%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU669^2 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n010.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 09:36:05 AM UTC 2026
% Result : Theorem 0.31s 1.33s
% Output : Refutation 0.31s
% Verified :
% SZS Type : Refutation
% Derivation depth : 22
% Number of leaves : 5
% Syntax : Number of formulae : 51 ( 39 unt; 0 typ; 0 def)
% Number of atoms : 341 ( 72 equ; 0 cnn)
% Maximal formula atoms : 4 ( 6 avg)
% Number of connectives : 644 ( 10 ~; 4 |; 29 &; 513 @)
% ( 0 <=>; 38 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 3 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 42 ( 42 >; 0 *; 0 +; 0 <<)
% Number of symbols : 19 ( 14 usr; 10 con; 0-3 aty)
% ( 42 !!; 8 ??; 0 @@+; 0 @@-)
% Number of variables : 197 ( 105 ^; 73 !; 19 ?; 197 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_1,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(func_def_4,type,
kpair: $i > $i > $i ).
thf(func_def_5,type,
cartprod: $i > $i > $i ).
thf(func_def_7,type,
dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_8,type,
vEPSILON:
!>[X0: $tType] : ( ( X0 > $o ) > X0 ) ).
thf(func_def_24,type,
sK2: $i > $i > $o ).
thf(func_def_28,type,
ph6:
!>[X0: $tType] : X0 ).
thf(f56,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f55]) ).
thf(f55,plain,
$true != $true,
inference(superposition,[status(thm)],[f20,f53]) ).
thf(f53,plain,
( ( in @ sK4 @ sK1 )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f51]) ).
thf(f51,plain,
( ( $false = $true )
| ( ( in @ sK4 @ sK1 )
= $true ) ),
inference(superposition,[status(thm)],[f49,f38]) ).
thf(f38,plain,
! [X2: $i,X3: $i,X1: $i,X4: $i] :
( ( ( in @ ( kpair @ X3 @ X1 ) @ ( cartprod @ X4 @ X2 ) )
= $false )
| ( ( in @ X3 @ X4 )
= $true ) ),
inference(binary_proxy_clausification,[status(thm)],[f37]) ).
thf(f37,plain,
! [X2: $i,X3: $i,X1: $i,X4: $i] :
( ( ( in @ ( kpair @ X3 @ X1 ) @ ( cartprod @ X4 @ X2 ) )
=> ( in @ X3 @ X4 ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f36]) ).
thf(f36,plain,
! [X2: $i,X3: $i,X1: $i,X4: $i] :
( ( ^ [Y0: $i] :
( ( in @ ( kpair @ X3 @ X1 ) @ ( cartprod @ Y0 @ X2 ) )
=> ( in @ X3 @ Y0 ) )
@ X4 )
= $true ),
inference(pi_clausification,[status(thm)],[f35]) ).
thf(f35,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ ( kpair @ X3 @ X1 ) @ ( cartprod @ Y0 @ X2 ) )
=> ( in @ X3 @ Y0 ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f34]) ).
thf(f34,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( in @ ( kpair @ Y0 @ X1 ) @ ( cartprod @ Y1 @ X2 ) )
=> ( in @ Y0 @ Y1 ) ) )
@ X3 )
= $true ),
inference(pi_clausification,[status(thm)],[f33]) ).
thf(f33,plain,
! [X2: $i,X1: $i] :
( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( in @ ( kpair @ Y0 @ X1 ) @ ( cartprod @ Y1 @ X2 ) )
=> ( in @ Y0 @ Y1 ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f32]) ).
thf(f32,plain,
! [X2: $i,X1: $i] :
( ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ ( kpair @ Y1 @ X1 ) @ ( cartprod @ Y2 @ Y0 ) )
=> ( in @ Y1 @ Y2 ) ) ) )
@ X2 )
= $true ),
inference(pi_clausification,[status(thm)],[f31]) ).
thf(f31,plain,
! [X1: $i] :
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ ( kpair @ Y1 @ X1 ) @ ( cartprod @ Y2 @ Y0 ) )
=> ( in @ Y1 @ Y2 ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f30]) ).
thf(f30,plain,
! [X1: $i] :
( ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( !! @ $i
@ ^ [Y3: $i] :
( ( in @ ( kpair @ Y2 @ Y0 ) @ ( cartprod @ Y3 @ Y1 ) )
=> ( in @ Y2 @ Y3 ) ) ) ) )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f28]) ).
thf(f28,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( !! @ $i
@ ^ [Y3: $i] :
( ( in @ ( kpair @ Y2 @ Y0 ) @ ( cartprod @ Y3 @ Y1 ) )
=> ( in @ Y2 @ Y3 ) ) ) ) ) )
= $true ),
inference(definition_unfolding,[status(thm)],[f25,f19]) ).
thf(f19,plain,
cartprodpairmemEL = $true,
inference(cnf_transformation,[status(thm)],[f18]) ).
thf(f18,plain,
( ( dsetconstrEL = $true )
& ( ( in @ ( kpair @ sK4 @ sK0 )
@ ( dpsetconstr @ sK1 @ sK3
@ ^ [Y0: $i,Y1: $i] : ( sK2 @ Y0 @ Y1 ) ) )
= $true )
& ( ( in @ sK4 @ sK1 )
!= $true )
& ( cartprodpairmemEL = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2,sK3,sK4])],[f16,f17]) ).
thf(f17,plain,
( ? [X0: $i,X1: $i,X2: $i > $i > $o,X3: $i,X4: $i] :
( ( ( in @ ( kpair @ X4 @ X0 )
@ ( dpsetconstr @ X1 @ X3
@ ^ [Y0: $i,Y1: $i] : ( X2 @ Y0 @ Y1 ) ) )
= $true )
& ( ( in @ X4 @ X1 )
!= $true ) )
=> ( ( ( in @ ( kpair @ sK4 @ sK0 )
@ ( dpsetconstr @ sK1 @ sK3
@ ^ [Y0: $i,Y1: $i] : ( sK2 @ Y0 @ Y1 ) ) )
= $true )
& ( ( in @ sK4 @ sK1 )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f16,plain,
( ( dsetconstrEL = $true )
& ? [X0: $i,X1: $i,X2: $i > $i > $o,X3: $i,X4: $i] :
( ( ( in @ ( kpair @ X4 @ X0 )
@ ( dpsetconstr @ X1 @ X3
@ ^ [Y0: $i,Y1: $i] : ( X2 @ Y0 @ Y1 ) ) )
= $true )
& ( ( in @ X4 @ X1 )
!= $true ) )
& ( cartprodpairmemEL = $true ) ),
inference(flattening,[status(thm)],[f15]) ).
