%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU629^2 : 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 : n013.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 09:35:45 AM UTC 2026
% Result : Theorem 1.39s 1.29s
% Output : Refutation 1.39s
% Verified :
% SZS Type : Refutation
% Derivation depth : 21
% Number of leaves : 7
% Syntax : Number of formulae : 46 ( 23 unt; 0 typ; 2 def)
% Number of atoms : 374 ( 62 equ; 0 cnn)
% Maximal formula atoms : 6 ( 8 avg)
% Number of connectives : 676 ( 21 ~; 15 |; 12 &; 562 @)
% ( 2 <=>; 39 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 4 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 17 ( 13 usr; 11 con; 0-2 aty)
% ( 25 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 98 ( 37 ^; 51 !; 10 ?; 98 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_2,type,
setadjoin: $i > $i > $i ).
thf(func_def_3,type,
powerset: $i > $i ).
thf(func_def_4,type,
binunion: $i > $i > $i ).
thf(func_def_5,type,
kpair: $i > $i > $i ).
thf(f57,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f44,f50,f56]) ).
thf(f56,plain,
~ spl4_1,
inference(avatar_contradiction_clause,[status(thm)],[f55]) ).
thf(f55,plain,
( $false
| ~ spl4_1 ),
inference(trivial_inequality_removal,[status(thm)],[f51]) ).
thf(f51,plain,
( ( $true = $false )
| ~ spl4_1 ),
inference(superposition,[status(thm)],[f39,f15]) ).
thf(f15,plain,
( ( in @ sK3 @ sK1 )
= $true ),
inference(cnf_transformation,[status(thm)],[f14]) ).
thf(f14,plain,
( ( ubforcartprodlem2 = $true )
& ( $true
= ( in @ sK0 @ sK2 ) )
& ( $true
!= ( in @ ( kpair @ sK0 @ sK3 ) @ ( powerset @ ( powerset @ ( binunion @ sK2 @ sK1 ) ) ) ) )
& ( ( in @ sK3 @ sK1 )
= $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2,sK3])],[f11,f13,f12]) ).
thf(f12,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ X2 )
= $true )
& ? [X3: $i] :
( ( $true
!= ( in @ ( kpair @ X0 @ X3 ) @ ( powerset @ ( powerset @ ( binunion @ X2 @ X1 ) ) ) ) )
& ( ( in @ X3 @ X1 )
= $true ) ) )
=> ( ( $true
= ( in @ sK0 @ sK2 ) )
& ? [X3: $i] :
( ( ( in @ ( kpair @ sK0 @ X3 ) @ ( powerset @ ( powerset @ ( binunion @ sK2 @ sK1 ) ) ) )
!= $true )
& ( ( in @ X3 @ sK1 )
= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f13,plain,
( ? [X3: $i] :
( ( ( in @ ( kpair @ sK0 @ X3 ) @ ( powerset @ ( powerset @ ( binunion @ sK2 @ sK1 ) ) ) )
!= $true )
& ( ( in @ X3 @ sK1 )
= $true ) )
=> ( ( $true
!= ( in @ ( kpair @ sK0 @ sK3 ) @ ( powerset @ ( powerset @ ( binunion @ sK2 @ sK1 ) ) ) ) )
& ( ( in @ sK3 @ sK1 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f11,plain,
( ( ubforcartprodlem2 = $true )
& ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ X2 )
= $true )
& ? [X3: $i] :
( ( $true
!= ( in @ ( kpair @ X0 @ X3 ) @ ( powerset @ ( powerset @ ( binunion @ X2 @ X1 ) ) ) ) )
& ( ( in @ X3 @ X1 )
= $true ) ) ) ),
inference(ennf_transformation,[status(thm)],[f10]) ).
thf(f10,plain,
~ ( ( ubforcartprodlem2 = $true )
=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ X2 )
= $true )
=> ! [X3: $i] :
( ( ( in @ X3 @ X1 )
= $true )
=> ( $true
= ( in @ ( kpair @ X0 @ X3 ) @ ( powerset @ ( powerset @ ( binunion @ X2 @ X1 ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f9]) ).
thf(f9,plain,
~ ( ubforcartprodlem2
=> ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X0 @ X2 )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( kpair @ X0 @ X3 ) @ ( powerset @ ( powerset @ ( binunion @ X2 @ X1 ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f4]) ).
thf(f4,negated_conjecture,
~ ( ubforcartprodlem2
=> ! [X0: $i,X3: $i,X2: $i] :
( ( in @ X0 @ X2 )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( in @ ( kpair @ X0 @ X1 ) @ ( powerset @ ( powerset @ ( binunion @ X2 @ X3 ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f3]) ).
thf(f3,conjecture,
( ubforcartprodlem2
=> ! [X0: $i,X3: $i,X2: $i] :
( ( in @ X0 @ X2 )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( in @ ( kpair @ X0 @ X1 ) @ ( powerset @ ( powerset @ ( binunion @ X2 @ X3 ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.FnxS89P0Mq/DTF2THF_24478.p',ubforcartprodlem3) ).
thf(f39,plain,
( ( ( in @ sK3 @ sK1 )
= $false )
| ~ spl4_1 ),
inference(avatar_component_clause,[status(thm)],[f37]) ).
