%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU676^2 : 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 : n016.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 09:36:12 AM UTC 2026
% Result : Theorem 1.66s 1.35s
% Output : Refutation 1.66s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 8
% Syntax : Number of formulae : 43 ( 12 unt; 0 typ; 0 def)
% Number of atoms : 534 ( 137 equ; 0 cnn)
% Maximal formula atoms : 8 ( 12 avg)
% Number of connectives : 618 ( 61 ~; 41 |; 41 &; 441 @)
% ( 4 <=>; 30 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 30 ( 27 usr; 15 con; 0-4 aty)
% Number of variables : 123 ( 0 ^; 86 !; 37 ?; 123 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_2,type,
setadjoin: $i > $i > $i ).
thf(func_def_3,type,
powerset: $i > $i ).
thf(func_def_4,type,
setunion: $i > $i ).
thf(func_def_5,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(func_def_6,type,
subset: $i > $i > $o ).
thf(func_def_7,type,
kpair: $i > $i > $i ).
thf(func_def_8,type,
cartprod: $i > $i > $i ).
thf(func_def_9,type,
singleton: $i > $o ).
thf(func_def_11,type,
ex1: $i > ( $i > $o ) > $o ).
thf(func_def_12,type,
breln: $i > $i > $i > $o ).
thf(func_def_13,type,
func: $i > $i > $i > $o ).
thf(func_def_14,type,
funcSet: $i > $i > $i ).
thf(func_def_15,type,
ap: $i > $i > $i > $i > $i ).
thf(f74,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f73,f68]) ).
thf(f68,plain,
( ( func @ sK8 @ sK9 @ sK7 )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f67]) ).
thf(f67,plain,
( ( ( func @ sK8 @ sK9 @ sK7 )
= $true )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f66,f54]) ).
thf(f54,plain,
( $true
= ( in @ sK7 @ ( funcSet @ sK8 @ sK9 ) ) ),
inference(cnf_transformation,[status(thm)],[f45]) ).
thf(f45,plain,
( ( infuncsetfunc = $true )
& ( ( in @ ( ap @ sK8 @ sK9 @ sK7 @ sK10 ) @ sK9 )
!= $true )
& ( $true
= ( in @ sK10 @ sK8 ) )
& ( $true
= ( in @ sK7 @ ( funcSet @ sK8 @ sK9 ) ) )
& ( app = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK7,sK8,sK9,sK10])],[f31,f44,f43]) ).
thf(f43,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( ( in @ ( ap @ X1 @ X2 @ X0 @ X3 ) @ X2 )
!= $true )
& ( ( in @ X3 @ X1 )
= $true ) )
& ( $true
= ( in @ X0 @ ( funcSet @ X1 @ X2 ) ) ) )
=> ( ? [X3: $i] :
( ( $true
!= ( in @ ( ap @ sK8 @ sK9 @ sK7 @ X3 ) @ sK9 ) )
& ( $true
= ( in @ X3 @ sK8 ) ) )
& ( $true
= ( in @ sK7 @ ( funcSet @ sK8 @ sK9 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f44,plain,
( ? [X3: $i] :
( ( $true
!= ( in @ ( ap @ sK8 @ sK9 @ sK7 @ X3 ) @ sK9 ) )
& ( $true
= ( in @ X3 @ sK8 ) ) )
=> ( ( ( in @ ( ap @ sK8 @ sK9 @ sK7 @ sK10 ) @ sK9 )
!= $true )
& ( $true
= ( in @ sK10 @ sK8 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f31,plain,
( ( infuncsetfunc = $true )
& ? [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( ( in @ ( ap @ X1 @ X2 @ X0 @ X3 ) @ X2 )
!= $true )
& ( ( in @ X3 @ X1 )
= $true ) )
& ( $true
= ( in @ X0 @ ( funcSet @ X1 @ X2 ) ) ) )
& ( app = $true ) ),
inference(flattening,[status(thm)],[f30]) ).
thf(f30,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( ( in @ ( ap @ X1 @ X2 @ X0 @ X3 ) @ X2 )
!= $true )
& ( ( in @ X3 @ X1 )
= $true ) )
& ( $true
= ( in @ X0 @ ( funcSet @ X1 @ X2 ) ) ) )
& ( infuncsetfunc = $true )
& ( app = $true ) ),
inference(ennf_transformation,[status(thm)],[f27]) ).
thf(f27,plain,
~ ( ( app = $true )
=> ( ( infuncsetfunc = $true )
=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X0 @ ( funcSet @ X1 @ X2 ) ) )
=> ! [X3: $i] :
( ( ( in @ X3 @ X1 )
= $true )
=> ( ( in @ ( ap @ X1 @ X2 @ X0 @ X3 ) @ X2 )
= $true ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f26]) ).
thf(f26,plain,
~ ( app
=> ( infuncsetfunc
=> ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X0 @ ( funcSet @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X0 @ X3 ) @ X2 ) ) ) ) ),
inference(rectify,[status(thm)],[f10]) ).
thf(f10,negated_conjecture,
~ ( app
=> ( infuncsetfunc
=> ! [X7: $i,X0: $i,X3: $i] :
( ( in @ X7 @ ( funcSet @ X0 @ X3 ) )
=> ! [X1: $i] :
( ( in @ X1 @ X0 )
=> ( in @ ( ap @ X0 @ X3 @ X7 @ X1 ) @ X3 ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f9]) ).
thf(f9,conjecture,
( app
=> ( infuncsetfunc
=> ! [X7: $i,X0: $i,X3: $i] :
( ( in @ X7 @ ( funcSet @ X0 @ X3 ) )
=> ! [X1: $i] :
( ( in @ X1 @ X0 )
=> ( in @ ( ap @ X0 @ X3 @ X7 @ X1 ) @ X3 ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.Q7saQnOX7g/DTF2THF_24156.p',ap2p) ).
thf(f66,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X3 @ ( funcSet @ X4 @ X5 ) )
!= $true )
| ( $true
= ( func @ X4 @ X5 @ X3 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f60]) ).
thf(f60,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X3 @ ( funcSet @ X4 @ X5 ) )
!= $true )
| ( $true
= ( func @ X4 @ X5 @ X3 ) )
| ( $true != $true ) ),
inference(definition_unfolding,[status(thm)],[f46,f57]) ).
thf(f57,plain,
infuncsetfunc = $true,
inference(cnf_transformation,[status(thm)],[f45]) ).
