%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU631^2 : 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 : n008.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 09:35:46 AM UTC 2026
% Result : Theorem 1.53s 1.30s
% Output : Refutation 1.53s
% Verified :
% SZS Type : Refutation
% Derivation depth : 24
% Number of leaves : 6
% Syntax : Number of formulae : 52 ( 28 unt; 0 typ; 0 def)
% Number of atoms : 500 ( 125 equ; 0 cnn)
% Maximal formula atoms : 8 ( 9 avg)
% Number of connectives : 758 ( 51 ~; 35 |; 49 &; 589 @)
% ( 2 <=>; 13 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 4 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 14 ( 14 >; 0 *; 0 +; 0 <<)
% Number of symbols : 22 ( 18 usr; 12 con; 0-2 aty)
% ( 0 !!; 19 ??; 0 @@+; 0 @@-)
% Number of variables : 159 ( 71 ^; 61 !; 27 ?; 159 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_2,type,
setadjoin: $i > $i > $i ).
thf(func_def_3,type,
powerset: $i > $i ).
thf(func_def_4,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(func_def_7,type,
binunion: $i > $i > $i ).
thf(func_def_8,type,
kpair: $i > $i > $i ).
thf(func_def_9,type,
cartprod: $i > $i > $i ).
thf(func_def_24,type,
sK4: $i > $o ).
thf(func_def_28,type,
ph7:
!>[X0: $tType] : X0 ).
thf(f57,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f56,f52]) ).
thf(f52,plain,
( sK1
= ( setadjoin @ ( setadjoin @ sK8 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK8 @ ( setadjoin @ sK9 @ emptyset ) ) @ emptyset ) ) ),
inference(equality_proxy_clausification,[status(thm)],[f50]) ).
thf(f50,plain,
( $true
= ( ( setadjoin @ ( setadjoin @ sK8 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK8 @ ( setadjoin @ sK9 @ emptyset ) ) @ emptyset ) )
= sK1 ) ),
inference(binary_proxy_clausification,[status(thm)],[f49]) ).
thf(f49,plain,
( $true
= ( ( in @ sK9 @ sK2 )
& ( ( setadjoin @ ( setadjoin @ sK8 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK8 @ ( setadjoin @ sK9 @ emptyset ) ) @ emptyset ) )
= sK1 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f48]) ).
thf(f48,plain,
( $true
= ( ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
& ( ( setadjoin @ ( setadjoin @ sK8 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK8 @ ( setadjoin @ Y0 @ emptyset ) ) @ emptyset ) )
= sK1 ) )
@ sK9 ) ),
inference(sigma_clausification,[status(thm)],[f46]) ).
thf(f46,plain,
( $true
= ( ?? @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
& ( ( setadjoin @ ( setadjoin @ sK8 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK8 @ ( setadjoin @ Y0 @ emptyset ) ) @ emptyset ) )
= sK1 ) ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f45]) ).
thf(f45,plain,
( $true
= ( ( in @ sK8 @ sK0 )
& ( ?? @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
& ( ( setadjoin @ ( setadjoin @ sK8 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK8 @ ( setadjoin @ Y0 @ emptyset ) ) @ emptyset ) )
= sK1 ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f44]) ).
thf(f44,plain,
( $true
= ( ^ [Y0: $i] :
( ( in @ Y0 @ sK0 )
& ( ?? @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK2 )
& ( ( setadjoin @ ( setadjoin @ Y0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y0 @ ( setadjoin @ Y1 @ emptyset ) ) @ emptyset ) )
= sK1 ) ) ) )
@ sK8 ) ),
inference(sigma_clausification,[status(thm)],[f43]) ).
thf(f43,plain,
( $true
= ( ?? @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK0 )
& ( ?? @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK2 )
& ( ( setadjoin @ ( setadjoin @ Y0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y0 @ ( setadjoin @ Y1 @ emptyset ) ) @ emptyset ) )
= sK1 ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f42]) ).
thf(f42,plain,
( $true
= ( ^ [Y0: $i] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK0 )
& ( ?? @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ sK2 )
& ( ( setadjoin @ ( setadjoin @ Y1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y1 @ ( setadjoin @ Y2 @ emptyset ) ) @ emptyset ) )
= Y0 ) ) ) ) )
@ sK1 ) ),
inference(trivial_inequality_removal,[status(thm)],[f41]) ).
thf(f41,plain,
( ( $true != $true )
| ( $true
= ( ^ [Y0: $i] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK0 )
& ( ?? @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ sK2 )
& ( ( setadjoin @ ( setadjoin @ Y1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y1 @ ( setadjoin @ Y2 @ emptyset ) ) @ emptyset ) )
= Y0 ) ) ) ) )
@ sK1 ) ) ),
inference(superposition,[status(thm)],[f38,f39]) ).
thf(f39,plain,
( $true
= ( in @ sK1
@ ( dsetconstr @ ( powerset @ ( powerset @ ( binunion @ sK0 @ sK2 ) ) )
@ ^ [Y0: $i] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK0 )
& ( ?? @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ sK2 )
& ( ( setadjoin @ ( setadjoin @ Y1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y1 @ ( setadjoin @ Y2 @ emptyset ) ) @ emptyset ) )
= Y0 ) ) ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f32]) ).
thf(f32,plain,
( $true
= ( in @ sK1
@ ( ^ [Y0: $i,Y1: $i] :
( dsetconstr @ ( powerset @ ( powerset @ ( binunion @ Y0 @ Y1 ) ) )
@ ^ [Y2: $i] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
& ( ?? @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y1 )
& ( ( ^ [Y5: $i,Y6: $i] : ( setadjoin @ ( setadjoin @ Y5 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y5 @ ( setadjoin @ Y6 @ emptyset ) ) @ emptyset ) )
@ Y3
@ Y4 )
= Y2 ) ) ) ) ) )
@ sK0
@ sK2 ) ) ),
inference(definition_unfolding,[status(thm)],[f24,f31]) ).
thf(f31,plain,
( cartprod
= ( ^ [Y0: $i,Y1: $i] :
( dsetconstr @ ( powerset @ ( powerset @ ( binunion @ Y0 @ Y1 ) ) )
@ ^ [Y2: $i] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
& ( ?? @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y1 )
& ( ( ^ [Y5: $i,Y6: $i] : ( setadjoin @ ( setadjoin @ Y5 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y5 @ ( setadjoin @ Y6 @ emptyset ) ) @ emptyset ) )
@ Y3
@ Y4 )
= Y2 ) ) ) ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f30,f29]) ).
thf(f29,plain,
( kpair
= ( ^ [Y0: $i,Y1: $i] : ( setadjoin @ ( setadjoin @ Y0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y0 @ ( setadjoin @ Y1 @ emptyset ) ) @ emptyset ) ) ) ),
inference(cnf_transformation,[status(thm)],[f14]) ).
thf(f14,plain,
( kpair
= ( ^ [Y0: $i,Y1: $i] : ( setadjoin @ ( setadjoin @ Y0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y0 @ ( setadjoin @ Y1 @ emptyset ) ) @ emptyset ) ) ) ),
inference(fool_elimination,[status(thm)],[f13]) ).
thf(f13,plain,
( kpair
= ( ^ [X0: $i,X1: $i] : ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) @ emptyset ) ) ) ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
( kpair
= ( ^ [X2: $i,X3: $i] : ( setadjoin @ ( setadjoin @ X2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.Yf04cq51GA/DTF2THF_30224.p',kpair) ).
