%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU628^2 : 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 09:35:43 AM UTC 2026
% Result : Theorem 1.58s 1.38s
% Output : Refutation 1.58s
% Verified :
% SZS Type : Refutation
% Derivation depth : 16
% Number of leaves : 8
% Syntax : Number of formulae : 44 ( 12 unt; 0 typ; 0 def)
% Number of atoms : 637 ( 142 equ; 0 cnn)
% Maximal formula atoms : 8 ( 14 avg)
% Number of connectives : 874 ( 64 ~; 41 |; 45 &; 690 @)
% ( 4 <=>; 30 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 21 ( 18 usr; 15 con; 0-2 aty)
% Number of variables : 114 ( 0 ^; 76 !; 38 ?; 114 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_2,type,
setadjoin: $i > $i > $i ).
thf(func_def_3,type,
powerset: $i > $i ).
thf(func_def_4,type,
subset: $i > $i > $o ).
thf(func_def_7,type,
binunion: $i > $i > $i ).
thf(f55,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f54,f30]) ).
thf(f30,plain,
( ( in @ sK0 @ sK2 )
= $true ),
inference(cnf_transformation,[status(thm)],[f19]) ).
thf(f19,plain,
( ( ubforcartprodlem1 = $true )
& ( ( in @ sK3 @ sK1 )
= $true )
& ( ( in @ ( setadjoin @ ( setadjoin @ sK0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK0 @ ( setadjoin @ sK3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ sK2 @ sK1 ) ) ) )
!= $true )
& ( ( in @ sK0 @ sK2 )
= $true )
& ( powersetI1 = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2,sK3])],[f16,f18,f17]) ).
thf(f17,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ X1 )
= $true )
& ( $true
!= ( in @ ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X0 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X2 @ X1 ) ) ) ) ) )
& ( $true
= ( in @ X0 @ X2 ) ) )
=> ( ? [X3: $i] :
( ( $true
= ( in @ X3 @ sK1 ) )
& ( $true
!= ( in @ ( setadjoin @ ( setadjoin @ sK0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK0 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ sK2 @ sK1 ) ) ) ) ) )
& ( ( in @ sK0 @ sK2 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f18,plain,
( ? [X3: $i] :
( ( $true
= ( in @ X3 @ sK1 ) )
& ( $true
!= ( in @ ( setadjoin @ ( setadjoin @ sK0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK0 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ sK2 @ sK1 ) ) ) ) ) )
=> ( ( ( in @ sK3 @ sK1 )
= $true )
& ( ( in @ ( setadjoin @ ( setadjoin @ sK0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK0 @ ( setadjoin @ sK3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ sK2 @ sK1 ) ) ) )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f16,plain,
( ( ubforcartprodlem1 = $true )
& ? [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ X1 )
= $true )
& ( $true
!= ( in @ ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X0 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X2 @ X1 ) ) ) ) ) )
& ( $true
= ( in @ X0 @ X2 ) ) )
& ( powersetI1 = $true ) ),
inference(rectify,[status(thm)],[f15]) ).
thf(f15,plain,
( ( ubforcartprodlem1 = $true )
& ? [X1: $i,X0: $i,X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ X0 )
= $true )
& ( ( in @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X2 @ X0 ) ) ) )
!= $true ) )
& ( ( in @ X1 @ X2 )
= $true ) )
& ( powersetI1 = $true ) ),
inference(flattening,[status(thm)],[f14]) ).
thf(f14,plain,
( ? [X1: $i,X0: $i,X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ X0 )
= $true )
& ( ( in @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X2 @ X0 ) ) ) )
!= $true ) )
& ( ( in @ X1 @ X2 )
= $true ) )
& ( ubforcartprodlem1 = $true )
& ( powersetI1 = $true ) ),
inference(ennf_transformation,[status(thm)],[f11]) ).
thf(f11,plain,
~ ( ( powersetI1 = $true )
=> ( ( ubforcartprodlem1 = $true )
=> ! [X0: $i,X2: $i,X1: $i] :
( ( ( in @ X1 @ X2 )
= $true )
=> ! [X3: $i] :
( ( ( in @ X3 @ X0 )
= $true )
=> ( ( in @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X2 @ X0 ) ) ) )
= $true ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f10]) ).
thf(f10,plain,
~ ( powersetI1
=> ( ubforcartprodlem1
=> ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X1 @ X2 )
=> ! [X3: $i] :
( ( in @ X3 @ X0 )
=> ( in @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X2 @ X0 ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f4]) ).
thf(f4,negated_conjecture,
~ ( powersetI1
=> ( ubforcartprodlem1
=> ! [X1: $i,X2: $i,X0: $i] :
( ( in @ X2 @ X0 )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ ( setadjoin @ X2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X0 @ X1 ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f3]) ).
thf(f3,conjecture,
( powersetI1
=> ( ubforcartprodlem1
=> ! [X1: $i,X2: $i,X0: $i] :
( ( in @ X2 @ X0 )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ ( setadjoin @ X2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X0 @ X1 ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.O9ONI9QTIu/DTF2THF_3818.p',ubforcartprodlem2) ).
thf(f54,plain,
( ( in @ sK0 @ sK2 )
!= $true ),
inference(subsumption_resolution,[status(thm)],[f53,f32]) ).
thf(f32,plain,
( ( in @ sK3 @ sK1 )
= $true ),
inference(cnf_transformation,[status(thm)],[f19]) ).
thf(f53,plain,
( ( ( in @ sK3 @ sK1 )
!= $true )
| ( ( in @ sK0 @ sK2 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f52]) ).
thf(f52,plain,
( ( ( in @ sK0 @ sK2 )
!= $true )
| ( ( in @ sK3 @ sK1 )
!= $true )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f51,f49]) ).
thf(f49,plain,
! [X6: $i,X7: $i,X4: $i,X5: $i] :
( ( ( subset @ ( setadjoin @ ( setadjoin @ X4 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X4 @ ( setadjoin @ X7 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X6 @ X5 ) ) )
= $true )
| ( $true
!= ( in @ X4 @ X6 ) )
| ( $true
!= ( in @ X7 @ X5 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f47]) ).
