%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU646^2 : 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 : n003.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 09:35:55 AM UTC 2026
% Result : Theorem 1.70s 1.45s
% Output : Refutation 1.70s
% Verified :
% SZS Type : Refutation
% Derivation depth : 15
% Number of leaves : 8
% Syntax : Number of formulae : 44 ( 11 unt; 0 typ; 0 def)
% Number of atoms : 395 ( 137 equ; 0 cnn)
% Maximal formula atoms : 10 ( 8 avg)
% Number of connectives : 541 ( 56 ~; 43 |; 42 &; 377 @)
% ( 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 : 20 ( 17 usr; 12 con; 0-3 aty)
% Number of variables : 134 ( 0 ^; 94 !; 40 ?; 134 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_2,type,
setadjoin: $i > $i > $i ).
thf(func_def_3,type,
kpair: $i > $i > $i ).
thf(func_def_5,type,
cartprod: $i > $i > $i ).
thf(func_def_7,type,
kfst: $i > $i ).
thf(func_def_17,type,
sK3: $i > $i > $i > $i ).
thf(func_def_18,type,
sK4: $i > $i > $i > $i ).
thf(f62,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f61,f35]) ).
thf(f35,plain,
( ( in @ ( kfst @ sK7 ) @ sK6 )
!= $true ),
inference(cnf_transformation,[status(thm)],[f24]) ).
thf(f24,plain,
( ( kfstpairEq = $true )
& ( $true
= ( in @ sK7 @ ( cartprod @ sK6 @ sK5 ) ) )
& ( ( in @ ( kfst @ sK7 ) @ sK6 )
!= $true )
& ( cartprodmempair1 = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK5,sK6,sK7])],[f15,f23]) ).
thf(f23,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
= $true )
& ( $true
!= ( in @ ( kfst @ X2 ) @ X1 ) ) )
=> ( ( $true
= ( in @ sK7 @ ( cartprod @ sK6 @ sK5 ) ) )
& ( ( in @ ( kfst @ sK7 ) @ sK6 )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f15,plain,
( ( kfstpairEq = $true )
& ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
= $true )
& ( $true
!= ( in @ ( kfst @ X2 ) @ X1 ) ) )
& ( cartprodmempair1 = $true ) ),
inference(flattening,[status(thm)],[f14]) ).
thf(f14,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
= $true )
& ( $true
!= ( in @ ( kfst @ X2 ) @ X1 ) ) )
& ( kfstpairEq = $true )
& ( cartprodmempair1 = $true ) ),
inference(ennf_transformation,[status(thm)],[f11]) ).
thf(f11,plain,
~ ( ( cartprodmempair1 = $true )
=> ( ( kfstpairEq = $true )
=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
= $true )
=> ( $true
= ( in @ ( kfst @ X2 ) @ X1 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f10]) ).
thf(f10,plain,
~ ( cartprodmempair1
=> ( kfstpairEq
=> ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
=> ( in @ ( kfst @ X2 ) @ X1 ) ) ) ),
inference(rectify,[status(thm)],[f5]) ).
thf(f5,negated_conjecture,
~ ( cartprodmempair1
=> ( kfstpairEq
=> ! [X3: $i,X2: $i,X4: $i] :
( ( in @ X4 @ ( cartprod @ X2 @ X3 ) )
=> ( in @ ( kfst @ X4 ) @ X2 ) ) ) ),
inference(negated_conjecture,[status(cth)],[f4]) ).
thf(f4,conjecture,
( cartprodmempair1
=> ( kfstpairEq
=> ! [X3: $i,X2: $i,X4: $i] :
( ( in @ X4 @ ( cartprod @ X2 @ X3 ) )
=> ( in @ ( kfst @ X4 ) @ X2 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.i6FyzDOOiI/DTF2THF_19631.p',cartprodfstin) ).
thf(f61,plain,
( ( in @ ( kfst @ sK7 ) @ sK6 )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f56]) ).
thf(f56,plain,
( ( ( in @ ( kfst @ sK7 ) @ sK6 )
= $true )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f54,f36]) ).
thf(f36,plain,
( $true
= ( in @ sK7 @ ( cartprod @ sK6 @ sK5 ) ) ),
inference(cnf_transformation,[status(thm)],[f24]) ).
thf(f54,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( $true
!= ( in @ X0 @ ( cartprod @ X1 @ X2 ) ) )
| ( $true
= ( in @ ( kfst @ X0 ) @ X1 ) ) ),
inference(duplicate_literal_removal,[status(thm)],[f53]) ).
thf(f53,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( $true
!= ( in @ X0 @ ( cartprod @ X1 @ X2 ) ) )
| ( $true
!= ( in @ X0 @ ( cartprod @ X1 @ X2 ) ) )
| ( $true
= ( in @ ( kfst @ X0 ) @ X1 ) ) ),
inference(superposition,[status(thm)],[f47,f51]) ).
thf(f51,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( ( sK3 @ X0 @ X1 @ X2 )
= ( kfst @ X0 ) )
| ( $true
!= ( in @ X0 @ ( cartprod @ X1 @ X2 ) ) ) ),
inference(superposition,[status(thm)],[f49,f50]) ).
thf(f50,plain,
! [X6: $i,X7: $i,X5: $i] :
( ( ( kpair @ ( sK3 @ X7 @ X6 @ X5 ) @ ( sK4 @ X7 @ X6 @ X5 ) )
= X7 )
| ( $true
!= ( in @ X7 @ ( cartprod @ X6 @ X5 ) ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f42]) ).
thf(f42,plain,
! [X6: $i,X7: $i,X5: $i] :
( ( $true != $true )
| ( $true
!= ( in @ X7 @ ( cartprod @ X6 @ X5 ) ) )
| ( ( kpair @ ( sK3 @ X7 @ X6 @ X5 ) @ ( sK4 @ X7 @ X6 @ X5 ) )
= X7 ) ),
inference(definition_unfolding,[status(thm)],[f31,f34]) ).
