%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEV299^5 : TPTP v9.2.1. Bugfixed v6.2.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n019.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 09:40:47 AM UTC 2026
% Result : Theorem 1.65s 1.31s
% Output : Refutation 1.65s
% Verified :
% SZS Type : Refutation
% Derivation depth : 20
% Number of leaves : 9
% Syntax : Number of formulae : 61 ( 25 unt; 0 typ; 3 def)
% Number of atoms : 411 ( 118 equ; 0 cnn)
% Maximal formula atoms : 10 ( 6 avg)
% Number of connectives : 593 ( 74 ~; 40 |; 67 &; 324 @)
% ( 3 <=>; 36 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 3 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 233 ( 233 >; 0 *; 0 +; 0 <<)
% Number of symbols : 16 ( 11 usr; 7 con; 0-2 aty)
% ( 14 !!; 35 ??; 0 @@+; 0 @@-)
% Number of variables : 151 ( 107 ^; 32 !; 12 ?; 151 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
cNAT: ( ( $i > $o ) > $o ) > $o ).
thf(func_def_1,type,
cSUCC: ( ( $i > $o ) > $o ) > ( $i > $o ) > $o ).
thf(func_def_2,type,
cZERO: ( $i > $o ) > $o ).
thf(func_def_4,type,
vEPSILON:
!>[X0: $tType] : ( ( X0 > $o ) > X0 ) ).
thf(func_def_17,type,
sK0: ( ( $i > $o ) > $o ) > $o ).
thf(func_def_18,type,
sK1: ( $i > $o ) > $o ).
thf(func_def_20,type,
sK3: ( ( ( $i > $o ) > $o ) > $o ) > ( $i > $o ) > $o ).
thf(func_def_21,type,
ph4:
!>[X0: $tType] : X0 ).
thf(f91,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f60,f80,f85,f89]) ).
thf(f89,plain,
~ spl2_3,
inference(avatar_contradiction_clause,[status(thm)],[f88]) ).
thf(f88,plain,
( $false
| ~ spl2_3 ),
inference(trivial_inequality_removal,[status(thm)],[f86]) ).
thf(f86,plain,
( ( $true = $false )
| ~ spl2_3 ),
inference(backward_demodulation,[status(thm)],[f41,f78]) ).
thf(f78,plain,
( ( ( sK0
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) )
= $false )
| ~ spl2_3 ),
inference(avatar_component_clause,[status(thm)],[f76]) ).
thf(f76,definition,
( spl2_3
<=> ( ( sK0
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) )
= $false ) ),
introduced(definition,[new_symbols(naming,[spl2_3])],[avatar_definition]) ).
thf(f41,plain,
( ( sK0
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f30]) ).
thf(f30,plain,
( $true
= ( sK0
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) ) ) ),
inference(definition_unfolding,[status(thm)],[f23,f20]) ).
thf(f20,plain,
( cZERO
= ( ^ [Y0: $i > $o] :
~ ( ?? @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f12]) ).
thf(f12,plain,
( cZERO
= ( ^ [Y0: $i > $o] :
~ ( ?? @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) ) ) ),
inference(fool_elimination,[status(thm)],[f11]) ).
thf(f11,plain,
( ( ^ [X0: $i > $o] :
~ ? [X1: $i] : ( X0 @ X1 ) )
= cZERO ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
( ( ^ [X0: $i > $o] :
~ ? [X1: $i] : ( X0 @ X1 ) )
= cZERO ),
file('/export/starexec/sandbox2/tmp/tmp.9Dt3z1JBh5/DTF2THF_27919.p',cZERO_def) ).
thf(f23,plain,
( ( sK0 @ cZERO )
= $true ),
inference(cnf_transformation,[status(thm)],[f19]) ).
thf(f19,plain,
( ! [X1: ( $i > $o ) > $o] :
( ( $true
!= ( sK0 @ X1 ) )
| ( $true
= ( sK0 @ ( cSUCC @ X1 ) ) ) )
& ( $true
!= ( sK0 @ sK1 ) )
& ( $true
= ( cNAT @ sK1 ) )
& ( ( sK0 @ cZERO )
= $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1])],[f16,f18,f17]) ).
thf(f17,plain,
( ? [X0: ( ( $i > $o ) > $o ) > $o] :
( ! [X1: ( $i > $o ) > $o] :
( ( ( X0 @ X1 )
!= $true )
| ( ( X0 @ ( cSUCC @ X1 ) )
= $true ) )
& ? [X2: ( $i > $o ) > $o] :
( ( ( X0 @ X2 )
!= $true )
& ( $true
= ( cNAT @ X2 ) ) )
& ( ( X0 @ cZERO )
= $true ) )
=> ( ! [X1: ( $i > $o ) > $o] :
( ( $true
!= ( sK0 @ X1 ) )
| ( $true
= ( sK0 @ ( cSUCC @ X1 ) ) ) )
& ? [X2: ( $i > $o ) > $o] :
( ( $true
!= ( sK0 @ X2 ) )
& ( $true
= ( cNAT @ X2 ) ) )
& ( ( sK0 @ cZERO )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f18,plain,
( ? [X2: ( $i > $o ) > $o] :
( ( $true
!= ( sK0 @ X2 ) )
& ( $true
= ( cNAT @ X2 ) ) )
=> ( ( $true
!= ( sK0 @ sK1 ) )
& ( $true
= ( cNAT @ sK1 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f16,plain,
? [X0: ( ( $i > $o ) > $o ) > $o] :
( ! [X1: ( $i > $o ) > $o] :
( ( ( X0 @ X1 )
!= $true )
| ( ( X0 @ ( cSUCC @ X1 ) )
= $true ) )
& ? [X2: ( $i > $o ) > $o] :
( ( ( X0 @ X2 )
!= $true )
& ( $true
= ( cNAT @ X2 ) ) )
& ( ( X0 @ cZERO )
= $true ) ),
inference(flattening,[status(thm)],[f15]) ).
thf(f15,plain,
? [X0: ( ( $i > $o ) > $o ) > $o] :
( ? [X2: ( $i > $o ) > $o] :
( ( ( X0 @ X2 )
!= $true )
& ( $true
= ( cNAT @ X2 ) ) )
& ! [X1: ( $i > $o ) > $o] :
( ( ( X0 @ X1 )
!= $true )
| ( ( X0 @ ( cSUCC @ X1 ) )
= $true ) )
& ( ( X0 @ cZERO )
= $true ) ),
inference(ennf_transformation,[status(thm)],[f8]) ).
thf(f8,plain,
~ ! [X0: ( ( $i > $o ) > $o ) > $o] :
( ( ! [X1: ( $i > $o ) > $o] :
( ( ( X0 @ X1 )
= $true )
=> ( ( X0 @ ( cSUCC @ X1 ) )
= $true ) )
& ( ( X0 @ cZERO )
= $true ) )
=> ! [X2: ( $i > $o ) > $o] :
( ( $true
= ( cNAT @ X2 ) )
=> ( ( X0 @ X2 )
= $true ) ) ),
inference(fool_elimination,[status(thm)],[f7]) ).
