%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU672^2 : 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 : 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 09:36:06 AM UTC 2026
% Result : Theorem 1.69s 1.47s
% Output : Refutation 1.69s
% Verified :
% SZS Type : Refutation
% Derivation depth : 23
% Number of leaves : 7
% Syntax : Number of formulae : 56 ( 36 unt; 0 typ; 0 def)
% Number of atoms : 505 ( 98 equ; 0 cnn)
% Maximal formula atoms : 6 ( 9 avg)
% Number of connectives : 742 ( 15 ~; 12 |; 35 &; 642 @)
% ( 0 <=>; 20 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 3 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 7 ( 7 >; 0 *; 0 +; 0 <<)
% Number of symbols : 22 ( 17 usr; 9 con; 0-3 aty)
% ( 6 !!; 12 ??; 0 @@+; 0 @@-)
% Number of variables : 180 ( 136 ^; 32 !; 12 ?; 180 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_2,type,
setadjoin: $i > $i > $i ).
thf(func_def_3,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(func_def_4,type,
subset: $i > $i > $o ).
thf(func_def_5,type,
kpair: $i > $i > $i ).
thf(func_def_6,type,
cartprod: $i > $i > $i ).
thf(func_def_7,type,
singleton: $i > $o ).
thf(func_def_9,type,
ex1: $i > ( $i > $o ) > $o ).
thf(func_def_10,type,
breln: $i > $i > $i > $o ).
thf(func_def_11,type,
func: $i > $i > $i > $o ).
thf(func_def_29,type,
sK5: $i > $i ).
thf(func_def_30,type,
ph6:
!>[X0: $tType] : X0 ).
thf(f62,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f61]) ).
thf(f61,plain,
$true = $false,
inference(backward_demodulation,[status(thm)],[f57,f60]) ).
thf(f60,plain,
( $false
= ( in @ ( sK5 @ sK3 ) @ ( setadjoin @ ( sK5 @ sK3 ) @ emptyset ) ) ),
inference(equality_resolution,[status(thm)],[f59]) ).
thf(f59,plain,
! [X1: $i] :
( ( ( setadjoin @ X1 @ emptyset )
!= ( setadjoin @ ( sK5 @ sK3 ) @ emptyset ) )
| ( $false
= ( in @ X1 @ ( setadjoin @ ( sK5 @ sK3 ) @ emptyset ) ) ) ),
inference(forward_demodulation,[status(thm)],[f58,f56]) ).
thf(f56,plain,
( ( setadjoin @ ( sK5 @ sK3 ) @ emptyset )
= ( dsetconstr @ sK0
@ ^ [Y0: $i] : ( in @ ( kpair @ sK3 @ Y0 ) @ sK1 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f53]) ).
thf(f53,plain,
( ( ( setadjoin @ ( sK5 @ sK3 ) @ emptyset )
= ( dsetconstr @ sK0
@ ^ [Y0: $i] : ( in @ ( kpair @ sK3 @ Y0 ) @ sK1 ) ) )
| ( $true = $false ) ),
inference(superposition,[status(thm)],[f48,f25]) ).
thf(f25,plain,
( $true
= ( in @ sK3 @ sK2 ) ),
inference(cnf_transformation,[status(thm)],[f21]) ).
thf(f21,plain,
( ( $true
= ( func @ sK2 @ sK0 @ sK1 ) )
& ( $true
!= ( singleton
@ ( dsetconstr @ sK0
@ ^ [Y0: $i] : ( in @ ( kpair @ sK3 @ Y0 ) @ sK1 ) ) ) )
& ( $true
= ( in @ sK3 @ sK2 ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2,sK3])],[f18,f20,f19]) ).
thf(f19,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( func @ X2 @ X0 @ X1 ) )
& ? [X3: $i] :
( ( $true
!= ( singleton
@ ( dsetconstr @ X0
@ ^ [Y0: $i] : ( in @ ( kpair @ X3 @ Y0 ) @ X1 ) ) ) )
& ( $true
= ( in @ X3 @ X2 ) ) ) )
=> ( ( $true
= ( func @ sK2 @ sK0 @ sK1 ) )
& ? [X3: $i] :
( ( $true
!= ( singleton
@ ( dsetconstr @ sK0
@ ^ [Y0: $i] : ( in @ ( kpair @ X3 @ Y0 ) @ sK1 ) ) ) )
& ( $true
= ( in @ X3 @ sK2 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f20,plain,
( ? [X3: $i] :
( ( $true
!= ( singleton
@ ( dsetconstr @ sK0
@ ^ [Y0: $i] : ( in @ ( kpair @ X3 @ Y0 ) @ sK1 ) ) ) )
& ( $true
= ( in @ X3 @ sK2 ) ) )
=> ( ( $true
!= ( singleton
@ ( dsetconstr @ sK0
@ ^ [Y0: $i] : ( in @ ( kpair @ sK3 @ Y0 ) @ sK1 ) ) ) )
& ( $true
= ( in @ sK3 @ sK2 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f18,plain,
? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( func @ X2 @ X0 @ X1 ) )
& ? [X3: $i] :
( ( $true
!= ( singleton
@ ( dsetconstr @ X0
@ ^ [Y0: $i] : ( in @ ( kpair @ X3 @ Y0 ) @ X1 ) ) ) )
& ( $true
= ( in @ X3 @ X2 ) ) ) ),
inference(ennf_transformation,[status(thm)],[f15]) ).
thf(f15,plain,
~ ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( func @ X2 @ X0 @ X1 ) )
=> ! [X3: $i] :
( ( $true
= ( in @ X3 @ X2 ) )
=> ( $true
= ( singleton
@ ( dsetconstr @ X0
@ ^ [Y0: $i] : ( in @ ( kpair @ X3 @ Y0 ) @ X1 ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f14]) ).
thf(f14,plain,
~ ! [X0: $i,X1: $i,X2: $i] :
( ( func @ X2 @ X0 @ X1 )
=> ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( singleton
@ ( dsetconstr @ X0
@ ^ [X4: $i] : ( in @ ( kpair @ X3 @ X4 ) @ X1 ) ) ) ) ),
inference(rectify,[status(thm)],[f6]) ).
thf(f6,negated_conjecture,
~ ! [X3: $i,X7: $i,X0: $i] :
( ( func @ X0 @ X3 @ X7 )
=> ! [X1: $i] :
( ( in @ X1 @ X0 )
=> ( singleton
@ ( dsetconstr @ X3
@ ^ [X6: $i] : ( in @ ( kpair @ X1 @ X6 ) @ X7 ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f5]) ).
thf(f5,conjecture,
! [X3: $i,X7: $i,X0: $i] :
( ( func @ X0 @ X3 @ X7 )
=> ! [X1: $i] :
( ( in @ X1 @ X0 )
=> ( singleton
@ ( dsetconstr @ X3
@ ^ [X6: $i] : ( in @ ( kpair @ X1 @ X6 ) @ X7 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.PnBmkGKoOW/DTF2THF_4224.p',funcImageSingleton) ).
thf(f48,plain,
! [X1: $i] :
( ( $false
= ( in @ X1 @ sK2 ) )
| ( ( dsetconstr @ sK0
@ ^ [Y0: $i] : ( in @ ( kpair @ X1 @ Y0 ) @ sK1 ) )
= ( setadjoin @ ( sK5 @ X1 ) @ emptyset ) ) ),
inference(equality_proxy_clausification,[status(thm)],[f47]) ).
thf(f47,plain,
! [X1: $i] :
( ( $true
= ( ( setadjoin @ ( sK5 @ X1 ) @ emptyset )
= ( dsetconstr @ sK0
@ ^ [Y0: $i] : ( in @ ( kpair @ X1 @ Y0 ) @ sK1 ) ) ) )
| ( $false
= ( in @ X1 @ sK2 ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f45]) ).
thf(f45,plain,
! [X1: $i] :
( ( $true
= ( ( ( setadjoin @ ( sK5 @ X1 ) @ emptyset )
= ( dsetconstr @ sK0
@ ^ [Y0: $i] : ( in @ ( kpair @ X1 @ Y0 ) @ sK1 ) ) )
& ( in @ ( sK5 @ X1 )
@ ( dsetconstr @ sK0
@ ^ [Y0: $i] : ( in @ ( kpair @ X1 @ Y0 ) @ sK1 ) ) ) ) )
| ( $false
= ( in @ X1 @ sK2 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f44]) ).
