%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU720^2 : 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 : n008.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 09:36:39 AM UTC 2026
% Result : Theorem 1.60s 1.32s
% Output : Refutation 1.60s
% Verified :
% SZS Type : Refutation
% Derivation depth : 23
% Number of leaves : 4
% Syntax : Number of formulae : 33 ( 18 unt; 0 typ; 0 def)
% Number of atoms : 304 ( 69 equ; 0 cnn)
% Maximal formula atoms : 8 ( 9 avg)
% Number of connectives : 418 ( 39 ~; 8 |; 22 &; 289 @)
% ( 0 <=>; 45 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 5 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 11 ( 7 usr; 7 con; 0-2 aty)
% ( 15 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 71 ( 18 ^; 42 !; 11 ?; 71 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_1,type,
powerset: $i > $i ).
thf(func_def_2,type,
setminus: $i > $i > $i ).
thf(f36,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f35]) ).
thf(f35,plain,
$false = $true,
inference(superposition,[status(thm)],[f18,f33]) ).
thf(f33,plain,
( $false
= ( in @ sK2 @ sK0 ) ),
inference(subsumption_resolution,[status(thm)],[f32,f17]) ).
thf(f17,plain,
( $true
!= ( in @ sK2 @ sK1 ) ),
inference(cnf_transformation,[status(thm)],[f14]) ).
thf(f14,plain,
( ( ( in @ sK1 @ ( powerset @ sK0 ) )
= $true )
& ( ( in @ sK2 @ sK0 )
= $true )
& ( $true
!= ( in @ sK2 @ sK1 ) )
& ( ( in @ sK2 @ ( setminus @ sK0 @ sK1 ) )
!= $true )
& ( setminusI = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2])],[f11,f13,f12]) ).
thf(f12,plain,
( ? [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
= $true )
& ? [X2: $i] :
( ( ( in @ X2 @ X0 )
= $true )
& ( ( in @ X2 @ X1 )
!= $true )
& ( ( in @ X2 @ ( setminus @ X0 @ X1 ) )
!= $true ) ) )
=> ( ( ( in @ sK1 @ ( powerset @ sK0 ) )
= $true )
& ? [X2: $i] :
( ( ( in @ X2 @ sK0 )
= $true )
& ( $true
!= ( in @ X2 @ sK1 ) )
& ( ( in @ X2 @ ( setminus @ sK0 @ sK1 ) )
!= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f13,plain,
( ? [X2: $i] :
( ( ( in @ X2 @ sK0 )
= $true )
& ( $true
!= ( in @ X2 @ sK1 ) )
& ( ( in @ X2 @ ( setminus @ sK0 @ sK1 ) )
!= $true ) )
=> ( ( ( in @ sK2 @ sK0 )
= $true )
& ( $true
!= ( in @ sK2 @ sK1 ) )
& ( ( in @ sK2 @ ( setminus @ sK0 @ sK1 ) )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f11,plain,
( ? [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
= $true )
& ? [X2: $i] :
( ( ( in @ X2 @ X0 )
= $true )
& ( ( in @ X2 @ X1 )
!= $true )
& ( ( in @ X2 @ ( setminus @ X0 @ X1 ) )
!= $true ) ) )
& ( setminusI = $true ) ),
inference(flattening,[status(thm)],[f10]) ).
thf(f10,plain,
( ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ( ( in @ X2 @ X1 )
!= $true )
& ( ( in @ X2 @ ( setminus @ X0 @ X1 ) )
!= $true )
& ( ( in @ X2 @ X0 )
= $true ) )
& ( ( in @ X1 @ ( powerset @ X0 ) )
= $true ) )
& ( setminusI = $true ) ),
inference(ennf_transformation,[status(thm)],[f9]) ).
thf(f9,plain,
~ ( ( setminusI = $true )
=> ! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
= $true )
=> ! [X2: $i] :
( ( ( in @ X2 @ X0 )
= $true )
=> ( ( ( in @ X2 @ ( setminus @ X0 @ X1 ) )
!= $true )
=> ( ( in @ X2 @ X1 )
= $true ) ) ) ) ),
inference(flattening,[status(thm)],[f8]) ).
