%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU610^2 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n002.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 09:35:36 AM UTC 2026
% Result : Theorem 0.32s 1.34s
% Output : Refutation 0.32s
% Verified :
% SZS Type : Refutation
% Derivation depth : 21
% Number of leaves : 12
% Syntax : Number of formulae : 72 ( 14 unt; 0 typ; 2 def)
% Number of atoms : 759 ( 222 equ; 0 cnn)
% Maximal formula atoms : 8 ( 10 avg)
% Number of connectives : 677 ( 107 ~; 86 |; 57 &; 374 @)
% ( 10 <=>; 43 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 5 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 23 ( 20 usr; 18 con; 0-2 aty)
% Number of variables : 143 ( 0 ^; 108 !; 35 ?; 143 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_5,type,
subset: $i > $i > $o ).
thf(func_def_7,type,
setminus: $i > $i > $i ).
thf(func_def_17,type,
sK6: $i > $i > $i ).
thf(f121,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f96,f102,f120]) ).
thf(f120,plain,
~ spl14_1,
inference(avatar_contradiction_clause,[status(thm)],[f119]) ).
thf(f119,plain,
( $false
| ~ spl14_1 ),
inference(subsumption_resolution,[status(thm)],[f118,f52]) ).
thf(f52,plain,
( $true
!= ( subset @ sK0 @ sK1 ) ),
inference(cnf_transformation,[status(thm)],[f28]) ).
thf(f28,plain,
( ( emptyset
= ( setminus @ sK0 @ sK1 ) )
& ( $true
!= ( subset @ sK0 @ sK1 ) )
& ( emptysetE = $true )
& ( subsetI2 = $true )
& ( in__Cong = $true )
& ( setminusI = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1])],[f26,f27]) ).
thf(f27,plain,
( ? [X0: $i,X1: $i] :
( ( emptyset
= ( setminus @ X0 @ X1 ) )
& ( $true
!= ( subset @ X0 @ X1 ) ) )
=> ( ( emptyset
= ( setminus @ sK0 @ sK1 ) )
& ( $true
!= ( subset @ sK0 @ sK1 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f26,plain,
( ? [X0: $i,X1: $i] :
( ( emptyset
= ( setminus @ X0 @ X1 ) )
& ( $true
!= ( subset @ X0 @ X1 ) ) )
& ( emptysetE = $true )
& ( subsetI2 = $true )
& ( in__Cong = $true )
& ( setminusI = $true ) ),
inference(rectify,[status(thm)],[f22]) ).
thf(f22,plain,
( ? [X1: $i,X0: $i] :
( ( emptyset
= ( setminus @ X1 @ X0 ) )
& ( ( subset @ X1 @ X0 )
!= $true ) )
& ( emptysetE = $true )
& ( subsetI2 = $true )
& ( in__Cong = $true )
& ( setminusI = $true ) ),
inference(flattening,[status(thm)],[f21]) ).
thf(f21,plain,
( ? [X1: $i,X0: $i] :
( ( emptyset
= ( setminus @ X1 @ X0 ) )
& ( ( subset @ X1 @ X0 )
!= $true ) )
& ( setminusI = $true )
& ( subsetI2 = $true )
& ( in__Cong = $true )
& ( emptysetE = $true ) ),
inference(ennf_transformation,[status(thm)],[f15]) ).
thf(f15,plain,
~ ( ( emptysetE = $true )
=> ( ( in__Cong = $true )
=> ( ( subsetI2 = $true )
=> ( ( setminusI = $true )
=> ! [X1: $i,X0: $i] :
( ( emptyset
= ( setminus @ X1 @ X0 ) )
=> ( ( subset @ X1 @ X0 )
= $true ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f14]) ).
thf(f14,plain,
~ ( emptysetE
=> ( in__Cong
=> ( subsetI2
=> ( setminusI
=> ! [X0: $i,X1: $i] :
( ( emptyset
= ( setminus @ X1 @ X0 ) )
=> ( subset @ X1 @ X0 ) ) ) ) ) ),
inference(rectify,[status(thm)],[f6]) ).
thf(f6,negated_conjecture,
~ ( emptysetE
=> ( in__Cong
=> ( subsetI2
=> ( setminusI
=> ! [X3: $i,X2: $i] :
( ( emptyset
= ( setminus @ X2 @ X3 ) )
=> ( subset @ X2 @ X3 ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f5]) ).
thf(f5,conjecture,
( emptysetE
=> ( in__Cong
=> ( subsetI2
=> ( setminusI
=> ! [X3: $i,X2: $i] :
( ( emptyset
= ( setminus @ X2 @ X3 ) )
=> ( subset @ X2 @ X3 ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.DIwqCHI51i/DTF2THF_30792.p',setminusSubset1) ).
thf(f118,plain,
( ( $true
= ( subset @ sK0 @ sK1 ) )
| ~ spl14_1 ),
inference(trivial_inequality_removal,[status(thm)],[f117]) ).
thf(f117,plain,
( ( $true
= ( subset @ sK0 @ sK1 ) )
| ( $true != $true )
| ~ spl14_1 ),
inference(duplicate_literal_removal,[status(thm)],[f116]) ).
thf(f116,plain,
( ( $true
= ( subset @ sK0 @ sK1 ) )
| ( $true
= ( subset @ sK0 @ sK1 ) )
| ( $true != $true )
| ~ spl14_1 ),
inference(superposition,[status(thm)],[f113,f89]) ).
thf(f89,plain,
! [X0: $i,X1: $i] :
( ( $true
= ( in @ ( sK6 @ X1 @ X0 ) @ X1 ) )
| ( ( subset @ X1 @ X0 )
= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f74]) ).
thf(f74,plain,
! [X0: $i,X1: $i] :
( ( ( subset @ X1 @ X0 )
= $true )
| ( $true != $true )
| ( $true
= ( in @ ( sK6 @ X1 @ X0 ) @ X1 ) ) ),
inference(definition_unfolding,[status(thm)],[f60,f50]) ).
thf(f50,plain,
subsetI2 = $true,
inference(cnf_transformation,[status(thm)],[f28]) ).
thf(f60,plain,
! [X0: $i,X1: $i] :
( ( ( subset @ X1 @ X0 )
= $true )
| ( $true
= ( in @ ( sK6 @ X1 @ X0 ) @ X1 ) )
| ( subsetI2 != $true ) ),
inference(cnf_transformation,[status(thm)],[f38]) ).
thf(f38,plain,
( ( ! [X0: $i,X1: $i] :
( ( ( subset @ X1 @ X0 )
= $true )
| ( ( $true
!= ( in @ ( sK6 @ X1 @ X0 ) @ X0 ) )
& ( $true
= ( in @ ( sK6 @ X1 @ X0 ) @ X1 ) ) ) )
| ( subsetI2 != $true ) )
& ( ( subsetI2 = $true )
| ( ( ( subset @ sK8 @ sK7 )
!= $true )
& ! [X5: $i] :
( ( ( in @ X5 @ sK7 )
= $true )
| ( $true
!= ( in @ X5 @ sK8 ) ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK6,sK7,sK8])],[f35,f37,f36]) ).
