Inductive pmap : Type :=
  | Empty
  | Branch : option T -> pmap -> pmap -> pmap.Inductive pmap : Type :=
  | Empty
  | Branch : option T -> pmap -> pmap -> pmap.Inductive pmap : Type :=
  | Empty
  | Branch : option T -> pmap -> pmap -> pmap.Definition pmap_here (m : pmap) : option T :=
    match m with
      | Empty => None
      | Branch d _ _ => d
    end.Definition pmap_left (m : pmap) : pmap :=
    match m with
      | Empty => Empty
      | Branch _ l _ => l
    end.Definition pmap_right (m : pmap) : pmap :=
    match m with
      | Empty => Empty
      | Branch _ _ r => r
    end.Fixpoint pmap_lookup (p : positive) (m : pmap) : option T :=
    match m with
      | Empty => None
      | Branch d l r =>
        match p with
          | xH => d
          | xO p => pmap_lookup p l
          | xI p => pmap_lookup p r
        end
    end.Fixpoint pmap_insert (p : positive) (v : T) (m : pmap) : pmap :=
    match p with
      | xH => Branch (Some v) (pmap_left m) (pmap_right m)
      | xO p =>
        Branch (pmap_here m) (pmap_insert p v (pmap_left m)) (pmap_right m)
      | xI p =>
        Branch (pmap_here m) (pmap_left m) (pmap_insert p v (pmap_right m))
    end.Definition branch (o : option T) (l r : pmap) : pmap :=
    match o , l , r with
      | None , Empty , Empty => Empty
      | _ , _ , _ => Branch o l r
    end.Fixpoint pmap_remove (p : positive) (m : pmap) : pmap :=
    match m with
      | Empty => Empty
      | Branch d l r =>
        match p with
          | xH => branch None l r
          | xO p => branch d (pmap_remove p l) r
          | xI p => branch d l (pmap_remove p r)
        end
    end.Definition pmap_empty : pmap := Empty.Fixpoint pmap_union (f m : pmap) : pmap :=
    match f with
      | Empty => m
      | Branch d l r =>
        Branch (match d with
                  | Some x => Some x
                  | None => pmap_here m
                end) (pmap_union l (pmap_left m)) (pmap_union r (pmap_right m))
    end.Global Instance Map_pmap : Map positive T pmap :=
  { empty := pmap_empty
  ; add := pmap_insert
  ; remove := pmap_remove
  ; lookup := pmap_lookup
  ; union := pmap_union
  }.

Lemma lookup_empty : forall k, pmap_lookup k Empty = None.