src/HOL/Probability/Information.thy
 author wenzelm Thu, 18 Apr 2013 17:07:01 +0200 changeset 51717 9e7d1c139569 parent 50419 3177d0374701 child 53015 a1119cf551e8 permissions -rw-r--r--
simplifier uses proper Proof.context instead of historic type simpset;
```
(*  Title:      HOL/Probability/Information.thy
Author:     Johannes Hölzl, TU München
Author:     Armin Heller, TU München
*)

theory Information
imports
Independent_Family
"~~/src/HOL/Library/Convex"
begin

lemma log_le: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> x \<le> y \<Longrightarrow> log a x \<le> log a y"
by (subst log_le_cancel_iff) auto

lemma log_less: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> x < y \<Longrightarrow> log a x < log a y"
by (subst log_less_cancel_iff) auto

lemma setsum_cartesian_product':
"(\<Sum>x\<in>A \<times> B. f x) = (\<Sum>x\<in>A. setsum (\<lambda>y. f (x, y)) B)"
unfolding setsum_cartesian_product by simp

lemma split_pairs:
"((A, B) = X) \<longleftrightarrow> (fst X = A \<and> snd X = B)" and
"(X = (A, B)) \<longleftrightarrow> (fst X = A \<and> snd X = B)" by auto

section "Information theory"

locale information_space = prob_space +
fixes b :: real assumes b_gt_1: "1 < b"

context information_space
begin

text {* Introduce some simplification rules for logarithm of base @{term b}. *}

lemma log_neg_const:
assumes "x \<le> 0"
shows "log b x = log b 0"
proof -
{ fix u :: real
have "x \<le> 0" by fact
also have "0 < exp u"
using exp_gt_zero .
finally have "exp u \<noteq> x"
by auto }
then show "log b x = log b 0"
qed

lemma log_mult_eq:
"log b (A * B) = (if 0 < A * B then log b \<bar>A\<bar> + log b \<bar>B\<bar> else log b 0)"
using log_mult[of b "\<bar>A\<bar>" "\<bar>B\<bar>"] b_gt_1 log_neg_const[of "A * B"]
by (auto simp: zero_less_mult_iff mult_le_0_iff)

lemma log_inverse_eq:
"log b (inverse B) = (if 0 < B then - log b B else log b 0)"
using log_inverse[of b B] log_neg_const[of "inverse B"] b_gt_1 by simp

lemma log_divide_eq:
"log b (A / B) = (if 0 < A * B then log b \<bar>A\<bar> - log b \<bar>B\<bar> else log b 0)"
unfolding divide_inverse log_mult_eq log_inverse_eq abs_inverse
by (auto simp: zero_less_mult_iff mult_le_0_iff)

lemmas log_simps = log_mult_eq log_inverse_eq log_divide_eq

end

subsection "Kullback\$-\$Leibler divergence"

text {* The Kullback\$-\$Leibler divergence is also known as relative entropy or
Kullback\$-\$Leibler distance. *}

definition
"entropy_density b M N = log b \<circ> real \<circ> RN_deriv M N"

definition
"KL_divergence b M N = integral\<^isup>L N (entropy_density b M N)"

lemma (in information_space) measurable_entropy_density:
assumes ac: "absolutely_continuous M N" "sets N = events"
shows "entropy_density b M N \<in> borel_measurable M"
proof -
from borel_measurable_RN_deriv[OF ac] b_gt_1 show ?thesis
unfolding entropy_density_def by auto
qed

lemma borel_measurable_RN_deriv_density[measurable (raw)]:
"f \<in> borel_measurable M \<Longrightarrow> RN_deriv M (density M f) \<in> borel_measurable M"
using borel_measurable_RN_deriv_density[of "\<lambda>x. max 0 (f x )" M]

lemma (in sigma_finite_measure) KL_density:
fixes f :: "'a \<Rightarrow> real"
assumes "1 < b"
assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
shows "KL_divergence b M (density M f) = (\<integral>x. f x * log b (f x) \<partial>M)"
unfolding KL_divergence_def
proof (subst integral_density)
show "entropy_density b M (density M (\<lambda>x. ereal (f x))) \<in> borel_measurable M"
using f
by (auto simp: comp_def entropy_density_def)
have "density M (RN_deriv M (density M f)) = density M f"
using f by (intro density_RN_deriv_density) auto
then have eq: "AE x in M. RN_deriv M (density M f) x = f x"
using f
by (intro density_unique)
(auto intro!: borel_measurable_log borel_measurable_RN_deriv_density simp: RN_deriv_density_nonneg)
show "(\<integral>x. f x * entropy_density b M (density M (\<lambda>x. ereal (f x))) x \<partial>M) = (\<integral>x. f x * log b (f x) \<partial>M)"
apply (intro integral_cong_AE)
using eq
apply eventually_elim
apply (auto simp: entropy_density_def)
done
qed fact+

lemma (in sigma_finite_measure) KL_density_density:
fixes f g :: "'a \<Rightarrow> real"
assumes "1 < b"
assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
assumes g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x"
assumes ac: "AE x in M. f x = 0 \<longrightarrow> g x = 0"
shows "KL_divergence b (density M f) (density M g) = (\<integral>x. g x * log b (g x / f x) \<partial>M)"
proof -
interpret Mf: sigma_finite_measure "density M f"
using f by (subst sigma_finite_iff_density_finite) auto
have "KL_divergence b (density M f) (density M g) =
KL_divergence b (density M f) (density (density M f) (\<lambda>x. g x / f x))"
using f g ac by (subst density_density_divide) simp_all
also have "\<dots> = (\<integral>x. (g x / f x) * log b (g x / f x) \<partial>density M f)"
using f g `1 < b` by (intro Mf.KL_density) (auto simp: AE_density divide_nonneg_nonneg)
also have "\<dots> = (\<integral>x. g x * log b (g x / f x) \<partial>M)"
using ac f g `1 < b` by (subst integral_density) (auto intro!: integral_cong_AE)
finally show ?thesis .
qed

lemma (in information_space) KL_gt_0:
fixes D :: "'a \<Rightarrow> real"
assumes "prob_space (density M D)"
assumes D: "D \<in> borel_measurable M" "AE x in M. 0 \<le> D x"
assumes int: "integrable M (\<lambda>x. D x * log b (D x))"
assumes A: "density M D \<noteq> M"
shows "0 < KL_divergence b M (density M D)"
proof -
interpret N: prob_space "density M D" by fact

obtain A where "A \<in> sets M" "emeasure (density M D) A \<noteq> emeasure M A"
using measure_eqI[of "density M D" M] `density M D \<noteq> M` by auto

let ?D_set = "{x\<in>space M. D x \<noteq> 0}"
have [simp, intro]: "?D_set \<in> sets M"
using D by auto

have D_neg: "(\<integral>\<^isup>+ x. ereal (- D x) \<partial>M) = 0"
using D by (subst positive_integral_0_iff_AE) auto

have "(\<integral>\<^isup>+ x. ereal (D x) \<partial>M) = emeasure (density M D) (space M)"
using D by (simp add: emeasure_density cong: positive_integral_cong)
then have D_pos: "(\<integral>\<^isup>+ x. ereal (D x) \<partial>M) = 1"
using N.emeasure_space_1 by simp

have "integrable M D" "integral\<^isup>L M D = 1"
using D D_pos D_neg unfolding integrable_def lebesgue_integral_def by simp_all

have "0 \<le> 1 - measure M ?D_set"
using prob_le_1 by (auto simp: field_simps)
also have "\<dots> = (\<integral> x. D x - indicator ?D_set x \<partial>M)"
using `integrable M D` `integral\<^isup>L M D = 1`
also have "\<dots> < (\<integral> x. D x * (ln b * log b (D x)) \<partial>M)"
proof (rule integral_less_AE)
show "integrable M (\<lambda>x. D x - indicator ?D_set x)"
using `integrable M D`
by (intro integral_diff integral_indicator) auto
next
from integral_cmult(1)[OF int, of "ln b"]
show "integrable M (\<lambda>x. D x * (ln b * log b (D x)))"
next
show "emeasure M {x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} \<noteq> 0"
proof
assume eq_0: "emeasure M {x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} = 0"
then have disj: "AE x in M. D x = 1 \<or> D x = 0"
using D(1) by (auto intro!: AE_I[OF subset_refl] sets.sets_Collect)

have "emeasure M {x\<in>space M. D x = 1} = (\<integral>\<^isup>+ x. indicator {x\<in>space M. D x = 1} x \<partial>M)"
using D(1) by auto
also have "\<dots> = (\<integral>\<^isup>+ x. ereal (D x) \<partial>M)"
using disj by (auto intro!: positive_integral_cong_AE simp: indicator_def one_ereal_def)
finally have "AE x in M. D x = 1"
using D D_pos by (intro AE_I_eq_1) auto
then have "(\<integral>\<^isup>+x. indicator A x\<partial>M) = (\<integral>\<^isup>+x. ereal (D x) * indicator A x\<partial>M)"
by (intro positive_integral_cong_AE) (auto simp: one_ereal_def[symmetric])
also have "\<dots> = density M D A"
using `A \<in> sets M` D by (simp add: emeasure_density)
finally show False using `A \<in> sets M` `emeasure (density M D) A \<noteq> emeasure M A` by simp
qed
show "{x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} \<in> sets M"
using D(1) by (auto intro: sets.sets_Collect_conj)

show "AE t in M. t \<in> {x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} \<longrightarrow>
D t - indicator ?D_set t \<noteq> D t * (ln b * log b (D t))"
using D(2)
proof (eventually_elim, safe)
fix t assume Dt: "t \<in> space M" "D t \<noteq> 1" "D t \<noteq> 0" "0 \<le> D t"
and eq: "D t - indicator ?D_set t = D t * (ln b * log b (D t))"

have "D t - 1 = D t - indicator ?D_set t"
using Dt by simp
also note eq
also have "D t * (ln b * log b (D t)) = - D t * ln (1 / D t)"
using b_gt_1 `D t \<noteq> 0` `0 \<le> D t`
by (simp add: log_def ln_div less_le)
finally have "ln (1 / D t) = 1 / D t - 1"
using `D t \<noteq> 0` by (auto simp: field_simps)
from ln_eq_minus_one[OF _ this] `D t \<noteq> 0` `0 \<le> D t` `D t \<noteq> 1`
show False by auto
qed

show "AE t in M. D t - indicator ?D_set t \<le> D t * (ln b * log b (D t))"
using D(2) AE_space
proof eventually_elim
fix t assume "t \<in> space M" "0 \<le> D t"
show "D t - indicator ?D_set t \<le> D t * (ln b * log b (D t))"
proof cases
assume asm: "D t \<noteq> 0"
then have "0 < D t" using `0 \<le> D t` by auto
then have "0 < 1 / D t" by auto
have "D t - indicator ?D_set t \<le> - D t * (1 / D t - 1)"
using asm `t \<in> space M` by (simp add: field_simps)
also have "- D t * (1 / D t - 1) \<le> - D t * ln (1 / D t)"
using ln_le_minus_one `0 < 1 / D t` by (intro mult_left_mono_neg) auto
also have "\<dots> = D t * (ln b * log b (D t))"
using `0 < D t` b_gt_1
finally show ?thesis by simp
qed simp
qed
qed
also have "\<dots> = (\<integral> x. ln b * (D x * log b (D x)) \<partial>M)"
also have "\<dots> = ln b * (\<integral> x. D x * log b (D x) \<partial>M)"
using int by (rule integral_cmult)
finally show ?thesis
using b_gt_1 D by (subst KL_density) (auto simp: zero_less_mult_iff)
qed

lemma (in sigma_finite_measure) KL_same_eq_0: "KL_divergence b M M = 0"
proof -
have "AE x in M. 1 = RN_deriv M M x"
proof (rule RN_deriv_unique)
show "(\<lambda>x. 1) \<in> borel_measurable M" "AE x in M. 0 \<le> (1 :: ereal)" by auto
show "density M (\<lambda>x. 1) = M"
apply (auto intro!: measure_eqI emeasure_density)
apply (subst emeasure_density)
apply auto
done
qed
then have "AE x in M. log b (real (RN_deriv M M x)) = 0"
by (elim AE_mp) simp
from integral_cong_AE[OF this]
have "integral\<^isup>L M (entropy_density b M M) = 0"
then show "KL_divergence b M M = 0"
unfolding KL_divergence_def
by auto
qed

lemma (in information_space) KL_eq_0_iff_eq:
fixes D :: "'a \<Rightarrow> real"
assumes "prob_space (density M D)"
assumes D: "D \<in> borel_measurable M" "AE x in M. 0 \<le> D x"
assumes int: "integrable M (\<lambda>x. D x * log b (D x))"
shows "KL_divergence b M (density M D) = 0 \<longleftrightarrow> density M D = M"
using KL_same_eq_0[of b] KL_gt_0[OF assms]
by (auto simp: less_le)

lemma (in information_space) KL_eq_0_iff_eq_ac:
fixes D :: "'a \<Rightarrow> real"
assumes "prob_space N"
assumes ac: "absolutely_continuous M N" "sets N = sets M"
assumes int: "integrable N (entropy_density b M N)"
shows "KL_divergence b M N = 0 \<longleftrightarrow> N = M"
proof -
interpret N: prob_space N by fact
have "finite_measure N" by unfold_locales
from real_RN_deriv[OF this ac] guess D . note D = this

have "N = density M (RN_deriv M N)"
using ac by (rule density_RN_deriv[symmetric])
also have "\<dots> = density M D"
using borel_measurable_RN_deriv[OF ac] D by (auto intro!: density_cong)
finally have N: "N = density M D" .

