%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : SET976+1 : TPTP v9.3.1. Released v3.2.0.
% Transfm : none
% Format : tptp:raw
% Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n002.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 08:37:23 AM UTC 2026
% Result : Theorem 0.18s 0.48s
% Output : CNFRefutation 0.18s
% Verified :
% SZS Type : Refutation
% Derivation depth : 14
% Number of leaves : 5
% Syntax : Number of formulae : 42 ( 11 unt; 0 def)
% Number of atoms : 96 ( 34 equ)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 92 ( 38 ~; 40 |; 9 &)
% ( 5 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 3 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 3 ( 1 usr; 1 prp; 0-2 aty)
% Number of functors : 8 ( 8 usr; 4 con; 0-2 aty)
% Number of variables : 67 ( 7 sgn 19 !; 4 ?)
% Comments :
%------------------------------------------------------------------------------
fof(commutativity_k2_tarski,axiom,
! [A,B] : unordered_pair(A,B) = unordered_pair(B,A),
file('theBenchmark.p',commutativity_k2_tarski) ).
fof(d1_tarski,axiom,
! [A,B] :
( B = singleton(A)
<=> ! [C] :
( in(C,B)
<=> C = A ) ),
file('theBenchmark.p',d1_tarski) ).
fof(d5_tarski,axiom,
! [A,B] : ordered_pair(A,B) = unordered_pair(unordered_pair(A,B),singleton(A)),
file('theBenchmark.p',d5_tarski) ).
fof(l55_zfmisc_1,axiom,
! [A,B,C,D] :
( in(ordered_pair(A,B),cartesian_product2(C,D))
<=> ( in(B,D)
& in(A,C) ) ),
file('theBenchmark.p',l55_zfmisc_1) ).
fof(t129_zfmisc_1,conjecture,
! [A,B,C,D] :
( in(ordered_pair(A,B),cartesian_product2(C,singleton(D)))
<=> ( B = D
& in(A,C) ) ),
file('theBenchmark.p',t129_zfmisc_1) ).
fof(t129_zfmisc_1_neg,negated_conjecture,
~ ! [A,B,C,D] :
( in(ordered_pair(A,B),cartesian_product2(C,singleton(D)))
<=> ( B = D
& in(A,C) ) ),
inference(assume_negation,[status(cth)],[t129_zfmisc_1]) ).
cnf(c_11,plain,
unordered_pair(A,B) = unordered_pair(B,A),
inference(clausify,[status(thm)],[commutativity_k2_tarski]) ).
cnf(c_12,plain,
( C = A
| ~ in(C,B)
| singleton(A) != B ),
inference(clausify,[status(thm)],[d1_tarski]) ).
cnf(c_13,plain,
( C != A
| in(C,B)
| singleton(A) != B ),
inference(clausify,[status(thm)],[d1_tarski]) ).
cnf(c_16,plain,
ordered_pair(A,B) = unordered_pair(unordered_pair(A,B),singleton(A)),
inference(clausify,[status(thm)],[d5_tarski]) ).
cnf(c_17,plain,
ordered_pair(A,B) = unordered_pair(singleton(A),unordered_pair(A,B)),
inference(paramod,[status(thm)],[c_11,c_16]) ).
cnf(c_20,plain,
( in(A,C)
| ~ in(ordered_pair(A,B),cartesian_product2(C,D)) ),
inference(clausify,[status(thm)],[l55_zfmisc_1]) ).
cnf(c_21,plain,
( in(A,C)
| ~ in(unordered_pair(singleton(A),unordered_pair(A,B)),cartesian_product2(C,D)) ),
inference(paramod,[status(thm)],[c_17,c_20]) ).
cnf(c_22,plain,
( in(B,D)
| ~ in(ordered_pair(A,B),cartesian_product2(C,D)) ),
inference(clausify,[status(thm)],[l55_zfmisc_1]) ).
cnf(c_23,plain,
( in(B,D)
| ~ in(unordered_pair(singleton(A),unordered_pair(A,B)),cartesian_product2(C,D)) ),
inference(paramod,[status(thm)],[c_17,c_22]) ).
