%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU670^2 : 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 : n020.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 1.68s 1.49s
% Output : Refutation 1.68s
% Verified :
% SZS Type : Refutation
% Derivation depth : 15
% Number of leaves : 7
% Syntax : Number of formulae : 46 ( 17 unt; 0 typ; 0 def)
% Number of atoms : 444 ( 120 equ; 0 cnn)
% Maximal formula atoms : 6 ( 9 avg)
% Number of connectives : 615 ( 46 ~; 33 |; 45 &; 458 @)
% ( 4 <=>; 21 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 38 ( 38 >; 0 *; 0 +; 0 <<)
% Number of symbols : 24 ( 20 usr; 15 con; 0-3 aty)
% ( 0 !!; 8 ??; 0 @@+; 0 @@-)
% Number of variables : 199 ( 66 ^; 93 !; 40 ?; 199 :)
% 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_20,type,
sK1: $i > $o ).
thf(func_def_26,type,
sK7: $i > $i > $o ).
thf(func_def_33,type,
ph13:
!>[X0: $tType] : X0 ).
thf(f55,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f54,f33]) ).
thf(f33,plain,
( $true
!= ( in @ sK6 @ sK4 ) ),
inference(cnf_transformation,[status(thm)],[f24]) ).
thf(f24,plain,
( ( cartprodpairmemER = $true )
& ( dsetconstrEL = $true )
& ( $true
= ( in @ ( kpair @ sK3 @ sK6 )
@ ( dpsetconstr @ sK5 @ sK4
@ ^ [Y0: $i,Y1: $i] : ( sK7 @ Y0 @ Y1 ) ) ) )
& ( $true
!= ( in @ sK6 @ sK4 ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK3,sK4,sK5,sK6,sK7])],[f18,f23]) ).
thf(f23,plain,
( ? [X0: $i,X1: $i,X2: $i,X3: $i,X4: $i > $i > $o] :
( ( $true
= ( in @ ( kpair @ X0 @ X3 )
@ ( dpsetconstr @ X2 @ X1
@ ^ [Y0: $i,Y1: $i] : ( X4 @ Y0 @ Y1 ) ) ) )
& ( ( in @ X3 @ X1 )
!= $true ) )
=> ( ( $true
= ( in @ ( kpair @ sK3 @ sK6 )
@ ( dpsetconstr @ sK5 @ sK4
@ ^ [Y0: $i,Y1: $i] : ( sK7 @ Y0 @ Y1 ) ) ) )
& ( $true
!= ( in @ sK6 @ sK4 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f18,plain,
( ( cartprodpairmemER = $true )
& ( dsetconstrEL = $true )
& ? [X0: $i,X1: $i,X2: $i,X3: $i,X4: $i > $i > $o] :
( ( $true
= ( in @ ( kpair @ X0 @ X3 )
@ ( dpsetconstr @ X2 @ X1
@ ^ [Y0: $i,Y1: $i] : ( X4 @ Y0 @ Y1 ) ) ) )
& ( ( in @ X3 @ X1 )
!= $true ) ) ),
inference(flattening,[status(thm)],[f17]) ).
thf(f17,plain,
( ? [X0: $i,X1: $i,X2: $i,X3: $i,X4: $i > $i > $o] :
( ( $true
= ( in @ ( kpair @ X0 @ X3 )
@ ( dpsetconstr @ X2 @ X1
@ ^ [Y0: $i,Y1: $i] : ( X4 @ Y0 @ Y1 ) ) ) )
& ( ( in @ X3 @ X1 )
!= $true ) )
& ( cartprodpairmemER = $true )
& ( dsetconstrEL = $true ) ),
inference(ennf_transformation,[status(thm)],[f10]) ).
thf(f10,plain,
~ ( ( dsetconstrEL = $true )
=> ( ( cartprodpairmemER = $true )
=> ! [X0: $i,X1: $i,X2: $i,X3: $i,X4: $i > $i > $o] :
( ( $true
= ( in @ ( kpair @ X0 @ X3 )
@ ( dpsetconstr @ X2 @ X1
@ ^ [Y0: $i,Y1: $i] : ( X4 @ Y0 @ Y1 ) ) ) )
=> ( ( in @ X3 @ X1 )
= $true ) ) ) ),
inference(fool_elimination,[status(thm)],[f9]) ).
thf(f9,plain,
~ ( dsetconstrEL
=> ( cartprodpairmemER
=> ! [X0: $i,X1: $i,X2: $i,X3: $i,X4: $i > $i > $o] :
( ( in @ ( kpair @ X0 @ X3 )
@ ( dpsetconstr @ X2 @ X1
@ ^ [X5: $i,X6: $i] : ( X4 @ X5 @ X6 ) ) )
=> ( in @ X3 @ X1 ) ) ) ),
inference(rectify,[status(thm)],[f5]) ).
thf(f5,negated_conjecture,
~ ( dsetconstrEL
=> ( cartprodpairmemER
=> ! [X2: $i,X4: $i,X0: $i,X3: $i,X1: $i > $i > $o] :
( ( in @ ( kpair @ X2 @ X3 )
@ ( dpsetconstr @ X0 @ X4
@ ^ [X6: $i,X5: $i] : ( X1 @ X6 @ X5 ) ) )
=> ( in @ X3 @ X4 ) ) ) ),
inference(negated_conjecture,[status(cth)],[f4]) ).
thf(f4,conjecture,
( dsetconstrEL
=> ( cartprodpairmemER
=> ! [X2: $i,X4: $i,X0: $i,X3: $i,X1: $i > $i > $o] :
( ( in @ ( kpair @ X2 @ X3 )
@ ( dpsetconstr @ X0 @ X4
@ ^ [X6: $i,X5: $i] : ( X1 @ X6 @ X5 ) ) )
=> ( in @ X3 @ X4 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.Kt2I8leLFE/DTF2THF_1246.p',dpsetconstrEL2) ).
thf(f54,plain,
( $true
= ( in @ sK6 @ sK4 ) ),
inference(trivial_inequality_removal,[status(thm)],[f53]) ).
thf(f53,plain,
( ( $true
= ( in @ sK6 @ sK4 ) )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f49,f52]) ).
thf(f52,plain,
( $true
= ( in @ ( kpair @ sK3 @ sK6 ) @ ( cartprod @ sK5 @ sK4 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f51]) ).
thf(f51,plain,
( ( $true != $true )
| ( $true
= ( in @ ( kpair @ sK3 @ sK6 ) @ ( cartprod @ sK5 @ sK4 ) ) ) ),
inference(superposition,[status(thm)],[f48,f50]) ).
thf(f50,plain,
( $true
= ( in @ ( kpair @ sK3 @ sK6 )
@ ( dsetconstr @ ( cartprod @ sK5 @ sK4 )
@ ^ [Y0: $i] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK5 )
& ( ?? @ $i
@ ^ [Y2: $i] :
( ( sK7 @ Y1 @ Y2 )
& ( ( kpair @ Y1 @ Y2 )
= Y0 )
& ( in @ Y2 @ sK4 ) ) ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f43]) ).