from absolutely_continuous_AE[OF ac(2,1) D(2)] D b_gt_1 ac measurable_entropy_density
have "integrable N (\<lambda>x. log b (D x))"
by (intro integrable_cong_AE[THEN iffD2, OF _ _ _ int])
(auto simp: N entropy_density_def)
with D b_gt_1 have "integrable M (\<lambda>x. D x * log b (D x))"
by (subst integral_density(2)[symmetric]) (auto simp: N[symmetric] comp_def)
with `prob_space N` D show ?thesis
unfolding N
by (intro KL_eq_0_iff_eq) auto
qed

lemma (in information_space) KL_nonneg:
assumes "prob_space (density M D)"
assumes D: "D \<in> borel_measurable M" "AE x in M. 0 \<le> D x"
assumes int: "integrable M (\<lambda>x. D x * log b (D x))"
shows "0 \<le> KL_divergence b M (density M D)"
using KL_gt_0[OF assms] by (cases "density M D = M") (auto simp: KL_same_eq_0)

lemma (in sigma_finite_measure) KL_density_density_nonneg:
fixes f g :: "'a \<Rightarrow> real"
assumes "1 < b"
assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" "prob_space (density M f)"
assumes g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x" "prob_space (density M g)"
assumes ac: "AE x in M. f x = 0 \<longrightarrow> g x = 0"
assumes int: "integrable M (\<lambda>x. g x * log b (g x / f x))"
shows "0 \<le> KL_divergence b (density M f) (density M g)"
proof -
interpret Mf: prob_space "density M f" by fact
interpret Mf: information_space "density M f" b by default fact
have eq: "density (density M f) (\<lambda>x. g x / f x) = density M g" (is "?DD = _")
using f g ac by (subst density_density_divide) simp_all

have "0 \<le> KL_divergence b (density M f) (density (density M f) (\<lambda>x. g x / f x))"
proof (rule Mf.KL_nonneg)
show "prob_space ?DD" unfolding eq by fact
from f g show "(\<lambda>x. g x / f x) \<in> borel_measurable (density M f)"
by auto
show "AE x in density M f. 0 \<le> g x / f x"
using f g by (auto simp: AE_density divide_nonneg_nonneg)
show "integrable (density M f) (\<lambda>x. g x / f x * log b (g x / f x))"
using `1 < b` f g ac
by (subst integral_density)
(auto intro!: integrable_cong_AE[THEN iffD2, OF _ _ _ int] measurable_If)
qed
also have "\<dots> = KL_divergence b (density M f) (density M g)"
using f g ac by (subst density_density_divide) simp_all
finally show ?thesis .
qed

subsection {* Finite Entropy *}

definition (in information_space)
"finite_entropy S X f \<longleftrightarrow> distributed M S X f \<and> integrable S (\<lambda>x. f x * log b (f x))"

lemma (in information_space) finite_entropy_simple_function:
assumes X: "simple_function M X"
shows "finite_entropy (count_space (X`space M)) X (\<lambda>a. measure M {x \<in> space M. X x = a})"
unfolding finite_entropy_def
proof
have [simp]: "finite (X ` space M)"
using X by (auto simp: simple_function_def)
then show "integrable (count_space (X ` space M))
(\<lambda>x. prob {xa \<in> space M. X xa = x} * log b (prob {xa \<in> space M. X xa = x}))"
by (rule integrable_count_space)
have d: "distributed M (count_space (X ` space M)) X (\<lambda>x. ereal (if x \<in> X`space M then prob {xa \<in> space M. X xa = x} else 0))"
by (rule distributed_simple_function_superset[OF X]) (auto intro!: arg_cong[where f=prob])
show "distributed M (count_space (X ` space M)) X (\<lambda>x. ereal (prob {xa \<in> space M. X xa = x}))"
by (rule distributed_cong_density[THEN iffD1, OF _ _ _ d]) auto
qed

lemma distributed_transform_AE:
assumes T: "T \<in> measurable P Q" "absolutely_continuous Q (distr P Q T)"
assumes g: "distributed M Q Y g"
shows "AE x in P. 0 \<le> g (T x)"
using g
apply (subst AE_distr_iff[symmetric, OF T(1)])
apply simp
apply (rule absolutely_continuous_AE[OF _ T(2)])
apply simp
done

lemma ac_fst:
assumes "sigma_finite_measure T"
shows "absolutely_continuous S (distr (S \<Otimes>\<^isub>M T) S fst)"
proof -
interpret sigma_finite_measure T by fact
{ fix A assume "A \<in> sets S" "emeasure S A = 0"
moreover then have "fst -` A \<inter> space (S \<Otimes>\<^isub>M T) = A \<times> space T"
by (auto simp: space_pair_measure dest!: sets.sets_into_space)
ultimately have "emeasure (S \<Otimes>\<^isub>M T) (fst -` A \<inter> space (S \<Otimes>\<^isub>M T)) = 0"
then show ?thesis
unfolding absolutely_continuous_def
apply (auto simp: null_sets_distr_iff)
apply (auto simp: null_sets_def intro!: measurable_sets)
done
qed

lemma ac_snd:
assumes "sigma_finite_measure T"
shows "absolutely_continuous T (distr (S \<Otimes>\<^isub>M T) T snd)"
proof -
interpret sigma_finite_measure T by fact
{ fix A assume "A \<in> sets T" "emeasure T A = 0"
moreover then have "snd -` A \<inter> space (S \<Otimes>\<^isub>M T) = space S \<times> A"
by (auto simp: space_pair_measure dest!: sets.sets_into_space)
ultimately have "emeasure (S \<Otimes>\<^isub>M T) (snd -` A \<inter> space (S \<Otimes>\<^isub>M T)) = 0"
then show ?thesis
unfolding absolutely_continuous_def
apply (auto simp: null_sets_distr_iff)
apply (auto simp: null_sets_def intro!: measurable_sets)
done
qed

lemma distributed_integrable:
"distributed M N X f \<Longrightarrow> g \<in> borel_measurable N \<Longrightarrow>
integrable N (\<lambda>x. f x * g x) \<longleftrightarrow> integrable M (\<lambda>x. g (X x))"
by (auto simp: distributed_real_AE
distributed_distr_eq_density[symmetric] integral_density[symmetric] integrable_distr_eq)

lemma distributed_transform_integrable:
assumes Px: "distributed M N X Px"
assumes "distributed M P Y Py"
assumes Y: "Y = (\<lambda>x. T (X x))" and T: "T \<in> measurable N P" and f: "f \<in> borel_measurable P"
shows "integrable P (\<lambda>x. Py x * f x) \<longleftrightarrow> integrable N (\<lambda>x. Px x * f (T x))"
proof -
have "integrable P (\<lambda>x. Py x * f x) \<longleftrightarrow> integrable M (\<lambda>x. f (Y x))"
by (rule distributed_integrable) fact+
also have "\<dots> \<longleftrightarrow> integrable M (\<lambda>x. f (T (X x)))"
using Y by simp
also have "\<dots> \<longleftrightarrow> integrable N (\<lambda>x. Px x * f (T x))"
using measurable_comp[OF T f] Px by (intro distributed_integrable[symmetric]) (auto simp: comp_def)
finally show ?thesis .
qed

lemma integrable_cong_AE_imp: "integrable M g \<Longrightarrow> f \<in> borel_measurable M \<Longrightarrow> (AE x in M. g x = f x) \<Longrightarrow> integrable M f"
using integrable_cong_AE by blast

lemma (in information_space) finite_entropy_integrable:
"finite_entropy S X Px \<Longrightarrow> integrable S (\<lambda>x. Px x * log b (Px x))"
unfolding finite_entropy_def by auto

lemma (in information_space) finite_entropy_distributed:
"finite_entropy S X Px \<Longrightarrow> distributed M S X Px"
unfolding finite_entropy_def by auto

lemma (in information_space) finite_entropy_integrable_transform:
assumes Fx: "finite_entropy S X Px"
assumes Fy: "distributed M T Y Py"
and "X = (\<lambda>x. f (Y x))"
and "f \<in> measurable T S"
shows "integrable T (\<lambda>x. Py x * log b (Px (f x)))"
using assms unfolding finite_entropy_def
using distributed_transform_integrable[of M T Y Py S X Px f "\<lambda>x. log b (Px x)"]
by auto

subsection {* Mutual Information *}

definition (in prob_space)
"mutual_information b S T X Y =
KL_divergence b (distr M S X \<Otimes>\<^isub>M distr M T Y) (distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)))"

lemma (in information_space) mutual_information_indep_vars:
fixes S T X Y
defines "P \<equiv> distr M S X \<Otimes>\<^isub>M distr M T Y"
defines "Q \<equiv> distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))"
shows "indep_var S X T Y \<longleftrightarrow>
(random_variable S X \<and> random_variable T Y \<and>
absolutely_continuous P Q \<and> integrable Q (entropy_density b P Q) \<and>
mutual_information b S T X Y = 0)"
unfolding indep_var_distribution_eq
proof safe
assume rv[measurable]: "random_variable S X" "random_variable T Y"

interpret X: prob_space "distr M S X"
by (rule prob_space_distr) fact
interpret Y: prob_space "distr M T Y"
by (rule prob_space_distr) fact
interpret XY: pair_prob_space "distr M S X" "distr M T Y" by default
interpret P: information_space P b unfolding P_def by default (rule b_gt_1)

interpret Q: prob_space Q unfolding Q_def
by (rule prob_space_distr) simp

{ assume "distr M S X \<Otimes>\<^isub>M distr M T Y = distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))"
then have [simp]: "Q = P"  unfolding Q_def P_def by simp

show ac: "absolutely_continuous P Q" by (simp add: absolutely_continuous_def)
then have ed: "entropy_density b P Q \<in> borel_measurable P"
by (rule P.measurable_entropy_density) simp

have "AE x in P. 1 = RN_deriv P Q x"
proof (rule P.RN_deriv_unique)
show "density P (\<lambda>x. 1) = Q"
unfolding `Q = P` by (intro measure_eqI) (auto simp: emeasure_density)
qed auto
then have ae_0: "AE x in P. entropy_density b P Q x = 0"
by eventually_elim (auto simp: entropy_density_def)
then have "integrable P (entropy_density b P Q) \<longleftrightarrow> integrable Q (\<lambda>x. 0)"
using ed unfolding `Q = P` by (intro integrable_cong_AE) auto
then show "integrable Q (entropy_density b P Q)" by simp

show "mutual_information b S T X Y = 0"
unfolding mutual_information_def KL_divergence_def P_def[symmetric] Q_def[symmetric] `Q = P`
using ae_0 by (simp cong: integral_cong_AE) }

{ assume ac: "absolutely_continuous P Q"
assume int: "integrable Q (entropy_density b P Q)"
assume I_eq_0: "mutual_information b S T X Y = 0"

have eq: "Q = P"
proof (rule P.KL_eq_0_iff_eq_ac[THEN iffD1])
show "prob_space Q" by unfold_locales
show "absolutely_continuous P Q" by fact
show "integrable Q (entropy_density b P Q)" by fact
show "sets Q = sets P" by (simp add: P_def Q_def sets_pair_measure)
show "KL_divergence b P Q = 0"
using I_eq_0 unfolding mutual_information_def by (simp add: P_def Q_def)
qed
then show "distr M S X \<Otimes>\<^isub>M distr M T Y = distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))"
unfolding P_def Q_def .. }
qed

abbreviation (in information_space)
mutual_information_Pow ("\<I>'(_ ; _')") where
"\<I>(X ; Y) \<equiv> mutual_information b (count_space (X`space M)) (count_space (Y`space M)) X Y"

lemma (in information_space)
fixes Pxy :: "'b \<times> 'c \<Rightarrow> real" and Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real"
assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
assumes Fx: "finite_entropy S X Px" and Fy: "finite_entropy T Y Py"
assumes Fxy: "finite_entropy (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
defines "f \<equiv> \<lambda>x. Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x)))"
shows mutual_information_distr': "mutual_information b S T X Y = integral\<^isup>L (S \<Otimes>\<^isub>M T) f" (is "?M = ?R")
and mutual_information_nonneg': "0 \<le> mutual_information b S T X Y"
proof -
have Px: "distributed M S X Px"
using Fx by (auto simp: finite_entropy_def)
have Py: "distributed M T Y Py"
using Fy by (auto simp: finite_entropy_def)
have Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
using Fxy by (auto simp: finite_entropy_def)

have X: "random_variable S X"
using Px by auto
have Y: "random_variable T Y"
using Py by auto
interpret S: sigma_finite_measure S by fact
interpret T: sigma_finite_measure T by fact
interpret ST: pair_sigma_finite S T ..
interpret X: prob_space "distr M S X" using X by (rule prob_space_distr)
interpret Y: prob_space "distr M T Y" using Y by (rule prob_space_distr)
interpret XY: pair_prob_space "distr M S X" "distr M T Y" ..
let ?P = "S \<Otimes>\<^isub>M T"
let ?D = "distr M ?P (\<lambda>x. (X x, Y x))"

{ fix A assume "A \<in> sets S"
with X Y have "emeasure (distr M S X) A = emeasure ?D (A \<times> space T)"
by (auto simp: emeasure_distr measurable_Pair measurable_space
intro!: arg_cong[where f="emeasure M"]) }
note marginal_eq1 = this
{ fix A assume "A \<in> sets T"
with X Y have "emeasure (distr M T Y) A = emeasure ?D (space S \<times> A)"
by (auto simp: emeasure_distr measurable_Pair measurable_space
intro!: arg_cong[where f="emeasure M"]) }
note marginal_eq2 = this

have eq: "(\<lambda>x. ereal (Px (fst x) * Py (snd x))) = (\<lambda>(x, y). ereal (Px x) * ereal (Py y))"
by auto

have distr_eq: "distr M S X \<Otimes>\<^isub>M distr M T Y = density ?P (\<lambda>x. ereal (Px (fst x) * Py (snd x)))"
unfolding Px(1)[THEN distributed_distr_eq_density] Py(1)[THEN distributed_distr_eq_density] eq
proof (subst pair_measure_density)
show "(\<lambda>x. ereal (Px x)) \<in> borel_measurable S" "(\<lambda>y. ereal (Py y)) \<in> borel_measurable T"
"AE x in S. 0 \<le> ereal (Px x)" "AE y in T. 0 \<le> ereal (Py y)"
using Px Py by (auto simp: distributed_def)
show "sigma_finite_measure (density T Py)" unfolding Py(1)[THEN distributed_distr_eq_density, symmetric] ..
qed (fact | simp)+

have M: "?M = KL_divergence b (density ?P (\<lambda>x. ereal (Px (fst x) * Py (snd x)))) (density ?P (\<lambda>x. ereal (Pxy x)))"
unfolding mutual_information_def distr_eq Pxy(1)[THEN distributed_distr_eq_density] ..

