src/HOL/Probability/Information.thy
 author hoelzl Wed Feb 02 12:34:45 2011 +0100 (2011-02-02) changeset 41689 3e39b0e730d6 parent 41661 baf1964bc468 child 41833 563bea92b2c0 permissions -rw-r--r--
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
changed syntax for simple_function, simple_integral, positive_integral, integral and RN_deriv.
introduced binder variants for simple_integral, positive_integral and integral.
```     1 theory Information
```
```     2 imports
```
```     3   Probability_Space
```
```     4   "~~/src/HOL/Library/Convex"
```
```     5   Lebesgue_Measure
```
```     6 begin
```
```     7
```
```     8 lemma log_le: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> x \<le> y \<Longrightarrow> log a x \<le> log a y"
```
```     9   by (subst log_le_cancel_iff) auto
```
```    10
```
```    11 lemma log_less: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> x < y \<Longrightarrow> log a x < log a y"
```
```    12   by (subst log_less_cancel_iff) auto
```
```    13
```
```    14 lemma setsum_cartesian_product':
```
```    15   "(\<Sum>x\<in>A \<times> B. f x) = (\<Sum>x\<in>A. setsum (\<lambda>y. f (x, y)) B)"
```
```    16   unfolding setsum_cartesian_product by simp
```
```    17
```
```    18 section "Convex theory"
```
```    19
```
```    20 lemma log_setsum:
```
```    21   assumes "finite s" "s \<noteq> {}"
```
```    22   assumes "b > 1"
```
```    23   assumes "(\<Sum> i \<in> s. a i) = 1"
```
```    24   assumes "\<And> i. i \<in> s \<Longrightarrow> a i \<ge> 0"
```
```    25   assumes "\<And> i. i \<in> s \<Longrightarrow> y i \<in> {0 <..}"
```
```    26   shows "log b (\<Sum> i \<in> s. a i * y i) \<ge> (\<Sum> i \<in> s. a i * log b (y i))"
```
```    27 proof -
```
```    28   have "convex_on {0 <..} (\<lambda> x. - log b x)"
```
```    29     by (rule minus_log_convex[OF `b > 1`])
```
```    30   hence "- log b (\<Sum> i \<in> s. a i * y i) \<le> (\<Sum> i \<in> s. a i * - log b (y i))"
```
```    31     using convex_on_setsum[of _ _ "\<lambda> x. - log b x"] assms pos_is_convex by fastsimp
```
```    32   thus ?thesis by (auto simp add:setsum_negf le_imp_neg_le)
```
```    33 qed
```
```    34
```
```    35 lemma log_setsum':
```
```    36   assumes "finite s" "s \<noteq> {}"
```
```    37   assumes "b > 1"
```
```    38   assumes "(\<Sum> i \<in> s. a i) = 1"
```
```    39   assumes pos: "\<And> i. i \<in> s \<Longrightarrow> 0 \<le> a i"
```
```    40           "\<And> i. \<lbrakk> i \<in> s ; 0 < a i \<rbrakk> \<Longrightarrow> 0 < y i"
```
```    41   shows "log b (\<Sum> i \<in> s. a i * y i) \<ge> (\<Sum> i \<in> s. a i * log b (y i))"
```
```    42 proof -
```
```    43   have "\<And>y. (\<Sum> i \<in> s - {i. a i = 0}. a i * y i) = (\<Sum> i \<in> s. a i * y i)"
```
```    44     using assms by (auto intro!: setsum_mono_zero_cong_left)
```
```    45   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))"
```
```    46   proof (rule log_setsum)
```
```    47     have "setsum a (s - {i. a i = 0}) = setsum a s"
```
```    48       using assms(1) by (rule setsum_mono_zero_cong_left) auto
```
```    49     thus sum_1: "setsum a (s - {i. a i = 0}) = 1"
```
```    50       "finite (s - {i. a i = 0})" using assms by simp_all
```
```    51
```
```    52     show "s - {i. a i = 0} \<noteq> {}"
```
```    53     proof
```
```    54       assume *: "s - {i. a i = 0} = {}"
```
```    55       hence "setsum a (s - {i. a i = 0}) = 0" by (simp add: * setsum_empty)
```
```    56       with sum_1 show False by simp
```
```    57     qed
```
```    58
```
```    59     fix i assume "i \<in> s - {i. a i = 0}"
```
```    60     hence "i \<in> s" "a i \<noteq> 0" by simp_all
```
```    61     thus "0 \<le> a i" "y i \<in> {0<..}" using pos[of i] by auto
```
```    62   qed fact+
```
```    63   ultimately show ?thesis by simp
```
```    64 qed
```
```    65
```
```    66 lemma log_setsum_divide:
```
```    67   assumes "finite S" and "S \<noteq> {}" and "1 < b"
```
```    68   assumes "(\<Sum>x\<in>S. g x) = 1"
```
```    69   assumes pos: "\<And>x. x \<in> S \<Longrightarrow> g x \<ge> 0" "\<And>x. x \<in> S \<Longrightarrow> f x \<ge> 0"
```
```    70   assumes g_pos: "\<And>x. \<lbrakk> x \<in> S ; 0 < g x \<rbrakk> \<Longrightarrow> 0 < f x"
```
```    71   shows "- (\<Sum>x\<in>S. g x * log b (g x / f x)) \<le> log b (\<Sum>x\<in>S. f x)"
```
```    72 proof -
```
```    73   have log_mono: "\<And>x y. 0 < x \<Longrightarrow> x \<le> y \<Longrightarrow> log b x \<le> log b y"
```
```    74     using `1 < b` by (subst log_le_cancel_iff) auto
```
```    75
```
```    76   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))"
```
```    77   proof (unfold setsum_negf[symmetric], rule setsum_cong)
```
```    78     fix x assume x: "x \<in> S"
```
```    79     show "- (g x * log b (g x / f x)) = g x * log b (f x / g x)"
```
```    80     proof (cases "g x = 0")
```
```    81       case False
```
```    82       with pos[OF x] g_pos[OF x] have "0 < f x" "0 < g x" by simp_all
```
```    83       thus ?thesis using `1 < b` by (simp add: log_divide field_simps)
```
```    84     qed simp
```
```    85   qed rule
```
```    86   also have "... \<le> log b (\<Sum>x\<in>S. g x * (f x / g x))"
```
```    87   proof (rule log_setsum')
```
```    88     fix x assume x: "x \<in> S" "0 < g x"
```
```    89     with g_pos[OF x] show "0 < f x / g x" by (safe intro!: divide_pos_pos)
```
```    90   qed fact+
```
```    91   also have "... = log b (\<Sum>x\<in>S - {x. g x = 0}. f x)" using `finite S`
```
```    92     by (auto intro!: setsum_mono_zero_cong_right arg_cong[where f="log b"]
```
```    93         split: split_if_asm)
```
```    94   also have "... \<le> log b (\<Sum>x\<in>S. f x)"
```
```    95   proof (rule log_mono)
```
```    96     have "0 = (\<Sum>x\<in>S - {x. g x = 0}. 0)" by simp
```
```    97     also have "... < (\<Sum>x\<in>S - {x. g x = 0}. f x)" (is "_ < ?sum")
```
```    98     proof (rule setsum_strict_mono)
```
```    99       show "finite (S - {x. g x = 0})" using `finite S` by simp
```
```   100       show "S - {x. g x = 0} \<noteq> {}"
```
```   101       proof
```
```   102         assume "S - {x. g x = 0} = {}"
```
```   103         hence "(\<Sum>x\<in>S. g x) = 0" by (subst setsum_0') auto
```
```   104         with `(\<Sum>x\<in>S. g x) = 1` show False by simp
```
```   105       qed
```
```   106       fix x assume "x \<in> S - {x. g x = 0}"
```
```   107       thus "0 < f x" using g_pos[of x] pos(1)[of x] by auto
```
```   108     qed
```
```   109     finally show "0 < ?sum" .
```
```   110     show "(\<Sum>x\<in>S - {x. g x = 0}. f x) \<le> (\<Sum>x\<in>S. f x)"
```
```   111       using `finite S` pos by (auto intro!: setsum_mono2)
```
```   112   qed
```
```   113   finally show ?thesis .
```
```   114 qed
```
```   115
```
```   116 lemma split_pairs:
```
```   117   "((A, B) = X) \<longleftrightarrow> (fst X = A \<and> snd X = B)" and
```
```   118   "(X = (A, B)) \<longleftrightarrow> (fst X = A \<and> snd X = B)" by auto
```
```   119
```
```   120 section "Information theory"
```
```   121
```
```   122 locale information_space = prob_space +
```
```   123   fixes b :: real assumes b_gt_1: "1 < b"
```
```   124
```
```   125 context information_space
```
```   126 begin
```
```   127
```
```   128 text {* Introduce some simplification rules for logarithm of base @{term b}. *}
```
```   129
```
```   130 lemma log_neg_const:
```
```   131   assumes "x \<le> 0"
```
```   132   shows "log b x = log b 0"
```
```   133 proof -
```
```   134   { fix u :: real
```
```   135     have "x \<le> 0" by fact
```
```   136     also have "0 < exp u"
```
```   137       using exp_gt_zero .
