%------------------------------------------------------------------------------
% File : Vampire-SAT---5.0.1
% Problem : SEU664^2 : TPTP v9.3.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : run_vampire /export/starexec/sandbox2/benchmark/theBenchmark.p 300 SAT
% Computer : n013.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Sep 30 08:32:54 AM UTC 2026
% Result : Theorem 0.19s 0.30s
% Output : Refutation 0.19s
% Verified :
% SZS Type : Refutation
% Derivation depth : 34
% Number of leaves : 4
% Syntax : Number of formulae : 88 ( 43 unt; 0 typ; 0 def)
% Number of atoms : 665 ( 211 equ; 0 cnn)
% Maximal formula atoms : 5 ( 7 avg)
% Number of connectives : 1193 ( 7 ~; 60 |; 43 &; 918 @)
% ( 0 <=>; 77 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 4 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 31 ( 26 usr; 10 con; 0-4 aty)
% ( 65 !!; 23 ??; 0 @@+; 0 @@-)
% Number of variables : 303 ( 101 ^; 192 !; 10 ?; 303 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
sTfun: ( $tType * $tType ) > $tType ).
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_1,type,
kpair: $i > $i > $i ).
thf(func_def_2,type,
cartprod: $i > $i > $i ).
thf(func_def_3,type,
cartprodmempair1: $o ).
thf(func_def_5,type,
kfst: $i > $i ).
thf(func_def_6,type,
ksnd: $i > $i ).
thf(func_def_7,type,
cartprodfstpairEq: $o ).
thf(func_def_8,type,
cartprodsndpairEq: $o ).
thf(func_def_9,type,
vPI:
!>[X0: $tType] : ( ( X0 > $o ) > $o ) ).
thf(func_def_10,type,
vIMP: $o > $o > $o ).
thf(func_def_11,type,
db2:
!>[X0: $tType] : X0 ).
thf(func_def_12,type,
db0:
!>[X0: $tType] : X0 ).
thf(func_def_13,type,
db1:
!>[X0: $tType] : X0 ).
thf(func_def_14,type,
vSIGMA:
!>[X0: $tType] : ( ( X0 > $o ) > $o ) ).
thf(func_def_15,type,
vAND: $o > $o > $o ).
thf(func_def_16,type,
db3:
!>[X0: $tType] : X0 ).
thf(func_def_17,type,
db4:
!>[X0: $tType] : X0 ).
thf(func_def_18,type,
vEQ:
!>[X0: $tType] : ( X0 > X0 > $o ) ).
thf(func_def_19,type,
vLAM:
!>[X0: $tType,X1: $tType] : ( X1 > X0 > X1 ) ).
thf(func_def_25,type,
vNOT: $o > $o ).
thf(func_def_26,type,
sK3: $i > $i > $i > $i ).
thf(func_def_27,type,
sK4: $i > $i > $i > $i ).
thf(f1,axiom,
( cartprodmempair1
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X2 @ ( cartprod @ X0 @ X1 ) )
=> ? [X3: $i] :
( ? [X4: $i] :
( ( X2
= ( kpair @ X3 @ X4 ) )
& ( in @ X4 @ X1 ) )
& ( in @ X3 @ X0 ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cartprodmempair1) ).
thf(f2,axiom,
( cartprodfstpairEq
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X2 @ X0 )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( kfst @ ( kpair @ X2 @ X3 ) )
= X2 ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cartprodfstpairEq) ).
thf(f3,axiom,
( cartprodsndpairEq
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X2 @ X0 )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( ksnd @ ( kpair @ X2 @ X3 ) )
= X3 ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cartprodsndpairEq) ).
thf(f4,conjecture,
( cartprodmempair1
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X2 @ ( cartprod @ X0 @ X1 ) )
=> ( ( kpair @ ( kfst @ X2 ) @ ( ksnd @ X2 ) )
= X2 ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cartprodpairsurjEq) ).
thf(f5,negated_conjecture,
~ ( cartprodmempair1
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X2 @ ( cartprod @ X0 @ X1 ) )
=> ( ( kpair @ ( kfst @ X2 ) @ ( ksnd @ X2 ) )
= X2 ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f4]) ).
thf(f6,plain,
( cartprodmempair1
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X2 @ ( cartprod @ X0 @ X1 ) )
=> ? [X3: $i] :
( ? [X4: $i] :
( ( X2
= ( kpair @ X3 @ X4 ) )
& ( in @ X4 @ X1 ) )
& ( in @ X3 @ X0 ) ) ) ) ),
inference(rectify,[],[f1]) ).