thf(f44,plain,
! [X1: $i] :
( ( $true
= ( ^ [Y0: $i] :
( ( ( setadjoin @ Y0 @ emptyset )
= ( dsetconstr @ sK0
@ ^ [Y1: $i] : ( in @ ( kpair @ X1 @ Y1 ) @ sK1 ) ) )
& ( in @ Y0
@ ( dsetconstr @ sK0
@ ^ [Y1: $i] : ( in @ ( kpair @ X1 @ Y1 ) @ sK1 ) ) ) )
@ ( sK5 @ X1 ) ) )
| ( $false
= ( in @ X1 @ sK2 ) ) ),
inference(sigma_clausification,[status(thm)],[f43]) ).
thf(f43,plain,
! [X1: $i] :
( ( $true
= ( ?? @ $i
@ ^ [Y0: $i] :
( ( ( setadjoin @ Y0 @ emptyset )
= ( dsetconstr @ sK0
@ ^ [Y1: $i] : ( in @ ( kpair @ X1 @ Y1 ) @ sK1 ) ) )
& ( in @ Y0
@ ( dsetconstr @ sK0
@ ^ [Y1: $i] : ( in @ ( kpair @ X1 @ Y1 ) @ sK1 ) ) ) ) ) )
| ( $false
= ( in @ X1 @ sK2 ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f42]) ).
thf(f42,plain,
! [X1: $i] :
( $true
= ( ( in @ X1 @ sK2 )
=> ( ?? @ $i
@ ^ [Y0: $i] :
( ( ( setadjoin @ Y0 @ emptyset )
= ( dsetconstr @ sK0
@ ^ [Y1: $i] : ( in @ ( kpair @ X1 @ Y1 ) @ sK1 ) ) )
& ( in @ Y0
@ ( dsetconstr @ sK0
@ ^ [Y1: $i] : ( in @ ( kpair @ X1 @ Y1 ) @ sK1 ) ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f41]) ).
thf(f41,plain,
! [X1: $i] :
( $true
= ( ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( ?? @ $i
@ ^ [Y1: $i] :
( ( ( setadjoin @ Y1 @ emptyset )
= ( dsetconstr @ sK0
@ ^ [Y2: $i] : ( in @ ( kpair @ Y0 @ Y2 ) @ sK1 ) ) )
& ( in @ Y1
@ ( dsetconstr @ sK0
@ ^ [Y2: $i] : ( in @ ( kpair @ Y0 @ Y2 ) @ sK1 ) ) ) ) ) )
@ X1 ) ),
inference(pi_clausification,[status(thm)],[f39]) ).
thf(f39,plain,
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( ?? @ $i
@ ^ [Y1: $i] :
( ( ( setadjoin @ Y1 @ emptyset )
= ( dsetconstr @ sK0
@ ^ [Y2: $i] : ( in @ ( kpair @ Y0 @ Y2 ) @ sK1 ) ) )
& ( in @ Y1
@ ( dsetconstr @ sK0
@ ^ [Y2: $i] : ( in @ ( kpair @ Y0 @ Y2 ) @ sK1 ) ) ) ) ) ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f38]) ).
thf(f38,plain,
( $true
= ( ( subset @ sK1 @ ( cartprod @ sK2 @ sK0 ) )
& ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( ?? @ $i
@ ^ [Y1: $i] :
( ( ( setadjoin @ Y1 @ emptyset )
= ( dsetconstr @ sK0
@ ^ [Y2: $i] : ( in @ ( kpair @ Y0 @ Y2 ) @ sK1 ) ) )
& ( in @ Y1
@ ( dsetconstr @ sK0
@ ^ [Y2: $i] : ( in @ ( kpair @ Y0 @ Y2 ) @ sK1 ) ) ) ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f31]) ).
thf(f31,plain,
( $true
= ( ^ [Y0: $i,Y1: $i,Y2: $i] :
( ( ^ [Y3: $i,Y4: $i,Y5: $i] : ( subset @ Y5 @ ( cartprod @ Y3 @ Y4 ) )
@ Y0
@ Y1
@ Y2 )
& ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( ^ [Y4: $i,Y5: $i > $o] :
( ^ [Y6: $i] :
( ?? @ $i
@ ^ [Y7: $i] :
( ( ( setadjoin @ Y7 @ emptyset )
= Y6 )
& ( in @ Y7 @ Y6 ) ) )
@ ( dsetconstr @ Y4
@ ^ [Y6: $i] : ( Y5 @ Y6 ) ) )
@ Y1
@ ^ [Y4: $i] : ( in @ ( kpair @ Y3 @ Y4 ) @ Y2 ) ) ) ) )
@ sK2
@ sK0
@ sK1 ) ),
inference(definition_unfolding,[status(thm)],[f27,f30]) ).
thf(f30,plain,
( func
= ( ^ [Y0: $i,Y1: $i,Y2: $i] :
( ( ^ [Y3: $i,Y4: $i,Y5: $i] : ( subset @ Y5 @ ( cartprod @ Y3 @ Y4 ) )
@ Y0
@ Y1
@ Y2 )
& ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( ^ [Y4: $i,Y5: $i > $o] :
( ^ [Y6: $i] :
( ?? @ $i
@ ^ [Y7: $i] :
( ( ( setadjoin @ Y7 @ emptyset )
= Y6 )
& ( in @ Y7 @ Y6 ) ) )
@ ( dsetconstr @ Y4
@ ^ [Y6: $i] : ( Y5 @ Y6 ) ) )
@ Y1
@ ^ [Y4: $i] : ( in @ ( kpair @ Y3 @ Y4 ) @ Y2 ) ) ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f24,f28,f29]) ).
thf(f29,plain,
( ex1
= ( ^ [Y0: $i,Y1: $i > $o] :
( ^ [Y2: $i] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( ( setadjoin @ Y3 @ emptyset )
= Y2 )
& ( in @ Y3 @ Y2 ) ) )
@ ( dsetconstr @ Y0
@ ^ [Y2: $i] : ( Y1 @ Y2 ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f22,f23]) ).
thf(f23,plain,
( singleton
= ( ^ [Y0: $i] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( ( setadjoin @ Y1 @ emptyset )
= Y0 )
& ( in @ Y1 @ Y0 ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f17]) ).
thf(f17,plain,
( singleton
= ( ^ [Y0: $i] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( ( setadjoin @ Y1 @ emptyset )
= Y0 )
& ( in @ Y1 @ Y0 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f16]) ).
thf(f16,plain,
( singleton
= ( ^ [X0: $i] :
? [X1: $i] :
( ( in @ X1 @ X0 )
& ( ( setadjoin @ X1 @ emptyset )
= X0 ) ) ) ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
( singleton
= ( ^ [X0: $i] :
? [X1: $i] :
( ( in @ X1 @ X0 )
& ( ( setadjoin @ X1 @ emptyset )
= X0 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.PnBmkGKoOW/DTF2THF_4224.p',singleton) ).
thf(f22,plain,
( ex1
= ( ^ [Y0: $i,Y1: $i > $o] :
( singleton
@ ( dsetconstr @ Y0
@ ^ [Y2: $i] : ( Y1 @ Y2 ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f9]) ).
thf(f9,plain,
( ex1
= ( ^ [Y0: $i,Y1: $i > $o] :
( singleton
@ ( dsetconstr @ Y0
@ ^ [Y2: $i] : ( Y1 @ Y2 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f8]) ).
thf(f8,plain,
( ( ^ [X0: $i,X1: $i > $o] :
( singleton
@ ( dsetconstr @ X0
@ ^ [X2: $i] : ( X1 @ X2 ) ) ) )
= ex1 ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
( ( ^ [X0: $i,X2: $i > $o] :
( singleton
@ ( dsetconstr @ X0
@ ^ [X1: $i] : ( X2 @ X1 ) ) ) )
= ex1 ),
file('/export/starexec/sandbox/tmp/tmp.PnBmkGKoOW/DTF2THF_4224.p',ex1) ).
