
diff -r d26348b667f2 -r 25153c08655e src/HOL/Probability/Information.thy
--- a/src/HOL/Probability/Information.thy	Mon May 03 14:35:10 2010 +0200
+++ b/src/HOL/Probability/Information.thy	Mon May 03 14:35:10 2010 +0200
@@ -2,168 +2,263 @@
 imports Probability_Space Product_Measure "../Multivariate_Analysis/Convex"
 begin
 
-lemma pos_neg_part_abs:
-  fixes f :: "'a \<Rightarrow> real"
-  shows "pos_part f x + neg_part f x = \<bar>f x\<bar>"
-unfolding real_abs_def pos_part_def neg_part_def by auto
+section "Convex theory"
 
-lemma pos_part_abs:
-  fixes f :: "'a \<Rightarrow> real"
-  shows "pos_part (\<lambda> x. \<bar>f x\<bar>) y = \<bar>f y\<bar>"
-unfolding pos_part_def real_abs_def by auto
-
-lemma neg_part_abs:
-  fixes f :: "'a \<Rightarrow> real"
-  shows "neg_part (\<lambda> x. \<bar>f x\<bar>) y = 0"
-unfolding neg_part_def real_abs_def by auto
+lemma log_setsum:
+  assumes "finite s" "s \<noteq> {}"
+  assumes "b > 1"
+  assumes "(\<Sum> i \<in> s. a i) = 1"
+  assumes "\<And> i. i \<in> s \<Longrightarrow> a i \<ge> 0"
+  assumes "\<And> i. i \<in> s \<Longrightarrow> y i \<in> {0 <..}"
+  shows "log b (\<Sum> i \<in> s. a i * y i) \<ge> (\<Sum> i \<in> s. a i * log b (y i))"
+proof -
+  have "convex_on {0 <..} (\<lambda> x. - log b x)"
+    by (rule minus_log_convex[OF `b > 1`])
+  hence "- log b (\<Sum> i \<in> s. a i * y i) \<le> (\<Sum> i \<in> s. a i * - log b (y i))"
+    using convex_on_setsum[of _ _ "\<lambda> x. - log b x"] assms pos_is_convex by fastsimp
+  thus ?thesis by (auto simp add:setsum_negf le_imp_neg_le)
+qed
 
-lemma (in measure_space) int_abs:
-  assumes "integrable f"
-  shows "integrable (\<lambda> x. \<bar>f x\<bar>)"
-using assms
+lemma log_setsum':
+  assumes "finite s" "s \<noteq> {}"
+  assumes "b > 1"
+  assumes "(\<Sum> i \<in> s. a i) = 1"
+  assumes pos: "\<And> i. i \<in> s \<Longrightarrow> 0 \<le> a i"
+          "\<And> i. \<lbrakk> i \<in> s ; 0 < a i \<rbrakk> \<Longrightarrow> 0 < y i"
+  shows "log b (\<Sum> i \<in> s. a i * y i) \<ge> (\<Sum> i \<in> s. a i * log b (y i))"
 proof -
-  from assms obtain p q where pq: "p \<in> nnfis (pos_part f)" "q \<in> nnfis (neg_part f)"
-    unfolding integrable_def by auto
-  hence "p + q \<in> nnfis (\<lambda> x. pos_part f x + neg_part f x)"
-    using nnfis_add by auto
-  hence "p + q \<in> nnfis (\<lambda> x. \<bar>f x\<bar>)" using pos_neg_part_abs[of f] by simp
-  thus ?thesis unfolding integrable_def
-    using ext[OF pos_part_abs[of f], of "\<lambda> y. y"]
-      ext[OF neg_part_abs[of f], of "\<lambda> y. y"]
-    using nnfis_0 by auto
+  have "\<And>y. (\<Sum> i \<in> s - {i. a i = 0}. a i * y i) = (\<Sum> i \<in> s. a i * y i)"
+    using assms by (auto intro!: setsum_mono_zero_cong_left)
+  moreover have "log b (\<Sum> i \<in> s - {i. a i = 0}. a i * y i) \<ge> (\<Sum> i \<in> s - {i. a i = 0}. a i * log b (y i))"
+  proof (rule log_setsum)
+    have "setsum a (s - {i. a i = 0}) = setsum a s"
+      using assms(1) by (rule setsum_mono_zero_cong_left) auto
+    thus sum_1: "setsum a (s - {i. a i = 0}) = 1"
+      "finite (s - {i. a i = 0})" using assms by simp_all
+
+    show "s - {i. a i = 0} \<noteq> {}"
+    proof
+      assume *: "s - {i. a i = 0} = {}"
+      hence "setsum a (s - {i. a i = 0}) = 0" by (simp add: * setsum_empty)
+      with sum_1 show False by simp
+qed
+
+    fix i assume "i \<in> s - {i. a i = 0}"
+    hence "i \<in> s" "a i \<noteq> 0" by simp_all
+    thus "0 \<le> a i" "y i \<in> {0<..}" using pos[of i] by auto
+  qed fact+
+  ultimately show ?thesis by simp
 qed
 
-lemma (in measure_space) measure_mono:
-  assumes "a \<subseteq> b" "a \<in> sets M" "b \<in> sets M"
-  shows "measure M a \<le> measure M b"
+section "Information theory"
+
+lemma (in finite_prob_space) sum_over_space_distrib:
+  "(\<Sum>x\<in>X`space M. distribution X {x}) = 1"
+  unfolding distribution_def prob_space[symmetric] using finite_space
+  by (subst measure_finitely_additive'')
+     (auto simp add: disjoint_family_on_def sets_eq_Pow intro!: arg_cong[where f=prob])
+
+locale finite_information_space = finite_prob_space +
+  fixes b :: real assumes b_gt_1: "1 < b"
+
+definition
+  "KL_divergence b M X Y =
+    measure_space.integral (M\<lparr>measure := X\<rparr>)
+                           (\<lambda>x. log b ((measure_space.RN_deriv (M \<lparr>measure := Y\<rparr> ) X) x))"
+
+lemma (in finite_prob_space) distribution_mono:
+  assumes "\<And>t. \<lbrakk> t \<in> space M ; X t \<in> x \<rbrakk> \<Longrightarrow> Y t \<in> y"
+  shows "distribution X x \<le> distribution Y y"
+  unfolding distribution_def
+  using assms by (auto simp: sets_eq_Pow intro!: measure_mono)
+
+lemma (in prob_space) distribution_remove_const:
+  shows "joint_distribution X (\<lambda>x. ()) {(x, ())} = distribution X {x}"
+  and "joint_distribution (\<lambda>x. ()) X {((), x)} = distribution X {x}"
+  and "joint_distribution X (\<lambda>x. (Y x, ())) {(x, y, ())} = joint_distribution X Y {(x, y)}"
+  and "joint_distribution X (\<lambda>x. ((), Y x)) {(x, (), y)} = joint_distribution X Y {(x, y)}"
+  and "distribution (\<lambda>x. ()) {()} = 1"
+  unfolding prob_space[symmetric]
+  by (auto intro!: arg_cong[where f=prob] simp: distribution_def)
+
+
+context finite_information_space
+begin
+
+lemma distribution_mono_gt_0:
+  assumes gt_0: "0 < distribution X x"
+  assumes *: "\<And>t. \<lbrakk> t \<in> space M ; X t \<in> x \<rbrakk> \<Longrightarrow> Y t \<in> y"
+  shows "0 < distribution Y y"
+  by (rule less_le_trans[OF gt_0 distribution_mono]) (rule *)
+
+lemma
+  assumes "0 \<le> A" and pos: "0 < A \<Longrightarrow> 0 < B" "0 < A \<Longrightarrow> 0 < C"
+  shows mult_log_mult: "A * log b (B * C) = A * log b B + A * log b C" (is "?mult")
+  and mult_log_divide: "A * log b (B / C) = A * log b B - A * log b C" (is "?div")
 proof -
-  have "b = a \<union> (b - a)" using assms by auto
-  moreover have "{} = a \<inter> (b - a)" by auto
-  ultimately have "measure M b = measure M a + measure M (b - a)"
-    using measure_additive[of a "b - a"] local.Diff[of b a] assms by auto
-  moreover have "measure M (b - a) \<ge> 0" using positive assms by auto
-  ultimately show "measure M a \<le> measure M b" by auto
+  have "?mult \<and> ?div"
+proof (cases "A = 0")
+  case False
+  hence "0 < A" using `0 \<le> A` by auto
+    with pos[OF this] show "?mult \<and> ?div" using b_gt_1
+      by (auto simp: log_divide log_mult field_simps)
+qed simp
+  thus ?mult and ?div by auto
 qed
 
