Section Iteration.
Variable T : Type.
Implicit Types m n : nat.
Implicit Types x y : T.
Definition iter n f x :=
let fix loop m := if m is i.+1 then f (loop i) else x in loop n.
Definition iteri n f x :=
let fix loop m := if m is i.+1 then f i (loop i) else x in loop n.
Definition iterop n op x :=
let f i y := if i is 0 then x else op x y in iteri n f.

Lemma mono_leqif f : {mono f : m n / m <= n} ->
forall m n C, (f m <= f n ?= iff C) = (m <= n ?= iff C).