => (in @ X3 @ X0))))),
% 1.65/1.34 inference(negated_conjecture,[status(cth)],[f3])).
% 1.65/1.34 thf(f3,conjecture,(
% 1.65/1.34 cartprodmempair1 => (setukpairinjL => ! [X4 : $i,X0 : $i,X1 : $i,X3 : $i] : ((in @ (kpair @ X3 @ X4) @ (cartprod @ X0 @ X1)) => (in @ X3 @ X0)))),
% 1.65/1.34 file('/export/starexec/sandbox/tmp/tmp.RLP1xBcmXt/DTF2THF_18578.p',cartprodpairmemEL)).
% 1.65/1.34 thf(f68,plain,(
% 1.65/1.34 ((in @ sK12 @ sK9) = $true)),
% 1.65/1.34 inference(equality_resolution,[status(thm)],[f66])).
% 1.65/1.34 thf(f66,plain,(
% 1.65/1.34 ( ! [X0 : $i,X1 : $i] : (((kpair @ X0 @ X1) != (kpair @ sK12 @ sK10)) | ((in @ X0 @ sK9) = $true)) )),
% 1.65/1.34 inference(subsumption_resolution,[status(thm)],[f64,f40])).
% 1.65/1.34 thf(f40,plain,(
% 1.65/1.34 ($true = (in @ (kpair @ sK12 @ sK10) @ (cartprod @ sK9 @ sK11)))),
% 1.65/1.34 inference(cnf_transformation,[status(thm)],[f28])).
% 1.65/1.34 thf(f64,plain,(
% 1.65/1.34 ( ! [X0 : $i,X1 : $i] : (((in @ X0 @ sK9) = $true) | ((kpair @ X0 @ X1) != (kpair @ sK12 @ sK10)) | ($true != (in @ (kpair @ sK12 @ sK10) @ (cartprod @ sK9 @ sK11)))) )),
% 1.65/1.34 inference(superposition,[status(thm)],[f52,f60])).
% 1.65/1.34 thf(f60,plain,(
% 1.65/1.34 ( ! [X0 : $i,X1 : $i] : (((sK7 @ sK11 @ sK9 @ (kpair @ sK12 @ sK10)) = X0) | ((kpair @ X0 @ X1) != (kpair @ sK12 @ sK10))) )),
% 1.65/1.34 inference(trivial_inequality_removal,[status(thm)],[f56])).
% 1.65/1.34 thf(f56,plain,(
% 1.65/1.34 ( ! [X0 : $i,X1 : $i] : (((sK7 @ sK11 @ sK9 @ (kpair @ sK12 @ sK10)) = X0) | ($true != $true) | ((kpair @ X0 @ X1) != (kpair @ sK12 @ sK10))) )),
% 1.65/1.34 inference(superposition,[status(thm)],[f54,f40])).
% 1.65/1.34 thf(f54,plain,(
% 1.65/1.34 ( ! [X2 : $i,X3 : $i,X0 : $i,X1 : $i,X4 : $i] : (($true != (in @ X2 @ (cartprod @ X1 @ X0))) | ((sK7 @ X0 @ X1 @ X2) = X3) | ((kpair @ X3 @ X4) != X2)) )),
% 1.65/1.34 inference(superposition,[status(thm)],[f50,f51])).
% 1.65/1.34 thf(f51,plain,(
% 1.65/1.34 ( ! [X6 : $i,X7 : $i,X5 : $i] : (((kpair @ (sK7 @ X7 @ X6 @ X5) @ (sK8 @ X7 @ X6 @ X5)) = X5) | ($true != (in @ X5 @ (cartprod @ X6 @ X7)))) )),
% 1.65/1.34 inference(trivial_inequality_removal,[status(thm)],[f48])).
% 1.65/1.34 thf(f48,plain,(
% 1.65/1.34 ( ! [X6 : $i,X7 : $i,X5 : $i] : (((kpair @ (sK7 @ X7 @ X6 @ X5) @ (sK8 @ X7 @ X6 @ X5)) = X5) | ($true != $true) | ($true != (in @ X5 @ (cartprod @ X6 @ X7)))) )),
% 1.65/1.34 inference(definition_unfolding,[status(thm)],[f32,f38])).
% 1.65/1.34 thf(f38,plain,(
% 1.65/1.34 (cartprodmempair1 = $true)),
% 1.65/1.34 inference(cnf_transformation,[status(thm)],[f28])).
