%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SYO113^5 : TPTP v9.2.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n001.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 09:53:24 AM UTC 2026
% Result : Theorem 1.63s 1.32s
% Output : Refutation 1.63s
% Verified :
% SZS Type : Refutation
% Derivation depth : 20
% Number of leaves : 10
% Syntax : Number of formulae : 66 ( 7 unt; 0 typ; 7 def)
% Number of atoms : 845 ( 213 equ; 0 cnn)
% Maximal formula atoms : 22 ( 12 avg)
% Number of connectives : 794 ( 121 ~; 123 |; 106 &; 409 @)
% ( 13 <=>; 22 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 5 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 17 ( 14 usr; 12 con; 0-2 aty)
% Number of variables : 72 ( 0 ^; 60 !; 12 ?; 72 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
cNUMBER: $i > $o ).
thf(func_def_1,type,
cODD: $i > $o ).
thf(func_def_2,type,
cEVEN: $i > $o ).
thf(func_def_3,type,
cS: $i > $i ).
thf(f98,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f56,f57,f58,f69,f75,f78,f80,f97]) ).
thf(f97,plain,
( ~ spl2_5
| spl2_7 ),
inference(avatar_contradiction_clause,[status(thm)],[f96]) ).
thf(f96,plain,
( $false
| ~ spl2_5
| spl2_7 ),
inference(subsumption_resolution,[status(thm)],[f95,f93]) ).
thf(f93,plain,
( ( $true
= ( cODD @ sK0 ) )
| ~ spl2_5
| spl2_7 ),
inference(subsumption_resolution,[status(thm)],[f92,f47]) ).
thf(f47,plain,
( ( ( cNUMBER @ sK0 )
= $true )
| ~ spl2_5 ),
inference(avatar_component_clause,[status(thm)],[f45]) ).
thf(f45,definition,
( spl2_5
<=> ( ( cNUMBER @ sK0 )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl2_5])],[avatar_definition]) ).
thf(f92,plain,
( ( $true
= ( cODD @ sK0 ) )
| ( ( cNUMBER @ sK0 )
!= $true )
| spl2_7 ),
inference(trivial_inequality_removal,[status(thm)],[f91]) ).
thf(f91,plain,
( ( ( cNUMBER @ sK0 )
!= $true )
| ( $true != $true )
| ( $true
= ( cODD @ sK0 ) )
| spl2_7 ),
inference(superposition,[status(thm)],[f90,f25]) ).
thf(f25,plain,
! [X0: $i] :
( ( ( cEVEN @ X0 )
= $true )
| ( ( cNUMBER @ X0 )
!= $true )
| ( ( cODD @ X0 )
= $true ) ),
inference(cnf_transformation,[status(thm)],[f13]) ).
thf(f13,plain,
( ! [X0: $i] :
( ( ( ( cNUMBER @ X0 )
= $true )
| ( ( ( cODD @ X0 )
!= $true )
& ( ( cEVEN @ X0 )
!= $true ) ) )
& ( ( ( cODD @ X0 )
= $true )
| ( ( cEVEN @ X0 )
= $true )
| ( ( cNUMBER @ X0 )
!= $true ) ) )
& ( ( cEVEN @ c0 )
= $true )
& ( ( ( cNUMBER @ c0 )
!= $true )
| ! [X1: $i] :
( ( ( cNUMBER @ X1 )
= $true )
& ( ( cNUMBER @ ( cS @ X1 ) )
= $true ) )
| ( ( ( cNUMBER @ ( cS @ sK0 ) )
= $true )
& ( ( cNUMBER @ sK0 )
= $true )
& ( ( ( cNUMBER @ ( cS @ sK0 ) )
!= $true )
| ( $true
!= ( cNUMBER @ ( cS @ ( cS @ sK0 ) ) ) ) ) )
| ( ( cNUMBER @ ( cS @ c0 ) )
!= $true ) )
& ( ( cNUMBER @ sK1 )
!= $true )
& ( ( cODD @ ( cS @ c0 ) )
= $true )
& ! [X4: $i] :
( ( $true
!= ( cODD @ X4 ) )
| ( ( cODD @ ( cS @ ( cS @ X4 ) ) )
= $true ) )
& ! [X5: $i] :
( ( ( cEVEN @ X5 )
!= $true )
| ( ( cEVEN @ ( cS @ ( cS @ X5 ) ) )
= $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1])],[f10,f12,f11]) ).
thf(f11,plain,
( ? [X2: $i] :
( ( ( cNUMBER @ ( cS @ X2 ) )
= $true )
& ( ( cNUMBER @ X2 )
= $true )
& ( ( ( cNUMBER @ ( cS @ X2 ) )
!= $true )
| ( ( cNUMBER @ ( cS @ ( cS @ X2 ) ) )
!= $true ) ) )
=> ( ( ( cNUMBER @ ( cS @ sK0 ) )
= $true )
& ( ( cNUMBER @ sK0 )
= $true )
& ( ( ( cNUMBER @ ( cS @ sK0 ) )
!= $true )
| ( $true
!= ( cNUMBER @ ( cS @ ( cS @ sK0 ) ) ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f12,plain,
( ? [X3: $i] :
( $true
!= ( cNUMBER @ X3 ) )
=> ( ( cNUMBER @ sK1 )
!= $true ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f10,plain,
( ! [X0: $i] :
( ( ( ( cNUMBER @ X0 )
= $true )
| ( ( ( cODD @ X0 )
!= $true )
& ( ( cEVEN @ X0 )
!= $true ) ) )
& ( ( ( cODD @ X0 )
= $true )
| ( ( cEVEN @ X0 )
= $true )
| ( ( cNUMBER @ X0 )
!= $true ) ) )
& ( ( cEVEN @ c0 )
= $true )
& ( ( ( cNUMBER @ c0 )
!= $true )
| ! [X1: $i] :
( ( ( cNUMBER @ X1 )
= $true )
& ( ( cNUMBER @ ( cS @ X1 ) )
= $true ) )
| ? [X2: $i] :
( ( ( cNUMBER @ ( cS @ X2 ) )
= $true )
& ( ( cNUMBER @ X2 )
= $true )
& ( ( ( cNUMBER @ ( cS @ X2 ) )
!= $true )
| ( ( cNUMBER @ ( cS @ ( cS @ X2 ) ) )
!= $true ) ) )
| ( ( cNUMBER @ ( cS @ c0 ) )
!= $true ) )
& ? [X3: $i] :
( $true
!= ( cNUMBER @ X3 ) )
& ( ( cODD @ ( cS @ c0 ) )
= $true )
& ! [X4: $i] :
( ( $true
!= ( cODD @ X4 ) )
| ( ( cODD @ ( cS @ ( cS @ X4 ) ) )
= $true ) )
& ! [X5: $i] :
( ( ( cEVEN @ X5 )
!= $true )
| ( ( cEVEN @ ( cS @ ( cS @ X5 ) ) )
= $true ) ) ),
inference(rectify,[status(thm)],[f9]) ).
thf(f9,plain,
( ! [X1: $i] :
( ( ( ( cNUMBER @ X1 )
= $true )
| ( ( ( cODD @ X1 )
!= $true )
& ( ( cEVEN @ X1 )
!= $true ) ) )
& ( ( ( cODD @ X1 )
= $true )
| ( ( cEVEN @ X1 )
= $true )
| ( ( cNUMBER @ X1 )
!= $true ) ) )
& ( ( cEVEN @ c0 )
= $true )
& ( ( ( cNUMBER @ c0 )
!= $true )
| ! [X3: $i] :
( ( $true
= ( cNUMBER @ X3 ) )
& ( ( cNUMBER @ ( cS @ X3 ) )
= $true ) )
| ? [X2: $i] :
( ( ( cNUMBER @ ( cS @ X2 ) )
= $true )
& ( ( cNUMBER @ X2 )
= $true )
& ( ( ( cNUMBER @ ( cS @ X2 ) )
!= $true )
| ( ( cNUMBER @ ( cS @ ( cS @ X2 ) ) )
!= $true ) ) )
| ( ( cNUMBER @ ( cS @ c0 ) )
!= $true ) )
& ? [X5: $i] :
( $true
!= ( cNUMBER @ X5 ) )
& ( ( cODD @ ( cS @ c0 ) )
= $true )
& ! [X4: $i] :
( ( $true
!= ( cODD @ X4 ) )
| ( ( cODD @ ( cS @ ( cS @ X4 ) ) )
= $true ) )
& ! [X0: $i] :
( ( ( cEVEN @ X0 )
!= $true )
| ( ( cEVEN @ ( cS @ ( cS @ X0 ) ) )
= $true ) ) ),
inference(flattening,[status(thm)],[f8]) ).
