%------------------------------------------------------------------------------
% File : Prover9---2026-6A
% Problem : GRA010+2 : TPTP v9.3.1. Bugfixed v3.2.0.
% Transfm : none
% Format : tptp:raw
% Command : prover9 -casc 300 -f /export/starexec/sandbox/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 07:19:23 AM UTC 2026
% Result : Theorem 0.09s 0.40s
% Output : CNFRefutation 0.09s
% Verified :
% SZS Type : Refutation
% Derivation depth : 10
% Number of leaves : 2
% Syntax : Number of formulae : 27 ( 8 unt; 0 def)
% Number of atoms : 95 ( 20 equ)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 100 ( 32 ~; 36 |; 24 &)
% ( 0 <=>; 8 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 7 ( 5 usr; 2 prp; 0-3 aty)
% Number of functors : 9 ( 9 usr; 5 con; 0-3 aty)
% Number of variables : 66 ( 3 sgn 40 !; 9 ?)
% Comments :
%------------------------------------------------------------------------------
fof(sequential_pairs_and_triangles,axiom,
! [P,V1,V2] :
( ( ! [E1,E2] :
( ( sequential(E1,E2)
& on_path(E2,P)
& on_path(E1,P) )
=> ? [E3] : triangle(E1,E2,E3) )
& path(V1,V2,P) )
=> number_of_in(sequential_pairs,P) = number_of_in(triangles,P) ),
file('theBenchmark.p',sequential_pairs_and_triangles) ).
fof(complete_means_sequential_pairs_and_triangles,conjecture,
( complete
=> ! [P,V1,V2] :
( ( ! [E1,E2] :
( ( sequential(E1,E2)
& on_path(E2,P)
& on_path(E1,P) )
=> ? [E3] : triangle(E1,E2,E3) )
& path(V1,V2,P) )
=> number_of_in(sequential_pairs,P) = number_of_in(triangles,P) ) ),
file('theBenchmark.p',complete_means_sequential_pairs_and_triangles) ).
fof(complete_means_sequential_pairs_and_triangles_neg,negated_conjecture,
~ ( complete
=> ! [P,V1,V2] :
( ( ! [E1,E2] :
( ( sequential(E1,E2)
& on_path(E2,P)
& on_path(E1,P) )
=> ? [E3] : triangle(E1,E2,E3) )
& path(V1,V2,P) )
=> number_of_in(sequential_pairs,P) = number_of_in(triangles,P) ) ),
inference(assume_negation,[status(cth)],[complete_means_sequential_pairs_and_triangles]) ).
fof(nnf_16,plain,
! [P,V1,V2] :
( number_of_in(triangles,P) = number_of_in(sequential_pairs,P)
| ? [E1,E2] :
( ! [E3] : ~ triangle(E1,E2,E3)
& sequential(E1,E2)
& on_path(E2,P)
& on_path(E1,P) )
| ~ path(V1,V2,P) ),
inference(fof_nnf,[status(thm)],[sequential_pairs_and_triangles]) ).
fof(sk_sequential_pairs_and_triangles_sk,plain,
! [P,V1,V2] :
( number_of_in(triangles,P) = number_of_in(sequential_pairs,P)
| ( ! [E3] : ~ triangle(sK0(P,V1,V2),sK1(P,V1,V2),E3)
& sequential(sK0(P,V1,V2),sK1(P,V1,V2))
& on_path(sK1(P,V1,V2),P)
& on_path(sK0(P,V1,V2),P) )
| ~ path(V1,V2,P) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK0,sK1]),skolemize(E1,sK0(P,V1,V2)),skolemize(E2,sK1(P,V1,V2))],[nnf_16]) ).
