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 split_activations_by_first_activations_at_rank rk p :
wf p ->
activations_at_rank rk p = flat_map (activations_at_rank rk) (first_activations_at_given_rank rk p).