thf(f7,plain,
~ ! [X0: ( ( $i > $o ) > $o ) > $o] :
( ( ! [X1: ( $i > $o ) > $o] :
( ( X0 @ X1 )
=> ( X0 @ ( cSUCC @ X1 ) ) )
& ( X0 @ cZERO ) )
=> ! [X2: ( $i > $o ) > $o] :
( ( cNAT @ X2 )
=> ( X0 @ X2 ) ) ),
inference(rectify,[status(thm)],[f5]) ).
thf(f5,negated_conjecture,
~ ! [X4: ( ( $i > $o ) > $o ) > $o] :
( ( ! [X5: ( $i > $o ) > $o] :
( ( X4 @ X5 )
=> ( X4 @ ( cSUCC @ X5 ) ) )
& ( X4 @ cZERO ) )
=> ! [X6: ( $i > $o ) > $o] :
( ( cNAT @ X6 )
=> ( X4 @ X6 ) ) ),
inference(negated_conjecture,[status(cth)],[f4]) ).
thf(f4,conjecture,
! [X4: ( ( $i > $o ) > $o ) > $o] :
( ( ! [X5: ( $i > $o ) > $o] :
( ( X4 @ X5 )
=> ( X4 @ ( cSUCC @ X5 ) ) )
& ( X4 @ cZERO ) )
=> ! [X6: ( $i > $o ) > $o] :
( ( cNAT @ X6 )
=> ( X4 @ X6 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.9Dt3z1JBh5/DTF2THF_27919.p',cINDUCTION) ).
thf(f85,plain,
~ spl2_1,
inference(avatar_contradiction_clause,[status(thm)],[f84]) ).
thf(f84,plain,
( $false
| ~ spl2_1 ),
inference(trivial_inequality_removal,[status(thm)],[f81]) ).
thf(f81,plain,
( ( $true != $true )
| ~ spl2_1 ),
inference(superposition,[status(thm)],[f25,f54]) ).
thf(f54,plain,
( ( $true
= ( sK0 @ sK1 ) )
| ~ spl2_1 ),
inference(avatar_component_clause,[status(thm)],[f52]) ).
thf(f52,definition,
( spl2_1
<=> ( $true
= ( sK0 @ sK1 ) ) ),
introduced(definition,[new_symbols(naming,[spl2_1])],[avatar_definition]) ).
thf(f25,plain,
( $true
!= ( sK0 @ sK1 ) ),
inference(cnf_transformation,[status(thm)],[f19]) ).
thf(f80,plain,
( spl2_1
| spl2_3
| ~ spl2_2 ),
inference(avatar_split_clause,[status(thm)],[f73,f56,f76,f52]) ).
thf(f56,definition,
( spl2_2
<=> ( $true
= ( sK0 @ ( sK3 @ sK0 ) ) ) ),
introduced(definition,[new_symbols(naming,[spl2_2])],[avatar_definition]) ).
thf(f73,plain,
( ( $true
= ( sK0 @ sK1 ) )
| ( ( sK0
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) )
= $false )
| ~ spl2_2 ),
inference(trivial_inequality_removal,[status(thm)],[f69]) ).
thf(f69,plain,
( ( $true
= ( sK0 @ sK1 ) )
| ( ( sK0
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) )
= $false )
| ( $true = $false )
| ~ spl2_2 ),
inference(superposition,[status(thm)],[f39,f62]) ).
thf(f62,plain,
( ( ( sK0
@ ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( sK3 @ sK0
@ ^ [Y2: $i] :
( ( Y1 != Y2 )
& ( Y0 @ Y2 ) ) )
& ( Y0 @ Y1 ) ) ) )
= $true )
| ~ spl2_2 ),
inference(trivial_inequality_removal,[status(thm)],[f61]) ).
thf(f61,plain,
( ( $true != $true )
| ( ( sK0
@ ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( sK3 @ sK0
@ ^ [Y2: $i] :
( ( Y1 != Y2 )
& ( Y0 @ Y2 ) ) )
& ( Y0 @ Y1 ) ) ) )
= $true )
| ~ spl2_2 ),
inference(superposition,[status(thm)],[f31,f58]) ).
thf(f58,plain,
( ( $true
= ( sK0 @ ( sK3 @ sK0 ) ) )
| ~ spl2_2 ),
inference(avatar_component_clause,[status(thm)],[f56]) ).
thf(f31,plain,
! [X1: ( $i > $o ) > $o] :
( ( $true
!= ( sK0 @ X1 ) )
| ( $true
= ( sK0
@ ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( X1
@ ^ [Y2: $i] :
( ( Y1 != Y2 )
& ( Y0 @ Y2 ) ) )
& ( Y0 @ Y1 ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f28]) ).
thf(f28,plain,
! [X1: ( $i > $o ) > $o] :
( ( $true
= ( sK0
@ ( ^ [Y0: ( $i > $o ) > $o,Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) )
& ( Y1 @ Y2 ) ) )
@ X1 ) ) )
| ( $true
!= ( sK0 @ X1 ) ) ),
inference(definition_unfolding,[status(thm)],[f26,f21]) ).
thf(f21,plain,
( cSUCC
= ( ^ [Y0: ( $i > $o ) > $o,Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) )
& ( Y1 @ Y2 ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f10]) ).
thf(f10,plain,
( cSUCC
= ( ^ [Y0: ( $i > $o ) > $o,Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) )
& ( Y1 @ Y2 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f9]) ).
thf(f9,plain,
( cSUCC
= ( ^ [X0: ( $i > $o ) > $o,X1: $i > $o] :
? [X2: $i] :
( ( X1 @ X2 )
& ( X0
@ ^ [X3: $i] :
( ( X1 @ X3 )
& ( X2 != X3 ) ) ) ) ) ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
( cSUCC
= ( ^ [X2: ( $i > $o ) > $o,X0: $i > $o] :
? [X1: $i] :
( ( X0 @ X1 )
& ( X2
@ ^ [X3: $i] :
( ( X0 @ X3 )
& ( X1 != X3 ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.9Dt3z1JBh5/DTF2THF_27919.p',cSUCC_def) ).
thf(f26,plain,
! [X1: ( $i > $o ) > $o] :
( ( $true
!= ( sK0 @ X1 ) )
| ( $true
= ( sK0 @ ( cSUCC @ X1 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f19]) ).
thf(f39,plain,
! [X1: ( ( $i > $o ) > $o ) > $o] :
( ( ( X1
@ ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( sK3 @ X1
@ ^ [Y2: $i] :
( ( Y1 != Y2 )
& ( Y0 @ Y2 ) ) )
& ( Y0 @ Y1 ) ) ) )
= $false )
| ( ( X1
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) )
= $false )
| ( $true
= ( X1 @ sK1 ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f38]) ).
thf(f38,plain,
! [X1: ( ( $i > $o ) > $o ) > $o] :
( ( ( X1
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) )
= $false )
| ( ( ( X1 @ ( sK3 @ X1 ) )
=> ( X1
@ ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( sK3 @ X1
@ ^ [Y2: $i] :
( ( Y1 != Y2 )
& ( Y0 @ Y2 ) ) )
& ( Y0 @ Y1 ) ) ) ) )
= $false )
| ( $true
= ( X1 @ sK1 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f37]) ).