from Px Py have f: "(\<lambda>x. Px (fst x) * Py (snd x)) \<in> borel_measurable ?P"
by (intro borel_measurable_times) (auto intro: distributed_real_measurable measurable_fst'' measurable_snd'')
have PxPy_nonneg: "AE x in ?P. 0 \<le> Px (fst x) * Py (snd x)"
proof (rule ST.AE_pair_measure)
show "{x \<in> space ?P. 0 \<le> Px (fst x) * Py (snd x)} \<in> sets ?P"
using f by auto
show "AE x in S. AE y in T. 0 \<le> Px (fst (x, y)) * Py (snd (x, y))"
using Px Py by (auto simp: zero_le_mult_iff dest!: distributed_real_AE)
qed

have "(AE x in ?P. Px (fst x) = 0 \<longrightarrow> Pxy x = 0)"
by (rule subdensity_real[OF measurable_fst Pxy Px]) auto
moreover
have "(AE x in ?P. Py (snd x) = 0 \<longrightarrow> Pxy x = 0)"
by (rule subdensity_real[OF measurable_snd Pxy Py]) auto
ultimately have ac: "AE x in ?P. Px (fst x) * Py (snd x) = 0 \<longrightarrow> Pxy x = 0"
by eventually_elim auto

show "?M = ?R"
unfolding M f_def
using b_gt_1 f PxPy_nonneg Pxy[THEN distributed_real_measurable] Pxy[THEN distributed_real_AE] ac
by (rule ST.KL_density_density)

have X: "X = fst \<circ> (\<lambda>x. (X x, Y x))" and Y: "Y = snd \<circ> (\<lambda>x. (X x, Y x))"
by auto

have "integrable (S \<Otimes>\<^isub>M T) (\<lambda>x. Pxy x * log b (Pxy x) - Pxy x * log b (Px (fst x)) - Pxy x * log b (Py (snd x)))"
using finite_entropy_integrable[OF Fxy]
using finite_entropy_integrable_transform[OF Fx Pxy, of fst]
using finite_entropy_integrable_transform[OF Fy Pxy, of snd]
by simp
moreover have "f \<in> borel_measurable (S \<Otimes>\<^isub>M T)"
unfolding f_def using Px Py Pxy
by (auto intro: distributed_real_measurable measurable_fst'' measurable_snd''
intro!: borel_measurable_times borel_measurable_log borel_measurable_divide)
ultimately have int: "integrable (S \<Otimes>\<^isub>M T) f"
apply (rule integrable_cong_AE_imp)
using
distributed_transform_AE[OF measurable_fst ac_fst, of T, OF T Px]
distributed_transform_AE[OF measurable_snd ac_snd, of _ _ _ _ S, OF T Py]
subdensity_real[OF measurable_fst Pxy Px X]
subdensity_real[OF measurable_snd Pxy Py Y]
distributed_real_AE[OF Pxy]
by eventually_elim
(auto simp: f_def log_divide_eq log_mult_eq field_simps zero_less_mult_iff mult_nonneg_nonneg)

show "0 \<le> ?M" unfolding M
proof (rule ST.KL_density_density_nonneg
[OF b_gt_1 f PxPy_nonneg _ Pxy[THEN distributed_real_measurable] Pxy[THEN distributed_real_AE] _ ac int[unfolded f_def]])
show "prob_space (density (S \<Otimes>\<^isub>M T) (\<lambda>x. ereal (Pxy x))) "
unfolding distributed_distr_eq_density[OF Pxy, symmetric]
using distributed_measurable[OF Pxy] by (rule prob_space_distr)
show "prob_space (density (S \<Otimes>\<^isub>M T) (\<lambda>x. ereal (Px (fst x) * Py (snd x))))"
unfolding distr_eq[symmetric] by unfold_locales
qed
qed

lemma (in information_space)
fixes Pxy :: "'b \<times> 'c \<Rightarrow> real" and Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real"
assumes "sigma_finite_measure S" "sigma_finite_measure T"
assumes Px: "distributed M S X Px" and Py: "distributed M T Y Py"
assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
defines "f \<equiv> \<lambda>x. Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x)))"
shows mutual_information_distr: "mutual_information b S T X Y = integral\<^isup>L (S \<Otimes>\<^isub>M T) f" (is "?M = ?R")
and mutual_information_nonneg: "integrable (S \<Otimes>\<^isub>M T) f \<Longrightarrow> 0 \<le> mutual_information b S T X Y"
proof -
have X: "random_variable S X"
using Px by (auto simp: distributed_def)
have Y: "random_variable T Y"
using Py by (auto simp: distributed_def)
interpret S: sigma_finite_measure S by fact
interpret T: sigma_finite_measure T by fact
interpret ST: pair_sigma_finite S T ..
interpret X: prob_space "distr M S X" using X by (rule prob_space_distr)
interpret Y: prob_space "distr M T Y" using Y by (rule prob_space_distr)
interpret XY: pair_prob_space "distr M S X" "distr M T Y" ..
let ?P = "S \<Otimes>\<^isub>M T"
let ?D = "distr M ?P (\<lambda>x. (X x, Y x))"

{ fix A assume "A \<in> sets S"
with X Y have "emeasure (distr M S X) A = emeasure ?D (A \<times> space T)"
by (auto simp: emeasure_distr measurable_Pair measurable_space
intro!: arg_cong[where f="emeasure M"]) }
note marginal_eq1 = this
{ fix A assume "A \<in> sets T"
with X Y have "emeasure (distr M T Y) A = emeasure ?D (space S \<times> A)"
by (auto simp: emeasure_distr measurable_Pair measurable_space
intro!: arg_cong[where f="emeasure M"]) }
note marginal_eq2 = this

have eq: "(\<lambda>x. ereal (Px (fst x) * Py (snd x))) = (\<lambda>(x, y). ereal (Px x) * ereal (Py y))"
by auto

have distr_eq: "distr M S X \<Otimes>\<^isub>M distr M T Y = density ?P (\<lambda>x. ereal (Px (fst x) * Py (snd x)))"
unfolding Px(1)[THEN distributed_distr_eq_density] Py(1)[THEN distributed_distr_eq_density] eq
proof (subst pair_measure_density)
show "(\<lambda>x. ereal (Px x)) \<in> borel_measurable S" "(\<lambda>y. ereal (Py y)) \<in> borel_measurable T"
"AE x in S. 0 \<le> ereal (Px x)" "AE y in T. 0 \<le> ereal (Py y)"
using Px Py by (auto simp: distributed_def)
show "sigma_finite_measure (density T Py)" unfolding Py(1)[THEN distributed_distr_eq_density, symmetric] ..
qed (fact | simp)+

have M: "?M = KL_divergence b (density ?P (\<lambda>x. ereal (Px (fst x) * Py (snd x)))) (density ?P (\<lambda>x. ereal (Pxy x)))"
unfolding mutual_information_def distr_eq Pxy(1)[THEN distributed_distr_eq_density] ..

from Px Py have f: "(\<lambda>x. Px (fst x) * Py (snd x)) \<in> borel_measurable ?P"
by (intro borel_measurable_times) (auto intro: distributed_real_measurable measurable_fst'' measurable_snd'')
have PxPy_nonneg: "AE x in ?P. 0 \<le> Px (fst x) * Py (snd x)"
proof (rule ST.AE_pair_measure)
show "{x \<in> space ?P. 0 \<le> Px (fst x) * Py (snd x)} \<in> sets ?P"
using f by auto
show "AE x in S. AE y in T. 0 \<le> Px (fst (x, y)) * Py (snd (x, y))"
using Px Py by (auto simp: zero_le_mult_iff dest!: distributed_real_AE)
qed

have "(AE x in ?P. Px (fst x) = 0 \<longrightarrow> Pxy x = 0)"
by (rule subdensity_real[OF measurable_fst Pxy Px]) auto
moreover
have "(AE x in ?P. Py (snd x) = 0 \<longrightarrow> Pxy x = 0)"
by (rule subdensity_real[OF measurable_snd Pxy Py]) auto
ultimately have ac: "AE x in ?P. Px (fst x) * Py (snd x) = 0 \<longrightarrow> Pxy x = 0"
by eventually_elim auto

show "?M = ?R"
unfolding M f_def
using b_gt_1 f PxPy_nonneg Pxy[THEN distributed_real_measurable] Pxy[THEN distributed_real_AE] ac
by (rule ST.KL_density_density)

assume int: "integrable (S \<Otimes>\<^isub>M T) f"
show "0 \<le> ?M" unfolding M
proof (rule ST.KL_density_density_nonneg
[OF b_gt_1 f PxPy_nonneg _ Pxy[THEN distributed_real_measurable] Pxy[THEN distributed_real_AE] _ ac int[unfolded f_def]])
show "prob_space (density (S \<Otimes>\<^isub>M T) (\<lambda>x. ereal (Pxy x))) "
unfolding distributed_distr_eq_density[OF Pxy, symmetric]
using distributed_measurable[OF Pxy] by (rule prob_space_distr)
show "prob_space (density (S \<Otimes>\<^isub>M T) (\<lambda>x. ereal (Px (fst x) * Py (snd x))))"
unfolding distr_eq[symmetric] by unfold_locales
qed
qed

lemma (in information_space)
fixes Pxy :: "'b \<times> 'c \<Rightarrow> real" and Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real"
assumes "sigma_finite_measure S" "sigma_finite_measure T"
assumes Px: "distributed M S X Px" and Py: "distributed M T Y Py"
assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
assumes ae: "AE x in S. AE y in T. Pxy (x, y) = Px x * Py y"
shows mutual_information_eq_0: "mutual_information b S T X Y = 0"
proof -
interpret S: sigma_finite_measure S by fact
interpret T: sigma_finite_measure T by fact
interpret ST: pair_sigma_finite S T ..

have "AE x in S \<Otimes>\<^isub>M T. Px (fst x) = 0 \<longrightarrow> Pxy x = 0"
by (rule subdensity_real[OF measurable_fst Pxy Px]) auto
moreover
have "AE x in S \<Otimes>\<^isub>M T. Py (snd x) = 0 \<longrightarrow> Pxy x = 0"
by (rule subdensity_real[OF measurable_snd Pxy Py]) auto
moreover
have "AE x in S \<Otimes>\<^isub>M T. Pxy x = Px (fst x) * Py (snd x)"
using distributed_real_measurable[OF Px] distributed_real_measurable[OF Py] distributed_real_measurable[OF Pxy]
by (intro ST.AE_pair_measure) (auto simp: ae intro!: measurable_snd'' measurable_fst'')
ultimately have "AE x in S \<Otimes>\<^isub>M T. Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x))) = 0"
by eventually_elim simp
then have "(\<integral>x. Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x))) \<partial>(S \<Otimes>\<^isub>M T)) = (\<integral>x. 0 \<partial>(S \<Otimes>\<^isub>M T))"
by (rule integral_cong_AE)
then show ?thesis
by (subst mutual_information_distr[OF assms(1-5)]) simp
qed

lemma (in information_space) mutual_information_simple_distributed:
assumes X: "simple_distributed M X Px" and Y: "simple_distributed M Y Py"
assumes XY: "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy"
shows "\<I>(X ; Y) = (\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x))`space M. Pxy (x, y) * log b (Pxy (x, y) / (Px x * Py y)))"
proof (subst mutual_information_distr[OF _ _ simple_distributed[OF X] simple_distributed[OF Y] simple_distributed_joint[OF XY]])
note fin = simple_distributed_joint_finite[OF XY, simp]
show "sigma_finite_measure (count_space (X ` space M))"
show "sigma_finite_measure (count_space (Y ` space M))"
let ?Pxy = "\<lambda>x. (if x \<in> (\<lambda>x. (X x, Y x)) ` space M then Pxy x else 0)"
let ?f = "\<lambda>x. ?Pxy x * log b (?Pxy x / (Px (fst x) * Py (snd x)))"
have "\<And>x. ?f x = (if x \<in> (\<lambda>x. (X x, Y x)) ` space M then Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x))) else 0)"
by auto
with fin show "(\<integral> x. ?f x \<partial>(count_space (X ` space M) \<Otimes>\<^isub>M count_space (Y ` space M))) =
(\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` space M. Pxy (x, y) * log b (Pxy (x, y) / (Px x * Py y)))"
by (auto simp add: pair_measure_count_space lebesgue_integral_count_space_finite setsum_cases split_beta'
intro!: setsum_cong)
qed

lemma (in information_space)
fixes Pxy :: "'b \<times> 'c \<Rightarrow> real" and Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real"
assumes Px: "simple_distributed M X Px" and Py: "simple_distributed M Y Py"
assumes Pxy: "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy"
assumes ae: "\<forall>x\<in>space M. Pxy (X x, Y x) = Px (X x) * Py (Y x)"
shows mutual_information_eq_0_simple: "\<I>(X ; Y) = 0"
proof (subst mutual_information_simple_distributed[OF Px Py Pxy])
have "(\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` space M. Pxy (x, y) * log b (Pxy (x, y) / (Px x * Py y))) =
(\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` space M. 0)"
by (intro setsum_cong) (auto simp: ae)
then show "(\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` space M.
