%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU810^2 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n015.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 09:37:24 AM UTC 2026
% Result : Theorem 1.42s 1.27s
% Output : Refutation 1.42s
% Verified :
% SZS Type : Refutation
% Derivation depth : 17
% Number of leaves : 8
% Syntax : Number of formulae : 48 ( 28 unt; 0 typ; 2 def)
% Number of atoms : 251 ( 64 equ; 0 cnn)
% Maximal formula atoms : 6 ( 5 avg)
% Number of connectives : 247 ( 17 ~; 10 |; 14 &; 160 @)
% ( 2 <=>; 25 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 3 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 16 ( 12 usr; 11 con; 0-2 aty)
% ( 19 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 61 ( 28 ^; 30 !; 3 ?; 61 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_2,type,
setadjoin: $i > $i > $i ).
thf(func_def_8,type,
omegaS: $i > $i ).
thf(func_def_9,type,
vEPSILON:
!>[X0: $tType] : ( ( X0 > $o ) > X0 ) ).
thf(f78,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f65,f71,f77]) ).
thf(f77,plain,
~ spl1_1,
inference(avatar_contradiction_clause,[status(thm)],[f76]) ).
thf(f76,plain,
( $false
| ~ spl1_1 ),
inference(equality_resolution,[status(thm)],[f73]) ).
thf(f73,plain,
( ! [X0: $o] : ( $false != X0 )
| ~ spl1_1 ),
inference(superposition,[status(thm)],[f7,f61]) ).
thf(f61,plain,
( ! [X2: $o] : ( $true = X2 )
| ~ spl1_1 ),
inference(avatar_component_clause,[status(thm)],[f60]) ).
thf(f60,definition,
( spl1_1
<=> ! [X2: $o] : ( $true = X2 ) ),
introduced(definition,[new_symbols(naming,[spl1_1])],[avatar_definition]) ).
thf(f7,plain,
$true != $false,
introduced(definition,[],[fool_axiom]) ).
thf(f71,plain,
~ spl1_2,
inference(avatar_contradiction_clause,[status(thm)],[f70]) ).
thf(f70,plain,
( $false
| ~ spl1_2 ),
inference(trivial_inequality_removal,[status(thm)],[f67]) ).
thf(f67,plain,
( ( $true = $false )
| ~ spl1_2 ),
inference(superposition,[status(thm)],[f43,f64]) ).
thf(f64,plain,
( ! [X1: $i] :
( ( in @ X1 @ ( setadjoin @ sK0 @ sK0 ) )
= $false )
| ~ spl1_2 ),
inference(avatar_component_clause,[status(thm)],[f63]) ).
thf(f63,definition,
( spl1_2
<=> ! [X1: $i] :
( ( in @ X1 @ ( setadjoin @ sK0 @ sK0 ) )
= $false ) ),
introduced(definition,[new_symbols(naming,[spl1_2])],[avatar_definition]) ).
thf(f43,plain,
! [X2: $i,X1: $i] :
( $true
= ( in @ X1 @ ( setadjoin @ X1 @ X2 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f42]) ).
thf(f42,plain,
! [X2: $i,X1: $i] :
( $true
= ( ^ [Y0: $i] : ( in @ X1 @ ( setadjoin @ X1 @ Y0 ) )
@ X2 ) ),
inference(pi_clausification,[status(thm)],[f41]) ).
thf(f41,plain,
! [X1: $i] :
( ( !! @ $i
@ ^ [Y0: $i] : ( in @ X1 @ ( setadjoin @ X1 @ Y0 ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f40]) ).
thf(f40,plain,
! [X1: $i] :
( $true
= ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] : ( in @ Y0 @ ( setadjoin @ Y0 @ Y1 ) ) )
@ X1 ) ),
inference(pi_clausification,[status(thm)],[f32]) ).
thf(f32,plain,
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] : ( in @ Y0 @ ( setadjoin @ Y0 @ Y1 ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f27,f25]) ).
thf(f25,plain,
setadjoinIL = $true,
inference(cnf_transformation,[status(thm)],[f20]) ).
thf(f20,plain,
( ( setadjoinIL = $true )
& ( emptysetE = $true )
& ( emptyset
= ( omegaS @ sK0 ) )
& ( $true
= ( in @ sK0 @ omega ) )
& ( in__Cong = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0])],[f18,f19]) ).
