%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : SET646+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 : n015.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 08:36:43 AM UTC 2026
% Result : Theorem 0.11s 0.45s
% Output : CNFRefutation 0.11s
% Verified :
% SZS Type : Refutation
% Derivation depth : 14
% Number of leaves : 10
% Syntax : Number of formulae : 61 ( 14 unt; 0 def)
% Number of atoms : 232 ( 0 equ)
% Maximal formula atoms : 18 ( 3 avg)
% Number of connectives : 283 ( 112 ~; 105 |; 27 &)
% ( 6 <=>; 33 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 5 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 5 ( 4 usr; 1 prp; 0-2 aty)
% Number of functors : 13 ( 13 usr; 5 con; 0-2 aty)
% Number of variables : 129 ( 2 sgn 45 !; 5 ?)
% Comments :
%------------------------------------------------------------------------------
fof(p1,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ( subset(singleton(B),C)
<=> member(B,C) ) ) ),
file('theBenchmark.p',p1) ).
fof(p2,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ! [D] :
( ilf_type(D,set_type)
=> ! [E] :
( ilf_type(E,set_type)
=> ( member(ordered_pair(B,C),cross_product(D,E))
<=> ( member(C,E)
& member(B,D) ) ) ) ) ) ),
file('theBenchmark.p',p2) ).
fof(p3,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',p3) ).
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(p15,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',p15) ).
fof(p17,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',p17) ).
fof(p18,axiom,
! [B] :
( ilf_type(B,set_type)
=> ( ilf_type(power_set(B),set_type)
& ~ empty(power_set(B)) ) ),
file('theBenchmark.p',p18) ).
fof(p19,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',p19) ).
fof(p25,axiom,
! [B] : ilf_type(B,set_type),
file('theBenchmark.p',p25) ).
fof(prove_relset_1_8,conjecture,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ! [D] :
( ilf_type(D,set_type)
=> ! [E] :
( ilf_type(E,set_type)
=> ( ( member(E,C)
& member(D,B) )
=> ilf_type(singleton(ordered_pair(D,E)),relation_type(B,C)) ) ) ) ) ),
file('theBenchmark.p',prove_relset_1_8) ).
fof(prove_relset_1_8_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,set_type)
=> ( ( member(E,C)
& member(D,B) )
=> ilf_type(singleton(ordered_pair(D,E)),relation_type(B,C)) ) ) ) ) ),
inference(assume_negation,[status(cth)],[prove_relset_1_8]) ).
fof(nnf_26,plain,
? [B] :
( ? [C] :
( ? [D] :
( ? [E] :
( ~ ilf_type(singleton(ordered_pair(D,E)),relation_type(B,C))
& member(E,C)
& member(D,B)
& ilf_type(E,set_type) )
& ilf_type(D,set_type) )
& ilf_type(C,set_type) )
& ilf_type(B,set_type) ),
inference(fof_nnf,[status(thm)],[prove_relset_1_8_neg]) ).
fof(sk_prove_relset_1_8_neg,plain,
( ~ ilf_type(singleton(ordered_pair(sK0,sK1)),relation_type(sK2,sK3))
& member(sK1,sK3)
& member(sK0,sK2)
& ilf_type(sK1,set_type)
& ilf_type(sK0,set_type)
& ilf_type(sK3,set_type)
& ilf_type(sK2,set_type) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK2,sK3,sK0,sK1]),skolemize(B,sK2),skolemize(C,sK3),skolemize(D,sK0),skolemize(E,sK1)],[nnf_26]) ).
cnf(c_42,negated_conjecture,
~ ilf_type(singleton(ordered_pair(sK0,sK1)),relation_type(sK2,sK3)),
inference(split_conjunct,[status(thm)],[sk_prove_relset_1_8_neg]) ).
cnf(c_43,negated_conjecture,
member(sK1,sK3),
inference(split_conjunct,[status(thm)],[sk_prove_relset_1_8_neg]) ).
cnf(c_44,negated_conjecture,
member(sK0,sK2),
inference(split_conjunct,[status(thm)],[sk_prove_relset_1_8_neg]) ).
cnf(c_49,plain,
ilf_type(A,set_type),
inference(clausify,[status(thm)],[p25]) ).
cnf(c_57,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)],[p19]) ).
cnf(c_529,plain,
( ~ member(B,A)
| ilf_type(B,member_type(A))
| ~ ilf_type(A,set_type)
| empty(A) ),
inference(resolve,[status(thm)],[c_49,c_57]) ).
cnf(c_58,plain,
( ~ member(B,A)
| ilf_type(B,member_type(A))
| empty(A) ),
inference(resolve,[status(thm)],[c_49,c_529]) ).
