(** [MapInter], [MapRngRestrTo], [MapRngRestrBy], [MapInverse] not implemented: need a decidable equality on [A]. *)
Fixpoint MapDelta (m:Map) : Map -> Map :=
match m with
| M0 => fun m':Map => m'
| M1 a y =>
fun m':Map =>
match MapGet m' a with
| None => MapPut m' a y
| _ => MapRemove m' a
end
| M2 m1 m2 =>
fun m':Map =>
match m' with
| M0 => m
| M1 a' y' =>
match MapGet m a' with
| None => MapPut m a' y'
| _ => MapRemove m a'
end
| M2 m'1 m'2 => makeM2 (MapDelta m1 m'1) (MapDelta m2 m'2)
end
end.

(* Specifications: *)
Lemma MapDelta_semantics_comm :
forall m m':Map, eqm (MapGet (MapDelta m m')) (MapGet (MapDelta m' m)).
Lemma MapDelta_semantics_1_1 :
forall (a:ad) (y:A) (m':Map) (a0:ad),
MapGet (M1 a y) a0 = None ->
MapGet m' a0 = None -> MapGet (MapDelta (M1 a y) m') a0 = None.
Lemma MapDelta_semantics_1 :
forall (m m':Map) (a:ad),
MapGet m a = None ->
MapGet m' a = None -> MapGet (MapDelta m m') a = None.
Lemma MapDelta_semantics_2_1 :
forall (a:ad) (y:A) (m':Map) (a0:ad) (y0:A),
MapGet (M1 a y) a0 = None ->
MapGet m' a0 = Some y0 -> MapGet (MapDelta (M1 a y) m') a0 = Some y0.
Lemma MapDelta_semantics_2_2 :
forall (a:ad) (y:A) (m':Map) (a0:ad) (y0:A),
MapGet (M1 a y) a0 = Some y0 ->
MapGet m' a0 = None -> MapGet (MapDelta (M1 a y) m') a0 = Some y0.
Lemma MapDelta_semantics_2 :
forall (m m':Map) (a:ad) (y:A),
MapGet m a = None ->
MapGet m' a = Some y -> MapGet (MapDelta m m') a = Some y.
Lemma MapDelta_semantics_3_1 :
forall (a0:ad) (y0:A) (m':Map) (a:ad) (y y':A),
MapGet (M1 a0 y0) a = Some y ->
MapGet m' a = Some y' -> MapGet (MapDelta (M1 a0 y0) m') a = None.
Lemma MapDelta_semantics_3 :
forall (m m':Map) (a:ad) (y y':A),
MapGet m a = Some y ->
MapGet m' a = Some y' -> MapGet (MapDelta m m') a = None.
Lemma MapDelta_semantics :
forall m m':Map,
eqm (MapGet (MapDelta m m'))
(fun a0:ad =>
match MapGet m a0, MapGet m' a0 with
| None, Some y' => Some y'
| Some y, None => Some y
| _, _ => None
end).