%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU662^2 : 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 09:36:02 AM UTC 2026
% Result : Theorem 1.50s 1.25s
% Output : Refutation 1.50s
% Verified :
% SZS Type : Refutation
% Derivation depth : 11
% Number of leaves : 5
% Syntax : Number of formulae : 23 ( 6 unt; 0 typ; 0 def)
% Number of atoms : 181 ( 65 equ; 0 cnn)
% Maximal formula atoms : 6 ( 7 avg)
% Number of connectives : 200 ( 22 ~; 8 |; 25 &; 125 @)
% ( 1 <=>; 19 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 6 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 14 ( 11 usr; 10 con; 0-2 aty)
% Number of variables : 58 ( 0 ^; 34 !; 24 ?; 58 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_1,type,
kpair: $i > $i > $i ).
thf(func_def_2,type,
kfst: $i > $i ).
thf(f46,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f34,f45]) ).
thf(f45,plain,
! [X0: $i,X1: $i] :
( ( kfst @ ( kpair @ X0 @ X1 ) )
= X0 ),
inference(trivial_inequality_removal,[status(thm)],[f43]) ).
thf(f43,plain,
! [X0: $i,X1: $i] :
( ( $true != $true )
| ( ( kfst @ ( kpair @ X0 @ X1 ) )
= X0 ) ),
inference(definition_unfolding,[status(thm)],[f38,f35]) ).
thf(f35,plain,
kfstpairEq = $true,
inference(cnf_transformation,[status(thm)],[f23]) ).
thf(f23,plain,
( ( cartprodmempaircEq = $true )
& ( kfstpairEq = $true )
& ( sK5
!= ( kfst @ ( kpair @ sK5 @ sK7 ) ) )
& ( $true
= ( in @ sK7 @ sK4 ) )
& ( $true
= ( in @ sK5 @ sK6 ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK4,sK5,sK6,sK7])],[f20,f22,f21]) ).
thf(f21,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( ( kfst @ ( kpair @ X1 @ X3 ) )
!= X1 )
& ( $true
= ( in @ X3 @ X0 ) ) )
& ( $true
= ( in @ X1 @ X2 ) ) )
=> ( ? [X3: $i] :
( ( ( kfst @ ( kpair @ sK5 @ X3 ) )
!= sK5 )
& ( $true
= ( in @ X3 @ sK4 ) ) )
& ( $true
= ( in @ sK5 @ sK6 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f22,plain,
( ? [X3: $i] :
( ( ( kfst @ ( kpair @ sK5 @ X3 ) )
!= sK5 )
& ( $true
= ( in @ X3 @ sK4 ) ) )
=> ( ( sK5
!= ( kfst @ ( kpair @ sK5 @ sK7 ) ) )
& ( $true
= ( in @ sK7 @ sK4 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f20,plain,
( ( cartprodmempaircEq = $true )
& ( kfstpairEq = $true )
& ? [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( ( kfst @ ( kpair @ X1 @ X3 ) )
!= X1 )
& ( $true
= ( in @ X3 @ X0 ) ) )
& ( $true
= ( in @ X1 @ X2 ) ) ) ),
inference(rectify,[status(thm)],[f14]) ).
thf(f14,plain,
( ( cartprodmempaircEq = $true )
& ( kfstpairEq = $true )
& ? [X1: $i,X2: $i,X0: $i] :
( ? [X3: $i] :
( ( ( kfst @ ( kpair @ X2 @ X3 ) )
!= X2 )
& ( $true
= ( in @ X3 @ X1 ) ) )
& ( $true
= ( in @ X2 @ X0 ) ) ) ),
inference(flattening,[status(thm)],[f13]) ).
thf(f13,plain,
( ? [X1: $i,X2: $i,X0: $i] :
( ? [X3: $i] :
( ( ( kfst @ ( kpair @ X2 @ X3 ) )
!= X2 )
& ( $true
= ( in @ X3 @ X1 ) ) )
& ( $true
= ( in @ X2 @ X0 ) ) )
& ( cartprodmempaircEq = $true )
& ( kfstpairEq = $true ) ),
inference(ennf_transformation,[status(thm)],[f9]) ).
