%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU526^2 : 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 : n022.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 09:35:05 AM UTC 2026
% Result : Theorem 1.62s 1.39s
% Output : Refutation 1.62s
% Verified :
% SZS Type : Refutation
% Derivation depth : 17
% Number of leaves : 12
% Syntax : Number of formulae : 61 ( 18 unt; 0 typ; 2 def)
% Number of atoms : 564 ( 181 equ; 0 cnn)
% Maximal formula atoms : 12 ( 9 avg)
% Number of connectives : 477 ( 85 ~; 63 |; 50 &; 236 @)
% ( 7 <=>; 36 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 5 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 17 ( 14 usr; 12 con; 0-2 aty)
% Number of variables : 115 ( 4 ^; 79 !; 32 ?; 115 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_5,type,
nonempty: $i > $o ).
thf(func_def_17,type,
sK5: $i > $i > $i ).
thf(func_def_18,type,
sK6: $i > $i > $i ).
thf(f93,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f73,f83,f90]) ).
thf(f90,plain,
~ spl7_1,
inference(avatar_contradiction_clause,[status(thm)],[f89]) ).
thf(f89,plain,
( $false
| ~ spl7_1 ),
inference(trivial_inequality_removal,[status(thm)],[f88]) ).
thf(f88,plain,
( ( $true != $true )
| ~ spl7_1 ),
inference(superposition,[status(thm)],[f38,f69]) ).
thf(f69,plain,
( ! [X1: $o] : ( $true = X1 )
| ~ spl7_1 ),
inference(avatar_component_clause,[status(thm)],[f68]) ).
thf(f68,definition,
( spl7_1
<=> ! [X1: $o] : ( $true = X1 ) ),
introduced(definition,[new_symbols(naming,[spl7_1])],[avatar_definition]) ).
thf(f38,plain,
! [X1: $i] :
( ( in @ X1 @ sK2 )
!= $true ),
inference(cnf_transformation,[status(thm)],[f26]) ).
thf(f26,plain,
( ( emptysetE = $true )
& ! [X1: $i] :
( ( in @ X1 @ sK2 )
!= $true )
& ( $true
= ( nonempty @ sK2 ) )
& ( setext = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK2])],[f19,f25]) ).
thf(f25,plain,
( ? [X0: $i] :
( ! [X1: $i] :
( ( in @ X1 @ X0 )
!= $true )
& ( $true
= ( nonempty @ X0 ) ) )
=> ( ! [X1: $i] :
( ( in @ X1 @ sK2 )
!= $true )
& ( $true
= ( nonempty @ sK2 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f19,plain,
( ( emptysetE = $true )
& ? [X0: $i] :
( ! [X1: $i] :
( ( in @ X1 @ X0 )
!= $true )
& ( $true
= ( nonempty @ X0 ) ) )
& ( setext = $true ) ),
inference(flattening,[status(thm)],[f18]) ).
thf(f18,plain,
( ? [X0: $i] :
( ! [X1: $i] :
( ( in @ X1 @ X0 )
!= $true )
& ( $true
= ( nonempty @ X0 ) ) )
& ( setext = $true )
& ( emptysetE = $true ) ),
inference(ennf_transformation,[status(thm)],[f14]) ).
thf(f14,plain,
~ ( ( emptysetE = $true )
=> ( ( setext = $true )
=> ! [X0: $i] :
( ( $true
= ( nonempty @ X0 ) )
=> ? [X1: $i] :
( ( in @ X1 @ X0 )
= $true ) ) ) ),
inference(true_and_false_elimination,[status(thm)],[f10]) ).
thf(f10,plain,
~ ( ( emptysetE = $true )
=> ( ( setext = $true )
=> ! [X0: $i] :
( ( $true
= ( nonempty @ X0 ) )
=> ? [X1: $i] :
( $true
& ( ( in @ X1 @ X0 )
= $true ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f9]) ).
thf(f9,plain,
~ ( emptysetE
=> ( setext
=> ! [X0: $i] :
( ( nonempty @ X0 )
=> ? [X1: $i] :
( $true
& ( in @ X1 @ X0 ) ) ) ) ),
inference(rectify,[status(thm)],[f5]) ).
thf(f5,negated_conjecture,
~ ( emptysetE
=> ( setext
=> ! [X2: $i] :
( ( nonempty @ X2 )
=> ? [X0: $i] :
( $true
& ( in @ X0 @ X2 ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f4]) ).
thf(f4,conjecture,
( emptysetE
=> ( setext
=> ! [X2: $i] :
( ( nonempty @ X2 )
=> ? [X0: $i] :
( $true
& ( in @ X0 @ X2 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.bXuGu7mRJV/DTF2THF_31515.p',nonemptyImpWitness) ).
thf(f83,plain,
~ spl7_2,
inference(avatar_contradiction_clause,[status(thm)],[f82]) ).
thf(f82,plain,
( $false
| ~ spl7_2 ),
inference(subsumption_resolution,[status(thm)],[f81,f65]) ).
thf(f65,plain,
emptyset != sK2,
inference(equality_proxy_clausification,[status(thm)],[f64]) ).
thf(f64,plain,
( $false
= ( emptyset = sK2 ) ),
inference(not_proxy_clausification,[status(thm)],[f63]) ).
thf(f63,plain,
( $true
= ( emptyset != sK2 ) ),
inference(beta_eta_normalization,[status(thm)],[f51]) ).
thf(f51,plain,
( $true
= ( ^ [Y0: $i] : ( emptyset != Y0 )
@ sK2 ) ),
inference(definition_unfolding,[status(thm)],[f37,f47]) ).
thf(f47,plain,
( nonempty
= ( ^ [Y0: $i] : ( emptyset != Y0 ) ) ),
inference(cnf_transformation,[status(thm)],[f13]) ).
thf(f13,plain,
( nonempty
= ( ^ [Y0: $i] : ( emptyset != Y0 ) ) ),
inference(fool_elimination,[status(thm)],[f3]) ).
thf(f3,axiom,
( nonempty
= ( ^ [X0: $i] : ( emptyset != X0 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.bXuGu7mRJV/DTF2THF_31515.p',nonempty) ).
thf(f37,plain,
( $true
= ( nonempty @ sK2 ) ),
inference(cnf_transformation,[status(thm)],[f26]) ).
thf(f81,plain,
( ( emptyset = sK2 )
| ~ spl7_2 ),
inference(trivial_inequality_removal,[status(thm)],[f79]) ).
thf(f79,plain,
( ( emptyset = sK2 )
| ( $true != $true )
| ~ spl7_2 ),
inference(superposition,[status(thm)],[f38,f77]) ).