thf(f8,plain,
( ! [X1: $i] :
( ( ( ( cNUMBER @ X1 )
= $true )
| ( ( ( cODD @ X1 )
!= $true )
& ( ( cEVEN @ X1 )
!= $true ) ) )
& ( ( ( cODD @ X1 )
= $true )
| ( ( cEVEN @ X1 )
= $true )
| ( ( cNUMBER @ X1 )
!= $true ) ) )
& ( ( cEVEN @ c0 )
= $true )
& ( ( ( cNUMBER @ c0 )
!= $true )
| ! [X3: $i] :
( ( $true
= ( cNUMBER @ X3 ) )
& ( ( cNUMBER @ ( cS @ X3 ) )
= $true ) )
| ? [X2: $i] :
( ( ( cNUMBER @ ( cS @ X2 ) )
= $true )
& ( ( cNUMBER @ X2 )
= $true )
& ( ( ( cNUMBER @ ( cS @ X2 ) )
!= $true )
| ( ( cNUMBER @ ( cS @ ( cS @ X2 ) ) )
!= $true ) ) )
| ( ( cNUMBER @ ( cS @ c0 ) )
!= $true ) )
& ? [X5: $i] :
( $true
!= ( cNUMBER @ X5 ) )
& ( ( cODD @ ( cS @ c0 ) )
= $true )
& ! [X4: $i] :
( ( $true
!= ( cODD @ X4 ) )
| ( ( cODD @ ( cS @ ( cS @ X4 ) ) )
= $true ) )
& ! [X0: $i] :
( ( ( cEVEN @ X0 )
!= $true )
| ( ( cEVEN @ ( cS @ ( cS @ X0 ) ) )
= $true ) ) ),
inference(nnf_transformation,[status(thm)],[f7]) ).
thf(f7,plain,
( ! [X1: $i] :
( ( ( cNUMBER @ X1 )
= $true )
<=> ( ( ( cODD @ X1 )
= $true )
| ( ( cEVEN @ X1 )
= $true ) ) )
& ( ( cEVEN @ c0 )
= $true )
& ( ( ( cNUMBER @ c0 )
!= $true )
| ! [X3: $i] :
( ( $true
= ( cNUMBER @ X3 ) )
& ( ( cNUMBER @ ( cS @ X3 ) )
= $true ) )
| ? [X2: $i] :
( ( ( cNUMBER @ ( cS @ X2 ) )
= $true )
& ( ( cNUMBER @ X2 )
= $true )
& ( ( ( cNUMBER @ ( cS @ X2 ) )
!= $true )
| ( ( cNUMBER @ ( cS @ ( cS @ X2 ) ) )
!= $true ) ) )
| ( ( cNUMBER @ ( cS @ c0 ) )
!= $true ) )
& ? [X5: $i] :
( $true
!= ( cNUMBER @ X5 ) )
& ( ( cODD @ ( cS @ c0 ) )
= $true )
& ! [X4: $i] :
( ( $true
!= ( cODD @ X4 ) )
| ( ( cODD @ ( cS @ ( cS @ X4 ) ) )
= $true ) )
& ! [X0: $i] :
( ( ( cEVEN @ X0 )
!= $true )
| ( ( cEVEN @ ( cS @ ( cS @ X0 ) ) )
= $true ) ) ),
inference(flattening,[status(thm)],[f6]) ).
thf(f6,plain,
( ? [X5: $i] :
( $true
!= ( cNUMBER @ X5 ) )
& ! [X4: $i] :
( ( $true
!= ( cODD @ X4 ) )
| ( ( cODD @ ( cS @ ( cS @ X4 ) ) )
= $true ) )
& ( ! [X3: $i] :
( ( $true
= ( cNUMBER @ X3 ) )
& ( ( cNUMBER @ ( cS @ X3 ) )
= $true ) )
| ( ( cNUMBER @ ( cS @ c0 ) )
!= $true )
| ? [X2: $i] :
( ( ( ( cNUMBER @ ( cS @ X2 ) )
!= $true )
| ( ( cNUMBER @ ( cS @ ( cS @ X2 ) ) )
!= $true ) )
& ( ( cNUMBER @ ( cS @ X2 ) )
= $true )
& ( ( cNUMBER @ X2 )
= $true ) )
| ( ( cNUMBER @ c0 )
!= $true ) )
& ! [X1: $i] :
( ( ( cNUMBER @ X1 )
= $true )
<=> ( ( ( cODD @ X1 )
= $true )
| ( ( cEVEN @ X1 )
= $true ) ) )
& ( ( cEVEN @ c0 )
= $true )
& ( ( cODD @ ( cS @ c0 ) )
= $true )
& ! [X0: $i] :
( ( ( cEVEN @ X0 )
!= $true )
| ( ( cEVEN @ ( cS @ ( cS @ X0 ) ) )
= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f5]) ).
thf(f5,plain,
~ ( ( ! [X4: $i] :
( ( $true
= ( cODD @ X4 ) )
=> ( ( cODD @ ( cS @ ( cS @ X4 ) ) )
= $true ) )
& ( ( ( ( cNUMBER @ ( cS @ c0 ) )
= $true )
& ! [X2: $i] :
( ( ( ( cNUMBER @ ( cS @ X2 ) )
= $true )
& ( ( cNUMBER @ X2 )
= $true ) )
=> ( ( ( cNUMBER @ ( cS @ ( cS @ X2 ) ) )
= $true )
& ( ( cNUMBER @ ( cS @ X2 ) )
= $true ) ) )
& ( ( cNUMBER @ c0 )
= $true ) )
=> ! [X3: $i] :
( ( $true
= ( cNUMBER @ X3 ) )
& ( ( cNUMBER @ ( cS @ X3 ) )
= $true ) ) )
& ! [X1: $i] :
( ( ( cNUMBER @ X1 )
= $true )
<=> ( ( ( cODD @ X1 )
= $true )
| ( ( cEVEN @ X1 )
= $true ) ) )
& ( ( cEVEN @ c0 )
= $true )
& ( ( cODD @ ( cS @ c0 ) )
= $true )
& ! [X0: $i] :
( ( ( cEVEN @ X0 )
= $true )
=> ( ( cEVEN @ ( cS @ ( cS @ X0 ) ) )
= $true ) ) )
=> ! [X5: $i] :
( $true
= ( cNUMBER @ X5 ) ) ),
inference(fool_elimination,[status(thm)],[f4]) ).
thf(f4,plain,
~ ( ( ( cEVEN @ c0 )
& ( cODD @ ( cS @ c0 ) )
& ! [X0: $i] :
( ( cEVEN @ X0 )
=> ( cEVEN @ ( cS @ ( cS @ X0 ) ) ) )
& ! [X1: $i] :
( ( cNUMBER @ X1 )
<=> ( ( cEVEN @ X1 )
| ( cODD @ X1 ) ) )
& ( ( ( cNUMBER @ ( cS @ c0 ) )
& ! [X2: $i] :
( ( ( cNUMBER @ ( cS @ X2 ) )
& ( cNUMBER @ X2 ) )
=> ( ( cNUMBER @ ( cS @ ( cS @ X2 ) ) )
& ( cNUMBER @ ( cS @ X2 ) ) ) )
& ( cNUMBER @ c0 ) )
=> ! [X3: $i] :
( ( cNUMBER @ X3 )
& ( cNUMBER @ ( cS @ X3 ) ) ) )
& ! [X4: $i] :
( ( cODD @ X4 )
=> ( cODD @ ( cS @ ( cS @ X4 ) ) ) ) )
=> ! [X5: $i] : ( cNUMBER @ X5 ) ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,negated_conjecture,
~ ( ( ( cEVEN @ c0 )
& ( cODD @ ( cS @ c0 ) )
& ! [X0: $i] :
( ( cEVEN @ X0 )
=> ( cEVEN @ ( cS @ ( cS @ X0 ) ) ) )
& ! [X0: $i] :
( ( cNUMBER @ X0 )
<=> ( ( cEVEN @ X0 )
| ( cODD @ X0 ) ) )
& ( ( ( cNUMBER @ ( cS @ c0 ) )
& ! [X1: $i] :
( ( ( cNUMBER @ ( cS @ X1 ) )
& ( cNUMBER @ X1 ) )
=> ( ( cNUMBER @ ( cS @ ( cS @ X1 ) ) )
& ( cNUMBER @ ( cS @ X1 ) ) ) )
& ( cNUMBER @ c0 ) )
=> ! [X1: $i] :
( ( cNUMBER @ X1 )
& ( cNUMBER @ ( cS @ X1 ) ) ) )
& ! [X0: $i] :
( ( cODD @ X0 )
=> ( cODD @ ( cS @ ( cS @ X0 ) ) ) ) )
=> ! [X0: $i] : ( cNUMBER @ X0 ) ),
inference(negated_conjecture,[status(cth)],[f1]) ).