thf(f34,plain,
cartprodmempair1 = $true,
inference(cnf_transformation,[status(thm)],[f24]) ).
thf(f31,plain,
! [X6: $i,X7: $i,X5: $i] :
( ( ( kpair @ ( sK3 @ X7 @ X6 @ X5 ) @ ( sK4 @ X7 @ X6 @ X5 ) )
= X7 )
| ( $true
!= ( in @ X7 @ ( cartprod @ X6 @ X5 ) ) )
| ( cartprodmempair1 != $true ) ),
inference(cnf_transformation,[status(thm)],[f22]) ).
thf(f22,plain,
( ( ( cartprodmempair1 = $true )
| ( ! [X3: $i] :
( ! [X4: $i] :
( ( ( kpair @ X3 @ X4 )
!= sK2 )
| ( $true
!= ( in @ X4 @ sK0 ) ) )
| ( $true
!= ( in @ X3 @ sK1 ) ) )
& ( ( in @ sK2 @ ( cartprod @ sK1 @ sK0 ) )
= $true ) ) )
& ( ! [X5: $i,X6: $i,X7: $i] :
( ( ( ( kpair @ ( sK3 @ X7 @ X6 @ X5 ) @ ( sK4 @ X7 @ X6 @ X5 ) )
= X7 )
& ( $true
= ( in @ ( sK4 @ X7 @ X6 @ X5 ) @ X5 ) )
& ( ( in @ ( sK3 @ X7 @ X6 @ X5 ) @ X6 )
= $true ) )
| ( $true
!= ( in @ X7 @ ( cartprod @ X6 @ X5 ) ) ) )
| ( cartprodmempair1 != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2,sK3,sK4])],[f18,f21,f20,f19]) ).
thf(f19,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ! [X3: $i] :
( ! [X4: $i] :
( ( ( kpair @ X3 @ X4 )
!= X2 )
| ( $true
!= ( in @ X4 @ X0 ) ) )
| ( $true
!= ( in @ X3 @ X1 ) ) )
& ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
= $true ) )
=> ( ! [X3: $i] :
( ! [X4: $i] :
( ( ( kpair @ X3 @ X4 )
!= sK2 )
| ( $true
!= ( in @ X4 @ sK0 ) ) )
| ( $true
!= ( in @ X3 @ sK1 ) ) )
& ( ( in @ sK2 @ ( cartprod @ sK1 @ sK0 ) )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f20,plain,
! [X5: $i,X6: $i,X7: $i] :
( ? [X8: $i] :
( ? [X9: $i] :
( ( ( kpair @ X8 @ X9 )
= X7 )
& ( $true
= ( in @ X9 @ X5 ) ) )
& ( ( in @ X8 @ X6 )
= $true ) )
=> ( ? [X9: $i] :
( ( ( kpair @ ( sK3 @ X7 @ X6 @ X5 ) @ X9 )
= X7 )
& ( $true
= ( in @ X9 @ X5 ) ) )
& ( ( in @ ( sK3 @ X7 @ X6 @ X5 ) @ X6 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f21,plain,
! [X5: $i,X6: $i,X7: $i] :
( ? [X9: $i] :
( ( ( kpair @ ( sK3 @ X7 @ X6 @ X5 ) @ X9 )
= X7 )
& ( $true
= ( in @ X9 @ X5 ) ) )
=> ( ( ( kpair @ ( sK3 @ X7 @ X6 @ X5 ) @ ( sK4 @ X7 @ X6 @ X5 ) )
= X7 )
& ( $true
= ( in @ ( sK4 @ X7 @ X6 @ X5 ) @ X5 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f18,plain,
( ( ( cartprodmempair1 = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ! [X3: $i] :
( ! [X4: $i] :
( ( ( kpair @ X3 @ X4 )
!= X2 )
| ( $true
!= ( in @ X4 @ X0 ) ) )
| ( $true
!= ( in @ X3 @ X1 ) ) )
& ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
= $true ) ) )
& ( ! [X5: $i,X6: $i,X7: $i] :
( ? [X8: $i] :
( ? [X9: $i] :
( ( ( kpair @ X8 @ X9 )
= X7 )
& ( $true
= ( in @ X9 @ X5 ) ) )
& ( ( in @ X8 @ X6 )
= $true ) )
| ( $true
!= ( in @ X7 @ ( cartprod @ X6 @ X5 ) ) ) )
| ( cartprodmempair1 != $true ) ) ),
inference(rectify,[status(thm)],[f17]) ).
thf(f17,plain,
( ( ( cartprodmempair1 = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ! [X3: $i] :
( ! [X4: $i] :
( ( ( kpair @ X3 @ X4 )
!= X2 )
| ( $true
!= ( in @ X4 @ X0 ) ) )
| ( $true
!= ( in @ X3 @ X1 ) ) )
& ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
= $true ) ) )
& ( ! [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ? [X4: $i] :
( ( ( kpair @ X3 @ X4 )
= X2 )
& ( $true
= ( in @ X4 @ X0 ) ) )
& ( $true
= ( in @ X3 @ X1 ) ) )
| ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
!= $true ) )
| ( cartprodmempair1 != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f16]) ).
thf(f16,plain,
( ( cartprodmempair1 = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ? [X4: $i] :
( ( ( kpair @ X3 @ X4 )
= X2 )
& ( $true
= ( in @ X4 @ X0 ) ) )
& ( $true
= ( in @ X3 @ X1 ) ) )
| ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
!= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f9]) ).
thf(f9,plain,
( ( cartprodmempair1 = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
= $true )
=> ? [X3: $i] :
( ? [X4: $i] :
( ( ( kpair @ X3 @ X4 )
= X2 )
& ( $true
= ( in @ X4 @ X0 ) ) )
& ( $true
= ( in @ X3 @ X1 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f8]) ).