thf(f77,plain,
( ! [X0: $i] :
( ( ( in @ ( sK5 @ X0 @ emptyset ) @ X0 )
= $true )
| ( emptyset = X0 ) )
| ~ spl7_2 ),
inference(trivial_inequality_removal,[status(thm)],[f76]) ).
thf(f76,plain,
( ! [X0: $i] :
( ( ( in @ ( sK5 @ X0 @ emptyset ) @ X0 )
= $true )
| ( $true != $true )
| ( emptyset = X0 ) )
| ~ spl7_2 ),
inference(superposition,[status(thm)],[f72,f60]) ).
thf(f60,plain,
! [X4: $i,X5: $i] :
( ( ( in @ ( sK6 @ X5 @ X4 ) @ X4 )
= $true )
| ( X4 = X5 )
| ( ( in @ ( sK5 @ X5 @ X4 ) @ X5 )
= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f57]) ).
thf(f57,plain,
! [X4: $i,X5: $i] :
( ( X4 = X5 )
| ( $true != $true )
| ( ( in @ ( sK5 @ X5 @ X4 ) @ X5 )
= $true )
| ( ( in @ ( sK6 @ X5 @ X4 ) @ X4 )
= $true ) ),
inference(definition_unfolding,[status(thm)],[f41,f36]) ).
thf(f36,plain,
setext = $true,
inference(cnf_transformation,[status(thm)],[f26]) ).
thf(f41,plain,
! [X4: $i,X5: $i] :
( ( ( in @ ( sK5 @ X5 @ X4 ) @ X5 )
= $true )
| ( X4 = X5 )
| ( ( in @ ( sK6 @ X5 @ X4 ) @ X4 )
= $true )
| ( setext != $true ) ),
inference(cnf_transformation,[status(thm)],[f32]) ).
thf(f32,plain,
( ( ( setext = $true )
| ( ! [X2: $i] :
( ( $true
= ( in @ X2 @ sK3 ) )
| ( $true
!= ( in @ X2 @ sK4 ) ) )
& ( sK4 != sK3 )
& ! [X3: $i] :
( ( $true
!= ( in @ X3 @ sK3 ) )
| ( $true
= ( in @ X3 @ sK4 ) ) ) ) )
& ( ! [X4: $i,X5: $i] :
( ( ( $true
!= ( in @ ( sK5 @ X5 @ X4 ) @ X4 ) )
& ( ( in @ ( sK5 @ X5 @ X4 ) @ X5 )
= $true ) )
| ( X4 = X5 )
| ( ( ( in @ ( sK6 @ X5 @ X4 ) @ X4 )
= $true )
& ( $true
!= ( in @ ( sK6 @ X5 @ X4 ) @ X5 ) ) ) )
| ( setext != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK3,sK4,sK5,sK6])],[f28,f31,f30,f29]) ).
thf(f29,plain,
( ? [X0: $i,X1: $i] :
( ! [X2: $i] :
( ( ( in @ X2 @ X0 )
= $true )
| ( ( in @ X2 @ X1 )
!= $true ) )
& ( X0 != X1 )
& ! [X3: $i] :
( ( $true
!= ( in @ X3 @ X0 ) )
| ( ( in @ X3 @ X1 )
= $true ) ) )
=> ( ! [X2: $i] :
( ( $true
= ( in @ X2 @ sK3 ) )
| ( $true
!= ( in @ X2 @ sK4 ) ) )
& ( sK4 != sK3 )
& ! [X3: $i] :
( ( $true
!= ( in @ X3 @ sK3 ) )
| ( $true
= ( in @ X3 @ sK4 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f30,plain,
! [X4: $i,X5: $i] :
( ? [X6: $i] :
( ( ( in @ X6 @ X4 )
!= $true )
& ( $true
= ( in @ X6 @ X5 ) ) )
=> ( ( $true
!= ( in @ ( sK5 @ X5 @ X4 ) @ X4 ) )
& ( ( in @ ( sK5 @ X5 @ X4 ) @ X5 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f31,plain,
! [X4: $i,X5: $i] :
( ? [X7: $i] :
( ( $true
= ( in @ X7 @ X4 ) )
& ( $true
!= ( in @ X7 @ X5 ) ) )
=> ( ( ( in @ ( sK6 @ X5 @ X4 ) @ X4 )
= $true )
& ( $true
!= ( in @ ( sK6 @ X5 @ X4 ) @ X5 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f28,plain,
( ( ( setext = $true )
| ? [X0: $i,X1: $i] :
( ! [X2: $i] :
( ( ( in @ X2 @ X0 )
= $true )
| ( ( in @ X2 @ X1 )
!= $true ) )
& ( X0 != X1 )
& ! [X3: $i] :
( ( $true
!= ( in @ X3 @ X0 ) )
| ( ( in @ X3 @ X1 )
= $true ) ) ) )
& ( ! [X4: $i,X5: $i] :
( ? [X6: $i] :
( ( ( in @ X6 @ X4 )
!= $true )
& ( $true
= ( in @ X6 @ X5 ) ) )
| ( X4 = X5 )
| ? [X7: $i] :
( ( $true
= ( in @ X7 @ X4 ) )
& ( $true
!= ( in @ X7 @ X5 ) ) ) )
| ( setext != $true ) ) ),
inference(rectify,[status(thm)],[f27]) ).
thf(f27,plain,
( ( ( setext = $true )
| ? [X0: $i,X1: $i] :
( ! [X2: $i] :
( ( ( in @ X2 @ X0 )
= $true )
| ( ( in @ X2 @ X1 )
!= $true ) )
& ( X0 != X1 )
& ! [X3: $i] :
( ( $true
!= ( in @ X3 @ X0 ) )
| ( ( in @ X3 @ X1 )
= $true ) ) ) )
& ( ! [X0: $i,X1: $i] :
( ? [X2: $i] :
( ( ( in @ X2 @ X0 )
!= $true )
& ( ( in @ X2 @ X1 )
= $true ) )
| ( X0 = X1 )
| ? [X3: $i] :
( ( $true
= ( in @ X3 @ X0 ) )
& ( ( in @ X3 @ X1 )
!= $true ) ) )
| ( setext != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f17]) ).
thf(f17,plain,
( ( setext = $true )
<=> ! [X0: $i,X1: $i] :
( ? [X2: $i] :
( ( ( in @ X2 @ X0 )
!= $true )
& ( ( in @ X2 @ X1 )
= $true ) )
| ( X0 = X1 )
| ? [X3: $i] :
( ( $true
= ( in @ X3 @ X0 ) )
& ( ( in @ X3 @ X1 )
!= $true ) ) ) ),
inference(flattening,[status(thm)],[f16]) ).