thf(f47,plain,
! [X6: $i,X7: $i,X4: $i,X5: $i] :
( ( $true
!= ( in @ X4 @ X6 ) )
| ( ( subset @ ( setadjoin @ ( setadjoin @ X4 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X4 @ ( setadjoin @ X7 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X6 @ X5 ) ) )
= $true )
| ( $true
!= ( in @ X7 @ X5 ) )
| ( $true != $true ) ),
inference(definition_unfolding,[status(thm)],[f37,f33]) ).
thf(f33,plain,
ubforcartprodlem1 = $true,
inference(cnf_transformation,[status(thm)],[f19]) ).
thf(f37,plain,
! [X6: $i,X7: $i,X4: $i,X5: $i] :
( ( $true
!= ( in @ X7 @ X5 ) )
| ( ( subset @ ( setadjoin @ ( setadjoin @ X4 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X4 @ ( setadjoin @ X7 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X6 @ X5 ) ) )
= $true )
| ( $true
!= ( in @ X4 @ X6 ) )
| ( ubforcartprodlem1 != $true ) ),
inference(cnf_transformation,[status(thm)],[f28]) ).
thf(f28,plain,
( ( ( ubforcartprodlem1 = $true )
| ( ( ( in @ sK9 @ sK7 )
= $true )
& ( ( subset @ ( setadjoin @ ( setadjoin @ sK6 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK6 @ ( setadjoin @ sK9 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ sK8 @ sK7 ) ) )
!= $true )
& ( ( in @ sK6 @ sK8 )
= $true ) ) )
& ( ! [X4: $i,X5: $i,X6: $i] :
( ! [X7: $i] :
( ( $true
!= ( in @ X7 @ X5 ) )
| ( ( subset @ ( setadjoin @ ( setadjoin @ X4 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X4 @ ( setadjoin @ X7 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X6 @ X5 ) ) )
= $true ) )
| ( $true
!= ( in @ X4 @ X6 ) ) )
| ( ubforcartprodlem1 != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK6,sK7,sK8,sK9])],[f25,f27,f26]) ).
thf(f26,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ X1 )
= $true )
& ( $true
!= ( subset @ ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X0 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X2 @ X1 ) ) ) ) )
& ( $true
= ( in @ X0 @ X2 ) ) )
=> ( ? [X3: $i] :
( ( ( in @ X3 @ sK7 )
= $true )
& ( ( subset @ ( setadjoin @ ( setadjoin @ sK6 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK6 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ sK8 @ sK7 ) ) )
!= $true ) )
& ( ( in @ sK6 @ sK8 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f27,plain,
( ? [X3: $i] :
( ( ( in @ X3 @ sK7 )
= $true )
& ( ( subset @ ( setadjoin @ ( setadjoin @ sK6 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK6 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ sK8 @ sK7 ) ) )
!= $true ) )
=> ( ( ( in @ sK9 @ sK7 )
= $true )
& ( ( subset @ ( setadjoin @ ( setadjoin @ sK6 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK6 @ ( setadjoin @ sK9 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ sK8 @ sK7 ) ) )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f25,plain,
( ( ( ubforcartprodlem1 = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ X1 )
= $true )
& ( $true
!= ( subset @ ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X0 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X2 @ X1 ) ) ) ) )
& ( $true
= ( in @ X0 @ X2 ) ) ) )
& ( ! [X4: $i,X5: $i,X6: $i] :
( ! [X7: $i] :
( ( $true
!= ( in @ X7 @ X5 ) )
| ( ( subset @ ( setadjoin @ ( setadjoin @ X4 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X4 @ ( setadjoin @ X7 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X6 @ X5 ) ) )
= $true ) )
| ( $true
!= ( in @ X4 @ X6 ) ) )
| ( ubforcartprodlem1 != $true ) ) ),
inference(rectify,[status(thm)],[f24]) ).
thf(f24,plain,
( ( ( ubforcartprodlem1 = $true )
| ? [X2: $i,X1: $i,X0: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ X1 )
= $true )
& ( ( subset @ ( setadjoin @ ( setadjoin @ X2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X0 @ X1 ) ) )
!= $true ) )
& ( ( in @ X2 @ X0 )
= $true ) ) )
& ( ! [X2: $i,X1: $i,X0: $i] :
( ! [X3: $i] :
( ( ( in @ X3 @ X1 )
!= $true )
| ( ( subset @ ( setadjoin @ ( setadjoin @ X2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X0 @ X1 ) ) )
= $true ) )
| ( ( in @ X2 @ X0 )
!= $true ) )
| ( ubforcartprodlem1 != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f13]) ).
thf(f13,plain,
( ( ubforcartprodlem1 = $true )
<=> ! [X2: $i,X1: $i,X0: $i] :
( ! [X3: $i] :
( ( ( in @ X3 @ X1 )
!= $true )
| ( ( subset @ ( setadjoin @ ( setadjoin @ X2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X0 @ X1 ) ) )
= $true ) )
| ( ( in @ X2 @ X0 )
!= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f7]) ).
thf(f7,plain,
( ( ubforcartprodlem1 = $true )
<=> ! [X2: $i,X1: $i,X0: $i] :
( ( ( in @ X2 @ X0 )
= $true )
=> ! [X3: $i] :
( ( ( in @ X3 @ X1 )
= $true )
=> ( ( subset @ ( setadjoin @ ( setadjoin @ X2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X0 @ X1 ) ) )
= $true ) ) ) ),
inference(fool_elimination,[status(thm)],[f6]) ).
thf(f6,plain,
! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X0 )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X0 @ X1 ) ) ) ) )
= ubforcartprodlem1 ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X0 )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X0 @ X1 ) ) ) ) )
= ubforcartprodlem1 ),
file('/export/starexec/sandbox/tmp/tmp.O9ONI9QTIu/DTF2THF_3818.p',ubforcartprodlem1) ).