thf(f8,plain,
! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
=> ? [X3: $i] :
( ( in @ X3 @ X1 )
& ? [X4: $i] :
( ( in @ X4 @ X0 )
& ( ( kpair @ X3 @ X4 )
= X2 ) ) ) )
= cartprodmempair1 ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
! [X3: $i,X2: $i,X4: $i] :
( ( ( in @ X4 @ ( cartprod @ X2 @ X3 ) )
=> ? [X0: $i] :
( ( in @ X0 @ X2 )
& ? [X1: $i] :
( ( in @ X1 @ X3 )
& ( ( kpair @ X0 @ X1 )
= X4 ) ) ) )
= cartprodmempair1 ),
file('/export/starexec/sandbox/tmp/tmp.i6FyzDOOiI/DTF2THF_19631.p',cartprodmempair1) ).
thf(f49,plain,
! [X2: $i,X3: $i] :
( ( kfst @ ( kpair @ X3 @ X2 ) )
= X3 ),
inference(trivial_inequality_removal,[status(thm)],[f46]) ).
thf(f46,plain,
! [X2: $i,X3: $i] :
( ( ( kfst @ ( kpair @ X3 @ X2 ) )
= X3 )
| ( $true != $true ) ),
inference(definition_unfolding,[status(thm)],[f38,f37]) ).
thf(f37,plain,
kfstpairEq = $true,
inference(cnf_transformation,[status(thm)],[f24]) ).
thf(f38,plain,
! [X2: $i,X3: $i] :
( ( ( kfst @ ( kpair @ X3 @ X2 ) )
= X3 )
| ( kfstpairEq != $true ) ),
inference(cnf_transformation,[status(thm)],[f28]) ).
thf(f28,plain,
( ( ( kfstpairEq = $true )
| ( sK9
!= ( kfst @ ( kpair @ sK9 @ sK8 ) ) ) )
& ( ! [X2: $i,X3: $i] :
( ( kfst @ ( kpair @ X3 @ X2 ) )
= X3 )
| ( kfstpairEq != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK8,sK9])],[f26,f27]) ).
thf(f27,plain,
( ? [X0: $i,X1: $i] :
( ( kfst @ ( kpair @ X1 @ X0 ) )
!= X1 )
=> ( sK9
!= ( kfst @ ( kpair @ sK9 @ sK8 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f26,plain,
( ( ( kfstpairEq = $true )
| ? [X0: $i,X1: $i] :
( ( kfst @ ( kpair @ X1 @ X0 ) )
!= X1 ) )
& ( ! [X2: $i,X3: $i] :
( ( kfst @ ( kpair @ X3 @ X2 ) )
= X3 )
| ( kfstpairEq != $true ) ) ),
inference(rectify,[status(thm)],[f25]) ).
thf(f25,plain,
( ( ( kfstpairEq = $true )
| ? [X0: $i,X1: $i] :
( ( kfst @ ( kpair @ X1 @ X0 ) )
!= X1 ) )
& ( ! [X0: $i,X1: $i] :
( ( kfst @ ( kpair @ X1 @ X0 ) )
= X1 )
| ( kfstpairEq != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f13]) ).
thf(f13,plain,
( ( kfstpairEq = $true )
<=> ! [X0: $i,X1: $i] :
( ( kfst @ ( kpair @ X1 @ X0 ) )
= X1 ) ),
inference(fool_elimination,[status(thm)],[f12]) ).
thf(f12,plain,
! [X0: $i,X1: $i] :
( ( ( kfst @ ( kpair @ X1 @ X0 ) )
= X1 )
= kfstpairEq ),
inference(rectify,[status(thm)],[f3]) ).
thf(f3,axiom,
! [X1: $i,X0: $i] :
( ( ( kfst @ ( kpair @ X0 @ X1 ) )
= X0 )
= kfstpairEq ),
file('/export/starexec/sandbox/tmp/tmp.i6FyzDOOiI/DTF2THF_19631.p',kfstpairEq) ).
thf(f47,plain,
! [X6: $i,X7: $i,X5: $i] :
( ( ( in @ ( sK3 @ X7 @ X6 @ X5 ) @ X6 )
= $true )
| ( $true
!= ( in @ X7 @ ( cartprod @ X6 @ X5 ) ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f44]) ).
thf(f44,plain,
! [X6: $i,X7: $i,X5: $i] :
( ( $true
!= ( in @ X7 @ ( cartprod @ X6 @ X5 ) ) )
| ( $true != $true )
| ( ( in @ ( sK3 @ X7 @ X6 @ X5 ) @ X6 )
= $true ) ),
inference(definition_unfolding,[status(thm)],[f29,f34]) ).
thf(f29,plain,
! [X6: $i,X7: $i,X5: $i] :
( ( ( in @ ( sK3 @ X7 @ X6 @ X5 ) @ X6 )
= $true )
| ( $true
!= ( in @ X7 @ ( cartprod @ X6 @ X5 ) ) )
| ( cartprodmempair1 != $true ) ),
inference(cnf_transformation,[status(thm)],[f22]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU646^2 : 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 : n003.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 12:29:40 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
% 0.33/0.48 ---- Original DTF file ---
% 0.33/0.48 thf(spec,logic,$$dhol).
% 0.33/0.48 %------------------------------------------------------------------------------
% 0.33/0.48 % File : SEU646^2 : TPTP v9.2.1. Released v3.7.0.
% 0.33/0.48 % Domain : Set Theory
% 0.33/0.48 % Problem : Ordered Pairs - Properties of Pairs
% 0.33/0.48 % Version : Especial > Reduced > Especial.