Pxy (x, y) * log b (Pxy (x, y) / (Px x * Py y))) = 0" by simp
qed

subsection {* Entropy *}

definition (in prob_space) entropy :: "real \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> real" where
"entropy b S X = - KL_divergence b S (distr M S X)"

abbreviation (in information_space)
entropy_Pow ("\<H>'(_')") where
"\<H>(X) \<equiv> entropy b (count_space (X`space M)) X"

lemma (in prob_space) distributed_RN_deriv:
assumes X: "distributed M S X Px"
shows "AE x in S. RN_deriv S (density S Px) x = Px x"
proof -
note D = distributed_measurable[OF X] distributed_borel_measurable[OF X] distributed_AE[OF X]
interpret X: prob_space "distr M S X"
using D(1) by (rule prob_space_distr)

have sf: "sigma_finite_measure (distr M S X)" by default
show ?thesis
using D
apply (subst eq_commute)
apply (intro RN_deriv_unique_sigma_finite)
apply (auto intro: divide_nonneg_nonneg measure_nonneg
simp: distributed_distr_eq_density[symmetric, OF X] sf)
done
qed

lemma (in information_space)
fixes X :: "'a \<Rightarrow> 'b"
assumes X: "distributed M MX X f"
shows entropy_distr: "entropy b MX X = - (\<integral>x. f x * log b (f x) \<partial>MX)" (is ?eq)
proof -
note D = distributed_measurable[OF X] distributed_borel_measurable[OF X] distributed_AE[OF X]
note ae = distributed_RN_deriv[OF X]

have ae_eq: "AE x in distr M MX X. log b (real (RN_deriv MX (distr M MX X) x)) =
log b (f x)"
unfolding distributed_distr_eq_density[OF X]
apply (subst AE_density)
using D apply simp
using ae apply eventually_elim
apply auto
done

have int_eq: "- (\<integral> x. log b (f x) \<partial>distr M MX X) = - (\<integral> x. f x * log b (f x) \<partial>MX)"
unfolding distributed_distr_eq_density[OF X]
using D
by (subst integral_density)
(auto simp: borel_measurable_ereal_iff)

show ?eq
unfolding entropy_def KL_divergence_def entropy_density_def comp_def
apply (subst integral_cong_AE)
apply (rule ae_eq)
apply (rule int_eq)
done
qed

lemma (in prob_space) distributed_imp_emeasure_nonzero:
assumes X: "distributed M MX X Px"
shows "emeasure MX {x \<in> space MX. Px x \<noteq> 0} \<noteq> 0"
proof
note Px = distributed_borel_measurable[OF X] distributed_AE[OF X]
interpret X: prob_space "distr M MX X"
using distributed_measurable[OF X] by (rule prob_space_distr)

assume "emeasure MX {x \<in> space MX. Px x \<noteq> 0} = 0"
with Px have "AE x in MX. Px x = 0"
by (intro AE_I[OF subset_refl]) (auto simp: borel_measurable_ereal_iff)
moreover
from X.emeasure_space_1 have "(\<integral>\<^isup>+x. Px x \<partial>MX) = 1"
unfolding distributed_distr_eq_density[OF X] using Px
by (subst (asm) emeasure_density)
(auto simp: borel_measurable_ereal_iff intro!: integral_cong cong: positive_integral_cong)
ultimately show False
qed

lemma (in information_space) entropy_le:
fixes Px :: "'b \<Rightarrow> real" and MX :: "'b measure"
assumes X: "distributed M MX X Px"
and fin: "emeasure MX {x \<in> space MX. Px x \<noteq> 0} \<noteq> \<infinity>"
and int: "integrable MX (\<lambda>x. - Px x * log b (Px x))"
shows "entropy b MX X \<le> log b (measure MX {x \<in> space MX. Px x \<noteq> 0})"
proof -
note Px = distributed_borel_measurable[OF X] distributed_AE[OF X]
interpret X: prob_space "distr M MX X"
using distributed_measurable[OF X] by (rule prob_space_distr)

have " - log b (measure MX {x \<in> space MX. Px x \<noteq> 0}) =
- log b (\<integral> x. indicator {x \<in> space MX. Px x \<noteq> 0} x \<partial>MX)"
using Px fin
by (subst integral_indicator) (auto simp: measure_def borel_measurable_ereal_iff)
also have "- log b (\<integral> x. indicator {x \<in> space MX. Px x \<noteq> 0} x \<partial>MX) = - log b (\<integral> x. 1 / Px x \<partial>distr M MX X)"
unfolding distributed_distr_eq_density[OF X] using Px
apply (intro arg_cong[where f="log b"] arg_cong[where f=uminus])
by (subst integral_density) (auto simp: borel_measurable_ereal_iff intro!: integral_cong)
also have "\<dots> \<le> (\<integral> x. - log b (1 / Px x) \<partial>distr M MX X)"
proof (rule X.jensens_inequality[of "\<lambda>x. 1 / Px x" "{0<..}" 0 1 "\<lambda>x. - log b x"])
show "AE x in distr M MX X. 1 / Px x \<in> {0<..}"
unfolding distributed_distr_eq_density[OF X]
using Px by (auto simp: AE_density)
have [simp]: "\<And>x. x \<in> space MX \<Longrightarrow> ereal (if Px x = 0 then 0 else 1) = indicator {x \<in> space MX. Px x \<noteq> 0} x"
by (auto simp: one_ereal_def)
have "(\<integral>\<^isup>+ x. max 0 (ereal (- (if Px x = 0 then 0 else 1))) \<partial>MX) = (\<integral>\<^isup>+ x. 0 \<partial>MX)"
by (intro positive_integral_cong) (auto split: split_max)
then show "integrable (distr M MX X) (\<lambda>x. 1 / Px x)"
unfolding distributed_distr_eq_density[OF X] using Px
by (auto simp: positive_integral_density integrable_def borel_measurable_ereal_iff fin positive_integral_max_0
cong: positive_integral_cong)
have "integrable MX (\<lambda>x. Px x * log b (1 / Px x)) =
integrable MX (\<lambda>x. - Px x * log b (Px x))"
using Px
by (intro integrable_cong_AE)
(auto simp: borel_measurable_ereal_iff log_divide_eq
intro!: measurable_If)
then show "integrable (distr M MX X) (\<lambda>x. - log b (1 / Px x))"
unfolding distributed_distr_eq_density[OF X]
using Px int
by (subst integral_density) (auto simp: borel_measurable_ereal_iff)
qed (auto simp: minus_log_convex[OF b_gt_1])
also have "\<dots> = (\<integral> x. log b (Px x) \<partial>distr M MX X)"
unfolding distributed_distr_eq_density[OF X] using Px
by (intro integral_cong_AE) (auto simp: AE_density log_divide_eq)
also have "\<dots> = - entropy b MX X"
unfolding distributed_distr_eq_density[OF X] using Px
by (subst entropy_distr[OF X]) (auto simp: borel_measurable_ereal_iff integral_density)
finally show ?thesis
by simp
qed

lemma (in information_space) entropy_le_space:
fixes Px :: "'b \<Rightarrow> real" and MX :: "'b measure"
assumes X: "distributed M MX X Px"
and fin: "finite_measure MX"
and int: "integrable MX (\<lambda>x. - Px x * log b (Px x))"
shows "entropy b MX X \<le> log b (measure MX (space MX))"
proof -
note Px = distributed_borel_measurable[OF X] distributed_AE[OF X]
interpret finite_measure MX by fact
have "entropy b MX X \<le> log b (measure MX {x \<in> space MX. Px x \<noteq> 0})"
using int X by (intro entropy_le) auto
also have "\<dots> \<le> log b (measure MX (space MX))"
using Px distributed_imp_emeasure_nonzero[OF X]
by (intro log_le)
(auto intro!: borel_measurable_ereal_iff finite_measure_mono b_gt_1
less_le[THEN iffD2] measure_nonneg simp: emeasure_eq_measure)
finally show ?thesis .
qed

lemma (in information_space) entropy_uniform:
assumes X: "distributed M MX X (\<lambda>x. indicator A x / measure MX A)" (is "distributed _ _ _ ?f")
shows "entropy b MX X = log b (measure MX A)"
proof (subst entropy_distr[OF X])
have [simp]: "emeasure MX A \<noteq> \<infinity>"
using uniform_distributed_params[OF X] by (auto simp add: measure_def)
have eq: "(\<integral> x. indicator A x / measure MX A * log b (indicator A x / measure MX A) \<partial>MX) =
(\<integral> x. (- log b (measure MX A) / measure MX A) * indicator A x \<partial>MX)"
using measure_nonneg[of MX A] uniform_distributed_params[OF X]
by (auto intro!: integral_cong split: split_indicator simp: log_divide_eq)
show "- (\<integral> x. indicator A x / measure MX A * log b (indicator A x / measure MX A) \<partial>MX) =
log b (measure MX A)"
unfolding eq using uniform_distributed_params[OF X]
by (subst lebesgue_integral_cmult) (auto simp: measure_def)
qed

lemma (in information_space) entropy_simple_distributed:
"simple_distributed M X f \<Longrightarrow> \<H>(X) = - (\<Sum>x\<in>X`space M. f x * log b (f x))"
by (subst entropy_distr[OF simple_distributed])

lemma (in information_space) entropy_le_card_not_0:
assumes X: "simple_distributed M X f"
shows "\<H>(X) \<le> log b (card (X ` space M \<inter> {x. f x \<noteq> 0}))"
proof -
let ?X = "count_space (X`space M)"
have "\<H>(X) \<le> log b (measure ?X {x \<in> space ?X. f x \<noteq> 0})"
by (rule entropy_le[OF simple_distributed[OF X]])
(simp_all add: simple_distributed_finite[OF X] subset_eq integrable_count_space emeasure_count_space)
also have "measure ?X {x \<in> space ?X. f x \<noteq> 0} = card (X ` space M \<inter> {x. f x \<noteq> 0})"
by (simp_all add: simple_distributed_finite[OF X] subset_eq emeasure_count_space measure_def Int_def)
finally show ?thesis .
qed

lemma (in information_space) entropy_le_card:
assumes X: "simple_distributed M X f"
shows "\<H>(X) \<le> log b (real (card (X ` space M)))"
proof -
let ?X = "count_space (X`space M)"
have "\<H>(X) \<le> log b (measure ?X (space ?X))"
by (rule entropy_le_space[OF simple_distributed[OF X]])
(simp_all add: simple_distributed_finite[OF X] subset_eq integrable_count_space emeasure_count_space finite_measure_count_space)
also have "measure ?X (space ?X) = card (X ` space M)"
by (simp_all add: simple_distributed_finite[OF X] subset_eq emeasure_count_space measure_def)
finally show ?thesis .
qed

subsection {* Conditional Mutual Information *}

definition (in prob_space)
"conditional_mutual_information b MX MY MZ X Y Z \<equiv>
mutual_information b MX (MY \<Otimes>\<^isub>M MZ) X (\<lambda>x. (Y x, Z x)) -
mutual_information b MX MZ X Z"

abbreviation (in information_space)
conditional_mutual_information_Pow ("\<I>'( _ ; _ | _ ')") where
"\<I>(X ; Y | Z) \<equiv> conditional_mutual_information b
(count_space (X ` space M)) (count_space (Y ` space M)) (count_space (Z ` space M)) X Y Z"

lemma (in information_space)
assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T" and P: "sigma_finite_measure P"
assumes Px[measurable]: "distributed M S X Px"
assumes Pz[measurable]: "distributed M P Z Pz"
assumes Pyz[measurable]: "distributed M (T \<Otimes>\<^isub>M P) (\<lambda>x. (Y x, Z x)) Pyz"
assumes Pxz[measurable]: "distributed M (S \<Otimes>\<^isub>M P) (\<lambda>x. (X x, Z x)) Pxz"
assumes Pxyz[measurable]: "distributed M (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) (\<lambda>x. (X x, Y x, Z x)) Pxyz"
assumes I1: "integrable (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) (\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Px x * Pyz (y, z))))"
assumes I2: "integrable (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) (\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxz (x, z) / (Px x * Pz z)))"
shows conditional_mutual_information_generic_eq: "conditional_mutual_information b S T P X Y Z
= (\<integral>(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Pxz (x, z) * (Pyz (y,z) / Pz z))) \<partial>(S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P))" (is "?eq")
and conditional_mutual_information_generic_nonneg: "0 \<le> conditional_mutual_information b S T P X Y Z" (is "?nonneg")
proof -
interpret S: sigma_finite_measure S by fact
interpret T: sigma_finite_measure T by fact
interpret P: sigma_finite_measure P by fact
interpret TP: pair_sigma_finite T P ..
interpret SP: pair_sigma_finite S P ..
interpret ST: pair_sigma_finite S T ..
interpret SPT: pair_sigma_finite "S \<Otimes>\<^isub>M P" T ..
interpret STP: pair_sigma_finite S "T \<Otimes>\<^isub>M P" ..
interpret TPS: pair_sigma_finite "T \<Otimes>\<^isub>M P" S ..
have TP: "sigma_finite_measure (T \<Otimes>\<^isub>M P)" ..
have SP: "sigma_finite_measure (S \<Otimes>\<^isub>M P)" ..
have YZ: "random_variable (T \<Otimes>\<^isub>M P) (\<lambda>x. (Y x, Z x))"
using Pyz by (simp add: distributed_measurable)

from Pxz Pxyz have distr_eq: "distr M (S \<Otimes>\<^isub>M P) (\<lambda>x. (X x, Z x)) =
distr (distr M (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) (\<lambda>x. (X x, Y x, Z x))) (S \<Otimes>\<^isub>M P) (\<lambda>(x, y, z). (x, z))"

have "mutual_information b S P X Z =
(\<integral>x. Pxz x * log b (Pxz x / (Px (fst x) * Pz (snd x))) \<partial>(S \<Otimes>\<^isub>M P))"
by (rule mutual_information_distr[OF S P Px Pz Pxz])
also have "\<dots> = (\<integral>(x,y,z). Pxyz (x,y,z) * log b (Pxz (x,z) / (Px x * Pz z)) \<partial>(S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P))"
using b_gt_1 Pxz Px Pz
by (subst distributed_transform_integral[OF Pxyz Pxz, where T="\<lambda>(x, y, z). (x, z)"]) (auto simp: split_beta')
finally have mi_eq:
"mutual_information b S P X Z = (\<integral>(x,y,z). Pxyz (x,y,z) * log b (Pxz (x,z) / (Px x * Pz z)) \<partial>(S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P))" .

have ae1: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. Px (fst x) = 0 \<longrightarrow> Pxyz x = 0"
by (intro subdensity_real[of fst, OF _ Pxyz Px]) auto
moreover have ae2: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. Pz (snd (snd x)) = 0 \<longrightarrow> Pxyz x = 0"
by (intro subdensity_real[of "\<lambda>x. snd (snd x)", OF _ Pxyz Pz]) auto
moreover have ae3: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. Pxz (fst x, snd (snd x)) = 0 \<longrightarrow> Pxyz x = 0"
by (intro subdensity_real[of "\<lambda>x. (fst x, snd (snd x))", OF _ Pxyz Pxz]) auto
moreover have ae4: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. Pyz (snd x) = 0 \<longrightarrow> Pxyz x = 0"
by (intro subdensity_real[of snd, OF _ Pxyz Pyz]) auto
moreover have ae5: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. 0 \<le> Px (fst x)"
using Px by (intro STP.AE_pair_measure) (auto simp: comp_def dest: distributed_real_AE)
moreover have ae6: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. 0 \<le> Pyz (snd x)"
using Pyz by (intro STP.AE_pair_measure) (auto simp: comp_def dest: distributed_real_AE)
moreover have ae7: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. 0 \<le> Pz (snd (snd x))"
using Pz Pz[THEN distributed_real_measurable]
by (auto intro!: TP.AE_pair_measure STP.AE_pair_measure AE_I2[of S] dest: distributed_real_AE)
moreover have ae8: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. 0 \<le> Pxz (fst x, snd (snd x))"
using Pxz[THEN distributed_real_AE, THEN SP.AE_pair]
by (auto intro!: TP.AE_pair_measure STP.AE_pair_measure)
moreover note Pxyz[THEN distributed_real_AE]
ultimately have ae: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P.