-lemma (in measure_space) integral_0:
-  fixes f :: "'a \<Rightarrow> real"
-  assumes "integrable f" "integral f = 0" "nonneg f" and borel: "f \<in> borel_measurable M"
-  shows "measure M ({x. f x \<noteq> 0} \<inter> space M) = 0"
-proof -
-  have "{x. f x \<noteq> 0} = {x. \<bar>f x\<bar> > 0}" by auto
-  moreover
-  { fix y assume "y \<in> {x. \<bar> f x \<bar> > 0}"
-    hence "\<bar> f y \<bar> > 0" by auto
-    hence "\<exists> n. \<bar>f y\<bar> \<ge> inverse (real (Suc n))"
-      using ex_inverse_of_nat_Suc_less[of "\<bar>f y\<bar>"] less_imp_le unfolding real_of_nat_def by auto
-    hence "y \<in> (\<Union> n. {x. \<bar>f x\<bar> \<ge> inverse (real (Suc n))})"
-      by auto }
-  moreover
-  { fix y assume "y \<in> (\<Union> n. {x. \<bar>f x\<bar> \<ge> inverse (real (Suc n))})"
-    then obtain n where n: "y \<in> {x. \<bar>f x\<bar> \<ge> inverse (real (Suc n))}" by auto
-    hence "\<bar>f y\<bar> \<ge> inverse (real (Suc n))" by auto
-    hence "\<bar>f y\<bar> > 0"
-      using real_of_nat_Suc_gt_zero
-        positive_imp_inverse_positive[of "real_of_nat (Suc n)"] by fastsimp
-    hence "y \<in> {x. \<bar>f x\<bar> > 0}" by auto }
-  ultimately have fneq0_UN: "{x. f x \<noteq> 0} = (\<Union> n. {x. \<bar>f x\<bar> \<ge> inverse (real (Suc n))})"
-    by blast
-  { fix n
-    have int_one: "integrable (\<lambda> x. \<bar>f x\<bar> ^ 1)" using int_abs assms by auto
-    have "measure M (f -` {inverse (real (Suc n))..} \<inter> space M)
-           \<le> integral (\<lambda> x. \<bar>f x\<bar> ^ 1) / (inverse (real (Suc n)) ^ 1)"
-      using markov_ineq[OF `integrable f` _ int_one] real_of_nat_Suc_gt_zero by auto
-    hence le0: "measure M (f -` {inverse (real (Suc n))..} \<inter> space M) \<le> 0"
-      using assms unfolding nonneg_def by auto
-    have "{x. f x \<ge> inverse (real (Suc n))} \<inter> space M \<in> sets M"
-      apply (subst Int_commute) unfolding Int_def
-      using borel[unfolded borel_measurable_ge_iff] by simp
-    hence m0: "measure M ({x. f x \<ge> inverse (real (Suc n))} \<inter> space M) = 0 \<and>
-      {x. f x \<ge> inverse (real (Suc n))} \<inter> space M \<in> sets M"
-      using positive le0 unfolding atLeast_def by fastsimp }
-  moreover hence "range (\<lambda> n. {x. f x \<ge> inverse (real (Suc n))} \<inter> space M) \<subseteq> sets M"
-    by auto
-  moreover
-  { fix n
-    have "inverse (real (Suc n)) \<ge> inverse (real (Suc (Suc n)))"
-      using less_imp_inverse_less real_of_nat_Suc_gt_zero[of n] by fastsimp
-    hence "\<And> x. f x \<ge> inverse (real (Suc n)) \<Longrightarrow> f x \<ge> inverse (real (Suc (Suc n)))" by (rule order_trans)
-    hence "{x. f x \<ge> inverse (real (Suc n))} \<inter> space M
-         \<subseteq> {x. f x \<ge> inverse (real (Suc (Suc n)))} \<inter> space M" by auto }
-  ultimately have "(\<lambda> x. 0) ----> measure M (\<Union> n. {x. f x \<ge> inverse (real (Suc n))} \<inter> space M)"
-    using monotone_convergence[of "\<lambda> n. {x. f x \<ge> inverse (real (Suc n))} \<inter> space M"]
-    unfolding o_def by (simp del: of_nat_Suc)
-  hence "measure M (\<Union> n. {x. f x \<ge> inverse (real (Suc n))} \<inter> space M) = 0"
-    using LIMSEQ_const[of 0] LIMSEQ_unique by simp
-  hence "measure M ((\<Union> n. {x. \<bar>f x\<bar> \<ge> inverse (real (Suc n))}) \<inter> space M) = 0"
-    using assms unfolding nonneg_def by auto
-  thus "measure M ({x. f x \<noteq> 0} \<inter> space M) = 0" using fneq0_UN by simp
+lemma split_pairs:
+  shows
+    "((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
+
+ML {*
+
+  (* tactic to solve equations of the form @{term "W * log b (X / (Y * Z)) = W * log b X - W * log b (Y * Z)"}
+     where @{term W} is a joint distribution of @{term X}, @{term Y}, and @{term Z}. *)
+
+  val mult_log_intros = [@{thm mult_log_divide}, @{thm mult_log_mult}]
+  val intros = [@{thm divide_pos_pos}, @{thm mult_pos_pos}, @{thm positive_distribution}]
+
+  val distribution_gt_0_tac = (rtac @{thm distribution_mono_gt_0}
+    THEN' assume_tac
+    THEN' clarsimp_tac (clasimpset_of @{context} addsimps2 @{thms split_pairs}))
+
+  val distr_mult_log_eq_tac = REPEAT_ALL_NEW (CHANGED o TRY o
+    (resolve_tac (mult_log_intros @ intros)
+      ORELSE' distribution_gt_0_tac
+      ORELSE' clarsimp_tac (clasimpset_of @{context})))
+
+  fun instanciate_term thy redex intro =
+    let
+      val intro_concl = Thm.concl_of intro
+
+      val lhs = intro_concl |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> fst
+
+      val m = SOME (Pattern.match thy (lhs, redex) (Vartab.empty, Vartab.empty))
+        handle Pattern.MATCH => NONE
+
+    in
+      Option.map (fn m => Envir.subst_term m intro_concl) m
+    end
+
+  fun mult_log_simproc simpset redex =
+  let
+    val ctxt = Simplifier.the_context simpset
+    val thy = ProofContext.theory_of ctxt
+    fun prove (SOME thm) = (SOME
+          (Goal.prove ctxt [] [] thm (K (distr_mult_log_eq_tac 1))
+           |> mk_meta_eq)
+            handle THM _ => NONE)
+      | prove NONE = NONE
+  in
+    get_first (instanciate_term thy (term_of redex) #> prove) mult_log_intros
+  end
+*}
+
+simproc_setup mult_log ("distribution X x * log b (A * B)" |
+                        "distribution X x * log b (A / B)") = {* K mult_log_simproc *}
+
+end
+
+lemma KL_divergence_eq_finite:
+  assumes u: "finite_measure_space (M\<lparr>measure := u\<rparr>)"
+  assumes v: "finite_measure_space (M\<lparr>measure := v\<rparr>)"
+  assumes u_0: "\<And>x. \<lbrakk> x \<in> space M ; v {x} = 0 \<rbrakk> \<Longrightarrow> u {x} = 0"
+  shows "KL_divergence b M u v = (\<Sum>x\<in>space M. u {x} * log b (u {x} / v {x}))" (is "_ = ?sum")
+proof (simp add: KL_divergence_def, subst finite_measure_space.integral_finite_singleton, simp_all add: u)
+  have ms_u: "measure_space (M\<lparr>measure := u\<rparr>)"
+    using u unfolding finite_measure_space_def by simp
+
+  show "(\<Sum>x \<in> space M. log b (measure_space.RN_deriv (M\<lparr>measure := v\<rparr>) u x) * u {x}) = ?sum"
+    apply (rule setsum_cong[OF refl])
+    apply simp
+    apply (safe intro!: arg_cong[where f="log b"] )
+    apply (subst finite_measure_space.RN_deriv_finite_singleton)
+    using assms ms_u by auto
 qed
 
-definition
-  "KL_divergence b M u v =
-    measure_space.integral (M\<lparr>measure := u\<rparr>)
-                           (\<lambda>x. log b ((measure_space.RN_deriv (M \<lparr>measure := v\<rparr> ) u) x))"
-
-lemma (in finite_prob_space) finite_measure_space:
-  shows "finite_measure_space \<lparr> space = X ` space M, sets = Pow (X ` space M), measure = distribution X\<rparr>"
-    (is "finite_measure_space ?S")
-proof (rule finite_Pow_additivity_sufficient, simp_all)
-  show "finite (X ` space M)" using finite_space by simp
-
-  show "positive ?S (distribution X)" unfolding distribution_def
-    unfolding positive_def using positive'[unfolded positive_def] sets_eq_Pow by auto
+lemma log_setsum_divide:
+  assumes "finite S" and "S \<noteq> {}" and "1 < b"
+  assumes "(\<Sum>x\<in>S. g x) = 1"
+  assumes pos: "\<And>x. x \<in> S \<Longrightarrow> g x \<ge> 0" "\<And>x. x \<in> S \<Longrightarrow> f x \<ge> 0"
+  assumes g_pos: "\<And>x. \<lbrakk> x \<in> S ; 0 < g x \<rbrakk> \<Longrightarrow> 0 < f x"
+  shows "- (\<Sum>x\<in>S. g x * log b (g x / f x)) \<le> log b (\<Sum>x\<in>S. f x)"
+proof -
+  have log_mono: "\<And>x y. 0 < x \<Longrightarrow> x \<le> y \<Longrightarrow> log b x \<le> log b y"
+    using `1 < b` by (subst log_le_cancel_iff) auto
 
-  show "additive ?S (distribution X)" unfolding additive_def distribution_def
-  proof (simp, safe)
-    fix x y
-    have x: "(X -` x) \<inter> space M \<in> sets M"
-      and y: "(X -` y) \<inter> space M \<in> sets M" using sets_eq_Pow by auto
-    assume "x \<inter> y = {}"
-    from additive[unfolded additive_def, rule_format, OF x y] this
-    have "prob (((X -` x) \<union> (X -` y)) \<inter> space M) =
-      prob ((X -` x) \<inter> space M) + prob ((X -` y) \<inter> space M)"
-      apply (subst Int_Un_distrib2)
-      by auto
-    thus "prob ((X -` x \<union> X -` y) \<inter> space M) = prob (X -` x \<inter> space M) + prob (X -` y \<inter> space M)"
-      by auto
+  have "- (\<Sum>x\<in>S. g x * log b (g x / f x)) = (\<Sum>x\<in>S. g x * log b (f x / g x))"
+  proof (unfold setsum_negf[symmetric], rule setsum_cong)
+    fix x assume x: "x \<in> S"
+    show "- (g x * log b (g x / f x)) = g x * log b (f x / g x)"
+    proof (cases "g x = 0")
+      case False
+      with pos[OF x] g_pos[OF x] have "0 < f x" "0 < g x" by simp_all
+      thus ?thesis using `1 < b` by (simp add: log_divide field_simps)
+    qed simp
+  qed rule
+  also have "... \<le> log b (\<Sum>x\<in>S. g x * (f x / g x))"
+  proof (rule log_setsum')
+    fix x assume x: "x \<in> S" "0 < g x"
+    with g_pos[OF x] show "0 < f x / g x" by (safe intro!: divide_pos_pos)
+  qed fact+
+  also have "... = log b (\<Sum>x\<in>S - {x. g x = 0}. f x)" using `finite S`
+    by (auto intro!: setsum_mono_zero_cong_right arg_cong[where f="log b"]
+        split: split_if_asm)
+  also have "... \<le> log b (\<Sum>x\<in>S. f x)"
+  proof (rule log_mono)
+    have "0 = (\<Sum>x\<in>S - {x. g x = 0}. 0)" by simp
+    also have "... < (\<Sum>x\<in>S - {x. g x = 0}. f x)" (is "_ < ?sum")
+    proof (rule setsum_strict_mono)
+      show "finite (S - {x. g x = 0})" using `finite S` by simp
+      show "S - {x. g x = 0} \<noteq> {}"
+      proof
+        assume "S - {x. g x = 0} = {}"
+        hence "(\<Sum>x\<in>S. g x) = 0" by (subst setsum_0') auto
+        with `(\<Sum>x\<in>S. g x) = 1` show False by simp
+      qed
+      fix x assume "x \<in> S - {x. g x = 0}"
+      thus "0 < f x" using g_pos[of x] pos(1)[of x] by auto
+    qed
+    finally show "0 < ?sum" .
+    show "(\<Sum>x\<in>S - {x. g x = 0}. f x) \<le> (\<Sum>x\<in>S. f x)"
+      using `finite S` pos by (auto intro!: setsum_mono2)
   qed
+  finally show ?thesis .
 qed
 
-lemma (in finite_prob_space) finite_prob_space:
-  "finite_prob_space \<lparr> space = X ` space M, sets = Pow (X ` space M), measure = distribution X\<rparr>"
-  (is "finite_prob_space ?S")
-  unfolding finite_prob_space_def prob_space_def prob_space_axioms_def
-proof safe
-  show "finite_measure_space ?S" by (rule finite_measure_space)
-  thus "measure_space ?S" by (simp add: finite_measure_space_def)
+lemma KL_divergence_positive_finite:
+  assumes u: "finite_prob_space (M\<lparr>measure := u\<rparr>)"
+  assumes v: "finite_prob_space (M\<lparr>measure := v\<rparr>)"
+  assumes u_0: "\<And>x. \<lbrakk> x \<in> space M ; v {x} = 0 \<rbrakk> \<Longrightarrow> u {x} = 0"
+  and "1 < b"
+  shows "0 \<le> KL_divergence b M u v"
+proof -
+  interpret u: finite_prob_space "M\<lparr>measure := u\<rparr>" using u .
+  interpret v: finite_prob_space "M\<lparr>measure := v\<rparr>" using v .
 
