%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : SET960+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 : n005.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 0.13s 0.42s
% Output : CNFRefutation 0.13s
% Verified :
% SZS Type : Refutation
% Derivation depth : 27
% Number of leaves : 5
% Syntax : Number of formulae : 60 ( 15 unt; 0 def)
% Number of atoms : 227 ( 145 equ)
% Maximal formula atoms : 28 ( 3 avg)
% Number of connectives : 262 ( 95 ~; 130 |; 32 &)
% ( 5 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 4 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 3 ( 1 usr; 1 prp; 0-2 aty)
% Number of functors : 13 ( 13 usr; 3 con; 0-4 aty)
% Number of variables : 129 ( 10 sgn 69 !; 10 ?)
% 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(d2_zfmisc_1,axiom,
! [A,B,C] :
( C = cartesian_product2(A,B)
<=> ! [D] :
( in(D,C)
<=> ? [E,F] :
( D = ordered_pair(E,F)
& in(F,B)
& in(E,A) ) ) ),
file('theBenchmark.p',d2_zfmisc_1) ).
fof(d5_tarski,axiom,
! [A,B] : ordered_pair(A,B) = unordered_pair(unordered_pair(A,B),singleton(A)),
file('theBenchmark.p',d5_tarski) ).
fof(t113_zfmisc_1,conjecture,
! [A,B] :
( cartesian_product2(A,B) = empty_set
<=> ( B = empty_set
| A = empty_set ) ),
file('theBenchmark.p',t113_zfmisc_1) ).
fof(t113_zfmisc_1_neg,negated_conjecture,
~ ! [A,B] :
( cartesian_product2(A,B) = empty_set
<=> ( B = empty_set
| A = empty_set ) ),
inference(assume_negation,[status(cth)],[t113_zfmisc_1]) ).
cnf(c_11,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_12,plain,
( ~ in(B,A)
| empty_set != A ),
inference(split_conjunct,[status(thm)],[sk_d1_xboole_0]) ).
cnf(c_13,plain,
( in(sK0(A),A)
| empty_set = A ),
inference(split_conjunct,[status(thm)],[sk_d1_xboole_0]) ).
fof(nnf_4,plain,
! [A,B,C] :
( ( ? [D] :
( ( ! [E,F] :
( ordered_pair(E,F) != D
| ~ in(F,B)
| ~ in(E,A) )
| ~ in(D,C) )
& ( ? [X0,X6] :
( ordered_pair(X0,X6) = D
& in(X6,B)
& in(X0,A) )
| in(D,C) ) )
| cartesian_product2(A,B) = C )
& ( ! [X1] :
( ( ! [X4,X5] :
( ordered_pair(X4,X5) != X1
| ~ in(X5,B)
| ~ in(X4,A) )
| in(X1,C) )
& ( ? [X2,X3] :
( ordered_pair(X2,X3) = X1
& in(X3,B)
& in(X2,A) )
| ~ in(X1,C) ) )
| cartesian_product2(A,B) != C ) ),
inference(fof_nnf,[status(thm)],[d2_zfmisc_1]) ).
fof(sk_d2_zfmisc_1_sk,plain,
! [A,B,C] :
( ( ( ( ! [E,F] :
( ordered_pair(E,F) != sK5(A,B,C)
| ~ in(F,B)
| ~ in(E,A) )
| ~ in(sK5(A,B,C),C) )
& ( ( ordered_pair(sK6(A,B,C),sK7(A,B,C)) = sK5(A,B,C)
& in(sK7(A,B,C),B)
& in(sK6(A,B,C),A) )
| in(sK5(A,B,C),C) ) )
| cartesian_product2(A,B) = C )
& ( ! [X1] :
( ( ! [X4,X5] :
( ordered_pair(X4,X5) != X1
| ~ in(X5,B)
| ~ in(X4,A) )
| in(X1,C) )
& ( ( ordered_pair(sK1(A,B,C,X1),sK2(A,B,C,X1)) = X1
& in(sK2(A,B,C,X1),B)
& in(sK1(A,B,C,X1),A) )
| ~ in(X1,C) ) )
| cartesian_product2(A,B) != C ) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK1,sK2,sK5,sK6,sK7]),skolemize(X2,sK1(A,B,C,X1)),skolemize(X3,sK2(A,B,C,X1)),skolemize(D,sK5(A,B,C)),skolemize(X0,sK6(A,B,C)),skolemize(X6,sK7(A,B,C))],[nnf_4]) ).