Pxyz x * log b (Pxyz x / (Px (fst x) * Pyz (snd x))) -
Pxyz x * log b (Pxz (fst x, snd (snd x)) / (Px (fst x) * Pz (snd (snd x)))) =
Pxyz x * log b (Pxyz x * Pz (snd (snd x)) / (Pxz (fst x, snd (snd x)) * Pyz (snd x))) "
proof eventually_elim
case (goal1 x)
show ?case
proof cases
assume "Pxyz x \<noteq> 0"
with goal1 have "0 < Px (fst x)" "0 < Pz (snd (snd x))" "0 < Pxz (fst x, snd (snd x))" "0 < Pyz (snd x)" "0 < Pxyz x"
by auto
then show ?thesis
using b_gt_1 by (simp add: log_simps mult_pos_pos less_imp_le field_simps)
qed simp
qed
with I1 I2 show ?eq
unfolding conditional_mutual_information_def
apply (subst mi_eq)
apply (subst mutual_information_distr[OF S TP Px Pyz Pxyz])
apply (subst integral_diff(2)[symmetric])
apply (auto intro!: integral_cong_AE simp: split_beta' simp del: integral_diff)
done

let ?P = "density (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) Pxyz"
interpret P: prob_space ?P
unfolding distributed_distr_eq_density[OF Pxyz, symmetric]
by (rule prob_space_distr) simp

let ?Q = "density (T \<Otimes>\<^isub>M P) Pyz"
interpret Q: prob_space ?Q
unfolding distributed_distr_eq_density[OF Pyz, symmetric]
by (rule prob_space_distr) simp

let ?f = "\<lambda>(x, y, z). Pxz (x, z) * (Pyz (y, z) / Pz z) / Pxyz (x, y, z)"

from subdensity_real[of snd, OF _ Pyz Pz]
have aeX1: "AE x in T \<Otimes>\<^isub>M P. Pz (snd x) = 0 \<longrightarrow> Pyz x = 0" by (auto simp: comp_def)
have aeX2: "AE x in T \<Otimes>\<^isub>M P. 0 \<le> Pz (snd x)"
using Pz by (intro TP.AE_pair_measure) (auto simp: comp_def dest: distributed_real_AE)

have aeX3: "AE y in T \<Otimes>\<^isub>M P. (\<integral>\<^isup>+ x. ereal (Pxz (x, snd y)) \<partial>S) = ereal (Pz (snd y))"
using Pz distributed_marginal_eq_joint2[OF P S Pz Pxz]
by (intro TP.AE_pair_measure) (auto dest: distributed_real_AE)

have "(\<integral>\<^isup>+ x. ?f x \<partial>?P) \<le> (\<integral>\<^isup>+ (x, y, z). Pxz (x, z) * (Pyz (y, z) / Pz z) \<partial>(S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P))"
apply (subst positive_integral_density)
apply simp
apply (rule distributed_AE[OF Pxyz])
apply auto []
apply (rule positive_integral_mono_AE)
using ae5 ae6 ae7 ae8
apply eventually_elim
apply (auto intro!: divide_nonneg_nonneg mult_nonneg_nonneg)
done
also have "\<dots> = (\<integral>\<^isup>+(y, z). \<integral>\<^isup>+ x. ereal (Pxz (x, z)) * ereal (Pyz (y, z) / Pz z) \<partial>S \<partial>T \<Otimes>\<^isub>M P)"
by (subst STP.positive_integral_snd_measurable[symmetric]) (auto simp add: split_beta')
also have "\<dots> = (\<integral>\<^isup>+x. ereal (Pyz x) * 1 \<partial>T \<Otimes>\<^isub>M P)"
apply (rule positive_integral_cong_AE)
using aeX1 aeX2 aeX3 distributed_AE[OF Pyz] AE_space
apply eventually_elim
proof (case_tac x, simp del: times_ereal.simps add: space_pair_measure)
fix a b assume "Pz b = 0 \<longrightarrow> Pyz (a, b) = 0" "0 \<le> Pz b" "a \<in> space T \<and> b \<in> space P"
"(\<integral>\<^isup>+ x. ereal (Pxz (x, b)) \<partial>S) = ereal (Pz b)" "0 \<le> Pyz (a, b)"
then show "(\<integral>\<^isup>+ x. ereal (Pxz (x, b)) * ereal (Pyz (a, b) / Pz b) \<partial>S) = ereal (Pyz (a, b))"
by (subst positive_integral_multc)
(auto intro!: divide_nonneg_nonneg split: prod.split)
qed
also have "\<dots> = 1"
using Q.emeasure_space_1 distributed_AE[OF Pyz] distributed_distr_eq_density[OF Pyz]
by (subst positive_integral_density[symmetric]) auto
finally have le1: "(\<integral>\<^isup>+ x. ?f x \<partial>?P) \<le> 1" .
also have "\<dots> < \<infinity>" by simp
finally have fin: "(\<integral>\<^isup>+ x. ?f x \<partial>?P) \<noteq> \<infinity>" by simp

have pos: "(\<integral>\<^isup>+ x. ?f x \<partial>?P) \<noteq> 0"
apply (subst positive_integral_density)
apply simp
apply (rule distributed_AE[OF Pxyz])
apply auto []
proof
let ?g = "\<lambda>x. ereal (if Pxyz x = 0 then 0 else Pxz (fst x, snd (snd x)) * Pyz (snd x) / Pz (snd (snd x)))"
assume "(\<integral>\<^isup>+ x. ?g x \<partial>(S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P)) = 0"
then have "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. ?g x \<le> 0"
by (intro positive_integral_0_iff_AE[THEN iffD1]) auto
then have "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. Pxyz x = 0"
using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 Pxyz[THEN distributed_real_AE]
by eventually_elim (auto split: split_if_asm simp: mult_le_0_iff divide_le_0_iff)
then have "(\<integral>\<^isup>+ x. ereal (Pxyz x) \<partial>S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) = 0"
by (subst positive_integral_cong_AE[of _ "\<lambda>x. 0"]) auto
with P.emeasure_space_1 show False
by (subst (asm) emeasure_density) (auto cong: positive_integral_cong)
qed

have neg: "(\<integral>\<^isup>+ x. - ?f x \<partial>?P) = 0"
apply (rule positive_integral_0_iff_AE[THEN iffD2])
apply simp
apply (subst AE_density)
apply simp
using ae5 ae6 ae7 ae8
apply eventually_elim
apply (auto intro!: mult_nonneg_nonneg divide_nonneg_nonneg)
done

have I3: "integrable (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) (\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Pxz (x, z) * (Pyz (y,z) / Pz z))))"
apply (rule integrable_cong_AE[THEN iffD1, OF _ _ _ integral_diff(1)[OF I1 I2]])
using ae
apply (auto simp: split_beta')
done

have "- log b 1 \<le> - log b (integral\<^isup>L ?P ?f)"
proof (intro le_imp_neg_le log_le[OF b_gt_1])
show "0 < integral\<^isup>L ?P ?f"
using neg pos fin positive_integral_positive[of ?P ?f]
by (cases "(\<integral>\<^isup>+ x. ?f x \<partial>?P)") (auto simp add: lebesgue_integral_def less_le split_beta')
show "integral\<^isup>L ?P ?f \<le> 1"
using neg le1 fin positive_integral_positive[of ?P ?f]
by (cases "(\<integral>\<^isup>+ x. ?f x \<partial>?P)") (auto simp add: lebesgue_integral_def split_beta' one_ereal_def)
qed
also have "- log b (integral\<^isup>L ?P ?f) \<le> (\<integral> x. - log b (?f x) \<partial>?P)"
proof (rule P.jensens_inequality[where a=0 and b=1 and I="{0<..}"])
show "AE x in ?P. ?f x \<in> {0<..}"
unfolding AE_density[OF distributed_borel_measurable[OF Pxyz]]
using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 Pxyz[THEN distributed_real_AE]
by eventually_elim (auto simp: divide_pos_pos mult_pos_pos)
show "integrable ?P ?f"
unfolding integrable_def
using fin neg by (auto simp: split_beta')
show "integrable ?P (\<lambda>x. - log b (?f x))"
apply (subst integral_density)
apply simp
apply (auto intro!: distributed_real_AE[OF Pxyz]) []
apply simp
apply (rule integrable_cong_AE[THEN iffD1, OF _ _ _ I3])
apply simp
apply simp
using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 Pxyz[THEN distributed_real_AE]
apply eventually_elim
apply (auto simp: log_divide_eq log_mult_eq zero_le_mult_iff zero_less_mult_iff zero_less_divide_iff field_simps)
done
qed (auto simp: b_gt_1 minus_log_convex)
also have "\<dots> = conditional_mutual_information b S T P X Y Z"
unfolding `?eq`
apply (subst integral_density)
apply simp
apply (auto intro!: distributed_real_AE[OF Pxyz]) []
apply simp
apply (intro integral_cong_AE)
using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 Pxyz[THEN distributed_real_AE]
apply eventually_elim
apply (auto simp: log_divide_eq zero_less_mult_iff zero_less_divide_iff field_simps)
done
finally show ?nonneg
by simp
qed

lemma (in information_space)
fixes Px :: "_ \<Rightarrow> real"
assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T" and P: "sigma_finite_measure P"
assumes Fx: "finite_entropy S X Px"
assumes Fz: "finite_entropy P Z Pz"
assumes Fyz: "finite_entropy (T \<Otimes>\<^isub>M P) (\<lambda>x. (Y x, Z x)) Pyz"
assumes Fxz: "finite_entropy (S \<Otimes>\<^isub>M P) (\<lambda>x. (X x, Z x)) Pxz"
assumes Fxyz: "finite_entropy (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) (\<lambda>x. (X x, Y x, Z x)) Pxyz"
shows conditional_mutual_information_generic_eq': "conditional_mutual_information b S T P X Y Z
= (\<integral>(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Pxz (x, z) * (Pyz (y,z) / Pz z))) \<partial>(S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P))" (is "?eq")
and conditional_mutual_information_generic_nonneg': "0 \<le> conditional_mutual_information b S T P X Y Z" (is "?nonneg")
proof -
note Px = Fx[THEN finite_entropy_distributed, measurable]
note Pz = Fz[THEN finite_entropy_distributed, measurable]
note Pyz = Fyz[THEN finite_entropy_distributed, measurable]
note Pxz = Fxz[THEN finite_entropy_distributed, measurable]
note Pxyz = Fxyz[THEN finite_entropy_distributed, measurable]

interpret S: sigma_finite_measure S by fact
interpret T: sigma_finite_measure T by fact
interpret P: sigma_finite_measure P by fact
interpret TP: pair_sigma_finite T P ..
interpret SP: pair_sigma_finite S P ..
interpret ST: pair_sigma_finite S T ..
interpret SPT: pair_sigma_finite "S \<Otimes>\<^isub>M P" T ..
interpret STP: pair_sigma_finite S "T \<Otimes>\<^isub>M P" ..
interpret TPS: pair_sigma_finite "T \<Otimes>\<^isub>M P" S ..
have TP: "sigma_finite_measure (T \<Otimes>\<^isub>M P)" ..
have SP: "sigma_finite_measure (S \<Otimes>\<^isub>M P)" ..

from Pxz Pxyz have distr_eq: "distr M (S \<Otimes>\<^isub>M P) (\<lambda>x. (X x, Z x)) =
distr (distr M (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) (\<lambda>x. (X x, Y x, Z x))) (S \<Otimes>\<^isub>M P) (\<lambda>(x, y, z). (x, z))"

have "mutual_information b S P X Z =
(\<integral>x. Pxz x * log b (Pxz x / (Px (fst x) * Pz (snd x))) \<partial>(S \<Otimes>\<^isub>M P))"
by (rule mutual_information_distr[OF S P Px Pz Pxz])
also have "\<dots> = (\<integral>(x,y,z). Pxyz (x,y,z) * log b (Pxz (x,z) / (Px x * Pz z)) \<partial>(S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P))"
using b_gt_1 Pxz Px Pz
by (subst distributed_transform_integral[OF Pxyz Pxz, where T="\<lambda>(x, y, z). (x, z)"])
(auto simp: split_beta')
finally have mi_eq:
"mutual_information b S P X Z = (\<integral>(x,y,z). Pxyz (x,y,z) * log b (Pxz (x,z) / (Px x * Pz z)) \<partial>(S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P))" .

have ae1: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. Px (fst x) = 0 \<longrightarrow> Pxyz x = 0"
by (intro subdensity_real[of fst, OF _ Pxyz Px]) auto
moreover have ae2: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. Pz (snd (snd x)) = 0 \<longrightarrow> Pxyz x = 0"
by (intro subdensity_real[of "\<lambda>x. snd (snd x)", OF _ Pxyz Pz]) auto
moreover have ae3: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. Pxz (fst x, snd (snd x)) = 0 \<longrightarrow> Pxyz x = 0"
by (intro subdensity_real[of "\<lambda>x. (fst x, snd (snd x))", OF _ Pxyz Pxz]) auto
moreover have ae4: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. Pyz (snd x) = 0 \<longrightarrow> Pxyz x = 0"
by (intro subdensity_real[of snd, OF _ Pxyz Pyz]) auto
moreover have ae5: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. 0 \<le> Px (fst x)"
using Px by (intro STP.AE_pair_measure) (auto dest: distributed_real_AE)
moreover have ae6: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. 0 \<le> Pyz (snd x)"
using Pyz by (intro STP.AE_pair_measure) (auto dest: distributed_real_AE)
moreover have ae7: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. 0 \<le> Pz (snd (snd x))"
using Pz Pz[THEN distributed_real_measurable] by (auto intro!: TP.AE_pair_measure STP.AE_pair_measure AE_I2[of S] dest: distributed_real_AE)
moreover have ae8: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. 0 \<le> Pxz (fst x, snd (snd x))"
using Pxz[THEN distributed_real_AE, THEN SP.AE_pair]
by (auto intro!: TP.AE_pair_measure STP.AE_pair_measure simp: comp_def)
moreover note ae9 = Pxyz[THEN distributed_real_AE]
ultimately have ae: "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P.