thf(f43,plain,
( $true
= ( in @ ( kpair @ sK3 @ sK6 )
@ ( ^ [Y0: $i,Y1: $i,Y2: $i > $i > $o] :
( dsetconstr @ ( cartprod @ Y0 @ Y1 )
@ ^ [Y3: $i] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y0 )
& ( ?? @ $i
@ ^ [Y5: $i] :
( ( Y2 @ Y4 @ Y5 )
& ( ( kpair @ Y4 @ Y5 )
= Y3 )
& ( in @ Y5 @ Y1 ) ) ) ) ) )
@ sK5
@ sK4
@ ^ [Y0: $i,Y1: $i] : ( sK7 @ Y0 @ Y1 ) ) ) ),
inference(definition_unfolding,[status(thm)],[f34,f29]) ).
thf(f29,plain,
( dpsetconstr
= ( ^ [Y0: $i,Y1: $i,Y2: $i > $i > $o] :
( dsetconstr @ ( cartprod @ Y0 @ Y1 )
@ ^ [Y3: $i] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y0 )
& ( ?? @ $i
@ ^ [Y5: $i] :
( ( Y2 @ Y4 @ Y5 )
& ( ( kpair @ Y4 @ Y5 )
= Y3 )
& ( in @ Y5 @ Y1 ) ) ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f12]) ).
thf(f12,plain,
( dpsetconstr
= ( ^ [Y0: $i,Y1: $i,Y2: $i > $i > $o] :
( dsetconstr @ ( cartprod @ Y0 @ Y1 )
@ ^ [Y3: $i] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y0 )
& ( ?? @ $i
@ ^ [Y5: $i] :
( ( Y2 @ Y4 @ Y5 )
& ( ( kpair @ Y4 @ Y5 )
= Y3 )
& ( in @ Y5 @ Y1 ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f11]) ).
thf(f11,plain,
( dpsetconstr
= ( ^ [X0: $i,X1: $i,X2: $i > $i > $o] :
( dsetconstr @ ( cartprod @ X0 @ X1 )
@ ^ [X3: $i] :
? [X4: $i] :
( ? [X5: $i] :
( ( in @ X5 @ X1 )
& ( ( kpair @ X4 @ X5 )
= X3 )
& ( X2 @ X4 @ X5 ) )
& ( in @ X4 @ X0 ) ) ) ) ),
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] :
( ? [X3: $i] :
( ( in @ X3 @ X4 )
& ( ( kpair @ X2 @ X3 )
= X5 )
& ( X1 @ X2 @ X3 ) )
& ( in @ X2 @ X0 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.Kt2I8leLFE/DTF2THF_1246.p',dpsetconstr) ).
thf(f34,plain,
( $true
= ( in @ ( kpair @ sK3 @ sK6 )
@ ( dpsetconstr @ sK5 @ sK4
@ ^ [Y0: $i,Y1: $i] : ( sK7 @ Y0 @ Y1 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f24]) ).
thf(f48,plain,
! [X3: $i,X4: $i > $o,X5: $i] :
( ( $true
!= ( in @ X3 @ ( dsetconstr @ X5 @ X4 ) ) )
| ( $true
= ( in @ X3 @ X5 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f47]) ).
thf(f47,plain,
! [X3: $i,X4: $i > $o,X5: $i] :
( ( $true
!= ( in @ X3
@ ( dsetconstr @ X5
@ ^ [Y0: $i] : ( X4 @ Y0 ) ) ) )
| ( $true
= ( in @ X3 @ X5 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f42]) ).
thf(f42,plain,
! [X3: $i,X4: $i > $o,X5: $i] :
( ( $true
= ( in @ X3 @ X5 ) )
| ( $true
!= ( in @ X3
@ ( dsetconstr @ X5
@ ^ [Y0: $i] : ( X4 @ Y0 ) ) ) )
| ( $true != $true ) ),
inference(definition_unfolding,[status(thm)],[f30,f35]) ).
thf(f35,plain,
dsetconstrEL = $true,
inference(cnf_transformation,[status(thm)],[f24]) ).
thf(f30,plain,
! [X3: $i,X4: $i > $o,X5: $i] :
( ( $true
= ( in @ X3 @ X5 ) )
| ( $true
!= ( in @ X3
@ ( dsetconstr @ X5
@ ^ [Y0: $i] : ( X4 @ Y0 ) ) ) )
| ( dsetconstrEL != $true ) ),
inference(cnf_transformation,[status(thm)],[f22]) ).
thf(f22,plain,
( ( ( dsetconstrEL = $true )
| ( ( $true
!= ( in @ sK0 @ sK2 ) )
& ( $true
= ( in @ sK0
@ ( dsetconstr @ sK2
@ ^ [Y0: $i] : ( sK1 @ Y0 ) ) ) ) ) )
& ( ! [X3: $i,X4: $i > $o,X5: $i] :
( ( $true
= ( in @ X3 @ X5 ) )
| ( $true
!= ( in @ X3
@ ( dsetconstr @ X5
@ ^ [Y0: $i] : ( X4 @ Y0 ) ) ) ) )
| ( dsetconstrEL != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2])],[f20,f21]) ).
thf(f21,plain,
( ? [X0: $i,X1: $i > $o,X2: $i] :
( ( ( in @ X0 @ X2 )
!= $true )
& ( ( in @ X0
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X1 @ Y0 ) ) )
= $true ) )
=> ( ( $true
!= ( in @ sK0 @ sK2 ) )
& ( $true
= ( in @ sK0
@ ( dsetconstr @ sK2
@ ^ [Y0: $i] : ( sK1 @ Y0 ) ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f20,plain,
( ( ( dsetconstrEL = $true )
| ? [X0: $i,X1: $i > $o,X2: $i] :
( ( ( in @ X0 @ X2 )
!= $true )
& ( ( in @ X0
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X1 @ Y0 ) ) )
= $true ) ) )
& ( ! [X3: $i,X4: $i > $o,X5: $i] :
( ( $true
= ( in @ X3 @ X5 ) )
| ( $true
!= ( in @ X3
@ ( dsetconstr @ X5
@ ^ [Y0: $i] : ( X4 @ Y0 ) ) ) ) )
| ( dsetconstrEL != $true ) ) ),
inference(rectify,[status(thm)],[f19]) ).
thf(f19,plain,
( ( ( dsetconstrEL = $true )
| ? [X0: $i,X1: $i > $o,X2: $i] :
( ( ( in @ X0 @ X2 )
!= $true )
& ( ( in @ X0
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X1 @ Y0 ) ) )
= $true ) ) )
& ( ! [X0: $i,X1: $i > $o,X2: $i] :
( ( ( in @ X0 @ X2 )
= $true )
| ( ( in @ X0
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X1 @ Y0 ) ) )
!= $true ) )
| ( dsetconstrEL != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f16]) ).