thf(f51,plain,
( $true
!= ( subset @ ( setadjoin @ ( setadjoin @ sK0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK0 @ ( setadjoin @ sK3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ sK2 @ sK1 ) ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f50]) ).
thf(f50,plain,
( ( $true
!= ( subset @ ( setadjoin @ ( setadjoin @ sK0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK0 @ ( setadjoin @ sK3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ sK2 @ sK1 ) ) ) )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f31,f48]) ).
thf(f48,plain,
! [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
| ( $true
!= ( subset @ X0 @ X1 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f41]) ).
thf(f41,plain,
! [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
| ( $true
!= ( subset @ X0 @ X1 ) )
| ( $true != $true ) ),
inference(definition_unfolding,[status(thm)],[f36,f29]) ).
thf(f29,plain,
powersetI1 = $true,
inference(cnf_transformation,[status(thm)],[f19]) ).
thf(f36,plain,
! [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
| ( $true
!= ( subset @ X0 @ X1 ) )
| ( powersetI1 != $true ) ),
inference(cnf_transformation,[status(thm)],[f23]) ).
thf(f23,plain,
( ( ! [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
| ( $true
!= ( subset @ X0 @ X1 ) ) )
| ( powersetI1 != $true ) )
& ( ( powersetI1 = $true )
| ( ( ( in @ sK4 @ ( powerset @ sK5 ) )
!= $true )
& ( $true
= ( subset @ sK4 @ sK5 ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK4,sK5])],[f21,f22]) ).
thf(f22,plain,
( ? [X2: $i,X3: $i] :
( ( ( in @ X2 @ ( powerset @ X3 ) )
!= $true )
& ( $true
= ( subset @ X2 @ X3 ) ) )
=> ( ( ( in @ sK4 @ ( powerset @ sK5 ) )
!= $true )
& ( $true
= ( subset @ sK4 @ sK5 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f21,plain,
( ( ! [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
| ( $true
!= ( subset @ X0 @ X1 ) ) )
| ( powersetI1 != $true ) )
& ( ( powersetI1 = $true )
| ? [X2: $i,X3: $i] :
( ( ( in @ X2 @ ( powerset @ X3 ) )
!= $true )
& ( $true
= ( subset @ X2 @ X3 ) ) ) ) ),
inference(rectify,[status(thm)],[f20]) ).
thf(f20,plain,
( ( ! [X1: $i,X0: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
= $true )
| ( ( subset @ X1 @ X0 )
!= $true ) )
| ( powersetI1 != $true ) )
& ( ( powersetI1 = $true )
| ? [X1: $i,X0: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
!= $true )
& ( ( subset @ X1 @ X0 )
= $true ) ) ) ),
inference(nnf_transformation,[status(thm)],[f12]) ).
thf(f12,plain,
( ! [X1: $i,X0: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
= $true )
| ( ( subset @ X1 @ X0 )
!= $true ) )
<=> ( powersetI1 = $true ) ),
inference(ennf_transformation,[status(thm)],[f9]) ).
thf(f9,plain,
( ( powersetI1 = $true )
<=> ! [X1: $i,X0: $i] :
( ( ( subset @ X1 @ X0 )
= $true )
=> ( ( in @ X1 @ ( powerset @ X0 ) )
= $true ) ) ),
inference(fool_elimination,[status(thm)],[f8]) ).
thf(f8,plain,
( powersetI1
= ( ! [X0: $i,X1: $i] :
( ( subset @ X1 @ X0 )
=> ( in @ X1 @ ( powerset @ X0 ) ) ) ) ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
( powersetI1
= ( ! [X0: $i,X1: $i] :
( ( subset @ X1 @ X0 )
=> ( in @ X1 @ ( powerset @ X0 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.O9ONI9QTIu/DTF2THF_3818.p',powersetI1) ).
thf(f31,plain,
( ( in @ ( setadjoin @ ( setadjoin @ sK0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK0 @ ( setadjoin @ sK3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ sK2 @ sK1 ) ) ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f19]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.09 % Problem : SEU628^2 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.10 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.11/0.30 % Computer : n005.cluster.edu
% 0.11/0.30 % Model : x86_64 x86_64
% 0.11/0.30 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.30 % Memory : 8042.1875MB
% 0.11/0.30 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.30 % CPULimit : 300
% 0.11/0.30 % WCLimit : 300
% 0.11/0.30 % DateTime : Tue Feb 24 12:28:44 EST 2026
% 0.11/0.30 % CPUTime :
% 0.11/0.30 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.39 ---- Original DTF file ---
% 0.17/0.39 thf(spec,logic,$$dhol).
% 0.17/0.39 %------------------------------------------------------------------------------
% 0.17/0.39 % File : SEU628^2 : TPTP v9.2.1. Released v3.7.0.
% 0.17/0.39 % Domain : Set Theory
% 0.17/0.39 % Problem : Ordered Pairs - Cartesian Products
% 0.17/0.39 % Version : Especial > Reduced > Especial.
% 0.17/0.39 % English : (! A:i.! B:i.! x:i.in x A -> (! y:i.in y B -> in (setadjoin
% 0.17/0.39 % (setadjoin x emptyset) (setadjoin (setadjoin x
% 0.17/0.39 % (setadjoin y emptyset)) emptyset)) (powerset (powerset
% 0.17/0.39 % (binunion A B)))))
% 0.17/0.39
% 0.17/0.39 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.17/0.39 % Source : [Bro08]
% 0.17/0.39 % Names : ZFC130l [Bro08]
% 0.17/0.39
% 0.17/0.39 % Status : Theorem
% 0.17/0.39 % Rating : 0.00 v8.2.0, 0.15 v8.1.0, 0.09 v7.5.0, 0.00 v6.0.0, 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 v3.7.0
% 0.17/0.39 % Syntax : Number of formulae : 11 ( 2 unt; 8 typ; 2 def)
% 0.17/0.39 % Number of atoms : 14 ( 2 equ; 0 cnn)
% 0.17/0.39 % Maximal formula atoms : 5 ( 4 avg)
% 0.17/0.39 % Number of connectives : 51 ( 0 ~; 0 |; 0 &; 44 @)
% 0.17/0.39 % ( 0 <=>; 7 =>; 0 <=; 0 <~>)
% 0.17/0.39 % Maximal formula depth : 17 ( 6 avg)
% 0.17/0.39 % Number of types : 2 ( 0 usr)
% 0.17/0.39 % Number of type conns : 9 ( 9 >; 0 *; 0 +; 0 <<)
% 0.17/0.39 % Number of symbols : 9 ( 8 usr; 3 con; 0-2 aty)
% 0.17/0.39 % Number of variables : 10 ( 0 ^; 10 !; 0 ?; 10 :)
% 0.17/0.39 % SPC : TH0_THM_EQU_NAR
% 0.17/0.39
% 0.17/0.39 % Comments : http://mathgate.info/detsetitem.php?id=185
% 0.17/0.39 %------------------------------------------------------------------------------
% 0.17/0.39 thf(in_type,type,
% 0.17/0.39 in: $i > $i > $o ).
