%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : SET640+3 : TPTP v9.3.1. Released v2.2.0.
% Transfm : none
% Format : tptp:raw
% Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n007.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 : Sun Sep 27 08:36:42 AM UTC 2026
% Result : Theorem 0.11s 0.38s
% Output : CNFRefutation 0.11s
% Verified :
% SZS Type : Refutation
% Derivation depth : 11
% Number of leaves : 9
% Syntax : Number of formulae : 55 ( 14 unt; 0 def)
% Number of atoms : 204 ( 0 equ)
% Maximal formula atoms : 18 ( 3 avg)
% Number of connectives : 244 ( 95 ~; 91 |; 23 &)
% ( 4 <=>; 31 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 5 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 5 ( 4 usr; 1 prp; 0-2 aty)
% Number of functors : 11 ( 11 usr; 5 con; 0-2 aty)
% Number of variables : 111 ( 2 sgn 42 !; 5 ?)
% Comments :
%------------------------------------------------------------------------------
fof(p1,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ! [D] :
( ilf_type(D,set_type)
=> ( ( subset(C,D)
& subset(B,C) )
=> subset(B,D) ) ) ) ),
file('theBenchmark.p',p1) ).
fof(p4,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ( ! [E] :
( ilf_type(E,relation_type(B,C))
=> ilf_type(E,subset_type(cross_product(B,C))) )
& ! [D] :
( ilf_type(D,subset_type(cross_product(B,C)))
=> ilf_type(D,relation_type(B,C)) ) ) ) ),
file('theBenchmark.p',p4) ).
fof(p6,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ( subset(B,C)
<=> ! [D] :
( ilf_type(D,set_type)
=> ( member(D,B)
=> member(D,C) ) ) ) ) ),
file('theBenchmark.p',p6) ).
fof(p8,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ( ilf_type(C,subset_type(B))
<=> ilf_type(C,member_type(power_set(B))) ) ) ),
file('theBenchmark.p',p8) ).
fof(p11,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ( member(B,power_set(C))
<=> ! [D] :
( ilf_type(D,set_type)
=> ( member(D,B)
=> member(D,C) ) ) ) ) ),
file('theBenchmark.p',p11) ).
fof(p12,axiom,
! [B] :
( ilf_type(B,set_type)
=> ( ilf_type(power_set(B),set_type)
& ~ empty(power_set(B)) ) ),
file('theBenchmark.p',p12) ).
fof(p13,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ( ilf_type(C,set_type)
& ~ empty(C) )
=> ( ilf_type(B,member_type(C))
<=> member(B,C) ) ) ),
file('theBenchmark.p',p13) ).
fof(p19,axiom,
! [B] : ilf_type(B,set_type),
file('theBenchmark.p',p19) ).
fof(prove_relset_1_2,conjecture,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ! [D] :
( ilf_type(D,set_type)
=> ! [E] :
( ilf_type(E,relation_type(C,D))
=> ( subset(B,E)
=> subset(B,cross_product(C,D)) ) ) ) ) ),
file('theBenchmark.p',prove_relset_1_2) ).
fof(prove_relset_1_2_neg,negated_conjecture,
~ ! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ! [D] :
( ilf_type(D,set_type)
=> ! [E] :
( ilf_type(E,relation_type(C,D))
=> ( subset(B,E)
=> subset(B,cross_product(C,D)) ) ) ) ) ),
inference(assume_negation,[status(cth)],[prove_relset_1_2]) ).
fof(nnf_20,plain,
? [B] :
( ? [C] :
( ? [D] :
( ? [E] :
( ~ subset(B,cross_product(C,D))
& subset(B,E)
& ilf_type(E,relation_type(C,D)) )
& ilf_type(D,set_type) )
& ilf_type(C,set_type) )
& ilf_type(B,set_type) ),
inference(fof_nnf,[status(thm)],[prove_relset_1_2_neg]) ).
fof(sk_prove_relset_1_2_neg,plain,
( ~ subset(sK0,cross_product(sK1,sK2))
& subset(sK0,sK3)
& ilf_type(sK3,relation_type(sK1,sK2))
& ilf_type(sK2,set_type)
& ilf_type(sK1,set_type)
& ilf_type(sK0,set_type) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK0,sK1,sK2,sK3]),skolemize(B,sK0),skolemize(C,sK1),skolemize(D,sK2),skolemize(E,sK3)],[nnf_20]) ).