thf(f16,plain,
( ( dsetconstrEL = $true )
<=> ! [X0: $i,X1: $i > $o,X2: $i] :
( ( ( in @ X0 @ X2 )
= $true )
| ( ( in @ X0
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X1 @ Y0 ) ) )
!= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f14]) ).
thf(f14,plain,
( ! [X0: $i,X1: $i > $o,X2: $i] :
( ( ( in @ X0
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X1 @ Y0 ) ) )
= $true )
=> ( ( in @ X0 @ X2 )
= $true ) )
<=> ( dsetconstrEL = $true ) ),
inference(fool_elimination,[status(thm)],[f13]) ).
thf(f13,plain,
! [X0: $i,X1: $i > $o,X2: $i] :
( ( ( in @ X0
@ ( dsetconstr @ X2
@ ^ [X3: $i] : ( X1 @ X3 ) ) )
=> ( in @ X0 @ X2 ) )
= dsetconstrEL ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
! [X2: $i,X1: $i > $o,X0: $i] :
( ( ( in @ X2
@ ( dsetconstr @ X0
@ ^ [X3: $i] : ( X1 @ X3 ) ) )
=> ( in @ X2 @ X0 ) )
= dsetconstrEL ),
file('/export/starexec/sandbox/tmp/tmp.Kt2I8leLFE/DTF2THF_1246.p',dsetconstrEL) ).
thf(f49,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i] :
( ( $true
!= ( in @ ( kpair @ X3 @ X1 ) @ ( cartprod @ X0 @ X2 ) ) )
| ( $true
= ( in @ X1 @ X2 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f44]) ).
thf(f44,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i] :
( ( $true
!= ( in @ ( kpair @ X3 @ X1 ) @ ( cartprod @ X0 @ X2 ) ) )
| ( $true
= ( in @ X1 @ X2 ) )
| ( $true != $true ) ),
inference(definition_unfolding,[status(thm)],[f39,f36]) ).
thf(f36,plain,
cartprodpairmemER = $true,
inference(cnf_transformation,[status(thm)],[f24]) ).
thf(f39,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i] :
( ( $true
!= ( in @ ( kpair @ X3 @ X1 ) @ ( cartprod @ X0 @ X2 ) ) )
| ( $true
= ( in @ X1 @ X2 ) )
| ( cartprodpairmemER != $true ) ),
inference(cnf_transformation,[status(thm)],[f28]) ).
thf(f28,plain,
( ( ! [X0: $i,X1: $i,X2: $i,X3: $i] :
( ( $true
!= ( in @ ( kpair @ X3 @ X1 ) @ ( cartprod @ X0 @ X2 ) ) )
| ( $true
= ( in @ X1 @ X2 ) ) )
| ( cartprodpairmemER != $true ) )
& ( ( cartprodpairmemER = $true )
| ( ( $true
= ( in @ ( kpair @ sK11 @ sK9 ) @ ( cartprod @ sK8 @ sK10 ) ) )
& ( $true
!= ( in @ sK9 @ sK10 ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK8,sK9,sK10,sK11])],[f26,f27]) ).
thf(f27,plain,
( ? [X4: $i,X5: $i,X6: $i,X7: $i] :
( ( ( in @ ( kpair @ X7 @ X5 ) @ ( cartprod @ X4 @ X6 ) )
= $true )
& ( $true
!= ( in @ X5 @ X6 ) ) )
=> ( ( $true
= ( in @ ( kpair @ sK11 @ sK9 ) @ ( cartprod @ sK8 @ sK10 ) ) )
& ( $true
!= ( in @ sK9 @ sK10 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f26,plain,
( ( ! [X0: $i,X1: $i,X2: $i,X3: $i] :
( ( $true
!= ( in @ ( kpair @ X3 @ X1 ) @ ( cartprod @ X0 @ X2 ) ) )
| ( $true
= ( in @ X1 @ X2 ) ) )
| ( cartprodpairmemER != $true ) )
& ( ( cartprodpairmemER = $true )
| ? [X4: $i,X5: $i,X6: $i,X7: $i] :
( ( ( in @ ( kpair @ X7 @ X5 ) @ ( cartprod @ X4 @ X6 ) )
= $true )
& ( $true
!= ( in @ X5 @ X6 ) ) ) ) ),
inference(rectify,[status(thm)],[f25]) ).
thf(f25,plain,
( ( ! [X0: $i,X1: $i,X2: $i,X3: $i] :
( ( $true
!= ( in @ ( kpair @ X3 @ X1 ) @ ( cartprod @ X0 @ X2 ) ) )
| ( $true
= ( in @ X1 @ X2 ) ) )
| ( cartprodpairmemER != $true ) )
& ( ( cartprodpairmemER = $true )
| ? [X0: $i,X1: $i,X2: $i,X3: $i] :
( ( $true
= ( in @ ( kpair @ X3 @ X1 ) @ ( cartprod @ X0 @ X2 ) ) )
& ( $true
!= ( in @ X1 @ X2 ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f15]) ).
thf(f15,plain,
( ! [X0: $i,X1: $i,X2: $i,X3: $i] :
( ( $true
!= ( in @ ( kpair @ X3 @ X1 ) @ ( cartprod @ X0 @ X2 ) ) )
| ( $true
= ( in @ X1 @ X2 ) ) )
<=> ( cartprodpairmemER = $true ) ),
inference(ennf_transformation,[status(thm)],[f8]) ).
thf(f8,plain,
( ! [X0: $i,X1: $i,X2: $i,X3: $i] :
( ( $true
= ( in @ ( kpair @ X3 @ X1 ) @ ( cartprod @ X0 @ X2 ) ) )
=> ( $true
= ( in @ X1 @ X2 ) ) )
<=> ( cartprodpairmemER = $true ) ),
inference(fool_elimination,[status(thm)],[f7]) ).
thf(f7,plain,
! [X0: $i,X1: $i,X2: $i,X3: $i] :
( ( ( in @ ( kpair @ X3 @ X1 ) @ ( cartprod @ X0 @ X2 ) )
=> ( in @ X1 @ X2 ) )
= cartprodpairmemER ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
! [X0: $i,X3: $i,X4: $i,X2: $i] :
( ( ( in @ ( kpair @ X2 @ X3 ) @ ( cartprod @ X0 @ X4 ) )
=> ( in @ X3 @ X4 ) )
= cartprodpairmemER ),
file('/export/starexec/sandbox/tmp/tmp.Kt2I8leLFE/DTF2THF_1246.p',cartprodpairmemER) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU670^2 : 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
% 0.15/0.33 % Computer : n020.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
% 0.15/0.34 % DateTime : Tue Feb 24 12:30:42 EST 2026
% 0.15/0.34 % CPUTime :
% 0.15/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.31/0.47 ---- Original DTF file ---
% 0.31/0.48 thf(spec,logic,$$dhol).
% 0.31/0.48 %------------------------------------------------------------------------------
% 0.31/0.48 % File : SEU670^2 : TPTP v9.2.1. Released v3.7.0.