thf(f46,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X3 @ ( funcSet @ X4 @ X5 ) )
!= $true )
| ( $true
= ( func @ X4 @ X5 @ X3 ) )
| ( infuncsetfunc != $true ) ),
inference(cnf_transformation,[status(thm)],[f37]) ).
thf(f37,plain,
( ( ( infuncsetfunc = $true )
| ( ( ( in @ sK0 @ ( funcSet @ sK1 @ sK2 ) )
= $true )
& ( $true
!= ( func @ sK1 @ sK2 @ sK0 ) ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X3 @ ( funcSet @ X4 @ X5 ) )
!= $true )
| ( $true
= ( func @ X4 @ X5 @ X3 ) ) )
| ( infuncsetfunc != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2])],[f35,f36]) ).
thf(f36,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X0 @ ( funcSet @ X1 @ X2 ) ) )
& ( ( func @ X1 @ X2 @ X0 )
!= $true ) )
=> ( ( ( in @ sK0 @ ( funcSet @ sK1 @ sK2 ) )
= $true )
& ( $true
!= ( func @ sK1 @ sK2 @ sK0 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f35,plain,
( ( ( infuncsetfunc = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X0 @ ( funcSet @ X1 @ X2 ) ) )
& ( ( func @ X1 @ X2 @ X0 )
!= $true ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X3 @ ( funcSet @ X4 @ X5 ) )
!= $true )
| ( $true
= ( func @ X4 @ X5 @ X3 ) ) )
| ( infuncsetfunc != $true ) ) ),
inference(rectify,[status(thm)],[f34]) ).
thf(f34,plain,
( ( ( infuncsetfunc = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X0 @ ( funcSet @ X1 @ X2 ) ) )
& ( ( func @ X1 @ X2 @ X0 )
!= $true ) ) )
& ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X0 @ ( funcSet @ X1 @ X2 ) ) )
| ( ( func @ X1 @ X2 @ X0 )
= $true ) )
| ( infuncsetfunc != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f33]) ).
thf(f33,plain,
( ( infuncsetfunc = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X0 @ ( funcSet @ X1 @ X2 ) ) )
| ( ( func @ X1 @ X2 @ X0 )
= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f15]) ).
thf(f15,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X0 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ( func @ X1 @ X2 @ X0 )
= $true ) )
<=> ( infuncsetfunc = $true ) ),
inference(fool_elimination,[status(thm)],[f14]) ).
thf(f14,plain,
( infuncsetfunc
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X0 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X0 ) ) ) ),
inference(rectify,[status(thm)],[f8]) ).
thf(f8,axiom,
( infuncsetfunc
= ( ! [X7: $i,X0: $i,X3: $i] :
( ( in @ X7 @ ( funcSet @ X0 @ X3 ) )
=> ( func @ X0 @ X3 @ X7 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.Q7saQnOX7g/DTF2THF_24156.p',infuncsetfunc) ).
thf(f73,plain,
( ( func @ sK8 @ sK9 @ sK7 )
!= $true ),
inference(subsumption_resolution,[status(thm)],[f72,f55]) ).
thf(f55,plain,
( $true
= ( in @ sK10 @ sK8 ) ),
inference(cnf_transformation,[status(thm)],[f45]) ).
thf(f72,plain,
( ( $true
!= ( in @ sK10 @ sK8 ) )
| ( ( func @ sK8 @ sK9 @ sK7 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f69]) ).
thf(f69,plain,
( ( ( func @ sK8 @ sK9 @ sK7 )
!= $true )
| ( $true != $true )
| ( $true
!= ( in @ sK10 @ sK8 ) ) ),
inference(superposition,[status(thm)],[f56,f65]) ).
thf(f65,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i] :
( ( $true
= ( in @ ( ap @ X0 @ X1 @ X2 @ X3 ) @ X1 ) )
| ( ( func @ X0 @ X1 @ X2 )
!= $true )
| ( ( in @ X3 @ X0 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f61]) ).
thf(f61,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i] :
( ( $true != $true )
| ( ( in @ X3 @ X0 )
!= $true )
| ( ( func @ X0 @ X1 @ X2 )
!= $true )
| ( $true
= ( in @ ( ap @ X0 @ X1 @ X2 @ X3 ) @ X1 ) ) ),
inference(definition_unfolding,[status(thm)],[f52,f53]) ).
thf(f53,plain,
app = $true,
inference(cnf_transformation,[status(thm)],[f45]) ).
thf(f52,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i] :
( ( ( func @ X0 @ X1 @ X2 )
!= $true )
| ( $true
= ( in @ ( ap @ X0 @ X1 @ X2 @ X3 ) @ X1 ) )
| ( ( in @ X3 @ X0 )
!= $true )
| ( app != $true ) ),
inference(cnf_transformation,[status(thm)],[f42]) ).
thf(f42,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( func @ X0 @ X1 @ X2 )
!= $true )
| ! [X3: $i] :
( ( $true
= ( in @ ( ap @ X0 @ X1 @ X2 @ X3 ) @ X1 ) )
| ( ( in @ X3 @ X0 )
!= $true ) ) )
| ( app != $true ) )
& ( ( app = $true )
| ( ( $true
= ( func @ sK3 @ sK4 @ sK5 ) )
& ( $true
!= ( in @ ( ap @ sK3 @ sK4 @ sK5 @ sK6 ) @ sK4 ) )
& ( ( in @ sK6 @ sK3 )
= $true ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK3,sK4,sK5,sK6])],[f39,f41,f40]) ).
thf(f40,plain,
( ? [X4: $i,X5: $i,X6: $i] :
( ( ( func @ X4 @ X5 @ X6 )
= $true )
& ? [X7: $i] :
( ( ( in @ ( ap @ X4 @ X5 @ X6 @ X7 ) @ X5 )
!= $true )
& ( ( in @ X7 @ X4 )
= $true ) ) )
=> ( ( $true
= ( func @ sK3 @ sK4 @ sK5 ) )
& ? [X7: $i] :
( ( ( in @ ( ap @ sK3 @ sK4 @ sK5 @ X7 ) @ sK4 )
!= $true )
& ( $true
= ( in @ X7 @ sK3 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f41,plain,
( ? [X7: $i] :
( ( ( in @ ( ap @ sK3 @ sK4 @ sK5 @ X7 ) @ sK4 )
!= $true )
& ( $true
= ( in @ X7 @ sK3 ) ) )
=> ( ( $true
!= ( in @ ( ap @ sK3 @ sK4 @ sK5 @ sK6 ) @ sK4 ) )
& ( ( in @ sK6 @ sK3 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f39,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( func @ X0 @ X1 @ X2 )
!= $true )
| ! [X3: $i] :
( ( $true
= ( in @ ( ap @ X0 @ X1 @ X2 @ X3 ) @ X1 ) )
| ( ( in @ X3 @ X0 )
!= $true ) ) )
| ( app != $true ) )
& ( ( app = $true )
| ? [X4: $i,X5: $i,X6: $i] :
( ( ( func @ X4 @ X5 @ X6 )
= $true )
& ? [X7: $i] :
( ( ( in @ ( ap @ X4 @ X5 @ X6 @ X7 ) @ X5 )
!= $true )
& ( ( in @ X7 @ X4 )
= $true ) ) ) ) ),
inference(rectify,[status(thm)],[f38]) ).