thf(f1,conjecture,
( ( ( cEVEN @ c0 )
& ( cODD @ ( cS @ c0 ) )
& ! [X0: $i] :
( ( cEVEN @ X0 )
=> ( cEVEN @ ( cS @ ( cS @ X0 ) ) ) )
& ! [X0: $i] :
( ( cNUMBER @ X0 )
<=> ( ( cEVEN @ X0 )
| ( cODD @ X0 ) ) )
& ( ( ( cNUMBER @ ( cS @ c0 ) )
& ! [X1: $i] :
( ( ( cNUMBER @ ( cS @ X1 ) )
& ( cNUMBER @ X1 ) )
=> ( ( cNUMBER @ ( cS @ ( cS @ X1 ) ) )
& ( cNUMBER @ ( cS @ X1 ) ) ) )
& ( cNUMBER @ c0 ) )
=> ! [X1: $i] :
( ( cNUMBER @ X1 )
& ( cNUMBER @ ( cS @ X1 ) ) ) )
& ! [X0: $i] :
( ( cODD @ X0 )
=> ( cODD @ ( cS @ ( cS @ X0 ) ) ) ) )
=> ! [X0: $i] : ( cNUMBER @ X0 ) ),
file('/export/starexec/sandbox/tmp/tmp.i1tU7BLO3J/DTF2THF_5105.p',cTHM350) ).
thf(f90,plain,
( ( ( cEVEN @ sK0 )
!= $true )
| spl2_7 ),
inference(trivial_inequality_removal,[status(thm)],[f89]) ).
thf(f89,plain,
( ( ( cEVEN @ sK0 )
!= $true )
| ( $true != $true )
| spl2_7 ),
inference(superposition,[status(thm)],[f84,f14]) ).
thf(f14,plain,
! [X5: $i] :
( ( ( cEVEN @ ( cS @ ( cS @ X5 ) ) )
= $true )
| ( ( cEVEN @ X5 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f13]) ).
thf(f84,plain,
( ( $true
!= ( cEVEN @ ( cS @ ( cS @ sK0 ) ) ) )
| spl2_7 ),
inference(trivial_inequality_removal,[status(thm)],[f82]) ).
thf(f82,plain,
( ( $true != $true )
| ( $true
!= ( cEVEN @ ( cS @ ( cS @ sK0 ) ) ) )
| spl2_7 ),
inference(superposition,[status(thm)],[f55,f26]) ).
thf(f26,plain,
! [X0: $i] :
( ( ( cNUMBER @ X0 )
= $true )
| ( ( cEVEN @ X0 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f13]) ).
thf(f55,plain,
( ( $true
!= ( cNUMBER @ ( cS @ ( cS @ sK0 ) ) ) )
| spl2_7 ),
inference(avatar_component_clause,[status(thm)],[f53]) ).
thf(f53,definition,
( spl2_7
<=> ( $true
= ( cNUMBER @ ( cS @ ( cS @ sK0 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[spl2_7])],[avatar_definition]) ).
thf(f95,plain,
( ( $true
!= ( cODD @ sK0 ) )
| spl2_7 ),
inference(trivial_inequality_removal,[status(thm)],[f94]) ).
thf(f94,plain,
( ( $true
!= ( cODD @ sK0 ) )
| ( $true != $true )
| spl2_7 ),
inference(superposition,[status(thm)],[f83,f15]) ).
thf(f15,plain,
! [X4: $i] :
( ( ( cODD @ ( cS @ ( cS @ X4 ) ) )
= $true )
| ( $true
!= ( cODD @ X4 ) ) ),
inference(cnf_transformation,[status(thm)],[f13]) ).
thf(f83,plain,
( ( ( cODD @ ( cS @ ( cS @ sK0 ) ) )
!= $true )
| spl2_7 ),
inference(trivial_inequality_removal,[status(thm)],[f81]) ).
thf(f81,plain,
( ( $true != $true )
| ( ( cODD @ ( cS @ ( cS @ sK0 ) ) )
!= $true )
| spl2_7 ),
inference(superposition,[status(thm)],[f55,f27]) ).
thf(f27,plain,
! [X0: $i] :
( ( ( cNUMBER @ X0 )
= $true )
| ( ( cODD @ X0 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f13]) ).
thf(f80,plain,
( ~ spl2_2
| spl2_7 ),
inference(avatar_contradiction_clause,[status(thm)],[f79]) ).
thf(f79,plain,
( $false
| ~ spl2_2
| spl2_7 ),
inference(subsumption_resolution,[status(thm)],[f55,f34]) ).
thf(f34,plain,
( ! [X1: $i] :
( ( cNUMBER @ ( cS @ X1 ) )
= $true )
| ~ spl2_2 ),
inference(avatar_component_clause,[status(thm)],[f33]) ).
thf(f33,definition,
( spl2_2
<=> ! [X1: $i] :
( ( cNUMBER @ ( cS @ X1 ) )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl2_2])],[avatar_definition]) ).
thf(f78,plain,
~ spl2_6,
inference(avatar_contradiction_clause,[status(thm)],[f77]) ).
thf(f77,plain,
( $false
| ~ spl2_6 ),
inference(trivial_inequality_removal,[status(thm)],[f76]) ).
thf(f76,plain,
( ( $true != $true )
| ~ spl2_6 ),
inference(superposition,[status(thm)],[f17,f50]) ).
thf(f50,plain,
( ! [X1: $i] :
( ( cNUMBER @ X1 )
= $true )
| ~ spl2_6 ),
inference(avatar_component_clause,[status(thm)],[f49]) ).
thf(f49,definition,
( spl2_6
<=> ! [X1: $i] :
( ( cNUMBER @ X1 )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl2_6])],[avatar_definition]) ).
thf(f17,plain,
( ( cNUMBER @ sK1 )
!= $true ),
inference(cnf_transformation,[status(thm)],[f13]) ).
thf(f75,plain,
spl2_4,
inference(avatar_contradiction_clause,[status(thm)],[f74]) ).
thf(f74,plain,
( $false
| spl2_4 ),
inference(subsumption_resolution,[status(thm)],[f72,f24]) ).
thf(f24,plain,
( ( cEVEN @ c0 )
= $true ),
inference(cnf_transformation,[status(thm)],[f13]) ).
thf(f72,plain,
( ( ( cEVEN @ c0 )
!= $true )
| spl2_4 ),
inference(trivial_inequality_removal,[status(thm)],[f71]) ).
thf(f71,plain,
( ( ( cEVEN @ c0 )
!= $true )
| ( $true != $true )
| spl2_4 ),
inference(superposition,[status(thm)],[f42,f26]) ).
thf(f42,plain,
( ( ( cNUMBER @ c0 )
!= $true )
| spl2_4 ),
inference(avatar_component_clause,[status(thm)],[f40]) ).
thf(f40,definition,
( spl2_4
<=> ( ( cNUMBER @ c0 )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl2_4])],[avatar_definition]) ).
thf(f69,plain,
spl2_1,
inference(avatar_contradiction_clause,[status(thm)],[f68]) ).
thf(f68,plain,
( $false
| spl2_1 ),
inference(subsumption_resolution,[status(thm)],[f67,f16]) ).
thf(f16,plain,
( ( cODD @ ( cS @ c0 ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f13]) ).
thf(f67,plain,
( ( ( cODD @ ( cS @ c0 ) )
!= $true )
| spl2_1 ),
inference(trivial_inequality_removal,[status(thm)],[f65]) ).
thf(f65,plain,
( ( $true != $true )
| ( ( cODD @ ( cS @ c0 ) )
!= $true )
| spl2_1 ),
inference(superposition,[status(thm)],[f31,f27]) ).
thf(f31,plain,
( ( ( cNUMBER @ ( cS @ c0 ) )
!= $true )
| spl2_1 ),
inference(avatar_component_clause,[status(thm)],[f29]) ).
thf(f29,definition,
( spl2_1
<=> ( ( cNUMBER @ ( cS @ c0 ) )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl2_1])],[avatar_definition]) ).
thf(f58,plain,
( spl2_5
| ~ spl2_1
| spl2_2
| ~ spl2_4 ),
inference(avatar_split_clause,[status(thm)],[f19,f40,f33,f29,f45]) ).
