%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : NUM386+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 : n001.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 08:16:54 AM UTC 2026
% Result : Theorem 0.86s 0.60s
% Output : CNFRefutation 0.86s
% Verified :
% SZS Type : Refutation
% Derivation depth : 21
% Number of leaves : 10
% Syntax : Number of formulae : 72 ( 23 unt; 0 def)
% Number of atoms : 183 ( 81 equ)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 177 ( 66 ~; 81 |; 23 &)
% ( 6 <=>; 1 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 3 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 6 ( 4 usr; 1 prp; 0-2 aty)
% Number of functors : 8 ( 8 usr; 4 con; 0-2 aty)
% Number of variables : 94 ( 12 sgn 35 !; 5 ?)
% Comments :
%------------------------------------------------------------------------------
fof(commutativity_k2_xboole_0,axiom,
! [A,B] : set_union2(A,B) = set_union2(B,A),
file('theBenchmark.p',commutativity_k2_xboole_0) ).
fof(d1_ordinal1,axiom,
! [A] : succ(A) = set_union2(A,singleton(A)),
file('theBenchmark.p',d1_ordinal1) ).
fof(d1_tarski,axiom,
! [A,B] :
( B = singleton(A)
<=> ! [C] :
( in(C,B)
<=> C = A ) ),
file('theBenchmark.p',d1_tarski) ).
fof(d2_xboole_0,axiom,
! [A,B,C] :
( C = set_union2(A,B)
<=> ! [D] :
( in(D,C)
<=> ( in(D,B)
| in(D,A) ) ) ),
file('theBenchmark.p',d2_xboole_0) ).
fof(fc4_relat_1,axiom,
( relation(empty_set)
& empty(empty_set) ),
file('theBenchmark.p',fc4_relat_1) ).
fof(idempotence_k2_xboole_0,axiom,
! [A,B] : set_union2(A,A) = A,
file('theBenchmark.p',idempotence_k2_xboole_0) ).
fof(rc2_funct_1,axiom,
? [A] :
( function(A)
& empty(A)
& relation(A) ),
file('theBenchmark.p',rc2_funct_1) ).
fof(t6_boole,axiom,
! [A] :
( empty(A)
=> A = empty_set ),
file('theBenchmark.p',t6_boole) ).
fof(t7_boole,axiom,
! [A,B] :
~ ( empty(B)
& in(A,B) ),
file('theBenchmark.p',t7_boole) ).
fof(t13_ordinal1,conjecture,
! [A,B] :
( in(A,succ(B))
<=> ( A = B
| in(A,B) ) ),
file('theBenchmark.p',t13_ordinal1) ).
fof(t13_ordinal1_neg,negated_conjecture,
~ ! [A,B] :
( in(A,succ(B))
<=> ( A = B
| in(A,B) ) ),
inference(assume_negation,[status(cth)],[t13_ordinal1]) ).
fof(nnf_33,plain,
? [A,B] :
( ( ( B != A
& ~ in(A,B) )
| ~ in(A,succ(B)) )
& ( B = A
| in(A,B)
| in(A,succ(B)) ) ),
inference(fof_nnf,[status(thm)],[t13_ordinal1_neg]) ).
fof(sk_t13_ordinal1_neg_sk,plain,
( ( ( sK1 != sK0
& ~ in(sK0,sK1) )
| ~ in(sK0,succ(sK1)) )
& ( sK1 = sK0
| in(sK0,sK1)
| in(sK0,succ(sK1)) ) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK0,sK1]),skolemize(A,sK0),skolemize(B,sK1)],[nnf_33]) ).
fof(sk_t13_ordinal1_neg,plain,
( ( sK1 != sK0
| ~ in(sK0,succ(sK1)) )
& ( ~ in(sK0,sK1)
| ~ in(sK0,succ(sK1)) )
& ( sK1 = sK0
| in(sK0,sK1)
| in(sK0,succ(sK1)) ) ),
inference(cnf_transformation,[status(thm)],[sk_t13_ordinal1_neg_sk]) ).
cnf(c_45,negated_conjecture,
( sK1 != sK0
| ~ in(sK0,succ(sK1)) ),
inference(split_conjunct,[status(thm)],[sk_t13_ordinal1_neg]) ).