thf(f16,plain,
( ! [X0: $i,X1: $i] :
( ( X0 = X1 )
| ? [X3: $i] :
( ( $true
= ( in @ X3 @ X0 ) )
& ( ( in @ X3 @ X1 )
!= $true ) )
| ? [X2: $i] :
( ( ( in @ X2 @ X0 )
!= $true )
& ( ( in @ X2 @ X1 )
= $true ) ) )
<=> ( setext = $true ) ),
inference(ennf_transformation,[status(thm)],[f12]) ).
thf(f12,plain,
( ! [X0: $i,X1: $i] :
( ! [X2: $i] :
( ( ( in @ X2 @ X1 )
= $true )
=> ( ( in @ X2 @ X0 )
= $true ) )
=> ( ! [X3: $i] :
( ( $true
= ( in @ X3 @ X0 ) )
=> ( ( in @ X3 @ X1 )
= $true ) )
=> ( X0 = X1 ) ) )
<=> ( setext = $true ) ),
inference(fool_elimination,[status(thm)],[f11]) ).
thf(f11,plain,
( setext
= ( ! [X0: $i,X1: $i] :
( ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ X0 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X0 )
=> ( in @ X3 @ X1 ) )
=> ( X0 = X1 ) ) ) ) ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
( setext
= ( ! [X3: $i,X2: $i] :
( ! [X0: $i] :
( ( in @ X0 @ X2 )
=> ( in @ X0 @ X3 ) )
=> ( ! [X0: $i] :
( ( in @ X0 @ X3 )
=> ( in @ X0 @ X2 ) )
=> ( X2 = X3 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.bXuGu7mRJV/DTF2THF_31515.p',setext) ).
thf(f72,plain,
( ! [X0: $i] :
( ( in @ X0 @ emptyset )
!= $true )
| ~ spl7_2 ),
inference(avatar_component_clause,[status(thm)],[f71]) ).
thf(f71,definition,
( spl7_2
<=> ! [X0: $i] :
( ( in @ X0 @ emptyset )
!= $true ) ),
introduced(definition,[new_symbols(naming,[spl7_2])],[avatar_definition]) ).
thf(f73,plain,
( spl7_1
| spl7_2 ),
inference(avatar_split_clause,[status(thm)],[f66,f71,f68]) ).
thf(f66,plain,
! [X0: $i,X1: $o] :
( ( $true = X1 )
| ( ( in @ X0 @ emptyset )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f48]) ).
thf(f48,plain,
! [X0: $i,X1: $o] :
( ( ( in @ X0 @ emptyset )
!= $true )
| ( $true = X1 )
| ( $true != $true ) ),
inference(definition_unfolding,[status(thm)],[f35,f39]) ).
thf(f39,plain,
emptysetE = $true,
inference(cnf_transformation,[status(thm)],[f26]) ).
thf(f35,plain,
! [X0: $i,X1: $o] :
( ( $true = X1 )
| ( ( in @ X0 @ emptyset )
!= $true )
| ( emptysetE != $true ) ),
inference(cnf_transformation,[status(thm)],[f24]) ).
thf(f24,plain,
( ( ! [X0: $i] :
( ! [X1: $o] : ( $true = X1 )
| ( ( in @ X0 @ emptyset )
!= $true ) )
| ( emptysetE != $true ) )
& ( ( emptysetE = $true )
| ( ( sK1 != $true )
& ( ( in @ sK0 @ emptyset )
= $true ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1])],[f21,f23,f22]) ).
thf(f22,plain,
( ? [X2: $i] :
( ? [X3: $o] : ( $true != X3 )
& ( $true
= ( in @ X2 @ emptyset ) ) )
=> ( ? [X3: $o] : ( $true != X3 )
& ( ( in @ sK0 @ emptyset )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f23,plain,
( ? [X3: $o] : ( $true != X3 )
=> ( sK1 != $true ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f21,plain,
( ( ! [X0: $i] :
( ! [X1: $o] : ( $true = X1 )
| ( ( in @ X0 @ emptyset )
!= $true ) )
| ( emptysetE != $true ) )
& ( ( emptysetE = $true )
| ? [X2: $i] :
( ? [X3: $o] : ( $true != X3 )
& ( $true
= ( in @ X2 @ emptyset ) ) ) ) ),
inference(rectify,[status(thm)],[f20]) ).
thf(f20,plain,
( ( ! [X0: $i] :
( ! [X1: $o] : ( $true = X1 )
| ( ( in @ X0 @ emptyset )
!= $true ) )
| ( emptysetE != $true ) )
& ( ( emptysetE = $true )
| ? [X0: $i] :
( ? [X1: $o] : ( $true != X1 )
& ( ( in @ X0 @ emptyset )
= $true ) ) ) ),
inference(nnf_transformation,[status(thm)],[f15]) ).
thf(f15,plain,
( ! [X0: $i] :
( ! [X1: $o] : ( $true = X1 )
| ( ( in @ X0 @ emptyset )
!= $true ) )
<=> ( emptysetE = $true ) ),
inference(ennf_transformation,[status(thm)],[f8]) ).
thf(f8,plain,
( ! [X0: $i] :
( ( ( in @ X0 @ emptyset )
= $true )
=> ! [X1: $o] : ( $true = X1 ) )
<=> ( emptysetE = $true ) ),
inference(fool_elimination,[status(thm)],[f7]) ).
thf(f7,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/sandbox/tmp/tmp.bXuGu7mRJV/DTF2THF_31515.p',emptysetE) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU526^2 : 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
% 0.18/0.34 % Computer : n022.cluster.edu
% 0.18/0.34 % Model : x86_64 x86_64
% 0.18/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.18/0.34 % Memory : 8042.1875MB
% 0.18/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.18/0.34 % CPULimit : 300
% 0.18/0.34 % WCLimit : 300
% 0.18/0.34 % DateTime : Tue Feb 24 12:25:51 EST 2026
% 0.18/0.34 % CPUTime :
% 0.18/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.31/0.46 ---- Original DTF file ---
% 0.31/0.47 thf(spec,logic,$$dhol).
% 0.31/0.47 %------------------------------------------------------------------------------
% 0.31/0.47 % File : SEU526^2 : TPTP v9.2.1. Released v3.7.0.
% 0.31/0.47 % Domain : Set Theory
% 0.31/0.47 % Problem : Preliminary Notions - Equality Laws
% 0.31/0.47 % Version : Especial > Reduced > Especial.
