%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : SET084+1 : TPTP v9.3.1. Bugfixed v5.4.0.
% Transfm : none
% Format : tptp:raw
% Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n011.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 08:35:41 AM UTC 2026
% Result : Theorem 0.81s 0.57s
% Output : CNFRefutation 0.81s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 10
% Syntax : Number of formulae : 57 ( 23 unt; 0 def)
% Number of atoms : 123 ( 32 equ)
% Maximal formula atoms : 6 ( 2 avg)
% Number of connectives : 105 ( 39 ~; 34 |; 23 &)
% ( 4 <=>; 5 =>; 0 <=; 0 <~>)
% Maximal formula depth : 7 ( 3 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 6 ( 4 usr; 1 prp; 0-2 aty)
% Number of functors : 12 ( 12 usr; 6 con; 0-2 aty)
% Number of variables : 92 ( 11 sgn 28 !; 6 ?)
% Comments :
%------------------------------------------------------------------------------
fof(subclass_defn,axiom,
! [X,Y] :
( subclass(X,Y)
<=> ! [U] :
( member(U,X)
=> member(U,Y) ) ),
file('theBenchmark.p',subclass_defn) ).
fof(class_elements_are_sets,axiom,
! [X] : subclass(X,universal_class),
file('theBenchmark.p',class_elements_are_sets) ).
fof(unordered_pair_defn,axiom,
! [U,X,Y] :
( member(U,unordered_pair(X,Y))
<=> ( ( U = Y
| U = X )
& member(U,universal_class) ) ),
file('theBenchmark.p',unordered_pair_defn) ).
fof(singleton_set_defn,axiom,
! [X] : singleton(X) = unordered_pair(X,X),
file('theBenchmark.p',singleton_set_defn) ).
fof(ordered_pair_defn,axiom,
! [X,Y] : ordered_pair(X,Y) = unordered_pair(singleton(X),unordered_pair(X,singleton(Y))),
file('theBenchmark.p',ordered_pair_defn) ).
fof(cross_product_defn,axiom,
! [U,V,X,Y] :
( member(ordered_pair(U,V),cross_product(X,Y))
<=> ( member(V,Y)
& member(U,X) ) ),
file('theBenchmark.p',cross_product_defn) ).
fof(inductive_defn,axiom,
! [X] :
( inductive(X)
<=> ( subclass(image(successor_relation,X),X)
& member(null_class,X) ) ),
file('theBenchmark.p',inductive_defn) ).
fof(infinity,axiom,
? [X] :
( ! [Y] :
( inductive(Y)
=> subclass(X,Y) )
& inductive(X)
& member(X,universal_class) ),
file('theBenchmark.p',infinity) ).
fof(regularity,axiom,
! [X] :
( X != null_class
=> ? [U] :
( disjoint(U,X)
& member(U,X)
& member(U,universal_class) ) ),
file('theBenchmark.p',regularity) ).
fof(singleton_identified_by_element2,conjecture,
! [X,Y] :
( ( member(Y,universal_class)
& singleton(X) = singleton(Y) )
=> X = Y ),
file('theBenchmark.p',singleton_identified_by_element2) ).
fof(singleton_identified_by_element2_neg,negated_conjecture,
~ ! [X,Y] :
( ( member(Y,universal_class)
& singleton(X) = singleton(Y) )
=> X = Y ),
inference(assume_negation,[status(cth)],[singleton_identified_by_element2]) ).
cnf(c_45,plain,
( member(null_class,A)
| ~ inductive(A) ),
inference(clausify,[status(thm)],[inductive_defn]) ).
fof(nnf_28,plain,
? [X] :
( ! [Y] :
( subclass(X,Y)
| ~ inductive(Y) )
& inductive(X)
& member(X,universal_class) ),
inference(fof_nnf,[status(thm)],[infinity]) ).
fof(sk_infinity,plain,
( ! [Y] :
( subclass(sK0,Y)
| ~ inductive(Y) )
& inductive(sK0)
& member(sK0,universal_class) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK0]),skolemize(X,sK0)],[nnf_28]) ).
cnf(c_46,plain,
inductive(sK0),
inference(split_conjunct,[status(thm)],[sk_infinity]) ).
cnf(c_62,plain,
member(null_class,sK0),
inference(resolve,[status(thm)],[c_46,c_45]) ).
fof(nnf_42,plain,
? [X,Y] :
( Y != X
& member(Y,universal_class)
& singleton(Y) = singleton(X) ),
inference(fof_nnf,[status(thm)],[singleton_identified_by_element2_neg]) ).
fof(sk_singleton_identified_by_element2_neg,plain,
( sK1 != sK2
& member(sK1,universal_class)
& singleton(sK1) = singleton(sK2) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK1,sK2]),skolemize(X,sK2),skolemize(Y,sK1)],[nnf_42]) ).