thf(f19,plain,
! [X1: $i] :
( ( ( cNUMBER @ c0 )
!= $true )
| ( ( cNUMBER @ sK0 )
= $true )
| ( ( cNUMBER @ ( cS @ X1 ) )
= $true )
| ( ( cNUMBER @ ( cS @ c0 ) )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f13]) ).
thf(f57,plain,
( spl2_3
| ~ spl2_4
| spl2_6
| ~ spl2_1 ),
inference(avatar_split_clause,[status(thm)],[f23,f29,f49,f40,f36]) ).
thf(f36,definition,
( spl2_3
<=> ( ( cNUMBER @ ( cS @ sK0 ) )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl2_3])],[avatar_definition]) ).
thf(f23,plain,
! [X1: $i] :
( ( ( cNUMBER @ c0 )
!= $true )
| ( ( cNUMBER @ X1 )
= $true )
| ( ( cNUMBER @ ( cS @ c0 ) )
!= $true )
| ( ( cNUMBER @ ( cS @ sK0 ) )
= $true ) ),
inference(cnf_transformation,[status(thm)],[f13]) ).
thf(f56,plain,
( ~ spl2_7
| ~ spl2_4
| ~ spl2_3
| ~ spl2_1
| spl2_6 ),
inference(avatar_split_clause,[status(thm)],[f21,f49,f29,f36,f40,f53]) ).
thf(f21,plain,
! [X1: $i] :
( ( ( cNUMBER @ ( cS @ sK0 ) )
!= $true )
| ( $true
!= ( cNUMBER @ ( cS @ ( cS @ sK0 ) ) ) )
| ( ( cNUMBER @ X1 )
= $true )
| ( ( cNUMBER @ c0 )
!= $true )
| ( ( cNUMBER @ ( cS @ c0 ) )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f13]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SYO113^5 : TPTP v9.2.1. Released v4.0.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.18/0.34 % Computer : n001.cluster.edu
% 0.18/0.34 % Model : x86_64 x86_64
% 0.18/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.18/0.34 % Memory : 8042.1875MB
% 0.18/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.18/0.34 % CPULimit : 300
% 0.18/0.34 % WCLimit : 300
% 0.18/0.34 % DateTime : Tue Feb 24 14:25:20 EST 2026
% 0.18/0.34 % CPUTime :
% 0.18/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.28/0.46 ---- Original DTF file ---
% 0.28/0.46 thf(spec,logic,$$dhol).
% 0.28/0.46 %------------------------------------------------------------------------------
% 0.28/0.46 % File : SYO113^5 : TPTP v9.2.1. Released v4.0.0.
% 0.28/0.46 % Domain : Syntactic
% 0.28/0.46 % Problem : TPS problem THM350
% 0.28/0.46 % Version : Especial.
% 0.28/0.46 % English :
% 0.28/0.46
% 0.28/0.46 % Refs : [Bro09] Brown (2009), Email to Geoff Sutcliffe
% 0.28/0.46 % Source : [Bro09]
% 0.28/0.46 % Names : tps_0484 [Bro09]
% 0.28/0.46 % : THM350 [TPS]
% 0.28/0.46
% 0.28/0.46 % Status : Theorem
% 0.28/0.46 % Rating : 0.17 v9.1.0, 0.12 v9.0.0, 0.17 v8.2.0, 0.18 v8.1.0, 0.17 v7.4.0, 0.22 v7.3.0, 0.20 v7.2.0, 0.25 v7.1.0, 0.29 v7.0.0, 0.25 v6.4.0, 0.29 v6.3.0, 0.33 v6.0.0, 0.17 v5.5.0, 0.00 v5.1.0, 0.25 v5.0.0, 0.00 v4.0.0
% 0.28/0.46 % Syntax : Number of formulae : 6 ( 0 unt; 5 typ; 0 def)
% 0.28/0.46 % Number of atoms : 18 ( 0 equ; 0 cnn)
% 0.28/0.46 % Maximal formula atoms : 18 ( 18 avg)
% 0.28/0.46 % Number of connectives : 46 ( 0 ~; 1 |; 10 &; 29 @)
% 0.28/0.46 % ( 1 <=>; 5 =>; 0 <=; 0 <~>)
% 0.28/0.46 % Maximal formula depth : 16 ( 16 avg)
% 0.28/0.46 % Number of types : 2 ( 0 usr)
% 0.28/0.46 % Number of type conns : 4 ( 4 >; 0 *; 0 +; 0 <<)
% 0.28/0.46 % Number of symbols : 5 ( 5 usr; 1 con; 0-1 aty)
% 0.28/0.46 % Number of variables : 6 ( 0 ^; 6 !; 0 ?; 6 :)
% 0.28/0.46 % SPC : TH0_THM_NEQ_NAR
% 0.28/0.46
% 0.28/0.46 % Comments : This problem is from the TPS library. Copyright (c) 2009 The TPS
% 0.28/0.46 % project in the Department of Mathematical Sciences at Carnegie
% 0.28/0.46 % Mellon University. Distributed under the Creative Commons copyleft
% 0.28/0.46 % license: http://creativecommons.org/licenses/by-sa/3.0/
% 0.28/0.46 %------------------------------------------------------------------------------
% 0.28/0.46 thf(cNUMBER,type,
% 0.28/0.46 cNUMBER: $i > $o ).
% 0.28/0.46
% 0.28/0.46 thf(cODD,type,
% 0.28/0.46 cODD: $i > $o ).
% 0.28/0.46
% 0.28/0.46 thf(cEVEN,type,
% 0.28/0.46 cEVEN: $i > $o ).
% 0.28/0.46
% 0.28/0.46 thf(cS,type,
% 0.28/0.46 cS: $i > $i ).
% 0.28/0.46
% 0.28/0.46 thf(c0,type,
% 0.28/0.46 c0: $i ).
% 0.28/0.46
% 0.28/0.46 thf(cTHM350,conjecture,
% 0.28/0.46 ( ( ( cEVEN @ c0 )
% 0.28/0.46 & ! [Xn: $i] :
% 0.28/0.46 ( ( cEVEN @ Xn )
% 0.28/0.46 => ( cEVEN @ ( cS @ ( cS @ Xn ) ) ) )
% 0.28/0.46 & ( cODD @ ( cS @ c0 ) )
% 0.28/0.46 & ! [Xn: $i] :
% 0.28/0.46 ( ( cODD @ Xn )
% 0.28/0.46 => ( cODD @ ( cS @ ( cS @ Xn ) ) ) )
% 0.28/0.46 & ( ( ( cNUMBER @ c0 )
% 0.28/0.46 & ( cNUMBER @ ( cS @ c0 ) )
% 0.28/0.46 & ! [Xx: $i] :
% 0.28/0.46 ( ( ( cNUMBER @ Xx )
% 0.28/0.46 & ( cNUMBER @ ( cS @ Xx ) ) )
% 0.28/0.46 => ( ( cNUMBER @ ( cS @ Xx ) )
% 0.28/0.46 & ( cNUMBER @ ( cS @ ( cS @ Xx ) ) ) ) ) )
% 0.28/0.46 => ! [Xx: $i] :
% 0.28/0.46 ( ( cNUMBER @ Xx )
% 0.28/0.46 & ( cNUMBER @ ( cS @ Xx ) ) ) )
% 0.28/0.46 & ! [Xn: $i] :
% 0.28/0.46 ( ( cNUMBER @ Xn )
% 0.28/0.46 <=> ( ( cEVEN @ Xn )
% 0.28/0.46 | ( cODD @ Xn ) ) ) )
% 0.28/0.46 => ! [Xn: $i] : ( cNUMBER @ Xn ) ) ).
% 0.28/0.46
% 0.28/0.46 %------------------------------------------------------------------------------
% 0.28/0.46 ------------------------
% 1.48/1.27 ---- Embedded in THF ---
% 1.48/1.27 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.48/1.27 %%% Generated on Tue Feb 24 14:25:21 EST 2026
% 1.48/1.27 %%% using '$$dhol' embedding, version 1.3.0.
% 1.48/1.27 %%% Logic specification used:
% 1.48/1.27 %%% thf(spec, logic, $$dhol).
% 1.48/1.27
% 1.48/1.27 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.i1tU7BLO3J/DTF2DTF5105.p
% See solution above
% 1.48/1.27 ------------------------
% 1.48/1.28 ---- Cleaned THF ---
% 1.48/1.28 thf(cNUMBER,type,
% 1.48/1.28 cNUMBER: $i > $o ).
% 1.48/1.28
% 1.48/1.28 thf(cODD,type,
% 1.48/1.28 cODD: $i > $o ).
