Module LBoolean <: SEMILATTICE_WITH_TOP.
Definition t := bool.
Definition eq (x y: t) := (x = y).
Definition eq_refl: forall x, eq x x := (@eq_refl t).
Definition eq_sym: forall x y, eq x y -> eq y x := (@eq_sym t).
Definition eq_trans: forall x y z, eq x y -> eq y z -> eq x z := (@eq_trans t).
Definition beq : t -> t -> bool := eqb.
Proof eqb_prop.
Definition ge (x y: t) : Prop := x = y \/ x = true.
Definition beq (x y: t) : bool :=
match x, y with
| None, None => true
| Some x1, Some y1 => L.beq x1 y1
| _, _ => false
end.
Definition ge (x y: t) : Prop :=
match x, y with
| None, _ => True
| _, None => False
| Some x1, Some y1 => L.ge x1 y1
end.
Definition bot : t := Some L.bot.
Definition lub (x y: t) : t :=
match x, y with
| None, _ => None
| _, None => None
| Some x1, Some y1 => Some (L.lub x1 y1)
end.

Lemma ge_lub_left: forall x y, ge (lub x y) x.