%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU660^2 : 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 : n001.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 1.63s 1.30s
% Output : Refutation 1.63s
% 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_12,type,
sK4: $i > $i > $i > $i ).
thf(func_def_13,type,
sK5: $i > $i > $i > $i ).
thf(f69,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f68,f31]) ).
thf(f31,plain,
( $true
!= ( in @ sK0 @ sK1 ) ),
inference(cnf_transformation,[status(thm)],[f18]) ).
thf(f18,plain,
( ( $true
= ( in @ ( kpair @ sK2 @ sK0 ) @ ( cartprod @ sK3 @ sK1 ) ) )
& ( $true
!= ( in @ sK0 @ sK1 ) )
& ( setukpairinjR = $true )
& ( cartprodmempair1 = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2,sK3])],[f16,f17]) ).
thf(f17,plain,
( ? [X0: $i,X1: $i,X2: $i,X3: $i] :
( ( $true
= ( in @ ( kpair @ X2 @ X0 ) @ ( cartprod @ X3 @ X1 ) ) )
& ( $true
!= ( in @ X0 @ X1 ) ) )
=> ( ( $true
= ( in @ ( kpair @ sK2 @ sK0 ) @ ( cartprod @ sK3 @ sK1 ) ) )
& ( $true
!= ( in @ sK0 @ sK1 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f16,plain,
( ? [X0: $i,X1: $i,X2: $i,X3: $i] :
( ( $true
= ( in @ ( kpair @ X2 @ X0 ) @ ( cartprod @ X3 @ X1 ) ) )
& ( $true
!= ( in @ X0 @ X1 ) ) )
& ( setukpairinjR = $true )
& ( cartprodmempair1 = $true ) ),
inference(rectify,[status(thm)],[f15]) ).
thf(f15,plain,
( ? [X1: $i,X3: $i,X2: $i,X0: $i] :
( ( $true
= ( in @ ( kpair @ X2 @ X1 ) @ ( cartprod @ X0 @ X3 ) ) )
& ( $true
!= ( in @ X1 @ X3 ) ) )
& ( setukpairinjR = $true )
& ( cartprodmempair1 = $true ) ),
inference(flattening,[status(thm)],[f14]) ).
thf(f14,plain,
( ? [X1: $i,X3: $i,X2: $i,X0: $i] :
( ( $true
= ( in @ ( kpair @ X2 @ X1 ) @ ( cartprod @ X0 @ X3 ) ) )
& ( $true
!= ( in @ X1 @ X3 ) ) )
& ( setukpairinjR = $true )
& ( cartprodmempair1 = $true ) ),
inference(ennf_transformation,[status(thm)],[f11]) ).
thf(f11,plain,
~ ( ( cartprodmempair1 = $true )
=> ( ( setukpairinjR = $true )
=> ! [X0: $i,X2: $i,X3: $i,X1: $i] :
( ( $true
= ( in @ ( kpair @ X2 @ X1 ) @ ( cartprod @ X0 @ X3 ) ) )
=> ( $true
= ( in @ X1 @ X3 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f10]) ).
thf(f10,plain,
~ ( cartprodmempair1
=> ( setukpairinjR
=> ! [X0: $i,X1: $i,X2: $i,X3: $i] :
( ( in @ ( kpair @ X2 @ X1 ) @ ( cartprod @ X0 @ X3 ) )
=> ( in @ X1 @ X3 ) ) ) ),
inference(rectify,[status(thm)],[f4]) ).
thf(f4,negated_conjecture,
~ ( cartprodmempair1
=> ( setukpairinjR
=> ! [X0: $i,X4: $i,X3: $i,X1: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X0 @ X1 ) )
=> ( in @ X4 @ X1 ) ) ) ),
inference(negated_conjecture,[status(cth)],[f3]) ).
thf(f3,conjecture,
( cartprodmempair1
=> ( setukpairinjR
=> ! [X0: $i,X4: $i,X3: $i,X1: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X0 @ X1 ) )
=> ( in @ X4 @ X1 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.3ERbz5YuW7/DTF2THF_28316.p',cartprodpairmemER) ).
thf(f68,plain,
( $true
= ( in @ sK0 @ sK1 ) ),
inference(equality_resolution,[status(thm)],[f66]) ).
thf(f66,plain,
! [X0: $i,X1: $i] :
( ( ( kpair @ sK2 @ sK0 )
!= ( kpair @ X1 @ X0 ) )
| ( $true
= ( in @ X0 @ sK1 ) ) ),
inference(subsumption_resolution,[status(thm)],[f64,f32]) ).
thf(f32,plain,
( $true
= ( in @ ( kpair @ sK2 @ sK0 ) @ ( cartprod @ sK3 @ sK1 ) ) ),
inference(cnf_transformation,[status(thm)],[f18]) ).
thf(f64,plain,
! [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ sK1 ) )
| ( ( kpair @ sK2 @ sK0 )
!= ( kpair @ X1 @ X0 ) )
| ( $true
!= ( in @ ( kpair @ sK2 @ sK0 ) @ ( cartprod @ sK3 @ sK1 ) ) ) ),
inference(superposition,[status(thm)],[f50,f60]) ).
thf(f60,plain,
! [X0: $i,X1: $i] :
( ( ( sK5 @ sK1 @ sK3 @ ( kpair @ sK2 @ sK0 ) )
= X1 )
| ( ( kpair @ X0 @ X1 )
!= ( kpair @ sK2 @ sK0 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f56]) ).
thf(f56,plain,
! [X0: $i,X1: $i] :
( ( ( sK5 @ sK1 @ sK3 @ ( kpair @ sK2 @ sK0 ) )
= X1 )
| ( ( kpair @ X0 @ X1 )
!= ( kpair @ sK2 @ sK0 ) )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f55,f32]) ).
thf(f55,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i,X4: $i] :
( ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
!= $true )
| ( ( kpair @ X3 @ X4 )
!= X2 )
| ( ( sK5 @ X0 @ X1 @ X2 )
= X4 ) ),
inference(superposition,[status(thm)],[f51,f49]) ).
