(** Here, we can see that Coq produced an [IH] that corresponds to [E'], the single recursive occurrence of [ev] in its own definition.  Since [E'] mentions [n'], the induction hypothesis talks about [n'], as opposed to [n] or some other number. *)
(** The equivalence between the second and third definitions of evenness now follows. *)

Theorem ev_even_iff : forall n,
ev n <-> exists k, n = double k.