thf(f8,plain,
~ ( ( setminusI = $true )
=> ! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
= $true )
=> ! [X2: $i] :
( ( ( in @ X2 @ X0 )
= $true )
=> ( ( ( in @ X2 @ ( setminus @ X0 @ X1 ) )
!= $true )
=> ( ( in @ X2 @ X1 )
= $true ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f7]) ).
thf(f7,plain,
~ ( setminusI
=> ! [X0: $i,X1: $i] :
( ( in @ X1 @ ( powerset @ X0 ) )
=> ! [X2: $i] :
( ( in @ X2 @ X0 )
=> ( ~ ( in @ X2 @ ( setminus @ X0 @ X1 ) )
=> ( in @ X2 @ X1 ) ) ) ) ),
inference(rectify,[status(thm)],[f3]) ).
thf(f3,negated_conjecture,
~ ( setminusI
=> ! [X0: $i,X3: $i] :
( ( in @ X3 @ ( powerset @ X0 ) )
=> ! [X2: $i] :
( ( in @ X2 @ X0 )
=> ( ~ ( in @ X2 @ ( setminus @ X0 @ X3 ) )
=> ( in @ X2 @ X3 ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f2]) ).
thf(f2,conjecture,
( setminusI
=> ! [X0: $i,X3: $i] :
( ( in @ X3 @ ( powerset @ X0 ) )
=> ! [X2: $i] :
( ( in @ X2 @ X0 )
=> ( ~ ( in @ X2 @ ( setminus @ X0 @ X3 ) )
=> ( in @ X2 @ X3 ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.Ssivfr5AOV/DTF2THF_9084.p',complementTE1) ).
thf(f32,plain,
( ( $false
= ( in @ sK2 @ sK0 ) )
| ( $true
= ( in @ sK2 @ sK1 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f31]) ).
thf(f31,plain,
( ( $false
= ( in @ sK2 @ sK0 ) )
| ( $true
= ( in @ sK2 @ sK1 ) )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f16,f30]) ).
thf(f30,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( $true
= ( in @ X1 @ ( setminus @ X2 @ X3 ) ) )
| ( $false
= ( in @ X1 @ X2 ) )
| ( ( in @ X1 @ X3 )
= $true ) ),
inference(not_proxy_clausification,[status(thm)],[f29]) ).
thf(f29,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( $true
= ( in @ X1 @ ( setminus @ X2 @ X3 ) ) )
| ( $false
= ( ~ ( in @ X1 @ X3 ) ) )
| ( $false
= ( in @ X1 @ X2 ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f28]) ).
thf(f28,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( ( ~ ( in @ X1 @ X3 )
=> ( in @ X1 @ ( setminus @ X2 @ X3 ) ) )
= $true )
| ( $false
= ( in @ X1 @ X2 ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f27]) ).
thf(f27,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( ( in @ X1 @ X2 )
=> ( ~ ( in @ X1 @ X3 )
=> ( in @ X1 @ ( setminus @ X2 @ X3 ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f26]) ).
thf(f26,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( ^ [Y0: $i] :
( ( in @ X1 @ X2 )
=> ( ~ ( in @ X1 @ Y0 )
=> ( in @ X1 @ ( setminus @ X2 @ Y0 ) ) ) )
@ X3 )
= $true ),
inference(pi_clausification,[status(thm)],[f25]) ).
thf(f25,plain,
! [X2: $i,X1: $i] :
( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ X1 @ X2 )
=> ( ~ ( in @ X1 @ Y0 )
=> ( in @ X1 @ ( setminus @ X2 @ Y0 ) ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f24]) ).
thf(f24,plain,
! [X2: $i,X1: $i] :
( ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( in @ X1 @ Y0 )
=> ( ~ ( in @ X1 @ Y1 )
=> ( in @ X1 @ ( setminus @ Y0 @ Y1 ) ) ) ) )
@ X2 )
= $true ),
inference(pi_clausification,[status(thm)],[f23]) ).
thf(f23,plain,
! [X1: $i] :
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( in @ X1 @ Y0 )
=> ( ~ ( in @ X1 @ Y1 )
=> ( in @ X1 @ ( setminus @ Y0 @ Y1 ) ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f22]) ).
thf(f22,plain,
! [X1: $i] :
( ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y0 @ Y1 )
=> ( ~ ( in @ Y0 @ Y2 )
=> ( in @ Y0 @ ( setminus @ Y1 @ Y2 ) ) ) ) ) )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f21]) ).