% 1.65/1.34 thf(f32,plain,(
% 1.65/1.34 ( ! [X6 : $i,X7 : $i,X5 : $i] : (((kpair @ (sK7 @ X7 @ X6 @ X5) @ (sK8 @ X7 @ X6 @ X5)) = X5) | ($true != (in @ X5 @ (cartprod @ X6 @ X7))) | (cartprodmempair1 != $true)) )),
% 1.65/1.34 inference(cnf_transformation,[status(thm)],[f25])).
% 1.65/1.34 thf(f25,plain,(
% 1.65/1.34 ((cartprodmempair1 = $true) | (! [X3 : $i] : (($true != (in @ X3 @ sK5)) | ! [X4 : $i] : (($true != (in @ X4 @ sK6)) | ((kpair @ X3 @ X4) != sK4))) & ($true = (in @ sK4 @ (cartprod @ sK5 @ sK6))))) & (! [X5 : $i,X6 : $i,X7 : $i] : ((((in @ (sK7 @ X7 @ X6 @ X5) @ X6) = $true) & (((in @ (sK8 @ X7 @ X6 @ X5) @ X7) = $true) & ((kpair @ (sK7 @ X7 @ X6 @ X5) @ (sK8 @ X7 @ X6 @ X5)) = X5))) | ($true != (in @ X5 @ (cartprod @ X6 @ X7)))) | (cartprodmempair1 != $true))),
% 1.65/1.34 inference(skolemisation,[status(esa),new_symbols(skolem,[sK4,sK5,sK6,sK7,sK8])],[f21,f24,f23,f22])).
% 1.65/1.34 thf(f22,plain,(
% 1.65/1.34 ? [X0 : $i,X1 : $i,X2 : $i] : (! [X3 : $i] : (((in @ X3 @ X1) != $true) | ! [X4 : $i] : (((in @ X4 @ X2) != $true) | ((kpair @ X3 @ X4) != X0))) & ($true = (in @ X0 @ (cartprod @ X1 @ X2)))) => (! [X3 : $i] : (($true != (in @ X3 @ sK5)) | ! [X4 : $i] : (($true != (in @ X4 @ sK6)) | ((kpair @ X3 @ X4) != sK4))) & ($true = (in @ sK4 @ (cartprod @ sK5 @ sK6))))),
% 1.65/1.34 introduced(definition,[],[choice_axiom])).
% 1.65/1.34 thf(f23,plain,(
% 1.65/1.34 ! [X5 : $i,X6 : $i,X7 : $i] : (? [X8 : $i] : (($true = (in @ X8 @ X6)) & ? [X9 : $i] : (((in @ X9 @ X7) = $true) & ((kpair @ X8 @ X9) = X5))) => (((in @ (sK7 @ X7 @ X6 @ X5) @ X6) = $true) & ? [X9 : $i] : (((in @ X9 @ X7) = $true) & ((kpair @ (sK7 @ X7 @ X6 @ X5) @ X9) = X5))))),
% 1.65/1.34 introduced(definition,[],[choice_axiom])).
% 1.65/1.34 thf(f24,plain,(
% 1.65/1.34 ! [X5 : $i,X6 : $i,X7 : $i] : (? [X9 : $i] : (((in @ X9 @ X7) = $true) & ((kpair @ (sK7 @ X7 @ X6 @ X5) @ X9) = X5)) => (((in @ (sK8 @ X7 @ X6 @ X5) @ X7) = $true) & ((kpair @ (sK7 @ X7 @ X6 @ X5) @ (sK8 @ X7 @ X6 @ X5)) = X5)))),
% 1.65/1.34 introduced(definition,[],[choice_axiom])).
% 1.65/1.34 thf(f21,plain,(
% 1.65/1.34 ((cartprodmempair1 = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (! [X3 : $i] : (((in @ X3 @ X1) != $true) | ! [X4 : $i] : (((in @ X4 @ X2) != $true) | ((kpair @ X3 @ X4) != X0))) & ($true = (in @ X0 @ (cartprod @ X1 @ X2))))) & (! [X5 : $i,X6 : $i,X7 : $i] : (? [X8 : $i] : (($true = (in @ X8 @ X6)) & ? [X9 : $i] : (((in @ X9 @ X7) = $true) & ((kpair @ X8 @ X9) = X5))) | ($true != (in @ X5 @ (cartprod @ X6 @ X7)))) | (cartprodmempair1 != $true))),
% 1.65/1.34 inference(rectify,[status(thm)],[f20])).