thf(f36,plain,
! [X0: $i,X1: $i] :
( ? [X2: $i] :
( ( $true
!= ( in @ X2 @ X0 ) )
& ( $true
= ( in @ X2 @ X1 ) ) )
=> ( ( $true
!= ( in @ ( sK6 @ X1 @ X0 ) @ X0 ) )
& ( $true
= ( in @ ( sK6 @ X1 @ X0 ) @ X1 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f37,plain,
( ? [X3: $i,X4: $i] :
( ( ( subset @ X4 @ X3 )
!= $true )
& ! [X5: $i] :
( ( ( in @ X5 @ X3 )
= $true )
| ( $true
!= ( in @ X5 @ X4 ) ) ) )
=> ( ( ( subset @ sK8 @ sK7 )
!= $true )
& ! [X5: $i] :
( ( ( in @ X5 @ sK7 )
= $true )
| ( $true
!= ( in @ X5 @ sK8 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f35,plain,
( ( ! [X0: $i,X1: $i] :
( ( ( subset @ X1 @ X0 )
= $true )
| ? [X2: $i] :
( ( $true
!= ( in @ X2 @ X0 ) )
& ( $true
= ( in @ X2 @ X1 ) ) ) )
| ( subsetI2 != $true ) )
& ( ( subsetI2 = $true )
| ? [X3: $i,X4: $i] :
( ( ( subset @ X4 @ X3 )
!= $true )
& ! [X5: $i] :
( ( ( in @ X5 @ X3 )
= $true )
| ( $true
!= ( in @ X5 @ X4 ) ) ) ) ) ),
inference(rectify,[status(thm)],[f34]) ).
thf(f34,plain,
( ( ! [X1: $i,X0: $i] :
( ( $true
= ( subset @ X0 @ X1 ) )
| ? [X2: $i] :
( ( $true
!= ( in @ X2 @ X1 ) )
& ( $true
= ( in @ X2 @ X0 ) ) ) )
| ( subsetI2 != $true ) )
& ( ( subsetI2 = $true )
| ? [X1: $i,X0: $i] :
( ( $true
!= ( subset @ X0 @ X1 ) )
& ! [X2: $i] :
( ( $true
= ( in @ X2 @ X1 ) )
| ( $true
!= ( in @ X2 @ X0 ) ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f23]) ).
thf(f23,plain,
( ! [X1: $i,X0: $i] :
( ( $true
= ( subset @ X0 @ X1 ) )
| ? [X2: $i] :
( ( $true
!= ( in @ X2 @ X1 ) )
& ( $true
= ( in @ X2 @ X0 ) ) ) )
<=> ( subsetI2 = $true ) ),
inference(ennf_transformation,[status(thm)],[f9]) ).
thf(f9,plain,
( ( subsetI2 = $true )
<=> ! [X1: $i,X0: $i] :
( ! [X2: $i] :
( ( $true
= ( in @ X2 @ X0 ) )
=> ( $true
= ( in @ X2 @ X1 ) ) )
=> ( $true
= ( subset @ X0 @ X1 ) ) ) ),
inference(fool_elimination,[status(thm)],[f8]) ).
thf(f8,plain,
! [X0: $i,X1: $i] :
( ( ! [X2: $i] :
( ( in @ X2 @ X0 )
=> ( in @ X2 @ X1 ) )
=> ( subset @ X0 @ X1 ) )
= subsetI2 ),
inference(rectify,[status(thm)],[f3]) ).
thf(f3,axiom,
! [X2: $i,X3: $i] :
( ( ! [X0: $i] :
( ( in @ X0 @ X2 )
=> ( in @ X0 @ X3 ) )
=> ( subset @ X2 @ X3 ) )
= subsetI2 ),
file('/export/starexec/sandbox2/tmp/tmp.DIwqCHI51i/DTF2THF_30792.p',subsetI2) ).
thf(f113,plain,
( ! [X0: $i] :
( ( ( in @ ( sK6 @ X0 @ sK1 ) @ sK0 )
!= $true )
| ( $true
= ( subset @ X0 @ sK1 ) ) )
| ~ spl14_1 ),
inference(trivial_inequality_removal,[status(thm)],[f112]) ).
thf(f112,plain,
( ! [X0: $i] :
( ( $true != $true )
| ( ( in @ ( sK6 @ X0 @ sK1 ) @ sK0 )
!= $true )
| ( $true
= ( subset @ X0 @ sK1 ) ) )
| ~ spl14_1 ),
inference(superposition,[status(thm)],[f88,f110]) ).
thf(f110,plain,
( ! [X0: $i] :
( ( $true
= ( in @ X0 @ sK1 ) )
| ( $true
!= ( in @ X0 @ sK0 ) ) )
| ~ spl14_1 ),
inference(trivial_inequality_removal,[status(thm)],[f108]) ).
thf(f108,plain,
( ! [X0: $i] :
( ( $true != $true )
| ( $true
= ( in @ X0 @ sK1 ) )
| ( $true
!= ( in @ X0 @ sK0 ) ) )
| ~ spl14_1 ),
inference(superposition,[status(thm)],[f92,f86]) ).
thf(f86,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( ( in @ X2 @ ( setminus @ X1 @ X0 ) )
= $true )
| ( $true
= ( in @ X2 @ X0 ) )
| ( $true
!= ( in @ X2 @ X1 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f77]) ).
thf(f77,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( ( in @ X2 @ ( setminus @ X1 @ X0 ) )
= $true )
| ( $true != $true )
| ( $true
!= ( in @ X2 @ X1 ) )
| ( $true
= ( in @ X2 @ X0 ) ) ),
inference(definition_unfolding,[status(thm)],[f65,f48]) ).
thf(f48,plain,
setminusI = $true,
inference(cnf_transformation,[status(thm)],[f28]) ).
thf(f65,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( $true
= ( in @ X2 @ X0 ) )
| ( ( in @ X2 @ ( setminus @ X1 @ X0 ) )
= $true )
| ( $true
!= ( in @ X2 @ X1 ) )
| ( setminusI != $true ) ),
inference(cnf_transformation,[status(thm)],[f42]) ).
thf(f42,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X2 @ X0 ) )
| ( ( in @ X2 @ ( setminus @ X1 @ X0 ) )
= $true )
| ( $true
!= ( in @ X2 @ X1 ) ) )
| ( setminusI != $true ) )
& ( ( setminusI = $true )
| ( ( $true
!= ( in @ sK11 @ sK9 ) )
& ( $true
!= ( in @ sK11 @ ( setminus @ sK10 @ sK9 ) ) )
& ( ( in @ sK11 @ sK10 )
= $true ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK9,sK10,sK11])],[f40,f41]) ).
thf(f41,plain,
( ? [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X5 @ X3 )
!= $true )
& ( $true
!= ( in @ X5 @ ( setminus @ X4 @ X3 ) ) )
& ( $true
= ( in @ X5 @ X4 ) ) )
=> ( ( $true
!= ( in @ sK11 @ sK9 ) )
& ( $true
!= ( in @ sK11 @ ( setminus @ sK10 @ sK9 ) ) )
& ( ( in @ sK11 @ sK10 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f40,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X2 @ X0 ) )
| ( ( in @ X2 @ ( setminus @ X1 @ X0 ) )
= $true )
| ( $true
!= ( in @ X2 @ X1 ) ) )
| ( setminusI != $true ) )
& ( ( setminusI = $true )
| ? [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X5 @ X3 )
!= $true )
& ( $true
!= ( in @ X5 @ ( setminus @ X4 @ X3 ) ) )
& ( $true
= ( in @ X5 @ X4 ) ) ) ) ),
inference(rectify,[status(thm)],[f39]) ).
thf(f39,plain,
( ( ! [X1: $i,X2: $i,X0: $i] :
( ( ( in @ X0 @ X1 )
= $true )
| ( ( in @ X0 @ ( setminus @ X2 @ X1 ) )
= $true )
| ( ( in @ X0 @ X2 )
!= $true ) )
| ( setminusI != $true ) )
& ( ( setminusI = $true )
| ? [X1: $i,X2: $i,X0: $i] :
( ( ( in @ X0 @ X1 )
!= $true )
& ( ( in @ X0 @ ( setminus @ X2 @ X1 ) )
!= $true )
& ( ( in @ X0 @ X2 )
= $true ) ) ) ),
inference(nnf_transformation,[status(thm)],[f25]) ).
thf(f25,plain,
( ! [X1: $i,X2: $i,X0: $i] :
( ( ( in @ X0 @ X1 )
= $true )
| ( ( in @ X0 @ ( setminus @ X2 @ X1 ) )
= $true )
| ( ( in @ X0 @ X2 )
!= $true ) )
<=> ( setminusI = $true ) ),
inference(flattening,[status(thm)],[f24]) ).