thf(f49,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( ( kpair @ ( sK4 @ X2 @ X1 @ X0 ) @ ( sK5 @ X2 @ X1 @ X0 ) )
= X0 )
| ( ( in @ X0 @ ( cartprod @ X1 @ X2 ) )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f43]) ).
thf(f43,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( $true != $true )
| ( ( kpair @ ( sK4 @ X2 @ X1 @ X0 ) @ ( sK5 @ X2 @ X1 @ X0 ) )
= X0 )
| ( ( in @ X0 @ ( cartprod @ X1 @ X2 ) )
!= $true ) ),
inference(definition_unfolding,[status(thm)],[f35,f29]) ).
thf(f29,plain,
cartprodmempair1 = $true,
inference(cnf_transformation,[status(thm)],[f18]) ).
thf(f35,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( ( in @ X0 @ ( cartprod @ X1 @ X2 ) )
!= $true )
| ( ( kpair @ ( sK4 @ X2 @ X1 @ X0 ) @ ( sK5 @ X2 @ X1 @ X0 ) )
= X0 )
| ( cartprodmempair1 != $true ) ),
inference(cnf_transformation,[status(thm)],[f24]) ).
thf(f24,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( cartprod @ X1 @ X2 ) )
!= $true )
| ( ( $true
= ( in @ ( sK4 @ X2 @ X1 @ X0 ) @ X1 ) )
& ( $true
= ( in @ ( sK5 @ X2 @ X1 @ X0 ) @ X2 ) )
& ( ( kpair @ ( sK4 @ X2 @ X1 @ X0 ) @ ( sK5 @ X2 @ X1 @ X0 ) )
= X0 ) ) )
| ( cartprodmempair1 != $true ) )
& ( ( cartprodmempair1 = $true )
| ( ( $true
= ( in @ sK6 @ ( cartprod @ sK7 @ sK8 ) ) )
& ! [X8: $i] :
( ( $true
!= ( in @ X8 @ sK7 ) )
| ! [X9: $i] :
( ( ( in @ X9 @ sK8 )
!= $true )
| ( sK6
!= ( kpair @ X8 @ X9 ) ) ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK4,sK5,sK6,sK7,sK8])],[f20,f23,f22,f21]) ).
thf(f21,plain,
! [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( $true
= ( in @ X3 @ X1 ) )
& ? [X4: $i] :
( ( $true
= ( in @ X4 @ X2 ) )
& ( ( kpair @ X3 @ X4 )
= X0 ) ) )
=> ( ( $true
= ( in @ ( sK4 @ X2 @ X1 @ X0 ) @ X1 ) )
& ? [X4: $i] :
( ( $true
= ( in @ X4 @ X2 ) )
& ( ( kpair @ ( sK4 @ X2 @ X1 @ X0 ) @ X4 )
= X0 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f22,plain,
! [X0: $i,X1: $i,X2: $i] :
( ? [X4: $i] :
( ( $true
= ( in @ X4 @ X2 ) )
& ( ( kpair @ ( sK4 @ X2 @ X1 @ X0 ) @ X4 )
= X0 ) )
=> ( ( $true
= ( in @ ( sK5 @ X2 @ X1 @ X0 ) @ X2 ) )
& ( ( kpair @ ( sK4 @ X2 @ X1 @ X0 ) @ ( sK5 @ X2 @ X1 @ X0 ) )
= X0 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f23,plain,
( ? [X5: $i,X6: $i,X7: $i] :
( ( $true
= ( in @ X5 @ ( cartprod @ X6 @ X7 ) ) )
& ! [X8: $i] :
( ( ( in @ X8 @ X6 )
!= $true )
| ! [X9: $i] :
( ( ( in @ X9 @ X7 )
!= $true )
| ( ( kpair @ X8 @ X9 )
!= X5 ) ) ) )
=> ( ( $true
= ( in @ sK6 @ ( cartprod @ sK7 @ sK8 ) ) )
& ! [X8: $i] :
( ( $true
!= ( in @ X8 @ sK7 ) )
| ! [X9: $i] :
( ( ( in @ X9 @ sK8 )
!= $true )
| ( sK6
!= ( kpair @ X8 @ X9 ) ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f20,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( cartprod @ X1 @ X2 ) )
!= $true )
| ? [X3: $i] :
( ( $true
= ( in @ X3 @ X1 ) )
& ? [X4: $i] :
( ( $true
= ( in @ X4 @ X2 ) )
& ( ( kpair @ X3 @ X4 )
= X0 ) ) ) )
| ( cartprodmempair1 != $true ) )
& ( ( cartprodmempair1 = $true )
| ? [X5: $i,X6: $i,X7: $i] :
( ( $true
= ( in @ X5 @ ( cartprod @ X6 @ X7 ) ) )
& ! [X8: $i] :
( ( ( in @ X8 @ X6 )
!= $true )
| ! [X9: $i] :
( ( ( in @ X9 @ X7 )
!= $true )
| ( ( kpair @ X8 @ X9 )
!= X5 ) ) ) ) ) ),
inference(rectify,[status(thm)],[f19]) ).
thf(f19,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( cartprod @ X1 @ X2 ) )
!= $true )
| ? [X3: $i] :
( ( $true
= ( in @ X3 @ X1 ) )
& ? [X4: $i] :
( ( $true
= ( in @ X4 @ X2 ) )
& ( ( kpair @ X3 @ X4 )
= X0 ) ) ) )
| ( cartprodmempair1 != $true ) )
& ( ( cartprodmempair1 = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( cartprod @ X1 @ X2 ) )
= $true )
& ! [X3: $i] :
( ( $true
!= ( in @ X3 @ X1 ) )
| ! [X4: $i] :
( ( $true
!= ( in @ X4 @ X2 ) )
| ( ( kpair @ X3 @ X4 )
!= X0 ) ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f12]) ).
thf(f12,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( cartprod @ X1 @ X2 ) )
!= $true )
| ? [X3: $i] :
( ( $true
= ( in @ X3 @ X1 ) )
& ? [X4: $i] :
( ( $true
= ( in @ X4 @ X2 ) )
& ( ( kpair @ X3 @ X4 )
= X0 ) ) ) )
<=> ( cartprodmempair1 = $true ) ),
inference(ennf_transformation,[status(thm)],[f7]) ).
