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 cbv_update_in_activations_InCBV proof_tree sub_proof_tree i s p c1 t c2 v: sub_proof_tree = cbv_update i s p c1 t c2 v -> InCBV sub_proof_tree proof_tree -> In sub_proof_tree (activations proof_tree).