%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU632^2 : 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 : n005.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 09:35:47 AM UTC 2026
% Result : Theorem 1.61s 1.26s
% Output : Refutation 1.61s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 8
% Syntax : Number of formulae : 39 ( 11 unt; 0 typ; 0 def)
% Number of atoms : 375 ( 122 equ; 0 cnn)
% Maximal formula atoms : 10 ( 9 avg)
% Number of connectives : 453 ( 48 ~; 35 |; 42 &; 305 @)
% ( 3 <=>; 20 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 6 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 21 ( 18 usr; 12 con; 0-3 aty)
% Number of variables : 119 ( 0 ^; 79 !; 40 ?; 119 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_2,type,
setadjoin: $i > $i > $i ).
thf(func_def_3,type,
setunion: $i > $i ).
thf(func_def_4,type,
iskpair: $i > $o ).
thf(func_def_6,type,
kpair: $i > $i > $i ).
thf(func_def_8,type,
cartprod: $i > $i > $i ).
thf(func_def_27,type,
sK8: $i > $i > $i > $i ).
thf(func_def_28,type,
sK9: $i > $i > $i > $i ).
thf(f62,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f60,f38]) ).
thf(f38,plain,
( ( iskpair @ sK4 )
!= $true ),
inference(cnf_transformation,[status(thm)],[f26]) ).
thf(f26,plain,
( ( ( iskpair @ sK4 )
!= $true )
& ( ( in @ sK4 @ ( cartprod @ sK3 @ sK2 ) )
= $true )
& ( kpairp = $true )
& ( cartprodmempair1 = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK2,sK3,sK4])],[f20,f25]) ).
thf(f25,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( iskpair @ X2 )
!= $true )
& ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
= $true ) )
=> ( ( ( iskpair @ sK4 )
!= $true )
& ( ( in @ sK4 @ ( cartprod @ sK3 @ sK2 ) )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f20,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( iskpair @ X2 )
!= $true )
& ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
= $true ) )
& ( kpairp = $true )
& ( cartprodmempair1 = $true ) ),
inference(flattening,[status(thm)],[f19]) ).
thf(f19,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( iskpair @ X2 )
!= $true )
& ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
= $true ) )
& ( cartprodmempair1 = $true )
& ( kpairp = $true ) ),
inference(ennf_transformation,[status(thm)],[f11]) ).
thf(f11,plain,
~ ( ( kpairp = $true )
=> ( ( cartprodmempair1 = $true )
=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
= $true )
=> ( ( iskpair @ X2 )
= $true ) ) ) ),
inference(fool_elimination,[status(thm)],[f10]) ).
thf(f10,plain,
~ ( kpairp
=> ( cartprodmempair1
=> ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
=> ( iskpair @ X2 ) ) ) ),
inference(rectify,[status(thm)],[f6]) ).
thf(f6,negated_conjecture,
~ ( kpairp
=> ( cartprodmempair1
=> ! [X3: $i,X0: $i,X4: $i] :
( ( in @ X4 @ ( cartprod @ X0 @ X3 ) )
=> ( iskpair @ X4 ) ) ) ),
inference(negated_conjecture,[status(cth)],[f5]) ).
thf(f5,conjecture,
( kpairp
=> ( cartprodmempair1
=> ! [X3: $i,X0: $i,X4: $i] :
( ( in @ X4 @ ( cartprod @ X0 @ X3 ) )
=> ( iskpair @ X4 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.q8E1UGFO71/DTF2THF_6523.p',cartprodmempair) ).
thf(f60,plain,
( ( iskpair @ sK4 )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f56]) ).
thf(f56,plain,
( ( $true != $true )
| ( ( iskpair @ sK4 )
= $true ) ),
inference(superposition,[status(thm)],[f55,f37]) ).
thf(f37,plain,
( ( in @ sK4 @ ( cartprod @ sK3 @ sK2 ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f26]) ).
thf(f55,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( ( in @ X0 @ ( cartprod @ X2 @ X1 ) )
!= $true )
| ( ( iskpair @ X0 )
= $true ) ),
inference(superposition,[status(thm)],[f53,f54]) ).
thf(f54,plain,
! [X6: $i,X7: $i,X5: $i] :
( ( ( kpair @ ( sK8 @ X7 @ X6 @ X5 ) @ ( sK9 @ X7 @ X6 @ X5 ) )
= X7 )
| ( ( in @ X7 @ ( cartprod @ X5 @ X6 ) )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f50]) ).
thf(f50,plain,
! [X6: $i,X7: $i,X5: $i] :
( ( $true != $true )
| ( ( in @ X7 @ ( cartprod @ X5 @ X6 ) )
!= $true )
| ( ( kpair @ ( sK8 @ X7 @ X6 @ X5 ) @ ( sK9 @ X7 @ X6 @ X5 ) )
= X7 ) ),
inference(definition_unfolding,[status(thm)],[f39,f35]) ).
thf(f35,plain,
cartprodmempair1 = $true,
inference(cnf_transformation,[status(thm)],[f26]) ).
