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_uniqueness p i s p0 c1 t c2 v lp1 lp2 : PPO_prog ->
let p' := (cbv_update i s p0 c1 t c2 v) in
wf p' ->
FA_path p p' lp1 -> FA_path p p' lp2 -> lp1 = lp2.