proposition: Lemma S_imp_SS h n : # L inF · h = ψ μF h → # L (e n) · h = Pow n IC z.
proof: Proof. intros Hh. induction n. - cbn. apply (InitialArrowUnique ILD). - rewrite e_comm, functor_comp, <- assoc, Hh. assert (H : # L (# F (e n)) · ψ μF h = ψ (iter_functor F n 0) (# L (e n) · h)). { apply pathsinv0, (eqtohomot (nat_trans_ax ψ _ _ (e n))). } now rewrite H, IHn. Qed.
imports: UniMath.Tactics.EnsureStructuredProofs UniMath.Foundations.PartD UniMath.MoreFoundations.Tactics UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Limits.Initial UniMath.CategoryTheory.Limits.Graphs.Colimits UniMath.CategoryTheory.FunctorAlgebras UniMath.CategoryTheory.Categories.HSET.Core UniMath.CategoryTheory.opp_precat UniMath.CategoryTheory.Chains.Chains UniMath.CategoryTheory.Chains.Adamek UniMath.CategoryTheory.yoneda UniMath.CategoryTheory.HorizontalComposition UniMath.CategoryTheory.whiskering