proposition: Lemma lower_spec h k: h ≪ k <-> (has_cup h -> has_cup k) /\ (has_bot h -> has_bot k) /\ (has_cap h -> has_cap k) /\ (has_top h -> has_top k) /\ (has_str h -> has_str k) /\ (has_cnv h -> has_cnv k) /\ (has_neg h -> has_neg k) /\ (has_div h -> has_div k).
proof: Proof. destruct h; destruct k. unfold lower. rewrite !landb_spec, !le_bool_spec. reflexivity. Qed.
imports: common