thf(f38,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( func @ X0 @ X1 @ X2 )
!= $true )
| ! [X3: $i] :
( ( $true
= ( in @ ( ap @ X0 @ X1 @ X2 @ X3 ) @ X1 ) )
| ( ( in @ X3 @ X0 )
!= $true ) ) )
| ( app != $true ) )
& ( ( app = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( func @ X0 @ X1 @ X2 )
= $true )
& ? [X3: $i] :
( ( $true
!= ( in @ ( ap @ X0 @ X1 @ X2 @ X3 ) @ X1 ) )
& ( ( in @ X3 @ X0 )
= $true ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f32]) ).
thf(f32,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( func @ X0 @ X1 @ X2 )
!= $true )
| ! [X3: $i] :
( ( $true
= ( in @ ( ap @ X0 @ X1 @ X2 @ X3 ) @ X1 ) )
| ( ( in @ X3 @ X0 )
!= $true ) ) )
<=> ( app = $true ) ),
inference(ennf_transformation,[status(thm)],[f13]) ).
thf(f13,plain,
( ( app = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( func @ X0 @ X1 @ X2 )
= $true )
=> ! [X3: $i] :
( ( ( in @ X3 @ X0 )
= $true )
=> ( $true
= ( in @ ( ap @ X0 @ X1 @ X2 @ X3 ) @ X1 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f12]) ).
thf(f12,plain,
( app
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( func @ X0 @ X1 @ X2 )
=> ! [X3: $i] :
( ( in @ X3 @ X0 )
=> ( in @ ( ap @ X0 @ X1 @ X2 @ X3 ) @ X1 ) ) ) ) ),
inference(rectify,[status(thm)],[f7]) ).
thf(f7,axiom,
( app
= ( ! [X0: $i,X3: $i,X7: $i] :
( ( func @ X0 @ X3 @ X7 )
=> ! [X1: $i] :
( ( in @ X1 @ X0 )
=> ( in @ ( ap @ X0 @ X3 @ X7 @ X1 ) @ X3 ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.Q7saQnOX7g/DTF2THF_24156.p',app) ).
thf(f56,plain,
( ( in @ ( ap @ sK8 @ sK9 @ sK7 @ sK10 ) @ sK9 )
!= $true ),
inference(cnf_transformation,[status(thm)],[f45]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.08 % Problem : SEU676^2 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.09 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.11/0.29 % Computer : n016.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 12:31:21 EST 2026
% 0.11/0.29 % CPUTime :
% 0.11/0.29 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.18/0.38 ---- Original DTF file ---
% 0.18/0.38 thf(spec,logic,$$dhol).
% 0.18/0.38 %------------------------------------------------------------------------------
% 0.18/0.38 % File : SEU676^2 : TPTP v9.2.1. Released v3.7.0.
% 0.18/0.38 % Domain : Set Theory
% 0.18/0.38 % Problem : Functions - Application
% 0.18/0.38 % Version : Especial > Reduced > Especial.
% 0.18/0.38 % English : (! A:i.! B:i.! f:i.in f (funcSet A B) -> (! x:i.in x A ->
% 0.18/0.38 % in (ap A B f x) B))
% 0.18/0.38
% 0.18/0.38 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.18/0.38 % Source : [Bro08]
% 0.18/0.38 % Names : ZFC178l [Bro08]
% 0.18/0.38
% 0.18/0.38 % Status : Theorem
% 0.18/0.38 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.20 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 v7.1.0, 0.25 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 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 v4.0.1, 0.67 v3.7.0
% 0.18/0.38 % Syntax : Number of formulae : 26 ( 8 unt; 17 typ; 8 def)
% 0.18/0.38 % Number of atoms : 34 ( 9 equ; 0 cnn)
% 0.18/0.38 % Maximal formula atoms : 5 ( 3 avg)
% 0.18/0.38 % Number of connectives : 78 ( 0 ~; 0 |; 2 &; 68 @)
% 0.18/0.38 % ( 0 <=>; 8 =>; 0 <=; 0 <~>)
% 0.18/0.38 % Maximal formula depth : 15 ( 3 avg)
% 0.18/0.38 % Number of types : 2 ( 0 usr)
% 0.18/0.38 % Number of type conns : 32 ( 32 >; 0 *; 0 +; 0 <<)
% 0.18/0.38 % Number of symbols : 18 ( 17 usr; 3 con; 0-4 aty)
% 0.18/0.38 % Number of variables : 32 ( 19 ^; 12 !; 1 ?; 32 :)
% 0.18/0.38 % SPC : TH0_THM_EQU_NAR
% 0.18/0.38
% 0.18/0.38 % Comments : http://mathgate.info/detsetitem.php?id=228
% 0.18/0.38 %------------------------------------------------------------------------------
% 0.18/0.38 thf(in_type,type,
% 0.18/0.38 in: $i > $i > $o ).
% 0.18/0.38
% 0.18/0.38 thf(emptyset_type,type,
% 0.18/0.38 emptyset: $i ).
% 0.18/0.38
% 0.18/0.38 thf(setadjoin_type,type,
% 0.18/0.38 setadjoin: $i > $i > $i ).
% 0.18/0.38
% 0.18/0.38 thf(powerset_type,type,
% 0.18/0.38 powerset: $i > $i ).
% 0.18/0.38
% 0.18/0.38 thf(setunion_type,type,
% 0.18/0.38 setunion: $i > $i ).
% 0.18/0.38
% 0.18/0.38 thf(dsetconstr_type,type,
% 0.18/0.38 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.18/0.38
% 0.18/0.38 thf(subset_type,type,
% 0.18/0.38 subset: $i > $i > $o ).