```
```   138     finally have "exp u \<noteq> x"
```
```   139       by auto }
```
```   140   then show "log b x = log b 0"
```
```   141     by (simp add: log_def ln_def)
```
```   142 qed
```
```   143
```
```   144 lemma log_mult_eq:
```
```   145   "log b (A * B) = (if 0 < A * B then log b \<bar>A\<bar> + log b \<bar>B\<bar> else log b 0)"
```
```   146   using log_mult[of b "\<bar>A\<bar>" "\<bar>B\<bar>"] b_gt_1 log_neg_const[of "A * B"]
```
```   147   by (auto simp: zero_less_mult_iff mult_le_0_iff)
```
```   148
```
```   149 lemma log_inverse_eq:
```
```   150   "log b (inverse B) = (if 0 < B then - log b B else log b 0)"
```
```   151   using log_inverse[of b B] log_neg_const[of "inverse B"] b_gt_1 by simp
```
```   152
```
```   153 lemma log_divide_eq:
```
```   154   "log b (A / B) = (if 0 < A * B then log b \<bar>A\<bar> - log b \<bar>B\<bar> else log b 0)"
```
```   155   unfolding divide_inverse log_mult_eq log_inverse_eq abs_inverse
```
```   156   by (auto simp: zero_less_mult_iff mult_le_0_iff)
```
```   157
```
```   158 lemmas log_simps = log_mult_eq log_inverse_eq log_divide_eq
```
```   159
```
```   160 end
```
```   161
```
```   162 subsection "Kullback\$-\$Leibler divergence"
```
```   163
```
```   164 text {* The Kullback\$-\$Leibler divergence is also known as relative entropy or
```
```   165 Kullback\$-\$Leibler distance. *}
```
```   166
```
```   167 definition
```
```   168   "KL_divergence b M \<nu> = \<integral>x. log b (real (RN_deriv M \<nu> x)) \<partial>M\<lparr>measure := \<nu>\<rparr>"
```
```   169
```
```   170 lemma (in sigma_finite_measure) KL_divergence_cong:
```
```   171   assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)" (is "measure_space ?\<nu>")
```
```   172   assumes [simp]: "sets N = sets M" "space N = space M"
```
```   173     "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A"
```
```   174     "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = \<nu>' A"
```
```   175   shows "KL_divergence b M \<nu> = KL_divergence b N \<nu>'"
```
```   176 proof -
```
```   177   interpret \<nu>: measure_space ?\<nu> by fact
```
```   178   have "KL_divergence b M \<nu> = \<integral>x. log b (real (RN_deriv N \<nu>' x)) \<partial>?\<nu>"
```
```   179     by (simp cong: RN_deriv_cong \<nu>.integral_cong add: KL_divergence_def)
```
```   180   also have "\<dots> = KL_divergence b N \<nu>'"
```
```   181     by (auto intro!: \<nu>.integral_cong_measure[symmetric] simp: KL_divergence_def)
```
```   182   finally show ?thesis .
```
```   183 qed
```
```   184
```
```   185 lemma (in finite_measure_space) KL_divergence_eq_finite:
```
```   186   assumes v: "finite_measure_space (M\<lparr>measure := \<nu>\<rparr>)"
```
```   187   assumes ac: "absolutely_continuous \<nu>"
```
```   188   shows "KL_divergence b M \<nu> = (\<Sum>x\<in>space M. real (\<nu> {x}) * log b (real (\<nu> {x}) / real (\<mu> {x})))" (is "_ = ?sum")
```
```   189 proof (simp add: KL_divergence_def finite_measure_space.integral_finite_singleton[OF v])
```
```   190   interpret v: finite_measure_space "M\<lparr>measure := \<nu>\<rparr>" by fact
```
```   191   have ms: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" by default
```
```   192   show "(\<Sum>x \<in> space M. log b (real (RN_deriv M \<nu> x)) * real (\<nu> {x})) = ?sum"
```
```   193     using RN_deriv_finite_measure[OF ms ac]
```
```   194     by (auto intro!: setsum_cong simp: field_simps real_of_pextreal_mult[symmetric])
```
```   195 qed
```
```   196
```
```   197 lemma (in finite_prob_space) KL_divergence_positive_finite:
```
```   198   assumes v: "finite_prob_space (M\<lparr>measure := \<nu>\<rparr>)"
```
```   199   assumes ac: "absolutely_continuous \<nu>"
```
```   200   and "1 < b"
```
```   201   shows "0 \<le> KL_divergence b M \<nu>"
```
```   202 proof -
```
```   203   interpret v: finite_prob_space "M\<lparr>measure := \<nu>\<rparr>" by fact
```
```   204   have ms: "finite_measure_space (M\<lparr>measure := \<nu>\<rparr>)" by default
```
```   205
```
```   206   have "- (KL_divergence b M \<nu>) \<le> log b (\<Sum>x\<in>space M. real (\<mu> {x}))"
```
```   207   proof (subst KL_divergence_eq_finite[OF ms ac], safe intro!: log_setsum_divide not_empty)
```
```   208     show "finite (space M)" using finite_space by simp
```
```   209     show "1 < b" by fact
```
```   210     show "(\<Sum>x\<in>space M. real (\<nu> {x})) = 1" using v.finite_sum_over_space_eq_1 by simp
```
```   211
```
```   212     fix x assume "x \<in> space M"
```
```   213     then have x: "{x} \<in> sets M" unfolding sets_eq_Pow by auto
```
```   214     { assume "0 < real (\<nu> {x})"
```
```   215       then have "\<nu> {x} \<noteq> 0" by auto
```
```   216       then have "\<mu> {x} \<noteq> 0"
```
```   217         using ac[unfolded absolutely_continuous_def, THEN bspec, of "{x}"] x by auto
```
```   218       thus "0 < prob {x}" using finite_measure[of "{x}"] x by auto }
```
```   219   qed auto
```
```   220   thus "0 \<le> KL_divergence b M \<nu>" using finite_sum_over_space_eq_1 by simp
```
```   221 qed
```
```   222
```
```   223 subsection {* Mutual Information *}
```
```   224
```
```   225 definition (in prob_space)
```
```   226   "mutual_information b S T X Y =
```
```   227     KL_divergence b (S\<lparr>measure := distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := distribution Y\<rparr>)
```
```   228       (joint_distribution X Y)"
```
```   229
```
```   230 definition (in prob_space)
```
```   231   "entropy b s X = mutual_information b s s X X"
```
```   232
```
```   233 abbreviation (in information_space)
```
```   234   mutual_information_Pow ("\<I>'(_ ; _')") where
```
```   235   "\<I>(X ; Y) \<equiv> mutual_information b
```
```   236     \<lparr> space = X`space M, sets = Pow (X`space M), measure = distribution X \<rparr>
```
```   237     \<lparr> space = Y`space M, sets = Pow (Y`space M), measure = distribution Y \<rparr> X Y"
```
```   238
```
```   239 lemma algebra_measure_update[simp]:
```
```   240   "algebra (M'\<lparr>measure := m\<rparr>) \<longleftrightarrow> algebra M'"
```
```   241   unfolding algebra_def by simp
```
```   242
```
```   243 lemma sigma_algebra_measure_update[simp]:
```
```   244   "sigma_algebra (M'\<lparr>measure := m\<rparr>) \<longleftrightarrow> sigma_algebra M'"
```
```   245   unfolding sigma_algebra_def sigma_algebra_axioms_def by simp
```
```   246
```
```   247 lemma finite_sigma_algebra_measure_update[simp]:
```
```   248   "finite_sigma_algebra (M'\<lparr>measure := m\<rparr>) \<longleftrightarrow> finite_sigma_algebra M'"
```
```   249   unfolding finite_sigma_algebra_def finite_sigma_algebra_axioms_def by simp
```
```   250
```
```   251 lemma (in prob_space) finite_variables_absolutely_continuous:
```
```   252   assumes X: "finite_random_variable S X" and Y: "finite_random_variable T Y"
```
```   253   shows "measure_space.absolutely_continuous
```
```   254     (S\<lparr>measure := distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := distribution Y\<rparr>)
```
```   255     (joint_distribution X Y)"
```
```   256 proof -
```
```   257   interpret X: finite_prob_space "S\<lparr>measure := distribution X\<rparr>"
```
```   258     using X by (rule distribution_finite_prob_space)
```
```   259   interpret Y: finite_prob_space "T\<lparr>measure := distribution Y\<rparr>"
```
```   260     using Y by (rule distribution_finite_prob_space)
```
```   261   interpret XY: pair_finite_prob_space
```
```   262     "S\<lparr>measure := distribution X\<rparr>" "T\<lparr> measure := distribution Y\<rparr>" by default
```
```   263   interpret P: finite_prob_space "XY.P\<lparr> measure := joint_distribution X Y\<rparr>"
```
```   264     using assms by (auto intro!: joint_distribution_finite_prob_space)
```
```   265   note rv = assms[THEN finite_random_variableD]
```
```   266   show "XY.absolutely_continuous (joint_distribution X Y)"
```
```   267   proof (rule XY.absolutely_continuousI)
```
```   268     show "finite_measure_space (XY.P\<lparr> measure := joint_distribution X Y\<rparr>)" by default
```
```   269     fix x assume "x \<in> space XY.P" and "XY.\<mu> {x} = 0"
```
```   270     then obtain a b where "(a, b) = x" and "a \<in> space S" "b \<in> space T"
```
```   271       and distr: "distribution X {a} * distribution Y {b} = 0"
```
```   272       by (cases x) (auto simp: space_pair_measure)
```
```   273     with X.sets_eq_Pow Y.sets_eq_Pow
```
```   274       joint_distribution_Times_le_fst[OF rv, of "{a}" "{b}"]
```
```   275       joint_distribution_Times_le_snd[OF rv, of "{a}" "{b}"]
```
```   276     have "joint_distribution X Y {x} \<le> distribution Y {b}"
```
```   277          "joint_distribution X Y {x} \<le> distribution X {a}"
```
```   278       by (auto simp del: X.sets_eq_Pow Y.sets_eq_Pow)
```
```   279     with distr show "joint_distribution X Y {x} = 0" by auto
```
```   280   qed
```
```   281 qed
```
```   282
```
```   283 lemma (in information_space)
```
```   284   assumes MX: "finite_random_variable MX X"
```
```   285   assumes MY: "finite_random_variable MY Y"
```
```   286   shows mutual_information_generic_eq:
```
```   287     "mutual_information b MX MY X Y = (\<Sum> (x,y) \<in> space MX \<times> space MY.
