%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : SET577+3 : TPTP v9.3.1. Released v2.2.0.
% Transfm : none
% Format : tptp:raw
% Command : prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n015.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 08:36:30 AM UTC 2026
% Result : Theorem 14.16s 2.19s
% Output : CNFRefutation 14.16s
% Verified :
% SZS Type : Refutation
% Derivation depth : 21
% Number of leaves : 4
% Syntax : Number of formulae : 43 ( 9 unt; 0 def)
% Number of atoms : 152 ( 34 equ)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 152 ( 43 ~; 84 |; 18 &)
% ( 5 <=>; 2 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 4 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 3 ( 1 usr; 1 prp; 0-2 aty)
% Number of functors : 5 ( 5 usr; 3 con; 0-2 aty)
% Number of variables : 76 ( 2 sgn 37 !; 4 ?)
% Comments :
%------------------------------------------------------------------------------
fof(union_defn,axiom,
! [B,C,D] :
( member(D,union(B,C))
<=> ( member(D,C)
| member(D,B) ) ),
file('theBenchmark.p',union_defn) ).
fof(commutativity_of_union,axiom,
! [B,C] : union(B,C) = union(C,B),
file('theBenchmark.p',commutativity_of_union) ).
fof(equal_member_defn,axiom,
! [B,C] :
( B = C
<=> ! [D] :
( member(D,B)
<=> member(D,C) ) ),
file('theBenchmark.p',equal_member_defn) ).
fof(prove_th18,conjecture,
! [B,C,D] :
( ! [E] :
( member(E,B)
<=> ( member(E,D)
| member(E,C) ) )
=> B = union(C,D) ),
file('theBenchmark.p',prove_th18) ).
fof(prove_th18_neg,negated_conjecture,
~ ! [B,C,D] :
( ! [E] :
( member(E,B)
<=> ( member(E,D)
| member(E,C) ) )
=> B = union(C,D) ),
inference(assume_negation,[status(cth)],[prove_th18]) ).
cnf(c_8,plain,
( member(A,C)
| member(A,B)
| ~ member(A,union(B,C)) ),
inference(clausify,[status(thm)],[union_defn]) ).
cnf(c_9,plain,
( ~ member(A,B)
| member(A,union(B,C)) ),
inference(clausify,[status(thm)],[union_defn]) ).
cnf(c_10,plain,
( ~ member(A,C)
| member(A,union(B,C)) ),
inference(clausify,[status(thm)],[union_defn]) ).
cnf(c_14,plain,
union(A,B) = union(B,A),
inference(clausify,[status(thm)],[commutativity_of_union]) ).
fof(nnf_6,plain,
! [B,C] :
( ( ? [D] :
( ( ~ member(D,C)
| ~ member(D,B) )
& ( member(D,C)
| member(D,B) ) )
| C = B )
& ( ! [X0] :
( ( ~ member(X0,C)
| member(X0,B) )
& ( member(X0,C)
| ~ member(X0,B) ) )
| C != B ) ),
inference(fof_nnf,[status(thm)],[equal_member_defn]) ).
fof(sk_equal_member_defn_sk,plain,
! [B,C] :
( ( ( ( ~ member(sK0(B,C),C)
| ~ member(sK0(B,C),B) )
& ( member(sK0(B,C),C)
| member(sK0(B,C),B) ) )
| C = B )
& ( ! [X0] :
( ( ~ member(X0,C)
| member(X0,B) )
& ( member(X0,C)
| ~ member(X0,B) ) )
| C != B ) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK0]),skolemize(D,sK0(B,C))],[nnf_6]) ).
fof(sk_equal_member_defn,plain,
( ! [VAR_0,VAR_1] :
( ~ member(sK0(VAR_1,VAR_0),VAR_0)
| ~ member(sK0(VAR_1,VAR_0),VAR_1)
| VAR_0 = VAR_1 )
& ! [VAR_0,VAR_1] :
( member(sK0(VAR_1,VAR_0),VAR_0)
| member(sK0(VAR_1,VAR_0),VAR_1)
| VAR_0 = VAR_1 )
& ! [VAR_0,VAR_1,VAR_2] :
( ~ member(VAR_2,VAR_0)
| member(VAR_2,VAR_1)
| VAR_0 != VAR_1 )
& ! [VAR_0,VAR_1,VAR_2] :
( member(VAR_2,VAR_0)
| ~ member(VAR_2,VAR_1)
| VAR_0 != VAR_1 ) ),
inference(cnf_transformation,[status(thm)],[sk_equal_member_defn_sk]) ).
cnf(c_21,plain,
( member(sK0(B,A),A)
| member(sK0(B,A),B)
| A = B ),
inference(split_conjunct,[status(thm)],[sk_equal_member_defn]) ).