thf(f37,plain,
! [X1: ( ( $i > $o ) > $o ) > $o] :
( ( ( ^ [Y0: ( $i > $o ) > $o] :
( ( X1 @ Y0 )
=> ( X1
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) )
& ( Y1 @ Y2 ) ) ) ) )
@ ( sK3 @ X1 ) )
= $false )
| ( $true
= ( X1 @ sK1 ) )
| ( ( X1
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) )
= $false ) ),
inference(sigma_clausification,[status(thm)],[f36]) ).
thf(f36,plain,
! [X1: ( ( $i > $o ) > $o ) > $o] :
( ( ( X1
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) )
= $false )
| ( $true
= ( X1 @ sK1 ) )
| ( ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( X1 @ Y0 )
=> ( X1
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) )
& ( Y1 @ Y2 ) ) ) ) ) )
= $false ) ),
inference(binary_proxy_clausification,[status(thm)],[f35]) ).
thf(f35,plain,
! [X1: ( ( $i > $o ) > $o ) > $o] :
( ( $true
= ( X1 @ sK1 ) )
| ( ( ( X1
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( X1 @ Y0 )
=> ( X1
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) )
& ( Y1 @ Y2 ) ) ) ) ) ) )
= $false ) ),
inference(binary_proxy_clausification,[status(thm)],[f34]) ).
thf(f34,plain,
! [X1: ( ( $i > $o ) > $o ) > $o] :
( $true
= ( ( ( X1
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y0: ( $i > $o ) > $o] :
( ( X1 @ Y0 )
=> ( X1
@ ^ [Y1: $i > $o] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( Y0
@ ^ [Y3: $i] :
( ( Y2 != Y3 )
& ( Y1 @ Y3 ) ) )
& ( Y1 @ Y2 ) ) ) ) ) ) )
=> ( X1 @ sK1 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f33]) ).
thf(f33,plain,
! [X1: ( ( $i > $o ) > $o ) > $o] :
( $true
= ( ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) )
=> ( Y0 @ sK1 ) )
@ X1 ) ),
inference(pi_clausification,[status(thm)],[f32]) ).
thf(f32,plain,
( $true
= ( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y0: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y0
@ ^ [Y1: $i > $o] :
~ ( ?? @ $i @ Y1 ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y1: ( $i > $o ) > $o] :
( ( Y0 @ Y1 )
=> ( Y0
@ ^ [Y2: $i > $o] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( Y1
@ ^ [Y4: $i] :
( ( Y3 != Y4 )
& ( Y2 @ Y4 ) ) )
& ( Y2 @ Y3 ) ) ) ) ) ) )
=> ( Y0 @ sK1 ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f29]) ).
thf(f29,plain,
( ( ^ [Y0: ( $i > $o ) > $o] :
( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i
@ ^ [Y3: $i] : ( Y2 @ Y3 ) ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ( ^ [Y3: ( $i > $o ) > $o,Y4: $i > $o] :
( ?? @ $i
@ ^ [Y5: $i] :
( ( Y3
@ ^ [Y6: $i] :
( ( Y5 != Y6 )
& ( Y4 @ Y6 ) ) )
& ( Y4 @ Y5 ) ) )
@ Y2 ) ) ) ) )
=> ( Y1 @ Y0 ) ) )
@ sK1 )
= $true ),
inference(definition_unfolding,[status(thm)],[f24,f27]) ).
thf(f27,plain,
( cNAT
= ( ^ [Y0: ( $i > $o ) > $o] :
( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y1
@ ^ [Y2: $i > $o] :
~ ( ?? @ $i
@ ^ [Y3: $i] : ( Y2 @ Y3 ) ) )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1
@ ( ^ [Y3: ( $i > $o ) > $o,Y4: $i > $o] :
( ?? @ $i
@ ^ [Y5: $i] :
( ( Y3
@ ^ [Y6: $i] :
( ( Y5 != Y6 )
& ( Y4 @ Y6 ) ) )
& ( Y4 @ Y5 ) ) )
@ Y2 ) ) ) ) )
=> ( Y1 @ Y0 ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f22,f20,f21]) ).
thf(f22,plain,
( cNAT
= ( ^ [Y0: ( $i > $o ) > $o] :
( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y1 @ cZERO )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1 @ ( cSUCC @ Y2 ) ) ) ) )
=> ( Y1 @ Y0 ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f14]) ).
thf(f14,plain,
( cNAT
= ( ^ [Y0: ( $i > $o ) > $o] :
( !! @ ( ( ( $i > $o ) > $o ) > $o )
@ ^ [Y1: ( ( $i > $o ) > $o ) > $o] :
( ( ( Y1 @ cZERO )
& ( !! @ ( ( $i > $o ) > $o )
@ ^ [Y2: ( $i > $o ) > $o] :
( ( Y1 @ Y2 )
=> ( Y1 @ ( cSUCC @ Y2 ) ) ) ) )
=> ( Y1 @ Y0 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f13]) ).
thf(f13,plain,
( ( ^ [X0: ( $i > $o ) > $o] :
! [X1: ( ( $i > $o ) > $o ) > $o] :
( ( ! [X2: ( $i > $o ) > $o] :
( ( X1 @ X2 )
=> ( X1 @ ( cSUCC @ X2 ) ) )
& ( X1 @ cZERO ) )
=> ( X1 @ X0 ) ) )
= cNAT ),
inference(rectify,[status(thm)],[f3]) ).
thf(f3,axiom,
( ( ^ [X2: ( $i > $o ) > $o] :
! [X0: ( ( $i > $o ) > $o ) > $o] :
( ( ! [X1: ( $i > $o ) > $o] :
( ( X0 @ X1 )
=> ( X0 @ ( cSUCC @ X1 ) ) )
& ( X0 @ cZERO ) )
=> ( X0 @ X2 ) ) )
= cNAT ),
file('/export/starexec/sandbox2/tmp/tmp.9Dt3z1JBh5/DTF2THF_27919.p',cNAT_def) ).
thf(f24,plain,
( $true
= ( cNAT @ sK1 ) ),
inference(cnf_transformation,[status(thm)],[f19]) ).
thf(f60,plain,
( spl2_1
| spl2_2 ),
inference(avatar_split_clause,[status(thm)],[f48,f56,f52]) ).
thf(f48,plain,
( ( $true
= ( sK0 @ ( sK3 @ sK0 ) ) )
| ( $true
= ( sK0 @ sK1 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f46]) ).
thf(f46,plain,
( ( $true
= ( sK0 @ ( sK3 @ sK0 ) ) )
| ( $true = $false )
| ( $true
= ( sK0 @ sK1 ) ) ),
inference(superposition,[status(thm)],[f41,f40]) ).
thf(f40,plain,
! [X1: ( ( $i > $o ) > $o ) > $o] :
( ( ( X1
@ ^ [Y0: $i > $o] :
~ ( ?? @ $i @ Y0 ) )
= $false )
| ( $true
= ( X1 @ sK1 ) )
| ( $true
= ( X1 @ ( sK3 @ X1 ) ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f38]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEV299^5 : TPTP v9.2.1. Bugfixed v6.2.0.