```
```   288       real (joint_distribution X Y {(x,y)}) *
```
```   289       log b (real (joint_distribution X Y {(x,y)}) /
```
```   290       (real (distribution X {x}) * real (distribution Y {y}))))"
```
```   291     (is ?sum)
```
```   292   and mutual_information_positive_generic:
```
```   293      "0 \<le> mutual_information b MX MY X Y" (is ?positive)
```
```   294 proof -
```
```   295   interpret X: finite_prob_space "MX\<lparr>measure := distribution X\<rparr>"
```
```   296     using MX by (rule distribution_finite_prob_space)
```
```   297   interpret Y: finite_prob_space "MY\<lparr>measure := distribution Y\<rparr>"
```
```   298     using MY by (rule distribution_finite_prob_space)
```
```   299   interpret XY: pair_finite_prob_space "MX\<lparr>measure := distribution X\<rparr>" "MY\<lparr>measure := distribution Y\<rparr>" by default
```
```   300   interpret P: finite_prob_space "XY.P\<lparr>measure := joint_distribution X Y\<rparr>"
```
```   301     using assms by (auto intro!: joint_distribution_finite_prob_space)
```
```   302
```
```   303   have P_ms: "finite_measure_space (XY.P\<lparr>measure :=joint_distribution X Y\<rparr>)" by default
```
```   304   have P_ps: "finite_prob_space (XY.P\<lparr>measure := joint_distribution X Y\<rparr>)" by default
```
```   305
```
```   306   show ?sum
```
```   307     unfolding Let_def mutual_information_def
```
```   308     by (subst XY.KL_divergence_eq_finite[OF P_ms finite_variables_absolutely_continuous[OF MX MY]])
```
```   309        (auto simp add: space_pair_measure setsum_cartesian_product' real_of_pextreal_mult[symmetric])
```
```   310
```
```   311   show ?positive
```
```   312     using XY.KL_divergence_positive_finite[OF P_ps finite_variables_absolutely_continuous[OF MX MY] b_gt_1]
```
```   313     unfolding mutual_information_def .
```
```   314 qed
```
```   315
```
```   316 lemma (in information_space) mutual_information_commute:
```
```   317   assumes X: "finite_random_variable S X" and Y: "finite_random_variable T Y"
```
```   318   shows "mutual_information b S T X Y = mutual_information b T S Y X"
```
```   319   unfolding mutual_information_generic_eq[OF X Y] mutual_information_generic_eq[OF Y X]
```
```   320   unfolding joint_distribution_commute_singleton[of X Y]
```
```   321   by (auto simp add: ac_simps intro!: setsum_reindex_cong[OF swap_inj_on])
```
```   322
```
```   323 lemma (in information_space) mutual_information_commute_simple:
```
```   324   assumes X: "simple_function M X" and Y: "simple_function M Y"
```
```   325   shows "\<I>(X;Y) = \<I>(Y;X)"
```
```   326   by (intro X Y simple_function_imp_finite_random_variable mutual_information_commute)
```
```   327
```
```   328 lemma (in information_space) mutual_information_eq:
```
```   329   assumes "simple_function M X" "simple_function M Y"
```
```   330   shows "\<I>(X;Y) = (\<Sum> (x,y) \<in> X ` space M \<times> Y ` space M.
```
```   331     real (distribution (\<lambda>x. (X x, Y x)) {(x,y)}) * log b (real (distribution (\<lambda>x. (X x, Y x)) {(x,y)}) /
```
```   332                                                    (real (distribution X {x}) * real (distribution Y {y}))))"
```
```   333   using assms by (simp add: mutual_information_generic_eq)
```
```   334
```
```   335 lemma (in information_space) mutual_information_generic_cong:
```
```   336   assumes X: "\<And>x. x \<in> space M \<Longrightarrow> X x = X' x"
```
```   337   assumes Y: "\<And>x. x \<in> space M \<Longrightarrow> Y x = Y' x"
```
```   338   shows "mutual_information b MX MY X Y = mutual_information b MX MY X' Y'"
```
```   339   unfolding mutual_information_def using X Y
```
```   340   by (simp cong: distribution_cong)
```
```   341
```
```   342 lemma (in information_space) mutual_information_cong:
```
```   343   assumes X: "\<And>x. x \<in> space M \<Longrightarrow> X x = X' x"
```
```   344   assumes Y: "\<And>x. x \<in> space M \<Longrightarrow> Y x = Y' x"
```
```   345   shows "\<I>(X; Y) = \<I>(X'; Y')"
```
```   346   unfolding mutual_information_def using X Y
```
```   347   by (simp cong: distribution_cong image_cong)
```
```   348
```
```   349 lemma (in information_space) mutual_information_positive:
```
```   350   assumes "simple_function M X" "simple_function M Y"
```
```   351   shows "0 \<le> \<I>(X;Y)"
```
```   352   using assms by (simp add: mutual_information_positive_generic)
```
```   353
```
```   354 subsection {* Entropy *}
```
```   355
```
```   356 abbreviation (in information_space)
```
```   357   entropy_Pow ("\<H>'(_')") where
```
```   358   "\<H>(X) \<equiv> entropy b \<lparr> space = X`space M, sets = Pow (X`space M), measure = distribution X \<rparr> X"
```
```   359
```
```   360 lemma (in information_space) entropy_generic_eq:
```
```   361   assumes MX: "finite_random_variable MX X"
```
```   362   shows "entropy b MX X = -(\<Sum> x \<in> space MX. real (distribution X {x}) * log b (real (distribution X {x})))"
```
```   363 proof -
```
```   364   interpret MX: finite_prob_space "MX\<lparr>measure := distribution X\<rparr>"
```
```   365     using MX by (rule distribution_finite_prob_space)
```
```   366   let "?X x" = "real (distribution X {x})"
```
```   367   let "?XX x y" = "real (joint_distribution X X {(x, y)})"
```
```   368   { fix x y
```
```   369     have "(\<lambda>x. (X x, X x)) -` {(x, y)} = (if x = y then X -` {x} else {})" by auto
```
```   370     then have "?XX x y * log b (?XX x y / (?X x * ?X y)) =
```
```   371         (if x = y then - ?X y * log b (?X y) else 0)"
```
```   372       unfolding distribution_def by (auto simp: log_simps zero_less_mult_iff) }
```
```   373   note remove_XX = this
```
```   374   show ?thesis
```
```   375     unfolding entropy_def mutual_information_generic_eq[OF MX MX]
```
```   376     unfolding setsum_cartesian_product[symmetric] setsum_negf[symmetric] remove_XX
```
```   377     using MX.finite_space by (auto simp: setsum_cases)
```
```   378 qed
```
```   379
```
```   380 lemma (in information_space) entropy_eq:
```
```   381   assumes "simple_function M X"
```
```   382   shows "\<H>(X) = -(\<Sum> x \<in> X ` space M. real (distribution X {x}) * log b (real (distribution X {x})))"
```
```   383   using assms by (simp add: entropy_generic_eq)
```
```   384
```
```   385 lemma (in information_space) entropy_positive:
```
```   386   "simple_function M X \<Longrightarrow> 0 \<le> \<H>(X)"
```
```   387   unfolding entropy_def by (simp add: mutual_information_positive)
```
```   388
```
```   389 lemma (in information_space) entropy_certainty_eq_0:
```
```   390   assumes "simple_function M X" and "x \<in> X ` space M" and "distribution X {x} = 1"
```
```   391   shows "\<H>(X) = 0"
```
```   392 proof -
```
```   393   let ?X = "\<lparr> space = X ` space M, sets = Pow (X ` space M), measure = distribution X\<rparr>"
```
```   394   note simple_function_imp_finite_random_variable[OF `simple_function M X`]
```
```   395   from distribution_finite_prob_space[OF this, of "\<lparr> measure = distribution X \<rparr>"]
```
```   396   interpret X: finite_prob_space ?X by simp
```
```   397   have "distribution X (X ` space M - {x}) = distribution X (X ` space M) - distribution X {x}"
```
```   398     using X.measure_compl[of "{x}"] assms by auto
```
```   399   also have "\<dots> = 0" using X.prob_space assms by auto
```
```   400   finally have X0: "distribution X (X ` space M - {x}) = 0" by auto
```
```   401   { fix y assume asm: "y \<noteq> x" "y \<in> X ` space M"
```
```   402     hence "{y} \<subseteq> X ` space M - {x}" by auto
```
```   403     from X.measure_mono[OF this] X0 asm
```
```   404     have "distribution X {y} = 0" by auto }
```
```   405   hence fi: "\<And> y. y \<in> X ` space M \<Longrightarrow> real (distribution X {y}) = (if x = y then 1 else 0)"
```
```   406     using assms by auto
```
