Local Notation "n == m" := (N1.eq n m) (at level 70, no associativity).Definition homomorphism (f : N1.t -> N2.t) : Prop :=
  f N1.zero == N2.zero /\ forall n, f (N1.succ n) == N2.succ (f n).Definition natural_isomorphism : N1.t -> N2.t :=
  N1.recursion N2.zero (fun (n : N1.t) (p : N2.t) => N2.succ p).#[global]
Instance natural_isomorphism_wd : Proper (N1.eq ==> N2.eq) natural_isomorphism.

Theorem natural_isomorphism_succ : forall n : N1.t, natural_isomorphism (N1.succ n) == N2.succ (natural_isomorphism n).