Definition activation_function (function_default : function) (p: cbv) :=
  match p with
    | cbv_update _ _ _ _ (fapply f _) _ _ => f
    | _                                   => function_default
  end.

Lemma FA_path_activation p p' i s p0 c1 t c2 v lp : let proof_tree := cbv_update i s p0 c1 t c2 v in wf proof_tree -> In p' (activations proof_tree) -> FA_path p p' lp -> In p (activations p').