proposition: Lemma functional_2_to_exponential_object_is_morphism : ∏ x m, mor_action_ax functional_2_to_exponential_object_morphism_data x m.
proof: Proof. intros a m. apply monoid_action_morphism_eq. intro x. apply functional_1_eq. unfold functional_2_to_exponential_object_morphism_data. unfold functional_2_to_monoid_action_morphism_data. unfold functional_2_to_monoid_action_morphism_data_term. cbn -[weqvecfun aaction]. set (v := weqvecfun _ [(pr1 a ; pr1 m ; pr11 x ; pr12 x)]). rewrite <- (pr_action _ (make_stn 4 0 (idpath true)) v : _ = pr1 a). rewrite <- (pr_action _ (make_stn 4 1 (idpath true)) v : _ = pr1 m). rewrite <- (pr_action _ (make_stn 4 2 (idpath true)) v : _ = pr11 x). rewrite <- (pr_action _ (make_stn 4 3 (idpath true)) v : _ = pr12 x). unfold compose, aaction. do 2 rewrite (move_action_through_vector_2 A _ _ _). do 2 rewrite <- comp_action. do 2 rewrite (move_action_through_vector_3 A _ _ _ _). do 2 rewrite <- comp_action. apply (maponpaths (λ x, aaction x v)). cbn -[weqvecfun]. apply compose_2_compose_λ. 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