thf(f37,definition,
( spl4_1
<=> ( ( in @ sK3 @ sK1 )
= $false ) ),
introduced(definition,[new_symbols(naming,[spl4_1])],[avatar_definition]) ).
thf(f50,plain,
~ spl4_2,
inference(avatar_contradiction_clause,[status(thm)],[f49]) ).
thf(f49,plain,
( $false
| ~ spl4_2 ),
inference(trivial_inequality_removal,[status(thm)],[f46]) ).
thf(f46,plain,
( ( $true = $false )
| ~ spl4_2 ),
inference(superposition,[status(thm)],[f17,f43]) ).
thf(f43,plain,
( ( ( in @ sK0 @ sK2 )
= $false )
| ~ spl4_2 ),
inference(avatar_component_clause,[status(thm)],[f41]) ).
thf(f41,definition,
( spl4_2
<=> ( ( in @ sK0 @ sK2 )
= $false ) ),
introduced(definition,[new_symbols(naming,[spl4_2])],[avatar_definition]) ).
thf(f17,plain,
( $true
= ( in @ sK0 @ sK2 ) ),
inference(cnf_transformation,[status(thm)],[f14]) ).
thf(f44,plain,
( spl4_1
| spl4_2 ),
inference(avatar_split_clause,[status(thm)],[f35,f41,f37]) ).
thf(f35,plain,
( ( ( in @ sK3 @ sK1 )
= $false )
| ( ( in @ sK0 @ sK2 )
= $false ) ),
inference(trivial_inequality_removal,[status(thm)],[f34]) ).
thf(f34,plain,
( ( $true != $true )
| ( ( in @ sK0 @ sK2 )
= $false )
| ( ( in @ sK3 @ sK1 )
= $false ) ),
inference(superposition,[status(thm)],[f23,f33]) ).
thf(f33,plain,
! [X2: $i,X3: $i,X1: $i,X4: $i] :
( ( ( in @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X2 @ X3 ) ) ) )
= $true )
| ( ( in @ X4 @ X3 )
= $false )
| ( $false
= ( in @ X1 @ X2 ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f32]) ).
thf(f32,plain,
! [X2: $i,X3: $i,X1: $i,X4: $i] :
( ( $true
= ( ( in @ X4 @ X3 )
=> ( in @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X2 @ X3 ) ) ) ) ) )
| ( $false
= ( in @ X1 @ X2 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f31]) ).
thf(f31,plain,
! [X2: $i,X3: $i,X1: $i,X4: $i] :
( ( $true
= ( ^ [Y0: $i] :
( ( in @ Y0 @ X3 )
=> ( in @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ Y0 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X2 @ X3 ) ) ) ) )
@ X4 ) )
| ( $false
= ( in @ X1 @ X2 ) ) ),
inference(pi_clausification,[status(thm)],[f30]) ).
thf(f30,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ X3 )
=> ( in @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ Y0 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X2 @ X3 ) ) ) ) ) )
= $true )
| ( $false
= ( in @ X1 @ X2 ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f29]) ).
thf(f29,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( ( in @ X1 @ X2 )
=> ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ X3 )
=> ( in @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ Y0 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X2 @ X3 ) ) ) ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f28]) ).
thf(f28,plain,
! [X2: $i,X3: $i,X1: $i] :
( $true
= ( ^ [Y0: $i] :
( ( in @ X1 @ X2 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ Y0 )
=> ( in @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ Y1 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X2 @ Y0 ) ) ) ) ) ) )
@ X3 ) ),
inference(pi_clausification,[status(thm)],[f27]) ).
thf(f27,plain,
! [X2: $i,X1: $i] :
( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ X1 @ X2 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ Y0 )
=> ( in @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ Y1 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X2 @ Y0 ) ) ) ) ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f26]) ).
thf(f26,plain,
! [X2: $i,X1: $i] :
( ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( in @ X1 @ Y0 )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y1 )
=> ( in @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ Y2 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ Y0 @ Y1 ) ) ) ) ) ) ) )
@ X2 )
= $true ),
inference(pi_clausification,[status(thm)],[f25]) ).
thf(f25,plain,
! [X1: $i] :
( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( in @ X1 @ Y0 )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y1 )
=> ( in @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ Y2 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ Y0 @ Y1 ) ) ) ) ) ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f24]) ).
thf(f24,plain,
! [X1: $i] :
( $true
= ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y0 @ Y1 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y2 )
=> ( in @ ( setadjoin @ ( setadjoin @ Y0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y0 @ ( setadjoin @ Y3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ Y1 @ Y2 ) ) ) ) ) ) ) ) )
@ X1 ) ),
inference(pi_clausification,[status(thm)],[f22]) ).
thf(f22,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y0 @ Y1 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y2 )
=> ( in @ ( setadjoin @ ( setadjoin @ Y0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y0 @ ( setadjoin @ Y3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ Y1 @ Y2 ) ) ) ) ) ) ) ) ) )
= $true ),
inference(definition_unfolding,[status(thm)],[f20,f18]) ).
thf(f18,plain,
ubforcartprodlem2 = $true,
inference(cnf_transformation,[status(thm)],[f14]) ).
