src/HOL/Probability/Information.thy
 author huffman Fri Aug 19 14:17:28 2011 -0700 (2011-08-19) changeset 44311 42c5cbf68052 parent 43920 cedb5cb948fd child 44890 22f665a2e91c permissions -rw-r--r--
new isCont theorems;
simplify some proofs.
```     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   Independent_Family
```
```    11   Radon_Nikodym
```
```    12   "~~/src/HOL/Library/Convex"
```
```    13 begin
```
```    14
```
```    15 lemma log_le: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> x \<le> y \<Longrightarrow> log a x \<le> log a y"
```
```    16   by (subst log_le_cancel_iff) auto
```
```    17
```
```    18 lemma log_less: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> x < y \<Longrightarrow> log a x < log a y"
```
```    19   by (subst log_less_cancel_iff) auto
```
```    20
```
```    21 lemma setsum_cartesian_product':
```
```    22   "(\<Sum>x\<in>A \<times> B. f x) = (\<Sum>x\<in>A. setsum (\<lambda>y. f (x, y)) B)"
```
```    23   unfolding setsum_cartesian_product by simp
```
```    24
```
```    25 section "Convex theory"
```
```    26
```
```    27 lemma log_setsum:
```
```    28   assumes "finite s" "s \<noteq> {}"
```
```    29   assumes "b > 1"
```
```    30   assumes "(\<Sum> i \<in> s. a i) = 1"
```
```    31   assumes "\<And> i. i \<in> s \<Longrightarrow> a i \<ge> 0"
```
```    32   assumes "\<And> i. i \<in> s \<Longrightarrow> y i \<in> {0 <..}"
```
```    33   shows "log b (\<Sum> i \<in> s. a i * y i) \<ge> (\<Sum> i \<in> s. a i * log b (y i))"
```
```    34 proof -
```
```    35   have "convex_on {0 <..} (\<lambda> x. - log b x)"
```
```    36     by (rule minus_log_convex[OF `b > 1`])
```
```    37   hence "- log b (\<Sum> i \<in> s. a i * y i) \<le> (\<Sum> i \<in> s. a i * - log b (y i))"
```
```    38     using convex_on_setsum[of _ _ "\<lambda> x. - log b x"] assms pos_is_convex by fastsimp
```
```    39   thus ?thesis by (auto simp add:setsum_negf le_imp_neg_le)
```
```    40 qed
```
```    41
```
```    42 lemma log_setsum':
```
```    43   assumes "finite s" "s \<noteq> {}"
```
```    44   assumes "b > 1"
```
```    45   assumes "(\<Sum> i \<in> s. a i) = 1"
```
```    46   assumes pos: "\<And> i. i \<in> s \<Longrightarrow> 0 \<le> a i"
```
```    47           "\<And> i. \<lbrakk> i \<in> s ; 0 < a i \<rbrakk> \<Longrightarrow> 0 < y i"
```
```    48   shows "log b (\<Sum> i \<in> s. a i * y i) \<ge> (\<Sum> i \<in> s. a i * log b (y i))"
```
```    49 proof -
```
```    50   have "\<And>y. (\<Sum> i \<in> s - {i. a i = 0}. a i * y i) = (\<Sum> i \<in> s. a i * y i)"
```
```    51     using assms by (auto intro!: setsum_mono_zero_cong_left)
```
```    52   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))"
```
```    53   proof (rule log_setsum)
```
```    54     have "setsum a (s - {i. a i = 0}) = setsum a s"
```
```    55       using assms(1) by (rule setsum_mono_zero_cong_left) auto
```
```    56     thus sum_1: "setsum a (s - {i. a i = 0}) = 1"
```
```    57       "finite (s - {i. a i = 0})" using assms by simp_all
```
```    58
```
```    59     show "s - {i. a i = 0} \<noteq> {}"
```
```    60     proof
```
```    61       assume *: "s - {i. a i = 0} = {}"
```
```    62       hence "setsum a (s - {i. a i = 0}) = 0" by (simp add: * setsum_empty)
```
```    63       with sum_1 show False by simp
```
```    64     qed
```
```    65
```
```    66     fix i assume "i \<in> s - {i. a i = 0}"
```
```    67     hence "i \<in> s" "a i \<noteq> 0" by simp_all
```
```    68     thus "0 \<le> a i" "y i \<in> {0<..}" using pos[of i] by auto
```
```    69   qed fact+
```
```    70   ultimately show ?thesis by simp
```
```    71 qed
```
```    72
```
```    73 lemma log_setsum_divide:
```
```    74   assumes "finite S" and "S \<noteq> {}" and "1 < b"
```
```    75   assumes "(\<Sum>x\<in>S. g x) = 1"
```
```    76   assumes pos: "\<And>x. x \<in> S \<Longrightarrow> g x \<ge> 0" "\<And>x. x \<in> S \<Longrightarrow> f x \<ge> 0"
```
```    77   assumes g_pos: "\<And>x. \<lbrakk> x \<in> S ; 0 < g x \<rbrakk> \<Longrightarrow> 0 < f x"
```
```    78   shows "- (\<Sum>x\<in>S. g x * log b (g x / f x)) \<le> log b (\<Sum>x\<in>S. f x)"
```
```    79 proof -
```
```    80   have log_mono: "\<And>x y. 0 < x \<Longrightarrow> x \<le> y \<Longrightarrow> log b x \<le> log b y"
```
```    81     using `1 < b` by (subst log_le_cancel_iff) auto
```
```    82
```
```    83   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))"
```
```    84   proof (unfold setsum_negf[symmetric], rule setsum_cong)
```
```    85     fix x assume x: "x \<in> S"
```
```    86     show "- (g x * log b (g x / f x)) = g x * log b (f x / g x)"
```
```    87     proof (cases "g x = 0")
```
```    88       case False
```
```    89       with pos[OF x] g_pos[OF x] have "0 < f x" "0 < g x" by simp_all
```
```    90       thus ?thesis using `1 < b` by (simp add: log_divide field_simps)
```
```    91     qed simp
```
```    92   qed rule
```
```    93   also have "... \<le> log b (\<Sum>x\<in>S. g x * (f x / g x))"
```
```    94   proof (rule log_setsum')
```
```    95     fix x assume x: "x \<in> S" "0 < g x"
```
```    96     with g_pos[OF x] show "0 < f x / g x" by (safe intro!: divide_pos_pos)
```
```    97   qed fact+
```
```    98   also have "... = log b (\<Sum>x\<in>S - {x. g x = 0}. f x)" using `finite S`
```
```    99     by (auto intro!: setsum_mono_zero_cong_right arg_cong[where f="log b"]
```
```   100         split: split_if_asm)
```
```   101   also have "... \<le> log b (\<Sum>x\<in>S. f x)"
```
```   102   proof (rule log_mono)
```
```   103     have "0 = (\<Sum>x\<in>S - {x. g x = 0}. 0)" by simp
```
```   104     also have "... < (\<Sum>x\<in>S - {x. g x = 0}. f x)" (is "_ < ?sum")
```
```   105     proof (rule setsum_strict_mono)
```
```   106       show "finite (S - {x. g x = 0})" using `finite S` by simp
```
```   107       show "S - {x. g x = 0} \<noteq> {}"
```
```   108       proof
```
```   109         assume "S - {x. g x = 0} = {}"
```
```   110         hence "(\<Sum>x\<in>S. g x) = 0" by (subst setsum_0') auto
```
```   111         with `(\<Sum>x\<in>S. g x) = 1` show False by simp
```
```   112       qed
```
```   113       fix x assume "x \<in> S - {x. g x = 0}"
```
```   114       thus "0 < f x" using g_pos[of x] pos(1)[of x] by auto
```
```   115     qed
```
```   116     finally show "0 < ?sum" .
```
```   117     show "(\<Sum>x\<in>S - {x. g x = 0}. f x) \<le> (\<Sum>x\<in>S. f x)"
```
```   118       using `finite S` pos by (auto intro!: setsum_mono2)
```
```   119   qed
```
```   120   finally show ?thesis .
```
```   121 qed
```
```   122
```
```   123 lemma split_pairs:
```
```   124   "((A, B) = X) \<longleftrightarrow> (fst X = A \<and> snd X = B)" and
```
```   125   "(X = (A, B)) \<longleftrightarrow> (fst X = A \<and> snd X = B)" by auto
```
```   126
```
```   127 section "Information theory"
```
```   128
```
```   129 locale information_space = prob_space +
```
```   130   fixes b :: real assumes b_gt_1: "1 < b"
```
```   131
```
```   132 context information_space
```
```   133 begin
```
```   134
```
```   135 text {* Introduce some simplification rules for logarithm of base @{term b}. *}
```
```   136
```
```   137 lemma log_neg_const:
```
```   138   assumes "x \<le> 0"
```
```   139   shows "log b x = log b 0"
```
```   140 proof -
```
```   141   { fix u :: real
```
```   142     have "x \<le> 0" by fact
```
```   143     also have "0 < exp u"
```
```   144       using exp_gt_zero .