fof(sk_sequential_pairs_and_triangles,plain,
( ! [VAR_0,VAR_1,VAR_2,VAR_3] :
( number_of_in(triangles,VAR_2) = number_of_in(sequential_pairs,VAR_2)
| ~ triangle(sK0(VAR_2,VAR_0,VAR_1),sK1(VAR_2,VAR_0,VAR_1),VAR_3)
| ~ path(VAR_0,VAR_1,VAR_2) )
& ! [VAR_0,VAR_1,VAR_2] :
( number_of_in(triangles,VAR_2) = number_of_in(sequential_pairs,VAR_2)
| sequential(sK0(VAR_2,VAR_0,VAR_1),sK1(VAR_2,VAR_0,VAR_1))
| ~ path(VAR_0,VAR_1,VAR_2) )
& ! [VAR_0,VAR_1,VAR_2] :
( number_of_in(triangles,VAR_2) = number_of_in(sequential_pairs,VAR_2)
| on_path(sK1(VAR_2,VAR_0,VAR_1),VAR_2)
| ~ path(VAR_0,VAR_1,VAR_2) )
& ! [VAR_0,VAR_1,VAR_2] :
( number_of_in(triangles,VAR_2) = number_of_in(sequential_pairs,VAR_2)
| on_path(sK0(VAR_2,VAR_0,VAR_1),VAR_2)
| ~ path(VAR_0,VAR_1,VAR_2) ) ),
inference(cnf_transformation,[status(thm)],[sk_sequential_pairs_and_triangles_sk]) ).
cnf(c_84,plain,
( number_of_in(triangles,C) = number_of_in(sequential_pairs,C)
| on_path(sK0(C,A,B),C)
| ~ path(A,B,C) ),
inference(split_conjunct,[status(thm)],[sk_sequential_pairs_and_triangles]) ).
cnf(c_85,plain,
( number_of_in(triangles,C) = number_of_in(sequential_pairs,C)
| on_path(sK1(C,A,B),C)
| ~ path(A,B,C) ),
inference(split_conjunct,[status(thm)],[sk_sequential_pairs_and_triangles]) ).
cnf(c_86,plain,
( number_of_in(triangles,C) = number_of_in(sequential_pairs,C)
| sequential(sK0(C,A,B),sK1(C,A,B))
| ~ path(A,B,C) ),
inference(split_conjunct,[status(thm)],[sk_sequential_pairs_and_triangles]) ).
cnf(c_87,plain,
( number_of_in(triangles,C) = number_of_in(sequential_pairs,C)
| ~ triangle(sK0(C,A,B),sK1(C,A,B),D)
| ~ path(A,B,C) ),
inference(split_conjunct,[status(thm)],[sk_sequential_pairs_and_triangles]) ).
fof(nnf_19,plain,
( ? [P,V1,V2] :
( number_of_in(triangles,P) != number_of_in(sequential_pairs,P)
& ! [E1,E2] :
( ? [E3] : triangle(E1,E2,E3)
| ~ sequential(E1,E2)
| ~ on_path(E2,P)
| ~ on_path(E1,P) )
& path(V1,V2,P) )
& complete ),
inference(fof_nnf,[status(thm)],[complete_means_sequential_pairs_and_triangles_neg]) ).
fof(sk_complete_means_sequential_pairs_and_triangles_neg,plain,
( number_of_in(triangles,sK4) != number_of_in(sequential_pairs,sK4)
& ! [E1,E2] :
( triangle(E1,E2,sK5(E1,E2))
| ~ sequential(E1,E2)
| ~ on_path(E2,sK4)
| ~ on_path(E1,sK4) )
& path(sK2,sK3,sK4)
& complete ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK2,sK3,sK4,sK5]),skolemize(P,sK4),skolemize(V1,sK2),skolemize(V2,sK3),skolemize(E3,sK5(E1,E2))],[nnf_19]) ).
cnf(c_90,negated_conjecture,
path(sK2,sK3,sK4),
inference(split_conjunct,[status(thm)],[sk_complete_means_sequential_pairs_and_triangles_neg]) ).
cnf(c_91,negated_conjecture,
( triangle(A,B,sK5(A,B))
| ~ sequential(A,B)
| ~ on_path(B,sK4)
| ~ on_path(A,sK4) ),
inference(split_conjunct,[status(thm)],[sk_complete_means_sequential_pairs_and_triangles_neg]) ).
cnf(c_92,negated_conjecture,
number_of_in(triangles,sK4) != number_of_in(sequential_pairs,sK4),
inference(split_conjunct,[status(thm)],[sk_complete_means_sequential_pairs_and_triangles_neg]) ).