thf(f9,plain,
~ ( ( kfstpairEq = $true )
=> ( ( cartprodmempaircEq = $true )
=> ! [X2: $i,X0: $i,X1: $i] :
( ( $true
= ( in @ X2 @ X0 ) )
=> ! [X3: $i] :
( ( $true
= ( in @ X3 @ X1 ) )
=> ( ( kfst @ ( kpair @ X2 @ X3 ) )
= X2 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f8]) ).
thf(f8,plain,
~ ( kfstpairEq
=> ( cartprodmempaircEq
=> ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X2 @ X0 )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( kfst @ ( kpair @ X2 @ X3 ) )
= X2 ) ) ) ) ),
inference(rectify,[status(thm)],[f4]) ).
thf(f4,negated_conjecture,
~ ( kfstpairEq
=> ( cartprodmempaircEq
=> ! [X2: $i,X3: $i,X0: $i] :
( ( in @ X0 @ X2 )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( ( kfst @ ( kpair @ X0 @ X1 ) )
= X0 ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f3]) ).
thf(f3,conjecture,
( kfstpairEq
=> ( cartprodmempaircEq
=> ! [X2: $i,X3: $i,X0: $i] :
( ( in @ X0 @ X2 )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( ( kfst @ ( kpair @ X0 @ X1 ) )
= X0 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.6INtkrR5XU/DTF2THF_28538.p',cartprodfstpairEq) ).
thf(f38,plain,
! [X0: $i,X1: $i] :
( ( ( kfst @ ( kpair @ X0 @ X1 ) )
= X0 )
| ( kfstpairEq != $true ) ),
inference(cnf_transformation,[status(thm)],[f27]) ).
thf(f27,plain,
( ( ! [X0: $i,X1: $i] :
( ( kfst @ ( kpair @ X0 @ X1 ) )
= X0 )
| ( kfstpairEq != $true ) )
& ( ( kfstpairEq = $true )
| ( ( kfst @ ( kpair @ sK8 @ sK9 ) )
!= sK8 ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK8,sK9])],[f25,f26]) ).
thf(f26,plain,
( ? [X2: $i,X3: $i] :
( ( kfst @ ( kpair @ X2 @ X3 ) )
!= X2 )
=> ( ( kfst @ ( kpair @ sK8 @ sK9 ) )
!= sK8 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f25,plain,
( ( ! [X0: $i,X1: $i] :
( ( kfst @ ( kpair @ X0 @ X1 ) )
= X0 )
| ( kfstpairEq != $true ) )
& ( ( kfstpairEq = $true )
| ? [X2: $i,X3: $i] :
( ( kfst @ ( kpair @ X2 @ X3 ) )
!= X2 ) ) ),
inference(rectify,[status(thm)],[f24]) ).
thf(f24,plain,
( ( ! [X1: $i,X0: $i] :
( ( kfst @ ( kpair @ X1 @ X0 ) )
= X1 )
| ( kfstpairEq != $true ) )
& ( ( kfstpairEq = $true )
| ? [X1: $i,X0: $i] :
( ( kfst @ ( kpair @ X1 @ X0 ) )
!= X1 ) ) ),
inference(nnf_transformation,[status(thm)],[f7]) ).
thf(f7,plain,
( ! [X1: $i,X0: $i] :
( ( kfst @ ( kpair @ X1 @ X0 ) )
= X1 )
<=> ( kfstpairEq = $true ) ),
inference(fool_elimination,[status(thm)],[f6]) ).
thf(f6,plain,
( kfstpairEq
= ( ! [X0: $i,X1: $i] :
( ( kfst @ ( kpair @ X1 @ X0 ) )
= X1 ) ) ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
( kfstpairEq
= ( ! [X1: $i,X0: $i] :
( ( kfst @ ( kpair @ X0 @ X1 ) )
= X0 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.6INtkrR5XU/DTF2THF_28538.p',kfstpairEq) ).
thf(f34,plain,
( sK5
!= ( kfst @ ( kpair @ sK5 @ sK7 ) ) ),
inference(cnf_transformation,[status(thm)],[f23]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU662^2 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.12 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.15/0.33 % Computer : n013.cluster.edu
% 0.15/0.33 % Model : x86_64 x86_64
% 0.15/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.33 % Memory : 8042.1875MB
% 0.15/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.33 % CPULimit : 300
% 0.15/0.33 % WCLimit : 300
% 0.15/0.33 % DateTime : Tue Feb 24 12:29:53 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
% 0.29/0.45 ---- Original DTF file ---
% 0.29/0.45 thf(spec,logic,$$dhol).