-  have "X -` X ` space M \<inter> space M = space M" by auto
-  thus "measure ?S (space ?S) = 1"
-    by (simp add: distribution_def prob_space)
-qed
+  have *: "space M \<noteq> {}" using u.not_empty by simp
 
-lemma (in finite_prob_space) finite_measure_space_image_prod:
-  "finite_measure_space \<lparr>space = X ` space M \<times> Y ` space M,
-    sets = Pow (X ` space M \<times> Y ` space M), measure_space.measure = distribution (\<lambda>x. (X x, Y x))\<rparr>"
-  (is "finite_measure_space ?Z")
-proof (rule finite_Pow_additivity_sufficient, simp_all)
-  show "finite (X ` space M \<times> Y ` space M)" using finite_space by simp
+  have "- (KL_divergence b M u v) \<le> log b (\<Sum>x\<in>space M. v {x})"
+  proof (subst KL_divergence_eq_finite, safe intro!: log_setsum_divide *)
+    show "finite_measure_space (M\<lparr>measure := u\<rparr>)"
+      "finite_measure_space (M\<lparr>measure := v\<rparr>)"
+       using u v unfolding finite_prob_space_eq by simp_all
 
-  let ?d = "distribution (\<lambda>x. (X x, Y x))"
+     show "finite (space M)" using u.finite_space by simp
+     show "1 < b" by fact
+     show "(\<Sum>x\<in>space M. u {x}) = 1" using u.sum_over_space_eq_1 by simp
 
-  show "positive ?Z ?d"
-    using sets_eq_Pow by (auto simp: positive_def distribution_def intro!: positive)
+     fix x assume x: "x \<in> space M"
+     thus pos: "0 \<le> u {x}" "0 \<le> v {x}"
+       using u.positive u.sets_eq_Pow v.positive v.sets_eq_Pow by simp_all
 
-  show "additive ?Z ?d" unfolding additive_def
-  proof safe
-    fix x y assume "x \<in> sets ?Z" and "y \<in> sets ?Z"
-    assume "x \<inter> y = {}"
-    thus "?d (x \<union> y) = ?d x + ?d y"
-      apply (simp add: distribution_def)
-      apply (subst measure_additive[unfolded sets_eq_Pow, simplified])
-      by (auto simp: Un_Int_distrib Un_Int_distrib2 intro!: arg_cong[where f=prob])
+     { assume "v {x} = 0" from u_0[OF x this] show "u {x} = 0" . }
+     { assume "0 < u {x}"
+       hence "v {x} \<noteq> 0" using u_0[OF x] by auto
+       with pos show "0 < v {x}" by simp }
   qed
+  thus "0 \<le> KL_divergence b M u v" using v.sum_over_space_eq_1 by simp
 qed
 
 definition (in prob_space)
@@ -174,346 +269,142 @@
     in
       KL_divergence b prod_space (joint_distribution X Y) (measure prod_space)"
 
-abbreviation (in finite_prob_space)
-  finite_mutual_information ("\<I>\<^bsub>_\<^esub>'(_ ; _')") where
-  "\<I>\<^bsub>b\<^esub>(X ; Y) \<equiv> mutual_information b
+abbreviation (in finite_information_space)
+  finite_mutual_information ("\<I>'(_ ; _')") where
+  "\<I>(X ; Y) \<equiv> mutual_information b
     \<lparr> space = X`space M, sets = Pow (X`space M) \<rparr>
     \<lparr> space = Y`space M, sets = Pow (Y`space M) \<rparr> X Y"
 
-abbreviation (in finite_prob_space)
-  finite_mutual_information_2 :: "('a \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'd) \<Rightarrow> real" ("\<I>'(_ ; _')") where
-  "\<I>(X ; Y) \<equiv> \<I>\<^bsub>2\<^esub>(X ; Y)"
+lemma (in finite_measure_space) measure_spaceI: "measure_space M"
+  by unfold_locales
 
-lemma (in prob_space) mutual_information_cong:
-  assumes [simp]: "space S1 = space S3" "sets S1 = sets S3"
-    "space S2 = space S4" "sets S2 = sets S4"
-  shows "mutual_information b S1 S2 X Y = mutual_information b S3 S4 X Y"
-  unfolding mutual_information_def by simp
+lemma prod_measure_times_finite:
+  assumes fms: "finite_measure_space M" "finite_measure_space M'" and a: "a \<in> space M \<times> space M'"
+  shows "prod_measure M M' {a} = measure M {fst a} * measure M' {snd a}"
+proof (cases a)
+  case (Pair b c)
+  hence a_eq: "{a} = {b} \<times> {c}" by simp
 
-lemma (in prob_space) joint_distribution:
-  "joint_distribution X Y = distribution (\<lambda>x. (X x, Y x))"
-  unfolding joint_distribution_def_raw distribution_def_raw ..
+  with fms[THEN finite_measure_space.measure_spaceI]
+    fms[THEN finite_measure_space.sets_eq_Pow] a Pair
+  show ?thesis unfolding a_eq
+    by (subst prod_measure_times) simp_all
+qed
 
-lemma (in finite_prob_space) finite_mutual_information_reduce:
-  "\<I>\<^bsub>b\<^esub>(X;Y) = (\<Sum> (x,y) \<in> X ` space M \<times> Y ` space M.
-    distribution (\<lambda>x. (X x, Y x)) {(x,y)} * log b (distribution (\<lambda>x. (X x, Y x)) {(x,y)} /
-                                                   (distribution X {x} * distribution Y {y})))"
-  (is "_ = setsum ?log ?prod")
-  unfolding Let_def mutual_information_def KL_divergence_def
-proof (subst finite_measure_space.integral_finite_singleton, simp_all add: joint_distribution)
-  let ?X = "\<lparr>space = X ` space M, sets = Pow (X ` space M), measure_space.measure = distribution X\<rparr>"
-  let ?Y = "\<lparr>space = Y ` space M, sets = Pow (Y ` space M), measure_space.measure = distribution Y\<rparr>"
-  let ?P = "prod_measure_space ?X ?Y"
+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
 
-  interpret X: finite_measure_space "?X" by (rule finite_measure_space)
-  moreover interpret Y: finite_measure_space "?Y" by (rule finite_measure_space)
-  ultimately have ms_X: "measure_space ?X" and ms_Y: "measure_space ?Y" by unfold_locales
-
-  interpret P: finite_measure_space "?P" by (rule finite_measure_space_finite_prod_measure) (fact+)
-
-  let ?P' = "measure_update (\<lambda>_. distribution (\<lambda>x. (X x, Y x))) ?P"
-  from finite_measure_space_image_prod[of X Y]
-    sigma_prod_sets_finite[OF X.finite_space Y.finite_space]
-  show "finite_measure_space ?P'"
-    by (simp add: X.sets_eq_Pow Y.sets_eq_Pow joint_distribution finite_measure_space_def prod_measure_space_def)
+lemma (in finite_information_space)
+  assumes MX: "finite_prob_space \<lparr> space = space MX, sets = sets MX, measure = distribution X\<rparr>"
+    (is "finite_prob_space ?MX")
+  assumes MY: "finite_prob_space \<lparr> space = space MY, sets = sets MY, measure = distribution Y\<rparr>"
+    (is "finite_prob_space ?MY")
+  and X_space: "X ` space M \<subseteq> space MX" and Y_space: "Y ` space M \<subseteq> space MY"
+  shows mutual_information_eq_generic:
+    "mutual_information b MX MY X Y = (\<Sum> (x,y) \<in> space MX \<times> space MY.
+      joint_distribution X Y {(x,y)} *
+      log b (joint_distribution X Y {(x,y)} /
+      (distribution X {x} * distribution Y {y})))"
+    (is "?equality")
+  and mutual_information_positive_generic:
+    "0 \<le> mutual_information b MX MY X Y" (is "?positive")
+proof -
+  let ?P = "prod_measure_space ?MX ?MY"
+  let ?measure = "joint_distribution X Y"
+  let ?P' = "measure_update (\<lambda>_. ?measure) ?P"
 
-  show "(\<Sum>x \<in> space ?P. log b (measure_space.RN_deriv ?P (distribution (\<lambda>x. (X x, Y x))) x) * distribution (\<lambda>x. (X x, Y x)) {x})
-    = setsum ?log ?prod"
-  proof (rule setsum_cong)
-    show "space ?P = ?prod" unfolding prod_measure_space_def by simp
-  next
-    fix x assume x: "x \<in> X ` space M \<times> Y ` space M"
-    then obtain d e where x_Pair: "x = (d, e)"
-      and d: "d \<in> X ` space M"
-      and e: "e \<in> Y ` space M" by auto
-
-    { fix x assume m_0: "measure ?P {x} = 0"
-      have "distribution (\<lambda>x. (X x, Y x)) {x} = 0"
-      proof (cases x)
-        case (Pair a b)
-        hence "(\<lambda>x. (X x, Y x)) -` {x} \<inter> space M = (X -` {a} \<inter> space M) \<inter> (Y -` {b} \<inter> space M)"
-          and x_prod: "{x} = {a} \<times> {b}" by auto
+  interpret X: finite_prob_space "?MX" using MX .
+  moreover interpret Y: finite_prob_space "?MY" using MY .
+  ultimately have ms_X: "measure_space ?MX"
+    and ms_Y: "measure_space ?MY" by unfold_locales
 
-        let ?PROD = "(\<lambda>x. (X x, Y x)) -` {x} \<inter> space M"
+  have fms_P: "finite_measure_space ?P"
+      by (rule finite_measure_space_finite_prod_measure) fact+
+
+  have fms_P': "finite_measure_space ?P'"
+      using finite_product_measure_space[of "space MX" "space MY"]
+        X.finite_space Y.finite_space sigma_prod_sets_finite[OF X.finite_space Y.finite_space]
+        X.sets_eq_Pow Y.sets_eq_Pow
+      by (simp add: prod_measure_space_def)
 
-        show ?thesis
-        proof (cases "{a} \<subseteq> X ` space M \<and> {b} \<subseteq> Y ` space M")
-          case False
-          hence "?PROD = {}"
-            unfolding Pair by auto
-          thus ?thesis by (auto simp: distribution_def)
-        next
-          have [intro]: "prob ?PROD \<le> 0 \<Longrightarrow> prob ?PROD = 0"
-            using sets_eq_Pow by (auto intro!: positive real_le_antisym[of _ 0])
+  { fix x assume "x \<in> space ?P"
+    hence x_in_MX: "{fst x} \<in> sets MX" using X.sets_eq_Pow
+      by (auto simp: prod_measure_space_def)
+
+    assume "measure ?P {x} = 0"
+    with prod_measure_times[OF ms_X ms_Y, of "{fst x}" "{snd x}"] x_in_MX
+    have "distribution X {fst x} = 0 \<or> distribution Y {snd x} = 0"
+      by (simp add: prod_measure_space_def)
+
+    hence "joint_distribution X Y {x} = 0"
+      by (cases x) (auto simp: distribution_order) }
+  note measure_0 = this
 