thf(f39,plain,
! [X6: $i,X7: $i,X5: $i] :
( ( ( in @ X7 @ ( cartprod @ X5 @ X6 ) )
!= $true )
| ( ( kpair @ ( sK8 @ X7 @ X6 @ X5 ) @ ( sK9 @ X7 @ X6 @ X5 ) )
= X7 )
| ( cartprodmempair1 != $true ) ),
inference(cnf_transformation,[status(thm)],[f32]) ).
thf(f32,plain,
( ( ( cartprodmempair1 = $true )
| ( ( $true
= ( in @ sK7 @ ( cartprod @ sK5 @ sK6 ) ) )
& ! [X3: $i] :
( ( ( in @ X3 @ sK5 )
!= $true )
| ! [X4: $i] :
( ( ( in @ X4 @ sK6 )
!= $true )
| ( ( kpair @ X3 @ X4 )
!= sK7 ) ) ) ) )
& ( ! [X5: $i,X6: $i,X7: $i] :
( ( ( in @ X7 @ ( cartprod @ X5 @ X6 ) )
!= $true )
| ( ( $true
= ( in @ ( sK8 @ X7 @ X6 @ X5 ) @ X5 ) )
& ( ( in @ ( sK9 @ X7 @ X6 @ X5 ) @ X6 )
= $true )
& ( ( kpair @ ( sK8 @ X7 @ X6 @ X5 ) @ ( sK9 @ X7 @ X6 @ X5 ) )
= X7 ) ) )
| ( cartprodmempair1 != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK5,sK6,sK7,sK8,sK9])],[f28,f31,f30,f29]) ).
thf(f29,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ ( cartprod @ X0 @ X1 ) )
= $true )
& ! [X3: $i] :
( ( ( in @ X3 @ X0 )
!= $true )
| ! [X4: $i] :
( ( ( in @ X4 @ X1 )
!= $true )
| ( ( kpair @ X3 @ X4 )
!= X2 ) ) ) )
=> ( ( $true
= ( in @ sK7 @ ( cartprod @ sK5 @ sK6 ) ) )
& ! [X3: $i] :
( ( ( in @ X3 @ sK5 )
!= $true )
| ! [X4: $i] :
( ( ( in @ X4 @ sK6 )
!= $true )
| ( ( kpair @ X3 @ X4 )
!= sK7 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f30,plain,
! [X5: $i,X6: $i,X7: $i] :
( ? [X8: $i] :
( ( ( in @ X8 @ X5 )
= $true )
& ? [X9: $i] :
( ( ( in @ X9 @ X6 )
= $true )
& ( ( kpair @ X8 @ X9 )
= X7 ) ) )
=> ( ( $true
= ( in @ ( sK8 @ X7 @ X6 @ X5 ) @ X5 ) )
& ? [X9: $i] :
( ( ( in @ X9 @ X6 )
= $true )
& ( ( kpair @ ( sK8 @ X7 @ X6 @ X5 ) @ X9 )
= X7 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f31,plain,
! [X5: $i,X6: $i,X7: $i] :
( ? [X9: $i] :
( ( ( in @ X9 @ X6 )
= $true )
& ( ( kpair @ ( sK8 @ X7 @ X6 @ X5 ) @ X9 )
= X7 ) )
=> ( ( ( in @ ( sK9 @ X7 @ X6 @ X5 ) @ X6 )
= $true )
& ( ( kpair @ ( sK8 @ X7 @ X6 @ X5 ) @ ( sK9 @ X7 @ X6 @ X5 ) )
= X7 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f28,plain,
( ( ( cartprodmempair1 = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ ( cartprod @ X0 @ X1 ) )
= $true )
& ! [X3: $i] :
( ( ( in @ X3 @ X0 )
!= $true )
| ! [X4: $i] :
( ( ( in @ X4 @ X1 )
!= $true )
| ( ( kpair @ X3 @ X4 )
!= X2 ) ) ) ) )
& ( ! [X5: $i,X6: $i,X7: $i] :
( ( ( in @ X7 @ ( cartprod @ X5 @ X6 ) )
!= $true )
| ? [X8: $i] :
( ( ( in @ X8 @ X5 )
= $true )
& ? [X9: $i] :
( ( ( in @ X9 @ X6 )
= $true )
& ( ( kpair @ X8 @ X9 )
= X7 ) ) ) )
| ( cartprodmempair1 != $true ) ) ),
inference(rectify,[status(thm)],[f27]) ).
thf(f27,plain,
( ( ( cartprodmempair1 = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ ( cartprod @ X0 @ X1 ) )
= $true )
& ! [X3: $i] :
( ( ( in @ X3 @ X0 )
!= $true )
| ! [X4: $i] :
( ( ( in @ X4 @ X1 )
!= $true )
| ( ( kpair @ X3 @ X4 )
!= X2 ) ) ) ) )
& ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ ( cartprod @ X0 @ X1 ) )
!= $true )
| ? [X3: $i] :
( ( ( in @ X3 @ X0 )
= $true )
& ? [X4: $i] :
( ( ( in @ X4 @ X1 )
= $true )
& ( ( kpair @ X3 @ X4 )
= X2 ) ) ) )
| ( cartprodmempair1 != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f18]) ).