thf(f21,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y0 @ Y1 )
=> ( ~ ( in @ Y0 @ Y2 )
=> ( in @ Y0 @ ( setminus @ Y1 @ Y2 ) ) ) ) ) ) )
= $true ),
inference(definition_unfolding,[status(thm)],[f20,f15]) ).
thf(f15,plain,
setminusI = $true,
inference(cnf_transformation,[status(thm)],[f14]) ).
thf(f20,plain,
( setminusI
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y0 @ Y1 )
=> ( ~ ( in @ Y0 @ Y2 )
=> ( in @ Y0 @ ( setminus @ Y1 @ Y2 ) ) ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f6]) ).
thf(f6,plain,
( setminusI
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y0 @ Y1 )
=> ( ~ ( in @ Y0 @ Y2 )
=> ( in @ Y0 @ ( setminus @ Y1 @ Y2 ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f5]) ).
thf(f5,plain,
! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X1 )
=> ( ~ ( in @ X2 @ X0 )
=> ( in @ X2 @ ( setminus @ X1 @ X0 ) ) ) )
= setminusI ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
! [X1: $i,X0: $i,X2: $i] :
( ( ( in @ X2 @ X0 )
=> ( ~ ( in @ X2 @ X1 )
=> ( in @ X2 @ ( setminus @ X0 @ X1 ) ) ) )
= setminusI ),
file('/export/starexec/sandbox2/tmp/tmp.Ssivfr5AOV/DTF2THF_9084.p',setminusI) ).
thf(f16,plain,
( ( in @ sK2 @ ( setminus @ sK0 @ sK1 ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f14]) ).
thf(f18,plain,
( ( in @ sK2 @ sK0 )
= $true ),
inference(cnf_transformation,[status(thm)],[f14]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU720^2 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.33 % Computer : n008.cluster.edu
% 0.16/0.33 % Model : x86_64 x86_64
% 0.16/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.33 % Memory : 8042.1875MB
% 0.16/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.33 % CPULimit : 300
% 0.16/0.33 % WCLimit : 300
% 0.16/0.33 % DateTime : Tue Feb 24 12:31:54 EST 2026
% 0.16/0.33 % CPUTime :
% 0.16/0.33 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 : SEU720^2 : TPTP v9.2.1. Released v3.7.0.
% 0.19/0.45 % Domain : Set Theory
% 0.19/0.45 % Problem : Typed Set Theory - Laws for Typed Sets
% 0.19/0.45 % Version : Especial > Reduced > Especial.
% 0.19/0.45 % English : (! A:i.! X:i.in X (powerset A) -> (! x:i.in x A ->
% 0.19/0.45 % ~(in x (setminus A X)) -> in x X))
% 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 : ZFC222l [Bro08]
% 0.19/0.45
% 0.19/0.45 % Status : Theorem
% 0.19/0.45 % Rating : 0.00 v8.2.0, 0.08 v8.1.0, 0.00 v7.4.0, 0.11 v7.2.0, 0.00 v7.1.0, 0.12 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.00 v6.1.0, 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
% 0.19/0.45 % Syntax : Number of formulae : 6 ( 1 unt; 4 typ; 1 def)
% 0.19/0.45 % Number of atoms : 10 ( 1 equ; 0 cnn)
% 0.19/0.45 % Maximal formula atoms : 5 ( 5 avg)
% 0.19/0.45 % Number of connectives : 27 ( 2 ~; 0 |; 0 &; 19 @)
% 0.19/0.45 % ( 0 <=>; 6 =>; 0 <=; 0 <~>)
% 0.19/0.45 % Maximal formula depth : 12 ( 7 avg)
% 0.19/0.45 % Number of types : 2 ( 0 usr)
% 0.19/0.45 % Number of type conns : 5 ( 5 >; 0 *; 0 +; 0 <<)
% 0.19/0.45 % Number of symbols : 5 ( 4 usr; 1 con; 0-2 aty)
% 0.19/0.45 % Number of variables : 6 ( 0 ^; 6 !; 0 ?; 6 :)
% 0.19/0.45 % SPC : TH0_THM_EQU_NAR
% 0.19/0.45
% 0.19/0.45 % Comments : http://mathgate.info/detsetitem.php?id=280
% 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(powerset_type,type,
% 0.19/0.45 powerset: $i > $i ).