% 0.12/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n019.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 12:43:24 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.19/0.46 ---- Original DTF file ---
% 0.19/0.46 thf(spec,logic,$$dhol).
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 % File : SEV299^5 : TPTP v9.2.1. Bugfixed v6.2.0.
% 0.19/0.46 % Domain : Set Theory
% 0.19/0.46 % Problem : TPS problem from TTTP-NATS-THMS
% 0.19/0.46 % Version : Especial.
% 0.19/0.46 % English :
% 0.19/0.46
% 0.19/0.46 % Refs : [Bro09] Brown (2009), Email to Geoff Sutcliffe
% 0.19/0.46 % Source : [Bro09]
% 0.19/0.46 % Names : tps_0787 [Bro09]
% 0.19/0.46
% 0.19/0.46 % Status : Theorem
% 0.19/0.46 % Rating : 0.22 v9.1.0, 0.12 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, 0.25 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0
% 0.19/0.46 % Syntax : Number of formulae : 7 ( 3 unt; 3 typ; 3 def)
% 0.19/0.46 % Number of atoms : 12 ( 4 equ; 0 cnn)
% 0.19/0.46 % Maximal formula atoms : 3 ( 3 avg)
% 0.19/0.46 % Number of connectives : 26 ( 2 ~; 0 |; 4 &; 15 @)
% 0.19/0.46 % ( 0 <=>; 5 =>; 0 <=; 0 <~>)
% 0.19/0.46 % Maximal formula depth : 8 ( 3 avg)
% 0.19/0.46 % Number of types : 2 ( 0 usr)
% 0.19/0.46 % Number of type conns : 28 ( 28 >; 0 *; 0 +; 0 <<)
% 0.19/0.46 % Number of symbols : 4 ( 3 usr; 0 con; 1-2 aty)
% 0.19/0.46 % Number of variables : 12 ( 5 ^; 5 !; 2 ?; 12 :)
% 0.19/0.46 % SPC : TH0_THM_EQU_NAR
% 0.19/0.46
% 0.19/0.46 % Comments : This problem is from the TPS library. Copyright (c) 2009 The TPS
% 0.19/0.46 % project in the Department of Mathematical Sciences at Carnegie
% 0.19/0.46 % Mellon University. Distributed under the Creative Commons copyleft
% 0.19/0.46 % license: http://creativecommons.org/licenses/by-sa/3.0/
% 0.19/0.46 % Bugfixes : v5.2.0 - Added missing type declarations.
% 0.19/0.46 % : v6.2.0 - Reordered definitions.
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 thf(cNAT_type,type,
% 0.19/0.46 cNAT: ( ( $i > $o ) > $o ) > $o ).
% 0.19/0.46
% 0.19/0.46 thf(cSUCC_type,type,
% 0.19/0.46 cSUCC: ( ( $i > $o ) > $o ) > ( $i > $o ) > $o ).
% 0.19/0.46
% 0.19/0.46 thf(cZERO_type,type,
% 0.19/0.46 cZERO: ( $i > $o ) > $o ).
% 0.19/0.46
% 0.19/0.46 thf(cZERO_def,definition,
% 0.19/0.46 ( cZERO
% 0.19/0.46 = ( ^ [Xp: $i > $o] :
% 0.19/0.46 ~ ? [Xx: $i] : ( Xp @ Xx ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cSUCC_def,definition,
% 0.19/0.46 ( cSUCC
% 0.19/0.46 = ( ^ [Xn: ( $i > $o ) > $o,Xp: $i > $o] :
% 0.19/0.46 ? [Xx: $i] :
% 0.19/0.46 ( ( Xp @ Xx )
% 0.19/0.46 & ( Xn
% 0.19/0.46 @ ^ [Xt: $i] :
% 0.19/0.46 ( ( Xt != Xx )
% 0.19/0.46 & ( Xp @ Xt ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cNAT_def,definition,
% 0.19/0.46 ( cNAT
% 0.19/0.46 = ( ^ [Xn: ( $i > $o ) > $o] :
% 0.19/0.46 ! [Xp: ( ( $i > $o ) > $o ) > $o] :
% 0.19/0.46 ( ( ( Xp @ cZERO )
% 0.19/0.46 & ! [Xx: ( $i > $o ) > $o] :
% 0.19/0.46 ( ( Xp @ Xx )
% 0.19/0.46 => ( Xp @ ( cSUCC @ Xx ) ) ) )
% 0.19/0.46 => ( Xp @ Xn ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cINDUCTION,conjecture,
% 0.19/0.46 ! [P: ( ( $i > $o ) > $o ) > $o] :
% 0.19/0.46 ( ( ( P @ cZERO )
% 0.19/0.46 & ! [X: ( $i > $o ) > $o] :
% 0.19/0.46 ( ( P @ X )
% 0.19/0.46 => ( P @ ( cSUCC @ X ) ) ) )
% 0.19/0.46 => ! [M: ( $i > $o ) > $o] :
% 0.19/0.46 ( ( cNAT @ M )
% 0.19/0.46 => ( P @ M ) ) ) ).
% 0.19/0.46
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 ------------------------
% 1.50/1.27 ---- Embedded in THF ---
% 1.50/1.27 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.50/1.27 %%% Generated on Tue Feb 24 12:43:25 EST 2026
% 1.50/1.27 %%% using '$$dhol' embedding, version 1.3.0.
% 1.50/1.27 %%% Logic specification used:
% 1.50/1.27 %%% thf(spec, logic, $$dhol).
% 1.50/1.27
% 1.50/1.27 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.9Dt3z1JBh5/DTF2DTF27919.p
% See solution above
% 1.50/1.27 ------------------------
% 1.50/1.28 ---- Cleaned THF ---
% 1.50/1.28 thf(cNAT_type,type,
% 1.50/1.28 cNAT: ( ( $i > $o ) > $o ) > $o ).
% 1.50/1.28
% 1.50/1.28 thf(cSUCC_type,type,
% 1.50/1.28 cSUCC: ( ( $i > $o ) > $o ) > ( $i > $o ) > $o ).
% 1.50/1.28
% 1.50/1.28 thf(cZERO_type,type,
% 1.50/1.28 cZERO: ( $i > $o ) > $o ).
% 1.50/1.28
% 1.50/1.28 thf(cZERO_def,definition,
% 1.50/1.28 ( cZERO
% 1.50/1.28 = ( ^ [Xp: $i > $o] :
% 1.50/1.28 ~ ? [Xx: $i] : ( Xp @ Xx ) ) ) ).
% 1.50/1.28
% 1.50/1.28 thf(cSUCC_def,definition,
% 1.50/1.28 ( cSUCC
% 1.50/1.28 = ( ^ [Xn: ( $i > $o ) > $o,Xp: $i > $o] :
% 1.50/1.28 ? [Xx: $i] :
% 1.50/1.28 ( ( Xp @ Xx )
% 1.50/1.28 & ( Xn
% 1.50/1.28 @ ^ [Xt: $i] :
% 1.50/1.28 ( ( Xt != Xx )
% 1.50/1.28 & ( Xp @ Xt ) ) ) ) ) ) ).
