%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : SET162+3 : TPTP v9.3.1. Released v2.2.0.
% Transfm : none
% Format : tptp:raw
% Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n026.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 08:35:56 AM UTC 2026
% Result : Theorem 0.19s 0.53s
% Output : CNFRefutation 0.19s
% Verified :
% SZS Type : Refutation
% Derivation depth : 11
% Number of leaves : 6
% Syntax : Number of formulae : 32 ( 17 unt; 0 def)
% Number of atoms : 65 ( 11 equ)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 61 ( 28 ~; 22 |; 7 &)
% ( 3 <=>; 1 =>; 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 : 52 ( 4 sgn 26 !; 2 ?)
% 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(empty_set_defn,axiom,
! [B] : ~ member(B,empty_set),
file('theBenchmark.p',empty_set_defn) ).
fof(equal_defn,axiom,
! [B,C] :
( B = C
<=> ( subset(C,B)
& subset(B,C) ) ),
file('theBenchmark.p',equal_defn) ).
fof(commutativity_of_union,axiom,
! [B,C] : union(B,C) = union(C,B),
file('theBenchmark.p',commutativity_of_union) ).
fof(subset_defn,axiom,
! [B,C] :
( subset(B,C)
<=> ! [D] :
( member(D,B)
=> member(D,C) ) ),
file('theBenchmark.p',subset_defn) ).
fof(prove_union_empty_set,conjecture,
! [B] : union(B,empty_set) = B,
file('theBenchmark.p',prove_union_empty_set) ).
fof(prove_union_empty_set_neg,negated_conjecture,
~ ! [B] : union(B,empty_set) = B,
inference(assume_negation,[status(cth)],[prove_union_empty_set]) ).
fof(nnf_9,plain,
? [B] : union(B,empty_set) != B,
inference(fof_nnf,[status(thm)],[prove_union_empty_set_neg]) ).
fof(sk_prove_union_empty_set_neg,plain,
union(sK0,empty_set) != sK0,
inference(skolemize,[status(esa),new_symbols(skolem,[sK0]),skolemize(B,sK0)],[nnf_9]) ).
cnf(c_13,negated_conjecture,
union(sK0,empty_set) != sK0,
inference(split_conjunct,[status(thm)],[sk_prove_union_empty_set_neg]) ).
fof(nnf_5,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(sK1(B,C),C)
& member(sK1(B,C),B) )
| subset(B,C) )
& ( ! [X0] :
( member(X0,C)
| ~ member(X0,B) )
| ~ subset(B,C) ) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK1]),skolemize(D,sK1(B,C))],[nnf_5]) ).
fof(sk_subset_defn,plain,
( ! [VAR_0,VAR_1] :
( ~ member(sK1(VAR_0,VAR_1),VAR_1)
| subset(VAR_0,VAR_1) )
& ! [VAR_0,VAR_1] :
( member(sK1(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_19,plain,
( ~ member(sK1(A,B),B)
| subset(A,B) ),
inference(split_conjunct,[status(thm)],[sk_subset_defn]) ).
cnf(c_20,plain,
( member(sK1(A,B),A)
| subset(A,B) ),
inference(split_conjunct,[status(thm)],[sk_subset_defn]) ).
cnf(c_22,plain,
union(A,B) = union(B,A),
inference(clausify,[status(thm)],[commutativity_of_union]) ).
cnf(c_23,plain,
( ~ subset(A,B)
| ~ subset(B,A)
| A = B ),
inference(clausify,[status(thm)],[equal_defn]) ).
cnf(c_26,plain,
~ member(A,empty_set),
inference(clausify,[status(thm)],[empty_set_defn]) ).
cnf(c_27,plain,
( ~ member(A,C)
| member(A,union(B,C)) ),
inference(clausify,[status(thm)],[union_defn]) ).
cnf(c_29,plain,
( member(A,C)
| member(A,B)
| ~ member(A,union(B,C)) ),
inference(clausify,[status(thm)],[union_defn]) ).
cnf(c_45,plain,
union(empty_set,sK0) != sK0,
inference(paramod,[status(thm)],[c_22,c_13]) ).
cnf(c_54,plain,
( subset(A,B)
| member(sK1(A,B),union(C,A)) ),
inference(resolve,[status(thm)],[c_27,c_20]) ).
cnf(c_645,plain,
( subset(A,union(B,A))
| subset(A,union(B,A)) ),
inference(resolve,[status(thm)],[c_54,c_19]) ).
cnf(c_603,plain,
subset(A,union(B,A)),
inference(copy,[status(thm)],[c_645]) ).
cnf(c_646,plain,
sK0 != union(empty_set,sK0),
inference(copy,[status(thm)],[c_45]) ).
cnf(c_647,plain,
( ~ subset(sK0,union(empty_set,sK0))
| ~ subset(union(empty_set,sK0),sK0) ),
inference(resolve,[status(thm)],[c_23,c_646]) ).
cnf(c_609,plain,
~ subset(union(empty_set,sK0),sK0),
inference(resolve,[status(thm)],[c_647,c_603]) ).
cnf(c_617,plain,
member(sK1(union(empty_set,sK0),sK0),union(empty_set,sK0)),
inference(resolve,[status(thm)],[c_609,c_20]) ).
cnf(c_618,plain,
~ member(sK1(union(empty_set,sK0),sK0),sK0),
inference(resolve,[status(thm)],[c_19,c_609]) ).
cnf(c_648,plain,
( member(A,B)
| ~ member(A,union(empty_set,B)) ),
inference(resolve,[status(thm)],[c_29,c_26]) ).
cnf(c_649,plain,
~ member(sK1(union(empty_set,sK0),sK0),union(empty_set,sK0)),
inference(resolve,[status(thm)],[c_648,c_618]) ).
cnf(c_644,plain,
$false,
inference(resolve,[status(thm)],[c_617,c_649]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.05 % Problem : SET162+3 : TPTP v9.3.1. Released v2.2.0.
% 0.00/0.06 % Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.19/0.43 % Computer : n026.cluster.edu
% 0.19/0.43 % Model : x86_64 x86_64
% 0.19/0.43 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.19/0.43 % Memory : 8046.5625MB
% 0.19/0.43 % OS : Linux 6.8.0-71-generic
% 0.19/0.43 % CPULimit : 300
% 0.19/0.43 % WCLimit : 300
% 0.19/0.43 % DateTime : Sat Sep 26 07:17:25 UTC 2026
% 0.19/0.44 % CPUTime :
% 0.19/0.44 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.19/0.44 % Process 385269 was started by sandbox on n026,
% 0.19/0.44 % Sat Sep 26 07:17:25 2026
% 0.19/0.44 % The command was "/export/starexec/sandbox/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p".
% 0.19/0.45
% 0.19/0.45 % From the command line: assign(max_seconds, 300).
% 0.19/0.53
% 0.19/0.53 % SZS status Theorem for theBenchmark
% 0.19/0.53
% 0.19/0.53 % Proof 1 at 0.04 (+ 0.02) seconds.
% 0.19/0.53 % Length of proof is 21.
% 0.19/0.53 % Level of proof is 6.
% 0.19/0.53 % Maximum clause weight is 24.000.
% 0.19/0.53 % Given clauses 71.
% 0.19/0.53
% 0.19/0.53 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------