% 0.19/0.45
% 0.19/0.45 thf(setminus_type,type,
% 0.19/0.45 setminus: $i > $i > $i ).
% 0.19/0.45
% 0.19/0.45 thf(setminusI_type,type,
% 0.19/0.45 setminusI: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setminusI,definition,
% 0.19/0.45 ( setminusI
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( ~ ( in @ Xx @ B )
% 0.19/0.45 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(complementTE1,conjecture,
% 0.19/0.45 ( setminusI
% 0.19/0.45 => ! [A: $i,X: $i] :
% 0.19/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.45 => ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 0.19/0.45 => ( in @ Xx @ X ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 %------------------------------------------------------------------------------
% 0.19/0.45 ------------------------
% 1.42/1.29 ---- Embedded in THF ---
% 1.42/1.29 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.42/1.29 %%% Generated on Tue Feb 24 12:31:55 EST 2026
% 1.42/1.29 %%% using '$$dhol' embedding, version 1.3.0.
% 1.42/1.29 %%% Logic specification used:
% 1.42/1.29 %%% thf(spec, logic, $$dhol).
% 1.42/1.29
% 1.42/1.29 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.Ssivfr5AOV/DTF2DTF9084.p
% See solution above
% 1.42/1.29 ------------------------
% 1.60/1.30 ---- Cleaned THF ---
% 1.60/1.30 thf(in_type,type,
% 1.60/1.30 in: $i > $i > $o ).
% 1.60/1.30
% 1.60/1.30 thf(powerset_type,type,
% 1.60/1.30 powerset: $i > $i ).
% 1.60/1.30
% 1.60/1.30 thf(setminus_type,type,
% 1.60/1.30 setminus: $i > $i > $i ).
% 1.60/1.30
% 1.60/1.30 thf(setminusI_type,type,
% 1.60/1.30 setminusI: $o ).
% 1.60/1.30
% 1.60/1.30 thf(setminusI,definition,
% 1.60/1.30 ( setminusI
% 1.60/1.30 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.60/1.30 ( ( in @ Xx @ A )
% 1.60/1.30 => ( ~ ( in @ Xx @ B )
% 1.60/1.30 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 1.60/1.30
% 1.60/1.30 thf(complementTE1,conjecture,
% 1.60/1.30 ( setminusI
% 1.60/1.30 => ! [A: $i,X: $i] :
% 1.60/1.30 ( ( in @ X @ ( powerset @ A ) )
% 1.60/1.30 => ! [Xx: $i] :
% 1.60/1.30 ( ( in @ Xx @ A )
% 1.60/1.30 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 1.60/1.30 => ( in @ Xx @ X ) ) ) ) ) ).
% 1.60/1.30 ------------------------
% 1.60/1.32 % (9232)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_9084 for (2999ds/18Mi)
% 1.60/1.32 % (9226)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_9084 for (2999ds/183Mi)
% 1.60/1.32 % (9231)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_9084 for (2999ds/275Mi)
% 1.60/1.32 % (9226)First to succeed.
% 1.60/1.32 % (9231)Also succeeded, but the first one will report.
% 1.60/1.32 % (9226)Refutation found. Thanks to Tanya!
% 1.60/1.32 % SZS status Theorem for DTF2THF_9084
% 1.60/1.32 % SZS output start Proof for DTF2THF_9084
% 1.60/1.32 thf(func_def_0, type, in: $i > $i > $o).
% 1.60/1.32 thf(func_def_1, type, powerset: $i > $i).
% 1.60/1.32 thf(func_def_2, type, setminus: $i > $i > $i).
% 1.60/1.32 thf(f36,plain,(
% 1.60/1.32 $false),
% 1.60/1.32 inference(trivial_inequality_removal,[status(thm)],[f35])).
% 1.60/1.32 thf(f35,plain,(
% 1.60/1.32 ($false = $true)),
% 1.60/1.32 inference(superposition,[status(thm)],[f18,f33])).
% 1.60/1.32 thf(f33,plain,(
% 1.60/1.32 ($false = (in @ sK2 @ sK0))),
% 1.60/1.32 inference(subsumption_resolution,[status(thm)],[f32,f17])).