% 0.17/0.39
% 0.17/0.39 thf(emptyset_type,type,
% 0.17/0.39 emptyset: $i ).
% 0.17/0.39
% 0.17/0.39 thf(setadjoin_type,type,
% 0.17/0.39 setadjoin: $i > $i > $i ).
% 0.17/0.39
% 0.17/0.39 thf(powerset_type,type,
% 0.17/0.39 powerset: $i > $i ).
% 0.17/0.39
% 0.17/0.39 thf(subset_type,type,
% 0.17/0.39 subset: $i > $i > $o ).
% 0.17/0.39
% 0.17/0.39 thf(powersetI1_type,type,
% 0.17/0.39 powersetI1: $o ).
% 0.17/0.39
% 0.17/0.39 thf(powersetI1,definition,
% 0.17/0.39 ( powersetI1
% 0.17/0.39 = ( ! [A: $i,B: $i] :
% 0.17/0.39 ( ( subset @ B @ A )
% 0.17/0.39 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(binunion_type,type,
% 0.17/0.39 binunion: $i > $i > $i ).
% 0.17/0.39
% 0.17/0.39 thf(ubforcartprodlem1_type,type,
% 0.17/0.39 ubforcartprodlem1: $o ).
% 0.17/0.39
% 0.17/0.39 thf(ubforcartprodlem1,definition,
% 0.17/0.39 ( ubforcartprodlem1
% 0.17/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 => ! [Xy: $i] :
% 0.17/0.39 ( ( in @ Xy @ B )
% 0.17/0.39 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(ubforcartprodlem2,conjecture,
% 0.17/0.39 ( powersetI1
% 0.17/0.39 => ( ubforcartprodlem1
% 0.17/0.39 => ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 => ! [Xy: $i] :
% 0.17/0.39 ( ( in @ Xy @ B )
% 0.17/0.39 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 %------------------------------------------------------------------------------
% 0.17/0.39 ------------------------
% 1.58/1.34 ---- Embedded in THF ---
% 1.58/1.34 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.58/1.34 %%% Generated on Tue Feb 24 12:28:45 EST 2026
% 1.58/1.34 %%% using '$$dhol' embedding, version 1.3.0.
% 1.58/1.34 %%% Logic specification used:
% 1.58/1.34 %%% thf(spec, logic, $$dhol).
% 1.58/1.34
% 1.58/1.34 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.O9ONI9QTIu/DTF2DTF3818.p
% See solution above
% 1.58/1.34 ------------------------
% 1.58/1.35 ---- Cleaned THF ---
% 1.58/1.35 thf(in_type,type,
% 1.58/1.35 in: $i > $i > $o ).
% 1.58/1.35
% 1.58/1.35 thf(emptyset_type,type,
% 1.58/1.35 emptyset: $i ).
% 1.58/1.35
% 1.58/1.35 thf(setadjoin_type,type,
% 1.58/1.35 setadjoin: $i > $i > $i ).
% 1.58/1.35
% 1.58/1.35 thf(powerset_type,type,
% 1.58/1.35 powerset: $i > $i ).
% 1.58/1.35
% 1.58/1.35 thf(subset_type,type,
% 1.58/1.35 subset: $i > $i > $o ).
% 1.58/1.35
% 1.58/1.35 thf(powersetI1_type,type,
% 1.58/1.35 powersetI1: $o ).
% 1.58/1.35
% 1.58/1.35 thf(powersetI1,definition,
% 1.58/1.35 ( powersetI1
% 1.58/1.35 = ( ! [A: $i,B: $i] :
% 1.58/1.35 ( ( subset @ B @ A )
% 1.58/1.35 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 1.58/1.35
% 1.58/1.35 thf(binunion_type,type,
% 1.58/1.35 binunion: $i > $i > $i ).
% 1.58/1.35
% 1.58/1.35 thf(ubforcartprodlem1_type,type,
% 1.58/1.35 ubforcartprodlem1: $o ).
% 1.58/1.35
% 1.58/1.35 thf(ubforcartprodlem1,definition,
% 1.58/1.35 ( ubforcartprodlem1
% 1.58/1.35 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.58/1.35 ( ( in @ Xx @ A )
% 1.58/1.35 => ! [Xy: $i] :
% 1.58/1.35 ( ( in @ Xy @ B )
% 1.58/1.35 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 1.58/1.35
% 1.58/1.35 thf(ubforcartprodlem2,conjecture,
% 1.58/1.35 ( powersetI1
% 1.58/1.35 => ( ubforcartprodlem1
% 1.58/1.35 => ! [A: $i,B: $i,Xx: $i] :
% 1.58/1.35 ( ( in @ Xx @ A )
% 1.58/1.35 => ! [Xy: $i] :
% 1.58/1.35 ( ( in @ Xy @ B )
% 1.58/1.35 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 1.58/1.35 ------------------------
% 1.58/1.37 % (3915)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_3818 for (2999ds/183Mi)
% 1.58/1.37 % (3920)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_3818 for (2999ds/275Mi)
% 1.58/1.37 % (3916)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_3818 for (2999ds/4Mi)
% 1.58/1.37 % (3919)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_3818 for (2999ds/2Mi)
% 1.58/1.37 % (3917)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_3818 for (2999ds/27Mi)
% 1.58/1.37 % (3921)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_3818 for (2999ds/18Mi)
% 1.58/1.37 % (3918)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_3818 for (2999ds/2Mi)
% 1.58/1.37 % (3918)Instruction limit reached!