Pxyz x * log b (Pxyz x / (Px (fst x) * Pyz (snd x))) -
Pxyz x * log b (Pxz (fst x, snd (snd x)) / (Px (fst x) * Pz (snd (snd x)))) =
Pxyz x * log b (Pxyz x * Pz (snd (snd x)) / (Pxz (fst x, snd (snd x)) * Pyz (snd x))) "
proof eventually_elim
case (goal1 x)
show ?case
proof cases
assume "Pxyz x \<noteq> 0"
with goal1 have "0 < Px (fst x)" "0 < Pz (snd (snd x))" "0 < Pxz (fst x, snd (snd x))" "0 < Pyz (snd x)" "0 < Pxyz x"
by auto
then show ?thesis
using b_gt_1 by (simp add: log_simps mult_pos_pos less_imp_le field_simps)
qed simp
qed

have "integrable (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P)
(\<lambda>x. Pxyz x * log b (Pxyz x) - Pxyz x * log b (Px (fst x)) - Pxyz x * log b (Pyz (snd x)))"
using finite_entropy_integrable[OF Fxyz]
using finite_entropy_integrable_transform[OF Fx Pxyz, of fst]
using finite_entropy_integrable_transform[OF Fyz Pxyz, of snd]
by simp
moreover have "(\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Px x * Pyz (y, z)))) \<in> borel_measurable (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P)"
using Pxyz Px Pyz by simp
ultimately have I1: "integrable (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) (\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Px x * Pyz (y, z))))"
apply (rule integrable_cong_AE_imp)
using ae1 ae4 ae5 ae6 ae9
by eventually_elim
(auto simp: log_divide_eq log_mult_eq mult_nonneg_nonneg field_simps zero_less_mult_iff)

have "integrable (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P)
(\<lambda>x. Pxyz x * log b (Pxz (fst x, snd (snd x))) - Pxyz x * log b (Px (fst x)) - Pxyz x * log b (Pz (snd (snd x))))"
using finite_entropy_integrable_transform[OF Fxz Pxyz, of "\<lambda>x. (fst x, snd (snd x))"]
using finite_entropy_integrable_transform[OF Fx Pxyz, of fst]
using finite_entropy_integrable_transform[OF Fz Pxyz, of "snd \<circ> snd"]
by simp
moreover have "(\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxz (x, z) / (Px x * Pz z))) \<in> borel_measurable (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P)"
using Pxyz Px Pz
by auto
ultimately have I2: "integrable (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) (\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxz (x, z) / (Px x * Pz z)))"
apply (rule integrable_cong_AE_imp)
using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 ae9
by eventually_elim
(auto simp: log_divide_eq log_mult_eq mult_nonneg_nonneg field_simps zero_less_mult_iff)

from ae I1 I2 show ?eq
unfolding conditional_mutual_information_def
apply (subst mi_eq)
apply (subst mutual_information_distr[OF S TP Px Pyz Pxyz])
apply (subst integral_diff(2)[symmetric])
apply (auto intro!: integral_cong_AE simp: split_beta' simp del: integral_diff)
done

let ?P = "density (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) Pxyz"
interpret P: prob_space ?P
unfolding distributed_distr_eq_density[OF Pxyz, symmetric] by (rule prob_space_distr) simp

let ?Q = "density (T \<Otimes>\<^isub>M P) Pyz"
interpret Q: prob_space ?Q
unfolding distributed_distr_eq_density[OF Pyz, symmetric] by (rule prob_space_distr) simp

let ?f = "\<lambda>(x, y, z). Pxz (x, z) * (Pyz (y, z) / Pz z) / Pxyz (x, y, z)"

from subdensity_real[of snd, OF _ Pyz Pz]
have aeX1: "AE x in T \<Otimes>\<^isub>M P. Pz (snd x) = 0 \<longrightarrow> Pyz x = 0" by (auto simp: comp_def)
have aeX2: "AE x in T \<Otimes>\<^isub>M P. 0 \<le> Pz (snd x)"
using Pz by (intro TP.AE_pair_measure) (auto dest: distributed_real_AE)

have aeX3: "AE y in T \<Otimes>\<^isub>M P. (\<integral>\<^isup>+ x. ereal (Pxz (x, snd y)) \<partial>S) = ereal (Pz (snd y))"
using Pz distributed_marginal_eq_joint2[OF P S Pz Pxz]
by (intro TP.AE_pair_measure) (auto dest: distributed_real_AE)
have "(\<integral>\<^isup>+ x. ?f x \<partial>?P) \<le> (\<integral>\<^isup>+ (x, y, z). Pxz (x, z) * (Pyz (y, z) / Pz z) \<partial>(S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P))"
apply (subst positive_integral_density)
apply (rule distributed_borel_measurable[OF Pxyz])
apply (rule distributed_AE[OF Pxyz])
apply simp
apply (rule positive_integral_mono_AE)
using ae5 ae6 ae7 ae8
apply eventually_elim
apply (auto intro!: divide_nonneg_nonneg mult_nonneg_nonneg)
done
also have "\<dots> = (\<integral>\<^isup>+(y, z). \<integral>\<^isup>+ x. ereal (Pxz (x, z)) * ereal (Pyz (y, z) / Pz z) \<partial>S \<partial>T \<Otimes>\<^isub>M P)"
by (subst STP.positive_integral_snd_measurable[symmetric])
also have "\<dots> = (\<integral>\<^isup>+x. ereal (Pyz x) * 1 \<partial>T \<Otimes>\<^isub>M P)"
apply (rule positive_integral_cong_AE)
using aeX1 aeX2 aeX3 distributed_AE[OF Pyz] AE_space
apply eventually_elim
proof (case_tac x, simp del: times_ereal.simps add: space_pair_measure)
fix a b assume "Pz b = 0 \<longrightarrow> Pyz (a, b) = 0" "0 \<le> Pz b" "a \<in> space T \<and> b \<in> space P"
"(\<integral>\<^isup>+ x. ereal (Pxz (x, b)) \<partial>S) = ereal (Pz b)" "0 \<le> Pyz (a, b)"
then show "(\<integral>\<^isup>+ x. ereal (Pxz (x, b)) * ereal (Pyz (a, b) / Pz b) \<partial>S) = ereal (Pyz (a, b))"
by (subst positive_integral_multc) (auto intro!: divide_nonneg_nonneg)
qed
also have "\<dots> = 1"
using Q.emeasure_space_1 distributed_AE[OF Pyz] distributed_distr_eq_density[OF Pyz]
by (subst positive_integral_density[symmetric]) auto
finally have le1: "(\<integral>\<^isup>+ x. ?f x \<partial>?P) \<le> 1" .
also have "\<dots> < \<infinity>" by simp
finally have fin: "(\<integral>\<^isup>+ x. ?f x \<partial>?P) \<noteq> \<infinity>" by simp

have pos: "(\<integral>\<^isup>+ x. ?f x \<partial>?P) \<noteq> 0"
apply (subst positive_integral_density)
apply simp
apply (rule distributed_AE[OF Pxyz])
apply simp
proof
let ?g = "\<lambda>x. ereal (if Pxyz x = 0 then 0 else Pxz (fst x, snd (snd x)) * Pyz (snd x) / Pz (snd (snd x)))"
assume "(\<integral>\<^isup>+ x. ?g x \<partial>(S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P)) = 0"
then have "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. ?g x \<le> 0"
by (intro positive_integral_0_iff_AE[THEN iffD1]) (auto intro!: borel_measurable_ereal measurable_If)
then have "AE x in S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P. Pxyz x = 0"
using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 Pxyz[THEN distributed_real_AE]
by eventually_elim (auto split: split_if_asm simp: mult_le_0_iff divide_le_0_iff)
then have "(\<integral>\<^isup>+ x. ereal (Pxyz x) \<partial>S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) = 0"
by (subst positive_integral_cong_AE[of _ "\<lambda>x. 0"]) auto
with P.emeasure_space_1 show False
by (subst (asm) emeasure_density) (auto cong: positive_integral_cong)
qed

have neg: "(\<integral>\<^isup>+ x. - ?f x \<partial>?P) = 0"
apply (rule positive_integral_0_iff_AE[THEN iffD2])
apply (auto simp: split_beta') []
apply (subst AE_density)
apply (auto simp: split_beta') []
using ae5 ae6 ae7 ae8
apply eventually_elim
apply (auto intro!: mult_nonneg_nonneg divide_nonneg_nonneg)
done

have I3: "integrable (S \<Otimes>\<^isub>M T \<Otimes>\<^isub>M P) (\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Pxz (x, z) * (Pyz (y,z) / Pz z))))"
apply (rule integrable_cong_AE[THEN iffD1, OF _ _ _ integral_diff(1)[OF I1 I2]])
using ae
apply (auto simp: split_beta')
done

have "- log b 1 \<le> - log b (integral\<^isup>L ?P ?f)"
proof (intro le_imp_neg_le log_le[OF b_gt_1])
show "0 < integral\<^isup>L ?P ?f"
using neg pos fin positive_integral_positive[of ?P ?f]
by (cases "(\<integral>\<^isup>+ x. ?f x \<partial>?P)") (auto simp add: lebesgue_integral_def less_le split_beta')
show "integral\<^isup>L ?P ?f \<le> 1"
using neg le1 fin positive_integral_positive[of ?P ?f]
by (cases "(\<integral>\<^isup>+ x. ?f x \<partial>?P)") (auto simp add: lebesgue_integral_def split_beta' one_ereal_def)
qed
also have "- log b (integral\<^isup>L ?P ?f) \<le> (\<integral> x. - log b (?f x) \<partial>?P)"
proof (rule P.jensens_inequality[where a=0 and b=1 and I="{0<..}"])
show "AE x in ?P. ?f x \<in> {0<..}"
unfolding AE_density[OF distributed_borel_measurable[OF Pxyz]]
using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 Pxyz[THEN distributed_real_AE]
by eventually_elim (auto simp: divide_pos_pos mult_pos_pos)
show "integrable ?P ?f"
unfolding integrable_def
using fin neg by (auto simp: split_beta')
show "integrable ?P (\<lambda>x. - log b (?f x))"
apply (subst integral_density)
apply simp
apply (auto intro!: distributed_real_AE[OF Pxyz]) []
apply simp
apply (rule integrable_cong_AE[THEN iffD1, OF _ _ _ I3])
apply simp
apply simp
using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 Pxyz[THEN distributed_real_AE]
apply eventually_elim
apply (auto simp: log_divide_eq log_mult_eq zero_le_mult_iff zero_less_mult_iff zero_less_divide_iff field_simps)
done
qed (auto simp: b_gt_1 minus_log_convex)
also have "\<dots> = conditional_mutual_information b S T P X Y Z"
unfolding `?eq`
apply (subst integral_density)
apply simp
apply (auto intro!: distributed_real_AE[OF Pxyz]) []
apply simp
apply (intro integral_cong_AE)
using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 Pxyz[THEN distributed_real_AE]
apply eventually_elim
apply (auto simp: log_divide_eq zero_less_mult_iff zero_less_divide_iff field_simps)
done
finally show ?nonneg
by simp
qed

lemma (in information_space) conditional_mutual_information_eq:
assumes Pz: "simple_distributed M Z Pz"
assumes Pyz: "simple_distributed M (\<lambda>x. (Y x, Z x)) Pyz"
assumes Pxz: "simple_distributed M (\<lambda>x. (X x, Z x)) Pxz"
assumes Pxyz: "simple_distributed M (\<lambda>x. (X x, Y x, Z x)) Pxyz"
shows "\<I>(X ; Y | Z) =
(\<Sum>(x, y, z)\<in>(\<lambda>x. (X x, Y x, Z x))`space M. Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Pxz (x, z) * (Pyz (y,z) / Pz z))))"
proof (subst conditional_mutual_information_generic_eq[OF _ _ _ _
simple_distributed[OF Pz] simple_distributed_joint[OF Pyz] simple_distributed_joint[OF Pxz]
simple_distributed_joint2[OF Pxyz]])
note simple_distributed_joint2_finite[OF Pxyz, simp]
show "sigma_finite_measure (count_space (X ` space M))"
show "sigma_finite_measure (count_space (Y ` space M))"
show "sigma_finite_measure (count_space (Z ` space M))"
have "count_space (X ` space M) \<Otimes>\<^isub>M count_space (Y ` space M) \<Otimes>\<^isub>M count_space (Z ` space M) =
count_space (X`space M \<times> Y`space M \<times> Z`space M)"
(is "?P = ?C")

let ?Px = "\<lambda>x. measure M (X -` {x} \<inter> space M)"
have "(\<lambda>x. (X x, Z x)) \<in> measurable M (count_space (X ` space M) \<Otimes>\<^isub>M count_space (Z ` space M))"
using simple_distributed_joint[OF Pxz] by (rule distributed_measurable)
from measurable_comp[OF this measurable_fst]
have "random_variable (count_space (X ` space M)) X"
then have "simple_function M X"
unfolding simple_function_def by (auto simp: measurable_count_space_eq2)
then have "simple_distributed M X ?Px"
by (rule simple_distributedI) auto
then show "distributed M (count_space (X ` space M)) X ?Px"
by (rule simple_distributed)

let ?f = "(\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x, Z x)) ` space M then Pxyz x else 0)"
let ?g = "(\<lambda>x. if x \<in> (\<lambda>x. (Y x, Z x)) ` space M then Pyz x else 0)"
let ?h = "(\<lambda>x. if x \<in> (\<lambda>x. (X x, Z x)) ` space M then Pxz x else 0)"
show
"integrable ?P (\<lambda>(x, y, z). ?f (x, y, z) * log b (?f (x, y, z) / (?Px x * ?g (y, z))))"
"integrable ?P (\<lambda>(x, y, z). ?f (x, y, z) * log b (?h (x, z) / (?Px x * Pz z)))"
by (auto intro!: integrable_count_space simp: pair_measure_count_space)
let ?i = "\<lambda>x y z. ?f (x, y, z) * log b (?f (x, y, z) / (?h (x, z) * (?g (y, z) / Pz z)))"
let ?j = "\<lambda>x y z. Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Pxz (x, z) * (Pyz (y,z) / Pz z)))"
have "(\<lambda>(x, y, z). ?i x y z) = (\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x, Z x)) ` space M then ?j (fst x) (fst (snd x)) (snd (snd x)) else 0)"
by (auto intro!: ext)
then show "(\<integral> (x, y, z). ?i x y z \<partial>?P) = (\<Sum>(x, y, z)\<in>(\<lambda>x. (X x, Y x, Z x)) ` space M. ?j x y z)"
by (auto intro!: setsum_cong simp add: `?P = ?C` lebesgue_integral_count_space_finite simple_distributed_finite setsum_cases split_beta')
qed

lemma (in information_space) conditional_mutual_information_nonneg:
assumes X: "simple_function M X" and Y: "simple_function M Y" and Z: "simple_function M Z"
shows "0 \<le> \<I>(X ; Y | Z)"
proof -
have [simp]: "count_space (X ` space M) \<Otimes>\<^isub>M count_space (Y ` space M) \<Otimes>\<^isub>M count_space (Z ` space M) =
count_space (X`space M \<times> Y`space M \<times> Z`space M)"
by (simp add: pair_measure_count_space X Y Z simple_functionD)
note sf = sigma_finite_measure_count_space_finite[OF simple_functionD(1)]
note sd = simple_distributedI[OF _ refl]
note sp = simple_function_Pair
show ?thesis
apply (rule conditional_mutual_information_generic_nonneg[OF sf[OF X] sf[OF Y] sf[OF Z]])
apply (rule simple_distributed[OF sd[OF X]])
apply (rule simple_distributed[OF sd[OF Z]])
apply (rule simple_distributed_joint[OF sd[OF sp[OF Y Z]]])
apply (rule simple_distributed_joint[OF sd[OF sp[OF X Z]]])
apply (rule simple_distributed_joint2[OF sd[OF sp[OF X sp[OF Y Z]]]])
apply (auto intro!: integrable_count_space simp: X Y Z simple_functionD)
done
qed

subsection {* Conditional Entropy *}

definition (in prob_space)
"conditional_entropy b S T X Y = - (\<integral>(x, y). log b (real (RN_deriv (S \<Otimes>\<^isub>M T) (distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))) (x, y)) /
real (RN_deriv T (distr M T Y) y)) \<partial>distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)))"

abbreviation (in information_space)
conditional_entropy_Pow ("\<H>'(_ | _')") where
"\<H>(X | Y) \<equiv> conditional_entropy b (count_space (X`space M)) (count_space (Y`space M)) X Y"

lemma (in information_space) conditional_entropy_generic_eq:
fixes Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real"
assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
assumes Py[measurable]: "distributed M T Y Py"
assumes Pxy[measurable]: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
shows "conditional_entropy b S T X Y = - (\<integral>(x, y). Pxy (x, y) * log b (Pxy (x, y) / Py y) \<partial>(S \<Otimes>\<^isub>M T))"
proof -
interpret S: sigma_finite_measure S by fact
interpret T: sigma_finite_measure T by fact
interpret ST: pair_sigma_finite S T ..

