Definition recursion {A} : A -> (N -> A -> A) -> N -> A :=
peano_rect (fun _ => A).
#[global]
Instance recursion_wd {A} (Aeq : relation A) :
Proper (Aeq==>(Logic.eq==>Aeq==>Aeq)==>Logic.eq==>Aeq) recursion.

(** Basic properties of comparison *)
Theorem compare_eq_iff n m : (n ?= m) = Eq <-> n = m.