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.

Lemma strong_induction_le (A : N -> Prop) : A 0 -> (forall n, (forall m, m <= n -> A m) -> A (succ n)) -> forall n, A n.