%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU694^2 : 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 : n007.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 09:36:27 AM UTC 2026
% Result : Theorem 1.64s 1.34s
% Output : Refutation 1.64s
% Verified :
% SZS Type : Refutation
% Derivation depth : 8
% Number of leaves : 3
% Syntax : Number of formulae : 11 ( 2 unt; 0 typ; 0 def)
% Number of atoms : 109 ( 26 equ; 0 cnn)
% Maximal formula atoms : 6 ( 9 avg)
% Number of connectives : 178 ( 11 ~; 0 |; 12 &; 145 @)
% ( 0 <=>; 10 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 7 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 10 ( 7 usr; 6 con; 0-4 aty)
% Number of variables : 30 ( 0 ^; 16 !; 14 ?; 30 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_1,type,
func: $i > $i > $i > $o ).
thf(func_def_2,type,
ap: $i > $i > $i > $i > $i ).
thf(f14,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f13]) ).
thf(f13,plain,
( ( ap @ sK1 @ sK2 @ sK0 @ sK3 )
!= ( ap @ sK1 @ sK2 @ sK0 @ sK3 ) ),
inference(cnf_transformation,[status(thm)],[f10]) ).
thf(f10,plain,
( ( ( ap @ sK1 @ sK2 @ sK0 @ sK3 )
!= ( ap @ sK1 @ sK2 @ sK0 @ sK3 ) )
& ( ( in @ sK3 @ sK1 )
= $true )
& ( ( func @ sK1 @ sK2 @ sK0 )
= $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2,sK3])],[f7,f9,f8]) ).
thf(f8,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( ( ap @ X1 @ X2 @ X0 @ X3 )
!= ( ap @ X1 @ X2 @ X0 @ X3 ) )
& ( ( in @ X3 @ X1 )
= $true ) )
& ( $true
= ( func @ X1 @ X2 @ X0 ) ) )
=> ( ? [X3: $i] :
( ( ( ap @ sK1 @ sK2 @ sK0 @ X3 )
!= ( ap @ sK1 @ sK2 @ sK0 @ X3 ) )
& ( $true
= ( in @ X3 @ sK1 ) ) )
& ( ( func @ sK1 @ sK2 @ sK0 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f9,plain,
( ? [X3: $i] :
( ( ( ap @ sK1 @ sK2 @ sK0 @ X3 )
!= ( ap @ sK1 @ sK2 @ sK0 @ X3 ) )
& ( $true
= ( in @ X3 @ sK1 ) ) )
=> ( ( ( ap @ sK1 @ sK2 @ sK0 @ sK3 )
!= ( ap @ sK1 @ sK2 @ sK0 @ sK3 ) )
& ( ( in @ sK3 @ sK1 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f7,plain,
? [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( ( ap @ X1 @ X2 @ X0 @ X3 )
!= ( ap @ X1 @ X2 @ X0 @ X3 ) )
& ( ( in @ X3 @ X1 )
= $true ) )
& ( $true
= ( func @ X1 @ X2 @ X0 ) ) ),
inference(rectify,[status(thm)],[f6]) ).
thf(f6,plain,
? [X1: $i,X0: $i,X2: $i] :
( ? [X3: $i] :
( ( ( ap @ X0 @ X2 @ X1 @ X3 )
!= ( ap @ X0 @ X2 @ X1 @ X3 ) )
& ( ( in @ X3 @ X0 )
= $true ) )
& ( $true
= ( func @ X0 @ X2 @ X1 ) ) ),
inference(ennf_transformation,[status(thm)],[f5]) ).
thf(f5,plain,
~ ! [X2: $i,X0: $i,X1: $i] :
( ( $true
= ( func @ X0 @ X2 @ X1 ) )
=> ! [X3: $i] :
( ( ( in @ X3 @ X0 )
= $true )
=> ( ( ap @ X0 @ X2 @ X1 @ X3 )
= ( ap @ X0 @ X2 @ X1 @ X3 ) ) ) ),
inference(fool_elimination,[status(thm)],[f4]) ).