fof(sk_d2_zfmisc_1,plain,
( ! [VAR_0,VAR_1,VAR_2,VAR_3,VAR_4] :
( ordered_pair(VAR_3,VAR_4) != sK5(VAR_0,VAR_1,VAR_2)
| ~ in(VAR_4,VAR_1)
| ~ in(VAR_3,VAR_0)
| ~ in(sK5(VAR_0,VAR_1,VAR_2),VAR_2)
| cartesian_product2(VAR_0,VAR_1) = VAR_2 )
& ! [VAR_0,VAR_1,VAR_2] :
( ordered_pair(sK6(VAR_0,VAR_1,VAR_2),sK7(VAR_0,VAR_1,VAR_2)) = sK5(VAR_0,VAR_1,VAR_2)
| in(sK5(VAR_0,VAR_1,VAR_2),VAR_2)
| cartesian_product2(VAR_0,VAR_1) = VAR_2 )
& ! [VAR_0,VAR_1,VAR_2] :
( in(sK7(VAR_0,VAR_1,VAR_2),VAR_1)
| in(sK5(VAR_0,VAR_1,VAR_2),VAR_2)
| cartesian_product2(VAR_0,VAR_1) = VAR_2 )
& ! [VAR_0,VAR_1,VAR_2] :
( in(sK6(VAR_0,VAR_1,VAR_2),VAR_0)
| in(sK5(VAR_0,VAR_1,VAR_2),VAR_2)
| cartesian_product2(VAR_0,VAR_1) = VAR_2 )
& ! [VAR_0,VAR_1,VAR_2,VAR_3,VAR_4,VAR_5] :
( ordered_pair(VAR_4,VAR_5) != VAR_3
| ~ in(VAR_5,VAR_1)
| ~ in(VAR_4,VAR_0)
| in(VAR_3,VAR_2)
| cartesian_product2(VAR_0,VAR_1) != VAR_2 )
& ! [VAR_0,VAR_1,VAR_2,VAR_3] :
( ordered_pair(sK1(VAR_0,VAR_1,VAR_2,VAR_3),sK2(VAR_0,VAR_1,VAR_2,VAR_3)) = VAR_3
| ~ in(VAR_3,VAR_2)
| cartesian_product2(VAR_0,VAR_1) != VAR_2 )
& ! [VAR_0,VAR_1,VAR_2,VAR_3] :
( in(sK2(VAR_0,VAR_1,VAR_2,VAR_3),VAR_1)
| ~ in(VAR_3,VAR_2)
| cartesian_product2(VAR_0,VAR_1) != VAR_2 )
& ! [VAR_0,VAR_1,VAR_2,VAR_3] :
( in(sK1(VAR_0,VAR_1,VAR_2,VAR_3),VAR_0)
| ~ in(VAR_3,VAR_2)
| cartesian_product2(VAR_0,VAR_1) != VAR_2 ) ),
inference(cnf_transformation,[status(thm)],[sk_d2_zfmisc_1_sk]) ).
cnf(c_14,plain,
( in(sK1(A,B,C,D),A)
| ~ in(D,C)
| cartesian_product2(A,B) != C ),
inference(split_conjunct,[status(thm)],[sk_d2_zfmisc_1]) ).
