%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : NUM811^5 : 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 09:24:49 AM UTC 2026
% Result : Theorem 1.65s 1.35s
% Output : Refutation 1.65s
% Verified :
% SZS Type : Refutation
% Derivation depth : 23
% Number of leaves : 4
% Syntax : Number of formulae : 59 ( 37 unt; 0 typ; 2 def)
% Number of atoms : 440 ( 51 equ; 0 cnn)
% Maximal formula atoms : 6 ( 7 avg)
% Number of connectives : 890 ( 24 ~; 17 |; 40 &; 613 @)
% ( 2 <=>; 65 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 2 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 25 ( 25 >; 0 *; 0 +; 0 <<)
% Number of symbols : 15 ( 10 usr; 11 con; 0-3 aty)
% ( 93 !!; 36 ??; 0 @@+; 0 @@-)
% Number of variables : 142 ( 97 ^; 42 !; 3 ?; 142 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_1,type,
cS: $i > $i ).
thf(func_def_2,type,
c_plus: $i > $i > $i > $o ).
thf(func_def_17,type,
ph1:
!>[X0: $tType] : X0 ).
thf(f142,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f86,f97,f141]) ).
thf(f141,plain,
~ spl0_1,
inference(avatar_contradiction_clause,[status(thm)],[f140]) ).
thf(f140,plain,
( $false
| ~ spl0_1 ),
inference(trivial_inequality_removal,[status(thm)],[f139]) ).
thf(f139,plain,
( ( $true = $false )
| ~ spl0_1 ),
inference(backward_demodulation,[status(thm)],[f127,f138]) ).
thf(f138,plain,
( ! [X1: $i] :
( ( c_plus @ ( cS @ sK4 ) @ sK2 @ X1 )
= $false )
| ~ spl0_1 ),
inference(pi_clausification,[status(thm)],[f116]) ).
thf(f116,plain,
( ( ( ?? @ $i @ ( c_plus @ ( cS @ sK4 ) @ sK2 ) )
= $false )
| ~ spl0_1 ),
inference(binary_proxy_clausification,[status(thm)],[f111]) ).
thf(f111,plain,
( ( ( ( ?? @ $i @ ( c_plus @ sK4 @ sK2 ) )
=> ( ?? @ $i @ ( c_plus @ ( cS @ sK4 ) @ sK2 ) ) )
= $false )
| ~ spl0_1 ),
inference(beta_eta_normalization,[status(thm)],[f107]) ).
thf(f107,plain,
( ( ( ^ [Y0: $i] :
( ( ?? @ $i @ ( c_plus @ Y0 @ sK2 ) )
=> ( ?? @ $i @ ( c_plus @ ( cS @ Y0 ) @ sK2 ) ) )
@ sK4 )
= $false )
| ~ spl0_1 ),
inference(sigma_clausification,[status(thm)],[f82]) ).
thf(f82,plain,
( ( ( !! @ $i
@ ^ [Y0: $i] :
( ( ?? @ $i @ ( c_plus @ Y0 @ sK2 ) )
=> ( ?? @ $i @ ( c_plus @ ( cS @ Y0 ) @ sK2 ) ) ) )
= $false )
| ~ spl0_1 ),
inference(avatar_component_clause,[status(thm)],[f81]) ).
thf(f81,definition,
( spl0_1
<=> ( ( !! @ $i
@ ^ [Y0: $i] :
( ( ?? @ $i @ ( c_plus @ Y0 @ sK2 ) )
=> ( ?? @ $i @ ( c_plus @ ( cS @ Y0 ) @ sK2 ) ) ) )
= $false ) ),
introduced(definition,[new_symbols(naming,[spl0_1])],[avatar_definition]) ).
thf(f127,plain,
( ( $true
= ( c_plus @ ( cS @ sK4 ) @ sK2 @ ( cS @ sK5 ) ) )
| ~ spl0_1 ),
inference(boolean_simplification,[status(thm)],[f126]) ).
thf(f126,plain,
( ( $true
= ( $true
=> ( c_plus @ ( cS @ sK4 ) @ sK2 @ ( cS @ sK5 ) ) ) )
| ~ spl0_1 ),
inference(superposition,[status(thm)],[f64,f122]) ).
thf(f122,plain,
( ( $true
= ( c_plus @ sK4 @ sK2 @ sK5 ) )
| ~ spl0_1 ),
inference(sigma_clausification,[status(thm)],[f117]) ).
thf(f117,plain,
( ( $true
= ( ?? @ $i @ ( c_plus @ sK4 @ sK2 ) ) )
| ~ spl0_1 ),
inference(binary_proxy_clausification,[status(thm)],[f111]) ).
thf(f64,plain,
! [X2: $i,X3: $i,X1: $i] :
( $true
= ( ( c_plus @ X1 @ X3 @ X2 )
=> ( c_plus @ ( cS @ X1 ) @ X3 @ ( cS @ X2 ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f62]) ).
thf(f62,plain,
! [X2: $i,X3: $i,X1: $i] :
( $true
= ( ^ [Y0: $i] :
( ( c_plus @ X1 @ Y0 @ X2 )
=> ( c_plus @ ( cS @ X1 ) @ Y0 @ ( cS @ X2 ) ) )
@ X3 ) ),
inference(pi_clausification,[status(thm)],[f61]) ).
thf(f61,plain,
! [X2: $i,X1: $i] :
( ( !! @ $i
@ ^ [Y0: $i] :
( ( c_plus @ X1 @ Y0 @ X2 )
=> ( c_plus @ ( cS @ X1 ) @ Y0 @ ( cS @ X2 ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f57]) ).
thf(f57,plain,
! [X2: $i,X1: $i] :
( $true
= ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( c_plus @ X1 @ Y1 @ Y0 )
=> ( c_plus @ ( cS @ X1 ) @ Y1 @ ( cS @ Y0 ) ) ) )
@ X2 ) ),
inference(pi_clausification,[status(thm)],[f30]) ).
thf(f30,plain,
! [X1: $i] :
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( c_plus @ X1 @ Y1 @ Y0 )
=> ( c_plus @ ( cS @ X1 ) @ Y1 @ ( cS @ Y0 ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f29]) ).
thf(f29,plain,
! [X1: $i] :
( $true
= ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( c_plus @ Y0 @ Y2 @ Y1 )
=> ( c_plus @ ( cS @ Y0 ) @ Y2 @ ( cS @ Y1 ) ) ) ) )
@ X1 ) ),
inference(pi_clausification,[status(thm)],[f25]) ).
thf(f25,plain,
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( c_plus @ Y0 @ Y2 @ Y1 )
=> ( c_plus @ ( cS @ Y0 ) @ Y2 @ ( cS @ Y1 ) ) ) ) ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f17]) ).
thf(f17,plain,
( $true
= ( ( !! @ ( $i > $o )
@ ^ [Y0: $i > $o] :
( ( ( !! @ $i
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
=> ( Y0 @ ( cS @ Y1 ) ) ) )
& ( Y0 @ c0 ) )
=> ( !! @ $i @ Y0 ) ) )
& ( !! @ $i
@ ^ [Y0: $i] : ( c_plus @ c0 @ Y0 @ Y0 ) )
& ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( c_plus @ Y0 @ Y2 @ Y1 )
=> ( c_plus @ ( cS @ Y0 ) @ Y2 @ ( cS @ Y1 ) ) ) ) ) ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f13]) ).