% 0.18/0.38
% 0.18/0.38 thf(kpair_type,type,
% 0.18/0.38 kpair: $i > $i > $i ).
% 0.18/0.38
% 0.18/0.38 thf(cartprod_type,type,
% 0.18/0.38 cartprod: $i > $i > $i ).
% 0.18/0.38
% 0.18/0.38 thf(singleton_type,type,
% 0.18/0.38 singleton: $i > $o ).
% 0.18/0.38
% 0.18/0.38 thf(singleton,definition,
% 0.18/0.38 ( singleton
% 0.18/0.38 = ( ^ [A: $i] :
% 0.18/0.38 ? [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ A )
% 0.18/0.38 & ( A
% 0.18/0.38 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(ex1_type,type,
% 0.18/0.38 ex1: $i > ( $i > $o ) > $o ).
% 0.18/0.38
% 0.18/0.38 thf(ex1,definition,
% 0.18/0.38 ( ex1
% 0.18/0.38 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.18/0.38 ( singleton
% 0.18/0.38 @ ( dsetconstr @ A
% 0.18/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(breln_type,type,
% 0.18/0.38 breln: $i > $i > $i > $o ).
% 0.18/0.38
% 0.18/0.38 thf(breln,definition,
% 0.18/0.38 ( breln
% 0.18/0.38 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(func_type,type,
% 0.18/0.38 func: $i > $i > $i > $o ).
% 0.18/0.38
% 0.18/0.38 thf(func,definition,
% 0.18/0.38 ( func
% 0.18/0.38 = ( ^ [A: $i,B: $i,R: $i] :
% 0.18/0.38 ( ( breln @ A @ B @ R )
% 0.18/0.38 & ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ A )
% 0.18/0.38 => ( ex1 @ B
% 0.18/0.38 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(funcSet_type,type,
% 0.18/0.38 funcSet: $i > $i > $i ).
% 0.18/0.38
% 0.18/0.38 thf(funcSet,definition,
% 0.18/0.38 ( funcSet
% 0.18/0.38 = ( ^ [A: $i,B: $i] :
% 0.18/0.38 ( dsetconstr @ ( powerset @ ( cartprod @ A @ B ) )
% 0.18/0.38 @ ^ [Xf: $i] : ( func @ A @ B @ Xf ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(ap_type,type,
% 0.18/0.38 ap: $i > $i > $i > $i > $i ).
% 0.18/0.38
% 0.18/0.38 thf(ap,definition,
% 0.18/0.38 ( ap
% 0.18/0.38 = ( ^ [A: $i,B: $i,Xf: $i,Xx: $i] :
% 0.18/0.38 ( setunion
% 0.18/0.38 @ ( dsetconstr @ B
% 0.18/0.38 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(app_type,type,
% 0.18/0.38 app: $o ).
% 0.18/0.38
% 0.18/0.38 thf(app,definition,
% 0.18/0.38 ( app
% 0.18/0.38 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.18/0.38 ( ( func @ A @ B @ Xf )
% 0.18/0.38 => ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ A )
% 0.18/0.38 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(infuncsetfunc_type,type,
% 0.18/0.38 infuncsetfunc: $o ).
% 0.18/0.38
% 0.18/0.38 thf(infuncsetfunc,definition,
% 0.18/0.38 ( infuncsetfunc
% 0.18/0.38 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.18/0.38 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.18/0.38 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(ap2p,conjecture,
% 0.18/0.38 ( app
% 0.18/0.38 => ( infuncsetfunc
% 0.18/0.38 => ! [A: $i,B: $i,Xf: $i] :
% 0.18/0.38 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.18/0.38 => ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ A )
% 0.18/0.38 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 %------------------------------------------------------------------------------
% 0.18/0.38 ------------------------
% 1.66/1.31 ---- Embedded in THF ---
% 1.66/1.31 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.66/1.31 %%% Generated on Tue Feb 24 12:31:22 EST 2026
% 1.66/1.31 %%% using '$$dhol' embedding, version 1.3.0.
% 1.66/1.31 %%% Logic specification used:
% 1.66/1.31 %%% thf(spec, logic, $$dhol).
% 1.66/1.31
% 1.66/1.31 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.Q7saQnOX7g/DTF2DTF24156.p
% See solution above
% 1.66/1.31 ------------------------
% 1.66/1.32 ---- Cleaned THF ---
% 1.66/1.32 thf(in_type,type,
% 1.66/1.32 in: $i > $i > $o ).
% 1.66/1.32
% 1.66/1.32 thf(emptyset_type,type,
% 1.66/1.32 emptyset: $i ).
% 1.66/1.32
% 1.66/1.32 thf(setadjoin_type,type,
% 1.66/1.32 setadjoin: $i > $i > $i ).
% 1.66/1.32
% 1.66/1.32 thf(powerset_type,type,
% 1.66/1.32 powerset: $i > $i ).
% 1.66/1.32
% 1.66/1.32 thf(setunion_type,type,
% 1.66/1.32 setunion: $i > $i ).
% 1.66/1.32
% 1.66/1.32 thf(dsetconstr_type,type,
% 1.66/1.32 dsetconstr: $i > ( $i > $o ) > $i ).
% 1.66/1.32
% 1.66/1.32 thf(subset_type,type,
% 1.66/1.32 subset: $i > $i > $o ).
% 1.66/1.32
% 1.66/1.32 thf(kpair_type,type,
% 1.66/1.32 kpair: $i > $i > $i ).
% 1.66/1.32
% 1.66/1.32 thf(cartprod_type,type,
% 1.66/1.32 cartprod: $i > $i > $i ).
% 1.66/1.32
% 1.66/1.32 thf(singleton_type,type,
% 1.66/1.32 singleton: $i > $o ).
% 1.66/1.32
% 1.66/1.32 thf(singleton,definition,
% 1.66/1.32 ( singleton
% 1.66/1.32 = ( ^ [A: $i] :
% 1.66/1.32 ? [Xx: $i] :
% 1.66/1.32 ( ( in @ Xx @ A )
% 1.66/1.32 & ( A
% 1.66/1.32 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 1.66/1.32
% 1.66/1.32 thf(ex1_type,type,
% 1.66/1.32 ex1: $i > ( $i > $o ) > $o ).
% 1.66/1.32
% 1.66/1.32 thf(ex1,definition,
% 1.66/1.32 ( ex1
% 1.66/1.32 = ( ^ [A: $i,Xphi: $i > $o] :
% 1.66/1.32 ( singleton
% 1.66/1.32 @ ( dsetconstr @ A
% 1.66/1.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 1.66/1.32
% 1.66/1.32 thf(breln_type,type,
% 1.66/1.32 breln: $i > $i > $i > $o ).
