%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU674^2 : 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 : n018.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 09:36:11 AM UTC 2026
% Result : Theorem 1.60s 1.37s
% Output : Refutation 1.60s
% Verified :
% SZS Type : Refutation
% Derivation depth : 14
% Number of leaves : 7
% Syntax : Number of formulae : 35 ( 14 unt; 0 typ; 0 def)
% Number of atoms : 402 ( 95 equ; 0 cnn)
% Maximal formula atoms : 8 ( 11 avg)
% Number of connectives : 534 ( 46 ~; 26 |; 27 &; 411 @)
% ( 2 <=>; 22 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 6 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 25 ( 22 usr; 11 con; 0-4 aty)
% Number of variables : 125 ( 43 ^; 58 !; 24 ?; 125 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_2,type,
setadjoin: $i > $i > $i ).
thf(func_def_3,type,
setunion: $i > $i ).
thf(func_def_4,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(func_def_5,type,
subset: $i > $i > $o ).
thf(func_def_6,type,
kpair: $i > $i > $i ).
thf(func_def_7,type,
cartprod: $i > $i > $i ).
thf(func_def_8,type,
singleton: $i > $o ).
thf(func_def_10,type,
ex1: $i > ( $i > $o ) > $o ).
thf(func_def_11,type,
breln: $i > $i > $i > $o ).
thf(func_def_12,type,
func: $i > $i > $i > $o ).
thf(func_def_14,type,
ap: $i > $i > $i > $i > $i ).
thf(func_def_36,type,
ph9:
!>[X0: $tType] : X0 ).
thf(f54,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f53,f41]) ).
thf(f41,plain,
( ( in @ sK7 @ sK5 )
= $true ),
inference(cnf_transformation,[status(thm)],[f33]) ).
thf(f33,plain,
( ( $true
!= ( in @ ( ap @ sK5 @ sK6 @ sK4 @ sK7 ) @ sK6 ) )
& ( ( in @ sK7 @ sK5 )
= $true )
& ( $true
= ( func @ sK5 @ sK6 @ sK4 ) )
& ( apProp = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK4,sK5,sK6,sK7])],[f24,f32,f31]) ).
thf(f31,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( ( in @ ( ap @ X1 @ X2 @ X0 @ X3 ) @ X2 )
!= $true )
& ( $true
= ( in @ X3 @ X1 ) ) )
& ( $true
= ( func @ X1 @ X2 @ X0 ) ) )
=> ( ? [X3: $i] :
( ( ( in @ ( ap @ sK5 @ sK6 @ sK4 @ X3 ) @ sK6 )
!= $true )
& ( $true
= ( in @ X3 @ sK5 ) ) )
& ( $true
= ( func @ sK5 @ sK6 @ sK4 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f32,plain,
( ? [X3: $i] :
( ( ( in @ ( ap @ sK5 @ sK6 @ sK4 @ X3 ) @ sK6 )
!= $true )
& ( $true
= ( in @ X3 @ sK5 ) ) )
=> ( ( $true
!= ( in @ ( ap @ sK5 @ sK6 @ sK4 @ sK7 ) @ sK6 ) )
& ( ( in @ sK7 @ sK5 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f24,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( ( in @ ( ap @ X1 @ X2 @ X0 @ X3 ) @ X2 )
!= $true )
& ( $true
= ( in @ X3 @ X1 ) ) )
& ( $true
= ( func @ X1 @ X2 @ X0 ) ) )
& ( apProp = $true ) ),
inference(ennf_transformation,[status(thm)],[f13]) ).
thf(f13,plain,
~ ( ( apProp = $true )
=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( func @ X1 @ X2 @ X0 ) )
=> ! [X3: $i] :
( ( $true
= ( in @ X3 @ X1 ) )
=> ( ( in @ ( ap @ X1 @ X2 @ X0 @ X3 ) @ X2 )
= $true ) ) ) ),
inference(fool_elimination,[status(thm)],[f12]) ).
thf(f12,plain,
~ ( apProp
=> ! [X0: $i,X1: $i,X2: $i] :
( ( func @ X1 @ X2 @ X0 )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X0 @ X3 ) @ X2 ) ) ) ),
inference(rectify,[status(thm)],[f8]) ).
thf(f8,negated_conjecture,
~ ( apProp
=> ! [X7: $i,X0: $i,X3: $i] :
( ( func @ X0 @ X3 @ X7 )
=> ! [X1: $i] :
( ( in @ X1 @ X0 )
=> ( in @ ( ap @ X0 @ X3 @ X7 @ X1 ) @ X3 ) ) ) ),
inference(negated_conjecture,[status(cth)],[f7]) ).
thf(f7,conjecture,
( apProp
=> ! [X7: $i,X0: $i,X3: $i] :
( ( func @ X0 @ X3 @ X7 )
=> ! [X1: $i] :
( ( in @ X1 @ X0 )
=> ( in @ ( ap @ X0 @ X3 @ X7 @ X1 ) @ X3 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.LmGkPjCsXB/DTF2THF_31356.p',app) ).
thf(f53,plain,
( ( in @ sK7 @ sK5 )
!= $true ),
inference(trivial_inequality_removal,[status(thm)],[f52]) ).
thf(f52,plain,
( ( ( in @ sK7 @ sK5 )
!= $true )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f51,f40]) ).
thf(f40,plain,
( $true
= ( func @ sK5 @ sK6 @ sK4 ) ),
inference(cnf_transformation,[status(thm)],[f33]) ).
thf(f51,plain,
! [X0: $i] :
( ( $true
!= ( func @ X0 @ sK6 @ sK4 ) )
| ( $true
!= ( in @ sK7 @ X0 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f50]) ).
thf(f50,plain,
! [X0: $i] :
( ( $true
!= ( in @ sK7 @ X0 ) )
| ( $true
!= ( func @ X0 @ sK6 @ sK4 ) )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f48,f49]) ).