thf(f15,plain,
( ? [X0: $i,X1: $i,X2: $i > $i > $o,X3: $i,X4: $i] :
( ( ( in @ ( kpair @ X4 @ X0 )
@ ( dpsetconstr @ X1 @ X3
@ ^ [Y0: $i,Y1: $i] : ( X2 @ Y0 @ Y1 ) ) )
= $true )
& ( ( in @ X4 @ X1 )
!= $true ) )
& ( cartprodpairmemEL = $true )
& ( dsetconstrEL = $true ) ),
inference(ennf_transformation,[status(thm)],[f8]) ).
thf(f8,plain,
~ ( ( dsetconstrEL = $true )
=> ( ( cartprodpairmemEL = $true )
=> ! [X0: $i,X1: $i,X2: $i > $i > $o,X3: $i,X4: $i] :
( ( ( in @ ( kpair @ X4 @ X0 )
@ ( dpsetconstr @ X1 @ X3
@ ^ [Y0: $i,Y1: $i] : ( X2 @ Y0 @ Y1 ) ) )
= $true )
=> ( ( in @ X4 @ X1 )
= $true ) ) ) ),
inference(fool_elimination,[status(thm)],[f7]) ).
thf(f7,plain,
~ ( dsetconstrEL
=> ( cartprodpairmemEL
=> ! [X0: $i,X1: $i,X2: $i > $i > $o,X3: $i,X4: $i] :
( ( in @ ( kpair @ X4 @ X0 )
@ ( dpsetconstr @ X1 @ X3
@ ^ [X5: $i,X6: $i] : ( X2 @ X5 @ X6 ) ) )
=> ( in @ X4 @ X1 ) ) ) ),
inference(rectify,[status(thm)],[f5]) ).
thf(f5,negated_conjecture,
~ ( dsetconstrEL
=> ( cartprodpairmemEL
=> ! [X3: $i,X0: $i,X1: $i > $i > $o,X4: $i,X2: $i] :
( ( in @ ( kpair @ X2 @ X3 )
@ ( dpsetconstr @ X0 @ X4
@ ^ [X6: $i,X5: $i] : ( X1 @ X6 @ X5 ) ) )
=> ( in @ X2 @ X0 ) ) ) ),
inference(negated_conjecture,[status(cth)],[f4]) ).
thf(f4,conjecture,
( dsetconstrEL
=> ( cartprodpairmemEL
=> ! [X3: $i,X0: $i,X1: $i > $i > $o,X4: $i,X2: $i] :
( ( in @ ( kpair @ X2 @ X3 )
@ ( dpsetconstr @ X0 @ X4
@ ^ [X6: $i,X5: $i] : ( X1 @ X6 @ X5 ) ) )
=> ( in @ X2 @ X0 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.j3kahdTx8R/DTF2THF_8102.p',dpsetconstrEL1) ).
thf(f25,plain,
( cartprodpairmemEL
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( !! @ $i
@ ^ [Y3: $i] :
( ( in @ ( kpair @ Y2 @ Y0 ) @ ( cartprod @ Y3 @ Y1 ) )
=> ( in @ Y2 @ Y3 ) ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f14]) ).
thf(f14,plain,
( cartprodpairmemEL
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( !! @ $i
@ ^ [Y3: $i] :
( ( in @ ( kpair @ Y2 @ Y0 ) @ ( cartprod @ Y3 @ Y1 ) )
=> ( in @ Y2 @ Y3 ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f13]) ).
thf(f13,plain,
( cartprodpairmemEL
= ( ! [X0: $i,X1: $i,X2: $i,X3: $i] :
( ( in @ ( kpair @ X1 @ X3 ) @ ( cartprod @ X0 @ X2 ) )
=> ( in @ X1 @ X0 ) ) ) ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
( cartprodpairmemEL
= ( ! [X0: $i,X2: $i,X4: $i,X3: $i] :
( ( in @ ( kpair @ X2 @ X3 ) @ ( cartprod @ X0 @ X4 ) )
=> ( in @ X2 @ X0 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.j3kahdTx8R/DTF2THF_8102.p',cartprodpairmemEL) ).
thf(f49,plain,
( ( in @ ( kpair @ sK4 @ sK0 ) @ ( cartprod @ sK1 @ sK3 ) )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f47]) ).
thf(f47,plain,
( ( ( in @ ( kpair @ sK4 @ sK0 ) @ ( cartprod @ sK1 @ sK3 ) )
= $true )
| ( $false = $true ) ),
inference(superposition,[status(thm)],[f29,f46]) ).
thf(f46,plain,
! [X2: $i,X3: $i,X1: $i > $o] :
( ( $false
= ( in @ X3 @ ( dsetconstr @ X2 @ X1 ) ) )
| ( ( in @ X3 @ X2 )
= $true ) ),
inference(binary_proxy_clausification,[status(thm)],[f45]) ).
thf(f45,plain,
! [X2: $i,X3: $i,X1: $i > $o] :
( ( ( in @ X3 @ ( dsetconstr @ X2 @ X1 ) )
=> ( in @ X3 @ X2 ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f44]) ).
thf(f44,plain,
! [X2: $i,X3: $i,X1: $i > $o] :
( ( ^ [Y0: $i] :
( ( in @ Y0 @ ( dsetconstr @ X2 @ X1 ) )
=> ( in @ Y0 @ X2 ) )
@ X3 )
= $true ),
inference(pi_clausification,[status(thm)],[f43]) ).
thf(f43,plain,
! [X2: $i,X1: $i > $o] :
( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ ( dsetconstr @ X2 @ X1 ) )
=> ( in @ Y0 @ X2 ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f42]) ).
thf(f42,plain,
! [X2: $i,X1: $i > $o] :
( ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ ( dsetconstr @ Y0 @ X1 ) )
=> ( in @ Y1 @ Y0 ) ) )
@ X2 )
= $true ),
inference(pi_clausification,[status(thm)],[f41]) ).
thf(f41,plain,
! [X1: $i > $o] :
( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ ( dsetconstr @ Y0 @ X1 ) )
=> ( in @ Y1 @ Y0 ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f40]) ).