thf(f7,plain,
( cartprodmempair1
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y0 @ ( cartprod @ Y2 @ Y1 ) )
=> ( ?? @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y2 )
& ( ?? @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y1 )
& ( ( kpair @ Y3 @ Y4 )
= Y0 ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[],[f6]) ).
thf(f8,plain,
( cartprodfstpairEq
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X2 @ X0 )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( kfst @ ( kpair @ X2 @ X3 ) )
= X2 ) ) ) ) ),
inference(rectify,[],[f2]) ).
thf(f9,plain,
( cartprodfstpairEq
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y0 @ Y2 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y1 )
=> ( ( kfst @ ( kpair @ Y0 @ Y3 ) )
= Y0 ) ) ) ) ) ) ) ),
inference(fool_elimination,[],[f8]) ).
thf(f10,plain,
( cartprodsndpairEq
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X2 @ X0 )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( ksnd @ ( kpair @ X2 @ X3 ) )
= X3 ) ) ) ) ),
inference(rectify,[],[f3]) ).
thf(f11,plain,
( cartprodsndpairEq
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y0 @ Y2 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y1 )
=> ( ( ksnd @ ( kpair @ Y0 @ Y3 ) )
= Y3 ) ) ) ) ) ) ) ),
inference(fool_elimination,[],[f10]) ).
thf(f12,plain,
~ ( cartprodmempair1
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X2 @ ( cartprod @ X0 @ X1 ) )
=> ( ( kpair @ ( kfst @ X2 ) @ ( ksnd @ X2 ) )
= X2 ) ) ) ) ),
inference(rectify,[],[f5]) ).
thf(f13,plain,
~ ( ( cartprodmempair1 = $true )
=> ( ( cartprodfstpairEq = $true )
=> ( ( cartprodsndpairEq = $true )
=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ ( cartprod @ X0 @ X1 ) )
= $true )
=> ( ( kpair @ ( kfst @ X2 ) @ ( ksnd @ X2 ) )
= X2 ) ) ) ) ),
inference(fool_elimination,[],[f12]) ).
thf(f14,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( kpair @ ( kfst @ X2 ) @ ( ksnd @ X2 ) )
!= X2 )
& ( ( in @ X2 @ ( cartprod @ X0 @ X1 ) )
= $true ) )
& ( cartprodsndpairEq = $true )
& ( cartprodfstpairEq = $true )
& ( cartprodmempair1 = $true ) ),
inference(ennf_transformation,[],[f13]) ).
thf(f15,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( kpair @ ( kfst @ X2 ) @ ( ksnd @ X2 ) )
!= X2 )
& ( ( in @ X2 @ ( cartprod @ X0 @ X1 ) )
= $true ) )
& ( cartprodsndpairEq = $true )
& ( cartprodfstpairEq = $true )
& ( cartprodmempair1 = $true ) ),
inference(flattening,[],[f14]) ).
thf(f16,plain,
( ( sK2
!= ( kpair @ ( kfst @ sK2 ) @ ( ksnd @ sK2 ) ) )
& ( $true
= ( in @ sK2 @ ( cartprod @ sK0 @ sK1 ) ) )
& ( cartprodsndpairEq = $true )
& ( cartprodfstpairEq = $true )
& ( cartprodmempair1 = $true ) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK0,sK1,sK2]),skolemize(X0,sK0),skolemize(X1,sK1),skolemize(X2,sK2)],[f15]) ).
thf(f17,plain,
( cartprodmempair1
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y0 @ ( cartprod @ Y2 @ Y1 ) )
=> ( ?? @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y2 )
& ( ?? @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y1 )
& ( ( kpair @ Y3 @ Y4 )
= Y0 ) ) ) ) ) ) ) ) ) ),
inference(cnf_transformation,[],[f7]) ).
thf(f18,plain,
( cartprodfstpairEq
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y0 @ Y2 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y1 )
=> ( ( kfst @ ( kpair @ Y0 @ Y3 ) )
= Y0 ) ) ) ) ) ) ) ),
inference(cnf_transformation,[],[f9]) ).
thf(f19,plain,
( cartprodsndpairEq
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y0 @ Y2 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y1 )
=> ( ( ksnd @ ( kpair @ Y0 @ Y3 ) )
= Y3 ) ) ) ) ) ) ) ),
inference(cnf_transformation,[],[f11]) ).
thf(f20,plain,
cartprodmempair1 = $true,
inference(cnf_transformation,[],[f16]) ).
thf(f21,plain,
cartprodfstpairEq = $true,
inference(cnf_transformation,[],[f16]) ).
