src/HOL/Probability/Information.thy
 author hoelzl Tue Mar 22 20:06:10 2011 +0100 (2011-03-22) changeset 42067 66c8281349ec parent 41981 cdf7693bbe08 child 42148 d596e7bb251f permissions -rw-r--r--
```     1 (*  Title:      HOL/Probability/Information.thy
```
```     2     Author:     Johannes Hölzl, TU München
```
```     3     Author:     Armin Heller, TU München
```
```     4 *)
```
```     5
```
```     6 header {*Information theory*}
```
```     7
```
```     8 theory Information
```
```     9 imports
```
```    10   Probability_Space
```
```    11   "~~/src/HOL/Library/Convex"
```
```    12 begin
```
```    13
```
```    14 lemma (in prob_space) not_zero_less_distribution[simp]:
```
```    15   "(\<not> 0 < distribution X A) \<longleftrightarrow> distribution X A = 0"
```
```    16   using distribution_positive[of X A] by arith
```
```    17
```
```    18 lemma log_le: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> x \<le> y \<Longrightarrow> log a x \<le> log a y"
```
```    19   by (subst log_le_cancel_iff) auto
```
```    20
```
```    21 lemma log_less: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> x < y \<Longrightarrow> log a x < log a y"
```
```    22   by (subst log_less_cancel_iff) auto
```
```    23
```
```    24 lemma setsum_cartesian_product':
```
```    25   "(\<Sum>x\<in>A \<times> B. f x) = (\<Sum>x\<in>A. setsum (\<lambda>y. f (x, y)) B)"
```
```    26   unfolding setsum_cartesian_product by simp
```
```    27
```
```    28 section "Convex theory"
```
```    29
```
```    30 lemma log_setsum:
```
```    31   assumes "finite s" "s \<noteq> {}"
```
```    32   assumes "b > 1"
```
```    33   assumes "(\<Sum> i \<in> s. a i) = 1"
```
```    34   assumes "\<And> i. i \<in> s \<Longrightarrow> a i \<ge> 0"
```
```    35   assumes "\<And> i. i \<in> s \<Longrightarrow> y i \<in> {0 <..}"
```
```    36   shows "log b (\<Sum> i \<in> s. a i * y i) \<ge> (\<Sum> i \<in> s. a i * log b (y i))"
```
```    37 proof -
```
```    38   have "convex_on {0 <..} (\<lambda> x. - log b x)"
```
```    39     by (rule minus_log_convex[OF `b > 1`])
```
```    40   hence "- log b (\<Sum> i \<in> s. a i * y i) \<le> (\<Sum> i \<in> s. a i * - log b (y i))"
```
```    41     using convex_on_setsum[of _ _ "\<lambda> x. - log b x"] assms pos_is_convex by fastsimp
```
```    42   thus ?thesis by (auto simp add:setsum_negf le_imp_neg_le)
```
```    43 qed
```
```    44
```
```    45 lemma log_setsum':
```
```    46   assumes "finite s" "s \<noteq> {}"
```
```    47   assumes "b > 1"
```
```    48   assumes "(\<Sum> i \<in> s. a i) = 1"
```
```    49   assumes pos: "\<And> i. i \<in> s \<Longrightarrow> 0 \<le> a i"
```
```    50           "\<And> i. \<lbrakk> i \<in> s ; 0 < a i \<rbrakk> \<Longrightarrow> 0 < y i"
```
```    51   shows "log b (\<Sum> i \<in> s. a i * y i) \<ge> (\<Sum> i \<in> s. a i * log b (y i))"
```
```    52 proof -
```
```    53   have "\<And>y. (\<Sum> i \<in> s - {i. a i = 0}. a i * y i) = (\<Sum> i \<in> s. a i * y i)"
```
```    54     using assms by (auto intro!: setsum_mono_zero_cong_left)
```
```    55   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))"
```
```    56   proof (rule log_setsum)
```
```    57     have "setsum a (s - {i. a i = 0}) = setsum a s"
```
```    58       using assms(1) by (rule setsum_mono_zero_cong_left) auto
```
```    59     thus sum_1: "setsum a (s - {i. a i = 0}) = 1"
```
```    60       "finite (s - {i. a i = 0})" using assms by simp_all
```
```    61
```
```    62     show "s - {i. a i = 0} \<noteq> {}"
```
```    63     proof
```
```    64       assume *: "s - {i. a i = 0} = {}"
```
```    65       hence "setsum a (s - {i. a i = 0}) = 0" by (simp add: * setsum_empty)
```
```    66       with sum_1 show False by simp
```
```    67     qed
```
```    68
```
```    69     fix i assume "i \<in> s - {i. a i = 0}"
```
```    70     hence "i \<in> s" "a i \<noteq> 0" by simp_all
```
```    71     thus "0 \<le> a i" "y i \<in> {0<..}" using pos[of i] by auto
```
```    72   qed fact+
```
```    73   ultimately show ?thesis by simp
```
```    74 qed
```
```    75
```
```    76 lemma log_setsum_divide:
```
```    77   assumes "finite S" and "S \<noteq> {}" and "1 < b"
```
```    78   assumes "(\<Sum>x\<in>S. g x) = 1"
```
```    79   assumes pos: "\<And>x. x \<in> S \<Longrightarrow> g x \<ge> 0" "\<And>x. x \<in> S \<Longrightarrow> f x \<ge> 0"
```
```    80   assumes g_pos: "\<And>x. \<lbrakk> x \<in> S ; 0 < g x \<rbrakk> \<Longrightarrow> 0 < f x"
```
```    81   shows "- (\<Sum>x\<in>S. g x * log b (g x / f x)) \<le> log b (\<Sum>x\<in>S. f x)"
```
```    82 proof -
```
```    83   have log_mono: "\<And>x y. 0 < x \<Longrightarrow> x \<le> y \<Longrightarrow> log b x \<le> log b y"
```
```    84     using `1 < b` by (subst log_le_cancel_iff) auto
```
```    85
```
```    86   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))"
```
```    87   proof (unfold setsum_negf[symmetric], rule setsum_cong)
```
```    88     fix x assume x: "x \<in> S"
```
```    89     show "- (g x * log b (g x / f x)) = g x * log b (f x / g x)"
```
```    90     proof (cases "g x = 0")
```
```    91       case False
```
```    92       with pos[OF x] g_pos[OF x] have "0 < f x" "0 < g x" by simp_all
```
```    93       thus ?thesis using `1 < b` by (simp add: log_divide field_simps)
```
```    94     qed simp
```
```    95   qed rule
```
```    96   also have "... \<le> log b (\<Sum>x\<in>S. g x * (f x / g x))"
```
```    97   proof (rule log_setsum')
```
```    98     fix x assume x: "x \<in> S" "0 < g x"
```
```    99     with g_pos[OF x] show "0 < f x / g x" by (safe intro!: divide_pos_pos)
```
```   100   qed fact+
```
```   101   also have "... = log b (\<Sum>x\<in>S - {x. g x = 0}. f x)" using `finite S`
```
```   102     by (auto intro!: setsum_mono_zero_cong_right arg_cong[where f="log b"]
```
```   103         split: split_if_asm)
```
```   104   also have "... \<le> log b (\<Sum>x\<in>S. f x)"
```
```   105   proof (rule log_mono)
```
```   106     have "0 = (\<Sum>x\<in>S - {x. g x = 0}. 0)" by simp
```
```   107     also have "... < (\<Sum>x\<in>S - {x. g x = 0}. f x)" (is "_ < ?sum")
```
```   108     proof (rule setsum_strict_mono)
```
```   109       show "finite (S - {x. g x = 0})" using `finite S` by simp
```
```   110       show "S - {x. g x = 0} \<noteq> {}"
```
```   111       proof
```
```   112         assume "S - {x. g x = 0} = {}"
```
```   113         hence "(\<Sum>x\<in>S. g x) = 0" by (subst setsum_0') auto
```
```   114         with `(\<Sum>x\<in>S. g x) = 1` show False by simp
```
```   115       qed
```
```   116       fix x assume "x \<in> S - {x. g x = 0}"
```
```   117       thus "0 < f x" using g_pos[of x] pos(1)[of x] by auto
```
```   118     qed
```
```   119     finally show "0 < ?sum" .
```
```   120     show "(\<Sum>x\<in>S - {x. g x = 0}. f x) \<le> (\<Sum>x\<in>S. f x)"
```
```   121       using `finite S` pos by (auto intro!: setsum_mono2)
```
```   122   qed
```
```   123   finally show ?thesis .
```
```   124 qed
```
```   125
```
```   126 lemma split_pairs:
```
```   127   "((A, B) = X) \<longleftrightarrow> (fst X = A \<and> snd X = B)" and
```
```   128   "(X = (A, B)) \<longleftrightarrow> (fst X = A \<and> snd X = B)" by auto
```
```   129
```
```   130 section "Information theory"
```
```   131
```
```   132 locale information_space = prob_space +
```
```   133   fixes b :: real assumes b_gt_1: "1 < b"
```
```   134
```
```   135 context information_space
```
```   136 begin
```
```   137
```
```   138 text {* Introduce some simplification rules for logarithm of base @{term b}. *}
```
```   139
```
```   140 lemma log_neg_const:
```
```   141   assumes "x \<le> 0"
```
```   142   shows "log b x = log b 0"
```
```   143 proof -
```
```   144   { fix u :: real
```
```   145     have "x \<le> 0" by fact
```
```   146     also have "0 < exp u"
```
```   147       using exp_gt_zero .