cnf(c_46,negated_conjecture,
( ~ in(sK0,sK1)
| ~ in(sK0,succ(sK1)) ),
inference(split_conjunct,[status(thm)],[sk_t13_ordinal1_neg]) ).
cnf(c_47,negated_conjecture,
( sK1 = sK0
| in(sK0,sK1)
| in(sK0,succ(sK1)) ),
inference(split_conjunct,[status(thm)],[sk_t13_ordinal1_neg]) ).
cnf(c_49,plain,
( ~ empty(B)
| ~ in(A,B) ),
inference(clausify,[status(thm)],[t7_boole]) ).
cnf(c_50,plain,
( empty_set = A
| ~ empty(A) ),
inference(clausify,[status(thm)],[t6_boole]) ).
fof(nnf_20,plain,
? [A] :
( function(A)
& empty(A)
& relation(A) ),
inference(fof_nnf,[status(thm)],[rc2_funct_1]) ).
fof(sk_rc2_funct_1,plain,
( function(sK2)
& empty(sK2)
& relation(sK2) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK2]),skolemize(A,sK2)],[nnf_20]) ).
cnf(c_59,plain,
empty(sK2),
inference(split_conjunct,[status(thm)],[sk_rc2_funct_1]) ).
cnf(c_65,plain,
set_union2(A,A) = A,
inference(clausify,[status(thm)],[idempotence_k2_xboole_0]) ).
cnf(c_67,plain,
empty(empty_set),
inference(clausify,[status(thm)],[fc4_relat_1]) ).
cnf(c_76,plain,
( ~ in(D,A)
| in(D,C)
| set_union2(A,B) != C ),
inference(clausify,[status(thm)],[d2_xboole_0]) ).
cnf(c_77,plain,
( in(D,B)
| in(D,A)
| ~ in(D,C)
| set_union2(A,B) != C ),
inference(clausify,[status(thm)],[d2_xboole_0]) ).
fof(nnf_7,plain,
! [A,B] :
( ( ? [C] :
( ( C != A
| ~ in(C,B) )
& ( C = A
| in(C,B) ) )
| singleton(A) = B )
& ( ! [X0] :
( ( X0 != A
| in(X0,B) )
& ( X0 = A
| ~ in(X0,B) ) )
| singleton(A) != B ) ),
inference(fof_nnf,[status(thm)],[d1_tarski]) ).
fof(sk_d1_tarski_sk,plain,
! [A,B] :
( ( ( ( sK3(A,B) != A
| ~ in(sK3(A,B),B) )
& ( sK3(A,B) = A
| in(sK3(A,B),B) ) )
| singleton(A) = B )
& ( ! [X0] :
( ( X0 != A
| in(X0,B) )
& ( X0 = A
| ~ in(X0,B) ) )
| singleton(A) != B ) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK3]),skolemize(C,sK3(A,B))],[nnf_7]) ).
fof(sk_d1_tarski,plain,
( ! [VAR_0,VAR_1] :
( sK3(VAR_0,VAR_1) != VAR_0
| ~ in(sK3(VAR_0,VAR_1),VAR_1)
| singleton(VAR_0) = VAR_1 )
& ! [VAR_0,VAR_1] :
( sK3(VAR_0,VAR_1) = VAR_0
| in(sK3(VAR_0,VAR_1),VAR_1)
| singleton(VAR_0) = VAR_1 )
& ! [VAR_0,VAR_1,VAR_2] :
( VAR_2 != VAR_0
| in(VAR_2,VAR_1)
| singleton(VAR_0) != VAR_1 )
& ! [VAR_0,VAR_1,VAR_2] :
( VAR_2 = VAR_0
| ~ in(VAR_2,VAR_1)
| singleton(VAR_0) != VAR_1 ) ),
inference(cnf_transformation,[status(thm)],[sk_d1_tarski_sk]) ).
cnf(c_79,plain,
( sK3(A,B) = A
| in(sK3(A,B),B)
| singleton(A) = B ),
inference(split_conjunct,[status(thm)],[sk_d1_tarski]) ).