thf(f30,plain,
( cartprod
= ( ^ [Y0: $i,Y1: $i] :
( dsetconstr @ ( powerset @ ( powerset @ ( binunion @ Y0 @ Y1 ) ) )
@ ^ [Y2: $i] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
& ( ?? @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y1 )
& ( ( kpair @ Y3 @ Y4 )
= Y2 ) ) ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f10]) ).
thf(f10,plain,
( cartprod
= ( ^ [Y0: $i,Y1: $i] :
( dsetconstr @ ( powerset @ ( powerset @ ( binunion @ Y0 @ Y1 ) ) )
@ ^ [Y2: $i] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
& ( ?? @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y1 )
& ( ( kpair @ Y3 @ Y4 )
= Y2 ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f9]) ).
thf(f9,plain,
( cartprod
= ( ^ [X0: $i,X1: $i] :
( dsetconstr @ ( powerset @ ( powerset @ ( binunion @ X0 @ X1 ) ) )
@ ^ [X2: $i] :
? [X3: $i] :
( ? [X4: $i] :
( ( ( kpair @ X3 @ X4 )
= X2 )
& ( in @ X4 @ X1 ) )
& ( in @ X3 @ X0 ) ) ) ) ),
inference(rectify,[status(thm)],[f3]) ).
thf(f3,axiom,
( cartprod
= ( ^ [X0: $i,X4: $i] :
( dsetconstr @ ( powerset @ ( powerset @ ( binunion @ X0 @ X4 ) ) )
@ ^ [X2: $i] :
? [X3: $i] :
( ? [X5: $i] :
( ( ( kpair @ X3 @ X5 )
= X2 )
& ( in @ X5 @ X4 ) )
& ( in @ X3 @ X0 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.Yf04cq51GA/DTF2THF_30224.p',cartprod) ).
thf(f24,plain,
( $true
= ( in @ sK1 @ ( cartprod @ sK0 @ sK2 ) ) ),
inference(cnf_transformation,[status(thm)],[f18]) ).
thf(f18,plain,
( ( dsetconstrER = $true )
& ( $true
= ( in @ sK1 @ ( cartprod @ sK0 @ sK2 ) ) )
& ! [X3: $i] :
( ! [X4: $i] :
( ( $true
!= ( in @ X4 @ sK2 ) )
| ( sK1
!= ( kpair @ X3 @ X4 ) ) )
| ( $true
!= ( in @ X3 @ sK0 ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2])],[f15,f17]) ).
thf(f17,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X1 @ ( cartprod @ X0 @ X2 ) ) )
& ! [X3: $i] :
( ! [X4: $i] :
( ( $true
!= ( in @ X4 @ X2 ) )
| ( ( kpair @ X3 @ X4 )
!= X1 ) )
| ( ( in @ X3 @ X0 )
!= $true ) ) )
=> ( ( $true
= ( in @ sK1 @ ( cartprod @ sK0 @ sK2 ) ) )
& ! [X3: $i] :
( ! [X4: $i] :
( ( $true
!= ( in @ X4 @ sK2 ) )
| ( sK1
!= ( kpair @ X3 @ X4 ) ) )
| ( $true
!= ( in @ X3 @ sK0 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f15,plain,
( ( dsetconstrER = $true )
& ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X1 @ ( cartprod @ X0 @ X2 ) ) )
& ! [X3: $i] :
( ! [X4: $i] :
( ( $true
!= ( in @ X4 @ X2 ) )
| ( ( kpair @ X3 @ X4 )
!= X1 ) )
| ( ( in @ X3 @ X0 )
!= $true ) ) ) ),
inference(ennf_transformation,[status(thm)],[f12]) ).
thf(f12,plain,
~ ( ( dsetconstrER = $true )
=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X1 @ ( cartprod @ X0 @ X2 ) ) )
=> ? [X3: $i] :
( ? [X4: $i] :
( ( ( kpair @ X3 @ X4 )
= X1 )
& ( $true
= ( in @ X4 @ X2 ) ) )
& ( ( in @ X3 @ X0 )
= $true ) ) ) ),
inference(fool_elimination,[status(thm)],[f11]) ).
thf(f11,plain,
~ ( dsetconstrER
=> ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X1 @ ( cartprod @ X0 @ X2 ) )
=> ? [X3: $i] :
( ? [X4: $i] :
( ( in @ X4 @ X2 )
& ( ( kpair @ X3 @ X4 )
= X1 ) )
& ( in @ X3 @ X0 ) ) ) ),
inference(rectify,[status(thm)],[f5]) ).
thf(f5,negated_conjecture,
~ ( dsetconstrER
=> ! [X0: $i,X6: $i,X4: $i] :
( ( in @ X6 @ ( cartprod @ X0 @ X4 ) )
=> ? [X2: $i] :
( ? [X3: $i] :
( ( in @ X3 @ X4 )
& ( ( kpair @ X2 @ X3 )
= X6 ) )
& ( in @ X2 @ X0 ) ) ) ),
inference(negated_conjecture,[status(cth)],[f4]) ).
thf(f4,conjecture,
( dsetconstrER
=> ! [X0: $i,X6: $i,X4: $i] :
( ( in @ X6 @ ( cartprod @ X0 @ X4 ) )
=> ? [X2: $i] :
( ? [X3: $i] :
( ( in @ X3 @ X4 )
& ( ( kpair @ X2 @ X3 )
= X6 ) )
& ( in @ X2 @ X0 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.Yf04cq51GA/DTF2THF_30224.p',cartprodmempair1) ).
thf(f38,plain,
! [X2: $i,X0: $i,X1: $i > $o] :
( ( $true
!= ( in @ X0 @ ( dsetconstr @ X2 @ X1 ) ) )
| ( $true
= ( X1 @ X0 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f37]) ).
thf(f37,plain,
! [X2: $i,X0: $i,X1: $i > $o] :
( ( $true
!= ( in @ X0
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X1 @ Y0 ) ) ) )
| ( $true
= ( X1 @ X0 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f34]) ).
thf(f34,plain,
! [X2: $i,X0: $i,X1: $i > $o] :
( ( $true != $true )
| ( $true
!= ( in @ X0
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X1 @ Y0 ) ) ) )
| ( $true
= ( X1 @ X0 ) ) ),
inference(definition_unfolding,[status(thm)],[f28,f25]) ).
thf(f25,plain,
dsetconstrER = $true,
inference(cnf_transformation,[status(thm)],[f18]) ).
thf(f28,plain,
! [X2: $i,X0: $i,X1: $i > $o] :
( ( $true
!= ( in @ X0
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X1 @ Y0 ) ) ) )
| ( $true
= ( X1 @ X0 ) )
| ( dsetconstrER != $true ) ),
inference(cnf_transformation,[status(thm)],[f22]) ).
thf(f22,plain,
( ( ! [X0: $i,X1: $i > $o,X2: $i] :
( ( $true
!= ( in @ X0
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X1 @ Y0 ) ) ) )
| ( $true
= ( X1 @ X0 ) ) )
| ( dsetconstrER != $true ) )
& ( ( dsetconstrER = $true )
| ( ( $true
= ( in @ sK3
@ ( dsetconstr @ sK5
@ ^ [Y0: $i] : ( sK4 @ Y0 ) ) ) )
& ( $true
!= ( sK4 @ sK3 ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK3,sK4,sK5])],[f20,f21]) ).