% 1.58/1.37 % (3918)------------------------------
% 1.58/1.37 % (3918)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.58/1.37 % (3918)Termination reason: Unknown
% 1.58/1.37 % (3918)Termination phase: Preprocessing 3
% 1.58/1.37 % (3919)Instruction limit reached!
% 1.58/1.37 % (3919)------------------------------
% 1.58/1.37 % (3919)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.58/1.37 % (3919)Termination reason: Unknown
% 1.58/1.37 % (3919)Termination phase: Saturation
% 1.58/1.37
% 1.58/1.37 % (3919)Memory used [KB]: 1023
% 1.58/1.37 % (3919)Time elapsed: 0.003 s
% 1.58/1.37 % (3919)Instructions burned: 3 (million)
% 1.58/1.37 % (3919)------------------------------
% 1.58/1.37 % (3919)------------------------------
% 1.58/1.37
% 1.58/1.37 % (3918)Memory used [KB]: 895
% 1.58/1.37 % (3918)Time elapsed: 0.003 s
% 1.58/1.37 % (3918)Instructions burned: 2 (million)
% 1.58/1.37 % (3918)------------------------------
% 1.58/1.37 % (3918)------------------------------
% 1.58/1.37 % (3916)Instruction limit reached!
% 1.58/1.37 % (3916)------------------------------
% 1.58/1.37 % (3916)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.58/1.37 % (3916)Termination reason: Unknown
% 1.58/1.37 % (3916)Termination phase: Saturation
% 1.58/1.37
% 1.58/1.37 % (3916)Memory used [KB]: 5500
% 1.58/1.37 % (3916)Time elapsed: 0.004 s
% 1.58/1.37 % (3916)Instructions burned: 4 (million)
% 1.58/1.37 % (3916)------------------------------
% 1.58/1.37 % (3916)------------------------------
% 1.58/1.37 % (3920)First to succeed.
% 1.58/1.38 % (3915)Also succeeded, but the first one will report.
% 1.58/1.38 % (3917)Also succeeded, but the first one will report.
% 1.58/1.38 % (3920)Refutation found. Thanks to Tanya!
% 1.58/1.38 % SZS status Theorem for DTF2THF_3818
% 1.58/1.38 % SZS output start Proof for DTF2THF_3818
% 1.58/1.38 thf(func_def_0, type, in: $i > $i > $o).
% 1.58/1.38 thf(func_def_2, type, setadjoin: $i > $i > $i).
% 1.58/1.38 thf(func_def_3, type, powerset: $i > $i).
% 1.58/1.38 thf(func_def_4, type, subset: $i > $i > $o).
% 1.58/1.38 thf(func_def_7, type, binunion: $i > $i > $i).
% 1.58/1.38 thf(f55,plain,(
% 1.58/1.38 $false),
% 1.58/1.38 inference(subsumption_resolution,[status(thm)],[f54,f30])).
% 1.58/1.38 thf(f30,plain,(
% 1.58/1.38 ((in @ sK0 @ sK2) = $true)),
% 1.58/1.38 inference(cnf_transformation,[status(thm)],[f19])).
% 1.58/1.38 thf(f19,plain,(
% 1.58/1.38 (ubforcartprodlem1 = $true) & ((((in @ sK3 @ sK1) = $true) & ((in @ (setadjoin @ (setadjoin @ sK0 @ emptyset) @ (setadjoin @ (setadjoin @ sK0 @ (setadjoin @ sK3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ sK2 @ sK1)))) != $true)) & ((in @ sK0 @ sK2) = $true)) & (powersetI1 = $true)),
% 1.58/1.38 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2,sK3])],[f16,f18,f17])).
% 1.58/1.38 thf(f17,plain,(
% 1.58/1.38 ? [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : (((in @ X3 @ X1) = $true) & ($true != (in @ (setadjoin @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X0 @ (setadjoin @ X3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ X2 @ X1)))))) & ($true = (in @ X0 @ X2))) => (? [X3 : $i] : (($true = (in @ X3 @ sK1)) & ($true != (in @ (setadjoin @ (setadjoin @ sK0 @ emptyset) @ (setadjoin @ (setadjoin @ sK0 @ (setadjoin @ X3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ sK2 @ sK1)))))) & ((in @ sK0 @ sK2) = $true))),
% 1.58/1.38 introduced(definition,[],[choice_axiom])).
% 1.58/1.38 thf(f18,plain,(
% 1.58/1.38 ? [X3 : $i] : (($true = (in @ X3 @ sK1)) & ($true != (in @ (setadjoin @ (setadjoin @ sK0 @ emptyset) @ (setadjoin @ (setadjoin @ sK0 @ (setadjoin @ X3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ sK2 @ sK1)))))) => (((in @ sK3 @ sK1) = $true) & ((in @ (setadjoin @ (setadjoin @ sK0 @ emptyset) @ (setadjoin @ (setadjoin @ sK0 @ (setadjoin @ sK3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ sK2 @ sK1)))) != $true))),
% 1.58/1.38 introduced(definition,[],[choice_axiom])).
% 1.58/1.38 thf(f16,plain,(
% 1.58/1.38 (ubforcartprodlem1 = $true) & ? [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : (((in @ X3 @ X1) = $true) & ($true != (in @ (setadjoin @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X0 @ (setadjoin @ X3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ X2 @ X1)))))) & ($true = (in @ X0 @ X2))) & (powersetI1 = $true)),
% 1.58/1.38 inference(rectify,[status(thm)],[f15])).