thf(f40,plain,
! [X1: $i > $o] :
( ( ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ ( dsetconstr @ Y1 @ Y0 ) )
=> ( in @ Y2 @ Y1 ) ) ) )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f39]) ).
thf(f39,plain,
( ( !! @ ( $i > $o )
@ ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ ( dsetconstr @ Y1 @ Y0 ) )
=> ( in @ Y2 @ Y1 ) ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f27]) ).
thf(f27,plain,
( ( !! @ ( $i > $o )
@ ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2
@ ( dsetconstr @ Y1
@ ^ [Y3: $i] : ( Y0 @ Y3 ) ) )
=> ( in @ Y2 @ Y1 ) ) ) ) )
= $true ),
inference(definition_unfolding,[status(thm)],[f24,f22]) ).
thf(f22,plain,
dsetconstrEL = $true,
inference(cnf_transformation,[status(thm)],[f18]) ).
thf(f24,plain,
( dsetconstrEL
= ( !! @ ( $i > $o )
@ ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2
@ ( dsetconstr @ Y1
@ ^ [Y3: $i] : ( Y0 @ Y3 ) ) )
=> ( in @ Y2 @ Y1 ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f12]) ).
thf(f12,plain,
( dsetconstrEL
= ( !! @ ( $i > $o )
@ ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2
@ ( dsetconstr @ Y1
@ ^ [Y3: $i] : ( Y0 @ Y3 ) ) )
=> ( in @ Y2 @ Y1 ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f11]) ).
thf(f11,plain,
( dsetconstrEL
= ( ! [X0: $i,X1: $i,X2: $i > $o] :
( ( in @ X0
@ ( dsetconstr @ X1
@ ^ [X3: $i] : ( X2 @ X3 ) ) )
=> ( in @ X0 @ X1 ) ) ) ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
( dsetconstrEL
= ( ! [X2: $i,X0: $i,X1: $i > $o] :
( ( in @ X2
@ ( dsetconstr @ X0
@ ^ [X3: $i] : ( X1 @ X3 ) ) )
=> ( in @ X2 @ X0 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.j3kahdTx8R/DTF2THF_8102.p',dsetconstrEL) ).
thf(f29,plain,
( ( in @ ( kpair @ sK4 @ sK0 )
@ ( dsetconstr @ ( cartprod @ sK1 @ sK3 )
@ ^ [Y0: $i] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( ?? @ $i
@ ^ [Y2: $i] :
( ( sK2 @ Y1 @ Y2 )
& ( in @ Y2 @ sK3 )
& ( ( kpair @ Y1 @ Y2 )
= Y0 ) ) )
& ( in @ Y1 @ sK1 ) ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f26]) ).
thf(f26,plain,
( ( in @ ( kpair @ sK4 @ sK0 )
@ ( ^ [Y0: $i,Y1: $i,Y2: $i > $i > $o] :
( dsetconstr @ ( cartprod @ Y0 @ Y1 )
@ ^ [Y3: $i] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( ?? @ $i
@ ^ [Y5: $i] :
( ( Y2 @ Y4 @ Y5 )
& ( in @ Y5 @ Y1 )
& ( ( kpair @ Y4 @ Y5 )
= Y3 ) ) )
& ( in @ Y4 @ Y0 ) ) ) )
@ sK1
@ sK3
@ ^ [Y0: $i,Y1: $i] : ( sK2 @ Y0 @ Y1 ) ) )
= $true ),
inference(definition_unfolding,[status(thm)],[f21,f23]) ).
thf(f23,plain,
( dpsetconstr
= ( ^ [Y0: $i,Y1: $i,Y2: $i > $i > $o] :
( dsetconstr @ ( cartprod @ Y0 @ Y1 )
@ ^ [Y3: $i] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( ?? @ $i
@ ^ [Y5: $i] :
( ( Y2 @ Y4 @ Y5 )
& ( in @ Y5 @ Y1 )
& ( ( kpair @ Y4 @ Y5 )
= Y3 ) ) )
& ( in @ Y4 @ Y0 ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f10]) ).
thf(f10,plain,
( dpsetconstr
= ( ^ [Y0: $i,Y1: $i,Y2: $i > $i > $o] :
( dsetconstr @ ( cartprod @ Y0 @ Y1 )
@ ^ [Y3: $i] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( ?? @ $i
@ ^ [Y5: $i] :
( ( Y2 @ Y4 @ Y5 )
& ( in @ Y5 @ Y1 )
& ( ( kpair @ Y4 @ Y5 )
= Y3 ) ) )
& ( in @ Y4 @ Y0 ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f9]) ).
thf(f9,plain,
( dpsetconstr
= ( ^ [X0: $i,X1: $i,X2: $i > $i > $o] :
( dsetconstr @ ( cartprod @ X0 @ X1 )
@ ^ [X3: $i] :
? [X4: $i] :
( ( in @ X4 @ X0 )
& ? [X5: $i] :
( ( ( kpair @ X4 @ X5 )
= X3 )
& ( in @ X5 @ X1 )
& ( X2 @ X4 @ X5 ) ) ) ) ) ),
inference(rectify,[status(thm)],[f3]) ).
thf(f3,axiom,
( dpsetconstr
= ( ^ [X0: $i,X4: $i,X1: $i > $i > $o] :
( dsetconstr @ ( cartprod @ X0 @ X4 )
@ ^ [X5: $i] :
? [X2: $i] :
( ( in @ X2 @ X0 )
& ? [X3: $i] :
( ( ( kpair @ X2 @ X3 )
= X5 )
& ( in @ X3 @ X4 )
& ( X1 @ X2 @ X3 ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.j3kahdTx8R/DTF2THF_8102.p',dpsetconstr) ).
thf(f21,plain,
( ( in @ ( kpair @ sK4 @ sK0 )
@ ( dpsetconstr @ sK1 @ sK3
@ ^ [Y0: $i,Y1: $i] : ( sK2 @ Y0 @ Y1 ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f18]) ).
thf(f20,plain,
( ( in @ sK4 @ sK1 )
!= $true ),
inference(cnf_transformation,[status(thm)],[f18]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU669^2 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.15/0.33 % Computer : n010.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 12:30:03 EST 2026
% 0.15/0.33 % CPUTime :
% 0.15/0.33 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.19/0.45 ---- Original DTF file ---
% 0.19/0.45 thf(spec,logic,$$dhol).