thf(f20,plain,
( ubforcartprodlem2
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y0 @ Y1 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y2 )
=> ( in @ ( setadjoin @ ( setadjoin @ Y0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y0 @ ( setadjoin @ Y3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ Y1 @ Y2 ) ) ) ) ) ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f7]) ).
thf(f7,plain,
( ubforcartprodlem2
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y0 @ Y1 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y2 )
=> ( in @ ( setadjoin @ ( setadjoin @ Y0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y0 @ ( setadjoin @ Y3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ Y1 @ Y2 ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f6]) ).
thf(f6,plain,
! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X1 )
=> ! [X3: $i] :
( ( in @ X3 @ X0 )
=> ( in @ ( setadjoin @ ( setadjoin @ X2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X0 ) ) ) ) ) )
= ubforcartprodlem2 ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
! [X3: $i,X2: $i,X0: $i] :
( ( ( in @ X0 @ X2 )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( in @ ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X2 @ X3 ) ) ) ) ) )
= ubforcartprodlem2 ),
file('/export/starexec/sandbox/tmp/tmp.FnxS89P0Mq/DTF2THF_24478.p',ubforcartprodlem2) ).
thf(f23,plain,
( $true
!= ( in @ ( setadjoin @ ( setadjoin @ sK0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK0 @ ( setadjoin @ sK3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ sK2 @ sK1 ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f21]) ).
thf(f21,plain,
( ( in
@ ( ^ [Y0: $i,Y1: $i] : ( setadjoin @ ( setadjoin @ Y0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y0 @ ( setadjoin @ Y1 @ emptyset ) ) @ emptyset ) )
@ sK0
@ sK3 )
@ ( powerset @ ( powerset @ ( binunion @ sK2 @ sK1 ) ) ) )
!= $true ),
inference(definition_unfolding,[status(thm)],[f16,f19]) ).
thf(f19,plain,
( kpair
= ( ^ [Y0: $i,Y1: $i] : ( setadjoin @ ( setadjoin @ Y0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y0 @ ( setadjoin @ Y1 @ emptyset ) ) @ emptyset ) ) ) ),
inference(cnf_transformation,[status(thm)],[f8]) ).
thf(f8,plain,
( kpair
= ( ^ [Y0: $i,Y1: $i] : ( setadjoin @ ( setadjoin @ Y0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y0 @ ( setadjoin @ Y1 @ emptyset ) ) @ emptyset ) ) ) ),
inference(fool_elimination,[status(thm)],[f1]) ).
thf(f1,axiom,
( ( ^ [X0: $i,X1: $i] : ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) @ emptyset ) ) )
= kpair ),
file('/export/starexec/sandbox/tmp/tmp.FnxS89P0Mq/DTF2THF_24478.p',kpair) ).
thf(f16,plain,
( $true
!= ( in @ ( kpair @ sK0 @ sK3 ) @ ( powerset @ ( powerset @ ( binunion @ sK2 @ sK1 ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f14]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.08 % Problem : SEU629^2 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.09 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.11/0.29 % Computer : n013.cluster.edu
% 0.11/0.29 % Model : x86_64 x86_64
% 0.11/0.29 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.29 % Memory : 8042.1875MB
% 0.11/0.29 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.29 % CPULimit : 300
% 0.11/0.29 % WCLimit : 300
% 0.11/0.29 % DateTime : Tue Feb 24 12:28:53 EST 2026
% 0.11/0.29 % CPUTime :
% 0.11/0.29 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.19/0.38 ---- Original DTF file ---
% 0.19/0.38 thf(spec,logic,$$dhol).
% 0.19/0.38 %------------------------------------------------------------------------------
% 0.19/0.38 % File : SEU629^2 : TPTP v9.2.1. Released v3.7.0.
% 0.19/0.38 % Domain : Set Theory
% 0.19/0.38 % Problem : Ordered Pairs - Cartesian Products
% 0.19/0.38 % Version : Especial > Reduced > Especial.
% 0.19/0.38 % English : (! A:i.! B:i.! x:i.in x A -> (! y:i.in y B -> in (kpair x y)
% 0.19/0.38 % (powerset (powerset (binunion A B)))))
% 0.19/0.38
% 0.19/0.38 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.19/0.38 % Source : [Bro08]
% 0.19/0.38 % Names : ZFC131l [Bro08]
% 0.19/0.38
% 0.19/0.38 % Status : Theorem
% 0.19/0.38 % Rating : 0.00 v8.2.0, 0.08 v8.1.0, 0.00 v6.0.0, 0.14 v5.5.0, 0.17 v5.4.0, 0.20 v5.3.0, 0.40 v5.2.0, 0.20 v4.1.0, 0.00 v3.7.0
% 0.19/0.38 % Syntax : Number of formulae : 10 ( 2 unt; 7 typ; 2 def)
% 0.19/0.38 % Number of atoms : 10 ( 2 equ; 0 cnn)
% 0.19/0.38 % Maximal formula atoms : 4 ( 3 avg)
% 0.19/0.38 % Number of connectives : 47 ( 0 ~; 0 |; 0 &; 42 @)
% 0.19/0.38 % ( 0 <=>; 5 =>; 0 <=; 0 <~>)
% 0.19/0.38 % Maximal formula depth : 13 ( 5 avg)
% 0.19/0.38 % Number of types : 2 ( 0 usr)
% 0.19/0.38 % Number of type conns : 9 ( 9 >; 0 *; 0 +; 0 <<)
% 0.19/0.38 % Number of symbols : 8 ( 7 usr; 2 con; 0-2 aty)
% 0.19/0.38 % Number of variables : 10 ( 2 ^; 8 !; 0 ?; 10 :)
% 0.19/0.38 % SPC : TH0_THM_EQU_NAR
% 0.19/0.38
% 0.19/0.38 % Comments : http://mathgate.info/detsetitem.php?id=186
% 0.19/0.38 %------------------------------------------------------------------------------
% 0.19/0.38 thf(in_type,type,
% 0.19/0.38 in: $i > $i > $o ).