cnf(c_62,plain,
( ~ empty(power_set(A))
| ~ ilf_type(A,set_type) ),
inference(clausify,[status(thm)],[p18]) ).
cnf(c_63,plain,
~ empty(power_set(A)),
inference(resolve,[status(thm)],[c_49,c_62]) ).
fof(nnf_17,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)],[p17]) ).
fof(sk_p17_sk,plain,
! [B] :
( ! [C] :
( ( ( ( ~ member(sK4(B,C),C)
& member(sK4(B,C),B)
& ilf_type(sK4(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,[sK4]),skolemize(D,sK4(B,C))],[nnf_17]) ).
fof(sk_p17,plain,
( ! [VAR_0,VAR_1] :
( ~ member(sK4(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(sK4(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(sK4(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_p17_sk]) ).
cnf(c_64,plain,
( ~ member(sK4(A,B),B)
| member(A,power_set(B))
| ~ ilf_type(B,set_type)
| ~ ilf_type(A,set_type) ),
inference(split_conjunct,[status(thm)],[sk_p17]) ).
cnf(c_530,plain,
( ~ member(sK4(B,A),A)
| member(B,power_set(A))
| ~ ilf_type(A,set_type) ),
inference(resolve,[status(thm)],[c_49,c_64]) ).
cnf(c_65,plain,
( ~ member(sK4(A,B),B)
| member(A,power_set(B)) ),
inference(resolve,[status(thm)],[c_49,c_530]) ).
cnf(c_66,plain,
( member(sK4(A,B),A)
| member(A,power_set(B))
| ~ ilf_type(B,set_type)
| ~ ilf_type(A,set_type) ),
inference(split_conjunct,[status(thm)],[sk_p17]) ).
cnf(c_531,plain,
( member(sK4(B,A),B)
| member(B,power_set(A))
| ~ ilf_type(A,set_type) ),
inference(resolve,[status(thm)],[c_49,c_66]) ).
cnf(c_67,plain,
( member(sK4(A,B),A)
| member(A,power_set(B)) ),
inference(resolve,[status(thm)],[c_49,c_531]) ).
cnf(c_78,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)],[p15]) ).
cnf(c_532,plain,
( member(C,A)
| ~ member(C,B)
| ~ ilf_type(C,set_type)
| ~ subset(B,A)
| ~ ilf_type(A,set_type) ),
inference(resolve,[status(thm)],[c_49,c_78]) ).
cnf(c_533,plain,
( member(C,B)
| ~ member(C,A)
| ~ ilf_type(C,set_type)
| ~ subset(A,B) ),
inference(resolve,[status(thm)],[c_49,c_532]) ).
cnf(c_79,plain,
( member(C,B)
| ~ member(C,A)
| ~ subset(A,B) ),
inference(resolve,[status(thm)],[c_49,c_533]) ).
cnf(c_91,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]) ).
cnf(c_534,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_49,c_91]) ).
cnf(c_92,plain,
( ~ ilf_type(A,member_type(power_set(B)))
| ilf_type(A,subset_type(B)) ),
inference(resolve,[status(thm)],[c_49,c_534]) ).
cnf(c_118,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)],[p3]) ).
cnf(c_535,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_49,c_118]) ).
cnf(c_119,plain,
( ilf_type(A,relation_type(B,C))
| ~ ilf_type(A,subset_type(cross_product(B,C))) ),
inference(resolve,[status(thm)],[c_49,c_535]) ).
cnf(c_120,plain,
( ~ member(B,D)
| ~ member(A,C)
| member(ordered_pair(A,B),cross_product(C,D))
| ~ ilf_type(D,set_type)
| ~ ilf_type(C,set_type)
| ~ ilf_type(B,set_type)
| ~ ilf_type(A,set_type) ),
inference(clausify,[status(thm)],[p2]) ).
cnf(c_536,plain,
( ~ member(A,C)
| ~ member(D,B)
| member(ordered_pair(D,A),cross_product(B,C))
| ~ ilf_type(C,set_type)
| ~ ilf_type(B,set_type)
| ~ ilf_type(A,set_type) ),
inference(resolve,[status(thm)],[c_49,c_120]) ).
cnf(c_537,plain,
( ~ member(D,B)
| ~ member(C,A)
| member(ordered_pair(C,D),cross_product(A,B))
| ~ ilf_type(B,set_type)
| ~ ilf_type(A,set_type) ),
inference(resolve,[status(thm)],[c_49,c_536]) ).
cnf(c_538,plain,
( ~ member(C,A)
| ~ member(B,D)
| member(ordered_pair(B,C),cross_product(D,A))
| ~ ilf_type(A,set_type) ),
inference(resolve,[status(thm)],[c_49,c_537]) ).
