proposition: Lemma Arity_to_functor_agrees (TC : Terminal C) (xs : list nat) : Arity_to_functor TC xs = Signature_Functor (Arity_to_Signature BPC BCC TC xs).
proof: Proof. induction xs as [[|n] xs]. - induction xs. apply idpath. - induction n as [|n IH]. + induction xs as [m []]. apply idpath. + induction xs as [m [k xs]]. assert (IHinst := IH (k,,xs)). change (S (S n),, m,, k,, xs) with (cons m (cons k (n,,xs))). unfold Arity_to_functor. do 2 rewrite map_cons. rewrite foldr1_cons. change (S n,, k,, xs) with (cons k (n,,xs)) in IHinst. etrans. { apply maponpaths. exact IHinst. } apply idpath. Defined.
imports: UniMath.Foundations.PartD UniMath.Combinatorics.Lists UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.whiskering UniMath.CategoryTheory.Limits.Graphs.Colimits UniMath.CategoryTheory.Limits.BinProducts UniMath.CategoryTheory.Limits.BinCoproducts UniMath.CategoryTheory.Limits.Coproducts UniMath.CategoryTheory.Limits.Terminal UniMath.CategoryTheory.Limits.Initial UniMath.CategoryTheory.FunctorAlgebras UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.exponentials UniMath.CategoryTheory.Chains.All UniMath.CategoryTheory.Monads.Monads UniMath.CategoryTheory.coslicecat UniMath.CategoryTheory.Categories.HSET.Core UniMath.CategoryTheory.Categories.HSET.Limits UniMath.CategoryTheory.Categories.HSET.Colimits UniMath.CategoryTheory.Categories.HSET.Structures UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors UniMath.CategoryTheory.Monoidal.Categories UniMath.CategoryTheory.Actegories.Actegories UniMath.CategoryTheory.Actegories.ConstructionOfActegories UniMath.CategoryTheory.Actegories.MorphismsOfActegories UniMath.CategoryTheory.Actegories.ConstructionOfActegoryMorphisms UniMath.CategoryTheory.Actegories.CoproductsInActegories UniMath.CategoryTheory.Monoidal.CategoriesOfMonoids UniMath.CategoryTheory.Monoidal.Examples.MonoidalPointedObjects UniMath.CategoryTheory.Monoidal.Examples.EndofunctorsMonoidalElementary UniMath.CategoryTheory.Monoidal.Examples.MonadsAsMonoidsElementary UniMath.CategoryTheory.DisplayedCats.Total UniMath.SubstitutionSystems.Signatures UniMath.SubstitutionSystems.SignatureExamples UniMath.SubstitutionSystems.SumOfSignatures UniMath.SubstitutionSystems.BinProductOfSignatures UniMath.SubstitutionSystems.SubstitutionSystems UniMath.SubstitutionSystems.LiftingInitial_alt UniMath.SubstitutionSystems.MonadsFromSubstitutionSystems UniMath.SubstitutionSystems.BindingSigToMonad UniMath.SubstitutionSystems.GeneralizedSubstitutionSystems UniMath.SubstitutionSystems.ConstructionOfGHSS UniMath.SubstitutionSystems.SigmaMonoids UniMath.SubstitutionSystems.Notation