thf(f21,plain,
( ? [X3: $i,X4: $i > $o,X5: $i] :
( ( $true
= ( in @ X3
@ ( dsetconstr @ X5
@ ^ [Y0: $i] : ( X4 @ Y0 ) ) ) )
& ( $true
!= ( X4 @ X3 ) ) )
=> ( ( $true
= ( in @ sK3
@ ( dsetconstr @ sK5
@ ^ [Y0: $i] : ( sK4 @ Y0 ) ) ) )
& ( $true
!= ( sK4 @ sK3 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f20,plain,
( ( ! [X0: $i,X1: $i > $o,X2: $i] :
( ( $true
!= ( in @ X0
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X1 @ Y0 ) ) ) )
| ( $true
= ( X1 @ X0 ) ) )
| ( dsetconstrER != $true ) )
& ( ( dsetconstrER = $true )
| ? [X3: $i,X4: $i > $o,X5: $i] :
( ( $true
= ( in @ X3
@ ( dsetconstr @ X5
@ ^ [Y0: $i] : ( X4 @ Y0 ) ) ) )
& ( $true
!= ( X4 @ X3 ) ) ) ) ),
inference(rectify,[status(thm)],[f19]) ).
thf(f19,plain,
( ( ! [X0: $i,X1: $i > $o,X2: $i] :
( ( $true
!= ( in @ X0
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X1 @ Y0 ) ) ) )
| ( $true
= ( X1 @ X0 ) ) )
| ( dsetconstrER != $true ) )
& ( ( dsetconstrER = $true )
| ? [X0: $i,X1: $i > $o,X2: $i] :
( ( $true
= ( in @ X0
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X1 @ Y0 ) ) ) )
& ( $true
!= ( X1 @ X0 ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f16]) ).
thf(f16,plain,
( ! [X0: $i,X1: $i > $o,X2: $i] :
( ( $true
!= ( in @ X0
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X1 @ Y0 ) ) ) )
| ( $true
= ( X1 @ X0 ) ) )
<=> ( dsetconstrER = $true ) ),
inference(ennf_transformation,[status(thm)],[f8]) ).
thf(f8,plain,
( ! [X0: $i,X1: $i > $o,X2: $i] :
( ( $true
= ( in @ X0
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X1 @ Y0 ) ) ) )
=> ( $true
= ( X1 @ X0 ) ) )
<=> ( dsetconstrER = $true ) ),
inference(fool_elimination,[status(thm)],[f7]) ).
thf(f7,plain,
! [X0: $i,X1: $i > $o,X2: $i] :
( ( ( in @ X0
@ ( dsetconstr @ X2
@ ^ [X3: $i] : ( X1 @ X3 ) ) )
=> ( X1 @ X0 ) )
= dsetconstrER ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
! [X2: $i,X1: $i > $o,X0: $i] :
( ( ( in @ X2
@ ( dsetconstr @ X0
@ ^ [X3: $i] : ( X1 @ X3 ) ) )
=> ( X1 @ X2 ) )
= dsetconstrER ),
file('/export/starexec/sandbox/tmp/tmp.Yf04cq51GA/DTF2THF_30224.p',dsetconstrER) ).
thf(f56,plain,
( sK1
!= ( setadjoin @ ( setadjoin @ sK8 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK8 @ ( setadjoin @ sK9 @ emptyset ) ) @ emptyset ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f55]) ).
thf(f55,plain,
( ( sK1
!= ( setadjoin @ ( setadjoin @ sK8 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK8 @ ( setadjoin @ sK9 @ emptyset ) ) @ emptyset ) ) )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f54,f47]) ).
thf(f47,plain,
( $true
= ( in @ sK8 @ sK0 ) ),
inference(binary_proxy_clausification,[status(thm)],[f45]) ).
thf(f54,plain,
! [X0: $i] :
( ( $true
!= ( in @ X0 @ sK0 ) )
| ( ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X0 @ ( setadjoin @ sK9 @ emptyset ) ) @ emptyset ) )
!= sK1 ) ),
inference(trivial_inequality_removal,[status(thm)],[f53]) ).
thf(f53,plain,
! [X0: $i] :
( ( $true != $true )
| ( $true
!= ( in @ X0 @ sK0 ) )
| ( ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X0 @ ( setadjoin @ sK9 @ emptyset ) ) @ emptyset ) )
!= sK1 ) ),
inference(superposition,[status(thm)],[f40,f51]) ).
thf(f51,plain,
( $true
= ( in @ sK9 @ sK2 ) ),
inference(binary_proxy_clausification,[status(thm)],[f49]) ).
thf(f40,plain,
! [X3: $i,X4: $i] :
( ( $true
!= ( in @ X4 @ sK2 ) )
| ( $true
!= ( in @ X3 @ sK0 ) )
| ( ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) )
!= sK1 ) ),
inference(beta_eta_normalization,[status(thm)],[f33]) ).
thf(f33,plain,
! [X3: $i,X4: $i] :
( ( $true
!= ( in @ X4 @ sK2 ) )
| ( ( ^ [Y0: $i,Y1: $i] : ( setadjoin @ ( setadjoin @ Y0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y0 @ ( setadjoin @ Y1 @ emptyset ) ) @ emptyset ) )
@ X3
@ X4 )
!= sK1 )
| ( $true
!= ( in @ X3 @ sK0 ) ) ),
inference(definition_unfolding,[status(thm)],[f23,f29]) ).
thf(f23,plain,
! [X3: $i,X4: $i] :
( ( $true
!= ( in @ X4 @ sK2 ) )
| ( sK1
!= ( kpair @ X3 @ X4 ) )
| ( $true
!= ( in @ X3 @ sK0 ) ) ),
inference(cnf_transformation,[status(thm)],[f18]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU631^2 : TPTP v9.2.1. Released v3.7.0.
% 0.13/0.12 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.33 % Computer : n008.cluster.edu
% 0.16/0.33 % Model : x86_64 x86_64
% 0.16/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.33 % Memory : 8042.1875MB
% 0.16/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.33 % CPULimit : 300
% 0.16/0.33 % WCLimit : 300
% 0.16/0.33 % DateTime : Tue Feb 24 12:29:24 EST 2026
% 0.16/0.33 % CPUTime :
% 0.16/0.33 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.20/0.44 ---- Original DTF file ---
% 0.20/0.44 thf(spec,logic,$$dhol).
% 0.20/0.44 %------------------------------------------------------------------------------
% 0.20/0.44 % File : SEU631^2 : TPTP v9.2.1. Released v3.7.0.
% 0.20/0.44 % Domain : Set Theory
% 0.20/0.44 % Problem : Ordered Pairs - Cartesian Products
% 0.20/0.44 % Version : Especial > Reduced > Especial.
