%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : SET600+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 : n010.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 08:36:34 AM UTC 2026
% Result : Theorem 0.16s 0.49s
% Output : CNFRefutation 0.16s
% Verified :
% SZS Type : Refutation
% Derivation depth : 21
% Number of leaves : 7
% Syntax : Number of formulae : 63 ( 9 unt; 0 def)
% Number of atoms : 179 ( 92 equ)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 191 ( 75 ~; 86 |; 21 &)
% ( 8 <=>; 1 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 4 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of predicates : 5 ( 3 usr; 1 prp; 0-2 aty)
% Number of functors : 6 ( 6 usr; 3 con; 0-2 aty)
% Number of variables : 82 ( 11 sgn 41 !; 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(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(subset_defn,axiom,
! [B,C] :
( subset(B,C)
<=> ! [D] :
( member(D,B)
=> member(D,C) ) ),
file('theBenchmark.p',subset_defn) ).
fof(empty_defn,axiom,
! [B] :
( empty(B)
<=> ! [C] : ~ member(C,B) ),
file('theBenchmark.p',empty_defn) ).
fof(equal_member_defn,axiom,
! [B,C] :
( B = C
<=> ! [D] :
( member(D,B)
<=> member(D,C) ) ),
file('theBenchmark.p',equal_member_defn) ).
fof(prove_th59,conjecture,
! [B,C] :
( union(B,C) = empty_set
<=> ( C = empty_set
& B = empty_set ) ),
file('theBenchmark.p',prove_th59) ).
fof(prove_th59_neg,negated_conjecture,
~ ! [B,C] :
( union(B,C) = empty_set
<=> ( C = empty_set
& B = empty_set ) ),
inference(assume_negation,[status(cth)],[prove_th59]) ).
fof(nnf_7,plain,
! [B] :
( ( ? [C] : member(C,B)
| empty(B) )
& ( ! [X0] : ~ member(X0,B)
| ~ empty(B) ) ),
inference(fof_nnf,[status(thm)],[empty_defn]) ).
fof(sk_empty_defn_sk,plain,
! [B] :
( ( member(sK0(B),B)
| empty(B) )
& ( ! [X0] : ~ member(X0,B)
| ~ empty(B) ) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK0]),skolemize(C,sK0(B))],[nnf_7]) ).
fof(sk_empty_defn,plain,
( ! [VAR_0] :
( member(sK0(VAR_0),VAR_0)
| empty(VAR_0) )
& ! [VAR_0,VAR_1] :
( ~ member(VAR_1,VAR_0)
| ~ empty(VAR_0) ) ),
inference(cnf_transformation,[status(thm)],[sk_empty_defn_sk]) ).
cnf(c_10,plain,
( member(sK0(A),A)
| empty(A) ),
inference(split_conjunct,[status(thm)],[sk_empty_defn]) ).
cnf(c_11,plain,
( ~ member(B,A)
| ~ empty(A) ),
inference(split_conjunct,[status(thm)],[sk_empty_defn]) ).
cnf(c_12,plain,
( ~ member(B,A)
| member(sK0(A),A) ),
inference(resolve,[status(thm)],[c_10,c_11]) ).
fof(nnf_9,plain,
? [B,C] :
( ( empty_set != C
| empty_set != B
| union(B,C) != empty_set )
& ( ( empty_set = C
& empty_set = B )
| union(B,C) = empty_set ) ),
inference(fof_nnf,[status(thm)],[prove_th59_neg]) ).
fof(sk_prove_th59_neg_sk,plain,
( ( empty_set != sK2
| empty_set != sK1
| union(sK1,sK2) != empty_set )
& ( ( empty_set = sK2
& empty_set = sK1 )
| union(sK1,sK2) = empty_set ) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK1,sK2]),skolemize(B,sK1),skolemize(C,sK2)],[nnf_9]) ).
fof(sk_prove_th59_neg,plain,
( ( empty_set != sK2
| empty_set != sK1
| union(sK1,sK2) != empty_set )
& ( empty_set = sK2
| union(sK1,sK2) = empty_set )
& ( empty_set = sK1
| union(sK1,sK2) = empty_set ) ),
inference(cnf_transformation,[status(thm)],[sk_prove_th59_neg_sk]) ).
cnf(c_13,negated_conjecture,
( empty_set != sK2
| empty_set != sK1
| union(sK1,sK2) != empty_set ),
inference(split_conjunct,[status(thm)],[sk_prove_th59_neg]) ).
