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.