% 0.19/0.38
% 0.19/0.38 thf(emptyset_type,type,
% 0.19/0.38 emptyset: $i ).
% 0.19/0.38
% 0.19/0.38 thf(setadjoin_type,type,
% 0.19/0.38 setadjoin: $i > $i > $i ).
% 0.19/0.38
% 0.19/0.38 thf(powerset_type,type,
% 0.19/0.38 powerset: $i > $i ).
% 0.19/0.38
% 0.19/0.38 thf(binunion_type,type,
% 0.19/0.38 binunion: $i > $i > $i ).
% 0.19/0.38
% 0.19/0.38 thf(kpair_type,type,
% 0.19/0.38 kpair: $i > $i > $i ).
% 0.19/0.38
% 0.19/0.38 thf(kpair,definition,
% 0.19/0.38 ( kpair
% 0.19/0.38 = ( ^ [Xx: $i,Xy: $i] : ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(ubforcartprodlem2_type,type,
% 0.19/0.38 ubforcartprodlem2: $o ).
% 0.19/0.38
% 0.19/0.38 thf(ubforcartprodlem2,definition,
% 0.19/0.38 ( ubforcartprodlem2
% 0.19/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 => ! [Xy: $i] :
% 0.19/0.38 ( ( in @ Xy @ B )
% 0.19/0.38 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(ubforcartprodlem3,conjecture,
% 0.19/0.38 ( ubforcartprodlem2
% 0.19/0.38 => ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 => ! [Xy: $i] :
% 0.19/0.38 ( ( in @ Xy @ B )
% 0.19/0.38 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 %------------------------------------------------------------------------------
% 0.19/0.38 ------------------------
% 1.39/1.26 ---- Embedded in THF ---
% 1.39/1.26 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.39/1.26 %%% Generated on Tue Feb 24 12:28:54 EST 2026
% 1.39/1.26 %%% using '$$dhol' embedding, version 1.3.0.
% 1.39/1.26 %%% Logic specification used:
% 1.39/1.26 %%% thf(spec, logic, $$dhol).
% 1.39/1.26
% 1.39/1.26 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.FnxS89P0Mq/DTF2DTF24478.p
% See solution above
% 1.39/1.26 ------------------------
% 1.39/1.27 ---- Cleaned THF ---
% 1.39/1.27 thf(in_type,type,
% 1.39/1.27 in: $i > $i > $o ).
% 1.39/1.27
% 1.39/1.27 thf(emptyset_type,type,
% 1.39/1.27 emptyset: $i ).
% 1.39/1.27
% 1.39/1.27 thf(setadjoin_type,type,
% 1.39/1.27 setadjoin: $i > $i > $i ).
% 1.39/1.27
% 1.39/1.27 thf(powerset_type,type,
% 1.39/1.27 powerset: $i > $i ).
% 1.39/1.27
% 1.39/1.27 thf(binunion_type,type,
% 1.39/1.27 binunion: $i > $i > $i ).
% 1.39/1.27
% 1.39/1.27 thf(kpair_type,type,
% 1.39/1.27 kpair: $i > $i > $i ).
% 1.39/1.27
% 1.39/1.27 thf(kpair,definition,
% 1.39/1.27 ( kpair
% 1.39/1.27 = ( ^ [Xx: $i,Xy: $i] : ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ).
% 1.39/1.27
% 1.39/1.27 thf(ubforcartprodlem2_type,type,
% 1.39/1.27 ubforcartprodlem2: $o ).
% 1.39/1.27
% 1.39/1.27 thf(ubforcartprodlem2,definition,
% 1.39/1.27 ( ubforcartprodlem2
% 1.39/1.27 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.39/1.27 ( ( in @ Xx @ A )
% 1.39/1.27 => ! [Xy: $i] :
% 1.39/1.27 ( ( in @ Xy @ B )
% 1.39/1.27 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 1.39/1.27
% 1.39/1.27 thf(ubforcartprodlem3,conjecture,
% 1.39/1.27 ( ubforcartprodlem2
% 1.39/1.27 => ! [A: $i,B: $i,Xx: $i] :
% 1.39/1.27 ( ( in @ Xx @ A )
% 1.39/1.27 => ! [Xy: $i] :
% 1.39/1.27 ( ( in @ Xy @ B )
% 1.39/1.27 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 1.39/1.27 ------------------------
% 1.39/1.29 % (24575)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_24478 for (2999ds/183Mi)
% 1.39/1.29 % (24577)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_24478 for (2999ds/27Mi)
% 1.39/1.29 % (24580)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_24478 for (2999ds/275Mi)
% 1.39/1.29 % (24579)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_24478 for (2999ds/2Mi)
% 1.39/1.29 % (24578)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_24478 for (2999ds/2Mi)
% 1.39/1.29 % (24576)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_24478 for (2999ds/4Mi)
% 1.39/1.29 % (24581)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_24478 for (2999ds/18Mi)
% 1.39/1.29 % (24578)Instruction limit reached!