thf(f19,plain,
( ? [X0: $i] :
( ( emptyset
= ( omegaS @ X0 ) )
& ( ( in @ X0 @ omega )
= $true ) )
=> ( ( emptyset
= ( omegaS @ sK0 ) )
& ( $true
= ( in @ sK0 @ omega ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f18,plain,
( ( setadjoinIL = $true )
& ( emptysetE = $true )
& ? [X0: $i] :
( ( emptyset
= ( omegaS @ X0 ) )
& ( ( in @ X0 @ omega )
= $true ) )
& ( in__Cong = $true ) ),
inference(flattening,[status(thm)],[f17]) ).
thf(f17,plain,
( ? [X0: $i] :
( ( emptyset
= ( omegaS @ X0 ) )
& ( ( in @ X0 @ omega )
= $true ) )
& ( in__Cong = $true )
& ( setadjoinIL = $true )
& ( emptysetE = $true ) ),
inference(ennf_transformation,[status(thm)],[f16]) ).
thf(f16,plain,
~ ( ( emptysetE = $true )
=> ( ( setadjoinIL = $true )
=> ( ( in__Cong = $true )
=> ! [X0: $i] :
( ( ( in @ X0 @ omega )
= $true )
=> ( emptyset
!= ( omegaS @ X0 ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f15]) ).
thf(f15,plain,
~ ( emptysetE
=> ( setadjoinIL
=> ( in__Cong
=> ! [X0: $i] :
( ( in @ X0 @ omega )
=> ( emptyset
!= ( omegaS @ X0 ) ) ) ) ) ),
inference(rectify,[status(thm)],[f6]) ).
thf(f6,negated_conjecture,
~ ( emptysetE
=> ( setadjoinIL
=> ( in__Cong
=> ! [X0: $i] :
( ( in @ X0 @ omega )
=> ( emptyset
!= ( omegaS @ X0 ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f5]) ).
thf(f5,conjecture,
( emptysetE
=> ( setadjoinIL
=> ( in__Cong
=> ! [X0: $i] :
( ( in @ X0 @ omega )
=> ( emptyset
!= ( omegaS @ X0 ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.R7K9szfKav/DTF2THF_25921.p',peano0notS) ).
thf(f27,plain,
( setadjoinIL
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] : ( in @ Y0 @ ( setadjoin @ Y0 @ Y1 ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f14]) ).
thf(f14,plain,
( setadjoinIL
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] : ( in @ Y0 @ ( setadjoin @ Y0 @ Y1 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f13]) ).
thf(f13,plain,
( setadjoinIL
= ( ! [X0: $i,X1: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X0 ) ) ) ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
( setadjoinIL
= ( ! [X2: $i,X0: $i] : ( in @ X0 @ ( setadjoin @ X0 @ X2 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.R7K9szfKav/DTF2THF_25921.p',setadjoinIL) ).
thf(f65,plain,
( spl1_1
| spl1_2 ),
inference(avatar_split_clause,[status(thm)],[f39,f63,f60]) ).
thf(f39,plain,
! [X2: $o,X1: $i] :
( ( $true = X2 )
| ( ( in @ X1 @ ( setadjoin @ sK0 @ sK0 ) )
= $false ) ),
inference(beta_eta_normalization,[status(thm)],[f38]) ).
thf(f38,plain,
! [X2: $o,X1: $i] :
( ( $true
= ( ^ [Y0: $o] : Y0
@ X2 ) )
| ( ( in @ X1 @ ( setadjoin @ sK0 @ sK0 ) )
= $false ) ),
inference(pi_clausification,[status(thm)],[f37]) ).
thf(f37,plain,
! [X1: $i] :
( ( $true
= ( !! @ $o
@ ^ [Y0: $o] : Y0 ) )
| ( ( in @ X1 @ ( setadjoin @ sK0 @ sK0 ) )
= $false ) ),
inference(binary_proxy_clausification,[status(thm)],[f36]) ).
thf(f36,plain,
! [X1: $i] :
( $true
= ( ( in @ X1 @ ( setadjoin @ sK0 @ sK0 ) )
=> ( !! @ $o
@ ^ [Y0: $o] : Y0 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f35]) ).
thf(f35,plain,
! [X1: $i] :
( $true
= ( ^ [Y0: $i] :
( ( in @ Y0 @ ( setadjoin @ sK0 @ sK0 ) )
=> ( !! @ $o
@ ^ [Y1: $o] : Y1 ) )
@ X1 ) ),
inference(pi_clausification,[status(thm)],[f34]) ).