thf(f24,plain,
( ( setminusI = $true )
<=> ! [X1: $i,X2: $i,X0: $i] :
( ( ( in @ X0 @ ( setminus @ X2 @ X1 ) )
= $true )
| ( ( in @ X0 @ X1 )
= $true )
| ( ( in @ X0 @ X2 )
!= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f18]) ).
thf(f18,plain,
( ( setminusI = $true )
<=> ! [X1: $i,X2: $i,X0: $i] :
( ( ( in @ X0 @ X2 )
= $true )
=> ( ( ( in @ X0 @ X1 )
!= $true )
=> ( ( in @ X0 @ ( setminus @ X2 @ X1 ) )
= $true ) ) ) ),
inference(flattening,[status(thm)],[f17]) ).
thf(f17,plain,
( ( setminusI = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ X2 )
= $true )
=> ( ( ( in @ X0 @ X1 )
!= $true )
=> ( ( in @ X0 @ ( setminus @ X2 @ X1 ) )
= $true ) ) ) ),
inference(fool_elimination,[status(thm)],[f16]) ).
thf(f16,plain,
( setminusI
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X0 @ X2 )
=> ( ~ ( in @ X0 @ X1 )
=> ( in @ X0 @ ( setminus @ X2 @ X1 ) ) ) ) ) ),
inference(rectify,[status(thm)],[f4]) ).
thf(f4,axiom,
( setminusI
= ( ! [X0: $i,X3: $i,X2: $i] :
( ( in @ X0 @ X2 )
=> ( ~ ( in @ X0 @ X3 )
=> ( in @ X0 @ ( setminus @ X2 @ X3 ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.DIwqCHI51i/DTF2THF_30792.p',setminusI) ).
thf(f92,plain,
( ! [X2: $i] :
( ( in @ X2 @ ( setminus @ sK0 @ sK1 ) )
!= $true )
| ~ spl14_1 ),
inference(avatar_component_clause,[status(thm)],[f91]) ).
thf(f91,definition,
( spl14_1
<=> ! [X2: $i] :
( ( in @ X2 @ ( setminus @ sK0 @ sK1 ) )
!= $true ) ),
introduced(definition,[new_symbols(naming,[spl14_1])],[avatar_definition]) ).
thf(f88,plain,
! [X0: $i,X1: $i] :
( ( $true
!= ( in @ ( sK6 @ X1 @ X0 ) @ X0 ) )
| ( ( subset @ X1 @ X0 )
= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f73]) ).
thf(f73,plain,
! [X0: $i,X1: $i] :
( ( $true != $true )
| ( $true
!= ( in @ ( sK6 @ X1 @ X0 ) @ X0 ) )
| ( ( subset @ X1 @ X0 )
= $true ) ),
inference(definition_unfolding,[status(thm)],[f61,f50]) ).
thf(f61,plain,
! [X0: $i,X1: $i] :
( ( ( subset @ X1 @ X0 )
= $true )
| ( $true
!= ( in @ ( sK6 @ X1 @ X0 ) @ X0 ) )
| ( subsetI2 != $true ) ),
inference(cnf_transformation,[status(thm)],[f38]) ).
thf(f102,plain,
~ spl14_2,
inference(avatar_contradiction_clause,[status(thm)],[f101]) ).
thf(f101,plain,
( $false
| ~ spl14_2 ),
inference(trivial_inequality_removal,[status(thm)],[f100]) ).
thf(f100,plain,
( ( $true != $true )
| ~ spl14_2 ),
inference(superposition,[status(thm)],[f52,f95]) ).
thf(f95,plain,
( ! [X3: $o] : ( $true = X3 )
| ~ spl14_2 ),
inference(avatar_component_clause,[status(thm)],[f94]) ).
thf(f94,definition,
( spl14_2
<=> ! [X3: $o] : ( $true = X3 ) ),
introduced(definition,[new_symbols(naming,[spl14_2])],[avatar_definition]) ).
thf(f96,plain,
( spl14_1
| spl14_2 ),
inference(avatar_split_clause,[status(thm)],[f87,f94,f91]) ).
thf(f87,plain,
! [X2: $i,X3: $o] :
( ( $true = X3 )
| ( ( in @ X2 @ ( setminus @ sK0 @ sK1 ) )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f83]) ).
thf(f83,plain,
! [X2: $i,X3: $o] :
( ( $true = X3 )
| ( ( in @ X2 @ ( setminus @ sK0 @ sK1 ) )
!= $true )
| ( $true != $true ) ),
inference(definition_unfolding,[status(thm)],[f66,f53,f51]) ).
thf(f51,plain,
emptysetE = $true,
inference(cnf_transformation,[status(thm)],[f28]) ).
thf(f53,plain,
( emptyset
= ( setminus @ sK0 @ sK1 ) ),
inference(cnf_transformation,[status(thm)],[f28]) ).
thf(f66,plain,
! [X2: $i,X3: $o] :
( ( $true = X3 )
| ( $true
!= ( in @ X2 @ emptyset ) )
| ( emptysetE != $true ) ),
inference(cnf_transformation,[status(thm)],[f47]) ).
thf(f47,plain,
( ( ( emptysetE = $true )
| ( ( $true != sK13 )
& ( $true
= ( in @ sK12 @ emptyset ) ) ) )
& ( ! [X2: $i] :
( ! [X3: $o] : ( $true = X3 )
| ( $true
!= ( in @ X2 @ emptyset ) ) )
| ( emptysetE != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK12,sK13])],[f44,f46,f45]) ).
thf(f45,plain,
( ? [X0: $i] :
( ? [X1: $o] : ( $true != X1 )
& ( ( in @ X0 @ emptyset )
= $true ) )
=> ( ? [X1: $o] : ( $true != X1 )
& ( $true
= ( in @ sK12 @ emptyset ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f46,plain,
( ? [X1: $o] : ( $true != X1 )
=> ( $true != sK13 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f44,plain,
( ( ( emptysetE = $true )
| ? [X0: $i] :
( ? [X1: $o] : ( $true != X1 )
& ( ( in @ X0 @ emptyset )
= $true ) ) )
& ( ! [X2: $i] :
( ! [X3: $o] : ( $true = X3 )
| ( $true
!= ( in @ X2 @ emptyset ) ) )
| ( emptysetE != $true ) ) ),
inference(rectify,[status(thm)],[f43]) ).
thf(f43,plain,
( ( ( emptysetE = $true )
| ? [X0: $i] :
( ? [X1: $o] : ( $true != X1 )
& ( ( in @ X0 @ emptyset )
= $true ) ) )
& ( ! [X0: $i] :
( ! [X1: $o] : ( $true = X1 )
| ( ( in @ X0 @ emptyset )
!= $true ) )
| ( emptysetE != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f19]) ).
thf(f19,plain,
( ( emptysetE = $true )
<=> ! [X0: $i] :
( ! [X1: $o] : ( $true = X1 )
| ( ( in @ X0 @ emptyset )
!= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f11]) ).
thf(f11,plain,
( ! [X0: $i] :
( ( ( in @ X0 @ emptyset )
= $true )
=> ! [X1: $o] : ( $true = X1 ) )
<=> ( emptysetE = $true ) ),
inference(fool_elimination,[status(thm)],[f10]) ).
thf(f10,plain,
! [X0: $i] :
( ( ( in @ X0 @ emptyset )
=> ! [X1: $o] : X1 )
= emptysetE ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
! [X0: $i] :
( ( ( in @ X0 @ emptyset )
=> ! [X1: $o] : X1 )
= emptysetE ),
file('/export/starexec/sandbox2/tmp/tmp.DIwqCHI51i/DTF2THF_30792.p',emptysetE) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.13 % Problem : SEU610^2 : TPTP v9.2.1. Released v3.7.0.