% 1.39/1.29 % (24578)------------------------------
% 1.39/1.29 % (24578)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.39/1.29 % (24578)Termination reason: Unknown
% 1.39/1.29 % (24578)Termination phase: Saturation
% 1.39/1.29 % (24579)Instruction limit reached!
% 1.39/1.29 % (24579)------------------------------
% 1.39/1.29 % (24579)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.39/1.29 % (24579)Termination reason: Unknown
% 1.39/1.29 % (24579)Termination phase: Saturation
% 1.39/1.29
% 1.39/1.29 % (24579)Memory used [KB]: 1023
% 1.39/1.29 % (24579)Time elapsed: 0.003 s
% 1.39/1.29 % (24579)Instructions burned: 3 (million)
% 1.39/1.29 % (24579)------------------------------
% 1.39/1.29 % (24579)------------------------------
% 1.39/1.29
% 1.39/1.29 % (24578)Memory used [KB]: 1023
% 1.39/1.29 % (24578)Time elapsed: 0.003 s
% 1.39/1.29 % (24578)Instructions burned: 3 (million)
% 1.39/1.29 % (24578)------------------------------
% 1.39/1.29 % (24578)------------------------------
% 1.39/1.29 % (24576)Instruction limit reached!
% 1.39/1.29 % (24576)------------------------------
% 1.39/1.29 % (24576)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.39/1.29 % (24576)Termination reason: Unknown
% 1.39/1.29 % (24576)Termination phase: Saturation
% 1.39/1.29
% 1.39/1.29 % (24576)Memory used [KB]: 5500
% 1.39/1.29 % (24576)Time elapsed: 0.004 s
% 1.39/1.29 % (24576)Instructions burned: 4 (million)
% 1.39/1.29 % (24576)------------------------------
% 1.39/1.29 % (24576)------------------------------
% 1.39/1.29 % (24580)Refutation not found, incomplete strategy
% 1.39/1.29 % (24580)------------------------------
% 1.39/1.29 % (24580)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.39/1.29 % (24580)Termination reason: Refutation not found, incomplete strategy
% 1.39/1.29
% 1.39/1.29
% 1.39/1.29 % (24580)Memory used [KB]: 5500
% 1.39/1.29 % (24580)Time elapsed: 0.004 s
% 1.39/1.29 % (24580)Instructions burned: 4 (million)
% 1.39/1.29 % (24580)------------------------------
% 1.39/1.29 % (24580)------------------------------
% 1.39/1.29 % (24575)First to succeed.
% 1.39/1.29 % (24577)Also succeeded, but the first one will report.
% 1.39/1.29 % (24575)Refutation found. Thanks to Tanya!
% 1.39/1.29 % SZS status Theorem for DTF2THF_24478
% 1.39/1.29 % SZS output start Proof for DTF2THF_24478
% 1.39/1.29 thf(func_def_0, type, in: $i > $i > $o).
% 1.39/1.29 thf(func_def_2, type, setadjoin: $i > $i > $i).
% 1.39/1.29 thf(func_def_3, type, powerset: $i > $i).
% 1.39/1.29 thf(func_def_4, type, binunion: $i > $i > $i).
% 1.39/1.29 thf(func_def_5, type, kpair: $i > $i > $i).
% 1.39/1.29 thf(f57,plain,(
% 1.39/1.29 $false),
% 1.39/1.29 inference(avatar_sat_refutation,[status(thm)],[f44,f50,f56])).
% 1.39/1.29 thf(f56,plain,(
% 1.39/1.29 ~spl4_1),
% 1.39/1.29 inference(avatar_contradiction_clause,[status(thm)],[f55])).
% 1.39/1.29 thf(f55,plain,(
% 1.39/1.29 $false | ~spl4_1),
% 1.39/1.29 inference(trivial_inequality_removal,[status(thm)],[f51])).
% 1.39/1.29 thf(f51,plain,(
% 1.39/1.29 ($true = $false) | ~spl4_1),
% 1.39/1.29 inference(superposition,[status(thm)],[f39,f15])).
% 1.39/1.29 thf(f15,plain,(
% 1.39/1.29 ((in @ sK3 @ sK1) = $true)),
% 1.39/1.29 inference(cnf_transformation,[status(thm)],[f14])).
% 1.39/1.29 thf(f14,plain,(
% 1.39/1.29 (ubforcartprodlem2 = $true) & (($true = (in @ sK0 @ sK2)) & (($true != (in @ (kpair @ sK0 @ sK3) @ (powerset @ (powerset @ (binunion @ sK2 @ sK1))))) & ((in @ sK3 @ sK1) = $true)))),
% 1.39/1.29 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2,sK3])],[f11,f13,f12])).