thf(f7,plain,
( ( cartprodmempair1 = $true )
<=> ! [X0: $i,X2: $i,X1: $i] :
( ( ( in @ X0 @ ( cartprod @ X1 @ X2 ) )
= $true )
=> ? [X3: $i] :
( ( $true
= ( in @ X3 @ X1 ) )
& ? [X4: $i] :
( ( $true
= ( in @ X4 @ X2 ) )
& ( ( kpair @ X3 @ X4 )
= X0 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f6]) ).
thf(f6,plain,
( cartprodmempair1
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X0 @ ( cartprod @ X1 @ X2 ) )
=> ? [X3: $i] :
( ? [X4: $i] :
( ( in @ X4 @ X2 )
& ( ( kpair @ X3 @ X4 )
= X0 ) )
& ( in @ X3 @ X1 ) ) ) ) ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
( cartprodmempair1
= ( ! [X2: $i,X0: $i,X1: $i] :
( ( in @ X2 @ ( cartprod @ X0 @ X1 ) )
=> ? [X3: $i] :
( ? [X4: $i] :
( ( in @ X4 @ X1 )
& ( ( kpair @ X3 @ X4 )
= X2 ) )
& ( in @ X3 @ X0 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.3ERbz5YuW7/DTF2THF_28316.p',cartprodmempair1) ).
thf(f51,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i] :
( ( ( kpair @ X0 @ X1 )
!= ( kpair @ X3 @ X2 ) )
| ( X1 = X2 ) ),
inference(trivial_inequality_removal,[status(thm)],[f46]) ).
thf(f46,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i] :
( ( X1 = X2 )
| ( $true != $true )
| ( ( kpair @ X0 @ X1 )
!= ( kpair @ X3 @ X2 ) ) ),
inference(definition_unfolding,[status(thm)],[f40,f30]) ).
thf(f30,plain,
setukpairinjR = $true,
inference(cnf_transformation,[status(thm)],[f18]) ).
thf(f40,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i] :
( ( X1 = X2 )
| ( ( kpair @ X0 @ X1 )
!= ( kpair @ X3 @ X2 ) )
| ( setukpairinjR != $true ) ),
inference(cnf_transformation,[status(thm)],[f28]) ).
thf(f28,plain,
( ( ! [X0: $i,X1: $i,X2: $i,X3: $i] :
( ( X1 = X2 )
| ( ( kpair @ X0 @ X1 )
!= ( kpair @ X3 @ X2 ) ) )
| ( setukpairinjR != $true ) )
& ( ( setukpairinjR = $true )
| ( ( sK10 != sK11 )
& ( ( kpair @ sK9 @ sK10 )
= ( kpair @ sK12 @ sK11 ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK9,sK10,sK11,sK12])],[f26,f27]) ).
thf(f27,plain,
( ? [X4: $i,X5: $i,X6: $i,X7: $i] :
( ( X5 != X6 )
& ( ( kpair @ X7 @ X6 )
= ( kpair @ X4 @ X5 ) ) )
=> ( ( sK10 != sK11 )
& ( ( kpair @ sK9 @ sK10 )
= ( kpair @ sK12 @ sK11 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f26,plain,
( ( ! [X0: $i,X1: $i,X2: $i,X3: $i] :
( ( X1 = X2 )
| ( ( kpair @ X0 @ X1 )
!= ( kpair @ X3 @ X2 ) ) )
| ( setukpairinjR != $true ) )
& ( ( setukpairinjR = $true )
| ? [X4: $i,X5: $i,X6: $i,X7: $i] :
( ( X5 != X6 )
& ( ( kpair @ X7 @ X6 )
= ( kpair @ X4 @ X5 ) ) ) ) ),
inference(rectify,[status(thm)],[f25]) ).
thf(f25,plain,
( ( ! [X3: $i,X0: $i,X2: $i,X1: $i] :
( ( X0 = X2 )
| ( ( kpair @ X1 @ X2 )
!= ( kpair @ X3 @ X0 ) ) )
| ( setukpairinjR != $true ) )
& ( ( setukpairinjR = $true )
| ? [X3: $i,X0: $i,X2: $i,X1: $i] :
( ( X0 != X2 )
& ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X0 ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f13]) ).
thf(f13,plain,
( ! [X3: $i,X0: $i,X2: $i,X1: $i] :
( ( X0 = X2 )
| ( ( kpair @ X1 @ X2 )
!= ( kpair @ X3 @ X0 ) ) )
<=> ( setukpairinjR = $true ) ),
inference(ennf_transformation,[status(thm)],[f9]) ).
thf(f9,plain,
( ( setukpairinjR = $true )
<=> ! [X0: $i,X2: $i,X1: $i,X3: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X0 ) )
=> ( X0 = X2 ) ) ),
inference(fool_elimination,[status(thm)],[f8]) ).
thf(f8,plain,
( setukpairinjR
= ( ! [X0: $i,X1: $i,X2: $i,X3: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X0 ) )
=> ( X0 = X2 ) ) ) ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
( setukpairinjR
= ( ! [X4: $i,X5: $i,X2: $i,X3: $i] :
( ( ( kpair @ X3 @ X4 )
= ( kpair @ X5 @ X2 ) )
=> ( X2 = X4 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.3ERbz5YuW7/DTF2THF_28316.p',setukpairinjR) ).
thf(f50,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( $true
= ( in @ ( sK5 @ X2 @ X1 @ X0 ) @ X2 ) )
| ( ( in @ X0 @ ( cartprod @ X1 @ X2 ) )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f42]) ).
thf(f42,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( ( in @ X0 @ ( cartprod @ X1 @ X2 ) )
!= $true )
| ( $true != $true )
| ( $true
= ( in @ ( sK5 @ X2 @ X1 @ X0 ) @ X2 ) ) ),
inference(definition_unfolding,[status(thm)],[f36,f29]) ).
thf(f36,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( ( in @ X0 @ ( cartprod @ X1 @ X2 ) )
!= $true )
| ( $true
= ( in @ ( sK5 @ X2 @ X1 @ X0 ) @ X2 ) )
| ( cartprodmempair1 != $true ) ),
inference(cnf_transformation,[status(thm)],[f24]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU660^2 : TPTP v9.2.1. Released v3.7.0.