```
```   148     finally have "exp u \<noteq> x"
```
```   149       by auto }
```
```   150   then show "log b x = log b 0"
```
```   151     by (simp add: log_def ln_def)
```
```   152 qed
```
```   153
```
```   154 lemma log_mult_eq:
```
```   155   "log b (A * B) = (if 0 < A * B then log b \<bar>A\<bar> + log b \<bar>B\<bar> else log b 0)"
```
```   156   using log_mult[of b "\<bar>A\<bar>" "\<bar>B\<bar>"] b_gt_1 log_neg_const[of "A * B"]
```
```   157   by (auto simp: zero_less_mult_iff mult_le_0_iff)
```
```   158
```
```   159 lemma log_inverse_eq:
```
```   160   "log b (inverse B) = (if 0 < B then - log b B else log b 0)"
```
```   161   using log_inverse[of b B] log_neg_const[of "inverse B"] b_gt_1 by simp
```
```   162
```
```   163 lemma log_divide_eq:
```
```   164   "log b (A / B) = (if 0 < A * B then log b \<bar>A\<bar> - log b \<bar>B\<bar> else log b 0)"
```
```   165   unfolding divide_inverse log_mult_eq log_inverse_eq abs_inverse
```
```   166   by (auto simp: zero_less_mult_iff mult_le_0_iff)
```
```   167
```
```   168 lemmas log_simps = log_mult_eq log_inverse_eq log_divide_eq
```
```   169
```
```   170 end
```
```   171
```
```   172 subsection "Kullback\$-\$Leibler divergence"
```
```   173
```
```   174 text {* The Kullback\$-\$Leibler divergence is also known as relative entropy or
```
```   175 Kullback\$-\$Leibler distance. *}
```
```   176
```
```   177 definition
```
```   178   "KL_divergence b M \<nu> = \<integral>x. log b (real (RN_deriv M \<nu> x)) \<partial>M\<lparr>measure := \<nu>\<rparr>"
```
```   179
```
```   180 lemma (in sigma_finite_measure) KL_divergence_vimage:
```
```   181   assumes T: "T \<in> measure_preserving M M'"
```
```   182     and T': "T' \<in> measure_preserving (M'\<lparr> measure := \<nu>' \<rparr>) (M\<lparr> measure := \<nu> \<rparr>)"
```
```   183     and inv: "\<And>x. x \<in> space M \<Longrightarrow> T' (T x) = x"
```
```   184     and inv': "\<And>x. x \<in> space M' \<Longrightarrow> T (T' x) = x"
```
```   185   and \<nu>': "measure_space (M'\<lparr>measure := \<nu>'\<rparr>)" "measure_space.absolutely_continuous M' \<nu>'"
```
```   186   and "1 < b"
```
```   187   shows "KL_divergence b M' \<nu>' = KL_divergence b M \<nu>"
```
```   188 proof -
```
```   189   interpret \<nu>': measure_space "M'\<lparr>measure := \<nu>'\<rparr>" by fact
```
```   190   have M: "measure_space (M\<lparr> measure := \<nu>\<rparr>)"
```
```   191     by (rule \<nu>'.measure_space_vimage[OF _ T'], simp) default
```
```   192   have "sigma_algebra (M'\<lparr> measure := \<nu>'\<rparr>)" by default
```
```   193   then have saM': "sigma_algebra M'" by simp
```
```   194   then interpret M': measure_space M' by (rule measure_space_vimage) fact
```
```   195   have ac: "absolutely_continuous \<nu>" unfolding absolutely_continuous_def
```
```   196   proof safe
```
```   197     fix N assume N: "N \<in> sets M" and N_0: "\<mu> N = 0"
```
```   198     then have N': "T' -` N \<inter> space M' \<in> sets M'"
```
```   199       using T' by (auto simp: measurable_def measure_preserving_def)
```
```   200     have "T -` (T' -` N \<inter> space M') \<inter> space M = N"
```
```   201       using inv T N sets_into_space[OF N] by (auto simp: measurable_def measure_preserving_def)
```
```   202     then have "measure M' (T' -` N \<inter> space M') = 0"
```
```   203       using measure_preservingD[OF T N'] N_0 by auto
```
```   204     with \<nu>'(2) N' show "\<nu> N = 0" using measure_preservingD[OF T', of N] N
```
```   205       unfolding M'.absolutely_continuous_def measurable_def by auto
```
```   206   qed
```
```   207
```
```   208   have sa: "sigma_algebra (M\<lparr>measure := \<nu>\<rparr>)" by simp default
```
```   209   have AE: "AE x. RN_deriv M' \<nu>' (T x) = RN_deriv M \<nu> x"
```
```   210     by (rule RN_deriv_vimage[OF T T' inv \<nu>'])
```
```   211   show ?thesis
```
```   212     unfolding KL_divergence_def
```
```   213   proof (subst \<nu>'.integral_vimage[OF sa T'])
```
```   214     show "(\<lambda>x. log b (real (RN_deriv M \<nu> x))) \<in> borel_measurable (M\<lparr>measure := \<nu>\<rparr>)"
```
```   215       by (auto intro!: RN_deriv[OF M ac] borel_measurable_log[OF _ `1 < b`])
```
```   216     have "(\<integral> x. log b (real (RN_deriv M' \<nu>' x)) \<partial>M'\<lparr>measure := \<nu>'\<rparr>) =
```
```   217       (\<integral> x. log b (real (RN_deriv M' \<nu>' (T (T' x)))) \<partial>M'\<lparr>measure := \<nu>'\<rparr>)" (is "?l = _")
```
```   218       using inv' by (auto intro!: \<nu>'.integral_cong)
```
```   219     also have "\<dots> = (\<integral> x. log b (real (RN_deriv M \<nu> (T' x))) \<partial>M'\<lparr>measure := \<nu>'\<rparr>)" (is "_ = ?r")
```
```   220       using M ac AE
```
```   221       by (intro \<nu>'.integral_cong_AE \<nu>'.almost_everywhere_vimage[OF sa T'] absolutely_continuous_AE[OF M])
```
```   222          (auto elim!: AE_mp)
```
```   223     finally show "?l = ?r" .
```
```   224   qed
```
```   225 qed
```
```   226
```
```   227 lemma (in sigma_finite_measure) KL_divergence_cong:
```
```   228   assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)" (is "measure_space ?\<nu>")
```
```   229   assumes [simp]: "sets N = sets M" "space N = space M"
```
```   230     "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A"
```
```   231     "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = \<nu>' A"
```
```   232   shows "KL_divergence b M \<nu> = KL_divergence b N \<nu>'"
```
```   233 proof -
```
```   234   interpret \<nu>: measure_space ?\<nu> by fact
```
```   235   have "KL_divergence b M \<nu> = \<integral>x. log b (real (RN_deriv N \<nu>' x)) \<partial>?\<nu>"
```
```   236     by (simp cong: RN_deriv_cong \<nu>.integral_cong add: KL_divergence_def)
```
```   237   also have "\<dots> = KL_divergence b N \<nu>'"
```
```   238     by (auto intro!: \<nu>.integral_cong_measure[symmetric] simp: KL_divergence_def)
```
```   239   finally show ?thesis .
```
```   240 qed
```
```   241
```
```   242 lemma (in finite_measure_space) KL_divergence_eq_finite:
```
```   243   assumes v: "finite_measure_space (M\<lparr>measure := \<nu>\<rparr>)"
```
```   244   assumes ac: "absolutely_continuous \<nu>"
```
```   245   shows "KL_divergence b M \<nu> = (\<Sum>x\<in>space M. real (\<nu> {x}) * log b (real (\<nu> {x}) / real (\<mu> {x})))" (is "_ = ?sum")
```
```   246 proof (simp add: KL_divergence_def finite_measure_space.integral_finite_singleton[OF v])
```
```   247   interpret v: finite_measure_space "M\<lparr>measure := \<nu>\<rparr>" by fact
```
```   248   have ms: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" by default
```
```   249   show "(\<Sum>x \<in> space M. log b (real (RN_deriv M \<nu> x)) * real (\<nu> {x})) = ?sum"
```
```   250     using RN_deriv_finite_measure[OF ms ac]
```
```   251     by (auto intro!: setsum_cong simp: field_simps)
```
```   252 qed
```
```   253
```
```   254 lemma (in finite_prob_space) KL_divergence_positive_finite:
```
```   255   assumes v: "finite_prob_space (M\<lparr>measure := \<nu>\<rparr>)"
```
```   256   assumes ac: "absolutely_continuous \<nu>"
```
```   257   and "1 < b"
```
```   258   shows "0 \<le> KL_divergence b M \<nu>"
```
```   259 proof -
```
```   260   interpret v: finite_prob_space "M\<lparr>measure := \<nu>\<rparr>" by fact
```
```   261   have ms: "finite_measure_space (M\<lparr>measure := \<nu>\<rparr>)" by default
```
```   262
```
```   263   have "- (KL_divergence b M \<nu>) \<le> log b (\<Sum>x\<in>space M. real (\<mu> {x}))"
```
```   264   proof (subst KL_divergence_eq_finite[OF ms ac], safe intro!: log_setsum_divide not_empty)
```
```   265     show "finite (space M)" using finite_space by simp
```
```   266     show "1 < b" by fact
```
```   267     show "(\<Sum>x\<in>space M. real (\<nu> {x})) = 1"
```
```   268       using v.finite_sum_over_space_eq_1 by (simp add: v.\<mu>'_def)
```
```   269
```
```   270     fix x assume "x \<in> space M"
```
```   271     then have x: "{x} \<in> sets M" unfolding sets_eq_Pow by auto
```
```   272     { assume "0 < real (\<nu> {x})"
```
```   273       then have "\<nu> {x} \<noteq> 0" by auto
```
```   274       then have "\<mu> {x} \<noteq> 0"
```
```   275         using ac[unfolded absolutely_continuous_def, THEN bspec, of "{x}"] x by auto
```
```   276       thus "0 < real (\<mu> {x})" using real_measure[OF x] by auto }
```
```   277     show "0 \<le> real (\<mu> {x})" "0 \<le> real (\<nu> {x})"
```