% 1.39/1.29 thf(f12,plain,(
% 1.39/1.29 ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ X2) = $true) & ? [X3 : $i] : (($true != (in @ (kpair @ X0 @ X3) @ (powerset @ (powerset @ (binunion @ X2 @ X1))))) & ((in @ X3 @ X1) = $true))) => (($true = (in @ sK0 @ sK2)) & ? [X3 : $i] : (((in @ (kpair @ sK0 @ X3) @ (powerset @ (powerset @ (binunion @ sK2 @ sK1)))) != $true) & ((in @ X3 @ sK1) = $true)))),
% 1.39/1.29 introduced(definition,[],[choice_axiom])).
% 1.39/1.29 thf(f13,plain,(
% 1.39/1.29 ? [X3 : $i] : (((in @ (kpair @ sK0 @ X3) @ (powerset @ (powerset @ (binunion @ sK2 @ sK1)))) != $true) & ((in @ X3 @ sK1) = $true)) => (($true != (in @ (kpair @ sK0 @ sK3) @ (powerset @ (powerset @ (binunion @ sK2 @ sK1))))) & ((in @ sK3 @ sK1) = $true))),
% 1.39/1.29 introduced(definition,[],[choice_axiom])).
% 1.39/1.29 thf(f11,plain,(
% 1.39/1.29 (ubforcartprodlem2 = $true) & ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ X2) = $true) & ? [X3 : $i] : (($true != (in @ (kpair @ X0 @ X3) @ (powerset @ (powerset @ (binunion @ X2 @ X1))))) & ((in @ X3 @ X1) = $true)))),
% 1.39/1.29 inference(ennf_transformation,[status(thm)],[f10])).
% 1.39/1.29 thf(f10,plain,(
% 1.39/1.29 ~((ubforcartprodlem2 = $true) => ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ X2) = $true) => ! [X3 : $i] : (((in @ X3 @ X1) = $true) => ($true = (in @ (kpair @ X0 @ X3) @ (powerset @ (powerset @ (binunion @ X2 @ X1))))))))),
% 1.39/1.29 inference(fool_elimination,[status(thm)],[f9])).
% 1.39/1.29 thf(f9,plain,(
% 1.39/1.29 ~(ubforcartprodlem2 => ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X0 @ X2) => ! [X3 : $i] : ((in @ X3 @ X1) => (in @ (kpair @ X0 @ X3) @ (powerset @ (powerset @ (binunion @ X2 @ X1)))))))),
% 1.39/1.29 inference(rectify,[status(thm)],[f4])).
% 1.39/1.29 thf(f4,negated_conjecture,(
% 1.39/1.29 ~(ubforcartprodlem2 => ! [X0 : $i,X3 : $i,X2 : $i] : ((in @ X0 @ X2) => ! [X1 : $i] : ((in @ X1 @ X3) => (in @ (kpair @ X0 @ X1) @ (powerset @ (powerset @ (binunion @ X2 @ X3)))))))),
% 1.39/1.29 inference(negated_conjecture,[status(cth)],[f3])).
% 1.39/1.29 thf(f3,conjecture,(
% 1.39/1.29 ubforcartprodlem2 => ! [X0 : $i,X3 : $i,X2 : $i] : ((in @ X0 @ X2) => ! [X1 : $i] : ((in @ X1 @ X3) => (in @ (kpair @ X0 @ X1) @ (powerset @ (powerset @ (binunion @ X2 @ X3))))))),
% 1.39/1.29 file('/export/starexec/sandbox/tmp/tmp.FnxS89P0Mq/DTF2THF_24478.p',ubforcartprodlem3)).
% 1.39/1.29 thf(f39,plain,(
% 1.39/1.29 ((in @ sK3 @ sK1) = $false) | ~spl4_1),
% 1.39/1.29 inference(avatar_component_clause,[status(thm)],[f37])).
% 1.39/1.29 thf(f37,definition,(
% 1.39/1.29 spl4_1 <=> ((in @ sK3 @ sK1) = $false)),
% 1.39/1.29 introduced(definition,[new_symbols(naming,[spl4_1])],[avatar_definition])).
% 1.39/1.29 thf(f50,plain,(
% 1.39/1.29 ~spl4_2),
% 1.39/1.29 inference(avatar_contradiction_clause,[status(thm)],[f49])).
% 1.39/1.29 thf(f49,plain,(
% 1.39/1.29 $false | ~spl4_2),
% 1.39/1.29 inference(trivial_inequality_removal,[status(thm)],[f46])).
% 1.39/1.29 thf(f46,plain,(
% 1.39/1.29 ($true = $false) | ~spl4_2),
% 1.39/1.29 inference(superposition,[status(thm)],[f17,f43])).
% 1.39/1.29 thf(f43,plain,(
% 1.39/1.29 ((in @ sK0 @ sK2) = $false) | ~spl4_2),
% 1.39/1.29 inference(avatar_component_clause,[status(thm)],[f41])).
% 1.39/1.29 thf(f41,definition,(
% 1.39/1.29 spl4_2 <=> ((in @ sK0 @ sK2) = $false)),
% 1.39/1.29 introduced(definition,[new_symbols(naming,[spl4_2])],[avatar_definition])).
% 1.39/1.29 thf(f17,plain,(
% 1.39/1.29 ($true = (in @ sK0 @ sK2))),
% 1.39/1.29 inference(cnf_transformation,[status(thm)],[f14])).