thf(f18,plain,
( ( cartprodmempair1 = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ ( cartprod @ X0 @ X1 ) )
!= $true )
| ? [X3: $i] :
( ( ( in @ X3 @ X0 )
= $true )
& ? [X4: $i] :
( ( ( in @ X4 @ X1 )
= $true )
& ( ( kpair @ X3 @ X4 )
= X2 ) ) ) ) ),
inference(ennf_transformation,[status(thm)],[f13]) ).
thf(f13,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ ( cartprod @ X0 @ X1 ) )
= $true )
=> ? [X3: $i] :
( ( ( in @ X3 @ X0 )
= $true )
& ? [X4: $i] :
( ( ( in @ X4 @ X1 )
= $true )
& ( ( kpair @ X3 @ X4 )
= X2 ) ) ) )
<=> ( cartprodmempair1 = $true ) ),
inference(fool_elimination,[status(thm)],[f12]) ).
thf(f12,plain,
( cartprodmempair1
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X2 @ ( cartprod @ X0 @ X1 ) )
=> ? [X3: $i] :
( ? [X4: $i] :
( ( ( kpair @ X3 @ X4 )
= X2 )
& ( in @ X4 @ X1 ) )
& ( in @ X3 @ X0 ) ) ) ) ),
inference(rectify,[status(thm)],[f4]) ).
thf(f4,axiom,
( cartprodmempair1
= ( ! [X0: $i,X3: $i,X4: $i] :
( ( in @ X4 @ ( cartprod @ X0 @ X3 ) )
=> ? [X1: $i] :
( ? [X2: $i] :
( ( ( kpair @ X1 @ X2 )
= X4 )
& ( in @ X2 @ X3 ) )
& ( in @ X1 @ X0 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.q8E1UGFO71/DTF2THF_6523.p',cartprodmempair1) ).
thf(f53,plain,
! [X0: $i,X1: $i] :
( ( iskpair @ ( kpair @ X1 @ X0 ) )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f44]) ).
thf(f44,plain,
! [X0: $i,X1: $i] :
( ( ( iskpair @ ( kpair @ X1 @ X0 ) )
= $true )
| ( $true != $true ) ),
inference(definition_unfolding,[status(thm)],[f34,f36]) ).
thf(f36,plain,
kpairp = $true,
inference(cnf_transformation,[status(thm)],[f26]) ).
thf(f34,plain,
! [X0: $i,X1: $i] :
( ( ( iskpair @ ( kpair @ X1 @ X0 ) )
= $true )
| ( kpairp != $true ) ),
inference(cnf_transformation,[status(thm)],[f24]) ).
thf(f24,plain,
( ( ! [X0: $i,X1: $i] :
( ( iskpair @ ( kpair @ X1 @ X0 ) )
= $true )
| ( kpairp != $true ) )
& ( ( kpairp = $true )
| ( ( iskpair @ ( kpair @ sK1 @ sK0 ) )
!= $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1])],[f22,f23]) ).
thf(f23,plain,
( ? [X2: $i,X3: $i] :
( ( iskpair @ ( kpair @ X3 @ X2 ) )
!= $true )
=> ( ( iskpair @ ( kpair @ sK1 @ sK0 ) )
!= $true ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f22,plain,
( ( ! [X0: $i,X1: $i] :
( ( iskpair @ ( kpair @ X1 @ X0 ) )
= $true )
| ( kpairp != $true ) )
& ( ( kpairp = $true )
| ? [X2: $i,X3: $i] :
( ( iskpair @ ( kpair @ X3 @ X2 ) )
!= $true ) ) ),
inference(rectify,[status(thm)],[f21]) ).
thf(f21,plain,
( ( ! [X0: $i,X1: $i] :
( ( iskpair @ ( kpair @ X1 @ X0 ) )
= $true )
| ( kpairp != $true ) )
& ( ( kpairp = $true )
| ? [X0: $i,X1: $i] :
( ( iskpair @ ( kpair @ X1 @ X0 ) )
!= $true ) ) ),
inference(nnf_transformation,[status(thm)],[f17]) ).
thf(f17,plain,
( ! [X0: $i,X1: $i] :
( ( iskpair @ ( kpair @ X1 @ X0 ) )
= $true )
<=> ( kpairp = $true ) ),
inference(fool_elimination,[status(thm)],[f16]) ).
thf(f16,plain,
! [X0: $i,X1: $i] :
( ( iskpair @ ( kpair @ X1 @ X0 ) )
= kpairp ),
inference(rectify,[status(thm)],[f3]) ).
thf(f3,axiom,
! [X2: $i,X1: $i] :
( ( iskpair @ ( kpair @ X1 @ X2 ) )
= kpairp ),
file('/export/starexec/sandbox/tmp/tmp.q8E1UGFO71/DTF2THF_6523.p',kpairp) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU632^2 : TPTP v9.2.1. Released v3.7.0.
% 0.13/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n005.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 12:29:14 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 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 : SEU632^2 : TPTP v9.2.1. Released v3.7.0.
