proposition: Lemma is_runit_I1 (a : A) : compose a (I1) = make_functional_1 a.
proof: Proof. pose (Ha := (pr_action _ (● 0 : stn 1) (weqvecfun 1 [(a)]))). refine (!_ @ maponpaths _ Ha). refine (!_ @ maponpaths (λ x, _ x _) Ha). unfold make_functional_1, compose, I1, aaction. epose (Hlift := !lift_constant_action (A := A) 1 _ (weqvecfun 1 [(a)])). refine (maponpaths (λ x, _ (_ [(_ ; x)])) Hlift @ !_). refine (maponpaths (λ x, _ (_ [(x ; _)])) Hlift @ !_). rewrite (move_action_through_vector_2 A (pr _) _ _). rewrite (move_action_through_vector_2 A _ (pr _) _). do 2 rewrite <- comp_action. apply (maponpaths (λ x, aaction x _)). rewrite <- (inflate_is_lift_constant I1_λ : _ = lift_constant (T := L) _ _). apply I1_runit_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