%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : SET961+1 : TPTP v9.3.1. Bugfixed v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n007.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 08:37:22 AM UTC 2026
% Result : Theorem 2.08s 0.71s
% Output : CNFRefutation 2.08s
% Verified :
% SZS Type : Refutation
% Derivation depth : 17
% Number of leaves : 6
% Syntax : Number of formulae : 63 ( 20 unt; 0 def)
% Number of atoms : 156 ( 68 equ)
% Maximal formula atoms : 6 ( 2 avg)
% Number of connectives : 141 ( 48 ~; 74 |; 13 &)
% ( 3 <=>; 3 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 3 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 3 ( 1 usr; 1 prp; 0-2 aty)
% Number of functors : 9 ( 9 usr; 3 con; 0-2 aty)
% Number of variables : 104 ( 13 sgn 32 !; 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_xboole_0,axiom,
! [A] :
( A = empty_set
<=> ! [B] : ~ in(B,A) ),
file('theBenchmark.p',d1_xboole_0) ).
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(t2_tarski,axiom,
! [A,B] :
( ! [C] :
( in(C,A)
<=> in(C,B) )
=> A = B ),
file('theBenchmark.p',t2_tarski) ).
fof(t114_zfmisc_1,conjecture,
! [A,B] :
( cartesian_product2(A,B) = cartesian_product2(B,A)
=> ( A = B
| B = empty_set
| A = empty_set ) ),
file('theBenchmark.p',t114_zfmisc_1) ).
fof(t114_zfmisc_1_neg,negated_conjecture,
~ ! [A,B] :
( cartesian_product2(A,B) = cartesian_product2(B,A)
=> ( A = B
| B = empty_set
| A = empty_set ) ),
inference(assume_negation,[status(cth)],[t114_zfmisc_1]) ).
cnf(c_12,plain,
unordered_pair(A,B) = unordered_pair(B,A),
inference(clausify,[status(thm)],[commutativity_k2_tarski]) ).
fof(nnf_3,plain,
! [A] :
( ( ? [B] : in(B,A)
| empty_set = A )
& ( ! [X0] : ~ in(X0,A)
| empty_set != A ) ),
inference(fof_nnf,[status(thm)],[d1_xboole_0]) ).
fof(sk_d1_xboole_0_sk,plain,
! [A] :
( ( in(sK0(A),A)
| empty_set = A )
& ( ! [X0] : ~ in(X0,A)
| empty_set != A ) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK0]),skolemize(B,sK0(A))],[nnf_3]) ).
fof(sk_d1_xboole_0,plain,
( ! [VAR_0] :
( in(sK0(VAR_0),VAR_0)
| empty_set = VAR_0 )
& ! [VAR_0,VAR_1] :
( ~ in(VAR_1,VAR_0)
| empty_set != VAR_0 ) ),
inference(cnf_transformation,[status(thm)],[sk_d1_xboole_0_sk]) ).
cnf(c_13,plain,
( ~ in(B,A)
| empty_set != A ),
inference(split_conjunct,[status(thm)],[sk_d1_xboole_0]) ).
cnf(c_14,plain,
( in(sK0(A),A)
| empty_set = A ),
inference(split_conjunct,[status(thm)],[sk_d1_xboole_0]) ).
cnf(c_15,plain,
ordered_pair(A,B) = unordered_pair(unordered_pair(A,B),singleton(A)),
inference(clausify,[status(thm)],[d5_tarski]) ).
cnf(c_16,plain,
ordered_pair(A,B) = unordered_pair(singleton(A),unordered_pair(A,B)),
inference(paramod,[status(thm)],[c_12,c_15]) ).
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_16,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_16,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_16,c_24]) ).
fof(nnf_9,plain,
! [A,B] :
( B = A
| ? [C] :
( ( ~ in(C,B)
| ~ in(C,A) )
& ( in(C,B)
| in(C,A) ) ) ),
inference(fof_nnf,[status(thm)],[t2_tarski]) ).
fof(sk_t2_tarski_sk,plain,
! [A,B] :
( B = A
| ( ( ~ in(sK1(A,B),B)
| ~ in(sK1(A,B),A) )
& ( in(sK1(A,B),B)
| in(sK1(A,B),A) ) ) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK1]),skolemize(C,sK1(A,B))],[nnf_9]) ).
fof(sk_t2_tarski,plain,
( ! [VAR_0,VAR_1] :
( VAR_1 = VAR_0
| ~ in(sK1(VAR_0,VAR_1),VAR_1)
| ~ in(sK1(VAR_0,VAR_1),VAR_0) )
& ! [VAR_0,VAR_1] :
( VAR_1 = VAR_0
| in(sK1(VAR_0,VAR_1),VAR_1)
| in(sK1(VAR_0,VAR_1),VAR_0) ) ),
inference(cnf_transformation,[status(thm)],[sk_t2_tarski_sk]) ).
cnf(c_28,plain,
( B = A
| in(sK1(A,B),B)
| in(sK1(A,B),A) ),
inference(split_conjunct,[status(thm)],[sk_t2_tarski]) ).