% 1.50/1.28
% 1.50/1.28 thf(cNAT_def,definition,
% 1.50/1.28 ( cNAT
% 1.50/1.28 = ( ^ [Xn: ( $i > $o ) > $o] :
% 1.50/1.28 ! [Xp: ( ( $i > $o ) > $o ) > $o] :
% 1.50/1.28 ( ( ( Xp @ cZERO )
% 1.50/1.28 & ! [Xx: ( $i > $o ) > $o] :
% 1.50/1.28 ( ( Xp @ Xx )
% 1.50/1.28 => ( Xp @ ( cSUCC @ Xx ) ) ) )
% 1.50/1.28 => ( Xp @ Xn ) ) ) ) ).
% 1.50/1.28
% 1.50/1.28 thf(cINDUCTION,conjecture,
% 1.50/1.28 ! [P: ( ( $i > $o ) > $o ) > $o] :
% 1.50/1.28 ( ( ( P @ cZERO )
% 1.50/1.28 & ! [X: ( $i > $o ) > $o] :
% 1.50/1.28 ( ( P @ X )
% 1.50/1.28 => ( P @ ( cSUCC @ X ) ) ) )
% 1.50/1.28 => ! [M: ( $i > $o ) > $o] :
% 1.50/1.28 ( ( cNAT @ M )
% 1.50/1.28 => ( P @ M ) ) ) ).
% 1.50/1.28 ------------------------
% 1.65/1.30 % (28078)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_27919 for (2999ds/2Mi)
% 1.65/1.30 % (28078)Instruction limit reached!
% 1.65/1.30 % (28078)------------------------------
% 1.65/1.30 % (28078)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.65/1.30 % (28078)Termination reason: Unknown
% 1.65/1.30 % (28078)Termination phase: Saturation
% 1.65/1.30
% 1.65/1.30 % (28078)Memory used [KB]: 1023
% 1.65/1.30 % (28078)Time elapsed: 0.003 s
% 1.65/1.30 % (28078)Instructions burned: 3 (million)
% 1.65/1.30 % (28078)------------------------------
% 1.65/1.30 % (28078)------------------------------
% 1.65/1.30 % (28076)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_27919 for (2999ds/27Mi)
% 1.65/1.31 % (28074)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_27919 for (2999ds/183Mi)
% 1.65/1.31 % (28079)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_27919 for (2999ds/275Mi)
% 1.65/1.31 % (28079)Refutation not found, incomplete strategy
% 1.65/1.31 % (28079)------------------------------
% 1.65/1.31 % (28079)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.65/1.31 % (28079)Termination reason: Refutation not found, incomplete strategy
% 1.65/1.31
% 1.65/1.31
% 1.65/1.31 % (28079)Memory used [KB]: 5500
% 1.65/1.31 % (28079)Time elapsed: 0.003 s
% 1.65/1.31 % (28079)Instructions burned: 3 (million)
% 1.65/1.31 % (28079)------------------------------
% 1.65/1.31 % (28079)------------------------------
% 1.65/1.31 % (28076)First to succeed.
% 1.65/1.31 % (28076)Refutation found. Thanks to Tanya!
% 1.65/1.31 % SZS status Theorem for DTF2THF_27919
% 1.65/1.31 % SZS output start Proof for DTF2THF_27919
% 1.65/1.31 thf(func_def_0, type, cNAT: (($i > $o) > $o) > $o).
% 1.65/1.31 thf(func_def_1, type, cSUCC: (($i > $o) > $o) > ($i > $o) > $o).
% 1.65/1.31 thf(func_def_2, type, cZERO: ($i > $o) > $o).
% 1.65/1.31 thf(func_def_4, type, vEPSILON: !>[X0: $tType]:((X0 > $o) > X0)).
% 1.65/1.31 thf(func_def_17, type, sK0: (($i > $o) > $o) > $o).
% 1.65/1.31 thf(func_def_18, type, sK1: ($i > $o) > $o).
% 1.65/1.31 thf(func_def_20, type, sK3: ((($i > $o) > $o) > $o) > ($i > $o) > $o).
% 1.65/1.31 thf(func_def_21, type, ph4: !>[X0: $tType]:(X0)).
% 1.65/1.31 thf(f91,plain,(
% 1.65/1.31 $false),
% 1.65/1.31 inference(avatar_sat_refutation,[status(thm)],[f60,f80,f85,f89])).
% 1.65/1.31 thf(f89,plain,(
% 1.65/1.31 ~spl2_3),
% 1.65/1.31 inference(avatar_contradiction_clause,[status(thm)],[f88])).
% 1.65/1.31 thf(f88,plain,(
% 1.65/1.31 $false | ~spl2_3),
% 1.65/1.31 inference(trivial_inequality_removal,[status(thm)],[f86])).
% 1.65/1.31 thf(f86,plain,(
% 1.65/1.31 ($true = $false) | ~spl2_3),
% 1.65/1.31 inference(backward_demodulation,[status(thm)],[f41,f78])).
% 1.65/1.31 thf(f78,plain,(
% 1.65/1.31 ((sK0 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0)))) = $false) | ~spl2_3),
% 1.65/1.31 inference(avatar_component_clause,[status(thm)],[f76])).
% 1.65/1.31 thf(f76,definition,(
% 1.65/1.31 spl2_3 <=> ((sK0 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0)))) = $false)),
% 1.65/1.31 introduced(definition,[new_symbols(naming,[spl2_3])],[avatar_definition])).
% 1.65/1.31 thf(f41,plain,(
% 1.65/1.31 ((sK0 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0)))) = $true)),
% 1.65/1.31 inference(beta_eta_normalization,[status(thm)],[f30])).
% 1.65/1.31 thf(f30,plain,(
% 1.65/1.31 ($true = (sK0 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ (^[Y1 : $i]: (Y0 @ Y1)))))))),
% 1.65/1.31 inference(definition_unfolding,[status(thm)],[f23,f20])).
% 1.65/1.31 thf(f20,plain,(
% 1.65/1.31 (cZERO = (^[Y0 : $i > $o]: (~ (?? @ $i @ (^[Y1 : $i]: (Y0 @ Y1))))))),
% 1.65/1.31 inference(cnf_transformation,[status(thm)],[f12])).
% 1.65/1.31 thf(f12,plain,(
% 1.65/1.31 (cZERO = (^[Y0 : $i > $o]: (~ (?? @ $i @ (^[Y1 : $i]: (Y0 @ Y1))))))),
% 1.65/1.31 inference(fool_elimination,[status(thm)],[f11])).
% 1.65/1.31 thf(f11,plain,(
% 1.65/1.31 ((^[X0 : $i > $o] : (~? [X1 : $i] : (X0 @ X1))) = cZERO)),
% 1.65/1.31 inference(rectify,[status(thm)],[f1])).
