proposition: Lemma independence_bound_on_entropy : `H P <= \sum_(i < n.+1) `H (fdist_nth P i).
proof: Proof. rewrite chain_rule_rV; apply leR_sumR => /= i _. case: ifPn => [/eqP|] i0. rewrite (_ : i = ord0); last exact/val_inj. rewrite head_of_fdist_rV_fdist_nth. by apply/RleP; rewrite lexx. apply: leR_trans; first exact: information_cant_hurt. rewrite fdistX1 fdist_take_nth. by apply/RleP; rewrite lexx. 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