thf(f28,plain,
( breln
= ( ^ [Y0: $i,Y1: $i,Y2: $i] : ( subset @ Y2 @ ( cartprod @ Y0 @ Y1 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f11]) ).
thf(f11,plain,
( breln
= ( ^ [Y0: $i,Y1: $i,Y2: $i] : ( subset @ Y2 @ ( cartprod @ Y0 @ Y1 ) ) ) ),
inference(fool_elimination,[status(thm)],[f10]) ).
thf(f10,plain,
( ( ^ [X0: $i,X1: $i,X2: $i] : ( subset @ X2 @ ( cartprod @ X0 @ X1 ) ) )
= breln ),
inference(rectify,[status(thm)],[f3]) ).
thf(f3,axiom,
( ( ^ [X0: $i,X3: $i,X4: $i] : ( subset @ X4 @ ( cartprod @ X0 @ X3 ) ) )
= breln ),
file('/export/starexec/sandbox/tmp/tmp.PnBmkGKoOW/DTF2THF_4224.p',breln) ).
thf(f24,plain,
( func
= ( ^ [Y0: $i,Y1: $i,Y2: $i] :
( ( breln @ Y0 @ Y1 @ Y2 )
& ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( ex1 @ Y1
@ ^ [Y4: $i] : ( in @ ( kpair @ Y3 @ Y4 ) @ Y2 ) ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f13]) ).
thf(f13,plain,
( func
= ( ^ [Y0: $i,Y1: $i,Y2: $i] :
( ( breln @ Y0 @ Y1 @ Y2 )
& ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( ex1 @ Y1
@ ^ [Y4: $i] : ( in @ ( kpair @ Y3 @ Y4 ) @ Y2 ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f12]) ).
thf(f12,plain,
( func
= ( ^ [X0: $i,X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X0 )
=> ( ex1 @ X1
@ ^ [X4: $i] : ( in @ ( kpair @ X3 @ X4 ) @ X2 ) ) )
& ( breln @ X0 @ X1 @ X2 ) ) ) ),
inference(rectify,[status(thm)],[f4]) ).
thf(f4,axiom,
( func
= ( ^ [X0: $i,X3: $i,X5: $i] :
( ! [X1: $i] :
( ( in @ X1 @ X0 )
=> ( ex1 @ X3
@ ^ [X6: $i] : ( in @ ( kpair @ X1 @ X6 ) @ X5 ) ) )
& ( breln @ X0 @ X3 @ X5 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.PnBmkGKoOW/DTF2THF_4224.p',func) ).
thf(f27,plain,
( $true
= ( func @ sK2 @ sK0 @ sK1 ) ),
inference(cnf_transformation,[status(thm)],[f21]) ).
thf(f58,plain,
! [X1: $i] :
( ( ( in @ X1
@ ( dsetconstr @ sK0
@ ^ [Y0: $i] : ( in @ ( kpair @ sK3 @ Y0 ) @ sK1 ) ) )
= $false )
| ( ( setadjoin @ X1 @ emptyset )
!= ( setadjoin @ ( sK5 @ sK3 ) @ emptyset ) ) ),
inference(backward_demodulation,[status(thm)],[f37,f56]) ).
thf(f37,plain,
! [X1: $i] :
( ( ( in @ X1
@ ( dsetconstr @ sK0
@ ^ [Y0: $i] : ( in @ ( kpair @ sK3 @ Y0 ) @ sK1 ) ) )
= $false )
| ( ( setadjoin @ X1 @ emptyset )
!= ( dsetconstr @ sK0
@ ^ [Y0: $i] : ( in @ ( kpair @ sK3 @ Y0 ) @ sK1 ) ) ) ),
inference(equality_proxy_clausification,[status(thm)],[f36]) ).
thf(f36,plain,
! [X1: $i] :
( ( ( in @ X1
@ ( dsetconstr @ sK0
@ ^ [Y0: $i] : ( in @ ( kpair @ sK3 @ Y0 ) @ sK1 ) ) )
= $false )
| ( ( ( setadjoin @ X1 @ emptyset )
= ( dsetconstr @ sK0
@ ^ [Y0: $i] : ( in @ ( kpair @ sK3 @ Y0 ) @ sK1 ) ) )
= $false ) ),
inference(binary_proxy_clausification,[status(thm)],[f35]) ).
thf(f35,plain,
! [X1: $i] :
( ( ( ( setadjoin @ X1 @ emptyset )
= ( dsetconstr @ sK0
@ ^ [Y0: $i] : ( in @ ( kpair @ sK3 @ Y0 ) @ sK1 ) ) )
& ( in @ X1
@ ( dsetconstr @ sK0
@ ^ [Y0: $i] : ( in @ ( kpair @ sK3 @ Y0 ) @ sK1 ) ) ) )
= $false ),
inference(beta_eta_normalization,[status(thm)],[f34]) ).
thf(f34,plain,
! [X1: $i] :
( ( ^ [Y0: $i] :
( ( ( setadjoin @ Y0 @ emptyset )
= ( dsetconstr @ sK0
@ ^ [Y1: $i] : ( in @ ( kpair @ sK3 @ Y1 ) @ sK1 ) ) )
& ( in @ Y0
@ ( dsetconstr @ sK0
@ ^ [Y1: $i] : ( in @ ( kpair @ sK3 @ Y1 ) @ sK1 ) ) ) )
@ X1 )
= $false ),
inference(pi_clausification,[status(thm)],[f33]) ).
thf(f33,plain,
( $true
!= ( ?? @ $i
@ ^ [Y0: $i] :
( ( ( setadjoin @ Y0 @ emptyset )
= ( dsetconstr @ sK0
@ ^ [Y1: $i] : ( in @ ( kpair @ sK3 @ Y1 ) @ sK1 ) ) )
& ( in @ Y0
@ ( dsetconstr @ sK0
@ ^ [Y1: $i] : ( in @ ( kpair @ sK3 @ Y1 ) @ sK1 ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f32]) ).
thf(f32,plain,
( $true
!= ( ^ [Y0: $i] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( ( setadjoin @ Y1 @ emptyset )
= Y0 )
& ( in @ Y1 @ Y0 ) ) )
@ ( dsetconstr @ sK0
@ ^ [Y0: $i] : ( in @ ( kpair @ sK3 @ Y0 ) @ sK1 ) ) ) ),
inference(definition_unfolding,[status(thm)],[f26,f23]) ).
thf(f26,plain,
( $true
!= ( singleton
@ ( dsetconstr @ sK0
@ ^ [Y0: $i] : ( in @ ( kpair @ sK3 @ Y0 ) @ sK1 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f21]) ).
thf(f57,plain,
( $true
= ( in @ ( sK5 @ sK3 ) @ ( setadjoin @ ( sK5 @ sK3 ) @ emptyset ) ) ),
inference(backward_demodulation,[status(thm)],[f52,f56]) ).
thf(f52,plain,
( $true
= ( in @ ( sK5 @ sK3 )
@ ( dsetconstr @ sK0
@ ^ [Y0: $i] : ( in @ ( kpair @ sK3 @ Y0 ) @ sK1 ) ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f49]) ).
thf(f49,plain,
( ( $true = $false )
| ( $true
= ( in @ ( sK5 @ sK3 )
@ ( dsetconstr @ sK0
@ ^ [Y0: $i] : ( in @ ( kpair @ sK3 @ Y0 ) @ sK1 ) ) ) ) ),
inference(superposition,[status(thm)],[f46,f25]) ).
thf(f46,plain,
! [X1: $i] :
( ( $false
= ( in @ X1 @ sK2 ) )
| ( $true
= ( in @ ( sK5 @ X1 )
@ ( dsetconstr @ sK0
@ ^ [Y0: $i] : ( in @ ( kpair @ X1 @ Y0 ) @ sK1 ) ) ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f45]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU672^2 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n002.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 12:30:48 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.19/0.48 ---- Original DTF file ---
% 0.19/0.48 thf(spec,logic,$$dhol).
% 0.19/0.48 %------------------------------------------------------------------------------
% 0.19/0.48 % File : SEU672^2 : TPTP v9.2.1. Released v3.7.0.