```   278       using real_measure[OF x] v.real_measure[of "{x}"] x by auto
```
```   279   qed
```
```   280   thus "0 \<le> KL_divergence b M \<nu>" using finite_sum_over_space_eq_1 by (simp add: \<mu>'_def)
```
```   281 qed
```
```   282
```
```   283 subsection {* Mutual Information *}
```
```   284
```
```   285 definition (in prob_space)
```
```   286   "mutual_information b S T X Y =
```
```   287     KL_divergence b (S\<lparr>measure := extreal\<circ>distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := extreal\<circ>distribution Y\<rparr>)
```
```   288       (extreal\<circ>joint_distribution X Y)"
```
```   289
```
```   290 definition (in prob_space)
```
```   291   "entropy b s X = mutual_information b s s X X"
```
```   292
```
```   293 abbreviation (in information_space)
```
```   294   mutual_information_Pow ("\<I>'(_ ; _')") where
```
```   295   "\<I>(X ; Y) \<equiv> mutual_information b
```
```   296     \<lparr> space = X`space M, sets = Pow (X`space M), measure = extreal\<circ>distribution X \<rparr>
```
```   297     \<lparr> space = Y`space M, sets = Pow (Y`space M), measure = extreal\<circ>distribution Y \<rparr> X Y"
```
```   298
```
```   299 lemma (in prob_space) finite_variables_absolutely_continuous:
```
```   300   assumes X: "finite_random_variable S X" and Y: "finite_random_variable T Y"
```
```   301   shows "measure_space.absolutely_continuous
```
```   302     (S\<lparr>measure := extreal\<circ>distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := extreal\<circ>distribution Y\<rparr>)
```
```   303     (extreal\<circ>joint_distribution X Y)"
```
```   304 proof -
```
```   305   interpret X: finite_prob_space "S\<lparr>measure := extreal\<circ>distribution X\<rparr>"
```
```   306     using X by (rule distribution_finite_prob_space)
```
```   307   interpret Y: finite_prob_space "T\<lparr>measure := extreal\<circ>distribution Y\<rparr>"
```
```   308     using Y by (rule distribution_finite_prob_space)
```
```   309   interpret XY: pair_finite_prob_space
```
```   310     "S\<lparr>measure := extreal\<circ>distribution X\<rparr>" "T\<lparr> measure := extreal\<circ>distribution Y\<rparr>" by default
```
```   311   interpret P: finite_prob_space "XY.P\<lparr> measure := extreal\<circ>joint_distribution X Y\<rparr>"
```
```   312     using assms by (auto intro!: joint_distribution_finite_prob_space)
```
```   313   note rv = assms[THEN finite_random_variableD]
```
```   314   show "XY.absolutely_continuous (extreal\<circ>joint_distribution X Y)"
```
```   315   proof (rule XY.absolutely_continuousI)
```
```   316     show "finite_measure_space (XY.P\<lparr> measure := extreal\<circ>joint_distribution X Y\<rparr>)" by default
```
```   317     fix x assume "x \<in> space XY.P" and "XY.\<mu> {x} = 0"
```
```   318     then obtain a b where "x = (a, b)"
```
```   319       and "distribution X {a} = 0 \<or> distribution Y {b} = 0"
```
```   320       by (cases x) (auto simp: space_pair_measure)
```
```   321     with finite_distribution_order(5,6)[OF X Y]
```
```   322     show "(extreal \<circ> joint_distribution X Y) {x} = 0" by auto
```
```   323   qed
```
```   324 qed
```
```   325
```
```   326 lemma (in information_space)
```
```   327   assumes MX: "finite_random_variable MX X"
```
```   328   assumes MY: "finite_random_variable MY Y"
```
```   329   shows mutual_information_generic_eq:
```
```   330     "mutual_information b MX MY X Y = (\<Sum> (x,y) \<in> space MX \<times> space MY.
```
```   331       joint_distribution X Y {(x,y)} *
```
```   332       log b (joint_distribution X Y {(x,y)} /
```
```   333       (distribution X {x} * distribution Y {y})))"
```
```   334     (is ?sum)
```
```   335   and mutual_information_positive_generic:
```
```   336      "0 \<le> mutual_information b MX MY X Y" (is ?positive)
```
```   337 proof -
```
```   338   interpret X: finite_prob_space "MX\<lparr>measure := extreal\<circ>distribution X\<rparr>"
```
```   339     using MX by (rule distribution_finite_prob_space)
```
```   340   interpret Y: finite_prob_space "MY\<lparr>measure := extreal\<circ>distribution Y\<rparr>"
```
```   341     using MY by (rule distribution_finite_prob_space)
```
```   342   interpret XY: pair_finite_prob_space "MX\<lparr>measure := extreal\<circ>distribution X\<rparr>" "MY\<lparr>measure := extreal\<circ>distribution Y\<rparr>" by default
```
```   343   interpret P: finite_prob_space "XY.P\<lparr>measure := extreal\<circ>joint_distribution X Y\<rparr>"
```
```   344     using assms by (auto intro!: joint_distribution_finite_prob_space)
```
```   345
```
```   346   have P_ms: "finite_measure_space (XY.P\<lparr>measure := extreal\<circ>joint_distribution X Y\<rparr>)" by default
```
```   347   have P_ps: "finite_prob_space (XY.P\<lparr>measure := extreal\<circ>joint_distribution X Y\<rparr>)" by default
```
```   348
```
```   349   show ?sum
```
```   350     unfolding Let_def mutual_information_def
```
```   351     by (subst XY.KL_divergence_eq_finite[OF P_ms finite_variables_absolutely_continuous[OF MX MY]])
```
```   352        (auto simp add: space_pair_measure setsum_cartesian_product')
```
```   353
```
```   354   show ?positive
```
```   355     using XY.KL_divergence_positive_finite[OF P_ps finite_variables_absolutely_continuous[OF MX MY] b_gt_1]
```
```   356     unfolding mutual_information_def .
```
```   357 qed
```
```   358
```
```   359 lemma (in information_space) mutual_information_commute_generic:
```
```   360   assumes X: "random_variable S X" and Y: "random_variable T Y"
```
```   361   assumes ac: "measure_space.absolutely_continuous
```
```   362     (S\<lparr>measure := extreal\<circ>distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := extreal\<circ>distribution Y\<rparr>) (extreal\<circ>joint_distribution X Y)"
```
```   363   shows "mutual_information b S T X Y = mutual_information b T S Y X"
```
```   364 proof -
```
```   365   let ?S = "S\<lparr>measure := extreal\<circ>distribution X\<rparr>" and ?T = "T\<lparr>measure := extreal\<circ>distribution Y\<rparr>"
```
```   366   interpret S: prob_space ?S using X by (rule distribution_prob_space)
```
```   367   interpret T: prob_space ?T using Y by (rule distribution_prob_space)
```
```   368   interpret P: pair_prob_space ?S ?T ..
```
```   369   interpret Q: pair_prob_space ?T ?S ..
```
```   370   show ?thesis
```
```   371     unfolding mutual_information_def
```
```   372   proof (intro Q.KL_divergence_vimage[OF Q.measure_preserving_swap _ _ _ _ ac b_gt_1])
```
```   373     show "(\<lambda>(x,y). (y,x)) \<in> measure_preserving
```
```   374       (P.P \<lparr> measure := extreal\<circ>joint_distribution X Y\<rparr>) (Q.P \<lparr> measure := extreal\<circ>joint_distribution Y X\<rparr>)"
```
```   375       using X Y unfolding measurable_def
```
```   376       unfolding measure_preserving_def using P.pair_sigma_algebra_swap_measurable
```
```   377       by (auto simp add: space_pair_measure distribution_def intro!: arg_cong[where f=\<mu>'])
```
```   378     have "prob_space (P.P\<lparr> measure := extreal\<circ>joint_distribution X Y\<rparr>)"
```
```   379       using X Y by (auto intro!: distribution_prob_space random_variable_pairI)
```
```   380     then show "measure_space (P.P\<lparr> measure := extreal\<circ>joint_distribution X Y\<rparr>)"
```
```   381       unfolding prob_space_def by simp
```
```   382   qed auto
```
```   383 qed
```
```   384
```
```   385 lemma (in information_space) mutual_information_commute:
```
```   386   assumes X: "finite_random_variable S X" and Y: "finite_random_variable T Y"
```
```   387   shows "mutual_information b S T X Y = mutual_information b T S Y X"
```
```   388   unfolding mutual_information_generic_eq[OF X Y] mutual_information_generic_eq[OF Y X]
```
```   389   unfolding joint_distribution_commute_singleton[of X Y]
```
```   390   by (auto simp add: ac_simps intro!: setsum_reindex_cong[OF swap_inj_on])
```
```   391
```
```   392 lemma (in information_space) mutual_information_commute_simple:
```
```   393   assumes X: "simple_function M X" and Y: "simple_function M Y"
```
```   394   shows "\<I>(X;Y) = \<I>(Y;X)"
```
```   395   by (intro mutual_information_commute X Y simple_function_imp_finite_random_variable)
```
```   396
```
```   397 lemma (in information_space) mutual_information_eq:
```
```   398   assumes "simple_function M X" "simple_function M Y"
```
```   399   shows "\<I>(X;Y) = (\<Sum> (x,y) \<in> X ` space M \<times> Y ` space M.