cnf(c_29,plain,
( B = A
| ~ in(sK1(A,B),B)
| ~ in(sK1(A,B),A) ),
inference(split_conjunct,[status(thm)],[sk_t2_tarski]) ).
fof(nnf_10,plain,
? [A,B] :
( B != A
& empty_set != B
& empty_set != A
& cartesian_product2(B,A) = cartesian_product2(A,B) ),
inference(fof_nnf,[status(thm)],[t114_zfmisc_1_neg]) ).
fof(sk_t114_zfmisc_1_neg,plain,
( sK2 != sK3
& empty_set != sK2
& empty_set != sK3
& cartesian_product2(sK2,sK3) = cartesian_product2(sK3,sK2) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK2,sK3]),skolemize(A,sK3),skolemize(B,sK2)],[nnf_10]) ).
cnf(c_30,negated_conjecture,
cartesian_product2(sK2,sK3) = cartesian_product2(sK3,sK2),
inference(split_conjunct,[status(thm)],[sk_t114_zfmisc_1_neg]) ).
cnf(c_31,negated_conjecture,
empty_set != sK3,
inference(split_conjunct,[status(thm)],[sk_t114_zfmisc_1_neg]) ).
cnf(c_32,negated_conjecture,
sK3 != empty_set,
inference(copy,[status(thm)],[c_31]) ).
cnf(c_33,negated_conjecture,
empty_set != sK2,
inference(split_conjunct,[status(thm)],[sk_t114_zfmisc_1_neg]) ).
cnf(c_34,negated_conjecture,
sK2 != empty_set,
inference(copy,[status(thm)],[c_33]) ).
cnf(c_35,negated_conjecture,
sK2 != sK3,
inference(split_conjunct,[status(thm)],[sk_t114_zfmisc_1_neg]) ).
cnf(c_38,plain,
~ in(A,empty_set),
inference(xxres,[status(thm)],[c_13]) ).
cnf(c_41,plain,
( in(A,C)
| ~ in(unordered_pair(singleton(A),unordered_pair(B,A)),cartesian_product2(C,D)) ),
inference(paramod,[status(thm)],[c_12,c_21]) ).
cnf(c_1329,plain,
( empty_set = A
| ~ in(B,C)
| in(unordered_pair(singleton(sK0(A)),unordered_pair(sK0(A),B)),cartesian_product2(A,C)) ),
inference(resolve,[status(thm)],[c_25,c_14]) ).
cnf(c_45,plain,
( empty_set = B
| empty_set = A
| in(unordered_pair(singleton(sK0(A)),unordered_pair(sK0(A),sK0(B))),cartesian_product2(A,B)) ),
inference(resolve,[status(thm)],[c_1329,c_14]) ).
cnf(c_1330,plain,
( B = A
| in(sK1(A,B),A)
| ~ in(C,D)
| in(unordered_pair(singleton(sK1(A,B)),unordered_pair(sK1(A,B),C)),cartesian_product2(B,D)) ),
inference(resolve,[status(thm)],[c_25,c_28]) ).
cnf(c_55,plain,
( D = C
| in(sK1(C,D),D)
| B = A
| in(sK1(A,B),A)
| in(unordered_pair(singleton(sK1(A,B)),unordered_pair(sK1(A,B),sK1(C,D))),cartesian_product2(B,C)) ),
inference(resolve,[status(thm)],[c_1330,c_28]) ).
cnf(c_70,plain,
( in(B,sK3)
| ~ in(unordered_pair(singleton(A),unordered_pair(A,B)),cartesian_product2(sK3,sK2)) ),
inference(paramod,[status(thm)],[c_30,c_23]) ).
cnf(c_128,plain,
( in(A,sK2)
| ~ in(unordered_pair(singleton(A),unordered_pair(B,A)),cartesian_product2(sK3,sK2)) ),
inference(paramod,[status(thm)],[c_30,c_41]) ).
cnf(c_1331,plain,
( empty_set = sK3
| empty_set = sK2
| in(unordered_pair(singleton(sK0(sK2)),unordered_pair(sK0(sK2),sK0(sK3))),cartesian_product2(sK3,sK2)) ),
inference(paramod,[status(thm)],[c_30,c_45]) ).
cnf(c_1332,plain,
( empty_set = sK3
| empty_set = sK2
| in(unordered_pair(singleton(sK0(sK2)),unordered_pair(sK0(sK3),sK0(sK2))),cartesian_product2(sK3,sK2)) ),
inference(paramod,[status(thm)],[c_12,c_1331]) ).
cnf(c_1333,plain,
( empty_set = sK3
| sK2 = empty_set
| in(unordered_pair(singleton(sK0(sK2)),unordered_pair(sK0(sK3),sK0(sK2))),cartesian_product2(sK3,sK2)) ),
inference(copy,[status(thm)],[c_1332]) ).
cnf(c_1334,plain,
( sK3 = empty_set
| sK2 = empty_set
| in(unordered_pair(singleton(sK0(sK2)),unordered_pair(sK0(sK3),sK0(sK2))),cartesian_product2(sK3,sK2)) ),
inference(copy,[status(thm)],[c_1333]) ).