% 1.66/1.32
% 1.66/1.32 thf(breln,definition,
% 1.66/1.32 ( breln
% 1.66/1.32 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 1.66/1.32
% 1.66/1.32 thf(func_type,type,
% 1.66/1.32 func: $i > $i > $i > $o ).
% 1.66/1.32
% 1.66/1.32 thf(func,definition,
% 1.66/1.32 ( func
% 1.66/1.32 = ( ^ [A: $i,B: $i,R: $i] :
% 1.66/1.32 ( ( breln @ A @ B @ R )
% 1.66/1.32 & ! [Xx: $i] :
% 1.66/1.32 ( ( in @ Xx @ A )
% 1.66/1.32 => ( ex1 @ B
% 1.66/1.32 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ).
% 1.66/1.32
% 1.66/1.32 thf(funcSet_type,type,
% 1.66/1.32 funcSet: $i > $i > $i ).
% 1.66/1.32
% 1.66/1.32 thf(funcSet,definition,
% 1.66/1.32 ( funcSet
% 1.66/1.32 = ( ^ [A: $i,B: $i] :
% 1.66/1.32 ( dsetconstr @ ( powerset @ ( cartprod @ A @ B ) )
% 1.66/1.32 @ ^ [Xf: $i] : ( func @ A @ B @ Xf ) ) ) ) ).
% 1.66/1.32
% 1.66/1.32 thf(ap_type,type,
% 1.66/1.32 ap: $i > $i > $i > $i > $i ).
% 1.66/1.32
% 1.66/1.32 thf(ap,definition,
% 1.66/1.32 ( ap
% 1.66/1.32 = ( ^ [A: $i,B: $i,Xf: $i,Xx: $i] :
% 1.66/1.32 ( setunion
% 1.66/1.32 @ ( dsetconstr @ B
% 1.66/1.32 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ).
% 1.66/1.32
% 1.66/1.32 thf(app_type,type,
% 1.66/1.32 app: $o ).
% 1.66/1.32
% 1.66/1.32 thf(app,definition,
% 1.66/1.32 ( app
% 1.66/1.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 1.66/1.32 ( ( func @ A @ B @ Xf )
% 1.66/1.32 => ! [Xx: $i] :
% 1.66/1.32 ( ( in @ Xx @ A )
% 1.66/1.32 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 1.66/1.32
% 1.66/1.32 thf(infuncsetfunc_type,type,
% 1.66/1.32 infuncsetfunc: $o ).
% 1.66/1.32
% 1.66/1.32 thf(infuncsetfunc,definition,
% 1.66/1.32 ( infuncsetfunc
% 1.66/1.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 1.66/1.32 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 1.66/1.32 => ( func @ A @ B @ Xf ) ) ) ) ).
% 1.66/1.32
% 1.66/1.32 thf(ap2p,conjecture,
% 1.66/1.32 ( app
% 1.66/1.32 => ( infuncsetfunc
% 1.66/1.32 => ! [A: $i,B: $i,Xf: $i] :
% 1.66/1.32 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 1.66/1.32 => ! [Xx: $i] :
% 1.66/1.32 ( ( in @ Xx @ A )
% 1.66/1.32 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 1.66/1.32 ------------------------
% 1.66/1.34 % (24255)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_24156 for (2999ds/27Mi)
% 1.66/1.34 % (24256)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_24156 for (2999ds/2Mi)
% 1.66/1.34 % (24257)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_24156 for (2999ds/2Mi)
% 1.66/1.34 % (24253)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_24156 for (2999ds/183Mi)
% 1.66/1.34 % (24259)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_24156 for (2999ds/18Mi)
% 1.66/1.34 % (24254)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_24156 for (2999ds/4Mi)
% 1.66/1.34 % (24258)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_24156 for (2999ds/275Mi)
% 1.66/1.34 % (24256)Instruction limit reached!
% 1.66/1.34 % (24256)------------------------------
% 1.66/1.34 % (24256)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.66/1.34 % (24256)Termination reason: Unknown
% 1.66/1.34 % (24256)Termination phase: Property scanning
% 1.66/1.34
% 1.66/1.34 % (24256)Memory used [KB]: 1023
% 1.66/1.34 % (24256)Time elapsed: 0.003 s
% 1.66/1.34 % (24256)Instructions burned: 3 (million)
% 1.66/1.34 % (24256)------------------------------
% 1.66/1.34 % (24256)------------------------------
% 1.66/1.34 % (24257)Instruction limit reached!
% 1.66/1.34 % (24257)------------------------------
% 1.66/1.34 % (24257)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.66/1.34 % (24257)Termination reason: Unknown
% 1.66/1.34 % (24257)Termination phase: Property scanning
% 1.66/1.34
% 1.66/1.34 % (24257)Memory used [KB]: 1023
% 1.66/1.34 % (24257)Time elapsed: 0.003 s
% 1.66/1.34 % (24257)Instructions burned: 3 (million)
% 1.66/1.34 % (24257)------------------------------
% 1.66/1.34 % (24257)------------------------------
% 1.66/1.34 % (24254)Instruction limit reached!
% 1.66/1.34 % (24254)------------------------------
% 1.66/1.34 % (24254)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.66/1.34 % (24254)Termination reason: Unknown
% 1.66/1.34 % (24254)Termination phase: Property scanning
% 1.66/1.34
% 1.66/1.34 % (24254)Memory used [KB]: 1023
% 1.66/1.34 % (24254)Time elapsed: 0.004 s
% 1.66/1.34 % (24254)Instructions burned: 4 (million)
% 1.66/1.34 % (24254)------------------------------
% 1.66/1.34 % (24254)------------------------------
% 1.66/1.35 % (24258)First to succeed.
% 1.66/1.35 % (24258)Refutation found. Thanks to Tanya!
% 1.66/1.35 % SZS status Theorem for DTF2THF_24156
% 1.66/1.35 % SZS output start Proof for DTF2THF_24156
% 1.66/1.35 thf(func_def_0, type, in: $i > $i > $o).
% 1.66/1.35 thf(func_def_2, type, setadjoin: $i > $i > $i).
% 1.66/1.35 thf(func_def_3, type, powerset: $i > $i).
% 1.66/1.35 thf(func_def_4, type, setunion: $i > $i).
% 1.66/1.35 thf(func_def_5, type, dsetconstr: $i > ($i > $o) > $i).