% 0.20/0.44 % English : (! A:i.! B:i.! u:i.in u (cartprod A B) -> (? x:i.in x A &
% 0.20/0.44 % (? y:i.in y B & u = kpair x y)))
% 0.20/0.44
% 0.20/0.44 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.20/0.44 % Source : [Bro08]
% 0.20/0.44 % Names : ZFC133l [Bro08]
% 0.20/0.44
% 0.20/0.44 % Status : Theorem
% 0.20/0.44 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.10 v8.2.0, 0.31 v8.1.0, 0.27 v7.5.0, 0.14 v7.4.0, 0.00 v7.1.0, 0.12 v7.0.0, 0.00 v6.2.0, 0.14 v6.1.0, 0.00 v6.0.0, 0.29 v5.5.0, 0.33 v5.4.0, 0.40 v5.3.0, 0.60 v4.1.0, 0.67 v3.7.0
% 0.20/0.44 % Syntax : Number of formulae : 13 ( 3 unt; 9 typ; 3 def)
% 0.20/0.44 % Number of atoms : 13 ( 5 equ; 0 cnn)
% 0.20/0.44 % Maximal formula atoms : 5 ( 3 avg)
% 0.20/0.44 % Number of connectives : 45 ( 0 ~; 0 |; 4 &; 38 @)
% 0.20/0.44 % ( 0 <=>; 3 =>; 0 <=; 0 <~>)
% 0.20/0.44 % Maximal formula depth : 12 ( 4 avg)
% 0.20/0.44 % Number of types : 2 ( 0 usr)
% 0.20/0.44 % Number of type conns : 15 ( 15 >; 0 *; 0 +; 0 <<)
% 0.20/0.44 % Number of symbols : 10 ( 9 usr; 2 con; 0-2 aty)
% 0.20/0.44 % Number of variables : 16 ( 6 ^; 6 !; 4 ?; 16 :)
% 0.20/0.44 % SPC : TH0_THM_EQU_NAR
% 0.20/0.44
% 0.20/0.44 % Comments : http://mathgate.info/detsetitem.php?id=188
% 0.20/0.44 %------------------------------------------------------------------------------
% 0.20/0.44 thf(in_type,type,
% 0.20/0.44 in: $i > $i > $o ).
% 0.20/0.44
% 0.20/0.44 thf(emptyset_type,type,
% 0.20/0.44 emptyset: $i ).
% 0.20/0.44
% 0.20/0.44 thf(setadjoin_type,type,
% 0.20/0.44 setadjoin: $i > $i > $i ).
% 0.20/0.44
% 0.20/0.44 thf(powerset_type,type,
% 0.20/0.44 powerset: $i > $i ).
% 0.20/0.44
% 0.20/0.44 thf(dsetconstr_type,type,
% 0.20/0.44 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.20/0.44
% 0.20/0.44 thf(dsetconstrER_type,type,
% 0.20/0.44 dsetconstrER: $o ).
% 0.20/0.44
% 0.20/0.44 thf(dsetconstrER,definition,
% 0.20/0.44 ( dsetconstrER
% 0.20/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.20/0.44 ( ( in @ Xx
% 0.20/0.44 @ ( dsetconstr @ A
% 0.20/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.20/0.44 => ( Xphi @ Xx ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(binunion_type,type,
% 0.20/0.44 binunion: $i > $i > $i ).
% 0.20/0.44
% 0.20/0.44 thf(kpair_type,type,
% 0.20/0.44 kpair: $i > $i > $i ).
% 0.20/0.44
% 0.20/0.44 thf(kpair,definition,
% 0.20/0.44 ( kpair
% 0.20/0.44 = ( ^ [Xx: $i,Xy: $i] : ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(cartprod_type,type,
% 0.20/0.44 cartprod: $i > $i > $i ).
% 0.20/0.44
% 0.20/0.44 thf(cartprod,definition,
% 0.20/0.44 ( cartprod
% 0.20/0.44 = ( ^ [A: $i,B: $i] :
% 0.20/0.44 ( dsetconstr @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) )
% 0.20/0.44 @ ^ [Xx: $i] :
% 0.20/0.44 ? [Xy: $i] :
% 0.20/0.44 ( ( in @ Xy @ A )
% 0.20/0.44 & ? [Xz: $i] :
% 0.20/0.44 ( ( in @ Xz @ B )
% 0.20/0.44 & ( Xx
% 0.20/0.44 = ( kpair @ Xy @ Xz ) ) ) ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(cartprodmempair1,conjecture,
% 0.20/0.44 ( dsetconstrER
% 0.20/0.44 => ! [A: $i,B: $i,Xu: $i] :
% 0.20/0.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.20/0.44 => ? [Xx: $i] :
% 0.20/0.44 ( ( in @ Xx @ A )
% 0.20/0.44 & ? [Xy: $i] :
% 0.20/0.44 ( ( in @ Xy @ B )
% 0.20/0.44 & ( Xu
% 0.20/0.44 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 %------------------------------------------------------------------------------
% 0.20/0.44 ------------------------
% 1.53/1.25 ---- Embedded in THF ---
% 1.53/1.25 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.53/1.25 %%% Generated on Tue Feb 24 12:29:25 EST 2026
% 1.53/1.25 %%% using '$$dhol' embedding, version 1.3.0.
% 1.53/1.25 %%% Logic specification used:
% 1.53/1.25 %%% thf(spec, logic, $$dhol).
% 1.53/1.25
% 1.53/1.25 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.Yf04cq51GA/DTF2DTF30224.p
% See solution above
% 1.53/1.25 ------------------------
% 1.53/1.26 ---- Cleaned THF ---
% 1.53/1.26 thf(in_type,type,
% 1.53/1.26 in: $i > $i > $o ).
% 1.53/1.26
% 1.53/1.26 thf(emptyset_type,type,
% 1.53/1.26 emptyset: $i ).
% 1.53/1.26
% 1.53/1.26 thf(setadjoin_type,type,
% 1.53/1.26 setadjoin: $i > $i > $i ).
% 1.53/1.26
% 1.53/1.26 thf(powerset_type,type,
% 1.53/1.26 powerset: $i > $i ).
% 1.53/1.26
% 1.53/1.26 thf(dsetconstr_type,type,
% 1.53/1.26 dsetconstr: $i > ( $i > $o ) > $i ).
% 1.53/1.26
% 1.53/1.26 thf(dsetconstrER_type,type,
% 1.53/1.26 dsetconstrER: $o ).
% 1.53/1.26
% 1.53/1.26 thf(dsetconstrER,definition,
% 1.53/1.26 ( dsetconstrER
% 1.53/1.26 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.53/1.26 ( ( in @ Xx
% 1.53/1.26 @ ( dsetconstr @ A
% 1.53/1.26 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 1.53/1.26 => ( Xphi @ Xx ) ) ) ) ).
% 1.53/1.26
% 1.53/1.26 thf(binunion_type,type,
% 1.53/1.26 binunion: $i > $i > $i ).
% 1.53/1.26
% 1.53/1.26 thf(kpair_type,type,
% 1.53/1.26 kpair: $i > $i > $i ).
% 1.53/1.26
% 1.53/1.26 thf(kpair,definition,
% 1.53/1.26 ( kpair
% 1.53/1.26 = ( ^ [Xx: $i,Xy: $i] : ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ).
% 1.53/1.26
% 1.53/1.26 thf(cartprod_type,type,
% 1.53/1.26 cartprod: $i > $i > $i ).
% 1.53/1.26
% 1.53/1.26 thf(cartprod,definition,
% 1.53/1.26 ( cartprod
% 1.53/1.26 = ( ^ [A: $i,B: $i] :
% 1.53/1.26 ( dsetconstr @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) )
% 1.53/1.26 @ ^ [Xx: $i] :
% 1.53/1.26 ? [Xy: $i] :
% 1.53/1.26 ( ( in @ Xy @ A )
% 1.53/1.26 & ? [Xz: $i] :
% 1.53/1.26 ( ( in @ Xz @ B )
% 1.53/1.26 & ( Xx
% 1.53/1.26 = ( kpair @ Xy @ Xz ) ) ) ) ) ) ) ).
