Definition lattice_bounded_def_1 (p : prechain bool) : Prop :=
  lattice_bounded_def_0 p ->
  forall m : Map bool, lattice_bounded_def_0 (concat bool p m).

Lemma lattice_bounded_2 : forall p : prechain bool, lattice_bounded_def_1 p -> forall m : Map bool, lattice_bounded_def_1 (concat bool p m).