%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEV272^5 : TPTP v9.2.1. Bugfixed v5.2.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n002.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 09:40:43 AM UTC 2026
% Result : Theorem 0.31s 1.33s
% Output : Refutation 0.31s
% Verified :
% SZS Type : Refutation
% Derivation depth : 7
% Number of leaves : 3
% Syntax : Number of formulae : 16 ( 16 unt; 0 typ; 0 def)
% Number of atoms : 78 ( 21 equ; 0 cnn)
% Maximal formula atoms : 1 ( 4 avg)
% Number of connectives : 238 ( 18 ~; 0 |; 26 &; 142 @)
% ( 0 <=>; 28 =>; 0 <=; 0 <~>)
% Maximal formula depth : 2 ( 1 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 162 ( 162 >; 0 *; 0 +; 0 <<)
% Number of symbols : 7 ( 3 usr; 3 con; 0-2 aty)
% ( 16 !!; 8 ??; 0 @@+; 0 @@-)
% Number of variables : 78 ( 62 ^; 12 !; 4 ?; 78 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
cSUCC: ( ( $i > $o ) > $o ) > ( $i > $o ) > $o ).
thf(func_def_1,type,
cZERO: ( $i > $o ) > $o ).
thf(func_def_3,type,
vEPSILON:
!>[X0: $tType] : ( ( X0 > $o ) > X0 ) ).
thf(f17,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f16]) ).
thf(f16,plain,
( ( ^ [Y0: ( $i > $o ) > $o] :
( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ( ^ [Y3: ( $i > $o ) > $o,Y4: $i > $o] :
( ?? @ $i
@ ^ [Y5: $i] :
( ( Y4 @ Y5 )
& ( Y3
@ ^ [Y6: $i] :
( ( Y5 != Y6 )
& ( Y4 @ Y6 ) ) ) ) )
@ Y2 ) ) ) )
& ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i
@ ^ [Y3: $i] : ( Y2 @ Y3 ) ) ) )
=> ( Y1 @ Y0 ) ) ) )
!= ( ^ [Y0: ( $i > $o ) > $o] :
( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ( ^ [Y3: ( $i > $o ) > $o,Y4: $i > $o] :
( ?? @ $i
@ ^ [Y5: $i] :
( ( Y4 @ Y5 )
& ( Y3
@ ^ [Y6: $i] :
( ( Y5 != Y6 )
& ( Y4 @ Y6 ) ) ) ) )
@ Y2 ) ) ) )
& ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i
@ ^ [Y3: $i] : ( Y2 @ Y3 ) ) ) )
=> ( Y1 @ Y0 ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f14,f13,f15,f13,f15]) ).
thf(f15,plain,
( cZERO
= ( ^ [Y0: $i > $o] :
~ ( ?? @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f9]) ).
thf(f9,plain,
( cZERO
= ( ^ [Y0: $i > $o] :
~ ( ?? @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) ) ) ),
inference(fool_elimination,[status(thm)],[f8]) ).
thf(f8,plain,
( cZERO
= ( ^ [X0: $i > $o] :
~ ? [X1: $i] : ( X0 @ X1 ) ) ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
( cZERO
= ( ^ [X1: $i > $o] :
~ ? [X2: $i] : ( X1 @ X2 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.f3zRvd6Ekv/DTF2THF_12958.p',cZERO_def) ).
thf(f13,plain,
( cSUCC
= ( ^ [Y0: ( $i > $o ) > $o,Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( Y0
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f7]) ).
thf(f7,plain,
( cSUCC
= ( ^ [Y0: ( $i > $o ) > $o,Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( Y0
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f6]) ).
