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import Verification.Tree
import Verification.BinarySearchTree
import Verification.Specifications
import AvlVerification
namespace Implementation
open Primitives
open avl_verification
open Tree (AVLTree AVLTree.set)
open Specifications (OrdSpecLinearOrderEq infallible ltOfRustOrder gtOfRustOrder)
variable (T: Type) (H: avl_verification.Ord T) [DecidableEq T] [LinearOrder T] (Ospec: OrdSpecLinearOrderEq H)
@[pspec]
def AVLTreeSet.find_loop_spec
(a: T) (t: Option (AVLNode T)):
BST.Invariant t -> a ∈ AVLTree.set t -> AVLTreeSet.find_loop _ H a t = Result.ok true := fun Hbst Hmem => by
match t with
| none => trivial
| some (AVLNode.mk b left right) =>
rw [AVLTreeSet.find_loop]
dsimp only
have : ∀ a b, ∃ o, H.cmp a b = .ok o := infallible H
progress keep Hordering as ⟨ ordering ⟩
cases ordering
all_goals dsimp only
. refine' AVLTreeSet.find_loop_spec a right (BST.right Hbst) (BST.right_pos Hbst Hmem _)
exact ltOfRustOrder _ _ _ Hordering
. refine' AVLTreeSet.find_loop_spec a left (BST.left Hbst) (BST.left_pos Hbst Hmem _)
exact gtOfRustOrder _ _ _ Hordering
def AVLTreeSet.find_spec
(a: T) (t: AVLTreeSet T):
BST.Invariant t.root -> a ∈ AVLTree.set t.root ->
t.find _ H a = Result.ok true := fun Hbst Hmem => by
rw [AVLTreeSet.find]; progress
end Implementation
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