%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : SET893+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 : n026.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 08:37:14 AM UTC 2026
% Result : Theorem 0.16s 0.46s
% Output : CNFRefutation 0.16s
% Verified :
% SZS Type : Refutation
% Derivation depth : 17
% Number of leaves : 5
% Syntax : Number of formulae : 47 ( 11 unt; 0 def)
% Number of atoms : 108 ( 57 equ)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 106 ( 45 ~; 47 |; 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 : 71 ( 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(t34_zfmisc_1,conjecture,
! [A,B,C,D] :
( in(ordered_pair(A,B),cartesian_product2(singleton(C),singleton(D)))
<=> ( B = D
& A = C ) ),
file('theBenchmark.p',t34_zfmisc_1) ).
fof(t34_zfmisc_1_neg,negated_conjecture,
~ ! [A,B,C,D] :
( in(ordered_pair(A,B),cartesian_product2(singleton(C),singleton(D)))
<=> ( B = D
& A = C ) ),
inference(assume_negation,[status(cth)],[t34_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
| C != A
| ~ in(ordered_pair(A,B),cartesian_product2(singleton(C),singleton(D))) )
& ( ( D = B
& C = A )
| in(ordered_pair(A,B),cartesian_product2(singleton(C),singleton(D))) ) ),
inference(fof_nnf,[status(thm)],[t34_zfmisc_1_neg]) ).
fof(sk_t34_zfmisc_1_neg_sk,plain,
( ( sK3 != sK1
| sK2 != sK0
| ~ in(ordered_pair(sK0,sK1),cartesian_product2(singleton(sK2),singleton(sK3))) )
& ( ( sK3 = sK1
& sK2 = sK0 )
| in(ordered_pair(sK0,sK1),cartesian_product2(singleton(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_t34_zfmisc_1_neg,plain,
( ( sK3 != sK1
| sK2 != sK0
| ~ in(ordered_pair(sK0,sK1),cartesian_product2(singleton(sK2),singleton(sK3))) )
& ( sK3 = sK1
| in(ordered_pair(sK0,sK1),cartesian_product2(singleton(sK2),singleton(sK3))) )
& ( sK2 = sK0
| in(ordered_pair(sK0,sK1),cartesian_product2(singleton(sK2),singleton(sK3))) ) ),
inference(cnf_transformation,[status(thm)],[sk_t34_zfmisc_1_neg_sk]) ).
cnf(c_28,negated_conjecture,
( sK2 = sK0
| in(ordered_pair(sK0,sK1),cartesian_product2(singleton(sK2),singleton(sK3))) ),
inference(split_conjunct,[status(thm)],[sk_t34_zfmisc_1_neg]) ).
cnf(c_29,negated_conjecture,
( sK2 = sK0
| in(unordered_pair(singleton(sK0),unordered_pair(sK0,sK1)),cartesian_product2(singleton(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(singleton(sK2),singleton(sK3))) ),
inference(split_conjunct,[status(thm)],[sk_t34_zfmisc_1_neg]) ).
cnf(c_31,negated_conjecture,
( sK3 = sK1
| in(unordered_pair(singleton(sK0),unordered_pair(sK0,sK1)),cartesian_product2(singleton(sK2),singleton(sK3))) ),
inference(paramod,[status(thm)],[c_17,c_30]) ).
cnf(c_32,negated_conjecture,
( sK3 != sK1
| sK2 != sK0
| ~ in(ordered_pair(sK0,sK1),cartesian_product2(singleton(sK2),singleton(sK3))) ),
inference(split_conjunct,[status(thm)],[sk_t34_zfmisc_1_neg]) ).
cnf(c_33,negated_conjecture,
( sK3 != sK1
| sK2 != sK0
| ~ in(unordered_pair(singleton(sK0),unordered_pair(sK0,sK1)),cartesian_product2(singleton(sK2),singleton(sK3))) ),
inference(paramod,[status(thm)],[c_17,c_32]) ).
cnf(c_157,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_157]) ).
cnf(c_55,plain,
( in(sK0,singleton(sK2))
| sK2 = sK0 ),
inference(resolve,[status(thm)],[c_29,c_21]) ).
cnf(c_62,plain,
( in(sK1,singleton(sK3))
| sK3 = sK1 ),
inference(resolve,[status(thm)],[c_31,c_23]) ).
cnf(c_158,plain,
( ~ in(B,C)
| in(unordered_pair(singleton(A),unordered_pair(A,B)),cartesian_product2(singleton(A),C)) ),
inference(resolve,[status(thm)],[c_25,c_41]) ).
cnf(c_86,plain,
in(unordered_pair(singleton(A),unordered_pair(A,B)),cartesian_product2(singleton(A),singleton(B))),
inference(resolve,[status(thm)],[c_158,c_41]) ).