% 1.48/1.28
% 1.48/1.28 thf(cEVEN,type,
% 1.48/1.28 cEVEN: $i > $o ).
% 1.48/1.28
% 1.48/1.28 thf(cS,type,
% 1.48/1.28 cS: $i > $i ).
% 1.48/1.28
% 1.48/1.28 thf(c0,type,
% 1.48/1.28 c0: $i ).
% 1.48/1.28
% 1.48/1.28 thf(cTHM350,conjecture,
% 1.48/1.28 ( ( ( cEVEN @ c0 )
% 1.48/1.28 & ! [Xn: $i] :
% 1.48/1.28 ( ( cEVEN @ Xn )
% 1.48/1.28 => ( cEVEN @ ( cS @ ( cS @ Xn ) ) ) )
% 1.48/1.28 & ( cODD @ ( cS @ c0 ) )
% 1.48/1.28 & ! [Xn: $i] :
% 1.48/1.28 ( ( cODD @ Xn )
% 1.48/1.28 => ( cODD @ ( cS @ ( cS @ Xn ) ) ) )
% 1.48/1.28 & ( ( ( cNUMBER @ c0 )
% 1.48/1.28 & ( cNUMBER @ ( cS @ c0 ) )
% 1.48/1.28 & ! [Xx: $i] :
% 1.48/1.28 ( ( ( cNUMBER @ Xx )
% 1.48/1.28 & ( cNUMBER @ ( cS @ Xx ) ) )
% 1.48/1.28 => ( ( cNUMBER @ ( cS @ Xx ) )
% 1.48/1.28 & ( cNUMBER @ ( cS @ ( cS @ Xx ) ) ) ) ) )
% 1.48/1.28 => ! [Xx: $i] :
% 1.48/1.28 ( ( cNUMBER @ Xx )
% 1.48/1.28 & ( cNUMBER @ ( cS @ Xx ) ) ) )
% 1.48/1.28 & ! [Xn: $i] :
% 1.48/1.28 ( ( cNUMBER @ Xn )
% 1.48/1.28 <=> ( ( cEVEN @ Xn )
% 1.48/1.28 | ( cODD @ Xn ) ) ) )
% 1.48/1.28 => ! [Xn: $i] : ( cNUMBER @ Xn ) ) ).
% 1.48/1.28 ------------------------
% 1.63/1.30 % (5393)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_5105 for (2999ds/2Mi)
% 1.63/1.30 % (5390)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_5105 for (2999ds/4Mi)
% 1.63/1.30 % (5393)Instruction limit reached!
% 1.63/1.30 % (5393)------------------------------
% 1.63/1.30 % (5393)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.63/1.30 % (5393)Termination reason: Unknown
% 1.63/1.30 % (5393)Termination phase: Saturation
% 1.63/1.30
% 1.63/1.30 % (5393)Memory used [KB]: 5500
% 1.63/1.30 % (5393)Time elapsed: 0.003 s
% 1.63/1.30 % (5393)Instructions burned: 3 (million)
% 1.63/1.30 % (5393)------------------------------
% 1.63/1.30 % (5393)------------------------------
% 1.63/1.30 % (5390)Instruction limit reached!
% 1.63/1.30 % (5390)------------------------------
% 1.63/1.30 % (5390)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.63/1.30 % (5390)Termination reason: Unknown
% 1.63/1.30 % (5390)Termination phase: Saturation
% 1.63/1.30
% 1.63/1.30 % (5390)Memory used [KB]: 5500
% 1.63/1.30 % (5390)Time elapsed: 0.003 s
% 1.63/1.30 % (5390)Instructions burned: 4 (million)
% 1.63/1.30 % (5390)------------------------------
% 1.63/1.30 % (5390)------------------------------
% 1.63/1.31 % (5389)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_5105 for (2999ds/183Mi)
% 1.63/1.31 % (5392)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_5105 for (2999ds/2Mi)
% 1.63/1.31 % (5391)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_5105 for (2999ds/27Mi)
% 1.63/1.31 % (5394)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_5105 for (2999ds/275Mi)
% 1.63/1.31 % (5395)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_5105 for (2999ds/18Mi)
% 1.63/1.31 % (5392)Instruction limit reached!
% 1.63/1.31 % (5392)------------------------------
% 1.63/1.31 % (5392)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.63/1.31 % (5392)Termination reason: Unknown
% 1.63/1.31 % (5392)Termination phase: Preprocessing 3
% 1.63/1.31
% 1.63/1.31 % (5392)Memory used [KB]: 895
% 1.63/1.31 % (5392)Time elapsed: 0.003 s
% 1.63/1.31 % (5392)Instructions burned: 2 (million)
% 1.63/1.31 % (5392)------------------------------
% 1.63/1.31 % (5392)------------------------------
% 1.63/1.31 % (5396)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_5105 for (2999ds/3Mi)
% 1.63/1.31 % (5396)Instruction limit reached!
% 1.63/1.31 % (5396)------------------------------
% 1.63/1.31 % (5396)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.63/1.31 % (5396)Termination reason: Unknown
% 1.63/1.31 % (5396)Termination phase: Saturation
% 1.63/1.31
% 1.63/1.31 % (5396)Memory used [KB]: 5500
% 1.63/1.31 % (5396)Time elapsed: 0.002 s
% 1.63/1.31 % (5396)Instructions burned: 3 (million)
% 1.63/1.31 % (5396)------------------------------
% 1.63/1.31 % (5396)------------------------------
% 1.63/1.31 % (5397)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_5105 for (2999ds/37Mi)
% 1.63/1.31 % (5389)First to succeed.
% 1.63/1.31 % (5394)Also succeeded, but the first one will report.
% 1.63/1.32 % (5389)Refutation found. Thanks to Tanya!
% 1.63/1.32 % SZS status Theorem for DTF2THF_5105
% 1.63/1.32 % SZS output start Proof for DTF2THF_5105
% 1.63/1.32 thf(func_def_0, type, cNUMBER: $i > $o).
% 1.63/1.32 thf(func_def_1, type, cODD: $i > $o).
% 1.63/1.32 thf(func_def_2, type, cEVEN: $i > $o).
% 1.63/1.32 thf(func_def_3, type, cS: $i > $i).
% 1.63/1.32 thf(f98,plain,(
% 1.63/1.32 $false),
% 1.63/1.32 inference(avatar_sat_refutation,[status(thm)],[f56,f57,f58,f69,f75,f78,f80,f97])).
% 1.63/1.32 thf(f97,plain,(
% 1.63/1.32 ~spl2_5 | spl2_7),
% 1.63/1.32 inference(avatar_contradiction_clause,[status(thm)],[f96])).
% 1.63/1.32 thf(f96,plain,(
% 1.63/1.32 $false | (~spl2_5 | spl2_7)),
% 1.63/1.32 inference(subsumption_resolution,[status(thm)],[f95,f93])).
% 1.63/1.32 thf(f93,plain,(
% 1.63/1.32 ($true = (cODD @ sK0)) | (~spl2_5 | spl2_7)),
% 1.63/1.32 inference(subsumption_resolution,[status(thm)],[f92,f47])).
% 1.63/1.32 thf(f47,plain,(
% 1.63/1.32 ((cNUMBER @ sK0) = $true) | ~spl2_5),
% 1.63/1.32 inference(avatar_component_clause,[status(thm)],[f45])).
% 1.63/1.32 thf(f45,definition,(
% 1.63/1.32 spl2_5 <=> ((cNUMBER @ sK0) = $true)),
% 1.63/1.32 introduced(definition,[new_symbols(naming,[spl2_5])],[avatar_definition])).
% 1.63/1.32 thf(f92,plain,(
% 1.63/1.32 ($true = (cODD @ sK0)) | ((cNUMBER @ sK0) != $true) | spl2_7),
% 1.63/1.32 inference(trivial_inequality_removal,[status(thm)],[f91])).
% 1.63/1.32 thf(f91,plain,(
% 1.63/1.32 ((cNUMBER @ sK0) != $true) | ($true != $true) | ($true = (cODD @ sK0)) | spl2_7),
% 1.63/1.32 inference(superposition,[status(thm)],[f90,f25])).
% 1.63/1.32 thf(f25,plain,(
% 1.63/1.32 ( ! [X0 : $i] : (((cEVEN @ X0) = $true) | ((cNUMBER @ X0) != $true) | ((cODD @ X0) = $true)) )),
% 1.63/1.32 inference(cnf_transformation,[status(thm)],[f13])).