% 0.29/0.45 %------------------------------------------------------------------------------
% 0.29/0.45 % File : SEU662^2 : TPTP v9.2.1. Released v3.7.0.
% 0.29/0.45 % Domain : Set Theory
% 0.29/0.45 % Problem : Ordered Pairs - Properties of Pairs
% 0.29/0.45 % Version : Especial > Reduced > Especial.
% 0.29/0.45 % English : (! A:i.! B:i.! x:i.in x A -> (! y:i.in y B ->
% 0.29/0.45 % kfst (kpair x y) = x))
% 0.29/0.45
% 0.29/0.45 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.29/0.45 % Source : [Bro08]
% 0.29/0.45 % Names : ZFC164l [Bro08]
% 0.29/0.45
% 0.29/0.45 % Status : Theorem
% 0.29/0.45 % 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
% 0.29/0.45 % Syntax : Number of formulae : 8 ( 2 unt; 5 typ; 2 def)
% 0.29/0.45 % Number of atoms : 13 ( 5 equ; 0 cnn)
% 0.29/0.45 % Maximal formula atoms : 5 ( 4 avg)
% 0.29/0.45 % Number of connectives : 24 ( 0 ~; 0 |; 0 &; 18 @)
% 0.29/0.45 % ( 0 <=>; 6 =>; 0 <=; 0 <~>)
% 0.29/0.45 % Maximal formula depth : 11 ( 4 avg)
% 0.29/0.45 % Number of types : 2 ( 0 usr)
% 0.29/0.45 % Number of type conns : 5 ( 5 >; 0 *; 0 +; 0 <<)
% 0.29/0.45 % Number of symbols : 6 ( 5 usr; 2 con; 0-2 aty)
% 0.29/0.45 % Number of variables : 10 ( 0 ^; 10 !; 0 ?; 10 :)
% 0.29/0.45 % SPC : TH0_THM_EQU_NAR
% 0.29/0.45
% 0.29/0.45 % Comments : http://mathgate.info/detsetitem.php?id=216
% 0.29/0.45 %------------------------------------------------------------------------------
% 0.29/0.45 thf(in_type,type,
% 0.29/0.45 in: $i > $i > $o ).
% 0.29/0.45
% 0.29/0.45 thf(kpair_type,type,
% 0.29/0.45 kpair: $i > $i > $i ).
% 0.29/0.45
% 0.29/0.45 thf(kfst_type,type,
% 0.29/0.45 kfst: $i > $i ).
% 0.29/0.45
% 0.29/0.45 thf(kfstpairEq_type,type,
% 0.29/0.45 kfstpairEq: $o ).
% 0.29/0.45
% 0.29/0.45 thf(kfstpairEq,definition,
% 0.29/0.45 ( kfstpairEq
% 0.29/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.45 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.29/0.45 = Xx ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(cartprodmempaircEq_type,type,
% 0.29/0.45 cartprodmempaircEq: $o ).
% 0.29/0.45
% 0.29/0.45 thf(cartprodmempaircEq,definition,
% 0.29/0.45 ( cartprodmempaircEq
% 0.29/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ! [Xy: $i] :
% 0.29/0.45 ( ( in @ Xy @ B )
% 0.29/0.45 => ( ( kpair @ Xx @ Xy )
% 0.29/0.45 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(cartprodfstpairEq,conjecture,
% 0.29/0.45 ( kfstpairEq
% 0.29/0.45 => ( cartprodmempaircEq
% 0.29/0.45 => ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ! [Xy: $i] :
% 0.29/0.45 ( ( in @ Xy @ B )
% 0.29/0.45 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.29/0.45 = Xx ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 %------------------------------------------------------------------------------
% 0.29/0.45 ------------------------
% 1.50/1.22 ---- Embedded in THF ---
% 1.50/1.22 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.50/1.22 %%% Generated on Tue Feb 24 12:29:54 EST 2026
% 1.50/1.22 %%% using '$$dhol' embedding, version 1.3.0.
% 1.50/1.22 %%% Logic specification used:
% 1.50/1.22 %%% thf(spec, logic, $$dhol).
% 1.50/1.22
% 1.50/1.22 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.6INtkrR5XU/DTF2DTF28538.p
% See solution above
% 1.50/1.22 ------------------------
% 1.50/1.23 ---- Cleaned THF ---
% 1.50/1.23 thf(in_type,type,
% 1.50/1.23 in: $i > $i > $o ).
