Fixpoint FA_path p p' (lp : list cbv) := match lp with
| [] => p = p'
| h :: t => In p (first_activations h) /\ FA_path h p' t
end.

Lemma FA_path_right p p' p'' lp :
FA_path p p' lp ->
(In p' (first_activations p'') <->
FA_path p p'' (lp ++ [p''])).