cnf(c_22,plain,
( ~ member(sK0(B,A),A)
| ~ member(sK0(B,A),B)
| A = B ),
inference(split_conjunct,[status(thm)],[sk_equal_member_defn]) ).
fof(nnf_7,plain,
? [B,C,D] :
( union(C,D) != B
& ! [E] :
( ( ( ~ member(E,D)
& ~ member(E,C) )
| member(E,B) )
& ( member(E,D)
| member(E,C)
| ~ member(E,B) ) ) ),
inference(fof_nnf,[status(thm)],[prove_th18_neg]) ).
fof(sk_prove_th18_neg_sk,plain,
( union(sK2,sK3) != sK1
& ! [E] :
( ( ( ~ member(E,sK3)
& ~ member(E,sK2) )
| member(E,sK1) )
& ( member(E,sK3)
| member(E,sK2)
| ~ member(E,sK1) ) ) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK1,sK2,sK3]),skolemize(B,sK1),skolemize(C,sK2),skolemize(D,sK3)],[nnf_7]) ).
fof(sk_prove_th18_neg,plain,
( union(sK2,sK3) != sK1
& ! [VAR_0] :
( ~ member(VAR_0,sK3)
| member(VAR_0,sK1) )
& ! [VAR_0] :
( ~ member(VAR_0,sK2)
| member(VAR_0,sK1) )
& ! [VAR_0] :
( member(VAR_0,sK3)
| member(VAR_0,sK2)
| ~ member(VAR_0,sK1) ) ),
inference(cnf_transformation,[status(thm)],[sk_prove_th18_neg_sk]) ).
cnf(c_23,negated_conjecture,
( member(A,sK3)
| member(A,sK2)
| ~ member(A,sK1) ),
inference(split_conjunct,[status(thm)],[sk_prove_th18_neg]) ).
cnf(c_24,negated_conjecture,
( ~ member(A,sK2)
| member(A,sK1) ),
inference(split_conjunct,[status(thm)],[sk_prove_th18_neg]) ).
cnf(c_25,negated_conjecture,
( ~ member(A,sK3)
| member(A,sK1) ),
inference(split_conjunct,[status(thm)],[sk_prove_th18_neg]) ).
cnf(c_26,negated_conjecture,
union(sK2,sK3) != sK1,
inference(split_conjunct,[status(thm)],[sk_prove_th18_neg]) ).
cnf(c_5494,plain,
( member(sK0(union(B,C),A),C)
| member(sK0(union(B,C),A),B)
| member(sK0(union(B,C),A),A)
| A = union(B,C) ),
inference(resolve,[status(thm)],[c_21,c_8]) ).
cnf(c_38,plain,
( member(sK0(union(A,B),C),B)
| member(sK0(union(A,B),C),A)
| member(sK0(union(A,B),C),C)
| union(A,B) = C ),
inference(copy,[status(thm)],[c_5494]) ).
cnf(c_221,plain,
( member(sK0(union(A,B),sK1),sK3)
| member(sK0(union(A,B),sK1),sK2)
| member(sK0(union(A,B),sK1),B)
| member(sK0(union(A,B),sK1),A)
| union(A,B) = sK1 ),
inference(resolve,[status(thm)],[c_38,c_23]) ).
cnf(c_231,plain,
( member(sK0(union(sK3,A),B),sK1)
| member(sK0(union(sK3,A),B),A)
| member(sK0(union(sK3,A),B),B)
| union(sK3,A) = B ),
inference(resolve,[status(thm)],[c_38,c_25]) ).
cnf(c_271,plain,
( member(sK0(union(sK2,A),sK1),sK3)
| member(sK0(union(sK2,A),sK1),A)
| member(sK0(union(sK2,A),sK1),sK2)
| union(sK2,A) = sK1 ),
inference(factor,[status(thm)],[c_221]) ).
cnf(c_5495,plain,
( member(sK0(union(sK2,sK3),sK1),sK3)
| member(sK0(union(sK2,sK3),sK1),sK2)
| union(sK2,sK3) = sK1 ),
inference(factor,[status(thm)],[c_271]) ).
cnf(c_374,negated_conjecture,
( member(sK0(union(sK2,sK3),sK1),sK3)
| member(sK0(union(sK2,sK3),sK1),sK2) ),
inference(resolve,[status(thm)],[c_26,c_5495]) ).
cnf(c_5496,negated_conjecture,
( member(sK0(union(sK3,sK2),A),sK1)
| member(sK0(union(sK3,sK2),A),sK1)
| member(sK0(union(sK3,sK2),A),A)
| union(sK3,sK2) = A ),
inference(resolve,[status(thm)],[c_231,c_24]) ).