% 0.19/0.48 % Domain : Set Theory
% 0.19/0.48 % Problem : Functions
% 0.19/0.48 % Version : Especial > Reduced > Especial.
% 0.19/0.48 % English : (! A:i.! B:i.! f:i.func A B f -> (! x:i.in x A -> singleton
% 0.19/0.48 % (dsetconstr B (^ y:i.in (kpair x y) f))))
% 0.19/0.48
% 0.19/0.48 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.19/0.48 % Source : [Bro08]
% 0.19/0.48 % Names : ZFC174l [Bro08]
% 0.19/0.48
% 0.19/0.48 % Status : Theorem
% 0.19/0.48 % Rating : 0.00 v8.2.0, 0.15 v8.1.0, 0.09 v7.5.0, 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, 0.14 v5.5.0, 0.33 v5.4.0, 0.40 v5.3.0, 0.60 v5.2.0, 0.40 v5.1.0, 0.20 v4.1.0, 0.00 v3.7.0
% 0.19/0.48 % Syntax : Number of formulae : 16 ( 4 unt; 11 typ; 4 def)
% 0.19/0.48 % Number of atoms : 20 ( 5 equ; 0 cnn)
% 0.19/0.48 % Maximal formula atoms : 4 ( 4 avg)
% 0.19/0.48 % Number of connectives : 40 ( 0 ~; 0 |; 2 &; 35 @)
% 0.19/0.48 % ( 0 <=>; 3 =>; 0 <=; 0 <~>)
% 0.19/0.48 % Maximal formula depth : 14 ( 4 avg)
% 0.19/0.48 % Number of types : 2 ( 0 usr)
% 0.19/0.48 % Number of type conns : 24 ( 24 >; 0 *; 0 +; 0 <<)
% 0.19/0.48 % Number of symbols : 12 ( 11 usr; 1 con; 0-3 aty)
% 0.19/0.48 % Number of variables : 18 ( 12 ^; 5 !; 1 ?; 18 :)
% 0.19/0.48 % SPC : TH0_THM_EQU_NAR
% 0.19/0.48
% 0.19/0.48 % Comments : http://mathgate.info/detsetitem.php?id=223
% 0.19/0.48 %------------------------------------------------------------------------------
% 0.19/0.48 thf(in_type,type,
% 0.19/0.48 in: $i > $i > $o ).
% 0.19/0.48
% 0.19/0.48 thf(emptyset_type,type,
% 0.19/0.48 emptyset: $i ).
% 0.19/0.48
% 0.19/0.48 thf(setadjoin_type,type,
% 0.19/0.48 setadjoin: $i > $i > $i ).
% 0.19/0.48
% 0.19/0.48 thf(dsetconstr_type,type,
% 0.19/0.48 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.19/0.48
% 0.19/0.48 thf(subset_type,type,
% 0.19/0.48 subset: $i > $i > $o ).
% 0.19/0.48
% 0.19/0.48 thf(kpair_type,type,
% 0.19/0.48 kpair: $i > $i > $i ).
% 0.19/0.48
% 0.19/0.48 thf(cartprod_type,type,
% 0.19/0.48 cartprod: $i > $i > $i ).
% 0.19/0.48
% 0.19/0.48 thf(singleton_type,type,
% 0.19/0.48 singleton: $i > $o ).
% 0.19/0.48
% 0.19/0.48 thf(singleton,definition,
% 0.19/0.48 ( singleton
% 0.19/0.48 = ( ^ [A: $i] :
% 0.19/0.48 ? [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 & ( A
% 0.19/0.48 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(ex1_type,type,
% 0.19/0.48 ex1: $i > ( $i > $o ) > $o ).
% 0.19/0.48
% 0.19/0.48 thf(ex1,definition,
% 0.19/0.48 ( ex1
% 0.19/0.48 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.19/0.48 ( singleton
% 0.19/0.48 @ ( dsetconstr @ A
% 0.19/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(breln_type,type,
% 0.19/0.48 breln: $i > $i > $i > $o ).
% 0.19/0.48
% 0.19/0.48 thf(breln,definition,
% 0.19/0.48 ( breln
% 0.19/0.48 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(func_type,type,
% 0.19/0.48 func: $i > $i > $i > $o ).
% 0.19/0.48
% 0.19/0.48 thf(func,definition,
% 0.19/0.48 ( func
% 0.19/0.48 = ( ^ [A: $i,B: $i,R: $i] :
% 0.19/0.48 ( ( breln @ A @ B @ R )
% 0.19/0.48 & ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( ex1 @ B
% 0.19/0.48 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(funcImageSingleton,conjecture,
% 0.19/0.48 ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.48 ( ( func @ A @ B @ Xf )
% 0.19/0.48 => ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( singleton
% 0.19/0.48 @ ( dsetconstr @ B
% 0.19/0.48 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 %------------------------------------------------------------------------------
% 0.19/0.48 ------------------------
% 1.69/1.44 ---- Embedded in THF ---
% 1.69/1.44 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.69/1.44 %%% Generated on Tue Feb 24 12:30:49 EST 2026
% 1.69/1.44 %%% using '$$dhol' embedding, version 1.3.0.
% 1.69/1.44 %%% Logic specification used:
% 1.69/1.44 %%% thf(spec, logic, $$dhol).
% 1.69/1.44
% 1.69/1.44 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.PnBmkGKoOW/DTF2DTF4224.p
% See solution above
% 1.69/1.44 ------------------------
% 1.69/1.45 ---- Cleaned THF ---
% 1.69/1.45 thf(in_type,type,
% 1.69/1.45 in: $i > $i > $o ).
% 1.69/1.45
% 1.69/1.45 thf(emptyset_type,type,
% 1.69/1.45 emptyset: $i ).
% 1.69/1.45
% 1.69/1.45 thf(setadjoin_type,type,
% 1.69/1.45 setadjoin: $i > $i > $i ).
% 1.69/1.45
% 1.69/1.45 thf(dsetconstr_type,type,
% 1.69/1.45 dsetconstr: $i > ( $i > $o ) > $i ).
% 1.69/1.45
% 1.69/1.45 thf(subset_type,type,
% 1.69/1.45 subset: $i > $i > $o ).
% 1.69/1.45
% 1.69/1.45 thf(kpair_type,type,
% 1.69/1.45 kpair: $i > $i > $i ).
% 1.69/1.45
% 1.69/1.45 thf(cartprod_type,type,
% 1.69/1.45 cartprod: $i > $i > $i ).
% 1.69/1.45
% 1.69/1.45 thf(singleton_type,type,
% 1.69/1.45 singleton: $i > $o ).
% 1.69/1.45
% 1.69/1.45 thf(singleton,definition,
% 1.69/1.45 ( singleton
% 1.69/1.45 = ( ^ [A: $i] :
% 1.69/1.45 ? [Xx: $i] :
% 1.69/1.45 ( ( in @ Xx @ A )
% 1.69/1.45 & ( A
% 1.69/1.45 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 1.69/1.45
% 1.69/1.45 thf(ex1_type,type,
% 1.69/1.45 ex1: $i > ( $i > $o ) > $o ).
% 1.69/1.45
% 1.69/1.45 thf(ex1,definition,
% 1.69/1.45 ( ex1
% 1.69/1.45 = ( ^ [A: $i,Xphi: $i > $o] :
% 1.69/1.45 ( singleton
% 1.69/1.45 @ ( dsetconstr @ A
% 1.69/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 1.69/1.45
% 1.69/1.45 thf(breln_type,type,
% 1.69/1.45 breln: $i > $i > $i > $o ).
% 1.69/1.45
% 1.69/1.45 thf(breln,definition,
% 1.69/1.45 ( breln
% 1.69/1.45 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 1.69/1.45
% 1.69/1.45 thf(func_type,type,
% 1.69/1.45 func: $i > $i > $i > $o ).
