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_strict i s p0 c1 t c2 v :
wf (cbv_update i s p0 c1 t c2 v) ->
~ cbv_update i s p0 c1 t c2 v ∈ activations p0.