cnf(c_121,plain,
( ~ member(B,D)
| ~ member(A,C)
| member(ordered_pair(A,B),cross_product(C,D)) ),
inference(resolve,[status(thm)],[c_49,c_538]) ).
cnf(c_126,plain,
( ~ member(A,B)
| subset(singleton(A),B)
| ~ ilf_type(B,set_type)
| ~ ilf_type(A,set_type) ),
inference(clausify,[status(thm)],[p1]) ).
cnf(c_539,plain,
( ~ member(B,A)
| subset(singleton(B),A)
| ~ ilf_type(A,set_type) ),
inference(resolve,[status(thm)],[c_49,c_126]) ).
cnf(c_127,plain,
( ~ member(A,B)
| subset(singleton(A),B) ),
inference(resolve,[status(thm)],[c_49,c_539]) ).
cnf(c_166,plain,
( member(A,power_set(C))
| member(sK4(A,C),B)
| ~ subset(A,B) ),
inference(resolve,[status(thm)],[c_79,c_67]) ).
cnf(c_224,negated_conjecture,
~ ilf_type(singleton(ordered_pair(sK0,sK1)),subset_type(cross_product(sK2,sK3))),
inference(resolve,[status(thm)],[c_119,c_42]) ).
cnf(c_278,plain,
~ ilf_type(singleton(ordered_pair(sK0,sK1)),member_type(power_set(cross_product(sK2,sK3)))),
inference(resolve,[status(thm)],[c_92,c_224]) ).
cnf(c_540,plain,
( ~ member(A,power_set(B))
| ilf_type(A,member_type(power_set(B))) ),
inference(resolve,[status(thm)],[c_58,c_63]) ).
cnf(c_279,plain,
~ member(singleton(ordered_pair(sK0,sK1)),power_set(cross_product(sK2,sK3))),
inference(resolve,[status(thm)],[c_540,c_278]) ).
cnf(c_284,plain,
~ member(sK4(singleton(ordered_pair(sK0,sK1)),cross_product(sK2,sK3)),cross_product(sK2,sK3)),
inference(resolve,[status(thm)],[c_65,c_279]) ).
cnf(c_541,plain,
( member(singleton(ordered_pair(sK0,sK1)),power_set(cross_product(sK2,sK3)))
| ~ subset(singleton(ordered_pair(sK0,sK1)),cross_product(sK2,sK3)) ),
inference(resolve,[status(thm)],[c_166,c_284]) ).
cnf(c_514,plain,
~ subset(singleton(ordered_pair(sK0,sK1)),cross_product(sK2,sK3)),
inference(resolve,[status(thm)],[c_541,c_279]) ).
cnf(c_522,plain,
~ member(ordered_pair(sK0,sK1),cross_product(sK2,sK3)),
inference(resolve,[status(thm)],[c_127,c_514]) ).
cnf(c_542,plain,
( ~ member(sK1,sK3)
| ~ member(sK0,sK2) ),
inference(resolve,[status(thm)],[c_121,c_522]) ).
cnf(c_543,negated_conjecture,
~ member(sK0,sK2),
inference(resolve,[status(thm)],[c_542,c_43]) ).
cnf(c_528,negated_conjecture,
$false,
inference(resolve,[status(thm)],[c_44,c_543]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : SET646+3 : TPTP v9.3.1. Released v2.2.0.
% 0.00/0.04 % Command : prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.11/0.36 % Computer : n015.cluster.edu
% 0.11/0.36 % Model : x86_64 x86_64
% 0.11/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.36 % Memory : 8046.5625MB
% 0.11/0.36 % OS : Linux 6.8.0-71-generic
% 0.11/0.36 % CPULimit : 300
% 0.11/0.36 % WCLimit : 300
% 0.11/0.36 % DateTime : Sat Sep 26 08:05:59 UTC 2026
% 0.11/0.37 % CPUTime :
% 0.11/0.37 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.11/0.37 % Process 3524946 was started by sandbox2 on n015,
% 0.11/0.37 % Sat Sep 26 08:05:59 2026
% 0.11/0.37 % The command was "/export/starexec/sandbox2/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p".
% 0.11/0.38
% 0.11/0.38 % From the command line: assign(max_seconds, 300).
% 0.11/0.45
% 0.11/0.45 % SZS status Theorem for theBenchmark
% 0.11/0.45
% 0.11/0.45 % Proof 1 at 0.04 (+ 0.02) seconds.
% 0.11/0.45 % Length of proof is 40.
% 0.11/0.45 % Level of proof is 9.
% 0.11/0.45 % Maximum clause weight is 32.000.
% 0.11/0.45 % Given clauses 101.
% 0.11/0.45
% 0.11/0.45 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------