% 0.11/0.14 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.17/0.35 % Computer : n002.cluster.edu
% 0.17/0.35 % Model : x86_64 x86_64
% 0.17/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.35 % Memory : 8042.1875MB
% 0.17/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Tue Feb 24 12:29:03 EST 2026
% 0.13/0.35 % CPUTime :
% 0.13/0.35 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.28/0.47 ---- Original DTF file ---
% 0.28/0.47 thf(spec,logic,$$dhol).
% 0.28/0.47 %------------------------------------------------------------------------------
% 0.28/0.47 % File : SEU610^2 : TPTP v9.2.1. Released v3.7.0.
% 0.28/0.47 % Domain : Set Theory
% 0.28/0.47 % Problem : Preliminary Notions - Operations on Sets - Set Difference
% 0.28/0.47 % Version : Especial > Reduced > Especial.
% 0.28/0.47 % English : (! A:i.! B:i.setminus A B = emptyset -> subset A B)
% 0.28/0.47
% 0.28/0.47 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.28/0.47 % Source : [Bro08]
% 0.28/0.47 % Names : ZFC112l [Bro08]
% 0.28/0.47
% 0.28/0.47 % Status : Theorem
% 0.28/0.47 % Rating : 0.00 v8.2.0, 0.08 v8.1.0, 0.00 v7.4.0, 0.11 v7.2.0, 0.00 v7.1.0, 0.25 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, 0.20 v4.1.0, 0.00 v4.0.1, 0.33 v3.7.0
% 0.28/0.47 % Syntax : Number of formulae : 13 ( 4 unt; 8 typ; 4 def)
% 0.28/0.47 % Number of atoms : 25 ( 7 equ; 0 cnn)
% 0.28/0.47 % Maximal formula atoms : 6 ( 5 avg)
% 0.28/0.47 % Number of connectives : 38 ( 1 ~; 0 |; 0 &; 24 @)
% 0.28/0.47 % ( 1 <=>; 12 =>; 0 <=; 0 <~>)
% 0.28/0.47 % Maximal formula depth : 10 ( 3 avg)
% 0.28/0.47 % Number of types : 2 ( 0 usr)
% 0.28/0.47 % Number of type conns : 6 ( 6 >; 0 *; 0 +; 0 <<)
% 0.28/0.47 % Number of symbols : 9 ( 8 usr; 5 con; 0-2 aty)
% 0.28/0.47 % Number of variables : 14 ( 0 ^; 14 !; 0 ?; 14 :)
% 0.28/0.47 % SPC : TH0_THM_EQU_NAR
% 0.28/0.47
% 0.28/0.47 % Comments : http://mathgate.info/detsetitem.php?id=465
% 0.28/0.47 %------------------------------------------------------------------------------
% 0.28/0.47 thf(in_type,type,
% 0.28/0.47 in: $i > $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(emptyset_type,type,
% 0.28/0.47 emptyset: $i ).
% 0.28/0.47
% 0.28/0.47 thf(emptysetE_type,type,
% 0.28/0.47 emptysetE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(emptysetE,definition,
% 0.28/0.47 ( emptysetE
% 0.28/0.47 = ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ emptyset )
% 0.28/0.47 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(in__Cong_type,type,
% 0.28/0.47 in__Cong: $o ).
% 0.28/0.47
% 0.28/0.47 thf(in__Cong,definition,
% 0.28/0.47 ( in__Cong
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( A = B )
% 0.28/0.47 => ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xx = Xy )
% 0.28/0.47 => ( ( in @ Xx @ A )
% 0.28/0.47 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subset_type,type,
% 0.28/0.47 subset: $i > $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(subsetI2_type,type,
% 0.28/0.47 subsetI2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subsetI2,definition,
% 0.28/0.47 ( subsetI2
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ Xx @ B ) )
% 0.28/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminus_type,type,
% 0.28/0.47 setminus: $i > $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(setminusI_type,type,
% 0.28/0.47 setminusI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusI,definition,
% 0.28/0.47 ( setminusI
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ~ ( in @ Xx @ B )
% 0.28/0.47 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusSubset1,conjecture,
% 0.28/0.47 ( emptysetE
% 0.28/0.47 => ( in__Cong
% 0.28/0.47 => ( subsetI2
% 0.28/0.47 => ( setminusI
% 0.28/0.47 => ! [A: $i,B: $i] :
% 0.28/0.47 ( ( ( setminus @ A @ B )
% 0.28/0.47 = emptyset )
% 0.28/0.47 => ( subset @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 %------------------------------------------------------------------------------
% 0.28/0.48 ------------------------
% 0.32/1.30 ---- Embedded in THF ---
% 0.32/1.30 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.32/1.30 %%% Generated on Tue Feb 24 12:29:04 EST 2026
% 0.32/1.30 %%% using '$$dhol' embedding, version 1.3.0.
% 0.32/1.30 %%% Logic specification used:
% 0.32/1.30 %%% thf(spec, logic, $$dhol).
% 0.32/1.30
% 0.32/1.30 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.DIwqCHI51i/DTF2DTF30792.p
% See solution above
% 0.32/1.30 ------------------------
% 0.32/1.31 ---- Cleaned THF ---
% 0.32/1.31 thf(in_type,type,
% 0.32/1.31 in: $i > $i > $o ).
% 0.32/1.31
% 0.32/1.31 thf(emptyset_type,type,
% 0.32/1.31 emptyset: $i ).
% 0.32/1.31
% 0.32/1.31 thf(emptysetE_type,type,
% 0.32/1.31 emptysetE: $o ).
% 0.32/1.31
% 0.32/1.31 thf(emptysetE,definition,
% 0.32/1.31 ( emptysetE
% 0.32/1.31 = ( ! [Xx: $i] :
% 0.32/1.31 ( ( in @ Xx @ emptyset )
% 0.32/1.31 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.32/1.31
% 0.32/1.31 thf(in__Cong_type,type,
% 0.32/1.31 in__Cong: $o ).
% 0.32/1.31
% 0.32/1.31 thf(in__Cong,definition,
% 0.32/1.31 ( in__Cong
% 0.32/1.31 = ( ! [A: $i,B: $i] :
% 0.32/1.31 ( ( A = B )
% 0.32/1.31 => ! [Xx: $i,Xy: $i] :
% 0.32/1.31 ( ( Xx = Xy )
% 0.32/1.31 => ( ( in @ Xx @ A )
% 0.32/1.31 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.32/1.31
% 0.32/1.31 thf(subset_type,type,
% 0.32/1.31 subset: $i > $i > $o ).
% 0.32/1.31
% 0.32/1.31 thf(subsetI2_type,type,
% 0.32/1.31 subsetI2: $o ).
% 0.32/1.31
% 0.32/1.31 thf(subsetI2,definition,
% 0.32/1.31 ( subsetI2
% 0.32/1.31 = ( ! [A: $i,B: $i] :
% 0.32/1.31 ( ! [Xx: $i] :
% 0.32/1.31 ( ( in @ Xx @ A )
% 0.32/1.31 => ( in @ Xx @ B ) )
% 0.32/1.31 => ( subset @ A @ B ) ) ) ) ).
% 0.32/1.31
% 0.32/1.31 thf(setminus_type,type,
% 0.32/1.31 setminus: $i > $i > $i ).
% 0.32/1.31
% 0.32/1.31 thf(setminusI_type,type,
% 0.32/1.31 setminusI: $o ).