-          case True
-          with prod_measure_times[OF ms_X ms_Y, simplified, of "{a}" "{b}"]
-          have "prob (X -` {a} \<inter> space M) = 0 \<or> prob (Y -` {b} \<inter> space M) = 0" (is "?X_0 \<or> ?Y_0") using m_0
-            by (simp add: prod_measure_space_def distribution_def Pair)
-          thus ?thesis
-          proof (rule disjE)
-            assume ?X_0
-            have "prob ?PROD \<le> prob (X -` {a} \<inter> space M)"
-              using sets_eq_Pow Pair by (auto intro!: measure_mono)
-            thus ?thesis using `?X_0` by (auto simp: distribution_def)
-          next
-            assume ?Y_0
-            have "prob ?PROD \<le> prob (Y -` {b} \<inter> space M)"
-              using sets_eq_Pow Pair by (auto intro!: measure_mono)
-            thus ?thesis using `?Y_0` by (auto simp: distribution_def)
-          qed
-        qed
-      qed }
-    note measure_zero_joint_distribution = this
+  show ?equality
+    unfolding Let_def mutual_information_def using fms_P fms_P' measure_0 MX MY
+    by (subst KL_divergence_eq_finite)
+       (simp_all add: prod_measure_space_def prod_measure_times_finite
+         finite_prob_space_eq setsum_cartesian_product')
 
-    show "log b (measure_space.RN_deriv ?P (distribution (\<lambda>x. (X x, Y x))) x) * distribution (\<lambda>x. (X x, Y x)) {x} = ?log x"
-    apply (cases "distribution (\<lambda>x. (X x, Y x)) {x} \<noteq> 0")
-    apply (subst P.RN_deriv_finite_singleton)
-    proof (simp_all add: x_Pair)
-      from `finite_measure_space ?P'` show "measure_space ?P'" by (simp add: finite_measure_space_def)
-    next
-      fix x assume m_0: "measure ?P {x} = 0" thus "distribution (\<lambda>x. (X x, Y x)) {x} = 0" by fact
-    next
-      show "(d,e) \<in> space ?P" unfolding prod_measure_space_def using x x_Pair by simp
-    next
-      assume jd_0: "distribution (\<lambda>x. (X x, Y x)) {(d, e)} \<noteq> 0"
-      show "measure ?P {(d,e)} \<noteq> 0"
-      proof
-        assume "measure ?P {(d,e)} = 0"
-        from measure_zero_joint_distribution[OF this] jd_0
-        show False by simp
-      qed
-    next
-      assume jd_0: "distribution (\<lambda>x. (X x, Y x)) {(d, e)} \<noteq> 0"
-      with prod_measure_times[OF ms_X ms_Y, simplified, of "{d}" "{e}"] d
-      show "log b (distribution (\<lambda>x. (X x, Y x)) {(d, e)} / measure ?P {(d, e)}) =
-        log b (distribution (\<lambda>x. (X x, Y x)) {(d, e)} / (distribution X {d} * distribution Y {e}))"
-        by (auto intro!: arg_cong[where f="log b"] simp: prod_measure_space_def)
-    qed
+  show ?positive
+    unfolding Let_def mutual_information_def using measure_0 b_gt_1
+  proof (safe intro!: KL_divergence_positive_finite, simp_all)
+    from ms_X ms_Y X.top Y.top X.prob_space Y.prob_space
+    have "measure ?P (space ?P) = 1"
+      by (simp add: prod_measure_space_def, subst prod_measure_times, simp_all)
+    with fms_P show "finite_prob_space ?P"
+      by (simp add: finite_prob_space_eq)
+
+    from ms_X ms_Y X.top Y.top X.prob_space Y.prob_space Y.not_empty X_space Y_space
+    have "measure ?P' (space ?P') = 1" unfolding prob_space[symmetric]
+      by (auto simp add: prod_measure_space_def distribution_def vimage_Times comp_def
+        intro!: arg_cong[where f=prob])
+    with fms_P' show "finite_prob_space ?P'"
+      by (simp add: finite_prob_space_eq)
   qed
 qed
 
-lemma (in finite_prob_space) distribution_log_split:
-  assumes "1 < b"
-  shows
-  "distribution (\<lambda>x. (X x, Z x)) {(X x, z)} * log b (distribution (\<lambda>x. (X x, Z x)) {(X x, z)} /
-                                                     (distribution X {X x} * distribution Z {z})) =
-   distribution (\<lambda>x. (X x, Z x)) {(X x, z)} * log b (distribution (\<lambda>x. (X x, Z x)) {(X x, z)} /
-                                                     distribution Z {z}) -
-   distribution (\<lambda>x. (X x, Z x)) {(X x, z)} * log b (distribution X {X x})"
-  (is "?lhs = ?rhs")
-proof (cases "distribution (\<lambda>x. (X x, Z x)) {(X x, z)} = 0")
-  case True thus ?thesis by simp
-next
-  case False
-
-  let ?dZ = "distribution Z"
-  let ?dX = "distribution X"
-  let ?dXZ = "distribution (\<lambda>x. (X x, Z x))"
-
-  have dist_nneg: "\<And>x X. 0 \<le> distribution X x"
-    unfolding distribution_def using sets_eq_Pow by (auto intro: positive)
-
-  have "?lhs = ?dXZ {(X x, z)} * (log b (?dXZ {(X x, z)} / ?dZ {z}) - log b (?dX {X x}))"
-  proof -
-    have pos_dXZ: "0 < ?dXZ {(X x, z)}"
-      using False dist_nneg[of "\<lambda>x. (X x, Z x)" "{(X x, z)}"] by auto
-    moreover
-    have "((\<lambda>x. (X x, Z x)) -` {(X x, z)}) \<inter> space M \<subseteq> (X -` {X x}) \<inter> space M" by auto
-    hence "?dXZ {(X x, z)} \<le> ?dX {X x}"
-      unfolding distribution_def
-      by (rule measure_mono) (simp_all add: sets_eq_Pow)
-    with pos_dXZ have "0 < ?dX {X x}" by (rule less_le_trans)
-    moreover
-    have "((\<lambda>x. (X x, Z x)) -` {(X x, z)}) \<inter> space M \<subseteq> (Z -` {z}) \<inter> space M" by auto
-    hence "?dXZ {(X x, z)} \<le> ?dZ {z}"
-      unfolding distribution_def
-      by (rule measure_mono) (simp_all add: sets_eq_Pow)
-    with pos_dXZ have "0 < ?dZ {z}" by (rule less_le_trans)
-    moreover have "0 < b" by (rule less_trans[OF _ `1 < b`]) simp
-    moreover have "b \<noteq> 1" by (rule ccontr) (insert `1 < b`, simp)
-    ultimately show ?thesis
-      using pos_dXZ
-      apply (subst (2) mult_commute)
-      apply (subst divide_divide_eq_left[symmetric])
-      apply (subst log_divide)
-      by (auto intro: divide_pos_pos)
-  qed
-  also have "... = ?rhs"
-    by (simp add: field_simps)
-  finally show ?thesis .
-qed
-
-lemma (in finite_prob_space) finite_mutual_information_reduce_prod:
-  "mutual_information b
-    \<lparr> space = X ` space M, sets = Pow (X ` space M) \<rparr>
-    \<lparr> space = Y ` space M \<times> Z ` space M, sets = Pow (Y ` space M \<times> Z ` space M) \<rparr>
-    X (\<lambda>x. (Y x,Z x)) =
-    (\<Sum> (x, y, z) \<in> X`space M \<times> Y`space M \<times> Z`space M.
-      distribution (\<lambda>x. (X x, Y x,Z x)) {(x, y, z)} *
-      log b (distribution (\<lambda>x. (X x, Y x,Z x)) {(x, y, z)} /
-              (distribution X {x} * distribution (\<lambda>x. (Y x,Z x)) {(y,z)})))" (is "_ = setsum ?log ?space")
-  unfolding Let_def mutual_information_def KL_divergence_def using finite_space
-proof (subst finite_measure_space.integral_finite_singleton,
-       simp_all add: prod_measure_space_def sigma_prod_sets_finite joint_distribution)
-  let ?sets = "Pow (X ` space M \<times> Y ` space M \<times> Z ` space M)"
-    and ?measure = "distribution (\<lambda>x. (X x, Y x, Z x))"
-  let ?P = "\<lparr> space = ?space, sets = ?sets, measure = ?measure\<rparr>"
-
-  show "finite_measure_space ?P"
-  proof (rule finite_Pow_additivity_sufficient, simp_all)
-    show "finite ?space" using finite_space by auto
-
-    show "positive ?P ?measure"
-      using sets_eq_Pow by (auto simp: positive_def distribution_def intro!: positive)
-
-    show "additive ?P ?measure"
-    proof (simp add: additive_def distribution_def, safe)
-      fix x y assume "x \<subseteq> ?space" and "y \<subseteq> ?space"
-      assume "x \<inter> y = {}"
-      thus "prob (((\<lambda>x. (X x, Y x, Z x)) -` x \<union> (\<lambda>x. (X x, Y x, Z x)) -` y) \<inter> space M) =
-            prob ((\<lambda>x. (X x, Y x, Z x)) -` x \<inter> space M) + prob ((\<lambda>x. (X x, Y x, Z x)) -` y \<inter> space M)"
-        apply (subst measure_additive[unfolded sets_eq_Pow, simplified])
-        by (auto simp: Un_Int_distrib Un_Int_distrib2 intro!: arg_cong[where f=prob])
-    qed
-  qed
+lemma (in finite_information_space) mutual_information_eq:
+  "\<I>(X;Y) = (\<Sum> (x,y) \<in> X ` space M \<times> Y ` space M.
+    distribution (\<lambda>x. (X x, Y x)) {(x,y)} * log b (distribution (\<lambda>x. (X x, Y x)) {(x,y)} /
+                                                   (distribution X {x} * distribution Y {y})))"
+  by (subst mutual_information_eq_generic) (simp_all add: finite_prob_space_of_images)
 
