proposition: Proposition cofree_comonoid_REL_comm : cofree_comonoid_REL_comult · sym_mon_braiding REL_sym_mon_closed_cat _ _ = cofree_comonoid_REL_comult.
proof: Proof. use funextsec ; intro x. use funextsec ; intro y. induction y as [ y₁ y₂ ]. use hPropUnivalence. - use factor_through_squash_hProp ; cbn. intros [ [ a₁ a₂ ] [ p₁ [ p₂ p₃ ] ] ] ; cbn in p₁, p₂, p₃. induction p₂, p₃. refine (_ @ p₁). apply (commax Mmonoid). - intro p ; cbn in p. refine (hinhpr ((y₂ ,, y₁) ,, _ ,, _ ,, _)) ; cbn. + refine (_ @ p). apply (commax Mmonoid). + apply idpath. + apply idpath. Qed.
imports: UniMath.MoreFoundations.All UniMath.Algebra.Monoids UniMath.Algebra.Free_Monoids_and_Groups UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.Categories.Rel UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.whiskering UniMath.CategoryTheory.Monads.Comonads UniMath.CategoryTheory.Monoidal.Categories UniMath.CategoryTheory.Monoidal.Functors UniMath.CategoryTheory.Monoidal.Adjunctions UniMath.CategoryTheory.Monoidal.FunctorCategories UniMath.CategoryTheory.Monoidal.Displayed.TotalMonoidal UniMath.CategoryTheory.Monoidal.Displayed.Symmetric UniMath.CategoryTheory.Monoidal.Structure.Cartesian UniMath.CategoryTheory.Monoidal.Structure.Symmetric UniMath.CategoryTheory.Monoidal.Structure.Closed UniMath.CategoryTheory.Monoidal.Comonoids.Category UniMath.CategoryTheory.Monoidal.Comonoids.Monoidal UniMath.CategoryTheory.Monoidal.Comonoids.Symmetric UniMath.CategoryTheory.Monoidal.Comonoids.CommComonoidsCartesian UniMath.CategoryTheory.Monoidal.Examples.Relations UniMath.Semantics.LinearLogic.LafontCategory