cnf(c_36,negated_conjecture,
~ subset(sK0,cross_product(sK1,sK2)),
inference(split_conjunct,[status(thm)],[sk_prove_relset_1_2_neg]) ).
cnf(c_37,negated_conjecture,
subset(sK0,sK3),
inference(split_conjunct,[status(thm)],[sk_prove_relset_1_2_neg]) ).
cnf(c_38,negated_conjecture,
ilf_type(sK3,relation_type(sK1,sK2)),
inference(split_conjunct,[status(thm)],[sk_prove_relset_1_2_neg]) ).
cnf(c_42,plain,
ilf_type(A,set_type),
inference(clausify,[status(thm)],[p19]) ).
cnf(c_52,plain,
( member(A,B)
| ~ ilf_type(A,member_type(B))
| ~ ilf_type(B,set_type)
| empty(B)
| ~ ilf_type(A,set_type) ),
inference(clausify,[status(thm)],[p13]) ).
cnf(c_373,plain,
( member(B,A)
| ~ ilf_type(B,member_type(A))
| ~ ilf_type(A,set_type)
| empty(A) ),
inference(resolve,[status(thm)],[c_42,c_52]) ).
cnf(c_53,plain,
( member(B,A)
| ~ ilf_type(B,member_type(A))
| empty(A) ),
inference(resolve,[status(thm)],[c_42,c_373]) ).
cnf(c_55,plain,
( ~ empty(power_set(A))
| ~ ilf_type(A,set_type) ),
inference(clausify,[status(thm)],[p12]) ).
cnf(c_56,plain,
~ empty(power_set(A)),
inference(resolve,[status(thm)],[c_42,c_55]) ).
cnf(c_62,plain,
( member(C,B)
| ~ member(C,A)
| ~ ilf_type(C,set_type)
| ~ member(A,power_set(B))
| ~ ilf_type(B,set_type)
| ~ ilf_type(A,set_type) ),
inference(clausify,[status(thm)],[p11]) ).
cnf(c_374,plain,
( member(C,A)
| ~ member(C,B)
| ~ ilf_type(C,set_type)
| ~ member(B,power_set(A))
| ~ ilf_type(A,set_type) ),
inference(resolve,[status(thm)],[c_42,c_62]) ).
cnf(c_375,plain,
( member(C,B)
| ~ member(C,A)
| ~ ilf_type(C,set_type)
| ~ member(A,power_set(B)) ),
inference(resolve,[status(thm)],[c_42,c_374]) ).
cnf(c_63,plain,
( member(C,B)
| ~ member(C,A)
| ~ member(A,power_set(B)) ),
inference(resolve,[status(thm)],[c_42,c_375]) ).
cnf(c_70,plain,
( ilf_type(B,member_type(power_set(A)))
| ~ ilf_type(B,subset_type(A))
| ~ ilf_type(B,set_type)
| ~ ilf_type(A,set_type) ),
inference(clausify,[status(thm)],[p8]) ).
cnf(c_376,plain,
( ilf_type(A,member_type(power_set(B)))
| ~ ilf_type(A,subset_type(B))
| ~ ilf_type(A,set_type) ),
inference(resolve,[status(thm)],[c_42,c_70]) ).
cnf(c_71,plain,
( ilf_type(A,member_type(power_set(B)))
| ~ ilf_type(A,subset_type(B)) ),
inference(resolve,[status(thm)],[c_42,c_376]) ).
fof(nnf_6,plain,
! [B] :
( ! [C] :
( ( ( ? [D] :
( ~ member(D,C)
& member(D,B)
& ilf_type(D,set_type) )
| subset(B,C) )
& ( ! [X0] :
( member(X0,C)
| ~ member(X0,B)
| ~ ilf_type(X0,set_type) )
| ~ subset(B,C) ) )
| ~ ilf_type(C,set_type) )
| ~ ilf_type(B,set_type) ),
inference(fof_nnf,[status(thm)],[p6]) ).
