Definition mpl_compat_8_def (s : state) : Prop :=
  forall (a c : ad) (pl l : prec_list),
  MapGet prec_list s c = Some l ->
  a <> c -> MapGet prec_list (union_mpl_0 a pl s) c = Some l.

Lemma mpl_compat_8 : forall (s : state) (a c : ad) (pl l : prec_list), MapGet prec_list s c = Some l -> a <> c -> MapGet prec_list (union_mpl_0 a pl s) c = Some l.