% 0.13/0.14 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.35 % Computer : n001.cluster.edu
% 0.17/0.35 % Model : x86_64 x86_64
% 0.17/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.35 % Memory : 8042.1875MB
% 0.17/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.35 % CPULimit : 300
% 0.17/0.35 % WCLimit : 300
% 0.17/0.35 % DateTime : Tue Feb 24 12:30:35 EST 2026
% 0.17/0.35 % CPUTime :
% 0.17/0.35 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.32/0.47 ---- Original DTF file ---
% 0.32/0.48 thf(spec,logic,$$dhol).
% 0.32/0.48 %------------------------------------------------------------------------------
% 0.32/0.48 % File : SEU660^2 : TPTP v9.2.1. Released v3.7.0.
% 0.32/0.48 % Domain : Set Theory
% 0.32/0.48 % Problem : Ordered Pairs - Properties of Pairs
% 0.32/0.48 % Version : Especial > Reduced > Especial.
% 0.32/0.48 % English : (! A:i.! B:i.! x:i.! y:i.in (kpair x y) (cartprod A B) -> in y B)
% 0.32/0.48
% 0.32/0.48 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.32/0.48 % Source : [Bro08]
% 0.32/0.48 % Names : ZFC162l [Bro08]
% 0.32/0.48
% 0.32/0.48 % Status : Theorem
% 0.32/0.48 % 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
% 0.32/0.48 % Syntax : Number of formulae : 8 ( 2 unt; 5 typ; 2 def)
% 0.32/0.48 % Number of atoms : 14 ( 5 equ; 0 cnn)
% 0.32/0.48 % Maximal formula atoms : 4 ( 4 avg)
% 0.32/0.48 % Number of connectives : 29 ( 0 ~; 0 |; 2 &; 22 @)
% 0.32/0.48 % ( 0 <=>; 5 =>; 0 <=; 0 <~>)
% 0.32/0.48 % Maximal formula depth : 12 ( 5 avg)
% 0.32/0.48 % Number of types : 2 ( 0 usr)
% 0.32/0.48 % Number of type conns : 6 ( 6 >; 0 *; 0 +; 0 <<)
% 0.32/0.48 % Number of symbols : 6 ( 5 usr; 2 con; 0-2 aty)
% 0.32/0.48 % Number of variables : 13 ( 0 ^; 11 !; 2 ?; 13 :)
% 0.32/0.48 % SPC : TH0_THM_EQU_NAR
% 0.32/0.48
% 0.32/0.48 % Comments : http://mathgate.info/detsetitem.php?id=338
% 0.32/0.48 %------------------------------------------------------------------------------
% 0.32/0.48 thf(in_type,type,
% 0.32/0.48 in: $i > $i > $o ).
% 0.32/0.48
% 0.32/0.48 thf(kpair_type,type,
% 0.32/0.48 kpair: $i > $i > $i ).
% 0.32/0.48
% 0.32/0.48 thf(cartprod_type,type,
% 0.32/0.48 cartprod: $i > $i > $i ).
% 0.32/0.48
% 0.32/0.48 thf(cartprodmempair1_type,type,
% 0.32/0.48 cartprodmempair1: $o ).
% 0.32/0.48
% 0.32/0.48 thf(cartprodmempair1,definition,
% 0.32/0.48 ( cartprodmempair1
% 0.32/0.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.32/0.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.32/0.48 => ? [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 & ? [Xy: $i] :
% 0.32/0.48 ( ( in @ Xy @ B )
% 0.32/0.48 & ( Xu
% 0.32/0.48 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(setukpairinjR_type,type,
% 0.32/0.48 setukpairinjR: $o ).
% 0.32/0.48
% 0.32/0.48 thf(setukpairinjR,definition,
% 0.32/0.48 ( setukpairinjR
% 0.32/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.32/0.48 ( ( ( kpair @ Xx @ Xy )
% 0.32/0.48 = ( kpair @ Xz @ Xu ) )
% 0.32/0.48 => ( Xy = Xu ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(cartprodpairmemER,conjecture,
% 0.32/0.48 ( cartprodmempair1
% 0.32/0.48 => ( setukpairinjR
% 0.32/0.48 => ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.32/0.48 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.32/0.48 => ( in @ Xy @ B ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 %------------------------------------------------------------------------------
% 0.32/0.48 ------------------------
% 1.47/1.27 ---- Embedded in THF ---
% 1.47/1.27 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.47/1.27 %%% Generated on Tue Feb 24 12:30:36 EST 2026
% 1.47/1.27 %%% using '$$dhol' embedding, version 1.3.0.
% 1.47/1.27 %%% Logic specification used:
% 1.47/1.27 %%% thf(spec, logic, $$dhol).
% 1.47/1.27
% 1.47/1.27 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.3ERbz5YuW7/DTF2DTF28316.p
% See solution above
% 1.47/1.27 ------------------------
% 1.47/1.28 ---- Cleaned THF ---
% 1.47/1.28 thf(in_type,type,
% 1.47/1.28 in: $i > $i > $o ).
% 1.47/1.28
% 1.47/1.28 thf(kpair_type,type,
% 1.47/1.28 kpair: $i > $i > $i ).
% 1.47/1.28
% 1.47/1.28 thf(cartprod_type,type,
% 1.47/1.28 cartprod: $i > $i > $i ).
% 1.47/1.28
% 1.47/1.28 thf(cartprodmempair1_type,type,
% 1.47/1.28 cartprodmempair1: $o ).
% 1.47/1.28
% 1.47/1.28 thf(cartprodmempair1,definition,
% 1.47/1.28 ( cartprodmempair1
% 1.47/1.28 = ( ! [A: $i,B: $i,Xu: $i] :
% 1.47/1.28 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 1.47/1.28 => ? [Xx: $i] :
% 1.47/1.28 ( ( in @ Xx @ A )
% 1.47/1.28 & ? [Xy: $i] :
% 1.47/1.28 ( ( in @ Xy @ B )
% 1.47/1.28 & ( Xu
% 1.47/1.28 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 1.47/1.28
% 1.47/1.28 thf(setukpairinjR_type,type,
% 1.47/1.28 setukpairinjR: $o ).
