%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : SET950+1 : TPTP v9.3.1. Released v3.2.0.
% Transfm : none
% Format : tptp:raw
% Command : prover9 -casc 300 -f /export/starexec/sandbox2/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 08:37:21 AM UTC 2026
% Result : Theorem 5.58s 1.15s
% Output : CNFRefutation 5.58s
% Verified :
% SZS Type : Refutation
% Derivation depth : 11
% Number of leaves : 5
% Syntax : Number of formulae : 36 ( 13 unt; 0 def)
% Number of atoms : 142 ( 50 equ)
% Maximal formula atoms : 28 ( 3 avg)
% Number of connectives : 178 ( 72 ~; 67 |; 35 &)
% ( 3 <=>; 1 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 5 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 4 ( 2 usr; 1 prp; 0-2 aty)
% Number of functors : 13 ( 13 usr; 4 con; 0-4 aty)
% Number of variables : 130 ( 0 sgn 75 !; 11 ?)
% Comments :
%------------------------------------------------------------------------------
fof(commutativity_k2_tarski,axiom,
! [A,B] : unordered_pair(A,B) = unordered_pair(B,A),
file('theBenchmark.p',commutativity_k2_tarski) ).
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(d3_tarski,axiom,
! [A,B] :
( subset(A,B)
<=> ! [C] :
( in(C,A)
=> in(C,B) ) ),
file('theBenchmark.p',d3_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(t103_zfmisc_1,conjecture,
! [A,B,C,D] :
~ ( ! [E,F] :
~ ( D = ordered_pair(E,F)
& in(F,C)
& in(E,B) )
& in(D,A)
& subset(A,cartesian_product2(B,C)) ),
file('theBenchmark.p',t103_zfmisc_1) ).
fof(t103_zfmisc_1_neg,negated_conjecture,
~ ! [A,B,C,D] :
~ ( ! [E,F] :
~ ( D = ordered_pair(E,F)
& in(F,C)
& in(E,B) )
& in(D,A)
& subset(A,cartesian_product2(B,C)) ),
inference(assume_negation,[status(cth)],[t103_zfmisc_1]) ).
cnf(c_12,plain,
( in(C,B)
| ~ in(C,A)
| ~ subset(A,B) ),
inference(clausify,[status(thm)],[d3_tarski]) ).
fof(nnf_10,plain,
? [A,B,C,D] :
( ! [E,F] :
( ordered_pair(E,F) != D
| ~ in(F,C)
| ~ in(E,B) )
& in(D,A)
& subset(A,cartesian_product2(B,C)) ),
inference(fof_nnf,[status(thm)],[t103_zfmisc_1_neg]) ).
fof(sk_t103_zfmisc_1_neg,plain,
( ! [E,F] :
( ordered_pair(E,F) != sK3
| ~ in(F,sK2)
| ~ in(E,sK1) )
& in(sK3,sK0)
& subset(sK0,cartesian_product2(sK1,sK2)) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK0,sK1,sK2,sK3]),skolemize(A,sK0),skolemize(B,sK1),skolemize(C,sK2),skolemize(D,sK3)],[nnf_10]) ).
cnf(c_15,negated_conjecture,
subset(sK0,cartesian_product2(sK1,sK2)),
inference(split_conjunct,[status(thm)],[sk_t103_zfmisc_1_neg]) ).
cnf(c_16,negated_conjecture,
( in(A,cartesian_product2(sK1,sK2))
| ~ in(A,sK0) ),
inference(resolve,[status(thm)],[c_15,c_12]) ).
cnf(c_19,negated_conjecture,
( ordered_pair(A,B) != sK3
| ~ in(B,sK2)
| ~ in(A,sK1) ),
inference(split_conjunct,[status(thm)],[sk_t103_zfmisc_1_neg]) ).
cnf(c_20,negated_conjecture,
in(sK3,sK0),
inference(split_conjunct,[status(thm)],[sk_t103_zfmisc_1_neg]) ).
cnf(c_24,plain,
ordered_pair(A,B) = unordered_pair(unordered_pair(A,B),singleton(A)),
inference(clausify,[status(thm)],[d5_tarski]) ).