% 0.20/0.46 % Domain : Set Theory
% 0.20/0.46 % Problem : Ordered Pairs - Cartesian Products
% 0.20/0.46 % Version : Especial > Reduced > Especial.
% 0.20/0.46 % English : (! A:i.! B:i.! u:i.in u (cartprod A B) -> iskpair u)
% 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 : ZFC134l [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, 0.20 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, 0.25 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.14 v6.1.0, 0.29 v5.5.0, 0.17 v5.4.0, 0.40 v5.3.0, 0.60 v5.2.0, 0.40 v5.1.0, 0.60 v5.0.0, 0.40 v4.1.0, 0.00 v4.0.0, 0.33 v3.7.0
% 0.20/0.46 % Syntax : Number of formulae : 14 ( 4 unt; 9 typ; 4 def)
% 0.20/0.46 % Number of atoms : 19 ( 6 equ; 0 cnn)
% 0.20/0.46 % Maximal formula atoms : 4 ( 3 avg)
% 0.20/0.46 % Number of connectives : 52 ( 0 ~; 0 |; 4 &; 44 @)
% 0.20/0.46 % ( 0 <=>; 4 =>; 0 <=; 0 <~>)
% 0.20/0.46 % Maximal formula depth : 10 ( 3 avg)
% 0.20/0.46 % Number of types : 2 ( 0 usr)
% 0.20/0.46 % Number of type conns : 10 ( 10 >; 0 *; 0 +; 0 <<)
% 0.20/0.46 % Number of symbols : 10 ( 9 usr; 3 con; 0-2 aty)
% 0.20/0.46 % Number of variables : 15 ( 3 ^; 8 !; 4 ?; 15 :)
% 0.20/0.46 % SPC : TH0_THM_EQU_NAR
% 0.20/0.46
% 0.20/0.46 % Comments : http://mathgate.info/detsetitem.php?id=189
% 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(emptyset_type,type,
% 0.20/0.46 emptyset: $i ).
% 0.20/0.46
% 0.20/0.46 thf(setadjoin_type,type,
% 0.20/0.46 setadjoin: $i > $i > $i ).
% 0.20/0.46
% 0.20/0.46 thf(setunion_type,type,
% 0.20/0.46 setunion: $i > $i ).
% 0.20/0.46
% 0.20/0.46 thf(iskpair_type,type,
% 0.20/0.46 iskpair: $i > $o ).
% 0.20/0.46
% 0.20/0.46 thf(iskpair,definition,
% 0.20/0.46 ( iskpair
% 0.20/0.46 = ( ^ [A: $i] :
% 0.20/0.46 ? [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.20/0.46 & ? [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ ( setunion @ A ) )
% 0.20/0.46 & ( A
% 0.20/0.46 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ).
% 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(kpair,definition,
% 0.20/0.46 ( kpair
% 0.20/0.46 = ( ^ [Xx: $i,Xy: $i] : ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(kpairp_type,type,
% 0.20/0.46 kpairp: $o ).
% 0.20/0.46
% 0.20/0.46 thf(kpairp,definition,
% 0.20/0.46 ( kpairp
% 0.20/0.46 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 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(cartprodmempair,conjecture,
% 0.20/0.46 ( kpairp
% 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 => ( iskpair @ Xu ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 %------------------------------------------------------------------------------
% 0.20/0.46 ------------------------
% 1.61/1.23 ---- Embedded in THF ---
% 1.61/1.23 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.61/1.23 %%% Generated on Tue Feb 24 12:29:15 EST 2026
% 1.61/1.23 %%% using '$$dhol' embedding, version 1.3.0.
% 1.61/1.23 %%% Logic specification used:
% 1.61/1.23 %%% thf(spec, logic, $$dhol).
% 1.61/1.23
% 1.61/1.23 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.q8E1UGFO71/DTF2DTF6523.p
% See solution above
% 1.61/1.23 ------------------------
% 1.61/1.24 ---- Cleaned THF ---
% 1.61/1.24 thf(in_type,type,
% 1.61/1.24 in: $i > $i > $o ).
% 1.61/1.24
% 1.61/1.24 thf(emptyset_type,type,
% 1.61/1.24 emptyset: $i ).
% 1.61/1.24
% 1.61/1.24 thf(setadjoin_type,type,
% 1.61/1.24 setadjoin: $i > $i > $i ).
% 1.61/1.24
% 1.61/1.24 thf(setunion_type,type,
% 1.61/1.24 setunion: $i > $i ).
% 1.61/1.24
% 1.61/1.24 thf(iskpair_type,type,
% 1.61/1.24 iskpair: $i > $o ).
% 1.61/1.24
% 1.61/1.24 thf(iskpair,definition,
% 1.61/1.24 ( iskpair
% 1.61/1.24 = ( ^ [A: $i] :
% 1.61/1.24 ? [Xx: $i] :
% 1.61/1.24 ( ( in @ Xx @ ( setunion @ A ) )
% 1.61/1.24 & ? [Xy: $i] :
% 1.61/1.24 ( ( in @ Xy @ ( setunion @ A ) )
% 1.61/1.24 & ( A
% 1.61/1.24 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ).