cnf(c_15,plain,
( in(sK2(A,B,C,D),B)
| ~ in(D,C)
| cartesian_product2(A,B) != C ),
inference(split_conjunct,[status(thm)],[sk_d2_zfmisc_1]) ).
cnf(c_17,plain,
( ordered_pair(E,F) != D
| ~ in(F,B)
| ~ in(E,A)
| in(D,C)
| cartesian_product2(A,B) != C ),
inference(split_conjunct,[status(thm)],[sk_d2_zfmisc_1]) ).
cnf(c_23,plain,
ordered_pair(A,B) = unordered_pair(unordered_pair(A,B),singleton(A)),
inference(clausify,[status(thm)],[d5_tarski]) ).
cnf(c_24,plain,
ordered_pair(A,B) = unordered_pair(singleton(A),unordered_pair(A,B)),
inference(paramod,[status(thm)],[c_11,c_23]) ).
fof(nnf_9,plain,
? [A,B] :
( ( ( empty_set != B
& empty_set != A )
| cartesian_product2(A,B) != empty_set )
& ( empty_set = B
| empty_set = A
| cartesian_product2(A,B) = empty_set ) ),
inference(fof_nnf,[status(thm)],[t113_zfmisc_1_neg]) ).
fof(sk_t113_zfmisc_1_neg_sk,plain,
( ( ( empty_set != sK4
& empty_set != sK3 )
| cartesian_product2(sK3,sK4) != empty_set )
& ( empty_set = sK4
| empty_set = sK3
| cartesian_product2(sK3,sK4) = empty_set ) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK3,sK4]),skolemize(A,sK3),skolemize(B,sK4)],[nnf_9]) ).
fof(sk_t113_zfmisc_1_neg,plain,
( ( empty_set != sK4
| cartesian_product2(sK3,sK4) != empty_set )
& ( empty_set != sK3
| cartesian_product2(sK3,sK4) != empty_set )
& ( empty_set = sK4
| empty_set = sK3
| cartesian_product2(sK3,sK4) = empty_set ) ),
inference(cnf_transformation,[status(thm)],[sk_t113_zfmisc_1_neg_sk]) ).
cnf(c_30,negated_conjecture,
( empty_set = sK4
| empty_set = sK3
| cartesian_product2(sK3,sK4) = empty_set ),
inference(split_conjunct,[status(thm)],[sk_t113_zfmisc_1_neg]) ).
cnf(c_283,negated_conjecture,
( empty_set = sK4
| sK3 = empty_set
| cartesian_product2(sK3,sK4) = empty_set ),
inference(copy,[status(thm)],[c_30]) ).
cnf(c_31,negated_conjecture,
( sK4 = empty_set
| sK3 = empty_set
| cartesian_product2(sK3,sK4) = empty_set ),
inference(copy,[status(thm)],[c_283]) ).
cnf(c_32,negated_conjecture,
( empty_set != sK3
| cartesian_product2(sK3,sK4) != empty_set ),
inference(split_conjunct,[status(thm)],[sk_t113_zfmisc_1_neg]) ).
cnf(c_33,negated_conjecture,
( sK3 != empty_set
| cartesian_product2(sK3,sK4) != empty_set ),
inference(copy,[status(thm)],[c_32]) ).
cnf(c_34,negated_conjecture,
( empty_set != sK4
| cartesian_product2(sK3,sK4) != empty_set ),
inference(split_conjunct,[status(thm)],[sk_t113_zfmisc_1_neg]) ).
cnf(c_35,negated_conjecture,
( sK4 != empty_set
| cartesian_product2(sK3,sK4) != empty_set ),
inference(copy,[status(thm)],[c_34]) ).
cnf(c_43,plain,
( unordered_pair(singleton(E),unordered_pair(E,F)) != D
| ~ in(F,B)
| ~ in(E,A)
| in(D,C)
| cartesian_product2(A,B) != C ),
inference(paramod,[status(thm)],[c_24,c_17]) ).