```
```   400     distribution (\<lambda>x. (X x, Y x)) {(x,y)} * log b (distribution (\<lambda>x. (X x, Y x)) {(x,y)} /
```
```   401                                                    (distribution X {x} * distribution Y {y})))"
```
```   402   using assms by (simp add: mutual_information_generic_eq)
```
```   403
```
```   404 lemma (in information_space) mutual_information_generic_cong:
```
```   405   assumes X: "\<And>x. x \<in> space M \<Longrightarrow> X x = X' x"
```
```   406   assumes Y: "\<And>x. x \<in> space M \<Longrightarrow> Y x = Y' x"
```
```   407   shows "mutual_information b MX MY X Y = mutual_information b MX MY X' Y'"
```
```   408   unfolding mutual_information_def using X Y
```
```   409   by (simp cong: distribution_cong)
```
```   410
```
```   411 lemma (in information_space) mutual_information_cong:
```
```   412   assumes X: "\<And>x. x \<in> space M \<Longrightarrow> X x = X' x"
```
```   413   assumes Y: "\<And>x. x \<in> space M \<Longrightarrow> Y x = Y' x"
```
```   414   shows "\<I>(X; Y) = \<I>(X'; Y')"
```
```   415   unfolding mutual_information_def using X Y
```
```   416   by (simp cong: distribution_cong image_cong)
```
```   417
```
```   418 lemma (in information_space) mutual_information_positive:
```
```   419   assumes "simple_function M X" "simple_function M Y"
```
```   420   shows "0 \<le> \<I>(X;Y)"
```
```   421   using assms by (simp add: mutual_information_positive_generic)
```
```   422
```
```   423 subsection {* Entropy *}
```
```   424
```
```   425 abbreviation (in information_space)
```
```   426   entropy_Pow ("\<H>'(_')") where
```
```   427   "\<H>(X) \<equiv> entropy b \<lparr> space = X`space M, sets = Pow (X`space M), measure = extreal\<circ>distribution X \<rparr> X"
```
```   428
```
```   429 lemma (in information_space) entropy_generic_eq:
```
```   430   fixes X :: "'a \<Rightarrow> 'c"
```
```   431   assumes MX: "finite_random_variable MX X"
```
```   432   shows "entropy b MX X = -(\<Sum> x \<in> space MX. distribution X {x} * log b (distribution X {x}))"
```
```   433 proof -
```
```   434   interpret MX: finite_prob_space "MX\<lparr>measure := extreal\<circ>distribution X\<rparr>"
```
```   435     using MX by (rule distribution_finite_prob_space)
```
```   436   let "?X x" = "distribution X {x}"
```
```   437   let "?XX x y" = "joint_distribution X X {(x, y)}"
```
```   438
```
```   439   { fix x y :: 'c
```
```   440     { assume "x \<noteq> y"
```
```   441       then have "(\<lambda>x. (X x, X x)) -` {(x,y)} \<inter> space M = {}" by auto
```
```   442       then have "joint_distribution X X {(x, y)} = 0" by (simp add: distribution_def) }
```
```   443     then have "?XX x y * log b (?XX x y / (?X x * ?X y)) =
```
```   444         (if x = y then - ?X y * log b (?X y) else 0)"
```
```   445       by (auto simp: log_simps zero_less_mult_iff) }
```
```   446   note remove_XX = this
```
```   447
```
```   448   show ?thesis
```
```   449     unfolding entropy_def mutual_information_generic_eq[OF MX MX]
```
```   450     unfolding setsum_cartesian_product[symmetric] setsum_negf[symmetric] remove_XX
```
```   451     using MX.finite_space by (auto simp: setsum_cases)
```
```   452 qed
```
```   453
```
```   454 lemma (in information_space) entropy_eq:
```
```   455   assumes "simple_function M X"
```
```   456   shows "\<H>(X) = -(\<Sum> x \<in> X ` space M. distribution X {x} * log b (distribution X {x}))"
```
```   457   using assms by (simp add: entropy_generic_eq)
```
```   458
```
```   459 lemma (in information_space) entropy_positive:
```
```   460   "simple_function M X \<Longrightarrow> 0 \<le> \<H>(X)"
```
```   461   unfolding entropy_def by (simp add: mutual_information_positive)
```
```   462
```
```   463 lemma (in information_space) entropy_certainty_eq_0:
```
```   464   assumes X: "simple_function M X" and "x \<in> X ` space M" and "distribution X {x} = 1"
```
```   465   shows "\<H>(X) = 0"
```
```   466 proof -
```
```   467   let ?X = "\<lparr> space = X ` space M, sets = Pow (X ` space M), measure = extreal\<circ>distribution X\<rparr>"
```
```   468   note simple_function_imp_finite_random_variable[OF `simple_function M X`]
```
```   469   from distribution_finite_prob_space[OF this, of "\<lparr> measure = extreal\<circ>distribution X \<rparr>"]
```
```   470   interpret X: finite_prob_space ?X by simp
```
```   471   have "distribution X (X ` space M - {x}) = distribution X (X ` space M) - distribution X {x}"
```
```   472     using X.measure_compl[of "{x}"] assms by auto
```
```   473   also have "\<dots> = 0" using X.prob_space assms by auto
```
```   474   finally have X0: "distribution X (X ` space M - {x}) = 0" by auto
```
```   475   { fix y assume *: "y \<in> X ` space M"
```
```   476     { assume asm: "y \<noteq> x"
```
```   477       with * have "{y} \<subseteq> X ` space M - {x}" by auto
```
```   478       from X.measure_mono[OF this] X0 asm *
```
```   479       have "distribution X {y} = 0"  by (auto intro: antisym) }
```
```   480     then have "distribution X {y} = (if x = y then 1 else 0)"
```
```   481       using assms by auto }
```
```   482   note fi = this
```
```   483   have y: "\<And>y. (if x = y then 1 else 0) * log b (if x = y then 1 else 0) = 0" by simp
```
```   484   show ?thesis unfolding entropy_eq[OF `simple_function M X`] by (auto simp: y fi)
```
```   485 qed
```
```   486
```
```   487 lemma (in information_space) entropy_le_card_not_0:
```
```   488   assumes X: "simple_function M X"
```
```   489   shows "\<H>(X) \<le> log b (card (X ` space M \<inter> {x. distribution X {x} \<noteq> 0}))"
```
```   490 proof -
```
```   491   let "?p x" = "distribution X {x}"
```
```   492   have "\<H>(X) = (\<Sum>x\<in>X`space M. ?p x * log b (1 / ?p x))"
```
```   493     unfolding entropy_eq[OF X] setsum_negf[symmetric]
```
```   494     by (auto intro!: setsum_cong simp: log_simps)
```
```   495   also have "\<dots> \<le> log b (\<Sum>x\<in>X`space M. ?p x * (1 / ?p x))"
```
```   496     using not_empty b_gt_1 `simple_function M X` sum_over_space_real_distribution[OF X]
```
```   497     by (intro log_setsum') (auto simp: simple_function_def)
```
```   498   also have "\<dots> = log b (\<Sum>x\<in>X`space M. if ?p x \<noteq> 0 then 1 else 0)"
```
```   499     by (intro arg_cong[where f="\<lambda>X. log b X"] setsum_cong) auto
```
```   500   finally show ?thesis
```
```   501     using `simple_function M X` by (auto simp: setsum_cases real_eq_of_nat simple_function_def)
```
```   502 qed
```
```   503
```
```   504 lemma (in prob_space) measure'_translate:
```
```   505   assumes X: "random_variable S X" and A: "A \<in> sets S"
```
```   506   shows "finite_measure.\<mu>' (S\<lparr> measure := extreal\<circ>distribution X \<rparr>) A = distribution X A"
```
```   507 proof -
```
```   508   interpret S: prob_space "S\<lparr> measure := extreal\<circ>distribution X \<rparr>"
```
```   509     using distribution_prob_space[OF X] .
```
```   510   from A show "S.\<mu>' A = distribution X A"
```
```   511     unfolding S.\<mu>'_def by (simp add: distribution_def_raw \<mu>'_def)
```
```   512 qed
```
```   513
```
```   514 lemma (in information_space) entropy_uniform_max:
```
```   515   assumes X: "simple_function M X"
```
```   516   assumes "\<And>x y. \<lbrakk> x \<in> X ` space M ; y \<in> X ` space M \<rbrakk> \<Longrightarrow> distribution X {x} = distribution X {y}"
```
```   517   shows "\<H>(X) = log b (real (card (X ` space M)))"
```
```   518 proof -
```
```   519   let ?X = "\<lparr> space = X ` space M, sets = Pow (X ` space M), measure = undefined\<rparr>\<lparr> measure := extreal\<circ>distribution X\<rparr>"
```
```   520   note frv = simple_function_imp_finite_random_variable[OF X]
```
```   521   from distribution_finite_prob_space[OF this, of "\<lparr> measure = extreal\<circ>distribution X \<rparr>"]
```
```   522   interpret X: finite_prob_space ?X by simp
```
```   523   note rv = finite_random_variableD[OF frv]
```
```   524   have card_gt0: "0 < card (X ` space M)" unfolding card_gt_0_iff
```
```   525     using `simple_function M X` not_empty by (auto simp: simple_function_def)
```
```   526   { fix x assume "x \<in> space ?X"
```
```   527     moreover then have "X.\<mu>' {x} = 1 / card (space ?X)"
```
```   528     proof (rule X.uniform_prob)
```
```   529       fix x y assume "x \<in> space ?X" "y \<in> space ?X"
```
```   530       with assms(2)[of x y] show "X.\<mu>' {x} = X.\<mu>' {y}"
```
```   531         by (subst (1 2) measure'_translate[OF rv]) auto
```
```   532     qed
```
```   533     ultimately have "distribution X {x} = 1 / card (space ?X)"
```
```   534       by (subst (asm) measure'_translate[OF rv]) auto }
```
```   535   thus ?thesis
```
```   536     using not_empty X.finite_space b_gt_1 card_gt0
```
```   537     by (simp add: entropy_eq[OF `simple_function M X`] real_eq_of_nat[symmetric] log_simps)
```
```   538 qed
```
```   539
```
```   540 lemma (in information_space) entropy_le_card:
```
```   541   assumes "simple_function M X"
```
```   542   shows "\<H>(X) \<le> log b (real (card (X ` space M)))"
```
```   543 proof cases
```
```   544   assume "X ` space M \<inter> {x. distribution X {x} \<noteq> 0} = {}"
```
```   545   then have "\<And>x. x\<in>X`space M \<Longrightarrow> distribution X {x} = 0" by auto
```
```   546   moreover
```
```   547   have "0 < card (X`space M)"
```
```   548     using `simple_function M X` not_empty
```
```   549     by (auto simp: card_gt_0_iff simple_function_def)
```
```   550   then have "log b 1 \<le> log b (real (card (X`space M)))"
```
```   551     using b_gt_1 by (intro log_le) auto
```
```   552   ultimately show ?thesis using assms by (simp add: entropy_eq)
```
```   553 next
```
```   554   assume False: "X ` space M \<inter> {x. distribution X {x} \<noteq> 0} \<noteq> {}"
```
```   555   have "card (X ` space M \<inter> {x. distribution X {x} \<noteq> 0}) \<le> card (X ` space M)"
```
```   556     (is "?A \<le> ?B") using assms not_empty by (auto intro!: card_mono simp: simple_function_def)
```
```   557   note entropy_le_card_not_0[OF assms]
```
```   558   also have "log b (real ?A) \<le> log b (real ?B)"
```
```   559     using b_gt_1 False not_empty `?A \<le> ?B` assms
```
```   560     by (auto intro!: log_le simp: card_gt_0_iff simp: simple_function_def)
```
```   561   finally show ?thesis .
