%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : SET656+3 : TPTP v9.3.1. Released v2.2.0.
% Transfm : none
% Format : tptp:raw
% Command : prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n003.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 08:36:45 AM UTC 2026
% Result : Theorem 0.12s 0.42s
% Output : CNFRefutation 0.12s
% Verified :
% SZS Type : Refutation
% Derivation depth : 15
% Number of leaves : 9
% Syntax : Number of formulae : 55 ( 12 unt; 0 def)
% Number of atoms : 206 ( 17 equ)
% Maximal formula atoms : 18 ( 3 avg)
% Number of connectives : 255 ( 104 ~; 103 |; 18 &)
% ( 4 <=>; 26 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 5 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 6 ( 4 usr; 1 prp; 0-2 aty)
% Number of functors : 11 ( 11 usr; 4 con; 0-2 aty)
% Number of variables : 108 ( 2 sgn 39 !; 4 ?)
% Comments :
%------------------------------------------------------------------------------
fof(p1,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ( subset(B,C)
=> intersection(B,C) = B ) ) ),
file('theBenchmark.p',p1) ).
fof(p6,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',p6) ).
fof(p13,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',p13) ).
fof(p17,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',p17) ).
fof(p19,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',p19) ).
fof(p20,axiom,
! [B] :
( ilf_type(B,set_type)
=> ( ilf_type(power_set(B),set_type)
& ~ empty(power_set(B)) ) ),
file('theBenchmark.p',p20) ).
fof(p21,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',p21) ).
fof(p30,axiom,
! [B] : ilf_type(B,set_type),
file('theBenchmark.p',p30) ).
fof(prove_relset_1_18,conjecture,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ! [D] :
( ilf_type(D,relation_type(B,C))
=> intersection(D,cross_product(B,C)) = D ) ) ),
file('theBenchmark.p',prove_relset_1_18) ).
fof(prove_relset_1_18_neg,negated_conjecture,
~ ! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ! [D] :
( ilf_type(D,relation_type(B,C))
=> intersection(D,cross_product(B,C)) = D ) ) ),
inference(assume_negation,[status(cth)],[prove_relset_1_18]) ).
cnf(c_34,plain,
( intersection(A,B) = A
| ~ subset(A,B)
| ~ ilf_type(B,set_type)
| ~ ilf_type(A,set_type) ),
inference(clausify,[status(thm)],[p1]) ).
fof(nnf_17,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)],[p17]) ).
fof(sk_p17_sk,plain,
! [B] :
( ! [C] :
( ( ( ( ~ member(sK0(B,C),C)
& member(sK0(B,C),B)
& ilf_type(sK0(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,[sK0]),skolemize(D,sK0(B,C))],[nnf_17]) ).
fof(sk_p17,plain,
( ! [VAR_0,VAR_1] :
( ~ member(sK0(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(sK0(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(sK0(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_p17_sk]) ).
cnf(c_35,plain,
( member(sK0(A,B),A)
| subset(A,B)
| ~ ilf_type(B,set_type)
| ~ ilf_type(A,set_type) ),
inference(split_conjunct,[status(thm)],[sk_p17]) ).
cnf(c_36,plain,
( ~ member(sK0(A,B),B)
| subset(A,B)
| ~ ilf_type(B,set_type)
| ~ ilf_type(A,set_type) ),
inference(split_conjunct,[status(thm)],[sk_p17]) ).
cnf(c_102,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)],[p6]) ).
cnf(c_113,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)],[p13]) ).
cnf(c_116,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)],[p19]) ).
cnf(c_120,plain,
( ~ empty(power_set(A))
| ~ ilf_type(A,set_type) ),
inference(clausify,[status(thm)],[p20]) ).
cnf(c_122,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)],[p21]) ).
cnf(c_135,plain,
ilf_type(A,set_type),
inference(clausify,[status(thm)],[p30]) ).
fof(nnf_31,plain,
? [B] :
( ? [C] :
( ? [D] :
( intersection(D,cross_product(B,C)) != D
& ilf_type(D,relation_type(B,C)) )
& ilf_type(C,set_type) )
& ilf_type(B,set_type) ),
inference(fof_nnf,[status(thm)],[prove_relset_1_18_neg]) ).