thf(f6,plain,
( ( ^ [X0: ( $i > $o ) > $o,X1: $i > $o] :
? [X2: $i] :
( ( X0
@ ^ [X3: $i] :
( ( X1 @ X3 )
& ( X2 != X3 ) ) )
& ( X1 @ X2 ) ) )
= cSUCC ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
( ( ^ [X0: ( $i > $o ) > $o,X1: $i > $o] :
? [X2: $i] :
( ( X0
@ ^ [X3: $i] :
( ( X1 @ X3 )
& ( X2 != X3 ) ) )
& ( X1 @ X2 ) ) )
= cSUCC ),
file('/export/starexec/sandbox/tmp/tmp.f3zRvd6Ekv/DTF2THF_12958.p',cSUCC_def) ).
thf(f14,plain,
( ( ^ [Y0: ( $i > $o ) > $o] :
( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1 @ ( cSUCC @ Y2 ) ) ) )
& ( Y1 @ cZERO ) )
=> ( Y1 @ Y0 ) ) ) )
!= ( ^ [Y0: ( $i > $o ) > $o] :
( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1 @ ( cSUCC @ Y2 ) ) ) )
& ( Y1 @ cZERO ) )
=> ( Y1 @ Y0 ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f12]) ).
thf(f12,plain,
( ( ^ [Y0: ( $i > $o ) > $o] :
( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1 @ ( cSUCC @ Y2 ) ) ) )
& ( Y1 @ cZERO ) )
=> ( Y1 @ Y0 ) ) ) )
!= ( ^ [Y0: ( $i > $o ) > $o] :
( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1 @ ( cSUCC @ Y2 ) ) ) )
& ( Y1 @ cZERO ) )
=> ( Y1 @ Y0 ) ) ) ) ),
inference(flattening,[status(thm)],[f11]) ).
thf(f11,plain,
( ( ^ [Y0: ( $i > $o ) > $o] :
( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1 @ ( cSUCC @ Y2 ) ) ) )
& ( Y1 @ cZERO ) )
=> ( Y1 @ Y0 ) ) ) )
!= ( ^ [Y0: ( $i > $o ) > $o] :
( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1 @ ( cSUCC @ Y2 ) ) ) )
& ( Y1 @ cZERO ) )
=> ( Y1 @ Y0 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f10]) ).
thf(f10,plain,
( ( ^ [X0: ( $i > $o ) > $o] :
! [X1: ( ( $i > $o ) > $o ) > $o] :
( ( ( X1 @ cZERO )
& ! [X2: ( $i > $o ) > $o] :
( ( X1 @ X2 )
=> ( X1 @ ( cSUCC @ X2 ) ) ) )
=> ( X1 @ X0 ) ) )
!= ( ^ [X3: ( $i > $o ) > $o] :
! [X4: ( ( $i > $o ) > $o ) > $o] :
( ( ( X4 @ cZERO )
& ! [X5: ( $i > $o ) > $o] :
( ( X4 @ X5 )
=> ( X4 @ ( cSUCC @ X5 ) ) ) )
=> ( X4 @ X3 ) ) ) ),
inference(rectify,[status(thm)],[f4]) ).
thf(f4,negated_conjecture,
( ( ^ [X4: ( $i > $o ) > $o] :
! [X5: ( ( $i > $o ) > $o ) > $o] :
( ( ( X5 @ cZERO )
& ! [X6: ( $i > $o ) > $o] :
( ( X5 @ X6 )
=> ( X5 @ ( cSUCC @ X6 ) ) ) )
=> ( X5 @ X4 ) ) )
!= ( ^ [X2: ( $i > $o ) > $o] :
! [X7: ( ( $i > $o ) > $o ) > $o] :
( ( ( X7 @ cZERO )
& ! [X6: ( $i > $o ) > $o] :
( ( X7 @ X6 )
=> ( X7 @ ( cSUCC @ X6 ) ) ) )
=> ( X7 @ X2 ) ) ) ),
inference(negated_conjecture,[status(cth)],[f3]) ).