% 0.19/0.45 %------------------------------------------------------------------------------
% 0.19/0.45 % File : SEU669^2 : TPTP v9.2.1. Released v3.7.0.
% 0.19/0.45 % Domain : Set Theory
% 0.19/0.45 % Problem : Ordered Pairs - Sets of Pairs
% 0.19/0.45 % Version : Especial > Reduced > Especial.
% 0.19/0.45 % English : (! A:i.! B:i.! phi:i>(i>o).! x:i.! y:i.in (kpair x y)
% 0.19/0.45 % (dpsetconstr A B (^ z,u:i.phi z u)) -> in x A)
% 0.19/0.45
% 0.19/0.45 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.19/0.45 % Source : [Bro08]
% 0.19/0.45 % Names : ZFC171l [Bro08]
% 0.19/0.45
% 0.19/0.45 % Status : Theorem
% 0.19/0.45 % 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, 0.11 v7.2.0, 0.12 v7.1.0, 0.25 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.29 v6.1.0, 0.14 v6.0.0, 0.29 v5.5.0, 0.33 v5.4.0, 0.40 v5.3.0, 0.60 v5.2.0, 0.40 v4.1.0, 0.33 v4.0.1, 0.67 v3.7.0
% 0.19/0.45 % Syntax : Number of formulae : 11 ( 3 unt; 7 typ; 3 def)
% 0.19/0.45 % Number of atoms : 16 ( 4 equ; 0 cnn)
% 0.19/0.45 % Maximal formula atoms : 4 ( 4 avg)
% 0.19/0.45 % Number of connectives : 46 ( 0 ~; 0 |; 3 &; 38 @)
% 0.19/0.45 % ( 0 <=>; 5 =>; 0 <=; 0 <~>)
% 0.19/0.45 % Maximal formula depth : 15 ( 5 avg)
% 0.19/0.45 % Number of types : 2 ( 0 usr)
% 0.19/0.45 % Number of type conns : 19 ( 19 >; 0 *; 0 +; 0 <<)
% 0.19/0.45 % Number of symbols : 8 ( 7 usr; 2 con; 0-3 aty)
% 0.19/0.45 % Number of variables : 21 ( 7 ^; 12 !; 2 ?; 21 :)
% 0.19/0.45 % SPC : TH0_THM_EQU_NAR
% 0.19/0.45
% 0.19/0.45 % Comments : http://mathgate.info/detsetitem.php?id=339
% 0.19/0.45 %------------------------------------------------------------------------------
% 0.19/0.45 thf(in_type,type,
% 0.19/0.45 in: $i > $i > $o ).
% 0.19/0.45
% 0.19/0.45 thf(dsetconstr_type,type,
% 0.19/0.45 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.19/0.45
% 0.19/0.45 thf(dsetconstrEL_type,type,
% 0.19/0.45 dsetconstrEL: $o ).
% 0.19/0.45
% 0.19/0.45 thf(dsetconstrEL,definition,
% 0.19/0.45 ( dsetconstrEL
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx
% 0.19/0.45 @ ( dsetconstr @ A
% 0.19/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.45 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(kpair_type,type,
% 0.19/0.45 kpair: $i > $i > $i ).
% 0.19/0.45
% 0.19/0.45 thf(cartprod_type,type,
% 0.19/0.45 cartprod: $i > $i > $i ).
% 0.19/0.45
% 0.19/0.45 thf(cartprodpairmemEL_type,type,
% 0.19/0.45 cartprodpairmemEL: $o ).
% 0.19/0.45
% 0.19/0.45 thf(cartprodpairmemEL,definition,
% 0.19/0.45 ( cartprodpairmemEL
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.19/0.45 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(dpsetconstr_type,type,
% 0.19/0.45 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.19/0.45
% 0.19/0.45 thf(dpsetconstr,definition,
% 0.19/0.45 ( dpsetconstr
% 0.19/0.45 = ( ^ [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.19/0.45 ( dsetconstr @ ( cartprod @ A @ B )
% 0.19/0.45 @ ^ [Xu: $i] :
% 0.19/0.45 ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 & ? [Xy: $i] :
% 0.19/0.45 ( ( in @ Xy @ B )
% 0.19/0.45 & ( Xphi @ Xx @ Xy )
% 0.19/0.45 & ( Xu
% 0.19/0.45 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(dpsetconstrEL1,conjecture,
% 0.19/0.45 ( dsetconstrEL
% 0.19/0.45 => ( cartprodpairmemEL
% 0.19/0.45 => ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( in @ ( kpair @ Xx @ Xy )
% 0.19/0.45 @ ( dpsetconstr @ A @ B
% 0.19/0.45 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.19/0.45 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 %------------------------------------------------------------------------------
% 0.19/0.45 ------------------------
% 0.31/1.30 ---- Embedded in THF ---
% 0.31/1.30 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.31/1.30 %%% Generated on Tue Feb 24 12:30:04 EST 2026
% 0.31/1.30 %%% using '$$dhol' embedding, version 1.3.0.
% 0.31/1.30 %%% Logic specification used:
% 0.31/1.30 %%% thf(spec, logic, $$dhol).
% 0.31/1.30
% 0.31/1.30 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.j3kahdTx8R/DTF2DTF8102.p
% See solution above
% 0.31/1.30 ------------------------
% 0.31/1.31 ---- Cleaned THF ---
% 0.31/1.31 thf(in_type,type,
% 0.31/1.31 in: $i > $i > $o ).
% 0.31/1.31
% 0.31/1.31 thf(dsetconstr_type,type,
% 0.31/1.31 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.31/1.31
% 0.31/1.31 thf(dsetconstrEL_type,type,
% 0.31/1.31 dsetconstrEL: $o ).
% 0.31/1.31
% 0.31/1.31 thf(dsetconstrEL,definition,
% 0.31/1.31 ( dsetconstrEL
% 0.31/1.31 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/1.31 ( ( in @ Xx
% 0.31/1.31 @ ( dsetconstr @ A
% 0.31/1.31 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.31/1.31 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/1.31
% 0.31/1.31 thf(kpair_type,type,
% 0.31/1.31 kpair: $i > $i > $i ).
% 0.31/1.31
% 0.31/1.31 thf(cartprod_type,type,
% 0.31/1.31 cartprod: $i > $i > $i ).
