proposition: Lemma rt_chain_iso_pair_rel_impl_total_rel {A : C} {v : vertex nat_graph} {iv : rt_chain_iso_v A v} {iSv : rt_chain_iso_v A (S v)} (H : rt_chain_iso_pair_rel iv iSv) : rt_chain_iso_pair_total_rel iv iSv.
proof: Proof. etrans. apply assoc. etrans. apply cancel_postcomposition. { etrans. apply assoc. apply cancel_postcomposition. exact (pathsinv0 (monoidal_leftunitorinvnat V _ _ _)). } etrans. apply assoc4. etrans. apply cancel_postcomposition, cancel_precomposition. exact (pathsinv0 (whiskerscommutes V (bifunctor_equalwhiskers V) _ _)). etrans. apply cancel_postcomposition, assoc. etrans. apply assoc'. etrans. apply cancel_precomposition. { etrans. exact (pathsinv0 (id_left _)). etrans. apply cancel_postcomposition. exact (pathsinv0 (pr2 (monoidal_associatorisolaw V _ _ _))). etrans. apply assoc'. apply cancel_precomposition. etrans. apply assoc. exact (pathsinv0 H). } etrans. apply assoc. etrans. apply assoc. apply cancel_postcomposition. apply pathsinv0. etrans. apply cancel_precomposition. apply (bifunctor_leftid V). etrans. apply id_right. etrans. apply (bifunctor_rightcomp V). etrans. apply cancel_postcomposition. apply (bifunctor_rightcomp V). apply cancel_postcomposition. cbn. etrans. apply cancel_postcomposition. exact (pathsinv0 (monoidal_triangle_identity'_inv V _ _)). etrans. apply assoc'. etrans. apply cancel_precomposition. exact (pathsinv0 (monoidal_associatorinvnatright V _ _ _ _ _)). apply assoc. 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