fof(sk_p6_sk,plain,
! [B] :
( ! [C] :
( ( ( ( ~ member(sK4(B,C),C)
& member(sK4(B,C),B)
& ilf_type(sK4(B,C),set_type) )
| subset(B,C) )
& ( ! [X0] :
( member(X0,C)
| ~ member(X0,B)
| ~ ilf_type(X0,set_type) )
| ~ subset(B,C) ) )
| ~ ilf_type(C,set_type) )
| ~ ilf_type(B,set_type) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK4]),skolemize(D,sK4(B,C))],[nnf_6]) ).
fof(sk_p6,plain,
( ! [VAR_0,VAR_1] :
( ~ member(sK4(VAR_0,VAR_1),VAR_1)
| subset(VAR_0,VAR_1)
| ~ ilf_type(VAR_1,set_type)
| ~ ilf_type(VAR_0,set_type) )
& ! [VAR_0,VAR_1] :
( member(sK4(VAR_0,VAR_1),VAR_0)
| subset(VAR_0,VAR_1)
| ~ ilf_type(VAR_1,set_type)
| ~ ilf_type(VAR_0,set_type) )
& ! [VAR_0,VAR_1] :
( ilf_type(sK4(VAR_0,VAR_1),set_type)
| subset(VAR_0,VAR_1)
| ~ ilf_type(VAR_1,set_type)
| ~ ilf_type(VAR_0,set_type) )
& ! [VAR_0,VAR_1,VAR_2] :
( member(VAR_2,VAR_1)
| ~ member(VAR_2,VAR_0)
| ~ ilf_type(VAR_2,set_type)
| ~ subset(VAR_0,VAR_1)
| ~ ilf_type(VAR_1,set_type)
| ~ ilf_type(VAR_0,set_type) ) ),
inference(cnf_transformation,[status(thm)],[sk_p6_sk]) ).
cnf(c_73,plain,
( ~ member(sK4(A,B),B)
| subset(A,B)
| ~ ilf_type(B,set_type)
| ~ ilf_type(A,set_type) ),
inference(split_conjunct,[status(thm)],[sk_p6]) ).
cnf(c_377,plain,
( ~ member(sK4(B,A),A)
| subset(B,A)
| ~ ilf_type(A,set_type) ),
inference(resolve,[status(thm)],[c_42,c_73]) ).
cnf(c_74,plain,
( ~ member(sK4(A,B),B)
| subset(A,B) ),
inference(resolve,[status(thm)],[c_42,c_377]) ).
cnf(c_75,plain,
( member(sK4(A,B),A)
| subset(A,B)
| ~ ilf_type(B,set_type)
| ~ ilf_type(A,set_type) ),
inference(split_conjunct,[status(thm)],[sk_p6]) ).
cnf(c_378,plain,
( member(sK4(B,A),B)
| subset(B,A)
| ~ ilf_type(A,set_type) ),
inference(resolve,[status(thm)],[c_42,c_75]) ).
cnf(c_76,plain,
( member(sK4(A,B),A)
| subset(A,B) ),
inference(resolve,[status(thm)],[c_42,c_378]) ).
cnf(c_82,plain,
( ilf_type(C,subset_type(cross_product(A,B)))
| ~ ilf_type(C,relation_type(A,B))
| ~ ilf_type(B,set_type)
| ~ ilf_type(A,set_type) ),
inference(clausify,[status(thm)],[p4]) ).
cnf(c_379,plain,
( ilf_type(B,subset_type(cross_product(C,A)))
| ~ ilf_type(B,relation_type(C,A))
| ~ ilf_type(A,set_type) ),
inference(resolve,[status(thm)],[c_42,c_82]) ).
cnf(c_83,plain,
( ilf_type(A,subset_type(cross_product(B,C)))
| ~ ilf_type(A,relation_type(B,C)) ),
inference(resolve,[status(thm)],[c_42,c_379]) ).
cnf(c_97,plain,
( subset(A,C)
| ~ subset(B,C)
| ~ subset(A,B)
| ~ ilf_type(C,set_type)
| ~ ilf_type(B,set_type)
| ~ ilf_type(A,set_type) ),
inference(clausify,[status(thm)],[p1]) ).