% 0.31/0.48 % Domain : Set Theory
% 0.31/0.48 % Problem : Ordered Pairs - Sets of Pairs
% 0.31/0.48 % Version : Especial > Reduced > Especial.
% 0.31/0.48 % English : (! A:i.! B:i.! phi:i>(i>o).! x:i.! y:i.in (kpair x y)
% 0.31/0.48 % (dpsetconstr A B (^ z,u:i.phi z u)) -> in y B)
% 0.31/0.48
% 0.31/0.48 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.31/0.48 % Source : [Bro08]
% 0.31/0.48 % Names : ZFC172l [Bro08]
% 0.31/0.48
% 0.31/0.48 % Status : Theorem
% 0.31/0.48 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.10 v8.2.0, 0.31 v8.1.0, 0.36 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
% 0.31/0.48 % Syntax : Number of formulae : 11 ( 3 unt; 7 typ; 3 def)
% 0.31/0.48 % Number of atoms : 16 ( 4 equ; 0 cnn)
% 0.31/0.48 % Maximal formula atoms : 4 ( 4 avg)
% 0.31/0.48 % Number of connectives : 46 ( 0 ~; 0 |; 3 &; 38 @)
% 0.31/0.48 % ( 0 <=>; 5 =>; 0 <=; 0 <~>)
% 0.31/0.48 % Maximal formula depth : 15 ( 5 avg)
% 0.31/0.48 % Number of types : 2 ( 0 usr)
% 0.31/0.48 % Number of type conns : 19 ( 19 >; 0 *; 0 +; 0 <<)
% 0.31/0.48 % Number of symbols : 8 ( 7 usr; 2 con; 0-3 aty)
% 0.31/0.48 % Number of variables : 21 ( 7 ^; 12 !; 2 ?; 21 :)
% 0.31/0.48 % SPC : TH0_THM_EQU_NAR
% 0.31/0.48
% 0.31/0.48 % Comments : http://mathgate.info/detsetitem.php?id=340
% 0.31/0.48 %------------------------------------------------------------------------------
% 0.31/0.48 thf(in_type,type,
% 0.31/0.48 in: $i > $i > $o ).
% 0.31/0.48
% 0.31/0.48 thf(dsetconstr_type,type,
% 0.31/0.48 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.31/0.48
% 0.31/0.48 thf(dsetconstrEL_type,type,
% 0.31/0.48 dsetconstrEL: $o ).
% 0.31/0.48
% 0.31/0.48 thf(dsetconstrEL,definition,
% 0.31/0.48 ( dsetconstrEL
% 0.31/0.48 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx
% 0.31/0.48 @ ( dsetconstr @ A
% 0.31/0.48 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.31/0.48 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(kpair_type,type,
% 0.31/0.48 kpair: $i > $i > $i ).
% 0.31/0.48
% 0.31/0.48 thf(cartprod_type,type,
% 0.31/0.48 cartprod: $i > $i > $i ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodpairmemER_type,type,
% 0.31/0.48 cartprodpairmemER: $o ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodpairmemER,definition,
% 0.31/0.48 ( cartprodpairmemER
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.31/0.48 => ( in @ Xy @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(dpsetconstr_type,type,
% 0.31/0.48 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.31/0.48
% 0.31/0.48 thf(dpsetconstr,definition,
% 0.31/0.48 ( dpsetconstr
% 0.31/0.48 = ( ^ [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.31/0.48 ( dsetconstr @ ( cartprod @ A @ B )
% 0.31/0.48 @ ^ [Xu: $i] :
% 0.31/0.48 ? [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 & ? [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ B )
% 0.31/0.48 & ( Xphi @ Xx @ Xy )
% 0.31/0.48 & ( Xu
% 0.31/0.48 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(dpsetconstrEL2,conjecture,
% 0.31/0.48 ( dsetconstrEL
% 0.31/0.48 => ( cartprodpairmemER
% 0.31/0.48 => ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( in @ ( kpair @ Xx @ Xy )
% 0.31/0.48 @ ( dpsetconstr @ A @ B
% 0.31/0.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.31/0.48 => ( in @ Xy @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 %------------------------------------------------------------------------------
% 0.31/0.48 ------------------------
% 1.68/1.45 ---- Embedded in THF ---
% 1.68/1.45 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.68/1.45 %%% Generated on Tue Feb 24 12:30:43 EST 2026
% 1.68/1.45 %%% using '$$dhol' embedding, version 1.3.0.
% 1.68/1.45 %%% Logic specification used:
% 1.68/1.45 %%% thf(spec, logic, $$dhol).
% 1.68/1.45
% 1.68/1.45 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.Kt2I8leLFE/DTF2DTF1246.p
% See solution above
% 1.68/1.45 ------------------------
% 1.68/1.46 ---- Cleaned THF ---
% 1.68/1.46 thf(in_type,type,
% 1.68/1.46 in: $i > $i > $o ).
% 1.68/1.46
% 1.68/1.46 thf(dsetconstr_type,type,
% 1.68/1.46 dsetconstr: $i > ( $i > $o ) > $i ).
% 1.68/1.46
% 1.68/1.46 thf(dsetconstrEL_type,type,
% 1.68/1.46 dsetconstrEL: $o ).
% 1.68/1.46
% 1.68/1.46 thf(dsetconstrEL,definition,
% 1.68/1.46 ( dsetconstrEL
% 1.68/1.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.68/1.46 ( ( in @ Xx
% 1.68/1.46 @ ( dsetconstr @ A
% 1.68/1.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 1.68/1.46 => ( in @ Xx @ A ) ) ) ) ).
% 1.68/1.46
% 1.68/1.46 thf(kpair_type,type,
% 1.68/1.46 kpair: $i > $i > $i ).
% 1.68/1.46
% 1.68/1.46 thf(cartprod_type,type,
% 1.68/1.46 cartprod: $i > $i > $i ).
% 1.68/1.46
% 1.68/1.46 thf(cartprodpairmemER_type,type,
% 1.68/1.46 cartprodpairmemER: $o ).
% 1.68/1.46
% 1.68/1.46 thf(cartprodpairmemER,definition,
% 1.68/1.46 ( cartprodpairmemER
% 1.68/1.46 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 1.68/1.46 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 1.68/1.46 => ( in @ Xy @ B ) ) ) ) ).