% 1.66/1.35 thf(func_def_6, type, subset: $i > $i > $o).
% 1.66/1.35 thf(func_def_7, type, kpair: $i > $i > $i).
% 1.66/1.35 thf(func_def_8, type, cartprod: $i > $i > $i).
% 1.66/1.35 thf(func_def_9, type, singleton: $i > $o).
% 1.66/1.35 thf(func_def_11, type, ex1: $i > ($i > $o) > $o).
% 1.66/1.35 thf(func_def_12, type, breln: $i > $i > $i > $o).
% 1.66/1.35 thf(func_def_13, type, func: $i > $i > $i > $o).
% 1.66/1.35 thf(func_def_14, type, funcSet: $i > $i > $i).
% 1.66/1.35 thf(func_def_15, type, ap: $i > $i > $i > $i > $i).
% 1.66/1.35 thf(f74,plain,(
% 1.66/1.35 $false),
% 1.66/1.35 inference(subsumption_resolution,[status(thm)],[f73,f68])).
% 1.66/1.35 thf(f68,plain,(
% 1.66/1.35 ((func @ sK8 @ sK9 @ sK7) = $true)),
% 1.66/1.35 inference(trivial_inequality_removal,[status(thm)],[f67])).
% 1.66/1.35 thf(f67,plain,(
% 1.66/1.35 ((func @ sK8 @ sK9 @ sK7) = $true) | ($true != $true)),
% 1.66/1.35 inference(superposition,[status(thm)],[f66,f54])).
% 1.66/1.35 thf(f54,plain,(
% 1.66/1.35 ($true = (in @ sK7 @ (funcSet @ sK8 @ sK9)))),
% 1.66/1.35 inference(cnf_transformation,[status(thm)],[f45])).
% 1.66/1.35 thf(f45,plain,(
% 1.66/1.35 (infuncsetfunc = $true) & ((((in @ (ap @ sK8 @ sK9 @ sK7 @ sK10) @ sK9) != $true) & ($true = (in @ sK10 @ sK8))) & ($true = (in @ sK7 @ (funcSet @ sK8 @ sK9)))) & (app = $true)),
% 1.66/1.35 inference(skolemisation,[status(esa),new_symbols(skolem,[sK7,sK8,sK9,sK10])],[f31,f44,f43])).
% 1.66/1.35 thf(f43,plain,(
% 1.66/1.35 ? [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : (((in @ (ap @ X1 @ X2 @ X0 @ X3) @ X2) != $true) & ((in @ X3 @ X1) = $true)) & ($true = (in @ X0 @ (funcSet @ X1 @ X2)))) => (? [X3 : $i] : (($true != (in @ (ap @ sK8 @ sK9 @ sK7 @ X3) @ sK9)) & ($true = (in @ X3 @ sK8))) & ($true = (in @ sK7 @ (funcSet @ sK8 @ sK9))))),
% 1.66/1.35 introduced(definition,[],[choice_axiom])).
% 1.66/1.35 thf(f44,plain,(
% 1.66/1.35 ? [X3 : $i] : (($true != (in @ (ap @ sK8 @ sK9 @ sK7 @ X3) @ sK9)) & ($true = (in @ X3 @ sK8))) => (((in @ (ap @ sK8 @ sK9 @ sK7 @ sK10) @ sK9) != $true) & ($true = (in @ sK10 @ sK8)))),
% 1.66/1.35 introduced(definition,[],[choice_axiom])).
% 1.66/1.35 thf(f31,plain,(
% 1.66/1.35 (infuncsetfunc = $true) & ? [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : (((in @ (ap @ X1 @ X2 @ X0 @ X3) @ X2) != $true) & ((in @ X3 @ X1) = $true)) & ($true = (in @ X0 @ (funcSet @ X1 @ X2)))) & (app = $true)),
% 1.66/1.35 inference(flattening,[status(thm)],[f30])).
% 1.66/1.35 thf(f30,plain,(
% 1.66/1.35 (? [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : (((in @ (ap @ X1 @ X2 @ X0 @ X3) @ X2) != $true) & ((in @ X3 @ X1) = $true)) & ($true = (in @ X0 @ (funcSet @ X1 @ X2)))) & (infuncsetfunc = $true)) & (app = $true)),
% 1.66/1.35 inference(ennf_transformation,[status(thm)],[f27])).
% 1.66/1.35 thf(f27,plain,(
% 1.66/1.35 ~((app = $true) => ((infuncsetfunc = $true) => ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X0 @ (funcSet @ X1 @ X2))) => ! [X3 : $i] : (((in @ X3 @ X1) = $true) => ((in @ (ap @ X1 @ X2 @ X0 @ X3) @ X2) = $true)))))),
% 1.66/1.35 inference(fool_elimination,[status(thm)],[f26])).
% 1.66/1.35 thf(f26,plain,(
% 1.66/1.35 ~(app => (infuncsetfunc => ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X0 @ (funcSet @ X1 @ X2)) => ! [X3 : $i] : ((in @ X3 @ X1) => (in @ (ap @ X1 @ X2 @ X0 @ X3) @ X2)))))),
% 1.66/1.35 inference(rectify,[status(thm)],[f10])).
% 1.66/1.35 thf(f10,negated_conjecture,(
% 1.66/1.35 ~(app => (infuncsetfunc => ! [X7 : $i,X0 : $i,X3 : $i] : ((in @ X7 @ (funcSet @ X0 @ X3)) => ! [X1 : $i] : ((in @ X1 @ X0) => (in @ (ap @ X0 @ X3 @ X7 @ X1) @ X3)))))),
% 1.66/1.35 inference(negated_conjecture,[status(cth)],[f9])).
% 1.66/1.35 thf(f9,conjecture,(
% 1.66/1.35 app => (infuncsetfunc => ! [X7 : $i,X0 : $i,X3 : $i] : ((in @ X7 @ (funcSet @ X0 @ X3)) => ! [X1 : $i] : ((in @ X1 @ X0) => (in @ (ap @ X0 @ X3 @ X7 @ X1) @ X3))))),
% 1.66/1.35 file('/export/starexec/sandbox2/tmp/tmp.Q7saQnOX7g/DTF2THF_24156.p',ap2p)).
% 1.66/1.35 thf(f66,plain,(
% 1.66/1.35 ( ! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X3 @ (funcSet @ X4 @ X5)) != $true) | ($true = (func @ X4 @ X5 @ X3))) )),
% 1.66/1.35 inference(trivial_inequality_removal,[status(thm)],[f60])).
