proposition: Lemma presheaf_to_L_preserves_abs (n : nat) (s : (presheaf_lambda_theory (S n) : hSet)) : presheaf_to_L n (abs s) = abs (presheaf_to_L (S n) s).
proof: Proof. refine (maponpaths (λ x, x _) (presheaf_mor_comp (P'' := theory_presheaf L) _ _ _) @ _). apply (maponpaths abs). refine (hom_weq_eq _ _ @ _). apply (maponpaths (λ x, pr1 s _ x)). refine (maponpaths (λ x, pr1 x _ _) (!z_iso_inv_after_z_iso (pow_iso _)) @ _). refine (maponpaths (λ x, x _) (presheaf_mor_comp (P'' := PO (S n)) _ _ _) @ _). apply (maponpaths (pow_f_inv (S n) _)). apply funextsec. intro i. do 2 refine (!_ @ maponpaths _ (homotweqinvweq stnweq i)). induction (invmap stnweq i) as [i' | i']; refine (maponpaths (λ x, (_ x : presheaf_morphism _ _) _ _) (homotinvweqweq stnweq _) @ _). - refine (maponpaths (λ x, x _) (presheaf_mor_comp (P'' := theory_presheaf L) _ _ _) @ _). refine (mor_op (ProductPr _ _ (pow n) _ : presheaf_morphism (PO _) (theory_presheaf L)) _ _ @ _). refine (maponpaths (λ x, x _ • _) (!presheaf_mor_comp (P'' := theory_presheaf L) _ _ _) @ _). refine (maponpaths (λ x, pr1 x _ _ • _) (ProductPrCommutes _ _ _ (pow _) _ _ _) @ _). apply pr_comp. - apply idpath. 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