% 1.39/1.29 thf(f44,plain,(
% 1.39/1.29 spl4_1 | spl4_2),
% 1.39/1.29 inference(avatar_split_clause,[status(thm)],[f35,f41,f37])).
% 1.39/1.29 thf(f35,plain,(
% 1.39/1.29 ((in @ sK3 @ sK1) = $false) | ((in @ sK0 @ sK2) = $false)),
% 1.39/1.29 inference(trivial_inequality_removal,[status(thm)],[f34])).
% 1.39/1.29 thf(f34,plain,(
% 1.39/1.29 ($true != $true) | ((in @ sK0 @ sK2) = $false) | ((in @ sK3 @ sK1) = $false)),
% 1.39/1.29 inference(superposition,[status(thm)],[f23,f33])).
% 1.39/1.29 thf(f33,plain,(
% 1.39/1.29 ( ! [X2 : $i,X3 : $i,X1 : $i,X4 : $i] : (((in @ (setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ X4 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ X2 @ X3)))) = $true) | ((in @ X4 @ X3) = $false) | ($false = (in @ X1 @ X2))) )),
% 1.39/1.29 inference(binary_proxy_clausification,[status(thm)],[f32])).
% 1.39/1.29 thf(f32,plain,(
% 1.39/1.29 ( ! [X2 : $i,X3 : $i,X1 : $i,X4 : $i] : (($true = ((in @ X4 @ X3) => (in @ (setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ X4 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ X2 @ X3)))))) | ($false = (in @ X1 @ X2))) )),
% 1.39/1.29 inference(beta_eta_normalization,[status(thm)],[f31])).
% 1.39/1.29 thf(f31,plain,(
% 1.39/1.29 ( ! [X2 : $i,X3 : $i,X1 : $i,X4 : $i] : (($true = ((^[Y0 : $i]: ((in @ Y0 @ X3) => (in @ (setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ Y0 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ X2 @ X3)))))) @ X4)) | ($false = (in @ X1 @ X2))) )),
% 1.39/1.29 inference(pi_clausification,[status(thm)],[f30])).
% 1.39/1.29 thf(f30,plain,(
% 1.39/1.29 ( ! [X2 : $i,X3 : $i,X1 : $i] : (((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ X3) => (in @ (setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ Y0 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ X2 @ X3))))))) = $true) | ($false = (in @ X1 @ X2))) )),
% 1.39/1.29 inference(binary_proxy_clausification,[status(thm)],[f29])).
% 1.39/1.29 thf(f29,plain,(
% 1.39/1.29 ( ! [X2 : $i,X3 : $i,X1 : $i] : ((((in @ X1 @ X2) => (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ X3) => (in @ (setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ Y0 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ X2 @ X3)))))))) = $true)) )),
% 1.39/1.29 inference(beta_eta_normalization,[status(thm)],[f28])).
% 1.39/1.29 thf(f28,plain,(
% 1.39/1.29 ( ! [X2 : $i,X3 : $i,X1 : $i] : (($true = ((^[Y0 : $i]: ((in @ X1 @ X2) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ Y0) => (in @ (setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ Y1 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ X2 @ Y0))))))))) @ X3))) )),
% 1.39/1.29 inference(pi_clausification,[status(thm)],[f27])).
% 1.39/1.29 thf(f27,plain,(
% 1.39/1.29 ( ! [X2 : $i,X1 : $i] : (((!! @ $i @ (^[Y0 : $i]: ((in @ X1 @ X2) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ Y0) => (in @ (setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ Y1 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ X2 @ Y0)))))))))) = $true)) )),
% 1.39/1.29 inference(beta_eta_normalization,[status(thm)],[f26])).
% 1.39/1.29 thf(f26,plain,(
% 1.39/1.29 ( ! [X2 : $i,X1 : $i] : ((((^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: ((in @ X1 @ Y0) => (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ Y1) => (in @ (setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ Y2 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ Y0 @ Y1))))))))))) @ X2) = $true)) )),
% 1.39/1.29 inference(pi_clausification,[status(thm)],[f25])).
% 1.39/1.29 thf(f25,plain,(
% 1.39/1.29 ( ! [X1 : $i] : (((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: ((in @ X1 @ Y0) => (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ Y1) => (in @ (setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ Y2 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ Y0 @ Y1)))))))))))) = $true)) )),
% 1.39/1.29 inference(beta_eta_normalization,[status(thm)],[f24])).
% 1.39/1.29 thf(f24,plain,(
% 1.39/1.29 ( ! [X1 : $i] : (($true = ((^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y0 @ Y1) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y2) => (in @ (setadjoin @ (setadjoin @ Y0 @ emptyset) @ (setadjoin @ (setadjoin @ Y0 @ (setadjoin @ Y3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ Y1 @ Y2))))))))))))) @ X1))) )),
% 1.39/1.29 inference(pi_clausification,[status(thm)],[f22])).
% 1.39/1.29 thf(f22,plain,(
% 1.39/1.29 ((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y0 @ Y1) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y2) => (in @ (setadjoin @ (setadjoin @ Y0 @ emptyset) @ (setadjoin @ (setadjoin @ Y0 @ (setadjoin @ Y3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ Y1 @ Y2)))))))))))))) = $true)),
% 1.39/1.29 inference(definition_unfolding,[status(thm)],[f20,f18])).
