%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : SET641+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 : n006.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 0.10s 5.39s
% Output : CNFRefutation 0.10s
% Verified :
% SZS Type : Refutation
% Derivation depth : 10
% Number of leaves : 8
% Syntax : Number of formulae : 47 ( 12 unt; 0 def)
% Number of atoms : 172 ( 0 equ)
% Maximal formula atoms : 18 ( 3 avg)
% Number of connectives : 206 ( 81 ~; 76 |; 20 &)
% ( 4 <=>; 25 =>; 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 : 10 ( 10 usr; 4 con; 0-2 aty)
% Number of variables : 86 ( 2 sgn 37 !; 4 ?)
% Comments :
%------------------------------------------------------------------------------
fof(p1,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',p1) ).
fof(p5,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',p5) ).
fof(p7,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',p7) ).
fof(p10,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',p10) ).
fof(p11,axiom,
! [B] :
( ilf_type(B,set_type)
=> ( ilf_type(power_set(B),set_type)
& ~ empty(power_set(B)) ) ),
file('theBenchmark.p',p11) ).
fof(p12,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',p12) ).
fof(p18,axiom,
! [B] : ilf_type(B,set_type),
file('theBenchmark.p',p18) ).
fof(prove_relset_1_3,conjecture,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ! [D] :
( ilf_type(D,set_type)
=> ( subset(B,cross_product(C,D))
=> ilf_type(B,relation_type(C,D)) ) ) ) ),
file('theBenchmark.p',prove_relset_1_3) ).
fof(prove_relset_1_3_neg,negated_conjecture,
~ ! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ! [D] :
( ilf_type(D,set_type)
=> ( subset(B,cross_product(C,D))
=> ilf_type(B,relation_type(C,D)) ) ) ) ),
inference(assume_negation,[status(cth)],[prove_relset_1_3]) ).
cnf(c_21,plain,
( member(C,B)
| ~ member(C,A)
| ~ ilf_type(C,set_type)
| ~ subset(A,B)
| ~ ilf_type(B,set_type)
| ~ ilf_type(A,set_type) ),
inference(clausify,[status(thm)],[p5]) ).
fof(nnf_19,plain,
? [B] :
( ? [C] :
( ? [D] :
( ~ ilf_type(B,relation_type(C,D))
& subset(B,cross_product(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_3_neg]) ).
fof(sk_prove_relset_1_3_neg,plain,
( ~ ilf_type(sK0,relation_type(sK1,sK2))
& subset(sK0,cross_product(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]),skolemize(B,sK0),skolemize(C,sK1),skolemize(D,sK2)],[nnf_19]) ).
cnf(c_25,negated_conjecture,
subset(sK0,cross_product(sK1,sK2)),
inference(split_conjunct,[status(thm)],[sk_prove_relset_1_3_neg]) ).
cnf(c_55,plain,
( ilf_type(C,relation_type(A,B))
| ~ ilf_type(C,subset_type(cross_product(A,B)))
| ~ ilf_type(B,set_type)
| ~ ilf_type(A,set_type) ),
inference(clausify,[status(thm)],[p1]) ).
cnf(c_67,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)],[p7]) ).
fof(nnf_10,plain,
! [B] :
( ! [C] :
( ( ( ? [D] :
( ~ member(D,C)
& member(D,B)
& ilf_type(D,set_type) )
| member(B,power_set(C)) )
& ( ! [X0] :
( member(X0,C)
| ~ member(X0,B)
| ~ ilf_type(X0,set_type) )
| ~ member(B,power_set(C)) ) )
| ~ ilf_type(C,set_type) )
| ~ ilf_type(B,set_type) ),
inference(fof_nnf,[status(thm)],[p10]) ).
fof(sk_p10_sk,plain,
! [B] :
( ! [C] :
( ( ( ( ~ member(sK3(B,C),C)
& member(sK3(B,C),B)
& ilf_type(sK3(B,C),set_type) )
| member(B,power_set(C)) )
& ( ! [X0] :
( member(X0,C)
| ~ member(X0,B)
| ~ ilf_type(X0,set_type) )
| ~ member(B,power_set(C)) ) )
| ~ ilf_type(C,set_type) )
| ~ ilf_type(B,set_type) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK3]),skolemize(D,sK3(B,C))],[nnf_10]) ).
