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 activations_at_rank_activation_rank proof_tree p:
In p (activations proof_tree) ->
In p (activations_at_rank (activation_rank rank p) proof_tree).