thf(f13,plain,
( ( ( ( !! @ ( $i > $o )
@ ^ [Y0: $i > $o] :
( ( ( !! @ $i
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
=> ( Y0 @ ( cS @ Y1 ) ) ) )
& ( Y0 @ c0 ) )
=> ( !! @ $i @ Y0 ) ) )
& ( !! @ $i
@ ^ [Y0: $i] : ( c_plus @ c0 @ Y0 @ Y0 ) )
& ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( c_plus @ Y0 @ Y2 @ Y1 )
=> ( c_plus @ ( cS @ Y0 ) @ Y2 @ ( cS @ Y1 ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] : ( ?? @ $i @ ( c_plus @ Y1 @ Y0 ) ) ) ) )
= $false ),
inference(not_proxy_clausification,[status(thm)],[f12]) ).
thf(f12,plain,
( $true
= ( ~ ( ( ( !! @ ( $i > $o )
@ ^ [Y0: $i > $o] :
( ( ( !! @ $i
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
=> ( Y0 @ ( cS @ Y1 ) ) ) )
& ( Y0 @ c0 ) )
=> ( !! @ $i @ Y0 ) ) )
& ( !! @ $i
@ ^ [Y0: $i] : ( c_plus @ c0 @ Y0 @ Y0 ) )
& ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( c_plus @ Y0 @ Y2 @ Y1 )
=> ( c_plus @ ( cS @ Y0 ) @ Y2 @ ( cS @ Y1 ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] : ( ?? @ $i @ ( c_plus @ Y1 @ Y0 ) ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f11]) ).
thf(f11,plain,
( $true
= ( ~ ( ( ( !! @ ( $i > $o )
@ ^ [Y0: $i > $o] :
( ( ( !! @ $i
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
=> ( Y0 @ ( cS @ Y1 ) ) ) )
& ( Y0 @ c0 ) )
=> ( !! @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) ) ) )
& ( !! @ $i
@ ^ [Y0: $i] : ( c_plus @ c0 @ Y0 @ Y0 ) )
& ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( c_plus @ Y0 @ Y2 @ Y1 )
=> ( c_plus @ ( cS @ Y0 ) @ Y2 @ ( cS @ Y1 ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ?? @ $i
@ ^ [Y2: $i] : ( c_plus @ Y1 @ Y0 @ Y2 ) ) ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f9,f10]) ).
thf(f10,plain,
( cIND
= ( !! @ ( $i > $o )
@ ^ [Y0: $i > $o] :
( ( ( !! @ $i
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
=> ( Y0 @ ( cS @ Y1 ) ) ) )
& ( Y0 @ c0 ) )
=> ( !! @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f8]) ).
thf(f8,plain,
( cIND
= ( !! @ ( $i > $o )
@ ^ [Y0: $i > $o] :
( ( ( !! @ $i
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
=> ( Y0 @ ( cS @ Y1 ) ) ) )
& ( Y0 @ c0 ) )
=> ( !! @ $i
@ ^ [Y1: $i] : ( Y0 @ Y1 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f7]) ).
thf(f7,plain,
! [X0: $i > $o] :
( ( ( ( X0 @ c0 )
& ! [X1: $i] :
( ( X0 @ X1 )
=> ( X0 @ ( cS @ X1 ) ) ) )
=> ! [X2: $i] : ( X0 @ X2 ) )
= cIND ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
! [X0: $i > $o] :
( ( ( ( X0 @ c0 )
& ! [X1: $i] :
( ( X0 @ X1 )
=> ( X0 @ ( cS @ X1 ) ) ) )
=> ! [X1: $i] : ( X0 @ X1 ) )
= cIND ),
file('/export/starexec/sandbox/tmp/tmp.Nts2X4mKFM/DTF2THF_23053.p',cIND_def) ).
thf(f9,plain,
( $true
= ( ~ ( ( cIND
& ( !! @ $i
@ ^ [Y0: $i] : ( c_plus @ c0 @ Y0 @ Y0 ) )
& ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( c_plus @ Y0 @ Y2 @ Y1 )
=> ( c_plus @ ( cS @ Y0 ) @ Y2 @ ( cS @ Y1 ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ?? @ $i
@ ^ [Y2: $i] : ( c_plus @ Y1 @ Y0 @ Y2 ) ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f6]) ).
thf(f6,plain,
( $true
= ( ~ ( ( cIND
& ( !! @ $i
@ ^ [Y0: $i] : ( c_plus @ c0 @ Y0 @ Y0 ) )
& ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( c_plus @ Y0 @ Y2 @ Y1 )
=> ( c_plus @ ( cS @ Y0 ) @ Y2 @ ( cS @ Y1 ) ) ) ) ) ) )
=> ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ?? @ $i
@ ^ [Y2: $i] : ( c_plus @ Y1 @ Y0 @ Y2 ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f5]) ).
thf(f5,plain,
~ ( ( ! [X0: $i,X1: $i,X2: $i] :
( ( c_plus @ X2 @ X0 @ X1 )
=> ( c_plus @ ( cS @ X2 ) @ X0 @ ( cS @ X1 ) ) )
& ! [X3: $i] : ( c_plus @ c0 @ X3 @ X3 )
& cIND )
=> ! [X4: $i,X5: $i] :
? [X6: $i] : ( c_plus @ X4 @ X5 @ X6 ) ),
inference(rectify,[status(thm)],[f3]) ).
thf(f3,negated_conjecture,
~ ( ( ! [X1: $i,X3: $i,X2: $i] :
( ( c_plus @ X2 @ X1 @ X3 )
=> ( c_plus @ ( cS @ X2 ) @ X1 @ ( cS @ X3 ) ) )
& ! [X1: $i] : ( c_plus @ c0 @ X1 @ X1 )
& cIND )
=> ! [X2: $i,X1: $i] :
? [X3: $i] : ( c_plus @ X2 @ X1 @ X3 ) ),
inference(negated_conjecture,[status(cth)],[f2]) ).
thf(f2,conjecture,
( ( ! [X1: $i,X3: $i,X2: $i] :
( ( c_plus @ X2 @ X1 @ X3 )
=> ( c_plus @ ( cS @ X2 ) @ X1 @ ( cS @ X3 ) ) )
& ! [X1: $i] : ( c_plus @ c0 @ X1 @ X1 )
& cIND )
=> ! [X2: $i,X1: $i] :
? [X3: $i] : ( c_plus @ X2 @ X1 @ X3 ) ),
file('/export/starexec/sandbox/tmp/tmp.Nts2X4mKFM/DTF2THF_23053.p',cTHM129) ).
thf(f97,plain,
~ spl0_2,
inference(avatar_contradiction_clause,[status(thm)],[f96]) ).
thf(f96,plain,
( $false
| ~ spl0_2 ),
inference(trivial_inequality_removal,[status(thm)],[f91]) ).
thf(f91,plain,
( ( $true = $false )
| ~ spl0_2 ),
inference(superposition,[status(thm)],[f36,f89]) ).
thf(f89,plain,
( ! [X1: $i] :
( ( c_plus @ c0 @ sK2 @ X1 )
= $false )
| ~ spl0_2 ),
inference(pi_clausification,[status(thm)],[f85]) ).
thf(f85,plain,
( ( ( ?? @ $i @ ( c_plus @ c0 @ sK2 ) )
= $false )
| ~ spl0_2 ),
inference(avatar_component_clause,[status(thm)],[f84]) ).
thf(f84,definition,
( spl0_2
<=> ( ( ?? @ $i @ ( c_plus @ c0 @ sK2 ) )
= $false ) ),
introduced(definition,[new_symbols(naming,[spl0_2])],[avatar_definition]) ).
thf(f36,plain,
! [X1: $i] :
( ( c_plus @ c0 @ X1 @ X1 )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f35]) ).
thf(f35,plain,
! [X1: $i] :
( $true
= ( ^ [Y0: $i] : ( c_plus @ c0 @ Y0 @ Y0 )
@ X1 ) ),
inference(pi_clausification,[status(thm)],[f31]) ).