thf(f34,plain,
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ ( setadjoin @ sK0 @ sK0 ) )
=> ( !! @ $o
@ ^ [Y1: $o] : Y1 ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f33]) ).
thf(f33,plain,
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0
@ ( ^ [Y1: $i] : ( setadjoin @ Y1 @ Y1 )
@ sK0 ) )
=> ( !! @ $o
@ ^ [Y1: $o] : Y1 ) ) ) ),
inference(definition_unfolding,[status(thm)],[f29,f24,f30]) ).
thf(f30,plain,
( emptyset
= ( ^ [Y0: $i] : ( setadjoin @ Y0 @ Y0 )
@ sK0 ) ),
inference(definition_unfolding,[status(thm)],[f23,f28]) ).
thf(f28,plain,
( omegaS
= ( ^ [Y0: $i] : ( setadjoin @ Y0 @ Y0 ) ) ),
inference(cnf_transformation,[status(thm)],[f8]) ).
thf(f8,plain,
( omegaS
= ( ^ [Y0: $i] : ( setadjoin @ Y0 @ Y0 ) ) ),
inference(fool_elimination,[status(thm)],[f4]) ).
thf(f4,axiom,
( ( ^ [X0: $i] : ( setadjoin @ X0 @ X0 ) )
= omegaS ),
file('/export/starexec/sandbox/tmp/tmp.R7K9szfKav/DTF2THF_25921.p',omegaS) ).
thf(f23,plain,
( emptyset
= ( omegaS @ sK0 ) ),
inference(cnf_transformation,[status(thm)],[f20]) ).
thf(f24,plain,
emptysetE = $true,
inference(cnf_transformation,[status(thm)],[f20]) ).
thf(f29,plain,
( emptysetE
= ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ emptyset )
=> ( !! @ $o
@ ^ [Y1: $o] : Y1 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f12]) ).
thf(f12,plain,
( emptysetE
= ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ emptyset )
=> ( !! @ $o
@ ^ [Y1: $o] : Y1 ) ) ) ),
inference(fool_elimination,[status(thm)],[f11]) ).
thf(f11,plain,
! [X0: $i] :
( ( ( in @ X0 @ emptyset )
=> ! [X1: $o] : X1 )
= emptysetE ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
! [X0: $i] :
( ( ( in @ X0 @ emptyset )
=> ! [X1: $o] : X1 )
= emptysetE ),
file('/export/starexec/sandbox/tmp/tmp.R7K9szfKav/DTF2THF_25921.p',emptysetE) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU810^2 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.12 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.15/0.33 % Computer : n015.cluster.edu
% 0.15/0.33 % Model : x86_64 x86_64
% 0.15/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.33 % Memory : 8042.1875MB
% 0.15/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.33 % CPULimit : 300
% 0.15/0.33 % WCLimit : 300
% 0.15/0.33 % DateTime : Tue Feb 24 12:35:01 EST 2026
% 0.15/0.33 % CPUTime :
% 0.15/0.33 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.19/0.47 ---- Original DTF file ---
% 0.19/0.47 thf(spec,logic,$$dhol).
% 0.19/0.47 %------------------------------------------------------------------------------
% 0.19/0.47 % File : SEU810^2 : TPTP v9.2.1. Released v3.7.0.
% 0.19/0.47 % Domain : Set Theory
% 0.19/0.47 % Problem : Omega and Peano
% 0.19/0.47 % Version : Especial > Reduced > Especial.