```
```   145     finally have "exp u \<noteq> x"
```
```   146       by auto }
```
```   147   then show "log b x = log b 0"
```
```   148     by (simp add: log_def ln_def)
```
```   149 qed
```
```   150
```
```   151 lemma log_mult_eq:
```
```   152   "log b (A * B) = (if 0 < A * B then log b \<bar>A\<bar> + log b \<bar>B\<bar> else log b 0)"
```
```   153   using log_mult[of b "\<bar>A\<bar>" "\<bar>B\<bar>"] b_gt_1 log_neg_const[of "A * B"]
```
```   154   by (auto simp: zero_less_mult_iff mult_le_0_iff)
```
```   155
```
```   156 lemma log_inverse_eq:
```
```   157   "log b (inverse B) = (if 0 < B then - log b B else log b 0)"
```
```   158   using log_inverse[of b B] log_neg_const[of "inverse B"] b_gt_1 by simp
```
```   159
```
```   160 lemma log_divide_eq:
```
```   161   "log b (A / B) = (if 0 < A * B then log b \<bar>A\<bar> - log b \<bar>B\<bar> else log b 0)"
```
```   162   unfolding divide_inverse log_mult_eq log_inverse_eq abs_inverse
```
```   163   by (auto simp: zero_less_mult_iff mult_le_0_iff)
```
```   164
```
```   165 lemmas log_simps = log_mult_eq log_inverse_eq log_divide_eq
```
```   166
```
```   167 end
```
```   168
```
```   169 subsection "Kullback\$-\$Leibler divergence"
```
```   170
```
```   171 text {* The Kullback\$-\$Leibler divergence is also known as relative entropy or
```
```   172 Kullback\$-\$Leibler distance. *}
```
```   173
```
```   174 definition
```
```   175   "entropy_density b M \<nu> = log b \<circ> real \<circ> RN_deriv M \<nu>"
```
```   176
```
```   177 definition
```
```   178   "KL_divergence b M \<nu> = integral\<^isup>L (M\<lparr>measure := \<nu>\<rparr>) (entropy_density b M \<nu>)"
```
```   179
```
```   180 lemma (in information_space) measurable_entropy_density:
```
```   181   assumes ps: "prob_space (M\<lparr>measure := \<nu>\<rparr>)"
```
```   182   assumes ac: "absolutely_continuous \<nu>"
```
```   183   shows "entropy_density b M \<nu> \<in> borel_measurable M"
```
```   184 proof -
```
```   185   interpret \<nu>: prob_space "M\<lparr>measure := \<nu>\<rparr>" by fact
```
```   186   have "measure_space (M\<lparr>measure := \<nu>\<rparr>)" by fact
```
```   187   from RN_deriv[OF this ac] b_gt_1 show ?thesis
```
```   188     unfolding entropy_density_def
```
```   189     by (intro measurable_comp) auto
```
```   190 qed
```
```   191
```
```   192 lemma (in information_space) KL_gt_0:
```
```   193   assumes ps: "prob_space (M\<lparr>measure := \<nu>\<rparr>)"
```
```   194   assumes ac: "absolutely_continuous \<nu>"
```
```   195   assumes int: "integrable (M\<lparr> measure := \<nu> \<rparr>) (entropy_density b M \<nu>)"
```
```   196   assumes A: "A \<in> sets M" "\<nu> A \<noteq> \<mu> A"
```
```   197   shows "0 < KL_divergence b M \<nu>"
```
```   198 proof -
```
```   199   interpret \<nu>: prob_space "M\<lparr>measure := \<nu>\<rparr>" by fact
```
```   200   have ms: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" by default
```
```   201   have fms: "finite_measure (M\<lparr>measure := \<nu>\<rparr>)" by default
```
```   202   note RN = RN_deriv[OF ms ac]
```
```   203
```
```   204   from real_RN_deriv[OF fms ac] guess D . note D = this
```
```   205   with absolutely_continuous_AE[OF ms] ac
```
```   206   have D\<nu>: "AE x in M\<lparr>measure := \<nu>\<rparr>. RN_deriv M \<nu> x = ereal (D x)"
```
```   207     by auto
```
```   208
```
```   209   def f \<equiv> "\<lambda>x. if D x = 0 then 1 else 1 / D x"
```
```   210   with D have f_borel: "f \<in> borel_measurable M"
```
```   211     by (auto intro!: measurable_If)
```
```   212
```
```   213   have "KL_divergence b M \<nu> = 1 / ln b * (\<integral> x. ln b * entropy_density b M \<nu> x \<partial>M\<lparr>measure := \<nu>\<rparr>)"
```
```   214     unfolding KL_divergence_def using int b_gt_1
```
```   215     by (simp add: integral_cmult)
```
```   216
```
```   217   { fix A assume "A \<in> sets M"
```
```   218     with RN D have "\<nu>.\<mu> A = (\<integral>\<^isup>+ x. ereal (D x) * indicator A x \<partial>M)"
```
```   219       by (auto intro!: positive_integral_cong_AE) }
```
```   220   note D_density = this
```
```   221
```
```   222   have ln_entropy: "(\<lambda>x. ln b * entropy_density b M \<nu> x) \<in> borel_measurable M"
```
```   223     using measurable_entropy_density[OF ps ac] by auto
```
```   224
```
```   225   have "integrable (M\<lparr>measure := \<nu>\<rparr>) (\<lambda>x. ln b * entropy_density b M \<nu> x)"
```
```   226     using int by auto
```
```   227   moreover have "integrable (M\<lparr>measure := \<nu>\<rparr>) (\<lambda>x. ln b * entropy_density b M \<nu> x) \<longleftrightarrow>
```
```   228       integrable M (\<lambda>x. D x * (ln b * entropy_density b M \<nu> x))"
```
```   229     using D D_density ln_entropy
```
```   230     by (intro integral_translated_density) auto
```
```   231   ultimately have M_int: "integrable M (\<lambda>x. D x * (ln b * entropy_density b M \<nu> x))"
```
```   232     by simp
```
```   233
```
```   234   have D_neg: "(\<integral>\<^isup>+ x. ereal (- D x) \<partial>M) = 0"
```
```   235     using D by (subst positive_integral_0_iff_AE) auto
```
```   236
```
```   237   have "(\<integral>\<^isup>+ x. ereal (D x) \<partial>M) = \<nu> (space M)"
```
```   238     using RN D by (auto intro!: positive_integral_cong_AE)
```
```   239   then have D_pos: "(\<integral>\<^isup>+ x. ereal (D x) \<partial>M) = 1"
```
```   240     using \<nu>.measure_space_1 by simp
```
```   241
```
```   242   have "integrable M D"
```
```   243     using D_pos D_neg D by (auto simp: integrable_def)
```
```   244
```
```   245   have "integral\<^isup>L M D = 1"
```
```   246     using D_pos D_neg by (auto simp: lebesgue_integral_def)
```
```   247
```
```   248   let ?D_set = "{x\<in>space M. D x \<noteq> 0}"
```
```   249   have [simp, intro]: "?D_set \<in> sets M"
```
```   250     using D by (auto intro: sets_Collect)
```
```   251
```
```   252   have "0 \<le> 1 - \<mu>' ?D_set"
```
```   253     using prob_le_1 by (auto simp: field_simps)
```
```   254   also have "\<dots> = (\<integral> x. D x - indicator ?D_set x \<partial>M)"
```
```   255     using `integrable M D` `integral\<^isup>L M D = 1`
```
```   256     by (simp add: \<mu>'_def)
```
```   257   also have "\<dots> < (\<integral> x. D x * (ln b * entropy_density b M \<nu> x) \<partial>M)"
```
```   258   proof (rule integral_less_AE)
```
```   259     show "integrable M (\<lambda>x. D x - indicator ?D_set x)"
```
```   260       using `integrable M D`
```
```   261       by (intro integral_diff integral_indicator) auto
```
```   262   next
```
```   263     show "integrable M (\<lambda>x. D x * (ln b * entropy_density b M \<nu> x))"
```
```   264       by fact
```
```   265   next
```
```   266     show "\<mu> {x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} \<noteq> 0"
```
```   267     proof
```
```   268       assume eq_0: "\<mu> {x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} = 0"
```
```   269       then have disj: "AE x. D x = 1 \<or> D x = 0"
```
```   270         using D(1) by (auto intro!: AE_I[OF subset_refl] sets_Collect)
```
```   271
```
```   272       have "\<mu> {x\<in>space M. D x = 1} = (\<integral>\<^isup>+ x. indicator {x\<in>space M. D x = 1} x \<partial>M)"
```
```   273         using D(1) by auto
```
```   274       also have "\<dots> = (\<integral>\<^isup>+ x. ereal (D x) * indicator {x\<in>space M. D x \<noteq> 0} x \<partial>M)"
```
```   275         using disj by (auto intro!: positive_integral_cong_AE simp: indicator_def one_ereal_def)
```
```   276       also have "\<dots> = \<nu> {x\<in>space M. D x \<noteq> 0}"
```
```   277         using D(1) D_density by auto
```
```   278       also have "\<dots> = \<nu> (space M)"
```
```   279         using D_density D(1) by (auto intro!: positive_integral_cong simp: indicator_def)
```
```   280       finally have "AE x. D x = 1"
```
```   281         using D(1) \<nu>.measure_space_1 by (intro AE_I_eq_1) auto
```
```   282       then have "(\<integral>\<^isup>+x. indicator A x\<partial>M) = (\<integral>\<^isup>+x. ereal (D x) * indicator A x\<partial>M)"
```
```   283         by (intro positive_integral_cong_AE) (auto simp: one_ereal_def[symmetric])
```
```   284       also have "\<dots> = \<nu> A"
```
```   285         using `A \<in> sets M` D_density by simp
```
```   286       finally show False using `A \<in> sets M` `\<nu> A \<noteq> \<mu> A` by simp
```
```   287     qed
```
```   288     show "{x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} \<in> sets M"
```
```   289       using D(1) by (auto intro: sets_Collect)
```
```   290
```
```   291     show "AE t. t \<in> {x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} \<longrightarrow>
```