fof(sk_p10,plain,
( ! [VAR_0,VAR_1] :
( ~ member(sK3(VAR_0,VAR_1),VAR_1)
| member(VAR_0,power_set(VAR_1))
| ~ ilf_type(VAR_1,set_type)
| ~ ilf_type(VAR_0,set_type) )
& ! [VAR_0,VAR_1] :
( member(sK3(VAR_0,VAR_1),VAR_0)
| member(VAR_0,power_set(VAR_1))
| ~ ilf_type(VAR_1,set_type)
| ~ ilf_type(VAR_0,set_type) )
& ! [VAR_0,VAR_1] :
( ilf_type(sK3(VAR_0,VAR_1),set_type)
| member(VAR_0,power_set(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)
| ~ member(VAR_0,power_set(VAR_1))
| ~ ilf_type(VAR_1,set_type)
| ~ ilf_type(VAR_0,set_type) ) ),
inference(cnf_transformation,[status(thm)],[sk_p10_sk]) ).
cnf(c_71,plain,
( member(sK3(A,B),A)
| member(A,power_set(B))
| ~ ilf_type(B,set_type)
| ~ ilf_type(A,set_type) ),
inference(split_conjunct,[status(thm)],[sk_p10]) ).
cnf(c_72,plain,
( ~ member(sK3(A,B),B)
| member(A,power_set(B))
| ~ ilf_type(B,set_type)
| ~ ilf_type(A,set_type) ),
inference(split_conjunct,[status(thm)],[sk_p10]) ).
cnf(c_73,plain,
( ~ empty(power_set(A))
| ~ ilf_type(A,set_type) ),
inference(clausify,[status(thm)],[p11]) ).
cnf(c_76,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)],[p12]) ).
cnf(c_81,plain,
ilf_type(A,set_type),
inference(clausify,[status(thm)],[p18]) ).
cnf(c_82,negated_conjecture,
~ ilf_type(sK0,relation_type(sK1,sK2)),
inference(split_conjunct,[status(thm)],[sk_prove_relset_1_3_neg]) ).
cnf(c_87,negated_conjecture,
( member(A,cross_product(sK1,sK2))
| ~ member(A,sK0)
| ~ ilf_type(A,set_type)
| ~ ilf_type(cross_product(sK1,sK2),set_type)
| ~ ilf_type(sK0,set_type) ),
inference(resolve,[status(thm)],[c_25,c_21]) ).
cnf(c_236,negated_conjecture,
( member(A,cross_product(sK1,sK2))
| ~ member(A,sK0)
| ~ ilf_type(A,set_type)
| ~ ilf_type(cross_product(sK1,sK2),set_type) ),
inference(resolve,[status(thm)],[c_81,c_87]) ).
cnf(c_237,negated_conjecture,
( member(A,cross_product(sK1,sK2))
| ~ member(A,sK0)
| ~ ilf_type(A,set_type) ),
inference(resolve,[status(thm)],[c_81,c_236]) ).
cnf(c_88,negated_conjecture,
( member(A,cross_product(sK1,sK2))
| ~ member(A,sK0) ),
inference(resolve,[status(thm)],[c_81,c_237]) ).
cnf(c_238,plain,
( ~ member(B,A)
| ilf_type(B,member_type(A))
| ~ ilf_type(A,set_type)
| empty(A) ),
inference(resolve,[status(thm)],[c_81,c_76]) ).
cnf(c_138,plain,
( ~ member(B,A)
| ilf_type(B,member_type(A))
| empty(A) ),
inference(resolve,[status(thm)],[c_81,c_238]) ).
cnf(c_140,plain,
~ empty(power_set(A)),
inference(resolve,[status(thm)],[c_81,c_73]) ).
cnf(c_239,plain,
( ~ member(sK3(B,A),A)
| member(B,power_set(A))
| ~ ilf_type(A,set_type) ),
inference(resolve,[status(thm)],[c_81,c_72]) ).
