Lemma commutative_symmetric_braiding_after_4_copies (m : commutative_comonoid M) : δ_{m} · (δ_{m} #⊗ δ_{m} · (α^{M}_{ m, m, m ⊗_{ M} m} · (m ⊗^{M}_{l} (αinv^{M}_{ m, m, m} · (sym_mon_braiding M m m ⊗^{M}_{r} m · α^{M}_{ m, m, m})) · αinv^{M}_{m, m, m ⊗_{ M} m}))) = δ_{m} · δ_{m} #⊗ δ_{m}. Proof. rewrite ! assoc. etrans. { apply maponpaths_2. rewrite ! (bifunctor_leftcomp M). rewrite ! assoc. apply maponpaths_2. exact (commutative_symmetric_braiding_after_4_copies' _). } etrans. { apply maponpaths_2. rewrite assoc'. apply maponpaths. rewrite <- (bifunctor_leftcomp M). apply maponpaths. apply (monoidal_associatorisolaw M m m m). } rewrite (bifunctor_leftid M). rewrite id_right. etrans. { rewrite assoc'. apply maponpaths. apply (monoidal_associatorisolaw M m m _). } apply id_right. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.DisplayedCats.Projection. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.CategoryTheory.DisplayedCats.Examples.Sigma. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Functors. Import BifunctorNotations. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Require Import UniMath.CategoryTheory.Monoidal.Structure.SymmetricDiagonal. Require Import UniMath.CategoryTheory.Monoidal.Structure.Cartesian. Require Import UniMath.CategoryTheory.Monoidal.Displayed.WhiskeredDisplayedBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Displayed.Monoidal. Require Import UniMath.CategoryTheory.Monoidal.Displayed.TotalMonoidal. Require Import UniMath.CategoryTheory.Monoidal.Displayed.Symmetric. Require Import UniMath.CategoryTheory.Monoidal.Examples.Sigma. Require Import UniMath.CategoryTheory.Monoidal.Examples.Fullsub. Require Import UniMath.CategoryTheory.Monoidal.Examples.DiagonalFunctor. Require Import UniMath.CategoryTheory.Monoidal.Examples.ConstantFunctor. Require Import UniMath.CategoryTheory.Monoidal.CategoriesOfMonoids. Require Import UniMath.CategoryTheory.Categories.Dialgebras. Import MonoidalNotations. Import ComonoidNotations. Import ComonoidNotations. Import ComonoidNotations. Import ComonoidNotations. Import ComonoidNotations.