% 1.66/1.35 thf(f60,plain,(
% 1.66/1.35 ( ! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X3 @ (funcSet @ X4 @ X5)) != $true) | ($true = (func @ X4 @ X5 @ X3)) | ($true != $true)) )),
% 1.66/1.35 inference(definition_unfolding,[status(thm)],[f46,f57])).
% 1.66/1.35 thf(f57,plain,(
% 1.66/1.35 (infuncsetfunc = $true)),
% 1.66/1.35 inference(cnf_transformation,[status(thm)],[f45])).
% 1.66/1.35 thf(f46,plain,(
% 1.66/1.35 ( ! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X3 @ (funcSet @ X4 @ X5)) != $true) | ($true = (func @ X4 @ X5 @ X3)) | (infuncsetfunc != $true)) )),
% 1.66/1.35 inference(cnf_transformation,[status(thm)],[f37])).
% 1.66/1.35 thf(f37,plain,(
% 1.66/1.35 ((infuncsetfunc = $true) | (((in @ sK0 @ (funcSet @ sK1 @ sK2)) = $true) & ($true != (func @ sK1 @ sK2 @ sK0)))) & (! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X3 @ (funcSet @ X4 @ X5)) != $true) | ($true = (func @ X4 @ X5 @ X3))) | (infuncsetfunc != $true))),
% 1.66/1.35 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2])],[f35,f36])).
% 1.66/1.35 thf(f36,plain,(
% 1.66/1.35 ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X0 @ (funcSet @ X1 @ X2))) & ((func @ X1 @ X2 @ X0) != $true)) => (((in @ sK0 @ (funcSet @ sK1 @ sK2)) = $true) & ($true != (func @ sK1 @ sK2 @ sK0)))),
% 1.66/1.35 introduced(definition,[],[choice_axiom])).
% 1.66/1.35 thf(f35,plain,(
% 1.66/1.35 ((infuncsetfunc = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X0 @ (funcSet @ X1 @ X2))) & ((func @ X1 @ X2 @ X0) != $true))) & (! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X3 @ (funcSet @ X4 @ X5)) != $true) | ($true = (func @ X4 @ X5 @ X3))) | (infuncsetfunc != $true))),
% 1.66/1.35 inference(rectify,[status(thm)],[f34])).
% 1.66/1.35 thf(f34,plain,(
% 1.66/1.35 ((infuncsetfunc = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X0 @ (funcSet @ X1 @ X2))) & ((func @ X1 @ X2 @ X0) != $true))) & (! [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X0 @ (funcSet @ X1 @ X2))) | ((func @ X1 @ X2 @ X0) = $true)) | (infuncsetfunc != $true))),
% 1.66/1.35 inference(nnf_transformation,[status(thm)],[f33])).
% 1.66/1.35 thf(f33,plain,(
% 1.66/1.35 (infuncsetfunc = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X0 @ (funcSet @ X1 @ X2))) | ((func @ X1 @ X2 @ X0) = $true))),
% 1.66/1.35 inference(ennf_transformation,[status(thm)],[f15])).
% 1.66/1.35 thf(f15,plain,(
% 1.66/1.35 ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X0 @ (funcSet @ X1 @ X2))) => ((func @ X1 @ X2 @ X0) = $true)) <=> (infuncsetfunc = $true)),
% 1.66/1.35 inference(fool_elimination,[status(thm)],[f14])).
% 1.66/1.35 thf(f14,plain,(
% 1.66/1.35 (infuncsetfunc = ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X0 @ (funcSet @ X1 @ X2)) => (func @ X1 @ X2 @ X0)))),
% 1.66/1.35 inference(rectify,[status(thm)],[f8])).
% 1.66/1.35 thf(f8,axiom,(
% 1.66/1.35 (infuncsetfunc = ! [X7 : $i,X0 : $i,X3 : $i] : ((in @ X7 @ (funcSet @ X0 @ X3)) => (func @ X0 @ X3 @ X7)))),
% 1.66/1.35 file('/export/starexec/sandbox2/tmp/tmp.Q7saQnOX7g/DTF2THF_24156.p',infuncsetfunc)).
% 1.66/1.35 thf(f73,plain,(
% 1.66/1.35 ((func @ sK8 @ sK9 @ sK7) != $true)),
% 1.66/1.35 inference(subsumption_resolution,[status(thm)],[f72,f55])).
% 1.66/1.35 thf(f55,plain,(
% 1.66/1.35 ($true = (in @ sK10 @ sK8))),
% 1.66/1.35 inference(cnf_transformation,[status(thm)],[f45])).
% 1.66/1.35 thf(f72,plain,(
% 1.66/1.35 ($true != (in @ sK10 @ sK8)) | ((func @ sK8 @ sK9 @ sK7) != $true)),
% 1.66/1.35 inference(trivial_inequality_removal,[status(thm)],[f69])).
% 1.66/1.35 thf(f69,plain,(
% 1.66/1.35 ((func @ sK8 @ sK9 @ sK7) != $true) | ($true != $true) | ($true != (in @ sK10 @ sK8))),
% 1.66/1.35 inference(superposition,[status(thm)],[f56,f65])).
% 1.66/1.35 thf(f65,plain,(
% 1.66/1.35 ( ! [X2 : $i,X3 : $i,X0 : $i,X1 : $i] : (($true = (in @ (ap @ X0 @ X1 @ X2 @ X3) @ X1)) | ((func @ X0 @ X1 @ X2) != $true) | ((in @ X3 @ X0) != $true)) )),
% 1.66/1.35 inference(trivial_inequality_removal,[status(thm)],[f61])).
% 1.66/1.35 thf(f61,plain,(
% 1.66/1.35 ( ! [X2 : $i,X3 : $i,X0 : $i,X1 : $i] : (($true != $true) | ((in @ X3 @ X0) != $true) | ((func @ X0 @ X1 @ X2) != $true) | ($true = (in @ (ap @ X0 @ X1 @ X2 @ X3) @ X1))) )),
% 1.66/1.35 inference(definition_unfolding,[status(thm)],[f52,f53])).
% 1.66/1.35 thf(f53,plain,(
% 1.66/1.35 (app = $true)),
% 1.66/1.35 inference(cnf_transformation,[status(thm)],[f45])).