% 0.32/1.31
% 0.32/1.31 thf(setminusI,definition,
% 0.32/1.31 ( setminusI
% 0.32/1.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/1.31 ( ( in @ Xx @ A )
% 0.32/1.31 => ( ~ ( in @ Xx @ B )
% 0.32/1.31 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.32/1.31
% 0.32/1.31 thf(setminusSubset1,conjecture,
% 0.32/1.31 ( emptysetE
% 0.32/1.31 => ( in__Cong
% 0.32/1.31 => ( subsetI2
% 0.32/1.31 => ( setminusI
% 0.32/1.31 => ! [A: $i,B: $i] :
% 0.32/1.31 ( ( ( setminus @ A @ B )
% 0.32/1.31 = emptyset )
% 0.32/1.31 => ( subset @ A @ B ) ) ) ) ) ) ).
% 0.32/1.31 ------------------------
% 0.32/1.33 % (30962)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_30792 for (3000ds/2Mi)
% 0.32/1.33 % (30962)Instruction limit reached!
% 0.32/1.33 % (30962)------------------------------
% 0.32/1.33 % (30962)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.33 % (30962)Termination reason: Unknown
% 0.32/1.33 % (30962)Termination phase: Property scanning
% 0.32/1.33
% 0.32/1.33 % (30962)Memory used [KB]: 1023
% 0.32/1.33 % (30962)Time elapsed: 0.002 s
% 0.32/1.33 % (30962)Instructions burned: 3 (million)
% 0.32/1.33 % (30962)------------------------------
% 0.32/1.33 % (30962)------------------------------
% 0.32/1.33 % (30963)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_30792 for (3000ds/275Mi)
% 0.32/1.33 % (30960)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_30792 for (3000ds/27Mi)
% 0.32/1.33 % (30959)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_30792 for (3000ds/4Mi)
% 0.32/1.33 % (30958)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_30792 for (3000ds/183Mi)
% 0.32/1.33 % (30961)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_30792 for (3000ds/2Mi)
% 0.32/1.33 % (30964)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_30792 for (3000ds/18Mi)
% 0.32/1.33 % (30961)Instruction limit reached!
% 0.32/1.33 % (30961)------------------------------
% 0.32/1.33 % (30961)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.33 % (30961)Termination reason: Unknown
% 0.32/1.33 % (30961)Termination phase: Property scanning
% 0.32/1.33
% 0.32/1.33 % (30961)Memory used [KB]: 895
% 0.32/1.33 % (30961)Time elapsed: 0.003 s
% 0.32/1.33 % (30961)Instructions burned: 2 (million)
% 0.32/1.33 % (30961)------------------------------
% 0.32/1.33 % (30961)------------------------------
% 0.32/1.34 % (30959)Instruction limit reached!
% 0.32/1.34 % (30959)------------------------------
% 0.32/1.34 % (30959)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.34 % (30959)Termination reason: Unknown
% 0.32/1.34 % (30959)Termination phase: Saturation
% 0.32/1.34
% 0.32/1.34 % (30959)Memory used [KB]: 5500
% 0.32/1.34 % (30959)Time elapsed: 0.005 s
% 0.32/1.34 % (30959)Instructions burned: 5 (million)
% 0.32/1.34 % (30959)------------------------------
% 0.32/1.34 % (30959)------------------------------
% 0.32/1.34 % (30971)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_30792 for (2999ds/3Mi)
% 0.32/1.34 % (30971)Instruction limit reached!
% 0.32/1.34 % (30971)------------------------------
% 0.32/1.34 % (30971)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.34 % (30971)Termination reason: Unknown
% 0.32/1.34 % (30971)Termination phase: Saturation
% 0.32/1.34
% 0.32/1.34 % (30971)Memory used [KB]: 5500
% 0.32/1.34 % (30971)Time elapsed: 0.003 s
% 0.32/1.34 % (30971)Instructions burned: 4 (million)
% 0.32/1.34 % (30971)------------------------------
% 0.32/1.34 % (30971)------------------------------
% 0.32/1.34 % (30963)First to succeed.
% 0.32/1.34 % (30964)Also succeeded, but the first one will report.
% 0.32/1.34 % (30963)Refutation found. Thanks to Tanya!
% 0.32/1.34 % SZS status Theorem for DTF2THF_30792
% 0.32/1.34 % SZS output start Proof for DTF2THF_30792
% 0.32/1.34 thf(func_def_0, type, in: $i > $i > $o).
% 0.32/1.34 thf(func_def_5, type, subset: $i > $i > $o).
% 0.32/1.34 thf(func_def_7, type, setminus: $i > $i > $i).
% 0.32/1.34 thf(func_def_17, type, sK6: $i > $i > $i).
% 0.32/1.34 thf(f121,plain,(
% 0.32/1.34 $false),
% 0.32/1.34 inference(avatar_sat_refutation,[status(thm)],[f96,f102,f120])).
% 0.32/1.34 thf(f120,plain,(
% 0.32/1.34 ~spl14_1),
% 0.32/1.34 inference(avatar_contradiction_clause,[status(thm)],[f119])).
% 0.32/1.34 thf(f119,plain,(
% 0.32/1.34 $false | ~spl14_1),
% 0.32/1.34 inference(subsumption_resolution,[status(thm)],[f118,f52])).
% 0.32/1.34 thf(f52,plain,(
% 0.32/1.34 ($true != (subset @ sK0 @ sK1))),
% 0.32/1.34 inference(cnf_transformation,[status(thm)],[f28])).
% 0.32/1.34 thf(f28,plain,(
% 0.32/1.34 ((emptyset = (setminus @ sK0 @ sK1)) & ($true != (subset @ sK0 @ sK1))) & (emptysetE = $true) & (subsetI2 = $true) & (in__Cong = $true) & (setminusI = $true)),
% 0.32/1.34 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1])],[f26,f27])).
% 0.32/1.34 thf(f27,plain,(
% 0.32/1.34 ? [X0 : $i,X1 : $i] : ((emptyset = (setminus @ X0 @ X1)) & ($true != (subset @ X0 @ X1))) => ((emptyset = (setminus @ sK0 @ sK1)) & ($true != (subset @ sK0 @ sK1)))),
% 0.32/1.34 introduced(definition,[],[choice_axiom])).
% 0.32/1.34 thf(f26,plain,(
% 0.32/1.34 ? [X0 : $i,X1 : $i] : ((emptyset = (setminus @ X0 @ X1)) & ($true != (subset @ X0 @ X1))) & (emptysetE = $true) & (subsetI2 = $true) & (in__Cong = $true) & (setminusI = $true)),
% 0.32/1.34 inference(rectify,[status(thm)],[f22])).
% 0.32/1.34 thf(f22,plain,(
% 0.32/1.34 ? [X1 : $i,X0 : $i] : ((emptyset = (setminus @ X1 @ X0)) & ((subset @ X1 @ X0) != $true)) & (emptysetE = $true) & (subsetI2 = $true) & (in__Cong = $true) & (setminusI = $true)),
% 0.32/1.34 inference(flattening,[status(thm)],[f21])).
% 0.32/1.34 thf(f21,plain,(
% 0.32/1.34 (((? [X1 : $i,X0 : $i] : ((emptyset = (setminus @ X1 @ X0)) & ((subset @ X1 @ X0) != $true)) & (setminusI = $true)) & (subsetI2 = $true)) & (in__Cong = $true)) & (emptysetE = $true)),
% 0.32/1.34 inference(ennf_transformation,[status(thm)],[f15])).
% 0.32/1.34 thf(f15,plain,(
% 0.32/1.34 ~((emptysetE = $true) => ((in__Cong = $true) => ((subsetI2 = $true) => ((setminusI = $true) => ! [X1 : $i,X0 : $i] : ((emptyset = (setminus @ X1 @ X0)) => ((subset @ X1 @ X0) = $true))))))),
% 0.32/1.34 inference(fool_elimination,[status(thm)],[f14])).
% 0.32/1.34 thf(f14,plain,(
% 0.32/1.34 ~(emptysetE => (in__Cong => (subsetI2 => (setminusI => ! [X0 : $i,X1 : $i] : ((emptyset = (setminus @ X1 @ X0)) => (subset @ X1 @ X0))))))),
% 0.32/1.34 inference(rectify,[status(thm)],[f6])).