thf(f3,conjecture,
( ( ^ [X4: ( $i > $o ) > $o] :
! [X5: ( ( $i > $o ) > $o ) > $o] :
( ( ( X5 @ cZERO )
& ! [X6: ( $i > $o ) > $o] :
( ( X5 @ X6 )
=> ( X5 @ ( cSUCC @ X6 ) ) ) )
=> ( X5 @ X4 ) ) )
= ( ^ [X2: ( $i > $o ) > $o] :
! [X7: ( ( $i > $o ) > $o ) > $o] :
( ( ( X7 @ cZERO )
& ! [X6: ( $i > $o ) > $o] :
( ( X7 @ X6 )
=> ( X7 @ ( cSUCC @ X6 ) ) ) )
=> ( X7 @ X2 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.f3zRvd6Ekv/DTF2THF_12958.p',cX6007A_pme) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEV272^5 : TPTP v9.2.1. Bugfixed v5.2.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.35 % Computer : n002.cluster.edu
% 0.17/0.35 % Model : x86_64 x86_64
% 0.17/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.35 % Memory : 8042.1875MB
% 0.17/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.35 % CPULimit : 300
% 0.17/0.35 % WCLimit : 300
% 0.17/0.35 % DateTime : Tue Feb 24 12:43:03 EST 2026
% 0.17/0.35 % CPUTime :
% 0.17/0.35 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.25/0.47 ---- Original DTF file ---
% 0.25/0.47 thf(spec,logic,$$dhol).
% 0.25/0.47 %------------------------------------------------------------------------------
% 0.25/0.47 % File : SEV272^5 : TPTP v9.2.1. Bugfixed v5.2.0.
% 0.25/0.47 % Domain : Set Theory (Sets of sets)
% 0.25/0.47 % Problem : TPS problem X6007A
% 0.25/0.47 % Version : Especial.
% 0.25/0.47 % English :
% 0.25/0.47
% 0.25/0.47 % Refs : [Bro09] Brown (2009), Email to Geoff Sutcliffe
% 0.25/0.47 % Source : [Bro09]
% 0.25/0.47 % Names : tps_0393 [Bro09]
% 0.25/0.47 % : X6007A [TPS]
% 0.25/0.47
% 0.25/0.47 % Status : Theorem
% 0.25/0.47 % Rating : 0.22 v9.1.0, 0.25 v9.0.0, 0.30 v8.2.0, 0.38 v8.1.0, 0.27 v7.5.0, 0.14 v7.4.0, 0.11 v7.2.0, 0.12 v7.1.0, 0.00 v6.2.0, 0.14 v6.1.0, 0.00 v6.0.0, 0.14 v5.5.0, 0.17 v5.4.0, 0.20 v5.2.0
% 0.25/0.47 % Syntax : Number of formulae : 5 ( 3 unt; 2 typ; 2 def)
% 0.25/0.47 % Number of atoms : 10 ( 4 equ; 0 cnn)
% 0.25/0.47 % Maximal formula atoms : 1 ( 3 avg)
% 0.25/0.47 % Number of connectives : 24 ( 2 ~; 0 |; 4 &; 14 @)
% 0.25/0.47 % ( 0 <=>; 4 =>; 0 <=; 0 <~>)
% 0.25/0.47 % Maximal formula depth : 1 ( 1 avg)
% 0.25/0.47 % Number of types : 2 ( 0 usr)
% 0.25/0.47 % Number of type conns : 25 ( 25 >; 0 *; 0 +; 0 <<)
% 0.25/0.47 % Number of symbols : 3 ( 2 usr; 0 con; 1-2 aty)
% 0.25/0.47 % Number of variables : 12 ( 6 ^; 4 !; 2 ?; 12 :)
% 0.25/0.47 % SPC : TH0_THM_EQU_NAR
% 0.25/0.47
% 0.25/0.47 % Comments : This problem is from the TPS library. Copyright (c) 2009 The TPS
% 0.25/0.47 % project in the Department of Mathematical Sciences at Carnegie
% 0.25/0.47 % Mellon University. Distributed under the Creative Commons copyleft
% 0.25/0.47 % license: http://creativecommons.org/licenses/by-sa/3.0/
% 0.25/0.47 % : Polymorphic definitions expanded.