% 1.65/1.34 thf(f20,plain,(
% 1.65/1.34 ((cartprodmempair1 = $true) | ? [X0 : $i,X2 : $i,X1 : $i] : (! [X3 : $i] : (($true != (in @ X3 @ X2)) | ! [X4 : $i] : (((in @ X4 @ X1) != $true) | ((kpair @ X3 @ X4) != X0))) & ((in @ X0 @ (cartprod @ X2 @ X1)) = $true))) & (! [X0 : $i,X2 : $i,X1 : $i] : (? [X3 : $i] : (($true = (in @ X3 @ X2)) & ? [X4 : $i] : (((in @ X4 @ X1) = $true) & ((kpair @ X3 @ X4) = X0))) | ((in @ X0 @ (cartprod @ X2 @ X1)) != $true)) | (cartprodmempair1 != $true))),
% 1.65/1.34 inference(nnf_transformation,[status(thm)],[f12])).
% 1.65/1.34 thf(f12,plain,(
% 1.65/1.34 (cartprodmempair1 = $true) <=> ! [X0 : $i,X2 : $i,X1 : $i] : (? [X3 : $i] : (($true = (in @ X3 @ X2)) & ? [X4 : $i] : (((in @ X4 @ X1) = $true) & ((kpair @ X3 @ X4) = X0))) | ((in @ X0 @ (cartprod @ X2 @ X1)) != $true))),
% 1.65/1.34 inference(ennf_transformation,[status(thm)],[f7])).
% 1.65/1.34 thf(f7,plain,(
% 1.65/1.34 (cartprodmempair1 = $true) <=> ! [X0 : $i,X2 : $i,X1 : $i] : (((in @ X0 @ (cartprod @ X2 @ X1)) = $true) => ? [X3 : $i] : (($true = (in @ X3 @ X2)) & ? [X4 : $i] : (((in @ X4 @ X1) = $true) & ((kpair @ X3 @ X4) = X0))))),
% 1.65/1.34 inference(fool_elimination,[status(thm)],[f6])).
% 1.65/1.34 thf(f6,plain,(
% 1.65/1.34 (! [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)),
% 1.65/1.34 inference(rectify,[status(thm)],[f1])).
% 1.65/1.34 thf(f1,axiom,(
% 1.65/1.34 (! [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)),
% 1.65/1.34 file('/export/starexec/sandbox/tmp/tmp.RLP1xBcmXt/DTF2THF_18578.p',cartprodmempair1)).
% 1.65/1.34 thf(f50,plain,(
% 1.65/1.34 ( ! [X2 : $i,X3 : $i,X0 : $i,X1 : $i] : (((kpair @ X1 @ X2) != (kpair @ X0 @ X3)) | (X0 = X1)) )),
% 1.65/1.34 inference(trivial_inequality_removal,[status(thm)],[f41])).
% 1.65/1.34 thf(f41,plain,(
% 1.65/1.34 ( ! [X2 : $i,X3 : $i,X0 : $i,X1 : $i] : (((kpair @ X1 @ X2) != (kpair @ X0 @ X3)) | ($true != $true) | (X0 = X1)) )),
% 1.65/1.34 inference(definition_unfolding,[status(thm)],[f31,f37])).
% 1.65/1.34 thf(f37,plain,(
% 1.65/1.34 (setukpairinjL = $true)),
% 1.65/1.34 inference(cnf_transformation,[status(thm)],[f28])).
% 1.65/1.34 thf(f31,plain,(
% 1.65/1.34 ( ! [X2 : $i,X3 : $i,X0 : $i,X1 : $i] : (((kpair @ X1 @ X2) != (kpair @ X0 @ X3)) | (X0 = X1) | (setukpairinjL != $true)) )),
% 1.65/1.34 inference(cnf_transformation,[status(thm)],[f19])).
% 1.65/1.34 thf(f19,plain,(
% 1.65/1.34 (! [X0 : $i,X1 : $i,X2 : $i,X3 : $i] : (((kpair @ X1 @ X2) != (kpair @ X0 @ X3)) | (X0 = X1)) | (setukpairinjL != $true)) & ((setukpairinjL = $true) | (((kpair @ sK0 @ sK3) = (kpair @ sK1 @ sK2)) & (sK1 != sK0)))),
% 1.65/1.34 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2,sK3])],[f17,f18])).
% 1.65/1.34 thf(f18,plain,(
% 1.65/1.34 ? [X4 : $i,X5 : $i,X6 : $i,X7 : $i] : (((kpair @ X4 @ X7) = (kpair @ X5 @ X6)) & (X4 != X5)) => (((kpair @ sK0 @ sK3) = (kpair @ sK1 @ sK2)) & (sK1 != sK0))),
% 1.65/1.34 introduced(definition,[],[choice_axiom])).
