Section Elt.
Variable elt:Type.
Implicit Types m r : t elt.
(** * Basic results about [MapsTo], [In], [lt_tree], [gt_tree], [height] *)
(** Facts about [MapsTo] and [In]. *)
Hint Resolve MapsTo_In : core.
Hint Immediate MapsTo_1 : core.
(** Results about [lt_tree] and [gt_tree] *)
Hint Resolve lt_leaf gt_leaf lt_tree_node gt_tree_node : core.
Hint Resolve lt_left lt_right gt_left gt_right : core.
Hint Resolve lt_tree_not_in lt_tree_trans gt_tree_not_in gt_tree_trans : core.
(** * Empty map *)
Definition Empty m := forall (a:key)(e:elt) , ~ MapsTo a e m.
(** * Emptyness test *)
(** * Membership *)
(** * Helper functions *)
Hint Resolve create_bst : core.
Hint Resolve bal_bst : core.
(** * Insertion *)
Hint Resolve add_bst : core.
(** * Extraction of minimum binding *)
Hint Resolve remove_min_bst : core.
Hint Resolve remove_min_gt_tree : core.

Lemma remove_min_find : forall l x e r h y,
bst (Node l x e r h) ->
find y (Node l x e r h) =
match X.compare y (remove_min l x e r)#2#1 with
| LT _ => None
| EQ _ => Some (remove_min l x e r)#2#2
| GT _ => find y (remove_min l x e r)#1
end.