proposition: Lemma Tinf_monoid_assoc_pointwise (v : vertex nat_graph) : αinv^{V}_{_, _, _} · free_monoid_coeq_sequence_diagram_on_Tinf_Tinf_map v ⊗^{V}_{r} Tinf · Tinf_monoid_mul = pair_diagram_lob (free_monoid_coeq_sequence_on I_{V} v) ⊗^{V}_{l} Tinf_monoid_mul · free_monoid_coeq_sequence_diagram_on_Tinf_Tinf_map v.
proof: Proof. induction v as [|v Hv]. - apply pathsinv0. etrans. apply (monoidal_leftunitornat V). apply cancel_postcomposition. apply pathsinv0. etrans. apply cancel_precomposition. exact (pathsinv0 (right_whisker_with_lunitor V _ _)). etrans. apply assoc. etrans. apply cancel_postcomposition. exact (pr2 (monoidal_associatorisolaw V _ _ _)). apply id_left. - induction v as [|v _]. * etrans. apply assoc'. etrans. apply cancel_precomposition. { etrans. apply cancel_postcomposition. apply (bifunctor_rightcomp V). etrans. apply assoc'. apply cancel_precomposition. exact (Tinf_mul_alg_map_commutes). } etrans. apply assoc. etrans. apply assoc. apply pathsinv0. etrans. apply assoc. apply cancel_postcomposition. etrans. exact (pathsinv0 (whiskerscommutes V (bifunctor_equalwhiskers V) _ _)). apply pathsinv0. etrans. apply assoc. apply cancel_postcomposition. etrans. apply assoc'. etrans. apply cancel_precomposition. exact (pathsinv0 (monoidal_associatornatright V _ _ _ _ _)). etrans. apply assoc. etrans. apply cancel_postcomposition. apply (monoidal_associatorisolaw V). apply id_left. * set (coeq := next_pair_diagram_coeq (free_monoid_coeq_sequence_on I_{V} v)). set (rt_coequalizer := rt_coeq_coequalizer (rt_coeq (Tinf ⊗_{V} Tinf)) coeq). apply pathsinv0. use (CoequalizerOutUnique' rt_coequalizer). etrans. { etrans. apply assoc. etrans. apply cancel_postcomposition. apply (whiskerscommutes V (bifunctor_equalwhiskers V)). etrans. apply assoc'. etrans. apply cancel_precomposition. exact (pr22 (free_monoid_coeq_sequence_on_Tinf_pd_Tinf_map (S v))). etrans. apply assoc. etrans. apply cancel_postcomposition. { etrans. apply assoc. etrans. apply cancel_postcomposition. exact (pathsinv0 (monoidal_associatornatleft V _ _ _ _ _)). etrans. apply assoc'. apply cancel_precomposition. etrans. exact (pathsinv0 (bifunctor_leftcomp V _ _ _ _ _ _)). apply maponpaths. exact (!Hv). } reflexivity. } etrans. apply cancel_postcomposition. etrans. apply cancel_precomposition. apply (bifunctor_leftcomp V). apply assoc. etrans. apply assoc'. apply pathsinv0. etrans. { etrans. apply assoc. apply cancel_postcomposition. etrans. apply assoc. etrans. apply cancel_postcomposition. apply (monoidal_associatorinvnatright V). etrans. apply assoc'. apply cancel_precomposition. etrans. exact (pathsinv0 (bifunctor_rightcomp V _ _ _ _ _ _)). etrans. apply maponpaths. exact (pr22 (free_monoid_coeq_sequence_on_Tinf_pd_Tinf_map (S v))). apply (bifunctor_rightcomp V). } etrans. apply cancel_postcomposition, assoc. etrans. apply assoc'. etrans. apply cancel_precomposition. exact Tinf_mul_alg_map_commutes. etrans. apply assoc. etrans. apply cancel_postcomposition, assoc. etrans. apply assoc'. apply cancel_postcomposition. apply Tinf_mul_assoc_pointwise_associator_mess. Qed.
imports: UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.Limits.Graphs.Coequalizers UniMath.CategoryTheory.Limits.Graphs.Colimits UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.whiskering UniMath.CategoryTheory.Chains.Chains UniMath.Combinatorics.StandardFiniteSets UniMath.CategoryTheory.Monoidal.Categories UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors UniMath.CategoryTheory.Monoidal.CategoriesOfMonoids UniMath.ModelCategories.Generated.GenericFreeMonoid UniMath.ModelCategories.Generated.MonoidalHelpers UniMath.ModelCategories.Helpers