% 1.65/1.34 thf(f17,plain,(
% 1.65/1.34 (! [X0 : $i,X1 : $i,X2 : $i,X3 : $i] : (((kpair @ X1 @ X2) != (kpair @ X0 @ X3)) | (X0 = X1)) | (setukpairinjL != $true)) & ((setukpairinjL = $true) | ? [X4 : $i,X5 : $i,X6 : $i,X7 : $i] : (((kpair @ X4 @ X7) = (kpair @ X5 @ X6)) & (X4 != X5)))),
% 1.65/1.34 inference(rectify,[status(thm)],[f16])).
% 1.65/1.34 thf(f16,plain,(
% 1.65/1.34 (! [X0 : $i,X1 : $i,X2 : $i,X3 : $i] : (((kpair @ X1 @ X2) != (kpair @ X0 @ X3)) | (X0 = X1)) | (setukpairinjL != $true)) & ((setukpairinjL = $true) | ? [X0 : $i,X1 : $i,X2 : $i,X3 : $i] : (((kpair @ X1 @ X2) = (kpair @ X0 @ X3)) & (X0 != X1)))),
% 1.65/1.34 inference(nnf_transformation,[status(thm)],[f15])).
% 1.65/1.34 thf(f15,plain,(
% 1.65/1.34 ! [X0 : $i,X1 : $i,X2 : $i,X3 : $i] : (((kpair @ X1 @ X2) != (kpair @ X0 @ X3)) | (X0 = X1)) <=> (setukpairinjL = $true)),
% 1.65/1.34 inference(ennf_transformation,[status(thm)],[f9])).
% 1.65/1.34 thf(f9,plain,(
% 1.65/1.34 (setukpairinjL = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i,X3 : $i] : (((kpair @ X1 @ X2) = (kpair @ X0 @ X3)) => (X0 = X1))),
% 1.65/1.34 inference(fool_elimination,[status(thm)],[f8])).
% 1.65/1.34 thf(f8,plain,(
% 1.65/1.34 (! [X0 : $i,X1 : $i,X2 : $i,X3 : $i] : (((kpair @ X1 @ X2) = (kpair @ X0 @ X3)) => (X0 = X1)) = setukpairinjL)),
% 1.65/1.34 inference(rectify,[status(thm)],[f2])).
% 1.65/1.34 thf(f2,axiom,(
% 1.65/1.34 (! [X3 : $i,X5 : $i,X2 : $i,X4 : $i] : (((kpair @ X3 @ X4) = (kpair @ X5 @ X2)) => (X3 = X5)) = setukpairinjL)),
% 1.65/1.34 file('/export/starexec/sandbox/tmp/tmp.RLP1xBcmXt/DTF2THF_18578.p',setukpairinjL)).
% 1.65/1.34 thf(f52,plain,(
% 1.65/1.34 ( ! [X6 : $i,X7 : $i,X5 : $i] : (((in @ (sK7 @ X7 @ X6 @ X5) @ X6) = $true) | ($true != (in @ X5 @ (cartprod @ X6 @ X7)))) )),
% 1.65/1.34 inference(trivial_inequality_removal,[status(thm)],[f46])).
% 1.65/1.34 thf(f46,plain,(
% 1.65/1.34 ( ! [X6 : $i,X7 : $i,X5 : $i] : (($true != (in @ X5 @ (cartprod @ X6 @ X7))) | ($true != $true) | ((in @ (sK7 @ X7 @ X6 @ X5) @ X6) = $true)) )),
% 1.65/1.34 inference(definition_unfolding,[status(thm)],[f34,f38])).
% 1.65/1.34 thf(f34,plain,(
% 1.65/1.34 ( ! [X6 : $i,X7 : $i,X5 : $i] : (((in @ (sK7 @ X7 @ X6 @ X5) @ X6) = $true) | ($true != (in @ X5 @ (cartprod @ X6 @ X7))) | (cartprodmempair1 != $true)) )),
% 1.65/1.34 inference(cnf_transformation,[status(thm)],[f25])).
% 1.65/1.34 % SZS output end Proof for DTF2THF_18578
% 1.65/1.34 % (18773)------------------------------
% 1.65/1.34 % (18773)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.65/1.34 % (18773)Termination reason: Refutation
% 1.65/1.34
% 1.65/1.34 % (18773)Memory used [KB]: 5500
% 1.65/1.34 % (18773)Time elapsed: 0.005 s
% 1.65/1.34 % (18773)Instructions burned: 9 (million)
% 1.65/1.34 % (18773)------------------------------
% 1.65/1.34 % (18773)------------------------------
% 1.65/1.34 % (18767)Success in time 0.013 s
%------------------------------------------------------------------------------