% 1.53/1.26
% 1.53/1.26 thf(cartprodmempair1,conjecture,
% 1.53/1.26 ( dsetconstrER
% 1.53/1.26 => ! [A: $i,B: $i,Xu: $i] :
% 1.53/1.26 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 1.53/1.26 => ? [Xx: $i] :
% 1.53/1.26 ( ( in @ Xx @ A )
% 1.53/1.26 & ? [Xy: $i] :
% 1.53/1.26 ( ( in @ Xy @ B )
% 1.53/1.26 & ( Xu
% 1.53/1.26 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 1.53/1.26 ------------------------
% 1.53/1.28 % (30506)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_30224 for (2999ds/275Mi)
% 1.53/1.28 % (30502)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_30224 for (2999ds/4Mi)
% 1.53/1.28 % (30506)Refutation not found, incomplete strategy
% 1.53/1.28 % (30506)------------------------------
% 1.53/1.28 % (30506)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.53/1.28 % (30506)Termination reason: Refutation not found, incomplete strategy
% 1.53/1.28
% 1.53/1.28
% 1.53/1.28 % (30506)Memory used [KB]: 5500
% 1.53/1.28 % (30506)Time elapsed: 0.003 s
% 1.53/1.28 % (30506)Instructions burned: 3 (million)
% 1.53/1.28 % (30506)------------------------------
% 1.53/1.28 % (30506)------------------------------
% 1.53/1.28 % (30502)Instruction limit reached!
% 1.53/1.28 % (30502)------------------------------
% 1.53/1.28 % (30502)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.53/1.28 % (30502)Termination reason: Unknown
% 1.53/1.28 % (30502)Termination phase: Saturation
% 1.53/1.28
% 1.53/1.28 % (30502)Memory used [KB]: 5500
% 1.53/1.28 % (30502)Time elapsed: 0.003 s
% 1.53/1.28 % (30502)Instructions burned: 5 (million)
% 1.53/1.28 % (30502)------------------------------
% 1.53/1.28 % (30502)------------------------------
% 1.53/1.29 % (30501)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_30224 for (2999ds/183Mi)
% 1.53/1.29 % (30503)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_30224 for (2999ds/27Mi)
% 1.53/1.29 % (30504)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_30224 for (2999ds/2Mi)
% 1.53/1.29 % (30505)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_30224 for (2999ds/2Mi)
% 1.53/1.29 % (30507)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_30224 for (2999ds/18Mi)
% 1.53/1.29 % (30504)Instruction limit reached!
% 1.53/1.29 % (30504)------------------------------
% 1.53/1.29 % (30504)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.53/1.29 % (30504)Termination reason: Unknown
% 1.53/1.29 % (30504)Termination phase: Property scanning
% 1.53/1.29
% 1.53/1.29 % (30504)Memory used [KB]: 895
% 1.53/1.29 % (30504)Time elapsed: 0.003 s
% 1.53/1.29 % (30504)Instructions burned: 2 (million)
% 1.53/1.29 % (30504)------------------------------
% 1.53/1.29 % (30504)------------------------------
% 1.53/1.29 % (30505)Instruction limit reached!
% 1.53/1.29 % (30505)------------------------------
% 1.53/1.29 % (30505)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.53/1.29 % (30505)Termination reason: Unknown
% 1.53/1.29 % (30505)Termination phase: shuffling
% 1.53/1.29
% 1.53/1.29 % (30505)Memory used [KB]: 895
% 1.53/1.29 % (30505)Time elapsed: 0.003 s
% 1.53/1.29 % (30505)Instructions burned: 2 (million)
% 1.53/1.29 % (30505)------------------------------
% 1.53/1.29 % (30505)------------------------------
% 1.53/1.29 % (30508)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_30224 for (2999ds/3Mi)
% 1.53/1.29 % (30509)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_30224 for (2999ds/37Mi)
% 1.53/1.29 % (30508)Instruction limit reached!
% 1.53/1.29 % (30508)------------------------------
% 1.53/1.29 % (30508)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.53/1.29 % (30508)Termination reason: Unknown
% 1.53/1.29 % (30508)Termination phase: Saturation
% 1.53/1.29
% 1.53/1.29 % (30508)Memory used [KB]: 5500
% 1.53/1.29 % (30508)Time elapsed: 0.002 s
% 1.53/1.29 % (30508)Instructions burned: 3 (million)
% 1.53/1.29 % (30508)------------------------------
% 1.53/1.29 % (30508)------------------------------
% 1.53/1.29 % (30507)First to succeed.
% 1.53/1.29 % (30503)Also succeeded, but the first one will report.
% 1.53/1.30 % (30507)Refutation found. Thanks to Tanya!
% 1.53/1.30 % SZS status Theorem for DTF2THF_30224
% 1.53/1.30 % SZS output start Proof for DTF2THF_30224
% 1.53/1.30 thf(func_def_0, type, in: $i > $i > $o).
% 1.53/1.30 thf(func_def_2, type, setadjoin: $i > $i > $i).
% 1.53/1.30 thf(func_def_3, type, powerset: $i > $i).
% 1.53/1.30 thf(func_def_4, type, dsetconstr: $i > ($i > $o) > $i).
% 1.53/1.30 thf(func_def_7, type, binunion: $i > $i > $i).
% 1.53/1.30 thf(func_def_8, type, kpair: $i > $i > $i).
% 1.53/1.30 thf(func_def_9, type, cartprod: $i > $i > $i).
% 1.53/1.30 thf(func_def_24, type, sK4: $i > $o).
% 1.53/1.30 thf(func_def_28, type, ph7: !>[X0: $tType]:(X0)).
% 1.53/1.30 thf(f57,plain,(
% 1.53/1.30 $false),
% 1.53/1.30 inference(subsumption_resolution,[status(thm)],[f56,f52])).
% 1.53/1.30 thf(f52,plain,(
% 1.53/1.30 (sK1 = (setadjoin @ (setadjoin @ sK8 @ emptyset) @ (setadjoin @ (setadjoin @ sK8 @ (setadjoin @ sK9 @ emptyset)) @ emptyset)))),
% 1.53/1.30 inference(equality_proxy_clausification,[status(thm)],[f50])).
% 1.53/1.30 thf(f50,plain,(
% 1.53/1.30 ($true = ((setadjoin @ (setadjoin @ sK8 @ emptyset) @ (setadjoin @ (setadjoin @ sK8 @ (setadjoin @ sK9 @ emptyset)) @ emptyset)) = sK1))),
% 1.53/1.30 inference(binary_proxy_clausification,[status(thm)],[f49])).
% 1.53/1.30 thf(f49,plain,(
% 1.53/1.30 ($true = ((in @ sK9 @ sK2) & ((setadjoin @ (setadjoin @ sK8 @ emptyset) @ (setadjoin @ (setadjoin @ sK8 @ (setadjoin @ sK9 @ emptyset)) @ emptyset)) = sK1)))),
% 1.53/1.30 inference(beta_eta_normalization,[status(thm)],[f48])).
% 1.53/1.30 thf(f48,plain,(
% 1.53/1.30 ($true = ((^[Y0 : $i]: ((in @ Y0 @ sK2) & ((setadjoin @ (setadjoin @ sK8 @ emptyset) @ (setadjoin @ (setadjoin @ sK8 @ (setadjoin @ Y0 @ emptyset)) @ emptyset)) = sK1))) @ sK9))),
% 1.53/1.30 inference(sigma_clausification,[status(thm)],[f46])).