% 1.58/1.38 thf(f15,plain,(
% 1.58/1.38 (ubforcartprodlem1 = $true) & ? [X1 : $i,X0 : $i,X2 : $i] : (? [X3 : $i] : (((in @ X3 @ X0) = $true) & ((in @ (setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ X3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ X2 @ X0)))) != $true)) & ((in @ X1 @ X2) = $true)) & (powersetI1 = $true)),
% 1.58/1.38 inference(flattening,[status(thm)],[f14])).
% 1.58/1.38 thf(f14,plain,(
% 1.58/1.38 (? [X1 : $i,X0 : $i,X2 : $i] : (? [X3 : $i] : (((in @ X3 @ X0) = $true) & ((in @ (setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ X3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ X2 @ X0)))) != $true)) & ((in @ X1 @ X2) = $true)) & (ubforcartprodlem1 = $true)) & (powersetI1 = $true)),
% 1.58/1.38 inference(ennf_transformation,[status(thm)],[f11])).
% 1.58/1.38 thf(f11,plain,(
% 1.58/1.38 ~((powersetI1 = $true) => ((ubforcartprodlem1 = $true) => ! [X0 : $i,X2 : $i,X1 : $i] : (((in @ X1 @ X2) = $true) => ! [X3 : $i] : (((in @ X3 @ X0) = $true) => ((in @ (setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ X3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ X2 @ X0)))) = $true)))))),
% 1.58/1.38 inference(fool_elimination,[status(thm)],[f10])).
% 1.58/1.38 thf(f10,plain,(
% 1.58/1.38 ~(powersetI1 => (ubforcartprodlem1 => ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X1 @ X2) => ! [X3 : $i] : ((in @ X3 @ X0) => (in @ (setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ X3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ X2 @ X0))))))))),
% 1.58/1.38 inference(rectify,[status(thm)],[f4])).
% 1.58/1.38 thf(f4,negated_conjecture,(
% 1.58/1.38 ~(powersetI1 => (ubforcartprodlem1 => ! [X1 : $i,X2 : $i,X0 : $i] : ((in @ X2 @ X0) => ! [X3 : $i] : ((in @ X3 @ X1) => (in @ (setadjoin @ (setadjoin @ X2 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ (setadjoin @ X3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ X0 @ X1))))))))),
% 1.58/1.38 inference(negated_conjecture,[status(cth)],[f3])).
% 1.58/1.38 thf(f3,conjecture,(
% 1.58/1.38 powersetI1 => (ubforcartprodlem1 => ! [X1 : $i,X2 : $i,X0 : $i] : ((in @ X2 @ X0) => ! [X3 : $i] : ((in @ X3 @ X1) => (in @ (setadjoin @ (setadjoin @ X2 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ (setadjoin @ X3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ X0 @ X1)))))))),
% 1.58/1.38 file('/export/starexec/sandbox/tmp/tmp.O9ONI9QTIu/DTF2THF_3818.p',ubforcartprodlem2)).
% 1.58/1.38 thf(f54,plain,(
% 1.58/1.38 ((in @ sK0 @ sK2) != $true)),
% 1.58/1.38 inference(subsumption_resolution,[status(thm)],[f53,f32])).
% 1.58/1.38 thf(f32,plain,(
% 1.58/1.38 ((in @ sK3 @ sK1) = $true)),
% 1.58/1.38 inference(cnf_transformation,[status(thm)],[f19])).
% 1.58/1.38 thf(f53,plain,(
% 1.58/1.38 ((in @ sK3 @ sK1) != $true) | ((in @ sK0 @ sK2) != $true)),
% 1.58/1.38 inference(trivial_inequality_removal,[status(thm)],[f52])).
% 1.58/1.38 thf(f52,plain,(
% 1.58/1.38 ((in @ sK0 @ sK2) != $true) | ((in @ sK3 @ sK1) != $true) | ($true != $true)),
% 1.58/1.38 inference(superposition,[status(thm)],[f51,f49])).
% 1.58/1.38 thf(f49,plain,(
% 1.58/1.38 ( ! [X6 : $i,X7 : $i,X4 : $i,X5 : $i] : (((subset @ (setadjoin @ (setadjoin @ X4 @ emptyset) @ (setadjoin @ (setadjoin @ X4 @ (setadjoin @ X7 @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ X6 @ X5))) = $true) | ($true != (in @ X4 @ X6)) | ($true != (in @ X7 @ X5))) )),
% 1.58/1.38 inference(trivial_inequality_removal,[status(thm)],[f47])).
% 1.58/1.38 thf(f47,plain,(
% 1.58/1.38 ( ! [X6 : $i,X7 : $i,X4 : $i,X5 : $i] : (($true != (in @ X4 @ X6)) | ((subset @ (setadjoin @ (setadjoin @ X4 @ emptyset) @ (setadjoin @ (setadjoin @ X4 @ (setadjoin @ X7 @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ X6 @ X5))) = $true) | ($true != (in @ X7 @ X5)) | ($true != $true)) )),
% 1.58/1.38 inference(definition_unfolding,[status(thm)],[f37,f33])).
% 1.58/1.38 thf(f33,plain,(
% 1.58/1.38 (ubforcartprodlem1 = $true)),
% 1.58/1.38 inference(cnf_transformation,[status(thm)],[f19])).
% 1.58/1.38 thf(f37,plain,(
% 1.58/1.38 ( ! [X6 : $i,X7 : $i,X4 : $i,X5 : $i] : (($true != (in @ X7 @ X5)) | ((subset @ (setadjoin @ (setadjoin @ X4 @ emptyset) @ (setadjoin @ (setadjoin @ X4 @ (setadjoin @ X7 @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ X6 @ X5))) = $true) | ($true != (in @ X4 @ X6)) | (ubforcartprodlem1 != $true)) )),
% 1.58/1.38 inference(cnf_transformation,[status(thm)],[f28])).