have "AE x in density (S \<Otimes>\<^isub>M T) (\<lambda>x. ereal (Pxy x)). Pxy x = real (RN_deriv (S \<Otimes>\<^isub>M T) (distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))) x)"
unfolding AE_density[OF distributed_borel_measurable, OF Pxy]
unfolding distributed_distr_eq_density[OF Pxy]
using distributed_RN_deriv[OF Pxy]
by auto
moreover
have "AE x in density (S \<Otimes>\<^isub>M T) (\<lambda>x. ereal (Pxy x)). Py (snd x) = real (RN_deriv T (distr M T Y) (snd x))"
unfolding AE_density[OF distributed_borel_measurable, OF Pxy]
unfolding distributed_distr_eq_density[OF Py]
apply (rule ST.AE_pair_measure)
apply (auto intro!: sets.sets_Collect borel_measurable_eq measurable_compose[OF _ distributed_real_measurable[OF Py]]
distributed_real_measurable[OF Pxy] distributed_real_AE[OF Py]
borel_measurable_real_of_ereal measurable_compose[OF _ borel_measurable_RN_deriv_density])
using distributed_RN_deriv[OF Py]
apply auto
done
ultimately
have "conditional_entropy b S T X Y = - (\<integral>x. Pxy x * log b (Pxy x / Py (snd x)) \<partial>(S \<Otimes>\<^isub>M T))"
unfolding conditional_entropy_def neg_equal_iff_equal
apply (subst integral_density(1)[symmetric])
apply (auto simp: distributed_real_measurable[OF Pxy] distributed_real_AE[OF Pxy]
measurable_compose[OF _ distributed_real_measurable[OF Py]]
distributed_distr_eq_density[OF Pxy]
intro!: integral_cong_AE)
done
then show ?thesis by (simp add: split_beta')
qed

lemma (in information_space) conditional_entropy_eq_entropy:
fixes Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real"
assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
assumes Py: "distributed M T Y Py"
assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
assumes I1: "integrable (S \<Otimes>\<^isub>M T) (\<lambda>x. Pxy x * log b (Pxy x))"
assumes I2: "integrable (S \<Otimes>\<^isub>M T) (\<lambda>x. Pxy x * log b (Py (snd x)))"
shows "conditional_entropy b S T X Y = entropy b (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) - entropy b T Y"
proof -
interpret S: sigma_finite_measure S by fact
interpret T: sigma_finite_measure T by fact
interpret ST: pair_sigma_finite S T ..

have "entropy b T Y = - (\<integral>y. Py y * log b (Py y) \<partial>T)"
by (rule entropy_distr[OF Py])
also have "\<dots> = - (\<integral>(x,y). Pxy (x,y) * log b (Py y) \<partial>(S \<Otimes>\<^isub>M T))"
using b_gt_1 Py[THEN distributed_real_measurable]
by (subst distributed_transform_integral[OF Pxy Py, where T=snd]) (auto intro!: integral_cong)
finally have e_eq: "entropy b T Y = - (\<integral>(x,y). Pxy (x,y) * log b (Py y) \<partial>(S \<Otimes>\<^isub>M T))" .

have ae2: "AE x in S \<Otimes>\<^isub>M T. Py (snd x) = 0 \<longrightarrow> Pxy x = 0"
by (intro subdensity_real[of snd, OF _ Pxy Py]) (auto intro: measurable_Pair)
moreover have ae4: "AE x in S \<Otimes>\<^isub>M T. 0 \<le> Py (snd x)"
using Py by (intro ST.AE_pair_measure) (auto simp: comp_def intro!: measurable_snd'' dest: distributed_real_AE distributed_real_measurable)
moreover note ae5 = Pxy[THEN distributed_real_AE]
ultimately have "AE x in S \<Otimes>\<^isub>M T. 0 \<le> Pxy x \<and> 0 \<le> Py (snd x) \<and>
(Pxy x = 0 \<or> (Pxy x \<noteq> 0 \<longrightarrow> 0 < Pxy x \<and> 0 < Py (snd x)))"
by eventually_elim auto
then have ae: "AE x in S \<Otimes>\<^isub>M T.
Pxy x * log b (Pxy x) - Pxy x * log b (Py (snd x)) = Pxy x * log b (Pxy x / Py (snd x))"
by eventually_elim (auto simp: log_simps mult_pos_pos field_simps b_gt_1)
have "conditional_entropy b S T X Y =
- (\<integral>x. Pxy x * log b (Pxy x) - Pxy x * log b (Py (snd x)) \<partial>(S \<Otimes>\<^isub>M T))"
unfolding conditional_entropy_generic_eq[OF S T Py Pxy] neg_equal_iff_equal
apply (intro integral_cong_AE)
using ae
apply eventually_elim
apply auto
done
also have "\<dots> = - (\<integral>x. Pxy x * log b (Pxy x) \<partial>(S \<Otimes>\<^isub>M T)) - - (\<integral>x.  Pxy x * log b (Py (snd x)) \<partial>(S \<Otimes>\<^isub>M T))"
by (simp add: integral_diff[OF I1 I2])
finally show ?thesis
unfolding conditional_entropy_generic_eq[OF S T Py Pxy] entropy_distr[OF Pxy] e_eq
qed

lemma (in information_space) conditional_entropy_eq_entropy_simple:
assumes X: "simple_function M X" and Y: "simple_function M Y"
shows "\<H>(X | Y) = entropy b (count_space (X`space M) \<Otimes>\<^isub>M count_space (Y`space M)) (\<lambda>x. (X x, Y x)) - \<H>(Y)"
proof -
have "count_space (X ` space M) \<Otimes>\<^isub>M count_space (Y ` space M) = count_space (X`space M \<times> Y`space M)"
(is "?P = ?C") using X Y by (simp add: simple_functionD pair_measure_count_space)
show ?thesis
by (rule conditional_entropy_eq_entropy sigma_finite_measure_count_space_finite
simple_functionD  X Y simple_distributed simple_distributedI[OF _ refl]
simple_distributed_joint simple_function_Pair integrable_count_space)+
(auto simp: `?P = ?C` intro!: integrable_count_space simple_functionD  X Y)
qed

lemma (in information_space) conditional_entropy_eq:
assumes Y: "simple_distributed M Y Py"
assumes XY: "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy"
shows "\<H>(X | Y) = - (\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` space M. Pxy (x, y) * log b (Pxy (x, y) / Py y))"
proof (subst conditional_entropy_generic_eq[OF _ _
simple_distributed[OF Y] simple_distributed_joint[OF XY]])
have "finite ((\<lambda>x. (X x, Y x))`space M)"
using XY unfolding simple_distributed_def by auto
from finite_imageI[OF this, of fst]
have [simp]: "finite (X`space M)"
note Y[THEN simple_distributed_finite, simp]
show "sigma_finite_measure (count_space (X ` space M))"
show "sigma_finite_measure (count_space (Y ` space M))"
let ?f = "(\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x)) ` space M then Pxy x else 0)"
have "count_space (X ` space M) \<Otimes>\<^isub>M count_space (Y ` space M) = count_space (X`space M \<times> Y`space M)"
(is "?P = ?C")
using Y by (simp add: simple_distributed_finite pair_measure_count_space)
have eq: "(\<lambda>(x, y). ?f (x, y) * log b (?f (x, y) / Py y)) =
(\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x)) ` space M then Pxy x * log b (Pxy x / Py (snd x)) else 0)"
by auto
from Y show "- (\<integral> (x, y). ?f (x, y) * log b (?f (x, y) / Py y) \<partial>?P) =
- (\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` space M. Pxy (x, y) * log b (Pxy (x, y) / Py y))"
by (auto intro!: setsum_cong simp add: `?P = ?C` lebesgue_integral_count_space_finite simple_distributed_finite eq setsum_cases split_beta')
qed

lemma (in information_space) conditional_mutual_information_eq_conditional_entropy:
assumes X: "simple_function M X" and Y: "simple_function M Y"
shows "\<I>(X ; X | Y) = \<H>(X | Y)"
proof -
def Py \<equiv> "\<lambda>x. if x \<in> Y`space M then measure M (Y -` {x} \<inter> space M) else 0"
def Pxy \<equiv> "\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x))`space M then measure M ((\<lambda>x. (X x, Y x)) -` {x} \<inter> space M) else 0"
def Pxxy \<equiv> "\<lambda>x. if x \<in> (\<lambda>x. (X x, X x, Y x))`space M then measure M ((\<lambda>x. (X x, X x, Y x)) -` {x} \<inter> space M) else 0"
let ?M = "X`space M \<times> X`space M \<times> Y`space M"

note XY = simple_function_Pair[OF X Y]
note XXY = simple_function_Pair[OF X XY]
have Py: "simple_distributed M Y Py"
using Y by (rule simple_distributedI) (auto simp: Py_def)
have Pxy: "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy"
using XY by (rule simple_distributedI) (auto simp: Pxy_def)
have Pxxy: "simple_distributed M (\<lambda>x. (X x, X x, Y x)) Pxxy"
using XXY by (rule simple_distributedI) (auto simp: Pxxy_def)
have eq: "(\<lambda>x. (X x, X x, Y x)) ` space M = (\<lambda>(x, y). (x, x, y)) ` (\<lambda>x. (X x, Y x)) ` space M"
by auto
have inj: "\<And>A. inj_on (\<lambda>(x, y). (x, x, y)) A"
by (auto simp: inj_on_def)
have Pxxy_eq: "\<And>x y. Pxxy (x, x, y) = Pxy (x, y)"
by (auto simp: Pxxy_def Pxy_def intro!: arg_cong[where f=prob])
have "AE x in count_space ((\<lambda>x. (X x, Y x))`space M). Py (snd x) = 0 \<longrightarrow> Pxy x = 0"
by (intro subdensity_real[of snd, OF _ Pxy[THEN simple_distributed] Py[THEN simple_distributed]]) (auto intro: measurable_Pair)
then show ?thesis
apply (subst conditional_mutual_information_eq[OF Py Pxy Pxy Pxxy])
apply (subst conditional_entropy_eq[OF Py Pxy])
apply (auto intro!: setsum_cong simp: Pxxy_eq setsum_negf[symmetric] eq setsum_reindex[OF inj]
log_simps zero_less_mult_iff zero_le_mult_iff field_simps mult_less_0_iff AE_count_space)
using Py[THEN simple_distributed, THEN distributed_real_AE] Pxy[THEN simple_distributed, THEN distributed_real_AE]
apply (auto simp add: not_le[symmetric] AE_count_space)
done
qed

lemma (in information_space) conditional_entropy_nonneg:
assumes X: "simple_function M X" and Y: "simple_function M Y" shows "0 \<le> \<H>(X | Y)"
using conditional_mutual_information_eq_conditional_entropy[OF X Y] conditional_mutual_information_nonneg[OF X X Y]
by simp

subsection {* Equalities *}

lemma (in information_space) mutual_information_eq_entropy_conditional_entropy_distr:
fixes Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real" and Pxy :: "('b \<times> 'c) \<Rightarrow> real"
assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
assumes Px: "distributed M S X Px" and Py: "distributed M T Y Py"
assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
assumes Ix: "integrable(S \<Otimes>\<^isub>M T) (\<lambda>x. Pxy x * log b (Px (fst x)))"
assumes Iy: "integrable(S \<Otimes>\<^isub>M T) (\<lambda>x. Pxy x * log b (Py (snd x)))"
assumes Ixy: "integrable(S \<Otimes>\<^isub>M T) (\<lambda>x. Pxy x * log b (Pxy x))"
shows  "mutual_information b S T X Y = entropy b S X + entropy b T Y - entropy b (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))"
proof -
have X: "entropy b S X = - (\<integral>x. Pxy x * log b (Px (fst x)) \<partial>(S \<Otimes>\<^isub>M T))"
using b_gt_1 Px[THEN distributed_real_measurable]
apply (subst entropy_distr[OF Px])
apply (subst distributed_transform_integral[OF Pxy Px, where T=fst])
apply (auto intro!: integral_cong)
done

have Y: "entropy b T Y = - (\<integral>x. Pxy x * log b (Py (snd x)) \<partial>(S \<Otimes>\<^isub>M T))"
using b_gt_1 Py[THEN distributed_real_measurable]
apply (subst entropy_distr[OF Py])
apply (subst distributed_transform_integral[OF Pxy Py, where T=snd])
apply (auto intro!: integral_cong)
done

interpret S: sigma_finite_measure S by fact
interpret T: sigma_finite_measure T by fact
interpret ST: pair_sigma_finite S T ..
have ST: "sigma_finite_measure (S \<Otimes>\<^isub>M T)" ..