% 0.19/0.47 % English : (! x:i.in x omega -> ~(omegaS x = emptyset))
% 0.19/0.47
% 0.19/0.47 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.19/0.47 % Source : [Bro08]
% 0.19/0.47 % Names : ZFC312l [Bro08]
% 0.19/0.47
% 0.19/0.47 % Status : Theorem
% 0.19/0.47 % Rating : 0.00 v8.2.0, 0.08 v8.1.0, 0.00 v7.1.0, 0.12 v7.0.0, 0.00 v6.2.0, 0.14 v5.5.0, 0.17 v5.4.0, 0.20 v5.3.0, 0.40 v5.2.0, 0.20 v4.1.0, 0.00 v3.7.0
% 0.19/0.47 % Syntax : Number of formulae : 13 ( 4 unt; 8 typ; 4 def)
% 0.19/0.47 % Number of atoms : 18 ( 7 equ; 0 cnn)
% 0.19/0.47 % Maximal formula atoms : 5 ( 3 avg)
% 0.19/0.47 % Number of connectives : 24 ( 1 ~; 0 |; 0 &; 15 @)
% 0.19/0.47 % ( 1 <=>; 7 =>; 0 <=; 0 <~>)
% 0.19/0.47 % Maximal formula depth : 8 ( 2 avg)
% 0.19/0.47 % Number of types : 2 ( 0 usr)
% 0.19/0.47 % Number of type conns : 5 ( 5 >; 0 *; 0 +; 0 <<)
% 0.19/0.47 % Number of symbols : 9 ( 8 usr; 5 con; 0-2 aty)
% 0.19/0.47 % Number of variables : 10 ( 1 ^; 9 !; 0 ?; 10 :)
% 0.19/0.47 % SPC : TH0_THM_EQU_NAR
% 0.19/0.47
% 0.19/0.47 % Comments : http://mathgate.info/detsetitem.php?id=443
% 0.19/0.47 %------------------------------------------------------------------------------
% 0.19/0.47 thf(in_type,type,
% 0.19/0.47 in: $i > $i > $o ).
% 0.19/0.47
% 0.19/0.47 thf(emptyset_type,type,
% 0.19/0.47 emptyset: $i ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoin_type,type,
% 0.19/0.47 setadjoin: $i > $i > $i ).
% 0.19/0.47
% 0.19/0.47 thf(omega_type,type,
% 0.19/0.47 omega: $i ).
% 0.19/0.47
% 0.19/0.47 thf(emptysetE_type,type,
% 0.19/0.47 emptysetE: $o ).
% 0.19/0.47
% 0.19/0.47 thf(emptysetE,definition,
% 0.19/0.47 ( emptysetE
% 0.19/0.47 = ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ emptyset )
% 0.19/0.47 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoinIL_type,type,
% 0.19/0.47 setadjoinIL: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoinIL,definition,
% 0.19/0.47 ( setadjoinIL
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(in__Cong_type,type,
% 0.19/0.47 in__Cong: $o ).
% 0.19/0.47
% 0.19/0.47 thf(in__Cong,definition,
% 0.19/0.47 ( in__Cong
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( A = B )
% 0.19/0.47 => ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( Xx = Xy )
% 0.19/0.47 => ( ( in @ Xx @ A )
% 0.19/0.47 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(omegaS_type,type,
% 0.19/0.47 omegaS: $i > $i ).
% 0.19/0.47
% 0.19/0.47 thf(omegaS,definition,
% 0.19/0.47 ( omegaS
% 0.19/0.47 = ( ^ [Xx: $i] : ( setadjoin @ Xx @ Xx ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(peano0notS,conjecture,
% 0.19/0.47 ( emptysetE
% 0.19/0.47 => ( setadjoinIL
% 0.19/0.47 => ( in__Cong
% 0.19/0.47 => ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ omega )
% 0.19/0.47 => ( ( omegaS @ Xx )
% 0.19/0.47 != emptyset ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 %------------------------------------------------------------------------------
% 0.19/0.47 ------------------------
% 1.42/1.25 ---- Embedded in THF ---
% 1.42/1.25 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.42/1.25 %%% Generated on Tue Feb 24 12:35:02 EST 2026
% 1.42/1.25 %%% using '$$dhol' embedding, version 1.3.0.
% 1.42/1.25 %%% Logic specification used:
% 1.42/1.25 %%% thf(spec, logic, $$dhol).
% 1.42/1.25
% 1.42/1.25 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.R7K9szfKav/DTF2DTF25921.p
% See solution above
% 1.42/1.25 ------------------------
% 1.42/1.26 ---- Cleaned THF ---
% 1.42/1.26 thf(in_type,type,
% 1.42/1.26 in: $i > $i > $o ).
% 1.42/1.26
% 1.42/1.26 thf(emptyset_type,type,
% 1.42/1.26 emptyset: $i ).
% 1.42/1.26
% 1.42/1.26 thf(setadjoin_type,type,
% 1.42/1.26 setadjoin: $i > $i > $i ).
% 1.42/1.26
% 1.42/1.26 thf(omega_type,type,
% 1.42/1.26 omega: $i ).
% 1.42/1.26
% 1.42/1.26 thf(emptysetE_type,type,
% 1.42/1.26 emptysetE: $o ).