```
```   562 qed
```
```   563
```
```   564 lemma (in information_space) entropy_commute:
```
```   565   assumes "simple_function M X" "simple_function M Y"
```
```   566   shows "\<H>(\<lambda>x. (X x, Y x)) = \<H>(\<lambda>x. (Y x, X x))"
```
```   567 proof -
```
```   568   have sf: "simple_function M (\<lambda>x. (X x, Y x))" "simple_function M (\<lambda>x. (Y x, X x))"
```
```   569     using assms by (auto intro: simple_function_Pair)
```
```   570   have *: "(\<lambda>x. (Y x, X x))`space M = (\<lambda>(a,b). (b,a))`(\<lambda>x. (X x, Y x))`space M"
```
```   571     by auto
```
```   572   have inj: "\<And>X. inj_on (\<lambda>(a,b). (b,a)) X"
```
```   573     by (auto intro!: inj_onI)
```
```   574   show ?thesis
```
```   575     unfolding sf[THEN entropy_eq] unfolding * setsum_reindex[OF inj]
```
```   576     by (simp add: joint_distribution_commute[of Y X] split_beta)
```
```   577 qed
```
```   578
```
```   579 lemma (in information_space) entropy_eq_cartesian_product:
```
```   580   assumes "simple_function M X" "simple_function M Y"
```
```   581   shows "\<H>(\<lambda>x. (X x, Y x)) = -(\<Sum>x\<in>X`space M. \<Sum>y\<in>Y`space M.
```
```   582     joint_distribution X Y {(x,y)} * log b (joint_distribution X Y {(x,y)}))"
```
```   583 proof -
```
```   584   have sf: "simple_function M (\<lambda>x. (X x, Y x))"
```
```   585     using assms by (auto intro: simple_function_Pair)
```
```   586   { fix x assume "x\<notin>(\<lambda>x. (X x, Y x))`space M"
```
```   587     then have "(\<lambda>x. (X x, Y x)) -` {x} \<inter> space M = {}" by auto
```
```   588     then have "joint_distribution X Y {x} = 0"
```
```   589       unfolding distribution_def by auto }
```
```   590   then show ?thesis using sf assms
```
```   591     unfolding entropy_eq[OF sf] neg_equal_iff_equal setsum_cartesian_product
```
```   592     by (auto intro!: setsum_mono_zero_cong_left simp: simple_function_def)
```
```   593 qed
```
```   594
```
```   595 subsection {* Conditional Mutual Information *}
```
```   596
```
```   597 definition (in prob_space)
```
```   598   "conditional_mutual_information b MX MY MZ X Y Z \<equiv>
```
```   599     mutual_information b MX (MY \<Otimes>\<^isub>M MZ) X (\<lambda>x. (Y x, Z x)) -
```
```   600     mutual_information b MX MZ X Z"
```
```   601
```
```   602 abbreviation (in information_space)
```
```   603   conditional_mutual_information_Pow ("\<I>'( _ ; _ | _ ')") where
```
```   604   "\<I>(X ; Y | Z) \<equiv> conditional_mutual_information b
```
```   605     \<lparr> space = X`space M, sets = Pow (X`space M), measure = extreal\<circ>distribution X \<rparr>
```
```   606     \<lparr> space = Y`space M, sets = Pow (Y`space M), measure = extreal\<circ>distribution Y \<rparr>
```
```   607     \<lparr> space = Z`space M, sets = Pow (Z`space M), measure = extreal\<circ>distribution Z \<rparr>
```
```   608     X Y Z"
```
```   609
```
```   610 lemma (in information_space) conditional_mutual_information_generic_eq:
```
```   611   assumes MX: "finite_random_variable MX X"
```
```   612     and MY: "finite_random_variable MY Y"
```
```   613     and MZ: "finite_random_variable MZ Z"
```
```   614   shows "conditional_mutual_information b MX MY MZ X Y Z = (\<Sum>(x, y, z) \<in> space MX \<times> space MY \<times> space MZ.
```
```   615              distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} *
```
```   616              log b (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} /
```
```   617     (joint_distribution X Z {(x, z)} * (joint_distribution Y Z {(y,z)} / distribution Z {z}))))"
```
```   618   (is "_ = (\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?XZ x z * (?YZ y z / ?Z z))))")
```
```   619 proof -
```
```   620   let ?X = "\<lambda>x. distribution X {x}"
```
```   621   note finite_var = MX MY MZ
```
```   622   note YZ = finite_random_variable_pairI[OF finite_var(2,3)]
```
```   623   note XYZ = finite_random_variable_pairI[OF MX YZ]
```
```   624   note XZ = finite_random_variable_pairI[OF finite_var(1,3)]
```
```   625   note ZX = finite_random_variable_pairI[OF finite_var(3,1)]
```
```   626   note YZX = finite_random_variable_pairI[OF finite_var(2) ZX]
```
```   627   note order1 =
```
```   628     finite_distribution_order(5,6)[OF finite_var(1) YZ]
```
```   629     finite_distribution_order(5,6)[OF finite_var(1,3)]
```
```   630
```
```   631   note random_var = finite_var[THEN finite_random_variableD]
```
```   632   note finite = finite_var(1) YZ finite_var(3) XZ YZX
```
```   633
```
```   634   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>
```
```   635           \<Longrightarrow> joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)} = 0"
```
```   636     unfolding joint_distribution_commute_singleton[of X]
```
```   637     unfolding joint_distribution_assoc_singleton[symmetric]
```
```   638     using finite_distribution_order(6)[OF finite_var(2) ZX]
```
```   639     by auto
```
```   640
```
```   641   have "(\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?XZ x z * (?YZ y z / ?Z z)))) =
```
```   642     (\<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))))"
```
```   643     (is "(\<Sum>(x, y, z)\<in>?S. ?L x y z) = (\<Sum>(x, y, z)\<in>?S. ?R x y z)")
```
```   644   proof (safe intro!: setsum_cong)
```
```   645     fix x y z assume space: "x \<in> space MX" "y \<in> space MY" "z \<in> space MZ"
```
```   646     show "?L x y z = ?R x y z"
```
```   647     proof cases
```
```   648       assume "?XYZ x y z \<noteq> 0"
```
```   649       with space have "0 < ?X x" "0 < ?Z z" "0 < ?XZ x z" "0 < ?YZ y z" "0 < ?XYZ x y z"
```
```   650         using order1 order2 by (auto simp: less_le)
```
```   651       with b_gt_1 show ?thesis
```
```   652         by (simp add: log_mult log_divide zero_less_mult_iff zero_less_divide_iff)
```
```   653     qed simp
```
```   654   qed
```
```   655   also have "\<dots> = (\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?X x * ?YZ y z))) -
```
```   656                   (\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XZ x z / (?X x * ?Z z)))"
```
```   657     by (auto simp add: setsum_subtractf[symmetric] field_simps intro!: setsum_cong)
```
```   658   also have "(\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XZ x z / (?X x * ?Z z))) =
```
```   659              (\<Sum>(x, z)\<in>space MX \<times> space MZ. ?XZ x z * log b (?XZ x z / (?X x * ?Z z)))"
```
```   660     unfolding setsum_cartesian_product[symmetric] setsum_commute[of _ _ "space MY"]
```
```   661               setsum_left_distrib[symmetric]
```
```   662     unfolding joint_distribution_commute_singleton[of X]
```
```   663     unfolding joint_distribution_assoc_singleton[symmetric]
```
```   664     using setsum_joint_distribution_singleton[OF finite_var(2) ZX]
```
```   665     by (intro setsum_cong refl) (simp add: space_pair_measure)
```
```   666   also have "(\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?X x * ?YZ y z))) -
```
```   667              (\<Sum>(x, z)\<in>space MX \<times> space MZ. ?XZ x z * log b (?XZ x z / (?X x * ?Z z))) =
```
```   668              conditional_mutual_information b MX MY MZ X Y Z"
```
```   669     unfolding conditional_mutual_information_def
```
```   670     unfolding mutual_information_generic_eq[OF finite_var(1,3)]
```
```   671     unfolding mutual_information_generic_eq[OF finite_var(1) YZ]
```
```   672     by (simp add: space_sigma space_pair_measure setsum_cartesian_product')
```
```   673   finally show ?thesis by simp
```
```   674 qed
```
```   675
```
```   676 lemma (in information_space) conditional_mutual_information_eq:
```
```   677   assumes "simple_function M X" "simple_function M Y" "simple_function M Z"
```
```   678   shows "\<I>(X;Y|Z) = (\<Sum>(x, y, z) \<in> X`space M \<times> Y`space M \<times> Z`space M.
