proposition: Lemma lambda_algebra_theory_has_beta : has_β lambda_algebra_theory.
proof: Proof. apply endomorphism_theory_has_β. assert ((functional_abs : monoid_action_cat _⟦_, _⟧) · functional_app = identity _). { apply monoid_action_morphism_eq. intro a. apply functional_2_eq. exact (!pr2 a). } refine (assoc' _ _ _ @ _). refine (maponpaths (λ x, _ · x) (assoc _ _ _ ) @ _). rewrite X. rewrite id_left. apply z_iso_after_z_iso_inv. Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.Algebra.Monoids UniMath.CategoryTheory.Categories.HSET.Core UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Limits.BinProducts UniMath.Combinatorics.StandardFiniteSets UniMath.Combinatorics.Tuples UniMath.Combinatorics.Vectors UniMath.AlgebraicTheories.AlgebraicTheories UniMath.AlgebraicTheories.Algebras UniMath.AlgebraicTheories.AlgebraMorphisms UniMath.AlgebraicTheories.Examples.LambdaCalculus UniMath.AlgebraicTheories.Examples.EndomorphismTheory UniMath.AlgebraicTheories.FundamentalTheorem.SpecificUtilities.LambdaTerms UniMath.AlgebraicTheories.FundamentalTheorem.CommonUtilities.MonoidActions UniMath.AlgebraicTheories.LambdaTheories UniMath.AlgebraicTheories.LambdaCalculus UniMath.AlgebraicTheories.LambdaTheoryCategory