thf(f49,plain,
! [X6: $i,X7: $i,X4: $i,X5: $i] :
( ( $true
= ( in
@ ( setunion
@ ( dsetconstr @ X5
@ ^ [Y0: $i] : ( in @ ( kpair @ X7 @ Y0 ) @ X4 ) ) )
@ X5 ) )
| ( $true
!= ( in @ X7 @ X6 ) )
| ( $true
!= ( func @ X6 @ X5 @ X4 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f46]) ).
thf(f46,plain,
! [X6: $i,X7: $i,X4: $i,X5: $i] :
( ( $true
!= ( in @ X7 @ X6 ) )
| ( $true != $true )
| ( $true
!= ( func @ X6 @ X5 @ X4 ) )
| ( $true
= ( in
@ ( setunion
@ ( dsetconstr @ X5
@ ^ [Y0: $i] : ( in @ ( kpair @ X7 @ Y0 ) @ X4 ) ) )
@ X5 ) ) ),
inference(definition_unfolding,[status(thm)],[f35,f39]) ).
thf(f39,plain,
apProp = $true,
inference(cnf_transformation,[status(thm)],[f33]) ).
thf(f35,plain,
! [X6: $i,X7: $i,X4: $i,X5: $i] :
( ( $true
!= ( func @ X6 @ X5 @ X4 ) )
| ( $true
!= ( in @ X7 @ X6 ) )
| ( $true
= ( in
@ ( setunion
@ ( dsetconstr @ X5
@ ^ [Y0: $i] : ( in @ ( kpair @ X7 @ Y0 ) @ X4 ) ) )
@ X5 ) )
| ( apProp != $true ) ),
inference(cnf_transformation,[status(thm)],[f30]) ).
thf(f30,plain,
( ( ( apProp = $true )
| ( ( ( func @ sK2 @ sK1 @ sK0 )
= $true )
& ( $true
= ( in @ sK3 @ sK2 ) )
& ( ( in
@ ( setunion
@ ( dsetconstr @ sK1
@ ^ [Y0: $i] : ( in @ ( kpair @ sK3 @ Y0 ) @ sK0 ) ) )
@ sK1 )
!= $true ) ) )
& ( ! [X4: $i,X5: $i,X6: $i] :
( ( $true
!= ( func @ X6 @ X5 @ X4 ) )
| ! [X7: $i] :
( ( $true
!= ( in @ X7 @ X6 ) )
| ( $true
= ( in
@ ( setunion
@ ( dsetconstr @ X5
@ ^ [Y0: $i] : ( in @ ( kpair @ X7 @ Y0 ) @ X4 ) ) )
@ X5 ) ) ) )
| ( apProp != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2,sK3])],[f27,f29,f28]) ).
thf(f28,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( func @ X2 @ X1 @ X0 )
= $true )
& ? [X3: $i] :
( ( ( in @ X3 @ X2 )
= $true )
& ( $true
!= ( in
@ ( setunion
@ ( dsetconstr @ X1
@ ^ [Y0: $i] : ( in @ ( kpair @ X3 @ Y0 ) @ X0 ) ) )
@ X1 ) ) ) )
=> ( ( ( func @ sK2 @ sK1 @ sK0 )
= $true )
& ? [X3: $i] :
( ( ( in @ X3 @ sK2 )
= $true )
& ( $true
!= ( in
@ ( setunion
@ ( dsetconstr @ sK1
@ ^ [Y0: $i] : ( in @ ( kpair @ X3 @ Y0 ) @ sK0 ) ) )
@ sK1 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f29,plain,
( ? [X3: $i] :
( ( ( in @ X3 @ sK2 )
= $true )
& ( $true
!= ( in
@ ( setunion
@ ( dsetconstr @ sK1
@ ^ [Y0: $i] : ( in @ ( kpair @ X3 @ Y0 ) @ sK0 ) ) )
@ sK1 ) ) )
=> ( ( $true
= ( in @ sK3 @ sK2 ) )
& ( ( in
@ ( setunion
@ ( dsetconstr @ sK1
@ ^ [Y0: $i] : ( in @ ( kpair @ sK3 @ Y0 ) @ sK0 ) ) )
@ sK1 )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f27,plain,
( ( ( apProp = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( func @ X2 @ X1 @ X0 )
= $true )
& ? [X3: $i] :
( ( ( in @ X3 @ X2 )
= $true )
& ( $true
!= ( in
@ ( setunion
@ ( dsetconstr @ X1
@ ^ [Y0: $i] : ( in @ ( kpair @ X3 @ Y0 ) @ X0 ) ) )
@ X1 ) ) ) ) )
& ( ! [X4: $i,X5: $i,X6: $i] :
( ( $true
!= ( func @ X6 @ X5 @ X4 ) )
| ! [X7: $i] :
( ( $true
!= ( in @ X7 @ X6 ) )
| ( $true
= ( in
@ ( setunion
@ ( dsetconstr @ X5
@ ^ [Y0: $i] : ( in @ ( kpair @ X7 @ Y0 ) @ X4 ) ) )
@ X5 ) ) ) )
| ( apProp != $true ) ) ),
inference(rectify,[status(thm)],[f26]) ).
thf(f26,plain,
( ( ( apProp = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( func @ X2 @ X1 @ X0 )
= $true )
& ? [X3: $i] :
( ( ( in @ X3 @ X2 )
= $true )
& ( $true
!= ( in
@ ( setunion
@ ( dsetconstr @ X1
@ ^ [Y0: $i] : ( in @ ( kpair @ X3 @ Y0 ) @ X0 ) ) )
@ X1 ) ) ) ) )
& ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( func @ X2 @ X1 @ X0 )
!= $true )
| ! [X3: $i] :
( ( ( in @ X3 @ X2 )
!= $true )
| ( $true
= ( in
@ ( setunion
@ ( dsetconstr @ X1
@ ^ [Y0: $i] : ( in @ ( kpair @ X3 @ Y0 ) @ X0 ) ) )
@ X1 ) ) ) )
| ( apProp != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f25]) ).