% 0.25/0.47 % Bugfixes : v5.2.0 - Added missing type declarations.
% 0.25/0.47 %------------------------------------------------------------------------------
% 0.25/0.47 thf(cSUCC_type,type,
% 0.25/0.47 cSUCC: ( ( $i > $o ) > $o ) > ( $i > $o ) > $o ).
% 0.25/0.47
% 0.25/0.47 thf(cZERO_type,type,
% 0.25/0.47 cZERO: ( $i > $o ) > $o ).
% 0.25/0.47
% 0.25/0.47 thf(cSUCC_def,definition,
% 0.25/0.47 ( cSUCC
% 0.25/0.47 = ( ^ [Xn: ( $i > $o ) > $o,Xp: $i > $o] :
% 0.25/0.47 ? [Xx: $i] :
% 0.25/0.47 ( ( Xp @ Xx )
% 0.25/0.47 & ( Xn
% 0.25/0.47 @ ^ [Xt: $i] :
% 0.25/0.47 ( ( Xt != Xx )
% 0.25/0.47 & ( Xp @ Xt ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 thf(cZERO_def,definition,
% 0.25/0.47 ( cZERO
% 0.25/0.47 = ( ^ [Xp: $i > $o] :
% 0.25/0.47 ~ ? [Xx: $i] : ( Xp @ Xx ) ) ) ).
% 0.25/0.47
% 0.25/0.47 thf(cX6007A_pme,conjecture,
% 0.25/0.47 ( ( ^ [N: ( $i > $o ) > $o] :
% 0.25/0.47 ! [P: ( ( $i > $o ) > $o ) > $o] :
% 0.25/0.47 ( ( ( P @ cZERO )
% 0.25/0.47 & ! [X: ( $i > $o ) > $o] :
% 0.25/0.47 ( ( P @ X )
% 0.25/0.47 => ( P @ ( cSUCC @ X ) ) ) )
% 0.25/0.47 => ( P @ N ) ) )
% 0.25/0.47 = ( ^ [Xx: ( $i > $o ) > $o] :
% 0.25/0.47 ! [S: ( ( $i > $o ) > $o ) > $o] :
% 0.25/0.47 ( ( ( S @ cZERO )
% 0.25/0.47 & ! [X: ( $i > $o ) > $o] :
% 0.25/0.47 ( ( S @ X )
% 0.25/0.47 => ( S @ ( cSUCC @ X ) ) ) )
% 0.25/0.47 => ( S @ Xx ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 %------------------------------------------------------------------------------
% 0.25/0.47 ------------------------
% 0.31/1.30 ---- Embedded in THF ---
% 0.31/1.30 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.31/1.30 %%% Generated on Tue Feb 24 12:43:04 EST 2026
% 0.31/1.30 %%% using '$$dhol' embedding, version 1.3.0.
% 0.31/1.30 %%% Logic specification used:
% 0.31/1.30 %%% thf(spec, logic, $$dhol).
% 0.31/1.30
% 0.31/1.30 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.f3zRvd6Ekv/DTF2DTF12958.p
% See solution above
% 0.31/1.30 ------------------------
% 0.31/1.31 ---- Cleaned THF ---
% 0.31/1.31 thf(cSUCC_type,type,
% 0.31/1.31 cSUCC: ( ( $i > $o ) > $o ) > ( $i > $o ) > $o ).
% 0.31/1.31
% 0.31/1.31 thf(cZERO_type,type,
% 0.31/1.31 cZERO: ( $i > $o ) > $o ).
% 0.31/1.31
% 0.31/1.31 thf(cSUCC_def,definition,
% 0.31/1.31 ( cSUCC
% 0.31/1.31 = ( ^ [Xn: ( $i > $o ) > $o,Xp: $i > $o] :
% 0.31/1.31 ? [Xx: $i] :
% 0.31/1.31 ( ( Xp @ Xx )
% 0.31/1.31 & ( Xn
% 0.31/1.31 @ ^ [Xt: $i] :
% 0.31/1.31 ( ( Xt != Xx )
% 0.31/1.31 & ( Xp @ Xt ) ) ) ) ) ) ).