```   407   have y: "\<And>y. (if x = y then 1 else 0) * log b (if x = y then 1 else 0) = 0" by simp
```
```   408   show ?thesis unfolding entropy_eq[OF `simple_function M X`] by (auto simp: y fi)
```
```   409 qed
```
```   410
```
```   411 lemma (in information_space) entropy_le_card_not_0:
```
```   412   assumes "simple_function M X"
```
```   413   shows "\<H>(X) \<le> log b (real (card (X ` space M \<inter> {x . distribution X {x} \<noteq> 0})))"
```
```   414 proof -
```
```   415   let "?d x" = "distribution X {x}"
```
```   416   let "?p x" = "real (?d x)"
```
```   417   have "\<H>(X) = (\<Sum>x\<in>X`space M. ?p x * log b (1 / ?p x))"
```
```   418     by (auto intro!: setsum_cong simp: entropy_eq[OF `simple_function M X`] setsum_negf[symmetric] log_simps not_less)
```
```   419   also have "\<dots> \<le> log b (\<Sum>x\<in>X`space M. ?p x * (1 / ?p x))"
```
```   420     apply (rule log_setsum')
```
```   421     using not_empty b_gt_1 `simple_function M X` sum_over_space_real_distribution
```
```   422     by (auto simp: simple_function_def)
```
```   423   also have "\<dots> = log b (\<Sum>x\<in>X`space M. if ?d x \<noteq> 0 then 1 else 0)"
```
```   424     using distribution_finite[OF `simple_function M X`[THEN simple_function_imp_random_variable], simplified]
```
```   425     by (intro arg_cong[where f="\<lambda>X. log b X"] setsum_cong) (auto simp: real_of_pextreal_eq_0)
```
```   426   finally show ?thesis
```
```   427     using `simple_function M X` by (auto simp: setsum_cases real_eq_of_nat simple_function_def)
```
```   428 qed
```
```   429
```
```   430 lemma (in information_space) entropy_uniform_max:
```
```   431   assumes "simple_function M X"
```
```   432   assumes "\<And>x y. \<lbrakk> x \<in> X ` space M ; y \<in> X ` space M \<rbrakk> \<Longrightarrow> distribution X {x} = distribution X {y}"
```
```   433   shows "\<H>(X) = log b (real (card (X ` space M)))"
```
```   434 proof -
```
```   435   let ?X = "\<lparr> space = X ` space M, sets = Pow (X ` space M), measure = distribution X\<rparr>"
```
```   436   note simple_function_imp_finite_random_variable[OF `simple_function M X`]
```
```   437   from distribution_finite_prob_space[OF this, of "\<lparr> measure = distribution X \<rparr>"]
```
```   438   interpret X: finite_prob_space ?X by simp
```
```   439   have card_gt0: "0 < card (X ` space M)" unfolding card_gt_0_iff
```
```   440     using `simple_function M X` not_empty by (auto simp: simple_function_def)
```
```   441   { fix x assume "x \<in> X ` space M"
```
```   442     hence "real (distribution X {x}) = 1 / real (card (X ` space M))"
```
```   443     proof (rule X.uniform_prob[simplified])
```
```   444       fix x y assume "x \<in> X`space M" "y \<in> X`space M"
```
```   445       from assms(2)[OF this] show "real (distribution X {x}) = real (distribution X {y})" by simp
```
```   446     qed }
```
```   447   thus ?thesis
```
```   448     using not_empty X.finite_space b_gt_1 card_gt0
```
```   449     by (simp add: entropy_eq[OF `simple_function M X`] real_eq_of_nat[symmetric] log_simps)
```
```   450 qed
```
```   451
```
```   452 lemma (in information_space) entropy_le_card:
```
```   453   assumes "simple_function M X"
```
```   454   shows "\<H>(X) \<le> log b (real (card (X ` space M)))"
```
```   455 proof cases
```
```   456   assume "X ` space M \<inter> {x. distribution X {x} \<noteq> 0} = {}"
```
```   457   then have "\<And>x. x\<in>X`space M \<Longrightarrow> distribution X {x} = 0" by auto
```
```   458   moreover
```
```   459   have "0 < card (X`space M)"
```
```   460     using `simple_function M X` not_empty
```
```   461     by (auto simp: card_gt_0_iff simple_function_def)
```
```   462   then have "log b 1 \<le> log b (real (card (X`space M)))"
```
```   463     using b_gt_1 by (intro log_le) auto
```
```   464   ultimately show ?thesis using assms by (simp add: entropy_eq)
```
```   465 next
```
```   466   assume False: "X ` space M \<inter> {x. distribution X {x} \<noteq> 0} \<noteq> {}"
```
```   467   have "card (X ` space M \<inter> {x. distribution X {x} \<noteq> 0}) \<le> card (X ` space M)"
```
```   468     (is "?A \<le> ?B") using assms not_empty by (auto intro!: card_mono simp: simple_function_def)
```
```   469   note entropy_le_card_not_0[OF assms]
```
```   470   also have "log b (real ?A) \<le> log b (real ?B)"
```
```   471     using b_gt_1 False not_empty `?A \<le> ?B` assms
```
```   472     by (auto intro!: log_le simp: card_gt_0_iff simp: simple_function_def)
```
```   473   finally show ?thesis .
```
```   474 qed
```
```   475
```
```   476 lemma (in information_space) entropy_commute:
```
```   477   assumes "simple_function M X" "simple_function M Y"
```
```   478   shows "\<H>(\<lambda>x. (X x, Y x)) = \<H>(\<lambda>x. (Y x, X x))"
```
```   479 proof -
```
```   480   have sf: "simple_function M (\<lambda>x. (X x, Y x))" "simple_function M (\<lambda>x. (Y x, X x))"
```
```   481     using assms by (auto intro: simple_function_Pair)
```
```   482   have *: "(\<lambda>x. (Y x, X x))`space M = (\<lambda>(a,b). (b,a))`(\<lambda>x. (X x, Y x))`space M"
```
```   483     by auto
```
```   484   have inj: "\<And>X. inj_on (\<lambda>(a,b). (b,a)) X"
```
```   485     by (auto intro!: inj_onI)
```
```   486   show ?thesis
```
```   487     unfolding sf[THEN entropy_eq] unfolding * setsum_reindex[OF inj]
```
```   488     by (simp add: joint_distribution_commute[of Y X] split_beta)
```
```   489 qed
```
```   490
```
```   491 lemma (in information_space) entropy_eq_cartesian_product:
```
```   492   assumes "simple_function M X" "simple_function M Y"
```
```   493   shows "\<H>(\<lambda>x. (X x, Y x)) = -(\<Sum>x\<in>X`space M. \<Sum>y\<in>Y`space M.
```
```   494     real (joint_distribution X Y {(x,y)}) *
```
```   495     log b (real (joint_distribution X Y {(x,y)})))"
```
```   496 proof -
```
```   497   have sf: "simple_function M (\<lambda>x. (X x, Y x))"
```
```   498     using assms by (auto intro: simple_function_Pair)
```
```   499   { fix x assume "x\<notin>(\<lambda>x. (X x, Y x))`space M"
```
```   500     then have "(\<lambda>x. (X x, Y x)) -` {x} \<inter> space M = {}" by auto
```
```   501     then have "joint_distribution X Y {x} = 0"
```
```   502       unfolding distribution_def by auto }
```
```   503   then show ?thesis using sf assms
```
```   504     unfolding entropy_eq[OF sf] neg_equal_iff_equal setsum_cartesian_product
```
```   505     by (auto intro!: setsum_mono_zero_cong_left simp: simple_function_def)
```
```   506 qed
```
```   507
```
```   508 subsection {* Conditional Mutual Information *}
```
```   509
```
```   510 definition (in prob_space)
```
```   511   "conditional_mutual_information b MX MY MZ X Y Z \<equiv>
```
```   512     mutual_information b MX (MY \<Otimes>\<^isub>M MZ) X (\<lambda>x. (Y x, Z x)) -
```
```   513     mutual_information b MX MZ X Z"
```
```   514
```
```   515 abbreviation (in information_space)
```
```   516   conditional_mutual_information_Pow ("\<I>'( _ ; _ | _ ')") where
```
```   517   "\<I>(X ; Y | Z) \<equiv> conditional_mutual_information b
```
```   518     \<lparr> space = X`space M, sets = Pow (X`space M), measure = distribution X \<rparr>
```
```   519     \<lparr> space = Y`space M, sets = Pow (Y`space M), measure = distribution Y \<rparr>
```
```   520     \<lparr> space = Z`space M, sets = Pow (Z`space M), measure = distribution Z \<rparr>
```
```   521     X Y Z"
```
```   522
```
```   523 lemma (in information_space) conditional_mutual_information_generic_eq:
```
```   524   assumes MX: "finite_random_variable MX X"
```
```   525     and MY: "finite_random_variable MY Y"
```
```   526     and MZ: "finite_random_variable MZ Z"
```
```   527   shows "conditional_mutual_information b MX MY MZ X Y Z = (\<Sum>(x, y, z) \<in> space MX \<times> space MY \<times> space MZ.