% 1.47/1.28
% 1.47/1.28 thf(setukpairinjR,definition,
% 1.47/1.28 ( setukpairinjR
% 1.47/1.28 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 1.47/1.28 ( ( ( kpair @ Xx @ Xy )
% 1.47/1.28 = ( kpair @ Xz @ Xu ) )
% 1.47/1.28 => ( Xy = Xu ) ) ) ) ).
% 1.47/1.28
% 1.47/1.28 thf(cartprodpairmemER,conjecture,
% 1.47/1.28 ( cartprodmempair1
% 1.47/1.28 => ( setukpairinjR
% 1.47/1.28 => ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 1.47/1.28 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 1.47/1.28 => ( in @ Xy @ B ) ) ) ) ).
% 1.47/1.28 ------------------------
% 1.63/1.30 % (28653)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_28316 for (2999ds/275Mi)
% 1.63/1.30 % (28653)First to succeed.
% 1.63/1.30 % (28653)Refutation found. Thanks to Tanya!
% 1.63/1.30 % SZS status Theorem for DTF2THF_28316
% 1.63/1.30 % SZS output start Proof for DTF2THF_28316
% 1.63/1.30 thf(func_def_0, type, in: $i > $i > $o).
% 1.63/1.30 thf(func_def_1, type, kpair: $i > $i > $i).
% 1.63/1.30 thf(func_def_2, type, cartprod: $i > $i > $i).
% 1.63/1.30 thf(func_def_12, type, sK4: $i > $i > $i > $i).
% 1.63/1.30 thf(func_def_13, type, sK5: $i > $i > $i > $i).
% 1.63/1.30 thf(f69,plain,(
% 1.63/1.30 $false),
% 1.63/1.30 inference(subsumption_resolution,[status(thm)],[f68,f31])).
% 1.63/1.30 thf(f31,plain,(
% 1.63/1.30 ($true != (in @ sK0 @ sK1))),
% 1.63/1.30 inference(cnf_transformation,[status(thm)],[f18])).
% 1.63/1.30 thf(f18,plain,(
% 1.63/1.30 (($true = (in @ (kpair @ sK2 @ sK0) @ (cartprod @ sK3 @ sK1))) & ($true != (in @ sK0 @ sK1))) & (setukpairinjR = $true) & (cartprodmempair1 = $true)),
% 1.63/1.30 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2,sK3])],[f16,f17])).
% 1.63/1.30 thf(f17,plain,(
% 1.63/1.30 ? [X0 : $i,X1 : $i,X2 : $i,X3 : $i] : (($true = (in @ (kpair @ X2 @ X0) @ (cartprod @ X3 @ X1))) & ($true != (in @ X0 @ X1))) => (($true = (in @ (kpair @ sK2 @ sK0) @ (cartprod @ sK3 @ sK1))) & ($true != (in @ sK0 @ sK1)))),
% 1.63/1.30 introduced(definition,[],[choice_axiom])).
% 1.63/1.30 thf(f16,plain,(
% 1.63/1.30 ? [X0 : $i,X1 : $i,X2 : $i,X3 : $i] : (($true = (in @ (kpair @ X2 @ X0) @ (cartprod @ X3 @ X1))) & ($true != (in @ X0 @ X1))) & (setukpairinjR = $true) & (cartprodmempair1 = $true)),
% 1.63/1.30 inference(rectify,[status(thm)],[f15])).
% 1.63/1.30 thf(f15,plain,(
% 1.63/1.30 ? [X1 : $i,X3 : $i,X2 : $i,X0 : $i] : (($true = (in @ (kpair @ X2 @ X1) @ (cartprod @ X0 @ X3))) & ($true != (in @ X1 @ X3))) & (setukpairinjR = $true) & (cartprodmempair1 = $true)),
% 1.63/1.30 inference(flattening,[status(thm)],[f14])).
% 1.63/1.30 thf(f14,plain,(
% 1.63/1.30 (? [X1 : $i,X3 : $i,X2 : $i,X0 : $i] : (($true = (in @ (kpair @ X2 @ X1) @ (cartprod @ X0 @ X3))) & ($true != (in @ X1 @ X3))) & (setukpairinjR = $true)) & (cartprodmempair1 = $true)),
% 1.63/1.30 inference(ennf_transformation,[status(thm)],[f11])).
% 1.63/1.30 thf(f11,plain,(
% 1.63/1.30 ~((cartprodmempair1 = $true) => ((setukpairinjR = $true) => ! [X0 : $i,X2 : $i,X3 : $i,X1 : $i] : (($true = (in @ (kpair @ X2 @ X1) @ (cartprod @ X0 @ X3))) => ($true = (in @ X1 @ X3)))))),
% 1.63/1.30 inference(fool_elimination,[status(thm)],[f10])).
% 1.63/1.30 thf(f10,plain,(
% 1.63/1.30 ~(cartprodmempair1 => (setukpairinjR => ! [X0 : $i,X1 : $i,X2 : $i,X3 : $i] : ((in @ (kpair @ X2 @ X1) @ (cartprod @ X0 @ X3)) => (in @ X1 @ X3))))),
% 1.63/1.30 inference(rectify,[status(thm)],[f4])).
% 1.63/1.30 thf(f4,negated_conjecture,(
% 1.63/1.30 ~(cartprodmempair1 => (setukpairinjR => ! [X0 : $i,X4 : $i,X3 : $i,X1 : $i] : ((in @ (kpair @ X3 @ X4) @ (cartprod @ X0 @ X1)) => (in @ X4 @ X1))))),
% 1.63/1.30 inference(negated_conjecture,[status(cth)],[f3])).