% 1.58/1.38 thf(f28,plain,(
% 1.58/1.38 ((ubforcartprodlem1 = $true) | ((((in @ sK9 @ sK7) = $true) & ((subset @ (setadjoin @ (setadjoin @ sK6 @ emptyset) @ (setadjoin @ (setadjoin @ sK6 @ (setadjoin @ sK9 @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ sK8 @ sK7))) != $true)) & ((in @ sK6 @ sK8) = $true))) & (! [X4 : $i,X5 : $i,X6 : $i] : (! [X7 : $i] : (($true != (in @ X7 @ X5)) | ((subset @ (setadjoin @ (setadjoin @ X4 @ emptyset) @ (setadjoin @ (setadjoin @ X4 @ (setadjoin @ X7 @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ X6 @ X5))) = $true)) | ($true != (in @ X4 @ X6))) | (ubforcartprodlem1 != $true))),
% 1.58/1.38 inference(skolemisation,[status(esa),new_symbols(skolem,[sK6,sK7,sK8,sK9])],[f25,f27,f26])).
% 1.58/1.38 thf(f26,plain,(
% 1.58/1.38 ? [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : (((in @ X3 @ X1) = $true) & ($true != (subset @ (setadjoin @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X0 @ (setadjoin @ X3 @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ X2 @ X1))))) & ($true = (in @ X0 @ X2))) => (? [X3 : $i] : (((in @ X3 @ sK7) = $true) & ((subset @ (setadjoin @ (setadjoin @ sK6 @ emptyset) @ (setadjoin @ (setadjoin @ sK6 @ (setadjoin @ X3 @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ sK8 @ sK7))) != $true)) & ((in @ sK6 @ sK8) = $true))),
% 1.58/1.38 introduced(definition,[],[choice_axiom])).
% 1.58/1.38 thf(f27,plain,(
% 1.58/1.38 ? [X3 : $i] : (((in @ X3 @ sK7) = $true) & ((subset @ (setadjoin @ (setadjoin @ sK6 @ emptyset) @ (setadjoin @ (setadjoin @ sK6 @ (setadjoin @ X3 @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ sK8 @ sK7))) != $true)) => (((in @ sK9 @ sK7) = $true) & ((subset @ (setadjoin @ (setadjoin @ sK6 @ emptyset) @ (setadjoin @ (setadjoin @ sK6 @ (setadjoin @ sK9 @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ sK8 @ sK7))) != $true))),
% 1.58/1.38 introduced(definition,[],[choice_axiom])).
% 1.58/1.38 thf(f25,plain,(
% 1.58/1.38 ((ubforcartprodlem1 = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : (((in @ X3 @ X1) = $true) & ($true != (subset @ (setadjoin @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X0 @ (setadjoin @ X3 @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ X2 @ X1))))) & ($true = (in @ X0 @ X2)))) & (! [X4 : $i,X5 : $i,X6 : $i] : (! [X7 : $i] : (($true != (in @ X7 @ X5)) | ((subset @ (setadjoin @ (setadjoin @ X4 @ emptyset) @ (setadjoin @ (setadjoin @ X4 @ (setadjoin @ X7 @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ X6 @ X5))) = $true)) | ($true != (in @ X4 @ X6))) | (ubforcartprodlem1 != $true))),
% 1.58/1.38 inference(rectify,[status(thm)],[f24])).
% 1.58/1.38 thf(f24,plain,(
% 1.58/1.38 ((ubforcartprodlem1 = $true) | ? [X2 : $i,X1 : $i,X0 : $i] : (? [X3 : $i] : (((in @ X3 @ X1) = $true) & ((subset @ (setadjoin @ (setadjoin @ X2 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ (setadjoin @ X3 @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ X0 @ X1))) != $true)) & ((in @ X2 @ X0) = $true))) & (! [X2 : $i,X1 : $i,X0 : $i] : (! [X3 : $i] : (((in @ X3 @ X1) != $true) | ((subset @ (setadjoin @ (setadjoin @ X2 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ (setadjoin @ X3 @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ X0 @ X1))) = $true)) | ((in @ X2 @ X0) != $true)) | (ubforcartprodlem1 != $true))),
% 1.58/1.38 inference(nnf_transformation,[status(thm)],[f13])).
% 1.58/1.38 thf(f13,plain,(
% 1.58/1.38 (ubforcartprodlem1 = $true) <=> ! [X2 : $i,X1 : $i,X0 : $i] : (! [X3 : $i] : (((in @ X3 @ X1) != $true) | ((subset @ (setadjoin @ (setadjoin @ X2 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ (setadjoin @ X3 @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ X0 @ X1))) = $true)) | ((in @ X2 @ X0) != $true))),
% 1.58/1.38 inference(ennf_transformation,[status(thm)],[f7])).
% 1.58/1.38 thf(f7,plain,(
% 1.58/1.38 (ubforcartprodlem1 = $true) <=> ! [X2 : $i,X1 : $i,X0 : $i] : (((in @ X2 @ X0) = $true) => ! [X3 : $i] : (((in @ X3 @ X1) = $true) => ((subset @ (setadjoin @ (setadjoin @ X2 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ (setadjoin @ X3 @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ X0 @ X1))) = $true)))),
% 1.58/1.38 inference(fool_elimination,[status(thm)],[f6])).
% 1.58/1.38 thf(f6,plain,(
% 1.58/1.38 (! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X2 @ X0) => ! [X3 : $i] : ((in @ X3 @ X1) => (subset @ (setadjoin @ (setadjoin @ X2 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ (setadjoin @ X3 @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ X0 @ X1))))) = ubforcartprodlem1)),
% 1.58/1.38 inference(rectify,[status(thm)],[f2])).
% 1.58/1.38 thf(f2,axiom,(
% 1.58/1.38 (! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X2 @ X0) => ! [X3 : $i] : ((in @ X3 @ X1) => (subset @ (setadjoin @ (setadjoin @ X2 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ (setadjoin @ X3 @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ X0 @ X1))))) = ubforcartprodlem1)),
% 1.58/1.38 file('/export/starexec/sandbox/tmp/tmp.O9ONI9QTIu/DTF2THF_3818.p',ubforcartprodlem1)).