% 0.31/1.31
% 0.31/1.31 thf(cartprodpairmemEL_type,type,
% 0.31/1.31 cartprodpairmemEL: $o ).
% 0.31/1.31
% 0.31/1.31 thf(cartprodpairmemEL,definition,
% 0.31/1.31 ( cartprodpairmemEL
% 0.31/1.31 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.31/1.31 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.31/1.31 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/1.31
% 0.31/1.31 thf(dpsetconstr_type,type,
% 0.31/1.31 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.31/1.31
% 0.31/1.31 thf(dpsetconstr,definition,
% 0.31/1.31 ( dpsetconstr
% 0.31/1.31 = ( ^ [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.31/1.31 ( dsetconstr @ ( cartprod @ A @ B )
% 0.31/1.31 @ ^ [Xu: $i] :
% 0.31/1.31 ? [Xx: $i] :
% 0.31/1.31 ( ( in @ Xx @ A )
% 0.31/1.31 & ? [Xy: $i] :
% 0.31/1.31 ( ( in @ Xy @ B )
% 0.31/1.31 & ( Xphi @ Xx @ Xy )
% 0.31/1.31 & ( Xu
% 0.31/1.31 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.31/1.31
% 0.31/1.31 thf(dpsetconstrEL1,conjecture,
% 0.31/1.31 ( dsetconstrEL
% 0.31/1.31 => ( cartprodpairmemEL
% 0.31/1.31 => ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.31/1.31 ( ( in @ ( kpair @ Xx @ Xy )
% 0.31/1.31 @ ( dpsetconstr @ A @ B
% 0.31/1.31 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.31/1.31 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/1.31 ------------------------
% 0.31/1.33 % (8281)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_8102 for (2999ds/27Mi)
% 0.31/1.33 % (8281)First to succeed.
% 0.31/1.33 % (8281)Refutation found. Thanks to Tanya!
% 0.31/1.33 % SZS status Theorem for DTF2THF_8102
% 0.31/1.33 % SZS output start Proof for DTF2THF_8102
% 0.31/1.33 thf(func_def_0, type, in: $i > $i > $o).
% 0.31/1.33 thf(func_def_1, type, dsetconstr: $i > ($i > $o) > $i).
% 0.31/1.33 thf(func_def_4, type, kpair: $i > $i > $i).
% 0.31/1.33 thf(func_def_5, type, cartprod: $i > $i > $i).
% 0.31/1.33 thf(func_def_7, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 0.31/1.33 thf(func_def_8, type, vEPSILON: !>[X0: $tType]:((X0 > $o) > X0)).
% 0.31/1.33 thf(func_def_24, type, sK2: $i > $i > $o).
% 0.31/1.33 thf(func_def_28, type, ph6: !>[X0: $tType]:(X0)).
% 0.31/1.33 thf(f56,plain,(
% 0.31/1.33 $false),
% 0.31/1.33 inference(trivial_inequality_removal,[status(thm)],[f55])).
% 0.31/1.33 thf(f55,plain,(
% 0.31/1.33 ($true != $true)),
% 0.31/1.33 inference(superposition,[status(thm)],[f20,f53])).
% 0.31/1.33 thf(f53,plain,(
% 0.31/1.33 ((in @ sK4 @ sK1) = $true)),
% 0.31/1.33 inference(trivial_inequality_removal,[status(thm)],[f51])).
% 0.31/1.33 thf(f51,plain,(
% 0.31/1.33 ($false = $true) | ((in @ sK4 @ sK1) = $true)),
% 0.31/1.33 inference(superposition,[status(thm)],[f49,f38])).
% 0.31/1.33 thf(f38,plain,(
% 0.31/1.33 ( ! [X2 : $i,X3 : $i,X1 : $i,X4 : $i] : (((in @ (kpair @ X3 @ X1) @ (cartprod @ X4 @ X2)) = $false) | ((in @ X3 @ X4) = $true)) )),
% 0.31/1.33 inference(binary_proxy_clausification,[status(thm)],[f37])).
% 0.31/1.33 thf(f37,plain,(
% 0.31/1.33 ( ! [X2 : $i,X3 : $i,X1 : $i,X4 : $i] : ((((in @ (kpair @ X3 @ X1) @ (cartprod @ X4 @ X2)) => (in @ X3 @ X4)) = $true)) )),
% 0.31/1.33 inference(beta_eta_normalization,[status(thm)],[f36])).
% 0.31/1.33 thf(f36,plain,(
% 0.31/1.33 ( ! [X2 : $i,X3 : $i,X1 : $i,X4 : $i] : ((((^[Y0 : $i]: ((in @ (kpair @ X3 @ X1) @ (cartprod @ Y0 @ X2)) => (in @ X3 @ Y0))) @ X4) = $true)) )),
% 0.31/1.33 inference(pi_clausification,[status(thm)],[f35])).
% 0.31/1.33 thf(f35,plain,(
% 0.31/1.33 ( ! [X2 : $i,X3 : $i,X1 : $i] : (((!! @ $i @ (^[Y0 : $i]: ((in @ (kpair @ X3 @ X1) @ (cartprod @ Y0 @ X2)) => (in @ X3 @ Y0)))) = $true)) )),
% 0.31/1.33 inference(beta_eta_normalization,[status(thm)],[f34])).
% 0.31/1.33 thf(f34,plain,(
% 0.31/1.33 ( ! [X2 : $i,X3 : $i,X1 : $i] : ((((^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: ((in @ (kpair @ Y0 @ X1) @ (cartprod @ Y1 @ X2)) => (in @ Y0 @ Y1))))) @ X3) = $true)) )),
% 0.31/1.33 inference(pi_clausification,[status(thm)],[f33])).
% 0.31/1.33 thf(f33,plain,(
% 0.31/1.33 ( ! [X2 : $i,X1 : $i] : (((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: ((in @ (kpair @ Y0 @ X1) @ (cartprod @ Y1 @ X2)) => (in @ Y0 @ Y1)))))) = $true)) )),
% 0.31/1.33 inference(beta_eta_normalization,[status(thm)],[f32])).
% 0.31/1.33 thf(f32,plain,(
% 0.31/1.33 ( ! [X2 : $i,X1 : $i] : ((((^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((in @ (kpair @ Y1 @ X1) @ (cartprod @ Y2 @ Y0)) => (in @ Y1 @ Y2))))))) @ X2) = $true)) )),
% 0.31/1.33 inference(pi_clausification,[status(thm)],[f31])).