```   292       D t - indicator ?D_set t \<noteq> D t * (ln b * entropy_density b M \<nu> t)"
```
```   293       using D(2)
```
```   294     proof (elim AE_mp, safe intro!: AE_I2)
```
```   295       fix t assume Dt: "t \<in> space M" "D t \<noteq> 1" "D t \<noteq> 0"
```
```   296         and RN: "RN_deriv M \<nu> t = ereal (D t)"
```
```   297         and eq: "D t - indicator ?D_set t = D t * (ln b * entropy_density b M \<nu> t)"
```
```   298
```
```   299       have "D t - 1 = D t - indicator ?D_set t"
```
```   300         using Dt by simp
```
```   301       also note eq
```
```   302       also have "D t * (ln b * entropy_density b M \<nu> t) = - D t * ln (1 / D t)"
```
```   303         using RN b_gt_1 `D t \<noteq> 0` `0 \<le> D t`
```
```   304         by (simp add: entropy_density_def log_def ln_div less_le)
```
```   305       finally have "ln (1 / D t) = 1 / D t - 1"
```
```   306         using `D t \<noteq> 0` by (auto simp: field_simps)
```
```   307       from ln_eq_minus_one[OF _ this] `D t \<noteq> 0` `0 \<le> D t` `D t \<noteq> 1`
```
```   308       show False by auto
```
```   309     qed
```
```   310
```
```   311     show "AE t. D t - indicator ?D_set t \<le> D t * (ln b * entropy_density b M \<nu> t)"
```
```   312       using D(2)
```
```   313     proof (elim AE_mp, intro AE_I2 impI)
```
```   314       fix t assume "t \<in> space M" and RN: "RN_deriv M \<nu> t = ereal (D t)"
```
```   315       show "D t - indicator ?D_set t \<le> D t * (ln b * entropy_density b M \<nu> t)"
```
```   316       proof cases
```
```   317         assume asm: "D t \<noteq> 0"
```
```   318         then have "0 < D t" using `0 \<le> D t` by auto
```
```   319         then have "0 < 1 / D t" by auto
```
```   320         have "D t - indicator ?D_set t \<le> - D t * (1 / D t - 1)"
```
```   321           using asm `t \<in> space M` by (simp add: field_simps)
```
```   322         also have "- D t * (1 / D t - 1) \<le> - D t * ln (1 / D t)"
```
```   323           using ln_le_minus_one `0 < 1 / D t` by (intro mult_left_mono_neg) auto
```
```   324         also have "\<dots> = D t * (ln b * entropy_density b M \<nu> t)"
```
```   325           using `0 < D t` RN b_gt_1
```
```   326           by (simp_all add: log_def ln_div entropy_density_def)
```
```   327         finally show ?thesis by simp
```
```   328       qed simp
```
```   329     qed
```
```   330   qed
```
```   331   also have "\<dots> = (\<integral> x. ln b * entropy_density b M \<nu> x \<partial>M\<lparr>measure := \<nu>\<rparr>)"
```
```   332     using D D_density ln_entropy
```
```   333     by (intro integral_translated_density[symmetric]) auto
```
```   334   also have "\<dots> = ln b * (\<integral> x. entropy_density b M \<nu> x \<partial>M\<lparr>measure := \<nu>\<rparr>)"
```
```   335     using int by (rule \<nu>.integral_cmult)
```
```   336   finally show "0 < KL_divergence b M \<nu>"
```
```   337     using b_gt_1 by (auto simp: KL_divergence_def zero_less_mult_iff)
```
```   338 qed
```
```   339
```
```   340 lemma (in sigma_finite_measure) KL_eq_0:
```
```   341   assumes eq: "\<forall>A\<in>sets M. \<nu> A = measure M A"
```
```   342   shows "KL_divergence b M \<nu> = 0"
```
```   343 proof -
```
```   344   have "AE x. 1 = RN_deriv M \<nu> x"
```
```   345   proof (rule RN_deriv_unique)
```
```   346     show "measure_space (M\<lparr>measure := \<nu>\<rparr>)"
```
```   347       using eq by (intro measure_space_cong) auto
```
```   348     show "absolutely_continuous \<nu>"
```
```   349       unfolding absolutely_continuous_def using eq by auto
```
```   350     show "(\<lambda>x. 1) \<in> borel_measurable M" "AE x. 0 \<le> (1 :: ereal)" by auto
```
```   351     fix A assume "A \<in> sets M"
```
```   352     with eq show "\<nu> A = \<integral>\<^isup>+ x. 1 * indicator A x \<partial>M" by simp
```
```   353   qed
```
```   354   then have "AE x. log b (real (RN_deriv M \<nu> x)) = 0"
```
```   355     by (elim AE_mp) simp
```
```   356   from integral_cong_AE[OF this]
```
```   357   have "integral\<^isup>L M (entropy_density b M \<nu>) = 0"
```
```   358     by (simp add: entropy_density_def comp_def)
```
```   359   with eq show "KL_divergence b M \<nu> = 0"
```
```   360     unfolding KL_divergence_def
```
```   361     by (subst integral_cong_measure) auto
```
```   362 qed
```
```   363
```
```   364 lemma (in information_space) KL_eq_0_imp:
```
```   365   assumes ps: "prob_space (M\<lparr>measure := \<nu>\<rparr>)"
```
```   366   assumes ac: "absolutely_continuous \<nu>"
```
```   367   assumes int: "integrable (M\<lparr> measure := \<nu> \<rparr>) (entropy_density b M \<nu>)"
```
```   368   assumes KL: "KL_divergence b M \<nu> = 0"
```
```   369   shows "\<forall>A\<in>sets M. \<nu> A = \<mu> A"
```
```   370   by (metis less_imp_neq KL_gt_0 assms)
```
```   371
```
```   372 lemma (in information_space) KL_ge_0:
```
```   373   assumes ps: "prob_space (M\<lparr>measure := \<nu>\<rparr>)"
```
```   374   assumes ac: "absolutely_continuous \<nu>"
```
```   375   assumes int: "integrable (M\<lparr> measure := \<nu> \<rparr>) (entropy_density b M \<nu>)"
```
```   376   shows "0 \<le> KL_divergence b M \<nu>"
```
```   377   using KL_eq_0 KL_gt_0[OF ps ac int]
```
```   378   by (cases "\<forall>A\<in>sets M. \<nu> A = measure M A") (auto simp: le_less)
```
```   379
```
```   380
```
```   381 lemma (in sigma_finite_measure) KL_divergence_vimage:
```
```   382   assumes T: "T \<in> measure_preserving M M'"
```
```   383     and T': "T' \<in> measure_preserving (M'\<lparr> measure := \<nu>' \<rparr>) (M\<lparr> measure := \<nu> \<rparr>)"
```
```   384     and inv: "\<And>x. x \<in> space M \<Longrightarrow> T' (T x) = x"
```
```   385     and inv': "\<And>x. x \<in> space M' \<Longrightarrow> T (T' x) = x"
```
```   386   and \<nu>': "measure_space (M'\<lparr>measure := \<nu>'\<rparr>)" "measure_space.absolutely_continuous M' \<nu>'"
```
```   387   and "1 < b"
```
```   388   shows "KL_divergence b M' \<nu>' = KL_divergence b M \<nu>"
```
```   389 proof -
```
```   390   interpret \<nu>': measure_space "M'\<lparr>measure := \<nu>'\<rparr>" by fact
```
```   391   have M: "measure_space (M\<lparr> measure := \<nu>\<rparr>)"
```
```   392     by (rule \<nu>'.measure_space_vimage[OF _ T'], simp) default
```
```   393   have "sigma_algebra (M'\<lparr> measure := \<nu>'\<rparr>)" by default
```
```   394   then have saM': "sigma_algebra M'" by simp
```
```   395   then interpret M': measure_space M' by (rule measure_space_vimage) fact
```
```   396   have ac: "absolutely_continuous \<nu>" unfolding absolutely_continuous_def
```
```   397   proof safe
```
```   398     fix N assume N: "N \<in> sets M" and N_0: "\<mu> N = 0"
```
```   399     then have N': "T' -` N \<inter> space M' \<in> sets M'"
```
```   400       using T' by (auto simp: measurable_def measure_preserving_def)
```
```   401     have "T -` (T' -` N \<inter> space M') \<inter> space M = N"
```
```   402       using inv T N sets_into_space[OF N] by (auto simp: measurable_def measure_preserving_def)
```
```   403     then have "measure M' (T' -` N \<inter> space M') = 0"
```
```   404       using measure_preservingD[OF T N'] N_0 by auto
```
```   405     with \<nu>'(2) N' show "\<nu> N = 0" using measure_preservingD[OF T', of N] N
```
```   406       unfolding M'.absolutely_continuous_def measurable_def by auto
```
```   407   qed
```
```   408
```
```   409   have sa: "sigma_algebra (M\<lparr>measure := \<nu>\<rparr>)" by simp default
```
```   410   have AE: "AE x. RN_deriv M' \<nu>' (T x) = RN_deriv M \<nu> x"
```
```   411     by (rule RN_deriv_vimage[OF T T' inv \<nu>'])
```
```   412   show ?thesis
```
```   413     unfolding KL_divergence_def entropy_density_def comp_def
```
```   414   proof (subst \<nu>'.integral_vimage[OF sa T'])
```
```   415     show "(\<lambda>x. log b (real (RN_deriv M \<nu> x))) \<in> borel_measurable (M\<lparr>measure := \<nu>\<rparr>)"
```
```   416       by (auto intro!: RN_deriv[OF M ac] borel_measurable_log[OF _ `1 < b`])
```
```   417     have "(\<integral> x. log b (real (RN_deriv M' \<nu>' x)) \<partial>M'\<lparr>measure := \<nu>'\<rparr>) =
```
```   418       (\<integral> x. log b (real (RN_deriv M' \<nu>' (T (T' x)))) \<partial>M'\<lparr>measure := \<nu>'\<rparr>)" (is "?l = _")
```
```   419       using inv' by (auto intro!: \<nu>'.integral_cong)
```
```   420     also have "\<dots> = (\<integral> x. log b (real (RN_deriv M \<nu> (T' x))) \<partial>M'\<lparr>measure := \<nu>'\<rparr>)" (is "_ = ?r")
```
```   421       using M ac AE
```
```   422       by (intro \<nu>'.integral_cong_AE \<nu>'.almost_everywhere_vimage[OF sa T'] absolutely_continuous_AE[OF M])
```
```   423          (auto elim!: AE_mp)
```
```   424     finally show "?l = ?r" .
