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 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.