% 1.58/1.38 thf(f51,plain,(
% 1.58/1.38 ($true != (subset @ (setadjoin @ (setadjoin @ sK0 @ emptyset) @ (setadjoin @ (setadjoin @ sK0 @ (setadjoin @ sK3 @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ sK2 @ sK1))))),
% 1.58/1.38 inference(trivial_inequality_removal,[status(thm)],[f50])).
% 1.58/1.38 thf(f50,plain,(
% 1.58/1.38 ($true != (subset @ (setadjoin @ (setadjoin @ sK0 @ emptyset) @ (setadjoin @ (setadjoin @ sK0 @ (setadjoin @ sK3 @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ sK2 @ sK1)))) | ($true != $true)),
% 1.58/1.38 inference(superposition,[status(thm)],[f31,f48])).
% 1.58/1.38 thf(f48,plain,(
% 1.58/1.38 ( ! [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) | ($true != (subset @ X0 @ X1))) )),
% 1.58/1.38 inference(trivial_inequality_removal,[status(thm)],[f41])).
% 1.58/1.38 thf(f41,plain,(
% 1.58/1.38 ( ! [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) | ($true != (subset @ X0 @ X1)) | ($true != $true)) )),
% 1.58/1.38 inference(definition_unfolding,[status(thm)],[f36,f29])).
% 1.58/1.38 thf(f29,plain,(
% 1.58/1.38 (powersetI1 = $true)),
% 1.58/1.38 inference(cnf_transformation,[status(thm)],[f19])).
% 1.58/1.38 thf(f36,plain,(
% 1.58/1.38 ( ! [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) | ($true != (subset @ X0 @ X1)) | (powersetI1 != $true)) )),
% 1.58/1.38 inference(cnf_transformation,[status(thm)],[f23])).
% 1.58/1.38 thf(f23,plain,(
% 1.58/1.38 (! [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) | ($true != (subset @ X0 @ X1))) | (powersetI1 != $true)) & ((powersetI1 = $true) | (((in @ sK4 @ (powerset @ sK5)) != $true) & ($true = (subset @ sK4 @ sK5))))),
% 1.58/1.38 inference(skolemisation,[status(esa),new_symbols(skolem,[sK4,sK5])],[f21,f22])).
% 1.58/1.38 thf(f22,plain,(
% 1.58/1.38 ? [X2 : $i,X3 : $i] : (((in @ X2 @ (powerset @ X3)) != $true) & ($true = (subset @ X2 @ X3))) => (((in @ sK4 @ (powerset @ sK5)) != $true) & ($true = (subset @ sK4 @ sK5)))),
% 1.58/1.38 introduced(definition,[],[choice_axiom])).
% 1.58/1.38 thf(f21,plain,(
% 1.58/1.38 (! [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) | ($true != (subset @ X0 @ X1))) | (powersetI1 != $true)) & ((powersetI1 = $true) | ? [X2 : $i,X3 : $i] : (((in @ X2 @ (powerset @ X3)) != $true) & ($true = (subset @ X2 @ X3))))),
% 1.58/1.38 inference(rectify,[status(thm)],[f20])).
% 1.58/1.38 thf(f20,plain,(
% 1.58/1.38 (! [X1 : $i,X0 : $i] : (((in @ X1 @ (powerset @ X0)) = $true) | ((subset @ X1 @ X0) != $true)) | (powersetI1 != $true)) & ((powersetI1 = $true) | ? [X1 : $i,X0 : $i] : (((in @ X1 @ (powerset @ X0)) != $true) & ((subset @ X1 @ X0) = $true)))),
% 1.58/1.38 inference(nnf_transformation,[status(thm)],[f12])).
% 1.58/1.38 thf(f12,plain,(
% 1.58/1.38 ! [X1 : $i,X0 : $i] : (((in @ X1 @ (powerset @ X0)) = $true) | ((subset @ X1 @ X0) != $true)) <=> (powersetI1 = $true)),
% 1.58/1.38 inference(ennf_transformation,[status(thm)],[f9])).
% 1.58/1.38 thf(f9,plain,(
% 1.58/1.38 (powersetI1 = $true) <=> ! [X1 : $i,X0 : $i] : (((subset @ X1 @ X0) = $true) => ((in @ X1 @ (powerset @ X0)) = $true))),
% 1.58/1.38 inference(fool_elimination,[status(thm)],[f8])).
% 1.58/1.38 thf(f8,plain,(
% 1.58/1.38 (powersetI1 = ! [X0 : $i,X1 : $i] : ((subset @ X1 @ X0) => (in @ X1 @ (powerset @ X0))))),
% 1.58/1.38 inference(rectify,[status(thm)],[f1])).
% 1.58/1.38 thf(f1,axiom,(
% 1.58/1.38 (powersetI1 = ! [X0 : $i,X1 : $i] : ((subset @ X1 @ X0) => (in @ X1 @ (powerset @ X0))))),
% 1.58/1.38 file('/export/starexec/sandbox/tmp/tmp.O9ONI9QTIu/DTF2THF_3818.p',powersetI1)).
% 1.58/1.38 thf(f31,plain,(
% 1.58/1.38 ((in @ (setadjoin @ (setadjoin @ sK0 @ emptyset) @ (setadjoin @ (setadjoin @ sK0 @ (setadjoin @ sK3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ sK2 @ sK1)))) != $true)),
% 1.58/1.38 inference(cnf_transformation,[status(thm)],[f19])).
% 1.58/1.38 % SZS output end Proof for DTF2THF_3818
% 1.58/1.38 % (3920)------------------------------
% 1.58/1.38 % (3920)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.58/1.38 % (3920)Termination reason: Refutation
% 1.58/1.38
% 1.58/1.38 % (3920)Memory used [KB]: 5500
% 1.58/1.38 % (3920)Time elapsed: 0.007 s
% 1.58/1.38 % (3920)Instructions burned: 5 (million)
% 1.58/1.38 % (3920)------------------------------
% 1.58/1.38 % (3920)------------------------------
% 1.58/1.38 % (3914)Success in time 0.02 s
%------------------------------------------------------------------------------