% 0.32/1.34 thf(f6,negated_conjecture,(
% 0.32/1.34 ~(emptysetE => (in__Cong => (subsetI2 => (setminusI => ! [X3 : $i,X2 : $i] : ((emptyset = (setminus @ X2 @ X3)) => (subset @ X2 @ X3))))))),
% 0.32/1.34 inference(negated_conjecture,[status(cth)],[f5])).
% 0.32/1.34 thf(f5,conjecture,(
% 0.32/1.34 emptysetE => (in__Cong => (subsetI2 => (setminusI => ! [X3 : $i,X2 : $i] : ((emptyset = (setminus @ X2 @ X3)) => (subset @ X2 @ X3)))))),
% 0.32/1.34 file('/export/starexec/sandbox2/tmp/tmp.DIwqCHI51i/DTF2THF_30792.p',setminusSubset1)).
% 0.32/1.34 thf(f118,plain,(
% 0.32/1.34 ($true = (subset @ sK0 @ sK1)) | ~spl14_1),
% 0.32/1.34 inference(trivial_inequality_removal,[status(thm)],[f117])).
% 0.32/1.34 thf(f117,plain,(
% 0.32/1.34 ($true = (subset @ sK0 @ sK1)) | ($true != $true) | ~spl14_1),
% 0.32/1.34 inference(duplicate_literal_removal,[status(thm)],[f116])).
% 0.32/1.34 thf(f116,plain,(
% 0.32/1.34 ($true = (subset @ sK0 @ sK1)) | ($true = (subset @ sK0 @ sK1)) | ($true != $true) | ~spl14_1),
% 0.32/1.34 inference(superposition,[status(thm)],[f113,f89])).
% 0.32/1.34 thf(f89,plain,(
% 0.32/1.34 ( ! [X0 : $i,X1 : $i] : (($true = (in @ (sK6 @ X1 @ X0) @ X1)) | ((subset @ X1 @ X0) = $true)) )),
% 0.32/1.34 inference(trivial_inequality_removal,[status(thm)],[f74])).
% 0.32/1.34 thf(f74,plain,(
% 0.32/1.34 ( ! [X0 : $i,X1 : $i] : (((subset @ X1 @ X0) = $true) | ($true != $true) | ($true = (in @ (sK6 @ X1 @ X0) @ X1))) )),
% 0.32/1.34 inference(definition_unfolding,[status(thm)],[f60,f50])).
% 0.32/1.34 thf(f50,plain,(
% 0.32/1.34 (subsetI2 = $true)),
% 0.32/1.34 inference(cnf_transformation,[status(thm)],[f28])).
% 0.32/1.34 thf(f60,plain,(
% 0.32/1.34 ( ! [X0 : $i,X1 : $i] : (((subset @ X1 @ X0) = $true) | ($true = (in @ (sK6 @ X1 @ X0) @ X1)) | (subsetI2 != $true)) )),
% 0.32/1.34 inference(cnf_transformation,[status(thm)],[f38])).
% 0.32/1.34 thf(f38,plain,(
% 0.32/1.34 (! [X0 : $i,X1 : $i] : (((subset @ X1 @ X0) = $true) | (($true != (in @ (sK6 @ X1 @ X0) @ X0)) & ($true = (in @ (sK6 @ X1 @ X0) @ X1)))) | (subsetI2 != $true)) & ((subsetI2 = $true) | (((subset @ sK8 @ sK7) != $true) & ! [X5 : $i] : (((in @ X5 @ sK7) = $true) | ($true != (in @ X5 @ sK8)))))),
% 0.32/1.34 inference(skolemisation,[status(esa),new_symbols(skolem,[sK6,sK7,sK8])],[f35,f37,f36])).
% 0.32/1.34 thf(f36,plain,(
% 0.32/1.34 ! [X0 : $i,X1 : $i] : (? [X2 : $i] : (($true != (in @ X2 @ X0)) & ($true = (in @ X2 @ X1))) => (($true != (in @ (sK6 @ X1 @ X0) @ X0)) & ($true = (in @ (sK6 @ X1 @ X0) @ X1))))),
% 0.32/1.34 introduced(definition,[],[choice_axiom])).
% 0.32/1.34 thf(f37,plain,(
% 0.32/1.34 ? [X3 : $i,X4 : $i] : (((subset @ X4 @ X3) != $true) & ! [X5 : $i] : (((in @ X5 @ X3) = $true) | ($true != (in @ X5 @ X4)))) => (((subset @ sK8 @ sK7) != $true) & ! [X5 : $i] : (((in @ X5 @ sK7) = $true) | ($true != (in @ X5 @ sK8))))),
% 0.32/1.34 introduced(definition,[],[choice_axiom])).
% 0.32/1.34 thf(f35,plain,(
% 0.32/1.34 (! [X0 : $i,X1 : $i] : (((subset @ X1 @ X0) = $true) | ? [X2 : $i] : (($true != (in @ X2 @ X0)) & ($true = (in @ X2 @ X1)))) | (subsetI2 != $true)) & ((subsetI2 = $true) | ? [X3 : $i,X4 : $i] : (((subset @ X4 @ X3) != $true) & ! [X5 : $i] : (((in @ X5 @ X3) = $true) | ($true != (in @ X5 @ X4)))))),
% 0.32/1.34 inference(rectify,[status(thm)],[f34])).
% 0.32/1.34 thf(f34,plain,(
% 0.32/1.34 (! [X1 : $i,X0 : $i] : (($true = (subset @ X0 @ X1)) | ? [X2 : $i] : (($true != (in @ X2 @ X1)) & ($true = (in @ X2 @ X0)))) | (subsetI2 != $true)) & ((subsetI2 = $true) | ? [X1 : $i,X0 : $i] : (($true != (subset @ X0 @ X1)) & ! [X2 : $i] : (($true = (in @ X2 @ X1)) | ($true != (in @ X2 @ X0)))))),
% 0.32/1.34 inference(nnf_transformation,[status(thm)],[f23])).
% 0.32/1.34 thf(f23,plain,(
% 0.32/1.34 ! [X1 : $i,X0 : $i] : (($true = (subset @ X0 @ X1)) | ? [X2 : $i] : (($true != (in @ X2 @ X1)) & ($true = (in @ X2 @ X0)))) <=> (subsetI2 = $true)),
% 0.32/1.34 inference(ennf_transformation,[status(thm)],[f9])).
% 0.32/1.34 thf(f9,plain,(
% 0.32/1.34 (subsetI2 = $true) <=> ! [X1 : $i,X0 : $i] : (! [X2 : $i] : (($true = (in @ X2 @ X0)) => ($true = (in @ X2 @ X1))) => ($true = (subset @ X0 @ X1)))),
% 0.32/1.34 inference(fool_elimination,[status(thm)],[f8])).
% 0.32/1.34 thf(f8,plain,(
% 0.32/1.34 (! [X0 : $i,X1 : $i] : (! [X2 : $i] : ((in @ X2 @ X0) => (in @ X2 @ X1)) => (subset @ X0 @ X1)) = subsetI2)),
% 0.32/1.34 inference(rectify,[status(thm)],[f3])).
% 0.32/1.34 thf(f3,axiom,(
% 0.32/1.34 (! [X2 : $i,X3 : $i] : (! [X0 : $i] : ((in @ X0 @ X2) => (in @ X0 @ X3)) => (subset @ X2 @ X3)) = subsetI2)),
% 0.32/1.34 file('/export/starexec/sandbox2/tmp/tmp.DIwqCHI51i/DTF2THF_30792.p',subsetI2)).
% 0.32/1.34 thf(f113,plain,(
% 0.32/1.34 ( ! [X0 : $i] : (((in @ (sK6 @ X0 @ sK1) @ sK0) != $true) | ($true = (subset @ X0 @ sK1))) ) | ~spl14_1),
% 0.32/1.34 inference(trivial_inequality_removal,[status(thm)],[f112])).
