Definition activation_function (function_default : function) (p: cbv) :=
  match p with
    | cbv_update _ _ _ _ (fapply f _) _ _ => f
    | _                                   => function_default
  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.