% 0.33/0.48 % English : (! A:i.! B:i.! u:i.in u (cartprod A B) -> in (kfst u) A)
% 0.33/0.48
% 0.33/0.48 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.33/0.48 % Source : [Bro08]
% 0.33/0.48 % Names : ZFC148l [Bro08]
% 0.33/0.48
% 0.33/0.48 % Status : Theorem
% 0.33/0.48 % 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, 0.40 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
% 0.33/0.48 % Syntax : Number of formulae : 12 ( 3 unt; 8 typ; 3 def)
% 0.33/0.48 % Number of atoms : 14 ( 5 equ; 0 cnn)
% 0.33/0.48 % Maximal formula atoms : 4 ( 3 avg)
% 0.33/0.48 % Number of connectives : 36 ( 0 ~; 0 |; 2 &; 30 @)
% 0.33/0.48 % ( 0 <=>; 4 =>; 0 <=; 0 <~>)
% 0.33/0.48 % Maximal formula depth : 10 ( 3 avg)
% 0.33/0.48 % Number of types : 2 ( 0 usr)
% 0.33/0.48 % Number of type conns : 9 ( 9 >; 0 *; 0 +; 0 <<)
% 0.33/0.48 % Number of symbols : 9 ( 8 usr; 3 con; 0-2 aty)
% 0.33/0.48 % Number of variables : 12 ( 2 ^; 8 !; 2 ?; 12 :)
% 0.33/0.48 % SPC : TH0_THM_EQU_NAR
% 0.33/0.48
% 0.33/0.48 % Comments : http://mathgate.info/detsetitem.php?id=203
% 0.33/0.48 %------------------------------------------------------------------------------
% 0.33/0.48 thf(in_type,type,
% 0.33/0.48 in: $i > $i > $o ).
% 0.33/0.48
% 0.33/0.48 thf(emptyset_type,type,
% 0.33/0.48 emptyset: $i ).
% 0.33/0.48
% 0.33/0.48 thf(setadjoin_type,type,
% 0.33/0.48 setadjoin: $i > $i > $i ).
% 0.33/0.48
% 0.33/0.48 thf(kpair_type,type,
% 0.33/0.48 kpair: $i > $i > $i ).
% 0.33/0.48
% 0.33/0.48 thf(kpair,definition,
% 0.33/0.48 ( kpair
% 0.33/0.48 = ( ^ [Xx: $i,Xy: $i] : ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ).
% 0.33/0.48
% 0.33/0.48 thf(cartprod_type,type,
% 0.33/0.48 cartprod: $i > $i > $i ).
% 0.33/0.48
% 0.33/0.48 thf(cartprodmempair1_type,type,
% 0.33/0.48 cartprodmempair1: $o ).
% 0.33/0.48
% 0.33/0.48 thf(cartprodmempair1,definition,
% 0.33/0.48 ( cartprodmempair1
% 0.33/0.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.33/0.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.33/0.48 => ? [Xx: $i] :
% 0.33/0.48 ( ( in @ Xx @ A )
% 0.33/0.48 & ? [Xy: $i] :
% 0.33/0.48 ( ( in @ Xy @ B )
% 0.33/0.48 & ( Xu
% 0.33/0.48 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.33/0.48
% 0.33/0.48 thf(kfst_type,type,
% 0.33/0.48 kfst: $i > $i ).
% 0.33/0.48
% 0.33/0.48 thf(kfstpairEq_type,type,
% 0.33/0.48 kfstpairEq: $o ).
% 0.33/0.48
% 0.33/0.48 thf(kfstpairEq,definition,
% 0.33/0.48 ( kfstpairEq
% 0.33/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.33/0.48 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.33/0.48 = Xx ) ) ) ).
% 0.33/0.48
% 0.33/0.48 thf(cartprodfstin,conjecture,
% 0.33/0.48 ( cartprodmempair1
% 0.33/0.48 => ( kfstpairEq
% 0.33/0.48 => ! [A: $i,B: $i,Xu: $i] :
% 0.33/0.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.33/0.48 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.33/0.48
% 0.33/0.48 %------------------------------------------------------------------------------
% 0.33/0.48 ------------------------
% 1.70/1.42 ---- Embedded in THF ---
% 1.70/1.42 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.70/1.42 %%% Generated on Tue Feb 24 12:29:41 EST 2026
% 1.70/1.42 %%% using '$$dhol' embedding, version 1.3.0.
% 1.70/1.42 %%% Logic specification used:
% 1.70/1.42 %%% thf(spec, logic, $$dhol).
% 1.70/1.42
% 1.70/1.42 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.i6FyzDOOiI/DTF2DTF19631.p
% See solution above
% 1.70/1.42 ------------------------
% 1.70/1.43 ---- Cleaned THF ---
% 1.70/1.43 thf(in_type,type,
% 1.70/1.43 in: $i > $i > $o ).
% 1.70/1.43
% 1.70/1.43 thf(emptyset_type,type,
% 1.70/1.43 emptyset: $i ).
% 1.70/1.43
% 1.70/1.43 thf(setadjoin_type,type,
% 1.70/1.43 setadjoin: $i > $i > $i ).
% 1.70/1.43
% 1.70/1.43 thf(kpair_type,type,
% 1.70/1.43 kpair: $i > $i > $i ).
% 1.70/1.43
% 1.70/1.43 thf(kpair,definition,
% 1.70/1.43 ( kpair
% 1.70/1.43 = ( ^ [Xx: $i,Xy: $i] : ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ).
% 1.70/1.43
% 1.70/1.43 thf(cartprod_type,type,
% 1.70/1.43 cartprod: $i > $i > $i ).
% 1.70/1.43
% 1.70/1.43 thf(cartprodmempair1_type,type,
% 1.70/1.43 cartprodmempair1: $o ).
% 1.70/1.43
% 1.70/1.43 thf(cartprodmempair1,definition,
% 1.70/1.43 ( cartprodmempair1
% 1.70/1.43 = ( ! [A: $i,B: $i,Xu: $i] :
% 1.70/1.43 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 1.70/1.43 => ? [Xx: $i] :
% 1.70/1.43 ( ( in @ Xx @ A )
% 1.70/1.43 & ? [Xy: $i] :
% 1.70/1.43 ( ( in @ Xy @ B )
% 1.70/1.43 & ( Xu
% 1.70/1.43 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 1.70/1.43
% 1.70/1.43 thf(kfst_type,type,
% 1.70/1.43 kfst: $i > $i ).