% 0.31/1.31
% 0.31/1.31 thf(cZERO_def,definition,
% 0.31/1.31 ( cZERO
% 0.31/1.31 = ( ^ [Xp: $i > $o] :
% 0.31/1.31 ~ ? [Xx: $i] : ( Xp @ Xx ) ) ) ).
% 0.31/1.31
% 0.31/1.31 thf(cX6007A_pme,conjecture,
% 0.31/1.31 ( ( ^ [N: ( $i > $o ) > $o] :
% 0.31/1.31 ! [P: ( ( $i > $o ) > $o ) > $o] :
% 0.31/1.31 ( ( ( P @ cZERO )
% 0.31/1.31 & ! [X: ( $i > $o ) > $o] :
% 0.31/1.31 ( ( P @ X )
% 0.31/1.31 => ( P @ ( cSUCC @ X ) ) ) )
% 0.31/1.31 => ( P @ N ) ) )
% 0.31/1.31 = ( ^ [Xx: ( $i > $o ) > $o] :
% 0.31/1.31 ! [S: ( ( $i > $o ) > $o ) > $o] :
% 0.31/1.31 ( ( ( S @ cZERO )
% 0.31/1.31 & ! [X: ( $i > $o ) > $o] :
% 0.31/1.31 ( ( S @ X )
% 0.31/1.31 => ( S @ ( cSUCC @ X ) ) ) )
% 0.31/1.31 => ( S @ Xx ) ) ) ) ).
% 0.31/1.31 ------------------------
% 0.31/1.33 % (13237)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_12958 for (2999ds/2Mi)
% 0.31/1.33 % (13237)Instruction limit reached!
% 0.31/1.33 % (13237)------------------------------
% 0.31/1.33 % (13237)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.31/1.33 % (13237)Termination reason: Unknown
% 0.31/1.33 % (13237)Termination phase: Clausification
% 0.31/1.33
% 0.31/1.33 % (13237)Memory used [KB]: 895
% 0.31/1.33 % (13237)Time elapsed: 0.002 s
% 0.31/1.33 % (13237)Instructions burned: 2 (million)
% 0.31/1.33 % (13237)------------------------------
% 0.31/1.33 % (13237)------------------------------
% 0.31/1.33 % (13235)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_12958 for (2999ds/27Mi)
% 0.31/1.33 % (13235)First to succeed.
% 0.31/1.33 % (13235)Refutation found. Thanks to Tanya!
% 0.31/1.33 % SZS status Theorem for DTF2THF_12958
% 0.31/1.33 % SZS output start Proof for DTF2THF_12958
% 0.31/1.33 thf(func_def_0, type, cSUCC: (($i > $o) > $o) > ($i > $o) > $o).
% 0.31/1.33 thf(func_def_1, type, cZERO: ($i > $o) > $o).
% 0.31/1.33 thf(func_def_3, type, vEPSILON: !>[X0: $tType]:((X0 > $o) > X0)).
% 0.31/1.33 thf(f17,plain,(
% 0.31/1.33 $false),
% 0.31/1.33 inference(trivial_inequality_removal,[status(thm)],[f16])).