cnf(c_163,negated_conjecture,
( number_of_in(triangles,sK4) = number_of_in(sequential_pairs,sK4)
| sequential(sK0(sK4,sK2,sK3),sK1(sK4,sK2,sK3)) ),
inference(resolve,[status(thm)],[c_90,c_86]) ).
cnf(c_131,negated_conjecture,
sequential(sK0(sK4,sK2,sK3),sK1(sK4,sK2,sK3)),
inference(resolve,[status(thm)],[c_92,c_163]) ).
cnf(c_164,negated_conjecture,
( number_of_in(triangles,sK4) = number_of_in(sequential_pairs,sK4)
| on_path(sK1(sK4,sK2,sK3),sK4) ),
inference(resolve,[status(thm)],[c_90,c_85]) ).
cnf(c_132,negated_conjecture,
on_path(sK1(sK4,sK2,sK3),sK4),
inference(resolve,[status(thm)],[c_92,c_164]) ).
cnf(c_165,negated_conjecture,
( number_of_in(triangles,sK4) = number_of_in(sequential_pairs,sK4)
| on_path(sK0(sK4,sK2,sK3),sK4) ),
inference(resolve,[status(thm)],[c_90,c_84]) ).
cnf(c_133,negated_conjecture,
on_path(sK0(sK4,sK2,sK3),sK4),
inference(resolve,[status(thm)],[c_92,c_165]) ).
cnf(c_166,negated_conjecture,
( number_of_in(triangles,sK4) = number_of_in(sequential_pairs,sK4)
| ~ triangle(sK0(sK4,sK2,sK3),sK1(sK4,sK2,sK3),A) ),
inference(resolve,[status(thm)],[c_87,c_90]) ).
cnf(c_150,negated_conjecture,
~ triangle(sK0(sK4,sK2,sK3),sK1(sK4,sK2,sK3),A),
inference(resolve,[status(thm)],[c_166,c_92]) ).
cnf(c_167,negated_conjecture,
( triangle(sK0(sK4,sK2,sK3),sK1(sK4,sK2,sK3),sK5(sK0(sK4,sK2,sK3),sK1(sK4,sK2,sK3)))
| ~ on_path(sK1(sK4,sK2,sK3),sK4)
| ~ on_path(sK0(sK4,sK2,sK3),sK4) ),
inference(resolve,[status(thm)],[c_131,c_91]) ).
cnf(c_168,negated_conjecture,
( triangle(sK0(sK4,sK2,sK3),sK1(sK4,sK2,sK3),sK5(sK0(sK4,sK2,sK3),sK1(sK4,sK2,sK3)))
| ~ on_path(sK1(sK4,sK2,sK3),sK4) ),
inference(resolve,[status(thm)],[c_133,c_167]) ).
cnf(c_169,negated_conjecture,
triangle(sK0(sK4,sK2,sK3),sK1(sK4,sK2,sK3),sK5(sK0(sK4,sK2,sK3),sK1(sK4,sK2,sK3))),
inference(resolve,[status(thm)],[c_132,c_168]) ).
cnf(c_162,negated_conjecture,
$false,
inference(resolve,[status(thm)],[c_150,c_169]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : GRA010+2 : TPTP v9.3.1. Bugfixed v3.2.0.
% 0.00/0.04 % Command : prover9 -casc 300 -f /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.09/0.37 % Computer : n010.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 : Fri Sep 25 13:49:01 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 909620 was started by sandbox on n010,
% 0.09/0.37 % Fri Sep 25 13:49:01 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).
% 0.09/0.40
% 0.09/0.40 % SZS status Theorem for theBenchmark
% 0.09/0.40
% 0.09/0.40 % Proof 1 at 0.01 (+ 0.01) seconds.
% 0.09/0.40 % Length of proof is 14.
% 0.09/0.40 % Level of proof is 3.
% 0.09/0.40 % Maximum clause weight is 31.000.
% 0.09/0.40 % Given clauses 76.
% 0.09/0.40
% 0.09/0.40 % SZS output start CNFRefutation for theBenchmark
% See solution above
%------------------------------------------------------------------------------