-  let ?X = "\<lparr>space = X ` space M, sets = Pow (X ` space M), measure = distribution X\<rparr>"
-  and ?YZ = "\<lparr>space = Y ` space M \<times> Z ` space M, sets = Pow (Y ` space M \<times> Z ` space M), measure = distribution (\<lambda>x. (Y x, Z x))\<rparr>"
-  let ?u = "prod_measure ?X ?YZ"
-
-  from finite_measure_space[of X] finite_measure_space_image_prod[of Y Z]
-  have ms_X: "measure_space ?X" and ms_YZ: "measure_space ?YZ"
-    by (simp_all add: finite_measure_space_def)
-
-  show "(\<Sum>x \<in> ?space. log b (measure_space.RN_deriv \<lparr>space=?space, sets=?sets, measure=?u\<rparr>
-    (distribution (\<lambda>x. (X x, Y x, Z x))) x) * distribution (\<lambda>x. (X x, Y x, Z x)) {x})
-    = setsum ?log ?space"
-  proof (rule setsum_cong)
-    fix x assume x: "x \<in> ?space"
-    then obtain d e f where x_Pair: "x = (d, e, f)"
-      and d: "d \<in> X ` space M"
-      and e: "e \<in> Y ` space M"
-      and f: "f \<in> Z ` space M" by auto
-
-    { fix x assume m_0: "?u {x} = 0"
-
-      let ?PROD = "(\<lambda>x. (X x, Y x, Z x)) -` {x} \<inter> space M"
-      obtain a b c where Pair: "x = (a, b, c)" by (cases x)
-      hence "?PROD = (X -` {a} \<inter> space M) \<inter> ((\<lambda>x. (Y x, Z x)) -` {(b, c)} \<inter> space M)"
-        and x_prod: "{x} = {a} \<times> {(b, c)}" by auto
-
-      have "distribution (\<lambda>x. (X x, Y x, Z x)) {x} = 0"
-      proof (cases "{a} \<subseteq> X ` space M")
-        case False
-        hence "?PROD = {}"
-          unfolding Pair by auto
-        thus ?thesis by (auto simp: distribution_def)
-      next
-        have [intro]: "prob ?PROD \<le> 0 \<Longrightarrow> prob ?PROD = 0"
-          using sets_eq_Pow by (auto intro!: positive real_le_antisym[of _ 0])
-
-        case True
-        with prod_measure_times[OF ms_X ms_YZ, simplified, of "{a}" "{(b,c)}"]
-        have "prob (X -` {a} \<inter> space M) = 0 \<or> prob ((\<lambda>x. (Y x, Z x)) -` {(b, c)} \<inter> space M) = 0"
-          (is "prob ?X = 0 \<or> prob ?Y = 0") using m_0
-          by (simp add: prod_measure_space_def distribution_def Pair)
-        thus ?thesis
-        proof (rule disjE)
-          assume "prob ?X = 0"
-          have "prob ?PROD \<le> prob ?X"
-            using sets_eq_Pow Pair by (auto intro!: measure_mono)
-          thus ?thesis using `prob ?X = 0` by (auto simp: distribution_def)
-        next
-          assume "prob ?Y = 0"
-          have "prob ?PROD \<le> prob ?Y"
-            using sets_eq_Pow Pair by (auto intro!: measure_mono)
-          thus ?thesis using `prob ?Y = 0` by (auto simp: distribution_def)
-        qed
-      qed }
-    note measure_zero_joint_distribution = this
-
-    from x_Pair d e f finite_space
-    show "log b (measure_space.RN_deriv \<lparr>space=?space, sets=?sets, measure=?u\<rparr>
-      (distribution (\<lambda>x. (X x, Y x, Z x))) x) * distribution (\<lambda>x. (X x, Y x, Z x)) {x} = ?log x"
-    apply (cases "distribution (\<lambda>x. (X x, Y x, Z x)) {x} \<noteq> 0")
-    apply (subst finite_measure_space.RN_deriv_finite_singleton)
-    proof simp_all
-      show "measure_space ?P" using `finite_measure_space ?P` by (simp add: finite_measure_space_def)
-
-      from finite_measure_space_finite_prod_measure[OF finite_measure_space[of X]
-        finite_measure_space_image_prod[of Y Z]] finite_space
-      show "finite_measure_space \<lparr>space=?space, sets=?sets, measure=?u\<rparr>"
-        by (simp add: prod_measure_space_def sigma_prod_sets_finite)
-    next
-      fix x assume "?u {x} = 0" thus "distribution (\<lambda>x. (X x, Y x, Z x)) {x} = 0" by fact
-    next
-      assume jd_0: "distribution (\<lambda>x. (X x, Y x, Z x)) {(d, e, f)} \<noteq> 0"
-      show "?u {(d,e,f)} \<noteq> 0"
-      proof
-        assume "?u {(d, e, f)} = 0"
-        from measure_zero_joint_distribution[OF this] jd_0
-        show False by simp
-      qed
-    next
-      assume jd_0: "distribution (\<lambda>x. (X x, Y x, Z x)) {(d, e, f)} \<noteq> 0"
-      with prod_measure_times[OF ms_X ms_YZ, simplified, of "{d}" "{(e,f)}"] d
-      show "log b (distribution (\<lambda>x. (X x, Y x, Z x)) {(d, e, f)} / ?u {(d, e, f)}) =
-        log b (distribution (\<lambda>x. (X x, Y x, Z x)) {(d, e, f)} / (distribution X {d} * distribution (\<lambda>x. (Y x, Z x)) {(e,f)}))"
-        by (auto intro!: arg_cong[where f="log b"] simp: prod_measure_space_def)
-    qed
-  qed simp
-qed
+lemma (in finite_information_space) mutual_information_positive: "0 \<le> \<I>(X;Y)"
+  by (subst mutual_information_positive_generic) (simp_all add: finite_prob_space_of_images)
 
 definition (in prob_space)
   "entropy b s X = mutual_information b s s X X"
 
-abbreviation (in finite_prob_space)
-  finite_entropy ("\<H>\<^bsub>_\<^esub>'(_')") where
-  "\<H>\<^bsub>b\<^esub>(X) \<equiv> entropy b \<lparr> space = X`space M, sets = Pow (X`space M) \<rparr> X"
-
-abbreviation (in finite_prob_space)
-  finite_entropy_2 ("\<H>'(_')") where
-  "\<H>(X) \<equiv> \<H>\<^bsub>2\<^esub>(X)"
+abbreviation (in finite_information_space)
+  finite_entropy ("\<H>'(_')") where
+  "\<H>(X) \<equiv> entropy b \<lparr> space = X`space M, sets = Pow (X`space M) \<rparr> X"
 
-lemma (in finite_prob_space) finite_entropy_reduce:
-  assumes "1 < b"
-  shows "\<H>\<^bsub>b\<^esub>(X) = -(\<Sum> x \<in> X ` space M. distribution X {x} * log b (distribution X {x}))"
+lemma (in finite_information_space) joint_distribution_remove[simp]:
+    "joint_distribution X X {(x, x)} = distribution X {x}"
+  unfolding distribution_def by (auto intro!: arg_cong[where f=prob])
+
+lemma (in finite_information_space) entropy_eq:
+  "\<H>(X) = -(\<Sum> x \<in> X ` space M. distribution X {x} * log b (distribution X {x}))"
 proof -
-  have fin: "finite (X ` space M)" using finite_space by simp
-
-  have If_mult_distr: "\<And>A B C D. If A B C * D = If A (B * D) (C * D)" by auto
-
+  { fix f
   { fix x y
     have "(\<lambda>x. (X x, X x)) -` {(x, y)} = (if x = y then X -` {x} else {})" by auto
-    hence "distribution (\<lambda>x. (X x, X x))  {(x,y)} = (if x = y then distribution X {x} else 0)"
+      hence "distribution (\<lambda>x. (X x, X x))  {(x,y)} * f x y = (if x = y then distribution X {x} * f x y else 0)"
       unfolding distribution_def by auto }
-  moreover
-  have "\<And>x. 0 \<le> distribution X x"
-    unfolding distribution_def using finite_space sets_eq_Pow by (auto intro: positive)
-  hence "\<And>x. distribution X x \<noteq> 0 \<Longrightarrow> 0 < distribution X x" by (auto simp: le_less)
-  ultimately
-  show ?thesis using `1 < b`
-    by (auto intro!: setsum_cong
-      simp: log_inverse If_mult_distr setsum_cases[OF fin] inverse_eq_divide[symmetric]
-        entropy_def setsum_negf[symmetric] joint_distribution finite_mutual_information_reduce
-        setsum_cartesian_product[symmetric])
+    hence "(\<Sum>(x, y) \<in> X ` space M \<times> X ` space M. joint_distribution X X {(x, y)} * f x y) =
+      (\<Sum>x \<in> X ` space M. distribution X {x} * f x x)"
+      unfolding setsum_cartesian_product' by (simp add: setsum_cases finite_space) }
+  note remove_cartesian_product = this
+
+  show ?thesis
+    unfolding entropy_def mutual_information_eq setsum_negf[symmetric] remove_cartesian_product
+    by (auto intro!: setsum_cong)
 qed
 
-lemma log_inj: assumes "1 < b" shows "inj_on (log b) {0 <..}"
-proof (rule inj_onI, simp)
-  fix x y assume pos: "0 < x" "0 < y" and *: "log b x = log b y"
-  show "x = y"
-  proof (cases rule: linorder_cases)
-    assume "x < y" hence "log b x < log b y"
-      using log_less_cancel_iff[OF `1 < b`] pos by simp
-    thus ?thesis using * by simp
-  next
-    assume "y < x" hence "log b y < log b x"
-      using log_less_cancel_iff[OF `1 < b`] pos by simp
-    thus ?thesis using * by simp
-  qed simp
-qed
+lemma (in finite_information_space) entropy_positive: "0 \<le> \<H>(X)"
+  unfolding entropy_def using mutual_information_positive .
 
 definition (in prob_space)
   "conditional_mutual_information b s1 s2 s3 X Y Z \<equiv>
@@ -524,160 +415,181 @@
       mutual_information b s1 prod_space X (\<lambda>x. (Y x, Z x)) -
       mutual_information b s1 s3 X Z"
 
-abbreviation (in finite_prob_space)
-  finite_conditional_mutual_information ("\<I>\<^bsub>_\<^esub>'( _ ; _ | _ ')") where
-  "\<I>\<^bsub>b\<^esub>(X ; Y | Z) \<equiv> conditional_mutual_information b
+abbreviation (in finite_information_space)
+  finite_conditional_mutual_information ("\<I>'( _ ; _ | _ ')") where
+  "\<I>(X ; Y | Z) \<equiv> conditional_mutual_information b
     \<lparr> space = X`space M, sets = Pow (X`space M) \<rparr>
     \<lparr> space = Y`space M, sets = Pow (Y`space M) \<rparr>
     \<lparr> space = Z`space M, sets = Pow (Z`space M) \<rparr>
     X Y Z"
 
-abbreviation (in finite_prob_space)
-  finite_conditional_mutual_information_2 ("\<I>'( _ ; _ | _ ')") where
-  "\<I>(X ; Y | Z) \<equiv> \<I>\<^bsub>2\<^esub>(X ; Y | Z)"
+lemma (in finite_information_space) setsum_distribution_gen:
+  assumes "Z -` {c} \<inter> space M = (\<Union>x \<in> X`space M. Y -` {f x}) \<inter> space M"
+  and "inj_on f (X`space M)"
+  shows "(\<Sum>x \<in> X`space M. distribution Y {f x}) = distribution Z {c}"
+  unfolding distribution_def assms
+  using finite_space assms
+  by (subst measure_finitely_additive'')
+     (auto simp add: disjoint_family_on_def sets_eq_Pow inj_on_def
+      intro!: arg_cong[where f=prob])
+
+lemma (in finite_information_space) setsum_distribution:
+  "(\<Sum>x \<in> X`space M. joint_distribution X Y {(x, y)}) = distribution Y {y}"
+  "(\<Sum>y \<in> Y`space M. joint_distribution X Y {(x, y)}) = distribution X {x}"
+  "(\<Sum>x \<in> X`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution Y Z {(y, z)}"
+  "(\<Sum>y \<in> Y`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution X Z {(x, z)}"
+  "(\<Sum>z \<in> Z`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution X Y {(x, y)}"
+  by (auto intro!: inj_onI setsum_distribution_gen)
 
