%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : SET635+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 : n011.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 08:36:41 AM UTC 2026
% Result : Theorem 0.09s 0.42s
% Output : CNFRefutation 0.09s
% Verified :
% SZS Type : Refutation
% Derivation depth : 8
% Number of leaves : 7
% Syntax : Number of formulae : 28 ( 26 unt; 0 def)
% Number of atoms : 30 ( 25 equ)
% Maximal formula atoms : 2 ( 1 avg)
% Number of connectives : 10 ( 8 ~; 1 |; 0 &)
% ( 1 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 5 ( 2 avg)
% Maximal term depth : 4 ( 2 avg)
% Number of predicates : 3 ( 1 usr; 1 prp; 0-2 aty)
% Number of functors : 7 ( 7 usr; 4 con; 0-2 aty)
% Number of variables : 51 ( 2 sgn 19 !; 3 ?)
% Comments :
%------------------------------------------------------------------------------
fof(intersection_is_subset,axiom,
! [B,C] : subset(intersection(B,C),B),
file('theBenchmark.p',intersection_is_subset) ).
fof(difference_empty_set,axiom,
! [B,C] :
( difference(B,C) = empty_set
<=> subset(B,C) ),
file('theBenchmark.p',difference_empty_set) ).
fof(union_empty_set,axiom,
! [B] : union(B,empty_set) = B,
file('theBenchmark.p',union_empty_set) ).
fof(difference_and_intersection_and_union,axiom,
! [B,C,D] : difference(B,intersection(C,D)) = union(difference(B,C),difference(B,D)),
file('theBenchmark.p',difference_and_intersection_and_union) ).
fof(difference_and_intersection,axiom,
! [B,C,D] : intersection(B,difference(C,D)) = difference(intersection(B,C),D),
file('theBenchmark.p',difference_and_intersection) ).
fof(commutativity_of_union,axiom,
! [B,C] : union(B,C) = union(C,B),
file('theBenchmark.p',commutativity_of_union) ).
fof(prove_th117,conjecture,
! [B,C,D] : intersection(B,difference(C,D)) = difference(intersection(B,C),intersection(B,D)),
file('theBenchmark.p',prove_th117) ).
fof(prove_th117_neg,negated_conjecture,
~ ! [B,C,D] : intersection(B,difference(C,D)) = difference(intersection(B,C),intersection(B,D)),
inference(assume_negation,[status(cth)],[prove_th117]) ).
fof(nnf_16,plain,
? [B,C,D] : difference(intersection(B,C),intersection(B,D)) != intersection(B,difference(C,D)),
inference(fof_nnf,[status(thm)],[prove_th117_neg]) ).
fof(sk_prove_th117_neg,plain,
difference(intersection(sK0,sK1),intersection(sK0,sK2)) != intersection(sK0,difference(sK1,sK2)),
inference(skolemize,[status(esa),new_symbols(skolem,[sK0,sK1,sK2]),skolemize(B,sK0),skolemize(C,sK1),skolemize(D,sK2)],[nnf_16]) ).
cnf(c_20,negated_conjecture,
difference(intersection(sK0,sK1),intersection(sK0,sK2)) != intersection(sK0,difference(sK1,sK2)),
inference(split_conjunct,[status(thm)],[sk_prove_th117_neg]) ).
cnf(c_21,negated_conjecture,
intersection(sK0,difference(sK1,sK2)) != difference(intersection(sK0,sK1),intersection(sK0,sK2)),
inference(copy,[status(thm)],[c_20]) ).
cnf(c_32,plain,
union(A,B) = union(B,A),
inference(clausify,[status(thm)],[commutativity_of_union]) ).
cnf(c_42,plain,
difference(intersection(A,B),C) = intersection(A,difference(B,C)),
inference(clausify,[status(thm)],[difference_and_intersection]) ).
cnf(c_43,plain,
intersection(A,difference(B,C)) = difference(intersection(A,B),C),
inference(copy,[status(thm)],[c_42]) ).
cnf(c_44,plain,
union(difference(A,B),difference(A,C)) = difference(A,intersection(B,C)),
inference(clausify,[status(thm)],[difference_and_intersection_and_union]) ).
cnf(c_45,plain,
union(A,empty_set) = A,
inference(clausify,[status(thm)],[union_empty_set]) ).
cnf(c_46,plain,
( ~ subset(A,B)
| difference(A,B) = empty_set ),
inference(clausify,[status(thm)],[difference_empty_set]) ).
cnf(c_48,plain,
subset(intersection(A,B),A),
inference(clausify,[status(thm)],[intersection_is_subset]) ).
cnf(c_223,negated_conjecture,
difference(intersection(sK0,sK1),sK2) != difference(intersection(sK0,sK1),intersection(sK0,sK2)),
inference(paramod,[status(thm)],[c_43,c_21]) ).
cnf(c_50,negated_conjecture,
difference(intersection(sK0,sK1),intersection(sK0,sK2)) != difference(intersection(sK0,sK1),sK2),
inference(copy,[status(thm)],[c_223]) ).
cnf(c_224,plain,
A = union(empty_set,A),
inference(paramod,[status(thm)],[c_45,c_32]) ).
cnf(c_100,plain,
union(empty_set,A) = A,
inference(copy,[status(thm)],[c_224]) ).
cnf(c_103,plain,
difference(intersection(A,B),A) = empty_set,
inference(resolve,[status(thm)],[c_48,c_46]) ).
cnf(c_225,plain,
union(empty_set,difference(intersection(A,B),C)) = difference(intersection(A,B),intersection(A,C)),
inference(paramod,[status(thm)],[c_103,c_44]) ).
cnf(c_226,plain,
difference(intersection(A,B),C) = difference(intersection(A,B),intersection(A,C)),
inference(paramod,[status(thm)],[c_100,c_225]) ).
cnf(c_221,plain,
difference(intersection(A,B),intersection(A,C)) = difference(intersection(A,B),C),
inference(copy,[status(thm)],[c_226]) ).
cnf(c_222,plain,
$false,
inference(resolve,[status(thm)],[c_221,c_50]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : SET635+3 : TPTP v9.3.1. Released v2.2.0.
% 0.00/0.04 % Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.09/0.37 % Computer : n011.cluster.edu
% 0.09/0.37 % Model : x86_64 x86_64
% 0.09/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.37 % Memory : 8046.5625MB
% 0.09/0.37 % OS : Linux 6.8.0-71-generic
% 0.09/0.37 % CPULimit : 300
% 0.09/0.37 % WCLimit : 300
% 0.09/0.37 % DateTime : Sat Sep 26 08:01:59 UTC 2026
% 0.09/0.37 % CPUTime :
% 0.09/0.37 % Prover9 (64) version 2026-6A, July 2026, CASC-J13.
% 0.09/0.37 % Process 4172490 was started by sandbox on n011,
% 0.09/0.37 % Sat Sep 26 08:01:59 2026
% 0.09/0.37 % The command was "/export/starexec/sandbox/solver/bin/prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p".
% 0.09/0.38
% 0.09/0.38 % From the command line: assign(max_seconds, 300).
% 0.09/0.42
% 0.09/0.42 % SZS status Theorem for theBenchmark
% 0.09/0.42
% 0.09/0.42 % Proof 1 at 0.02 (+ 0.01) seconds.
% 0.09/0.42 % Length of proof is 21.
% 0.09/0.42 % Level of proof is 4.
% 0.09/0.42 % Maximum clause weight is 13.000.
% 0.09/0.42 % Given clauses 51.
% 0.09/0.42
% 0.09/0.42 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------