% 1.69/1.45
% 1.69/1.45 thf(func,definition,
% 1.69/1.45 ( func
% 1.69/1.45 = ( ^ [A: $i,B: $i,R: $i] :
% 1.69/1.45 ( ( breln @ A @ B @ R )
% 1.69/1.45 & ! [Xx: $i] :
% 1.69/1.45 ( ( in @ Xx @ A )
% 1.69/1.45 => ( ex1 @ B
% 1.69/1.45 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ).
% 1.69/1.45
% 1.69/1.45 thf(funcImageSingleton,conjecture,
% 1.69/1.45 ! [A: $i,B: $i,Xf: $i] :
% 1.69/1.45 ( ( func @ A @ B @ Xf )
% 1.69/1.45 => ! [Xx: $i] :
% 1.69/1.45 ( ( in @ Xx @ A )
% 1.69/1.45 => ( singleton
% 1.69/1.45 @ ( dsetconstr @ B
% 1.69/1.45 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ).
% 1.69/1.45 ------------------------
% 1.69/1.47 % (4323)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_4224 for (2999ds/27Mi)
% 1.69/1.47 % (4321)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_4224 for (2999ds/183Mi)
% 1.69/1.47 % (4324)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_4224 for (2999ds/2Mi)
% 1.69/1.47 % (4322)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_4224 for (2999ds/4Mi)
% 1.69/1.47 % (4325)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_4224 for (2999ds/2Mi)
% 1.69/1.47 % (4327)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_4224 for (2999ds/18Mi)
% 1.69/1.47 % (4326)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_4224 for (2999ds/275Mi)
% 1.69/1.47 % (4324)Instruction limit reached!
% 1.69/1.47 % (4324)------------------------------
% 1.69/1.47 % (4324)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.69/1.47 % (4324)Termination reason: Unknown
% 1.69/1.47 % (4325)Instruction limit reached!
% 1.69/1.47 % (4325)------------------------------
% 1.69/1.47 % (4325)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.69/1.47 % (4325)Termination reason: Unknown
% 1.69/1.47 % (4325)Termination phase: Property scanning
% 1.69/1.47
% 1.69/1.47 % (4325)Memory used [KB]: 1023
% 1.69/1.47 % (4325)Time elapsed: 0.003 s
% 1.69/1.47 % (4325)Instructions burned: 3 (million)
% 1.69/1.47 % (4325)------------------------------
% 1.69/1.47 % (4325)------------------------------
% 1.69/1.47 % (4324)Termination phase: Function definition elimination
% 1.69/1.47
% 1.69/1.47 % (4324)Memory used [KB]: 1023
% 1.69/1.47 % (4324)Time elapsed: 0.003 s
% 1.69/1.47 % (4324)Instructions burned: 2 (million)
% 1.69/1.47 % (4324)------------------------------
% 1.69/1.47 % (4324)------------------------------
% 1.69/1.47 % (4326)Refutation not found, incomplete strategy
% 1.69/1.47 % (4326)------------------------------
% 1.69/1.47 % (4326)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.69/1.47 % (4326)Termination reason: Refutation not found, incomplete strategy
% 1.69/1.47
% 1.69/1.47
% 1.69/1.47 % (4326)Memory used [KB]: 5500
% 1.69/1.47 % (4326)Time elapsed: 0.003 s
% 1.69/1.47 % (4322)Instruction limit reached!
% 1.69/1.47 % (4322)------------------------------
% 1.69/1.47 % (4322)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.69/1.47 % (4322)Termination reason: Unknown
% 1.69/1.47 % (4322)Termination phase: Saturation
% 1.69/1.47
% 1.69/1.47 % (4322)Memory used [KB]: 5500
% 1.69/1.47 % (4322)Time elapsed: 0.004 s
% 1.69/1.47 % (4322)Instructions burned: 4 (million)
% 1.69/1.47 % (4322)------------------------------
% 1.69/1.47 % (4322)------------------------------
% 1.69/1.47 % (4326)Instructions burned: 3 (million)
% 1.69/1.47 % (4326)------------------------------
% 1.69/1.47 % (4326)------------------------------
% 1.69/1.47 % (4327)First to succeed.
% 1.69/1.47 % (4323)Also succeeded, but the first one will report.
% 1.69/1.47 % (4327)Refutation found. Thanks to Tanya!
% 1.69/1.47 % SZS status Theorem for DTF2THF_4224
% 1.69/1.47 % SZS output start Proof for DTF2THF_4224
% 1.69/1.47 thf(func_def_0, type, in: $i > $i > $o).
% 1.69/1.47 thf(func_def_2, type, setadjoin: $i > $i > $i).
% 1.69/1.47 thf(func_def_3, type, dsetconstr: $i > ($i > $o) > $i).
% 1.69/1.47 thf(func_def_4, type, subset: $i > $i > $o).
% 1.69/1.47 thf(func_def_5, type, kpair: $i > $i > $i).
% 1.69/1.47 thf(func_def_6, type, cartprod: $i > $i > $i).
% 1.69/1.47 thf(func_def_7, type, singleton: $i > $o).
% 1.69/1.47 thf(func_def_9, type, ex1: $i > ($i > $o) > $o).
% 1.69/1.47 thf(func_def_10, type, breln: $i > $i > $i > $o).
% 1.69/1.47 thf(func_def_11, type, func: $i > $i > $i > $o).
% 1.69/1.47 thf(func_def_29, type, sK5: $i > $i).
% 1.69/1.47 thf(func_def_30, type, ph6: !>[X0: $tType]:(X0)).
% 1.69/1.47 thf(f62,plain,(
% 1.69/1.47 $false),
% 1.69/1.47 inference(trivial_inequality_removal,[status(thm)],[f61])).
% 1.69/1.47 thf(f61,plain,(
% 1.69/1.47 ($true = $false)),
% 1.69/1.47 inference(backward_demodulation,[status(thm)],[f57,f60])).
% 1.69/1.47 thf(f60,plain,(
% 1.69/1.47 ($false = (in @ (sK5 @ sK3) @ (setadjoin @ (sK5 @ sK3) @ emptyset)))),
% 1.69/1.47 inference(equality_resolution,[status(thm)],[f59])).
% 1.69/1.47 thf(f59,plain,(
% 1.69/1.47 ( ! [X1 : $i] : (((setadjoin @ X1 @ emptyset) != (setadjoin @ (sK5 @ sK3) @ emptyset)) | ($false = (in @ X1 @ (setadjoin @ (sK5 @ sK3) @ emptyset)))) )),
% 1.69/1.47 inference(forward_demodulation,[status(thm)],[f58,f56])).
% 1.69/1.47 thf(f56,plain,(
% 1.69/1.47 ((setadjoin @ (sK5 @ sK3) @ emptyset) = (dsetconstr @ sK0 @ (^[Y0 : $i]: (in @ (kpair @ sK3 @ Y0) @ sK1))))),
% 1.69/1.47 inference(trivial_inequality_removal,[status(thm)],[f53])).
% 1.69/1.47 thf(f53,plain,(
% 1.69/1.47 ((setadjoin @ (sK5 @ sK3) @ emptyset) = (dsetconstr @ sK0 @ (^[Y0 : $i]: (in @ (kpair @ sK3 @ Y0) @ sK1)))) | ($true = $false)),
% 1.69/1.47 inference(superposition,[status(thm)],[f48,f25])).
% 1.69/1.47 thf(f25,plain,(
% 1.69/1.47 ($true = (in @ sK3 @ sK2))),
% 1.69/1.47 inference(cnf_transformation,[status(thm)],[f21])).
% 1.69/1.47 thf(f21,plain,(
% 1.69/1.47 ($true = (func @ sK2 @ sK0 @ sK1)) & (($true != (singleton @ (dsetconstr @ sK0 @ (^[Y0 : $i]: (in @ (kpair @ sK3 @ Y0) @ sK1))))) & ($true = (in @ sK3 @ sK2)))),
% 1.69/1.47 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2,sK3])],[f18,f20,f19])).
% 1.69/1.47 thf(f19,plain,(
% 1.69/1.47 ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (func @ X2 @ X0 @ X1)) & ? [X3 : $i] : (($true != (singleton @ (dsetconstr @ X0 @ (^[Y0 : $i]: (in @ (kpair @ X3 @ Y0) @ X1))))) & ($true = (in @ X3 @ X2)))) => (($true = (func @ sK2 @ sK0 @ sK1)) & ? [X3 : $i] : (($true != (singleton @ (dsetconstr @ sK0 @ (^[Y0 : $i]: (in @ (kpair @ X3 @ Y0) @ sK1))))) & ($true = (in @ X3 @ sK2))))),
% 1.69/1.47 introduced(definition,[],[choice_axiom])).
