Definition dummy {X Y} (P : X -> Prop) (Q : Y -> Prop) : X -> Y -> Prop :=
  fun x y => P x /\ Q y.

Lemma poram_split_dummy {X Y} (Pre Pre' Post Post' : state -> Prop) (m : Poram X) (m' : Poram Y) : poram_lift Pre Post triv m -> poram_lift Pre' Post' triv m' -> poram_lift2 (dummy Pre Pre') (dummy Post Post') triv2 m m'.