% 1.68/1.46
% 1.68/1.46 thf(dpsetconstr_type,type,
% 1.68/1.46 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 1.68/1.46
% 1.68/1.46 thf(dpsetconstr,definition,
% 1.68/1.46 ( dpsetconstr
% 1.68/1.46 = ( ^ [A: $i,B: $i,Xphi: $i > $i > $o] :
% 1.68/1.46 ( dsetconstr @ ( cartprod @ A @ B )
% 1.68/1.46 @ ^ [Xu: $i] :
% 1.68/1.46 ? [Xx: $i] :
% 1.68/1.46 ( ( in @ Xx @ A )
% 1.68/1.46 & ? [Xy: $i] :
% 1.68/1.46 ( ( in @ Xy @ B )
% 1.68/1.46 & ( Xphi @ Xx @ Xy )
% 1.68/1.46 & ( Xu
% 1.68/1.46 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 1.68/1.46
% 1.68/1.46 thf(dpsetconstrEL2,conjecture,
% 1.68/1.46 ( dsetconstrEL
% 1.68/1.46 => ( cartprodpairmemER
% 1.68/1.46 => ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 1.68/1.46 ( ( in @ ( kpair @ Xx @ Xy )
% 1.68/1.46 @ ( dpsetconstr @ A @ B
% 1.68/1.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 1.68/1.46 => ( in @ Xy @ B ) ) ) ) ).
% 1.68/1.46 ------------------------
% 1.68/1.48 % (1400)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_1246 for (2999ds/2Mi)
% 1.68/1.48 % (1399)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_1246 for (2999ds/2Mi)
% 1.68/1.48 % (1398)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_1246 for (2999ds/27Mi)
% 1.68/1.48 % (1396)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_1246 for (2999ds/183Mi)
% 1.68/1.48 % (1397)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_1246 for (2999ds/4Mi)
% 1.68/1.48 % (1401)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_1246 for (2999ds/275Mi)
% 1.68/1.48 % (1402)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_1246 for (2999ds/18Mi)
% 1.68/1.48 % (1399)Instruction limit reached!
% 1.68/1.48 % (1399)------------------------------
% 1.68/1.48 % (1399)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.68/1.48 % (1399)Termination reason: Unknown
% 1.68/1.48 % (1399)Termination phase: Property scanning
% 1.68/1.48 % (1400)Instruction limit reached!
% 1.68/1.48 % (1400)------------------------------
% 1.68/1.48 % (1400)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.68/1.48 % (1400)Termination reason: Unknown
% 1.68/1.48 % (1400)Termination phase: Function definition elimination
% 1.68/1.48
% 1.68/1.48 % (1400)Memory used [KB]: 1023
% 1.68/1.48 % (1400)Time elapsed: 0.003 s
% 1.68/1.48 % (1400)Instructions burned: 3 (million)
% 1.68/1.48 % (1400)------------------------------
% 1.68/1.48 % (1400)------------------------------
% 1.68/1.48
% 1.68/1.48 % (1399)Memory used [KB]: 1023
% 1.68/1.48 % (1399)Time elapsed: 0.003 s
% 1.68/1.48 % (1399)Instructions burned: 3 (million)
% 1.68/1.48 % (1399)------------------------------
% 1.68/1.48 % (1399)------------------------------
% 1.68/1.48 % (1397)Instruction limit reached!
% 1.68/1.48 % (1397)------------------------------
% 1.68/1.48 % (1397)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.68/1.48 % (1397)Termination reason: Unknown
% 1.68/1.48 % (1397)Termination phase: Saturation
% 1.68/1.48
% 1.68/1.48 % (1397)Memory used [KB]: 5500
% 1.68/1.48 % (1397)Time elapsed: 0.004 s
% 1.68/1.48 % (1397)Instructions burned: 4 (million)
% 1.68/1.48 % (1397)------------------------------
% 1.68/1.48 % (1397)------------------------------
% 1.68/1.48 % (1401)Refutation not found, incomplete strategy
% 1.68/1.48 % (1401)------------------------------
% 1.68/1.48 % (1401)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.68/1.48 % (1401)Termination reason: Refutation not found, incomplete strategy
% 1.68/1.48
% 1.68/1.48
% 1.68/1.48 % (1401)Memory used [KB]: 5500
% 1.68/1.48 % (1401)Time elapsed: 0.004 s
% 1.68/1.48 % (1401)Instructions burned: 4 (million)
% 1.68/1.48 % (1401)------------------------------
% 1.68/1.48 % (1401)------------------------------
% 1.68/1.48 % (1402)First to succeed.
% 1.68/1.49 % (1396)Also succeeded, but the first one will report.
% 1.68/1.49 % (1402)Refutation found. Thanks to Tanya!
% 1.68/1.49 % SZS status Theorem for DTF2THF_1246
% 1.68/1.49 % SZS output start Proof for DTF2THF_1246
% 1.68/1.49 thf(func_def_0, type, in: $i > $i > $o).
% 1.68/1.49 thf(func_def_1, type, dsetconstr: $i > ($i > $o) > $i).
% 1.68/1.49 thf(func_def_4, type, kpair: $i > $i > $i).
% 1.68/1.49 thf(func_def_5, type, cartprod: $i > $i > $i).
% 1.68/1.49 thf(func_def_7, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 1.68/1.49 thf(func_def_20, type, sK1: $i > $o).
% 1.68/1.49 thf(func_def_26, type, sK7: $i > $i > $o).
% 1.68/1.49 thf(func_def_33, type, ph13: !>[X0: $tType]:(X0)).
% 1.68/1.49 thf(f55,plain,(
% 1.68/1.49 $false),
% 1.68/1.49 inference(subsumption_resolution,[status(thm)],[f54,f33])).
% 1.68/1.49 thf(f33,plain,(
% 1.68/1.49 ($true != (in @ sK6 @ sK4))),
% 1.68/1.49 inference(cnf_transformation,[status(thm)],[f24])).
% 1.68/1.49 thf(f24,plain,(
% 1.68/1.49 (cartprodpairmemER = $true) & (dsetconstrEL = $true) & (($true = (in @ (kpair @ sK3 @ sK6) @ (dpsetconstr @ sK5 @ sK4 @ (^[Y0 : $i]: ((^[Y1 : $i]: (sK7 @ Y0 @ Y1))))))) & ($true != (in @ sK6 @ sK4)))),
% 1.68/1.49 inference(skolemisation,[status(esa),new_symbols(skolem,[sK3,sK4,sK5,sK6,sK7])],[f18,f23])).
% 1.68/1.49 thf(f23,plain,(
% 1.68/1.49 ? [X0 : $i,X1 : $i,X2 : $i,X3 : $i,X4 : $i > $i > $o] : (($true = (in @ (kpair @ X0 @ X3) @ (dpsetconstr @ X2 @ X1 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X4 @ Y0 @ Y1))))))) & ((in @ X3 @ X1) != $true)) => (($true = (in @ (kpair @ sK3 @ sK6) @ (dpsetconstr @ sK5 @ sK4 @ (^[Y0 : $i]: ((^[Y1 : $i]: (sK7 @ Y0 @ Y1))))))) & ($true != (in @ sK6 @ sK4)))),
% 1.68/1.49 introduced(definition,[],[choice_axiom])).
% 1.68/1.49 thf(f18,plain,(
% 1.68/1.49 (cartprodpairmemER = $true) & (dsetconstrEL = $true) & ? [X0 : $i,X1 : $i,X2 : $i,X3 : $i,X4 : $i > $i > $o] : (($true = (in @ (kpair @ X0 @ X3) @ (dpsetconstr @ X2 @ X1 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X4 @ Y0 @ Y1))))))) & ((in @ X3 @ X1) != $true))),
% 1.68/1.49 inference(flattening,[status(thm)],[f17])).