% 1.53/1.30 thf(f46,plain,(
% 1.53/1.30 ($true = (?? @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) & ((setadjoin @ (setadjoin @ sK8 @ emptyset) @ (setadjoin @ (setadjoin @ sK8 @ (setadjoin @ Y0 @ emptyset)) @ emptyset)) = sK1)))))),
% 1.53/1.30 inference(binary_proxy_clausification,[status(thm)],[f45])).
% 1.53/1.30 thf(f45,plain,(
% 1.53/1.30 ($true = ((in @ sK8 @ sK0) & (?? @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) & ((setadjoin @ (setadjoin @ sK8 @ emptyset) @ (setadjoin @ (setadjoin @ sK8 @ (setadjoin @ Y0 @ emptyset)) @ emptyset)) = sK1))))))),
% 1.53/1.30 inference(beta_eta_normalization,[status(thm)],[f44])).
% 1.53/1.30 thf(f44,plain,(
% 1.53/1.30 ($true = ((^[Y0 : $i]: ((in @ Y0 @ sK0) & (?? @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK2) & ((setadjoin @ (setadjoin @ Y0 @ emptyset) @ (setadjoin @ (setadjoin @ Y0 @ (setadjoin @ Y1 @ emptyset)) @ emptyset)) = sK1)))))) @ sK8))),
% 1.53/1.30 inference(sigma_clausification,[status(thm)],[f43])).
% 1.53/1.30 thf(f43,plain,(
% 1.53/1.30 ($true = (?? @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK0) & (?? @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK2) & ((setadjoin @ (setadjoin @ Y0 @ emptyset) @ (setadjoin @ (setadjoin @ Y0 @ (setadjoin @ Y1 @ emptyset)) @ emptyset)) = sK1))))))))),
% 1.53/1.30 inference(beta_eta_normalization,[status(thm)],[f42])).
% 1.53/1.30 thf(f42,plain,(
% 1.53/1.30 ($true = ((^[Y0 : $i]: (?? @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK0) & (?? @ $i @ (^[Y2 : $i]: ((in @ Y2 @ sK2) & ((setadjoin @ (setadjoin @ Y1 @ emptyset) @ (setadjoin @ (setadjoin @ Y1 @ (setadjoin @ Y2 @ emptyset)) @ emptyset)) = Y0)))))))) @ sK1))),
% 1.53/1.30 inference(trivial_inequality_removal,[status(thm)],[f41])).
% 1.53/1.30 thf(f41,plain,(
% 1.53/1.30 ($true != $true) | ($true = ((^[Y0 : $i]: (?? @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK0) & (?? @ $i @ (^[Y2 : $i]: ((in @ Y2 @ sK2) & ((setadjoin @ (setadjoin @ Y1 @ emptyset) @ (setadjoin @ (setadjoin @ Y1 @ (setadjoin @ Y2 @ emptyset)) @ emptyset)) = Y0)))))))) @ sK1))),
% 1.53/1.30 inference(superposition,[status(thm)],[f38,f39])).
% 1.53/1.30 thf(f39,plain,(
% 1.53/1.30 ($true = (in @ sK1 @ (dsetconstr @ (powerset @ (powerset @ (binunion @ sK0 @ sK2))) @ (^[Y0 : $i]: (?? @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK0) & (?? @ $i @ (^[Y2 : $i]: ((in @ Y2 @ sK2) & ((setadjoin @ (setadjoin @ Y1 @ emptyset) @ (setadjoin @ (setadjoin @ Y1 @ (setadjoin @ Y2 @ emptyset)) @ emptyset)) = Y0)))))))))))),
% 1.53/1.30 inference(beta_eta_normalization,[status(thm)],[f32])).
% 1.53/1.30 thf(f32,plain,(
% 1.53/1.30 ($true = (in @ sK1 @ ((^[Y0 : $i]: ((^[Y1 : $i]: (dsetconstr @ (powerset @ (powerset @ (binunion @ Y0 @ Y1))) @ (^[Y2 : $i]: (?? @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) & (?? @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y1) & (((^[Y5 : $i]: ((^[Y6 : $i]: (setadjoin @ (setadjoin @ Y5 @ emptyset) @ (setadjoin @ (setadjoin @ Y5 @ (setadjoin @ Y6 @ emptyset)) @ emptyset))))) @ Y3 @ Y4) = Y2)))))))))))) @ sK0 @ sK2)))),
% 1.53/1.30 inference(definition_unfolding,[status(thm)],[f24,f31])).
% 1.53/1.30 thf(f31,plain,(
% 1.53/1.30 (cartprod = (^[Y0 : $i]: ((^[Y1 : $i]: (dsetconstr @ (powerset @ (powerset @ (binunion @ Y0 @ Y1))) @ (^[Y2 : $i]: (?? @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) & (?? @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y1) & (((^[Y5 : $i]: ((^[Y6 : $i]: (setadjoin @ (setadjoin @ Y5 @ emptyset) @ (setadjoin @ (setadjoin @ Y5 @ (setadjoin @ Y6 @ emptyset)) @ emptyset))))) @ Y3 @ Y4) = Y2)))))))))))))),
% 1.53/1.30 inference(definition_unfolding,[status(thm)],[f30,f29])).
% 1.53/1.30 thf(f29,plain,(
% 1.53/1.30 (kpair = (^[Y0 : $i]: ((^[Y1 : $i]: (setadjoin @ (setadjoin @ Y0 @ emptyset) @ (setadjoin @ (setadjoin @ Y0 @ (setadjoin @ Y1 @ emptyset)) @ emptyset))))))),
% 1.53/1.30 inference(cnf_transformation,[status(thm)],[f14])).
% 1.53/1.30 thf(f14,plain,(
% 1.53/1.30 (kpair = (^[Y0 : $i]: ((^[Y1 : $i]: (setadjoin @ (setadjoin @ Y0 @ emptyset) @ (setadjoin @ (setadjoin @ Y0 @ (setadjoin @ Y1 @ emptyset)) @ emptyset))))))),
% 1.53/1.30 inference(fool_elimination,[status(thm)],[f13])).
% 1.53/1.30 thf(f13,plain,(
% 1.53/1.30 (kpair = (^[X0 : $i, X1 : $i] : (setadjoin @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset)) @ emptyset))))),
% 1.53/1.30 inference(rectify,[status(thm)],[f2])).
% 1.53/1.30 thf(f2,axiom,(
% 1.53/1.30 (kpair = (^[X2 : $i, X3 : $i] : (setadjoin @ (setadjoin @ X2 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ (setadjoin @ X3 @ emptyset)) @ emptyset))))),
% 1.53/1.30 file('/export/starexec/sandbox/tmp/tmp.Yf04cq51GA/DTF2THF_30224.p',kpair)).
% 1.53/1.30 thf(f30,plain,(
% 1.53/1.30 (cartprod = (^[Y0 : $i]: ((^[Y1 : $i]: (dsetconstr @ (powerset @ (powerset @ (binunion @ Y0 @ Y1))) @ (^[Y2 : $i]: (?? @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) & (?? @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y1) & ((kpair @ Y3 @ Y4) = Y2)))))))))))))),
% 1.53/1.30 inference(cnf_transformation,[status(thm)],[f10])).
% 1.53/1.30 thf(f10,plain,(
% 1.53/1.30 (cartprod = (^[Y0 : $i]: ((^[Y1 : $i]: (dsetconstr @ (powerset @ (powerset @ (binunion @ Y0 @ Y1))) @ (^[Y2 : $i]: (?? @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) & (?? @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y1) & ((kpair @ Y3 @ Y4) = Y2)))))))))))))),
% 1.53/1.30 inference(fool_elimination,[status(thm)],[f9])).