% 1.70/1.43
% 1.70/1.43 thf(kfstpairEq_type,type,
% 1.70/1.43 kfstpairEq: $o ).
% 1.70/1.43
% 1.70/1.43 thf(kfstpairEq,definition,
% 1.70/1.43 ( kfstpairEq
% 1.70/1.43 = ( ! [Xx: $i,Xy: $i] :
% 1.70/1.43 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 1.70/1.43 = Xx ) ) ) ).
% 1.70/1.43
% 1.70/1.43 thf(cartprodfstin,conjecture,
% 1.70/1.43 ( cartprodmempair1
% 1.70/1.43 => ( kfstpairEq
% 1.70/1.43 => ! [A: $i,B: $i,Xu: $i] :
% 1.70/1.43 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 1.70/1.43 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 1.70/1.43 ------------------------
% 1.70/1.45 % (19729)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_19631 for (2999ds/4Mi)
% 1.70/1.45 % (19734)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_19631 for (2999ds/18Mi)
% 1.70/1.45 % (19728)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_19631 for (2999ds/183Mi)
% 1.70/1.45 % (19732)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_19631 for (2999ds/2Mi)
% 1.70/1.45 % (19731)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_19631 for (2999ds/2Mi)
% 1.70/1.45 % (19733)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_19631 for (2999ds/275Mi)
% 1.70/1.45 % (19730)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_19631 for (2999ds/27Mi)
% 1.70/1.45 % (19731)Instruction limit reached!
% 1.70/1.45 % (19731)------------------------------
% 1.70/1.45 % (19731)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.70/1.45 % (19731)Termination reason: Unknown
% 1.70/1.45 % (19732)Instruction limit reached!
% 1.70/1.45 % (19732)------------------------------
% 1.70/1.45 % (19732)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.70/1.45 % (19731)Termination phase: Function definition elimination
% 1.70/1.45
% 1.70/1.45 % (19731)Memory used [KB]: 1023
% 1.70/1.45 % (19731)Time elapsed: 0.003 s
% 1.70/1.45 % (19731)Instructions burned: 2 (million)
% 1.70/1.45 % (19731)------------------------------
% 1.70/1.45 % (19731)------------------------------
% 1.70/1.45 % (19732)Termination reason: Unknown
% 1.70/1.45 % (19732)Termination phase: Property scanning
% 1.70/1.45
% 1.70/1.45 % (19732)Memory used [KB]: 1023
% 1.70/1.45 % (19732)Time elapsed: 0.003 s
% 1.70/1.45 % (19732)Instructions burned: 3 (million)
% 1.70/1.45 % (19732)------------------------------
% 1.70/1.45 % (19732)------------------------------
% 1.70/1.45 % (19729)Instruction limit reached!
% 1.70/1.45 % (19729)------------------------------
% 1.70/1.45 % (19729)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.70/1.45 % (19729)Termination reason: Unknown
% 1.70/1.45 % (19729)Termination phase: Saturation
% 1.70/1.45
% 1.70/1.45 % (19729)Memory used [KB]: 5500
% 1.70/1.45 % (19729)Time elapsed: 0.004 s
% 1.70/1.45 % (19729)Instructions burned: 4 (million)
% 1.70/1.45 % (19729)------------------------------
% 1.70/1.45 % (19729)------------------------------
% 1.70/1.45 % (19733)First to succeed.
% 1.70/1.45 % (19734)Also succeeded, but the first one will report.
% 1.70/1.45 % (19733)Refutation found. Thanks to Tanya!
% 1.70/1.45 % SZS status Theorem for DTF2THF_19631
% 1.70/1.45 % SZS output start Proof for DTF2THF_19631
% 1.70/1.45 thf(func_def_0, type, in: $i > $i > $o).
% 1.70/1.45 thf(func_def_2, type, setadjoin: $i > $i > $i).
% 1.70/1.45 thf(func_def_3, type, kpair: $i > $i > $i).
% 1.70/1.45 thf(func_def_5, type, cartprod: $i > $i > $i).
% 1.70/1.45 thf(func_def_7, type, kfst: $i > $i).
% 1.70/1.45 thf(func_def_17, type, sK3: $i > $i > $i > $i).
% 1.70/1.45 thf(func_def_18, type, sK4: $i > $i > $i > $i).
% 1.70/1.45 thf(f62,plain,(
% 1.70/1.45 $false),
% 1.70/1.45 inference(subsumption_resolution,[status(thm)],[f61,f35])).
% 1.70/1.45 thf(f35,plain,(
% 1.70/1.45 ((in @ (kfst @ sK7) @ sK6) != $true)),
% 1.70/1.45 inference(cnf_transformation,[status(thm)],[f24])).
% 1.70/1.45 thf(f24,plain,(
% 1.70/1.45 (kfstpairEq = $true) & (($true = (in @ sK7 @ (cartprod @ sK6 @ sK5))) & ((in @ (kfst @ sK7) @ sK6) != $true)) & (cartprodmempair1 = $true)),
% 1.70/1.45 inference(skolemisation,[status(esa),new_symbols(skolem,[sK5,sK6,sK7])],[f15,f23])).
% 1.70/1.45 thf(f23,plain,(
% 1.70/1.45 ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ (cartprod @ X1 @ X0)) = $true) & ($true != (in @ (kfst @ X2) @ X1))) => (($true = (in @ sK7 @ (cartprod @ sK6 @ sK5))) & ((in @ (kfst @ sK7) @ sK6) != $true))),
% 1.70/1.45 introduced(definition,[],[choice_axiom])).
% 1.70/1.45 thf(f15,plain,(
% 1.70/1.45 (kfstpairEq = $true) & ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ (cartprod @ X1 @ X0)) = $true) & ($true != (in @ (kfst @ X2) @ X1))) & (cartprodmempair1 = $true)),
% 1.70/1.45 inference(flattening,[status(thm)],[f14])).