% 1.39/1.29 thf(f18,plain,(
% 1.39/1.29 (ubforcartprodlem2 = $true)),
% 1.39/1.29 inference(cnf_transformation,[status(thm)],[f14])).
% 1.39/1.29 thf(f20,plain,(
% 1.39/1.29 (ubforcartprodlem2 = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y0 @ Y1) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y2) => (in @ (setadjoin @ (setadjoin @ Y0 @ emptyset) @ (setadjoin @ (setadjoin @ Y0 @ (setadjoin @ Y3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ Y1 @ Y2)))))))))))))))),
% 1.39/1.29 inference(cnf_transformation,[status(thm)],[f7])).
% 1.39/1.29 thf(f7,plain,(
% 1.39/1.29 (ubforcartprodlem2 = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y0 @ Y1) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y2) => (in @ (setadjoin @ (setadjoin @ Y0 @ emptyset) @ (setadjoin @ (setadjoin @ Y0 @ (setadjoin @ Y3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ Y1 @ Y2)))))))))))))))),
% 1.39/1.29 inference(fool_elimination,[status(thm)],[f6])).
% 1.39/1.29 thf(f6,plain,(
% 1.39/1.29 (! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X2 @ X1) => ! [X3 : $i] : ((in @ X3 @ X0) => (in @ (setadjoin @ (setadjoin @ X2 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ (setadjoin @ X3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ X1 @ X0)))))) = ubforcartprodlem2)),
% 1.39/1.29 inference(rectify,[status(thm)],[f2])).
% 1.39/1.29 thf(f2,axiom,(
% 1.39/1.29 (! [X3 : $i,X2 : $i,X0 : $i] : ((in @ X0 @ X2) => ! [X1 : $i] : ((in @ X1 @ X3) => (in @ (setadjoin @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ X2 @ X3)))))) = ubforcartprodlem2)),
% 1.39/1.29 file('/export/starexec/sandbox/tmp/tmp.FnxS89P0Mq/DTF2THF_24478.p',ubforcartprodlem2)).
% 1.39/1.29 thf(f23,plain,(
% 1.39/1.29 ($true != (in @ (setadjoin @ (setadjoin @ sK0 @ emptyset) @ (setadjoin @ (setadjoin @ sK0 @ (setadjoin @ sK3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ sK2 @ sK1)))))),
% 1.39/1.29 inference(beta_eta_normalization,[status(thm)],[f21])).
% 1.39/1.29 thf(f21,plain,(
% 1.39/1.29 ((in @ ((^[Y0 : $i]: ((^[Y1 : $i]: (setadjoin @ (setadjoin @ Y0 @ emptyset) @ (setadjoin @ (setadjoin @ Y0 @ (setadjoin @ Y1 @ emptyset)) @ emptyset))))) @ sK0 @ sK3) @ (powerset @ (powerset @ (binunion @ sK2 @ sK1)))) != $true)),
% 1.39/1.29 inference(definition_unfolding,[status(thm)],[f16,f19])).
% 1.39/1.29 thf(f19,plain,(
% 1.39/1.29 (kpair = (^[Y0 : $i]: ((^[Y1 : $i]: (setadjoin @ (setadjoin @ Y0 @ emptyset) @ (setadjoin @ (setadjoin @ Y0 @ (setadjoin @ Y1 @ emptyset)) @ emptyset))))))),
% 1.39/1.29 inference(cnf_transformation,[status(thm)],[f8])).
% 1.39/1.29 thf(f8,plain,(
% 1.39/1.29 (kpair = (^[Y0 : $i]: ((^[Y1 : $i]: (setadjoin @ (setadjoin @ Y0 @ emptyset) @ (setadjoin @ (setadjoin @ Y0 @ (setadjoin @ Y1 @ emptyset)) @ emptyset))))))),
% 1.39/1.29 inference(fool_elimination,[status(thm)],[f1])).
% 1.39/1.29 thf(f1,axiom,(
% 1.39/1.29 ((^[X0 : $i, X1 : $i] : (setadjoin @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset)) @ emptyset))) = kpair)),
% 1.39/1.29 file('/export/starexec/sandbox/tmp/tmp.FnxS89P0Mq/DTF2THF_24478.p',kpair)).
% 1.39/1.29 thf(f16,plain,(
% 1.39/1.29 ($true != (in @ (kpair @ sK0 @ sK3) @ (powerset @ (powerset @ (binunion @ sK2 @ sK1)))))),
% 1.39/1.29 inference(cnf_transformation,[status(thm)],[f14])).
% 1.39/1.29 % SZS output end Proof for DTF2THF_24478
% 1.39/1.29 % (24575)------------------------------
% 1.39/1.29 % (24575)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.39/1.29 % (24575)Termination reason: Refutation
% 1.39/1.29
% 1.39/1.29 % (24575)Memory used [KB]: 5500
% 1.39/1.29 % (24575)Time elapsed: 0.007 s
% 1.39/1.29 % (24575)Instructions burned: 6 (million)
% 1.39/1.29 % (24575)------------------------------
% 1.39/1.29 % (24575)------------------------------
% 1.39/1.29 % (24574)Success in time 0.01 s
%------------------------------------------------------------------------------