% 1.63/1.30 thf(f3,conjecture,(
% 1.63/1.30 cartprodmempair1 => (setukpairinjR => ! [X0 : $i,X4 : $i,X3 : $i,X1 : $i] : ((in @ (kpair @ X3 @ X4) @ (cartprod @ X0 @ X1)) => (in @ X4 @ X1)))),
% 1.63/1.30 file('/export/starexec/sandbox/tmp/tmp.3ERbz5YuW7/DTF2THF_28316.p',cartprodpairmemER)).
% 1.63/1.30 thf(f68,plain,(
% 1.63/1.30 ($true = (in @ sK0 @ sK1))),
% 1.63/1.30 inference(equality_resolution,[status(thm)],[f66])).
% 1.63/1.30 thf(f66,plain,(
% 1.63/1.30 ( ! [X0 : $i,X1 : $i] : (((kpair @ sK2 @ sK0) != (kpair @ X1 @ X0)) | ($true = (in @ X0 @ sK1))) )),
% 1.63/1.30 inference(subsumption_resolution,[status(thm)],[f64,f32])).
% 1.63/1.30 thf(f32,plain,(
% 1.63/1.30 ($true = (in @ (kpair @ sK2 @ sK0) @ (cartprod @ sK3 @ sK1)))),
% 1.63/1.30 inference(cnf_transformation,[status(thm)],[f18])).
% 1.63/1.30 thf(f64,plain,(
% 1.63/1.30 ( ! [X0 : $i,X1 : $i] : (($true = (in @ X0 @ sK1)) | ((kpair @ sK2 @ sK0) != (kpair @ X1 @ X0)) | ($true != (in @ (kpair @ sK2 @ sK0) @ (cartprod @ sK3 @ sK1)))) )),
% 1.63/1.30 inference(superposition,[status(thm)],[f50,f60])).
% 1.63/1.30 thf(f60,plain,(
% 1.63/1.30 ( ! [X0 : $i,X1 : $i] : (((sK5 @ sK1 @ sK3 @ (kpair @ sK2 @ sK0)) = X1) | ((kpair @ X0 @ X1) != (kpair @ sK2 @ sK0))) )),
% 1.63/1.30 inference(trivial_inequality_removal,[status(thm)],[f56])).
% 1.63/1.30 thf(f56,plain,(
% 1.63/1.30 ( ! [X0 : $i,X1 : $i] : (((sK5 @ sK1 @ sK3 @ (kpair @ sK2 @ sK0)) = X1) | ((kpair @ X0 @ X1) != (kpair @ sK2 @ sK0)) | ($true != $true)) )),
% 1.63/1.30 inference(superposition,[status(thm)],[f55,f32])).
% 1.63/1.30 thf(f55,plain,(
% 1.63/1.30 ( ! [X2 : $i,X3 : $i,X0 : $i,X1 : $i,X4 : $i] : (((in @ X2 @ (cartprod @ X1 @ X0)) != $true) | ((kpair @ X3 @ X4) != X2) | ((sK5 @ X0 @ X1 @ X2) = X4)) )),
% 1.63/1.30 inference(superposition,[status(thm)],[f51,f49])).
% 1.63/1.30 thf(f49,plain,(
% 1.63/1.30 ( ! [X2 : $i,X0 : $i,X1 : $i] : (((kpair @ (sK4 @ X2 @ X1 @ X0) @ (sK5 @ X2 @ X1 @ X0)) = X0) | ((in @ X0 @ (cartprod @ X1 @ X2)) != $true)) )),
% 1.63/1.30 inference(trivial_inequality_removal,[status(thm)],[f43])).
% 1.63/1.30 thf(f43,plain,(
% 1.63/1.30 ( ! [X2 : $i,X0 : $i,X1 : $i] : (($true != $true) | ((kpair @ (sK4 @ X2 @ X1 @ X0) @ (sK5 @ X2 @ X1 @ X0)) = X0) | ((in @ X0 @ (cartprod @ X1 @ X2)) != $true)) )),
% 1.63/1.30 inference(definition_unfolding,[status(thm)],[f35,f29])).
% 1.63/1.30 thf(f29,plain,(
% 1.63/1.30 (cartprodmempair1 = $true)),
% 1.63/1.30 inference(cnf_transformation,[status(thm)],[f18])).
% 1.63/1.30 thf(f35,plain,(
% 1.63/1.30 ( ! [X2 : $i,X0 : $i,X1 : $i] : (((in @ X0 @ (cartprod @ X1 @ X2)) != $true) | ((kpair @ (sK4 @ X2 @ X1 @ X0) @ (sK5 @ X2 @ X1 @ X0)) = X0) | (cartprodmempair1 != $true)) )),
% 1.63/1.30 inference(cnf_transformation,[status(thm)],[f24])).
% 1.63/1.30 thf(f24,plain,(
% 1.63/1.30 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ (cartprod @ X1 @ X2)) != $true) | (($true = (in @ (sK4 @ X2 @ X1 @ X0) @ X1)) & (($true = (in @ (sK5 @ X2 @ X1 @ X0) @ X2)) & ((kpair @ (sK4 @ X2 @ X1 @ X0) @ (sK5 @ X2 @ X1 @ X0)) = X0)))) | (cartprodmempair1 != $true)) & ((cartprodmempair1 = $true) | (($true = (in @ sK6 @ (cartprod @ sK7 @ sK8))) & ! [X8 : $i] : (($true != (in @ X8 @ sK7)) | ! [X9 : $i] : (((in @ X9 @ sK8) != $true) | (sK6 != (kpair @ X8 @ X9))))))),
% 1.63/1.30 inference(skolemisation,[status(esa),new_symbols(skolem,[sK4,sK5,sK6,sK7,sK8])],[f20,f23,f22,f21])).
% 1.63/1.30 thf(f21,plain,(
% 1.63/1.30 ! [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : (($true = (in @ X3 @ X1)) & ? [X4 : $i] : (($true = (in @ X4 @ X2)) & ((kpair @ X3 @ X4) = X0))) => (($true = (in @ (sK4 @ X2 @ X1 @ X0) @ X1)) & ? [X4 : $i] : (($true = (in @ X4 @ X2)) & ((kpair @ (sK4 @ X2 @ X1 @ X0) @ X4) = X0))))),
% 1.63/1.30 introduced(definition,[],[choice_axiom])).