thf(f31,plain,
( $true
= ( !! @ $i
@ ^ [Y0: $i] : ( c_plus @ c0 @ Y0 @ Y0 ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f28]) ).
thf(f28,plain,
( $true
= ( ( !! @ ( $i > $o )
@ ^ [Y0: $i > $o] :
( ( ( !! @ $i
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
=> ( Y0 @ ( cS @ Y1 ) ) ) )
& ( Y0 @ c0 ) )
=> ( !! @ $i @ Y0 ) ) )
& ( !! @ $i
@ ^ [Y0: $i] : ( c_plus @ c0 @ Y0 @ Y0 ) ) ) ),
inference(boolean_simplification,[status(thm)],[f27]) ).
thf(f27,plain,
( $true
= ( ( !! @ ( $i > $o )
@ ^ [Y0: $i > $o] :
( ( ( !! @ $i
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
=> ( Y0 @ ( cS @ Y1 ) ) ) )
& ( Y0 @ c0 ) )
=> ( !! @ $i @ Y0 ) ) )
& ( !! @ $i
@ ^ [Y0: $i] : ( c_plus @ c0 @ Y0 @ Y0 ) )
& $true ) ),
inference(backward_demodulation,[status(thm)],[f17,f25]) ).
thf(f86,plain,
( spl0_1
| spl0_2 ),
inference(avatar_split_clause,[status(thm)],[f79,f84,f81]) ).
thf(f79,plain,
( ( ( ?? @ $i @ ( c_plus @ c0 @ sK2 ) )
= $false )
| ( ( !! @ $i
@ ^ [Y0: $i] :
( ( ?? @ $i @ ( c_plus @ Y0 @ sK2 ) )
=> ( ?? @ $i @ ( c_plus @ ( cS @ Y0 ) @ sK2 ) ) ) )
= $false ) ),
inference(binary_proxy_clausification,[status(thm)],[f72]) ).
thf(f72,plain,
( ( ( !! @ $i
@ ^ [Y0: $i] :
( ( ?? @ $i @ ( c_plus @ Y0 @ sK2 ) )
=> ( ?? @ $i @ ( c_plus @ ( cS @ Y0 ) @ sK2 ) ) ) )
& ( ?? @ $i @ ( c_plus @ c0 @ sK2 ) ) )
= $false ),
inference(not_proxy_clausification,[status(thm)],[f47]) ).
thf(f47,plain,
( $true
= ( ~ ( ( !! @ $i
@ ^ [Y0: $i] :
( ( ?? @ $i @ ( c_plus @ Y0 @ sK2 ) )
=> ( ?? @ $i @ ( c_plus @ ( cS @ Y0 ) @ sK2 ) ) ) )
& ( ?? @ $i @ ( c_plus @ c0 @ sK2 ) ) ) ) ),
inference(boolean_simplification,[status(thm)],[f46]) ).
thf(f46,plain,
( $true
= ( ( ( !! @ $i
@ ^ [Y0: $i] :
( ( ?? @ $i @ ( c_plus @ Y0 @ sK2 ) )
=> ( ?? @ $i @ ( c_plus @ ( cS @ Y0 ) @ sK2 ) ) ) )
& ( ?? @ $i @ ( c_plus @ c0 @ sK2 ) ) )
=> $false ) ),
inference(beta_eta_normalization,[status(thm)],[f42]) ).
thf(f42,plain,
( $true
= ( ( ( !! @ $i
@ ^ [Y0: $i] :
( ( ^ [Y1: $i] : ( ?? @ $i @ ( c_plus @ Y1 @ sK2 ) )
@ Y0 )
=> ( ^ [Y1: $i] : ( ?? @ $i @ ( c_plus @ Y1 @ sK2 ) )
@ ( cS @ Y0 ) ) ) )
& ( ^ [Y0: $i] : ( ?? @ $i @ ( c_plus @ Y0 @ sK2 ) )
@ c0 ) )
=> $false ) ),
inference(superposition,[status(thm)],[f38,f21]) ).
thf(f21,plain,
( ( !! @ $i
@ ^ [Y0: $i] : ( ?? @ $i @ ( c_plus @ Y0 @ sK2 ) ) )
= $false ),
inference(beta_eta_normalization,[status(thm)],[f20]) ).
thf(f20,plain,
( ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] : ( ?? @ $i @ ( c_plus @ Y1 @ Y0 ) ) )
@ sK2 )
= $false ),
inference(sigma_clausification,[status(thm)],[f19]) ).
thf(f19,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] : ( ?? @ $i @ ( c_plus @ Y1 @ Y0 ) ) ) )
= $false ),
inference(boolean_simplification,[status(thm)],[f18]) ).
thf(f18,plain,
( ( $true
=> ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] : ( ?? @ $i @ ( c_plus @ Y1 @ Y0 ) ) ) ) )
= $false ),
inference(backward_demodulation,[status(thm)],[f13,f17]) ).
thf(f38,plain,
! [X1: $i > $o] :
( ( ( ( !! @ $i
@ ^ [Y0: $i] :
( ( X1 @ Y0 )
=> ( X1 @ ( cS @ Y0 ) ) ) )
& ( X1 @ c0 ) )
=> ( !! @ $i @ X1 ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f37]) ).
thf(f37,plain,
! [X1: $i > $o] :
( $true
= ( ^ [Y0: $i > $o] :
( ( ( !! @ $i
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
=> ( Y0 @ ( cS @ Y1 ) ) ) )
& ( Y0 @ c0 ) )
=> ( !! @ $i @ Y0 ) )
@ X1 ) ),
inference(pi_clausification,[status(thm)],[f34]) ).
thf(f34,plain,
( $true
= ( !! @ ( $i > $o )
@ ^ [Y0: $i > $o] :
( ( ( !! @ $i
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
=> ( Y0 @ ( cS @ Y1 ) ) ) )
& ( Y0 @ c0 ) )
=> ( !! @ $i @ Y0 ) ) ) ),
inference(boolean_simplification,[status(thm)],[f33]) ).
thf(f33,plain,
( $true
= ( ( !! @ ( $i > $o )
@ ^ [Y0: $i > $o] :
( ( ( !! @ $i
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
=> ( Y0 @ ( cS @ Y1 ) ) ) )
& ( Y0 @ c0 ) )
=> ( !! @ $i @ Y0 ) ) )
& $true ) ),
inference(backward_demodulation,[status(thm)],[f28,f31]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : NUM811^5 : 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
% 0.15/0.34 % Computer : n002.cluster.edu
% 0.15/0.34 % Model : x86_64 x86_64
% 0.15/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.34 % Memory : 8042.1875MB
% 0.15/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.34 % CPULimit : 300
% 0.15/0.34 % WCLimit : 300
% 0.15/0.34 % DateTime : Tue Feb 24 11:15:03 EST 2026
% 0.15/0.34 % CPUTime :
% 0.15/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.30/0.46 ---- Original DTF file ---
% 0.30/0.46 thf(spec,logic,$$dhol).
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 % File : NUM811^5 : TPTP v9.2.1. Bugfixed v5.2.0.
% 0.30/0.46 % Domain : Number Theory (Induction on naturals)
% 0.30/0.46 % Problem : TPS problem THM129
% 0.30/0.46 % Version : Especial.
% 0.30/0.46 % English : Induction theorem for addition.