```
```   679              distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} *
```
```   680              log b (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} /
```
```   681     (joint_distribution X Z {(x, z)} * joint_distribution Y Z {(y,z)} / distribution Z {z})))"
```
```   682   by (subst conditional_mutual_information_generic_eq[OF assms[THEN simple_function_imp_finite_random_variable]])
```
```   683      simp
```
```   684
```
```   685 lemma (in information_space) conditional_mutual_information_eq_mutual_information:
```
```   686   assumes X: "simple_function M X" and Y: "simple_function M Y"
```
```   687   shows "\<I>(X ; Y) = \<I>(X ; Y | (\<lambda>x. ()))"
```
```   688 proof -
```
```   689   have [simp]: "(\<lambda>x. ()) ` space M = {()}" using not_empty by auto
```
```   690   have C: "simple_function M (\<lambda>x. ())" by auto
```
```   691   show ?thesis
```
```   692     unfolding conditional_mutual_information_eq[OF X Y C]
```
```   693     unfolding mutual_information_eq[OF X Y]
```
```   694     by (simp add: setsum_cartesian_product' distribution_remove_const)
```
```   695 qed
```
```   696
```
```   697 lemma (in prob_space) distribution_unit[simp]: "distribution (\<lambda>x. ()) {()} = 1"
```
```   698   unfolding distribution_def using prob_space by auto
```
```   699
```
```   700 lemma (in prob_space) joint_distribution_unit[simp]: "distribution (\<lambda>x. (X x, ())) {(a, ())} = distribution X {a}"
```
```   701   unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
```
```   702
```
```   703 lemma (in prob_space) setsum_distribution:
```
```   704   assumes X: "finite_random_variable MX X" shows "(\<Sum>a\<in>space MX. distribution X {a}) = 1"
```
```   705   using setsum_joint_distribution[OF assms, of "\<lparr> space = UNIV, sets = Pow UNIV \<rparr>" "\<lambda>x. ()" "{()}"]
```
```   706   using sigma_algebra_Pow[of "UNIV::unit set" "()"] by simp
```
```   707
```
```   708 lemma (in prob_space) setsum_real_distribution:
```
```   709   fixes MX :: "('c, 'd) measure_space_scheme"
```
```   710   assumes X: "finite_random_variable MX X" shows "(\<Sum>a\<in>space MX. distribution X {a}) = 1"
```
```   711   using setsum_joint_distribution[OF assms, of "\<lparr> space = UNIV, sets = Pow UNIV, measure = undefined \<rparr>" "\<lambda>x. ()" "{()}"]
```
```   712   using sigma_algebra_Pow[of "UNIV::unit set" "\<lparr> measure = undefined \<rparr>"]
```
```   713   by auto
```
```   714
```
```   715 lemma (in information_space) conditional_mutual_information_generic_positive:
```
```   716   assumes X: "finite_random_variable MX X" and Y: "finite_random_variable MY Y" and Z: "finite_random_variable MZ Z"
```
```   717   shows "0 \<le> conditional_mutual_information b MX MY MZ X Y Z"
```
```   718 proof (cases "space MX \<times> space MY \<times> space MZ = {}")
```
```   719   case True show ?thesis
```
```   720     unfolding conditional_mutual_information_generic_eq[OF assms] True
```
```   721     by simp
```
```   722 next
```
```   723   case False
```
```   724   let ?dXYZ = "distribution (\<lambda>x. (X x, Y x, Z x))"
```
```   725   let ?dXZ = "joint_distribution X Z"
```
```   726   let ?dYZ = "joint_distribution Y Z"
```
```   727   let ?dX = "distribution X"
```
```   728   let ?dZ = "distribution Z"
```
```   729   let ?M = "space MX \<times> space MY \<times> space MZ"
```
```   730
```
```   731   note YZ = finite_random_variable_pairI[OF Y Z]
```
```   732   note XZ = finite_random_variable_pairI[OF X Z]
```
```   733   note ZX = finite_random_variable_pairI[OF Z X]
```
```   734   note YZ = finite_random_variable_pairI[OF Y Z]
```
```   735   note XYZ = finite_random_variable_pairI[OF X YZ]
```
```   736   note finite = Z YZ XZ XYZ
```
```   737   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>
```
```   738           \<Longrightarrow> joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)} = 0"
```
```   739     unfolding joint_distribution_commute_singleton[of X]
```
```   740     unfolding joint_distribution_assoc_singleton[symmetric]
```
```   741     using finite_distribution_order(6)[OF Y ZX]
```
```   742     by auto
```
```   743
```
```   744   note order = order
```
```   745     finite_distribution_order(5,6)[OF X YZ]
```
```   746     finite_distribution_order(5,6)[OF Y Z]
```
```   747
```
```   748   have "- conditional_mutual_information b MX MY MZ X Y Z = - (\<Sum>(x, y, z) \<in> ?M. ?dXYZ {(x, y, z)} *
```
```   749     log b (?dXYZ {(x, y, z)} / (?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z})))"
```
```   750     unfolding conditional_mutual_information_generic_eq[OF assms] neg_equal_iff_equal by auto
```
```   751   also have "\<dots> \<le> log b (\<Sum>(x, y, z) \<in> ?M. ?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z})"
```
```   752     unfolding split_beta'
```
```   753   proof (rule log_setsum_divide)
```
```   754     show "?M \<noteq> {}" using False by simp
```
```   755     show "1 < b" using b_gt_1 .
```
```   756
```
```   757     show "finite ?M" using assms
```
```   758       unfolding finite_sigma_algebra_def finite_sigma_algebra_axioms_def by auto
```
```   759
```
```   760     show "(\<Sum>x\<in>?M. ?dXYZ {(fst x, fst (snd x), snd (snd x))}) = 1"
```
```   761       unfolding setsum_cartesian_product'
```
```   762       unfolding setsum_commute[of _ "space MY"]
```
```   763       unfolding setsum_commute[of _ "space MZ"]
```
```   764       by (simp_all add: space_pair_measure
```
```   765                         setsum_joint_distribution_singleton[OF X YZ]
```
```   766                         setsum_joint_distribution_singleton[OF Y Z]
```
```   767                         setsum_distribution[OF Z])
```
```   768
```
```   769     fix x assume "x \<in> ?M"
```
```   770     let ?x = "(fst x, fst (snd x), snd (snd x))"
```
```   771
```
```   772     show "0 \<le> ?dXYZ {?x}"
```
```   773       "0 \<le> ?dXZ {(fst x, snd (snd x))} * ?dYZ {(fst (snd x), snd (snd x))} / ?dZ {snd (snd x)}"
```
```   774      by (simp_all add: mult_nonneg_nonneg divide_nonneg_nonneg)
```
```   775
```
```   776     assume *: "0 < ?dXYZ {?x}"
```
```   777     with `x \<in> ?M` finite order show "0 < ?dXZ {(fst x, snd (snd x))} * ?dYZ {(fst (snd x), snd (snd x))} / ?dZ {snd (snd x)}"
```
```   778       by (cases x) (auto simp add: zero_le_mult_iff zero_le_divide_iff less_le)
```
```   779   qed
```
```   780   also have "(\<Sum>(x, y, z) \<in> ?M. ?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z}) = (\<Sum>z\<in>space MZ. ?dZ {z})"
```
```   781     apply (simp add: setsum_cartesian_product')
```
```   782     apply (subst setsum_commute)
```
```   783     apply (subst (2) setsum_commute)
```
```   784     by (auto simp: setsum_divide_distrib[symmetric] setsum_product[symmetric]
```
```   785                    setsum_joint_distribution_singleton[OF X Z]
```
```   786                    setsum_joint_distribution_singleton[OF Y Z]
```
```   787           intro!: setsum_cong)
```
```   788   also have "log b (\<Sum>z\<in>space MZ. ?dZ {z}) = 0"
```
```   789     unfolding setsum_real_distribution[OF Z] by simp
```
```   790   finally show ?thesis by simp
```
```   791 qed
```
```   792
```
```   793 lemma (in information_space) conditional_mutual_information_positive:
```
```   794   assumes "simple_function M X" and "simple_function M Y" and "simple_function M Z"
```
```   795   shows "0 \<le> \<I>(X;Y|Z)"
```
```   796   by (rule conditional_mutual_information_generic_positive[OF assms[THEN simple_function_imp_finite_random_variable]])
```
```   797
```
```   798 subsection {* Conditional Entropy *}
```
```   799
```
```   800 definition (in prob_space)
```
```   801   "conditional_entropy b S T X Y = conditional_mutual_information b S S T X X Y"
```
```   802
```
```   803 abbreviation (in information_space)
```
```   804   conditional_entropy_Pow ("\<H>'(_ | _')") where
```
```   805   "\<H>(X | Y) \<equiv> conditional_entropy b
```
```   806     \<lparr> space = X`space M, sets = Pow (X`space M), measure = extreal\<circ>distribution X \<rparr>
```
```   807     \<lparr> space = Y`space M, sets = Pow (Y`space M), measure = extreal\<circ>distribution Y \<rparr> X Y"
```
```   808
```
```   809 lemma (in information_space) conditional_entropy_positive:
```
```   810   "simple_function M X \<Longrightarrow> simple_function M Y \<Longrightarrow> 0 \<le> \<H>(X | Y)"
```
```   811   unfolding conditional_entropy_def by (auto intro!: conditional_mutual_information_positive)
```
```   812
```
```   813 lemma (in information_space) conditional_entropy_generic_eq:
```
```   814   fixes MX :: "('c, 'd) measure_space_scheme" and MY :: "('e, 'f) measure_space_scheme"
```
```   815   assumes MX: "finite_random_variable MX X"
```
```   816   assumes MZ: "finite_random_variable MZ Z"
```
```   817   shows "conditional_entropy b MX MZ X Z =
```
```   818      - (\<Sum>(x, z)\<in>space MX \<times> space MZ.