```
```   528              real (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)}) *
```
```   529              log b (real (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)}) /
```
```   530     (real (joint_distribution X Z {(x, z)}) * real (joint_distribution Y Z {(y,z)} / distribution Z {z}))))"
```
```   531   (is "_ = (\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?XZ x z * ?YZdZ y z)))")
```
```   532 proof -
```
```   533   let ?YZ = "\<lambda>y z. real (joint_distribution Y Z {(y, z)})"
```
```   534   let ?X = "\<lambda>x. real (distribution X {x})"
```
```   535   let ?Z = "\<lambda>z. real (distribution Z {z})"
```
```   536
```
```   537   txt {* This proof is actually quiet easy, however we need to show that the
```
```   538     distributions are finite and the joint distributions are zero when one of
```
```   539     the variables distribution is also zero. *}
```
```   540
```
```   541   note finite_var = MX MY MZ
```
```   542   note random_var = finite_var[THEN finite_random_variableD]
```
```   543
```
```   544   note space_simps = space_pair_measure space_sigma algebra.simps
```
```   545
```
```   546   note YZ = finite_random_variable_pairI[OF finite_var(2,3)]
```
```   547   note XZ = finite_random_variable_pairI[OF finite_var(1,3)]
```
```   548   note ZX = finite_random_variable_pairI[OF finite_var(3,1)]
```
```   549   note YZX = finite_random_variable_pairI[OF finite_var(2) ZX]
```
```   550   note order1 =
```
```   551     finite_distribution_order(5,6)[OF finite_var(1) YZ, simplified space_simps]
```
```   552     finite_distribution_order(5,6)[OF finite_var(1,3), simplified space_simps]
```
```   553
```
```   554   note finite = finite_var(1) YZ finite_var(3) XZ YZX
```
```   555   note finite[THEN finite_distribution_finite, simplified space_simps, simp]
```
```   556
```
```   557   have order2: "\<And>x y z. \<lbrakk>x \<in> space MX; y \<in> space MY; z \<in> space MZ; joint_distribution X Z {(x, z)} = 0\<rbrakk>
```
```   558           \<Longrightarrow> joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)} = 0"
```
```   559     unfolding joint_distribution_commute_singleton[of X]
```
```   560     unfolding joint_distribution_assoc_singleton[symmetric]
```
```   561     using finite_distribution_order(6)[OF finite_var(2) ZX]
```
```   562     by (auto simp: space_simps)
```
```   563
```
```   564   have "(\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?XZ x z * ?YZdZ y z))) =
```
```   565     (\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * (log b (?XYZ x y z / (?X x * ?YZ y z)) - log b (?XZ x z / (?X x * ?Z z))))"
```
```   566     (is "(\<Sum>(x, y, z)\<in>?S. ?L x y z) = (\<Sum>(x, y, z)\<in>?S. ?R x y z)")
```
```   567   proof (safe intro!: setsum_cong)
```
```   568     fix x y z assume space: "x \<in> space MX" "y \<in> space MY" "z \<in> space MZ"
```
```   569     then have *: "?XYZ x y z / (?XZ x z * ?YZdZ y z) =
```
```   570       (?XYZ x y z / (?X x * ?YZ y z)) / (?XZ x z / (?X x * ?Z z))"
```
```   571       using order1(3)
```
```   572       by (auto simp: real_of_pextreal_mult[symmetric] real_of_pextreal_eq_0)
```
```   573     show "?L x y z = ?R x y z"
```
```   574     proof cases
```
```   575       assume "?XYZ x y z \<noteq> 0"
```
```   576       with space b_gt_1 order1 order2 show ?thesis unfolding *
```
```   577         by (subst log_divide)
```
```   578            (auto simp: zero_less_divide_iff zero_less_real_of_pextreal
```
```   579                        real_of_pextreal_eq_0 zero_less_mult_iff)
```
```   580     qed simp
```
```   581   qed
```
```   582   also have "\<dots> = (\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?X x * ?YZ y z))) -
```
```   583                   (\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XZ x z / (?X x * ?Z z)))"
```
```   584     by (auto simp add: setsum_subtractf[symmetric] field_simps intro!: setsum_cong)
```
```   585   also have "(\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XZ x z / (?X x * ?Z z))) =
```
```   586              (\<Sum>(x, z)\<in>space MX \<times> space MZ. ?XZ x z * log b (?XZ x z / (?X x * ?Z z)))"
```
```   587     unfolding setsum_cartesian_product[symmetric] setsum_commute[of _ _ "space MY"]
```
```   588               setsum_left_distrib[symmetric]
```
```   589     unfolding joint_distribution_commute_singleton[of X]
```
```   590     unfolding joint_distribution_assoc_singleton[symmetric]
```
```   591     using setsum_real_joint_distribution_singleton[OF finite_var(2) ZX, unfolded space_simps]
```
```   592     by (intro setsum_cong refl) simp
```
```   593   also have "(\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?X x * ?YZ y z))) -
```
```   594              (\<Sum>(x, z)\<in>space MX \<times> space MZ. ?XZ x z * log b (?XZ x z / (?X x * ?Z z))) =
```
```   595              conditional_mutual_information b MX MY MZ X Y Z"
```
```   596     unfolding conditional_mutual_information_def
```
```   597     unfolding mutual_information_generic_eq[OF finite_var(1,3)]
```
```   598     unfolding mutual_information_generic_eq[OF finite_var(1) YZ]
```
```   599     by (simp add: space_sigma space_pair_measure setsum_cartesian_product')
```
```   600   finally show ?thesis by simp
```
```   601 qed
```
```   602
```
```   603 lemma (in information_space) conditional_mutual_information_eq:
```
```   604   assumes "simple_function M X" "simple_function M Y" "simple_function M Z"
```
```   605   shows "\<I>(X;Y|Z) = (\<Sum>(x, y, z) \<in> X`space M \<times> Y`space M \<times> Z`space M.
```
```   606              real (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)}) *
```
```   607              log b (real (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)}) /
```
```   608     (real (joint_distribution X Z {(x, z)}) * real (joint_distribution Y Z {(y,z)} / distribution Z {z}))))"
```
```   609   using conditional_mutual_information_generic_eq[OF assms[THEN simple_function_imp_finite_random_variable]]
```
```   610   by simp
```
```   611
```
```   612 lemma (in information_space) conditional_mutual_information_eq_mutual_information:
```
```   613   assumes X: "simple_function M X" and Y: "simple_function M Y"
```
```   614   shows "\<I>(X ; Y) = \<I>(X ; Y | (\<lambda>x. ()))"
```
```   615 proof -
```
```   616   have [simp]: "(\<lambda>x. ()) ` space M = {()}" using not_empty by auto
```
```   617   have C: "simple_function M (\<lambda>x. ())" by auto
```
```   618   show ?thesis
```
```   619     unfolding conditional_mutual_information_eq[OF X Y C]
```
```   620     unfolding mutual_information_eq[OF X Y]
```
```   621     by (simp add: setsum_cartesian_product' distribution_remove_const)
```
```   622 qed
```
```   623
```
```   624 lemma (in prob_space) distribution_unit[simp]: "distribution (\<lambda>x. ()) {()} = 1"
```
```   625   unfolding distribution_def using measure_space_1 by auto
```
```   626
```
```   627 lemma (in prob_space) joint_distribution_unit[simp]: "distribution (\<lambda>x. (X x, ())) {(a, ())} = distribution X {a}"
```
```   628   unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>])
```
```   629
```
```   630 lemma (in prob_space) setsum_distribution:
```
```   631   assumes X: "finite_random_variable MX X" shows "(\<Sum>a\<in>space MX. distribution X {a}) = 1"
```
```   632   using setsum_joint_distribution[OF assms, of "\<lparr> space = UNIV, sets = Pow UNIV \<rparr>" "\<lambda>x. ()" "{()}"]
```
```   633   using sigma_algebra_Pow[of "UNIV::unit set" "()"] by simp
```
```   634
```
```   635 lemma (in prob_space) setsum_real_distribution:
```
```   636   fixes MX :: "('c, 'd) measure_space_scheme"
```
```   637   assumes X: "finite_random_variable MX X" shows "(\<Sum>a\<in>space MX. real (distribution X {a})) = 1"
```
```   638   using setsum_real_joint_distribution[OF assms, of "\<lparr> space = UNIV, sets = Pow UNIV, measure = undefined \<rparr>" "\<lambda>x. ()" "{()}"]
```
```   639   using sigma_algebra_Pow[of "UNIV::unit set" "\<lparr> measure = undefined \<rparr>"] by simp
```
```   640
```
```   641 lemma (in information_space) conditional_mutual_information_generic_positive:
```
```   642   assumes "finite_random_variable MX X" and "finite_random_variable MY Y" and "finite_random_variable MZ Z"
```
```   643   shows "0 \<le> conditional_mutual_information b MX MY MZ X Y Z"
```
```   644 proof (cases "space MX \<times> space MY \<times> space MZ = {}")
```
```   645   case True show ?thesis
```
```   646     unfolding conditional_mutual_information_generic_eq[OF assms] True
```
```   647     by simp
```
```   648 next
```
```   649   case False
```
```   650   let "?dXYZ A" = "real (distribution (\<lambda>x. (X x, Y x, Z x)) A)"
```
```   651   let "?dXZ A" = "real (joint_distribution X Z A)"
```
```   652   let "?dYZ A" = "real (joint_distribution Y Z A)"
```
```   653   let "?dX A" = "real (distribution X A)"
```
```   654   let "?dZ A" = "real (distribution Z A)"
```
```   655   let ?M = "space MX \<times> space MY \<times> space MZ"
```
```   656
```
```   657   have split_beta: "\<And>f. split f = (\<lambda>x. f (fst x) (snd x))" by (simp add: fun_eq_iff)
```
```   658
```
```   659   note space_simps = space_pair_measure space_sigma algebra.simps
```
```   660
```
```   661   note finite_var = assms
```
```   662   note YZ = finite_random_variable_pairI[OF finite_var(2,3)]
```
```   663   note XZ = finite_random_variable_pairI[OF finite_var(1,3)]
```
```   664   note ZX = finite_random_variable_pairI[OF finite_var(3,1)]
```
```   665   note YZ = finite_random_variable_pairI[OF finite_var(2,3)]
```
```   666   note XYZ = finite_random_variable_pairI[OF finite_var(1) YZ]
```
```   667   note finite = finite_var(3) YZ XZ XYZ
```
```   668   note finite = finite[THEN finite_distribution_finite, simplified space_simps]
```
```   669
```
```   670   have order: "\<And>x y z. \<lbrakk>x \<in> space MX; y \<in> space MY; z \<in> space MZ; joint_distribution X Z {(x, z)} = 0\<rbrakk>
```
```   671           \<Longrightarrow> joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)} = 0"
```
```   672     unfolding joint_distribution_commute_singleton[of X]
```
```   673     unfolding joint_distribution_assoc_singleton[symmetric]
```
```   674     using finite_distribution_order(6)[OF finite_var(2) ZX]
```
```   675     by (auto simp: space_simps)
```
```   676
```
```   677   note order = order
```
```   678     finite_distribution_order(5,6)[OF finite_var(1) YZ, simplified space_simps]
```
```   679     finite_distribution_order(5,6)[OF finite_var(2,3), simplified space_simps]
```
```   680
```
```   681   have "- conditional_mutual_information b MX MY MZ X Y Z = - (\<Sum>(x, y, z) \<in> ?M. ?dXYZ {(x, y, z)} *
```
```   682     log b (?dXYZ {(x, y, z)} / (?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z})))"
```
```   683     unfolding conditional_mutual_information_generic_eq[OF assms] neg_equal_iff_equal
```
```   684     by (intro setsum_cong) (auto intro!: arg_cong[where f="log b"] simp: real_of_pextreal_mult[symmetric])
```
```   685   also have "\<dots> \<le> log b (\<Sum>(x, y, z) \<in> ?M. ?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z})"
```
```   686     unfolding split_beta
```
```   687   proof (rule log_setsum_divide)
```
```   688     show "?M \<noteq> {}" using False by simp
```
```   689     show "1 < b" using b_gt_1 .