```
```   425   qed
```
```   426 qed
```
```   427
```
```   428 lemma (in sigma_finite_measure) KL_divergence_cong:
```
```   429   assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)" (is "measure_space ?\<nu>")
```
```   430   assumes [simp]: "sets N = sets M" "space N = space M"
```
```   431     "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A"
```
```   432     "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = \<nu>' A"
```
```   433   shows "KL_divergence b M \<nu> = KL_divergence b N \<nu>'"
```
```   434 proof -
```
```   435   interpret \<nu>: measure_space ?\<nu> by fact
```
```   436   have "KL_divergence b M \<nu> = \<integral>x. log b (real (RN_deriv N \<nu>' x)) \<partial>?\<nu>"
```
```   437     by (simp cong: RN_deriv_cong \<nu>.integral_cong add: KL_divergence_def entropy_density_def)
```
```   438   also have "\<dots> = KL_divergence b N \<nu>'"
```
```   439     by (auto intro!: \<nu>.integral_cong_measure[symmetric] simp: KL_divergence_def entropy_density_def comp_def)
```
```   440   finally show ?thesis .
```
```   441 qed
```
```   442
```
```   443 lemma (in finite_measure_space) KL_divergence_eq_finite:
```
```   444   assumes v: "finite_measure_space (M\<lparr>measure := \<nu>\<rparr>)"
```
```   445   assumes ac: "absolutely_continuous \<nu>"
```
```   446   shows "KL_divergence b M \<nu> = (\<Sum>x\<in>space M. real (\<nu> {x}) * log b (real (\<nu> {x}) / real (\<mu> {x})))" (is "_ = ?sum")
```
```   447 proof (simp add: KL_divergence_def finite_measure_space.integral_finite_singleton[OF v] entropy_density_def)
```
```   448   interpret v: finite_measure_space "M\<lparr>measure := \<nu>\<rparr>" by fact
```
```   449   have ms: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" by default
```
```   450   show "(\<Sum>x \<in> space M. log b (real (RN_deriv M \<nu> x)) * real (\<nu> {x})) = ?sum"
```
```   451     using RN_deriv_finite_measure[OF ms ac]
```
```   452     by (auto intro!: setsum_cong simp: field_simps)
```
```   453 qed
```
```   454
```
```   455 lemma (in finite_prob_space) KL_divergence_positive_finite:
```
```   456   assumes v: "finite_prob_space (M\<lparr>measure := \<nu>\<rparr>)"
```
```   457   assumes ac: "absolutely_continuous \<nu>"
```
```   458   and "1 < b"
```
```   459   shows "0 \<le> KL_divergence b M \<nu>"
```
```   460 proof -
```
```   461   interpret information_space M by default fact
```
```   462   interpret v: finite_prob_space "M\<lparr>measure := \<nu>\<rparr>" by fact
```
```   463   have ps: "prob_space (M\<lparr>measure := \<nu>\<rparr>)" by default
```
```   464   from KL_ge_0[OF this ac v.integral_finite_singleton(1)] show ?thesis .
```
```   465 qed
```
```   466
```
```   467 subsection {* Mutual Information *}
```
```   468
```
```   469 definition (in prob_space)
```
```   470   "mutual_information b S T X Y =
```
```   471     KL_divergence b (S\<lparr>measure := ereal\<circ>distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := ereal\<circ>distribution Y\<rparr>)
```
```   472       (ereal\<circ>joint_distribution X Y)"
```
```   473
```
```   474 lemma (in information_space)
```
```   475   fixes S T X Y
```
```   476   defines "P \<equiv> S\<lparr>measure := ereal\<circ>distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := ereal\<circ>distribution Y\<rparr>"
```
```   477   shows "indep_var S X T Y \<longleftrightarrow>
```
```   478     (random_variable S X \<and> random_variable T Y \<and>
```
```   479       measure_space.absolutely_continuous P (ereal\<circ>joint_distribution X Y) \<and>
```
```   480       integrable (P\<lparr>measure := (ereal\<circ>joint_distribution X Y)\<rparr>)
```
```   481         (entropy_density b P (ereal\<circ>joint_distribution X Y)) \<and>
```
```   482      mutual_information b S T X Y = 0)"
```
```   483 proof safe
```
```   484   assume indep: "indep_var S X T Y"
```
```   485   then have "random_variable S X" "random_variable T Y"
```
```   486     by (blast dest: indep_var_rv1 indep_var_rv2)+
```
```   487   then show "sigma_algebra S" "X \<in> measurable M S" "sigma_algebra T" "Y \<in> measurable M T"
```
```   488     by blast+
```
```   489
```
```   490   interpret X: prob_space "S\<lparr>measure := ereal\<circ>distribution X\<rparr>"
```
```   491     by (rule distribution_prob_space) fact
```
```   492   interpret Y: prob_space "T\<lparr>measure := ereal\<circ>distribution Y\<rparr>"
```
```   493     by (rule distribution_prob_space) fact
```
```   494   interpret XY: pair_prob_space "S\<lparr>measure := ereal\<circ>distribution X\<rparr>" "T\<lparr>measure := ereal\<circ>distribution Y\<rparr>" by default
```
```   495   interpret XY: information_space XY.P b by default (rule b_gt_1)
```
```   496
```
```   497   let ?J = "XY.P\<lparr> measure := (ereal\<circ>joint_distribution X Y) \<rparr>"
```
```   498   { fix A assume "A \<in> sets XY.P"
```
```   499     then have "ereal (joint_distribution X Y A) = XY.\<mu> A"
```
```   500       using indep_var_distributionD[OF indep]
```
```   501       by (simp add: XY.P.finite_measure_eq) }
```
```   502   note j_eq = this
```
```   503
```
```   504   interpret J: prob_space ?J
```
```   505     using j_eq by (intro XY.prob_space_cong) auto
```
```   506
```
```   507   have ac: "XY.absolutely_continuous (ereal\<circ>joint_distribution X Y)"
```
```   508     by (simp add: XY.absolutely_continuous_def j_eq)
```
```   509   then show "measure_space.absolutely_continuous P (ereal\<circ>joint_distribution X Y)"
```
```   510     unfolding P_def .