fof(nnf_3,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) != sK7(A,B,C)
| ~ in(F,B)
| ~ in(E,A) )
| ~ in(sK7(A,B,C),C) )
& ( ( ordered_pair(sK8(A,B,C),sK9(A,B,C)) = sK7(A,B,C)
& in(sK9(A,B,C),B)
& in(sK8(A,B,C),A) )
| in(sK7(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(sK4(A,B,C,X1),sK5(A,B,C,X1)) = X1
& in(sK5(A,B,C,X1),B)
& in(sK4(A,B,C,X1),A) )
| ~ in(X1,C) ) )
| cartesian_product2(A,B) != C ) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK4,sK5,sK7,sK8,sK9]),skolemize(X2,sK4(A,B,C,X1)),skolemize(X3,sK5(A,B,C,X1)),skolemize(D,sK7(A,B,C)),skolemize(X0,sK8(A,B,C)),skolemize(X6,sK9(A,B,C))],[nnf_3]) ).
fof(sk_d2_zfmisc_1,plain,
( ! [VAR_0,VAR_1,VAR_2,VAR_3,VAR_4] :
( ordered_pair(VAR_3,VAR_4) != sK7(VAR_0,VAR_1,VAR_2)
| ~ in(VAR_4,VAR_1)
| ~ in(VAR_3,VAR_0)
| ~ in(sK7(VAR_0,VAR_1,VAR_2),VAR_2)
| cartesian_product2(VAR_0,VAR_1) = VAR_2 )
& ! [VAR_0,VAR_1,VAR_2] :
( ordered_pair(sK8(VAR_0,VAR_1,VAR_2),sK9(VAR_0,VAR_1,VAR_2)) = sK7(VAR_0,VAR_1,VAR_2)
| in(sK7(VAR_0,VAR_1,VAR_2),VAR_2)
| cartesian_product2(VAR_0,VAR_1) = VAR_2 )
& ! [VAR_0,VAR_1,VAR_2] :
( in(sK9(VAR_0,VAR_1,VAR_2),VAR_1)
| in(sK7(VAR_0,VAR_1,VAR_2),VAR_2)
| cartesian_product2(VAR_0,VAR_1) = VAR_2 )
& ! [VAR_0,VAR_1,VAR_2] :
( in(sK8(VAR_0,VAR_1,VAR_2),VAR_0)
| in(sK7(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(sK4(VAR_0,VAR_1,VAR_2,VAR_3),sK5(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(sK5(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(sK4(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_33,plain,
( ordered_pair(sK4(A,B,C,D),sK5(A,B,C,D)) = D
| ~ in(D,C)
| cartesian_product2(A,B) != C ),
inference(split_conjunct,[status(thm)],[sk_d2_zfmisc_1]) ).
cnf(c_34,plain,
( unordered_pair(unordered_pair(sK4(A,B,C,D),sK5(A,B,C,D)),singleton(sK4(A,B,C,D))) = D
| ~ in(D,C)
| cartesian_product2(A,B) != C ),
inference(paramod,[status(thm)],[c_24,c_33]) ).
cnf(c_35,plain,
( in(sK5(A,B,C,D),B)
| ~ in(D,C)
| cartesian_product2(A,B) != C ),
inference(split_conjunct,[status(thm)],[sk_d2_zfmisc_1]) ).
cnf(c_36,plain,
( in(sK4(A,B,C,D),A)
| ~ in(D,C)
| cartesian_product2(A,B) != C ),
inference(split_conjunct,[status(thm)],[sk_d2_zfmisc_1]) ).
cnf(c_37,plain,
unordered_pair(A,B) = unordered_pair(B,A),
inference(clausify,[status(thm)],[commutativity_k2_tarski]) ).
cnf(c_1232,negated_conjecture,
( unordered_pair(unordered_pair(A,B),singleton(A)) != sK3
| ~ in(B,sK2)
| ~ in(A,sK1) ),
inference(paramod,[status(thm)],[c_24,c_19]) ).
cnf(c_40,negated_conjecture,
( unordered_pair(singleton(A),unordered_pair(A,B)) != sK3
| ~ in(B,sK2)
| ~ in(A,sK1) ),
inference(paramod,[status(thm)],[c_37,c_1232]) ).
cnf(c_45,plain,
( unordered_pair(singleton(sK4(A,B,C,D)),unordered_pair(sK4(A,B,C,D),sK5(A,B,C,D))) = D
| ~ in(D,C)
| cartesian_product2(A,B) != C ),
inference(paramod,[status(thm)],[c_37,c_34]) ).
