%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : SET977+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 : n004.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 0.16s 10.43s
% Output : CNFRefutation 0.16s
% Verified :
% SZS Type : Refutation
% Derivation depth : 14
% Number of leaves : 4
% Syntax : Number of formulae : 29 ( 11 unt; 0 def)
% Number of atoms : 61 ( 36 equ)
% Maximal formula atoms : 4 ( 2 avg)
% Number of connectives : 61 ( 29 ~; 18 |; 9 &)
% ( 3 <=>; 2 =>; 0 <=; 0 <~>)
% Maximal formula depth : 7 ( 3 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 3 ( 1 usr; 1 prp; 0-2 aty)
% Number of functors : 7 ( 7 usr; 3 con; 0-2 aty)
% Number of variables : 45 ( 4 sgn 21 !; 3 ?)
% Comments :
%------------------------------------------------------------------------------
fof(d1_xboole_0,axiom,
! [A] :
( A = empty_set
<=> ! [B] : ~ in(B,A) ),
file('theBenchmark.p',d1_xboole_0) ).
fof(t128_zfmisc_1,axiom,
! [A,B,C,D] :
( in(ordered_pair(A,B),cartesian_product2(singleton(C),D))
<=> ( in(B,D)
& A = C ) ),
file('theBenchmark.p',t128_zfmisc_1) ).
fof(t129_zfmisc_1,axiom,
! [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(t130_zfmisc_1,conjecture,
! [A,B] :
( A != empty_set
=> ( cartesian_product2(A,singleton(B)) != empty_set
& cartesian_product2(singleton(B),A) != empty_set ) ),
file('theBenchmark.p',t130_zfmisc_1) ).
fof(t130_zfmisc_1_neg,negated_conjecture,
~ ! [A,B] :
( A != empty_set
=> ( cartesian_product2(A,singleton(B)) != empty_set
& cartesian_product2(singleton(B),A) != empty_set ) ),
inference(assume_negation,[status(cth)],[t130_zfmisc_1]) ).
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_22,plain,
( ~ in(B,D)
| C != A
| in(ordered_pair(A,B),cartesian_product2(singleton(C),D)) ),
inference(clausify,[status(thm)],[t128_zfmisc_1]) ).
cnf(c_25,plain,
( D != B
| ~ in(A,C)
| in(ordered_pair(A,B),cartesian_product2(C,singleton(D))) ),
inference(clausify,[status(thm)],[t129_zfmisc_1]) ).
fof(nnf_10,plain,
? [A,B] :
( ( cartesian_product2(A,singleton(B)) = empty_set
| cartesian_product2(singleton(B),A) = empty_set )
& empty_set != A ),
inference(fof_nnf,[status(thm)],[t130_zfmisc_1_neg]) ).
fof(sk_t130_zfmisc_1_neg,plain,
( ( cartesian_product2(sK1,singleton(sK2)) = empty_set
| cartesian_product2(singleton(sK2),sK1) = empty_set )
& empty_set != sK1 ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK1,sK2]),skolemize(A,sK1),skolemize(B,sK2)],[nnf_10]) ).
cnf(c_26,negated_conjecture,
empty_set != sK1,
inference(split_conjunct,[status(thm)],[sk_t130_zfmisc_1_neg]) ).
cnf(c_27,negated_conjecture,
sK1 != empty_set,
inference(copy,[status(thm)],[c_26]) ).
cnf(c_28,negated_conjecture,
( cartesian_product2(sK1,singleton(sK2)) = empty_set
| cartesian_product2(singleton(sK2),sK1) = empty_set ),
inference(split_conjunct,[status(thm)],[sk_t130_zfmisc_1_neg]) ).
cnf(c_160,plain,
( in(sK0(A),A)
| A = empty_set ),
inference(copy,[status(thm)],[c_14]) ).
cnf(c_37,negated_conjecture,
in(sK0(sK1),sK1),
inference(resolve,[status(thm)],[c_27,c_160]) ).
cnf(c_53,plain,
( B != A
| in(ordered_pair(sK0(sK1),A),cartesian_product2(sK1,singleton(B))) ),
inference(resolve,[status(thm)],[c_37,c_25]) ).
cnf(c_54,plain,
( B != A
| in(ordered_pair(A,sK0(sK1)),cartesian_product2(singleton(B),sK1)) ),
inference(resolve,[status(thm)],[c_37,c_22]) ).
cnf(c_70,plain,
in(ordered_pair(sK0(sK1),A),cartesian_product2(sK1,singleton(A))),
inference(xxres,[status(thm)],[c_53]) ).
cnf(c_161,plain,
empty_set != cartesian_product2(sK1,singleton(A)),
inference(resolve,[status(thm)],[c_70,c_13]) ).
cnf(c_131,plain,
cartesian_product2(sK1,singleton(A)) != empty_set,
inference(copy,[status(thm)],[c_161]) ).
cnf(c_143,negated_conjecture,
cartesian_product2(singleton(sK2),sK1) = empty_set,
inference(resolve,[status(thm)],[c_131,c_28]) ).
cnf(c_162,negated_conjecture,
empty_set = cartesian_product2(singleton(sK2),sK1),
inference(copy,[status(thm)],[c_143]) ).
cnf(c_144,negated_conjecture,
~ in(A,cartesian_product2(singleton(sK2),sK1)),
inference(resolve,[status(thm)],[c_13,c_162]) ).
cnf(c_158,plain,
in(ordered_pair(A,sK0(sK1)),cartesian_product2(singleton(A),sK1)),
inference(xxres,[status(thm)],[c_54]) ).
cnf(c_159,plain,
$false,
inference(resolve,[status(thm)],[c_158,c_144]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.04 % Problem : SET977+1 : TPTP v9.3.1. Released v3.2.0.
% 0.00/0.05 % Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.10/10.39 % Computer : n004.cluster.edu
% 0.10/10.39 % Model : x86_64 x86_64
% 0.10/10.39 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/10.39 % Memory : 8046.5625MB
% 0.10/10.39 % OS : Linux 6.8.0-71-generic
% 0.10/10.39 % CPULimit : 300
% 0.10/10.39 % WCLimit : 300
% 0.10/10.39 % DateTime : Sat Sep 26 08:29:17 UTC 2026
% 0.10/10.40 % CPUTime :
% 0.16/10.40 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.16/10.40 % Process 1243092 was started by sandbox on n004,
% 0.16/10.40 % Sat Sep 26 08:29:17 2026
% 0.16/10.40 % The command was "/export/starexec/sandbox/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p".
% 0.16/10.40
% 0.16/10.40 % From the command line: assign(max_seconds, 300).
% 0.16/10.43
% 0.16/10.43 % SZS status Theorem for theBenchmark
% 0.16/10.43
% 0.16/10.43 % Proof 1 at 0.01 (+ 0.00) seconds.
% 0.16/10.43 % Length of proof is 20.
% 0.16/10.43 % Level of proof is 9.
% 0.16/10.43 % Maximum clause weight is 29.000.
% 0.16/10.43 % Given clauses 41.
% 0.16/10.43
% 0.16/10.43 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------