cnf(c_24,plain,
( ~ in(B,D)
| ~ in(A,C)
| in(ordered_pair(A,B),cartesian_product2(C,D)) ),
inference(clausify,[status(thm)],[l55_zfmisc_1]) ).
cnf(c_25,plain,
( ~ in(B,D)
| ~ in(A,C)
| in(unordered_pair(singleton(A),unordered_pair(A,B)),cartesian_product2(C,D)) ),
inference(paramod,[status(thm)],[c_17,c_24]) ).
fof(nnf_9,plain,
? [A,B,C,D] :
( ( D != B
| ~ in(A,C)
| ~ in(ordered_pair(A,B),cartesian_product2(C,singleton(D))) )
& ( ( D = B
& in(A,C) )
| in(ordered_pair(A,B),cartesian_product2(C,singleton(D))) ) ),
inference(fof_nnf,[status(thm)],[t129_zfmisc_1_neg]) ).
fof(sk_t129_zfmisc_1_neg_sk,plain,
( ( sK3 != sK1
| ~ in(sK0,sK2)
| ~ in(ordered_pair(sK0,sK1),cartesian_product2(sK2,singleton(sK3))) )
& ( ( sK3 = sK1
& in(sK0,sK2) )
| in(ordered_pair(sK0,sK1),cartesian_product2(sK2,singleton(sK3))) ) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK0,sK1,sK2,sK3]),skolemize(A,sK0),skolemize(B,sK1),skolemize(C,sK2),skolemize(D,sK3)],[nnf_9]) ).
fof(sk_t129_zfmisc_1_neg,plain,
( ( sK3 != sK1
| ~ in(sK0,sK2)
| ~ in(ordered_pair(sK0,sK1),cartesian_product2(sK2,singleton(sK3))) )
& ( sK3 = sK1
| in(ordered_pair(sK0,sK1),cartesian_product2(sK2,singleton(sK3))) )
& ( in(sK0,sK2)
| in(ordered_pair(sK0,sK1),cartesian_product2(sK2,singleton(sK3))) ) ),
inference(cnf_transformation,[status(thm)],[sk_t129_zfmisc_1_neg_sk]) ).
cnf(c_28,negated_conjecture,
( in(sK0,sK2)
| in(ordered_pair(sK0,sK1),cartesian_product2(sK2,singleton(sK3))) ),
inference(split_conjunct,[status(thm)],[sk_t129_zfmisc_1_neg]) ).
cnf(c_29,negated_conjecture,
( in(sK0,sK2)
| in(unordered_pair(singleton(sK0),unordered_pair(sK0,sK1)),cartesian_product2(sK2,singleton(sK3))) ),
inference(paramod,[status(thm)],[c_17,c_28]) ).
cnf(c_30,negated_conjecture,
( sK3 = sK1
| in(ordered_pair(sK0,sK1),cartesian_product2(sK2,singleton(sK3))) ),
inference(split_conjunct,[status(thm)],[sk_t129_zfmisc_1_neg]) ).
cnf(c_31,negated_conjecture,
( sK3 = sK1
| in(unordered_pair(singleton(sK0),unordered_pair(sK0,sK1)),cartesian_product2(sK2,singleton(sK3))) ),
inference(paramod,[status(thm)],[c_17,c_30]) ).
cnf(c_32,negated_conjecture,
( sK3 != sK1
| ~ in(sK0,sK2)
| ~ in(ordered_pair(sK0,sK1),cartesian_product2(sK2,singleton(sK3))) ),
inference(split_conjunct,[status(thm)],[sk_t129_zfmisc_1_neg]) ).
cnf(c_33,negated_conjecture,
( sK3 != sK1
| ~ in(sK0,sK2)
| ~ in(unordered_pair(singleton(sK0),unordered_pair(sK0,sK1)),cartesian_product2(sK2,singleton(sK3))) ),
inference(paramod,[status(thm)],[c_17,c_32]) ).
cnf(c_114,plain,
( A != B
| in(A,singleton(B)) ),
inference(xxres,[status(thm)],[c_13]) ).
cnf(c_41,plain,
in(A,singleton(A)),
inference(xxres,[status(thm)],[c_114]) ).
cnf(c_115,negated_conjecture,
( in(sK0,sK2)
| in(sK0,sK2) ),
inference(resolve,[status(thm)],[c_29,c_21]) ).