thf(f22,plain,
cartprodsndpairEq = $true,
inference(cnf_transformation,[],[f16]) ).
thf(f23,plain,
( $true
= ( in @ sK2 @ ( cartprod @ sK0 @ sK1 ) ) ),
inference(cnf_transformation,[],[f16]) ).
thf(f24,plain,
( sK2
!= ( kpair @ ( kfst @ sK2 ) @ ( ksnd @ sK2 ) ) ),
inference(cnf_transformation,[],[f16]) ).
thf(f27,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y0 @ ( cartprod @ Y2 @ Y1 ) )
=> ( ?? @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y2 )
& ( ?? @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y1 )
& ( ( kpair @ Y3 @ Y4 )
= Y0 ) ) ) ) ) ) ) ) )
= $true ),
inference(definition_unfolding,[],[f17,f20]) ).
thf(f28,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y0 @ Y2 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y1 )
=> ( ( kfst @ ( kpair @ Y0 @ Y3 ) )
= Y0 ) ) ) ) ) ) )
= $true ),
inference(definition_unfolding,[],[f18,f21]) ).
thf(f29,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y0 @ Y2 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y1 )
=> ( ( ksnd @ ( kpair @ Y0 @ Y3 ) )
= Y3 ) ) ) ) ) ) )
= $true ),
inference(definition_unfolding,[],[f19,f22]) ).
thf(f30,plain,
! [X1: $i] :
( $true
= ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y0 @ Y2 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y1 )
=> ( ( ksnd @ ( kpair @ Y0 @ Y3 ) )
= Y3 ) ) ) ) ) )
@ X1 ) ),
inference(pi_proxy_clausification,[],[f29]) ).
thf(f31,plain,
! [X1: $i] :
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( in @ X1 @ Y1 )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( ( ksnd @ ( kpair @ X1 @ Y2 ) )
= Y2 ) ) ) ) ) ) ),
inference(beta-eta_normalization,[],[f30]) ).
thf(f32,plain,
! [X2: $i,X1: $i] :
( $true
= ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( in @ X1 @ Y1 )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( ( ksnd @ ( kpair @ X1 @ Y2 ) )
= Y2 ) ) ) ) )
@ X2 ) ),
inference(pi_proxy_clausification,[],[f31]) ).
thf(f33,plain,
! [X2: $i,X1: $i] :
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ X1 @ Y0 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ X2 )
=> ( ( ksnd @ ( kpair @ X1 @ Y1 ) )
= Y1 ) ) ) ) ) ),
inference(beta-eta_normalization,[],[f32]) ).
thf(f34,plain,
! [X2: $i,X3: $i,X1: $i] :
( $true
= ( ^ [Y0: $i] :
( ( in @ X1 @ Y0 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ X2 )
=> ( ( ksnd @ ( kpair @ X1 @ Y1 ) )
= Y1 ) ) ) )
@ X3 ) ),
inference(pi_proxy_clausification,[],[f33]) ).
thf(f35,plain,
! [X2: $i,X3: $i,X1: $i] :
( $true
= ( ( in @ X1 @ X3 )
=> ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ X2 )
=> ( ( ksnd @ ( kpair @ X1 @ Y0 ) )
= Y0 ) ) ) ) ),
inference(beta-eta_normalization,[],[f34]) ).
thf(f36,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( $false
= ( in @ X1 @ X3 ) )
| ( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ X2 )
=> ( ( ksnd @ ( kpair @ X1 @ Y0 ) )
= Y0 ) ) ) ) ),
inference(imp_proxy_clausification,[],[f35]) ).
thf(f37,plain,
! [X2: $i,X3: $i,X1: $i,X4: $i] :
( ( $false
= ( in @ X1 @ X3 ) )
| ( $true
= ( ^ [Y0: $i] :
( ( in @ Y0 @ X2 )
=> ( ( ksnd @ ( kpair @ X1 @ Y0 ) )
= Y0 ) )
@ X4 ) ) ),
inference(pi_proxy_clausification,[],[f36]) ).
thf(f38,plain,
! [X2: $i,X3: $i,X1: $i,X4: $i] :
( ( $false
= ( in @ X1 @ X3 ) )
| ( $true
= ( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X1 @ X4 ) )
= X4 ) ) ) ),
inference(beta-eta_normalization,[],[f37]) ).
thf(f39,plain,
! [X2: $i,X3: $i,X1: $i,X4: $i] :
( ( $false
= ( in @ X1 @ X3 ) )
| ( $false
= ( in @ X4 @ X2 ) )
| ( $true
= ( ( ksnd @ ( kpair @ X1 @ X4 ) )
= X4 ) ) ),
inference(imp_proxy_clausification,[],[f38]) ).