```
```   511
```
```   512   have ed: "entropy_density b XY.P (ereal\<circ>joint_distribution X Y) \<in> borel_measurable XY.P"
```
```   513     by (rule XY.measurable_entropy_density) (default | fact)+
```
```   514
```
```   515   have "AE x in XY.P. 1 = RN_deriv XY.P (ereal\<circ>joint_distribution X Y) x"
```
```   516   proof (rule XY.RN_deriv_unique[OF _ ac])
```
```   517     show "measure_space ?J" by default
```
```   518     fix A assume "A \<in> sets XY.P"
```
```   519     then show "(ereal\<circ>joint_distribution X Y) A = (\<integral>\<^isup>+ x. 1 * indicator A x \<partial>XY.P)"
```
```   520       by (simp add: j_eq)
```
```   521   qed (insert XY.measurable_const[of 1 borel], auto)
```
```   522   then have ae_XY: "AE x in XY.P. entropy_density b XY.P (ereal\<circ>joint_distribution X Y) x = 0"
```
```   523     by (elim XY.AE_mp) (simp add: entropy_density_def)
```
```   524   have ae_J: "AE x in ?J. entropy_density b XY.P (ereal\<circ>joint_distribution X Y) x = 0"
```
```   525   proof (rule XY.absolutely_continuous_AE)
```
```   526     show "measure_space ?J" by default
```
```   527     show "XY.absolutely_continuous (measure ?J)"
```
```   528       using ac by simp
```
```   529   qed (insert ae_XY, simp_all)
```
```   530   then show "integrable (P\<lparr>measure := (ereal\<circ>joint_distribution X Y)\<rparr>)
```
```   531         (entropy_density b P (ereal\<circ>joint_distribution X Y))"
```
```   532     unfolding P_def
```
```   533     using ed XY.measurable_const[of 0 borel]
```
```   534     by (subst J.integrable_cong_AE) auto
```
```   535
```
```   536   show "mutual_information b S T X Y = 0"
```
```   537     unfolding mutual_information_def KL_divergence_def P_def
```
```   538     by (subst J.integral_cong_AE[OF ae_J]) simp
```
```   539 next
```
```   540   assume "sigma_algebra S" "X \<in> measurable M S" "sigma_algebra T" "Y \<in> measurable M T"
```
```   541   then have rvs: "random_variable S X" "random_variable T Y" by blast+
```
```   542
```
```   543   interpret X: prob_space "S\<lparr>measure := ereal\<circ>distribution X\<rparr>"
```
```   544     by (rule distribution_prob_space) fact
```
```   545   interpret Y: prob_space "T\<lparr>measure := ereal\<circ>distribution Y\<rparr>"
```
```   546     by (rule distribution_prob_space) fact
```
```   547   interpret XY: pair_prob_space "S\<lparr>measure := ereal\<circ>distribution X\<rparr>" "T\<lparr>measure := ereal\<circ>distribution Y\<rparr>" by default
```
```   548   interpret XY: information_space XY.P b by default (rule b_gt_1)
```
```   549
```
```   550   let ?J = "XY.P\<lparr> measure := (ereal\<circ>joint_distribution X Y) \<rparr>"
```
```   551   interpret J: prob_space ?J
```
```   552     using rvs by (intro joint_distribution_prob_space) auto
```
```   553
```
```   554   assume ac: "measure_space.absolutely_continuous P (ereal\<circ>joint_distribution X Y)"
```
```   555   assume int: "integrable (P\<lparr>measure := (ereal\<circ>joint_distribution X Y)\<rparr>)
```
```   556         (entropy_density b P (ereal\<circ>joint_distribution X Y))"
```
```   557   assume I_eq_0: "mutual_information b S T X Y = 0"
```
```   558
```
```   559   have eq: "\<forall>A\<in>sets XY.P. (ereal \<circ> joint_distribution X Y) A = XY.\<mu> A"
```
```   560   proof (rule XY.KL_eq_0_imp)
```
```   561     show "prob_space ?J" by default
```
```   562     show "XY.absolutely_continuous (ereal\<circ>joint_distribution X Y)"
```
```   563       using ac by (simp add: P_def)
```
```   564     show "integrable ?J (entropy_density b XY.P (ereal\<circ>joint_distribution X Y))"
```
```   565       using int by (simp add: P_def)
```
```   566     show "KL_divergence b XY.P (ereal\<circ>joint_distribution X Y) = 0"
```
```   567       using I_eq_0 unfolding mutual_information_def by (simp add: P_def)
```
```   568   qed
```
```   569
```
```   570   { fix S X assume "sigma_algebra S"
```
```   571     interpret S: sigma_algebra S by fact
```
```   572     have "Int_stable \<lparr>space = space M, sets = {X -` A \<inter> space M |A. A \<in> sets S}\<rparr>"
```
```   573     proof (safe intro!: Int_stableI)
```
```   574       fix A B assume "A \<in> sets S" "B \<in> sets S"
```
```   575       then show "\<exists>C. (X -` A \<inter> space M) \<inter> (X -` B \<inter> space M) = (X -` C \<inter> space M) \<and> C \<in> sets S"
```
```   576         by (intro exI[of _ "A \<inter> B"]) auto
```
```   577     qed }
```
```   578   note Int_stable = this
```
```   579
```
```   580   show "indep_var S X T Y" unfolding indep_var_eq
```
```   581   proof (intro conjI indep_set_sigma_sets Int_stable)
```
```   582     show "indep_set {X -` A \<inter> space M |A. A \<in> sets S} {Y -` A \<inter> space M |A. A \<in> sets T}"
```
```   583     proof (safe intro!: indep_setI)
```
```   584       { fix A assume "A \<in> sets S" then show "X -` A \<inter> space M \<in> sets M"
```
```   585         using `X \<in> measurable M S` by (auto intro: measurable_sets) }
```
```   586       { fix A assume "A \<in> sets T" then show "Y -` A \<inter> space M \<in> sets M"
```
```   587         using `Y \<in> measurable M T` by (auto intro: measurable_sets) }
```
```   588     next
```
```   589       fix A B assume ab: "A \<in> sets S" "B \<in> sets T"
```
```   590       have "ereal (prob ((X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M))) =
```
```   591         ereal (joint_distribution X Y (A \<times> B))"
```
```   592         unfolding distribution_def
```
```   593         by (intro arg_cong[where f="\<lambda>C. ereal (prob C)"]) auto
```
```   594       also have "\<dots> = XY.\<mu> (A \<times> B)"
```
```   595         using ab eq by (auto simp: XY.finite_measure_eq)
```
```   596       also have "\<dots> = ereal (distribution X A) * ereal (distribution Y B)"
```
```   597         using ab by (simp add: XY.pair_measure_times)
```
```   598       finally show "prob ((X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M)) =
```
```   599         prob (X -` A \<inter> space M) * prob (Y -` B \<inter> space M)"
```
```   600         unfolding distribution_def by simp
```
```   601     qed
```
```   602   qed fact+
```
```   603 qed
```
```   604
```
```   605 lemma (in information_space) mutual_information_commute_generic:
```
```   606   assumes X: "random_variable S X" and Y: "random_variable T Y"
```
```   607   assumes ac: "measure_space.absolutely_continuous
```
```   608     (S\<lparr>measure := ereal\<circ>distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := ereal\<circ>distribution Y\<rparr>) (ereal\<circ>joint_distribution X Y)"
```
```   609   shows "mutual_information b S T X Y = mutual_information b T S Y X"
```
```   610 proof -
```
```   611   let ?S = "S\<lparr>measure := ereal\<circ>distribution X\<rparr>" and ?T = "T\<lparr>measure := ereal\<circ>distribution Y\<rparr>"
```
```   612   interpret S: prob_space ?S using X by (rule distribution_prob_space)
```
```   613   interpret T: prob_space ?T using Y by (rule distribution_prob_space)
```
```   614   interpret P: pair_prob_space ?S ?T ..
```
```   615   interpret Q: pair_prob_space ?T ?S ..
```
```   616   show ?thesis
```
```   617     unfolding mutual_information_def
```
```   618   proof (intro Q.KL_divergence_vimage[OF Q.measure_preserving_swap _ _ _ _ ac b_gt_1])
```
```   619     show "(\<lambda>(x,y). (y,x)) \<in> measure_preserving
```
```   620       (P.P \<lparr> measure := ereal\<circ>joint_distribution X Y\<rparr>) (Q.P \<lparr> measure := ereal\<circ>joint_distribution Y X\<rparr>)"
```
```   621       using X Y unfolding measurable_def
```
```   622       unfolding measure_preserving_def using P.pair_sigma_algebra_swap_measurable
```
```   623       by (auto simp add: space_pair_measure distribution_def intro!: arg_cong[where f=\<mu>'])
```
```   624     have "prob_space (P.P\<lparr> measure := ereal\<circ>joint_distribution X Y\<rparr>)"
```
```   625       using X Y by (auto intro!: distribution_prob_space random_variable_pairI)
```
```   626     then show "measure_space (P.P\<lparr> measure := ereal\<circ>joint_distribution X Y\<rparr>)"
```
```   627       unfolding prob_space_def by simp
```
```   628   qed auto
```
```   629 qed
```
```   630
```
```   631 definition (in prob_space)
```
```   632   "entropy b s X = mutual_information b s s X X"
```
```   633
```
```   634 abbreviation (in information_space)
```
```   635   mutual_information_Pow ("\<I>'(_ ; _')") where
```
```   636   "\<I>(X ; Y) \<equiv> mutual_information b
```
```   637     \<lparr> space = X`space M, sets = Pow (X`space M), measure = ereal\<circ>distribution X \<rparr>
```
```   638     \<lparr> space = Y`space M, sets = Pow (Y`space M), measure = ereal\<circ>distribution Y \<rparr> X Y"
```
```   639
```
```   640 lemma (in prob_space) finite_variables_absolutely_continuous:
```
```   641   assumes X: "finite_random_variable S X" and Y: "finite_random_variable T Y"
```
```   642   shows "measure_space.absolutely_continuous
```
```   643     (S\<lparr>measure := ereal\<circ>distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := ereal\<circ>distribution Y\<rparr>)
```
```   644     (ereal\<circ>joint_distribution X Y)"
```
```   645 proof -
```
```   646   interpret X: finite_prob_space "S\<lparr>measure := ereal\<circ>distribution X\<rparr>"
```
```   647     using X by (rule distribution_finite_prob_space)
```
```   648   interpret Y: finite_prob_space "T\<lparr>measure := ereal\<circ>distribution Y\<rparr>"
```
```   649     using Y by (rule distribution_finite_prob_space)
```
```   650   interpret XY: pair_finite_prob_space
```
```   651     "S\<lparr>measure := ereal\<circ>distribution X\<rparr>" "T\<lparr> measure := ereal\<circ>distribution Y\<rparr>" by default
```
```   652   interpret P: finite_prob_space "XY.P\<lparr> measure := ereal\<circ>joint_distribution X Y\<rparr>"
```
```   653     using assms by (auto intro!: joint_distribution_finite_prob_space)
```
```   654   note rv = assms[THEN finite_random_variableD]
```
```   655   show "XY.absolutely_continuous (ereal\<circ>joint_distribution X Y)"
```
```   656   proof (rule XY.absolutely_continuousI)
```
```   657     show "finite_measure_space (XY.P\<lparr> measure := ereal\<circ>joint_distribution X Y\<rparr>)" by default
```
```   658     fix x assume "x \<in> space XY.P" and "XY.\<mu> {x} = 0"
```
```   659     then obtain a b where "x = (a, b)"
```
```   660       and "distribution X {a} = 0 \<or> distribution Y {b} = 0"
```
```   661       by (cases x) (auto simp: space_pair_measure)
```
```   662     with finite_distribution_order(5,6)[OF X Y]
```
```   663     show "(ereal \<circ> joint_distribution X Y) {x} = 0" by auto
```
```   664   qed
```
```   665 qed
```
```   666
```
```   667 lemma (in information_space)
```
```   668   assumes MX: "finite_random_variable MX X"
```
```   669   assumes MY: "finite_random_variable MY Y"
```
```   670   shows mutual_information_generic_eq:
```
```   671     "mutual_information b MX MY X Y = (\<Sum> (x,y) \<in> space MX \<times> space MY.