cnf(c_80,plain,
( C != A
| in(C,B)
| singleton(A) != B ),
inference(split_conjunct,[status(thm)],[sk_d1_tarski]) ).
cnf(c_81,plain,
( C = A
| ~ in(C,B)
| singleton(A) != B ),
inference(split_conjunct,[status(thm)],[sk_d1_tarski]) ).
cnf(c_82,plain,
succ(A) = set_union2(A,singleton(A)),
inference(clausify,[status(thm)],[d1_ordinal1]) ).
cnf(c_83,plain,
set_union2(A,B) = set_union2(B,A),
inference(clausify,[status(thm)],[commutativity_k2_xboole_0]) ).
cnf(c_94,negated_conjecture,
( sK1 = sK0
| in(sK0,sK1)
| in(sK0,set_union2(sK1,singleton(sK1))) ),
inference(paramod,[status(thm)],[c_82,c_47]) ).
cnf(c_95,negated_conjecture,
( ~ in(sK0,sK1)
| ~ in(sK0,set_union2(sK1,singleton(sK1))) ),
inference(paramod,[status(thm)],[c_82,c_46]) ).
cnf(c_96,negated_conjecture,
( sK1 != sK0
| ~ in(sK0,set_union2(sK1,singleton(sK1))) ),
inference(paramod,[status(thm)],[c_82,c_45]) ).
cnf(c_1436,plain,
empty_set = sK2,
inference(resolve,[status(thm)],[c_59,c_50]) ).
cnf(c_98,plain,
sK2 = empty_set,
inference(copy,[status(thm)],[c_1436]) ).
cnf(c_1437,plain,
~ in(A,sK2),
inference(resolve,[status(thm)],[c_49,c_59]) ).
cnf(c_99,plain,
~ in(A,empty_set),
inference(paramod,[status(thm)],[c_98,c_1437]) ).
cnf(c_1438,plain,
( sK3(B,A) = B
| singleton(B) = A
| ~ empty(A) ),
inference(resolve,[status(thm)],[c_49,c_79]) ).
cnf(c_380,plain,
( sK3(A,empty_set) = A
| singleton(A) = empty_set ),
inference(resolve,[status(thm)],[c_1438,c_67]) ).
cnf(c_1439,plain,
A = set_union2(A,A),
inference(copy,[status(thm)],[c_65]) ).
cnf(c_1440,plain,
( A != B
| in(A,set_union2(singleton(B),singleton(B))) ),
inference(resolve,[status(thm)],[c_80,c_1439]) ).
cnf(c_1441,plain,
in(set_union2(A,A),set_union2(singleton(A),singleton(A))),
inference(resolve,[status(thm)],[c_1440,c_65]) ).
cnf(c_1442,plain,
in(A,set_union2(singleton(A),singleton(A))),
inference(paramod,[status(thm)],[c_65,c_1441]) ).
cnf(c_386,plain,
in(A,singleton(A)),
inference(paramod,[status(thm)],[c_65,c_1442]) ).
cnf(c_1443,plain,
( in(A,B)
| in(A,C)
| ~ in(A,set_union2(B,C)) ),
inference(resolve,[status(thm)],[c_77,c_83]) ).
cnf(c_1444,negated_conjecture,
( sK1 = sK0
| in(sK0,sK1)
| in(sK0,sK1)
| in(sK0,singleton(sK1)) ),
inference(resolve,[status(thm)],[c_1443,c_94]) ).
cnf(c_493,negated_conjecture,
( sK1 = sK0
| in(sK0,sK1)
| in(sK0,singleton(sK1)) ),
inference(copy,[status(thm)],[c_1444]) ).
cnf(c_1445,plain,
( B != A
| singleton(A) != empty_set ),
inference(resolve,[status(thm)],[c_80,c_99]) ).
cnf(c_510,plain,
singleton(A) != empty_set,
inference(resolve,[status(thm)],[c_1445,c_65]) ).
cnf(c_511,plain,
sK3(A,empty_set) = A,
inference(resolve,[status(thm)],[c_510,c_380]) ).