% 1.69/1.47 thf(f20,plain,(
% 1.69/1.47 ? [X3 : $i] : (($true != (singleton @ (dsetconstr @ sK0 @ (^[Y0 : $i]: (in @ (kpair @ X3 @ Y0) @ sK1))))) & ($true = (in @ X3 @ sK2))) => (($true != (singleton @ (dsetconstr @ sK0 @ (^[Y0 : $i]: (in @ (kpair @ sK3 @ Y0) @ sK1))))) & ($true = (in @ sK3 @ sK2)))),
% 1.69/1.47 introduced(definition,[],[choice_axiom])).
% 1.69/1.47 thf(f18,plain,(
% 1.69/1.47 ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (func @ X2 @ X0 @ X1)) & ? [X3 : $i] : (($true != (singleton @ (dsetconstr @ X0 @ (^[Y0 : $i]: (in @ (kpair @ X3 @ Y0) @ X1))))) & ($true = (in @ X3 @ X2))))),
% 1.69/1.47 inference(ennf_transformation,[status(thm)],[f15])).
% 1.69/1.47 thf(f15,plain,(
% 1.69/1.47 ~! [X0 : $i,X1 : $i,X2 : $i] : (($true = (func @ X2 @ X0 @ X1)) => ! [X3 : $i] : (($true = (in @ X3 @ X2)) => ($true = (singleton @ (dsetconstr @ X0 @ (^[Y0 : $i]: (in @ (kpair @ X3 @ Y0) @ X1)))))))),
% 1.69/1.47 inference(fool_elimination,[status(thm)],[f14])).
% 1.69/1.47 thf(f14,plain,(
% 1.69/1.47 ~! [X0 : $i,X1 : $i,X2 : $i] : ((func @ X2 @ X0 @ X1) => ! [X3 : $i] : ((in @ X3 @ X2) => (singleton @ (dsetconstr @ X0 @ (^[X4 : $i] : (in @ (kpair @ X3 @ X4) @ X1))))))),
% 1.69/1.47 inference(rectify,[status(thm)],[f6])).
% 1.69/1.47 thf(f6,negated_conjecture,(
% 1.69/1.47 ~! [X3 : $i,X7 : $i,X0 : $i] : ((func @ X0 @ X3 @ X7) => ! [X1 : $i] : ((in @ X1 @ X0) => (singleton @ (dsetconstr @ X3 @ (^[X6 : $i] : (in @ (kpair @ X1 @ X6) @ X7))))))),
% 1.69/1.47 inference(negated_conjecture,[status(cth)],[f5])).
% 1.69/1.47 thf(f5,conjecture,(
% 1.69/1.47 ! [X3 : $i,X7 : $i,X0 : $i] : ((func @ X0 @ X3 @ X7) => ! [X1 : $i] : ((in @ X1 @ X0) => (singleton @ (dsetconstr @ X3 @ (^[X6 : $i] : (in @ (kpair @ X1 @ X6) @ X7))))))),
% 1.69/1.47 file('/export/starexec/sandbox/tmp/tmp.PnBmkGKoOW/DTF2THF_4224.p',funcImageSingleton)).
% 1.69/1.47 thf(f48,plain,(
% 1.69/1.47 ( ! [X1 : $i] : (($false = (in @ X1 @ sK2)) | ((dsetconstr @ sK0 @ (^[Y0 : $i]: (in @ (kpair @ X1 @ Y0) @ sK1))) = (setadjoin @ (sK5 @ X1) @ emptyset))) )),
% 1.69/1.47 inference(equality_proxy_clausification,[status(thm)],[f47])).
% 1.69/1.47 thf(f47,plain,(
% 1.69/1.47 ( ! [X1 : $i] : (($true = ((setadjoin @ (sK5 @ X1) @ emptyset) = (dsetconstr @ sK0 @ (^[Y0 : $i]: (in @ (kpair @ X1 @ Y0) @ sK1))))) | ($false = (in @ X1 @ sK2))) )),
% 1.69/1.47 inference(binary_proxy_clausification,[status(thm)],[f45])).
% 1.69/1.47 thf(f45,plain,(
% 1.69/1.47 ( ! [X1 : $i] : (($true = (((setadjoin @ (sK5 @ X1) @ emptyset) = (dsetconstr @ sK0 @ (^[Y0 : $i]: (in @ (kpair @ X1 @ Y0) @ sK1)))) & (in @ (sK5 @ X1) @ (dsetconstr @ sK0 @ (^[Y0 : $i]: (in @ (kpair @ X1 @ Y0) @ sK1)))))) | ($false = (in @ X1 @ sK2))) )),
% 1.69/1.47 inference(beta_eta_normalization,[status(thm)],[f44])).
% 1.69/1.47 thf(f44,plain,(
% 1.69/1.47 ( ! [X1 : $i] : (($true = ((^[Y0 : $i]: (((setadjoin @ Y0 @ emptyset) = (dsetconstr @ sK0 @ (^[Y1 : $i]: (in @ (kpair @ X1 @ Y1) @ sK1)))) & (in @ Y0 @ (dsetconstr @ sK0 @ (^[Y1 : $i]: (in @ (kpair @ X1 @ Y1) @ sK1)))))) @ (sK5 @ X1))) | ($false = (in @ X1 @ sK2))) )),
% 1.69/1.47 inference(sigma_clausification,[status(thm)],[f43])).
% 1.69/1.47 thf(f43,plain,(
% 1.69/1.47 ( ! [X1 : $i] : (($true = (?? @ $i @ (^[Y0 : $i]: (((setadjoin @ Y0 @ emptyset) = (dsetconstr @ sK0 @ (^[Y1 : $i]: (in @ (kpair @ X1 @ Y1) @ sK1)))) & (in @ Y0 @ (dsetconstr @ sK0 @ (^[Y1 : $i]: (in @ (kpair @ X1 @ Y1) @ sK1)))))))) | ($false = (in @ X1 @ sK2))) )),
% 1.69/1.47 inference(binary_proxy_clausification,[status(thm)],[f42])).
% 1.69/1.47 thf(f42,plain,(
% 1.69/1.47 ( ! [X1 : $i] : (($true = ((in @ X1 @ sK2) => (?? @ $i @ (^[Y0 : $i]: (((setadjoin @ Y0 @ emptyset) = (dsetconstr @ sK0 @ (^[Y1 : $i]: (in @ (kpair @ X1 @ Y1) @ sK1)))) & (in @ Y0 @ (dsetconstr @ sK0 @ (^[Y1 : $i]: (in @ (kpair @ X1 @ Y1) @ sK1)))))))))) )),
% 1.69/1.47 inference(beta_eta_normalization,[status(thm)],[f41])).
% 1.69/1.47 thf(f41,plain,(
% 1.69/1.47 ( ! [X1 : $i] : (($true = ((^[Y0 : $i]: ((in @ Y0 @ sK2) => (?? @ $i @ (^[Y1 : $i]: (((setadjoin @ Y1 @ emptyset) = (dsetconstr @ sK0 @ (^[Y2 : $i]: (in @ (kpair @ Y0 @ Y2) @ sK1)))) & (in @ Y1 @ (dsetconstr @ sK0 @ (^[Y2 : $i]: (in @ (kpair @ Y0 @ Y2) @ sK1))))))))) @ X1))) )),
% 1.69/1.47 inference(pi_clausification,[status(thm)],[f39])).
% 1.69/1.47 thf(f39,plain,(
% 1.69/1.47 ($true = (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (?? @ $i @ (^[Y1 : $i]: (((setadjoin @ Y1 @ emptyset) = (dsetconstr @ sK0 @ (^[Y2 : $i]: (in @ (kpair @ Y0 @ Y2) @ sK1)))) & (in @ Y1 @ (dsetconstr @ sK0 @ (^[Y2 : $i]: (in @ (kpair @ Y0 @ Y2) @ sK1)))))))))))),
% 1.69/1.47 inference(binary_proxy_clausification,[status(thm)],[f38])).
