Proposition conditional_distribution_1_eq {x y : C} (p : I_{C} --> x ⊗ y) : p = p · proj1 · ⟨identity x, p|1⟩. Proof. unfold conditional_distribution_1. etrans. { apply (conditional_eq p). } rewrite <- pairing_eq. rewrite copy_tensor_x_I_mor_comp. unfold pairing. rewrite assoc. 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.