% 0.31/1.33 thf(f16,plain,(
% 0.31/1.33 ((^[Y0 : ($i > $o) > $o]: (!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ ((^[Y3 : ($i > $o) > $o]: ((^[Y4 : $i > $o]: (?? @ $i @ (^[Y5 : $i]: ((Y4 @ Y5) & (Y3 @ (^[Y6 : $i]: ((~ (Y5 = Y6)) & (Y4 @ Y6)))))))))) @ Y2))))) & (Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ (^[Y3 : $i]: (Y2 @ Y3))))))) => (Y1 @ Y0))))) != (^[Y0 : ($i > $o) > $o]: (!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ ((^[Y3 : ($i > $o) > $o]: ((^[Y4 : $i > $o]: (?? @ $i @ (^[Y5 : $i]: ((Y4 @ Y5) & (Y3 @ (^[Y6 : $i]: ((~ (Y5 = Y6)) & (Y4 @ Y6)))))))))) @ Y2))))) & (Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ (^[Y3 : $i]: (Y2 @ Y3))))))) => (Y1 @ Y0))))))),
% 0.31/1.33 inference(definition_unfolding,[status(thm)],[f14,f13,f15,f13,f15])).
% 0.31/1.33 thf(f15,plain,(
% 0.31/1.33 (cZERO = (^[Y0 : $i > $o]: (~ (?? @ $i @ (^[Y1 : $i]: (Y0 @ Y1))))))),
% 0.31/1.33 inference(cnf_transformation,[status(thm)],[f9])).
% 0.31/1.33 thf(f9,plain,(
% 0.31/1.33 (cZERO = (^[Y0 : $i > $o]: (~ (?? @ $i @ (^[Y1 : $i]: (Y0 @ Y1))))))),
% 0.31/1.33 inference(fool_elimination,[status(thm)],[f8])).
% 0.31/1.33 thf(f8,plain,(
% 0.31/1.33 (cZERO = (^[X0 : $i > $o] : (~? [X1 : $i] : (X0 @ X1))))),
% 0.31/1.33 inference(rectify,[status(thm)],[f2])).
% 0.31/1.33 thf(f2,axiom,(
% 0.31/1.33 (cZERO = (^[X1 : $i > $o] : (~? [X2 : $i] : (X1 @ X2))))),
% 0.31/1.33 file('/export/starexec/sandbox/tmp/tmp.f3zRvd6Ekv/DTF2THF_12958.p',cZERO_def)).
% 0.31/1.33 thf(f13,plain,(
% 0.31/1.33 (cSUCC = (^[Y0 : ($i > $o) > $o]: ((^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) & (Y0 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3)))))))))))),
% 0.31/1.33 inference(cnf_transformation,[status(thm)],[f7])).
% 0.31/1.33 thf(f7,plain,(
% 0.31/1.33 (cSUCC = (^[Y0 : ($i > $o) > $o]: ((^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y1 @ Y2) & (Y0 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3)))))))))))),
% 0.31/1.33 inference(fool_elimination,[status(thm)],[f6])).
% 0.31/1.33 thf(f6,plain,(
% 0.31/1.33 ((^[X0 : ($i > $o) > $o, X1 : $i > $o] : (? [X2 : $i] : ((X0 @ (^[X3 : $i] : ((X1 @ X3) & (X2 != X3)))) & (X1 @ X2)))) = cSUCC)),
% 0.31/1.33 inference(rectify,[status(thm)],[f1])).
% 0.31/1.33 thf(f1,axiom,(
% 0.31/1.33 ((^[X0 : ($i > $o) > $o, X1 : $i > $o] : (? [X2 : $i] : ((X0 @ (^[X3 : $i] : ((X1 @ X3) & (X2 != X3)))) & (X1 @ X2)))) = cSUCC)),
% 0.31/1.33 file('/export/starexec/sandbox/tmp/tmp.f3zRvd6Ekv/DTF2THF_12958.p',cSUCC_def)).
% 0.31/1.33 thf(f14,plain,(
% 0.31/1.33 ((^[Y0 : ($i > $o) > $o]: (!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (cSUCC @ Y2))))) & (Y1 @ cZERO)) => (Y1 @ Y0))))) != (^[Y0 : ($i > $o) > $o]: (!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (cSUCC @ Y2))))) & (Y1 @ cZERO)) => (Y1 @ Y0))))))),
% 0.31/1.33 inference(cnf_transformation,[status(thm)],[f12])).