% 1.61/1.24
% 1.61/1.24 thf(kpair_type,type,
% 1.61/1.24 kpair: $i > $i > $i ).
% 1.61/1.24
% 1.61/1.24 thf(kpair,definition,
% 1.61/1.24 ( kpair
% 1.61/1.24 = ( ^ [Xx: $i,Xy: $i] : ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ).
% 1.61/1.24
% 1.61/1.24 thf(kpairp_type,type,
% 1.61/1.24 kpairp: $o ).
% 1.61/1.24
% 1.61/1.24 thf(kpairp,definition,
% 1.61/1.24 ( kpairp
% 1.61/1.24 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 1.61/1.24
% 1.61/1.24 thf(cartprod_type,type,
% 1.61/1.24 cartprod: $i > $i > $i ).
% 1.61/1.24
% 1.61/1.24 thf(cartprodmempair1_type,type,
% 1.61/1.24 cartprodmempair1: $o ).
% 1.61/1.24
% 1.61/1.24 thf(cartprodmempair1,definition,
% 1.61/1.24 ( cartprodmempair1
% 1.61/1.24 = ( ! [A: $i,B: $i,Xu: $i] :
% 1.61/1.24 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 1.61/1.24 => ? [Xx: $i] :
% 1.61/1.24 ( ( in @ Xx @ A )
% 1.61/1.24 & ? [Xy: $i] :
% 1.61/1.24 ( ( in @ Xy @ B )
% 1.61/1.24 & ( Xu
% 1.61/1.24 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 1.61/1.24
% 1.61/1.24 thf(cartprodmempair,conjecture,
% 1.61/1.24 ( kpairp
% 1.61/1.24 => ( cartprodmempair1
% 1.61/1.24 => ! [A: $i,B: $i,Xu: $i] :
% 1.61/1.24 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 1.61/1.24 => ( iskpair @ Xu ) ) ) ) ).
% 1.61/1.24 ------------------------
% 1.61/1.26 % (6711)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_6523 for (3000ds/183Mi)
% 1.61/1.26 % (6716)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_6523 for (3000ds/275Mi)
% 1.61/1.26 % (6716)First to succeed.
% 1.61/1.26 % (6716)Refutation found. Thanks to Tanya!
% 1.61/1.26 % SZS status Theorem for DTF2THF_6523
% 1.61/1.26 % SZS output start Proof for DTF2THF_6523
% 1.61/1.26 thf(func_def_0, type, in: $i > $i > $o).
% 1.61/1.26 thf(func_def_2, type, setadjoin: $i > $i > $i).
% 1.61/1.26 thf(func_def_3, type, setunion: $i > $i).
% 1.61/1.26 thf(func_def_4, type, iskpair: $i > $o).
% 1.61/1.26 thf(func_def_6, type, kpair: $i > $i > $i).
% 1.61/1.26 thf(func_def_8, type, cartprod: $i > $i > $i).
% 1.61/1.26 thf(func_def_27, type, sK8: $i > $i > $i > $i).
% 1.61/1.26 thf(func_def_28, type, sK9: $i > $i > $i > $i).
% 1.61/1.26 thf(f62,plain,(
% 1.61/1.26 $false),
% 1.61/1.26 inference(subsumption_resolution,[status(thm)],[f60,f38])).
% 1.61/1.26 thf(f38,plain,(
% 1.61/1.26 ((iskpair @ sK4) != $true)),
% 1.61/1.26 inference(cnf_transformation,[status(thm)],[f26])).
% 1.61/1.26 thf(f26,plain,(
% 1.61/1.26 (((iskpair @ sK4) != $true) & ((in @ sK4 @ (cartprod @ sK3 @ sK2)) = $true)) & (kpairp = $true) & (cartprodmempair1 = $true)),
% 1.61/1.26 inference(skolemisation,[status(esa),new_symbols(skolem,[sK2,sK3,sK4])],[f20,f25])).
% 1.61/1.26 thf(f25,plain,(
% 1.61/1.26 ? [X0 : $i,X1 : $i,X2 : $i] : (((iskpair @ X2) != $true) & ((in @ X2 @ (cartprod @ X1 @ X0)) = $true)) => (((iskpair @ sK4) != $true) & ((in @ sK4 @ (cartprod @ sK3 @ sK2)) = $true))),
% 1.61/1.26 introduced(definition,[],[choice_axiom])).
% 1.61/1.26 thf(f20,plain,(
% 1.61/1.26 ? [X0 : $i,X1 : $i,X2 : $i] : (((iskpair @ X2) != $true) & ((in @ X2 @ (cartprod @ X1 @ X0)) = $true)) & (kpairp = $true) & (cartprodmempair1 = $true)),
% 1.61/1.26 inference(flattening,[status(thm)],[f19])).
% 1.61/1.26 thf(f19,plain,(
% 1.61/1.26 (? [X0 : $i,X1 : $i,X2 : $i] : (((iskpair @ X2) != $true) & ((in @ X2 @ (cartprod @ X1 @ X0)) = $true)) & (cartprodmempair1 = $true)) & (kpairp = $true)),
% 1.61/1.26 inference(ennf_transformation,[status(thm)],[f11])).