% 0.30/0.46
% 0.30/0.46 % Refs : [Bro09] Brown (2009), Email to Geoff Sutcliffe
% 0.30/0.46 % Source : [Bro09]
% 0.30/0.46 % Names : tps_0360 [Bro09]
% 0.30/0.46 % : tps_0882 [Bro09]
% 0.30/0.46 % : THM129 [TPS]
% 0.30/0.46
% 0.30/0.46 % Status : Theorem
% 0.30/0.46 % Rating : 0.78 v9.1.0, 0.75 v9.0.0, 0.80 v8.2.0, 0.85 v8.1.0, 0.82 v7.5.0, 0.71 v7.4.0, 0.67 v7.2.0, 0.62 v7.0.0, 0.71 v6.4.0, 0.67 v6.3.0, 0.80 v6.2.0, 0.71 v6.1.0, 0.57 v5.5.0, 0.83 v5.4.0, 0.80 v5.2.0
% 0.30/0.46 % Syntax : Number of formulae : 6 ( 1 unt; 4 typ; 1 def)
% 0.30/0.46 % Number of atoms : 7 ( 1 equ; 0 cnn)
% 0.30/0.46 % Maximal formula atoms : 5 ( 3 avg)
% 0.30/0.46 % Number of connectives : 26 ( 0 ~; 0 |; 3 &; 19 @)
% 0.30/0.46 % ( 0 <=>; 4 =>; 0 <=; 0 <~>)
% 0.30/0.46 % Maximal formula depth : 12 ( 7 avg)
% 0.30/0.46 % Number of types : 2 ( 0 usr)
% 0.30/0.46 % Number of type conns : 5 ( 5 >; 0 *; 0 +; 0 <<)
% 0.30/0.46 % Number of symbols : 5 ( 4 usr; 2 con; 0-3 aty)
% 0.30/0.46 % Number of variables : 10 ( 0 ^; 9 !; 1 ?; 10 :)
% 0.30/0.46 % SPC : TH0_THM_EQU_NAR
% 0.30/0.46
% 0.30/0.46 % Comments : This problem is from the TPS library. Copyright (c) 2009 The TPS
% 0.30/0.46 % project in the Department of Mathematical Sciences at Carnegie
% 0.30/0.46 % Mellon University. Distributed under the Creative Commons copyleft
% 0.30/0.46 % license: http://creativecommons.org/licenses/by-sa/3.0/
% 0.30/0.46 % :
% 0.30/0.46 % Bugfixes : v5.2.0 - Added missing type declarations.
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 thf(c0_type,type,
% 0.30/0.46 c0: $i ).
% 0.30/0.46
% 0.30/0.46 thf(cS_type,type,
% 0.30/0.46 cS: $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(c_plus_type,type,
% 0.30/0.46 c_plus: $i > $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(cIND_type,type,
% 0.30/0.46 cIND: $o ).
% 0.30/0.46
% 0.30/0.46 thf(cIND_def,definition,
% 0.30/0.46 ( cIND
% 0.30/0.46 = ( ! [Xp: $i > $o] :
% 0.30/0.46 ( ( ( Xp @ c0 )
% 0.30/0.46 & ! [Xx: $i] :
% 0.30/0.46 ( ( Xp @ Xx )
% 0.30/0.46 => ( Xp @ ( cS @ Xx ) ) ) )
% 0.30/0.46 => ! [Xx: $i] : ( Xp @ Xx ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cTHM129,conjecture,
% 0.30/0.46 ( ( cIND
% 0.30/0.46 & ! [Xx: $i] : ( c_plus @ c0 @ Xx @ Xx )
% 0.30/0.46 & ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/0.46 ( ( c_plus @ Xy @ Xx @ Xz )
% 0.30/0.46 => ( c_plus @ ( cS @ Xy ) @ Xx @ ( cS @ Xz ) ) ) )
% 0.30/0.46 => ! [Xy: $i,Xx: $i] :
% 0.30/0.46 ? [Xz: $i] : ( c_plus @ Xy @ Xx @ Xz ) ) ).
% 0.30/0.46
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 ------------------------
% 1.47/1.29 ---- Embedded in THF ---
% 1.47/1.29 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.47/1.29 %%% Generated on Tue Feb 24 11:15:04 EST 2026
% 1.47/1.29 %%% using '$$dhol' embedding, version 1.3.0.
% 1.47/1.29 %%% Logic specification used:
% 1.47/1.29 %%% thf(spec, logic, $$dhol).
% 1.47/1.29
% 1.47/1.29 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.Nts2X4mKFM/DTF2DTF23053.p
% See solution above
% 1.47/1.29 ------------------------
% 1.65/1.30 ---- Cleaned THF ---
% 1.65/1.30 thf(c0_type,type,
% 1.65/1.30 c0: $i ).
% 1.65/1.30
% 1.65/1.30 thf(cS_type,type,
% 1.65/1.30 cS: $i > $i ).
% 1.65/1.30
% 1.65/1.30 thf(c_plus_type,type,
% 1.65/1.30 c_plus: $i > $i > $i > $o ).
% 1.65/1.30
% 1.65/1.30 thf(cIND_type,type,
% 1.65/1.30 cIND: $o ).
% 1.65/1.30
% 1.65/1.30 thf(cIND_def,definition,
% 1.65/1.30 ( cIND
% 1.65/1.30 = ( ! [Xp: $i > $o] :
% 1.65/1.30 ( ( ( Xp @ c0 )
% 1.65/1.30 & ! [Xx: $i] :
% 1.65/1.30 ( ( Xp @ Xx )
% 1.65/1.30 => ( Xp @ ( cS @ Xx ) ) ) )
% 1.65/1.30 => ! [Xx: $i] : ( Xp @ Xx ) ) ) ) ).
% 1.65/1.30
% 1.65/1.30 thf(cTHM129,conjecture,
% 1.65/1.30 ( ( cIND
% 1.65/1.30 & ! [Xx: $i] : ( c_plus @ c0 @ Xx @ Xx )
% 1.65/1.30 & ! [Xx: $i,Xy: $i,Xz: $i] :
% 1.65/1.30 ( ( c_plus @ Xy @ Xx @ Xz )
% 1.65/1.30 => ( c_plus @ ( cS @ Xy ) @ Xx @ ( cS @ Xz ) ) ) )
% 1.65/1.30 => ! [Xy: $i,Xx: $i] :
% 1.65/1.30 ? [Xz: $i] : ( c_plus @ Xy @ Xx @ Xz ) ) ).
% 1.65/1.30 ------------------------
% 1.65/1.31 % (23329)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_23053 for (2999ds/2Mi)
% 1.65/1.32 % (23329)Instruction limit reached!
% 1.65/1.32 % (23329)------------------------------
% 1.65/1.32 % (23329)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.65/1.32 % (23329)Termination reason: Unknown
% 1.65/1.32 % (23329)Termination phase: Function definition elimination
% 1.65/1.32
% 1.65/1.32 % (23329)Memory used [KB]: 895
% 1.65/1.32 % (23329)Time elapsed: 0.002 s
% 1.65/1.32 % (23329)Instructions burned: 2 (million)
% 1.65/1.32 % (23329)------------------------------
% 1.65/1.32 % (23329)------------------------------
% 1.65/1.32 % (23328)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_23053 for (2999ds/27Mi)
% 1.65/1.32 % (23326)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_23053 for (2999ds/183Mi)
% 1.65/1.32 % (23327)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_23053 for (2999ds/4Mi)
% 1.65/1.32 % (23330)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_23053 for (2999ds/2Mi)
% 1.65/1.32 % (23331)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_23053 for (2999ds/275Mi)
% 1.65/1.32 % (23332)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_23053 for (2999ds/18Mi)
% 1.65/1.32 % (23330)Instruction limit reached!