cnf(c_1446,plain,
( ~ in(A,C)
| in(A,set_union2(B,C)) ),
inference(resolve,[status(thm)],[c_76,c_83]) ).
cnf(c_790,plain,
in(A,set_union2(B,singleton(A))),
inference(resolve,[status(thm)],[c_1446,c_386]) ).
cnf(c_1447,plain,
( A = B
| ~ in(A,singleton(B)) ),
inference(xxres,[status(thm)],[c_81]) ).
cnf(c_1448,plain,
( sK1 = sK0
| in(sK0,sK1)
| sK0 = sK1 ),
inference(resolve,[status(thm)],[c_1447,c_493]) ).
cnf(c_1449,plain,
( sK1 = sK0
| in(sK0,sK1)
| sK1 = sK0 ),
inference(copy,[status(thm)],[c_1448]) ).
cnf(c_1323,plain,
( in(sK0,sK1)
| sK1 = sK0 ),
inference(copy,[status(thm)],[c_1449]) ).
cnf(c_1450,plain,
A = sK3(A,empty_set),
inference(copy,[status(thm)],[c_511]) ).
cnf(c_1451,plain,
( ~ in(A,B)
| in(A,sK3(set_union2(B,C),empty_set)) ),
inference(resolve,[status(thm)],[c_76,c_1450]) ).
cnf(c_1452,plain,
( sK1 = sK0
| in(sK0,sK3(set_union2(sK1,A),empty_set)) ),
inference(resolve,[status(thm)],[c_1451,c_1323]) ).
cnf(c_1332,plain,
( sK1 = sK0
| in(sK0,set_union2(sK1,A)) ),
inference(paramod,[status(thm)],[c_511,c_1452]) ).
cnf(c_1453,negated_conjecture,
( sK1 = sK0
| ~ in(sK0,sK1) ),
inference(resolve,[status(thm)],[c_95,c_1332]) ).
cnf(c_1454,negated_conjecture,
( sK1 = sK0
| sK1 = sK0 ),
inference(resolve,[status(thm)],[c_1453,c_1323]) ).
cnf(c_1434,negated_conjecture,
sK1 = sK0,
inference(copy,[status(thm)],[c_1454]) ).
cnf(c_1455,negated_conjecture,
( sK1 != sK0
| ~ in(sK0,set_union2(sK0,singleton(sK1))) ),
inference(paramod,[status(thm)],[c_1434,c_96]) ).
cnf(c_1456,negated_conjecture,
( sK1 != sK0
| ~ in(sK0,set_union2(sK0,singleton(sK0))) ),
inference(paramod,[status(thm)],[c_1434,c_1455]) ).
cnf(c_1457,negated_conjecture,
( sK0 != sK0
| ~ in(sK0,set_union2(sK0,singleton(sK0))) ),
inference(paramod,[status(thm)],[c_1434,c_1456]) ).
cnf(c_1458,negated_conjecture,
~ in(sK0,set_union2(sK0,singleton(sK0))),
inference(copy,[status(thm)],[c_1457]) ).
cnf(c_1435,negated_conjecture,
$false,
inference(resolve,[status(thm)],[c_790,c_1458]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : NUM386+1 : TPTP v9.3.1. Released v3.2.0.
% 0.00/0.03 % Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.08/0.35 % Computer : n001.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 02:43:41 UTC 2026
% 0.02/0.36 % CPUTime :
% 0.02/0.36 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.02/0.36 % Process 729813 was started by sandbox on n001,
% 0.02/0.36 % Sat Sep 26 02:43:41 2026
% 0.02/0.36 % The command was "/export/starexec/sandbox/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p".
% 0.02/0.37
% 0.02/0.37 % From the command line: assign(max_seconds, 300).
% 0.86/0.60
% 0.86/0.60 % SZS status Theorem for theBenchmark
% 0.86/0.60
% 0.86/0.60 % Proof 1 at 0.15 (+ 0.07) seconds.
% 0.86/0.60 % Length of proof is 40.
% 0.86/0.60 % Level of proof is 8.
% 0.86/0.60 % Maximum clause weight is 27.000.
% 0.86/0.60 % Given clauses 176.
% 0.86/0.60
% 0.86/0.60 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------