% 1.60/1.32 thf(f17,plain,(
% 1.60/1.32 ($true != (in @ sK2 @ sK1))),
% 1.60/1.32 inference(cnf_transformation,[status(thm)],[f14])).
% 1.60/1.32 thf(f14,plain,(
% 1.60/1.32 (((in @ sK1 @ (powerset @ sK0)) = $true) & (((in @ sK2 @ sK0) = $true) & ($true != (in @ sK2 @ sK1)) & ((in @ sK2 @ (setminus @ sK0 @ sK1)) != $true))) & (setminusI = $true)),
% 1.60/1.32 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2])],[f11,f13,f12])).
% 1.60/1.32 thf(f12,plain,(
% 1.60/1.32 ? [X0 : $i,X1 : $i] : (((in @ X1 @ (powerset @ X0)) = $true) & ? [X2 : $i] : (((in @ X2 @ X0) = $true) & ((in @ X2 @ X1) != $true) & ((in @ X2 @ (setminus @ X0 @ X1)) != $true))) => (((in @ sK1 @ (powerset @ sK0)) = $true) & ? [X2 : $i] : (((in @ X2 @ sK0) = $true) & ($true != (in @ X2 @ sK1)) & ((in @ X2 @ (setminus @ sK0 @ sK1)) != $true)))),
% 1.60/1.32 introduced(definition,[],[choice_axiom])).
% 1.60/1.32 thf(f13,plain,(
% 1.60/1.32 ? [X2 : $i] : (((in @ X2 @ sK0) = $true) & ($true != (in @ X2 @ sK1)) & ((in @ X2 @ (setminus @ sK0 @ sK1)) != $true)) => (((in @ sK2 @ sK0) = $true) & ($true != (in @ sK2 @ sK1)) & ((in @ sK2 @ (setminus @ sK0 @ sK1)) != $true))),
% 1.60/1.32 introduced(definition,[],[choice_axiom])).
% 1.60/1.32 thf(f11,plain,(
% 1.60/1.32 ? [X0 : $i,X1 : $i] : (((in @ X1 @ (powerset @ X0)) = $true) & ? [X2 : $i] : (((in @ X2 @ X0) = $true) & ((in @ X2 @ X1) != $true) & ((in @ X2 @ (setminus @ X0 @ X1)) != $true))) & (setminusI = $true)),
% 1.60/1.32 inference(flattening,[status(thm)],[f10])).
% 1.60/1.32 thf(f10,plain,(
% 1.60/1.32 ? [X0 : $i,X1 : $i] : (? [X2 : $i] : ((((in @ X2 @ X1) != $true) & ((in @ X2 @ (setminus @ X0 @ X1)) != $true)) & ((in @ X2 @ X0) = $true)) & ((in @ X1 @ (powerset @ X0)) = $true)) & (setminusI = $true)),
% 1.60/1.32 inference(ennf_transformation,[status(thm)],[f9])).
% 1.60/1.32 thf(f9,plain,(
% 1.60/1.32 ~((setminusI = $true) => ! [X0 : $i,X1 : $i] : (((in @ X1 @ (powerset @ X0)) = $true) => ! [X2 : $i] : (((in @ X2 @ X0) = $true) => (((in @ X2 @ (setminus @ X0 @ X1)) != $true) => ((in @ X2 @ X1) = $true)))))),
% 1.60/1.32 inference(flattening,[status(thm)],[f8])).
% 1.60/1.32 thf(f8,plain,(
% 1.60/1.32 ~((setminusI = $true) => ! [X0 : $i,X1 : $i] : (((in @ X1 @ (powerset @ X0)) = $true) => ! [X2 : $i] : (((in @ X2 @ X0) = $true) => (~((in @ X2 @ (setminus @ X0 @ X1)) = $true) => ((in @ X2 @ X1) = $true)))))),
% 1.60/1.32 inference(fool_elimination,[status(thm)],[f7])).
% 1.60/1.32 thf(f7,plain,(
% 1.60/1.32 ~(setminusI => ! [X0 : $i,X1 : $i] : ((in @ X1 @ (powerset @ X0)) => ! [X2 : $i] : ((in @ X2 @ X0) => (~(in @ X2 @ (setminus @ X0 @ X1)) => (in @ X2 @ X1)))))),
% 1.60/1.32 inference(rectify,[status(thm)],[f3])).