have XY: "entropy b (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) = - (\<integral>x. Pxy x * log b (Pxy x) \<partial>(S \<Otimes>\<^isub>M T))"
by (subst entropy_distr[OF Pxy]) (auto intro!: integral_cong)

have "AE x in S \<Otimes>\<^isub>M T. Px (fst x) = 0 \<longrightarrow> Pxy x = 0"
by (intro subdensity_real[of fst, OF _ Pxy Px]) (auto intro: measurable_Pair)
moreover have "AE x in S \<Otimes>\<^isub>M T. Py (snd x) = 0 \<longrightarrow> Pxy x = 0"
by (intro subdensity_real[of snd, OF _ Pxy Py]) (auto intro: measurable_Pair)
moreover have "AE x in S \<Otimes>\<^isub>M T. 0 \<le> Px (fst x)"
using Px by (intro ST.AE_pair_measure) (auto simp: comp_def intro!: measurable_fst'' dest: distributed_real_AE distributed_real_measurable)
moreover have "AE x in S \<Otimes>\<^isub>M T. 0 \<le> Py (snd x)"
using Py by (intro ST.AE_pair_measure) (auto simp: comp_def intro!: measurable_snd'' dest: distributed_real_AE distributed_real_measurable)
moreover note Pxy[THEN distributed_real_AE]
ultimately have "AE x in S \<Otimes>\<^isub>M T. Pxy x * log b (Pxy x) - Pxy x * log b (Px (fst x)) - Pxy x * log b (Py (snd x)) =
Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x)))"
(is "AE x in _. ?f x = ?g x")
proof eventually_elim
case (goal1 x)
show ?case
proof cases
assume "Pxy x \<noteq> 0"
with goal1 have "0 < Px (fst x)" "0 < Py (snd x)" "0 < Pxy x"
by auto
then show ?thesis
using b_gt_1 by (simp add: log_simps mult_pos_pos less_imp_le field_simps)
qed simp
qed

have "entropy b S X + entropy b T Y - entropy b (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) = integral\<^isup>L (S \<Otimes>\<^isub>M T) ?f"
unfolding X Y XY
apply (subst integral_diff)
apply (intro integral_diff Ixy Ix Iy)+
apply (subst integral_diff)
apply (intro integral_diff Ixy Ix Iy)+
done
also have "\<dots> = integral\<^isup>L (S \<Otimes>\<^isub>M T) ?g"
using `AE x in _. ?f x = ?g x` by (rule integral_cong_AE)
also have "\<dots> = mutual_information b S T X Y"
by (rule mutual_information_distr[OF S T Px Py Pxy, symmetric])
finally show ?thesis ..
qed

lemma (in information_space) mutual_information_eq_entropy_conditional_entropy':
fixes Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real" and Pxy :: "('b \<times> 'c) \<Rightarrow> real"
assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
assumes Px: "distributed M S X Px" and Py: "distributed M T Y Py"
assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
assumes Ix: "integrable(S \<Otimes>\<^isub>M T) (\<lambda>x. Pxy x * log b (Px (fst x)))"
assumes Iy: "integrable(S \<Otimes>\<^isub>M T) (\<lambda>x. Pxy x * log b (Py (snd x)))"
assumes Ixy: "integrable(S \<Otimes>\<^isub>M T) (\<lambda>x. Pxy x * log b (Pxy x))"
shows  "mutual_information b S T X Y = entropy b S X - conditional_entropy b S T X Y"
using
mutual_information_eq_entropy_conditional_entropy_distr[OF S T Px Py Pxy Ix Iy Ixy]
conditional_entropy_eq_entropy[OF S T Py Pxy Ixy Iy]
by simp

lemma (in information_space) mutual_information_eq_entropy_conditional_entropy:
assumes sf_X: "simple_function M X" and sf_Y: "simple_function M Y"
shows  "\<I>(X ; Y) = \<H>(X) - \<H>(X | Y)"
proof -
have X: "simple_distributed M X (\<lambda>x. measure M (X -` {x} \<inter> space M))"
using sf_X by (rule simple_distributedI) auto
have Y: "simple_distributed M Y (\<lambda>x. measure M (Y -` {x} \<inter> space M))"
using sf_Y by (rule simple_distributedI) auto
have sf_XY: "simple_function M (\<lambda>x. (X x, Y x))"
using sf_X sf_Y by (rule simple_function_Pair)
then have XY: "simple_distributed M (\<lambda>x. (X x, Y x)) (\<lambda>x. measure M ((\<lambda>x. (X x, Y x)) -` {x} \<inter> space M))"
by (rule simple_distributedI) auto
from simple_distributed_joint_finite[OF this, simp]
have eq: "count_space (X ` space M) \<Otimes>\<^isub>M count_space (Y ` space M) = count_space (X ` space M \<times> Y ` space M)"

have "\<I>(X ; Y) = \<H>(X) + \<H>(Y) - entropy b (count_space (X`space M) \<Otimes>\<^isub>M count_space (Y`space M)) (\<lambda>x. (X x, Y x))"
using sigma_finite_measure_count_space_finite sigma_finite_measure_count_space_finite simple_distributed[OF X] simple_distributed[OF Y] simple_distributed_joint[OF XY]
by (rule mutual_information_eq_entropy_conditional_entropy_distr) (auto simp: eq integrable_count_space)
then show ?thesis
unfolding conditional_entropy_eq_entropy_simple[OF sf_X sf_Y] by simp
qed

lemma (in information_space) mutual_information_nonneg_simple:
assumes sf_X: "simple_function M X" and sf_Y: "simple_function M Y"
shows  "0 \<le> \<I>(X ; Y)"
proof -
have X: "simple_distributed M X (\<lambda>x. measure M (X -` {x} \<inter> space M))"
using sf_X by (rule simple_distributedI) auto
have Y: "simple_distributed M Y (\<lambda>x. measure M (Y -` {x} \<inter> space M))"
using sf_Y by (rule simple_distributedI) auto

have sf_XY: "simple_function M (\<lambda>x. (X x, Y x))"
using sf_X sf_Y by (rule simple_function_Pair)
then have XY: "simple_distributed M (\<lambda>x. (X x, Y x)) (\<lambda>x. measure M ((\<lambda>x. (X x, Y x)) -` {x} \<inter> space M))"
by (rule simple_distributedI) auto

from simple_distributed_joint_finite[OF this, simp]
have eq: "count_space (X ` space M) \<Otimes>\<^isub>M count_space (Y ` space M) = count_space (X ` space M \<times> Y ` space M)"

show ?thesis
by (rule mutual_information_nonneg[OF _ _ simple_distributed[OF X] simple_distributed[OF Y] simple_distributed_joint[OF XY]])
qed

lemma (in information_space) conditional_entropy_less_eq_entropy:
assumes X: "simple_function M X" and Z: "simple_function M Z"
shows "\<H>(X | Z) \<le> \<H>(X)"
proof -
have "0 \<le> \<I>(X ; Z)" using X Z by (rule mutual_information_nonneg_simple)
also have "\<I>(X ; Z) = \<H>(X) - \<H>(X | Z)" using mutual_information_eq_entropy_conditional_entropy[OF assms] .
finally show ?thesis by auto
qed

lemma (in information_space)
fixes Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real" and Pxy :: "('b \<times> 'c) \<Rightarrow> real"
assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
assumes Px: "finite_entropy S X Px" and Py: "finite_entropy T Y Py"
assumes Pxy: "finite_entropy (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
shows "conditional_entropy b S T X Y \<le> entropy b S X"
proof -

have "0 \<le> mutual_information b S T X Y"
by (rule mutual_information_nonneg') fact+
also have "\<dots> = entropy b S X - conditional_entropy b S T X Y"
apply (rule mutual_information_eq_entropy_conditional_entropy')
using assms
by (auto intro!: finite_entropy_integrable finite_entropy_distributed
finite_entropy_integrable_transform[OF Px]
finite_entropy_integrable_transform[OF Py])
finally show ?thesis by auto
qed

lemma (in information_space) entropy_chain_rule:
assumes X: "simple_function M X" and Y: "simple_function M Y"
shows  "\<H>(\<lambda>x. (X x, Y x)) = \<H>(X) + \<H>(Y|X)"
proof -
note XY = simple_distributedI[OF simple_function_Pair[OF X Y] refl]
note YX = simple_distributedI[OF simple_function_Pair[OF Y X] refl]
note simple_distributed_joint_finite[OF this, simp]
let ?f = "\<lambda>x. prob ((\<lambda>x. (X x, Y x)) -` {x} \<inter> space M)"
let ?g = "\<lambda>x. prob ((\<lambda>x. (Y x, X x)) -` {x} \<inter> space M)"
let ?h = "\<lambda>x. if x \<in> (\<lambda>x. (Y x, X x)) ` space M then prob ((\<lambda>x. (Y x, X x)) -` {x} \<inter> space M) else 0"
have "\<H>(\<lambda>x. (X x, Y x)) = - (\<Sum>x\<in>(\<lambda>x. (X x, Y x)) ` space M. ?f x * log b (?f x))"
using XY by (rule entropy_simple_distributed)
also have "\<dots> = - (\<Sum>x\<in>(\<lambda>(x, y). (y, x)) ` (\<lambda>x. (X x, Y x)) ` space M. ?g x * log b (?g x))"
by (subst (2) setsum_reindex) (auto simp: inj_on_def intro!: setsum_cong arg_cong[where f="\<lambda>A. prob A * log b (prob A)"])
also have "\<dots> = - (\<Sum>x\<in>(\<lambda>x. (Y x, X x)) ` space M. ?h x * log b (?h x))"
by (auto intro!: setsum_cong)
also have "\<dots> = entropy b (count_space (Y ` space M) \<Otimes>\<^isub>M count_space (X ` space M)) (\<lambda>x. (Y x, X x))"
by (subst entropy_distr[OF simple_distributed_joint[OF YX]])
(auto simp: pair_measure_count_space sigma_finite_measure_count_space_finite lebesgue_integral_count_space_finite
cong del: setsum_cong  intro!: setsum_mono_zero_left)
finally have "\<H>(\<lambda>x. (X x, Y x)) = entropy b (count_space (Y ` space M) \<Otimes>\<^isub>M count_space (X ` space M)) (\<lambda>x. (Y x, X x))" .
then show ?thesis
unfolding conditional_entropy_eq_entropy_simple[OF Y X] by simp
qed

lemma (in information_space) entropy_partition:
assumes X: "simple_function M X"
shows "\<H>(X) = \<H>(f \<circ> X) + \<H>(X|f \<circ> X)"
proof -
note fX = simple_function_compose[OF X, of f]
have eq: "(\<lambda>x. ((f \<circ> X) x, X x)) ` space M = (\<lambda>x. (f x, x)) ` X ` space M" by auto
have inj: "\<And>A. inj_on (\<lambda>x. (f x, x)) A"
by (auto simp: inj_on_def)
show ?thesis
apply (subst entropy_chain_rule[symmetric, OF fX X])
apply (subst entropy_simple_distributed[OF simple_distributedI[OF simple_function_Pair[OF fX X] refl]])
apply (subst entropy_simple_distributed[OF simple_distributedI[OF X refl]])
unfolding eq
apply (subst setsum_reindex[OF inj])
apply (auto intro!: setsum_cong arg_cong[where f="\<lambda>A. prob A * log b (prob A)"])
done
qed

corollary (in information_space) entropy_data_processing:
assumes X: "simple_function M X" shows "\<H>(f \<circ> X) \<le> \<H>(X)"
proof -
note fX = simple_function_compose[OF X, of f]
from X have "\<H>(X) = \<H>(f\<circ>X) + \<H>(X|f\<circ>X)" by (rule entropy_partition)
then show "\<H>(f \<circ> X) \<le> \<H>(X)"
by (auto intro: conditional_entropy_nonneg[OF X fX])
qed

corollary (in information_space) entropy_of_inj:
assumes X: "simple_function M X" and inj: "inj_on f (X`space M)"
shows "\<H>(f \<circ> X) = \<H>(X)"
proof (rule antisym)
show "\<H>(f \<circ> X) \<le> \<H>(X)" using entropy_data_processing[OF X] .
next
have sf: "simple_function M (f \<circ> X)"
using X by auto
have "\<H>(X) = \<H>(the_inv_into (X`space M) f \<circ> (f \<circ> X))"
unfolding o_assoc
apply (subst entropy_simple_distributed[OF simple_distributedI[OF X refl]])
apply (subst entropy_simple_distributed[OF simple_distributedI[OF simple_function_compose[OF X]], where f="\<lambda>x. prob (X -` {x} \<inter> space M)"])
apply (auto intro!: setsum_cong arg_cong[where f=prob] image_eqI simp: the_inv_into_f_f[OF inj] comp_def)
done
also have "... \<le> \<H>(f \<circ> X)"
using entropy_data_processing[OF sf] .
finally show "\<H>(X) \<le> \<H>(f \<circ> X)" .
qed

end
```