thf(f40,plain,
! [X2: $i,X3: $i,X1: $i,X4: $i] :
( ( $false
= ( in @ X1 @ X3 ) )
| ( $false
= ( in @ X4 @ X2 ) )
| ( ( ksnd @ ( kpair @ X1 @ X4 ) )
= X4 ) ),
inference(equality_proxy_clausification,[],[f39]) ).
thf(f41,plain,
! [X1: $i] :
( $true
= ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y0 @ Y2 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y1 )
=> ( ( kfst @ ( kpair @ Y0 @ Y3 ) )
= Y0 ) ) ) ) ) )
@ X1 ) ),
inference(pi_proxy_clausification,[],[f28]) ).
thf(f42,plain,
! [X1: $i] :
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( in @ X1 @ Y1 )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( ( kfst @ ( kpair @ X1 @ Y2 ) )
= X1 ) ) ) ) ) ) ),
inference(beta-eta_normalization,[],[f41]) ).
thf(f43,plain,
! [X2: $i,X1: $i] :
( $true
= ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( in @ X1 @ Y1 )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( ( kfst @ ( kpair @ X1 @ Y2 ) )
= X1 ) ) ) ) )
@ X2 ) ),
inference(pi_proxy_clausification,[],[f42]) ).
thf(f44,plain,
! [X2: $i,X1: $i] :
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ X1 @ Y0 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ X2 )
=> ( ( kfst @ ( kpair @ X1 @ Y1 ) )
= X1 ) ) ) ) ) ),
inference(beta-eta_normalization,[],[f43]) ).
thf(f45,plain,
! [X2: $i,X3: $i,X1: $i] :
( $true
= ( ^ [Y0: $i] :
( ( in @ X1 @ Y0 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ X2 )
=> ( ( kfst @ ( kpair @ X1 @ Y1 ) )
= X1 ) ) ) )
@ X3 ) ),
inference(pi_proxy_clausification,[],[f44]) ).
thf(f46,plain,
! [X2: $i,X3: $i,X1: $i] :
( $true
= ( ( in @ X1 @ X3 )
=> ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ X2 )
=> ( ( kfst @ ( kpair @ X1 @ Y0 ) )
= X1 ) ) ) ) ),
inference(beta-eta_normalization,[],[f45]) ).
thf(f47,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( $false
= ( in @ X1 @ X3 ) )
| ( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ X2 )
=> ( ( kfst @ ( kpair @ X1 @ Y0 ) )
= X1 ) ) ) ) ),
inference(imp_proxy_clausification,[],[f46]) ).
thf(f48,plain,
! [X2: $i,X3: $i,X1: $i,X4: $i] :
( ( $false
= ( in @ X1 @ X3 ) )
| ( $true
= ( ^ [Y0: $i] :
( ( in @ Y0 @ X2 )
=> ( ( kfst @ ( kpair @ X1 @ Y0 ) )
= X1 ) )
@ X4 ) ) ),
inference(pi_proxy_clausification,[],[f47]) ).
thf(f49,plain,
! [X2: $i,X3: $i,X1: $i,X4: $i] :
( ( $false
= ( in @ X1 @ X3 ) )
| ( $true
= ( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X1 @ X4 ) )
= X1 ) ) ) ),
inference(beta-eta_normalization,[],[f48]) ).
thf(f50,plain,
! [X2: $i,X3: $i,X1: $i,X4: $i] :
( ( $false
= ( in @ X1 @ X3 ) )
| ( $false
= ( in @ X4 @ X2 ) )
| ( $true
= ( ( kfst @ ( kpair @ X1 @ X4 ) )
= X1 ) ) ),
inference(imp_proxy_clausification,[],[f49]) ).
thf(f51,plain,
! [X2: $i,X3: $i,X1: $i,X4: $i] :
( ( $false
= ( in @ X1 @ X3 ) )
| ( $false
= ( in @ X4 @ X2 ) )
| ( ( kfst @ ( kpair @ X1 @ X4 ) )
= X1 ) ),
inference(equality_proxy_clausification,[],[f50]) ).
thf(f52,plain,
! [X1: $i] :
( $true
= ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y0 @ ( cartprod @ Y2 @ Y1 ) )
=> ( ?? @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y2 )
& ( ?? @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y1 )
& ( ( kpair @ Y3 @ Y4 )
= Y0 ) ) ) ) ) ) ) )
@ X1 ) ),
inference(pi_proxy_clausification,[],[f27]) ).