-lemma image_pair_eq_Sigma:
-  "(\<lambda>x. (f x, g x)) ` A = Sigma (f ` A) (\<lambda>x. g ` (f -` {x} \<inter> A))"
-proof (safe intro!: imageI vimageI, simp_all)
-  fix a b assume *: "a \<in> A" "b \<in> A" and eq: "f a = f b"
-  show "(f b, g a) \<in> (\<lambda>x. (f x, g x)) ` A" unfolding eq[symmetric]
-    using * by auto
+lemma (in finite_information_space) conditional_mutual_information_eq_sum:
+   "\<I>(X ; Y | Z) =
+     (\<Sum>(x, y, z)\<in>X ` space M \<times> (\<lambda>x. (Y x, Z x)) ` space M.
+             distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} *
+             log b (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)}/
+        distribution (\<lambda>x. (Y x, Z x)) {(y, z)})) -
+     (\<Sum>(x, z)\<in>X ` space M \<times> Z ` space M.
+        distribution (\<lambda>x. (X x, Z x)) {(x,z)} * log b (distribution (\<lambda>x. (X x, Z x)) {(x,z)} / distribution Z {z}))"
+  (is "_ = ?rhs")
+proof -
+  have setsum_product:
+    "\<And>f x. (\<Sum>v\<in>(\<lambda>x. (Y x, Z x)) ` space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x,v)} * f v)
+      = (\<Sum>v\<in>Y ` space M \<times> Z ` space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x,v)} * f v)"
+  proof (safe intro!: setsum_mono_zero_cong_left imageI)
+    fix x y z f
+    assume *: "(Y y, Z z) \<notin> (\<lambda>x. (Y x, Z x)) ` space M" and "y \<in> space M" "z \<in> space M"
+    hence "(\<lambda>x. (X x, Y x, Z x)) -` {(x, Y y, Z z)} \<inter> space M = {}"
+    proof safe
+      fix x' assume x': "x' \<in> space M" and eq: "Y x' = Y y" "Z x' = Z z"
+      have "(Y y, Z z) \<in> (\<lambda>x. (Y x, Z x)) ` space M" using eq[symmetric] x' by auto
+      thus "x' \<in> {}" using * by auto
+    qed
+    thus "joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, Y y, Z z)} * f (Y y) (Z z) = 0"
+      unfolding distribution_def by simp
+  qed (simp add: finite_space)
+
+  thus ?thesis
+    unfolding conditional_mutual_information_def Let_def mutual_information_eq
+    apply (subst mutual_information_eq_generic)
+    by (auto simp add: prod_measure_space_def sigma_prod_sets_finite finite_space
+        finite_prob_space_of_images finite_product_prob_space_of_images
+        setsum_cartesian_product' setsum_product setsum_subtractf setsum_addf
+        setsum_left_distrib[symmetric] setsum_distribution
+      cong: setsum_cong)
 qed
 
-lemma inj_on_swap: "inj_on (\<lambda>(x,y). (y,x)) A" by (auto intro!: inj_onI)
-
-lemma (in finite_prob_space) finite_conditional_mutual_information_reduce:
-  assumes "1 < b"
-  shows "\<I>\<^bsub>b\<^esub>(X ; Y | Z) =
-	- (\<Sum> (x, z) \<in> (X ` space M \<times> Z ` space M).
-             distribution (\<lambda>x. (X x, Z x)) {(x,z)} * log b (distribution (\<lambda>x. (X x, Z x)) {(x,z)} / distribution Z {z}))
-	+ (\<Sum> (x, y, z) \<in> (X ` space M \<times> (\<lambda>x. (Y x, Z x)) ` space M).
+lemma (in finite_information_space) conditional_mutual_information_eq:
+  "\<I>(X ; Y | Z) = (\<Sum>(x, y, z) \<in> X ` space M \<times> Y ` space M \<times> Z ` space M.
              distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} *
              log b (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)}/
-             distribution (\<lambda>x. (Y x, Z x)) {(y, z)}))" (is "_ = ?rhs")
-unfolding conditional_mutual_information_def Let_def using finite_space
-apply (simp add: prod_measure_space_def sigma_prod_sets_finite)
-apply (subst mutual_information_cong[of _ "\<lparr>space = X ` space M, sets = Pow (X ` space M)\<rparr>"
-  _ "\<lparr>space = Y ` space M \<times> Z ` space M, sets = Pow (Y ` space M \<times> Z ` space M)\<rparr>"], simp_all)
-apply (subst finite_mutual_information_reduce_prod, simp_all)
-apply (subst finite_mutual_information_reduce, simp_all)
+    (joint_distribution X Z {(x, z)} * joint_distribution Y Z {(y,z)} / distribution Z {z})))"
+  unfolding conditional_mutual_information_def Let_def mutual_information_eq
+    apply (subst mutual_information_eq_generic)
+  by (auto simp add: prod_measure_space_def sigma_prod_sets_finite finite_space
+      finite_prob_space_of_images finite_product_prob_space_of_images
+      setsum_cartesian_product' setsum_product setsum_subtractf setsum_addf
+      setsum_left_distrib[symmetric] setsum_distribution setsum_commute[where A="Y`space M"]
+    cong: setsum_cong)
+
+lemma (in finite_information_space) conditional_mutual_information_eq_mutual_information:
+  "\<I>(X ; Y) = \<I>(X ; Y | (\<lambda>x. ()))"
+proof -
+  have [simp]: "(\<lambda>x. ()) ` space M = {()}" using not_empty by auto
+
+  show ?thesis
+    unfolding conditional_mutual_information_eq mutual_information_eq
+    by (simp add: setsum_cartesian_product' distribution_remove_const)
+qed
+
+lemma (in finite_information_space) conditional_mutual_information_positive:
+  "0 \<le> \<I>(X ; Y | Z)"
 proof -
   let ?dXYZ = "distribution (\<lambda>x. (X x, Y x, Z x))"
-  let ?dXZ = "distribution (\<lambda>x. (X x, Z x))"
-  let ?dYZ = "distribution (\<lambda>x. (Y x, Z x))"
+  let ?dXZ = "joint_distribution X Z"
+  let ?dYZ = "joint_distribution Y Z"
   let ?dX = "distribution X"
-  let ?dY = "distribution Y"
   let ?dZ = "distribution Z"
+  let ?M = "X ` space M \<times> Y ` space M \<times> Z ` space M"
+
+  have split_beta: "\<And>f. split f = (\<lambda>x. f (fst x) (snd x))" by (simp add: expand_fun_eq)
 
-  have If_mult_distr: "\<And>A B C D. If A B C * D = If A (B * D) (C * D)" by auto
-  { fix x y
-    have "(\<lambda>x. (X x, Y x, Z x)) -` {(X x, y)} \<inter> space M =
-      (if y \<in> (\<lambda>x. (Y x, Z x)) ` space M then (\<lambda>x. (X x, Y x, Z x)) -` {(X x, y)} \<inter> space M else {})" by auto
-    hence "?dXYZ {(X x, y)} = (if y \<in> (\<lambda>x. (Y x, Z x)) ` space M then ?dXYZ {(X x, y)} else 0)"
-      unfolding distribution_def by auto }
-  note split_measure = this
-
-  have sets: "Y ` space M \<times> Z ` space M \<inter> (\<lambda>x. (Y x, Z x)) ` space M = (\<lambda>x. (Y x, Z x)) ` space M" by auto
-
-  have cong: "\<And>A B C D. \<lbrakk> A = C ; B = D \<rbrakk> \<Longrightarrow> A + B = C + D" by auto
-
-  { fix A f have "setsum f A = setsum (\<lambda>(x, y). f (y, x)) ((\<lambda>(x, y). (y, x)) ` A)"
-    using setsum_reindex[OF inj_on_swap, of "\<lambda>(x, y). f (y, x)" A] by (simp add: split_twice) }
-  note setsum_reindex_swap = this
-
-  { fix A B f assume *: "finite A" "\<forall>x\<in>A. finite (B x)"
-    have "(\<Sum>x\<in>Sigma A B. f x) = (\<Sum>x\<in>A. setsum (\<lambda>y. f (x, y)) (B x))"
-      unfolding setsum_Sigma[OF *] by simp }
-  note setsum_Sigma = this
+  have "- (\<Sum>(x, y, z) \<in> ?M. ?dXYZ {(x, y, z)} *
+    log b (?dXYZ {(x, y, z)} / (?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z})))
+    \<le> log b (\<Sum>(x, y, z) \<in> ?M. ?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z})"
+    unfolding split_beta
+  proof (rule log_setsum_divide)
+    show "?M \<noteq> {}" using not_empty by simp
+    show "1 < b" using b_gt_1 .
 
-  { fix x
-    have "(\<Sum>z\<in>Z ` space M. ?dXZ {(X x, z)}) = (\<Sum>yz\<in>(\<lambda>x. (Y x, Z x)) ` space M. ?dXYZ {(X x, yz)})"
-      apply (subst setsum_reindex_swap)
-      apply (simp add: image_image distribution_def)
-      unfolding image_pair_eq_Sigma
-      apply (subst setsum_Sigma)
-      using finite_space apply simp_all
-      apply (rule setsum_cong[OF refl])
-      apply (subst measure_finitely_additive'')
-      by (auto simp add: disjoint_family_on_def sets_eq_Pow intro!: arg_cong[where f=prob]) }
+    fix x assume "x \<in> ?M"
+    show "0 \<le> ?dXYZ {(fst x, fst (snd x), snd (snd x))}" using positive_distribution .
+    show "0 \<le> ?dXZ {(fst x, snd (snd x))} * ?dYZ {(fst (snd x), snd (snd x))} / ?dZ {snd (snd x)}"
+      by (auto intro!: mult_nonneg_nonneg positive_distribution simp: zero_le_divide_iff)
 
-  thus "(\<Sum>(x, y, z)\<in>X ` space M \<times> Y ` space M \<times> Z ` space M.
-      ?dXYZ {(x, y, z)} * log b (?dXYZ {(x, y, z)} / (?dX {x} * ?dYZ {(y, z)}))) -
-    (\<Sum>(x, y)\<in>X ` space M \<times> Z ` space M.
-      ?dXZ {(x, y)} * log b (?dXZ {(x, y)} / (?dX {x} * ?dZ {y}))) =
-  - (\<Sum> (x, z) \<in> (X ` space M \<times> Z ` space M).
-      ?dXZ {(x,z)} * log b (?dXZ {(x,z)} / ?dZ {z})) +
-    (\<Sum> (x, y, z) \<in> (X ` space M \<times> (\<lambda>x. (Y x, Z x)) ` space M).
-      ?dXYZ {(x, y, z)} * log b (?dXYZ {(x, y, z)} / ?dYZ {(y, z)}))"
-    using finite_space
-    apply (auto simp: setsum_cartesian_product[symmetric] setsum_negf[symmetric]
-                      setsum_addf[symmetric] diff_minus
-      intro!: setsum_cong[OF refl])
-    apply (subst split_measure)
-    apply (simp add: If_mult_distr setsum_cases sets distribution_log_split[OF assms, of X])
-    apply (subst add_commute)
-    by (simp add: setsum_subtractf setsum_negf field_simps setsum_right_distrib[symmetric] sets_eq_Pow)
+    assume *: "0 < ?dXYZ {(fst x, fst (snd x), snd (snd x))}"
+    thus "0 < ?dXZ {(fst x, snd (snd x))} * ?dYZ {(fst (snd x), snd (snd x))} / ?dZ {snd (snd x)}"
+      by (auto intro!: divide_pos_pos mult_pos_pos
+           intro: distribution_order(6) distribution_mono_gt_0)
+  qed (simp_all add: setsum_cartesian_product' sum_over_space_distrib setsum_distribution finite_space)
+  also have "(\<Sum>(x, y, z) \<in> ?M. ?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z}) = (\<Sum>z\<in>Z`space M. ?dZ {z})"
+    apply (simp add: setsum_cartesian_product')
+    apply (subst setsum_commute)
+    apply (subst (2) setsum_commute)
+    by (auto simp: setsum_divide_distrib[symmetric] setsum_product[symmetric] setsum_distribution
+          intro!: setsum_cong)
+  finally show ?thesis
+    unfolding conditional_mutual_information_eq sum_over_space_distrib by simp
 qed
 
+
 definition (in prob_space)
   "conditional_entropy b S T X Y = conditional_mutual_information b S S T X X Y"
 
-abbreviation (in finite_prob_space)
-  finite_conditional_entropy ("\<H>\<^bsub>_\<^esub>'(_ | _')") where
-  "\<H>\<^bsub>b\<^esub>(X | Y) \<equiv> conditional_entropy b
+abbreviation (in finite_information_space)
+  finite_conditional_entropy ("\<H>'(_ | _')") where
+  "\<H>(X | Y) \<equiv> conditional_entropy b
     \<lparr> space = X`space M, sets = Pow (X`space M) \<rparr>
     \<lparr> space = Y`space M, sets = Pow (Y`space M) \<rparr> X Y"
 
-abbreviation (in finite_prob_space)
-  finite_conditional_entropy_2 ("\<H>'(_ | _')") where
-  "\<H>(X | Y) \<equiv> \<H>\<^bsub>2\<^esub>(X | Y)"
+lemma (in finite_information_space) conditional_entropy_positive:
+  "0 \<le> \<H>(X | Y)" unfolding conditional_entropy_def using conditional_mutual_information_positive .
 
