(** 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.

(* Specifications: *)
Lemma newMap_semantics : eqm (MapGet newMap) (fun a:ad => None).
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).
Lemma M1_semantics_1 : forall (a:ad) (y:A), MapGet (M1 a y) a = Some y.
Lemma M1_semantics_2 : forall (a a':ad) (y:A), Neqb a a' = false -> MapGet (M1 a y) a' = None.
Lemma Map2_semantics_1 : forall m m':Map, eqm (MapGet m) (fun a:ad => MapGet (M2 m m') (Ndouble 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 (Ndouble a)).
Lemma Map2_semantics_2 : forall m m':Map, eqm (MapGet m') (fun a:ad => MapGet (M2 m m') (Ndouble_plus_one a)).
Lemma Map2_semantics_2_eq : forall (m m':Map) (f:ad -> option A), eqm (MapGet (M2 m m')) f -> eqm (MapGet m') (fun a:ad => f (Ndouble_plus_one a)).
Lemma MapGet_M2_bit_0_0 : forall a:ad, Nbit0 a = false -> forall m m':Map, MapGet (M2 m m') a = MapGet m (Ndiv2 a).
Lemma MapGet_M2_bit_0_1 : forall a:ad, Nbit0 a = true -> forall m m':Map, MapGet (M2 m m') a = MapGet m' (Ndiv2 a).
Lemma MapGet_M2_bit_0_if : forall (m m':Map) (a:ad), MapGet (M2 m m') a = (if Nbit0 a then MapGet m' (Ndiv2 a) else MapGet m (Ndiv2 a)).
Lemma MapGet_M2_bit_0 : forall (m m' m'':Map) (a:ad), (if Nbit0 a then MapGet (M2 m' m) a else MapGet (M2 m m'') a) = MapGet m (Ndiv2 a).
Lemma Map2_semantics_3 : forall m m':Map, eqm (MapGet (M2 m m')) (fun a:ad => match Nbit0 a with | false => MapGet m (Ndiv2 a) | true => MapGet m' (Ndiv2 a) end).
Lemma Map2_semantics_3_eq : forall (m m':Map) (f f':ad -> option A), eqm (MapGet m) f -> eqm (MapGet m') f' -> eqm (MapGet (M2 m m')) (fun a:ad => match Nbit0 a with | false => f (Ndiv2 a) | true => f' (Ndiv2 a) end).