%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : SET666+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 : n018.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 08:36:46 AM UTC 2026
% Result : Theorem 0.12s 0.39s
% Output : CNFRefutation 0.12s
% Verified :
% SZS Type : Refutation
% Derivation depth : 12
% Number of leaves : 10
% Syntax : Number of formulae : 54 ( 12 unt; 0 def)
% Number of atoms : 183 ( 0 equ)
% Maximal formula atoms : 18 ( 3 avg)
% Number of connectives : 223 ( 94 ~; 88 |; 14 &)
% ( 5 <=>; 22 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 4 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 5 ( 4 usr; 1 prp; 0-2 aty)
% Number of functors : 10 ( 10 usr; 2 con; 0-2 aty)
% Number of variables : 96 ( 2 sgn 36 !; 2 ?)
% 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(p3,axiom,
! [B] :
( ilf_type(B,set_type)
=> subset(identity_relation_of(B),cross_product(B,B)) ),
file('theBenchmark.p',p3) ).
fof(p6,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ( ilf_type(C,identity_relation_of_type(B))
<=> ilf_type(C,relation_type(B,B)) ) ) ),
file('theBenchmark.p',p6) ).
fof(p12,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',p12) ).
fof(p14,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',p14) ).
fof(p16,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',p16) ).
fof(p17,axiom,
! [B] :
( ilf_type(B,set_type)
=> ( ilf_type(power_set(B),set_type)
& ~ empty(power_set(B)) ) ),
file('theBenchmark.p',p17) ).
fof(p18,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',p18) ).
fof(p24,axiom,
! [B] : ilf_type(B,set_type),
file('theBenchmark.p',p24) ).
fof(prove_relset_1_29,conjecture,
! [B] :
( ilf_type(B,set_type)
=> ilf_type(identity_relation_of(B),identity_relation_of_type(B)) ),
file('theBenchmark.p',prove_relset_1_29) ).
fof(prove_relset_1_29_neg,negated_conjecture,
~ ! [B] :
( ilf_type(B,set_type)
=> ilf_type(identity_relation_of(B),identity_relation_of_type(B)) ),
inference(assume_negation,[status(cth)],[prove_relset_1_29]) ).
cnf(c_26,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)],[p14]) ).
cnf(c_27,plain,
( subset(identity_relation_of(A),cross_product(A,A))
| ~ ilf_type(A,set_type) ),
inference(clausify,[status(thm)],[p3]) ).
cnf(c_84,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_92,plain,
( ~ ilf_type(B,relation_type(A,A))
| ilf_type(B,identity_relation_of_type(A))
| ~ ilf_type(B,set_type)
| ~ ilf_type(A,set_type) ),
inference(clausify,[status(thm)],[p6]) ).
cnf(c_98,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)],[p12]) ).
fof(nnf_16,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)],[p16]) ).
fof(sk_p16_sk,plain,
! [B] :
( ! [C] :
( ( ( ( ~ member(sK0(B,C),C)
& member(sK0(B,C),B)
& ilf_type(sK0(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,[sK0]),skolemize(D,sK0(B,C))],[nnf_16]) ).
fof(sk_p16,plain,
( ! [VAR_0,VAR_1] :
( ~ member(sK0(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(sK0(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(sK0(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_p16_sk]) ).
cnf(c_102,plain,
( member(sK0(A,B),A)
| member(A,power_set(B))
| ~ ilf_type(B,set_type)
| ~ ilf_type(A,set_type) ),
inference(split_conjunct,[status(thm)],[sk_p16]) ).
cnf(c_103,plain,
( ~ member(sK0(A,B),B)
| member(A,power_set(B))
| ~ ilf_type(B,set_type)
| ~ ilf_type(A,set_type) ),
inference(split_conjunct,[status(thm)],[sk_p16]) ).
cnf(c_104,plain,
( ~ empty(power_set(A))
| ~ ilf_type(A,set_type) ),
inference(clausify,[status(thm)],[p17]) ).
cnf(c_107,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)],[p18]) ).
cnf(c_112,plain,
ilf_type(A,set_type),
inference(clausify,[status(thm)],[p24]) ).
fof(nnf_25,plain,
? [B] :
( ~ ilf_type(identity_relation_of(B),identity_relation_of_type(B))
& ilf_type(B,set_type) ),
inference(fof_nnf,[status(thm)],[prove_relset_1_29_neg]) ).
fof(sk_prove_relset_1_29_neg,plain,
( ~ ilf_type(identity_relation_of(sK1),identity_relation_of_type(sK1))
& ilf_type(sK1,set_type) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK1]),skolemize(B,sK1)],[nnf_25]) ).
cnf(c_113,negated_conjecture,
~ ilf_type(identity_relation_of(sK1),identity_relation_of_type(sK1)),
inference(split_conjunct,[status(thm)],[sk_prove_relset_1_29_neg]) ).
cnf(c_114,plain,
( ~ ilf_type(A,set_type)
| member(B,cross_product(A,A))
| ~ member(B,identity_relation_of(A))
| ~ ilf_type(B,set_type)
| ~ ilf_type(cross_product(A,A),set_type)
| ~ ilf_type(identity_relation_of(A),set_type) ),
inference(resolve,[status(thm)],[c_26,c_27]) ).
cnf(c_340,plain,
( ~ ilf_type(A,set_type)
| member(B,cross_product(A,A))
| ~ member(B,identity_relation_of(A))
| ~ ilf_type(B,set_type)
| ~ ilf_type(cross_product(A,A),set_type) ),
inference(resolve,[status(thm)],[c_112,c_114]) ).