% 1.61/1.26 thf(f11,plain,(
% 1.61/1.26 ~((kpairp = $true) => ((cartprodmempair1 = $true) => ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ (cartprod @ X1 @ X0)) = $true) => ((iskpair @ X2) = $true))))),
% 1.61/1.26 inference(fool_elimination,[status(thm)],[f10])).
% 1.61/1.26 thf(f10,plain,(
% 1.61/1.26 ~(kpairp => (cartprodmempair1 => ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X2 @ (cartprod @ X1 @ X0)) => (iskpair @ X2))))),
% 1.61/1.26 inference(rectify,[status(thm)],[f6])).
% 1.61/1.26 thf(f6,negated_conjecture,(
% 1.61/1.26 ~(kpairp => (cartprodmempair1 => ! [X3 : $i,X0 : $i,X4 : $i] : ((in @ X4 @ (cartprod @ X0 @ X3)) => (iskpair @ X4))))),
% 1.61/1.26 inference(negated_conjecture,[status(cth)],[f5])).
% 1.61/1.26 thf(f5,conjecture,(
% 1.61/1.26 kpairp => (cartprodmempair1 => ! [X3 : $i,X0 : $i,X4 : $i] : ((in @ X4 @ (cartprod @ X0 @ X3)) => (iskpair @ X4)))),
% 1.61/1.26 file('/export/starexec/sandbox/tmp/tmp.q8E1UGFO71/DTF2THF_6523.p',cartprodmempair)).
% 1.61/1.26 thf(f60,plain,(
% 1.61/1.26 ((iskpair @ sK4) = $true)),
% 1.61/1.26 inference(trivial_inequality_removal,[status(thm)],[f56])).
% 1.61/1.26 thf(f56,plain,(
% 1.61/1.26 ($true != $true) | ((iskpair @ sK4) = $true)),
% 1.61/1.26 inference(superposition,[status(thm)],[f55,f37])).
% 1.61/1.26 thf(f37,plain,(
% 1.61/1.26 ((in @ sK4 @ (cartprod @ sK3 @ sK2)) = $true)),
% 1.61/1.26 inference(cnf_transformation,[status(thm)],[f26])).
% 1.61/1.26 thf(f55,plain,(
% 1.61/1.26 ( ! [X2 : $i,X0 : $i,X1 : $i] : (((in @ X0 @ (cartprod @ X2 @ X1)) != $true) | ((iskpair @ X0) = $true)) )),
% 1.61/1.26 inference(superposition,[status(thm)],[f53,f54])).
% 1.61/1.26 thf(f54,plain,(
% 1.61/1.26 ( ! [X6 : $i,X7 : $i,X5 : $i] : (((kpair @ (sK8 @ X7 @ X6 @ X5) @ (sK9 @ X7 @ X6 @ X5)) = X7) | ((in @ X7 @ (cartprod @ X5 @ X6)) != $true)) )),
% 1.61/1.26 inference(trivial_inequality_removal,[status(thm)],[f50])).
% 1.61/1.26 thf(f50,plain,(
% 1.61/1.26 ( ! [X6 : $i,X7 : $i,X5 : $i] : (($true != $true) | ((in @ X7 @ (cartprod @ X5 @ X6)) != $true) | ((kpair @ (sK8 @ X7 @ X6 @ X5) @ (sK9 @ X7 @ X6 @ X5)) = X7)) )),
% 1.61/1.26 inference(definition_unfolding,[status(thm)],[f39,f35])).
% 1.61/1.26 thf(f35,plain,(
% 1.61/1.26 (cartprodmempair1 = $true)),
% 1.61/1.26 inference(cnf_transformation,[status(thm)],[f26])).
% 1.61/1.26 thf(f39,plain,(
% 1.61/1.26 ( ! [X6 : $i,X7 : $i,X5 : $i] : (((in @ X7 @ (cartprod @ X5 @ X6)) != $true) | ((kpair @ (sK8 @ X7 @ X6 @ X5) @ (sK9 @ X7 @ X6 @ X5)) = X7) | (cartprodmempair1 != $true)) )),
% 1.61/1.26 inference(cnf_transformation,[status(thm)],[f32])).
% 1.61/1.26 thf(f32,plain,(
% 1.61/1.26 ((cartprodmempair1 = $true) | (($true = (in @ sK7 @ (cartprod @ sK5 @ sK6))) & ! [X3 : $i] : (((in @ X3 @ sK5) != $true) | ! [X4 : $i] : (((in @ X4 @ sK6) != $true) | ((kpair @ X3 @ X4) != sK7))))) & (! [X5 : $i,X6 : $i,X7 : $i] : (((in @ X7 @ (cartprod @ X5 @ X6)) != $true) | (($true = (in @ (sK8 @ X7 @ X6 @ X5) @ X5)) & (((in @ (sK9 @ X7 @ X6 @ X5) @ X6) = $true) & ((kpair @ (sK8 @ X7 @ X6 @ X5) @ (sK9 @ X7 @ X6 @ X5)) = X7)))) | (cartprodmempair1 != $true))),
% 1.61/1.26 inference(skolemisation,[status(esa),new_symbols(skolem,[sK5,sK6,sK7,sK8,sK9])],[f28,f31,f30,f29])).