% 1.65/1.32 % (23330)------------------------------
% 1.65/1.32 % (23330)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.65/1.32 % (23330)Termination reason: Unknown
% 1.65/1.32 % (23330)Termination phase: Property scanning
% 1.65/1.32
% 1.65/1.32 % (23330)Memory used [KB]: 895
% 1.65/1.32 % (23330)Time elapsed: 0.003 s
% 1.65/1.32 % (23330)Instructions burned: 2 (million)
% 1.65/1.32 % (23330)------------------------------
% 1.65/1.32 % (23330)------------------------------
% 1.65/1.32 % (23333)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_23053 for (2999ds/3Mi)
% 1.65/1.33 % (23327)Instruction limit reached!
% 1.65/1.33 % (23327)------------------------------
% 1.65/1.33 % (23327)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.65/1.33 % (23327)Termination reason: Unknown
% 1.65/1.33 % (23327)Termination phase: Saturation
% 1.65/1.33
% 1.65/1.33 % (23327)Memory used [KB]: 5500
% 1.65/1.33 % (23327)Time elapsed: 0.005 s
% 1.65/1.33 % (23327)Instructions burned: 4 (million)
% 1.65/1.33 % (23327)------------------------------
% 1.65/1.33 % (23327)------------------------------
% 1.65/1.33 % (23333)Instruction limit reached!
% 1.65/1.33 % (23333)------------------------------
% 1.65/1.33 % (23333)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.65/1.33 % (23333)Termination reason: Unknown
% 1.65/1.33 % (23333)Termination phase: Saturation
% 1.65/1.33
% 1.65/1.33 % (23333)Memory used [KB]: 5500
% 1.65/1.33 % (23333)Time elapsed: 0.003 s
% 1.65/1.33 % (23333)Instructions burned: 5 (million)
% 1.65/1.33 % (23333)------------------------------
% 1.65/1.33 % (23333)------------------------------
% 1.65/1.33 % (23328)Instruction limit reached!
% 1.65/1.33 % (23328)------------------------------
% 1.65/1.33 % (23328)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.65/1.33 % (23328)Termination reason: Unknown
% 1.65/1.33 % (23328)Termination phase: Saturation
% 1.65/1.33
% 1.65/1.33 % (23328)Memory used [KB]: 5628
% 1.65/1.33 % (23328)Time elapsed: 0.015 s
% 1.65/1.33 % (23328)Instructions burned: 29 (million)
% 1.65/1.33 % (23328)------------------------------
% 1.65/1.33 % (23328)------------------------------
% 1.65/1.33 % (23332)Instruction limit reached!
% 1.65/1.33 % (23332)------------------------------
% 1.65/1.33 % (23332)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.65/1.33 % (23332)Termination reason: Unknown
% 1.65/1.33 % (23332)Termination phase: Saturation
% 1.65/1.33
% 1.65/1.33 % (23332)Memory used [KB]: 5628
% 1.65/1.33 % (23332)Time elapsed: 0.014 s
% 1.65/1.33 % (23332)Instructions burned: 18 (million)
% 1.65/1.33 % (23332)------------------------------
% 1.65/1.33 % (23332)------------------------------
% 1.65/1.34 % (23336)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_23053 for (2999ds/3Mi)
% 1.65/1.34 % (23336)Refutation not found, incomplete strategy
% 1.65/1.34 % (23336)------------------------------
% 1.65/1.34 % (23336)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.65/1.34 % (23336)Termination reason: Refutation not found, incomplete strategy
% 1.65/1.34
% 1.65/1.34
% 1.65/1.34 % (23336)Memory used [KB]: 5500
% 1.65/1.34 % (23336)Time elapsed: 0.002 s
% 1.65/1.34 % (23336)Instructions burned: 2 (million)
% 1.65/1.34 % (23336)------------------------------
% 1.65/1.34 % (23336)------------------------------
% 1.65/1.34 % (23334)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_23053 for (2999ds/37Mi)
% 1.65/1.34 % (23337)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_23053 for (2999ds/1041Mi)
% 1.65/1.34 % (23335)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_23053 for (2999ds/15Mi)
% 1.65/1.35 % (23339)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_23053 for (2999ds/16Mi)
% 1.65/1.35 % (23338)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_23053 for (2999ds/7Mi)
% 1.65/1.35 % (23337)First to succeed.
% 1.65/1.35 % (23335)Instruction limit reached!
% 1.65/1.35 % (23335)------------------------------
% 1.65/1.35 % (23335)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.65/1.35 % (23335)Termination reason: Unknown
% 1.65/1.35 % (23335)Termination phase: Saturation
% 1.65/1.35
% 1.65/1.35 % (23335)Memory used [KB]: 5628
% 1.65/1.35 % (23335)Time elapsed: 0.012 s
% 1.65/1.35 % (23335)Instructions burned: 15 (million)
% 1.65/1.35 % (23335)------------------------------
% 1.65/1.35 % (23335)------------------------------
% 1.65/1.35 % (23339)Instruction limit reached!
% 1.65/1.35 % (23339)------------------------------
% 1.65/1.35 % (23339)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.65/1.35 % (23339)Termination reason: Unknown
% 1.65/1.35 % (23339)Termination phase: Saturation
% 1.65/1.35
% 1.65/1.35 % (23339)Memory used [KB]: 5756
% 1.65/1.35 % (23339)Time elapsed: 0.008 s
% 1.65/1.35 % (23339)Instructions burned: 17 (million)
% 1.65/1.35 % (23339)------------------------------
% 1.65/1.35 % (23339)------------------------------
% 1.65/1.35 % (23337)Refutation found. Thanks to Tanya!
% 1.65/1.35 % SZS status Theorem for DTF2THF_23053
% 1.65/1.35 % SZS output start Proof for DTF2THF_23053
% 1.65/1.35 thf(func_def_1, type, cS: $i > $i).
% 1.65/1.35 thf(func_def_2, type, c_plus: $i > $i > $i > $o).
% 1.65/1.35 thf(func_def_17, type, ph1: !>[X0: $tType]:(X0)).
% 1.65/1.35 thf(f142,plain,(
% 1.65/1.35 $false),
% 1.65/1.35 inference(avatar_sat_refutation,[status(thm)],[f86,f97,f141])).
% 1.65/1.35 thf(f141,plain,(
% 1.65/1.35 ~spl0_1),
% 1.65/1.35 inference(avatar_contradiction_clause,[status(thm)],[f140])).
% 1.65/1.35 thf(f140,plain,(
% 1.65/1.35 $false | ~spl0_1),
% 1.65/1.35 inference(trivial_inequality_removal,[status(thm)],[f139])).
% 1.65/1.35 thf(f139,plain,(
% 1.65/1.35 ($true = $false) | ~spl0_1),
% 1.65/1.35 inference(backward_demodulation,[status(thm)],[f127,f138])).
% 1.65/1.35 thf(f138,plain,(
% 1.65/1.35 ( ! [X1 : $i] : (((c_plus @ (cS @ sK4) @ sK2 @ X1) = $false)) ) | ~spl0_1),
% 1.65/1.35 inference(pi_clausification,[status(thm)],[f116])).
% 1.65/1.35 thf(f116,plain,(
% 1.65/1.35 ((?? @ $i @ (c_plus @ (cS @ sK4) @ sK2)) = $false) | ~spl0_1),
% 1.65/1.35 inference(binary_proxy_clausification,[status(thm)],[f111])).
% 1.65/1.35 thf(f111,plain,(
% 1.65/1.35 (((?? @ $i @ (c_plus @ sK4 @ sK2)) => (?? @ $i @ (c_plus @ (cS @ sK4) @ sK2))) = $false) | ~spl0_1),
% 1.65/1.35 inference(beta_eta_normalization,[status(thm)],[f107])).