% 1.60/1.32 thf(f3,negated_conjecture,(
% 1.60/1.32 ~(setminusI => ! [X0 : $i,X3 : $i] : ((in @ X3 @ (powerset @ X0)) => ! [X2 : $i] : ((in @ X2 @ X0) => (~(in @ X2 @ (setminus @ X0 @ X3)) => (in @ X2 @ X3)))))),
% 1.60/1.32 inference(negated_conjecture,[status(cth)],[f2])).
% 1.60/1.32 thf(f2,conjecture,(
% 1.60/1.32 setminusI => ! [X0 : $i,X3 : $i] : ((in @ X3 @ (powerset @ X0)) => ! [X2 : $i] : ((in @ X2 @ X0) => (~(in @ X2 @ (setminus @ X0 @ X3)) => (in @ X2 @ X3))))),
% 1.60/1.32 file('/export/starexec/sandbox2/tmp/tmp.Ssivfr5AOV/DTF2THF_9084.p',complementTE1)).
% 1.60/1.32 thf(f32,plain,(
% 1.60/1.32 ($false = (in @ sK2 @ sK0)) | ($true = (in @ sK2 @ sK1))),
% 1.60/1.32 inference(trivial_inequality_removal,[status(thm)],[f31])).
% 1.60/1.32 thf(f31,plain,(
% 1.60/1.32 ($false = (in @ sK2 @ sK0)) | ($true = (in @ sK2 @ sK1)) | ($true != $true)),
% 1.60/1.32 inference(superposition,[status(thm)],[f16,f30])).
% 1.60/1.32 thf(f30,plain,(
% 1.60/1.32 ( ! [X2 : $i,X3 : $i,X1 : $i] : (($true = (in @ X1 @ (setminus @ X2 @ X3))) | ($false = (in @ X1 @ X2)) | ((in @ X1 @ X3) = $true)) )),
% 1.60/1.32 inference(not_proxy_clausification,[status(thm)],[f29])).
% 1.60/1.32 thf(f29,plain,(
% 1.60/1.32 ( ! [X2 : $i,X3 : $i,X1 : $i] : (($true = (in @ X1 @ (setminus @ X2 @ X3))) | ($false = (~ (in @ X1 @ X3))) | ($false = (in @ X1 @ X2))) )),
% 1.60/1.32 inference(binary_proxy_clausification,[status(thm)],[f28])).
% 1.60/1.32 thf(f28,plain,(
% 1.60/1.32 ( ! [X2 : $i,X3 : $i,X1 : $i] : ((((~ (in @ X1 @ X3)) => (in @ X1 @ (setminus @ X2 @ X3))) = $true) | ($false = (in @ X1 @ X2))) )),
% 1.60/1.32 inference(binary_proxy_clausification,[status(thm)],[f27])).
% 1.60/1.32 thf(f27,plain,(
% 1.60/1.32 ( ! [X2 : $i,X3 : $i,X1 : $i] : ((((in @ X1 @ X2) => ((~ (in @ X1 @ X3)) => (in @ X1 @ (setminus @ X2 @ X3)))) = $true)) )),
% 1.60/1.32 inference(beta_eta_normalization,[status(thm)],[f26])).
% 1.60/1.32 thf(f26,plain,(
% 1.60/1.32 ( ! [X2 : $i,X3 : $i,X1 : $i] : ((((^[Y0 : $i]: ((in @ X1 @ X2) => ((~ (in @ X1 @ Y0)) => (in @ X1 @ (setminus @ X2 @ Y0))))) @ X3) = $true)) )),
% 1.60/1.32 inference(pi_clausification,[status(thm)],[f25])).
% 1.60/1.32 thf(f25,plain,(
% 1.60/1.32 ( ! [X2 : $i,X1 : $i] : (((!! @ $i @ (^[Y0 : $i]: ((in @ X1 @ X2) => ((~ (in @ X1 @ Y0)) => (in @ X1 @ (setminus @ X2 @ Y0)))))) = $true)) )),
% 1.60/1.32 inference(beta_eta_normalization,[status(thm)],[f24])).
