%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : SET185+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 : n009.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 08:35:59 AM UTC 2026
% Result : Theorem 0.13s 0.38s
% Output : CNFRefutation 0.13s
% Verified :
% SZS Type : Refutation
% Derivation depth : 11
% Number of leaves : 4
% Syntax : Number of formulae : 31 ( 9 unt; 0 def)
% Number of atoms : 73 ( 8 equ)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 69 ( 27 ~; 27 |; 9 &)
% ( 3 <=>; 3 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 3 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 4 ( 2 usr; 1 prp; 0-2 aty)
% Number of functors : 4 ( 4 usr; 2 con; 0-2 aty)
% Number of variables : 54 ( 3 sgn 25 !; 3 ?)
% 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(subset_defn,axiom,
! [B,C] :
( subset(B,C)
<=> ! [D] :
( member(D,B)
=> member(D,C) ) ),
file('theBenchmark.p',subset_defn) ).
fof(equal_defn,axiom,
! [B,C] :
( B = C
<=> ( subset(C,B)
& subset(B,C) ) ),
file('theBenchmark.p',equal_defn) ).
fof(prove_subset_union,conjecture,
! [B,C] :
( subset(B,C)
=> union(B,C) = C ),
file('theBenchmark.p',prove_subset_union) ).
fof(prove_subset_union_neg,negated_conjecture,
~ ! [B,C] :
( subset(B,C)
=> union(B,C) = C ),
inference(assume_negation,[status(cth)],[prove_subset_union]) ).
cnf(c_8,plain,
( member(A,C)
| 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]) ).
fof(nnf_2,plain,
! [B,C] :
( ( ? [D] :
( ~ member(D,C)
& member(D,B) )
| subset(B,C) )
& ( ! [X0] :
( member(X0,C)
| ~ member(X0,B) )
| ~ subset(B,C) ) ),
inference(fof_nnf,[status(thm)],[subset_defn]) ).
fof(sk_subset_defn_sk,plain,
! [B,C] :
( ( ( ~ member(sK0(B,C),C)
& member(sK0(B,C),B) )
| subset(B,C) )
& ( ! [X0] :
( member(X0,C)
| ~ member(X0,B) )
| ~ subset(B,C) ) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK0]),skolemize(D,sK0(B,C))],[nnf_2]) ).
fof(sk_subset_defn,plain,
( ! [VAR_0,VAR_1] :
( ~ member(sK0(VAR_0,VAR_1),VAR_1)
| subset(VAR_0,VAR_1) )
& ! [VAR_0,VAR_1] :
( member(sK0(VAR_0,VAR_1),VAR_0)
| subset(VAR_0,VAR_1) )
& ! [VAR_0,VAR_1,VAR_2] :
( member(VAR_2,VAR_1)
| ~ member(VAR_2,VAR_0)
| ~ subset(VAR_0,VAR_1) ) ),
inference(cnf_transformation,[status(thm)],[sk_subset_defn_sk]) ).
cnf(c_11,plain,
( member(C,B)
| ~ member(C,A)
| ~ subset(A,B) ),
inference(split_conjunct,[status(thm)],[sk_subset_defn]) ).
cnf(c_12,plain,
( member(sK0(A,B),A)
| subset(A,B) ),
inference(split_conjunct,[status(thm)],[sk_subset_defn]) ).
cnf(c_13,plain,
( ~ member(sK0(A,B),B)
| subset(A,B) ),
inference(split_conjunct,[status(thm)],[sk_subset_defn]) ).
cnf(c_16,plain,
( ~ subset(A,B)
| ~ subset(B,A)
| A = B ),
inference(clausify,[status(thm)],[equal_defn]) ).
fof(nnf_7,plain,
? [B,C] :
( union(B,C) != C
& subset(B,C) ),
inference(fof_nnf,[status(thm)],[prove_subset_union_neg]) ).
fof(sk_prove_subset_union_neg,plain,
( union(sK1,sK2) != sK2
& subset(sK1,sK2) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK1,sK2]),skolemize(B,sK1),skolemize(C,sK2)],[nnf_7]) ).