thf(f4,plain,
~ ! [X0: $i,X1: $i,X2: $i] :
( ( func @ X0 @ X2 @ X1 )
=> ! [X3: $i] :
( ( in @ X3 @ X0 )
=> ( ( ap @ X0 @ X2 @ X1 @ X3 )
= ( ap @ X0 @ X2 @ X1 @ X3 ) ) ) ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,negated_conjecture,
~ ! [X0: $i,X2: $i,X1: $i] :
( ( func @ X0 @ X1 @ X2 )
=> ! [X3: $i] :
( ( in @ X3 @ X0 )
=> ( ( ap @ X0 @ X1 @ X2 @ X3 )
= ( ap @ X0 @ X1 @ X2 @ X3 ) ) ) ),
inference(negated_conjecture,[status(cth)],[f1]) ).
thf(f1,conjecture,
! [X0: $i,X2: $i,X1: $i] :
( ( func @ X0 @ X1 @ X2 )
=> ! [X3: $i] :
( ( in @ X3 @ X0 )
=> ( ( ap @ X0 @ X1 @ X2 @ X3 )
= ( ap @ X0 @ X1 @ X2 @ X3 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.Z3xuX8F1WD/DTF2THF_8143.p',ap2apEq2) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU694^2 : TPTP v9.2.1. Released v3.7.0.
% 0.11/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n007.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 12:30:23 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.19/0.46 ---- Original DTF file ---
% 0.19/0.47 thf(spec,logic,$$dhol).
% 0.19/0.47 %------------------------------------------------------------------------------
% 0.19/0.47 % File : SEU694^2 : TPTP v9.2.1. Released v3.7.0.
% 0.19/0.47 % Domain : Set Theory
% 0.19/0.47 % Problem : Functions - Extensionality and Beta Reduction
% 0.19/0.47 % Version : Especial > Reduced > Especial.
% 0.19/0.47 % English : (! A:i.! B:i.! f:i.func A B f -> (! x:i.in x A ->
% 0.19/0.47 % ap A B f x = ap A B f x))
% 0.19/0.47
% 0.19/0.47 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.19/0.47 % Source : [Bro08]
% 0.19/0.47 % Names : ZFC196l [Bro08]
% 0.19/0.47
% 0.19/0.47 % Status : Theorem
% 0.19/0.47 % Rating : 0.00 v6.0.0, 0.14 v5.5.0, 0.17 v5.4.0, 0.20 v5.3.0, 0.40 v5.2.0, 0.20 v4.1.0, 0.00 v3.7.0
% 0.19/0.47 % Syntax : Number of formulae : 4 ( 0 unt; 3 typ; 0 def)
% 0.19/0.47 % Number of atoms : 3 ( 1 equ; 0 cnn)
% 0.19/0.47 % Maximal formula atoms : 3 ( 3 avg)
% 0.19/0.47 % Number of connectives : 15 ( 0 ~; 0 |; 0 &; 13 @)
% 0.19/0.47 % ( 0 <=>; 2 =>; 0 <=; 0 <~>)
% 0.19/0.47 % Maximal formula depth : 9 ( 9 avg)
% 0.19/0.47 % Number of types : 2 ( 0 usr)
% 0.19/0.47 % Number of type conns : 9 ( 9 >; 0 *; 0 +; 0 <<)
% 0.19/0.47 % Number of symbols : 4 ( 3 usr; 0 con; 2-4 aty)
% 0.19/0.47 % Number of variables : 4 ( 0 ^; 4 !; 0 ?; 4 :)
% 0.19/0.47 % SPC : TH0_THM_EQU_NAR
% 0.19/0.47
% 0.19/0.47 % Comments : http://mathgate.info/detsetitem.php?id=358
% 0.19/0.47 %------------------------------------------------------------------------------
% 0.19/0.47 thf(in,type,
% 0.19/0.47 in: $i > $i > $o ).
% 0.19/0.47
% 0.19/0.47 thf(func,type,
% 0.19/0.47 func: $i > $i > $i > $o ).
% 0.19/0.47
% 0.19/0.47 thf(ap,type,
% 0.19/0.47 ap: $i > $i > $i > $i > $i ).
