%------------------------------------------------------------------------------
% File     : GRA001+0 : TPTP v9.0.0. Bugfixed v3.2.0.
% Domain   : Graph Theory
% Axioms   : Directed graphs and paths
% Version  : [TPTP] axioms : Especial.
% English  :

% Refs     :
% Source   : [TPTP]
% Names    :

% Status   : Satisfiable
% Syntax   : Number of formulae    :   12 (   0 unt;   0 def)
%            Number of atoms       :   72 (  21 equ)
%            Maximal formula atoms :    9 (   6 avg)
%            Number of connectives :   66 (   6   ~;   3   |;  38   &)
%                                         (   2 <=>;  12  =>;   2  <=;   3 <~>)
%            Maximal formula depth :   13 (  10 avg)
%            Maximal term depth    :    2 (   1 avg)
%            Number of predicates  :   11 (  10 usr;   1 prp; 0-3 aty)
%            Number of functors    :    5 (   5 usr;   1 con; 0-2 aty)
%            Number of variables   :   48 (  39   !;   9   ?)
% SPC      : 

% Comments :
% Bugfixes : v3.2.0 - Added formula edge_ends_are_vertices.
%------------------------------------------------------------------------------
fof(no_loops,axiom,
    ! [E] :
      ( edge(E)
     => head_of(E) != tail_of(E) ) ).

fof(edge_ends_are_vertices,axiom,
    ! [E] :
      ( edge(E)
     => ( vertex(head_of(E))
        & vertex(tail_of(E)) ) ) ).

fof(complete_properties,axiom,
    ( complete
   => ! [V1,V2] :
        ( ( vertex(V1)
          & vertex(V2)
          & V1 != V2 )
       => ? [E] :
            ( edge(E)
            & ( ( V1 = head_of(E)
                & V2 = tail_of(E) )
            <~> ( V2 = head_of(E)
                & V1 = tail_of(E) ) ) ) ) ) ).

fof(path_defn,axiom,
    ! [V1,V2,P] :
      ( path(V1,V2,P)
     <= ( vertex(V1)
        & vertex(V2)
        & ? [E] :
            ( edge(E)
            & V1 = tail_of(E)
            & ( ( V2 = head_of(E)
                & P = path_cons(E,empty) )
              | ? [TP] :
                  ( path(head_of(E),V2,TP)
                  & P = path_cons(E,TP) ) ) ) ) ) ).

fof(path_properties,axiom,
    ! [V1,V2,P] :
      ( path(V1,V2,P)
     => ( vertex(V1)
        & vertex(V2)
        & ? [E] :
            ( edge(E)
            & V1 = tail_of(E)
            & ( ( V2 = head_of(E)
                & P = path_cons(E,empty) )
            <~> ? [TP] :
                  ( path(head_of(E),V2,TP)
                  & P = path_cons(E,TP) ) ) ) ) ) ).

fof(on_path_properties,axiom,
    ! [V1,V2,P,E] :
      ( ( path(V1,V2,P)
        & on_path(E,P) )
     => ( edge(E)
        & in_path(head_of(E),P)
        & in_path(tail_of(E),P) ) ) ).

fof(in_path_properties,axiom,
    ! [V1,V2,P,V] :
      ( ( path(V1,V2,P)
        & in_path(V,P) )
     => ( vertex(V)
        & ? [E] :
            ( on_path(E,P)
            & ( V = head_of(E)
              | V = tail_of(E) ) ) ) ) ).

fof(sequential_defn,axiom,
    ! [E1,E2] :
      ( sequential(E1,E2)
    <=> ( edge(E1)
        & edge(E2)
        & E1 != E2
        & head_of(E1) = tail_of(E2) ) ) ).

fof(precedes_defn,axiom,
    ! [P,V1,V2] :
      ( path(V1,V2,P)
     => ! [E1,E2] :
          ( precedes(E1,E2,P)
         <= ( on_path(E1,P)
            & on_path(E2,P)
            & ( sequential(E1,E2)
              | ? [E3] :
                  ( sequential(E1,E3)
                  & precedes(E3,E2,P) ) ) ) ) ) ).

fof(precedes_properties,axiom,
    ! [P,V1,V2] :
      ( path(V1,V2,P)
     => ! [E1,E2] :
          ( precedes(E1,E2,P)
         => ( on_path(E1,P)
            & on_path(E2,P)
            & ( sequential(E1,E2)
            <~> ? [E3] :
                  ( sequential(E1,E3)
                  & precedes(E3,E2,P) ) ) ) ) ) ).

fof(shortest_path_defn,axiom,
    ! [V1,V2,SP] :
      ( shortest_path(V1,V2,SP)
    <=> ( path(V1,V2,SP)
        & V1 != V2
        & ! [P] :
            ( path(V1,V2,P)
           => less_or_equal(length_of(SP),length_of(P)) ) ) ) ).

fof(shortest_path_properties,axiom,
    ! [V1,V2,E1,E2,P] :
      ( ( shortest_path(V1,V2,P)
        & precedes(E1,E2,P) )
     => ( ~ ? [E3] :
              ( tail_of(E3) = tail_of(E1)
              & head_of(E3) = head_of(E2) )
        & ~ precedes(E2,E1,P) ) ) ).

%------------------------------------------------------------------------------