cnf(c_23,negated_conjecture,
subset(sK1,sK2),
inference(split_conjunct,[status(thm)],[sk_prove_subset_union_neg]) ).
cnf(c_24,negated_conjecture,
union(sK1,sK2) != sK2,
inference(split_conjunct,[status(thm)],[sk_prove_subset_union_neg]) ).
cnf(c_27,plain,
( member(sK0(A,B),union(C,A))
| subset(A,B) ),
inference(resolve,[status(thm)],[c_12,c_10]) ).
cnf(c_29,plain,
( member(sK0(union(A,B),C),B)
| member(sK0(union(A,B),C),A)
| subset(union(A,B),C) ),
inference(resolve,[status(thm)],[c_12,c_8]) ).
cnf(c_63,plain,
( member(A,sK2)
| ~ member(A,sK1) ),
inference(resolve,[status(thm)],[c_23,c_11]) ).
cnf(c_169,plain,
( subset(A,union(B,A))
| subset(A,union(B,A)) ),
inference(resolve,[status(thm)],[c_27,c_13]) ).
cnf(c_77,plain,
subset(A,union(B,A)),
inference(copy,[status(thm)],[c_169]) ).
cnf(c_170,negated_conjecture,
sK2 != union(sK1,sK2),
inference(copy,[status(thm)],[c_24]) ).
cnf(c_171,negated_conjecture,
( ~ subset(sK2,union(sK1,sK2))
| ~ subset(union(sK1,sK2),sK2) ),
inference(resolve,[status(thm)],[c_16,c_170]) ).
cnf(c_99,negated_conjecture,
~ subset(union(sK1,sK2),sK2),
inference(resolve,[status(thm)],[c_171,c_77]) ).
cnf(c_105,plain,
~ member(sK0(union(sK1,sK2),sK2),sK2),
inference(resolve,[status(thm)],[c_13,c_99]) ).
cnf(c_172,plain,
( member(sK0(union(sK1,sK2),sK2),sK2)
| member(sK0(union(sK1,sK2),sK2),sK1) ),
inference(resolve,[status(thm)],[c_29,c_99]) ).
cnf(c_111,plain,
member(sK0(union(sK1,sK2),sK2),sK1),
inference(resolve,[status(thm)],[c_105,c_172]) ).
cnf(c_173,plain,
~ member(sK0(union(sK1,sK2),sK2),sK1),
inference(resolve,[status(thm)],[c_63,c_105]) ).
cnf(c_168,plain,
$false,
inference(resolve,[status(thm)],[c_111,c_173]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : SET185+3 : TPTP v9.3.1. Released v2.2.0.
% 0.00/0.03 % Command : prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.08/0.35 % Computer : n009.cluster.edu
% 0.08/0.35 % Model : x86_64 x86_64
% 0.08/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.08/0.35 % Memory : 8046.5625MB
% 0.08/0.35 % OS : Linux 6.8.0-71-generic
% 0.08/0.35 % CPULimit : 300
% 0.08/0.35 % WCLimit : 300
% 0.08/0.35 % DateTime : Sat Sep 26 07:18:00 UTC 2026
% 0.08/0.35 % CPUTime :
% 0.08/0.35 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.08/0.35 % Process 3669801 was started by sandbox2 on n009,
% 0.08/0.35 % Sat Sep 26 07:18:00 2026
% 0.08/0.35 % The command was "/export/starexec/sandbox2/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p".
% 0.08/0.36
% 0.08/0.36 % From the command line: assign(max_seconds, 300).
% 0.13/0.38
% 0.13/0.38 % SZS status Theorem for theBenchmark
% 0.13/0.38
% 0.13/0.38 % Proof 1 at 0.01 (+ 0.01) seconds.
% 0.13/0.38 % Length of proof is 20.
% 0.13/0.38 % Level of proof is 7.
% 0.13/0.38 % Maximum clause weight is 34.000.
% 0.13/0.38 % Given clauses 32.
% 0.13/0.38
% 0.13/0.38 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------