% 1.70/1.45 thf(f14,plain,(
% 1.70/1.45 (? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ (cartprod @ X1 @ X0)) = $true) & ($true != (in @ (kfst @ X2) @ X1))) & (kfstpairEq = $true)) & (cartprodmempair1 = $true)),
% 1.70/1.45 inference(ennf_transformation,[status(thm)],[f11])).
% 1.70/1.45 thf(f11,plain,(
% 1.70/1.45 ~((cartprodmempair1 = $true) => ((kfstpairEq = $true) => ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ (cartprod @ X1 @ X0)) = $true) => ($true = (in @ (kfst @ X2) @ X1)))))),
% 1.70/1.45 inference(fool_elimination,[status(thm)],[f10])).
% 1.70/1.45 thf(f10,plain,(
% 1.70/1.45 ~(cartprodmempair1 => (kfstpairEq => ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X2 @ (cartprod @ X1 @ X0)) => (in @ (kfst @ X2) @ X1))))),
% 1.70/1.45 inference(rectify,[status(thm)],[f5])).
% 1.70/1.45 thf(f5,negated_conjecture,(
% 1.70/1.45 ~(cartprodmempair1 => (kfstpairEq => ! [X3 : $i,X2 : $i,X4 : $i] : ((in @ X4 @ (cartprod @ X2 @ X3)) => (in @ (kfst @ X4) @ X2))))),
% 1.70/1.45 inference(negated_conjecture,[status(cth)],[f4])).
% 1.70/1.45 thf(f4,conjecture,(
% 1.70/1.45 cartprodmempair1 => (kfstpairEq => ! [X3 : $i,X2 : $i,X4 : $i] : ((in @ X4 @ (cartprod @ X2 @ X3)) => (in @ (kfst @ X4) @ X2)))),
% 1.70/1.45 file('/export/starexec/sandbox/tmp/tmp.i6FyzDOOiI/DTF2THF_19631.p',cartprodfstin)).
% 1.70/1.45 thf(f61,plain,(
% 1.70/1.45 ((in @ (kfst @ sK7) @ sK6) = $true)),
% 1.70/1.45 inference(trivial_inequality_removal,[status(thm)],[f56])).
% 1.70/1.45 thf(f56,plain,(
% 1.70/1.45 ((in @ (kfst @ sK7) @ sK6) = $true) | ($true != $true)),
% 1.70/1.45 inference(superposition,[status(thm)],[f54,f36])).
% 1.70/1.45 thf(f36,plain,(
% 1.70/1.45 ($true = (in @ sK7 @ (cartprod @ sK6 @ sK5)))),
% 1.70/1.45 inference(cnf_transformation,[status(thm)],[f24])).
% 1.70/1.45 thf(f54,plain,(
% 1.70/1.45 ( ! [X2 : $i,X0 : $i,X1 : $i] : (($true != (in @ X0 @ (cartprod @ X1 @ X2))) | ($true = (in @ (kfst @ X0) @ X1))) )),
% 1.70/1.45 inference(duplicate_literal_removal,[status(thm)],[f53])).
% 1.70/1.45 thf(f53,plain,(
% 1.70/1.45 ( ! [X2 : $i,X0 : $i,X1 : $i] : (($true != (in @ X0 @ (cartprod @ X1 @ X2))) | ($true != (in @ X0 @ (cartprod @ X1 @ X2))) | ($true = (in @ (kfst @ X0) @ X1))) )),
% 1.70/1.45 inference(superposition,[status(thm)],[f47,f51])).
% 1.70/1.45 thf(f51,plain,(
% 1.70/1.45 ( ! [X2 : $i,X0 : $i,X1 : $i] : (((sK3 @ X0 @ X1 @ X2) = (kfst @ X0)) | ($true != (in @ X0 @ (cartprod @ X1 @ X2)))) )),
% 1.70/1.45 inference(superposition,[status(thm)],[f49,f50])).
% 1.70/1.45 thf(f50,plain,(
% 1.70/1.45 ( ! [X6 : $i,X7 : $i,X5 : $i] : (((kpair @ (sK3 @ X7 @ X6 @ X5) @ (sK4 @ X7 @ X6 @ X5)) = X7) | ($true != (in @ X7 @ (cartprod @ X6 @ X5)))) )),
% 1.70/1.45 inference(trivial_inequality_removal,[status(thm)],[f42])).
% 1.70/1.45 thf(f42,plain,(
% 1.70/1.45 ( ! [X6 : $i,X7 : $i,X5 : $i] : (($true != $true) | ($true != (in @ X7 @ (cartprod @ X6 @ X5))) | ((kpair @ (sK3 @ X7 @ X6 @ X5) @ (sK4 @ X7 @ X6 @ X5)) = X7)) )),
% 1.70/1.45 inference(definition_unfolding,[status(thm)],[f31,f34])).
% 1.70/1.45 thf(f34,plain,(
% 1.70/1.45 (cartprodmempair1 = $true)),
% 1.70/1.45 inference(cnf_transformation,[status(thm)],[f24])).
% 1.70/1.45 thf(f31,plain,(
% 1.70/1.45 ( ! [X6 : $i,X7 : $i,X5 : $i] : (((kpair @ (sK3 @ X7 @ X6 @ X5) @ (sK4 @ X7 @ X6 @ X5)) = X7) | ($true != (in @ X7 @ (cartprod @ X6 @ X5))) | (cartprodmempair1 != $true)) )),
% 1.70/1.45 inference(cnf_transformation,[status(thm)],[f22])).