% 1.50/1.23
% 1.50/1.23 thf(kpair_type,type,
% 1.50/1.23 kpair: $i > $i > $i ).
% 1.50/1.23
% 1.50/1.23 thf(kfst_type,type,
% 1.50/1.23 kfst: $i > $i ).
% 1.50/1.23
% 1.50/1.23 thf(kfstpairEq_type,type,
% 1.50/1.23 kfstpairEq: $o ).
% 1.50/1.23
% 1.50/1.23 thf(kfstpairEq,definition,
% 1.50/1.23 ( kfstpairEq
% 1.50/1.23 = ( ! [Xx: $i,Xy: $i] :
% 1.50/1.23 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 1.50/1.23 = Xx ) ) ) ).
% 1.50/1.23
% 1.50/1.23 thf(cartprodmempaircEq_type,type,
% 1.50/1.23 cartprodmempaircEq: $o ).
% 1.50/1.23
% 1.50/1.23 thf(cartprodmempaircEq,definition,
% 1.50/1.23 ( cartprodmempaircEq
% 1.50/1.23 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.50/1.23 ( ( in @ Xx @ A )
% 1.50/1.23 => ! [Xy: $i] :
% 1.50/1.23 ( ( in @ Xy @ B )
% 1.50/1.23 => ( ( kpair @ Xx @ Xy )
% 1.50/1.23 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 1.50/1.23
% 1.50/1.23 thf(cartprodfstpairEq,conjecture,
% 1.50/1.23 ( kfstpairEq
% 1.50/1.23 => ( cartprodmempaircEq
% 1.50/1.23 => ! [A: $i,B: $i,Xx: $i] :
% 1.50/1.23 ( ( in @ Xx @ A )
% 1.50/1.23 => ! [Xy: $i] :
% 1.50/1.23 ( ( in @ Xy @ B )
% 1.50/1.23 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 1.50/1.23 = Xx ) ) ) ) ) ).
% 1.50/1.23 ------------------------
% 1.50/1.24 % (28809)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_28538 for (2999ds/4Mi)
% 1.50/1.24 % (28814)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_28538 for (2999ds/18Mi)
% 1.50/1.25 % (28809)Instruction limit reached!
% 1.50/1.25 % (28809)------------------------------
% 1.50/1.25 % (28809)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.50/1.25 % (28809)Termination reason: Unknown
% 1.50/1.25 % (28809)Termination phase: Saturation
% 1.50/1.25
% 1.50/1.25 % (28809)Memory used [KB]: 5500
% 1.50/1.25 % (28809)Time elapsed: 0.003 s
% 1.50/1.25 % (28809)Instructions burned: 4 (million)
% 1.50/1.25 % (28809)------------------------------
% 1.50/1.25 % (28809)------------------------------
% 1.50/1.25 % (28814)First to succeed.
% 1.50/1.25 % (28814)Refutation found. Thanks to Tanya!
% 1.50/1.25 % SZS status Theorem for DTF2THF_28538
% 1.50/1.25 % SZS output start Proof for DTF2THF_28538
% 1.50/1.25 thf(func_def_0, type, in: $i > $i > $o).
% 1.50/1.25 thf(func_def_1, type, kpair: $i > $i > $i).
% 1.50/1.25 thf(func_def_2, type, kfst: $i > $i).
% 1.50/1.25 thf(f46,plain,(
% 1.50/1.25 $false),
% 1.50/1.25 inference(subsumption_resolution,[status(thm)],[f34,f45])).
% 1.50/1.25 thf(f45,plain,(
% 1.50/1.25 ( ! [X0 : $i,X1 : $i] : (((kfst @ (kpair @ X0 @ X1)) = X0)) )),
% 1.50/1.25 inference(trivial_inequality_removal,[status(thm)],[f43])).
% 1.50/1.25 thf(f43,plain,(
% 1.50/1.25 ( ! [X0 : $i,X1 : $i] : (($true != $true) | ((kfst @ (kpair @ X0 @ X1)) = X0)) )),
% 1.50/1.25 inference(definition_unfolding,[status(thm)],[f38,f35])).
% 1.50/1.25 thf(f35,plain,(
% 1.50/1.25 (kfstpairEq = $true)),
% 1.50/1.25 inference(cnf_transformation,[status(thm)],[f23])).