-lemma (in finite_prob_space) finite_conditional_entropy_reduce:
-  assumes "1 < b"
-  shows "\<H>\<^bsub>b\<^esub>(X | Z) =
+lemma (in finite_information_space) conditional_entropy_eq:
+  "\<H>(X | Z) =
      - (\<Sum>(x, z)\<in>X ` space M \<times> Z ` space M.
          joint_distribution X Z {(x, z)} *
          log b (joint_distribution X Z {(x, z)} / distribution Z {z}))"
 proof -
   have *: "\<And>x y z. (\<lambda>x. (X x, X x, Z x)) -` {(x, y, z)} = (if x = y then (\<lambda>x. (X x, Z x)) -` {(x, z)} else {})" by auto
   show ?thesis
-    unfolding finite_conditional_mutual_information_reduce[OF assms]
-      conditional_entropy_def joint_distribution_def distribution_def *
+    unfolding conditional_mutual_information_eq_sum
+      conditional_entropy_def distribution_def *
     by (auto intro!: setsum_0')
 qed
 
-lemma (in finite_prob_space) finite_mutual_information_eq_entropy_conditional_entropy:
-  assumes "1 < b" shows "\<I>\<^bsub>b\<^esub>(X ; Z) = \<H>\<^bsub>b\<^esub>(X) - \<H>\<^bsub>b\<^esub>(X | Z)" (is "mutual_information b ?X ?Z X Z = _")
-  unfolding finite_mutual_information_reduce
-    finite_entropy_reduce[OF assms]
-    finite_conditional_entropy_reduce[OF assms]
-    joint_distribution diff_minus_eq_add
+lemma (in finite_information_space) mutual_information_eq_entropy_conditional_entropy:
+  "\<I>(X ; Z) = \<H>(X) - \<H>(X | Z)"
+  unfolding mutual_information_eq entropy_eq conditional_entropy_eq
   using finite_space
-  apply (auto simp add: setsum_addf[symmetric] setsum_subtractf
-      setsum_Sigma[symmetric] distribution_log_split[OF assms] setsum_negf[symmetric]
-    intro!: setsum_cong[OF refl])
-  apply (simp add: setsum_negf setsum_left_distrib[symmetric])
-proof (rule disjI2)
-  let ?dX = "distribution X"
-  and ?dXZ = "distribution (\<lambda>x. (X x, Z x))"
+  by (auto simp add: setsum_addf setsum_subtractf setsum_cartesian_product'
+      setsum_left_distrib[symmetric] setsum_addf setsum_distribution)
 
-  fix x assume "x \<in> space M"
-  have "\<And>z. (\<lambda>x. (X x, Z x)) -` {(X x, z)} \<inter> space M = (X -` {X x} \<inter> space M) \<inter> (Z -` {z} \<inter> space M)" by auto
-  thus "(\<Sum>z\<in>Z ` space M. distribution (\<lambda>x. (X x, Z x)) {(X x, z)}) = distribution X {X x}"
-    unfolding distribution_def
-    apply (subst prob_real_sum_image_fn[where e="X -` {X x} \<inter> space M" and s = "Z`space M" and f="\<lambda>z. Z -` {z} \<inter> space M"])
-    using finite_space sets_eq_Pow by auto
+lemma (in finite_information_space) conditional_entropy_less_eq_entropy:
+  "\<H>(X | Z) \<le> \<H>(X)"
+proof -
+  have "\<I>(X ; Z) = \<H>(X) - \<H>(X | Z)" using mutual_information_eq_entropy_conditional_entropy .
+  with mutual_information_positive[of X Z] entropy_positive[of X]
+  show ?thesis by auto
 qed
 
 (* -------------Entropy of a RV with a certain event is zero---------------- *)
 
-lemma (in finite_prob_space) finite_entropy_certainty_eq_0:
-  assumes "x \<in> X ` space M" and "distribution X {x} = 1" and "b > 1"
-  shows "\<H>\<^bsub>b\<^esub>(X) = 0"
+lemma (in finite_information_space) finite_entropy_certainty_eq_0:
+  assumes "x \<in> X ` space M" and "distribution X {x} = 1"
+  shows "\<H>(X) = 0"
 proof -
   interpret X: finite_prob_space "\<lparr> space = X ` space M,
     sets = Pow (X ` space M),
-    measure = distribution X\<rparr>" by (rule finite_prob_space)
+    measure = distribution X\<rparr>" by (rule finite_prob_space_of_images)
 
   have "distribution X (X ` space M - {x}) = distribution X (X ` space M) - distribution X {x}"
     using X.measure_compl[of "{x}"] assms by auto
@@ -694,34 +606,18 @@
 
   have y: "\<And>y. (if x = y then 1 else 0) * log b (if x = y then 1 else 0) = 0" by simp
 
-  show ?thesis
-    unfolding finite_entropy_reduce[OF `b > 1`] by (auto simp: y fi)
+  show ?thesis unfolding entropy_eq by (auto simp: y fi)
 qed
 (* --------------- upper bound on entropy for a rv ------------------------- *)
 
-lemma log_setsum:
-  assumes "finite s" "s \<noteq> {}"
-  assumes "b > 1"
-  assumes "(\<Sum> i \<in> s. a i) = 1"
-  assumes "\<And> i. i \<in> s \<Longrightarrow> a i \<ge> 0"
-  assumes "\<And> i. i \<in> s \<Longrightarrow> y i \<in> {0 <..}"
-  shows "log b (\<Sum> i \<in> s. a i * y i) \<ge> (\<Sum> i \<in> s. a i * log b (y i))"
-proof -
-  have "convex_on {0 <..} (\<lambda> x. - log b x)"
-    by (rule minus_log_convex[OF `b > 1`])
-  hence "- log b (\<Sum> i \<in> s. a i * y i) \<le> (\<Sum> i \<in> s. a i * - log b (y i))"
-    using convex_on_setsum[of _ _ "\<lambda> x. - log b x"] assms pos_is_convex by fastsimp
-  thus ?thesis by (auto simp add:setsum_negf le_imp_neg_le)
-qed
-
-lemma (in finite_prob_space) finite_entropy_le_card:
-  assumes "1 < b"
-  shows "\<H>\<^bsub>b\<^esub>(X) \<le> log b (real (card (X ` space M \<inter> {x . distribution X {x} \<noteq> 0})))"
+lemma (in finite_information_space) finite_entropy_le_card:
+  "\<H>(X) \<le> log b (real (card (X ` space M \<inter> {x . distribution X {x} \<noteq> 0})))"
 proof -
   interpret X: finite_prob_space "\<lparr>space = X ` space M,
                                     sets = Pow (X ` space M),
                                  measure = distribution X\<rparr>"
-    using finite_prob_space by auto
+    using finite_prob_space_of_images by auto
+
   have triv: "\<And> x. (if distribution X {x} \<noteq> 0 then distribution X {x} else 0) = distribution X {x}"
     by auto
   hence sum1: "(\<Sum> x \<in> X ` space M \<inter> {y. distribution X {y} \<noteq> 0}. distribution X {x}) = 1"
@@ -753,7 +649,7 @@
     also have "\<dots> = (if distribution X {x} \<noteq> 0
                     then distribution X {x} * log b (inverse (distribution X {x}))
                     else 0)"
-      using log_inverse `1 < b` X.positive[of "{x}"] asm by auto
+      using log_inverse b_gt_1 X.positive[of "{x}"] asm by auto
     finally have "- distribution X {x} * log b (distribution X {x})
                  = (if distribution X {x} \<noteq> 0 
                     then distribution X {x} * log b (inverse (distribution X {x}))
@@ -769,7 +665,7 @@
     unfolding setsum_restrict_set[OF finite_imageI[OF finite_space, of X]] by auto
   also have "\<dots> \<le> log b (\<Sum> x \<in> X ` space M \<inter> {y. distribution X {y} \<noteq> 0}.
                           distribution X {x} * (inverse (distribution X {x})))"
-    apply (subst log_setsum[OF _ _ `b > 1` sum1, 
+    apply (subst log_setsum[OF _ _ b_gt_1 sum1, 
      unfolded greaterThan_iff, OF _ _ _]) using pos sets_eq_Pow
       X.finite_space assms X.positive not_empty by auto
   also have "\<dots> = log b (\<Sum> x \<in> X ` space M \<inter> {y. distribution X {y} \<noteq> 0}. 1)"
@@ -778,7 +674,7 @@
     by auto
   finally have "- (\<Sum>x\<in>X ` space M. distribution X {x} * log b (distribution X {x}))
                \<le> log b (real_of_nat (card (X ` space M \<inter> {y. distribution X {y} \<noteq> 0})))" by simp
-  thus ?thesis unfolding finite_entropy_reduce[OF assms] real_eq_of_nat by auto
+  thus ?thesis unfolding entropy_eq real_eq_of_nat by auto
 qed
 
 (* --------------- entropy is maximal for a uniform rv --------------------- *)
@@ -808,34 +704,31 @@
     by (auto simp:field_simps)
 qed
 
-lemma (in finite_prob_space) finite_entropy_uniform_max:
-  assumes "b > 1"
+lemma (in finite_information_space) finite_entropy_uniform_max:
   assumes "\<And>x y. \<lbrakk> x \<in> X ` space M ; y \<in> X ` space M \<rbrakk> \<Longrightarrow> distribution X {x} = distribution X {y}"
-  shows "\<H>\<^bsub>b\<^esub>(X) = log b (real (card (X ` space M)))"
+  shows "\<H>(X) = log b (real (card (X ` space M)))"
 proof -
   interpret X: finite_prob_space "\<lparr>space = X ` space M,
                                     sets = Pow (X ` space M),
                                  measure = distribution X\<rparr>"
-    using finite_prob_space by auto
+    using finite_prob_space_of_images by auto
+
   { fix x assume xasm: "x \<in> X ` space M"
     hence card_gt0: "real (card (X ` space M)) > 0"
       using card_gt_0_iff X.finite_space by auto
     from xasm have "\<And> y. y \<in> X ` space M \<Longrightarrow> distribution X {y} = distribution X {x}"
       using assms by blast
     hence "- (\<Sum>x\<in>X ` space M. distribution X {x} * log b (distribution X {x}))
-         = - (\<Sum> y \<in> X ` space M. distribution X {x} * log b (distribution X {x}))"
-      by auto
-    also have "\<dots> = - real_of_nat (card (X ` space M)) * distribution X {x} * log b (distribution X {x})"
-      by auto
+         = - real (card (X ` space M)) * distribution X {x} * log b (distribution X {x})"
+      unfolding real_eq_of_nat by auto
     also have "\<dots> = - real (card (X ` space M)) * (1 / real (card (X ` space M))) * log b (1 / real (card (X ` space M)))"
-      unfolding real_eq_of_nat[symmetric]
-      by (auto simp: X.uniform_prob[simplified, OF xasm assms(2)])
+      by (auto simp: X.uniform_prob[simplified, OF xasm assms])
     also have "\<dots> = log b (real (card (X ` space M)))"
       unfolding inverse_eq_divide[symmetric]
-      using card_gt0 log_inverse `b > 1` 
+      using card_gt0 log_inverse b_gt_1
       by (auto simp add:field_simps card_gt0)
     finally have ?thesis
-      unfolding finite_entropy_reduce[OF `b > 1`] by auto }
+      unfolding entropy_eq by auto }
   moreover
   { assume "X ` space M = {}"
     hence "distribution X (X ` space M) = 0"
@@ -844,4 +737,199 @@
   ultimately show ?thesis by auto
 qed
 