% 0.31/1.33 thf(f12,plain,(
% 0.31/1.33 ((^[Y0 : ($i > $o) > $o]: (!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (cSUCC @ Y2))))) & (Y1 @ cZERO)) => (Y1 @ Y0))))) != (^[Y0 : ($i > $o) > $o]: (!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (cSUCC @ Y2))))) & (Y1 @ cZERO)) => (Y1 @ Y0))))))),
% 0.31/1.33 inference(flattening,[status(thm)],[f11])).
% 0.31/1.33 thf(f11,plain,(
% 0.31/1.33 ~((^[Y0 : ($i > $o) > $o]: (!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (cSUCC @ Y2))))) & (Y1 @ cZERO)) => (Y1 @ Y0))))) = (^[Y0 : ($i > $o) > $o]: (!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (cSUCC @ Y2))))) & (Y1 @ cZERO)) => (Y1 @ Y0))))))),
% 0.31/1.33 inference(fool_elimination,[status(thm)],[f10])).
% 0.31/1.33 thf(f10,plain,(
% 0.31/1.33 ~((^[X0 : ($i > $o) > $o] : (! [X1 : (($i > $o) > $o) > $o] : (((X1 @ cZERO) & ! [X2 : ($i > $o) > $o] : ((X1 @ X2) => (X1 @ (cSUCC @ X2)))) => (X1 @ X0)))) = (^[X3 : ($i > $o) > $o] : (! [X4 : (($i > $o) > $o) > $o] : (((X4 @ cZERO) & ! [X5 : ($i > $o) > $o] : ((X4 @ X5) => (X4 @ (cSUCC @ X5)))) => (X4 @ X3)))))),
% 0.31/1.33 inference(rectify,[status(thm)],[f4])).
% 0.31/1.33 thf(f4,negated_conjecture,(
% 0.31/1.33 ~((^[X4 : ($i > $o) > $o] : (! [X5 : (($i > $o) > $o) > $o] : (((X5 @ cZERO) & ! [X6 : ($i > $o) > $o] : ((X5 @ X6) => (X5 @ (cSUCC @ X6)))) => (X5 @ X4)))) = (^[X2 : ($i > $o) > $o] : (! [X7 : (($i > $o) > $o) > $o] : (((X7 @ cZERO) & ! [X6 : ($i > $o) > $o] : ((X7 @ X6) => (X7 @ (cSUCC @ X6)))) => (X7 @ X2)))))),
% 0.31/1.33 inference(negated_conjecture,[status(cth)],[f3])).
% 0.31/1.33 thf(f3,conjecture,(
% 0.31/1.33 ((^[X4 : ($i > $o) > $o] : (! [X5 : (($i > $o) > $o) > $o] : (((X5 @ cZERO) & ! [X6 : ($i > $o) > $o] : ((X5 @ X6) => (X5 @ (cSUCC @ X6)))) => (X5 @ X4)))) = (^[X2 : ($i > $o) > $o] : (! [X7 : (($i > $o) > $o) > $o] : (((X7 @ cZERO) & ! [X6 : ($i > $o) > $o] : ((X7 @ X6) => (X7 @ (cSUCC @ X6)))) => (X7 @ X2)))))),
% 0.31/1.33 file('/export/starexec/sandbox/tmp/tmp.f3zRvd6Ekv/DTF2THF_12958.p',cX6007A_pme)).
% 0.31/1.33 % SZS output end Proof for DTF2THF_12958
% 0.31/1.33 % (13235)------------------------------
% 0.31/1.33 % (13235)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.31/1.33 % (13235)Termination reason: Refutation
% 0.31/1.33
% 0.31/1.33 % (13235)Memory used [KB]: 5500
% 0.31/1.33 % (13235)Time elapsed: 0.002 s
% 0.31/1.33 % (13235)Instructions burned: 3 (million)
% 0.31/1.33 % (13235)------------------------------
% 0.31/1.33 % (13235)------------------------------
% 0.31/1.33 % (13232)Success in time 0.013 s
%------------------------------------------------------------------------------