% 0.31/0.47 % English : (! A:i.nonempty A -> (? x:i.in x A & true))
% 0.31/0.47
% 0.31/0.47 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.31/0.47 % Source : [Bro08]
% 0.31/0.47 % Names : ZFC028l [Bro08]
% 0.31/0.47
% 0.31/0.47 % Status : Theorem
% 0.31/0.47 % Rating : 0.00 v9.0.0, 0.10 v8.2.0, 0.08 v8.1.0, 0.00 v7.4.0, 0.11 v7.2.0, 0.00 v7.1.0, 0.12 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.00 v6.1.0, 0.14 v5.5.0, 0.17 v5.4.0, 0.20 v5.3.0, 0.40 v5.2.0, 0.20 v4.1.0, 0.00 v3.7.0
% 0.31/0.47 % Syntax : Number of formulae : 9 ( 3 unt; 5 typ; 3 def)
% 0.31/0.47 % Number of atoms : 18 ( 5 equ; 0 cnn)
% 0.31/0.47 % Maximal formula atoms : 5 ( 4 avg)
% 0.31/0.47 % Number of connectives : 23 ( 1 ~; 0 |; 1 &; 13 @)
% 0.31/0.47 % ( 0 <=>; 8 =>; 0 <=; 0 <~>)
% 0.31/0.47 % Maximal formula depth : 9 ( 3 avg)
% 0.31/0.47 % Number of types : 2 ( 0 usr)
% 0.31/0.47 % Number of type conns : 3 ( 3 >; 0 *; 0 +; 0 <<)
% 0.31/0.47 % Number of symbols : 7 ( 5 usr; 4 con; 0-2 aty)
% 0.31/0.47 % Number of variables : 9 ( 1 ^; 7 !; 1 ?; 9 :)
% 0.31/0.47 % SPC : TH0_THM_EQU_NAR
% 0.31/0.47
% 0.31/0.47 % Comments : http://mathgate.info/detsetitem.php?id=467
% 0.31/0.47 %------------------------------------------------------------------------------
% 0.31/0.47 thf(in_type,type,
% 0.31/0.47 in: $i > $i > $o ).
% 0.31/0.47
% 0.31/0.47 thf(emptyset_type,type,
% 0.31/0.47 emptyset: $i ).
% 0.31/0.47
% 0.31/0.47 thf(emptysetE_type,type,
% 0.31/0.47 emptysetE: $o ).
% 0.31/0.47
% 0.31/0.47 thf(emptysetE,definition,
% 0.31/0.47 ( emptysetE
% 0.31/0.47 = ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ emptyset )
% 0.31/0.47 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setext_type,type,
% 0.31/0.47 setext: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setext,definition,
% 0.31/0.47 ( setext
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( in @ Xx @ B ) )
% 0.31/0.47 => ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ B )
% 0.31/0.47 => ( in @ Xx @ A ) )
% 0.31/0.47 => ( A = B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(nonempty_type,type,
% 0.31/0.47 nonempty: $i > $o ).
% 0.31/0.47
% 0.31/0.47 thf(nonempty,definition,
% 0.31/0.47 ( nonempty
% 0.31/0.47 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(nonemptyImpWitness,conjecture,
% 0.31/0.47 ( emptysetE
% 0.31/0.47 => ( setext
% 0.31/0.47 => ! [A: $i] :
% 0.31/0.47 ( ( nonempty @ A )
% 0.31/0.47 => ? [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 & $true ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 %------------------------------------------------------------------------------
% 0.31/0.47 ------------------------
% 1.62/1.35 ---- Embedded in THF ---
% 1.62/1.35 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.62/1.35 %%% Generated on Tue Feb 24 12:25:52 EST 2026
% 1.62/1.35 %%% using '$$dhol' embedding, version 1.3.0.
% 1.62/1.35 %%% Logic specification used:
% 1.62/1.35 %%% thf(spec, logic, $$dhol).
% 1.62/1.35
% 1.62/1.35 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.bXuGu7mRJV/DTF2DTF31515.p
% See solution above
% 1.62/1.35 ------------------------
% 1.62/1.36 ---- Cleaned THF ---
% 1.62/1.36 thf(in_type,type,
% 1.62/1.36 in: $i > $i > $o ).
% 1.62/1.36
% 1.62/1.36 thf(emptyset_type,type,
% 1.62/1.36 emptyset: $i ).
% 1.62/1.36
% 1.62/1.36 thf(emptysetE_type,type,
% 1.62/1.36 emptysetE: $o ).
% 1.62/1.36
% 1.62/1.36 thf(emptysetE,definition,
% 1.62/1.36 ( emptysetE
% 1.62/1.36 = ( ! [Xx: $i] :
% 1.62/1.36 ( ( in @ Xx @ emptyset )
% 1.62/1.36 => ! [Xphi: $o] : Xphi ) ) ) ).
% 1.62/1.36
% 1.62/1.36 thf(setext_type,type,
% 1.62/1.36 setext: $o ).
% 1.62/1.36
% 1.62/1.36 thf(setext,definition,
% 1.62/1.36 ( setext
% 1.62/1.36 = ( ! [A: $i,B: $i] :
% 1.62/1.36 ( ! [Xx: $i] :
% 1.62/1.36 ( ( in @ Xx @ A )
% 1.62/1.36 => ( in @ Xx @ B ) )
% 1.62/1.36 => ( ! [Xx: $i] :
% 1.62/1.36 ( ( in @ Xx @ B )
% 1.62/1.36 => ( in @ Xx @ A ) )
% 1.62/1.36 => ( A = B ) ) ) ) ) ).
% 1.62/1.36
% 1.62/1.36 thf(nonempty_type,type,
% 1.62/1.36 nonempty: $i > $o ).
% 1.62/1.36
% 1.62/1.36 thf(nonempty,definition,
% 1.62/1.36 ( nonempty
% 1.62/1.36 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 1.62/1.36
% 1.62/1.36 thf(nonemptyImpWitness,conjecture,
% 1.62/1.36 ( emptysetE
% 1.62/1.36 => ( setext
% 1.62/1.36 => ! [A: $i] :
% 1.62/1.36 ( ( nonempty @ A )
% 1.62/1.36 => ? [Xx: $i] :
% 1.62/1.36 ( ( in @ Xx @ A )
% 1.62/1.36 & $true ) ) ) ) ).
% 1.62/1.36 ------------------------
% 1.62/1.38 % (31634)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_31515 for (2999ds/2Mi)
% 1.62/1.38 % (31634)Instruction limit reached!
