(** At any given moment, Coq has constructed a term with a
    "hole" (indicated by [?Goal] here, and so on), and it knows what
    type of evidence is needed to fill this hole.  

    Each hole corresponds to a subgoal, and the proof is
    finished when there are no more subgoals.  At this point, the
    evidence we've built stored in the global context under the name
    given in the [Theorem] command. *)

(** Tactic proofs are useful and convenient, but they are not
    essential: in principle, we can always construct the required
    evidence by hand, as shown above. Then we can use [Definition]
    (rather than [Theorem]) to give a global name directly to a
    piece of evidence. *)

Definition ev_4''' : ev 4 :=
  ev_SS 2 (ev_SS 0 ev_0).

Theorem ev_8 : ev 8.