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+import Verification.Tree
+import Verification.BinarySearchTree
+import Verification.Specifications
+
+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) [LinearOrder T] (Ospec: OrdSpecLinearOrderEq H)
+
+@[pspec]
+theorem AVLTreeSet.insert_loop_spec_local (p: T -> Prop)
+  (Hcmp_spec: ∀ a b, ∃ o, H.cmp a b = Result.ok o)
+  (a: T) (t: Option (AVLNode T)):
+  ∃ added t_new, AVLTreeSet.insert_loop T H a t = Result.ok ⟨ added, t_new ⟩
+--  ∧ AVLTree.set t'.2 = insert a (AVLTree.set t)
+  ∧ (BST.ForallNode p t -> p a -> BST.ForallNode p t_new)
+  := by match t with
+  | none =>
+    simp [AVLTreeSet.insert_loop, AVLTree.set, setOf]
+    intros _ Hpa
+    constructor; all_goals try simp [BST.ForallNode.none]
+    exact Hpa
+  | some (AVLNode.mk b left right) =>
+    rw [AVLTreeSet.insert_loop]
+    simp only []
+    progress keep Hordering as ⟨ ordering ⟩
+    cases ordering
+    all_goals simp only []
+    {
+      progress keep Htree as ⟨ added₁, right₁, Hnode ⟩
+      refine' ⟨ added₁, ⟨ some (AVLNode.mk b left right₁), _ ⟩ ⟩
+      simp only [true_and]
+      intros Hptree Hpa
+      constructor
+      exact Hptree.left
+      exact Hptree.label
+      exact Hnode Hptree.right Hpa
+    }
+    {
+      simp; tauto
+    }
+    {
+      -- TODO: investigate wlog.
+      -- Symmetric case of left.
+      progress keep Htree as ⟨ added₁, left₁, Hnode ⟩
+      refine' ⟨ added₁, ⟨ some (AVLNode.mk b left₁ right), _ ⟩ ⟩
+      simp only [true_and]
+      intros Hptree Hpa
+      constructor
+      exact Hnode Hptree.left Hpa
+      exact Hptree.label
+      exact Hptree.right
+    }
+
+@[pspec]
+lemma AVLTreeSet.insert_loop_spec_global
+  (a: T) (t: Option (AVLNode T))
+  :
+  BST.Invariant t -> ∃ added t_new, AVLTreeSet.insert_loop T H a t = Result.ok ⟨ added, t_new ⟩
+  ∧ BST.Invariant t_new ∧ AVLTree.set t_new = {a} ∪ AVLTree.set t := by 
+  intro Hbst
+  match t with
+  | none => simp [AVLTreeSet.insert_loop, AVLTree.set, setOf]
+  | some (AVLNode.mk b left right) =>
+    rw [AVLTreeSet.insert_loop]
+    simp only []
+    have : ∀ a b, ∃ o, H.cmp a b = .ok o := infallible H
+    progress keep Hordering as ⟨ ordering ⟩
+    cases ordering
+    all_goals simp only []
+    {
+      have ⟨ added₂, right₂, ⟨ H_result, ⟨ H_bst, H_set ⟩ ⟩ ⟩ := AVLTreeSet.insert_loop_spec_global a right (BST.right Hbst)
+      progress keep Htree with AVLTreeSet.insert_loop_spec_local as ⟨ added₁, right₁, Hnode ⟩
+      exact (fun x => b < x)
+      rewrite [Htree] at H_result; simp at H_result
+      refine' ⟨ added₁, ⟨ some (AVLNode.mk b left right₁), _ ⟩ ⟩
+      simp only [true_and]
+      split_conjs
+      cases' Hbst with _ _ _ H_forall_left H_forall_right H_bst_left H_bst_right
+      constructor
+      exact H_forall_left
+      apply Hnode; exact H_forall_right
+      exact (ltOfRustOrder H b a Hordering)
+      exact H_bst_left
+      convert H_bst
+      exact H_result.2
+      simp [AVLTree.set_some]
+      rewrite [H_result.2, H_set]
+      simp
+    }
+    {
+      simp; split_conjs
+      . tauto
+      . simp [Ospec.equivalence _ _ Hordering]
+    }
+    {
+      have ⟨ added₂, left₂, ⟨ H_result, ⟨ H_bst, H_set ⟩ ⟩ ⟩ := AVLTreeSet.insert_loop_spec_global a left (BST.left Hbst)
+      progress keep Htree with AVLTreeSet.insert_loop_spec_local as ⟨ added₁, left₁, Hnode ⟩
+      exact (fun x => x < b)
+      rewrite [Htree] at H_result; simp at H_result
+      refine' ⟨ added₁, ⟨ some (AVLNode.mk b left₁ right), _ ⟩ ⟩
+      simp only [true_and]
+      split_conjs
+      cases' Hbst with _ _ _ H_forall_left H_forall_right H_bst_left H_bst_right
+      constructor
+      apply Hnode; exact H_forall_left
+      exact (gtOfRustOrder H b a Hordering)
+      exact H_forall_right
+      convert H_bst
+      exact H_result.2
+      exact H_bst_right
+      simp [AVLTree.set_some]
+      rewrite [H_result.2, H_set]
+      simp [Set.singleton_union, Set.insert_comm, Set.insert_union]
+    }
+
+@[pspec]
+def AVLTreeSet.insert_spec 
+  (a: T) (t: AVLTreeSet T):
+  BST.Invariant t.root -> (∃ t' added,t.insert _ H a = Result.ok (added, t')
+  -- it's still a binary search tree.
+  ∧ BST.Invariant t'.root
+  ∧ AVLTree.set t'.root = {a} ∪ AVLTree.set t.root)
+  := by
+  rw [AVLTreeSet.insert]; intro Hbst
+  progress keep h as ⟨ t', Hset ⟩; 
+  simp; assumption
+
+end Implementation
+