thf(f25,plain,
( ( apProp = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( func @ X2 @ X1 @ X0 )
!= $true )
| ! [X3: $i] :
( ( ( in @ X3 @ X2 )
!= $true )
| ( $true
= ( in
@ ( setunion
@ ( dsetconstr @ X1
@ ^ [Y0: $i] : ( in @ ( kpair @ X3 @ Y0 ) @ X0 ) ) )
@ X1 ) ) ) ) ),
inference(ennf_transformation,[status(thm)],[f19]) ).
thf(f19,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( func @ X2 @ X1 @ X0 )
= $true )
=> ! [X3: $i] :
( ( ( in @ X3 @ X2 )
= $true )
=> ( $true
= ( in
@ ( setunion
@ ( dsetconstr @ X1
@ ^ [Y0: $i] : ( in @ ( kpair @ X3 @ Y0 ) @ X0 ) ) )
@ X1 ) ) ) )
<=> ( apProp = $true ) ),
inference(fool_elimination,[status(thm)],[f18]) ).
thf(f18,plain,
( apProp
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( func @ X2 @ X1 @ X0 )
=> ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X1
@ ^ [X4: $i] : ( in @ ( kpair @ X3 @ X4 ) @ X0 ) ) )
@ X1 ) ) ) ) ),
inference(rectify,[status(thm)],[f5]) ).
thf(f5,axiom,
( apProp
= ( ! [X7: $i,X3: $i,X0: $i] :
( ( func @ X0 @ X3 @ X7 )
=> ! [X1: $i] :
( ( in @ X1 @ X0 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X3
@ ^ [X6: $i] : ( in @ ( kpair @ X1 @ X6 ) @ X7 ) ) )
@ X3 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.LmGkPjCsXB/DTF2THF_31356.p',apProp) ).
thf(f48,plain,
( ( in
@ ( setunion
@ ( dsetconstr @ sK6
@ ^ [Y0: $i] : ( in @ ( kpair @ sK7 @ Y0 ) @ sK4 ) ) )
@ sK6 )
!= $true ),
inference(beta_eta_normalization,[status(thm)],[f47]) ).
thf(f47,plain,
( ( in
@ ( ^ [Y0: $i,Y1: $i,Y2: $i,Y3: $i] :
( setunion
@ ( dsetconstr @ Y1
@ ^ [Y4: $i] : ( in @ ( kpair @ Y3 @ Y4 ) @ Y2 ) ) )
@ sK5
@ sK6
@ sK4
@ sK7 )
@ sK6 )
!= $true ),
inference(definition_unfolding,[status(thm)],[f42,f34]) ).
thf(f34,plain,
( ap
= ( ^ [Y0: $i,Y1: $i,Y2: $i,Y3: $i] :
( setunion
@ ( dsetconstr @ Y1
@ ^ [Y4: $i] : ( in @ ( kpair @ Y3 @ Y4 ) @ Y2 ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f21]) ).
thf(f21,plain,
( ap
= ( ^ [Y0: $i,Y1: $i,Y2: $i,Y3: $i] :
( setunion
@ ( dsetconstr @ Y1
@ ^ [Y4: $i] : ( in @ ( kpair @ Y3 @ Y4 ) @ Y2 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f20]) ).
thf(f20,plain,
( ap
= ( ^ [X0: $i,X1: $i,X2: $i,X3: $i] :
( setunion
@ ( dsetconstr @ X1
@ ^ [X4: $i] : ( in @ ( kpair @ X3 @ X4 ) @ X2 ) ) ) ) ),
inference(rectify,[status(thm)],[f6]) ).
thf(f6,axiom,
( ap
= ( ^ [X0: $i,X3: $i,X7: $i,X1: $i] :
( setunion
@ ( dsetconstr @ X3
@ ^ [X6: $i] : ( in @ ( kpair @ X1 @ X6 ) @ X7 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.LmGkPjCsXB/DTF2THF_31356.p',ap) ).
thf(f42,plain,
( $true
!= ( in @ ( ap @ sK5 @ sK6 @ sK4 @ sK7 ) @ sK6 ) ),
inference(cnf_transformation,[status(thm)],[f33]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.09 % Problem : SEU674^2 : 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
% 0.11/0.29 % Computer : n018.cluster.edu
% 0.11/0.29 % Model : x86_64 x86_64
% 0.11/0.29 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.29 % Memory : 8042.1875MB
% 0.11/0.29 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.29 % CPULimit : 300
% 0.11/0.29 % WCLimit : 300
% 0.11/0.29 % DateTime : Tue Feb 24 12:30:31 EST 2026
% 0.11/0.29 % CPUTime :
% 0.11/0.29 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.38 ---- Original DTF file ---
% 0.17/0.39 thf(spec,logic,$$dhol).
% 0.17/0.39 %------------------------------------------------------------------------------
% 0.17/0.39 % File : SEU674^2 : TPTP v9.2.1. Released v3.7.0.
% 0.17/0.39 % Domain : Set Theory
% 0.17/0.39 % Problem : Functions - Application
% 0.17/0.39 % Version : Especial > Reduced > Especial.
% 0.17/0.39 % English : (! A:i.! B:i.! f:i.func A B f -> (! x:i.in x A ->
% 0.17/0.39 % in (ap A B f x) 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 : ZFC176l [Bro08]
% 0.17/0.39
% 0.17/0.39 % Status : Theorem
% 0.17/0.39 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.10 v8.2.0, 0.31 v8.1.0, 0.27 v7.5.0, 0.14 v7.4.0, 0.11 v7.2.0, 0.12 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.29 v6.1.0, 0.14 v6.0.0, 0.29 v5.5.0, 0.33 v5.4.0, 0.40 v5.3.0, 0.60 v5.2.0, 0.40 v4.1.0, 0.33 v4.0.0, 0.67 v3.7.0
% 0.17/0.39 % Syntax : Number of formulae : 21 ( 6 unt; 14 typ; 6 def)
% 0.17/0.39 % Number of atoms : 28 ( 7 equ; 0 cnn)
% 0.17/0.39 % Maximal formula atoms : 4 ( 4 avg)
% 0.17/0.39 % Number of connectives : 63 ( 0 ~; 0 |; 2 &; 55 @)
% 0.17/0.39 % ( 0 <=>; 6 =>; 0 <=; 0 <~>)
% 0.17/0.39 % Maximal formula depth : 14 ( 3 avg)
% 0.17/0.39 % Number of types : 2 ( 0 usr)
% 0.17/0.39 % Number of type conns : 29 ( 29 >; 0 *; 0 +; 0 <<)
% 0.17/0.39 % Number of symbols : 15 ( 14 usr; 2 con; 0-4 aty)
% 0.17/0.39 % Number of variables : 27 ( 17 ^; 9 !; 1 ?; 27 :)
% 0.17/0.39 % SPC : TH0_THM_EQU_NAR
% 0.17/0.39
% 0.17/0.39 % Comments : http://mathgate.info/detsetitem.php?id=226
% 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(setunion_type,type,
% 0.17/0.39 setunion: $i > $i ).
