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_refl p p' lp : PPO_prog -> wf p' -> FA_path p p' lp -> In p' (activations p) -> lp = [].