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 le_max_active_size proof_tree p : In p (activations proof_tree) -> term_size (proj_left p) + value_size (proj_right p) <= max_active_size proof_tree.