% 1.70/1.45 thf(f22,plain,(
% 1.70/1.45 ((cartprodmempair1 = $true) | (! [X3 : $i] : (! [X4 : $i] : (((kpair @ X3 @ X4) != sK2) | ($true != (in @ X4 @ sK0))) | ($true != (in @ X3 @ sK1))) & ((in @ sK2 @ (cartprod @ sK1 @ sK0)) = $true))) & (! [X5 : $i,X6 : $i,X7 : $i] : (((((kpair @ (sK3 @ X7 @ X6 @ X5) @ (sK4 @ X7 @ X6 @ X5)) = X7) & ($true = (in @ (sK4 @ X7 @ X6 @ X5) @ X5))) & ((in @ (sK3 @ X7 @ X6 @ X5) @ X6) = $true)) | ($true != (in @ X7 @ (cartprod @ X6 @ X5)))) | (cartprodmempair1 != $true))),
% 1.70/1.45 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2,sK3,sK4])],[f18,f21,f20,f19])).
% 1.70/1.45 thf(f19,plain,(
% 1.70/1.45 ? [X0 : $i,X1 : $i,X2 : $i] : (! [X3 : $i] : (! [X4 : $i] : (((kpair @ X3 @ X4) != X2) | ($true != (in @ X4 @ X0))) | ($true != (in @ X3 @ X1))) & ((in @ X2 @ (cartprod @ X1 @ X0)) = $true)) => (! [X3 : $i] : (! [X4 : $i] : (((kpair @ X3 @ X4) != sK2) | ($true != (in @ X4 @ sK0))) | ($true != (in @ X3 @ sK1))) & ((in @ sK2 @ (cartprod @ sK1 @ sK0)) = $true))),
% 1.70/1.45 introduced(definition,[],[choice_axiom])).
% 1.70/1.45 thf(f20,plain,(
% 1.70/1.45 ! [X5 : $i,X6 : $i,X7 : $i] : (? [X8 : $i] : (? [X9 : $i] : (((kpair @ X8 @ X9) = X7) & ($true = (in @ X9 @ X5))) & ((in @ X8 @ X6) = $true)) => (? [X9 : $i] : (((kpair @ (sK3 @ X7 @ X6 @ X5) @ X9) = X7) & ($true = (in @ X9 @ X5))) & ((in @ (sK3 @ X7 @ X6 @ X5) @ X6) = $true)))),
% 1.70/1.45 introduced(definition,[],[choice_axiom])).
% 1.70/1.45 thf(f21,plain,(
% 1.70/1.45 ! [X5 : $i,X6 : $i,X7 : $i] : (? [X9 : $i] : (((kpair @ (sK3 @ X7 @ X6 @ X5) @ X9) = X7) & ($true = (in @ X9 @ X5))) => (((kpair @ (sK3 @ X7 @ X6 @ X5) @ (sK4 @ X7 @ X6 @ X5)) = X7) & ($true = (in @ (sK4 @ X7 @ X6 @ X5) @ X5))))),
% 1.70/1.45 introduced(definition,[],[choice_axiom])).
% 1.70/1.45 thf(f18,plain,(
% 1.70/1.45 ((cartprodmempair1 = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (! [X3 : $i] : (! [X4 : $i] : (((kpair @ X3 @ X4) != X2) | ($true != (in @ X4 @ X0))) | ($true != (in @ X3 @ X1))) & ((in @ X2 @ (cartprod @ X1 @ X0)) = $true))) & (! [X5 : $i,X6 : $i,X7 : $i] : (? [X8 : $i] : (? [X9 : $i] : (((kpair @ X8 @ X9) = X7) & ($true = (in @ X9 @ X5))) & ((in @ X8 @ X6) = $true)) | ($true != (in @ X7 @ (cartprod @ X6 @ X5)))) | (cartprodmempair1 != $true))),
% 1.70/1.45 inference(rectify,[status(thm)],[f17])).
% 1.70/1.45 thf(f17,plain,(
% 1.70/1.45 ((cartprodmempair1 = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (! [X3 : $i] : (! [X4 : $i] : (((kpair @ X3 @ X4) != X2) | ($true != (in @ X4 @ X0))) | ($true != (in @ X3 @ X1))) & ((in @ X2 @ (cartprod @ X1 @ X0)) = $true))) & (! [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : (? [X4 : $i] : (((kpair @ X3 @ X4) = X2) & ($true = (in @ X4 @ X0))) & ($true = (in @ X3 @ X1))) | ((in @ X2 @ (cartprod @ X1 @ X0)) != $true)) | (cartprodmempair1 != $true))),
% 1.70/1.45 inference(nnf_transformation,[status(thm)],[f16])).
% 1.70/1.45 thf(f16,plain,(
% 1.70/1.45 (cartprodmempair1 = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : (? [X4 : $i] : (((kpair @ X3 @ X4) = X2) & ($true = (in @ X4 @ X0))) & ($true = (in @ X3 @ X1))) | ((in @ X2 @ (cartprod @ X1 @ X0)) != $true))),
% 1.70/1.45 inference(ennf_transformation,[status(thm)],[f9])).
% 1.70/1.45 thf(f9,plain,(
% 1.70/1.45 (cartprodmempair1 = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ (cartprod @ X1 @ X0)) = $true) => ? [X3 : $i] : (? [X4 : $i] : (((kpair @ X3 @ X4) = X2) & ($true = (in @ X4 @ X0))) & ($true = (in @ X3 @ X1))))),
% 1.70/1.45 inference(fool_elimination,[status(thm)],[f8])).
% 1.70/1.45 thf(f8,plain,(
% 1.70/1.45 (! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X2 @ (cartprod @ X1 @ X0)) => ? [X3 : $i] : ((in @ X3 @ X1) & ? [X4 : $i] : ((in @ X4 @ X0) & ((kpair @ X3 @ X4) = X2)))) = cartprodmempair1)),
% 1.70/1.45 inference(rectify,[status(thm)],[f2])).
% 1.70/1.45 thf(f2,axiom,(
% 1.70/1.45 (! [X3 : $i,X2 : $i,X4 : $i] : ((in @ X4 @ (cartprod @ X2 @ X3)) => ? [X0 : $i] : ((in @ X0 @ X2) & ? [X1 : $i] : ((in @ X1 @ X3) & ((kpair @ X0 @ X1) = X4)))) = cartprodmempair1)),
% 1.70/1.45 file('/export/starexec/sandbox/tmp/tmp.i6FyzDOOiI/DTF2THF_19631.p',cartprodmempair1)).