thf(f53,plain,
! [X1: $i] :
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( in @ X1 @ ( cartprod @ Y1 @ Y0 ) )
=> ( ?? @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y1 )
& ( ?? @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
& ( ( kpair @ Y2 @ Y3 )
= X1 ) ) ) ) ) ) ) ) ),
inference(beta-eta_normalization,[],[f52]) ).
thf(f54,plain,
! [X2: $i,X1: $i] :
( $true
= ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( in @ X1 @ ( cartprod @ Y1 @ Y0 ) )
=> ( ?? @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y1 )
& ( ?? @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
& ( ( kpair @ Y2 @ Y3 )
= X1 ) ) ) ) ) ) )
@ X2 ) ),
inference(pi_proxy_clausification,[],[f53]) ).
thf(f55,plain,
! [X2: $i,X1: $i] :
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ X1 @ ( cartprod @ Y0 @ X2 ) )
=> ( ?? @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ Y0 )
& ( ?? @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ X2 )
& ( ( kpair @ Y1 @ Y2 )
= X1 ) ) ) ) ) ) ) ),
inference(beta-eta_normalization,[],[f54]) ).
thf(f56,plain,
! [X2: $i,X3: $i,X1: $i] :
( $true
= ( ^ [Y0: $i] :
( ( in @ X1 @ ( cartprod @ Y0 @ X2 ) )
=> ( ?? @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ Y0 )
& ( ?? @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ X2 )
& ( ( kpair @ Y1 @ Y2 )
= X1 ) ) ) ) ) )
@ X3 ) ),
inference(pi_proxy_clausification,[],[f55]) ).
thf(f57,plain,
! [X2: $i,X3: $i,X1: $i] :
( $true
= ( ( in @ X1 @ ( cartprod @ X3 @ X2 ) )
=> ( ?? @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ X3 )
& ( ?? @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ X2 )
& ( ( kpair @ Y0 @ Y1 )
= X1 ) ) ) ) ) ) ),
inference(beta-eta_normalization,[],[f56]) ).
thf(f58,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( $false
= ( in @ X1 @ ( cartprod @ X3 @ X2 ) ) )
| ( $true
= ( ?? @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ X3 )
& ( ?? @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ X2 )
& ( ( kpair @ Y0 @ Y1 )
= X1 ) ) ) ) ) ) ),
inference(imp_proxy_clausification,[],[f57]) ).
thf(f59,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( $false
= ( in @ X1 @ ( cartprod @ X3 @ X2 ) ) )
| ( $true
= ( ^ [Y0: $i] :
( ( in @ Y0 @ X3 )
& ( ?? @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ X2 )
& ( ( kpair @ Y0 @ Y1 )
= X1 ) ) ) )
@ ( sK3 @ X3 @ X1 @ X2 ) ) ) ),
inference(sigma_proxy_clausification,[],[f58]) ).
thf(f60,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( $false
= ( in @ X1 @ ( cartprod @ X3 @ X2 ) ) )
| ( $true
= ( ( in @ ( sK3 @ X3 @ X1 @ X2 ) @ X3 )
& ( ?? @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ X2 )
& ( ( kpair @ ( sK3 @ X3 @ X1 @ X2 ) @ Y0 )
= X1 ) ) ) ) ) ),
inference(beta-eta_normalization,[],[f59]) ).
thf(f61,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( $false
= ( in @ X1 @ ( cartprod @ X3 @ X2 ) ) )
| ( $true
= ( ?? @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ X2 )
& ( ( kpair @ ( sK3 @ X3 @ X1 @ X2 ) @ Y0 )
= X1 ) ) ) ) ),
inference(and_proxy_clausification,[],[f60]) ).
thf(f62,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( $true
= ( in @ ( sK3 @ X3 @ X1 @ X2 ) @ X3 ) )
| ( $false
= ( in @ X1 @ ( cartprod @ X3 @ X2 ) ) ) ),
inference(and_proxy_clausification,[],[f60]) ).
thf(f63,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( $false
= ( in @ X1 @ ( cartprod @ X3 @ X2 ) ) )
| ( $true
= ( ^ [Y0: $i] :
( ( in @ Y0 @ X2 )
& ( ( kpair @ ( sK3 @ X3 @ X1 @ X2 ) @ Y0 )
= X1 ) )
@ ( sK4 @ X3 @ X1 @ X2 ) ) ) ),
inference(sigma_proxy_clausification,[],[f61]) ).