% 0.17/0.39
% 0.17/0.39 thf(dsetconstr_type,type,
% 0.17/0.39 dsetconstr: $i > ( $i > $o ) > $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(kpair_type,type,
% 0.17/0.39 kpair: $i > $i > $i ).
% 0.17/0.39
% 0.17/0.39 thf(cartprod_type,type,
% 0.17/0.39 cartprod: $i > $i > $i ).
% 0.17/0.39
% 0.17/0.39 thf(singleton_type,type,
% 0.17/0.39 singleton: $i > $o ).
% 0.17/0.39
% 0.17/0.39 thf(singleton,definition,
% 0.17/0.39 ( singleton
% 0.17/0.39 = ( ^ [A: $i] :
% 0.17/0.39 ? [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 & ( A
% 0.17/0.39 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(ex1_type,type,
% 0.17/0.39 ex1: $i > ( $i > $o ) > $o ).
% 0.17/0.39
% 0.17/0.39 thf(ex1,definition,
% 0.17/0.39 ( ex1
% 0.17/0.39 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.17/0.39 ( singleton
% 0.17/0.39 @ ( dsetconstr @ A
% 0.17/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(breln_type,type,
% 0.17/0.39 breln: $i > $i > $i > $o ).
% 0.17/0.39
% 0.17/0.39 thf(breln,definition,
% 0.17/0.39 ( breln
% 0.17/0.39 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(func_type,type,
% 0.17/0.39 func: $i > $i > $i > $o ).
% 0.17/0.39
% 0.17/0.39 thf(func,definition,
% 0.17/0.39 ( func
% 0.17/0.39 = ( ^ [A: $i,B: $i,R: $i] :
% 0.17/0.39 ( ( breln @ A @ B @ R )
% 0.17/0.39 & ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 => ( ex1 @ B
% 0.17/0.39 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(apProp_type,type,
% 0.17/0.39 apProp: $o ).
% 0.17/0.39
% 0.17/0.39 thf(apProp,definition,
% 0.17/0.39 ( apProp
% 0.17/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.39 ( ( func @ A @ B @ Xf )
% 0.17/0.39 => ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 => ( in
% 0.17/0.39 @ ( setunion
% 0.17/0.39 @ ( dsetconstr @ B
% 0.17/0.39 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.17/0.39 @ B ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(ap_type,type,
% 0.17/0.39 ap: $i > $i > $i > $i > $i ).
% 0.17/0.39
% 0.17/0.39 thf(ap,definition,
% 0.17/0.39 ( ap
% 0.17/0.39 = ( ^ [A: $i,B: $i,Xf: $i,Xx: $i] :
% 0.17/0.39 ( setunion
% 0.17/0.39 @ ( dsetconstr @ B
% 0.17/0.39 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(app,conjecture,
% 0.17/0.39 ( apProp
% 0.17/0.39 => ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.39 ( ( func @ A @ B @ Xf )
% 0.17/0.39 => ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 %------------------------------------------------------------------------------
% 0.17/0.39 ------------------------
% 1.60/1.33 ---- Embedded in THF ---
% 1.60/1.33 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.60/1.33 %%% Generated on Tue Feb 24 12:30:31 EST 2026
% 1.60/1.33 %%% using '$$dhol' embedding, version 1.3.0.
% 1.60/1.33 %%% Logic specification used:
% 1.60/1.33 %%% thf(spec, logic, $$dhol).
% 1.60/1.33
% 1.60/1.33 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.LmGkPjCsXB/DTF2DTF31356.p
% See solution above
% 1.60/1.33 ------------------------
% 1.60/1.34 ---- Cleaned THF ---
% 1.60/1.34 thf(in_type,type,
% 1.60/1.34 in: $i > $i > $o ).
% 1.60/1.34
% 1.60/1.34 thf(emptyset_type,type,
% 1.60/1.34 emptyset: $i ).
% 1.60/1.34
% 1.60/1.34 thf(setadjoin_type,type,
% 1.60/1.34 setadjoin: $i > $i > $i ).
% 1.60/1.34
% 1.60/1.34 thf(setunion_type,type,
% 1.60/1.34 setunion: $i > $i ).
% 1.60/1.34
% 1.60/1.34 thf(dsetconstr_type,type,
% 1.60/1.34 dsetconstr: $i > ( $i > $o ) > $i ).
% 1.60/1.34
% 1.60/1.34 thf(subset_type,type,
% 1.60/1.34 subset: $i > $i > $o ).
% 1.60/1.34
% 1.60/1.34 thf(kpair_type,type,
% 1.60/1.34 kpair: $i > $i > $i ).
% 1.60/1.34
% 1.60/1.34 thf(cartprod_type,type,
% 1.60/1.34 cartprod: $i > $i > $i ).
% 1.60/1.34
% 1.60/1.34 thf(singleton_type,type,
% 1.60/1.34 singleton: $i > $o ).
% 1.60/1.34
% 1.60/1.34 thf(singleton,definition,
% 1.60/1.34 ( singleton
% 1.60/1.34 = ( ^ [A: $i] :
% 1.60/1.34 ? [Xx: $i] :
% 1.60/1.34 ( ( in @ Xx @ A )
% 1.60/1.34 & ( A
% 1.60/1.34 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 1.60/1.34
% 1.60/1.34 thf(ex1_type,type,
% 1.60/1.34 ex1: $i > ( $i > $o ) > $o ).