cnf(c_178,negated_conjecture,
( empty_set != sK2
| sK1 != empty_set
| union(sK1,sK2) != empty_set ),
inference(copy,[status(thm)],[c_13]) ).
cnf(c_14,negated_conjecture,
( sK2 != empty_set
| sK1 != empty_set
| union(sK1,sK2) != empty_set ),
inference(copy,[status(thm)],[c_178]) ).
cnf(c_15,negated_conjecture,
( empty_set = sK2
| union(sK1,sK2) = empty_set ),
inference(split_conjunct,[status(thm)],[sk_prove_th59_neg]) ).
cnf(c_16,negated_conjecture,
( sK2 = empty_set
| union(sK1,sK2) = empty_set ),
inference(copy,[status(thm)],[c_15]) ).
cnf(c_17,negated_conjecture,
( empty_set = sK1
| union(sK1,sK2) = empty_set ),
inference(split_conjunct,[status(thm)],[sk_prove_th59_neg]) ).
cnf(c_18,negated_conjecture,
( sK1 = empty_set
| union(sK1,sK2) = empty_set ),
inference(copy,[status(thm)],[c_17]) ).
fof(nnf_8,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(sK3(B,C),C)
| ~ member(sK3(B,C),B) )
& ( member(sK3(B,C),C)
| member(sK3(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,[sK3]),skolemize(D,sK3(B,C))],[nnf_8]) ).
fof(sk_equal_member_defn,plain,
( ! [VAR_0,VAR_1] :
( ~ member(sK3(VAR_1,VAR_0),VAR_0)
| ~ member(sK3(VAR_1,VAR_0),VAR_1)
| VAR_0 = VAR_1 )
& ! [VAR_0,VAR_1] :
( member(sK3(VAR_1,VAR_0),VAR_0)
| member(sK3(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_20,plain,
( member(sK3(B,A),A)
| member(sK3(B,A),B)
| A = B ),
inference(split_conjunct,[status(thm)],[sk_equal_member_defn]) ).
cnf(c_26,plain,
( member(C,B)
| ~ member(C,A)
| ~ subset(A,B) ),
inference(clausify,[status(thm)],[subset_defn]) ).
cnf(c_29,plain,
( subset(A,B)
| A != B ),
inference(clausify,[status(thm)],[equal_defn]) ).
cnf(c_31,plain,
~ member(A,empty_set),
inference(clausify,[status(thm)],[empty_set_defn]) ).
cnf(c_32,plain,
( ~ member(A,C)
| member(A,union(B,C)) ),
inference(clausify,[status(thm)],[union_defn]) ).
cnf(c_33,plain,
( ~ member(A,B)
| member(A,union(B,C)) ),
inference(clausify,[status(thm)],[union_defn]) ).
cnf(c_34,plain,
( member(A,C)
| member(A,B)
| ~ member(A,union(B,C)) ),
inference(clausify,[status(thm)],[union_defn]) ).
cnf(c_36,plain,
( member(sK0(B),B)
| member(sK3(B,A),A)
| A = B ),
inference(resolve,[status(thm)],[c_20,c_12]) ).
cnf(c_48,plain,
( sK1 = empty_set
| subset(union(sK1,sK2),empty_set) ),
inference(resolve,[status(thm)],[c_29,c_18]) ).
cnf(c_49,plain,
( sK2 = empty_set
| subset(union(sK1,sK2),empty_set) ),
inference(resolve,[status(thm)],[c_29,c_16]) ).
cnf(c_179,plain,
( member(A,B)
| ~ member(A,union(empty_set,B)) ),
inference(resolve,[status(thm)],[c_34,c_31]) ).
cnf(c_64,plain,
~ member(A,union(empty_set,empty_set)),
inference(resolve,[status(thm)],[c_179,c_31]) ).
cnf(c_77,plain,
( member(sK0(A),A)
| empty_set = A ),
inference(resolve,[status(thm)],[c_36,c_31]) ).
cnf(c_180,plain,
empty_set = union(empty_set,empty_set),
inference(resolve,[status(thm)],[c_77,c_64]) ).
cnf(c_117,plain,
union(empty_set,empty_set) = empty_set,
inference(copy,[status(thm)],[c_180]) ).
cnf(c_120,plain,
( member(sK0(A),union(A,B))
| empty_set = A ),
inference(resolve,[status(thm)],[c_77,c_33]) ).
cnf(c_121,plain,
( member(sK0(A),union(B,A))
| empty_set = A ),
inference(resolve,[status(thm)],[c_77,c_32]) ).