cnf(c_55,negated_conjecture,
in(sK0,sK2),
inference(copy,[status(thm)],[c_115]) ).
cnf(c_56,negated_conjecture,
( sK3 != sK1
| ~ in(unordered_pair(singleton(sK0),unordered_pair(sK0,sK1)),cartesian_product2(sK2,singleton(sK3))) ),
inference(resolve,[status(thm)],[c_55,c_33]) ).
cnf(c_59,plain,
( in(sK1,singleton(sK3))
| sK3 = sK1 ),
inference(resolve,[status(thm)],[c_31,c_23]) ).
cnf(c_116,plain,
( ~ in(A,B)
| in(unordered_pair(singleton(sK0),unordered_pair(sK0,A)),cartesian_product2(sK2,B)) ),
inference(resolve,[status(thm)],[c_25,c_55]) ).
cnf(c_93,plain,
in(unordered_pair(singleton(sK0),unordered_pair(sK0,A)),cartesian_product2(sK2,singleton(A))),
inference(resolve,[status(thm)],[c_116,c_41]) ).
cnf(c_117,plain,
( A = B
| ~ in(A,singleton(B)) ),
inference(xxres,[status(thm)],[c_12]) ).
cnf(c_118,plain,
( sK3 = sK1
| sK1 = sK3 ),
inference(resolve,[status(thm)],[c_117,c_59]) ).
cnf(c_119,plain,
( sK3 = sK1
| sK3 = sK1 ),
inference(copy,[status(thm)],[c_118]) ).
cnf(c_112,plain,
sK3 = sK1,
inference(copy,[status(thm)],[c_119]) ).
cnf(c_120,negated_conjecture,
( sK3 != sK1
| ~ in(unordered_pair(singleton(sK0),unordered_pair(sK0,sK1)),cartesian_product2(sK2,singleton(sK1))) ),
inference(paramod,[status(thm)],[c_112,c_56]) ).
cnf(c_121,negated_conjecture,
( sK1 != sK1
| ~ in(unordered_pair(singleton(sK0),unordered_pair(sK0,sK1)),cartesian_product2(sK2,singleton(sK1))) ),
inference(paramod,[status(thm)],[c_112,c_120]) ).
cnf(c_122,negated_conjecture,
~ in(unordered_pair(singleton(sK0),unordered_pair(sK0,sK1)),cartesian_product2(sK2,singleton(sK1))),
inference(copy,[status(thm)],[c_121]) ).
cnf(c_113,negated_conjecture,
$false,
inference(resolve,[status(thm)],[c_93,c_122]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.05 % Problem : SET976+1 : TPTP v9.3.1. Released v3.2.0.
% 0.00/0.07 % Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.18/0.43 % Computer : n002.cluster.edu
% 0.18/0.43 % Model : x86_64 x86_64
% 0.18/0.43 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.18/0.43 % Memory : 8046.5625MB
% 0.18/0.43 % OS : Linux 6.8.0-71-generic
% 0.18/0.43 % CPULimit : 300
% 0.18/0.43 % WCLimit : 300
% 0.18/0.43 % DateTime : Sat Sep 26 08:31:06 UTC 2026
% 0.18/0.44 % CPUTime :
% 0.18/0.44 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.18/0.44 % Process 1260631 was started by sandbox on n002,
% 0.18/0.44 % Sat Sep 26 08:31:06 2026
% 0.18/0.44 % The command was "/export/starexec/sandbox/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p".
% 0.18/0.45
% 0.18/0.45 % From the command line: assign(max_seconds, 300).
% 0.18/0.48
% 0.18/0.48 % Assigning unary symbol singleton kb_weight 0 and highest precedence (14).
% 0.18/0.48
% 0.18/0.48 % SZS status Theorem for theBenchmark
% 0.18/0.48
% 0.18/0.48 % Proof 1 at 0.01 (+ 0.01) seconds.
% 0.18/0.48 % Length of proof is 29.
% 0.18/0.48 % Level of proof is 6.
% 0.18/0.48 % Maximum clause weight is 37.000.
% 0.18/0.48 % Given clauses 26.
% 0.18/0.48
% 0.18/0.48 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------