% 1.60/1.34
% 1.60/1.34 thf(ex1,definition,
% 1.60/1.34 ( ex1
% 1.60/1.34 = ( ^ [A: $i,Xphi: $i > $o] :
% 1.60/1.34 ( singleton
% 1.60/1.34 @ ( dsetconstr @ A
% 1.60/1.34 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 1.60/1.34
% 1.60/1.34 thf(breln_type,type,
% 1.60/1.34 breln: $i > $i > $i > $o ).
% 1.60/1.34
% 1.60/1.34 thf(breln,definition,
% 1.60/1.34 ( breln
% 1.60/1.34 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 1.60/1.34
% 1.60/1.34 thf(func_type,type,
% 1.60/1.34 func: $i > $i > $i > $o ).
% 1.60/1.34
% 1.60/1.34 thf(func,definition,
% 1.60/1.34 ( func
% 1.60/1.34 = ( ^ [A: $i,B: $i,R: $i] :
% 1.60/1.34 ( ( breln @ A @ B @ R )
% 1.60/1.34 & ! [Xx: $i] :
% 1.60/1.34 ( ( in @ Xx @ A )
% 1.60/1.34 => ( ex1 @ B
% 1.60/1.34 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ).
% 1.60/1.34
% 1.60/1.34 thf(apProp_type,type,
% 1.60/1.34 apProp: $o ).
% 1.60/1.34
% 1.60/1.34 thf(apProp,definition,
% 1.60/1.34 ( apProp
% 1.60/1.34 = ( ! [A: $i,B: $i,Xf: $i] :
% 1.60/1.34 ( ( func @ A @ B @ Xf )
% 1.60/1.34 => ! [Xx: $i] :
% 1.60/1.34 ( ( in @ Xx @ A )
% 1.60/1.34 => ( in
% 1.60/1.34 @ ( setunion
% 1.60/1.34 @ ( dsetconstr @ B
% 1.60/1.34 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 1.60/1.34 @ B ) ) ) ) ) ).
% 1.60/1.34
% 1.60/1.34 thf(ap_type,type,
% 1.60/1.34 ap: $i > $i > $i > $i > $i ).
% 1.60/1.34
% 1.60/1.34 thf(ap,definition,
% 1.60/1.34 ( ap
% 1.60/1.34 = ( ^ [A: $i,B: $i,Xf: $i,Xx: $i] :
% 1.60/1.34 ( setunion
% 1.60/1.34 @ ( dsetconstr @ B
% 1.60/1.34 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ).
% 1.60/1.34
% 1.60/1.34 thf(app,conjecture,
% 1.60/1.34 ( apProp
% 1.60/1.34 => ! [A: $i,B: $i,Xf: $i] :
% 1.60/1.34 ( ( func @ A @ B @ Xf )
% 1.60/1.34 => ! [Xx: $i] :
% 1.60/1.34 ( ( in @ Xx @ A )
% 1.60/1.34 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ).
% 1.60/1.34 ------------------------
% 1.60/1.36 % (31458)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_31356 for (2999ds/2Mi)
% 1.60/1.36 % (31455)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_31356 for (2999ds/4Mi)
% 1.60/1.36 % (31456)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_31356 for (2999ds/27Mi)
% 1.60/1.36 % (31454)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_31356 for (2999ds/183Mi)
% 1.60/1.36 % (31457)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_31356 for (2999ds/2Mi)
% 1.60/1.36 % (31459)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_31356 for (2999ds/275Mi)
% 1.60/1.36 % (31460)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_31356 for (2999ds/18Mi)
% 1.60/1.37 % (31457)Instruction limit reached!
% 1.60/1.37 % (31457)------------------------------
% 1.60/1.37 % (31457)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.60/1.37 % (31457)Termination reason: Unknown
% 1.60/1.37 % (31457)Termination phase: Property scanning
% 1.60/1.37 % (31458)Instruction limit reached!
% 1.60/1.37 % (31458)------------------------------
% 1.60/1.37 % (31458)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.60/1.37 % (31458)Termination reason: Unknown
% 1.60/1.37 % (31458)Termination phase: Property scanning
% 1.60/1.37
% 1.60/1.37 % (31458)Memory used [KB]: 1023
% 1.60/1.37 % (31458)Time elapsed: 0.003 s
% 1.60/1.37 % (31458)Instructions burned: 3 (million)
% 1.60/1.37 % (31458)------------------------------
% 1.60/1.37 % (31458)------------------------------
% 1.60/1.37
% 1.60/1.37 % (31457)Memory used [KB]: 1023
% 1.60/1.37 % (31457)Time elapsed: 0.003 s
% 1.60/1.37 % (31457)Instructions burned: 2 (million)
% 1.60/1.37 % (31457)------------------------------
% 1.60/1.37 % (31457)------------------------------
% 1.60/1.37 % (31455)Instruction limit reached!
% 1.60/1.37 % (31455)------------------------------
% 1.60/1.37 % (31455)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.60/1.37 % (31455)Termination reason: Unknown
% 1.60/1.37 % (31455)Termination phase: Property scanning
% 1.60/1.37
% 1.60/1.37 % (31455)Memory used [KB]: 1023
% 1.60/1.37 % (31455)Time elapsed: 0.004 s
% 1.60/1.37 % (31455)Instructions burned: 4 (million)
% 1.60/1.37 % (31455)------------------------------
% 1.60/1.37 % (31455)------------------------------
% 1.60/1.37 % (31459)First to succeed.
% 1.60/1.37 % (31459)Refutation found. Thanks to Tanya!
% 1.60/1.37 % SZS status Theorem for DTF2THF_31356
% 1.60/1.37 % SZS output start Proof for DTF2THF_31356
% 1.60/1.37 thf(func_def_0, type, in: $i > $i > $o).
% 1.60/1.37 thf(func_def_2, type, setadjoin: $i > $i > $i).
% 1.60/1.37 thf(func_def_3, type, setunion: $i > $i).
% 1.60/1.37 thf(func_def_4, type, dsetconstr: $i > ($i > $o) > $i).