cnf(c_64,negated_conjecture,
sK1 != sK2,
inference(split_conjunct,[status(thm)],[sk_singleton_identified_by_element2_neg]) ).
cnf(c_65,negated_conjecture,
member(sK1,universal_class),
inference(split_conjunct,[status(thm)],[sk_singleton_identified_by_element2_neg]) ).
cnf(c_66,negated_conjecture,
singleton(sK1) = singleton(sK2),
inference(split_conjunct,[status(thm)],[sk_singleton_identified_by_element2_neg]) ).
fof(nnf_39,plain,
! [X] :
( ? [U] :
( disjoint(U,X)
& member(U,X)
& member(U,universal_class) )
| null_class = X ),
inference(fof_nnf,[status(thm)],[regularity]) ).
fof(sk_regularity_sk,plain,
! [X] :
( ( disjoint(sK3(X),X)
& member(sK3(X),X)
& member(sK3(X),universal_class) )
| null_class = X ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK3]),skolemize(U,sK3(X))],[nnf_39]) ).
fof(sk_regularity,plain,
( ! [VAR_0] :
( disjoint(sK3(VAR_0),VAR_0)
| null_class = VAR_0 )
& ! [VAR_0] :
( member(sK3(VAR_0),VAR_0)
| null_class = VAR_0 )
& ! [VAR_0] :
( member(sK3(VAR_0),universal_class)
| null_class = VAR_0 ) ),
inference(cnf_transformation,[status(thm)],[sk_regularity_sk]) ).
cnf(c_70,plain,
( member(sK3(A),universal_class)
| null_class = A ),
inference(split_conjunct,[status(thm)],[sk_regularity]) ).
cnf(c_125,plain,
( ~ member(B,D)
| ~ member(A,C)
| member(ordered_pair(A,B),cross_product(C,D)) ),
inference(clausify,[status(thm)],[cross_product_defn]) ).
cnf(c_126,plain,
( member(B,D)
| ~ member(ordered_pair(A,B),cross_product(C,D)) ),
inference(clausify,[status(thm)],[cross_product_defn]) ).
cnf(c_128,plain,
ordered_pair(A,B) = unordered_pair(singleton(A),unordered_pair(A,singleton(B))),
inference(clausify,[status(thm)],[ordered_pair_defn]) ).
cnf(c_129,plain,
singleton(A) = unordered_pair(A,A),
inference(clausify,[status(thm)],[singleton_set_defn]) ).
cnf(c_132,plain,
( A != B
| ~ member(A,universal_class)
| member(A,unordered_pair(B,C)) ),
inference(clausify,[status(thm)],[unordered_pair_defn]) ).
cnf(c_133,plain,
( A = C
| A = B
| ~ member(A,unordered_pair(B,C)) ),
inference(clausify,[status(thm)],[unordered_pair_defn]) ).
cnf(c_138,plain,
subclass(A,universal_class),
inference(clausify,[status(thm)],[class_elements_are_sets]) ).
cnf(c_141,plain,
( member(C,B)
| ~ member(C,A)
| ~ subclass(A,B) ),
inference(clausify,[status(thm)],[subclass_defn]) ).
cnf(c_1653,plain,
( member(B,D)
| ~ member(unordered_pair(singleton(A),unordered_pair(A,singleton(B))),cross_product(C,D)) ),
inference(paramod,[status(thm)],[c_128,c_126]) ).
cnf(c_1654,plain,
( member(B,D)
| ~ member(unordered_pair(unordered_pair(A,A),unordered_pair(A,singleton(B))),cross_product(C,D)) ),
inference(paramod,[status(thm)],[c_129,c_1653]) ).
cnf(c_173,plain,
( member(B,D)
| ~ member(unordered_pair(unordered_pair(A,A),unordered_pair(A,unordered_pair(B,B))),cross_product(C,D)) ),
inference(paramod,[status(thm)],[c_129,c_1654]) ).
cnf(c_1655,plain,
( ~ member(B,D)
| ~ member(A,C)
| member(unordered_pair(singleton(A),unordered_pair(A,singleton(B))),cross_product(C,D)) ),
inference(paramod,[status(thm)],[c_128,c_125]) ).
cnf(c_1656,plain,
( ~ member(B,D)
| ~ member(A,C)
| member(unordered_pair(unordered_pair(A,A),unordered_pair(A,singleton(B))),cross_product(C,D)) ),
inference(paramod,[status(thm)],[c_129,c_1655]) ).
cnf(c_174,plain,
( ~ member(B,D)
| ~ member(A,C)
| member(unordered_pair(unordered_pair(A,A),unordered_pair(A,unordered_pair(B,B))),cross_product(C,D)) ),
inference(paramod,[status(thm)],[c_129,c_1656]) ).