% 1.62/1.38 % (31634)------------------------------
% 1.62/1.38 % (31634)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.62/1.38 % (31634)Termination reason: Unknown
% 1.62/1.38 % (31634)Termination phase: Saturation
% 1.62/1.38
% 1.62/1.38 % (31634)Memory used [KB]: 895
% 1.62/1.38 % (31634)Time elapsed: 0.003 s
% 1.62/1.38 % (31634)Instructions burned: 2 (million)
% 1.62/1.38 % (31634)------------------------------
% 1.62/1.38 % (31634)------------------------------
% 1.62/1.38 % (31631)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_31515 for (2999ds/4Mi)
% 1.62/1.38 % (31630)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_31515 for (2999ds/183Mi)
% 1.62/1.38 % (31636)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_31515 for (2999ds/18Mi)
% 1.62/1.38 % (31632)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_31515 for (2999ds/27Mi)
% 1.62/1.38 % (31633)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_31515 for (2999ds/2Mi)
% 1.62/1.38 % (31635)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_31515 for (2999ds/275Mi)
% 1.62/1.38 % (31633)Instruction limit reached!
% 1.62/1.38 % (31633)------------------------------
% 1.62/1.38 % (31633)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.62/1.38 % (31633)Termination reason: Unknown
% 1.62/1.38 % (31633)Termination phase: Clausification
% 1.62/1.38
% 1.62/1.38 % (31633)Memory used [KB]: 895
% 1.62/1.38 % (31633)Time elapsed: 0.003 s
% 1.62/1.38 % (31633)Instructions burned: 2 (million)
% 1.62/1.38 % (31633)------------------------------
% 1.62/1.38 % (31633)------------------------------
% 1.62/1.38 % (31631)Instruction limit reached!
% 1.62/1.38 % (31631)------------------------------
% 1.62/1.38 % (31631)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.62/1.38 % (31631)Termination reason: Unknown
% 1.62/1.38 % (31631)Termination phase: Saturation
% 1.62/1.38
% 1.62/1.38 % (31631)Memory used [KB]: 5500
% 1.62/1.38 % (31631)Time elapsed: 0.005 s
% 1.62/1.38 % (31631)Instructions burned: 4 (million)
% 1.62/1.38 % (31631)------------------------------
% 1.62/1.38 % (31631)------------------------------
% 1.62/1.39 % (31636)First to succeed.
% 1.62/1.39 % (31635)Refutation not found, incomplete strategy
% 1.62/1.39 % (31635)------------------------------
% 1.62/1.39 % (31635)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.62/1.39 % (31635)Termination reason: Refutation not found, incomplete strategy
% 1.62/1.39
% 1.62/1.39
% 1.62/1.39 % (31635)Memory used [KB]: 5500
% 1.62/1.39 % (31635)Time elapsed: 0.007 s
% 1.62/1.39 % (31635)Instructions burned: 7 (million)
% 1.62/1.39 % (31635)------------------------------
% 1.62/1.39 % (31635)------------------------------
% 1.62/1.39 % (31637)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_31515 for (2999ds/3Mi)
% 1.62/1.39 % (31630)Also succeeded, but the first one will report.
% 1.62/1.39 % (31637)Instruction limit reached!
% 1.62/1.39 % (31637)------------------------------
% 1.62/1.39 % (31637)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.62/1.39 % (31637)Termination reason: Unknown
% 1.62/1.39 % (31637)Termination phase: Saturation
% 1.62/1.39
% 1.62/1.39 % (31637)Memory used [KB]: 5500
% 1.62/1.39 % (31637)Time elapsed: 0.003 s
% 1.62/1.39 % (31637)Instructions burned: 3 (million)
% 1.62/1.39 % (31637)------------------------------
% 1.62/1.39 % (31637)------------------------------
% 1.62/1.39 % (31636)Refutation found. Thanks to Tanya!
% 1.62/1.39 % SZS status Theorem for DTF2THF_31515
% 1.62/1.39 % SZS output start Proof for DTF2THF_31515
% 1.62/1.39 thf(func_def_0, type, in: $i > $i > $o).
% 1.62/1.39 thf(func_def_5, type, nonempty: $i > $o).
% 1.62/1.39 thf(func_def_17, type, sK5: $i > $i > $i).
% 1.62/1.39 thf(func_def_18, type, sK6: $i > $i > $i).
% 1.62/1.39 thf(f93,plain,(
% 1.62/1.39 $false),
% 1.62/1.39 inference(avatar_sat_refutation,[status(thm)],[f73,f83,f90])).
% 1.62/1.39 thf(f90,plain,(
% 1.62/1.39 ~spl7_1),
% 1.62/1.39 inference(avatar_contradiction_clause,[status(thm)],[f89])).
% 1.62/1.39 thf(f89,plain,(
% 1.62/1.39 $false | ~spl7_1),
% 1.62/1.39 inference(trivial_inequality_removal,[status(thm)],[f88])).
% 1.62/1.39 thf(f88,plain,(
% 1.62/1.39 ($true != $true) | ~spl7_1),
% 1.62/1.39 inference(superposition,[status(thm)],[f38,f69])).
% 1.62/1.39 thf(f69,plain,(
% 1.62/1.39 ( ! [X1 : $o] : (($true = X1)) ) | ~spl7_1),
% 1.62/1.39 inference(avatar_component_clause,[status(thm)],[f68])).
% 1.62/1.39 thf(f68,definition,(
% 1.62/1.39 spl7_1 <=> ! [X1 : $o] : ($true = X1)),
% 1.62/1.39 introduced(definition,[new_symbols(naming,[spl7_1])],[avatar_definition])).
% 1.62/1.39 thf(f38,plain,(
% 1.62/1.39 ( ! [X1 : $i] : (((in @ X1 @ sK2) != $true)) )),
% 1.62/1.39 inference(cnf_transformation,[status(thm)],[f26])).
% 1.62/1.39 thf(f26,plain,(
% 1.62/1.39 (emptysetE = $true) & (! [X1 : $i] : ((in @ X1 @ sK2) != $true) & ($true = (nonempty @ sK2))) & (setext = $true)),
% 1.62/1.39 inference(skolemisation,[status(esa),new_symbols(skolem,[sK2])],[f19,f25])).
% 1.62/1.39 thf(f25,plain,(
% 1.62/1.39 ? [X0 : $i] : (! [X1 : $i] : ((in @ X1 @ X0) != $true) & ($true = (nonempty @ X0))) => (! [X1 : $i] : ((in @ X1 @ sK2) != $true) & ($true = (nonempty @ sK2)))),
% 1.62/1.39 introduced(definition,[],[choice_axiom])).
