Proposition conditional_distribution_2_eq {x y : C} (p : I_{C} --> x ⊗ y) : p = p · proj2 · ⟨p|2, identity y⟩. Proof. apply cancel_braiding. unfold conditional_distribution_2. etrans. { apply conditional_distribution_1_eq. } refine (!_). do 2 refine (assoc' _ _ _ @ _). rewrite pairing_sym_mon_braiding. rewrite !assoc'. apply maponpaths. refine (_ @ assoc' _ _ _). apply maponpaths_2. rewrite sym_mon_braiding_proj1. 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.