Section MapDefs.
(** We now define maps from ad to A. *)
Variable A : Type.
Inductive Map :=
| M0 : Map
| M1 : ad -> A -> Map
| M2 : Map -> Map -> Map.
(** The semantics of maps is given by the function [MapGet]. The semantics of a map [m] is a partial, finite function from [ad] to [A]: *)
Fixpoint MapGet (m:Map) : ad -> option A :=
match m with
| M0 => fun a:ad => None
| M1 x y => fun a:ad => if N.eqb x a then Some y else None
| M2 m1 m2 =>
fun a:ad =>
match a with
| N0 => MapGet m1 N0
| Npos xH => MapGet m2 N0
| Npos (xO p) => MapGet m1 (Npos p)
| Npos (xI p) => MapGet m2 (Npos p)
end
end.
Definition newMap := M0.
Definition MapSingleton := M1.
Definition eqm (g g':ad -> option A) := forall a:ad, g a = g' a.

Lemma Map2_semantics_1_eq :
forall (m m':Map) (f:ad -> option A),
eqm (MapGet (M2 m m')) f -> eqm (MapGet m) (fun a:ad => f (N.double a)).