cnf(c_5497,negated_conjecture,
( member(sK0(union(sK3,sK2),A),sK1)
| member(sK0(union(sK3,sK2),A),sK1)
| member(sK0(union(sK3,sK2),A),A)
| union(sK2,sK3) = A ),
inference(paramod,[status(thm)],[c_14,c_5496]) ).
cnf(c_5498,negated_conjecture,
( member(sK0(union(sK3,sK2),A),sK1)
| member(sK0(union(sK3,sK2),A),sK1)
| member(sK0(union(sK2,sK3),A),A)
| union(sK2,sK3) = A ),
inference(paramod,[status(thm)],[c_14,c_5497]) ).
cnf(c_5499,negated_conjecture,
( member(sK0(union(sK3,sK2),A),sK1)
| member(sK0(union(sK2,sK3),A),sK1)
| member(sK0(union(sK2,sK3),A),A)
| union(sK2,sK3) = A ),
inference(paramod,[status(thm)],[c_14,c_5498]) ).
cnf(c_5500,negated_conjecture,
( member(sK0(union(sK2,sK3),A),sK1)
| member(sK0(union(sK2,sK3),A),sK1)
| member(sK0(union(sK2,sK3),A),A)
| union(sK2,sK3) = A ),
inference(paramod,[status(thm)],[c_14,c_5499]) ).
cnf(c_5383,negated_conjecture,
( member(sK0(union(sK2,sK3),A),sK1)
| member(sK0(union(sK2,sK3),A),A)
| union(sK2,sK3) = A ),
inference(copy,[status(thm)],[c_5500]) ).
cnf(c_5501,plain,
( member(sK0(union(sK2,sK3),sK1),sK1)
| union(sK2,sK3) = sK1 ),
inference(factor,[status(thm)],[c_5383]) ).
cnf(c_5411,negated_conjecture,
member(sK0(union(sK2,sK3),sK1),sK1),
inference(resolve,[status(thm)],[c_26,c_5501]) ).
cnf(c_5502,plain,
( ~ member(sK0(union(sK2,sK3),sK1),union(sK2,sK3))
| sK1 = union(sK2,sK3) ),
inference(resolve,[status(thm)],[c_5411,c_22]) ).
cnf(c_5503,plain,
( ~ member(sK0(union(sK2,sK3),sK1),union(sK2,sK3))
| union(sK2,sK3) = sK1 ),
inference(copy,[status(thm)],[c_5502]) ).
cnf(c_5462,negated_conjecture,
~ member(sK0(union(sK2,sK3),sK1),union(sK2,sK3)),
inference(resolve,[status(thm)],[c_26,c_5503]) ).
cnf(c_5482,plain,
~ member(sK0(union(sK2,sK3),sK1),sK3),
inference(resolve,[status(thm)],[c_10,c_5462]) ).
cnf(c_5483,plain,
~ member(sK0(union(sK2,sK3),sK1),sK2),
inference(resolve,[status(thm)],[c_9,c_5462]) ).
cnf(c_5504,negated_conjecture,
member(sK0(union(sK2,sK3),sK1),sK3),
inference(resolve,[status(thm)],[c_5483,c_374]) ).
cnf(c_5493,negated_conjecture,
$false,
inference(resolve,[status(thm)],[c_5482,c_5504]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : SET577+3 : TPTP v9.3.1. Released v2.2.0.
% 0.00/0.04 % Command : prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.10/0.37 % Computer : n015.cluster.edu
% 0.10/0.37 % Model : x86_64 x86_64
% 0.10/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.37 % Memory : 8046.5625MB
% 0.10/0.37 % OS : Linux 6.8.0-71-generic
% 0.10/0.37 % CPULimit : 300
% 0.10/0.37 % WCLimit : 300
% 0.10/0.37 % DateTime : Sat Sep 26 08:00:13 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 3512780 was started by sandbox2 on n015,
% 0.10/0.37 % Sat Sep 26 08:00:13 2026
% 0.10/0.37 % The command was "/export/starexec/sandbox2/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p".
% 0.14/0.38
% 0.14/0.38 % From the command line: assign(max_seconds, 300).
% 14.16/2.19
% 14.16/2.19 % SZS status Theorem for theBenchmark
% 14.16/2.19
% 14.16/2.19 % Proof 1 at 1.58 (+ 0.21) seconds.
% 14.16/2.19 % Length of proof is 25.
% 14.16/2.19 % Level of proof is 8.
% 14.16/2.19 % Maximum clause weight is 48.000.
% 14.16/2.19 % Given clauses 451.
% 14.16/2.19
% 14.16/2.19 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------