proposition: Lemma presheaf_to_L_preserves_app (n : nat) (s : (presheaf_lambda_theory n : hSet)) : presheaf_to_L (S n) (app s) = app (presheaf_to_L n s).
proof: Proof. refine (invmap_hom_weq_eq _ _ @ _). match goal with | [ |- pr1 ?x1 _ _ • _ = _ ] => pose (x1' := x1) end. refine (maponpaths (λ x, pr1 x1' _ x • _) (aux2 n) @ _). refine (maponpaths (λ x, x • _) (mor_op (x1' : presheaf_morphism (PO n) (pr1 theory_presheaf_exponentiable _ : presheaf L)) _ _) @ _). refine (comp_comp L (l := S n) (m := S (S n)) _ _ _ @ _). refine (maponpaths (λ x, x _ • _) (presheaf_mor_comp (P'' := plus_1_presheaf (theory_presheaf L)) _ _ _) @ _). refine (_ @ comp_pr _ _). apply maponpaths. apply funextfun. intro i. refine (_ @ maponpaths _ (homotweqinvweq stnweq i)). induction (invmap stnweq i) as [i' | i']. - refine (maponpaths (λ x, x • _) (pr_comp _ _ _) @ _). refine (pr_comp _ _ _ @ _). refine (extend_tuple_i _ _ _ _ (dni_last_lt _) @ _). apply maponpaths. apply stn_eq. apply di_eq1. exact (stnlt (dni lastelement i')). - refine (pr_comp _ _ _ @ _). exact (maponpaths _ (homotinvweqweq _ _)). Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.Categories.HSET.Core UniMath.CategoryTheory.Categories.HSET.Limits UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Examples.Cartesian UniMath.CategoryTheory.DisplayedCats.Examples.Sigma UniMath.CategoryTheory.DisplayedCats.Fiber UniMath.CategoryTheory.DisplayedCats.Total UniMath.CategoryTheory.exponentials UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.Limits.BinProducts UniMath.CategoryTheory.Limits.Graphs.Colimits UniMath.CategoryTheory.Limits.Graphs.Limits UniMath.CategoryTheory.Limits.Products UniMath.CategoryTheory.Limits.Terminal UniMath.Combinatorics.StandardFiniteSets UniMath.Combinatorics.Tuples UniMath.AlgebraicTheories.AlgebraicTheories UniMath.AlgebraicTheories.AlgebraicTheoryCategory UniMath.AlgebraicTheories.AlgebraicTheoryCategoryCore UniMath.AlgebraicTheories.AlgebraicTheoryMorphisms UniMath.AlgebraicTheories.Examples.EndomorphismTheory UniMath.AlgebraicTheories.Examples.Plus1Presheaf UniMath.AlgebraicTheories.LambdaTheories UniMath.AlgebraicTheories.LambdaTheoryCategory UniMath.AlgebraicTheories.LambdaTheoryCategoryCore UniMath.AlgebraicTheories.LambdaTheoryMorphisms UniMath.AlgebraicTheories.PresheafCategory UniMath.AlgebraicTheories.PresheafCategoryCore UniMath.AlgebraicTheories.PresheafMorphisms UniMath.AlgebraicTheories.Presheaves