% 1.68/1.49 thf(f17,plain,(
% 1.68/1.49 (? [X0 : $i,X1 : $i,X2 : $i,X3 : $i,X4 : $i > $i > $o] : (($true = (in @ (kpair @ X0 @ X3) @ (dpsetconstr @ X2 @ X1 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X4 @ Y0 @ Y1))))))) & ((in @ X3 @ X1) != $true)) & (cartprodpairmemER = $true)) & (dsetconstrEL = $true)),
% 1.68/1.49 inference(ennf_transformation,[status(thm)],[f10])).
% 1.68/1.49 thf(f10,plain,(
% 1.68/1.49 ~((dsetconstrEL = $true) => ((cartprodpairmemER = $true) => ! [X0 : $i,X1 : $i,X2 : $i,X3 : $i,X4 : $i > $i > $o] : (($true = (in @ (kpair @ X0 @ X3) @ (dpsetconstr @ X2 @ X1 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X4 @ Y0 @ Y1))))))) => ((in @ X3 @ X1) = $true))))),
% 1.68/1.49 inference(fool_elimination,[status(thm)],[f9])).
% 1.68/1.49 thf(f9,plain,(
% 1.68/1.49 ~(dsetconstrEL => (cartprodpairmemER => ! [X0 : $i,X1 : $i,X2 : $i,X3 : $i,X4 : $i > $i > $o] : ((in @ (kpair @ X0 @ X3) @ (dpsetconstr @ X2 @ X1 @ (^[X5 : $i, X6 : $i] : (X4 @ X5 @ X6)))) => (in @ X3 @ X1))))),
% 1.68/1.49 inference(rectify,[status(thm)],[f5])).
% 1.68/1.49 thf(f5,negated_conjecture,(
% 1.68/1.49 ~(dsetconstrEL => (cartprodpairmemER => ! [X2 : $i,X4 : $i,X0 : $i,X3 : $i,X1 : $i > $i > $o] : ((in @ (kpair @ X2 @ X3) @ (dpsetconstr @ X0 @ X4 @ (^[X6 : $i, X5 : $i] : (X1 @ X6 @ X5)))) => (in @ X3 @ X4))))),
% 1.68/1.49 inference(negated_conjecture,[status(cth)],[f4])).
% 1.68/1.49 thf(f4,conjecture,(
% 1.68/1.49 dsetconstrEL => (cartprodpairmemER => ! [X2 : $i,X4 : $i,X0 : $i,X3 : $i,X1 : $i > $i > $o] : ((in @ (kpair @ X2 @ X3) @ (dpsetconstr @ X0 @ X4 @ (^[X6 : $i, X5 : $i] : (X1 @ X6 @ X5)))) => (in @ X3 @ X4)))),
% 1.68/1.49 file('/export/starexec/sandbox/tmp/tmp.Kt2I8leLFE/DTF2THF_1246.p',dpsetconstrEL2)).
% 1.68/1.49 thf(f54,plain,(
% 1.68/1.49 ($true = (in @ sK6 @ sK4))),
% 1.68/1.49 inference(trivial_inequality_removal,[status(thm)],[f53])).
% 1.68/1.49 thf(f53,plain,(
% 1.68/1.49 ($true = (in @ sK6 @ sK4)) | ($true != $true)),
% 1.68/1.49 inference(superposition,[status(thm)],[f49,f52])).
% 1.68/1.49 thf(f52,plain,(
% 1.68/1.49 ($true = (in @ (kpair @ sK3 @ sK6) @ (cartprod @ sK5 @ sK4)))),
% 1.68/1.49 inference(trivial_inequality_removal,[status(thm)],[f51])).
% 1.68/1.49 thf(f51,plain,(
% 1.68/1.49 ($true != $true) | ($true = (in @ (kpair @ sK3 @ sK6) @ (cartprod @ sK5 @ sK4)))),
% 1.68/1.49 inference(superposition,[status(thm)],[f48,f50])).
% 1.68/1.49 thf(f50,plain,(
% 1.68/1.49 ($true = (in @ (kpair @ sK3 @ sK6) @ (dsetconstr @ (cartprod @ sK5 @ sK4) @ (^[Y0 : $i]: (?? @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK5) & (?? @ $i @ (^[Y2 : $i]: (((sK7 @ Y1 @ Y2) & ((kpair @ Y1 @ Y2) = Y0)) & (in @ Y2 @ sK4)))))))))))),
% 1.68/1.49 inference(beta_eta_normalization,[status(thm)],[f43])).
% 1.68/1.49 thf(f43,plain,(
% 1.68/1.49 ($true = (in @ (kpair @ sK3 @ sK6) @ ((^[Y0 : $i]: ((^[Y1 : $i]: ((^[Y2 : $i > $i > $o]: (dsetconstr @ (cartprod @ Y0 @ Y1) @ (^[Y3 : $i]: (?? @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y0) & (?? @ $i @ (^[Y5 : $i]: (((Y2 @ Y4 @ Y5) & ((kpair @ Y4 @ Y5) = Y3)) & (in @ Y5 @ Y1)))))))))))))) @ sK5 @ sK4 @ (^[Y0 : $i]: ((^[Y1 : $i]: (sK7 @ Y0 @ Y1)))))))),
% 1.68/1.49 inference(definition_unfolding,[status(thm)],[f34,f29])).
% 1.68/1.49 thf(f29,plain,(
% 1.68/1.49 (dpsetconstr = (^[Y0 : $i]: ((^[Y1 : $i]: ((^[Y2 : $i > $i > $o]: (dsetconstr @ (cartprod @ Y0 @ Y1) @ (^[Y3 : $i]: (?? @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y0) & (?? @ $i @ (^[Y5 : $i]: (((Y2 @ Y4 @ Y5) & ((kpair @ Y4 @ Y5) = Y3)) & (in @ Y5 @ Y1)))))))))))))))),
% 1.68/1.49 inference(cnf_transformation,[status(thm)],[f12])).
% 1.68/1.49 thf(f12,plain,(
% 1.68/1.49 (dpsetconstr = (^[Y0 : $i]: ((^[Y1 : $i]: ((^[Y2 : $i > $i > $o]: (dsetconstr @ (cartprod @ Y0 @ Y1) @ (^[Y3 : $i]: (?? @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y0) & (?? @ $i @ (^[Y5 : $i]: (((Y2 @ Y4 @ Y5) & ((kpair @ Y4 @ Y5) = Y3)) & (in @ Y5 @ Y1)))))))))))))))),
% 1.68/1.49 inference(fool_elimination,[status(thm)],[f11])).