% 1.65/1.35 thf(f107,plain,(
% 1.65/1.35 (((^[Y0 : $i]: ((?? @ $i @ (c_plus @ Y0 @ sK2)) => (?? @ $i @ (c_plus @ (cS @ Y0) @ sK2)))) @ sK4) = $false) | ~spl0_1),
% 1.65/1.35 inference(sigma_clausification,[status(thm)],[f82])).
% 1.65/1.35 thf(f82,plain,(
% 1.65/1.35 ((!! @ $i @ (^[Y0 : $i]: ((?? @ $i @ (c_plus @ Y0 @ sK2)) => (?? @ $i @ (c_plus @ (cS @ Y0) @ sK2))))) = $false) | ~spl0_1),
% 1.65/1.35 inference(avatar_component_clause,[status(thm)],[f81])).
% 1.65/1.35 thf(f81,definition,(
% 1.65/1.35 spl0_1 <=> ((!! @ $i @ (^[Y0 : $i]: ((?? @ $i @ (c_plus @ Y0 @ sK2)) => (?? @ $i @ (c_plus @ (cS @ Y0) @ sK2))))) = $false)),
% 1.65/1.35 introduced(definition,[new_symbols(naming,[spl0_1])],[avatar_definition])).
% 1.65/1.35 thf(f127,plain,(
% 1.65/1.35 ($true = (c_plus @ (cS @ sK4) @ sK2 @ (cS @ sK5))) | ~spl0_1),
% 1.65/1.35 inference(boolean_simplification,[status(thm)],[f126])).
% 1.65/1.35 thf(f126,plain,(
% 1.65/1.35 ($true = ($true => (c_plus @ (cS @ sK4) @ sK2 @ (cS @ sK5)))) | ~spl0_1),
% 1.65/1.35 inference(superposition,[status(thm)],[f64,f122])).
% 1.65/1.35 thf(f122,plain,(
% 1.65/1.35 ($true = (c_plus @ sK4 @ sK2 @ sK5)) | ~spl0_1),
% 1.65/1.35 inference(sigma_clausification,[status(thm)],[f117])).
% 1.65/1.35 thf(f117,plain,(
% 1.65/1.35 ($true = (?? @ $i @ (c_plus @ sK4 @ sK2))) | ~spl0_1),
% 1.65/1.35 inference(binary_proxy_clausification,[status(thm)],[f111])).
% 1.65/1.35 thf(f64,plain,(
% 1.65/1.35 ( ! [X2 : $i,X3 : $i,X1 : $i] : (($true = ((c_plus @ X1 @ X3 @ X2) => (c_plus @ (cS @ X1) @ X3 @ (cS @ X2))))) )),
% 1.65/1.35 inference(beta_eta_normalization,[status(thm)],[f62])).
% 1.65/1.35 thf(f62,plain,(
% 1.65/1.35 ( ! [X2 : $i,X3 : $i,X1 : $i] : (($true = ((^[Y0 : $i]: ((c_plus @ X1 @ Y0 @ X2) => (c_plus @ (cS @ X1) @ Y0 @ (cS @ X2)))) @ X3))) )),
% 1.65/1.35 inference(pi_clausification,[status(thm)],[f61])).
% 1.65/1.35 thf(f61,plain,(
% 1.65/1.35 ( ! [X2 : $i,X1 : $i] : (((!! @ $i @ (^[Y0 : $i]: ((c_plus @ X1 @ Y0 @ X2) => (c_plus @ (cS @ X1) @ Y0 @ (cS @ X2))))) = $true)) )),
% 1.65/1.35 inference(beta_eta_normalization,[status(thm)],[f57])).
% 1.65/1.35 thf(f57,plain,(
% 1.65/1.35 ( ! [X2 : $i,X1 : $i] : (($true = ((^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: ((c_plus @ X1 @ Y1 @ Y0) => (c_plus @ (cS @ X1) @ Y1 @ (cS @ Y0)))))) @ X2))) )),
% 1.65/1.35 inference(pi_clausification,[status(thm)],[f30])).
% 1.65/1.35 thf(f30,plain,(
% 1.65/1.35 ( ! [X1 : $i] : (($true = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: ((c_plus @ X1 @ Y1 @ Y0) => (c_plus @ (cS @ X1) @ Y1 @ (cS @ Y0))))))))) )),
% 1.65/1.35 inference(beta_eta_normalization,[status(thm)],[f29])).
% 1.65/1.35 thf(f29,plain,(
% 1.65/1.35 ( ! [X1 : $i] : (($true = ((^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((c_plus @ Y0 @ Y2 @ Y1) => (c_plus @ (cS @ Y0) @ Y2 @ (cS @ Y1)))))))) @ X1))) )),
% 1.65/1.35 inference(pi_clausification,[status(thm)],[f25])).
% 1.65/1.35 thf(f25,plain,(
% 1.65/1.35 ($true = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((c_plus @ Y0 @ Y2 @ Y1) => (c_plus @ (cS @ Y0) @ Y2 @ (cS @ Y1))))))))))),
% 1.65/1.35 inference(binary_proxy_clausification,[status(thm)],[f17])).
% 1.65/1.35 thf(f17,plain,(
% 1.65/1.35 ($true = (((!! @ ($i > $o) @ (^[Y0 : $i > $o]: (((!! @ $i @ (^[Y1 : $i]: ((Y0 @ Y1) => (Y0 @ (cS @ Y1))))) & (Y0 @ c0)) => (!! @ $i @ Y0)))) & (!! @ $i @ (^[Y0 : $i]: (c_plus @ c0 @ Y0 @ Y0)))) & (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((c_plus @ Y0 @ Y2 @ Y1) => (c_plus @ (cS @ Y0) @ Y2 @ (cS @ Y1)))))))))))),
% 1.65/1.35 inference(binary_proxy_clausification,[status(thm)],[f13])).
% 1.65/1.35 thf(f13,plain,(
% 1.65/1.35 (((((!! @ ($i > $o) @ (^[Y0 : $i > $o]: (((!! @ $i @ (^[Y1 : $i]: ((Y0 @ Y1) => (Y0 @ (cS @ Y1))))) & (Y0 @ c0)) => (!! @ $i @ Y0)))) & (!! @ $i @ (^[Y0 : $i]: (c_plus @ c0 @ Y0 @ Y0)))) & (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((c_plus @ Y0 @ Y2 @ Y1) => (c_plus @ (cS @ Y0) @ Y2 @ (cS @ Y1)))))))))) => (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (?? @ $i @ (c_plus @ Y1 @ Y0))))))) = $false)),
% 1.65/1.35 inference(not_proxy_clausification,[status(thm)],[f12])).
% 1.65/1.35 thf(f12,plain,(
% 1.65/1.35 ($true = (~ ((((!! @ ($i > $o) @ (^[Y0 : $i > $o]: (((!! @ $i @ (^[Y1 : $i]: ((Y0 @ Y1) => (Y0 @ (cS @ Y1))))) & (Y0 @ c0)) => (!! @ $i @ Y0)))) & (!! @ $i @ (^[Y0 : $i]: (c_plus @ c0 @ Y0 @ Y0)))) & (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((c_plus @ Y0 @ Y2 @ Y1) => (c_plus @ (cS @ Y0) @ Y2 @ (cS @ Y1)))))))))) => (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (?? @ $i @ (c_plus @ Y1 @ Y0)))))))))),
% 1.65/1.35 inference(beta_eta_normalization,[status(thm)],[f11])).