% 0.19/0.47
% 0.19/0.47 thf(ap2apEq2,conjecture,
% 0.19/0.47 ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.47 ( ( func @ A @ B @ Xf )
% 0.19/0.47 => ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.19/0.47 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 %------------------------------------------------------------------------------
% 0.19/0.47 ------------------------
% 1.64/1.32 ---- Embedded in THF ---
% 1.64/1.32 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.64/1.32 %%% Generated on Tue Feb 24 12:30:24 EST 2026
% 1.64/1.32 %%% using '$$dhol' embedding, version 1.3.0.
% 1.64/1.32 %%% Logic specification used:
% 1.64/1.32 %%% thf(spec, logic, $$dhol).
% 1.64/1.32
% 1.64/1.32 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.Z3xuX8F1WD/DTF2DTF8143.p
% See solution above
% 1.64/1.32 ------------------------
% 1.64/1.33 ---- Cleaned THF ---
% 1.64/1.33 thf(in,type,
% 1.64/1.33 in: $i > $i > $o ).
% 1.64/1.33
% 1.64/1.33 thf(func,type,
% 1.64/1.33 func: $i > $i > $i > $o ).
% 1.64/1.33
% 1.64/1.33 thf(ap,type,
% 1.64/1.33 ap: $i > $i > $i > $i > $i ).
% 1.64/1.33
% 1.64/1.33 thf(ap2apEq2,conjecture,
% 1.64/1.33 ! [A: $i,B: $i,Xf: $i] :
% 1.64/1.33 ( ( func @ A @ B @ Xf )
% 1.64/1.33 => ! [Xx: $i] :
% 1.64/1.33 ( ( in @ Xx @ A )
% 1.64/1.33 => ( ( ap @ A @ B @ Xf @ Xx )
% 1.64/1.33 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ).
% 1.64/1.33 ------------------------
% 1.64/1.34 % (8290)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_8143 for (3000ds/18Mi)
% 1.64/1.34 % (8290)First to succeed.
% 1.64/1.34 % (8289)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_8143 for (3000ds/275Mi)
% 1.64/1.34 % (8284)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_8143 for (3000ds/183Mi)
% 1.64/1.34 % (8289)Also succeeded, but the first one will report.
% 1.64/1.34 % (8290)Refutation found. Thanks to Tanya!
% 1.64/1.34 % SZS status Theorem for DTF2THF_8143
% 1.64/1.34 % SZS output start Proof for DTF2THF_8143
% 1.64/1.34 thf(func_def_0, type, in: $i > $i > $o).
% 1.64/1.34 thf(func_def_1, type, func: $i > $i > $i > $o).
% 1.64/1.34 thf(func_def_2, type, ap: $i > $i > $i > $i > $i).
% 1.64/1.34 thf(f14,plain,(
% 1.64/1.34 $false),
% 1.64/1.34 inference(trivial_inequality_removal,[status(thm)],[f13])).
% 1.64/1.34 thf(f13,plain,(
% 1.64/1.34 ((ap @ sK1 @ sK2 @ sK0 @ sK3) != (ap @ sK1 @ sK2 @ sK0 @ sK3))),
% 1.64/1.34 inference(cnf_transformation,[status(thm)],[f10])).
% 1.64/1.34 thf(f10,plain,(
% 1.64/1.34 (((ap @ sK1 @ sK2 @ sK0 @ sK3) != (ap @ sK1 @ sK2 @ sK0 @ sK3)) & ((in @ sK3 @ sK1) = $true)) & ((func @ sK1 @ sK2 @ sK0) = $true)),
% 1.64/1.34 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2,sK3])],[f7,f9,f8])).
% 1.64/1.34 thf(f8,plain,(
% 1.64/1.34 ? [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : (((ap @ X1 @ X2 @ X0 @ X3) != (ap @ X1 @ X2 @ X0 @ X3)) & ((in @ X3 @ X1) = $true)) & ($true = (func @ X1 @ X2 @ X0))) => (? [X3 : $i] : (((ap @ sK1 @ sK2 @ sK0 @ X3) != (ap @ sK1 @ sK2 @ sK0 @ X3)) & ($true = (in @ X3 @ sK1))) & ((func @ sK1 @ sK2 @ sK0) = $true))),
% 1.64/1.34 introduced(definition,[],[choice_axiom])).