thf(f64,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( $false
= ( in @ X1 @ ( cartprod @ X3 @ X2 ) ) )
| ( $true
= ( ( in @ ( sK4 @ X3 @ X1 @ X2 ) @ X2 )
& ( ( kpair @ ( sK3 @ X3 @ X1 @ X2 ) @ ( sK4 @ X3 @ X1 @ X2 ) )
= X1 ) ) ) ),
inference(beta-eta_normalization,[],[f63]) ).
thf(f65,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( $false
= ( in @ X1 @ ( cartprod @ X3 @ X2 ) ) )
| ( $true
= ( ( kpair @ ( sK3 @ X3 @ X1 @ X2 ) @ ( sK4 @ X3 @ X1 @ X2 ) )
= X1 ) ) ),
inference(and_proxy_clausification,[],[f64]) ).
thf(f66,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( $true
= ( in @ ( sK4 @ X3 @ X1 @ X2 ) @ X2 ) )
| ( $false
= ( in @ X1 @ ( cartprod @ X3 @ X2 ) ) ) ),
inference(and_proxy_clausification,[],[f64]) ).
thf(f67,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( ( kpair @ ( sK3 @ X3 @ X1 @ X2 ) @ ( sK4 @ X3 @ X1 @ X2 ) )
= X1 )
| ( $false
= ( in @ X1 @ ( cartprod @ X3 @ X2 ) ) ) ),
inference(equality_proxy_clausification,[],[f65]) ).
thf(f151,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i,X4: $i] :
( ( $true = $false )
| ( $false
= ( in @ X3 @ X4 ) )
| ( ( ksnd @ ( kpair @ ( sK3 @ X0 @ X1 @ X2 ) @ X3 ) )
= X3 )
| ( $false
= ( in @ X1 @ ( cartprod @ X0 @ X2 ) ) ) ),
inference(constrained_superposition,[],[f40,f62]) ).
thf(f153,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i,X4: $i] :
( ( $true = $false )
| ( $false
= ( in @ X3 @ X4 ) )
| ( ( sK3 @ X0 @ X1 @ X2 )
= ( kfst @ ( kpair @ ( sK3 @ X0 @ X1 @ X2 ) @ X3 ) ) )
| ( $false
= ( in @ X1 @ ( cartprod @ X0 @ X2 ) ) ) ),
inference(constrained_superposition,[],[f51,f62]) ).
thf(f164,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i,X4: $i] :
( ( $false
= ( in @ X3 @ X4 ) )
| ( ( sK3 @ X0 @ X1 @ X2 )
= ( kfst @ ( kpair @ ( sK3 @ X0 @ X1 @ X2 ) @ X3 ) ) )
| ( $false
= ( in @ X1 @ ( cartprod @ X0 @ X2 ) ) ) ),
inference(trivial_inequality_removal,[],[f153]) ).
thf(f166,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i,X4: $i] :
( ( $false
= ( in @ X3 @ X4 ) )
| ( ( ksnd @ ( kpair @ ( sK3 @ X0 @ X1 @ X2 ) @ X3 ) )
= X3 )
| ( $false
= ( in @ X1 @ ( cartprod @ X0 @ X2 ) ) ) ),
inference(trivial_inequality_removal,[],[f151]) ).
thf(f239,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i] :
( ( ( sK4 @ X1 @ X0 @ X2 )
= ( ksnd @ X0 ) )
| ( $false
= ( in @ ( sK4 @ X1 @ X0 @ X2 ) @ X3 ) )
| ( $false
= ( in @ X0 @ ( cartprod @ X1 @ X2 ) ) )
| ( $false
= ( in @ X0 @ ( cartprod @ X1 @ X2 ) ) ) ),
inference(constrained_superposition,[],[f166,f67]) ).
thf(f242,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i] :
( ( $false
= ( in @ ( sK4 @ X1 @ X0 @ X2 ) @ X3 ) )
| ( ( sK4 @ X1 @ X0 @ X2 )
= ( ksnd @ X0 ) )
| ( $false
= ( in @ X0 @ ( cartprod @ X1 @ X2 ) ) ) ),
inference(duplicate_literal_removal,[],[f239]) ).
thf(f256,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i] :
( ( ( sK3 @ X1 @ X0 @ X2 )
= ( kfst @ X0 ) )
| ( $false
= ( in @ ( sK4 @ X1 @ X0 @ X2 ) @ X3 ) )
| ( $false
= ( in @ X0 @ ( cartprod @ X1 @ X2 ) ) )
| ( $false
= ( in @ X0 @ ( cartprod @ X1 @ X2 ) ) ) ),
inference(constrained_superposition,[],[f164,f67]) ).