+definition "subvimage A f g \<longleftrightarrow> (\<forall>x \<in> A. f -` {f x} \<inter> A \<subseteq> g -` {g x} \<inter> A)"
+
+lemma subvimageI:
+  assumes "\<And>x y. \<lbrakk> x \<in> A ; y \<in> A ; f x = f y \<rbrakk> \<Longrightarrow> g x = g y"
+  shows "subvimage A f g"
+  using assms unfolding subvimage_def by blast
+
+lemma subvimageE[consumes 1]:
+  assumes "subvimage A f g"
+  obtains "\<And>x y. \<lbrakk> x \<in> A ; y \<in> A ; f x = f y \<rbrakk> \<Longrightarrow> g x = g y"
+  using assms unfolding subvimage_def by blast
+
+lemma subvimageD:
+  "\<lbrakk> subvimage A f g ; x \<in> A ; y \<in> A ; f x = f y \<rbrakk> \<Longrightarrow> g x = g y"
+  using assms unfolding subvimage_def by blast
+
+lemma subvimage_subset:
+  "\<lbrakk> subvimage B f g ; A \<subseteq> B \<rbrakk> \<Longrightarrow> subvimage A f g"
+  unfolding subvimage_def by auto
+
+lemma subvimage_idem[intro]: "subvimage A g g"
+  by (safe intro!: subvimageI)
+
+lemma subvimage_comp_finer[intro]:
+  assumes svi: "subvimage A g h"
+  shows "subvimage A g (f \<circ> h)"
+proof (rule subvimageI, simp)
+  fix x y assume "x \<in> A" "y \<in> A" "g x = g y"
+  from svi[THEN subvimageD, OF this]
+  show "f (h x) = f (h y)" by simp
+qed
+
+lemma subvimage_comp_gran:
+  assumes svi: "subvimage A g h"
+  assumes inj: "inj_on f (g ` A)"
+  shows "subvimage A (f \<circ> g) h"
+  by (rule subvimageI) (auto intro!: subvimageD[OF svi] simp: inj_on_iff[OF inj])
+
+lemma subvimage_comp:
+  assumes svi: "subvimage (f ` A) g h"
+  shows "subvimage A (g \<circ> f) (h \<circ> f)"
+  by (rule subvimageI) (auto intro!: svi[THEN subvimageD])
+
+lemma subvimage_trans:
+  assumes fg: "subvimage A f g"
+  assumes gh: "subvimage A g h"
+  shows "subvimage A f h"
+  by (rule subvimageI) (auto intro!: fg[THEN subvimageD] gh[THEN subvimageD])
+
+lemma subvimage_translator:
+  assumes svi: "subvimage A f g"
+  shows "\<exists>h. \<forall>x \<in> A. h (f x)  = g x"
+proof (safe intro!: exI[of _ "\<lambda>x. (THE z. z \<in> (g ` (f -` {x} \<inter> A)))"])
+  fix x assume "x \<in> A"
+  show "(THE x'. x' \<in> (g ` (f -` {f x} \<inter> A))) = g x"
+    by (rule theI2[of _ "g x"])
+      (insert `x \<in> A`, auto intro!: svi[THEN subvimageD])
+qed
+
+lemma subvimage_translator_image:
+  assumes svi: "subvimage A f g"
+  shows "\<exists>h. h ` f ` A = g ` A"
+proof -
+  from subvimage_translator[OF svi]
+  obtain h where "\<And>x. x \<in> A \<Longrightarrow> h (f x) = g x" by auto
+  thus ?thesis
+    by (auto intro!: exI[of _ h]
+      simp: image_compose[symmetric] comp_def cong: image_cong)
+qed
+
+lemma subvimage_finite:
+  assumes svi: "subvimage A f g" and fin: "finite (f`A)"
+  shows "finite (g`A)"
+proof -
+  from subvimage_translator_image[OF svi]
+  obtain h where "g`A = h`f`A" by fastsimp
+  with fin show "finite (g`A)" by simp
+qed
+
+lemma subvimage_disj:
+  assumes svi: "subvimage A f g"
+  shows "f -` {x} \<inter> A \<subseteq> g -` {y} \<inter> A \<or>
+      f -` {x} \<inter> g -` {y} \<inter> A = {}" (is "?sub \<or> ?dist")
+proof (rule disjCI)
+  assume "\<not> ?dist"
+  then obtain z where "z \<in> A" and "x = f z" and "y = g z" by auto
+  thus "?sub" using svi unfolding subvimage_def by auto
+qed
+
+lemma setsum_image_split:
+  assumes svi: "subvimage A f g" and fin: "finite (f ` A)"
+  shows "(\<Sum>x\<in>f`A. h x) = (\<Sum>y\<in>g`A. \<Sum>x\<in>f`(g -` {y} \<inter> A). h x)"
+    (is "?lhs = ?rhs")
+proof -
+  have "f ` A =
+      snd ` (SIGMA x : g ` A. f ` (g -` {x} \<inter> A))"
+      (is "_ = snd ` ?SIGMA")
+    unfolding image_split_eq_Sigma[symmetric]
+    by (simp add: image_compose[symmetric] comp_def)
+  moreover
+  have snd_inj: "inj_on snd ?SIGMA"
+    unfolding image_split_eq_Sigma[symmetric]
+    by (auto intro!: inj_onI subvimageD[OF svi])
+  ultimately
+  have "(\<Sum>x\<in>f`A. h x) = (\<Sum>(x,y)\<in>?SIGMA. h y)"
+    by (auto simp: setsum_reindex intro: setsum_cong)
+  also have "... = ?rhs"
+    using subvimage_finite[OF svi fin] fin
+    apply (subst setsum_Sigma[symmetric])
+    by (auto intro!: finite_subset[of _ "f`A"])
+  finally show ?thesis .
+qed
+
+lemma (in finite_information_space) entropy_partition:
+  assumes svi: "subvimage (space M) X P"
+  shows "\<H>(X) = \<H>(P) + \<H>(X|P)"
+proof -
+  have "(\<Sum>x\<in>X ` space M. distribution X {x} * log b (distribution X {x})) =
+    (\<Sum>y\<in>P `space M. \<Sum>x\<in>X ` space M.
+    joint_distribution X P {(x, y)} * log b (joint_distribution X P {(x, y)}))"
+  proof (subst setsum_image_split[OF svi],
+      safe intro!: finite_imageI finite_space setsum_mono_zero_cong_left imageI)
+    fix p x assume in_space: "p \<in> space M" "x \<in> space M"
+    assume "joint_distribution X P {(X x, P p)} * log b (joint_distribution X P {(X x, P p)}) \<noteq> 0"
+    hence "(\<lambda>x. (X x, P x)) -` {(X x, P p)} \<inter> space M \<noteq> {}" by (auto simp: distribution_def)
+    with svi[unfolded subvimage_def, rule_format, OF `x \<in> space M`]
+    show "x \<in> P -` {P p}" by auto
+  next
+    fix p x assume in_space: "p \<in> space M" "x \<in> space M"
+    assume "P x = P p"
+    from this[symmetric] svi[unfolded subvimage_def, rule_format, OF `x \<in> space M`]
+    have "X -` {X x} \<inter> space M \<subseteq> P -` {P p} \<inter> space M"
+      by auto
+    hence "(\<lambda>x. (X x, P x)) -` {(X x, P p)} \<inter> space M = X -` {X x} \<inter> space M"
+      by auto
+    thus "distribution X {X x} * log b (distribution X {X x}) =
+          joint_distribution X P {(X x, P p)} *
+          log b (joint_distribution X P {(X x, P p)})"
+      by (auto simp: distribution_def)
+  qed
+  thus ?thesis
+  unfolding entropy_eq conditional_entropy_eq
+    by (simp add: setsum_cartesian_product' setsum_subtractf setsum_distribution
+      setsum_left_distrib[symmetric] setsum_commute[where B="P`space M"])
+qed
+
+corollary (in finite_information_space) entropy_data_processing:
+  "\<H>(f \<circ> X) \<le> \<H>(X)"
+  by (subst (2) entropy_partition[of _ "f \<circ> X"]) (auto intro: conditional_entropy_positive)
+
+lemma (in prob_space) distribution_cong:
+  assumes "\<And>x. x \<in> space M \<Longrightarrow> X x = Y x"
+  shows "distribution X = distribution Y"
+  unfolding distribution_def expand_fun_eq
+  using assms by (auto intro!: arg_cong[where f=prob])
+
+lemma (in prob_space) joint_distribution_cong:
+  assumes "\<And>x. x \<in> space M \<Longrightarrow> X x = X' x"
+  assumes "\<And>x. x \<in> space M \<Longrightarrow> Y x = Y' x"
+  shows "joint_distribution X Y = joint_distribution X' Y'"
+  unfolding distribution_def expand_fun_eq
+  using assms by (auto intro!: arg_cong[where f=prob])
+
+lemma image_cong:
+  "\<lbrakk> \<And>x. x \<in> S \<Longrightarrow> X x = X' x \<rbrakk> \<Longrightarrow> X ` S = X' ` S"
+  by (auto intro!: image_eqI)
+
+lemma (in finite_information_space) mutual_information_cong:
+  assumes X: "\<And>x. x \<in> space M \<Longrightarrow> X x = X' x"
+  assumes Y: "\<And>x. x \<in> space M \<Longrightarrow> Y x = Y' x"
+  shows "\<I>(X ; Y) = \<I>(X' ; Y')"
+proof -
+  have "X ` space M = X' ` space M" using X by (rule image_cong)
+  moreover have "Y ` space M = Y' ` space M" using Y by (rule image_cong)
+  ultimately show ?thesis
+  unfolding mutual_information_eq
+    using
+      assms[THEN distribution_cong]
+      joint_distribution_cong[OF assms]
+    by (auto intro!: setsum_cong)
+qed
+
+corollary (in finite_information_space) entropy_of_inj:
+  assumes "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 .
+next
+  have "\<H>(X) = \<H>(the_inv_into (X`space M) f \<circ> (f \<circ> X))"
+    by (auto intro!: mutual_information_cong simp: entropy_def the_inv_into_f_f[OF assms])
+  also have "... \<le> \<H>(f \<circ> X)"
+    using entropy_data_processing .
+  finally show "\<H>(X) \<le> \<H>(f \<circ> X)" .
+qed
+
 end




