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)).

Theorem gi:
forall (A: Type) (i: positive) (x: A), get i (init x) = x.