Definition elt := X.t.Definition elt_eq := X.eq.Definition t : Type -> Type := PMap.t.Definition init (A: Type) (x: A) := PMap.init x.Definition get (A: Type) (i: X.t) (m: t A) := PMap.get (X.index i) m.Definition set (A: Type) (i: X.t) (v: A) (m: t A) := PMap.set (X.index i) v m.Definition map (A B: Type) (f: A -> B) (m: t A) : t B := PMap.map f m.

(* Specifications: *)
Lemma gi: forall (A: Type) (x: A) (i: X.t), get i (init x) = x.
Lemma gss: forall (A: Type) (i: X.t) (x: A) (m: t A), get i (set i x m) = x.
Lemma gso: forall (A: Type) (i j: X.t) (x: A) (m: t A), i <> j -> get i (set j x m) = get i m.
Lemma gsspec: forall (A: Type) (i j: X.t) (x: A) (m: t A), get i (set j x m) = if X.eq i j then x else get i m.
Lemma gmap: forall (A B: Type) (f: A -> B) (i: X.t) (m: t A), get i (map f m) = f(get i m).
Lemma set2: forall (A: Type) (i: elt) (x y: A) (m: t A), set i y (set i x m) = set i y m.
Lemma gi: forall (A: Type) (x: A) (i: X.t), get i (init x) = x.
Lemma gss: forall (A: Type) (i: X.t) (x: A) (m: t A), get i (set i x m) = x.
Lemma gso: forall (A: Type) (i j: X.t) (x: A) (m: t A), i <> j -> get i (set j x m) = get i m.
Lemma gsspec: forall (A: Type) (i j: X.t) (x: A) (m: t A), get i (set j x m) = if X.eq i j then x else get i m.
Lemma gmap: forall (A B: Type) (f: A -> B) (i: X.t) (m: t A), get i (map f m) = f(get i m).
Lemma set2: forall (A: Type) (i: elt) (x y: A) (m: t A), set i y (set i x m) = set i y m.