% 1.66/1.35 thf(f52,plain,(
% 1.66/1.35 ( ! [X2 : $i,X3 : $i,X0 : $i,X1 : $i] : (((func @ X0 @ X1 @ X2) != $true) | ($true = (in @ (ap @ X0 @ X1 @ X2 @ X3) @ X1)) | ((in @ X3 @ X0) != $true) | (app != $true)) )),
% 1.66/1.35 inference(cnf_transformation,[status(thm)],[f42])).
% 1.66/1.35 thf(f42,plain,(
% 1.66/1.35 (! [X0 : $i,X1 : $i,X2 : $i] : (((func @ X0 @ X1 @ X2) != $true) | ! [X3 : $i] : (($true = (in @ (ap @ X0 @ X1 @ X2 @ X3) @ X1)) | ((in @ X3 @ X0) != $true))) | (app != $true)) & ((app = $true) | (($true = (func @ sK3 @ sK4 @ sK5)) & (($true != (in @ (ap @ sK3 @ sK4 @ sK5 @ sK6) @ sK4)) & ((in @ sK6 @ sK3) = $true))))),
% 1.66/1.35 inference(skolemisation,[status(esa),new_symbols(skolem,[sK3,sK4,sK5,sK6])],[f39,f41,f40])).
% 1.66/1.35 thf(f40,plain,(
% 1.66/1.35 ? [X4 : $i,X5 : $i,X6 : $i] : (((func @ X4 @ X5 @ X6) = $true) & ? [X7 : $i] : (((in @ (ap @ X4 @ X5 @ X6 @ X7) @ X5) != $true) & ((in @ X7 @ X4) = $true))) => (($true = (func @ sK3 @ sK4 @ sK5)) & ? [X7 : $i] : (((in @ (ap @ sK3 @ sK4 @ sK5 @ X7) @ sK4) != $true) & ($true = (in @ X7 @ sK3))))),
% 1.66/1.35 introduced(definition,[],[choice_axiom])).
% 1.66/1.35 thf(f41,plain,(
% 1.66/1.35 ? [X7 : $i] : (((in @ (ap @ sK3 @ sK4 @ sK5 @ X7) @ sK4) != $true) & ($true = (in @ X7 @ sK3))) => (($true != (in @ (ap @ sK3 @ sK4 @ sK5 @ sK6) @ sK4)) & ((in @ sK6 @ sK3) = $true))),
% 1.66/1.35 introduced(definition,[],[choice_axiom])).
% 1.66/1.35 thf(f39,plain,(
% 1.66/1.35 (! [X0 : $i,X1 : $i,X2 : $i] : (((func @ X0 @ X1 @ X2) != $true) | ! [X3 : $i] : (($true = (in @ (ap @ X0 @ X1 @ X2 @ X3) @ X1)) | ((in @ X3 @ X0) != $true))) | (app != $true)) & ((app = $true) | ? [X4 : $i,X5 : $i,X6 : $i] : (((func @ X4 @ X5 @ X6) = $true) & ? [X7 : $i] : (((in @ (ap @ X4 @ X5 @ X6 @ X7) @ X5) != $true) & ((in @ X7 @ X4) = $true))))),
% 1.66/1.35 inference(rectify,[status(thm)],[f38])).
% 1.66/1.35 thf(f38,plain,(
% 1.66/1.35 (! [X0 : $i,X1 : $i,X2 : $i] : (((func @ X0 @ X1 @ X2) != $true) | ! [X3 : $i] : (($true = (in @ (ap @ X0 @ X1 @ X2 @ X3) @ X1)) | ((in @ X3 @ X0) != $true))) | (app != $true)) & ((app = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((func @ X0 @ X1 @ X2) = $true) & ? [X3 : $i] : (($true != (in @ (ap @ X0 @ X1 @ X2 @ X3) @ X1)) & ((in @ X3 @ X0) = $true))))),
% 1.66/1.35 inference(nnf_transformation,[status(thm)],[f32])).
% 1.66/1.35 thf(f32,plain,(
% 1.66/1.35 ! [X0 : $i,X1 : $i,X2 : $i] : (((func @ X0 @ X1 @ X2) != $true) | ! [X3 : $i] : (($true = (in @ (ap @ X0 @ X1 @ X2 @ X3) @ X1)) | ((in @ X3 @ X0) != $true))) <=> (app = $true)),
% 1.66/1.35 inference(ennf_transformation,[status(thm)],[f13])).
% 1.66/1.35 thf(f13,plain,(
% 1.66/1.35 (app = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((func @ X0 @ X1 @ X2) = $true) => ! [X3 : $i] : (((in @ X3 @ X0) = $true) => ($true = (in @ (ap @ X0 @ X1 @ X2 @ X3) @ X1))))),
% 1.66/1.35 inference(fool_elimination,[status(thm)],[f12])).
% 1.66/1.35 thf(f12,plain,(
% 1.66/1.35 (app = ! [X0 : $i,X1 : $i,X2 : $i] : ((func @ X0 @ X1 @ X2) => ! [X3 : $i] : ((in @ X3 @ X0) => (in @ (ap @ X0 @ X1 @ X2 @ X3) @ X1))))),
% 1.66/1.35 inference(rectify,[status(thm)],[f7])).
% 1.66/1.35 thf(f7,axiom,(
% 1.66/1.35 (app = ! [X0 : $i,X3 : $i,X7 : $i] : ((func @ X0 @ X3 @ X7) => ! [X1 : $i] : ((in @ X1 @ X0) => (in @ (ap @ X0 @ X3 @ X7 @ X1) @ X3))))),
% 1.66/1.35 file('/export/starexec/sandbox2/tmp/tmp.Q7saQnOX7g/DTF2THF_24156.p',app)).
% 1.66/1.35 thf(f56,plain,(
% 1.66/1.35 ((in @ (ap @ sK8 @ sK9 @ sK7 @ sK10) @ sK9) != $true)),
% 1.66/1.35 inference(cnf_transformation,[status(thm)],[f45])).
% 1.66/1.35 % SZS output end Proof for DTF2THF_24156
% 1.66/1.35 % (24258)------------------------------
% 1.66/1.35 % (24258)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.66/1.35 % (24258)Termination reason: Refutation
% 1.66/1.35
% 1.66/1.35 % (24258)Memory used [KB]: 5628
% 1.66/1.35 % (24258)Time elapsed: 0.007 s
% 1.66/1.35 % (24258)Instructions burned: 5 (million)
% 1.66/1.35 % (24258)------------------------------
% 1.66/1.35 % (24258)------------------------------
% 1.66/1.35 % (24252)Success in time 0.021 s
%------------------------------------------------------------------------------