cnf(c_46,plain,
~ in(A,empty_set),
inference(xxres,[status(thm)],[c_12]) ).
cnf(c_284,negated_conjecture,
( sK4 = empty_set
| sK3 = empty_set
| unordered_pair(singleton(B),unordered_pair(B,C)) != A
| ~ in(C,sK4)
| ~ in(B,sK3)
| in(A,empty_set) ),
inference(resolve,[status(thm)],[c_43,c_31]) ).
cnf(c_285,negated_conjecture,
( empty_set = sK3
| sK4 = empty_set
| sK3 = empty_set
| unordered_pair(singleton(sK0(sK3)),unordered_pair(sK0(sK3),B)) != A
| ~ in(B,sK4)
| in(A,empty_set) ),
inference(resolve,[status(thm)],[c_284,c_13]) ).
cnf(c_286,negated_conjecture,
( empty_set = sK4
| empty_set = sK3
| sK4 = empty_set
| sK3 = empty_set
| unordered_pair(singleton(sK0(sK3)),unordered_pair(sK0(sK3),sK0(sK4))) != A
| in(A,empty_set) ),
inference(resolve,[status(thm)],[c_285,c_13]) ).
cnf(c_287,negated_conjecture,
( empty_set = sK4
| empty_set = sK3
| sK4 = empty_set
| sK3 = empty_set
| in(unordered_pair(unordered_pair(sK0(sK3),sK0(sK4)),singleton(sK0(sK3))),empty_set) ),
inference(resolve,[status(thm)],[c_286,c_11]) ).
cnf(c_288,negated_conjecture,
( empty_set = sK4
| empty_set = sK3
| sK4 = empty_set
| sK3 = empty_set
| in(unordered_pair(singleton(sK0(sK3)),unordered_pair(sK0(sK3),sK0(sK4))),empty_set) ),
inference(paramod,[status(thm)],[c_11,c_287]) ).
cnf(c_289,negated_conjecture,
( empty_set = sK4
| sK3 = empty_set
| sK4 = empty_set
| sK3 = empty_set
| in(unordered_pair(singleton(sK0(sK3)),unordered_pair(sK0(sK3),sK0(sK4))),empty_set) ),
inference(copy,[status(thm)],[c_288]) ).
cnf(c_290,negated_conjecture,
( sK4 = empty_set
| sK3 = empty_set
| sK4 = empty_set
| sK3 = empty_set
| in(unordered_pair(singleton(sK0(sK3)),unordered_pair(sK0(sK3),sK0(sK4))),empty_set) ),
inference(copy,[status(thm)],[c_289]) ).
cnf(c_291,negated_conjecture,
( sK4 = empty_set
| sK4 = empty_set
| sK3 = empty_set
| in(unordered_pair(singleton(sK0(sK3)),unordered_pair(sK0(sK3),sK0(sK4))),empty_set) ),
inference(copy,[status(thm)],[c_290]) ).
cnf(c_292,negated_conjecture,
( sK4 = empty_set
| sK3 = empty_set
| in(unordered_pair(singleton(sK0(sK3)),unordered_pair(sK0(sK3),sK0(sK4))),empty_set) ),
inference(copy,[status(thm)],[c_291]) ).
cnf(c_200,negated_conjecture,
( sK4 = empty_set
| sK3 = empty_set ),
inference(resolve,[status(thm)],[c_46,c_292]) ).
cnf(c_293,plain,
( in(sK2(B,C,cartesian_product2(B,C),A),C)
| ~ in(A,cartesian_product2(B,C)) ),
inference(xxres,[status(thm)],[c_15]) ).
cnf(c_258,plain,
~ in(A,cartesian_product2(B,empty_set)),
inference(resolve,[status(thm)],[c_293,c_46]) ).
cnf(c_294,plain,
( in(sK1(B,C,cartesian_product2(B,C),A),B)
| ~ in(A,cartesian_product2(B,C)) ),
inference(xxres,[status(thm)],[c_14]) ).