% 1.42/1.26
% 1.42/1.26 thf(emptysetE,definition,
% 1.42/1.26 ( emptysetE
% 1.42/1.26 = ( ! [Xx: $i] :
% 1.42/1.26 ( ( in @ Xx @ emptyset )
% 1.42/1.26 => ! [Xphi: $o] : Xphi ) ) ) ).
% 1.42/1.26
% 1.42/1.26 thf(setadjoinIL_type,type,
% 1.42/1.26 setadjoinIL: $o ).
% 1.42/1.26
% 1.42/1.26 thf(setadjoinIL,definition,
% 1.42/1.26 ( setadjoinIL
% 1.42/1.26 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 1.42/1.26
% 1.42/1.26 thf(in__Cong_type,type,
% 1.42/1.26 in__Cong: $o ).
% 1.42/1.26
% 1.42/1.26 thf(in__Cong,definition,
% 1.42/1.26 ( in__Cong
% 1.42/1.26 = ( ! [A: $i,B: $i] :
% 1.42/1.26 ( ( A = B )
% 1.42/1.26 => ! [Xx: $i,Xy: $i] :
% 1.42/1.26 ( ( Xx = Xy )
% 1.42/1.26 => ( ( in @ Xx @ A )
% 1.42/1.26 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 1.42/1.26
% 1.42/1.26 thf(omegaS_type,type,
% 1.42/1.26 omegaS: $i > $i ).
% 1.42/1.26
% 1.42/1.26 thf(omegaS,definition,
% 1.42/1.26 ( omegaS
% 1.42/1.26 = ( ^ [Xx: $i] : ( setadjoin @ Xx @ Xx ) ) ) ).
% 1.42/1.26
% 1.42/1.26 thf(peano0notS,conjecture,
% 1.42/1.26 ( emptysetE
% 1.42/1.26 => ( setadjoinIL
% 1.42/1.26 => ( in__Cong
% 1.42/1.26 => ! [Xx: $i] :
% 1.42/1.26 ( ( in @ Xx @ omega )
% 1.42/1.26 => ( ( omegaS @ Xx )
% 1.42/1.26 != emptyset ) ) ) ) ) ).
% 1.42/1.26 ------------------------
% 1.42/1.27 % (26073)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_25921 for (3000ds/27Mi)
% 1.42/1.27 % (26073)First to succeed.
% 1.42/1.27 % (26073)Refutation found. Thanks to Tanya!
% 1.42/1.27 % SZS status Theorem for DTF2THF_25921
% 1.42/1.27 % SZS output start Proof for DTF2THF_25921
% 1.42/1.27 thf(func_def_0, type, in: $i > $i > $o).
% 1.42/1.27 thf(func_def_2, type, setadjoin: $i > $i > $i).
% 1.42/1.27 thf(func_def_8, type, omegaS: $i > $i).
% 1.42/1.27 thf(func_def_9, type, vEPSILON: !>[X0: $tType]:((X0 > $o) > X0)).
% 1.42/1.27 thf(f78,plain,(
% 1.42/1.27 $false),
% 1.42/1.27 inference(avatar_sat_refutation,[status(thm)],[f65,f71,f77])).
% 1.42/1.27 thf(f77,plain,(
% 1.42/1.27 ~spl1_1),
% 1.42/1.27 inference(avatar_contradiction_clause,[status(thm)],[f76])).
% 1.42/1.27 thf(f76,plain,(
% 1.42/1.27 $false | ~spl1_1),
% 1.42/1.27 inference(equality_resolution,[status(thm)],[f73])).
% 1.42/1.27 thf(f73,plain,(
% 1.42/1.27 ( ! [X0 : $o] : (($false != X0)) ) | ~spl1_1),
% 1.42/1.27 inference(superposition,[status(thm)],[f7,f61])).
% 1.42/1.27 thf(f61,plain,(
% 1.42/1.27 ( ! [X2 : $o] : (($true = X2)) ) | ~spl1_1),
% 1.42/1.27 inference(avatar_component_clause,[status(thm)],[f60])).
% 1.42/1.27 thf(f60,definition,(
% 1.42/1.27 spl1_1 <=> ! [X2 : $o] : ($true = X2)),
% 1.42/1.27 introduced(definition,[new_symbols(naming,[spl1_1])],[avatar_definition])).
% 1.42/1.27 thf(f7,plain,(
% 1.42/1.27 ($true != $false)),
% 1.42/1.27 introduced(definition,[],[fool_axiom])).