% 1.63/1.32 thf(f13,plain,(
% 1.63/1.32 ! [X0 : $i] : ((((cNUMBER @ X0) = $true) | (((cODD @ X0) != $true) & ((cEVEN @ X0) != $true))) & (((cODD @ X0) = $true) | ((cEVEN @ X0) = $true) | ((cNUMBER @ X0) != $true))) & ((cEVEN @ c0) = $true) & (((cNUMBER @ c0) != $true) | ! [X1 : $i] : (((cNUMBER @ X1) = $true) & ((cNUMBER @ (cS @ X1)) = $true)) | (((cNUMBER @ (cS @ sK0)) = $true) & ((cNUMBER @ sK0) = $true) & (((cNUMBER @ (cS @ sK0)) != $true) | ($true != (cNUMBER @ (cS @ (cS @ sK0)))))) | ((cNUMBER @ (cS @ c0)) != $true)) & ((cNUMBER @ sK1) != $true) & ((cODD @ (cS @ c0)) = $true) & ! [X4 : $i] : (($true != (cODD @ X4)) | ((cODD @ (cS @ (cS @ X4))) = $true)) & ! [X5 : $i] : (((cEVEN @ X5) != $true) | ((cEVEN @ (cS @ (cS @ X5))) = $true))),
% 1.63/1.32 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1])],[f10,f12,f11])).
% 1.63/1.32 thf(f11,plain,(
% 1.63/1.32 ? [X2 : $i] : (((cNUMBER @ (cS @ X2)) = $true) & ((cNUMBER @ X2) = $true) & (((cNUMBER @ (cS @ X2)) != $true) | ((cNUMBER @ (cS @ (cS @ X2))) != $true))) => (((cNUMBER @ (cS @ sK0)) = $true) & ((cNUMBER @ sK0) = $true) & (((cNUMBER @ (cS @ sK0)) != $true) | ($true != (cNUMBER @ (cS @ (cS @ sK0))))))),
% 1.63/1.32 introduced(definition,[],[choice_axiom])).
% 1.63/1.32 thf(f12,plain,(
% 1.63/1.32 ? [X3 : $i] : ($true != (cNUMBER @ X3)) => ((cNUMBER @ sK1) != $true)),
% 1.63/1.32 introduced(definition,[],[choice_axiom])).
% 1.63/1.32 thf(f10,plain,(
% 1.63/1.32 ! [X0 : $i] : ((((cNUMBER @ X0) = $true) | (((cODD @ X0) != $true) & ((cEVEN @ X0) != $true))) & (((cODD @ X0) = $true) | ((cEVEN @ X0) = $true) | ((cNUMBER @ X0) != $true))) & ((cEVEN @ c0) = $true) & (((cNUMBER @ c0) != $true) | ! [X1 : $i] : (((cNUMBER @ X1) = $true) & ((cNUMBER @ (cS @ X1)) = $true)) | ? [X2 : $i] : (((cNUMBER @ (cS @ X2)) = $true) & ((cNUMBER @ X2) = $true) & (((cNUMBER @ (cS @ X2)) != $true) | ((cNUMBER @ (cS @ (cS @ X2))) != $true))) | ((cNUMBER @ (cS @ c0)) != $true)) & ? [X3 : $i] : ($true != (cNUMBER @ X3)) & ((cODD @ (cS @ c0)) = $true) & ! [X4 : $i] : (($true != (cODD @ X4)) | ((cODD @ (cS @ (cS @ X4))) = $true)) & ! [X5 : $i] : (((cEVEN @ X5) != $true) | ((cEVEN @ (cS @ (cS @ X5))) = $true))),
% 1.63/1.32 inference(rectify,[status(thm)],[f9])).
% 1.63/1.32 thf(f9,plain,(
% 1.63/1.32 ! [X1 : $i] : ((((cNUMBER @ X1) = $true) | (((cODD @ X1) != $true) & ((cEVEN @ X1) != $true))) & (((cODD @ X1) = $true) | ((cEVEN @ X1) = $true) | ((cNUMBER @ X1) != $true))) & ((cEVEN @ c0) = $true) & (((cNUMBER @ c0) != $true) | ! [X3 : $i] : (($true = (cNUMBER @ X3)) & ((cNUMBER @ (cS @ X3)) = $true)) | ? [X2 : $i] : (((cNUMBER @ (cS @ X2)) = $true) & ((cNUMBER @ X2) = $true) & (((cNUMBER @ (cS @ X2)) != $true) | ((cNUMBER @ (cS @ (cS @ X2))) != $true))) | ((cNUMBER @ (cS @ c0)) != $true)) & ? [X5 : $i] : ($true != (cNUMBER @ X5)) & ((cODD @ (cS @ c0)) = $true) & ! [X4 : $i] : (($true != (cODD @ X4)) | ((cODD @ (cS @ (cS @ X4))) = $true)) & ! [X0 : $i] : (((cEVEN @ X0) != $true) | ((cEVEN @ (cS @ (cS @ X0))) = $true))),
% 1.63/1.32 inference(flattening,[status(thm)],[f8])).
% 1.63/1.32 thf(f8,plain,(
% 1.63/1.32 ! [X1 : $i] : ((((cNUMBER @ X1) = $true) | (((cODD @ X1) != $true) & ((cEVEN @ X1) != $true))) & ((((cODD @ X1) = $true) | ((cEVEN @ X1) = $true)) | ((cNUMBER @ X1) != $true))) & ((cEVEN @ c0) = $true) & (((cNUMBER @ c0) != $true) | ! [X3 : $i] : (($true = (cNUMBER @ X3)) & ((cNUMBER @ (cS @ X3)) = $true)) | ? [X2 : $i] : (((cNUMBER @ (cS @ X2)) = $true) & ((cNUMBER @ X2) = $true) & (((cNUMBER @ (cS @ X2)) != $true) | ((cNUMBER @ (cS @ (cS @ X2))) != $true))) | ((cNUMBER @ (cS @ c0)) != $true)) & ? [X5 : $i] : ($true != (cNUMBER @ X5)) & ((cODD @ (cS @ c0)) = $true) & ! [X4 : $i] : (($true != (cODD @ X4)) | ((cODD @ (cS @ (cS @ X4))) = $true)) & ! [X0 : $i] : (((cEVEN @ X0) != $true) | ((cEVEN @ (cS @ (cS @ X0))) = $true))),
% 1.63/1.32 inference(nnf_transformation,[status(thm)],[f7])).
% 1.63/1.32 thf(f7,plain,(
% 1.63/1.32 ! [X1 : $i] : (((cNUMBER @ X1) = $true) <=> (((cODD @ X1) = $true) | ((cEVEN @ X1) = $true))) & ((cEVEN @ c0) = $true) & (((cNUMBER @ c0) != $true) | ! [X3 : $i] : (($true = (cNUMBER @ X3)) & ((cNUMBER @ (cS @ X3)) = $true)) | ? [X2 : $i] : (((cNUMBER @ (cS @ X2)) = $true) & ((cNUMBER @ X2) = $true) & (((cNUMBER @ (cS @ X2)) != $true) | ((cNUMBER @ (cS @ (cS @ X2))) != $true))) | ((cNUMBER @ (cS @ c0)) != $true)) & ? [X5 : $i] : ($true != (cNUMBER @ X5)) & ((cODD @ (cS @ c0)) = $true) & ! [X4 : $i] : (($true != (cODD @ X4)) | ((cODD @ (cS @ (cS @ X4))) = $true)) & ! [X0 : $i] : (((cEVEN @ X0) != $true) | ((cEVEN @ (cS @ (cS @ X0))) = $true))),
% 1.63/1.32 inference(flattening,[status(thm)],[f6])).
% 1.63/1.32 thf(f6,plain,(
% 1.63/1.32 ? [X5 : $i] : ($true != (cNUMBER @ X5)) & (! [X4 : $i] : (($true != (cODD @ X4)) | ((cODD @ (cS @ (cS @ X4))) = $true)) & (! [X3 : $i] : (($true = (cNUMBER @ X3)) & ((cNUMBER @ (cS @ X3)) = $true)) | (((cNUMBER @ (cS @ c0)) != $true) | ? [X2 : $i] : ((((cNUMBER @ (cS @ X2)) != $true) | ((cNUMBER @ (cS @ (cS @ X2))) != $true)) & (((cNUMBER @ (cS @ X2)) = $true) & ((cNUMBER @ X2) = $true))) | ((cNUMBER @ c0) != $true))) & ! [X1 : $i] : (((cNUMBER @ X1) = $true) <=> (((cODD @ X1) = $true) | ((cEVEN @ X1) = $true))) & ((cEVEN @ c0) = $true) & ((cODD @ (cS @ c0)) = $true) & ! [X0 : $i] : (((cEVEN @ X0) != $true) | ((cEVEN @ (cS @ (cS @ X0))) = $true)))),
% 1.63/1.32 inference(ennf_transformation,[status(thm)],[f5])).
