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, 0 insertions, 81 deletions

diff --git a/AvlVerification/Tree.lean b/AvlVerification/Tree.lean
deleted file mode 100644
index 8a043a1..0000000
--- a/AvlVerification/Tree.lean
+++ /dev/null
@@ -1,81 +0,0 @@
-import «AvlVerification»
-
-namespace Tree
-
-variable {T: Type}
-
-open avl_verification
-
--- Otherwise, Lean cannot prove termination by itself.
-@[reducible]
-def AVLTree (U: Type) := Option (AVLNode U)
-def AVLTree.nil: AVLTree T := none
-
-def AVLTree.val (t: AVLTree T): Option T := match t with
-| none => none
-| some (AVLNode.mk x _ _) => some x
-
-def AVLTree.left (t: AVLTree T): AVLTree T := match t with
-| none => none
-| some (AVLNode.mk _ left _) => left
-
-def AVLTree.right (t: AVLTree T): AVLTree T := match t with
-| none => none
-| some (AVLNode.mk _ _ right) => right
-
-def AVLNode.left (t: AVLNode T): AVLTree T := match t with
-| AVLNode.mk _ left _ => left
-
-def AVLNode.right (t: AVLNode T): AVLTree T := match t with
-| AVLNode.mk _ _ right => right
-
-def AVLNode.val (t: AVLNode T): T := match t with
-| AVLNode.mk x _ _ => x
-
-mutual
-def AVLTree.height_node (tree: AVLNode T): Nat := match tree with
-| AVLNode.mk y left right => 1 + AVLTree.height left + AVLTree.height right
-
-def AVLTree.height (tree: AVLTree T): Nat := match tree with 
-| none => 0
-| some n => 1 + AVLTree.height_node n
-end
-
-def AVLTreeSet.nil: AVLTreeSet T := { root := AVLTree.nil }
-
--- Automatic synthesization of this seems possible at the Lean level?
-instance: Inhabited (AVLTree T) where
-  default := AVLTree.nil
-
-instance: Inhabited (AVLTreeSet T) where
-  default := AVLTreeSet.nil
-
-instance: Coe (Option (AVLNode T)) (AVLTree T) where
-  coe x := x
-
--- TODO: ideally, it would be nice if we could generalize
--- this to any `BinaryTree` typeclass.
-
-def AVLTree.mem (tree: AVLTree T) (x: T) :=
-  match tree with
-  | none => False
-  | some (AVLNode.mk y left right) => x = y ∨ AVLTree.mem left x ∨ AVLTree.mem right x
-
-@[simp]
-def AVLTree.mem_none: AVLTree.mem none = ({}: Set T) := rfl
-
-@[simp]
-def AVLTree.mem_some {x: T} {left right: AVLTree T}: AVLTree.mem (some (AVLNode.mk x left right)) = (({x}: Set T) ∪ AVLTree.mem left ∪ AVLTree.mem right) := by
-  ext y
-  rw [AVLTree.mem]
-  simp [Set.insert_union]
-  simp [Set.insert_def, Set.setOf_set, Set.mem_def, Set.union_def]
-
--- TODO(reinforcement): ∪ is actually disjoint if we prove this is a binary search tree.
-
-def AVLTree.set (t: AVLTree T): Set T := _root_.setOf (AVLTree.mem t)
-
-@[simp]
-def AVLTree.set_some {x: T} {left right: AVLTree T}: AVLTree.set (some (AVLNode.mk x left right)) = {x} ∪ AVLTree.set left ∪ AVLTree.set right := sorry
-
-end Tree