fof(sk_prove_relset_1_18_neg,plain,
( intersection(sK1,cross_product(sK2,sK3)) != sK1
& ilf_type(sK1,relation_type(sK2,sK3))
& ilf_type(sK3,set_type)
& ilf_type(sK2,set_type) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK2,sK3,sK1]),skolemize(B,sK2),skolemize(C,sK3),skolemize(D,sK1)],[nnf_31]) ).
cnf(c_136,negated_conjecture,
ilf_type(sK1,relation_type(sK2,sK3)),
inference(split_conjunct,[status(thm)],[sk_prove_relset_1_18_neg]) ).
cnf(c_137,negated_conjecture,
intersection(sK1,cross_product(sK2,sK3)) != sK1,
inference(split_conjunct,[status(thm)],[sk_prove_relset_1_18_neg]) ).
cnf(c_140,plain,
( intersection(A,B) = A
| ~ ilf_type(B,set_type)
| ~ ilf_type(A,set_type)
| member(sK0(A,B),A)
| ~ ilf_type(B,set_type)
| ~ ilf_type(A,set_type) ),
inference(resolve,[status(thm)],[c_35,c_34]) ).
cnf(c_818,plain,
( intersection(A,B) = A
| ~ ilf_type(B,set_type)
| member(sK0(A,B),A)
| ~ ilf_type(B,set_type)
| ~ ilf_type(A,set_type) ),
inference(copy,[status(thm)],[c_140]) ).
cnf(c_819,plain,
( intersection(A,B) = A
| member(sK0(A,B),A)
| ~ ilf_type(B,set_type)
| ~ ilf_type(A,set_type) ),
inference(copy,[status(thm)],[c_818]) ).
cnf(c_820,plain,
( intersection(B,A) = B
| member(sK0(B,A),B)
| ~ ilf_type(A,set_type) ),
inference(resolve,[status(thm)],[c_135,c_819]) ).
cnf(c_141,plain,
( intersection(A,B) = A
| member(sK0(A,B),A) ),
inference(resolve,[status(thm)],[c_135,c_820]) ).
cnf(c_144,plain,
( intersection(A,B) = A
| ~ ilf_type(B,set_type)
| ~ ilf_type(A,set_type)
| ~ member(sK0(A,B),B)
| ~ ilf_type(B,set_type)
| ~ ilf_type(A,set_type) ),
inference(resolve,[status(thm)],[c_36,c_34]) ).
cnf(c_821,plain,
( intersection(A,B) = A
| ~ ilf_type(B,set_type)
| ~ member(sK0(A,B),B)
| ~ ilf_type(B,set_type)
| ~ ilf_type(A,set_type) ),
inference(copy,[status(thm)],[c_144]) ).
cnf(c_822,plain,
( intersection(A,B) = A
| ~ member(sK0(A,B),B)
| ~ ilf_type(B,set_type)
| ~ ilf_type(A,set_type) ),
inference(copy,[status(thm)],[c_821]) ).
cnf(c_823,plain,
( intersection(B,A) = B
| ~ member(sK0(B,A),A)
| ~ ilf_type(A,set_type) ),
inference(resolve,[status(thm)],[c_135,c_822]) ).
cnf(c_145,plain,
( intersection(A,B) = A
| ~ member(sK0(A,B),B) ),
inference(resolve,[status(thm)],[c_135,c_823]) ).
cnf(c_824,plain,
( member(B,A)
| ~ ilf_type(B,member_type(A))
| ~ ilf_type(A,set_type)
| empty(A) ),
inference(resolve,[status(thm)],[c_135,c_122]) ).
cnf(c_230,plain,
( member(B,A)
| ~ ilf_type(B,member_type(A))
| empty(A) ),
inference(resolve,[status(thm)],[c_135,c_824]) ).
cnf(c_231,plain,
~ empty(power_set(A)),
inference(resolve,[status(thm)],[c_135,c_120]) ).
