#[global]
Instance : Proper (eq==>eq==>iff) le.#[global]
Instance max_compat : Proper (eq==>eq==>eq) max.

(* Specifications: *)
Lemma max_unicity n m p : ((n < m /\ p == m) \/ (m <= n /\ p == n)) -> p == max n m.
Lemma max_unicity_ext f : (forall n m, (n < m /\ f n m == m) \/ (m <= n /\ f n m == n)) -> (forall n m, f n m == max n m).
Lemma max_mono f : (Proper (eq ==> eq) f) -> (Proper (le ==> le) f) -> forall x y, max (f x) (f y) == f (max x y).
Lemma max_id n : max n n == n.