```
```   690
```
```   691     show "finite ?M" using assms
```
```   692       unfolding finite_sigma_algebra_def finite_sigma_algebra_axioms_def by auto
```
```   693
```
```   694     show "(\<Sum>x\<in>?M. ?dXYZ {(fst x, fst (snd x), snd (snd x))}) = 1"
```
```   695       unfolding setsum_cartesian_product'
```
```   696       unfolding setsum_commute[of _ "space MY"]
```
```   697       unfolding setsum_commute[of _ "space MZ"]
```
```   698       by (simp_all add: space_pair_measure
```
```   699         setsum_real_joint_distribution_singleton[OF `finite_random_variable MX X` YZ]
```
```   700         setsum_real_joint_distribution_singleton[OF `finite_random_variable MY Y` finite_var(3)]
```
```   701         setsum_real_distribution[OF `finite_random_variable MZ Z`])
```
```   702
```
```   703     fix x assume "x \<in> ?M"
```
```   704     let ?x = "(fst x, fst (snd x), snd (snd x))"
```
```   705
```
```   706     show "0 \<le> ?dXYZ {?x}" using real_pextreal_nonneg .
```
```   707     show "0 \<le> ?dXZ {(fst x, snd (snd x))} * ?dYZ {(fst (snd x), snd (snd x))} / ?dZ {snd (snd x)}"
```
```   708      by (simp add: real_pextreal_nonneg mult_nonneg_nonneg divide_nonneg_nonneg)
```
```   709
```
```   710     assume *: "0 < ?dXYZ {?x}"
```
```   711     with `x \<in> ?M` show "0 < ?dXZ {(fst x, snd (snd x))} * ?dYZ {(fst (snd x), snd (snd x))} / ?dZ {snd (snd x)}"
```
```   712       using finite order
```
```   713       by (cases x)
```
```   714          (auto simp add: zero_less_real_of_pextreal zero_less_mult_iff zero_less_divide_iff)
```
```   715   qed
```
```   716   also have "(\<Sum>(x, y, z) \<in> ?M. ?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z}) = (\<Sum>z\<in>space MZ. ?dZ {z})"
```
```   717     apply (simp add: setsum_cartesian_product')
```
```   718     apply (subst setsum_commute)
```
```   719     apply (subst (2) setsum_commute)
```
```   720     by (auto simp: setsum_divide_distrib[symmetric] setsum_product[symmetric]
```
```   721                    setsum_real_joint_distribution_singleton[OF finite_var(1,3)]
```
```   722                    setsum_real_joint_distribution_singleton[OF finite_var(2,3)]
```
```   723           intro!: setsum_cong)
```
```   724   also have "log b (\<Sum>z\<in>space MZ. ?dZ {z}) = 0"
```
```   725     unfolding setsum_real_distribution[OF finite_var(3)] by simp
```
```   726   finally show ?thesis by simp
```
```   727 qed
```
```   728
```
```   729 lemma (in information_space) conditional_mutual_information_positive:
```
```   730   assumes "simple_function M X" and "simple_function M Y" and "simple_function M Z"
```
```   731   shows "0 \<le> \<I>(X;Y|Z)"
```
```   732   by (rule conditional_mutual_information_generic_positive[OF assms[THEN simple_function_imp_finite_random_variable]])
```
```   733
```
```   734 subsection {* Conditional Entropy *}
```
```   735
```
```   736 definition (in prob_space)
```
```   737   "conditional_entropy b S T X Y = conditional_mutual_information b S S T X X Y"
```
```   738
```
```   739 abbreviation (in information_space)
```
```   740   conditional_entropy_Pow ("\<H>'(_ | _')") where
```
```   741   "\<H>(X | Y) \<equiv> conditional_entropy b
```
```   742     \<lparr> space = X`space M, sets = Pow (X`space M), measure = distribution X \<rparr>
```
```   743     \<lparr> space = Y`space M, sets = Pow (Y`space M), measure = distribution Y \<rparr> X Y"
```
```   744
```
```   745 lemma (in information_space) conditional_entropy_positive:
```
```   746   "simple_function M X \<Longrightarrow> simple_function M Y \<Longrightarrow> 0 \<le> \<H>(X | Y)"
```
```   747   unfolding conditional_entropy_def by (auto intro!: conditional_mutual_information_positive)
```
```   748
```
```   749 lemma (in measure_space) empty_measureI: "A = {} \<Longrightarrow> \<mu> A = 0" by simp
```
```   750
```
```   751 lemma (in information_space) conditional_entropy_generic_eq:
```
```   752   fixes MX :: "('c, 'd) measure_space_scheme" and MY :: "('e, 'f) measure_space_scheme"
```
```   753   assumes MX: "finite_random_variable MX X"
```
```   754   assumes MZ: "finite_random_variable MZ Z"
```
```   755   shows "conditional_entropy b MX MZ X Z =
```
```   756      - (\<Sum>(x, z)\<in>space MX \<times> space MZ.
```
```   757          real (joint_distribution X Z {(x, z)}) *
```
```   758          log b (real (joint_distribution X Z {(x, z)}) / real (distribution Z {z})))"
```
```   759 proof -
```
```   760   interpret MX: finite_sigma_algebra MX using MX by simp
```
```   761   interpret MZ: finite_sigma_algebra MZ using MZ by simp
```
```   762   let "?XXZ x y z" = "joint_distribution X (\<lambda>x. (X x, Z x)) {(x, y, z)}"
```
```   763   let "?XZ x z" = "joint_distribution X Z {(x, z)}"
```
```   764   let "?Z z" = "distribution Z {z}"
```
```   765   let "?f x y z" = "log b (real (?XXZ x y z) / (real (?XZ x z) * real (?XZ y z / ?Z z)))"
```
```   766   { fix x z have "?XXZ x x z = ?XZ x z"
```
```   767       unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>]) }
```
```   768   note this[simp]
```
```   769   { fix x x' :: 'c and z assume "x' \<noteq> x"
```
```   770     then have "?XXZ x x' z = 0"
```
```   771       by (auto simp: distribution_def intro!: arg_cong[where f=\<mu>] empty_measureI) }
```
```   772   note this[simp]
```
```   773   { fix x x' z assume *: "x \<in> space MX" "z \<in> space MZ"
```
```   774     then have "(\<Sum>x'\<in>space MX. real (?XXZ x x' z) * ?f x x' z)
```
```   775       = (\<Sum>x'\<in>space MX. if x = x' then real (?XZ x z) * ?f x x z else 0)"
```
```   776       by (auto intro!: setsum_cong)
```
```   777     also have "\<dots> = real (?XZ x z) * ?f x x z"
```
```   778       using `x \<in> space MX` by (simp add: setsum_cases[OF MX.finite_space])
```
```   779     also have "\<dots> = real (?XZ x z) * log b (real (?Z z) / real (?XZ x z))"
```
```   780       by (auto simp: real_of_pextreal_mult[symmetric])
```
```   781     also have "\<dots> = - real (?XZ x z) * log b (real (?XZ x z) / real (?Z z))"
```
```   782       using assms[THEN finite_distribution_finite]
```
```   783       using finite_distribution_order(6)[OF MX MZ]
```
```   784       by (auto simp: log_simps field_simps zero_less_mult_iff zero_less_real_of_pextreal real_of_pextreal_eq_0)
```
```   785     finally have "(\<Sum>x'\<in>space MX. real (?XXZ x x' z) * ?f x x' z) =
```
```   786       - real (?XZ x z) * log b (real (?XZ x z) / real (?Z z))" . }
```
```   787   note * = this
```
```   788   show ?thesis
```
```   789     unfolding conditional_entropy_def
```
```   790     unfolding conditional_mutual_information_generic_eq[OF MX MX MZ]
```
```   791     by (auto simp: setsum_cartesian_product' setsum_negf[symmetric]
```
```   792                    setsum_commute[of _ "space MZ"] *   simp del: divide_pextreal_def
```
```   793              intro!: setsum_cong)
```
```   794 qed
```
```   795
```
```   796 lemma (in information_space) conditional_entropy_eq:
```
```   797   assumes "simple_function M X" "simple_function M Z"
```
```   798   shows "\<H>(X | Z) =
```
```   799      - (\<Sum>(x, z)\<in>X ` space M \<times> Z ` space M.