% 1.60/1.37 thf(func_def_5, type, subset: $i > $i > $o).
% 1.60/1.37 thf(func_def_6, type, kpair: $i > $i > $i).
% 1.60/1.37 thf(func_def_7, type, cartprod: $i > $i > $i).
% 1.60/1.37 thf(func_def_8, type, singleton: $i > $o).
% 1.60/1.37 thf(func_def_10, type, ex1: $i > ($i > $o) > $o).
% 1.60/1.37 thf(func_def_11, type, breln: $i > $i > $i > $o).
% 1.60/1.37 thf(func_def_12, type, func: $i > $i > $i > $o).
% 1.60/1.37 thf(func_def_14, type, ap: $i > $i > $i > $i > $i).
% 1.60/1.37 thf(func_def_36, type, ph9: !>[X0: $tType]:(X0)).
% 1.60/1.37 thf(f54,plain,(
% 1.60/1.37 $false),
% 1.60/1.37 inference(subsumption_resolution,[status(thm)],[f53,f41])).
% 1.60/1.37 thf(f41,plain,(
% 1.60/1.37 ((in @ sK7 @ sK5) = $true)),
% 1.60/1.37 inference(cnf_transformation,[status(thm)],[f33])).
% 1.60/1.37 thf(f33,plain,(
% 1.60/1.37 ((($true != (in @ (ap @ sK5 @ sK6 @ sK4 @ sK7) @ sK6)) & ((in @ sK7 @ sK5) = $true)) & ($true = (func @ sK5 @ sK6 @ sK4))) & (apProp = $true)),
% 1.60/1.37 inference(skolemisation,[status(esa),new_symbols(skolem,[sK4,sK5,sK6,sK7])],[f24,f32,f31])).
% 1.60/1.37 thf(f31,plain,(
% 1.60/1.37 ? [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : (((in @ (ap @ X1 @ X2 @ X0 @ X3) @ X2) != $true) & ($true = (in @ X3 @ X1))) & ($true = (func @ X1 @ X2 @ X0))) => (? [X3 : $i] : (((in @ (ap @ sK5 @ sK6 @ sK4 @ X3) @ sK6) != $true) & ($true = (in @ X3 @ sK5))) & ($true = (func @ sK5 @ sK6 @ sK4)))),
% 1.60/1.37 introduced(definition,[],[choice_axiom])).
% 1.60/1.37 thf(f32,plain,(
% 1.60/1.37 ? [X3 : $i] : (((in @ (ap @ sK5 @ sK6 @ sK4 @ X3) @ sK6) != $true) & ($true = (in @ X3 @ sK5))) => (($true != (in @ (ap @ sK5 @ sK6 @ sK4 @ sK7) @ sK6)) & ((in @ sK7 @ sK5) = $true))),
% 1.60/1.37 introduced(definition,[],[choice_axiom])).
% 1.60/1.37 thf(f24,plain,(
% 1.60/1.37 ? [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : (((in @ (ap @ X1 @ X2 @ X0 @ X3) @ X2) != $true) & ($true = (in @ X3 @ X1))) & ($true = (func @ X1 @ X2 @ X0))) & (apProp = $true)),
% 1.60/1.37 inference(ennf_transformation,[status(thm)],[f13])).
% 1.60/1.37 thf(f13,plain,(
% 1.60/1.37 ~((apProp = $true) => ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (func @ X1 @ X2 @ X0)) => ! [X3 : $i] : (($true = (in @ X3 @ X1)) => ((in @ (ap @ X1 @ X2 @ X0 @ X3) @ X2) = $true))))),
% 1.60/1.37 inference(fool_elimination,[status(thm)],[f12])).
% 1.60/1.37 thf(f12,plain,(
% 1.60/1.37 ~(apProp => ! [X0 : $i,X1 : $i,X2 : $i] : ((func @ X1 @ X2 @ X0) => ! [X3 : $i] : ((in @ X3 @ X1) => (in @ (ap @ X1 @ X2 @ X0 @ X3) @ X2))))),
% 1.60/1.37 inference(rectify,[status(thm)],[f8])).
% 1.60/1.37 thf(f8,negated_conjecture,(
% 1.60/1.37 ~(apProp => ! [X7 : $i,X0 : $i,X3 : $i] : ((func @ X0 @ X3 @ X7) => ! [X1 : $i] : ((in @ X1 @ X0) => (in @ (ap @ X0 @ X3 @ X7 @ X1) @ X3))))),
% 1.60/1.37 inference(negated_conjecture,[status(cth)],[f7])).
% 1.60/1.37 thf(f7,conjecture,(
% 1.60/1.37 apProp => ! [X7 : $i,X0 : $i,X3 : $i] : ((func @ X0 @ X3 @ X7) => ! [X1 : $i] : ((in @ X1 @ X0) => (in @ (ap @ X0 @ X3 @ X7 @ X1) @ X3)))),
% 1.60/1.37 file('/export/starexec/sandbox/tmp/tmp.LmGkPjCsXB/DTF2THF_31356.p',app)).
% 1.60/1.37 thf(f53,plain,(
% 1.60/1.37 ((in @ sK7 @ sK5) != $true)),
% 1.60/1.37 inference(trivial_inequality_removal,[status(thm)],[f52])).
% 1.60/1.37 thf(f52,plain,(
% 1.60/1.37 ((in @ sK7 @ sK5) != $true) | ($true != $true)),
% 1.60/1.37 inference(superposition,[status(thm)],[f51,f40])).
% 1.60/1.37 thf(f40,plain,(
% 1.60/1.37 ($true = (func @ sK5 @ sK6 @ sK4))),
% 1.60/1.37 inference(cnf_transformation,[status(thm)],[f33])).
% 1.60/1.37 thf(f51,plain,(
% 1.60/1.37 ( ! [X0 : $i] : (($true != (func @ X0 @ sK6 @ sK4)) | ($true != (in @ sK7 @ X0))) )),
% 1.60/1.37 inference(trivial_inequality_removal,[status(thm)],[f50])).
