(* to make the proof faster, we avoid unfolding of those definitions *)
Definition Jspec'_juicy_mem_equiv_def CS ext_link :=
  ext_spec_stable juicy_mem_equiv (JE_spec _ ( @OK_spec (Concurrent_Espec unit CS ext_link))).Definition Jspec'_hered_def CS ext_link :=
   ext_spec_stable age (JE_spec _ ( @OK_spec (Concurrent_Espec unit CS ext_link))).

Lemma cond_approx_eq_app n A P1 P2 phi : cond_approx_eq n A P1 P2 -> (level phi < n)%nat -> forall ts y z, app_pred (P1 ts (fmap (rmaps.dependent_type_functor_rec ts A) (approx n) (approx n) y) z) phi -> app_pred (P2 ts (fmap (rmaps.dependent_type_functor_rec ts A) (approx n) (approx n) y) z) phi.