Proposition bayesian_inverse_eq_l {y : C} (f : x --> y) : p · f · ⟨bayesian_inverse f, identity _⟩ = p · ⟨identity _, f⟩. Proof. symmetry. assert(e : is_bayesian_inverse p f (bayesian_inverse f)). { apply bayesian_inverse_eq. } unfold is_bayesian_inverse in e. rewrite <- e. reflexivity. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Structure.Cartesian. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Require Import UniMath.CategoryTheory.MarkovCategories.MarkovCategory. Require Import UniMath.CategoryTheory.MarkovCategories.Determinism. Require Import UniMath.CategoryTheory.MarkovCategories.InformationFlowAxioms. Require Import UniMath.CategoryTheory.MarkovCategories.AlmostSurely. Import MonoidalNotations.