```
```   672       joint_distribution X Y {(x,y)} *
```
```   673       log b (joint_distribution X Y {(x,y)} /
```
```   674       (distribution X {x} * distribution Y {y})))"
```
```   675     (is ?sum)
```
```   676   and mutual_information_positive_generic:
```
```   677      "0 \<le> mutual_information b MX MY X Y" (is ?positive)
```
```   678 proof -
```
```   679   interpret X: finite_prob_space "MX\<lparr>measure := ereal\<circ>distribution X\<rparr>"
```
```   680     using MX by (rule distribution_finite_prob_space)
```
```   681   interpret Y: finite_prob_space "MY\<lparr>measure := ereal\<circ>distribution Y\<rparr>"
```
```   682     using MY by (rule distribution_finite_prob_space)
```
```   683   interpret XY: pair_finite_prob_space "MX\<lparr>measure := ereal\<circ>distribution X\<rparr>" "MY\<lparr>measure := ereal\<circ>distribution Y\<rparr>" by default
```
```   684   interpret P: finite_prob_space "XY.P\<lparr>measure := ereal\<circ>joint_distribution X Y\<rparr>"
```
```   685     using assms by (auto intro!: joint_distribution_finite_prob_space)
```
```   686
```
```   687   have P_ms: "finite_measure_space (XY.P\<lparr>measure := ereal\<circ>joint_distribution X Y\<rparr>)" by default
```
```   688   have P_ps: "finite_prob_space (XY.P\<lparr>measure := ereal\<circ>joint_distribution X Y\<rparr>)" by default
```
```   689
```
```   690   show ?sum
```
```   691     unfolding Let_def mutual_information_def
```
```   692     by (subst XY.KL_divergence_eq_finite[OF P_ms finite_variables_absolutely_continuous[OF MX MY]])
```
```   693        (auto simp add: space_pair_measure setsum_cartesian_product')
```
```   694
```
```   695   show ?positive
```
```   696     using XY.KL_divergence_positive_finite[OF P_ps finite_variables_absolutely_continuous[OF MX MY] b_gt_1]
```
```   697     unfolding mutual_information_def .
```
```   698 qed
```
```   699
```
```   700 lemma (in information_space) mutual_information_commute:
```
```   701   assumes X: "finite_random_variable S X" and Y: "finite_random_variable T Y"
```
```   702   shows "mutual_information b S T X Y = mutual_information b T S Y X"
```
```   703   unfolding mutual_information_generic_eq[OF X Y] mutual_information_generic_eq[OF Y X]
```
```   704   unfolding joint_distribution_commute_singleton[of X Y]
```
```   705   by (auto simp add: ac_simps intro!: setsum_reindex_cong[OF swap_inj_on])
```
```   706
```
```   707 lemma (in information_space) mutual_information_commute_simple:
```
```   708   assumes X: "simple_function M X" and Y: "simple_function M Y"
```
```   709   shows "\<I>(X;Y) = \<I>(Y;X)"
```
```   710   by (intro mutual_information_commute X Y simple_function_imp_finite_random_variable)
```
```   711
```
```   712 lemma (in information_space) mutual_information_eq:
```
```   713   assumes "simple_function M X" "simple_function M Y"
```
```   714   shows "\<I>(X;Y) = (\<Sum> (x,y) \<in> X ` space M \<times> Y ` space M.
```
```   715     distribution (\<lambda>x. (X x, Y x)) {(x,y)} * log b (distribution (\<lambda>x. (X x, Y x)) {(x,y)} /
```
```   716                                                    (distribution X {x} * distribution Y {y})))"
```
```   717   using assms by (simp add: mutual_information_generic_eq)
```
```   718
```
```   719 lemma (in information_space) mutual_information_generic_cong:
```
```   720   assumes X: "\<And>x. x \<in> space M \<Longrightarrow> X x = X' x"
```
```   721   assumes Y: "\<And>x. x \<in> space M \<Longrightarrow> Y x = Y' x"
```
```   722   shows "mutual_information b MX MY X Y = mutual_information b MX MY X' Y'"
```
```   723   unfolding mutual_information_def using X Y
```
```   724   by (simp cong: distribution_cong)
```
```   725
```
```   726 lemma (in information_space) mutual_information_cong:
```
```   727   assumes X: "\<And>x. x \<in> space M \<Longrightarrow> X x = X' x"
```
```   728   assumes Y: "\<And>x. x \<in> space M \<Longrightarrow> Y x = Y' x"
```
```   729   shows "\<I>(X; Y) = \<I>(X'; Y')"
```
```   730   unfolding mutual_information_def using X Y
```
```   731   by (simp cong: distribution_cong image_cong)
```
```   732
```
```   733 lemma (in information_space) mutual_information_positive:
```
```   734   assumes "simple_function M X" "simple_function M Y"
```
```   735   shows "0 \<le> \<I>(X;Y)"
```
```   736   using assms by (simp add: mutual_information_positive_generic)
```
```   737
```
```   738 subsection {* Entropy *}
```
```   739
```
```   740 abbreviation (in information_space)
```
```   741   entropy_Pow ("\<H>'(_')") where
```
```   742   "\<H>(X) \<equiv> entropy b \<lparr> space = X`space M, sets = Pow (X`space M), measure = ereal\<circ>distribution X \<rparr> X"
```
```   743
```
```   744 lemma (in information_space) entropy_generic_eq:
```
```   745   fixes X :: "'a \<Rightarrow> 'c"
```
```   746   assumes MX: "finite_random_variable MX X"
```
```   747   shows "entropy b MX X = -(\<Sum> x \<in> space MX. distribution X {x} * log b (distribution X {x}))"
```
```   748 proof -
```
```   749   interpret MX: finite_prob_space "MX\<lparr>measure := ereal\<circ>distribution X\<rparr>"
```
```   750     using MX by (rule distribution_finite_prob_space)
```
```   751   let "?X x" = "distribution X {x}"
```
```   752   let "?XX x y" = "joint_distribution X X {(x, y)}"
```
```   753
```
```   754   { fix x y :: 'c
```
```   755     { assume "x \<noteq> y"
```
```   756       then have "(\<lambda>x. (X x, X x)) -` {(x,y)} \<inter> space M = {}" by auto
```
```   757       then have "joint_distribution X X {(x, y)} = 0" by (simp add: distribution_def) }
```
```   758     then have "?XX x y * log b (?XX x y / (?X x * ?X y)) =
```
```   759         (if x = y then - ?X y * log b (?X y) else 0)"
```
```   760       by (auto simp: log_simps zero_less_mult_iff) }
```
```   761   note remove_XX = this
```
```   762
```
```   763   show ?thesis
```
```   764     unfolding entropy_def mutual_information_generic_eq[OF MX MX]
```
```   765     unfolding setsum_cartesian_product[symmetric] setsum_negf[symmetric] remove_XX
```
```   766     using MX.finite_space by (auto simp: setsum_cases)
```
```   767 qed
```
```   768
```
```   769 lemma (in information_space) entropy_eq:
```
```   770   assumes "simple_function M X"
```
```   771   shows "\<H>(X) = -(\<Sum> x \<in> X ` space M. distribution X {x} * log b (distribution X {x}))"
```
```   772   using assms by (simp add: entropy_generic_eq)
```
```   773
```
```   774 lemma (in information_space) entropy_positive:
```
```   775   "simple_function M X \<Longrightarrow> 0 \<le> \<H>(X)"
```
```   776   unfolding entropy_def by (simp add: mutual_information_positive)
```
```   777
```
```   778 lemma (in information_space) entropy_certainty_eq_0:
```
```   779   assumes X: "simple_function M X" and "x \<in> X ` space M" and "distribution X {x} = 1"
```
```   780   shows "\<H>(X) = 0"
```
```   781 proof -
```
```   782   let ?X = "\<lparr> space = X ` space M, sets = Pow (X ` space M), measure = ereal\<circ>distribution X\<rparr>"
```
```   783   note simple_function_imp_finite_random_variable[OF `simple_function M X`]
```
```   784   from distribution_finite_prob_space[OF this, of "\<lparr> measure = ereal\<circ>distribution X \<rparr>"]
```
```   785   interpret X: finite_prob_space ?X by simp
```
```   786   have "distribution X (X ` space M - {x}) = distribution X (X ` space M) - distribution X {x}"
```
```   787     using X.measure_compl[of "{x}"] assms by auto
```
```   788   also have "\<dots> = 0" using X.prob_space assms by auto
```
```   789   finally have X0: "distribution X (X ` space M - {x}) = 0" by auto
```
```   790   { fix y assume *: "y \<in> X ` space M"
```
```   791     { assume asm: "y \<noteq> x"
```
```   792       with * have "{y} \<subseteq> X ` space M - {x}" by auto
```
```   793       from X.measure_mono[OF this] X0 asm *
```
```   794       have "distribution X {y} = 0"  by (auto intro: antisym) }
```
```   795     then have "distribution X {y} = (if x = y then 1 else 0)"
```
```   796       using assms by auto }
```
```   797   note fi = this
```
```   798   have y: "\<And>y. (if x = y then 1 else 0) * log b (if x = y then 1 else 0) = 0" by simp
```
```   799   show ?thesis unfolding entropy_eq[OF `simple_function M X`] by (auto simp: y fi)
```
```   800 qed
```
```   801
```
```   802 lemma (in information_space) entropy_le_card_not_0:
```
```   803   assumes X: "simple_function M X"
```
```   804   shows "\<H>(X) \<le> log b (card (X ` space M \<inter> {x. distribution X {x} \<noteq> 0}))"
```
```   805 proof -
```
```   806   let "?p x" = "distribution X {x}"
```
```   807   have "\<H>(X) = (\<Sum>x\<in>X`space M. ?p x * log b (1 / ?p x))"
```
```   808     unfolding entropy_eq[OF X] setsum_negf[symmetric]
```
```   809     by (auto intro!: setsum_cong simp: log_simps)
```
```   810   also have "\<dots> \<le> log b (\<Sum>x\<in>X`space M. ?p x * (1 / ?p x))"
```
```   811     using not_empty b_gt_1 `simple_function M X` sum_over_space_real_distribution[OF X]
```
```   812     by (intro log_setsum') (auto simp: simple_function_def)
```
```   813   also have "\<dots> = log b (\<Sum>x\<in>X`space M. if ?p x \<noteq> 0 then 1 else 0)"
```
```   814     by (intro arg_cong[where f="\<lambda>X. log b X"] setsum_cong) auto
```
```   815   finally show ?thesis
```
```   816     using `simple_function M X` by (auto simp: setsum_cases real_eq_of_nat simple_function_def)
```
```   817 qed
```
```   818
```
```   819 lemma (in prob_space) measure'_translate:
```
```   820   assumes X: "random_variable S X" and A: "A \<in> sets S"
```
```   821   shows "finite_measure.\<mu>' (S\<lparr> measure := ereal\<circ>distribution X \<rparr>) A = distribution X A"
```
```   822 proof -
```
```   823   interpret S: prob_space "S\<lparr> measure := ereal\<circ>distribution X \<rparr>"
```
```   824     using distribution_prob_space[OF X] .