cnf(c_380,plain,
( subset(C,B)
| ~ subset(A,B)
| ~ subset(C,A)
| ~ ilf_type(B,set_type)
| ~ ilf_type(A,set_type) ),
inference(resolve,[status(thm)],[c_42,c_97]) ).
cnf(c_381,plain,
( subset(B,A)
| ~ subset(C,A)
| ~ subset(B,C)
| ~ ilf_type(A,set_type) ),
inference(resolve,[status(thm)],[c_42,c_380]) ).
cnf(c_98,plain,
( subset(A,C)
| ~ subset(B,C)
| ~ subset(A,B) ),
inference(resolve,[status(thm)],[c_42,c_381]) ).
cnf(c_134,plain,
ilf_type(sK3,subset_type(cross_product(sK1,sK2))),
inference(resolve,[status(thm)],[c_83,c_38]) ).
cnf(c_382,negated_conjecture,
( subset(sK0,A)
| ~ subset(sK3,A) ),
inference(resolve,[status(thm)],[c_98,c_37]) ).
cnf(c_150,negated_conjecture,
~ subset(sK3,cross_product(sK1,sK2)),
inference(resolve,[status(thm)],[c_382,c_36]) ).
cnf(c_160,plain,
member(sK4(sK3,cross_product(sK1,sK2)),sK3),
inference(resolve,[status(thm)],[c_150,c_76]) ).
cnf(c_161,plain,
~ member(sK4(sK3,cross_product(sK1,sK2)),cross_product(sK1,sK2)),
inference(resolve,[status(thm)],[c_74,c_150]) ).
cnf(c_245,plain,
ilf_type(sK3,member_type(power_set(cross_product(sK1,sK2)))),
inference(resolve,[status(thm)],[c_134,c_71]) ).
cnf(c_383,plain,
( member(sK4(sK3,cross_product(sK1,sK2)),A)
| ~ member(sK3,power_set(A)) ),
inference(resolve,[status(thm)],[c_63,c_160]) ).
cnf(c_362,plain,
~ member(sK3,power_set(cross_product(sK1,sK2))),
inference(resolve,[status(thm)],[c_383,c_161]) ).
cnf(c_384,plain,
( member(A,power_set(B))
| ~ ilf_type(A,member_type(power_set(B))) ),
inference(resolve,[status(thm)],[c_53,c_56]) ).
cnf(c_385,plain,
~ ilf_type(sK3,member_type(power_set(cross_product(sK1,sK2)))),
inference(resolve,[status(thm)],[c_384,c_362]) ).
cnf(c_372,plain,
$false,
inference(resolve,[status(thm)],[c_245,c_385]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : SET640+3 : TPTP v9.3.1. Released v2.2.0.
% 0.00/0.03 % Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.07/0.34 % Computer : n007.cluster.edu
% 0.07/0.34 % Model : x86_64 x86_64
% 0.07/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.07/0.34 % Memory : 8046.5625MB
% 0.07/0.34 % OS : Linux 6.8.0-71-generic
% 0.07/0.34 % CPULimit : 300
% 0.07/0.34 % WCLimit : 300
% 0.07/0.34 % DateTime : Sat Sep 26 08:00:09 UTC 2026
% 0.07/0.34 % CPUTime :
% 0.07/0.34 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.07/0.34 % Process 3222038 was started by sandbox on n007,
% 0.07/0.34 % Sat Sep 26 08:00:09 2026
% 0.07/0.34 % The command was "/export/starexec/sandbox/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p".
% 0.07/0.35
% 0.07/0.35 % From the command line: assign(max_seconds, 300).
% 0.11/0.38
% 0.11/0.38 % SZS status Theorem for theBenchmark
% 0.11/0.38
% 0.11/0.38 % Proof 1 at 0.01 (+ 0.00) seconds.
% 0.11/0.38 % Length of proof is 36.
% 0.11/0.38 % Level of proof is 6.
% 0.11/0.38 % Maximum clause weight is 11.000.
% 0.11/0.38 % Given clauses 89.
% 0.11/0.38
% 0.11/0.38 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------