% 1.65/1.31 thf(f1,axiom,(
% 1.65/1.31 ((^[X0 : $i > $o] : (~? [X1 : $i] : (X0 @ X1))) = cZERO)),
% 1.65/1.31 file('/export/starexec/sandbox2/tmp/tmp.9Dt3z1JBh5/DTF2THF_27919.p',cZERO_def)).
% 1.65/1.31 thf(f23,plain,(
% 1.65/1.31 ((sK0 @ cZERO) = $true)),
% 1.65/1.31 inference(cnf_transformation,[status(thm)],[f19])).
% 1.65/1.31 thf(f19,plain,(
% 1.65/1.31 ! [X1 : ($i > $o) > $o] : (($true != (sK0 @ X1)) | ($true = (sK0 @ (cSUCC @ X1)))) & (($true != (sK0 @ sK1)) & ($true = (cNAT @ sK1))) & ((sK0 @ cZERO) = $true)),
% 1.65/1.31 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1])],[f16,f18,f17])).
% 1.65/1.31 thf(f17,plain,(
% 1.65/1.31 ? [X0 : (($i > $o) > $o) > $o] : (! [X1 : ($i > $o) > $o] : (((X0 @ X1) != $true) | ((X0 @ (cSUCC @ X1)) = $true)) & ? [X2 : ($i > $o) > $o] : (((X0 @ X2) != $true) & ($true = (cNAT @ X2))) & ((X0 @ cZERO) = $true)) => (! [X1 : ($i > $o) > $o] : (($true != (sK0 @ X1)) | ($true = (sK0 @ (cSUCC @ X1)))) & ? [X2 : ($i > $o) > $o] : (($true != (sK0 @ X2)) & ($true = (cNAT @ X2))) & ((sK0 @ cZERO) = $true))),
% 1.65/1.31 introduced(definition,[],[choice_axiom])).
% 1.65/1.31 thf(f18,plain,(
% 1.65/1.31 ? [X2 : ($i > $o) > $o] : (($true != (sK0 @ X2)) & ($true = (cNAT @ X2))) => (($true != (sK0 @ sK1)) & ($true = (cNAT @ sK1)))),
% 1.65/1.31 introduced(definition,[],[choice_axiom])).
% 1.65/1.31 thf(f16,plain,(
% 1.65/1.31 ? [X0 : (($i > $o) > $o) > $o] : (! [X1 : ($i > $o) > $o] : (((X0 @ X1) != $true) | ((X0 @ (cSUCC @ X1)) = $true)) & ? [X2 : ($i > $o) > $o] : (((X0 @ X2) != $true) & ($true = (cNAT @ X2))) & ((X0 @ cZERO) = $true))),
% 1.65/1.31 inference(flattening,[status(thm)],[f15])).
% 1.65/1.31 thf(f15,plain,(
% 1.65/1.31 ? [X0 : (($i > $o) > $o) > $o] : (? [X2 : ($i > $o) > $o] : (((X0 @ X2) != $true) & ($true = (cNAT @ X2))) & (! [X1 : ($i > $o) > $o] : (((X0 @ X1) != $true) | ((X0 @ (cSUCC @ X1)) = $true)) & ((X0 @ cZERO) = $true)))),
% 1.65/1.31 inference(ennf_transformation,[status(thm)],[f8])).
% 1.65/1.31 thf(f8,plain,(
% 1.65/1.31 ~! [X0 : (($i > $o) > $o) > $o] : ((! [X1 : ($i > $o) > $o] : (((X0 @ X1) = $true) => ((X0 @ (cSUCC @ X1)) = $true)) & ((X0 @ cZERO) = $true)) => ! [X2 : ($i > $o) > $o] : (($true = (cNAT @ X2)) => ((X0 @ X2) = $true)))),
% 1.65/1.31 inference(fool_elimination,[status(thm)],[f7])).
% 1.65/1.31 thf(f7,plain,(
% 1.65/1.31 ~! [X0 : (($i > $o) > $o) > $o] : ((! [X1 : ($i > $o) > $o] : ((X0 @ X1) => (X0 @ (cSUCC @ X1))) & (X0 @ cZERO)) => ! [X2 : ($i > $o) > $o] : ((cNAT @ X2) => (X0 @ X2)))),
% 1.65/1.31 inference(rectify,[status(thm)],[f5])).
% 1.65/1.31 thf(f5,negated_conjecture,(
% 1.65/1.31 ~! [X4 : (($i > $o) > $o) > $o] : ((! [X5 : ($i > $o) > $o] : ((X4 @ X5) => (X4 @ (cSUCC @ X5))) & (X4 @ cZERO)) => ! [X6 : ($i > $o) > $o] : ((cNAT @ X6) => (X4 @ X6)))),
% 1.65/1.31 inference(negated_conjecture,[status(cth)],[f4])).
% 1.65/1.31 thf(f4,conjecture,(
% 1.65/1.31 ! [X4 : (($i > $o) > $o) > $o] : ((! [X5 : ($i > $o) > $o] : ((X4 @ X5) => (X4 @ (cSUCC @ X5))) & (X4 @ cZERO)) => ! [X6 : ($i > $o) > $o] : ((cNAT @ X6) => (X4 @ X6)))),
% 1.65/1.31 file('/export/starexec/sandbox2/tmp/tmp.9Dt3z1JBh5/DTF2THF_27919.p',cINDUCTION)).
% 1.65/1.31 thf(f85,plain,(
% 1.65/1.31 ~spl2_1),
% 1.65/1.31 inference(avatar_contradiction_clause,[status(thm)],[f84])).
% 1.65/1.31 thf(f84,plain,(
% 1.65/1.31 $false | ~spl2_1),
% 1.65/1.31 inference(trivial_inequality_removal,[status(thm)],[f81])).
% 1.65/1.31 thf(f81,plain,(
% 1.65/1.31 ($true != $true) | ~spl2_1),
% 1.65/1.31 inference(superposition,[status(thm)],[f25,f54])).
% 1.65/1.31 thf(f54,plain,(
% 1.65/1.31 ($true = (sK0 @ sK1)) | ~spl2_1),
% 1.65/1.31 inference(avatar_component_clause,[status(thm)],[f52])).
% 1.65/1.31 thf(f52,definition,(
% 1.65/1.31 spl2_1 <=> ($true = (sK0 @ sK1))),
% 1.65/1.31 introduced(definition,[new_symbols(naming,[spl2_1])],[avatar_definition])).
% 1.65/1.31 thf(f25,plain,(
% 1.65/1.31 ($true != (sK0 @ sK1))),
% 1.65/1.31 inference(cnf_transformation,[status(thm)],[f19])).
% 1.65/1.31 thf(f80,plain,(
% 1.65/1.31 spl2_1 | spl2_3 | ~spl2_2),
% 1.65/1.31 inference(avatar_split_clause,[status(thm)],[f73,f56,f76,f52])).
% 1.65/1.31 thf(f56,definition,(
% 1.65/1.31 spl2_2 <=> ($true = (sK0 @ (sK3 @ sK0)))),
% 1.65/1.31 introduced(definition,[new_symbols(naming,[spl2_2])],[avatar_definition])).