cnf(c_159,plain,
( A = B
| ~ in(A,singleton(B)) ),
inference(xxres,[status(thm)],[c_12]) ).
cnf(c_160,plain,
( sK2 = sK0
| sK0 = sK2 ),
inference(resolve,[status(thm)],[c_159,c_55]) ).
cnf(c_161,plain,
( sK2 = sK0
| sK2 = sK0 ),
inference(copy,[status(thm)],[c_160]) ).
cnf(c_126,plain,
sK2 = sK0,
inference(copy,[status(thm)],[c_161]) ).
cnf(c_162,negated_conjecture,
( sK3 != sK1
| sK2 != sK0
| ~ in(unordered_pair(singleton(sK0),unordered_pair(sK0,sK1)),cartesian_product2(singleton(sK0),singleton(sK3))) ),
inference(paramod,[status(thm)],[c_126,c_33]) ).
cnf(c_163,negated_conjecture,
( sK3 != sK1
| sK0 != sK0
| ~ in(unordered_pair(singleton(sK0),unordered_pair(sK0,sK1)),cartesian_product2(singleton(sK0),singleton(sK3))) ),
inference(paramod,[status(thm)],[c_126,c_162]) ).
cnf(c_135,negated_conjecture,
( sK3 != sK1
| ~ in(unordered_pair(singleton(sK0),unordered_pair(sK0,sK1)),cartesian_product2(singleton(sK0),singleton(sK3))) ),
inference(copy,[status(thm)],[c_163]) ).
cnf(c_164,plain,
( A = B
| ~ in(A,singleton(B)) ),
inference(xxres,[status(thm)],[c_12]) ).
cnf(c_165,plain,
( sK3 = sK1
| sK1 = sK3 ),
inference(resolve,[status(thm)],[c_164,c_62]) ).
cnf(c_166,plain,
( sK3 = sK1
| sK3 = sK1 ),
inference(copy,[status(thm)],[c_165]) ).
cnf(c_155,plain,
sK3 = sK1,
inference(copy,[status(thm)],[c_166]) ).
cnf(c_167,negated_conjecture,
( sK3 != sK1
| ~ in(unordered_pair(singleton(sK0),unordered_pair(sK0,sK1)),cartesian_product2(singleton(sK0),singleton(sK1))) ),
inference(paramod,[status(thm)],[c_155,c_135]) ).
cnf(c_168,negated_conjecture,
( sK1 != sK1
| ~ in(unordered_pair(singleton(sK0),unordered_pair(sK0,sK1)),cartesian_product2(singleton(sK0),singleton(sK1))) ),
inference(paramod,[status(thm)],[c_155,c_167]) ).
cnf(c_169,negated_conjecture,
~ in(unordered_pair(singleton(sK0),unordered_pair(sK0,sK1)),cartesian_product2(singleton(sK0),singleton(sK1))),
inference(copy,[status(thm)],[c_168]) ).
cnf(c_156,negated_conjecture,
$false,
inference(resolve,[status(thm)],[c_86,c_169]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.05 % Problem : SET893+1 : TPTP v9.3.1. Released v3.2.0.
% 0.00/0.06 % Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.16/0.42 % Computer : n026.cluster.edu
% 0.16/0.42 % Model : x86_64 x86_64
% 0.16/0.42 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.42 % Memory : 8046.5625MB
% 0.16/0.42 % OS : Linux 6.8.0-71-generic
% 0.16/0.42 % CPULimit : 300
% 0.16/0.42 % WCLimit : 300
% 0.16/0.42 % DateTime : Sat Sep 26 08:27:11 UTC 2026
% 0.16/0.42 % CPUTime :
% 0.16/0.43 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.16/0.43 % Process 546478 was started by sandbox on n026,
% 0.16/0.43 % Sat Sep 26 08:27:11 2026
% 0.16/0.43 % The command was "/export/starexec/sandbox/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p".
% 0.16/0.43
% 0.16/0.43 % From the command line: assign(max_seconds, 300).
% 0.16/0.46
% 0.16/0.46 % Assigning unary symbol singleton kb_weight 0 and highest precedence (14).
% 0.16/0.46
% 0.16/0.46 % SZS status Theorem for theBenchmark
% 0.16/0.46
% 0.16/0.46 % Proof 1 at 0.01 (+ 0.01) seconds.
% 0.16/0.46 % Length of proof is 30.
% 0.16/0.46 % Level of proof is 7.
% 0.16/0.46 % Maximum clause weight is 38.000.
% 0.16/0.46 % Given clauses 36.
% 0.16/0.46
% 0.16/0.46 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------