% 1.63/1.30 thf(f22,plain,(
% 1.63/1.30 ! [X0 : $i,X1 : $i,X2 : $i] : (? [X4 : $i] : (($true = (in @ X4 @ X2)) & ((kpair @ (sK4 @ X2 @ X1 @ X0) @ X4) = X0)) => (($true = (in @ (sK5 @ X2 @ X1 @ X0) @ X2)) & ((kpair @ (sK4 @ X2 @ X1 @ X0) @ (sK5 @ X2 @ X1 @ X0)) = X0)))),
% 1.63/1.30 introduced(definition,[],[choice_axiom])).
% 1.63/1.30 thf(f23,plain,(
% 1.63/1.30 ? [X5 : $i,X6 : $i,X7 : $i] : (($true = (in @ X5 @ (cartprod @ X6 @ X7))) & ! [X8 : $i] : (((in @ X8 @ X6) != $true) | ! [X9 : $i] : (((in @ X9 @ X7) != $true) | ((kpair @ X8 @ X9) != X5)))) => (($true = (in @ sK6 @ (cartprod @ sK7 @ sK8))) & ! [X8 : $i] : (($true != (in @ X8 @ sK7)) | ! [X9 : $i] : (((in @ X9 @ sK8) != $true) | (sK6 != (kpair @ X8 @ X9)))))),
% 1.63/1.30 introduced(definition,[],[choice_axiom])).
% 1.63/1.30 thf(f20,plain,(
% 1.63/1.30 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ (cartprod @ X1 @ X2)) != $true) | ? [X3 : $i] : (($true = (in @ X3 @ X1)) & ? [X4 : $i] : (($true = (in @ X4 @ X2)) & ((kpair @ X3 @ X4) = X0)))) | (cartprodmempair1 != $true)) & ((cartprodmempair1 = $true) | ? [X5 : $i,X6 : $i,X7 : $i] : (($true = (in @ X5 @ (cartprod @ X6 @ X7))) & ! [X8 : $i] : (((in @ X8 @ X6) != $true) | ! [X9 : $i] : (((in @ X9 @ X7) != $true) | ((kpair @ X8 @ X9) != X5)))))),
% 1.63/1.30 inference(rectify,[status(thm)],[f19])).
% 1.63/1.30 thf(f19,plain,(
% 1.63/1.30 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ (cartprod @ X1 @ X2)) != $true) | ? [X3 : $i] : (($true = (in @ X3 @ X1)) & ? [X4 : $i] : (($true = (in @ X4 @ X2)) & ((kpair @ X3 @ X4) = X0)))) | (cartprodmempair1 != $true)) & ((cartprodmempair1 = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ (cartprod @ X1 @ X2)) = $true) & ! [X3 : $i] : (($true != (in @ X3 @ X1)) | ! [X4 : $i] : (($true != (in @ X4 @ X2)) | ((kpair @ X3 @ X4) != X0)))))),
% 1.63/1.30 inference(nnf_transformation,[status(thm)],[f12])).
% 1.63/1.30 thf(f12,plain,(
% 1.63/1.30 ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ (cartprod @ X1 @ X2)) != $true) | ? [X3 : $i] : (($true = (in @ X3 @ X1)) & ? [X4 : $i] : (($true = (in @ X4 @ X2)) & ((kpair @ X3 @ X4) = X0)))) <=> (cartprodmempair1 = $true)),
% 1.63/1.30 inference(ennf_transformation,[status(thm)],[f7])).
% 1.63/1.30 thf(f7,plain,(
% 1.63/1.30 (cartprodmempair1 = $true) <=> ! [X0 : $i,X2 : $i,X1 : $i] : (((in @ X0 @ (cartprod @ X1 @ X2)) = $true) => ? [X3 : $i] : (($true = (in @ X3 @ X1)) & ? [X4 : $i] : (($true = (in @ X4 @ X2)) & ((kpair @ X3 @ X4) = X0))))),
% 1.63/1.30 inference(fool_elimination,[status(thm)],[f6])).
% 1.63/1.30 thf(f6,plain,(
% 1.63/1.30 (cartprodmempair1 = ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X0 @ (cartprod @ X1 @ X2)) => ? [X3 : $i] : (? [X4 : $i] : ((in @ X4 @ X2) & ((kpair @ X3 @ X4) = X0)) & (in @ X3 @ X1))))),
% 1.63/1.30 inference(rectify,[status(thm)],[f1])).
% 1.63/1.30 thf(f1,axiom,(
% 1.63/1.30 (cartprodmempair1 = ! [X2 : $i,X0 : $i,X1 : $i] : ((in @ X2 @ (cartprod @ X0 @ X1)) => ? [X3 : $i] : (? [X4 : $i] : ((in @ X4 @ X1) & ((kpair @ X3 @ X4) = X2)) & (in @ X3 @ X0))))),
% 1.63/1.30 file('/export/starexec/sandbox/tmp/tmp.3ERbz5YuW7/DTF2THF_28316.p',cartprodmempair1)).
% 1.63/1.30 thf(f51,plain,(
% 1.63/1.30 ( ! [X2 : $i,X3 : $i,X0 : $i,X1 : $i] : (((kpair @ X0 @ X1) != (kpair @ X3 @ X2)) | (X1 = X2)) )),
% 1.63/1.30 inference(trivial_inequality_removal,[status(thm)],[f46])).
% 1.63/1.30 thf(f46,plain,(
% 1.63/1.30 ( ! [X2 : $i,X3 : $i,X0 : $i,X1 : $i] : ((X1 = X2) | ($true != $true) | ((kpair @ X0 @ X1) != (kpair @ X3 @ X2))) )),
% 1.63/1.30 inference(definition_unfolding,[status(thm)],[f40,f30])).
% 1.63/1.30 thf(f30,plain,(
% 1.63/1.30 (setukpairinjR = $true)),
% 1.63/1.30 inference(cnf_transformation,[status(thm)],[f18])).