% 1.64/1.34 thf(f9,plain,(
% 1.64/1.34 ? [X3 : $i] : (((ap @ sK1 @ sK2 @ sK0 @ X3) != (ap @ sK1 @ sK2 @ sK0 @ X3)) & ($true = (in @ X3 @ sK1))) => (((ap @ sK1 @ sK2 @ sK0 @ sK3) != (ap @ sK1 @ sK2 @ sK0 @ sK3)) & ((in @ sK3 @ sK1) = $true))),
% 1.64/1.34 introduced(definition,[],[choice_axiom])).
% 1.64/1.34 thf(f7,plain,(
% 1.64/1.34 ? [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : (((ap @ X1 @ X2 @ X0 @ X3) != (ap @ X1 @ X2 @ X0 @ X3)) & ((in @ X3 @ X1) = $true)) & ($true = (func @ X1 @ X2 @ X0)))),
% 1.64/1.34 inference(rectify,[status(thm)],[f6])).
% 1.64/1.34 thf(f6,plain,(
% 1.64/1.34 ? [X1 : $i,X0 : $i,X2 : $i] : (? [X3 : $i] : (((ap @ X0 @ X2 @ X1 @ X3) != (ap @ X0 @ X2 @ X1 @ X3)) & ((in @ X3 @ X0) = $true)) & ($true = (func @ X0 @ X2 @ X1)))),
% 1.64/1.34 inference(ennf_transformation,[status(thm)],[f5])).
% 1.64/1.34 thf(f5,plain,(
% 1.64/1.34 ~! [X2 : $i,X0 : $i,X1 : $i] : (($true = (func @ X0 @ X2 @ X1)) => ! [X3 : $i] : (((in @ X3 @ X0) = $true) => ((ap @ X0 @ X2 @ X1 @ X3) = (ap @ X0 @ X2 @ X1 @ X3))))),
% 1.64/1.34 inference(fool_elimination,[status(thm)],[f4])).
% 1.64/1.34 thf(f4,plain,(
% 1.64/1.34 ~! [X0 : $i,X1 : $i,X2 : $i] : ((func @ X0 @ X2 @ X1) => ! [X3 : $i] : ((in @ X3 @ X0) => ((ap @ X0 @ X2 @ X1 @ X3) = (ap @ X0 @ X2 @ X1 @ X3))))),
% 1.64/1.34 inference(rectify,[status(thm)],[f2])).
% 1.64/1.34 thf(f2,negated_conjecture,(
% 1.64/1.34 ~! [X0 : $i,X2 : $i,X1 : $i] : ((func @ X0 @ X1 @ X2) => ! [X3 : $i] : ((in @ X3 @ X0) => ((ap @ X0 @ X1 @ X2 @ X3) = (ap @ X0 @ X1 @ X2 @ X3))))),
% 1.64/1.34 inference(negated_conjecture,[status(cth)],[f1])).
% 1.64/1.34 thf(f1,conjecture,(
% 1.64/1.34 ! [X0 : $i,X2 : $i,X1 : $i] : ((func @ X0 @ X1 @ X2) => ! [X3 : $i] : ((in @ X3 @ X0) => ((ap @ X0 @ X1 @ X2 @ X3) = (ap @ X0 @ X1 @ X2 @ X3))))),
% 1.64/1.34 file('/export/starexec/sandbox2/tmp/tmp.Z3xuX8F1WD/DTF2THF_8143.p',ap2apEq2)).
% 1.64/1.34 % SZS output end Proof for DTF2THF_8143
% 1.64/1.34 % (8290)------------------------------
% 1.64/1.34 % (8290)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.64/1.34 % (8290)Termination reason: Refutation
% 1.64/1.34
% 1.64/1.34 % (8290)Memory used [KB]: 5373
% 1.64/1.34 % (8290)Time elapsed: 0.002 s
% 1.64/1.34 % (8290)------------------------------
% 1.64/1.34 % (8290)------------------------------
% 1.64/1.34 % (8283)Success in time 0.011 s
%------------------------------------------------------------------------------