% 1.70/1.45 thf(f49,plain,(
% 1.70/1.45 ( ! [X2 : $i,X3 : $i] : (((kfst @ (kpair @ X3 @ X2)) = X3)) )),
% 1.70/1.45 inference(trivial_inequality_removal,[status(thm)],[f46])).
% 1.70/1.45 thf(f46,plain,(
% 1.70/1.45 ( ! [X2 : $i,X3 : $i] : (((kfst @ (kpair @ X3 @ X2)) = X3) | ($true != $true)) )),
% 1.70/1.45 inference(definition_unfolding,[status(thm)],[f38,f37])).
% 1.70/1.45 thf(f37,plain,(
% 1.70/1.45 (kfstpairEq = $true)),
% 1.70/1.45 inference(cnf_transformation,[status(thm)],[f24])).
% 1.70/1.45 thf(f38,plain,(
% 1.70/1.45 ( ! [X2 : $i,X3 : $i] : (((kfst @ (kpair @ X3 @ X2)) = X3) | (kfstpairEq != $true)) )),
% 1.70/1.45 inference(cnf_transformation,[status(thm)],[f28])).
% 1.70/1.45 thf(f28,plain,(
% 1.70/1.45 ((kfstpairEq = $true) | (sK9 != (kfst @ (kpair @ sK9 @ sK8)))) & (! [X2 : $i,X3 : $i] : ((kfst @ (kpair @ X3 @ X2)) = X3) | (kfstpairEq != $true))),
% 1.70/1.45 inference(skolemisation,[status(esa),new_symbols(skolem,[sK8,sK9])],[f26,f27])).
% 1.70/1.45 thf(f27,plain,(
% 1.70/1.45 ? [X0 : $i,X1 : $i] : ((kfst @ (kpair @ X1 @ X0)) != X1) => (sK9 != (kfst @ (kpair @ sK9 @ sK8)))),
% 1.70/1.45 introduced(definition,[],[choice_axiom])).
% 1.70/1.45 thf(f26,plain,(
% 1.70/1.45 ((kfstpairEq = $true) | ? [X0 : $i,X1 : $i] : ((kfst @ (kpair @ X1 @ X0)) != X1)) & (! [X2 : $i,X3 : $i] : ((kfst @ (kpair @ X3 @ X2)) = X3) | (kfstpairEq != $true))),
% 1.70/1.45 inference(rectify,[status(thm)],[f25])).
% 1.70/1.45 thf(f25,plain,(
% 1.70/1.45 ((kfstpairEq = $true) | ? [X0 : $i,X1 : $i] : ((kfst @ (kpair @ X1 @ X0)) != X1)) & (! [X0 : $i,X1 : $i] : ((kfst @ (kpair @ X1 @ X0)) = X1) | (kfstpairEq != $true))),
% 1.70/1.45 inference(nnf_transformation,[status(thm)],[f13])).
% 1.70/1.45 thf(f13,plain,(
% 1.70/1.45 (kfstpairEq = $true) <=> ! [X0 : $i,X1 : $i] : ((kfst @ (kpair @ X1 @ X0)) = X1)),
% 1.70/1.45 inference(fool_elimination,[status(thm)],[f12])).
% 1.70/1.45 thf(f12,plain,(
% 1.70/1.45 (! [X0 : $i,X1 : $i] : ((kfst @ (kpair @ X1 @ X0)) = X1) = kfstpairEq)),
% 1.70/1.45 inference(rectify,[status(thm)],[f3])).
% 1.70/1.45 thf(f3,axiom,(
% 1.70/1.45 (! [X1 : $i,X0 : $i] : ((kfst @ (kpair @ X0 @ X1)) = X0) = kfstpairEq)),
% 1.70/1.45 file('/export/starexec/sandbox/tmp/tmp.i6FyzDOOiI/DTF2THF_19631.p',kfstpairEq)).
% 1.70/1.45 thf(f47,plain,(
% 1.70/1.45 ( ! [X6 : $i,X7 : $i,X5 : $i] : (((in @ (sK3 @ X7 @ X6 @ X5) @ X6) = $true) | ($true != (in @ X7 @ (cartprod @ X6 @ X5)))) )),
% 1.70/1.45 inference(trivial_inequality_removal,[status(thm)],[f44])).
% 1.70/1.45 thf(f44,plain,(
% 1.70/1.45 ( ! [X6 : $i,X7 : $i,X5 : $i] : (($true != (in @ X7 @ (cartprod @ X6 @ X5))) | ($true != $true) | ((in @ (sK3 @ X7 @ X6 @ X5) @ X6) = $true)) )),
% 1.70/1.45 inference(definition_unfolding,[status(thm)],[f29,f34])).
% 1.70/1.45 thf(f29,plain,(
% 1.70/1.45 ( ! [X6 : $i,X7 : $i,X5 : $i] : (((in @ (sK3 @ X7 @ X6 @ X5) @ X6) = $true) | ($true != (in @ X7 @ (cartprod @ X6 @ X5))) | (cartprodmempair1 != $true)) )),
% 1.70/1.45 inference(cnf_transformation,[status(thm)],[f22])).
% 1.70/1.45 % SZS output end Proof for DTF2THF_19631
% 1.70/1.45 % (19733)------------------------------
% 1.70/1.45 % (19733)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.70/1.45 % (19733)Termination reason: Refutation
% 1.70/1.45
% 1.70/1.45 % (19733)Memory used [KB]: 5500
% 1.70/1.45 % (19733)Time elapsed: 0.007 s
% 1.70/1.45 % (19733)Instructions burned: 6 (million)
% 1.70/1.45 % (19733)------------------------------
% 1.70/1.45 % (19733)------------------------------
% 1.70/1.45 % (19727)Success in time 0.022 s
%------------------------------------------------------------------------------