cnf(c_259,plain,
~ in(A,cartesian_product2(empty_set,B)),
inference(resolve,[status(thm)],[c_294,c_46]) ).
cnf(c_295,plain,
empty_set = cartesian_product2(A,empty_set),
inference(resolve,[status(thm)],[c_258,c_13]) ).
cnf(c_265,plain,
cartesian_product2(A,empty_set) = empty_set,
inference(copy,[status(thm)],[c_295]) ).
cnf(c_296,plain,
empty_set = cartesian_product2(empty_set,A),
inference(resolve,[status(thm)],[c_259,c_13]) ).
cnf(c_272,plain,
cartesian_product2(empty_set,A) = empty_set,
inference(copy,[status(thm)],[c_296]) ).
cnf(c_297,plain,
( sK4 != empty_set
| cartesian_product2(sK3,empty_set) != empty_set
| sK3 = empty_set ),
inference(paramod,[status(thm)],[c_200,c_35]) ).
cnf(c_298,plain,
( sK4 != empty_set
| empty_set != empty_set
| sK3 = empty_set ),
inference(paramod,[status(thm)],[c_265,c_297]) ).
cnf(c_279,plain,
( sK4 != empty_set
| sK3 = empty_set ),
inference(copy,[status(thm)],[c_298]) ).
cnf(c_299,plain,
( sK3 = empty_set
| sK3 = empty_set ),
inference(resolve,[status(thm)],[c_279,c_200]) ).
cnf(c_280,plain,
sK3 = empty_set,
inference(copy,[status(thm)],[c_299]) ).
cnf(c_300,negated_conjecture,
( sK3 != empty_set
| cartesian_product2(empty_set,sK4) != empty_set ),
inference(paramod,[status(thm)],[c_280,c_33]) ).
cnf(c_301,negated_conjecture,
( sK3 != empty_set
| empty_set != empty_set ),
inference(paramod,[status(thm)],[c_272,c_300]) ).
cnf(c_302,negated_conjecture,
( empty_set != empty_set
| empty_set != empty_set ),
inference(paramod,[status(thm)],[c_280,c_301]) ).
cnf(c_303,negated_conjecture,
empty_set != empty_set,
inference(copy,[status(thm)],[c_302]) ).
cnf(c_282,negated_conjecture,
$false,
inference(copy,[status(thm)],[c_303]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : SET960+1 : TPTP v9.3.1. Released v3.2.0.
% 0.00/0.04 % Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.08/0.34 % Computer : n005.cluster.edu
% 0.08/0.34 % Model : x86_64 x86_64
% 0.08/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.08/0.34 % Memory : 8046.5625MB
% 0.08/0.34 % OS : Linux 6.8.0-71-generic
% 0.08/0.34 % CPULimit : 300
% 0.08/0.34 % WCLimit : 300
% 0.08/0.34 % DateTime : Sat Sep 26 08:27:31 UTC 2026
% 0.08/0.34 % CPUTime :
% 0.08/0.35 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.08/0.35 % Process 1691651 was started by sandbox on n005,
% 0.08/0.35 % Sat Sep 26 08:27:31 2026
% 0.08/0.35 % The command was "/export/starexec/sandbox/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p".
% 0.08/0.35
% 0.08/0.35 % From the command line: assign(max_seconds, 300).
% 0.13/0.42
% 0.13/0.42 % SZS status Theorem for theBenchmark
% 0.13/0.42
% 0.13/0.42 % Proof 1 at 0.04 (+ 0.01) seconds.
% 0.13/0.42 % Length of proof is 29.
% 0.13/0.42 % Level of proof is 7.
% 0.13/0.42 % Maximum clause weight is 42.000.
% 0.13/0.42 % Given clauses 32.
% 0.13/0.42
% 0.13/0.42 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------