```
```   800          real (joint_distribution X Z {(x, z)}) *
```
```   801          log b (real (joint_distribution X Z {(x, z)}) / real (distribution Z {z})))"
```
```   802   using conditional_entropy_generic_eq[OF assms[THEN simple_function_imp_finite_random_variable]]
```
```   803   by simp
```
```   804
```
```   805 lemma (in information_space) conditional_entropy_eq_ce_with_hypothesis:
```
```   806   assumes X: "simple_function M X" and Y: "simple_function M Y"
```
```   807   shows "\<H>(X | Y) =
```
```   808     -(\<Sum>y\<in>Y`space M. real (distribution Y {y}) *
```
```   809       (\<Sum>x\<in>X`space M. real (joint_distribution X Y {(x,y)}) / real (distribution Y {(y)}) *
```
```   810               log b (real (joint_distribution X Y {(x,y)}) / real (distribution Y {(y)}))))"
```
```   811   unfolding conditional_entropy_eq[OF assms]
```
```   812   using finite_distribution_finite[OF finite_random_variable_pairI[OF assms[THEN simple_function_imp_finite_random_variable]]]
```
```   813   using finite_distribution_order(5,6)[OF assms[THEN simple_function_imp_finite_random_variable]]
```
```   814   using finite_distribution_finite[OF Y[THEN simple_function_imp_finite_random_variable]]
```
```   815   by (auto simp: setsum_cartesian_product'  setsum_commute[of _ "Y`space M"] setsum_right_distrib real_of_pextreal_eq_0
```
```   816            intro!: setsum_cong)
```
```   817
```
```   818 lemma (in information_space) conditional_entropy_eq_cartesian_product:
```
```   819   assumes "simple_function M X" "simple_function M Y"
```
```   820   shows "\<H>(X | Y) = -(\<Sum>x\<in>X`space M. \<Sum>y\<in>Y`space M.
```
```   821     real (joint_distribution X Y {(x,y)}) *
```
```   822     log b (real (joint_distribution X Y {(x,y)}) / real (distribution Y {y})))"
```
```   823   unfolding conditional_entropy_eq[OF assms]
```
```   824   by (auto intro!: setsum_cong simp: setsum_cartesian_product')
```
```   825
```
```   826 subsection {* Equalities *}
```
```   827
```
```   828 lemma (in information_space) mutual_information_eq_entropy_conditional_entropy:
```
```   829   assumes X: "simple_function M X" and Z: "simple_function M Z"
```
```   830   shows  "\<I>(X ; Z) = \<H>(X) - \<H>(X | Z)"
```
```   831 proof -
```
```   832   let "?XZ x z" = "real (joint_distribution X Z {(x, z)})"
```
```   833   let "?Z z" = "real (distribution Z {z})"
```
```   834   let "?X x" = "real (distribution X {x})"
```
```   835   note fX = X[THEN simple_function_imp_finite_random_variable]
```
```   836   note fZ = Z[THEN simple_function_imp_finite_random_variable]
```
```   837   note fX[THEN finite_distribution_finite, simp] and fZ[THEN finite_distribution_finite, simp]
```
```   838   note finite_distribution_order[OF fX fZ, simp]
```
```   839   { fix x z assume "x \<in> X`space M" "z \<in> Z`space M"
```
```   840     have "?XZ x z * log b (?XZ x z / (?X x * ?Z z)) =
```
```   841           ?XZ x z * log b (?XZ x z / ?Z z) - ?XZ x z * log b (?X x)"
```
```   842       by (auto simp: log_simps real_of_pextreal_mult[symmetric] zero_less_mult_iff
```
```   843                      zero_less_real_of_pextreal field_simps real_of_pextreal_eq_0 abs_mult) }
```
```   844   note * = this
```
```   845   show ?thesis
```
```   846     unfolding entropy_eq[OF X] conditional_entropy_eq[OF X Z] mutual_information_eq[OF X Z]
```
```   847     using setsum_real_joint_distribution_singleton[OF fZ fX, unfolded joint_distribution_commute_singleton[of Z X]]
```
```   848     by (simp add: * setsum_cartesian_product' setsum_subtractf setsum_left_distrib[symmetric]
```
```   849                      setsum_real_distribution)
```
```   850 qed
```
```   851
```
```   852 lemma (in information_space) conditional_entropy_less_eq_entropy:
```
```   853   assumes X: "simple_function M X" and Z: "simple_function M Z"
```
```   854   shows "\<H>(X | Z) \<le> \<H>(X)"
```
```   855 proof -
```
```   856   have "\<I>(X ; Z) = \<H>(X) - \<H>(X | Z)" using mutual_information_eq_entropy_conditional_entropy[OF assms] .
```
```   857   with mutual_information_positive[OF X Z] entropy_positive[OF X]
```
```   858   show ?thesis by auto
```
```   859 qed
```
```   860
```
```   861 lemma (in information_space) entropy_chain_rule:
```
```   862   assumes X: "simple_function M X" and Y: "simple_function M Y"
```
```   863   shows  "\<H>(\<lambda>x. (X x, Y x)) = \<H>(X) + \<H>(Y|X)"
```
```   864 proof -
```
```   865   let "?XY x y" = "real (joint_distribution X Y {(x, y)})"
```
```   866   let "?Y y" = "real (distribution Y {y})"
```
```   867   let "?X x" = "real (distribution X {x})"
```
```   868   note fX = X[THEN simple_function_imp_finite_random_variable]
```
```   869   note fY = Y[THEN simple_function_imp_finite_random_variable]
```
```   870   note fX[THEN finite_distribution_finite, simp] and fY[THEN finite_distribution_finite, simp]
```
```   871   note finite_distribution_order[OF fX fY, simp]
```
```   872   { fix x y assume "x \<in> X`space M" "y \<in> Y`space M"
```
```   873     have "?XY x y * log b (?XY x y / ?X x) =
```
```   874           ?XY x y * log b (?XY x y) - ?XY x y * log b (?X x)"
```
```   875       by (auto simp: log_simps real_of_pextreal_mult[symmetric] zero_less_mult_iff
```
```   876                      zero_less_real_of_pextreal field_simps real_of_pextreal_eq_0 abs_mult) }
```
```   877   note * = this
```
```   878   show ?thesis
```
```   879     using setsum_real_joint_distribution_singleton[OF fY fX]
```
```   880     unfolding entropy_eq[OF X] conditional_entropy_eq_cartesian_product[OF Y X] entropy_eq_cartesian_product[OF X Y]
```
```   881     unfolding joint_distribution_commute_singleton[of Y X] setsum_commute[of _ "X`space M"]
```
```   882     by (simp add: * setsum_subtractf setsum_left_distrib[symmetric])
```
```   883 qed
```
```   884
```
```   885 section {* Partitioning *}
```
```   886
```
```   887 definition "subvimage A f g \<longleftrightarrow> (\<forall>x \<in> A. f -` {f x} \<inter> A \<subseteq> g -` {g x} \<inter> A)"
```
```   888
```
```   889 lemma subvimageI:
```
```   890   assumes "\<And>x y. \<lbrakk> x \<in> A ; y \<in> A ; f x = f y \<rbrakk> \<Longrightarrow> g x = g y"
```
```   891   shows "subvimage A f g"
```
```   892   using assms unfolding subvimage_def by blast
```
```   893
```
```   894 lemma subvimageE[consumes 1]:
```
```   895   assumes "subvimage A f g"
```
```   896   obtains "\<And>x y. \<lbrakk> x \<in> A ; y \<in> A ; f x = f y \<rbrakk> \<Longrightarrow> g x = g y"
```
```   897   using assms unfolding subvimage_def by blast
```
```   898
```
```   899 lemma subvimageD:
```
```   900   "\<lbrakk> subvimage A f g ; x \<in> A ; y \<in> A ; f x = f y \<rbrakk> \<Longrightarrow> g x = g y"
```
```   901   using assms unfolding subvimage_def by blast
```
```   902
```
```   903 lemma subvimage_subset:
```
```   904   "\<lbrakk> subvimage B f g ; A \<subseteq> B \<rbrakk> \<Longrightarrow> subvimage A f g"
```
```   905   unfolding subvimage_def by auto
```
```   906
```
```   907 lemma subvimage_idem[intro]: "subvimage A g g"
```
```   908   by (safe intro!: subvimageI)
```
```   909
```
```   910 lemma subvimage_comp_finer[intro]:
```
```   911   assumes svi: "subvimage A g h"
```
```   912   shows "subvimage A g (f \<circ> h)"
```
```   913 proof (rule subvimageI, simp)
```
```   914   fix x y assume "x \<in> A" "y \<in> A" "g x = g y"
```
```   915   from svi[THEN subvimageD, OF this]
```
```   916   show "f (h x) = f (h y)" by simp
```
```   917 qed
```
```   918
```
```   919 lemma subvimage_comp_gran:
```
```   920   assumes svi: "subvimage A g h"
```
```   921   assumes inj: "inj_on f (g ` A)"
```
```   922   shows "subvimage A (f \<circ> g) h"
```
```   923   by (rule subvimageI) (auto intro!: subvimageD[OF svi] simp: inj_on_iff[OF inj])
```
```   924
```
```   925 lemma subvimage_comp:
```
```   926   assumes svi: "subvimage (f ` A) g h"
```
```   927   shows "subvimage A (g \<circ> f) (h \<circ> f)"
```
```   928   by (rule subvimageI) (auto intro!: svi[THEN subvimageD])
```
```   929
```
```   930 lemma subvimage_trans:
```
```   931   assumes fg: "subvimage A f g"
```
```   932   assumes gh: "subvimage A g h"
```
```   933   shows "subvimage A f h"
