import «AvlVerification»

namespace Tree

variable {T: Type}

open avl_verification

-- Otherwise, Lean cannot prove termination by itself.
@[reducible]
def AVLTree (U: Type) := Option (AVLNode U)
def AVLTree.nil: AVLTree T := none

def AVLTree.val (t: AVLTree T): Option T := match t with
| none => none
| some (AVLNode.mk x _ _) => some x

def AVLTree.left (t: AVLTree T): AVLTree T := match t with
| none => none
| some (AVLNode.mk _ left _) => left

def AVLTree.right (t: AVLTree T): AVLTree T := match t with
| none => none
| some (AVLNode.mk _ _ right) => right

def AVLNode.left (t: AVLNode T): AVLTree T := match t with
| AVLNode.mk _ left _ => left

def AVLNode.right (t: AVLNode T): AVLTree T := match t with
| AVLNode.mk _ _ right => right

def AVLNode.val (t: AVLNode T): T := match t with
| AVLNode.mk x _ _ => x

mutual
def AVLTree.height_node (tree: AVLNode T): Nat := match tree with
| AVLNode.mk y left right => 1 + AVLTree.height left + AVLTree.height right

def AVLTree.height (tree: AVLTree T): Nat := match tree with 
| none => 0
| some n => 1 + AVLTree.height_node n
end

def AVLTreeSet.nil: AVLTreeSet T := { root := AVLTree.nil }

-- Automatic synthesization of this seems possible at the Lean level?
instance: Inhabited (AVLTree T) where
  default := AVLTree.nil

instance: Inhabited (AVLTreeSet T) where
  default := AVLTreeSet.nil

instance: Coe (Option (AVLNode T)) (AVLTree T) where
  coe x := x

-- TODO: ideally, it would be nice if we could generalize
-- this to any `BinaryTree` typeclass.

def AVLTree.mem (tree: AVLTree T) (x: T) :=
  match tree with
  | none => False
  | some (AVLNode.mk y left right) => x = y ∨ AVLTree.mem left x ∨ AVLTree.mem right x

@[simp]
def AVLTree.mem_none: AVLTree.mem none = ({}: Set T) := rfl

@[simp]
def AVLTree.mem_some {x: T} {left right: AVLTree T}: AVLTree.mem (some (AVLNode.mk x left right)) = (({x}: Set T) ∪ AVLTree.mem left ∪ AVLTree.mem right) := by
  ext y
  rw [AVLTree.mem]
  simp [Set.insert_union]
  simp [Set.insert_def, Set.setOf_set, Set.mem_def, Set.union_def]

-- TODO(reinforcement): ∪ is actually disjoint if we prove this is a binary search tree.

def AVLTree.set (t: AVLTree T): Set T := _root_.setOf (AVLTree.mem t)

def AVLTree.setOf_left_incl (t: AVLTree T): t.left.set ⊂ t.set := sorry
def AVLTree.setOf_right_incl (t: AVLTree T): t.right.set ⊂ t.set := sorry

end Tree