%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : SET587+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 : n006.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 08:36:32 AM UTC 2026
% Result : Theorem 0.13s 0.54s
% Output : CNFRefutation 0.13s
% Verified :
% SZS Type : Refutation
% Derivation depth : 21
% Number of leaves : 6
% Syntax : Number of formulae : 55 ( 8 unt; 0 def)
% Number of atoms : 162 ( 34 equ)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 169 ( 62 ~; 78 |; 21 &)
% ( 7 <=>; 1 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 4 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 5 ( 3 usr; 1 prp; 0-2 aty)
% Number of functors : 7 ( 7 usr; 3 con; 0-2 aty)
% Number of variables : 100 ( 5 sgn 52 !; 5 ?)
% Comments :
%------------------------------------------------------------------------------
fof(difference_defn,axiom,
! [B,C,D] :
( member(D,difference(B,C))
<=> ( ~ member(D,C)
& member(D,B) ) ),
file('theBenchmark.p',difference_defn) ).
fof(empty_set_defn,axiom,
! [B] : ~ member(B,empty_set),
file('theBenchmark.p',empty_set_defn) ).
fof(subset_defn,axiom,
! [B,C] :
( subset(B,C)
<=> ! [D] :
( member(D,B)
=> member(D,C) ) ),
file('theBenchmark.p',subset_defn) ).
fof(equal_member_defn,axiom,
! [B,C] :
( B = C
<=> ! [D] :
( member(D,B)
<=> member(D,C) ) ),
file('theBenchmark.p',equal_member_defn) ).
fof(empty_defn,axiom,
! [B] :
( empty(B)
<=> ! [C] : ~ member(C,B) ),
file('theBenchmark.p',empty_defn) ).
fof(prove_difference_empty_set,conjecture,
! [B,C] :
( difference(B,C) = empty_set
<=> subset(B,C) ),
file('theBenchmark.p',prove_difference_empty_set) ).
fof(prove_difference_empty_set_neg,negated_conjecture,
~ ! [B,C] :
( difference(B,C) = empty_set
<=> subset(B,C) ),
inference(assume_negation,[status(cth)],[prove_difference_empty_set]) ).
fof(nnf_8,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_8]) ).
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] :
( ( ~ subset(B,C)
| difference(B,C) != empty_set )
& ( subset(B,C)
| difference(B,C) = empty_set ) ),
inference(fof_nnf,[status(thm)],[prove_difference_empty_set_neg]) ).
fof(sk_prove_difference_empty_set_neg,plain,
( ( ~ subset(sK1,sK2)
| difference(sK1,sK2) != empty_set )
& ( subset(sK1,sK2)
| difference(sK1,sK2) = empty_set ) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK1,sK2]),skolemize(B,sK1),skolemize(C,sK2)],[nnf_9]) ).
cnf(c_13,negated_conjecture,
( ~ subset(sK1,sK2)
| difference(sK1,sK2) != empty_set ),
inference(split_conjunct,[status(thm)],[sk_prove_difference_empty_set_neg]) ).
cnf(c_14,negated_conjecture,
( subset(sK1,sK2)
| difference(sK1,sK2) = empty_set ),
inference(split_conjunct,[status(thm)],[sk_prove_difference_empty_set_neg]) ).
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(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_6]) ).
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_17,plain,
( member(sK3(B,A),A)
| member(sK3(B,A),B)
| A = B ),
inference(split_conjunct,[status(thm)],[sk_equal_member_defn]) ).
fof(nnf_4,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(sK4(B,C),C)
& member(sK4(B,C),B) )
| subset(B,C) )
& ( ! [X0] :
( member(X0,C)
| ~ member(X0,B) )
| ~ subset(B,C) ) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK4]),skolemize(D,sK4(B,C))],[nnf_4]) ).
fof(sk_subset_defn,plain,
( ! [VAR_0,VAR_1] :
( ~ member(sK4(VAR_0,VAR_1),VAR_1)
| subset(VAR_0,VAR_1) )
& ! [VAR_0,VAR_1] :
( member(sK4(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_23,plain,
( ~ member(sK4(A,B),B)
| subset(A,B) ),
inference(split_conjunct,[status(thm)],[sk_subset_defn]) ).
cnf(c_24,plain,
( member(sK4(A,B),A)
| subset(A,B) ),
inference(split_conjunct,[status(thm)],[sk_subset_defn]) ).
cnf(c_25,plain,
( member(C,B)
| ~ member(C,A)
| ~ subset(A,B) ),
inference(split_conjunct,[status(thm)],[sk_subset_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,B)
| member(A,difference(B,C)) ),
inference(clausify,[status(thm)],[difference_defn]) ).
cnf(c_28,plain,
( ~ member(A,C)
| ~ member(A,difference(B,C)) ),
inference(clausify,[status(thm)],[difference_defn]) ).
cnf(c_29,plain,
( member(A,B)
| ~ member(A,difference(B,C)) ),
inference(clausify,[status(thm)],[difference_defn]) ).