% 1.65/1.31 thf(f73,plain,(
% 1.65/1.31 ($true = (sK0 @ sK1)) | ((sK0 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0)))) = $false) | ~spl2_2),
% 1.65/1.31 inference(trivial_inequality_removal,[status(thm)],[f69])).
% 1.65/1.31 thf(f69,plain,(
% 1.65/1.31 ($true = (sK0 @ sK1)) | ((sK0 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0)))) = $false) | ($true = $false) | ~spl2_2),
% 1.65/1.31 inference(superposition,[status(thm)],[f39,f62])).
% 1.65/1.31 thf(f62,plain,(
% 1.65/1.31 ((sK0 @ (^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((sK3 @ sK0 @ (^[Y2 : $i]: ((~ (Y1 = Y2)) & (Y0 @ Y2)))) & (Y0 @ Y1)))))) = $true) | ~spl2_2),
% 1.65/1.31 inference(trivial_inequality_removal,[status(thm)],[f61])).
% 1.65/1.31 thf(f61,plain,(
% 1.65/1.31 ($true != $true) | ((sK0 @ (^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((sK3 @ sK0 @ (^[Y2 : $i]: ((~ (Y1 = Y2)) & (Y0 @ Y2)))) & (Y0 @ Y1)))))) = $true) | ~spl2_2),
% 1.65/1.31 inference(superposition,[status(thm)],[f31,f58])).
% 1.65/1.31 thf(f58,plain,(
% 1.65/1.31 ($true = (sK0 @ (sK3 @ sK0))) | ~spl2_2),
% 1.65/1.31 inference(avatar_component_clause,[status(thm)],[f56])).
% 1.65/1.31 thf(f31,plain,(
% 1.65/1.31 ( ! [X1 : ($i > $o) > $o] : (($true != (sK0 @ X1)) | ($true = (sK0 @ (^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((X1 @ (^[Y2 : $i]: ((~ (Y1 = Y2)) & (Y0 @ Y2)))) & (Y0 @ Y1)))))))) )),
% 1.65/1.31 inference(beta_eta_normalization,[status(thm)],[f28])).
% 1.65/1.31 thf(f28,plain,(
% 1.65/1.31 ( ! [X1 : ($i > $o) > $o] : (($true = (sK0 @ ((^[Y0 : ($i > $o) > $o]: ((^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3)))) & (Y1 @ Y2))))))) @ X1))) | ($true != (sK0 @ X1))) )),
% 1.65/1.31 inference(definition_unfolding,[status(thm)],[f26,f21])).
% 1.65/1.31 thf(f21,plain,(
% 1.65/1.31 (cSUCC = (^[Y0 : ($i > $o) > $o]: ((^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3)))) & (Y1 @ Y2))))))))),
% 1.65/1.31 inference(cnf_transformation,[status(thm)],[f10])).
% 1.65/1.31 thf(f10,plain,(
% 1.65/1.31 (cSUCC = (^[Y0 : ($i > $o) > $o]: ((^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3)))) & (Y1 @ Y2))))))))),
% 1.65/1.31 inference(fool_elimination,[status(thm)],[f9])).
% 1.65/1.31 thf(f9,plain,(
% 1.65/1.31 (cSUCC = (^[X0 : ($i > $o) > $o, X1 : $i > $o] : (? [X2 : $i] : ((X1 @ X2) & (X0 @ (^[X3 : $i] : ((X1 @ X3) & (X2 != X3))))))))),
% 1.65/1.31 inference(rectify,[status(thm)],[f2])).
% 1.65/1.31 thf(f2,axiom,(
% 1.65/1.31 (cSUCC = (^[X2 : ($i > $o) > $o, X0 : $i > $o] : (? [X1 : $i] : ((X0 @ X1) & (X2 @ (^[X3 : $i] : ((X0 @ X3) & (X1 != X3))))))))),
% 1.65/1.31 file('/export/starexec/sandbox2/tmp/tmp.9Dt3z1JBh5/DTF2THF_27919.p',cSUCC_def)).
% 1.65/1.31 thf(f26,plain,(
% 1.65/1.31 ( ! [X1 : ($i > $o) > $o] : (($true != (sK0 @ X1)) | ($true = (sK0 @ (cSUCC @ X1)))) )),
% 1.65/1.31 inference(cnf_transformation,[status(thm)],[f19])).
% 1.65/1.31 thf(f39,plain,(
% 1.65/1.31 ( ! [X1 : (($i > $o) > $o) > $o] : (((X1 @ (^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((sK3 @ X1 @ (^[Y2 : $i]: ((~ (Y1 = Y2)) & (Y0 @ Y2)))) & (Y0 @ Y1)))))) = $false) | ((X1 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0)))) = $false) | ($true = (X1 @ sK1))) )),
% 1.65/1.31 inference(binary_proxy_clausification,[status(thm)],[f38])).
% 1.65/1.31 thf(f38,plain,(
% 1.65/1.31 ( ! [X1 : (($i > $o) > $o) > $o] : (((X1 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0)))) = $false) | (((X1 @ (sK3 @ X1)) => (X1 @ (^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((sK3 @ X1 @ (^[Y2 : $i]: ((~ (Y1 = Y2)) & (Y0 @ Y2)))) & (Y0 @ Y1))))))) = $false) | ($true = (X1 @ sK1))) )),
% 1.65/1.31 inference(beta_eta_normalization,[status(thm)],[f37])).
% 1.65/1.31 thf(f37,plain,(
% 1.65/1.31 ( ! [X1 : (($i > $o) > $o) > $o] : ((((^[Y0 : ($i > $o) > $o]: ((X1 @ Y0) => (X1 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3)))) & (Y1 @ Y2)))))))) @ (sK3 @ X1)) = $false) | ($true = (X1 @ sK1)) | ((X1 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0)))) = $false)) )),
% 1.65/1.31 inference(sigma_clausification,[status(thm)],[f36])).
% 1.65/1.31 thf(f36,plain,(
% 1.65/1.31 ( ! [X1 : (($i > $o) > $o) > $o] : (((X1 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0)))) = $false) | ($true = (X1 @ sK1)) | ((!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: ((X1 @ Y0) => (X1 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3)))) & (Y1 @ Y2))))))))) = $false)) )),
% 1.65/1.31 inference(binary_proxy_clausification,[status(thm)],[f35])).
% 1.65/1.31 thf(f35,plain,(
% 1.65/1.31 ( ! [X1 : (($i > $o) > $o) > $o] : (($true = (X1 @ sK1)) | (((X1 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0)))) & (!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: ((X1 @ Y0) => (X1 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3)))) & (Y1 @ Y2)))))))))) = $false)) )),
% 1.65/1.31 inference(binary_proxy_clausification,[status(thm)],[f34])).
% 1.65/1.31 thf(f34,plain,(
% 1.65/1.31 ( ! [X1 : (($i > $o) > $o) > $o] : (($true = (((X1 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0)))) & (!! @ (($i > $o) > $o) @ (^[Y0 : ($i > $o) > $o]: ((X1 @ Y0) => (X1 @ (^[Y1 : $i > $o]: (?? @ $i @ (^[Y2 : $i]: ((Y0 @ (^[Y3 : $i]: ((~ (Y2 = Y3)) & (Y1 @ Y3)))) & (Y1 @ Y2)))))))))) => (X1 @ sK1)))) )),
% 1.65/1.31 inference(beta_eta_normalization,[status(thm)],[f33])).