% 1.63/1.30 thf(f40,plain,(
% 1.63/1.30 ( ! [X2 : $i,X3 : $i,X0 : $i,X1 : $i] : ((X1 = X2) | ((kpair @ X0 @ X1) != (kpair @ X3 @ X2)) | (setukpairinjR != $true)) )),
% 1.63/1.30 inference(cnf_transformation,[status(thm)],[f28])).
% 1.63/1.30 thf(f28,plain,(
% 1.63/1.30 (! [X0 : $i,X1 : $i,X2 : $i,X3 : $i] : ((X1 = X2) | ((kpair @ X0 @ X1) != (kpair @ X3 @ X2))) | (setukpairinjR != $true)) & ((setukpairinjR = $true) | ((sK10 != sK11) & ((kpair @ sK9 @ sK10) = (kpair @ sK12 @ sK11))))),
% 1.63/1.30 inference(skolemisation,[status(esa),new_symbols(skolem,[sK9,sK10,sK11,sK12])],[f26,f27])).
% 1.63/1.30 thf(f27,plain,(
% 1.63/1.30 ? [X4 : $i,X5 : $i,X6 : $i,X7 : $i] : ((X5 != X6) & ((kpair @ X7 @ X6) = (kpair @ X4 @ X5))) => ((sK10 != sK11) & ((kpair @ sK9 @ sK10) = (kpair @ sK12 @ sK11)))),
% 1.63/1.30 introduced(definition,[],[choice_axiom])).
% 1.63/1.30 thf(f26,plain,(
% 1.63/1.30 (! [X0 : $i,X1 : $i,X2 : $i,X3 : $i] : ((X1 = X2) | ((kpair @ X0 @ X1) != (kpair @ X3 @ X2))) | (setukpairinjR != $true)) & ((setukpairinjR = $true) | ? [X4 : $i,X5 : $i,X6 : $i,X7 : $i] : ((X5 != X6) & ((kpair @ X7 @ X6) = (kpair @ X4 @ X5))))),
% 1.63/1.30 inference(rectify,[status(thm)],[f25])).
% 1.63/1.30 thf(f25,plain,(
% 1.63/1.30 (! [X3 : $i,X0 : $i,X2 : $i,X1 : $i] : ((X0 = X2) | ((kpair @ X1 @ X2) != (kpair @ X3 @ X0))) | (setukpairinjR != $true)) & ((setukpairinjR = $true) | ? [X3 : $i,X0 : $i,X2 : $i,X1 : $i] : ((X0 != X2) & ((kpair @ X1 @ X2) = (kpair @ X3 @ X0))))),
% 1.63/1.30 inference(nnf_transformation,[status(thm)],[f13])).
% 1.63/1.30 thf(f13,plain,(
% 1.63/1.30 ! [X3 : $i,X0 : $i,X2 : $i,X1 : $i] : ((X0 = X2) | ((kpair @ X1 @ X2) != (kpair @ X3 @ X0))) <=> (setukpairinjR = $true)),
% 1.63/1.30 inference(ennf_transformation,[status(thm)],[f9])).
% 1.63/1.30 thf(f9,plain,(
% 1.63/1.30 (setukpairinjR = $true) <=> ! [X0 : $i,X2 : $i,X1 : $i,X3 : $i] : (((kpair @ X1 @ X2) = (kpair @ X3 @ X0)) => (X0 = X2))),
% 1.63/1.30 inference(fool_elimination,[status(thm)],[f8])).
% 1.63/1.30 thf(f8,plain,(
% 1.63/1.30 (setukpairinjR = ! [X0 : $i,X1 : $i,X2 : $i,X3 : $i] : (((kpair @ X1 @ X2) = (kpair @ X3 @ X0)) => (X0 = X2)))),
% 1.63/1.30 inference(rectify,[status(thm)],[f2])).
% 1.63/1.30 thf(f2,axiom,(
% 1.63/1.30 (setukpairinjR = ! [X4 : $i,X5 : $i,X2 : $i,X3 : $i] : (((kpair @ X3 @ X4) = (kpair @ X5 @ X2)) => (X2 = X4)))),
% 1.63/1.30 file('/export/starexec/sandbox/tmp/tmp.3ERbz5YuW7/DTF2THF_28316.p',setukpairinjR)).
% 1.63/1.30 thf(f50,plain,(
% 1.63/1.30 ( ! [X2 : $i,X0 : $i,X1 : $i] : (($true = (in @ (sK5 @ X2 @ X1 @ X0) @ X2)) | ((in @ X0 @ (cartprod @ X1 @ X2)) != $true)) )),
% 1.63/1.30 inference(trivial_inequality_removal,[status(thm)],[f42])).
% 1.63/1.30 thf(f42,plain,(
% 1.63/1.30 ( ! [X2 : $i,X0 : $i,X1 : $i] : (((in @ X0 @ (cartprod @ X1 @ X2)) != $true) | ($true != $true) | ($true = (in @ (sK5 @ X2 @ X1 @ X0) @ X2))) )),
% 1.63/1.30 inference(definition_unfolding,[status(thm)],[f36,f29])).
% 1.63/1.30 thf(f36,plain,(
% 1.63/1.30 ( ! [X2 : $i,X0 : $i,X1 : $i] : (((in @ X0 @ (cartprod @ X1 @ X2)) != $true) | ($true = (in @ (sK5 @ X2 @ X1 @ X0) @ X2)) | (cartprodmempair1 != $true)) )),
% 1.63/1.30 inference(cnf_transformation,[status(thm)],[f24])).
% 1.63/1.30 % SZS output end Proof for DTF2THF_28316
% 1.63/1.30 % (28653)------------------------------
% 1.63/1.30 % (28653)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.63/1.30 % (28653)Termination reason: Refutation
% 1.63/1.30
% 1.63/1.30 % (28653)Memory used [KB]: 5500
% 1.63/1.30 % (28653)Time elapsed: 0.005 s
% 1.63/1.30 % (28653)Instructions burned: 8 (million)
% 1.63/1.30 % (28653)------------------------------
% 1.63/1.30 % (28653)------------------------------
% 1.63/1.30 % (28647)Success in time 0.015 s
%------------------------------------------------------------------------------