% 1.62/1.39 thf(f19,plain,(
% 1.62/1.39 (emptysetE = $true) & ? [X0 : $i] : (! [X1 : $i] : ((in @ X1 @ X0) != $true) & ($true = (nonempty @ X0))) & (setext = $true)),
% 1.62/1.39 inference(flattening,[status(thm)],[f18])).
% 1.62/1.39 thf(f18,plain,(
% 1.62/1.39 (? [X0 : $i] : (! [X1 : $i] : ((in @ X1 @ X0) != $true) & ($true = (nonempty @ X0))) & (setext = $true)) & (emptysetE = $true)),
% 1.62/1.39 inference(ennf_transformation,[status(thm)],[f14])).
% 1.62/1.39 thf(f14,plain,(
% 1.62/1.39 ~((emptysetE = $true) => ((setext = $true) => ! [X0 : $i] : (($true = (nonempty @ X0)) => ? [X1 : $i] : ((in @ X1 @ X0) = $true))))),
% 1.62/1.39 inference(true_and_false_elimination,[status(thm)],[f10])).
% 1.62/1.39 thf(f10,plain,(
% 1.62/1.39 ~((emptysetE = $true) => ((setext = $true) => ! [X0 : $i] : (($true = (nonempty @ X0)) => ? [X1 : $i] : ($true & ((in @ X1 @ X0) = $true)))))),
% 1.62/1.39 inference(fool_elimination,[status(thm)],[f9])).
% 1.62/1.39 thf(f9,plain,(
% 1.62/1.39 ~(emptysetE => (setext => ! [X0 : $i] : ((nonempty @ X0) => ? [X1 : $i] : ($true & (in @ X1 @ X0)))))),
% 1.62/1.39 inference(rectify,[status(thm)],[f5])).
% 1.62/1.39 thf(f5,negated_conjecture,(
% 1.62/1.39 ~(emptysetE => (setext => ! [X2 : $i] : ((nonempty @ X2) => ? [X0 : $i] : ($true & (in @ X0 @ X2)))))),
% 1.62/1.39 inference(negated_conjecture,[status(cth)],[f4])).
% 1.62/1.39 thf(f4,conjecture,(
% 1.62/1.39 emptysetE => (setext => ! [X2 : $i] : ((nonempty @ X2) => ? [X0 : $i] : ($true & (in @ X0 @ X2))))),
% 1.62/1.39 file('/export/starexec/sandbox/tmp/tmp.bXuGu7mRJV/DTF2THF_31515.p',nonemptyImpWitness)).
% 1.62/1.39 thf(f83,plain,(
% 1.62/1.39 ~spl7_2),
% 1.62/1.39 inference(avatar_contradiction_clause,[status(thm)],[f82])).
% 1.62/1.39 thf(f82,plain,(
% 1.62/1.39 $false | ~spl7_2),
% 1.62/1.39 inference(subsumption_resolution,[status(thm)],[f81,f65])).
% 1.62/1.39 thf(f65,plain,(
% 1.62/1.39 (emptyset != sK2)),
% 1.62/1.39 inference(equality_proxy_clausification,[status(thm)],[f64])).
% 1.62/1.39 thf(f64,plain,(
% 1.62/1.39 ($false = (emptyset = sK2))),
% 1.62/1.39 inference(not_proxy_clausification,[status(thm)],[f63])).
% 1.62/1.39 thf(f63,plain,(
% 1.62/1.39 ($true = (~ (emptyset = sK2)))),
% 1.62/1.39 inference(beta_eta_normalization,[status(thm)],[f51])).
% 1.62/1.39 thf(f51,plain,(
% 1.62/1.39 ($true = ((^[Y0 : $i]: (~ (emptyset = Y0))) @ sK2))),
% 1.62/1.39 inference(definition_unfolding,[status(thm)],[f37,f47])).
% 1.62/1.39 thf(f47,plain,(
% 1.62/1.39 (nonempty = (^[Y0 : $i]: (~ (emptyset = Y0))))),
% 1.62/1.39 inference(cnf_transformation,[status(thm)],[f13])).
% 1.62/1.39 thf(f13,plain,(
% 1.62/1.39 (nonempty = (^[Y0 : $i]: (~ (emptyset = Y0))))),
% 1.62/1.39 inference(fool_elimination,[status(thm)],[f3])).
% 1.62/1.39 thf(f3,axiom,(
% 1.62/1.39 (nonempty = (^[X0 : $i] : ((emptyset != X0))))),
% 1.62/1.39 file('/export/starexec/sandbox/tmp/tmp.bXuGu7mRJV/DTF2THF_31515.p',nonempty)).
% 1.62/1.39 thf(f37,plain,(
% 1.62/1.39 ($true = (nonempty @ sK2))),
% 1.62/1.39 inference(cnf_transformation,[status(thm)],[f26])).
% 1.62/1.39 thf(f81,plain,(
% 1.62/1.39 (emptyset = sK2) | ~spl7_2),
% 1.62/1.39 inference(trivial_inequality_removal,[status(thm)],[f79])).
% 1.62/1.39 thf(f79,plain,(
% 1.62/1.39 (emptyset = sK2) | ($true != $true) | ~spl7_2),
% 1.62/1.39 inference(superposition,[status(thm)],[f38,f77])).
% 1.62/1.39 thf(f77,plain,(
% 1.62/1.39 ( ! [X0 : $i] : (((in @ (sK5 @ X0 @ emptyset) @ X0) = $true) | (emptyset = X0)) ) | ~spl7_2),
% 1.62/1.39 inference(trivial_inequality_removal,[status(thm)],[f76])).
% 1.62/1.39 thf(f76,plain,(
% 1.62/1.39 ( ! [X0 : $i] : (((in @ (sK5 @ X0 @ emptyset) @ X0) = $true) | ($true != $true) | (emptyset = X0)) ) | ~spl7_2),
% 1.62/1.39 inference(superposition,[status(thm)],[f72,f60])).
% 1.62/1.39 thf(f60,plain,(
% 1.62/1.39 ( ! [X4 : $i,X5 : $i] : (((in @ (sK6 @ X5 @ X4) @ X4) = $true) | (X4 = X5) | ((in @ (sK5 @ X5 @ X4) @ X5) = $true)) )),
% 1.62/1.39 inference(trivial_inequality_removal,[status(thm)],[f57])).
% 1.62/1.39 thf(f57,plain,(
% 1.62/1.39 ( ! [X4 : $i,X5 : $i] : ((X4 = X5) | ($true != $true) | ((in @ (sK5 @ X5 @ X4) @ X5) = $true) | ((in @ (sK6 @ X5 @ X4) @ X4) = $true)) )),
% 1.62/1.39 inference(definition_unfolding,[status(thm)],[f41,f36])).