cnf(c_825,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_135,c_116]) ).
cnf(c_826,plain,
( member(C,B)
| ~ member(C,A)
| ~ ilf_type(C,set_type)
| ~ member(A,power_set(B)) ),
inference(resolve,[status(thm)],[c_135,c_825]) ).
cnf(c_234,plain,
( member(C,B)
| ~ member(C,A)
| ~ member(A,power_set(B)) ),
inference(resolve,[status(thm)],[c_135,c_826]) ).
cnf(c_827,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_135,c_113]) ).
cnf(c_237,plain,
( ilf_type(A,member_type(power_set(B)))
| ~ ilf_type(A,subset_type(B)) ),
inference(resolve,[status(thm)],[c_135,c_827]) ).
cnf(c_828,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_135,c_102]) ).
cnf(c_240,plain,
( ilf_type(A,subset_type(cross_product(B,C)))
| ~ ilf_type(A,relation_type(B,C)) ),
inference(resolve,[status(thm)],[c_135,c_828]) ).
cnf(c_260,plain,
member(sK0(sK1,cross_product(sK2,sK3)),sK1),
inference(resolve,[status(thm)],[c_141,c_137]) ).
cnf(c_262,negated_conjecture,
~ member(sK0(sK1,cross_product(sK2,sK3)),cross_product(sK2,sK3)),
inference(resolve,[status(thm)],[c_145,c_137]) ).
cnf(c_362,plain,
ilf_type(sK1,subset_type(cross_product(sK2,sK3))),
inference(resolve,[status(thm)],[c_240,c_136]) ).
cnf(c_829,plain,
( member(sK0(sK1,cross_product(sK2,sK3)),A)
| ~ member(sK1,power_set(A)) ),
inference(resolve,[status(thm)],[c_234,c_260]) ).
cnf(c_629,plain,
~ member(sK1,power_set(cross_product(sK2,sK3))),
inference(resolve,[status(thm)],[c_829,c_262]) ).
cnf(c_830,plain,
( member(A,power_set(B))
| ~ ilf_type(A,member_type(power_set(B))) ),
inference(resolve,[status(thm)],[c_230,c_231]) ).
cnf(c_649,plain,
~ ilf_type(sK1,member_type(power_set(cross_product(sK2,sK3)))),
inference(resolve,[status(thm)],[c_830,c_629]) ).
cnf(c_831,plain,
~ ilf_type(sK1,subset_type(cross_product(sK2,sK3))),
inference(resolve,[status(thm)],[c_237,c_649]) ).
cnf(c_817,plain,
$false,
inference(resolve,[status(thm)],[c_362,c_831]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : SET656+3 : TPTP v9.3.1. Released v2.2.0.
% 0.00/0.03 % Command : prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.08/0.35 % Computer : n003.cluster.edu
% 0.08/0.35 % Model : x86_64 x86_64
% 0.08/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.08/0.35 % Memory : 8046.5625MB
% 0.08/0.35 % OS : Linux 6.8.0-71-generic
% 0.08/0.35 % CPULimit : 300
% 0.08/0.35 % WCLimit : 300
% 0.08/0.35 % DateTime : Sat Sep 26 08:05:09 UTC 2026
% 0.08/0.35 % CPUTime :
% 0.08/0.35 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.08/0.35 % Process 2394979 was started by sandbox2 on n003,
% 0.08/0.35 % Sat Sep 26 08:05:09 2026
% 0.08/0.35 % The command was "/export/starexec/sandbox2/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p".
% 0.08/0.36
% 0.08/0.36 % From the command line: assign(max_seconds, 300).
% 0.12/0.42
% 0.12/0.42 % SZS status Theorem for theBenchmark
% 0.12/0.42
% 0.12/0.42 % Proof 1 at 0.03 (+ 0.01) seconds.
% 0.12/0.42 % Length of proof is 35.
% 0.12/0.42 % Level of proof is 7.
% 0.12/0.42 % Maximum clause weight is 32.000.
% 0.12/0.42 % Given clauses 96.
% 0.12/0.42
% 0.12/0.42 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------