```
```   825   from A show "S.\<mu>' A = distribution X A"
```
```   826     unfolding S.\<mu>'_def by (simp add: distribution_def_raw \<mu>'_def)
```
```   827 qed
```
```   828
```
```   829 lemma (in information_space) entropy_uniform_max:
```
```   830   assumes X: "simple_function M X"
```
```   831   assumes "\<And>x y. \<lbrakk> x \<in> X ` space M ; y \<in> X ` space M \<rbrakk> \<Longrightarrow> distribution X {x} = distribution X {y}"
```
```   832   shows "\<H>(X) = log b (real (card (X ` space M)))"
```
```   833 proof -
```
```   834   let ?X = "\<lparr> space = X ` space M, sets = Pow (X ` space M), measure = undefined\<rparr>\<lparr> measure := ereal\<circ>distribution X\<rparr>"
```
```   835   note frv = simple_function_imp_finite_random_variable[OF X]
```
```   836   from distribution_finite_prob_space[OF this, of "\<lparr> measure = ereal\<circ>distribution X \<rparr>"]
```
```   837   interpret X: finite_prob_space ?X by simp
```
```   838   note rv = finite_random_variableD[OF frv]
```
```   839   have card_gt0: "0 < card (X ` space M)" unfolding card_gt_0_iff
```
```   840     using `simple_function M X` not_empty by (auto simp: simple_function_def)
```
```   841   { fix x assume "x \<in> space ?X"
```
```   842     moreover then have "X.\<mu>' {x} = 1 / card (space ?X)"
```
```   843     proof (rule X.uniform_prob)
```
```   844       fix x y assume "x \<in> space ?X" "y \<in> space ?X"
```
```   845       with assms(2)[of x y] show "X.\<mu>' {x} = X.\<mu>' {y}"
```
```   846         by (subst (1 2) measure'_translate[OF rv]) auto
```
```   847     qed
```
```   848     ultimately have "distribution X {x} = 1 / card (space ?X)"
```
```   849       by (subst (asm) measure'_translate[OF rv]) auto }
```
```   850   thus ?thesis
```
```   851     using not_empty X.finite_space b_gt_1 card_gt0
```
```   852     by (simp add: entropy_eq[OF `simple_function M X`] real_eq_of_nat[symmetric] log_simps)
```
```   853 qed
```
```   854
```
```   855 lemma (in information_space) entropy_le_card:
```
```   856   assumes "simple_function M X"
```
```   857   shows "\<H>(X) \<le> log b (real (card (X ` space M)))"
```
```   858 proof cases
```
```   859   assume "X ` space M \<inter> {x. distribution X {x} \<noteq> 0} = {}"
```
```   860   then have "\<And>x. x\<in>X`space M \<Longrightarrow> distribution X {x} = 0" by auto
```
```   861   moreover
```
```   862   have "0 < card (X`space M)"
```
```   863     using `simple_function M X` not_empty
```
```   864     by (auto simp: card_gt_0_iff simple_function_def)
```
```   865   then have "log b 1 \<le> log b (real (card (X`space M)))"
```
```   866     using b_gt_1 by (intro log_le) auto
```
```   867   ultimately show ?thesis using assms by (simp add: entropy_eq)
```
```   868 next
```
```   869   assume False: "X ` space M \<inter> {x. distribution X {x} \<noteq> 0} \<noteq> {}"
```
```   870   have "card (X ` space M \<inter> {x. distribution X {x} \<noteq> 0}) \<le> card (X ` space M)"
```
```   871     (is "?A \<le> ?B") using assms not_empty by (auto intro!: card_mono simp: simple_function_def)
```
```   872   note entropy_le_card_not_0[OF assms]
```
```   873   also have "log b (real ?A) \<le> log b (real ?B)"
```
```   874     using b_gt_1 False not_empty `?A \<le> ?B` assms
```
```   875     by (auto intro!: log_le simp: card_gt_0_iff simp: simple_function_def)
```
```   876   finally show ?thesis .
```
```   877 qed
```
```   878
```
```   879 lemma (in information_space) entropy_commute:
```
```   880   assumes "simple_function M X" "simple_function M Y"
```
```   881   shows "\<H>(\<lambda>x. (X x, Y x)) = \<H>(\<lambda>x. (Y x, X x))"
```
```   882 proof -
```
```   883   have sf: "simple_function M (\<lambda>x. (X x, Y x))" "simple_function M (\<lambda>x. (Y x, X x))"
```
```   884     using assms by (auto intro: simple_function_Pair)
```
```   885   have *: "(\<lambda>x. (Y x, X x))`space M = (\<lambda>(a,b). (b,a))`(\<lambda>x. (X x, Y x))`space M"
```
```   886     by auto
```
```   887   have inj: "\<And>X. inj_on (\<lambda>(a,b). (b,a)) X"
```
```   888     by (auto intro!: inj_onI)
```
```   889   show ?thesis
```
```   890     unfolding sf[THEN entropy_eq] unfolding * setsum_reindex[OF inj]
```
```   891     by (simp add: joint_distribution_commute[of Y X] split_beta)
```
```   892 qed
```
```   893
```
```   894 lemma (in information_space) entropy_eq_cartesian_product:
```
```   895   assumes "simple_function M X" "simple_function M Y"
```
```   896   shows "\<H>(\<lambda>x. (X x, Y x)) = -(\<Sum>x\<in>X`space M. \<Sum>y\<in>Y`space M.
```
```   897     joint_distribution X Y {(x,y)} * log b (joint_distribution X Y {(x,y)}))"
```
```   898 proof -
```
```   899   have sf: "simple_function M (\<lambda>x. (X x, Y x))"
```
```   900     using assms by (auto intro: simple_function_Pair)
```
```   901   { fix x assume "x\<notin>(\<lambda>x. (X x, Y x))`space M"
```
```   902     then have "(\<lambda>x. (X x, Y x)) -` {x} \<inter> space M = {}" by auto
```
```   903     then have "joint_distribution X Y {x} = 0"
```
```   904       unfolding distribution_def by auto }
```
```   905   then show ?thesis using sf assms
```
```   906     unfolding entropy_eq[OF sf] neg_equal_iff_equal setsum_cartesian_product
```
```   907     by (auto intro!: setsum_mono_zero_cong_left simp: simple_function_def)
```
```   908 qed
```
```   909
```
```   910 subsection {* Conditional Mutual Information *}
```
```   911
```
```   912 definition (in prob_space)
```
```   913   "conditional_mutual_information b MX MY MZ X Y Z \<equiv>
```
```   914     mutual_information b MX (MY \<Otimes>\<^isub>M MZ) X (\<lambda>x. (Y x, Z x)) -
```
```   915     mutual_information b MX MZ X Z"
```
```   916
```
```   917 abbreviation (in information_space)
```
```   918   conditional_mutual_information_Pow ("\<I>'( _ ; _ | _ ')") where
```
```   919   "\<I>(X ; Y | Z) \<equiv> conditional_mutual_information b
```
```   920     \<lparr> space = X`space M, sets = Pow (X`space M), measure = ereal\<circ>distribution X \<rparr>
```
```   921     \<lparr> space = Y`space M, sets = Pow (Y`space M), measure = ereal\<circ>distribution Y \<rparr>
```
```   922     \<lparr> space = Z`space M, sets = Pow (Z`space M), measure = ereal\<circ>distribution Z \<rparr>
```
```   923     X Y Z"
```
```   924
```
```   925 lemma (in information_space) conditional_mutual_information_generic_eq:
```
```   926   assumes MX: "finite_random_variable MX X"
```
```   927     and MY: "finite_random_variable MY Y"
```
```   928     and MZ: "finite_random_variable MZ Z"
```
```   929   shows "conditional_mutual_information b MX MY MZ X Y Z = (\<Sum>(x, y, z) \<in> space MX \<times> space MY \<times> space MZ.
