proposition: Proposition lifting_linear_category_laws : linear_category_laws lifting_linear_category_data.
proof: Proof. repeat split. - intros X Y f. exact (nat_trans_ax lift_poset_comult X Y f). - intros X Y f. refine (nat_trans_ax lift_poset_counit X Y f @ _). apply id_right. - intros X. use eq_mor_hset_struct. intro x ; cbn in x. induction x as [ x | ]. + cbn ; apply idpath. + cbn ; apply idpath. - intros X. use eq_mor_hset_struct. intro x ; cbn in x. induction x as [ x | ]. + cbn ; apply idpath. + cbn ; apply idpath. - intros X. use eq_mor_hset_struct. intro x ; cbn in x. induction x as [ x | ]. + cbn ; apply idpath. + cbn ; apply idpath. - intros X. use eq_mor_hset_struct. intro x ; cbn in x. induction x as [ x | ]. + cbn ; apply idpath. + cbn ; apply idpath. - intros X. rewrite <-tensor_mor_left. rewrite <-tensor_mor_right. apply pathsinv0, (comonoid_to_law_assoc _ (lift_commutative_comonoid X)). - intros X. rewrite <-tensor_mor_right. etrans. 2: { apply (comonoid_to_law_unit_left _ (lift_commutative_comonoid X)). } apply idpath. - intros X. apply (commutative_comonoid_is_commutative _ (lift_commutative_comonoid X)). - intros X Y. use eq_mor_hset_struct. use setquotunivprop' ; [ intro ; apply setproperty | ]. intros xy. induction xy as [ x y ]. induction x as [ x | ], y as [ y | ] ; simpl ; apply idpath. - use eq_mor_hset_struct. intro x. induction x as [ | ]. + cbn. apply idpath. + use iscompsetquotpr ; cbn. refine (hinhpr (inr _)). split. * unfold product_point_coordinate ; cbn. exact (inl (idpath _)). * unfold product_point_coordinate ; cbn. exact (inr (idpath _)). - intros X Y. use eq_mor_hset_struct. use setquotunivprop' ; [ intro ; apply setproperty | ]. intros xy. induction xy as [ x y ]. induction x as [ x | ], y as [ y | ] ; cbn. + apply idpath. + apply idpath. + apply idpath. + apply idpath. - use eq_mor_hset_struct. intro x. induction x as [ | ] ; cbn. + apply idpath. + apply idpath. Qed.
imports: UniMath.MoreFoundations.All UniMath.OrderTheory.Posets UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.DisplayedCats.Structures.CartesianStructure UniMath.CategoryTheory.DisplayedCats.Structures.StructureLimitsAndColimits UniMath.CategoryTheory.DisplayedCats.Structures.StructuresSmashProduct UniMath.CategoryTheory.DisplayedCats.Examples.PointedPosetStrict UniMath.CategoryTheory.Monads.Comonads UniMath.CategoryTheory.Monoidal.Categories UniMath.CategoryTheory.Monoidal.Functors UniMath.CategoryTheory.Monoidal.FunctorCategories UniMath.CategoryTheory.Monoidal.Structure.Symmetric UniMath.CategoryTheory.Monoidal.Structure.Closed UniMath.CategoryTheory.Monoidal.Examples.SmashProductMonoidal UniMath.CategoryTheory.Monoidal.Examples.PosetsMonoidal UniMath.CategoryTheory.Monoidal.Examples.LiftPoset UniMath.Semantics.LinearLogic.LinearCategory UniMath.CategoryTheory.Monoidal.Comonoids.Category