TPTP Problem File: GRA005+1.p
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- Solve Problem
%--------------------------------------------------------------------------
% File : GRA005+1 : TPTP v8.2.0. Bugfixed v3.2.0.
% Domain : Graph Theory
% Problem : Maximal shortest path length in terms of triangles
% Version : Especial.
% English : In a shortest path P from V1 to V2 with edges E1 and E2, E1
% preceding E2, there is no edge from the tail of E1 to the
% head of E2.
% Refs :
% Source : [TPTP]
% Names :
% Status : Theorem
% Rating : 0.17 v7.5.0, 0.19 v7.4.0, 0.17 v7.3.0, 0.14 v7.1.0, 0.13 v7.0.0, 0.10 v6.4.0, 0.15 v6.3.0, 0.12 v6.2.0, 0.20 v6.1.0, 0.30 v6.0.0, 0.22 v5.4.0, 0.29 v5.3.0, 0.33 v5.2.0, 0.20 v5.1.0, 0.24 v5.0.0, 0.21 v4.1.0, 0.26 v4.0.1, 0.35 v4.0.0, 0.33 v3.7.0, 0.30 v3.5.0, 0.32 v3.4.0, 0.26 v3.3.0, 0.29 v3.2.0
% Syntax : Number of formulae : 18 ( 1 unt; 0 def)
% Number of atoms : 95 ( 26 equ)
% Maximal formula atoms : 9 ( 5 avg)
% Number of connectives : 84 ( 7 ~; 3 |; 49 &)
% ( 3 <=>; 17 =>; 2 <=; 3 <~>)
% Maximal formula depth : 13 ( 9 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 12 ( 11 usr; 1 prp; 0-3 aty)
% Number of functors : 12 ( 12 usr; 6 con; 0-2 aty)
% Number of variables : 71 ( 60 !; 11 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments :
% Bugfixes : v3.2.0 - Bugfix to GRA001+0.ax
%--------------------------------------------------------------------------
%----Include axioms for directed graphs and paths
include('Axioms/GRA001+0.ax').
%--------------------------------------------------------------------------
fof(triangle_defn,axiom,
! [E1,E2,E3] :
( triangle(E1,E2,E3)
<=> ( edge(E1)
& edge(E2)
& edge(E3)
& sequential(E1,E2)
& sequential(E2,E3)
& sequential(E3,E1) ) ) ).
fof(length_defn,axiom,
! [V1,V2,P] :
( path(V1,V2,P)
=> length_of(P) = number_of_in(edges,P) ) ).
fof(path_length_sequential_pairs,axiom,
! [V1,V2,P] :
( path(V1,V2,P)
=> number_of_in(sequential_pairs,P) = minus(length_of(P),n1) ) ).
fof(sequential_pairs_and_triangles,axiom,
! [P,V1,V2] :
( ( path(V1,V2,P)
& ! [E1,E2] :
( ( on_path(E1,P)
& on_path(E2,P)
& sequential(E1,E2) )
=> ? [E3] : triangle(E1,E2,E3) ) )
=> number_of_in(sequential_pairs,P) = number_of_in(triangles,P) ) ).
fof(graph_has_them_all,axiom,
! [Things,InThese] : less_or_equal(number_of_in(Things,InThese),number_of_in(Things,graph)) ).
fof(no_short_cut_edge,conjecture,
! [V1,V2,E1,E2,P] :
( ( shortest_path(V1,V2,P)
& precedes(E1,E2,P) )
=> ~ ? [E3] :
( edge(E3)
& tail_of(E3) = tail_of(E1)
& head_of(E3) = head_of(E2) ) ) ).
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