Module Import Raw := Raw I X.Module Import II:=MoreInt(I).Ltac omega_max := i2z_refl; lia.Implicit Types m r : t elt.(** * AVL trees *)

(** [avl s] : [s] is a properly balanced AVL tree,
    i.e. for any node the heights of the two children
    differ by at most 2 *)

Inductive avl : t elt -> Prop :=
  | RBLeaf : avl (Leaf _)
  | RBNode : forall x e l r h,
      avl l ->
      avl r ->
      -(2) <= height l - height r <= 2 ->
      h = max (height l) (height r) + 1 ->
      avl (Node l x e r h).(** * AVL trees *)

(** [avl s] : [s] is a properly balanced AVL tree,
    i.e. for any node the heights of the two children
    differ by at most 2 *)

Inductive avl : t elt -> Prop :=
  | RBLeaf : avl (Leaf _)
  | RBNode : forall x e l r h,
      avl l ->
      avl r ->
      -(2) <= height l - height r <= 2 ->
      h = max (height l) (height r) + 1 ->
      avl (Node l x e r h).(** * AVL trees *)

(** [avl s] : [s] is a properly balanced AVL tree,
    i.e. for any node the heights of the two children
    differ by at most 2 *)

Inductive avl : t elt -> Prop :=
  | RBLeaf : avl (Leaf _)
  | RBNode : forall x e l r h,
      avl l ->
      avl r ->
      -(2) <= height l - height r <= 2 ->
      h = max (height l) (height r) + 1 ->
      avl (Node l x e r h).

Lemma height_non_negative : forall (s : t elt), avl s -> height s >= 0.