cnf(c_1657,negated_conjecture,
unordered_pair(sK1,sK1) = singleton(sK2),
inference(paramod,[status(thm)],[c_129,c_66]) ).
cnf(c_194,negated_conjecture,
unordered_pair(sK1,sK1) = unordered_pair(sK2,sK2),
inference(paramod,[status(thm)],[c_129,c_1657]) ).
cnf(c_1658,negated_conjecture,
sK2 != sK1,
inference(copy,[status(thm)],[c_64]) ).
cnf(c_1659,negated_conjecture,
( sK2 = A
| ~ member(sK2,unordered_pair(sK1,A)) ),
inference(resolve,[status(thm)],[c_133,c_1658]) ).
cnf(c_1660,negated_conjecture,
sK2 != sK1,
inference(copy,[status(thm)],[c_64]) ).
cnf(c_1661,negated_conjecture,
~ member(sK2,unordered_pair(sK1,sK1)),
inference(resolve,[status(thm)],[c_1659,c_1660]) ).
cnf(c_394,negated_conjecture,
~ member(sK2,unordered_pair(sK2,sK2)),
inference(paramod,[status(thm)],[c_194,c_1661]) ).
cnf(c_455,plain,
( member(A,universal_class)
| ~ member(A,B) ),
inference(resolve,[status(thm)],[c_141,c_138]) ).
cnf(c_1662,negated_conjecture,
( ~ member(A,B)
| member(unordered_pair(unordered_pair(A,A),unordered_pair(A,unordered_pair(sK1,sK1))),cross_product(B,universal_class)) ),
inference(resolve,[status(thm)],[c_174,c_65]) ).
cnf(c_536,negated_conjecture,
( ~ member(A,B)
| member(unordered_pair(unordered_pair(A,A),unordered_pair(A,unordered_pair(sK2,sK2))),cross_product(B,universal_class)) ),
inference(paramod,[status(thm)],[c_194,c_1662]) ).
cnf(c_887,plain,
member(null_class,universal_class),
inference(resolve,[status(thm)],[c_455,c_62]) ).
cnf(c_1663,plain,
( sK2 != sK2
| ~ member(sK2,universal_class) ),
inference(resolve,[status(thm)],[c_132,c_394]) ).
cnf(c_1626,plain,
~ member(sK2,universal_class),
inference(xxres,[status(thm)],[c_1663]) ).
cnf(c_1646,plain,
~ member(unordered_pair(unordered_pair(A,A),unordered_pair(A,unordered_pair(sK2,sK2))),cross_product(B,universal_class)),
inference(resolve,[status(thm)],[c_173,c_1626]) ).
cnf(c_1664,plain,
( ~ member(null_class,universal_class)
| member(sK3(sK2),universal_class) ),
inference(paramod,[status(thm)],[c_70,c_1626]) ).
cnf(c_1649,plain,
member(sK3(sK2),universal_class),
inference(resolve,[status(thm)],[c_887,c_1664]) ).
cnf(c_1651,negated_conjecture,
~ member(A,B),
inference(resolve,[status(thm)],[c_1646,c_536]) ).
cnf(c_1652,plain,
$false,
inference(resolve,[status(thm)],[c_1651,c_1649]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : SET084+1 : TPTP v9.3.1. Bugfixed v5.4.0.
% 0.00/0.04 % Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.12/0.37 % Computer : n011.cluster.edu
% 0.12/0.37 % Model : x86_64 x86_64
% 0.12/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.37 % Memory : 8046.5625MB
% 0.12/0.37 % OS : Linux 6.8.0-71-generic
% 0.12/0.37 % CPULimit : 300
% 0.12/0.37 % WCLimit : 300
% 0.12/0.37 % DateTime : Sat Sep 26 07:04:28 UTC 2026
% 0.12/0.37 % CPUTime :
% 0.12/0.37 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.12/0.37 % Process 4038405 was started by sandbox on n011,
% 0.12/0.37 % Sat Sep 26 07:04:28 2026
% 0.12/0.37 % The command was "/export/starexec/sandbox/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p".
% 0.12/0.38
% 0.12/0.38 % From the command line: assign(max_seconds, 300).
% 0.81/0.57
% 0.81/0.57 % SZS status Theorem for theBenchmark
% 0.81/0.57
% 0.81/0.57 % Proof 1 at 0.12 (+ 0.05) seconds.
% 0.81/0.57 % Length of proof is 37.
% 0.81/0.57 % Level of proof is 7.
% 0.81/0.57 % Maximum clause weight is 33.000.
% 0.81/0.57 % Given clauses 192.
% 0.81/0.57
% 0.81/0.57 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------