% 1.68/1.49 thf(f11,plain,(
% 1.68/1.49 (dpsetconstr = (^[X0 : $i, X1 : $i, X2 : $i > $i > $o] : (dsetconstr @ (cartprod @ X0 @ X1) @ (^[X3 : $i] : (? [X4 : $i] : (? [X5 : $i] : ((in @ X5 @ X1) & ((kpair @ X4 @ X5) = X3) & (X2 @ X4 @ X5)) & (in @ X4 @ X0)))))))),
% 1.68/1.49 inference(rectify,[status(thm)],[f3])).
% 1.68/1.49 thf(f3,axiom,(
% 1.68/1.49 (dpsetconstr = (^[X0 : $i, X4 : $i, X1 : $i > $i > $o] : (dsetconstr @ (cartprod @ X0 @ X4) @ (^[X5 : $i] : (? [X2 : $i] : (? [X3 : $i] : ((in @ X3 @ X4) & ((kpair @ X2 @ X3) = X5) & (X1 @ X2 @ X3)) & (in @ X2 @ X0)))))))),
% 1.68/1.49 file('/export/starexec/sandbox/tmp/tmp.Kt2I8leLFE/DTF2THF_1246.p',dpsetconstr)).
% 1.68/1.49 thf(f34,plain,(
% 1.68/1.49 ($true = (in @ (kpair @ sK3 @ sK6) @ (dpsetconstr @ sK5 @ sK4 @ (^[Y0 : $i]: ((^[Y1 : $i]: (sK7 @ Y0 @ Y1)))))))),
% 1.68/1.49 inference(cnf_transformation,[status(thm)],[f24])).
% 1.68/1.49 thf(f48,plain,(
% 1.68/1.49 ( ! [X3 : $i,X4 : $i > $o,X5 : $i] : (($true != (in @ X3 @ (dsetconstr @ X5 @ X4))) | ($true = (in @ X3 @ X5))) )),
% 1.68/1.49 inference(beta_eta_normalization,[status(thm)],[f47])).
% 1.68/1.49 thf(f47,plain,(
% 1.68/1.49 ( ! [X3 : $i,X4 : $i > $o,X5 : $i] : (($true != (in @ X3 @ (dsetconstr @ X5 @ (^[Y0 : $i]: (X4 @ Y0))))) | ($true = (in @ X3 @ X5))) )),
% 1.68/1.49 inference(trivial_inequality_removal,[status(thm)],[f42])).
% 1.68/1.49 thf(f42,plain,(
% 1.68/1.49 ( ! [X3 : $i,X4 : $i > $o,X5 : $i] : (($true = (in @ X3 @ X5)) | ($true != (in @ X3 @ (dsetconstr @ X5 @ (^[Y0 : $i]: (X4 @ Y0))))) | ($true != $true)) )),
% 1.68/1.49 inference(definition_unfolding,[status(thm)],[f30,f35])).
% 1.68/1.49 thf(f35,plain,(
% 1.68/1.49 (dsetconstrEL = $true)),
% 1.68/1.49 inference(cnf_transformation,[status(thm)],[f24])).
% 1.68/1.49 thf(f30,plain,(
% 1.68/1.49 ( ! [X3 : $i,X4 : $i > $o,X5 : $i] : (($true = (in @ X3 @ X5)) | ($true != (in @ X3 @ (dsetconstr @ X5 @ (^[Y0 : $i]: (X4 @ Y0))))) | (dsetconstrEL != $true)) )),
% 1.68/1.49 inference(cnf_transformation,[status(thm)],[f22])).
% 1.68/1.49 thf(f22,plain,(
% 1.68/1.49 ((dsetconstrEL = $true) | (($true != (in @ sK0 @ sK2)) & ($true = (in @ sK0 @ (dsetconstr @ sK2 @ (^[Y0 : $i]: (sK1 @ Y0))))))) & (! [X3 : $i,X4 : $i > $o,X5 : $i] : (($true = (in @ X3 @ X5)) | ($true != (in @ X3 @ (dsetconstr @ X5 @ (^[Y0 : $i]: (X4 @ Y0)))))) | (dsetconstrEL != $true))),
% 1.68/1.49 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2])],[f20,f21])).
% 1.68/1.49 thf(f21,plain,(
% 1.68/1.49 ? [X0 : $i,X1 : $i > $o,X2 : $i] : (((in @ X0 @ X2) != $true) & ((in @ X0 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X1 @ Y0)))) = $true)) => (($true != (in @ sK0 @ sK2)) & ($true = (in @ sK0 @ (dsetconstr @ sK2 @ (^[Y0 : $i]: (sK1 @ Y0))))))),
% 1.68/1.49 introduced(definition,[],[choice_axiom])).
% 1.68/1.49 thf(f20,plain,(
% 1.68/1.49 ((dsetconstrEL = $true) | ? [X0 : $i,X1 : $i > $o,X2 : $i] : (((in @ X0 @ X2) != $true) & ((in @ X0 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X1 @ Y0)))) = $true))) & (! [X3 : $i,X4 : $i > $o,X5 : $i] : (($true = (in @ X3 @ X5)) | ($true != (in @ X3 @ (dsetconstr @ X5 @ (^[Y0 : $i]: (X4 @ Y0)))))) | (dsetconstrEL != $true))),
% 1.68/1.49 inference(rectify,[status(thm)],[f19])).
% 1.68/1.49 thf(f19,plain,(
% 1.68/1.49 ((dsetconstrEL = $true) | ? [X0 : $i,X1 : $i > $o,X2 : $i] : (((in @ X0 @ X2) != $true) & ((in @ X0 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X1 @ Y0)))) = $true))) & (! [X0 : $i,X1 : $i > $o,X2 : $i] : (((in @ X0 @ X2) = $true) | ((in @ X0 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X1 @ Y0)))) != $true)) | (dsetconstrEL != $true))),
% 1.68/1.49 inference(nnf_transformation,[status(thm)],[f16])).
% 1.68/1.49 thf(f16,plain,(
% 1.68/1.49 (dsetconstrEL = $true) <=> ! [X0 : $i,X1 : $i > $o,X2 : $i] : (((in @ X0 @ X2) = $true) | ((in @ X0 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X1 @ Y0)))) != $true))),
% 1.68/1.49 inference(ennf_transformation,[status(thm)],[f14])).
% 1.68/1.49 thf(f14,plain,(
% 1.68/1.49 ! [X0 : $i,X1 : $i > $o,X2 : $i] : (((in @ X0 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X1 @ Y0)))) = $true) => ((in @ X0 @ X2) = $true)) <=> (dsetconstrEL = $true)),
% 1.68/1.49 inference(fool_elimination,[status(thm)],[f13])).
% 1.68/1.49 thf(f13,plain,(
% 1.68/1.49 (! [X0 : $i,X1 : $i > $o,X2 : $i] : ((in @ X0 @ (dsetconstr @ X2 @ (^[X3 : $i] : (X1 @ X3)))) => (in @ X0 @ X2)) = dsetconstrEL)),
% 1.68/1.49 inference(rectify,[status(thm)],[f1])).