% 1.65/1.35 thf(f11,plain,(
% 1.65/1.35 ($true = (~ ((((!! @ ($i > $o) @ (^[Y0 : $i > $o]: (((!! @ $i @ (^[Y1 : $i]: ((Y0 @ Y1) => (Y0 @ (cS @ Y1))))) & (Y0 @ c0)) => (!! @ $i @ (^[Y1 : $i]: (Y0 @ Y1)))))) & (!! @ $i @ (^[Y0 : $i]: (c_plus @ c0 @ Y0 @ Y0)))) & (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((c_plus @ Y0 @ Y2 @ Y1) => (c_plus @ (cS @ Y0) @ Y2 @ (cS @ Y1)))))))))) => (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (?? @ $i @ (^[Y2 : $i]: (c_plus @ Y1 @ Y0 @ Y2))))))))))),
% 1.65/1.35 inference(definition_unfolding,[status(thm)],[f9,f10])).
% 1.65/1.35 thf(f10,plain,(
% 1.65/1.35 (cIND = (!! @ ($i > $o) @ (^[Y0 : $i > $o]: (((!! @ $i @ (^[Y1 : $i]: ((Y0 @ Y1) => (Y0 @ (cS @ Y1))))) & (Y0 @ c0)) => (!! @ $i @ (^[Y1 : $i]: (Y0 @ Y1)))))))),
% 1.65/1.35 inference(cnf_transformation,[status(thm)],[f8])).
% 1.65/1.35 thf(f8,plain,(
% 1.65/1.35 (cIND = (!! @ ($i > $o) @ (^[Y0 : $i > $o]: (((!! @ $i @ (^[Y1 : $i]: ((Y0 @ Y1) => (Y0 @ (cS @ Y1))))) & (Y0 @ c0)) => (!! @ $i @ (^[Y1 : $i]: (Y0 @ Y1)))))))),
% 1.65/1.35 inference(fool_elimination,[status(thm)],[f7])).
% 1.65/1.35 thf(f7,plain,(
% 1.65/1.35 (! [X0 : $i > $o] : (((X0 @ c0) & ! [X1 : $i] : ((X0 @ X1) => (X0 @ (cS @ X1)))) => ! [X2 : $i] : (X0 @ X2)) = cIND)),
% 1.65/1.35 inference(rectify,[status(thm)],[f1])).
% 1.65/1.35 thf(f1,axiom,(
% 1.65/1.35 (! [X0 : $i > $o] : (((X0 @ c0) & ! [X1 : $i] : ((X0 @ X1) => (X0 @ (cS @ X1)))) => ! [X1 : $i] : (X0 @ X1)) = cIND)),
% 1.65/1.35 file('/export/starexec/sandbox/tmp/tmp.Nts2X4mKFM/DTF2THF_23053.p',cIND_def)).
% 1.65/1.35 thf(f9,plain,(
% 1.65/1.35 ($true = (~ (((cIND & (!! @ $i @ (^[Y0 : $i]: (c_plus @ c0 @ Y0 @ Y0)))) & (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((c_plus @ Y0 @ Y2 @ Y1) => (c_plus @ (cS @ Y0) @ Y2 @ (cS @ Y1)))))))))) => (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (?? @ $i @ (^[Y2 : $i]: (c_plus @ Y1 @ Y0 @ Y2))))))))))),
% 1.65/1.35 inference(cnf_transformation,[status(thm)],[f6])).
% 1.65/1.35 thf(f6,plain,(
% 1.65/1.35 ($true = (~ (((cIND & (!! @ $i @ (^[Y0 : $i]: (c_plus @ c0 @ Y0 @ Y0)))) & (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((c_plus @ Y0 @ Y2 @ Y1) => (c_plus @ (cS @ Y0) @ Y2 @ (cS @ Y1)))))))))) => (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (?? @ $i @ (^[Y2 : $i]: (c_plus @ Y1 @ Y0 @ Y2))))))))))),
% 1.65/1.35 inference(fool_elimination,[status(thm)],[f5])).
% 1.65/1.35 thf(f5,plain,(
% 1.65/1.35 ~((! [X0 : $i,X1 : $i,X2 : $i] : ((c_plus @ X2 @ X0 @ X1) => (c_plus @ (cS @ X2) @ X0 @ (cS @ X1))) & ! [X3 : $i] : (c_plus @ c0 @ X3 @ X3) & cIND) => ! [X4 : $i,X5 : $i] : ? [X6 : $i] : (c_plus @ X4 @ X5 @ X6))),
% 1.65/1.35 inference(rectify,[status(thm)],[f3])).
% 1.65/1.35 thf(f3,negated_conjecture,(
% 1.65/1.35 ~((! [X1 : $i,X3 : $i,X2 : $i] : ((c_plus @ X2 @ X1 @ X3) => (c_plus @ (cS @ X2) @ X1 @ (cS @ X3))) & ! [X1 : $i] : (c_plus @ c0 @ X1 @ X1) & cIND) => ! [X2 : $i,X1 : $i] : ? [X3 : $i] : (c_plus @ X2 @ X1 @ X3))),
% 1.65/1.35 inference(negated_conjecture,[status(cth)],[f2])).
% 1.65/1.35 thf(f2,conjecture,(
% 1.65/1.35 (! [X1 : $i,X3 : $i,X2 : $i] : ((c_plus @ X2 @ X1 @ X3) => (c_plus @ (cS @ X2) @ X1 @ (cS @ X3))) & ! [X1 : $i] : (c_plus @ c0 @ X1 @ X1) & cIND) => ! [X2 : $i,X1 : $i] : ? [X3 : $i] : (c_plus @ X2 @ X1 @ X3)),
% 1.65/1.35 file('/export/starexec/sandbox/tmp/tmp.Nts2X4mKFM/DTF2THF_23053.p',cTHM129)).
% 1.65/1.35 thf(f97,plain,(
% 1.65/1.35 ~spl0_2),
% 1.65/1.35 inference(avatar_contradiction_clause,[status(thm)],[f96])).
% 1.65/1.35 thf(f96,plain,(
% 1.65/1.35 $false | ~spl0_2),
% 1.65/1.35 inference(trivial_inequality_removal,[status(thm)],[f91])).
% 1.65/1.35 thf(f91,plain,(
% 1.65/1.35 ($true = $false) | ~spl0_2),
% 1.65/1.35 inference(superposition,[status(thm)],[f36,f89])).
% 1.65/1.35 thf(f89,plain,(
% 1.65/1.35 ( ! [X1 : $i] : (((c_plus @ c0 @ sK2 @ X1) = $false)) ) | ~spl0_2),
% 1.65/1.35 inference(pi_clausification,[status(thm)],[f85])).
% 1.65/1.35 thf(f85,plain,(
% 1.65/1.35 ((?? @ $i @ (c_plus @ c0 @ sK2)) = $false) | ~spl0_2),
% 1.65/1.35 inference(avatar_component_clause,[status(thm)],[f84])).
% 1.65/1.35 thf(f84,definition,(
% 1.65/1.35 spl0_2 <=> ((?? @ $i @ (c_plus @ c0 @ sK2)) = $false)),
% 1.65/1.35 introduced(definition,[new_symbols(naming,[spl0_2])],[avatar_definition])).
% 1.65/1.35 thf(f36,plain,(
% 1.65/1.35 ( ! [X1 : $i] : (((c_plus @ c0 @ X1 @ X1) = $true)) )),
% 1.65/1.35 inference(beta_eta_normalization,[status(thm)],[f35])).
% 1.65/1.35 thf(f35,plain,(
% 1.65/1.35 ( ! [X1 : $i] : (($true = ((^[Y0 : $i]: (c_plus @ c0 @ Y0 @ Y0)) @ X1))) )),
% 1.65/1.35 inference(pi_clausification,[status(thm)],[f31])).