cnf(c_1335,negated_conjecture,
( sK3 = empty_set
| in(unordered_pair(singleton(sK0(sK2)),unordered_pair(sK0(sK3),sK0(sK2))),cartesian_product2(sK3,sK2)) ),
inference(resolve,[status(thm)],[c_34,c_1334]) ).
cnf(c_326,negated_conjecture,
in(unordered_pair(singleton(sK0(sK2)),unordered_pair(sK0(sK3),sK0(sK2))),cartesian_product2(sK3,sK2)),
inference(resolve,[status(thm)],[c_32,c_1335]) ).
cnf(c_412,plain,
in(sK0(sK2),sK2),
inference(resolve,[status(thm)],[c_326,c_128]) ).
cnf(c_1336,plain,
( in(sK1(sK2,B),sK3)
| B = sK2
| in(sK1(sK2,B),B)
| sK3 = A
| in(sK1(A,sK3),A) ),
inference(resolve,[status(thm)],[c_55,c_70]) ).
cnf(c_1079,plain,
( in(sK1(sK2,B),sK3)
| sK2 = B
| in(sK1(sK2,B),B)
| sK3 = A
| in(sK1(A,sK3),A) ),
inference(copy,[status(thm)],[c_1336]) ).
cnf(c_1337,plain,
( sK2 = sK3
| in(sK1(sK2,sK3),sK3)
| sK3 = A
| in(sK1(A,sK3),A) ),
inference(factor,[status(thm)],[c_1079]) ).
cnf(c_1116,negated_conjecture,
( in(sK1(sK2,sK3),sK3)
| sK3 = A
| in(sK1(A,sK3),A) ),
inference(resolve,[status(thm)],[c_35,c_1337]) ).
cnf(c_1338,plain,
( in(sK1(sK2,sK3),sK3)
| sK3 = empty_set ),
inference(resolve,[status(thm)],[c_1116,c_38]) ).
cnf(c_1206,negated_conjecture,
in(sK1(sK2,sK3),sK3),
inference(resolve,[status(thm)],[c_32,c_1338]) ).
cnf(c_1339,plain,
( sK3 = sK2
| ~ in(sK1(sK2,sK3),sK2) ),
inference(resolve,[status(thm)],[c_1206,c_29]) ).
cnf(c_1340,plain,
( sK2 = sK3
| ~ in(sK1(sK2,sK3),sK2) ),
inference(copy,[status(thm)],[c_1339]) ).
cnf(c_1278,negated_conjecture,
~ in(sK1(sK2,sK3),sK2),
inference(resolve,[status(thm)],[c_35,c_1340]) ).
cnf(c_1341,plain,
( ~ in(A,B)
| in(unordered_pair(singleton(sK1(sK2,sK3)),unordered_pair(sK1(sK2,sK3),A)),cartesian_product2(sK3,B)) ),
inference(resolve,[status(thm)],[c_25,c_1206]) ).
cnf(c_1342,plain,
in(unordered_pair(singleton(sK1(sK2,sK3)),unordered_pair(sK1(sK2,sK3),sK0(sK2))),cartesian_product2(sK3,sK2)),
inference(resolve,[status(thm)],[c_1341,c_412]) ).
cnf(c_1313,plain,
in(unordered_pair(singleton(sK1(sK2,sK3)),unordered_pair(sK0(sK2),sK1(sK2,sK3))),cartesian_product2(sK3,sK2)),
inference(paramod,[status(thm)],[c_12,c_1342]) ).
cnf(c_1327,plain,
~ in(unordered_pair(singleton(sK1(sK2,sK3)),unordered_pair(A,sK1(sK2,sK3))),cartesian_product2(sK3,sK2)),
inference(resolve,[status(thm)],[c_128,c_1278]) ).
cnf(c_1328,plain,
$false,
inference(resolve,[status(thm)],[c_1327,c_1313]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : SET961+1 : TPTP v9.3.1. Bugfixed v4.0.0.
% 0.00/0.04 % Command : prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.11/0.36 % Computer : n007.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 08:25:23 UTC 2026
% 0.11/0.36 % CPUTime :
% 0.11/0.36 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.11/0.36 % Process 3291999 was started by sandbox2 on n007,
% 0.11/0.36 % Sat Sep 26 08:25:23 2026
% 0.11/0.36 % The command was "/export/starexec/sandbox2/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p".
% 0.14/0.37
% 0.14/0.37 % From the command line: assign(max_seconds, 300).
% 2.08/0.71
% 2.08/0.71 % SZS status Theorem for theBenchmark
% 2.08/0.71
% 2.08/0.71 % Proof 1 at 0.21 (+ 0.08) seconds.
% 2.08/0.71 % Length of proof is 40.
% 2.08/0.71 % Level of proof is 10.
% 2.08/0.71 % Maximum clause weight is 57.000.
% 2.08/0.71 % Given clauses 140.
% 2.08/0.71
% 2.08/0.71 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------