% 1.69/1.47 thf(f38,plain,(
% 1.69/1.47 ($true = ((subset @ sK1 @ (cartprod @ sK2 @ sK0)) & (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (?? @ $i @ (^[Y1 : $i]: (((setadjoin @ Y1 @ emptyset) = (dsetconstr @ sK0 @ (^[Y2 : $i]: (in @ (kpair @ Y0 @ Y2) @ sK1)))) & (in @ Y1 @ (dsetconstr @ sK0 @ (^[Y2 : $i]: (in @ (kpair @ Y0 @ Y2) @ sK1))))))))))))),
% 1.69/1.47 inference(beta_eta_normalization,[status(thm)],[f31])).
% 1.69/1.47 thf(f31,plain,(
% 1.69/1.47 ($true = ((^[Y0 : $i]: ((^[Y1 : $i]: ((^[Y2 : $i]: (((^[Y3 : $i]: ((^[Y4 : $i]: ((^[Y5 : $i]: (subset @ Y5 @ (cartprod @ Y3 @ Y4))))))) @ Y0 @ Y1 @ Y2) & (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) => ((^[Y4 : $i]: ((^[Y5 : $i > $o]: ((^[Y6 : $i]: (?? @ $i @ (^[Y7 : $i]: (((setadjoin @ Y7 @ emptyset) = Y6) & (in @ Y7 @ Y6))))) @ (dsetconstr @ Y4 @ (^[Y6 : $i]: (Y5 @ Y6))))))) @ Y1 @ (^[Y4 : $i]: (in @ (kpair @ Y3 @ Y4) @ Y2)))))))))))) @ sK2 @ sK0 @ sK1))),
% 1.69/1.47 inference(definition_unfolding,[status(thm)],[f27,f30])).
% 1.69/1.47 thf(f30,plain,(
% 1.69/1.47 (func = (^[Y0 : $i]: ((^[Y1 : $i]: ((^[Y2 : $i]: (((^[Y3 : $i]: ((^[Y4 : $i]: ((^[Y5 : $i]: (subset @ Y5 @ (cartprod @ Y3 @ Y4))))))) @ Y0 @ Y1 @ Y2) & (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) => ((^[Y4 : $i]: ((^[Y5 : $i > $o]: ((^[Y6 : $i]: (?? @ $i @ (^[Y7 : $i]: (((setadjoin @ Y7 @ emptyset) = Y6) & (in @ Y7 @ Y6))))) @ (dsetconstr @ Y4 @ (^[Y6 : $i]: (Y5 @ Y6))))))) @ Y1 @ (^[Y4 : $i]: (in @ (kpair @ Y3 @ Y4) @ Y2)))))))))))))),
% 1.69/1.47 inference(definition_unfolding,[status(thm)],[f24,f28,f29])).
% 1.69/1.47 thf(f29,plain,(
% 1.69/1.47 (ex1 = (^[Y0 : $i]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (?? @ $i @ (^[Y3 : $i]: (((setadjoin @ Y3 @ emptyset) = Y2) & (in @ Y3 @ Y2))))) @ (dsetconstr @ Y0 @ (^[Y2 : $i]: (Y1 @ Y2))))))))),
% 1.69/1.47 inference(definition_unfolding,[status(thm)],[f22,f23])).
% 1.69/1.47 thf(f23,plain,(
% 1.69/1.47 (singleton = (^[Y0 : $i]: (?? @ $i @ (^[Y1 : $i]: (((setadjoin @ Y1 @ emptyset) = Y0) & (in @ Y1 @ Y0))))))),
% 1.69/1.47 inference(cnf_transformation,[status(thm)],[f17])).
% 1.69/1.47 thf(f17,plain,(
% 1.69/1.47 (singleton = (^[Y0 : $i]: (?? @ $i @ (^[Y1 : $i]: (((setadjoin @ Y1 @ emptyset) = Y0) & (in @ Y1 @ Y0))))))),
% 1.69/1.47 inference(fool_elimination,[status(thm)],[f16])).
% 1.69/1.47 thf(f16,plain,(
% 1.69/1.47 (singleton = (^[X0 : $i] : (? [X1 : $i] : ((in @ X1 @ X0) & ((setadjoin @ X1 @ emptyset) = X0)))))),
% 1.69/1.47 inference(rectify,[status(thm)],[f1])).
% 1.69/1.47 thf(f1,axiom,(
% 1.69/1.47 (singleton = (^[X0 : $i] : (? [X1 : $i] : ((in @ X1 @ X0) & ((setadjoin @ X1 @ emptyset) = X0)))))),
% 1.69/1.47 file('/export/starexec/sandbox/tmp/tmp.PnBmkGKoOW/DTF2THF_4224.p',singleton)).
% 1.69/1.47 thf(f22,plain,(
% 1.69/1.47 (ex1 = (^[Y0 : $i]: ((^[Y1 : $i > $o]: (singleton @ (dsetconstr @ Y0 @ (^[Y2 : $i]: (Y1 @ Y2))))))))),
% 1.69/1.47 inference(cnf_transformation,[status(thm)],[f9])).
% 1.69/1.47 thf(f9,plain,(
% 1.69/1.47 (ex1 = (^[Y0 : $i]: ((^[Y1 : $i > $o]: (singleton @ (dsetconstr @ Y0 @ (^[Y2 : $i]: (Y1 @ Y2))))))))),
% 1.69/1.47 inference(fool_elimination,[status(thm)],[f8])).
% 1.69/1.47 thf(f8,plain,(
% 1.69/1.47 ((^[X0 : $i, X1 : $i > $o] : (singleton @ (dsetconstr @ X0 @ (^[X2 : $i] : (X1 @ X2))))) = ex1)),
% 1.69/1.47 inference(rectify,[status(thm)],[f2])).
% 1.69/1.47 thf(f2,axiom,(
% 1.69/1.47 ((^[X0 : $i, X2 : $i > $o] : (singleton @ (dsetconstr @ X0 @ (^[X1 : $i] : (X2 @ X1))))) = ex1)),
% 1.69/1.47 file('/export/starexec/sandbox/tmp/tmp.PnBmkGKoOW/DTF2THF_4224.p',ex1)).
% 1.69/1.47 thf(f28,plain,(
% 1.69/1.47 (breln = (^[Y0 : $i]: ((^[Y1 : $i]: ((^[Y2 : $i]: (subset @ Y2 @ (cartprod @ Y0 @ Y1))))))))),
% 1.69/1.47 inference(cnf_transformation,[status(thm)],[f11])).
% 1.69/1.47 thf(f11,plain,(
% 1.69/1.47 (breln = (^[Y0 : $i]: ((^[Y1 : $i]: ((^[Y2 : $i]: (subset @ Y2 @ (cartprod @ Y0 @ Y1))))))))),
% 1.69/1.47 inference(fool_elimination,[status(thm)],[f10])).
% 1.69/1.47 thf(f10,plain,(
% 1.69/1.47 ((^[X0 : $i, X1 : $i, X2 : $i] : (subset @ X2 @ (cartprod @ X0 @ X1))) = breln)),
% 1.69/1.47 inference(rectify,[status(thm)],[f3])).
% 1.69/1.47 thf(f3,axiom,(
% 1.69/1.47 ((^[X0 : $i, X3 : $i, X4 : $i] : (subset @ X4 @ (cartprod @ X0 @ X3))) = breln)),
% 1.69/1.47 file('/export/starexec/sandbox/tmp/tmp.PnBmkGKoOW/DTF2THF_4224.p',breln)).
% 1.69/1.47 thf(f24,plain,(
% 1.69/1.47 (func = (^[Y0 : $i]: ((^[Y1 : $i]: ((^[Y2 : $i]: ((breln @ Y0 @ Y1 @ Y2) & (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) => (ex1 @ Y1 @ (^[Y4 : $i]: (in @ (kpair @ Y3 @ Y4) @ Y2)))))))))))))),
% 1.69/1.47 inference(cnf_transformation,[status(thm)],[f13])).