% 1.42/1.27 thf(f71,plain,(
% 1.42/1.27 ~spl1_2),
% 1.42/1.27 inference(avatar_contradiction_clause,[status(thm)],[f70])).
% 1.42/1.27 thf(f70,plain,(
% 1.42/1.27 $false | ~spl1_2),
% 1.42/1.27 inference(trivial_inequality_removal,[status(thm)],[f67])).
% 1.42/1.27 thf(f67,plain,(
% 1.42/1.27 ($true = $false) | ~spl1_2),
% 1.42/1.27 inference(superposition,[status(thm)],[f43,f64])).
% 1.42/1.27 thf(f64,plain,(
% 1.42/1.27 ( ! [X1 : $i] : (((in @ X1 @ (setadjoin @ sK0 @ sK0)) = $false)) ) | ~spl1_2),
% 1.42/1.27 inference(avatar_component_clause,[status(thm)],[f63])).
% 1.42/1.27 thf(f63,definition,(
% 1.42/1.27 spl1_2 <=> ! [X1 : $i] : ((in @ X1 @ (setadjoin @ sK0 @ sK0)) = $false)),
% 1.42/1.27 introduced(definition,[new_symbols(naming,[spl1_2])],[avatar_definition])).
% 1.42/1.27 thf(f43,plain,(
% 1.42/1.27 ( ! [X2 : $i,X1 : $i] : (($true = (in @ X1 @ (setadjoin @ X1 @ X2)))) )),
% 1.42/1.27 inference(beta_eta_normalization,[status(thm)],[f42])).
% 1.42/1.27 thf(f42,plain,(
% 1.42/1.27 ( ! [X2 : $i,X1 : $i] : (($true = ((^[Y0 : $i]: (in @ X1 @ (setadjoin @ X1 @ Y0))) @ X2))) )),
% 1.42/1.27 inference(pi_clausification,[status(thm)],[f41])).
% 1.42/1.27 thf(f41,plain,(
% 1.42/1.27 ( ! [X1 : $i] : (((!! @ $i @ (^[Y0 : $i]: (in @ X1 @ (setadjoin @ X1 @ Y0)))) = $true)) )),
% 1.42/1.27 inference(beta_eta_normalization,[status(thm)],[f40])).
% 1.42/1.27 thf(f40,plain,(
% 1.42/1.27 ( ! [X1 : $i] : (($true = ((^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (in @ Y0 @ (setadjoin @ Y0 @ Y1))))) @ X1))) )),
% 1.42/1.27 inference(pi_clausification,[status(thm)],[f32])).
% 1.42/1.27 thf(f32,plain,(
% 1.42/1.27 ($true = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (in @ Y0 @ (setadjoin @ Y0 @ Y1)))))))),
% 1.42/1.27 inference(definition_unfolding,[status(thm)],[f27,f25])).
% 1.42/1.27 thf(f25,plain,(
% 1.42/1.27 (setadjoinIL = $true)),
% 1.42/1.27 inference(cnf_transformation,[status(thm)],[f20])).
% 1.42/1.27 thf(f20,plain,(
% 1.42/1.27 (setadjoinIL = $true) & (emptysetE = $true) & ((emptyset = (omegaS @ sK0)) & ($true = (in @ sK0 @ omega))) & (in__Cong = $true)),
% 1.42/1.27 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0])],[f18,f19])).
% 1.42/1.27 thf(f19,plain,(
% 1.42/1.27 ? [X0 : $i] : ((emptyset = (omegaS @ X0)) & ((in @ X0 @ omega) = $true)) => ((emptyset = (omegaS @ sK0)) & ($true = (in @ sK0 @ omega)))),
% 1.42/1.27 introduced(definition,[],[choice_axiom])).
% 1.42/1.27 thf(f18,plain,(
% 1.42/1.27 (setadjoinIL = $true) & (emptysetE = $true) & ? [X0 : $i] : ((emptyset = (omegaS @ X0)) & ((in @ X0 @ omega) = $true)) & (in__Cong = $true)),
% 1.42/1.27 inference(flattening,[status(thm)],[f17])).
% 1.42/1.27 thf(f17,plain,(
% 1.42/1.27 ((? [X0 : $i] : ((emptyset = (omegaS @ X0)) & ((in @ X0 @ omega) = $true)) & (in__Cong = $true)) & (setadjoinIL = $true)) & (emptysetE = $true)),
% 1.42/1.27 inference(ennf_transformation,[status(thm)],[f16])).
