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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 -> ∃ b, AVLTreeSet.find_loop _ H a t = Result.ok b ∧ (b ↔ a ∈ AVLTree.set t) := fun Hbst => by
rewrite [AVLTreeSet.find_loop]
match t with
| none => use false; simp [AVLTree.set]; tauto
| some (AVLNode.mk b left right) =>
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
. convert (AVLTreeSet.find_loop_spec a right (BST.right Hbst)) using 4
apply Iff.intro
-- We apply a localization theorem here.
. intro Hmem; exact (BST.right_pos Hbst Hmem (ltOfRustOrder _ _ _ Hordering))
. intro Hmem; simp [AVLTree.set_some]; right; assumption
. simp [Ospec.equivalence _ _ Hordering]
. convert (AVLTreeSet.find_loop_spec a left (BST.left Hbst)) using 4
apply Iff.intro
-- We apply a localization theorem here.
. intro Hmem; exact (BST.left_pos Hbst Hmem (gtOfRustOrder _ _ _ Hordering))
. intro Hmem; simp [AVLTree.set_some]; left; right; assumption
def AVLTreeSet.find_spec
(a: T) (t: AVLTreeSet T):
BST.Invariant t.root -> ∃ b, t.find _ H a = Result.ok b ∧ (b ↔ a ∈ AVLTree.set t.root) := fun Hbst => by
rw [AVLTreeSet.find]
progress; simp only [Result.ok.injEq, exists_eq_left']; assumption
end Implementation
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