% 1.63/1.32 thf(f5,plain,(
% 1.63/1.32 ~((! [X4 : $i] : (($true = (cODD @ X4)) => ((cODD @ (cS @ (cS @ X4))) = $true)) & ((((cNUMBER @ (cS @ c0)) = $true) & ! [X2 : $i] : ((((cNUMBER @ (cS @ X2)) = $true) & ((cNUMBER @ X2) = $true)) => (((cNUMBER @ (cS @ (cS @ X2))) = $true) & ((cNUMBER @ (cS @ X2)) = $true))) & ((cNUMBER @ c0) = $true)) => ! [X3 : $i] : (($true = (cNUMBER @ X3)) & ((cNUMBER @ (cS @ X3)) = $true))) & ! [X1 : $i] : (((cNUMBER @ X1) = $true) <=> (((cODD @ X1) = $true) | ((cEVEN @ X1) = $true))) & ((cEVEN @ c0) = $true) & ((cODD @ (cS @ c0)) = $true) & ! [X0 : $i] : (((cEVEN @ X0) = $true) => ((cEVEN @ (cS @ (cS @ X0))) = $true))) => ! [X5 : $i] : ($true = (cNUMBER @ X5)))),
% 1.63/1.32 inference(fool_elimination,[status(thm)],[f4])).
% 1.63/1.32 thf(f4,plain,(
% 1.63/1.32 ~(((cEVEN @ c0) & (cODD @ (cS @ c0)) & ! [X0 : $i] : ((cEVEN @ X0) => (cEVEN @ (cS @ (cS @ X0)))) & ! [X1 : $i] : ((cNUMBER @ X1) <=> ((cEVEN @ X1) | (cODD @ X1))) & (((cNUMBER @ (cS @ c0)) & ! [X2 : $i] : (((cNUMBER @ (cS @ X2)) & (cNUMBER @ X2)) => ((cNUMBER @ (cS @ (cS @ X2))) & (cNUMBER @ (cS @ X2)))) & (cNUMBER @ c0)) => ! [X3 : $i] : ((cNUMBER @ X3) & (cNUMBER @ (cS @ X3)))) & ! [X4 : $i] : ((cODD @ X4) => (cODD @ (cS @ (cS @ X4))))) => ! [X5 : $i] : (cNUMBER @ X5))),
% 1.63/1.32 inference(rectify,[status(thm)],[f2])).
% 1.63/1.32 thf(f2,negated_conjecture,(
% 1.63/1.32 ~(((cEVEN @ c0) & (cODD @ (cS @ c0)) & ! [X0 : $i] : ((cEVEN @ X0) => (cEVEN @ (cS @ (cS @ X0)))) & ! [X0 : $i] : ((cNUMBER @ X0) <=> ((cEVEN @ X0) | (cODD @ X0))) & (((cNUMBER @ (cS @ c0)) & ! [X1 : $i] : (((cNUMBER @ (cS @ X1)) & (cNUMBER @ X1)) => ((cNUMBER @ (cS @ (cS @ X1))) & (cNUMBER @ (cS @ X1)))) & (cNUMBER @ c0)) => ! [X1 : $i] : ((cNUMBER @ X1) & (cNUMBER @ (cS @ X1)))) & ! [X0 : $i] : ((cODD @ X0) => (cODD @ (cS @ (cS @ X0))))) => ! [X0 : $i] : (cNUMBER @ X0))),
% 1.63/1.32 inference(negated_conjecture,[status(cth)],[f1])).
% 1.63/1.32 thf(f1,conjecture,(
% 1.63/1.32 ((cEVEN @ c0) & (cODD @ (cS @ c0)) & ! [X0 : $i] : ((cEVEN @ X0) => (cEVEN @ (cS @ (cS @ X0)))) & ! [X0 : $i] : ((cNUMBER @ X0) <=> ((cEVEN @ X0) | (cODD @ X0))) & (((cNUMBER @ (cS @ c0)) & ! [X1 : $i] : (((cNUMBER @ (cS @ X1)) & (cNUMBER @ X1)) => ((cNUMBER @ (cS @ (cS @ X1))) & (cNUMBER @ (cS @ X1)))) & (cNUMBER @ c0)) => ! [X1 : $i] : ((cNUMBER @ X1) & (cNUMBER @ (cS @ X1)))) & ! [X0 : $i] : ((cODD @ X0) => (cODD @ (cS @ (cS @ X0))))) => ! [X0 : $i] : (cNUMBER @ X0)),
% 1.63/1.32 file('/export/starexec/sandbox/tmp/tmp.i1tU7BLO3J/DTF2THF_5105.p',cTHM350)).
% 1.63/1.32 thf(f90,plain,(
% 1.63/1.32 ((cEVEN @ sK0) != $true) | spl2_7),
% 1.63/1.32 inference(trivial_inequality_removal,[status(thm)],[f89])).
% 1.63/1.32 thf(f89,plain,(
% 1.63/1.32 ((cEVEN @ sK0) != $true) | ($true != $true) | spl2_7),
% 1.63/1.32 inference(superposition,[status(thm)],[f84,f14])).
% 1.63/1.32 thf(f14,plain,(
% 1.63/1.32 ( ! [X5 : $i] : (((cEVEN @ (cS @ (cS @ X5))) = $true) | ((cEVEN @ X5) != $true)) )),
% 1.63/1.32 inference(cnf_transformation,[status(thm)],[f13])).
% 1.63/1.32 thf(f84,plain,(
% 1.63/1.32 ($true != (cEVEN @ (cS @ (cS @ sK0)))) | spl2_7),
% 1.63/1.32 inference(trivial_inequality_removal,[status(thm)],[f82])).
% 1.63/1.32 thf(f82,plain,(
% 1.63/1.32 ($true != $true) | ($true != (cEVEN @ (cS @ (cS @ sK0)))) | spl2_7),
% 1.63/1.32 inference(superposition,[status(thm)],[f55,f26])).
% 1.63/1.32 thf(f26,plain,(
% 1.63/1.32 ( ! [X0 : $i] : (((cNUMBER @ X0) = $true) | ((cEVEN @ X0) != $true)) )),
% 1.63/1.32 inference(cnf_transformation,[status(thm)],[f13])).
% 1.63/1.32 thf(f55,plain,(
% 1.63/1.32 ($true != (cNUMBER @ (cS @ (cS @ sK0)))) | spl2_7),
% 1.63/1.32 inference(avatar_component_clause,[status(thm)],[f53])).
% 1.63/1.32 thf(f53,definition,(
% 1.63/1.32 spl2_7 <=> ($true = (cNUMBER @ (cS @ (cS @ sK0))))),
% 1.63/1.32 introduced(definition,[new_symbols(naming,[spl2_7])],[avatar_definition])).
% 1.63/1.32 thf(f95,plain,(
% 1.63/1.32 ($true != (cODD @ sK0)) | spl2_7),
% 1.63/1.32 inference(trivial_inequality_removal,[status(thm)],[f94])).
% 1.63/1.32 thf(f94,plain,(
% 1.63/1.32 ($true != (cODD @ sK0)) | ($true != $true) | spl2_7),
% 1.63/1.32 inference(superposition,[status(thm)],[f83,f15])).
% 1.63/1.32 thf(f15,plain,(
% 1.63/1.32 ( ! [X4 : $i] : (((cODD @ (cS @ (cS @ X4))) = $true) | ($true != (cODD @ X4))) )),
% 1.63/1.32 inference(cnf_transformation,[status(thm)],[f13])).
% 1.63/1.32 thf(f83,plain,(
% 1.63/1.32 ((cODD @ (cS @ (cS @ sK0))) != $true) | spl2_7),
% 1.63/1.32 inference(trivial_inequality_removal,[status(thm)],[f81])).
% 1.63/1.32 thf(f81,plain,(
% 1.63/1.32 ($true != $true) | ((cODD @ (cS @ (cS @ sK0))) != $true) | spl2_7),
% 1.63/1.32 inference(superposition,[status(thm)],[f55,f27])).
% 1.63/1.32 thf(f27,plain,(
% 1.63/1.32 ( ! [X0 : $i] : (((cNUMBER @ X0) = $true) | ((cODD @ X0) != $true)) )),
% 1.63/1.32 inference(cnf_transformation,[status(thm)],[f13])).
