Section Iteration.
Variable T : Type.
Implicit Types m n : nat.
Implicit Types x y : T.
Implicit Types S : {pred 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 leq_exp2r m n e : e > 0 -> (m ^ e <= n ^ e) = (m <= n).