From 2ff68510aabc63e250f98264e0642557015de4e2 Mon Sep 17 00:00:00 2001
From: Raito Bezarius
Date: Thu, 18 Apr 2024 15:38:36 +0200
Subject: feat: outline of `find` proof

Signed-off-by: Raito Bezarius <masterancpp@gmail.com>
---
 AvlVerification/Find.lean | 42 ++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 42 insertions(+)
 create mode 100644 AvlVerification/Find.lean

(limited to 'AvlVerification/Find.lean')

diff --git a/AvlVerification/Find.lean b/AvlVerification/Find.lean
new file mode 100644
index 0000000..b729dab
--- /dev/null
+++ b/AvlVerification/Find.lean
@@ -0,0 +1,42 @@
+import AvlVerification.Tree
+import AvlVerification.BinarySearchTree
+import AvlVerification.Specifications
+
+namespace Implementation
+
+open Primitives
+open avl_verification
+open Tree (AVLTree AVLTree.set)
+open Specifications (OrdSpecDualityEq ordOfOrdSpec ltOfRustOrder gtOfRustOrder)
+
+variable (T: Type) (H: avl_verification.Ord T) (Ospec: @OrdSpecDualityEq T 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 := Ospec.infallible
+    progress keep Hordering as ⟨ ordering ⟩
+    cases ordering
+    all_goals dsimp only
+    . refine' AVLTreeSet.find_loop_spec a right (BST.right Hbst) _
+      -- b < a
+      -- Hbst fournit que a ∈ right
+      sorry
+    . refine' AVLTreeSet.find_loop_spec a left (BST.left Hbst) _
+      -- symmétrie du précédent.
+      sorry
+
+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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