Corollary activations_inCBV: forall p pi:cbv,
   In p (activations pi) -> InCBV p pi.Definition functions_of_prog : list function :=
  map (fun x => match x with | rule_intro f _ _ => f end) prog.

Lemma activation_is_function : forall proof_tree p, In p (activations proof_tree) -> exists i s p' c1 t c2 v, p = cbv_update i s p' c1 t c2 v.