% 1.68/1.49 thf(f1,axiom,(
% 1.68/1.49 (! [X2 : $i,X1 : $i > $o,X0 : $i] : ((in @ X2 @ (dsetconstr @ X0 @ (^[X3 : $i] : (X1 @ X3)))) => (in @ X2 @ X0)) = dsetconstrEL)),
% 1.68/1.49 file('/export/starexec/sandbox/tmp/tmp.Kt2I8leLFE/DTF2THF_1246.p',dsetconstrEL)).
% 1.68/1.49 thf(f49,plain,(
% 1.68/1.49 ( ! [X2 : $i,X3 : $i,X0 : $i,X1 : $i] : (($true != (in @ (kpair @ X3 @ X1) @ (cartprod @ X0 @ X2))) | ($true = (in @ X1 @ X2))) )),
% 1.68/1.49 inference(trivial_inequality_removal,[status(thm)],[f44])).
% 1.68/1.49 thf(f44,plain,(
% 1.68/1.49 ( ! [X2 : $i,X3 : $i,X0 : $i,X1 : $i] : (($true != (in @ (kpair @ X3 @ X1) @ (cartprod @ X0 @ X2))) | ($true = (in @ X1 @ X2)) | ($true != $true)) )),
% 1.68/1.49 inference(definition_unfolding,[status(thm)],[f39,f36])).
% 1.68/1.49 thf(f36,plain,(
% 1.68/1.49 (cartprodpairmemER = $true)),
% 1.68/1.49 inference(cnf_transformation,[status(thm)],[f24])).
% 1.68/1.49 thf(f39,plain,(
% 1.68/1.49 ( ! [X2 : $i,X3 : $i,X0 : $i,X1 : $i] : (($true != (in @ (kpair @ X3 @ X1) @ (cartprod @ X0 @ X2))) | ($true = (in @ X1 @ X2)) | (cartprodpairmemER != $true)) )),
% 1.68/1.49 inference(cnf_transformation,[status(thm)],[f28])).
% 1.68/1.49 thf(f28,plain,(
% 1.68/1.49 (! [X0 : $i,X1 : $i,X2 : $i,X3 : $i] : (($true != (in @ (kpair @ X3 @ X1) @ (cartprod @ X0 @ X2))) | ($true = (in @ X1 @ X2))) | (cartprodpairmemER != $true)) & ((cartprodpairmemER = $true) | (($true = (in @ (kpair @ sK11 @ sK9) @ (cartprod @ sK8 @ sK10))) & ($true != (in @ sK9 @ sK10))))),
% 1.68/1.49 inference(skolemisation,[status(esa),new_symbols(skolem,[sK8,sK9,sK10,sK11])],[f26,f27])).
% 1.68/1.49 thf(f27,plain,(
% 1.68/1.49 ? [X4 : $i,X5 : $i,X6 : $i,X7 : $i] : (((in @ (kpair @ X7 @ X5) @ (cartprod @ X4 @ X6)) = $true) & ($true != (in @ X5 @ X6))) => (($true = (in @ (kpair @ sK11 @ sK9) @ (cartprod @ sK8 @ sK10))) & ($true != (in @ sK9 @ sK10)))),
% 1.68/1.49 introduced(definition,[],[choice_axiom])).
% 1.68/1.49 thf(f26,plain,(
% 1.68/1.49 (! [X0 : $i,X1 : $i,X2 : $i,X3 : $i] : (($true != (in @ (kpair @ X3 @ X1) @ (cartprod @ X0 @ X2))) | ($true = (in @ X1 @ X2))) | (cartprodpairmemER != $true)) & ((cartprodpairmemER = $true) | ? [X4 : $i,X5 : $i,X6 : $i,X7 : $i] : (((in @ (kpair @ X7 @ X5) @ (cartprod @ X4 @ X6)) = $true) & ($true != (in @ X5 @ X6))))),
% 1.68/1.49 inference(rectify,[status(thm)],[f25])).
% 1.68/1.49 thf(f25,plain,(
% 1.68/1.49 (! [X0 : $i,X1 : $i,X2 : $i,X3 : $i] : (($true != (in @ (kpair @ X3 @ X1) @ (cartprod @ X0 @ X2))) | ($true = (in @ X1 @ X2))) | (cartprodpairmemER != $true)) & ((cartprodpairmemER = $true) | ? [X0 : $i,X1 : $i,X2 : $i,X3 : $i] : (($true = (in @ (kpair @ X3 @ X1) @ (cartprod @ X0 @ X2))) & ($true != (in @ X1 @ X2))))),
% 1.68/1.49 inference(nnf_transformation,[status(thm)],[f15])).
% 1.68/1.49 thf(f15,plain,(
% 1.68/1.49 ! [X0 : $i,X1 : $i,X2 : $i,X3 : $i] : (($true != (in @ (kpair @ X3 @ X1) @ (cartprod @ X0 @ X2))) | ($true = (in @ X1 @ X2))) <=> (cartprodpairmemER = $true)),
% 1.68/1.49 inference(ennf_transformation,[status(thm)],[f8])).
% 1.68/1.49 thf(f8,plain,(
% 1.68/1.49 ! [X0 : $i,X1 : $i,X2 : $i,X3 : $i] : (($true = (in @ (kpair @ X3 @ X1) @ (cartprod @ X0 @ X2))) => ($true = (in @ X1 @ X2))) <=> (cartprodpairmemER = $true)),
% 1.68/1.49 inference(fool_elimination,[status(thm)],[f7])).
% 1.68/1.49 thf(f7,plain,(
% 1.68/1.49 (! [X0 : $i,X1 : $i,X2 : $i,X3 : $i] : ((in @ (kpair @ X3 @ X1) @ (cartprod @ X0 @ X2)) => (in @ X1 @ X2)) = cartprodpairmemER)),
% 1.68/1.49 inference(rectify,[status(thm)],[f2])).
% 1.68/1.49 thf(f2,axiom,(
% 1.68/1.49 (! [X0 : $i,X3 : $i,X4 : $i,X2 : $i] : ((in @ (kpair @ X2 @ X3) @ (cartprod @ X0 @ X4)) => (in @ X3 @ X4)) = cartprodpairmemER)),
% 1.68/1.49 file('/export/starexec/sandbox/tmp/tmp.Kt2I8leLFE/DTF2THF_1246.p',cartprodpairmemER)).
% 1.68/1.49 % SZS output end Proof for DTF2THF_1246
% 1.68/1.49 % (1402)------------------------------
% 1.68/1.49 % (1402)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.68/1.49 % (1402)Termination reason: Refutation
% 1.68/1.49
% 1.68/1.49 % (1402)Memory used [KB]: 5500
% 1.68/1.49 % (1402)Time elapsed: 0.007 s
% 1.68/1.49 % (1402)Instructions burned: 6 (million)
% 1.68/1.49 % (1402)------------------------------
% 1.68/1.49 % (1402)------------------------------
% 1.68/1.49 % (1395)Success in time 0.021 s
%------------------------------------------------------------------------------