(** 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 Neqb 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 MapSingleton_semantics : forall (a:ad) (y:A), eqm (MapGet (MapSingleton a y)) (fun a':ad => if Neqb a a' then Some y else None).