cnf(c_32,plain,
( member(sK0(B),B)
| member(sK3(B,A),A)
| A = B ),
inference(resolve,[status(thm)],[c_17,c_12]) ).
cnf(c_43,plain,
( difference(sK1,sK2) != empty_set
| member(sK4(sK1,sK2),sK1) ),
inference(resolve,[status(thm)],[c_24,c_13]) ).
cnf(c_44,plain,
( member(sK4(B,C),B)
| member(A,C)
| ~ member(A,B) ),
inference(resolve,[status(thm)],[c_25,c_24]) ).
cnf(c_45,plain,
( difference(sK1,sK2) = empty_set
| member(A,sK2)
| ~ member(A,sK1) ),
inference(resolve,[status(thm)],[c_25,c_14]) ).
cnf(c_86,plain,
( member(sK0(A),A)
| empty_set = A ),
inference(resolve,[status(thm)],[c_32,c_26]) ).
cnf(c_521,plain,
( member(sK0(difference(A,B)),A)
| empty_set = difference(A,B) ),
inference(resolve,[status(thm)],[c_86,c_29]) ).
cnf(c_123,plain,
( member(sK0(difference(A,B)),A)
| difference(A,B) = empty_set ),
inference(copy,[status(thm)],[c_521]) ).
cnf(c_522,plain,
( ~ member(sK0(difference(A,B)),B)
| empty_set = difference(A,B) ),
inference(resolve,[status(thm)],[c_86,c_28]) ).
cnf(c_124,plain,
( ~ member(sK0(difference(A,B)),B)
| difference(A,B) = empty_set ),
inference(copy,[status(thm)],[c_522]) ).
cnf(c_205,plain,
( difference(sK1,sK2) = empty_set
| member(sK0(difference(sK1,A)),sK2)
| difference(sK1,A) = empty_set ),
inference(resolve,[status(thm)],[c_123,c_45]) ).
cnf(c_213,plain,
( member(sK0(difference(sK1,sK2)),sK2)
| difference(sK1,sK2) = empty_set ),
inference(factor,[status(thm)],[c_205]) ).
cnf(c_523,plain,
( difference(sK1,sK2) = empty_set
| difference(sK1,sK2) = empty_set ),
inference(resolve,[status(thm)],[c_213,c_124]) ).
cnf(c_308,plain,
difference(sK1,sK2) = empty_set,
inference(copy,[status(thm)],[c_523]) ).
cnf(c_524,plain,
( empty_set != empty_set
| member(sK4(sK1,sK2),sK1) ),
inference(paramod,[status(thm)],[c_308,c_43]) ).
cnf(c_309,plain,
member(sK4(sK1,sK2),sK1),
inference(copy,[status(thm)],[c_524]) ).
cnf(c_525,negated_conjecture,
( ~ subset(sK1,sK2)
| empty_set != empty_set ),
inference(paramod,[status(thm)],[c_308,c_13]) ).
cnf(c_310,negated_conjecture,
~ subset(sK1,sK2),
inference(copy,[status(thm)],[c_525]) ).
cnf(c_316,plain,
( member(sK4(sK1,A),sK1)
| member(sK4(sK1,sK2),A) ),
inference(resolve,[status(thm)],[c_309,c_44]) ).
cnf(c_324,negated_conjecture,
~ member(sK4(sK1,sK2),sK2),
inference(resolve,[status(thm)],[c_23,c_310]) ).
cnf(c_518,plain,
( member(sK4(sK1,A),B)
| member(sK4(sK1,A),difference(sK1,B))
| member(sK4(sK1,sK2),A) ),
inference(resolve,[status(thm)],[c_316,c_27]) ).
cnf(c_526,plain,
( member(sK4(sK1,sK2),difference(sK1,sK2))
| member(sK4(sK1,sK2),sK2) ),
inference(factor,[status(thm)],[c_518]) ).
cnf(c_527,plain,
( member(sK4(sK1,sK2),empty_set)
| member(sK4(sK1,sK2),sK2) ),
inference(paramod,[status(thm)],[c_308,c_526]) ).
cnf(c_528,plain,
member(sK4(sK1,sK2),empty_set),
inference(resolve,[status(thm)],[c_324,c_527]) ).
cnf(c_520,plain,
$false,
inference(resolve,[status(thm)],[c_26,c_528]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : SET587+3 : TPTP v9.3.1. Released v2.2.0.
% 0.00/0.04 % Command : prover9 -casc 300 -f /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.08/0.35 % Computer : n006.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 07:57:42 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 770115 was started by sandbox2 on n006,
% 0.08/0.35 % Sat Sep 26 07:57:42 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).
% 0.13/0.54
% 0.13/0.54 % SZS status Theorem for theBenchmark
% 0.13/0.54
% 0.13/0.54 % Proof 1 at 0.08 (+ 0.03) seconds.
% 0.13/0.54 % Length of proof is 35.
% 0.13/0.54 % Level of proof is 12.
% 0.13/0.54 % Maximum clause weight is 17.000.
% 0.13/0.54 % Given clauses 132.
% 0.13/0.54
% 0.13/0.54 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------