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).