cnf(c_141,plain,
( ~ member(sK3(A,B),B)
| member(A,power_set(B)) ),
inference(resolve,[status(thm)],[c_81,c_239]) ).
cnf(c_240,plain,
( member(sK3(B,A),B)
| member(B,power_set(A))
| ~ ilf_type(A,set_type) ),
inference(resolve,[status(thm)],[c_81,c_71]) ).
cnf(c_142,plain,
( member(sK3(A,B),A)
| member(A,power_set(B)) ),
inference(resolve,[status(thm)],[c_81,c_240]) ).
cnf(c_241,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_81,c_67]) ).
cnf(c_145,plain,
( ~ ilf_type(A,member_type(power_set(B)))
| ilf_type(A,subset_type(B)) ),
inference(resolve,[status(thm)],[c_81,c_241]) ).
cnf(c_242,plain,
( ilf_type(B,relation_type(C,A))
| ~ ilf_type(B,subset_type(cross_product(C,A)))
| ~ ilf_type(A,set_type) ),
inference(resolve,[status(thm)],[c_81,c_55]) ).
cnf(c_153,plain,
( ilf_type(A,relation_type(B,C))
| ~ ilf_type(A,subset_type(cross_product(B,C))) ),
inference(resolve,[status(thm)],[c_81,c_242]) ).
cnf(c_205,negated_conjecture,
~ ilf_type(sK0,subset_type(cross_product(sK1,sK2))),
inference(resolve,[status(thm)],[c_153,c_82]) ).
cnf(c_231,plain,
~ ilf_type(sK0,member_type(power_set(cross_product(sK1,sK2)))),
inference(resolve,[status(thm)],[c_145,c_205]) ).
cnf(c_243,plain,
( ~ member(A,power_set(B))
| ilf_type(A,member_type(power_set(B))) ),
inference(resolve,[status(thm)],[c_138,c_140]) ).
cnf(c_232,plain,
~ member(sK0,power_set(cross_product(sK1,sK2))),
inference(resolve,[status(thm)],[c_243,c_231]) ).
cnf(c_233,plain,
member(sK3(sK0,cross_product(sK1,sK2)),sK0),
inference(resolve,[status(thm)],[c_232,c_142]) ).
cnf(c_234,plain,
~ member(sK3(sK0,cross_product(sK1,sK2)),cross_product(sK1,sK2)),
inference(resolve,[status(thm)],[c_141,c_232]) ).
cnf(c_244,negated_conjecture,
~ member(sK3(sK0,cross_product(sK1,sK2)),sK0),
inference(resolve,[status(thm)],[c_88,c_234]) ).
cnf(c_235,negated_conjecture,
$false,
inference(resolve,[status(thm)],[c_233,c_244]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : SET641+3 : TPTP v9.3.1. Released v2.2.0.
% 0.00/0.04 % Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.10/5.37 % Computer : n006.cluster.edu
% 0.10/5.37 % Model : x86_64 x86_64
% 0.10/5.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/5.37 % Memory : 8046.5625MB
% 0.10/5.37 % OS : Linux 6.8.0-71-generic
% 0.10/5.37 % CPULimit : 300
% 0.10/5.37 % WCLimit : 300
% 0.10/5.37 % DateTime : Sat Sep 26 08:02:17 UTC 2026
% 0.10/5.37 % CPUTime :
% 0.10/5.37 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.10/5.37 % Process 779753 was started by sandbox on n006,
% 0.10/5.37 % Sat Sep 26 08:02:17 2026
% 0.10/5.37 % The command was "/export/starexec/sandbox/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p".
% 0.10/5.38
% 0.10/5.38 % From the command line: assign(max_seconds, 300).
% 0.10/5.39
% 0.10/5.39 % SZS status Theorem for theBenchmark
% 0.10/5.39
% 0.10/5.39 % Proof 1 at 0.01 (+ 0.00) seconds.
% 0.10/5.39 % Length of proof is 32.
% 0.10/5.39 % Level of proof is 7.
% 0.10/5.39 % Maximum clause weight is 32.000.
% 0.10/5.39 % Given clauses 36.
% 0.10/5.39
% 0.10/5.39 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------