```
```   819          joint_distribution X Z {(x, z)} * log b (joint_distribution X Z {(x, z)} / distribution Z {z}))"
```
```   820 proof -
```
```   821   interpret MX: finite_sigma_algebra MX using MX by simp
```
```   822   interpret MZ: finite_sigma_algebra MZ using MZ by simp
```
```   823   let "?XXZ x y z" = "joint_distribution X (\<lambda>x. (X x, Z x)) {(x, y, z)}"
```
```   824   let "?XZ x z" = "joint_distribution X Z {(x, z)}"
```
```   825   let "?Z z" = "distribution Z {z}"
```
```   826   let "?f x y z" = "log b (?XXZ x y z * ?Z z / (?XZ x z * ?XZ y z))"
```
```   827   { fix x z have "?XXZ x x z = ?XZ x z"
```
```   828       unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>']) }
```
```   829   note this[simp]
```
```   830   { fix x x' :: 'c and z assume "x' \<noteq> x"
```
```   831     then have "?XXZ x x' z = 0"
```
```   832       by (auto simp: distribution_def empty_measure'[symmetric]
```
```   833                simp del: empty_measure' intro!: arg_cong[where f=\<mu>']) }
```
```   834   note this[simp]
```
```   835   { fix x x' z assume *: "x \<in> space MX" "z \<in> space MZ"
```
```   836     then have "(\<Sum>x'\<in>space MX. ?XXZ x x' z * ?f x x' z)
```
```   837       = (\<Sum>x'\<in>space MX. if x = x' then ?XZ x z * ?f x x z else 0)"
```
```   838       by (auto intro!: setsum_cong)
```
```   839     also have "\<dots> = ?XZ x z * ?f x x z"
```
```   840       using `x \<in> space MX` by (simp add: setsum_cases[OF MX.finite_space])
```
```   841     also have "\<dots> = ?XZ x z * log b (?Z z / ?XZ x z)" by auto
```
```   842     also have "\<dots> = - ?XZ x z * log b (?XZ x z / ?Z z)"
```
```   843       using finite_distribution_order(6)[OF MX MZ]
```
```   844       by (auto simp: log_simps field_simps zero_less_mult_iff)
```
```   845     finally have "(\<Sum>x'\<in>space MX. ?XXZ x x' z * ?f x x' z) = - ?XZ x z * log b (?XZ x z / ?Z z)" . }
```
```   846   note * = this
```
```   847   show ?thesis
```
```   848     unfolding conditional_entropy_def
```
```   849     unfolding conditional_mutual_information_generic_eq[OF MX MX MZ]
```
```   850     by (auto simp: setsum_cartesian_product' setsum_negf[symmetric]
```
```   851                    setsum_commute[of _ "space MZ"] *
```
```   852              intro!: setsum_cong)
```
```   853 qed
```
```   854
```
```   855 lemma (in information_space) conditional_entropy_eq:
```
```   856   assumes "simple_function M X" "simple_function M Z"
```
```   857   shows "\<H>(X | Z) =
```
```   858      - (\<Sum>(x, z)\<in>X ` space M \<times> Z ` space M.
```
```   859          joint_distribution X Z {(x, z)} *
```
```   860          log b (joint_distribution X Z {(x, z)} / distribution Z {z}))"
```
```   861   by (subst conditional_entropy_generic_eq[OF assms[THEN simple_function_imp_finite_random_variable]])
```
```   862      simp
```
```   863
```
```   864 lemma (in information_space) conditional_entropy_eq_ce_with_hypothesis:
```
```   865   assumes X: "simple_function M X" and Y: "simple_function M Y"
```
```   866   shows "\<H>(X | Y) =
```
```   867     -(\<Sum>y\<in>Y`space M. distribution Y {y} *
```
```   868       (\<Sum>x\<in>X`space M. joint_distribution X Y {(x,y)} / distribution Y {(y)} *
```
```   869               log b (joint_distribution X Y {(x,y)} / distribution Y {(y)})))"
```
```   870   unfolding conditional_entropy_eq[OF assms]
```
```   871   using finite_distribution_order(5,6)[OF assms[THEN simple_function_imp_finite_random_variable]]
```
```   872   by (auto simp: setsum_cartesian_product'  setsum_commute[of _ "Y`space M"] setsum_right_distrib
```
```   873            intro!: setsum_cong)
```
```   874
```
```   875 lemma (in information_space) conditional_entropy_eq_cartesian_product:
```
```   876   assumes "simple_function M X" "simple_function M Y"
```
```   877   shows "\<H>(X | Y) = -(\<Sum>x\<in>X`space M. \<Sum>y\<in>Y`space M.
```
```   878     joint_distribution X Y {(x,y)} *
```
```   879     log b (joint_distribution X Y {(x,y)} / distribution Y {y}))"
```
```   880   unfolding conditional_entropy_eq[OF assms]
```
```   881   by (auto intro!: setsum_cong simp: setsum_cartesian_product')
```
```   882
```
```   883 subsection {* Equalities *}
```
```   884
```
```   885 lemma (in information_space) mutual_information_eq_entropy_conditional_entropy:
```
```   886   assumes X: "simple_function M X" and Z: "simple_function M Z"
```
```   887   shows  "\<I>(X ; Z) = \<H>(X) - \<H>(X | Z)"
```
```   888 proof -
```
```   889   let "?XZ x z" = "joint_distribution X Z {(x, z)}"
```
```   890   let "?Z z" = "distribution Z {z}"
```
```   891   let "?X x" = "distribution X {x}"
```
```   892   note fX = X[THEN simple_function_imp_finite_random_variable]
```
```   893   note fZ = Z[THEN simple_function_imp_finite_random_variable]
```
```   894   note finite_distribution_order[OF fX fZ, simp]
```
```   895   { fix x z assume "x \<in> X`space M" "z \<in> Z`space M"
```
```   896     have "?XZ x z * log b (?XZ x z / (?X x * ?Z z)) =
```
```   897           ?XZ x z * log b (?XZ x z / ?Z z) - ?XZ x z * log b (?X x)"
```
```   898       by (auto simp: log_simps zero_le_mult_iff field_simps less_le) }
```
```   899   note * = this
```
```   900   show ?thesis
```
```   901     unfolding entropy_eq[OF X] conditional_entropy_eq[OF X Z] mutual_information_eq[OF X Z]
```
```   902     using setsum_joint_distribution_singleton[OF fZ fX, unfolded joint_distribution_commute_singleton[of Z X]]
```
```   903     by (simp add: * setsum_cartesian_product' setsum_subtractf setsum_left_distrib[symmetric]
```
```   904                      setsum_distribution)
```
```   905 qed
```
```   906
```
```   907 lemma (in information_space) conditional_entropy_less_eq_entropy:
```
```   908   assumes X: "simple_function M X" and Z: "simple_function M Z"
```
```   909   shows "\<H>(X | Z) \<le> \<H>(X)"
```
```   910 proof -
```
```   911   have "\<I>(X ; Z) = \<H>(X) - \<H>(X | Z)" using mutual_information_eq_entropy_conditional_entropy[OF assms] .
```
```   912   with mutual_information_positive[OF X Z] entropy_positive[OF X]
```
```   913   show ?thesis by auto
```
```   914 qed
```
```   915
```
```   916 lemma (in information_space) entropy_chain_rule:
```
```   917   assumes X: "simple_function M X" and Y: "simple_function M Y"
```
```   918   shows  "\<H>(\<lambda>x. (X x, Y x)) = \<H>(X) + \<H>(Y|X)"
```
```   919 proof -
```
```   920   let "?XY x y" = "joint_distribution X Y {(x, y)}"
```
```   921   let "?Y y" = "distribution Y {y}"
```
```   922   let "?X x" = "distribution X {x}"
```
```   923   note fX = X[THEN simple_function_imp_finite_random_variable]
```
```   924   note fY = Y[THEN simple_function_imp_finite_random_variable]
```
```   925   note finite_distribution_order[OF fX fY, simp]
```
```   926   { fix x y assume "x \<in> X`space M" "y \<in> Y`space M"
```
```   927     have "?XY x y * log b (?XY x y / ?X x) =
```
```   928           ?XY x y * log b (?XY x y) - ?XY x y * log b (?X x)"
```
```   929       by (auto simp: log_simps zero_le_mult_iff field_simps less_le) }
```
```   930   note * = this
```
```   931   show ?thesis
```
```   932     using setsum_joint_distribution_singleton[OF fY fX]
```
```   933     unfolding entropy_eq[OF X] conditional_entropy_eq_cartesian_product[OF Y X] entropy_eq_cartesian_product[OF X Y]
```
```   934     unfolding joint_distribution_commute_singleton[of Y X] setsum_commute[of _ "X`space M"]
```
```   935     by (simp add: * setsum_subtractf setsum_left_distrib[symmetric])
```
```   936 qed
```
```   937
```
```   938 section {* Partitioning *}
```
```   939
```
```   940 definition "subvimage A f g \<longleftrightarrow> (\<forall>x \<in> A. f -` {f x} \<inter> A \<subseteq> g -` {g x} \<inter> A)"
```
```   941
```
```   942 lemma subvimageI:
```
```   943   assumes "\<And>x y. \<lbrakk> x \<in> A ; y \<in> A ; f x = f y \<rbrakk> \<Longrightarrow> g x = g y"
```
```   944   shows "subvimage A f g"
```
```   945   using assms unfolding subvimage_def by blast
```
```   946
```
```   947 lemma subvimageE[consumes 1]:
```
```   948   assumes "subvimage A f g"
```
```   949   obtains "\<And>x y. \<lbrakk> x \<in> A ; y \<in> A ; f x = f y \<rbrakk> \<Longrightarrow> g x = g y"
```
```   950   using assms unfolding subvimage_def by blast
```
```   951
```
```   952 lemma subvimageD:
```
```   953   "\<lbrakk> subvimage A f g ; x \<in> A ; y \<in> A ; f x = f y \<rbrakk> \<Longrightarrow> g x = g y"
```
```   954   using assms unfolding subvimage_def by blast
```
```   955
```
```   956 lemma subvimage_subset:
```
```   957   "\<lbrakk> subvimage B f g ; A \<subseteq> B \<rbrakk> \<Longrightarrow> subvimage A f g"
```
```   958   unfolding subvimage_def by auto
```
```   959
```