% 0.31/1.33 thf(f31,plain,(
% 0.31/1.33 ( ! [X1 : $i] : (($true = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((in @ (kpair @ Y1 @ X1) @ (cartprod @ Y2 @ Y0)) => (in @ Y1 @ Y2)))))))))) )),
% 0.31/1.33 inference(beta_eta_normalization,[status(thm)],[f30])).
% 0.31/1.33 thf(f30,plain,(
% 0.31/1.33 ( ! [X1 : $i] : ((((^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: ((in @ (kpair @ Y2 @ Y0) @ (cartprod @ Y3 @ Y1)) => (in @ Y2 @ Y3))))))))) @ X1) = $true)) )),
% 0.31/1.33 inference(pi_clausification,[status(thm)],[f28])).
% 0.31/1.33 thf(f28,plain,(
% 0.31/1.33 ((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: ((in @ (kpair @ Y2 @ Y0) @ (cartprod @ Y3 @ Y1)) => (in @ Y2 @ Y3)))))))))) = $true)),
% 0.31/1.33 inference(definition_unfolding,[status(thm)],[f25,f19])).
% 0.31/1.33 thf(f19,plain,(
% 0.31/1.33 (cartprodpairmemEL = $true)),
% 0.31/1.33 inference(cnf_transformation,[status(thm)],[f18])).
% 0.31/1.33 thf(f18,plain,(
% 0.31/1.33 (dsetconstrEL = $true) & (((in @ (kpair @ sK4 @ sK0) @ (dpsetconstr @ sK1 @ sK3 @ (^[Y0 : $i]: ((^[Y1 : $i]: (sK2 @ Y0 @ Y1)))))) = $true) & ((in @ sK4 @ sK1) != $true)) & (cartprodpairmemEL = $true)),
% 0.31/1.33 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2,sK3,sK4])],[f16,f17])).
% 0.31/1.33 thf(f17,plain,(
% 0.31/1.33 ? [X0 : $i,X1 : $i,X2 : $i > $i > $o,X3 : $i,X4 : $i] : (((in @ (kpair @ X4 @ X0) @ (dpsetconstr @ X1 @ X3 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X2 @ Y0 @ Y1)))))) = $true) & ((in @ X4 @ X1) != $true)) => (((in @ (kpair @ sK4 @ sK0) @ (dpsetconstr @ sK1 @ sK3 @ (^[Y0 : $i]: ((^[Y1 : $i]: (sK2 @ Y0 @ Y1)))))) = $true) & ((in @ sK4 @ sK1) != $true))),
% 0.31/1.33 introduced(definition,[],[choice_axiom])).
% 0.31/1.33 thf(f16,plain,(
% 0.31/1.33 (dsetconstrEL = $true) & ? [X0 : $i,X1 : $i,X2 : $i > $i > $o,X3 : $i,X4 : $i] : (((in @ (kpair @ X4 @ X0) @ (dpsetconstr @ X1 @ X3 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X2 @ Y0 @ Y1)))))) = $true) & ((in @ X4 @ X1) != $true)) & (cartprodpairmemEL = $true)),
% 0.31/1.33 inference(flattening,[status(thm)],[f15])).
% 0.31/1.33 thf(f15,plain,(
% 0.31/1.33 (? [X0 : $i,X1 : $i,X2 : $i > $i > $o,X3 : $i,X4 : $i] : (((in @ (kpair @ X4 @ X0) @ (dpsetconstr @ X1 @ X3 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X2 @ Y0 @ Y1)))))) = $true) & ((in @ X4 @ X1) != $true)) & (cartprodpairmemEL = $true)) & (dsetconstrEL = $true)),
% 0.31/1.33 inference(ennf_transformation,[status(thm)],[f8])).
% 0.31/1.33 thf(f8,plain,(
% 0.31/1.33 ~((dsetconstrEL = $true) => ((cartprodpairmemEL = $true) => ! [X0 : $i,X1 : $i,X2 : $i > $i > $o,X3 : $i,X4 : $i] : (((in @ (kpair @ X4 @ X0) @ (dpsetconstr @ X1 @ X3 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X2 @ Y0 @ Y1)))))) = $true) => ((in @ X4 @ X1) = $true))))),
% 0.31/1.33 inference(fool_elimination,[status(thm)],[f7])).
% 0.31/1.33 thf(f7,plain,(
% 0.31/1.33 ~(dsetconstrEL => (cartprodpairmemEL => ! [X0 : $i,X1 : $i,X2 : $i > $i > $o,X3 : $i,X4 : $i] : ((in @ (kpair @ X4 @ X0) @ (dpsetconstr @ X1 @ X3 @ (^[X5 : $i, X6 : $i] : (X2 @ X5 @ X6)))) => (in @ X4 @ X1))))),
% 0.31/1.33 inference(rectify,[status(thm)],[f5])).
% 0.31/1.33 thf(f5,negated_conjecture,(
% 0.31/1.33 ~(dsetconstrEL => (cartprodpairmemEL => ! [X3 : $i,X0 : $i,X1 : $i > $i > $o,X4 : $i,X2 : $i] : ((in @ (kpair @ X2 @ X3) @ (dpsetconstr @ X0 @ X4 @ (^[X6 : $i, X5 : $i] : (X1 @ X6 @ X5)))) => (in @ X2 @ X0))))),
% 0.31/1.33 inference(negated_conjecture,[status(cth)],[f4])).
% 0.31/1.33 thf(f4,conjecture,(
% 0.31/1.33 dsetconstrEL => (cartprodpairmemEL => ! [X3 : $i,X0 : $i,X1 : $i > $i > $o,X4 : $i,X2 : $i] : ((in @ (kpair @ X2 @ X3) @ (dpsetconstr @ X0 @ X4 @ (^[X6 : $i, X5 : $i] : (X1 @ X6 @ X5)))) => (in @ X2 @ X0)))),
% 0.31/1.33 file('/export/starexec/sandbox2/tmp/tmp.j3kahdTx8R/DTF2THF_8102.p',dpsetconstrEL1)).