thf(f261,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i] :
( ( $false
= ( in @ ( sK4 @ X1 @ X0 @ X2 ) @ X3 ) )
| ( ( sK3 @ X1 @ X0 @ X2 )
= ( kfst @ X0 ) )
| ( $false
= ( in @ X0 @ ( cartprod @ X1 @ X2 ) ) ) ),
inference(duplicate_literal_removal,[],[f256]) ).
thf(f277,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( $true = $false )
| ( $false
= ( in @ X1 @ ( cartprod @ X0 @ X2 ) ) )
| ( ( sK4 @ X0 @ X1 @ X2 )
= ( ksnd @ X1 ) )
| ( $false
= ( in @ X1 @ ( cartprod @ X0 @ X2 ) ) ) ),
inference(constrained_superposition,[],[f66,f242]) ).
thf(f278,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( $true = $false )
| ( $false
= ( in @ X1 @ ( cartprod @ X0 @ X2 ) ) )
| ( ( sK4 @ X0 @ X1 @ X2 )
= ( ksnd @ X1 ) ) ),
inference(duplicate_literal_removal,[],[f277]) ).
thf(f279,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( $false
= ( in @ X1 @ ( cartprod @ X0 @ X2 ) ) )
| ( ( sK4 @ X0 @ X1 @ X2 )
= ( ksnd @ X1 ) ) ),
inference(trivial_inequality_removal,[],[f278]) ).
thf(f287,plain,
( ( $true = $false )
| ( ( ksnd @ sK2 )
= ( sK4 @ sK0 @ sK2 @ sK1 ) ) ),
inference(constrained_superposition,[],[f23,f279]) ).
thf(f292,plain,
( ( ksnd @ sK2 )
= ( sK4 @ sK0 @ sK2 @ sK1 ) ),
inference(trivial_inequality_removal,[],[f287]) ).
thf(f299,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( $true = $false )
| ( $false
= ( in @ X1 @ ( cartprod @ X0 @ X2 ) ) )
| ( ( sK3 @ X0 @ X1 @ X2 )
= ( kfst @ X1 ) )
| ( $false
= ( in @ X1 @ ( cartprod @ X0 @ X2 ) ) ) ),
inference(constrained_superposition,[],[f66,f261]) ).
thf(f300,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( $true = $false )
| ( $false
= ( in @ X1 @ ( cartprod @ X0 @ X2 ) ) )
| ( ( sK3 @ X0 @ X1 @ X2 )
= ( kfst @ X1 ) ) ),
inference(duplicate_literal_removal,[],[f299]) ).
thf(f301,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( $false
= ( in @ X1 @ ( cartprod @ X0 @ X2 ) ) )
| ( ( sK3 @ X0 @ X1 @ X2 )
= ( kfst @ X1 ) ) ),
inference(trivial_inequality_removal,[],[f300]) ).
thf(f384,plain,
( ( $true = $false )
| ( ( kfst @ sK2 )
= ( sK3 @ sK0 @ sK2 @ sK1 ) ) ),
inference(constrained_superposition,[],[f23,f301]) ).
thf(f389,plain,
( ( kfst @ sK2 )
= ( sK3 @ sK0 @ sK2 @ sK1 ) ),
inference(trivial_inequality_removal,[],[f384]) ).
thf(f395,plain,
( ( sK2
= ( kpair @ ( kfst @ sK2 ) @ ( sK4 @ sK0 @ sK2 @ sK1 ) ) )
| ( ( in @ sK2 @ ( cartprod @ sK0 @ sK1 ) )
= $false ) ),
inference(constrained_superposition,[],[f67,f389]) ).
thf(f438,plain,
( ( sK2
= ( kpair @ ( kfst @ sK2 ) @ ( ksnd @ sK2 ) ) )
| ( ( in @ sK2 @ ( cartprod @ sK0 @ sK1 ) )
= $false ) ),
inference(forward_demodulation,[],[f395,f292]) ).
thf(f441,plain,
( ( in @ sK2 @ ( cartprod @ sK0 @ sK1 ) )
= $false ),
inference(forward_subsumption_resolution,[],[f438,f24]) ).
thf(f460,plain,
$true = $false,
inference(constrained_superposition,[],[f23,f441]) ).
thf(f461,plain,
$false,
inference(trivial_inequality_removal,[],[f460]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : SEU664^2 : TPTP v9.3.1. Released v3.7.0.