% 0.32/1.34 thf(f112,plain,(
% 0.32/1.34 ( ! [X0 : $i] : (($true != $true) | ((in @ (sK6 @ X0 @ sK1) @ sK0) != $true) | ($true = (subset @ X0 @ sK1))) ) | ~spl14_1),
% 0.32/1.34 inference(superposition,[status(thm)],[f88,f110])).
% 0.32/1.34 thf(f110,plain,(
% 0.32/1.34 ( ! [X0 : $i] : (($true = (in @ X0 @ sK1)) | ($true != (in @ X0 @ sK0))) ) | ~spl14_1),
% 0.32/1.34 inference(trivial_inequality_removal,[status(thm)],[f108])).
% 0.32/1.34 thf(f108,plain,(
% 0.32/1.34 ( ! [X0 : $i] : (($true != $true) | ($true = (in @ X0 @ sK1)) | ($true != (in @ X0 @ sK0))) ) | ~spl14_1),
% 0.32/1.34 inference(superposition,[status(thm)],[f92,f86])).
% 0.32/1.34 thf(f86,plain,(
% 0.32/1.34 ( ! [X2 : $i,X0 : $i,X1 : $i] : (((in @ X2 @ (setminus @ X1 @ X0)) = $true) | ($true = (in @ X2 @ X0)) | ($true != (in @ X2 @ X1))) )),
% 0.32/1.34 inference(trivial_inequality_removal,[status(thm)],[f77])).
% 0.32/1.34 thf(f77,plain,(
% 0.32/1.34 ( ! [X2 : $i,X0 : $i,X1 : $i] : (((in @ X2 @ (setminus @ X1 @ X0)) = $true) | ($true != $true) | ($true != (in @ X2 @ X1)) | ($true = (in @ X2 @ X0))) )),
% 0.32/1.34 inference(definition_unfolding,[status(thm)],[f65,f48])).
% 0.32/1.34 thf(f48,plain,(
% 0.32/1.34 (setminusI = $true)),
% 0.32/1.34 inference(cnf_transformation,[status(thm)],[f28])).
% 0.32/1.34 thf(f65,plain,(
% 0.32/1.34 ( ! [X2 : $i,X0 : $i,X1 : $i] : (($true = (in @ X2 @ X0)) | ((in @ X2 @ (setminus @ X1 @ X0)) = $true) | ($true != (in @ X2 @ X1)) | (setminusI != $true)) )),
% 0.32/1.34 inference(cnf_transformation,[status(thm)],[f42])).
% 0.32/1.34 thf(f42,plain,(
% 0.32/1.34 (! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X2 @ X0)) | ((in @ X2 @ (setminus @ X1 @ X0)) = $true) | ($true != (in @ X2 @ X1))) | (setminusI != $true)) & ((setminusI = $true) | (($true != (in @ sK11 @ sK9)) & ($true != (in @ sK11 @ (setminus @ sK10 @ sK9))) & ((in @ sK11 @ sK10) = $true)))),
% 0.32/1.34 inference(skolemisation,[status(esa),new_symbols(skolem,[sK9,sK10,sK11])],[f40,f41])).
% 0.32/1.34 thf(f41,plain,(
% 0.32/1.34 ? [X3 : $i,X4 : $i,X5 : $i] : (((in @ X5 @ X3) != $true) & ($true != (in @ X5 @ (setminus @ X4 @ X3))) & ($true = (in @ X5 @ X4))) => (($true != (in @ sK11 @ sK9)) & ($true != (in @ sK11 @ (setminus @ sK10 @ sK9))) & ((in @ sK11 @ sK10) = $true))),
% 0.32/1.34 introduced(definition,[],[choice_axiom])).
% 0.32/1.34 thf(f40,plain,(
% 0.32/1.34 (! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X2 @ X0)) | ((in @ X2 @ (setminus @ X1 @ X0)) = $true) | ($true != (in @ X2 @ X1))) | (setminusI != $true)) & ((setminusI = $true) | ? [X3 : $i,X4 : $i,X5 : $i] : (((in @ X5 @ X3) != $true) & ($true != (in @ X5 @ (setminus @ X4 @ X3))) & ($true = (in @ X5 @ X4))))),
% 0.32/1.34 inference(rectify,[status(thm)],[f39])).
% 0.32/1.34 thf(f39,plain,(
% 0.32/1.34 (! [X1 : $i,X2 : $i,X0 : $i] : (((in @ X0 @ X1) = $true) | ((in @ X0 @ (setminus @ X2 @ X1)) = $true) | ((in @ X0 @ X2) != $true)) | (setminusI != $true)) & ((setminusI = $true) | ? [X1 : $i,X2 : $i,X0 : $i] : (((in @ X0 @ X1) != $true) & ((in @ X0 @ (setminus @ X2 @ X1)) != $true) & ((in @ X0 @ X2) = $true)))),
% 0.32/1.34 inference(nnf_transformation,[status(thm)],[f25])).
% 0.32/1.34 thf(f25,plain,(
% 0.32/1.34 ! [X1 : $i,X2 : $i,X0 : $i] : (((in @ X0 @ X1) = $true) | ((in @ X0 @ (setminus @ X2 @ X1)) = $true) | ((in @ X0 @ X2) != $true)) <=> (setminusI = $true)),
% 0.32/1.34 inference(flattening,[status(thm)],[f24])).
% 0.32/1.34 thf(f24,plain,(
% 0.32/1.34 (setminusI = $true) <=> ! [X1 : $i,X2 : $i,X0 : $i] : ((((in @ X0 @ (setminus @ X2 @ X1)) = $true) | ((in @ X0 @ X1) = $true)) | ((in @ X0 @ X2) != $true))),
% 0.32/1.34 inference(ennf_transformation,[status(thm)],[f18])).
% 0.32/1.34 thf(f18,plain,(
% 0.32/1.34 (setminusI = $true) <=> ! [X1 : $i,X2 : $i,X0 : $i] : (((in @ X0 @ X2) = $true) => (((in @ X0 @ X1) != $true) => ((in @ X0 @ (setminus @ X2 @ X1)) = $true)))),
% 0.32/1.34 inference(flattening,[status(thm)],[f17])).
% 0.32/1.34 thf(f17,plain,(
% 0.32/1.34 (setminusI = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ X2) = $true) => (~((in @ X0 @ X1) = $true) => ((in @ X0 @ (setminus @ X2 @ X1)) = $true)))),
% 0.32/1.34 inference(fool_elimination,[status(thm)],[f16])).
% 0.32/1.34 thf(f16,plain,(
% 0.32/1.34 (setminusI = ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X0 @ X2) => (~(in @ X0 @ X1) => (in @ X0 @ (setminus @ X2 @ X1)))))),
% 0.32/1.34 inference(rectify,[status(thm)],[f4])).
% 0.32/1.34 thf(f4,axiom,(
% 0.32/1.34 (setminusI = ! [X0 : $i,X3 : $i,X2 : $i] : ((in @ X0 @ X2) => (~(in @ X0 @ X3) => (in @ X0 @ (setminus @ X2 @ X3)))))),
% 0.32/1.34 file('/export/starexec/sandbox2/tmp/tmp.DIwqCHI51i/DTF2THF_30792.p',setminusI)).
% 0.32/1.34 thf(f92,plain,(
% 0.32/1.34 ( ! [X2 : $i] : (((in @ X2 @ (setminus @ sK0 @ sK1)) != $true)) ) | ~spl14_1),
% 0.32/1.34 inference(avatar_component_clause,[status(thm)],[f91])).
% 0.32/1.34 thf(f91,definition,(
% 0.32/1.34 spl14_1 <=> ! [X2 : $i] : ((in @ X2 @ (setminus @ sK0 @ sK1)) != $true)),
% 0.32/1.34 introduced(definition,[new_symbols(naming,[spl14_1])],[avatar_definition])).