cnf(c_54,plain,
in(sK3,cartesian_product2(sK1,sK2)),
inference(resolve,[status(thm)],[c_20,c_16]) ).
cnf(c_1233,plain,
( unordered_pair(singleton(sK4(B,C,cartesian_product2(B,C),A)),unordered_pair(sK4(B,C,cartesian_product2(B,C),A),sK5(B,C,cartesian_product2(B,C),A))) = A
| ~ in(A,cartesian_product2(B,C)) ),
inference(xxres,[status(thm)],[c_45]) ).
cnf(c_298,plain,
unordered_pair(singleton(sK4(sK1,sK2,cartesian_product2(sK1,sK2),sK3)),unordered_pair(sK4(sK1,sK2,cartesian_product2(sK1,sK2),sK3),sK5(sK1,sK2,cartesian_product2(sK1,sK2),sK3))) = sK3,
inference(resolve,[status(thm)],[c_1233,c_54]) ).
cnf(c_1234,plain,
( in(sK4(B,C,cartesian_product2(B,C),A),B)
| ~ in(A,cartesian_product2(B,C)) ),
inference(xxres,[status(thm)],[c_36]) ).
cnf(c_299,plain,
in(sK4(sK1,sK2,cartesian_product2(sK1,sK2),sK3),sK1),
inference(resolve,[status(thm)],[c_1234,c_54]) ).
cnf(c_1235,plain,
( in(sK5(B,C,cartesian_product2(B,C),A),C)
| ~ in(A,cartesian_product2(B,C)) ),
inference(xxres,[status(thm)],[c_35]) ).
cnf(c_300,plain,
in(sK5(sK1,sK2,cartesian_product2(sK1,sK2),sK3),sK2),
inference(resolve,[status(thm)],[c_1235,c_54]) ).
cnf(c_1236,negated_conjecture,
( unordered_pair(singleton(sK4(sK1,sK2,cartesian_product2(sK1,sK2),sK3)),unordered_pair(sK4(sK1,sK2,cartesian_product2(sK1,sK2),sK3),A)) != sK3
| ~ in(A,sK2) ),
inference(resolve,[status(thm)],[c_40,c_299]) ).
cnf(c_1237,negated_conjecture,
unordered_pair(singleton(sK4(sK1,sK2,cartesian_product2(sK1,sK2),sK3)),unordered_pair(sK4(sK1,sK2,cartesian_product2(sK1,sK2),sK3),sK5(sK1,sK2,cartesian_product2(sK1,sK2),sK3))) != sK3,
inference(resolve,[status(thm)],[c_1236,c_300]) ).
cnf(c_1238,negated_conjecture,
sK3 != sK3,
inference(paramod,[status(thm)],[c_298,c_1237]) ).
cnf(c_1231,negated_conjecture,
$false,
inference(copy,[status(thm)],[c_1238]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : SET950+1 : TPTP v9.3.1. Released v3.2.0.
% 0.00/0.04 % Command : prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.10/0.36 % Computer : n002.cluster.edu
% 0.10/0.36 % Model : x86_64 x86_64
% 0.10/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.36 % Memory : 8046.5625MB
% 0.10/0.36 % OS : Linux 6.8.0-71-generic
% 0.10/0.36 % CPULimit : 300
% 0.10/0.36 % WCLimit : 300
% 0.10/0.36 % DateTime : Sat Sep 26 08:29:35 UTC 2026
% 0.10/0.37 % CPUTime :
% 0.10/0.37 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.10/0.37 % Process 1256453 was started by sandbox2 on n002,
% 0.10/0.37 % Sat Sep 26 08:29:35 2026
% 0.10/0.37 % The command was "/export/starexec/sandbox2/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p".
% 0.10/0.37
% 0.10/0.37 % From the command line: assign(max_seconds, 300).
% 5.58/1.15
% 5.58/1.15 % Assigning unary symbol singleton kb_weight 0 and highest precedence (19).
% 5.58/1.15
% 5.58/1.15 % SZS status Theorem for theBenchmark
% 5.58/1.15
% 5.58/1.15 % Proof 1 at 0.49 (+ 0.09) seconds.
% 5.58/1.15 % Length of proof is 23.
% 5.58/1.15 % Level of proof is 5.
% 5.58/1.15 % Maximum clause weight is 56.000.
% 5.58/1.15 % Given clauses 69.
% 5.58/1.15
% 5.58/1.15 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------