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

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