% 0.00/0.06 % Command : run_vampire /export/starexec/sandbox2/benchmark/theBenchmark.p 300 SAT
% 0.08/0.20 % Computer : n013.cluster.edu
% 0.08/0.20 % Model : x86_64 x86_64
% 0.08/0.20 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.08/0.20 % Memory : 8046.5625MB
% 0.08/0.20 % OS : Linux 6.8.0-71-generic
% 0.08/0.20 % CPULimit : 300
% 0.08/0.20 % WCLimit : 300
% 0.08/0.20 % DateTime : Tue Sep 29 14:03:53 UTC 2026
% 0.08/0.20 % CPUTime :
% 0.08/0.20 Running run_vampire /export/starexec/sandbox2/benchmark/theBenchmark.p 300 SAT
% 0.08/0.23 Running first-order model finding
% 0.08/0.23 Running: /export/starexec/sandbox2/solver/bin/vampire-ho --input_syntax tptp --output_axiom_names on --mode casc --intent sat -m 16384 --cores 7 -t 300 /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.19/0.30 % (2169532)Will run a generic schedule for satisfiability detection.
% 0.19/0.30 % (2169541)ott+31_1_sil=16000:lcm=predicate:bce=on:newcnf=on:random_seed=885014219:i=116_2999 on theBenchmark for (2999ds/116Mi)
% 0.19/0.30 % (2169541)WARNING: Not using 'newcnf' as currently not compatible with higher-order inputs.
% 0.19/0.30 % (2169538)% WARNING: option uhcvi not known.
% 0.19/0.30 % (2169537)fmb+10_1_sas=cadical:bce=on:rp=on:random_seed=2401347782_2999 on theBenchmark for (2999ds/0Mi)
% 0.19/0.30 % (2169538)dis+11_61:31_drc=ordering:lsd=5:bsr=unit_only:rp=on:newcnf=on:random_seed=2084748795:i=135531:add=off:rawr=on_2999 on theBenchmark for (2999ds/135531Mi)
% 0.19/0.30 % (2169539)dis+10_161_sil=256000:plsq=on:plsqr=61199697,1048576:gs=on:alpa=true:sac=on:slsq=on:cn=on:random_seed=2105952340:i=88024:add=on:rawr=on_2999 on theBenchmark for (2999ds/88024Mi)
% 0.19/0.30 % (2169540)dis+10_1_sil=32000:sp=arity:random_seed=3508419267:i=103:fgj=on_2999 on theBenchmark for (2999ds/103Mi)
% 0.19/0.30 % (2169542)ott+1_1_to=lpo:sil=16000:sp=reverse_arity:erd=off:random_seed=1575275764:i=131_2999 on theBenchmark for (2999ds/131Mi)
% 0.19/0.30 % (2169543)ott-3_16_to=lpo:sil=16000:sp=arity:fd=off:rp=on:random_seed=1424944242:i=159:bs=unit_only:nicw=on:fsr=off:amm=off_2999 on theBenchmark for (2999ds/159Mi)
% 0.19/0.30 % Exception at run slice level
% 0.19/0.30 User error: % (2169538)WARNING: Not using 'newcnf' as currently not compatible with higher-order inputs.
% 0.19/0.30 Finite model building is currently not compatible with polymorphism or higher-order constructs
% 0.19/0.30 % (2169538)WARNING: Look ahead literal selection is not currently compatible with higher-order. Ignoring request to use
% 0.19/0.30 % (2169541) found proof, printing to "/export/starexec/sandbox2/tmp/vampire-proof-2169532-2169541"...
% 0.19/0.30 % (2169541)...printing done.
% 0.19/0.30 % (2169541)Refutation found. Thanks to Tanya!
% 0.19/0.30 % SZS status Theorem for theBenchmark
% 0.19/0.30 % SZS output start Proof for theBenchmark
% See solution above
% 0.19/0.30 % (2169541)------------------------------
% 0.19/0.30 % (2169541)Version: Vampire 5.0.1 (Release build, commit 5ef7c2677 on 2026-07-16 16:54:09 +0200)
% 0.19/0.30 % (2169541)Linked with Z3 4.14.0.0 3c47fd96cf5645d0c42b2c819d9e9a84380aa721 z3-4.8.4-9178-g3c47fd96c
% 0.19/0.30 % (2169541)CaDiCaL version: 2.1.3
% 0.19/0.30 % (2169541)Termination reason: Refutation
% 0.19/0.30 % (2169541)Time elapsed: 0.026 s
% 0.19/0.30 % (2169541)Peak memory usage: 12 MB
% 0.19/0.30 % (2169541)Instructions burned: 93 (million)
% 0.19/0.30 % (2169532)Success in time 0.063 s
% 0.19/0.30 % Vampire exiting
%------------------------------------------------------------------------------