proposition: Lemma cond_entropyE : cond_entropy = - \sum_(a in A) \sum_(b in B) PQ (a, b) * log (\Pr_QP [ [set b] | [set a]]).
proof: Proof. rewrite /cond_entropy big_morph_oppR /=; apply eq_bigr => a _. rewrite /cond_entropy1 mulRN big_distrr /=; congr (- _); apply eq_bigr => b _. rewrite mulRA; congr (_ * _). by rewrite mulRC -(Pr_set1 P a) -jproduct_rule setX1 fdistXE Pr_set1. Qed.
imports: mathcomp(all_ssreflect all_algebra fingroup perm matrix) Reals mathcomp(Rstruct) ssrR Reals_ext realType_ext ssr_ext ssralg_ext bigop_ext logb ln_facts fdist jfdist_cond proba binary_entropy_function divergence