End PTree.
(** * An implementation of maps over type [positive] *)
Module PMap <: MAP.
Definition elt := positive.
Definition elt_eq := peq.
Definition t (A : Type) : Type := (A * PTree.t A)%type.
Definition init (A : Type) (x : A) :=
(x, PTree.empty A).
Definition get (A : Type) (i : positive) (m : t A) :=
match PTree.get i (snd m) with
| Some x => x
| None => fst m
end.
Definition set (A : Type) (i : positive) (x : A) (m : t A) :=
(fst m, PTree.set i x (snd m)).

(* Specifications: *)
Theorem gi:
forall (A: Type) (i: positive) (x: A), get i (init x) = x.
Theorem gss:
forall (A: Type) (i: positive) (x: A) (m: t A), get i (set i x m) = x.
Theorem gso:
forall (A: Type) (i j: positive) (x: A) (m: t A),
i <> j -> get i (set j x m) = get i m.
Theorem gsspec:
forall (A: Type) (i j: positive) (x: A) (m: t A),
get i (set j x m) = if peq i j then x else get i m.
Theorem gsident:
forall (A: Type) (i j: positive) (m: t A),
get j (set i (get i m) m) = get j m.