% 1.63/1.32 thf(f80,plain,(
% 1.63/1.32 ~spl2_2 | spl2_7),
% 1.63/1.32 inference(avatar_contradiction_clause,[status(thm)],[f79])).
% 1.63/1.32 thf(f79,plain,(
% 1.63/1.32 $false | (~spl2_2 | spl2_7)),
% 1.63/1.32 inference(subsumption_resolution,[status(thm)],[f55,f34])).
% 1.63/1.32 thf(f34,plain,(
% 1.63/1.32 ( ! [X1 : $i] : (((cNUMBER @ (cS @ X1)) = $true)) ) | ~spl2_2),
% 1.63/1.32 inference(avatar_component_clause,[status(thm)],[f33])).
% 1.63/1.32 thf(f33,definition,(
% 1.63/1.32 spl2_2 <=> ! [X1 : $i] : ((cNUMBER @ (cS @ X1)) = $true)),
% 1.63/1.32 introduced(definition,[new_symbols(naming,[spl2_2])],[avatar_definition])).
% 1.63/1.32 thf(f78,plain,(
% 1.63/1.32 ~spl2_6),
% 1.63/1.32 inference(avatar_contradiction_clause,[status(thm)],[f77])).
% 1.63/1.32 thf(f77,plain,(
% 1.63/1.32 $false | ~spl2_6),
% 1.63/1.32 inference(trivial_inequality_removal,[status(thm)],[f76])).
% 1.63/1.32 thf(f76,plain,(
% 1.63/1.32 ($true != $true) | ~spl2_6),
% 1.63/1.32 inference(superposition,[status(thm)],[f17,f50])).
% 1.63/1.32 thf(f50,plain,(
% 1.63/1.32 ( ! [X1 : $i] : (((cNUMBER @ X1) = $true)) ) | ~spl2_6),
% 1.63/1.32 inference(avatar_component_clause,[status(thm)],[f49])).
% 1.63/1.32 thf(f49,definition,(
% 1.63/1.32 spl2_6 <=> ! [X1 : $i] : ((cNUMBER @ X1) = $true)),
% 1.63/1.32 introduced(definition,[new_symbols(naming,[spl2_6])],[avatar_definition])).
% 1.63/1.32 thf(f17,plain,(
% 1.63/1.32 ((cNUMBER @ sK1) != $true)),
% 1.63/1.32 inference(cnf_transformation,[status(thm)],[f13])).
% 1.63/1.32 thf(f75,plain,(
% 1.63/1.32 spl2_4),
% 1.63/1.32 inference(avatar_contradiction_clause,[status(thm)],[f74])).
% 1.63/1.32 thf(f74,plain,(
% 1.63/1.32 $false | spl2_4),
% 1.63/1.32 inference(subsumption_resolution,[status(thm)],[f72,f24])).
% 1.63/1.32 thf(f24,plain,(
% 1.63/1.32 ((cEVEN @ c0) = $true)),
% 1.63/1.32 inference(cnf_transformation,[status(thm)],[f13])).
% 1.63/1.32 thf(f72,plain,(
% 1.63/1.32 ((cEVEN @ c0) != $true) | spl2_4),
% 1.63/1.32 inference(trivial_inequality_removal,[status(thm)],[f71])).
% 1.63/1.32 thf(f71,plain,(
% 1.63/1.32 ((cEVEN @ c0) != $true) | ($true != $true) | spl2_4),
% 1.63/1.32 inference(superposition,[status(thm)],[f42,f26])).
% 1.63/1.32 thf(f42,plain,(
% 1.63/1.32 ((cNUMBER @ c0) != $true) | spl2_4),
% 1.63/1.32 inference(avatar_component_clause,[status(thm)],[f40])).
% 1.63/1.32 thf(f40,definition,(
% 1.63/1.32 spl2_4 <=> ((cNUMBER @ c0) = $true)),
% 1.63/1.32 introduced(definition,[new_symbols(naming,[spl2_4])],[avatar_definition])).
% 1.63/1.32 thf(f69,plain,(
% 1.63/1.32 spl2_1),
% 1.63/1.32 inference(avatar_contradiction_clause,[status(thm)],[f68])).
% 1.63/1.32 thf(f68,plain,(
% 1.63/1.32 $false | spl2_1),
% 1.63/1.32 inference(subsumption_resolution,[status(thm)],[f67,f16])).
% 1.63/1.32 thf(f16,plain,(
% 1.63/1.32 ((cODD @ (cS @ c0)) = $true)),
% 1.63/1.32 inference(cnf_transformation,[status(thm)],[f13])).
% 1.63/1.32 thf(f67,plain,(
% 1.63/1.32 ((cODD @ (cS @ c0)) != $true) | spl2_1),
% 1.63/1.32 inference(trivial_inequality_removal,[status(thm)],[f65])).
% 1.63/1.32 thf(f65,plain,(
% 1.63/1.32 ($true != $true) | ((cODD @ (cS @ c0)) != $true) | spl2_1),
% 1.63/1.32 inference(superposition,[status(thm)],[f31,f27])).
% 1.63/1.32 thf(f31,plain,(
% 1.63/1.32 ((cNUMBER @ (cS @ c0)) != $true) | spl2_1),
% 1.63/1.32 inference(avatar_component_clause,[status(thm)],[f29])).
% 1.63/1.32 thf(f29,definition,(
% 1.63/1.32 spl2_1 <=> ((cNUMBER @ (cS @ c0)) = $true)),
% 1.63/1.32 introduced(definition,[new_symbols(naming,[spl2_1])],[avatar_definition])).
% 1.63/1.32 thf(f58,plain,(
% 1.63/1.32 spl2_5 | ~spl2_1 | spl2_2 | ~spl2_4),
% 1.63/1.32 inference(avatar_split_clause,[status(thm)],[f19,f40,f33,f29,f45])).
% 1.63/1.32 thf(f19,plain,(
% 1.63/1.32 ( ! [X1 : $i] : (((cNUMBER @ c0) != $true) | ((cNUMBER @ sK0) = $true) | ((cNUMBER @ (cS @ X1)) = $true) | ((cNUMBER @ (cS @ c0)) != $true)) )),
% 1.63/1.32 inference(cnf_transformation,[status(thm)],[f13])).
% 1.63/1.32 thf(f57,plain,(
% 1.63/1.32 spl2_3 | ~spl2_4 | spl2_6 | ~spl2_1),
% 1.63/1.32 inference(avatar_split_clause,[status(thm)],[f23,f29,f49,f40,f36])).
% 1.63/1.32 thf(f36,definition,(
% 1.63/1.32 spl2_3 <=> ((cNUMBER @ (cS @ sK0)) = $true)),
% 1.63/1.32 introduced(definition,[new_symbols(naming,[spl2_3])],[avatar_definition])).
% 1.63/1.32 thf(f23,plain,(
% 1.63/1.32 ( ! [X1 : $i] : (((cNUMBER @ c0) != $true) | ((cNUMBER @ X1) = $true) | ((cNUMBER @ (cS @ c0)) != $true) | ((cNUMBER @ (cS @ sK0)) = $true)) )),
% 1.63/1.32 inference(cnf_transformation,[status(thm)],[f13])).
% 1.63/1.32 thf(f56,plain,(
% 1.63/1.32 ~spl2_7 | ~spl2_4 | ~spl2_3 | ~spl2_1 | spl2_6),
% 1.63/1.32 inference(avatar_split_clause,[status(thm)],[f21,f49,f29,f36,f40,f53])).
% 1.63/1.32 thf(f21,plain,(
% 1.63/1.32 ( ! [X1 : $i] : (((cNUMBER @ (cS @ sK0)) != $true) | ($true != (cNUMBER @ (cS @ (cS @ sK0)))) | ((cNUMBER @ X1) = $true) | ((cNUMBER @ c0) != $true) | ((cNUMBER @ (cS @ c0)) != $true)) )),
% 1.63/1.32 inference(cnf_transformation,[status(thm)],[f13])).
% 1.63/1.32 % SZS output end Proof for DTF2THF_5105
% 1.63/1.32 % (5389)------------------------------
% 1.63/1.32 % (5389)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.63/1.32 % (5389)Termination reason: Refutation
% 1.63/1.32
% 1.63/1.32 % (5389)Memory used [KB]: 5628
% 1.63/1.32 % (5389)Time elapsed: 0.010 s
% 1.63/1.32 % (5389)Instructions burned: 8 (million)
% 1.63/1.32 % (5389)------------------------------
% 1.63/1.32 % (5389)------------------------------
% 1.63/1.32 % (5388)Success in time 0.026 s
%------------------------------------------------------------------------------