% 1.61/1.26 thf(f29,plain,(
% 1.61/1.26 ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ (cartprod @ X0 @ X1)) = $true) & ! [X3 : $i] : (((in @ X3 @ X0) != $true) | ! [X4 : $i] : (((in @ X4 @ X1) != $true) | ((kpair @ X3 @ X4) != X2)))) => (($true = (in @ sK7 @ (cartprod @ sK5 @ sK6))) & ! [X3 : $i] : (((in @ X3 @ sK5) != $true) | ! [X4 : $i] : (((in @ X4 @ sK6) != $true) | ((kpair @ X3 @ X4) != sK7))))),
% 1.61/1.26 introduced(definition,[],[choice_axiom])).
% 1.61/1.26 thf(f30,plain,(
% 1.61/1.26 ! [X5 : $i,X6 : $i,X7 : $i] : (? [X8 : $i] : (((in @ X8 @ X5) = $true) & ? [X9 : $i] : (((in @ X9 @ X6) = $true) & ((kpair @ X8 @ X9) = X7))) => (($true = (in @ (sK8 @ X7 @ X6 @ X5) @ X5)) & ? [X9 : $i] : (((in @ X9 @ X6) = $true) & ((kpair @ (sK8 @ X7 @ X6 @ X5) @ X9) = X7))))),
% 1.61/1.26 introduced(definition,[],[choice_axiom])).
% 1.61/1.26 thf(f31,plain,(
% 1.61/1.26 ! [X5 : $i,X6 : $i,X7 : $i] : (? [X9 : $i] : (((in @ X9 @ X6) = $true) & ((kpair @ (sK8 @ X7 @ X6 @ X5) @ X9) = X7)) => (((in @ (sK9 @ X7 @ X6 @ X5) @ X6) = $true) & ((kpair @ (sK8 @ X7 @ X6 @ X5) @ (sK9 @ X7 @ X6 @ X5)) = X7)))),
% 1.61/1.26 introduced(definition,[],[choice_axiom])).
% 1.61/1.26 thf(f28,plain,(
% 1.61/1.26 ((cartprodmempair1 = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ (cartprod @ X0 @ X1)) = $true) & ! [X3 : $i] : (((in @ X3 @ X0) != $true) | ! [X4 : $i] : (((in @ X4 @ X1) != $true) | ((kpair @ X3 @ X4) != X2))))) & (! [X5 : $i,X6 : $i,X7 : $i] : (((in @ X7 @ (cartprod @ X5 @ X6)) != $true) | ? [X8 : $i] : (((in @ X8 @ X5) = $true) & ? [X9 : $i] : (((in @ X9 @ X6) = $true) & ((kpair @ X8 @ X9) = X7)))) | (cartprodmempair1 != $true))),
% 1.61/1.26 inference(rectify,[status(thm)],[f27])).
% 1.61/1.26 thf(f27,plain,(
% 1.61/1.26 ((cartprodmempair1 = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ (cartprod @ X0 @ X1)) = $true) & ! [X3 : $i] : (((in @ X3 @ X0) != $true) | ! [X4 : $i] : (((in @ X4 @ X1) != $true) | ((kpair @ X3 @ X4) != X2))))) & (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ (cartprod @ X0 @ X1)) != $true) | ? [X3 : $i] : (((in @ X3 @ X0) = $true) & ? [X4 : $i] : (((in @ X4 @ X1) = $true) & ((kpair @ X3 @ X4) = X2)))) | (cartprodmempair1 != $true))),
% 1.61/1.26 inference(nnf_transformation,[status(thm)],[f18])).
% 1.61/1.26 thf(f18,plain,(
% 1.61/1.26 (cartprodmempair1 = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ (cartprod @ X0 @ X1)) != $true) | ? [X3 : $i] : (((in @ X3 @ X0) = $true) & ? [X4 : $i] : (((in @ X4 @ X1) = $true) & ((kpair @ X3 @ X4) = X2))))),
% 1.61/1.26 inference(ennf_transformation,[status(thm)],[f13])).
% 1.61/1.26 thf(f13,plain,(
% 1.61/1.26 ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ (cartprod @ X0 @ X1)) = $true) => ? [X3 : $i] : (((in @ X3 @ X0) = $true) & ? [X4 : $i] : (((in @ X4 @ X1) = $true) & ((kpair @ X3 @ X4) = X2)))) <=> (cartprodmempair1 = $true)),
% 1.61/1.26 inference(fool_elimination,[status(thm)],[f12])).
% 1.61/1.26 thf(f12,plain,(
% 1.61/1.26 (cartprodmempair1 = ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X2 @ (cartprod @ X0 @ X1)) => ? [X3 : $i] : (? [X4 : $i] : (((kpair @ X3 @ X4) = X2) & (in @ X4 @ X1)) & (in @ X3 @ X0))))),
% 1.61/1.26 inference(rectify,[status(thm)],[f4])).