cnf(c_341,plain,
( ~ ilf_type(B,set_type)
| member(A,cross_product(B,B))
| ~ member(A,identity_relation_of(B))
| ~ ilf_type(A,set_type) ),
inference(resolve,[status(thm)],[c_112,c_340]) ).
cnf(c_342,plain,
( ~ ilf_type(B,set_type)
| member(A,cross_product(B,B))
| ~ member(A,identity_relation_of(B)) ),
inference(resolve,[status(thm)],[c_112,c_341]) ).
cnf(c_115,plain,
( member(A,cross_product(B,B))
| ~ member(A,identity_relation_of(B)) ),
inference(resolve,[status(thm)],[c_112,c_342]) ).
cnf(c_343,plain,
( ~ member(B,A)
| ilf_type(B,member_type(A))
| ~ ilf_type(A,set_type)
| empty(A) ),
inference(resolve,[status(thm)],[c_112,c_107]) ).
cnf(c_167,plain,
( ~ member(B,A)
| ilf_type(B,member_type(A))
| empty(A) ),
inference(resolve,[status(thm)],[c_112,c_343]) ).
cnf(c_169,plain,
~ empty(power_set(A)),
inference(resolve,[status(thm)],[c_112,c_104]) ).
cnf(c_344,plain,
( ~ member(sK0(B,A),A)
| member(B,power_set(A))
| ~ ilf_type(A,set_type) ),
inference(resolve,[status(thm)],[c_112,c_103]) ).
cnf(c_170,plain,
( ~ member(sK0(A,B),B)
| member(A,power_set(B)) ),
inference(resolve,[status(thm)],[c_112,c_344]) ).
cnf(c_345,plain,
( member(sK0(B,A),B)
| member(B,power_set(A))
| ~ ilf_type(A,set_type) ),
inference(resolve,[status(thm)],[c_112,c_102]) ).
cnf(c_171,plain,
( member(sK0(A,B),A)
| member(A,power_set(B)) ),
inference(resolve,[status(thm)],[c_112,c_345]) ).
cnf(c_346,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_112,c_98]) ).
cnf(c_174,plain,
( ~ ilf_type(A,member_type(power_set(B)))
| ilf_type(A,subset_type(B)) ),
inference(resolve,[status(thm)],[c_112,c_346]) ).
cnf(c_347,plain,
( ~ ilf_type(A,relation_type(B,B))
| ilf_type(A,identity_relation_of_type(B))
| ~ ilf_type(A,set_type) ),
inference(resolve,[status(thm)],[c_112,c_92]) ).
cnf(c_177,plain,
( ~ ilf_type(A,relation_type(B,B))
| ilf_type(A,identity_relation_of_type(B)) ),
inference(resolve,[status(thm)],[c_112,c_347]) ).
cnf(c_348,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_112,c_84]) ).
cnf(c_185,plain,
( ilf_type(A,relation_type(B,C))
| ~ ilf_type(A,subset_type(cross_product(B,C))) ),
inference(resolve,[status(thm)],[c_112,c_348]) ).
cnf(c_214,negated_conjecture,
~ ilf_type(identity_relation_of(sK1),relation_type(sK1,sK1)),
inference(resolve,[status(thm)],[c_177,c_113]) ).
cnf(c_273,plain,
~ ilf_type(identity_relation_of(sK1),subset_type(cross_product(sK1,sK1))),
inference(resolve,[status(thm)],[c_185,c_214]) ).
cnf(c_274,plain,
~ ilf_type(identity_relation_of(sK1),member_type(power_set(cross_product(sK1,sK1)))),
inference(resolve,[status(thm)],[c_174,c_273]) ).
cnf(c_349,plain,
( ~ member(A,power_set(B))
| ilf_type(A,member_type(power_set(B))) ),
inference(resolve,[status(thm)],[c_167,c_169]) ).
cnf(c_275,plain,
~ member(identity_relation_of(sK1),power_set(cross_product(sK1,sK1))),
inference(resolve,[status(thm)],[c_349,c_274]) ).
cnf(c_277,plain,
~ member(sK0(identity_relation_of(sK1),cross_product(sK1,sK1)),cross_product(sK1,sK1)),
inference(resolve,[status(thm)],[c_170,c_275]) ).
cnf(c_338,plain,
~ member(sK0(identity_relation_of(sK1),cross_product(sK1,sK1)),identity_relation_of(sK1)),
inference(resolve,[status(thm)],[c_115,c_277]) ).
cnf(c_350,plain,
member(identity_relation_of(sK1),power_set(cross_product(sK1,sK1))),
inference(resolve,[status(thm)],[c_338,c_171]) ).
cnf(c_339,plain,
$false,
inference(resolve,[status(thm)],[c_275,c_350]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : SET666+3 : TPTP v9.3.1. Released v2.2.0.
% 0.00/0.04 % Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.08/0.35 % Computer : n018.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 08:06:06 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 4140881 was started by sandbox on n018,
% 0.08/0.35 % Sat Sep 26 08:06:06 2026
% 0.08/0.35 % The command was "/export/starexec/sandbox/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p".
% 0.12/0.36
% 0.12/0.36 % From the command line: assign(max_seconds, 300).
% 0.12/0.39
% 0.12/0.39 % SZS status Theorem for theBenchmark
% 0.12/0.39
% 0.12/0.39 % Proof 1 at 0.01 (+ 0.01) seconds.
% 0.12/0.39 % Length of proof is 37.
% 0.12/0.39 % Level of proof is 9.
% 0.12/0.39 % Maximum clause weight is 32.000.
% 0.12/0.39 % Given clauses 58.
% 0.12/0.39
% 0.12/0.39 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------