proposition: Lemma subset_lattice_is_bounded : bounded_latticeop subset_lattice (emptysubtype X) (totalsubtype X).
proof: Proof. use make_dirprod. - intros x; cbn. apply (invweq (hsubtype_univalence _ _)), subtype_equal_cond; use make_dirprod. + intros ? in_union. refine (hinhuniv _ in_union); clear in_union; intro in_union. induction in_union as [b bin]; induction b; cbn in bin. * induction bin. * assumption. + intros ? ?; apply hinhpr; exists false; assumption. - intros x; cbn. apply (invweq (hsubtype_univalence _ _)), subtype_equal_cond; use make_dirprod. + intros ? in_intersection; exact (in_intersection false). + intros ? ? b; induction b; cbn. * exact tt. * assumption. Qed.
imports: UniMath.Foundations.Preamble UniMath.Foundations.PartA UniMath.Foundations.Sets UniMath.MoreFoundations.Sets UniMath.MoreFoundations.Propositions UniMath.MoreFoundations.Subtypes UniMath.Algebra.BinaryOperations UniMath.OrderTheory.Lattice.Lattice UniMath.OrderTheory.Lattice.Bounded UniMath.OrderTheory.Lattice.Distributive UniMath.OrderTheory.Lattice.Complement UniMath.OrderTheory.Lattice.Boolean