% 1.50/1.25 thf(f23,plain,(
% 1.50/1.25 (cartprodmempaircEq = $true) & (kfstpairEq = $true) & (((sK5 != (kfst @ (kpair @ sK5 @ sK7))) & ($true = (in @ sK7 @ sK4))) & ($true = (in @ sK5 @ sK6)))),
% 1.50/1.25 inference(skolemisation,[status(esa),new_symbols(skolem,[sK4,sK5,sK6,sK7])],[f20,f22,f21])).
% 1.50/1.25 thf(f21,plain,(
% 1.50/1.25 ? [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : (((kfst @ (kpair @ X1 @ X3)) != X1) & ($true = (in @ X3 @ X0))) & ($true = (in @ X1 @ X2))) => (? [X3 : $i] : (((kfst @ (kpair @ sK5 @ X3)) != sK5) & ($true = (in @ X3 @ sK4))) & ($true = (in @ sK5 @ sK6)))),
% 1.50/1.25 introduced(definition,[],[choice_axiom])).
% 1.50/1.25 thf(f22,plain,(
% 1.50/1.25 ? [X3 : $i] : (((kfst @ (kpair @ sK5 @ X3)) != sK5) & ($true = (in @ X3 @ sK4))) => ((sK5 != (kfst @ (kpair @ sK5 @ sK7))) & ($true = (in @ sK7 @ sK4)))),
% 1.50/1.25 introduced(definition,[],[choice_axiom])).
% 1.50/1.25 thf(f20,plain,(
% 1.50/1.25 (cartprodmempaircEq = $true) & (kfstpairEq = $true) & ? [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : (((kfst @ (kpair @ X1 @ X3)) != X1) & ($true = (in @ X3 @ X0))) & ($true = (in @ X1 @ X2)))),
% 1.50/1.25 inference(rectify,[status(thm)],[f14])).
% 1.50/1.25 thf(f14,plain,(
% 1.50/1.25 (cartprodmempaircEq = $true) & (kfstpairEq = $true) & ? [X1 : $i,X2 : $i,X0 : $i] : (? [X3 : $i] : (((kfst @ (kpair @ X2 @ X3)) != X2) & ($true = (in @ X3 @ X1))) & ($true = (in @ X2 @ X0)))),
% 1.50/1.25 inference(flattening,[status(thm)],[f13])).
% 1.50/1.25 thf(f13,plain,(
% 1.50/1.25 (? [X1 : $i,X2 : $i,X0 : $i] : (? [X3 : $i] : (((kfst @ (kpair @ X2 @ X3)) != X2) & ($true = (in @ X3 @ X1))) & ($true = (in @ X2 @ X0))) & (cartprodmempaircEq = $true)) & (kfstpairEq = $true)),
% 1.50/1.25 inference(ennf_transformation,[status(thm)],[f9])).
% 1.50/1.25 thf(f9,plain,(
% 1.50/1.25 ~((kfstpairEq = $true) => ((cartprodmempaircEq = $true) => ! [X2 : $i,X0 : $i,X1 : $i] : (($true = (in @ X2 @ X0)) => ! [X3 : $i] : (($true = (in @ X3 @ X1)) => ((kfst @ (kpair @ X2 @ X3)) = X2)))))),
% 1.50/1.25 inference(fool_elimination,[status(thm)],[f8])).
% 1.50/1.25 thf(f8,plain,(
% 1.50/1.25 ~(kfstpairEq => (cartprodmempaircEq => ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X2 @ X0) => ! [X3 : $i] : ((in @ X3 @ X1) => ((kfst @ (kpair @ X2 @ X3)) = X2)))))),
% 1.50/1.25 inference(rectify,[status(thm)],[f4])).
% 1.50/1.25 thf(f4,negated_conjecture,(
% 1.50/1.25 ~(kfstpairEq => (cartprodmempaircEq => ! [X2 : $i,X3 : $i,X0 : $i] : ((in @ X0 @ X2) => ! [X1 : $i] : ((in @ X1 @ X3) => ((kfst @ (kpair @ X0 @ X1)) = X0)))))),
% 1.50/1.25 inference(negated_conjecture,[status(cth)],[f3])).
