Notation Poption := (Poption A).Inductive Poption {A} : Type:=
  PSome : A -> Poption
| PNone : Poption.Inductive Poption {A} : Type:=
  PSome : A -> Poption
| PNone : Poption.Notation Tree := (Tree A).Inductive Tree {A} : Type :=
   Tempty : Tree
 | Branch0 : Tree -> Tree -> Tree
 | Branch1 : A -> Tree -> Tree -> Tree.Inductive Tree {A} : Type :=
   Tempty : Tree
 | Branch0 : Tree -> Tree -> Tree
 | Branch1 : A -> Tree -> Tree -> Tree.Inductive Tree {A} : Type :=
   Tempty : Tree
 | Branch0 : Tree -> Tree -> Tree
 | Branch1 : A -> Tree -> Tree -> Tree.Fixpoint Tget (p:positive) (T:Tree) {struct p} : Poption :=
  match T with
    Tempty => PNone
  | Branch0 T1 T2 =>
    match p with
      xI pp => Tget pp T2
    | xO pp => Tget pp T1
    | xH => PNone
    end
  | Branch1 a T1 T2 =>
    match p with
      xI pp => Tget pp T2
    | xO pp => Tget pp T1
    | xH => PSome a
    end
end.Fixpoint Tadd (p:positive) (a:A) (T:Tree) {struct p}: Tree :=
 match T with
   | Tempty =>
       match p with
       | xI pp => Branch0 Tempty (Tadd pp a Tempty)
       | xO pp => Branch0 (Tadd pp a Tempty) Tempty
       | xH => Branch1 a Tempty Tempty
       end
   | Branch0 T1 T2 =>
       match p with
       | xI pp => Branch0 T1 (Tadd pp a T2)
       | xO pp => Branch0 (Tadd pp a T1) T2
       | xH => Branch1 a T1 T2
       end
   | Branch1 b T1 T2 =>
       match p with
       | xI pp => Branch1 b T1 (Tadd pp a T2)
       | xO pp => Branch1 b (Tadd pp a T1) T2
       | xH => Branch1 a T1 T2
       end
   end.Definition mkBranch0 (T1 T2:Tree) :=
  match T1,T2 with
    Tempty ,Tempty => Tempty
  | _,_ => Branch0 T1 T2
  end.Fixpoint Tremove (p:positive) (T:Tree) {struct p}: Tree :=
   match T with
      | Tempty => Tempty
      | Branch0 T1 T2 =>
        match p with
        | xI pp => mkBranch0 T1 (Tremove pp T2)
        | xO pp => mkBranch0 (Tremove pp T1) T2
        | xH => T
        end
      | Branch1 b T1 T2 =>
        match p with
        | xI pp => Branch1 b T1 (Tremove pp T2)
        | xO pp => Branch1 b (Tremove pp T1) T2
        | xH => mkBranch0 T1 T2
        end
      end.

Theorem Tget_Tadd: forall i j a T, Tget i (Tadd j a T) = match (i ?= j) with Eq => PSome a | Lt => Tget i T | Gt => Tget i T end.