(** A boolean version of [le] over [nat]. See now [Nat.leb] and its properties. In scope [nat_scope], the notation for [Nat.leb] is "<=?" *)
Notation leb_iff := Nat.leb_le (only parsing).

(* Specifications: *)
Lemma leb_iff_conv m n : (n <=? m) = false <-> m < n.
Lemma leb_correct m n : m <= n -> (m <=? n) = true.
Lemma leb_complete m n : (m <=? n) = true -> m <= n.
Lemma leb_correct_conv m n : m < n -> (n <=? m) = false.
Lemma leb_complete_conv m n : (n <=? m) = false -> m < n.
Lemma leb_compare n m : (n <=? m) = true <-> (n ?= m) <> Gt.