% 1.42/1.27 thf(f16,plain,(
% 1.42/1.27 ~((emptysetE = $true) => ((setadjoinIL = $true) => ((in__Cong = $true) => ! [X0 : $i] : (((in @ X0 @ omega) = $true) => (emptyset != (omegaS @ X0))))))),
% 1.42/1.27 inference(fool_elimination,[status(thm)],[f15])).
% 1.42/1.27 thf(f15,plain,(
% 1.42/1.27 ~(emptysetE => (setadjoinIL => (in__Cong => ! [X0 : $i] : ((in @ X0 @ omega) => (emptyset != (omegaS @ X0))))))),
% 1.42/1.27 inference(rectify,[status(thm)],[f6])).
% 1.42/1.27 thf(f6,negated_conjecture,(
% 1.42/1.27 ~(emptysetE => (setadjoinIL => (in__Cong => ! [X0 : $i] : ((in @ X0 @ omega) => (emptyset != (omegaS @ X0))))))),
% 1.42/1.27 inference(negated_conjecture,[status(cth)],[f5])).
% 1.42/1.27 thf(f5,conjecture,(
% 1.42/1.27 emptysetE => (setadjoinIL => (in__Cong => ! [X0 : $i] : ((in @ X0 @ omega) => (emptyset != (omegaS @ X0)))))),
% 1.42/1.27 file('/export/starexec/sandbox/tmp/tmp.R7K9szfKav/DTF2THF_25921.p',peano0notS)).
% 1.42/1.27 thf(f27,plain,(
% 1.42/1.27 (setadjoinIL = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (in @ Y0 @ (setadjoin @ Y0 @ Y1)))))))),
% 1.42/1.27 inference(cnf_transformation,[status(thm)],[f14])).
% 1.42/1.27 thf(f14,plain,(
% 1.42/1.27 (setadjoinIL = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (in @ Y0 @ (setadjoin @ Y0 @ Y1)))))))),
% 1.42/1.27 inference(fool_elimination,[status(thm)],[f13])).
% 1.42/1.27 thf(f13,plain,(
% 1.42/1.27 (setadjoinIL = ! [X0 : $i,X1 : $i] : (in @ X1 @ (setadjoin @ X1 @ X0)))),
% 1.42/1.27 inference(rectify,[status(thm)],[f2])).
% 1.42/1.27 thf(f2,axiom,(
% 1.42/1.27 (setadjoinIL = ! [X2 : $i,X0 : $i] : (in @ X0 @ (setadjoin @ X0 @ X2)))),
% 1.42/1.27 file('/export/starexec/sandbox/tmp/tmp.R7K9szfKav/DTF2THF_25921.p',setadjoinIL)).
% 1.42/1.27 thf(f65,plain,(
% 1.42/1.27 spl1_1 | spl1_2),
% 1.42/1.27 inference(avatar_split_clause,[status(thm)],[f39,f63,f60])).
% 1.42/1.27 thf(f39,plain,(
% 1.42/1.27 ( ! [X2 : $o,X1 : $i] : (($true = X2) | ((in @ X1 @ (setadjoin @ sK0 @ sK0)) = $false)) )),
% 1.42/1.27 inference(beta_eta_normalization,[status(thm)],[f38])).
% 1.42/1.27 thf(f38,plain,(
% 1.42/1.27 ( ! [X2 : $o,X1 : $i] : (($true = ((^[Y0 : $o]: (Y0)) @ X2)) | ((in @ X1 @ (setadjoin @ sK0 @ sK0)) = $false)) )),
% 1.42/1.27 inference(pi_clausification,[status(thm)],[f37])).
% 1.42/1.27 thf(f37,plain,(
% 1.42/1.27 ( ! [X1 : $i] : (($true = (!! @ $o @ (^[Y0 : $o]: (Y0)))) | ((in @ X1 @ (setadjoin @ sK0 @ sK0)) = $false)) )),
% 1.42/1.27 inference(binary_proxy_clausification,[status(thm)],[f36])).
% 1.42/1.27 thf(f36,plain,(
% 1.42/1.27 ( ! [X1 : $i] : (($true = ((in @ X1 @ (setadjoin @ sK0 @ sK0)) => (!! @ $o @ (^[Y0 : $o]: (Y0)))))) )),
% 1.42/1.27 inference(beta_eta_normalization,[status(thm)],[f35])).