% 1.60/1.37 thf(f50,plain,(
% 1.60/1.37 ( ! [X0 : $i] : (($true != (in @ sK7 @ X0)) | ($true != (func @ X0 @ sK6 @ sK4)) | ($true != $true)) )),
% 1.60/1.37 inference(superposition,[status(thm)],[f48,f49])).
% 1.60/1.37 thf(f49,plain,(
% 1.60/1.37 ( ! [X6 : $i,X7 : $i,X4 : $i,X5 : $i] : (($true = (in @ (setunion @ (dsetconstr @ X5 @ (^[Y0 : $i]: (in @ (kpair @ X7 @ Y0) @ X4)))) @ X5)) | ($true != (in @ X7 @ X6)) | ($true != (func @ X6 @ X5 @ X4))) )),
% 1.60/1.37 inference(trivial_inequality_removal,[status(thm)],[f46])).
% 1.60/1.37 thf(f46,plain,(
% 1.60/1.37 ( ! [X6 : $i,X7 : $i,X4 : $i,X5 : $i] : (($true != (in @ X7 @ X6)) | ($true != $true) | ($true != (func @ X6 @ X5 @ X4)) | ($true = (in @ (setunion @ (dsetconstr @ X5 @ (^[Y0 : $i]: (in @ (kpair @ X7 @ Y0) @ X4)))) @ X5))) )),
% 1.60/1.37 inference(definition_unfolding,[status(thm)],[f35,f39])).
% 1.60/1.37 thf(f39,plain,(
% 1.60/1.37 (apProp = $true)),
% 1.60/1.37 inference(cnf_transformation,[status(thm)],[f33])).
% 1.60/1.37 thf(f35,plain,(
% 1.60/1.37 ( ! [X6 : $i,X7 : $i,X4 : $i,X5 : $i] : (($true != (func @ X6 @ X5 @ X4)) | ($true != (in @ X7 @ X6)) | ($true = (in @ (setunion @ (dsetconstr @ X5 @ (^[Y0 : $i]: (in @ (kpair @ X7 @ Y0) @ X4)))) @ X5)) | (apProp != $true)) )),
% 1.60/1.37 inference(cnf_transformation,[status(thm)],[f30])).
% 1.60/1.37 thf(f30,plain,(
% 1.60/1.37 ((apProp = $true) | (((func @ sK2 @ sK1 @ sK0) = $true) & (($true = (in @ sK3 @ sK2)) & ((in @ (setunion @ (dsetconstr @ sK1 @ (^[Y0 : $i]: (in @ (kpair @ sK3 @ Y0) @ sK0)))) @ sK1) != $true)))) & (! [X4 : $i,X5 : $i,X6 : $i] : (($true != (func @ X6 @ X5 @ X4)) | ! [X7 : $i] : (($true != (in @ X7 @ X6)) | ($true = (in @ (setunion @ (dsetconstr @ X5 @ (^[Y0 : $i]: (in @ (kpair @ X7 @ Y0) @ X4)))) @ X5)))) | (apProp != $true))),
% 1.60/1.37 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2,sK3])],[f27,f29,f28])).
% 1.60/1.37 thf(f28,plain,(
% 1.60/1.37 ? [X0 : $i,X1 : $i,X2 : $i] : (((func @ X2 @ X1 @ X0) = $true) & ? [X3 : $i] : (((in @ X3 @ X2) = $true) & ($true != (in @ (setunion @ (dsetconstr @ X1 @ (^[Y0 : $i]: (in @ (kpair @ X3 @ Y0) @ X0)))) @ X1)))) => (((func @ sK2 @ sK1 @ sK0) = $true) & ? [X3 : $i] : (((in @ X3 @ sK2) = $true) & ($true != (in @ (setunion @ (dsetconstr @ sK1 @ (^[Y0 : $i]: (in @ (kpair @ X3 @ Y0) @ sK0)))) @ sK1))))),
% 1.60/1.37 introduced(definition,[],[choice_axiom])).
% 1.60/1.37 thf(f29,plain,(
% 1.60/1.37 ? [X3 : $i] : (((in @ X3 @ sK2) = $true) & ($true != (in @ (setunion @ (dsetconstr @ sK1 @ (^[Y0 : $i]: (in @ (kpair @ X3 @ Y0) @ sK0)))) @ sK1))) => (($true = (in @ sK3 @ sK2)) & ((in @ (setunion @ (dsetconstr @ sK1 @ (^[Y0 : $i]: (in @ (kpair @ sK3 @ Y0) @ sK0)))) @ sK1) != $true))),
% 1.60/1.37 introduced(definition,[],[choice_axiom])).
% 1.60/1.37 thf(f27,plain,(
% 1.60/1.37 ((apProp = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((func @ X2 @ X1 @ X0) = $true) & ? [X3 : $i] : (((in @ X3 @ X2) = $true) & ($true != (in @ (setunion @ (dsetconstr @ X1 @ (^[Y0 : $i]: (in @ (kpair @ X3 @ Y0) @ X0)))) @ X1))))) & (! [X4 : $i,X5 : $i,X6 : $i] : (($true != (func @ X6 @ X5 @ X4)) | ! [X7 : $i] : (($true != (in @ X7 @ X6)) | ($true = (in @ (setunion @ (dsetconstr @ X5 @ (^[Y0 : $i]: (in @ (kpair @ X7 @ Y0) @ X4)))) @ X5)))) | (apProp != $true))),
% 1.60/1.37 inference(rectify,[status(thm)],[f26])).
% 1.60/1.37 thf(f26,plain,(
% 1.60/1.37 ((apProp = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((func @ X2 @ X1 @ X0) = $true) & ? [X3 : $i] : (((in @ X3 @ X2) = $true) & ($true != (in @ (setunion @ (dsetconstr @ X1 @ (^[Y0 : $i]: (in @ (kpair @ X3 @ Y0) @ X0)))) @ X1))))) & (! [X0 : $i,X1 : $i,X2 : $i] : (((func @ X2 @ X1 @ X0) != $true) | ! [X3 : $i] : (((in @ X3 @ X2) != $true) | ($true = (in @ (setunion @ (dsetconstr @ X1 @ (^[Y0 : $i]: (in @ (kpair @ X3 @ Y0) @ X0)))) @ X1)))) | (apProp != $true))),
% 1.60/1.37 inference(nnf_transformation,[status(thm)],[f25])).