% 0.31/1.33 thf(f25,plain,(
% 0.31/1.33 (cartprodpairmemEL = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: ((in @ (kpair @ Y2 @ Y0) @ (cartprod @ Y3 @ Y1)) => (in @ Y2 @ Y3)))))))))))),
% 0.31/1.33 inference(cnf_transformation,[status(thm)],[f14])).
% 0.31/1.33 thf(f14,plain,(
% 0.31/1.33 (cartprodpairmemEL = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: ((in @ (kpair @ Y2 @ Y0) @ (cartprod @ Y3 @ Y1)) => (in @ Y2 @ Y3)))))))))))),
% 0.31/1.33 inference(fool_elimination,[status(thm)],[f13])).
% 0.31/1.33 thf(f13,plain,(
% 0.31/1.33 (cartprodpairmemEL = ! [X0 : $i,X1 : $i,X2 : $i,X3 : $i] : ((in @ (kpair @ X1 @ X3) @ (cartprod @ X0 @ X2)) => (in @ X1 @ X0)))),
% 0.31/1.33 inference(rectify,[status(thm)],[f2])).
% 0.31/1.33 thf(f2,axiom,(
% 0.31/1.33 (cartprodpairmemEL = ! [X0 : $i,X2 : $i,X4 : $i,X3 : $i] : ((in @ (kpair @ X2 @ X3) @ (cartprod @ X0 @ X4)) => (in @ X2 @ X0)))),
% 0.31/1.33 file('/export/starexec/sandbox2/tmp/tmp.j3kahdTx8R/DTF2THF_8102.p',cartprodpairmemEL)).
% 0.31/1.33 thf(f49,plain,(
% 0.31/1.33 ((in @ (kpair @ sK4 @ sK0) @ (cartprod @ sK1 @ sK3)) = $true)),
% 0.31/1.33 inference(trivial_inequality_removal,[status(thm)],[f47])).
% 0.31/1.33 thf(f47,plain,(
% 0.31/1.33 ((in @ (kpair @ sK4 @ sK0) @ (cartprod @ sK1 @ sK3)) = $true) | ($false = $true)),
% 0.31/1.33 inference(superposition,[status(thm)],[f29,f46])).
% 0.31/1.33 thf(f46,plain,(
% 0.31/1.33 ( ! [X2 : $i,X3 : $i,X1 : $i > $o] : (($false = (in @ X3 @ (dsetconstr @ X2 @ X1))) | ((in @ X3 @ X2) = $true)) )),
% 0.31/1.33 inference(binary_proxy_clausification,[status(thm)],[f45])).
% 0.31/1.33 thf(f45,plain,(
% 0.31/1.33 ( ! [X2 : $i,X3 : $i,X1 : $i > $o] : ((((in @ X3 @ (dsetconstr @ X2 @ X1)) => (in @ X3 @ X2)) = $true)) )),
% 0.31/1.33 inference(beta_eta_normalization,[status(thm)],[f44])).
% 0.31/1.33 thf(f44,plain,(
% 0.31/1.33 ( ! [X2 : $i,X3 : $i,X1 : $i > $o] : ((((^[Y0 : $i]: ((in @ Y0 @ (dsetconstr @ X2 @ X1)) => (in @ Y0 @ X2))) @ X3) = $true)) )),
% 0.31/1.33 inference(pi_clausification,[status(thm)],[f43])).
% 0.31/1.33 thf(f43,plain,(
% 0.31/1.33 ( ! [X2 : $i,X1 : $i > $o] : (((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ (dsetconstr @ X2 @ X1)) => (in @ Y0 @ X2)))) = $true)) )),
% 0.31/1.33 inference(beta_eta_normalization,[status(thm)],[f42])).
% 0.31/1.33 thf(f42,plain,(
% 0.31/1.33 ( ! [X2 : $i,X1 : $i > $o] : ((((^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ (dsetconstr @ Y0 @ X1)) => (in @ Y1 @ Y0))))) @ X2) = $true)) )),
% 0.31/1.33 inference(pi_clausification,[status(thm)],[f41])).
% 0.31/1.33 thf(f41,plain,(
% 0.31/1.33 ( ! [X1 : $i > $o] : (((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ (dsetconstr @ Y0 @ X1)) => (in @ Y1 @ Y0)))))) = $true)) )),
% 0.31/1.33 inference(beta_eta_normalization,[status(thm)],[f40])).
% 0.31/1.33 thf(f40,plain,(
% 0.31/1.33 ( ! [X1 : $i > $o] : ((((^[Y0 : $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ (dsetconstr @ Y1 @ Y0)) => (in @ Y2 @ Y1))))))) @ X1) = $true)) )),
% 0.31/1.33 inference(pi_clausification,[status(thm)],[f39])).
% 0.31/1.33 thf(f39,plain,(
% 0.31/1.33 ((!! @ ($i > $o) @ (^[Y0 : $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ (dsetconstr @ Y1 @ Y0)) => (in @ Y2 @ Y1)))))))) = $true)),
% 0.31/1.33 inference(beta_eta_normalization,[status(thm)],[f27])).
% 0.31/1.33 thf(f27,plain,(
% 0.31/1.33 ((!! @ ($i > $o) @ (^[Y0 : $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ (dsetconstr @ Y1 @ (^[Y3 : $i]: (Y0 @ Y3)))) => (in @ Y2 @ Y1)))))))) = $true)),
% 0.31/1.33 inference(definition_unfolding,[status(thm)],[f24,f22])).
% 0.31/1.33 thf(f22,plain,(
% 0.31/1.33 (dsetconstrEL = $true)),
% 0.31/1.33 inference(cnf_transformation,[status(thm)],[f18])).
% 0.31/1.33 thf(f24,plain,(
% 0.31/1.33 (dsetconstrEL = (!! @ ($i > $o) @ (^[Y0 : $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ (dsetconstr @ Y1 @ (^[Y3 : $i]: (Y0 @ Y3)))) => (in @ Y2 @ Y1)))))))))),
% 0.31/1.33 inference(cnf_transformation,[status(thm)],[f12])).
% 0.31/1.33 thf(f12,plain,(
% 0.31/1.33 (dsetconstrEL = (!! @ ($i > $o) @ (^[Y0 : $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ (dsetconstr @ Y1 @ (^[Y3 : $i]: (Y0 @ Y3)))) => (in @ Y2 @ Y1)))))))))),
% 0.31/1.33 inference(fool_elimination,[status(thm)],[f11])).