% 1.69/1.47 thf(f13,plain,(
% 1.69/1.47 (func = (^[Y0 : $i]: ((^[Y1 : $i]: ((^[Y2 : $i]: ((breln @ Y0 @ Y1 @ Y2) & (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y0) => (ex1 @ Y1 @ (^[Y4 : $i]: (in @ (kpair @ Y3 @ Y4) @ Y2)))))))))))))),
% 1.69/1.47 inference(fool_elimination,[status(thm)],[f12])).
% 1.69/1.47 thf(f12,plain,(
% 1.69/1.47 (func = (^[X0 : $i, X1 : $i, X2 : $i] : (! [X3 : $i] : ((in @ X3 @ X0) => (ex1 @ X1 @ (^[X4 : $i] : (in @ (kpair @ X3 @ X4) @ X2)))) & (breln @ X0 @ X1 @ X2))))),
% 1.69/1.47 inference(rectify,[status(thm)],[f4])).
% 1.69/1.47 thf(f4,axiom,(
% 1.69/1.47 (func = (^[X0 : $i, X3 : $i, X5 : $i] : (! [X1 : $i] : ((in @ X1 @ X0) => (ex1 @ X3 @ (^[X6 : $i] : (in @ (kpair @ X1 @ X6) @ X5)))) & (breln @ X0 @ X3 @ X5))))),
% 1.69/1.47 file('/export/starexec/sandbox/tmp/tmp.PnBmkGKoOW/DTF2THF_4224.p',func)).
% 1.69/1.47 thf(f27,plain,(
% 1.69/1.47 ($true = (func @ sK2 @ sK0 @ sK1))),
% 1.69/1.47 inference(cnf_transformation,[status(thm)],[f21])).
% 1.69/1.47 thf(f58,plain,(
% 1.69/1.47 ( ! [X1 : $i] : (((in @ X1 @ (dsetconstr @ sK0 @ (^[Y0 : $i]: (in @ (kpair @ sK3 @ Y0) @ sK1)))) = $false) | ((setadjoin @ X1 @ emptyset) != (setadjoin @ (sK5 @ sK3) @ emptyset))) )),
% 1.69/1.47 inference(backward_demodulation,[status(thm)],[f37,f56])).
% 1.69/1.47 thf(f37,plain,(
% 1.69/1.47 ( ! [X1 : $i] : (((in @ X1 @ (dsetconstr @ sK0 @ (^[Y0 : $i]: (in @ (kpair @ sK3 @ Y0) @ sK1)))) = $false) | ((setadjoin @ X1 @ emptyset) != (dsetconstr @ sK0 @ (^[Y0 : $i]: (in @ (kpair @ sK3 @ Y0) @ sK1))))) )),
% 1.69/1.47 inference(equality_proxy_clausification,[status(thm)],[f36])).
% 1.69/1.47 thf(f36,plain,(
% 1.69/1.47 ( ! [X1 : $i] : (((in @ X1 @ (dsetconstr @ sK0 @ (^[Y0 : $i]: (in @ (kpair @ sK3 @ Y0) @ sK1)))) = $false) | (((setadjoin @ X1 @ emptyset) = (dsetconstr @ sK0 @ (^[Y0 : $i]: (in @ (kpair @ sK3 @ Y0) @ sK1)))) = $false)) )),
% 1.69/1.47 inference(binary_proxy_clausification,[status(thm)],[f35])).
% 1.69/1.47 thf(f35,plain,(
% 1.69/1.47 ( ! [X1 : $i] : (((((setadjoin @ X1 @ emptyset) = (dsetconstr @ sK0 @ (^[Y0 : $i]: (in @ (kpair @ sK3 @ Y0) @ sK1)))) & (in @ X1 @ (dsetconstr @ sK0 @ (^[Y0 : $i]: (in @ (kpair @ sK3 @ Y0) @ sK1))))) = $false)) )),
% 1.69/1.47 inference(beta_eta_normalization,[status(thm)],[f34])).
% 1.69/1.47 thf(f34,plain,(
% 1.69/1.47 ( ! [X1 : $i] : ((((^[Y0 : $i]: (((setadjoin @ Y0 @ emptyset) = (dsetconstr @ sK0 @ (^[Y1 : $i]: (in @ (kpair @ sK3 @ Y1) @ sK1)))) & (in @ Y0 @ (dsetconstr @ sK0 @ (^[Y1 : $i]: (in @ (kpair @ sK3 @ Y1) @ sK1)))))) @ X1) = $false)) )),
% 1.69/1.47 inference(pi_clausification,[status(thm)],[f33])).
% 1.69/1.47 thf(f33,plain,(
% 1.69/1.47 ($true != (?? @ $i @ (^[Y0 : $i]: (((setadjoin @ Y0 @ emptyset) = (dsetconstr @ sK0 @ (^[Y1 : $i]: (in @ (kpair @ sK3 @ Y1) @ sK1)))) & (in @ Y0 @ (dsetconstr @ sK0 @ (^[Y1 : $i]: (in @ (kpair @ sK3 @ Y1) @ sK1))))))))),
% 1.69/1.47 inference(beta_eta_normalization,[status(thm)],[f32])).
% 1.69/1.47 thf(f32,plain,(
% 1.69/1.47 ($true != ((^[Y0 : $i]: (?? @ $i @ (^[Y1 : $i]: (((setadjoin @ Y1 @ emptyset) = Y0) & (in @ Y1 @ Y0))))) @ (dsetconstr @ sK0 @ (^[Y0 : $i]: (in @ (kpair @ sK3 @ Y0) @ sK1)))))),
% 1.69/1.47 inference(definition_unfolding,[status(thm)],[f26,f23])).
% 1.69/1.47 thf(f26,plain,(
% 1.69/1.47 ($true != (singleton @ (dsetconstr @ sK0 @ (^[Y0 : $i]: (in @ (kpair @ sK3 @ Y0) @ sK1)))))),
% 1.69/1.47 inference(cnf_transformation,[status(thm)],[f21])).
% 1.69/1.47 thf(f57,plain,(
% 1.69/1.47 ($true = (in @ (sK5 @ sK3) @ (setadjoin @ (sK5 @ sK3) @ emptyset)))),
% 1.69/1.47 inference(backward_demodulation,[status(thm)],[f52,f56])).
% 1.69/1.47 thf(f52,plain,(
% 1.69/1.47 ($true = (in @ (sK5 @ sK3) @ (dsetconstr @ sK0 @ (^[Y0 : $i]: (in @ (kpair @ sK3 @ Y0) @ sK1)))))),
% 1.69/1.47 inference(trivial_inequality_removal,[status(thm)],[f49])).
% 1.69/1.47 thf(f49,plain,(
% 1.69/1.47 ($true = $false) | ($true = (in @ (sK5 @ sK3) @ (dsetconstr @ sK0 @ (^[Y0 : $i]: (in @ (kpair @ sK3 @ Y0) @ sK1)))))),
% 1.69/1.47 inference(superposition,[status(thm)],[f46,f25])).
% 1.69/1.47 thf(f46,plain,(
% 1.69/1.47 ( ! [X1 : $i] : (($false = (in @ X1 @ sK2)) | ($true = (in @ (sK5 @ X1) @ (dsetconstr @ sK0 @ (^[Y0 : $i]: (in @ (kpair @ X1 @ Y0) @ sK1)))))) )),
% 1.69/1.47 inference(binary_proxy_clausification,[status(thm)],[f45])).
% 1.69/1.47 % SZS output end Proof for DTF2THF_4224
% 1.69/1.47 % (4327)------------------------------
% 1.69/1.47 % (4327)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.69/1.47 % (4327)Termination reason: Refutation
% 1.69/1.47
% 1.69/1.47 % (4327)Memory used [KB]: 5628
% 1.69/1.47 % (4327)Time elapsed: 0.008 s
% 1.69/1.47 % (4327)Instructions burned: 8 (million)
% 1.69/1.47 % (4327)------------------------------
% 1.69/1.47 % (4327)------------------------------
% 1.69/1.47 % (4320)Success in time 0.023 s
%------------------------------------------------------------------------------