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_irrefl i s p0 c1 t c2 v p:
PPO_prog ->
wf (cbv_update i s p0 c1 t c2 v) ->
(cbv_update i s p0 c1 t c2 v) ∈ activations p ->
~ In p (first_activations  (cbv_update i s p0 c1 t c2 v)).