% 1.53/1.30 thf(f9,plain,(
% 1.53/1.30 (cartprod = (^[X0 : $i, X1 : $i] : (dsetconstr @ (powerset @ (powerset @ (binunion @ X0 @ X1))) @ (^[X2 : $i] : (? [X3 : $i] : (? [X4 : $i] : (((kpair @ X3 @ X4) = X2) & (in @ X4 @ X1)) & (in @ X3 @ X0)))))))),
% 1.53/1.30 inference(rectify,[status(thm)],[f3])).
% 1.53/1.30 thf(f3,axiom,(
% 1.53/1.30 (cartprod = (^[X0 : $i, X4 : $i] : (dsetconstr @ (powerset @ (powerset @ (binunion @ X0 @ X4))) @ (^[X2 : $i] : (? [X3 : $i] : (? [X5 : $i] : (((kpair @ X3 @ X5) = X2) & (in @ X5 @ X4)) & (in @ X3 @ X0)))))))),
% 1.53/1.30 file('/export/starexec/sandbox/tmp/tmp.Yf04cq51GA/DTF2THF_30224.p',cartprod)).
% 1.53/1.30 thf(f24,plain,(
% 1.53/1.30 ($true = (in @ sK1 @ (cartprod @ sK0 @ sK2)))),
% 1.53/1.30 inference(cnf_transformation,[status(thm)],[f18])).
% 1.53/1.30 thf(f18,plain,(
% 1.53/1.30 (dsetconstrER = $true) & (($true = (in @ sK1 @ (cartprod @ sK0 @ sK2))) & ! [X3 : $i] : (! [X4 : $i] : (($true != (in @ X4 @ sK2)) | (sK1 != (kpair @ X3 @ X4))) | ($true != (in @ X3 @ sK0))))),
% 1.53/1.30 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2])],[f15,f17])).
% 1.53/1.30 thf(f17,plain,(
% 1.53/1.30 ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X1 @ (cartprod @ X0 @ X2))) & ! [X3 : $i] : (! [X4 : $i] : (($true != (in @ X4 @ X2)) | ((kpair @ X3 @ X4) != X1)) | ((in @ X3 @ X0) != $true))) => (($true = (in @ sK1 @ (cartprod @ sK0 @ sK2))) & ! [X3 : $i] : (! [X4 : $i] : (($true != (in @ X4 @ sK2)) | (sK1 != (kpair @ X3 @ X4))) | ($true != (in @ X3 @ sK0))))),
% 1.53/1.30 introduced(definition,[],[choice_axiom])).
% 1.53/1.30 thf(f15,plain,(
% 1.53/1.30 (dsetconstrER = $true) & ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X1 @ (cartprod @ X0 @ X2))) & ! [X3 : $i] : (! [X4 : $i] : (($true != (in @ X4 @ X2)) | ((kpair @ X3 @ X4) != X1)) | ((in @ X3 @ X0) != $true)))),
% 1.53/1.30 inference(ennf_transformation,[status(thm)],[f12])).
% 1.53/1.30 thf(f12,plain,(
% 1.53/1.30 ~((dsetconstrER = $true) => ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X1 @ (cartprod @ X0 @ X2))) => ? [X3 : $i] : (? [X4 : $i] : (((kpair @ X3 @ X4) = X1) & ($true = (in @ X4 @ X2))) & ((in @ X3 @ X0) = $true))))),
% 1.53/1.30 inference(fool_elimination,[status(thm)],[f11])).
% 1.53/1.30 thf(f11,plain,(
% 1.53/1.30 ~(dsetconstrER => ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X1 @ (cartprod @ X0 @ X2)) => ? [X3 : $i] : (? [X4 : $i] : ((in @ X4 @ X2) & ((kpair @ X3 @ X4) = X1)) & (in @ X3 @ X0))))),
% 1.53/1.30 inference(rectify,[status(thm)],[f5])).
% 1.53/1.30 thf(f5,negated_conjecture,(
% 1.53/1.30 ~(dsetconstrER => ! [X0 : $i,X6 : $i,X4 : $i] : ((in @ X6 @ (cartprod @ X0 @ X4)) => ? [X2 : $i] : (? [X3 : $i] : ((in @ X3 @ X4) & ((kpair @ X2 @ X3) = X6)) & (in @ X2 @ X0))))),
% 1.53/1.30 inference(negated_conjecture,[status(cth)],[f4])).
% 1.53/1.30 thf(f4,conjecture,(
% 1.53/1.30 dsetconstrER => ! [X0 : $i,X6 : $i,X4 : $i] : ((in @ X6 @ (cartprod @ X0 @ X4)) => ? [X2 : $i] : (? [X3 : $i] : ((in @ X3 @ X4) & ((kpair @ X2 @ X3) = X6)) & (in @ X2 @ X0)))),
% 1.53/1.30 file('/export/starexec/sandbox/tmp/tmp.Yf04cq51GA/DTF2THF_30224.p',cartprodmempair1)).
% 1.53/1.30 thf(f38,plain,(
% 1.53/1.30 ( ! [X2 : $i,X0 : $i,X1 : $i > $o] : (($true != (in @ X0 @ (dsetconstr @ X2 @ X1))) | ($true = (X1 @ X0))) )),
% 1.53/1.30 inference(beta_eta_normalization,[status(thm)],[f37])).
% 1.53/1.30 thf(f37,plain,(
% 1.53/1.30 ( ! [X2 : $i,X0 : $i,X1 : $i > $o] : (($true != (in @ X0 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X1 @ Y0))))) | ($true = (X1 @ X0))) )),
% 1.53/1.30 inference(trivial_inequality_removal,[status(thm)],[f34])).
% 1.53/1.30 thf(f34,plain,(
% 1.53/1.30 ( ! [X2 : $i,X0 : $i,X1 : $i > $o] : (($true != $true) | ($true != (in @ X0 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X1 @ Y0))))) | ($true = (X1 @ X0))) )),
% 1.53/1.30 inference(definition_unfolding,[status(thm)],[f28,f25])).
% 1.53/1.30 thf(f25,plain,(
% 1.53/1.30 (dsetconstrER = $true)),
% 1.53/1.30 inference(cnf_transformation,[status(thm)],[f18])).
% 1.53/1.30 thf(f28,plain,(
% 1.53/1.30 ( ! [X2 : $i,X0 : $i,X1 : $i > $o] : (($true != (in @ X0 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X1 @ Y0))))) | ($true = (X1 @ X0)) | (dsetconstrER != $true)) )),
% 1.53/1.30 inference(cnf_transformation,[status(thm)],[f22])).
% 1.53/1.30 thf(f22,plain,(
% 1.53/1.30 (! [X0 : $i,X1 : $i > $o,X2 : $i] : (($true != (in @ X0 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X1 @ Y0))))) | ($true = (X1 @ X0))) | (dsetconstrER != $true)) & ((dsetconstrER = $true) | (($true = (in @ sK3 @ (dsetconstr @ sK5 @ (^[Y0 : $i]: (sK4 @ Y0))))) & ($true != (sK4 @ sK3))))),
% 1.53/1.30 inference(skolemisation,[status(esa),new_symbols(skolem,[sK3,sK4,sK5])],[f20,f21])).
