Local Hint Immediate In_1 : core.
#[global]
Instance In_compat : Proper (X.eq==>eq==>iff) InT.

(* Specifications: *)
Lemma In_node_iff :
forall c l x r y,
InT y (Node c l x r) <-> InT y l \/ X.eq y x \/ InT y r.
Lemma In_leaf_iff : forall x, InT x Leaf <-> False.
(** Results about [lt_tree] and [gt_tree] *)
Lemma lt_leaf : forall x : elt, lt_tree x Leaf.
Lemma gt_leaf : forall x : elt, gt_tree x Leaf.
Lemma lt_tree_node :
forall (x y : elt) (l r : tree) (i : Info.t),
lt_tree x l -> lt_tree x r -> X.lt y x -> lt_tree x (Node i l y r).
Lemma gt_tree_node :
forall (x y : elt) (l r : tree) (i : Info.t),
gt_tree x l -> gt_tree x r -> X.lt x y -> gt_tree x (Node i l y r).