% 0.31/1.33 thf(f11,plain,(
% 0.31/1.33 (dsetconstrEL = ! [X0 : $i,X1 : $i,X2 : $i > $o] : ((in @ X0 @ (dsetconstr @ X1 @ (^[X3 : $i] : (X2 @ X3)))) => (in @ X0 @ X1)))),
% 0.31/1.33 inference(rectify,[status(thm)],[f1])).
% 0.31/1.33 thf(f1,axiom,(
% 0.31/1.33 (dsetconstrEL = ! [X2 : $i,X0 : $i,X1 : $i > $o] : ((in @ X2 @ (dsetconstr @ X0 @ (^[X3 : $i] : (X1 @ X3)))) => (in @ X2 @ X0)))),
% 0.31/1.33 file('/export/starexec/sandbox2/tmp/tmp.j3kahdTx8R/DTF2THF_8102.p',dsetconstrEL)).
% 0.31/1.33 thf(f29,plain,(
% 0.31/1.33 ((in @ (kpair @ sK4 @ sK0) @ (dsetconstr @ (cartprod @ sK1 @ sK3) @ (^[Y0 : $i]: (?? @ $i @ (^[Y1 : $i]: ((?? @ $i @ (^[Y2 : $i]: (((sK2 @ Y1 @ Y2) & (in @ Y2 @ sK3)) & ((kpair @ Y1 @ Y2) = Y0)))) & (in @ Y1 @ sK1))))))) = $true)),
% 0.31/1.33 inference(beta_eta_normalization,[status(thm)],[f26])).
% 0.31/1.33 thf(f26,plain,(
% 0.31/1.33 ((in @ (kpair @ sK4 @ sK0) @ ((^[Y0 : $i]: ((^[Y1 : $i]: ((^[Y2 : $i > $i > $o]: (dsetconstr @ (cartprod @ Y0 @ Y1) @ (^[Y3 : $i]: (?? @ $i @ (^[Y4 : $i]: ((?? @ $i @ (^[Y5 : $i]: (((Y2 @ Y4 @ Y5) & (in @ Y5 @ Y1)) & ((kpair @ Y4 @ Y5) = Y3)))) & (in @ Y4 @ Y0))))))))))) @ sK1 @ sK3 @ (^[Y0 : $i]: ((^[Y1 : $i]: (sK2 @ Y0 @ Y1)))))) = $true)),
% 0.31/1.33 inference(definition_unfolding,[status(thm)],[f21,f23])).
% 0.31/1.33 thf(f23,plain,(
% 0.31/1.33 (dpsetconstr = (^[Y0 : $i]: ((^[Y1 : $i]: ((^[Y2 : $i > $i > $o]: (dsetconstr @ (cartprod @ Y0 @ Y1) @ (^[Y3 : $i]: (?? @ $i @ (^[Y4 : $i]: ((?? @ $i @ (^[Y5 : $i]: (((Y2 @ Y4 @ Y5) & (in @ Y5 @ Y1)) & ((kpair @ Y4 @ Y5) = Y3)))) & (in @ Y4 @ Y0))))))))))))),
% 0.31/1.33 inference(cnf_transformation,[status(thm)],[f10])).
% 0.31/1.33 thf(f10,plain,(
% 0.31/1.33 (dpsetconstr = (^[Y0 : $i]: ((^[Y1 : $i]: ((^[Y2 : $i > $i > $o]: (dsetconstr @ (cartprod @ Y0 @ Y1) @ (^[Y3 : $i]: (?? @ $i @ (^[Y4 : $i]: ((?? @ $i @ (^[Y5 : $i]: (((Y2 @ Y4 @ Y5) & (in @ Y5 @ Y1)) & ((kpair @ Y4 @ Y5) = Y3)))) & (in @ Y4 @ Y0))))))))))))),
% 0.31/1.33 inference(fool_elimination,[status(thm)],[f9])).
% 0.31/1.33 thf(f9,plain,(
% 0.31/1.33 (dpsetconstr = (^[X0 : $i, X1 : $i, X2 : $i > $i > $o] : (dsetconstr @ (cartprod @ X0 @ X1) @ (^[X3 : $i] : (? [X4 : $i] : ((in @ X4 @ X0) & ? [X5 : $i] : (((kpair @ X4 @ X5) = X3) & (in @ X5 @ X1) & (X2 @ X4 @ X5))))))))),
% 0.31/1.33 inference(rectify,[status(thm)],[f3])).
% 0.31/1.33 thf(f3,axiom,(
% 0.31/1.33 (dpsetconstr = (^[X0 : $i, X4 : $i, X1 : $i > $i > $o] : (dsetconstr @ (cartprod @ X0 @ X4) @ (^[X5 : $i] : (? [X2 : $i] : ((in @ X2 @ X0) & ? [X3 : $i] : (((kpair @ X2 @ X3) = X5) & (in @ X3 @ X4) & (X1 @ X2 @ X3))))))))),
% 0.31/1.33 file('/export/starexec/sandbox2/tmp/tmp.j3kahdTx8R/DTF2THF_8102.p',dpsetconstr)).
% 0.31/1.33 thf(f21,plain,(
% 0.31/1.33 ((in @ (kpair @ sK4 @ sK0) @ (dpsetconstr @ sK1 @ sK3 @ (^[Y0 : $i]: ((^[Y1 : $i]: (sK2 @ Y0 @ Y1)))))) = $true)),
% 0.31/1.33 inference(cnf_transformation,[status(thm)],[f18])).
% 0.31/1.33 thf(f20,plain,(
% 0.31/1.33 ((in @ sK4 @ sK1) != $true)),
% 0.31/1.33 inference(cnf_transformation,[status(thm)],[f18])).
% 0.31/1.33 % SZS output end Proof for DTF2THF_8102
% 0.31/1.33 % (8281)------------------------------
% 0.31/1.33 % (8281)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.31/1.33 % (8281)Termination reason: Refutation
% 0.31/1.33
% 0.31/1.33 % (8281)Memory used [KB]: 5500
% 0.31/1.33 % (8281)Time elapsed: 0.005 s
% 0.31/1.33 % (8281)Instructions burned: 6 (million)
% 0.31/1.33 % (8281)------------------------------
% 0.31/1.33 % (8281)------------------------------
% 0.31/1.33 % (8278)Success in time 0.015 s
%------------------------------------------------------------------------------