```
```   934   by (rule subvimageI) (auto intro!: fg[THEN subvimageD] gh[THEN subvimageD])
```
```   935
```
```   936 lemma subvimage_translator:
```
```   937   assumes svi: "subvimage A f g"
```
```   938   shows "\<exists>h. \<forall>x \<in> A. h (f x)  = g x"
```
```   939 proof (safe intro!: exI[of _ "\<lambda>x. (THE z. z \<in> (g ` (f -` {x} \<inter> A)))"])
```
```   940   fix x assume "x \<in> A"
```
```   941   show "(THE x'. x' \<in> (g ` (f -` {f x} \<inter> A))) = g x"
```
```   942     by (rule theI2[of _ "g x"])
```
```   943       (insert `x \<in> A`, auto intro!: svi[THEN subvimageD])
```
```   944 qed
```
```   945
```
```   946 lemma subvimage_translator_image:
```
```   947   assumes svi: "subvimage A f g"
```
```   948   shows "\<exists>h. h ` f ` A = g ` A"
```
```   949 proof -
```
```   950   from subvimage_translator[OF svi]
```
```   951   obtain h where "\<And>x. x \<in> A \<Longrightarrow> h (f x) = g x" by auto
```
```   952   thus ?thesis
```
```   953     by (auto intro!: exI[of _ h]
```
```   954       simp: image_compose[symmetric] comp_def cong: image_cong)
```
```   955 qed
```
```   956
```
```   957 lemma subvimage_finite:
```
```   958   assumes svi: "subvimage A f g" and fin: "finite (f`A)"
```
```   959   shows "finite (g`A)"
```
```   960 proof -
```
```   961   from subvimage_translator_image[OF svi]
```
```   962   obtain h where "g`A = h`f`A" by fastsimp
```
```   963   with fin show "finite (g`A)" by simp
```
```   964 qed
```
```   965
```
```   966 lemma subvimage_disj:
```
```   967   assumes svi: "subvimage A f g"
```
```   968   shows "f -` {x} \<inter> A \<subseteq> g -` {y} \<inter> A \<or>
```
```   969       f -` {x} \<inter> g -` {y} \<inter> A = {}" (is "?sub \<or> ?dist")
```
```   970 proof (rule disjCI)
```
```   971   assume "\<not> ?dist"
```
```   972   then obtain z where "z \<in> A" and "x = f z" and "y = g z" by auto
```
```   973   thus "?sub" using svi unfolding subvimage_def by auto
```
```   974 qed
```
```   975
```
```   976 lemma setsum_image_split:
```
```   977   assumes svi: "subvimage A f g" and fin: "finite (f ` A)"
```
```   978   shows "(\<Sum>x\<in>f`A. h x) = (\<Sum>y\<in>g`A. \<Sum>x\<in>f`(g -` {y} \<inter> A). h x)"
```
```   979     (is "?lhs = ?rhs")
```
```   980 proof -
```
```   981   have "f ` A =
```
```   982       snd ` (SIGMA x : g ` A. f ` (g -` {x} \<inter> A))"
```
```   983       (is "_ = snd ` ?SIGMA")
```
```   984     unfolding image_split_eq_Sigma[symmetric]
```
```   985     by (simp add: image_compose[symmetric] comp_def)
```
```   986   moreover
```
```   987   have snd_inj: "inj_on snd ?SIGMA"
```
```   988     unfolding image_split_eq_Sigma[symmetric]
```
```   989     by (auto intro!: inj_onI subvimageD[OF svi])
```
```   990   ultimately
```
```   991   have "(\<Sum>x\<in>f`A. h x) = (\<Sum>(x,y)\<in>?SIGMA. h y)"
```
```   992     by (auto simp: setsum_reindex intro: setsum_cong)
```
```   993   also have "... = ?rhs"
```
```   994     using subvimage_finite[OF svi fin] fin
```
```   995     apply (subst setsum_Sigma[symmetric])
```
```   996     by (auto intro!: finite_subset[of _ "f`A"])
```
```   997   finally show ?thesis .
```
```   998 qed
```
```   999
```
```  1000 lemma (in information_space) entropy_partition:
```
```  1001   assumes sf: "simple_function M X" "simple_function M P"
```
```  1002   assumes svi: "subvimage (space M) X P"
```
```  1003   shows "\<H>(X) = \<H>(P) + \<H>(X|P)"
```
```  1004 proof -
```
```  1005   let "?XP x p" = "real (joint_distribution X P {(x, p)})"
```
```  1006   let "?X x" = "real (distribution X {x})"
```
```  1007   let "?P p" = "real (distribution P {p})"
```
```  1008   note fX = sf(1)[THEN simple_function_imp_finite_random_variable]
```
```  1009   note fP = sf(2)[THEN simple_function_imp_finite_random_variable]
```
```  1010   note fX[THEN finite_distribution_finite, simp] and fP[THEN finite_distribution_finite, simp]
```
```  1011   note finite_distribution_order[OF fX fP, simp]
```
```  1012   have "(\<Sum>x\<in>X ` space M. real (distribution X {x}) * log b (real (distribution X {x}))) =
```
```  1013     (\<Sum>y\<in>P `space M. \<Sum>x\<in>X ` space M.
```
```  1014     real (joint_distribution X P {(x, y)}) * log b (real (joint_distribution X P {(x, y)})))"
```
```  1015   proof (subst setsum_image_split[OF svi],
```
```  1016       safe intro!: setsum_mono_zero_cong_left imageI)
```
```  1017     show "finite (X ` space M)" "finite (X ` space M)" "finite (P ` space M)"
```
```  1018       using sf unfolding simple_function_def by auto
```
```  1019   next
```
```  1020     fix p x assume in_space: "p \<in> space M" "x \<in> space M"
```
```  1021     assume "real (joint_distribution X P {(X x, P p)}) * log b (real (joint_distribution X P {(X x, P p)})) \<noteq> 0"
```
```  1022     hence "(\<lambda>x. (X x, P x)) -` {(X x, P p)} \<inter> space M \<noteq> {}" by (auto simp: distribution_def)
```
```  1023     with svi[unfolded subvimage_def, rule_format, OF `x \<in> space M`]
```
```  1024     show "x \<in> P -` {P p}" by auto
```
```  1025   next
```
```  1026     fix p x assume in_space: "p \<in> space M" "x \<in> space M"
```
```  1027     assume "P x = P p"
```
```  1028     from this[symmetric] svi[unfolded subvimage_def, rule_format, OF `x \<in> space M`]
```
```  1029     have "X -` {X x} \<inter> space M \<subseteq> P -` {P p} \<inter> space M"
```
```  1030       by auto
```
```  1031     hence "(\<lambda>x. (X x, P x)) -` {(X x, P p)} \<inter> space M = X -` {X x} \<inter> space M"
```
```  1032       by auto
```
```  1033     thus "real (distribution X {X x}) * log b (real (distribution X {X x})) =
```
```  1034           real (joint_distribution X P {(X x, P p)}) *
```
```  1035           log b (real (joint_distribution X P {(X x, P p)}))"
```
```  1036       by (auto simp: distribution_def)
```
```  1037   qed
```
```  1038   moreover have "\<And>x y. real (joint_distribution X P {(x, y)}) *
```
```  1039       log b (real (joint_distribution X P {(x, y)}) / real (distribution P {y})) =
```
```  1040       real (joint_distribution X P {(x, y)}) * log b (real (joint_distribution X P {(x, y)})) -
```
```  1041       real (joint_distribution X P {(x, y)}) * log b (real (distribution P {y}))"
```
```  1042     by (auto simp add: log_simps zero_less_mult_iff field_simps)
```
```  1043   ultimately show ?thesis
```
```  1044     unfolding sf[THEN entropy_eq] conditional_entropy_eq[OF sf]
```
```  1045     using setsum_real_joint_distribution_singleton[OF fX fP]
```
```  1046     by (simp add: setsum_cartesian_product' setsum_subtractf setsum_real_distribution
```
```  1047       setsum_left_distrib[symmetric] setsum_commute[where B="P`space M"])
```
```  1048 qed
```
```  1049
```
```  1050 corollary (in information_space) entropy_data_processing:
```
```  1051   assumes X: "simple_function M X" shows "\<H>(f \<circ> X) \<le> \<H>(X)"
```
```  1052 proof -
```
```  1053   note X
```
```  1054   moreover have fX: "simple_function M (f \<circ> X)" using X by auto
```
```  1055   moreover have "subvimage (space M) X (f \<circ> X)" by auto
```
```  1056   ultimately have "\<H>(X) = \<H>(f\<circ>X) + \<H>(X|f\<circ>X)" by (rule entropy_partition)
```
```  1057   then show "\<H>(f \<circ> X) \<le> \<H>(X)"
```
```  1058     by (auto intro: conditional_entropy_positive[OF X fX])
```
```  1059 qed
```
```  1060
```
```  1061 corollary (in information_space) entropy_of_inj:
```
```  1062   assumes X: "simple_function M X" and inj: "inj_on f (X`space M)"
```
```  1063   shows "\<H>(f \<circ> X) = \<H>(X)"
```
```  1064 proof (rule antisym)
```
```  1065   show "\<H>(f \<circ> X) \<le> \<H>(X)" using entropy_data_processing[OF X] .
```
```  1066 next
```
```  1067   have sf: "simple_function M (f \<circ> X)"
```
```  1068     using X by auto
```
```  1069   have "\<H>(X) = \<H>(the_inv_into (X`space M) f \<circ> (f \<circ> X))"
```
```  1070     by (auto intro!: mutual_information_cong simp: entropy_def the_inv_into_f_f[OF inj])
```
```  1071   also have "... \<le> \<H>(f \<circ> X)"
```
```  1072     using entropy_data_processing[OF sf] .
```
```  1073   finally show "\<H>(X) \<le> \<H>(f \<circ> X)" .
```
```  1074 qed
```
```  1075
```
```  1076 end
```