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(** * Delocations  *)
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(** Let (X, A) and (Y, B) be two graphs.  A delocation from (X, A) to (Y, B) is a function f : X->Y such that:  1. (x,x':X)    (A x' x) -> (B (f x') (f x))  2. (x:X; y':Y) (B y' (f x)) -> (Ex x':X | (A x' x) /\ y'=(f x'))  Intuitively, a delocation is a morphism of graphs in which any edge of the target graph (Y, B) which points to an element of the image of f can be tracked back as an edge in the source graph via f. This notion is formalized as follows: *)
Definition deloc (X : Typ1) (A : Rel X) (Y : Typ1)
(B : Rel Y) (f : X -> Y) : Prop :=
and (forall x x' : X, A x' x -> B (f x') (f x))
(forall (x : X) (y' : Y),
B y' (f x) -> ex2 X (fun x' => A x' x) (fun x' => eq Y y' (f x'))).
(** The notion of delocation f:(X,A)->(Y,B) is interesting since it automatically induces a bisimulation from the pointed graph (X, A, x) to the pointed graph (Y, B, (f x)) for any vertex x:X in the source graph (X,A).  This property is stated by the following lemma: *)

Lemma EQV_deloc :
forall (X : Typ1) (A : Rel X) (Y : Typ1) (B : Rel Y) (f : X -> Y),
deloc X A Y B f -> forall x : X, EQV X A x Y B (f x).