% 1.60/1.32 thf(f24,plain,(
% 1.60/1.32 ( ! [X2 : $i,X1 : $i] : ((((^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: ((in @ X1 @ Y0) => ((~ (in @ X1 @ Y1)) => (in @ X1 @ (setminus @ Y0 @ Y1))))))) @ X2) = $true)) )),
% 1.60/1.32 inference(pi_clausification,[status(thm)],[f23])).
% 1.60/1.32 thf(f23,plain,(
% 1.60/1.32 ( ! [X1 : $i] : (($true = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: ((in @ X1 @ Y0) => ((~ (in @ X1 @ Y1)) => (in @ X1 @ (setminus @ Y0 @ Y1)))))))))) )),
% 1.60/1.32 inference(beta_eta_normalization,[status(thm)],[f22])).
% 1.60/1.32 thf(f22,plain,(
% 1.60/1.32 ( ! [X1 : $i] : ((((^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y0 @ Y1) => ((~ (in @ Y0 @ Y2)) => (in @ Y0 @ (setminus @ Y1 @ Y2))))))))) @ X1) = $true)) )),
% 1.60/1.32 inference(pi_clausification,[status(thm)],[f21])).
% 1.60/1.32 thf(f21,plain,(
% 1.60/1.32 ((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y0 @ Y1) => ((~ (in @ Y0 @ Y2)) => (in @ Y0 @ (setminus @ Y1 @ Y2)))))))))) = $true)),
% 1.60/1.32 inference(definition_unfolding,[status(thm)],[f20,f15])).
% 1.60/1.32 thf(f15,plain,(
% 1.60/1.32 (setminusI = $true)),
% 1.60/1.32 inference(cnf_transformation,[status(thm)],[f14])).
% 1.60/1.32 thf(f20,plain,(
% 1.60/1.32 (setminusI = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y0 @ Y1) => ((~ (in @ Y0 @ Y2)) => (in @ Y0 @ (setminus @ Y1 @ Y2)))))))))))),
% 1.60/1.32 inference(cnf_transformation,[status(thm)],[f6])).
% 1.60/1.32 thf(f6,plain,(
% 1.60/1.32 (setminusI = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y0 @ Y1) => ((~ (in @ Y0 @ Y2)) => (in @ Y0 @ (setminus @ Y1 @ Y2)))))))))))),
% 1.60/1.32 inference(fool_elimination,[status(thm)],[f5])).
% 1.60/1.32 thf(f5,plain,(
% 1.60/1.32 (! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X2 @ X1) => (~(in @ X2 @ X0) => (in @ X2 @ (setminus @ X1 @ X0)))) = setminusI)),
% 1.60/1.32 inference(rectify,[status(thm)],[f1])).
% 1.60/1.32 thf(f1,axiom,(
% 1.60/1.32 (! [X1 : $i,X0 : $i,X2 : $i] : ((in @ X2 @ X0) => (~(in @ X2 @ X1) => (in @ X2 @ (setminus @ X0 @ X1)))) = setminusI)),
% 1.60/1.32 file('/export/starexec/sandbox2/tmp/tmp.Ssivfr5AOV/DTF2THF_9084.p',setminusI)).
% 1.60/1.32 thf(f16,plain,(
% 1.60/1.32 ((in @ sK2 @ (setminus @ sK0 @ sK1)) != $true)),
% 1.60/1.32 inference(cnf_transformation,[status(thm)],[f14])).
% 1.60/1.32 thf(f18,plain,(
% 1.60/1.32 ((in @ sK2 @ sK0) = $true)),
% 1.60/1.32 inference(cnf_transformation,[status(thm)],[f14])).
% 1.60/1.32 % SZS output end Proof for DTF2THF_9084
% 1.60/1.32 % (9226)------------------------------
% 1.60/1.32 % (9226)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.60/1.32 % (9226)Termination reason: Refutation
% 1.60/1.32
% 1.60/1.32 % (9226)Memory used [KB]: 5500
% 1.60/1.32 % (9226)Time elapsed: 0.004 s
% 1.60/1.32 % (9226)Instructions burned: 3 (million)
% 1.60/1.32 % (9226)------------------------------
% 1.60/1.32 % (9226)------------------------------
% 1.60/1.32 % (9225)Success in time 0.004 s
%------------------------------------------------------------------------------