% 1.60/1.37 thf(f25,plain,(
% 1.60/1.37 (apProp = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((func @ X2 @ X1 @ X0) != $true) | ! [X3 : $i] : (((in @ X3 @ X2) != $true) | ($true = (in @ (setunion @ (dsetconstr @ X1 @ (^[Y0 : $i]: (in @ (kpair @ X3 @ Y0) @ X0)))) @ X1))))),
% 1.60/1.37 inference(ennf_transformation,[status(thm)],[f19])).
% 1.60/1.37 thf(f19,plain,(
% 1.60/1.37 ! [X0 : $i,X1 : $i,X2 : $i] : (((func @ X2 @ X1 @ X0) = $true) => ! [X3 : $i] : (((in @ X3 @ X2) = $true) => ($true = (in @ (setunion @ (dsetconstr @ X1 @ (^[Y0 : $i]: (in @ (kpair @ X3 @ Y0) @ X0)))) @ X1)))) <=> (apProp = $true)),
% 1.60/1.37 inference(fool_elimination,[status(thm)],[f18])).
% 1.60/1.37 thf(f18,plain,(
% 1.60/1.37 (apProp = ! [X0 : $i,X1 : $i,X2 : $i] : ((func @ X2 @ X1 @ X0) => ! [X3 : $i] : ((in @ X3 @ X2) => (in @ (setunion @ (dsetconstr @ X1 @ (^[X4 : $i] : (in @ (kpair @ X3 @ X4) @ X0)))) @ X1))))),
% 1.60/1.37 inference(rectify,[status(thm)],[f5])).
% 1.60/1.37 thf(f5,axiom,(
% 1.60/1.37 (apProp = ! [X7 : $i,X3 : $i,X0 : $i] : ((func @ X0 @ X3 @ X7) => ! [X1 : $i] : ((in @ X1 @ X0) => (in @ (setunion @ (dsetconstr @ X3 @ (^[X6 : $i] : (in @ (kpair @ X1 @ X6) @ X7)))) @ X3))))),
% 1.60/1.37 file('/export/starexec/sandbox/tmp/tmp.LmGkPjCsXB/DTF2THF_31356.p',apProp)).
% 1.60/1.37 thf(f48,plain,(
% 1.60/1.37 ((in @ (setunion @ (dsetconstr @ sK6 @ (^[Y0 : $i]: (in @ (kpair @ sK7 @ Y0) @ sK4)))) @ sK6) != $true)),
% 1.60/1.37 inference(beta_eta_normalization,[status(thm)],[f47])).
% 1.60/1.37 thf(f47,plain,(
% 1.60/1.37 ((in @ ((^[Y0 : $i]: ((^[Y1 : $i]: ((^[Y2 : $i]: ((^[Y3 : $i]: (setunion @ (dsetconstr @ Y1 @ (^[Y4 : $i]: (in @ (kpair @ Y3 @ Y4) @ Y2))))))))))) @ sK5 @ sK6 @ sK4 @ sK7) @ sK6) != $true)),
% 1.60/1.37 inference(definition_unfolding,[status(thm)],[f42,f34])).
% 1.60/1.37 thf(f34,plain,(
% 1.60/1.37 (ap = (^[Y0 : $i]: ((^[Y1 : $i]: ((^[Y2 : $i]: ((^[Y3 : $i]: (setunion @ (dsetconstr @ Y1 @ (^[Y4 : $i]: (in @ (kpair @ Y3 @ Y4) @ Y2))))))))))))),
% 1.60/1.37 inference(cnf_transformation,[status(thm)],[f21])).
% 1.60/1.37 thf(f21,plain,(
% 1.60/1.37 (ap = (^[Y0 : $i]: ((^[Y1 : $i]: ((^[Y2 : $i]: ((^[Y3 : $i]: (setunion @ (dsetconstr @ Y1 @ (^[Y4 : $i]: (in @ (kpair @ Y3 @ Y4) @ Y2))))))))))))),
% 1.60/1.37 inference(fool_elimination,[status(thm)],[f20])).
% 1.60/1.37 thf(f20,plain,(
% 1.60/1.37 (ap = (^[X0 : $i, X1 : $i, X2 : $i, X3 : $i] : (setunion @ (dsetconstr @ X1 @ (^[X4 : $i] : (in @ (kpair @ X3 @ X4) @ X2))))))),
% 1.60/1.37 inference(rectify,[status(thm)],[f6])).
% 1.60/1.37 thf(f6,axiom,(
% 1.60/1.37 (ap = (^[X0 : $i, X3 : $i, X7 : $i, X1 : $i] : (setunion @ (dsetconstr @ X3 @ (^[X6 : $i] : (in @ (kpair @ X1 @ X6) @ X7))))))),
% 1.60/1.37 file('/export/starexec/sandbox/tmp/tmp.LmGkPjCsXB/DTF2THF_31356.p',ap)).
% 1.60/1.37 thf(f42,plain,(
% 1.60/1.37 ($true != (in @ (ap @ sK5 @ sK6 @ sK4 @ sK7) @ sK6))),
% 1.60/1.37 inference(cnf_transformation,[status(thm)],[f33])).
% 1.60/1.37 % SZS output end Proof for DTF2THF_31356
% 1.60/1.37 % (31459)------------------------------
% 1.60/1.37 % (31459)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.60/1.37 % (31459)Termination reason: Refutation
% 1.60/1.37
% 1.60/1.37 % (31459)Memory used [KB]: 5500
% 1.60/1.37 % (31459)Time elapsed: 0.006 s
% 1.60/1.37 % (31459)Instructions burned: 5 (million)
% 1.60/1.37 % (31459)------------------------------
% 1.60/1.37 % (31459)------------------------------
% 1.60/1.37 % (31453)Success in time 0.021 s
%------------------------------------------------------------------------------