% 1.42/1.27 thf(f35,plain,(
% 1.42/1.27 ( ! [X1 : $i] : (($true = ((^[Y0 : $i]: ((in @ Y0 @ (setadjoin @ sK0 @ sK0)) => (!! @ $o @ (^[Y1 : $o]: (Y1))))) @ X1))) )),
% 1.42/1.27 inference(pi_clausification,[status(thm)],[f34])).
% 1.42/1.27 thf(f34,plain,(
% 1.42/1.27 ($true = (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ (setadjoin @ sK0 @ sK0)) => (!! @ $o @ (^[Y1 : $o]: (Y1)))))))),
% 1.42/1.27 inference(beta_eta_normalization,[status(thm)],[f33])).
% 1.42/1.27 thf(f33,plain,(
% 1.42/1.27 ($true = (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ ((^[Y1 : $i]: (setadjoin @ Y1 @ Y1)) @ sK0)) => (!! @ $o @ (^[Y1 : $o]: (Y1)))))))),
% 1.42/1.27 inference(definition_unfolding,[status(thm)],[f29,f24,f30])).
% 1.42/1.27 thf(f30,plain,(
% 1.42/1.27 (emptyset = ((^[Y0 : $i]: (setadjoin @ Y0 @ Y0)) @ sK0))),
% 1.42/1.27 inference(definition_unfolding,[status(thm)],[f23,f28])).
% 1.42/1.27 thf(f28,plain,(
% 1.42/1.27 (omegaS = (^[Y0 : $i]: (setadjoin @ Y0 @ Y0)))),
% 1.42/1.27 inference(cnf_transformation,[status(thm)],[f8])).
% 1.42/1.27 thf(f8,plain,(
% 1.42/1.27 (omegaS = (^[Y0 : $i]: (setadjoin @ Y0 @ Y0)))),
% 1.42/1.27 inference(fool_elimination,[status(thm)],[f4])).
% 1.42/1.27 thf(f4,axiom,(
% 1.42/1.27 ((^[X0 : $i] : (setadjoin @ X0 @ X0)) = omegaS)),
% 1.42/1.27 file('/export/starexec/sandbox/tmp/tmp.R7K9szfKav/DTF2THF_25921.p',omegaS)).
% 1.42/1.27 thf(f23,plain,(
% 1.42/1.27 (emptyset = (omegaS @ sK0))),
% 1.42/1.27 inference(cnf_transformation,[status(thm)],[f20])).
% 1.42/1.27 thf(f24,plain,(
% 1.42/1.27 (emptysetE = $true)),
% 1.42/1.27 inference(cnf_transformation,[status(thm)],[f20])).
% 1.42/1.27 thf(f29,plain,(
% 1.42/1.27 (emptysetE = (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ emptyset) => (!! @ $o @ (^[Y1 : $o]: (Y1)))))))),
% 1.42/1.27 inference(cnf_transformation,[status(thm)],[f12])).
% 1.42/1.27 thf(f12,plain,(
% 1.42/1.27 (emptysetE = (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ emptyset) => (!! @ $o @ (^[Y1 : $o]: (Y1)))))))),
% 1.42/1.27 inference(fool_elimination,[status(thm)],[f11])).
% 1.42/1.27 thf(f11,plain,(
% 1.42/1.27 (! [X0 : $i] : ((in @ X0 @ emptyset) => ! [X1 : $o] : X1) = emptysetE)),
% 1.42/1.27 inference(rectify,[status(thm)],[f1])).
% 1.42/1.27 thf(f1,axiom,(
% 1.42/1.27 (! [X0 : $i] : ((in @ X0 @ emptyset) => ! [X1 : $o] : X1) = emptysetE)),
% 1.42/1.27 file('/export/starexec/sandbox/tmp/tmp.R7K9szfKav/DTF2THF_25921.p',emptysetE)).
% 1.42/1.27 % SZS output end Proof for DTF2THF_25921
% 1.42/1.27 % (26073)------------------------------
% 1.42/1.27 % (26073)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.42/1.27 % (26073)Termination reason: Refutation
% 1.42/1.27
% 1.42/1.27 % (26073)Memory used [KB]: 5500
% 1.42/1.27 % (26073)Time elapsed: 0.004 s
% 1.42/1.27 % (26073)Instructions burned: 4 (million)
% 1.42/1.27 % (26073)------------------------------
% 1.42/1.27 % (26073)------------------------------
% 1.42/1.27 % (26070)Success in time 0.013 s
%------------------------------------------------------------------------------