% 1.62/1.39 thf(f36,plain,(
% 1.62/1.39 (setext = $true)),
% 1.62/1.39 inference(cnf_transformation,[status(thm)],[f26])).
% 1.62/1.39 thf(f41,plain,(
% 1.62/1.39 ( ! [X4 : $i,X5 : $i] : (((in @ (sK5 @ X5 @ X4) @ X5) = $true) | (X4 = X5) | ((in @ (sK6 @ X5 @ X4) @ X4) = $true) | (setext != $true)) )),
% 1.62/1.39 inference(cnf_transformation,[status(thm)],[f32])).
% 1.62/1.39 thf(f32,plain,(
% 1.62/1.39 ((setext = $true) | (! [X2 : $i] : (($true = (in @ X2 @ sK3)) | ($true != (in @ X2 @ sK4))) & (sK4 != sK3) & ! [X3 : $i] : (($true != (in @ X3 @ sK3)) | ($true = (in @ X3 @ sK4))))) & (! [X4 : $i,X5 : $i] : ((($true != (in @ (sK5 @ X5 @ X4) @ X4)) & ((in @ (sK5 @ X5 @ X4) @ X5) = $true)) | (X4 = X5) | (((in @ (sK6 @ X5 @ X4) @ X4) = $true) & ($true != (in @ (sK6 @ X5 @ X4) @ X5)))) | (setext != $true))),
% 1.62/1.39 inference(skolemisation,[status(esa),new_symbols(skolem,[sK3,sK4,sK5,sK6])],[f28,f31,f30,f29])).
% 1.62/1.39 thf(f29,plain,(
% 1.62/1.39 ? [X0 : $i,X1 : $i] : (! [X2 : $i] : (((in @ X2 @ X0) = $true) | ((in @ X2 @ X1) != $true)) & (X0 != X1) & ! [X3 : $i] : (($true != (in @ X3 @ X0)) | ((in @ X3 @ X1) = $true))) => (! [X2 : $i] : (($true = (in @ X2 @ sK3)) | ($true != (in @ X2 @ sK4))) & (sK4 != sK3) & ! [X3 : $i] : (($true != (in @ X3 @ sK3)) | ($true = (in @ X3 @ sK4))))),
% 1.62/1.39 introduced(definition,[],[choice_axiom])).
% 1.62/1.39 thf(f30,plain,(
% 1.62/1.39 ! [X4 : $i,X5 : $i] : (? [X6 : $i] : (((in @ X6 @ X4) != $true) & ($true = (in @ X6 @ X5))) => (($true != (in @ (sK5 @ X5 @ X4) @ X4)) & ((in @ (sK5 @ X5 @ X4) @ X5) = $true)))),
% 1.62/1.39 introduced(definition,[],[choice_axiom])).
% 1.62/1.39 thf(f31,plain,(
% 1.62/1.39 ! [X4 : $i,X5 : $i] : (? [X7 : $i] : (($true = (in @ X7 @ X4)) & ($true != (in @ X7 @ X5))) => (((in @ (sK6 @ X5 @ X4) @ X4) = $true) & ($true != (in @ (sK6 @ X5 @ X4) @ X5))))),
% 1.62/1.39 introduced(definition,[],[choice_axiom])).
% 1.62/1.39 thf(f28,plain,(
% 1.62/1.39 ((setext = $true) | ? [X0 : $i,X1 : $i] : (! [X2 : $i] : (((in @ X2 @ X0) = $true) | ((in @ X2 @ X1) != $true)) & (X0 != X1) & ! [X3 : $i] : (($true != (in @ X3 @ X0)) | ((in @ X3 @ X1) = $true)))) & (! [X4 : $i,X5 : $i] : (? [X6 : $i] : (((in @ X6 @ X4) != $true) & ($true = (in @ X6 @ X5))) | (X4 = X5) | ? [X7 : $i] : (($true = (in @ X7 @ X4)) & ($true != (in @ X7 @ X5)))) | (setext != $true))),
% 1.62/1.39 inference(rectify,[status(thm)],[f27])).
% 1.62/1.39 thf(f27,plain,(
% 1.62/1.39 ((setext = $true) | ? [X0 : $i,X1 : $i] : (! [X2 : $i] : (((in @ X2 @ X0) = $true) | ((in @ X2 @ X1) != $true)) & (X0 != X1) & ! [X3 : $i] : (($true != (in @ X3 @ X0)) | ((in @ X3 @ X1) = $true)))) & (! [X0 : $i,X1 : $i] : (? [X2 : $i] : (((in @ X2 @ X0) != $true) & ((in @ X2 @ X1) = $true)) | (X0 = X1) | ? [X3 : $i] : (($true = (in @ X3 @ X0)) & ((in @ X3 @ X1) != $true))) | (setext != $true))),
% 1.62/1.39 inference(nnf_transformation,[status(thm)],[f17])).
% 1.62/1.39 thf(f17,plain,(
% 1.62/1.39 (setext = $true) <=> ! [X0 : $i,X1 : $i] : (? [X2 : $i] : (((in @ X2 @ X0) != $true) & ((in @ X2 @ X1) = $true)) | (X0 = X1) | ? [X3 : $i] : (($true = (in @ X3 @ X0)) & ((in @ X3 @ X1) != $true)))),
% 1.62/1.39 inference(flattening,[status(thm)],[f16])).
% 1.62/1.39 thf(f16,plain,(
% 1.62/1.39 ! [X0 : $i,X1 : $i] : (((X0 = X1) | ? [X3 : $i] : (($true = (in @ X3 @ X0)) & ((in @ X3 @ X1) != $true))) | ? [X2 : $i] : (((in @ X2 @ X0) != $true) & ((in @ X2 @ X1) = $true))) <=> (setext = $true)),
% 1.62/1.39 inference(ennf_transformation,[status(thm)],[f12])).
% 1.62/1.39 thf(f12,plain,(
% 1.62/1.39 ! [X0 : $i,X1 : $i] : (! [X2 : $i] : (((in @ X2 @ X1) = $true) => ((in @ X2 @ X0) = $true)) => (! [X3 : $i] : (($true = (in @ X3 @ X0)) => ((in @ X3 @ X1) = $true)) => (X0 = X1))) <=> (setext = $true)),
% 1.62/1.39 inference(fool_elimination,[status(thm)],[f11])).