% 1.53/1.30 thf(f21,plain,(
% 1.53/1.30 ? [X3 : $i,X4 : $i > $o,X5 : $i] : (($true = (in @ X3 @ (dsetconstr @ X5 @ (^[Y0 : $i]: (X4 @ Y0))))) & ($true != (X4 @ X3))) => (($true = (in @ sK3 @ (dsetconstr @ sK5 @ (^[Y0 : $i]: (sK4 @ Y0))))) & ($true != (sK4 @ sK3)))),
% 1.53/1.30 introduced(definition,[],[choice_axiom])).
% 1.53/1.30 thf(f20,plain,(
% 1.53/1.30 (! [X0 : $i,X1 : $i > $o,X2 : $i] : (($true != (in @ X0 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X1 @ Y0))))) | ($true = (X1 @ X0))) | (dsetconstrER != $true)) & ((dsetconstrER = $true) | ? [X3 : $i,X4 : $i > $o,X5 : $i] : (($true = (in @ X3 @ (dsetconstr @ X5 @ (^[Y0 : $i]: (X4 @ Y0))))) & ($true != (X4 @ X3))))),
% 1.53/1.30 inference(rectify,[status(thm)],[f19])).
% 1.53/1.30 thf(f19,plain,(
% 1.53/1.30 (! [X0 : $i,X1 : $i > $o,X2 : $i] : (($true != (in @ X0 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X1 @ Y0))))) | ($true = (X1 @ X0))) | (dsetconstrER != $true)) & ((dsetconstrER = $true) | ? [X0 : $i,X1 : $i > $o,X2 : $i] : (($true = (in @ X0 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X1 @ Y0))))) & ($true != (X1 @ X0))))),
% 1.53/1.30 inference(nnf_transformation,[status(thm)],[f16])).
% 1.53/1.30 thf(f16,plain,(
% 1.53/1.30 ! [X0 : $i,X1 : $i > $o,X2 : $i] : (($true != (in @ X0 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X1 @ Y0))))) | ($true = (X1 @ X0))) <=> (dsetconstrER = $true)),
% 1.53/1.30 inference(ennf_transformation,[status(thm)],[f8])).
% 1.53/1.30 thf(f8,plain,(
% 1.53/1.30 ! [X0 : $i,X1 : $i > $o,X2 : $i] : (($true = (in @ X0 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X1 @ Y0))))) => ($true = (X1 @ X0))) <=> (dsetconstrER = $true)),
% 1.53/1.30 inference(fool_elimination,[status(thm)],[f7])).
% 1.53/1.30 thf(f7,plain,(
% 1.53/1.30 (! [X0 : $i,X1 : $i > $o,X2 : $i] : ((in @ X0 @ (dsetconstr @ X2 @ (^[X3 : $i] : (X1 @ X3)))) => (X1 @ X0)) = dsetconstrER)),
% 1.53/1.30 inference(rectify,[status(thm)],[f1])).
% 1.53/1.30 thf(f1,axiom,(
% 1.53/1.30 (! [X2 : $i,X1 : $i > $o,X0 : $i] : ((in @ X2 @ (dsetconstr @ X0 @ (^[X3 : $i] : (X1 @ X3)))) => (X1 @ X2)) = dsetconstrER)),
% 1.53/1.30 file('/export/starexec/sandbox/tmp/tmp.Yf04cq51GA/DTF2THF_30224.p',dsetconstrER)).
% 1.53/1.30 thf(f56,plain,(
% 1.53/1.30 (sK1 != (setadjoin @ (setadjoin @ sK8 @ emptyset) @ (setadjoin @ (setadjoin @ sK8 @ (setadjoin @ sK9 @ emptyset)) @ emptyset)))),
% 1.53/1.30 inference(trivial_inequality_removal,[status(thm)],[f55])).
% 1.53/1.30 thf(f55,plain,(
% 1.53/1.30 (sK1 != (setadjoin @ (setadjoin @ sK8 @ emptyset) @ (setadjoin @ (setadjoin @ sK8 @ (setadjoin @ sK9 @ emptyset)) @ emptyset))) | ($true != $true)),
% 1.53/1.30 inference(superposition,[status(thm)],[f54,f47])).
% 1.53/1.30 thf(f47,plain,(
% 1.53/1.30 ($true = (in @ sK8 @ sK0))),
% 1.53/1.30 inference(binary_proxy_clausification,[status(thm)],[f45])).
% 1.53/1.30 thf(f54,plain,(
% 1.53/1.30 ( ! [X0 : $i] : (($true != (in @ X0 @ sK0)) | ((setadjoin @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X0 @ (setadjoin @ sK9 @ emptyset)) @ emptyset)) != sK1)) )),
% 1.53/1.30 inference(trivial_inequality_removal,[status(thm)],[f53])).
% 1.53/1.30 thf(f53,plain,(
% 1.53/1.30 ( ! [X0 : $i] : (($true != $true) | ($true != (in @ X0 @ sK0)) | ((setadjoin @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X0 @ (setadjoin @ sK9 @ emptyset)) @ emptyset)) != sK1)) )),
% 1.53/1.30 inference(superposition,[status(thm)],[f40,f51])).
% 1.53/1.30 thf(f51,plain,(
% 1.53/1.30 ($true = (in @ sK9 @ sK2))),
% 1.53/1.30 inference(binary_proxy_clausification,[status(thm)],[f49])).
% 1.53/1.30 thf(f40,plain,(
% 1.53/1.30 ( ! [X3 : $i,X4 : $i] : (($true != (in @ X4 @ sK2)) | ($true != (in @ X3 @ sK0)) | ((setadjoin @ (setadjoin @ X3 @ emptyset) @ (setadjoin @ (setadjoin @ X3 @ (setadjoin @ X4 @ emptyset)) @ emptyset)) != sK1)) )),
% 1.53/1.30 inference(beta_eta_normalization,[status(thm)],[f33])).
% 1.53/1.30 thf(f33,plain,(
% 1.53/1.30 ( ! [X3 : $i,X4 : $i] : (($true != (in @ X4 @ sK2)) | (((^[Y0 : $i]: ((^[Y1 : $i]: (setadjoin @ (setadjoin @ Y0 @ emptyset) @ (setadjoin @ (setadjoin @ Y0 @ (setadjoin @ Y1 @ emptyset)) @ emptyset))))) @ X3 @ X4) != sK1) | ($true != (in @ X3 @ sK0))) )),
% 1.53/1.30 inference(definition_unfolding,[status(thm)],[f23,f29])).
% 1.53/1.30 thf(f23,plain,(
% 1.53/1.30 ( ! [X3 : $i,X4 : $i] : (($true != (in @ X4 @ sK2)) | (sK1 != (kpair @ X3 @ X4)) | ($true != (in @ X3 @ sK0))) )),
% 1.53/1.30 inference(cnf_transformation,[status(thm)],[f18])).
% 1.53/1.30 % SZS output end Proof for DTF2THF_30224
% 1.53/1.30 % (30507)------------------------------
% 1.53/1.30 % (30507)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.53/1.30 % (30507)Termination reason: Refutation
% 1.53/1.30
% 1.53/1.30 % (30507)Memory used [KB]: 5628
% 1.53/1.30 % (30507)Time elapsed: 0.009 s
% 1.53/1.30 % (30507)Instructions burned: 7 (million)
% 1.53/1.30 % (30507)------------------------------
% 1.53/1.30 % (30507)------------------------------
% 1.53/1.30 % (30500)Success in time 0.024 s
%------------------------------------------------------------------------------