% 0.32/1.34 thf(f88,plain,(
% 0.32/1.34 ( ! [X0 : $i,X1 : $i] : (($true != (in @ (sK6 @ X1 @ X0) @ X0)) | ((subset @ X1 @ X0) = $true)) )),
% 0.32/1.34 inference(trivial_inequality_removal,[status(thm)],[f73])).
% 0.32/1.34 thf(f73,plain,(
% 0.32/1.34 ( ! [X0 : $i,X1 : $i] : (($true != $true) | ($true != (in @ (sK6 @ X1 @ X0) @ X0)) | ((subset @ X1 @ X0) = $true)) )),
% 0.32/1.34 inference(definition_unfolding,[status(thm)],[f61,f50])).
% 0.32/1.34 thf(f61,plain,(
% 0.32/1.34 ( ! [X0 : $i,X1 : $i] : (((subset @ X1 @ X0) = $true) | ($true != (in @ (sK6 @ X1 @ X0) @ X0)) | (subsetI2 != $true)) )),
% 0.32/1.34 inference(cnf_transformation,[status(thm)],[f38])).
% 0.32/1.34 thf(f102,plain,(
% 0.32/1.34 ~spl14_2),
% 0.32/1.34 inference(avatar_contradiction_clause,[status(thm)],[f101])).
% 0.32/1.34 thf(f101,plain,(
% 0.32/1.34 $false | ~spl14_2),
% 0.32/1.34 inference(trivial_inequality_removal,[status(thm)],[f100])).
% 0.32/1.34 thf(f100,plain,(
% 0.32/1.34 ($true != $true) | ~spl14_2),
% 0.32/1.34 inference(superposition,[status(thm)],[f52,f95])).
% 0.32/1.34 thf(f95,plain,(
% 0.32/1.34 ( ! [X3 : $o] : (($true = X3)) ) | ~spl14_2),
% 0.32/1.34 inference(avatar_component_clause,[status(thm)],[f94])).
% 0.32/1.34 thf(f94,definition,(
% 0.32/1.34 spl14_2 <=> ! [X3 : $o] : ($true = X3)),
% 0.32/1.34 introduced(definition,[new_symbols(naming,[spl14_2])],[avatar_definition])).
% 0.32/1.34 thf(f96,plain,(
% 0.32/1.34 spl14_1 | spl14_2),
% 0.32/1.34 inference(avatar_split_clause,[status(thm)],[f87,f94,f91])).
% 0.32/1.34 thf(f87,plain,(
% 0.32/1.34 ( ! [X2 : $i,X3 : $o] : (($true = X3) | ((in @ X2 @ (setminus @ sK0 @ sK1)) != $true)) )),
% 0.32/1.34 inference(trivial_inequality_removal,[status(thm)],[f83])).
% 0.32/1.34 thf(f83,plain,(
% 0.32/1.34 ( ! [X2 : $i,X3 : $o] : (($true = X3) | ((in @ X2 @ (setminus @ sK0 @ sK1)) != $true) | ($true != $true)) )),
% 0.32/1.34 inference(definition_unfolding,[status(thm)],[f66,f53,f51])).
% 0.32/1.34 thf(f51,plain,(
% 0.32/1.34 (emptysetE = $true)),
% 0.32/1.34 inference(cnf_transformation,[status(thm)],[f28])).
% 0.32/1.34 thf(f53,plain,(
% 0.32/1.34 (emptyset = (setminus @ sK0 @ sK1))),
% 0.32/1.34 inference(cnf_transformation,[status(thm)],[f28])).
% 0.32/1.34 thf(f66,plain,(
% 0.32/1.34 ( ! [X2 : $i,X3 : $o] : (($true = X3) | ($true != (in @ X2 @ emptyset)) | (emptysetE != $true)) )),
% 0.32/1.34 inference(cnf_transformation,[status(thm)],[f47])).
% 0.32/1.34 thf(f47,plain,(
% 0.32/1.34 ((emptysetE = $true) | (($true != sK13) & ($true = (in @ sK12 @ emptyset)))) & (! [X2 : $i] : (! [X3 : $o] : ($true = X3) | ($true != (in @ X2 @ emptyset))) | (emptysetE != $true))),
% 0.32/1.34 inference(skolemisation,[status(esa),new_symbols(skolem,[sK12,sK13])],[f44,f46,f45])).
% 0.32/1.34 thf(f45,plain,(
% 0.32/1.34 ? [X0 : $i] : (? [X1 : $o] : ($true != X1) & ((in @ X0 @ emptyset) = $true)) => (? [X1 : $o] : ($true != X1) & ($true = (in @ sK12 @ emptyset)))),
% 0.32/1.34 introduced(definition,[],[choice_axiom])).
% 0.32/1.34 thf(f46,plain,(
% 0.32/1.34 ? [X1 : $o] : ($true != X1) => ($true != sK13)),
% 0.32/1.34 introduced(definition,[],[choice_axiom])).
% 0.32/1.34 thf(f44,plain,(
% 0.32/1.34 ((emptysetE = $true) | ? [X0 : $i] : (? [X1 : $o] : ($true != X1) & ((in @ X0 @ emptyset) = $true))) & (! [X2 : $i] : (! [X3 : $o] : ($true = X3) | ($true != (in @ X2 @ emptyset))) | (emptysetE != $true))),
% 0.32/1.34 inference(rectify,[status(thm)],[f43])).
% 0.32/1.34 thf(f43,plain,(
% 0.32/1.34 ((emptysetE = $true) | ? [X0 : $i] : (? [X1 : $o] : ($true != X1) & ((in @ X0 @ emptyset) = $true))) & (! [X0 : $i] : (! [X1 : $o] : ($true = X1) | ((in @ X0 @ emptyset) != $true)) | (emptysetE != $true))),
% 0.32/1.34 inference(nnf_transformation,[status(thm)],[f19])).
% 0.32/1.34 thf(f19,plain,(
% 0.32/1.34 (emptysetE = $true) <=> ! [X0 : $i] : (! [X1 : $o] : ($true = X1) | ((in @ X0 @ emptyset) != $true))),
% 0.32/1.34 inference(ennf_transformation,[status(thm)],[f11])).
% 0.32/1.34 thf(f11,plain,(
% 0.32/1.34 ! [X0 : $i] : (((in @ X0 @ emptyset) = $true) => ! [X1 : $o] : ($true = X1)) <=> (emptysetE = $true)),
% 0.32/1.34 inference(fool_elimination,[status(thm)],[f10])).
% 0.32/1.34 thf(f10,plain,(
% 0.32/1.34 (! [X0 : $i] : ((in @ X0 @ emptyset) => ! [X1 : $o] : X1) = emptysetE)),
% 0.32/1.34 inference(rectify,[status(thm)],[f1])).
% 0.32/1.34 thf(f1,axiom,(
% 0.32/1.34 (! [X0 : $i] : ((in @ X0 @ emptyset) => ! [X1 : $o] : X1) = emptysetE)),
% 0.32/1.34 file('/export/starexec/sandbox2/tmp/tmp.DIwqCHI51i/DTF2THF_30792.p',emptysetE)).
% 0.32/1.34 % SZS output end Proof for DTF2THF_30792
% 0.32/1.34 % (30963)------------------------------
% 0.32/1.34 % (30963)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.34 % (30963)Termination reason: Refutation
% 0.32/1.34
% 0.32/1.34 % (30963)Memory used [KB]: 5628
% 0.32/1.34 % (30963)Time elapsed: 0.010 s
% 0.32/1.34 % (30963)Instructions burned: 8 (million)
% 0.32/1.34 % (30963)------------------------------
% 0.32/1.34 % (30963)------------------------------
% 0.32/1.34 % (30956)Success in time 0.025 s
%------------------------------------------------------------------------------