feat: close `find` / `insert` proofs

After a complete 180 with the Order theory, we close the goals of find
and insert and we give an example of U32 order that we will upstream to
Aeneas directly.

Signed-off-by: Raito Bezarius <masterancpp@gmail.com>

1 files changed, 40 insertions, 0 deletions

diff --git a/Verification/Find.lean b/Verification/Find.lean
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index 0000000..764a685
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+++ b/Verification/Find.lean
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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
+