% 1.65/1.31 thf(f33,plain,(
% 1.65/1.31 ( ! [X1 : (($i > $o) > $o) > $o] : (($true = ((^[Y0 : (($i > $o) > $o) > $o]: (((Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))) & (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4)))) & (Y2 @ Y3)))))))))) => (Y0 @ sK1))) @ X1))) )),
% 1.65/1.31 inference(pi_clausification,[status(thm)],[f32])).
% 1.65/1.31 thf(f32,plain,(
% 1.65/1.31 ($true = (!! @ ((($i > $o) > $o) > $o) @ (^[Y0 : (($i > $o) > $o) > $o]: (((Y0 @ (^[Y1 : $i > $o]: (~ (?? @ $i @ Y1)))) & (!! @ (($i > $o) > $o) @ (^[Y1 : ($i > $o) > $o]: ((Y0 @ Y1) => (Y0 @ (^[Y2 : $i > $o]: (?? @ $i @ (^[Y3 : $i]: ((Y1 @ (^[Y4 : $i]: ((~ (Y3 = Y4)) & (Y2 @ Y4)))) & (Y2 @ Y3)))))))))) => (Y0 @ sK1)))))),
% 1.65/1.31 inference(beta_eta_normalization,[status(thm)],[f29])).
% 1.65/1.31 thf(f29,plain,(
% 1.65/1.31 (((^[Y0 : ($i > $o) > $o]: (!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ (^[Y3 : $i]: (Y2 @ Y3)))))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ ((^[Y3 : ($i > $o) > $o]: ((^[Y4 : $i > $o]: (?? @ $i @ (^[Y5 : $i]: ((Y3 @ (^[Y6 : $i]: ((~ (Y5 = Y6)) & (Y4 @ Y6)))) & (Y4 @ Y5))))))) @ Y2)))))) => (Y1 @ Y0))))) @ sK1) = $true)),
% 1.65/1.31 inference(definition_unfolding,[status(thm)],[f24,f27])).
% 1.65/1.31 thf(f27,plain,(
% 1.65/1.31 (cNAT = (^[Y0 : ($i > $o) > $o]: (!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((Y1 @ (^[Y2 : $i > $o]: (~ (?? @ $i @ (^[Y3 : $i]: (Y2 @ Y3)))))) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ ((^[Y3 : ($i > $o) > $o]: ((^[Y4 : $i > $o]: (?? @ $i @ (^[Y5 : $i]: ((Y3 @ (^[Y6 : $i]: ((~ (Y5 = Y6)) & (Y4 @ Y6)))) & (Y4 @ Y5))))))) @ Y2)))))) => (Y1 @ Y0))))))),
% 1.65/1.31 inference(definition_unfolding,[status(thm)],[f22,f20,f21])).
% 1.65/1.31 thf(f22,plain,(
% 1.65/1.31 (cNAT = (^[Y0 : ($i > $o) > $o]: (!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((Y1 @ cZERO) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (cSUCC @ Y2)))))) => (Y1 @ Y0))))))),
% 1.65/1.31 inference(cnf_transformation,[status(thm)],[f14])).
% 1.65/1.31 thf(f14,plain,(
% 1.65/1.31 (cNAT = (^[Y0 : ($i > $o) > $o]: (!! @ ((($i > $o) > $o) > $o) @ (^[Y1 : (($i > $o) > $o) > $o]: (((Y1 @ cZERO) & (!! @ (($i > $o) > $o) @ (^[Y2 : ($i > $o) > $o]: ((Y1 @ Y2) => (Y1 @ (cSUCC @ Y2)))))) => (Y1 @ Y0))))))),
% 1.65/1.31 inference(fool_elimination,[status(thm)],[f13])).
% 1.65/1.31 thf(f13,plain,(
% 1.65/1.31 ((^[X0 : ($i > $o) > $o] : (! [X1 : (($i > $o) > $o) > $o] : ((! [X2 : ($i > $o) > $o] : ((X1 @ X2) => (X1 @ (cSUCC @ X2))) & (X1 @ cZERO)) => (X1 @ X0)))) = cNAT)),
% 1.65/1.31 inference(rectify,[status(thm)],[f3])).
% 1.65/1.31 thf(f3,axiom,(
% 1.65/1.31 ((^[X2 : ($i > $o) > $o] : (! [X0 : (($i > $o) > $o) > $o] : ((! [X1 : ($i > $o) > $o] : ((X0 @ X1) => (X0 @ (cSUCC @ X1))) & (X0 @ cZERO)) => (X0 @ X2)))) = cNAT)),
% 1.65/1.31 file('/export/starexec/sandbox2/tmp/tmp.9Dt3z1JBh5/DTF2THF_27919.p',cNAT_def)).
% 1.65/1.31 thf(f24,plain,(
% 1.65/1.31 ($true = (cNAT @ sK1))),
% 1.65/1.31 inference(cnf_transformation,[status(thm)],[f19])).
% 1.65/1.31 thf(f60,plain,(
% 1.65/1.31 spl2_1 | spl2_2),
% 1.65/1.31 inference(avatar_split_clause,[status(thm)],[f48,f56,f52])).
% 1.65/1.31 thf(f48,plain,(
% 1.65/1.31 ($true = (sK0 @ (sK3 @ sK0))) | ($true = (sK0 @ sK1))),
% 1.65/1.31 inference(trivial_inequality_removal,[status(thm)],[f46])).
% 1.65/1.31 thf(f46,plain,(
% 1.65/1.31 ($true = (sK0 @ (sK3 @ sK0))) | ($true = $false) | ($true = (sK0 @ sK1))),
% 1.65/1.31 inference(superposition,[status(thm)],[f41,f40])).
% 1.65/1.31 thf(f40,plain,(
% 1.65/1.31 ( ! [X1 : (($i > $o) > $o) > $o] : (((X1 @ (^[Y0 : $i > $o]: (~ (?? @ $i @ Y0)))) = $false) | ($true = (X1 @ sK1)) | ($true = (X1 @ (sK3 @ X1)))) )),
% 1.65/1.31 inference(binary_proxy_clausification,[status(thm)],[f38])).
% 1.65/1.31 % SZS output end Proof for DTF2THF_27919
% 1.65/1.31 % (28076)------------------------------
% 1.65/1.31 % (28076)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.65/1.31 % (28076)Termination reason: Refutation
% 1.65/1.31
% 1.65/1.31 % (28076)Memory used [KB]: 5628
% 1.65/1.31 % (28076)Time elapsed: 0.010 s
% 1.65/1.31 % (28076)Instructions burned: 10 (million)
% 1.65/1.31 % (28076)------------------------------
% 1.65/1.31 % (28076)------------------------------
% 1.65/1.31 % (28073)Success in time 0.024 s
%------------------------------------------------------------------------------