% 1.62/1.39 thf(f11,plain,(
% 1.62/1.39 (setext = ! [X0 : $i,X1 : $i] : (! [X2 : $i] : ((in @ X2 @ X1) => (in @ X2 @ X0)) => (! [X3 : $i] : ((in @ X3 @ X0) => (in @ X3 @ X1)) => (X0 = X1))))),
% 1.62/1.39 inference(rectify,[status(thm)],[f2])).
% 1.62/1.39 thf(f2,axiom,(
% 1.62/1.39 (setext = ! [X3 : $i,X2 : $i] : (! [X0 : $i] : ((in @ X0 @ X2) => (in @ X0 @ X3)) => (! [X0 : $i] : ((in @ X0 @ X3) => (in @ X0 @ X2)) => (X2 = X3))))),
% 1.62/1.39 file('/export/starexec/sandbox/tmp/tmp.bXuGu7mRJV/DTF2THF_31515.p',setext)).
% 1.62/1.39 thf(f72,plain,(
% 1.62/1.39 ( ! [X0 : $i] : (((in @ X0 @ emptyset) != $true)) ) | ~spl7_2),
% 1.62/1.39 inference(avatar_component_clause,[status(thm)],[f71])).
% 1.62/1.39 thf(f71,definition,(
% 1.62/1.39 spl7_2 <=> ! [X0 : $i] : ((in @ X0 @ emptyset) != $true)),
% 1.62/1.39 introduced(definition,[new_symbols(naming,[spl7_2])],[avatar_definition])).
% 1.62/1.39 thf(f73,plain,(
% 1.62/1.39 spl7_1 | spl7_2),
% 1.62/1.39 inference(avatar_split_clause,[status(thm)],[f66,f71,f68])).
% 1.62/1.39 thf(f66,plain,(
% 1.62/1.39 ( ! [X0 : $i,X1 : $o] : (($true = X1) | ((in @ X0 @ emptyset) != $true)) )),
% 1.62/1.39 inference(trivial_inequality_removal,[status(thm)],[f48])).
% 1.62/1.39 thf(f48,plain,(
% 1.62/1.39 ( ! [X0 : $i,X1 : $o] : (((in @ X0 @ emptyset) != $true) | ($true = X1) | ($true != $true)) )),
% 1.62/1.39 inference(definition_unfolding,[status(thm)],[f35,f39])).
% 1.62/1.39 thf(f39,plain,(
% 1.62/1.39 (emptysetE = $true)),
% 1.62/1.39 inference(cnf_transformation,[status(thm)],[f26])).
% 1.62/1.39 thf(f35,plain,(
% 1.62/1.39 ( ! [X0 : $i,X1 : $o] : (($true = X1) | ((in @ X0 @ emptyset) != $true) | (emptysetE != $true)) )),
% 1.62/1.39 inference(cnf_transformation,[status(thm)],[f24])).
% 1.62/1.39 thf(f24,plain,(
% 1.62/1.39 (! [X0 : $i] : (! [X1 : $o] : ($true = X1) | ((in @ X0 @ emptyset) != $true)) | (emptysetE != $true)) & ((emptysetE = $true) | ((sK1 != $true) & ((in @ sK0 @ emptyset) = $true)))),
% 1.62/1.39 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1])],[f21,f23,f22])).
% 1.62/1.39 thf(f22,plain,(
% 1.62/1.39 ? [X2 : $i] : (? [X3 : $o] : ($true != X3) & ($true = (in @ X2 @ emptyset))) => (? [X3 : $o] : ($true != X3) & ((in @ sK0 @ emptyset) = $true))),
% 1.62/1.39 introduced(definition,[],[choice_axiom])).
% 1.62/1.39 thf(f23,plain,(
% 1.62/1.39 ? [X3 : $o] : ($true != X3) => (sK1 != $true)),
% 1.62/1.39 introduced(definition,[],[choice_axiom])).
% 1.62/1.39 thf(f21,plain,(
% 1.62/1.39 (! [X0 : $i] : (! [X1 : $o] : ($true = X1) | ((in @ X0 @ emptyset) != $true)) | (emptysetE != $true)) & ((emptysetE = $true) | ? [X2 : $i] : (? [X3 : $o] : ($true != X3) & ($true = (in @ X2 @ emptyset))))),
% 1.62/1.39 inference(rectify,[status(thm)],[f20])).
% 1.62/1.39 thf(f20,plain,(
% 1.62/1.39 (! [X0 : $i] : (! [X1 : $o] : ($true = X1) | ((in @ X0 @ emptyset) != $true)) | (emptysetE != $true)) & ((emptysetE = $true) | ? [X0 : $i] : (? [X1 : $o] : ($true != X1) & ((in @ X0 @ emptyset) = $true)))),
% 1.62/1.39 inference(nnf_transformation,[status(thm)],[f15])).
% 1.62/1.39 thf(f15,plain,(
% 1.62/1.39 ! [X0 : $i] : (! [X1 : $o] : ($true = X1) | ((in @ X0 @ emptyset) != $true)) <=> (emptysetE = $true)),
% 1.62/1.39 inference(ennf_transformation,[status(thm)],[f8])).
% 1.62/1.39 thf(f8,plain,(
% 1.62/1.39 ! [X0 : $i] : (((in @ X0 @ emptyset) = $true) => ! [X1 : $o] : ($true = X1)) <=> (emptysetE = $true)),
% 1.62/1.39 inference(fool_elimination,[status(thm)],[f7])).
% 1.62/1.39 thf(f7,plain,(
% 1.62/1.39 (! [X0 : $i] : ((in @ X0 @ emptyset) => ! [X1 : $o] : X1) = emptysetE)),
% 1.62/1.39 inference(rectify,[status(thm)],[f1])).
% 1.62/1.39 thf(f1,axiom,(
% 1.62/1.39 (! [X0 : $i] : ((in @ X0 @ emptyset) => ! [X1 : $o] : X1) = emptysetE)),
% 1.62/1.39 file('/export/starexec/sandbox/tmp/tmp.bXuGu7mRJV/DTF2THF_31515.p',emptysetE)).
% 1.62/1.39 % SZS output end Proof for DTF2THF_31515
% 1.62/1.39 % (31636)------------------------------
% 1.62/1.39 % (31636)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.62/1.39 % (31636)Termination reason: Refutation
% 1.62/1.39
% 1.62/1.39 % (31636)Memory used [KB]: 5500
% 1.62/1.39 % (31636)Time elapsed: 0.007 s
% 1.62/1.39 % (31636)Instructions burned: 5 (million)
% 1.62/1.39 % (31636)------------------------------
% 1.62/1.39 % (31636)------------------------------
% 1.62/1.39 % (31629)Success in time 0.02 s
%------------------------------------------------------------------------------