% 1.65/1.35 thf(f31,plain,(
% 1.65/1.35 ($true = (!! @ $i @ (^[Y0 : $i]: (c_plus @ c0 @ Y0 @ Y0))))),
% 1.65/1.35 inference(binary_proxy_clausification,[status(thm)],[f28])).
% 1.65/1.35 thf(f28,plain,(
% 1.65/1.35 ($true = ((!! @ ($i > $o) @ (^[Y0 : $i > $o]: (((!! @ $i @ (^[Y1 : $i]: ((Y0 @ Y1) => (Y0 @ (cS @ Y1))))) & (Y0 @ c0)) => (!! @ $i @ Y0)))) & (!! @ $i @ (^[Y0 : $i]: (c_plus @ c0 @ Y0 @ Y0)))))),
% 1.65/1.35 inference(boolean_simplification,[status(thm)],[f27])).
% 1.65/1.35 thf(f27,plain,(
% 1.65/1.35 ($true = (((!! @ ($i > $o) @ (^[Y0 : $i > $o]: (((!! @ $i @ (^[Y1 : $i]: ((Y0 @ Y1) => (Y0 @ (cS @ Y1))))) & (Y0 @ c0)) => (!! @ $i @ Y0)))) & (!! @ $i @ (^[Y0 : $i]: (c_plus @ c0 @ Y0 @ Y0)))) & $true))),
% 1.65/1.35 inference(backward_demodulation,[status(thm)],[f17,f25])).
% 1.65/1.35 thf(f86,plain,(
% 1.65/1.35 spl0_1 | spl0_2),
% 1.65/1.35 inference(avatar_split_clause,[status(thm)],[f79,f84,f81])).
% 1.65/1.35 thf(f79,plain,(
% 1.65/1.35 ((?? @ $i @ (c_plus @ c0 @ sK2)) = $false) | ((!! @ $i @ (^[Y0 : $i]: ((?? @ $i @ (c_plus @ Y0 @ sK2)) => (?? @ $i @ (c_plus @ (cS @ Y0) @ sK2))))) = $false)),
% 1.65/1.35 inference(binary_proxy_clausification,[status(thm)],[f72])).
% 1.65/1.35 thf(f72,plain,(
% 1.65/1.35 (((!! @ $i @ (^[Y0 : $i]: ((?? @ $i @ (c_plus @ Y0 @ sK2)) => (?? @ $i @ (c_plus @ (cS @ Y0) @ sK2))))) & (?? @ $i @ (c_plus @ c0 @ sK2))) = $false)),
% 1.65/1.35 inference(not_proxy_clausification,[status(thm)],[f47])).
% 1.65/1.35 thf(f47,plain,(
% 1.65/1.35 ($true = (~ ((!! @ $i @ (^[Y0 : $i]: ((?? @ $i @ (c_plus @ Y0 @ sK2)) => (?? @ $i @ (c_plus @ (cS @ Y0) @ sK2))))) & (?? @ $i @ (c_plus @ c0 @ sK2)))))),
% 1.65/1.35 inference(boolean_simplification,[status(thm)],[f46])).
% 1.65/1.35 thf(f46,plain,(
% 1.65/1.35 ($true = (((!! @ $i @ (^[Y0 : $i]: ((?? @ $i @ (c_plus @ Y0 @ sK2)) => (?? @ $i @ (c_plus @ (cS @ Y0) @ sK2))))) & (?? @ $i @ (c_plus @ c0 @ sK2))) => $false))),
% 1.65/1.35 inference(beta_eta_normalization,[status(thm)],[f42])).
% 1.65/1.35 thf(f42,plain,(
% 1.65/1.35 ($true = (((!! @ $i @ (^[Y0 : $i]: (((^[Y1 : $i]: (?? @ $i @ (c_plus @ Y1 @ sK2))) @ Y0) => ((^[Y1 : $i]: (?? @ $i @ (c_plus @ Y1 @ sK2))) @ (cS @ Y0))))) & ((^[Y0 : $i]: (?? @ $i @ (c_plus @ Y0 @ sK2))) @ c0)) => $false))),
% 1.65/1.35 inference(superposition,[status(thm)],[f38,f21])).
% 1.65/1.35 thf(f21,plain,(
% 1.65/1.35 ((!! @ $i @ (^[Y0 : $i]: (?? @ $i @ (c_plus @ Y0 @ sK2)))) = $false)),
% 1.65/1.35 inference(beta_eta_normalization,[status(thm)],[f20])).
% 1.65/1.35 thf(f20,plain,(
% 1.65/1.35 (((^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (?? @ $i @ (c_plus @ Y1 @ Y0))))) @ sK2) = $false)),
% 1.65/1.35 inference(sigma_clausification,[status(thm)],[f19])).
% 1.65/1.35 thf(f19,plain,(
% 1.65/1.35 ((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (?? @ $i @ (c_plus @ Y1 @ Y0)))))) = $false)),
% 1.65/1.35 inference(boolean_simplification,[status(thm)],[f18])).
% 1.65/1.35 thf(f18,plain,(
% 1.65/1.35 (($true => (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (?? @ $i @ (c_plus @ Y1 @ Y0))))))) = $false)),
% 1.65/1.35 inference(backward_demodulation,[status(thm)],[f13,f17])).
% 1.65/1.35 thf(f38,plain,(
% 1.65/1.35 ( ! [X1 : $i > $o] : (((((!! @ $i @ (^[Y0 : $i]: ((X1 @ Y0) => (X1 @ (cS @ Y0))))) & (X1 @ c0)) => (!! @ $i @ X1)) = $true)) )),
% 1.65/1.35 inference(beta_eta_normalization,[status(thm)],[f37])).
% 1.65/1.36 thf(f37,plain,(
% 1.65/1.36 ( ! [X1 : $i > $o] : (($true = ((^[Y0 : $i > $o]: (((!! @ $i @ (^[Y1 : $i]: ((Y0 @ Y1) => (Y0 @ (cS @ Y1))))) & (Y0 @ c0)) => (!! @ $i @ Y0))) @ X1))) )),
% 1.65/1.36 inference(pi_clausification,[status(thm)],[f34])).
% 1.65/1.36 thf(f34,plain,(
% 1.65/1.36 ($true = (!! @ ($i > $o) @ (^[Y0 : $i > $o]: (((!! @ $i @ (^[Y1 : $i]: ((Y0 @ Y1) => (Y0 @ (cS @ Y1))))) & (Y0 @ c0)) => (!! @ $i @ Y0)))))),
% 1.65/1.36 inference(boolean_simplification,[status(thm)],[f33])).
% 1.65/1.36 thf(f33,plain,(
% 1.65/1.36 ($true = ((!! @ ($i > $o) @ (^[Y0 : $i > $o]: (((!! @ $i @ (^[Y1 : $i]: ((Y0 @ Y1) => (Y0 @ (cS @ Y1))))) & (Y0 @ c0)) => (!! @ $i @ Y0)))) & $true))),
% 1.65/1.36 inference(backward_demodulation,[status(thm)],[f28,f31])).
% 1.65/1.36 % SZS output end Proof for DTF2THF_23053
% 1.65/1.36 % (23337)------------------------------
% 1.65/1.36 % (23337)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.65/1.36 % (23337)Termination reason: Refutation
% 1.65/1.36
% 1.65/1.36 % (23337)Memory used [KB]: 5628
% 1.65/1.36 % (23337)Time elapsed: 0.014 s
% 1.65/1.36 % (23337)Instructions burned: 27 (million)
% 1.65/1.36 % (23337)------------------------------
% 1.65/1.36 % (23337)------------------------------
% 1.65/1.36 % (23325)Success in time 0.046 s
%------------------------------------------------------------------------------