```   960 lemma subvimage_idem[intro]: "subvimage A g g"
```
```   961   by (safe intro!: subvimageI)
```
```   962
```
```   963 lemma subvimage_comp_finer[intro]:
```
```   964   assumes svi: "subvimage A g h"
```
```   965   shows "subvimage A g (f \<circ> h)"
```
```   966 proof (rule subvimageI, simp)
```
```   967   fix x y assume "x \<in> A" "y \<in> A" "g x = g y"
```
```   968   from svi[THEN subvimageD, OF this]
```
```   969   show "f (h x) = f (h y)" by simp
```
```   970 qed
```
```   971
```
```   972 lemma subvimage_comp_gran:
```
```   973   assumes svi: "subvimage A g h"
```
```   974   assumes inj: "inj_on f (g ` A)"
```
```   975   shows "subvimage A (f \<circ> g) h"
```
```   976   by (rule subvimageI) (auto intro!: subvimageD[OF svi] simp: inj_on_iff[OF inj])
```
```   977
```
```   978 lemma subvimage_comp:
```
```   979   assumes svi: "subvimage (f ` A) g h"
```
```   980   shows "subvimage A (g \<circ> f) (h \<circ> f)"
```
```   981   by (rule subvimageI) (auto intro!: svi[THEN subvimageD])
```
```   982
```
```   983 lemma subvimage_trans:
```
```   984   assumes fg: "subvimage A f g"
```
```   985   assumes gh: "subvimage A g h"
```
```   986   shows "subvimage A f h"
```
```   987   by (rule subvimageI) (auto intro!: fg[THEN subvimageD] gh[THEN subvimageD])
```
```   988
```
```   989 lemma subvimage_translator:
```
```   990   assumes svi: "subvimage A f g"
```
```   991   shows "\<exists>h. \<forall>x \<in> A. h (f x)  = g x"
```
```   992 proof (safe intro!: exI[of _ "\<lambda>x. (THE z. z \<in> (g ` (f -` {x} \<inter> A)))"])
```
```   993   fix x assume "x \<in> A"
```
```   994   show "(THE x'. x' \<in> (g ` (f -` {f x} \<inter> A))) = g x"
```
```   995     by (rule theI2[of _ "g x"])
```
```   996       (insert `x \<in> A`, auto intro!: svi[THEN subvimageD])
```
```   997 qed
```
```   998
```
```   999 lemma subvimage_translator_image:
```
```  1000   assumes svi: "subvimage A f g"
```
```  1001   shows "\<exists>h. h ` f ` A = g ` A"
```
```  1002 proof -
```
```  1003   from subvimage_translator[OF svi]
```
```  1004   obtain h where "\<And>x. x \<in> A \<Longrightarrow> h (f x) = g x" by auto
```
```  1005   thus ?thesis
```
```  1006     by (auto intro!: exI[of _ h]
```
```  1007       simp: image_compose[symmetric] comp_def cong: image_cong)
```
```  1008 qed
```
```  1009
```
```  1010 lemma subvimage_finite:
```
```  1011   assumes svi: "subvimage A f g" and fin: "finite (f`A)"
```
```  1012   shows "finite (g`A)"
```
```  1013 proof -
```
```  1014   from subvimage_translator_image[OF svi]
```
```  1015   obtain h where "g`A = h`f`A" by fastsimp
```
```  1016   with fin show "finite (g`A)" by simp
```
```  1017 qed
```
```  1018
```
```  1019 lemma subvimage_disj:
```
```  1020   assumes svi: "subvimage A f g"
```
```  1021   shows "f -` {x} \<inter> A \<subseteq> g -` {y} \<inter> A \<or>
```
```  1022       f -` {x} \<inter> g -` {y} \<inter> A = {}" (is "?sub \<or> ?dist")
```
```  1023 proof (rule disjCI)
```
```  1024   assume "\<not> ?dist"
```
```  1025   then obtain z where "z \<in> A" and "x = f z" and "y = g z" by auto
```
```  1026   thus "?sub" using svi unfolding subvimage_def by auto
```
```  1027 qed
```
```  1028
```
```  1029 lemma setsum_image_split:
```
```  1030   assumes svi: "subvimage A f g" and fin: "finite (f ` A)"
```
```  1031   shows "(\<Sum>x\<in>f`A. h x) = (\<Sum>y\<in>g`A. \<Sum>x\<in>f`(g -` {y} \<inter> A). h x)"
```
```  1032     (is "?lhs = ?rhs")
```
```  1033 proof -
```
```  1034   have "f ` A =
```
```  1035       snd ` (SIGMA x : g ` A. f ` (g -` {x} \<inter> A))"
```
```  1036       (is "_ = snd ` ?SIGMA")
```
```  1037     unfolding image_split_eq_Sigma[symmetric]
```
```  1038     by (simp add: image_compose[symmetric] comp_def)
```
```  1039   moreover
```
```  1040   have snd_inj: "inj_on snd ?SIGMA"
```
```  1041     unfolding image_split_eq_Sigma[symmetric]
```
```  1042     by (auto intro!: inj_onI subvimageD[OF svi])
```
```  1043   ultimately
```
```  1044   have "(\<Sum>x\<in>f`A. h x) = (\<Sum>(x,y)\<in>?SIGMA. h y)"
```
```  1045     by (auto simp: setsum_reindex intro: setsum_cong)
```
```  1046   also have "... = ?rhs"
```
```  1047     using subvimage_finite[OF svi fin] fin
```
```  1048     apply (subst setsum_Sigma[symmetric])
```
```  1049     by (auto intro!: finite_subset[of _ "f`A"])
```
```  1050   finally show ?thesis .
```
```  1051 qed
```
```  1052
```
```  1053 lemma (in information_space) entropy_partition:
```
```  1054   assumes sf: "simple_function M X" "simple_function M P"
```
```  1055   assumes svi: "subvimage (space M) X P"
```
```  1056   shows "\<H>(X) = \<H>(P) + \<H>(X|P)"
```
```  1057 proof -
```
```  1058   let "?XP x p" = "joint_distribution X P {(x, p)}"
```
```  1059   let "?X x" = "distribution X {x}"
```
```  1060   let "?P p" = "distribution P {p}"
```
```  1061   note fX = sf(1)[THEN simple_function_imp_finite_random_variable]
```
```  1062   note fP = sf(2)[THEN simple_function_imp_finite_random_variable]
```
```  1063   note finite_distribution_order[OF fX fP, simp]
```
```  1064   have "(\<Sum>x\<in>X ` space M. ?X x * log b (?X x)) =
```
```  1065     (\<Sum>y\<in>P `space M. \<Sum>x\<in>X ` space M. ?XP x y * log b (?XP x y))"
```
```  1066   proof (subst setsum_image_split[OF svi],
```
```  1067       safe intro!: setsum_mono_zero_cong_left imageI)
```
```  1068     show "finite (X ` space M)" "finite (X ` space M)" "finite (P ` space M)"
```
```  1069       using sf unfolding simple_function_def by auto
```
```  1070   next
```
```  1071     fix p x assume in_space: "p \<in> space M" "x \<in> space M"
```
```  1072     assume "?XP (X x) (P p) * log b (?XP (X x) (P p)) \<noteq> 0"
```
```  1073     hence "(\<lambda>x. (X x, P x)) -` {(X x, P p)} \<inter> space M \<noteq> {}" by (auto simp: distribution_def)
```
```  1074     with svi[unfolded subvimage_def, rule_format, OF `x \<in> space M`]
```
```  1075     show "x \<in> P -` {P p}" by auto
```
```  1076   next
```
```  1077     fix p x assume in_space: "p \<in> space M" "x \<in> space M"
```
```  1078     assume "P x = P p"
```
```  1079     from this[symmetric] svi[unfolded subvimage_def, rule_format, OF `x \<in> space M`]
```
```  1080     have "X -` {X x} \<inter> space M \<subseteq> P -` {P p} \<inter> space M"
```
```  1081       by auto
```
```  1082     hence "(\<lambda>x. (X x, P x)) -` {(X x, P p)} \<inter> space M = X -` {X x} \<inter> space M"
```
```  1083       by auto
```
```  1084     thus "?X (X x) * log b (?X (X x)) = ?XP (X x) (P p) * log b (?XP (X x) (P p))"
```
```  1085       by (auto simp: distribution_def)
```
```  1086   qed
```
```  1087   moreover have "\<And>x y. ?XP x y * log b (?XP x y / ?P y) =
```
```  1088       ?XP x y * log b (?XP x y) - ?XP x y * log b (?P y)"
```
```  1089     by (auto simp add: log_simps zero_less_mult_iff field_simps)
```
```  1090   ultimately show ?thesis
```
```  1091     unfolding sf[THEN entropy_eq] conditional_entropy_eq[OF sf]
```
```  1092     using setsum_joint_distribution_singleton[OF fX fP]
```
```  1093     by (simp add: setsum_cartesian_product' setsum_subtractf setsum_distribution
```
```  1094       setsum_left_distrib[symmetric] setsum_commute[where B="P`space M"])
```
```  1095 qed
```
```  1096
```
```  1097 corollary (in information_space) entropy_data_processing:
```
```  1098   assumes X: "simple_function M X" shows "\<H>(f \<circ> X) \<le> \<H>(X)"
```
```  1099 proof -
```
```  1100   note X
```
```  1101   moreover have fX: "simple_function M (f \<circ> X)" using X by auto
```
```  1102   moreover have "subvimage (space M) X (f \<circ> X)" by auto
```
```  1103   ultimately have "\<H>(X) = \<H>(f\<circ>X) + \<H>(X|f\<circ>X)" by (rule entropy_partition)
```
```  1104   then show "\<H>(f \<circ> X) \<le> \<H>(X)"
```
```  1105     by (auto intro: conditional_entropy_positive[OF X fX])
```
```  1106 qed
```
```  1107
```
```  1108 corollary (in information_space) entropy_of_inj:
```
```  1109   assumes X: "simple_function M X" and inj: "inj_on f (X`space M)"
```
```  1110   shows "\<H>(f \<circ> X) = \<H>(X)"
```
```  1111 proof (rule antisym)
```
```  1112   show "\<H>(f \<circ> X) \<le> \<H>(X)" using entropy_data_processing[OF X] .
```
```  1113 next
```
```  1114   have sf: "simple_function M (f \<circ> X)"
```
```  1115     using X by auto
```
```  1116   have "\<H>(X) = \<H>(the_inv_into (X`space M) f \<circ> (f \<circ> X))"
```
```  1117     by (auto intro!: mutual_information_cong simp: entropy_def the_inv_into_f_f[OF inj])
```
```  1118   also have "... \<le> \<H>(f \<circ> X)"
```
```  1119     using entropy_data_processing[OF sf] .
```
```  1120   finally show "\<H>(X) \<le> \<H>(f \<circ> X)" .
```
```  1121 qed
```
```  1122
```
```  1123 end
```