% 1.50/1.25 thf(f3,conjecture,(
% 1.50/1.25 kfstpairEq => (cartprodmempaircEq => ! [X2 : $i,X3 : $i,X0 : $i] : ((in @ X0 @ X2) => ! [X1 : $i] : ((in @ X1 @ X3) => ((kfst @ (kpair @ X0 @ X1)) = X0))))),
% 1.50/1.25 file('/export/starexec/sandbox/tmp/tmp.6INtkrR5XU/DTF2THF_28538.p',cartprodfstpairEq)).
% 1.50/1.25 thf(f38,plain,(
% 1.50/1.25 ( ! [X0 : $i,X1 : $i] : (((kfst @ (kpair @ X0 @ X1)) = X0) | (kfstpairEq != $true)) )),
% 1.50/1.25 inference(cnf_transformation,[status(thm)],[f27])).
% 1.50/1.25 thf(f27,plain,(
% 1.50/1.25 (! [X0 : $i,X1 : $i] : ((kfst @ (kpair @ X0 @ X1)) = X0) | (kfstpairEq != $true)) & ((kfstpairEq = $true) | ((kfst @ (kpair @ sK8 @ sK9)) != sK8))),
% 1.50/1.25 inference(skolemisation,[status(esa),new_symbols(skolem,[sK8,sK9])],[f25,f26])).
% 1.50/1.25 thf(f26,plain,(
% 1.50/1.25 ? [X2 : $i,X3 : $i] : ((kfst @ (kpair @ X2 @ X3)) != X2) => ((kfst @ (kpair @ sK8 @ sK9)) != sK8)),
% 1.50/1.25 introduced(definition,[],[choice_axiom])).
% 1.50/1.25 thf(f25,plain,(
% 1.50/1.25 (! [X0 : $i,X1 : $i] : ((kfst @ (kpair @ X0 @ X1)) = X0) | (kfstpairEq != $true)) & ((kfstpairEq = $true) | ? [X2 : $i,X3 : $i] : ((kfst @ (kpair @ X2 @ X3)) != X2))),
% 1.50/1.25 inference(rectify,[status(thm)],[f24])).
% 1.50/1.25 thf(f24,plain,(
% 1.50/1.25 (! [X1 : $i,X0 : $i] : ((kfst @ (kpair @ X1 @ X0)) = X1) | (kfstpairEq != $true)) & ((kfstpairEq = $true) | ? [X1 : $i,X0 : $i] : ((kfst @ (kpair @ X1 @ X0)) != X1))),
% 1.50/1.25 inference(nnf_transformation,[status(thm)],[f7])).
% 1.50/1.25 thf(f7,plain,(
% 1.50/1.25 ! [X1 : $i,X0 : $i] : ((kfst @ (kpair @ X1 @ X0)) = X1) <=> (kfstpairEq = $true)),
% 1.50/1.25 inference(fool_elimination,[status(thm)],[f6])).
% 1.50/1.25 thf(f6,plain,(
% 1.50/1.25 (kfstpairEq = ! [X0 : $i,X1 : $i] : ((kfst @ (kpair @ X1 @ X0)) = X1))),
% 1.50/1.25 inference(rectify,[status(thm)],[f1])).
% 1.50/1.25 thf(f1,axiom,(
% 1.50/1.25 (kfstpairEq = ! [X1 : $i,X0 : $i] : ((kfst @ (kpair @ X0 @ X1)) = X0))),
% 1.50/1.25 file('/export/starexec/sandbox/tmp/tmp.6INtkrR5XU/DTF2THF_28538.p',kfstpairEq)).
% 1.50/1.25 thf(f34,plain,(
% 1.50/1.25 (sK5 != (kfst @ (kpair @ sK5 @ sK7)))),
% 1.50/1.25 inference(cnf_transformation,[status(thm)],[f23])).
% 1.50/1.25 % SZS output end Proof for DTF2THF_28538
% 1.50/1.25 % (28814)------------------------------
% 1.50/1.25 % (28814)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.50/1.25 % (28814)Termination reason: Refutation
% 1.50/1.25
% 1.50/1.25 % (28814)Memory used [KB]: 5500
% 1.50/1.25 % (28814)Time elapsed: 0.002 s
% 1.50/1.25 % (28814)Instructions burned: 2 (million)
% 1.50/1.25 % (28814)------------------------------
% 1.50/1.25 % (28814)------------------------------
% 1.50/1.25 % (28807)Success in time 0.013 s
%------------------------------------------------------------------------------