% 1.61/1.26 thf(f4,axiom,(
% 1.61/1.26 (cartprodmempair1 = ! [X0 : $i,X3 : $i,X4 : $i] : ((in @ X4 @ (cartprod @ X0 @ X3)) => ? [X1 : $i] : (? [X2 : $i] : (((kpair @ X1 @ X2) = X4) & (in @ X2 @ X3)) & (in @ X1 @ X0))))),
% 1.61/1.26 file('/export/starexec/sandbox/tmp/tmp.q8E1UGFO71/DTF2THF_6523.p',cartprodmempair1)).
% 1.61/1.26 thf(f53,plain,(
% 1.61/1.26 ( ! [X0 : $i,X1 : $i] : (((iskpair @ (kpair @ X1 @ X0)) = $true)) )),
% 1.61/1.26 inference(trivial_inequality_removal,[status(thm)],[f44])).
% 1.61/1.26 thf(f44,plain,(
% 1.61/1.26 ( ! [X0 : $i,X1 : $i] : (((iskpair @ (kpair @ X1 @ X0)) = $true) | ($true != $true)) )),
% 1.61/1.26 inference(definition_unfolding,[status(thm)],[f34,f36])).
% 1.61/1.26 thf(f36,plain,(
% 1.61/1.26 (kpairp = $true)),
% 1.61/1.26 inference(cnf_transformation,[status(thm)],[f26])).
% 1.61/1.26 thf(f34,plain,(
% 1.61/1.26 ( ! [X0 : $i,X1 : $i] : (((iskpair @ (kpair @ X1 @ X0)) = $true) | (kpairp != $true)) )),
% 1.61/1.26 inference(cnf_transformation,[status(thm)],[f24])).
% 1.61/1.26 thf(f24,plain,(
% 1.61/1.26 (! [X0 : $i,X1 : $i] : ((iskpair @ (kpair @ X1 @ X0)) = $true) | (kpairp != $true)) & ((kpairp = $true) | ((iskpair @ (kpair @ sK1 @ sK0)) != $true))),
% 1.61/1.26 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1])],[f22,f23])).
% 1.61/1.26 thf(f23,plain,(
% 1.61/1.26 ? [X2 : $i,X3 : $i] : ((iskpair @ (kpair @ X3 @ X2)) != $true) => ((iskpair @ (kpair @ sK1 @ sK0)) != $true)),
% 1.61/1.26 introduced(definition,[],[choice_axiom])).
% 1.61/1.26 thf(f22,plain,(
% 1.61/1.26 (! [X0 : $i,X1 : $i] : ((iskpair @ (kpair @ X1 @ X0)) = $true) | (kpairp != $true)) & ((kpairp = $true) | ? [X2 : $i,X3 : $i] : ((iskpair @ (kpair @ X3 @ X2)) != $true))),
% 1.61/1.26 inference(rectify,[status(thm)],[f21])).
% 1.61/1.26 thf(f21,plain,(
% 1.61/1.26 (! [X0 : $i,X1 : $i] : ((iskpair @ (kpair @ X1 @ X0)) = $true) | (kpairp != $true)) & ((kpairp = $true) | ? [X0 : $i,X1 : $i] : ((iskpair @ (kpair @ X1 @ X0)) != $true))),
% 1.61/1.26 inference(nnf_transformation,[status(thm)],[f17])).
% 1.61/1.26 thf(f17,plain,(
% 1.61/1.26 ! [X0 : $i,X1 : $i] : ((iskpair @ (kpair @ X1 @ X0)) = $true) <=> (kpairp = $true)),
% 1.61/1.26 inference(fool_elimination,[status(thm)],[f16])).
% 1.61/1.26 thf(f16,plain,(
% 1.61/1.26 (! [X0 : $i,X1 : $i] : (iskpair @ (kpair @ X1 @ X0)) = kpairp)),
% 1.61/1.26 inference(rectify,[status(thm)],[f3])).
% 1.61/1.26 thf(f3,axiom,(
% 1.61/1.26 (! [X2 : $i,X1 : $i] : (iskpair @ (kpair @ X1 @ X2)) = kpairp)),
% 1.61/1.26 file('/export/starexec/sandbox/tmp/tmp.q8E1UGFO71/DTF2THF_6523.p',kpairp)).
% 1.61/1.26 % SZS output end Proof for DTF2THF_6523
% 1.61/1.26 % (6716)------------------------------
% 1.61/1.26 % (6716)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.61/1.26 % (6716)Termination reason: Refutation
% 1.61/1.26
% 1.61/1.26 % (6716)Memory used [KB]: 5500
% 1.61/1.26 % (6716)Time elapsed: 0.004 s
% 1.61/1.26 % (6716)Instructions burned: 5 (million)
% 1.61/1.26 % (6716)------------------------------
% 1.61/1.26 % (6716)------------------------------
% 1.61/1.26 % (6710)Success in time 0.004 s
%------------------------------------------------------------------------------