cnf(c_181,plain,
( member(A,empty_set)
| ~ member(A,union(sK1,sK2))
| sK1 = empty_set ),
inference(resolve,[status(thm)],[c_48,c_26]) ).
cnf(c_124,plain,
( ~ member(A,union(sK1,sK2))
| sK1 = empty_set ),
inference(resolve,[status(thm)],[c_31,c_181]) ).
cnf(c_182,plain,
( member(A,empty_set)
| ~ member(A,union(sK1,sK2))
| sK2 = empty_set ),
inference(resolve,[status(thm)],[c_49,c_26]) ).
cnf(c_137,plain,
( ~ member(A,union(sK1,sK2))
| sK2 = empty_set ),
inference(resolve,[status(thm)],[c_31,c_182]) ).
cnf(c_183,plain,
( sK1 = empty_set
| empty_set = sK1 ),
inference(resolve,[status(thm)],[c_120,c_124]) ).
cnf(c_184,plain,
( sK1 = empty_set
| sK1 = empty_set ),
inference(copy,[status(thm)],[c_183]) ).
cnf(c_158,plain,
sK1 = empty_set,
inference(copy,[status(thm)],[c_184]) ).
cnf(c_166,plain,
( ~ member(A,union(empty_set,sK2))
| sK2 = empty_set ),
inference(paramod,[status(thm)],[c_158,c_137]) ).
cnf(c_185,negated_conjecture,
( sK2 != empty_set
| sK1 != empty_set
| union(empty_set,sK2) != empty_set ),
inference(paramod,[status(thm)],[c_158,c_14]) ).
cnf(c_186,negated_conjecture,
( sK2 != empty_set
| empty_set != empty_set
| union(empty_set,sK2) != empty_set ),
inference(paramod,[status(thm)],[c_158,c_185]) ).
cnf(c_170,negated_conjecture,
( sK2 != empty_set
| union(empty_set,sK2) != empty_set ),
inference(copy,[status(thm)],[c_186]) ).
cnf(c_187,plain,
( sK2 = empty_set
| empty_set = sK2 ),
inference(resolve,[status(thm)],[c_121,c_166]) ).
cnf(c_188,plain,
( sK2 = empty_set
| sK2 = empty_set ),
inference(copy,[status(thm)],[c_187]) ).
cnf(c_172,plain,
sK2 = empty_set,
inference(copy,[status(thm)],[c_188]) ).
cnf(c_189,negated_conjecture,
( sK2 != empty_set
| union(empty_set,empty_set) != empty_set ),
inference(paramod,[status(thm)],[c_172,c_170]) ).
cnf(c_190,negated_conjecture,
( sK2 != empty_set
| empty_set != empty_set ),
inference(paramod,[status(thm)],[c_117,c_189]) ).
cnf(c_191,negated_conjecture,
( empty_set != empty_set
| empty_set != empty_set ),
inference(paramod,[status(thm)],[c_172,c_190]) ).
cnf(c_192,negated_conjecture,
empty_set != empty_set,
inference(copy,[status(thm)],[c_191]) ).
cnf(c_177,negated_conjecture,
$false,
inference(copy,[status(thm)],[c_192]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.05 % Problem : SET600+3 : TPTP v9.3.1. Released v2.2.0.
% 0.00/0.06 % Command : prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.16/0.42 % Computer : n010.cluster.edu
% 0.16/0.42 % Model : x86_64 x86_64
% 0.16/0.42 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.42 % Memory : 8046.5625MB
% 0.16/0.42 % OS : Linux 6.8.0-71-generic
% 0.16/0.42 % CPULimit : 300
% 0.16/0.42 % WCLimit : 300
% 0.16/0.42 % DateTime : Sat Sep 26 07:59:00 UTC 2026
% 0.16/0.43 % CPUTime :
% 0.16/0.43 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.16/0.43 % Process 2855190 was started by sandbox2 on n010,
% 0.16/0.43 % Sat Sep 26 07:59:00 2026
% 0.16/0.43 % The command was "/export/starexec/sandbox2/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p".
% 0.16/0.44
% 0.16/0.44 % From the command line: assign(max_seconds, 300).
% 0.16/0.49
% 0.16/0.49 % SZS status Theorem for theBenchmark
% 0.16/0.49
% 0.16/0.49 % Proof 1 at 0.03 (+ 0.01) seconds.
% 0.16/0.49 % Length of proof is 38.
% 0.16/0.49 % Level of proof is 9.
% 0.16/0.49 % Maximum clause weight is 13.000.
% 0.16/0.49 % Given clauses 49.
% 0.16/0.49
% 0.16/0.49 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------