```
```   930              distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} *
```
```   931              log b (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} /
```
```   932     (joint_distribution X Z {(x, z)} * (joint_distribution Y Z {(y,z)} / distribution Z {z}))))"
```
```   933   (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))))")
```
```   934 proof -
```
```   935   let ?X = "\<lambda>x. distribution X {x}"
```
```   936   note finite_var = MX MY MZ
```
```   937   note YZ = finite_random_variable_pairI[OF finite_var(2,3)]
```
```   938   note XYZ = finite_random_variable_pairI[OF MX YZ]
```
```   939   note XZ = finite_random_variable_pairI[OF finite_var(1,3)]
```
```   940   note ZX = finite_random_variable_pairI[OF finite_var(3,1)]
```
```   941   note YZX = finite_random_variable_pairI[OF finite_var(2) ZX]
```
```   942   note order1 =
```
```   943     finite_distribution_order(5,6)[OF finite_var(1) YZ]
```
```   944     finite_distribution_order(5,6)[OF finite_var(1,3)]
```
```   945
```
```   946   note random_var = finite_var[THEN finite_random_variableD]
```
```   947   note finite = finite_var(1) YZ finite_var(3) XZ YZX
```
```   948
```
```   949   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>
```
```   950           \<Longrightarrow> joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)} = 0"
```
```   951     unfolding joint_distribution_commute_singleton[of X]
```
```   952     unfolding joint_distribution_assoc_singleton[symmetric]
```
```   953     using finite_distribution_order(6)[OF finite_var(2) ZX]
```
```   954     by auto
```
```   955
```
```   956   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)))) =
```
```   957     (\<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))))"
```
```   958     (is "(\<Sum>(x, y, z)\<in>?S. ?L x y z) = (\<Sum>(x, y, z)\<in>?S. ?R x y z)")
```
```   959   proof (safe intro!: setsum_cong)
```
```   960     fix x y z assume space: "x \<in> space MX" "y \<in> space MY" "z \<in> space MZ"
```
```   961     show "?L x y z = ?R x y z"
```
```   962     proof cases
```
```   963       assume "?XYZ x y z \<noteq> 0"
```
```   964       with space have "0 < ?X x" "0 < ?Z z" "0 < ?XZ x z" "0 < ?YZ y z" "0 < ?XYZ x y z"
```
```   965         using order1 order2 by (auto simp: less_le)
```
```   966       with b_gt_1 show ?thesis
```
```   967         by (simp add: log_mult log_divide zero_less_mult_iff zero_less_divide_iff)
```
```   968     qed simp
```
```   969   qed
```
```   970   also have "\<dots> = (\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?X x * ?YZ y z))) -
```
```   971                   (\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XZ x z / (?X x * ?Z z)))"
```
```   972     by (auto simp add: setsum_subtractf[symmetric] field_simps intro!: setsum_cong)
```
```   973   also have "(\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XZ x z / (?X x * ?Z z))) =
```
```   974              (\<Sum>(x, z)\<in>space MX \<times> space MZ. ?XZ x z * log b (?XZ x z / (?X x * ?Z z)))"
```
```   975     unfolding setsum_cartesian_product[symmetric] setsum_commute[of _ _ "space MY"]
```
```   976               setsum_left_distrib[symmetric]
```
```   977     unfolding joint_distribution_commute_singleton[of X]
```
```   978     unfolding joint_distribution_assoc_singleton[symmetric]
```
```   979     using setsum_joint_distribution_singleton[OF finite_var(2) ZX]
```
```   980     by (intro setsum_cong refl) (simp add: space_pair_measure)
```
```   981   also have "(\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?X x * ?YZ y z))) -
```
```   982              (\<Sum>(x, z)\<in>space MX \<times> space MZ. ?XZ x z * log b (?XZ x z / (?X x * ?Z z))) =
```
```   983              conditional_mutual_information b MX MY MZ X Y Z"
```
```   984     unfolding conditional_mutual_information_def
```
```   985     unfolding mutual_information_generic_eq[OF finite_var(1,3)]
```
```   986     unfolding mutual_information_generic_eq[OF finite_var(1) YZ]
```
```   987     by (simp add: space_sigma space_pair_measure setsum_cartesian_product')
```
```   988   finally show ?thesis by simp
```
```   989 qed
```
```   990
```
```   991 lemma (in information_space) conditional_mutual_information_eq:
```
```   992   assumes "simple_function M X" "simple_function M Y" "simple_function M Z"
```
```   993   shows "\<I>(X;Y|Z) = (\<Sum>(x, y, z) \<in> X`space M \<times> Y`space M \<times> Z`space M.
```
```   994              distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} *
```
```   995              log b (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} /
```
```   996     (joint_distribution X Z {(x, z)} * joint_distribution Y Z {(y,z)} / distribution Z {z})))"
```
```   997   by (subst conditional_mutual_information_generic_eq[OF assms[THEN simple_function_imp_finite_random_variable]])
```
```   998      simp
```
```   999
```
```  1000 lemma (in information_space) conditional_mutual_information_eq_mutual_information:
```
```  1001   assumes X: "simple_function M X" and Y: "simple_function M Y"
```
```  1002   shows "\<I>(X ; Y) = \<I>(X ; Y | (\<lambda>x. ()))"
```
```  1003 proof -
```
```  1004   have [simp]: "(\<lambda>x. ()) ` space M = {()}" using not_empty by auto
```
```  1005   have C: "simple_function M (\<lambda>x. ())" by auto
```
```  1006   show ?thesis
```
```  1007     unfolding conditional_mutual_information_eq[OF X Y C]
```
```  1008     unfolding mutual_information_eq[OF X Y]
```
```  1009     by (simp add: setsum_cartesian_product' distribution_remove_const)
```
```  1010 qed
```
```  1011
```
```  1012 lemma (in prob_space) distribution_unit[simp]: "distribution (\<lambda>x. ()) {()} = 1"
```
```  1013   unfolding distribution_def using prob_space by auto
```
```  1014
```
```  1015 lemma (in prob_space) joint_distribution_unit[simp]: "distribution (\<lambda>x. (X x, ())) {(a, ())} = distribution X {a}"
```
```  1016   unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
```
```  1017
```
```  1018 lemma (in prob_space) setsum_distribution:
```
```  1019   assumes X: "finite_random_variable MX X" shows "(\<Sum>a\<in>space MX. distribution X {a}) = 1"
```
```  1020   using setsum_joint_distribution[OF assms, of "\<lparr> space = UNIV, sets = Pow UNIV \<rparr>" "\<lambda>x. ()" "{()}"]
```
```  1021   using sigma_algebra_Pow[of "UNIV::unit set" "()"] by simp
```
```  1022
```
```  1023 lemma (in prob_space) setsum_real_distribution:
```
```  1024   fixes MX :: "('c, 'd) measure_space_scheme"
```
```  1025   assumes X: "finite_random_variable MX X" shows "(\<Sum>a\<in>space MX. distribution X {a}) = 1"
```
```  1026   using setsum_joint_distribution[OF assms, of "\<lparr> space = UNIV, sets = Pow UNIV, measure = undefined \<rparr>" "\<lambda>x. ()" "{()}"]
```
```  1027   using sigma_algebra_Pow[of "UNIV::unit set" "\<lparr> measure = undefined \<rparr>"]
```
```  1028   by auto
```
```  1029
```
```  1030 lemma (in information_space) conditional_mutual_information_generic_positive:
```
```  1031   assumes X: "finite_random_variable MX X" and Y: "finite_random_variable MY Y" and Z: "finite_random_variable MZ Z"
```
```  1032   shows "0 \<le> conditional_mutual_information b MX MY MZ X Y Z"
```
```  1033 proof (cases "space MX \<times> space MY \<times> space MZ = {}")
```
```  1034   case True show ?thesis
```
```  1035     unfolding conditional_mutual_information_generic_eq[OF assms] True
```
```  1036     by simp
```
```  1037 next
```
```  1038   case False
```
```  1039   let ?dXYZ = "distribution (\<lambda>x. (X x, Y x, Z x))"
```
```  1040   let ?dXZ = "joint_distribution X Z"
```
```  1041   let ?dYZ = "joint_distribution Y Z"
```
```  1042   let ?dX = "distribution X"
```
```  1043   let ?dZ = "distribution Z"
```
```  1044   let ?M = "space MX \<times> space MY \<times> space MZ"
```
```  1045
```
```  1046   note YZ = finite_random_variable_pairI[OF Y Z]
```
```  1047   note XZ = finite_random_variable_pairI[OF X Z]
```
```  1048   note ZX = finite_random_variable_pairI[OF Z X]
```
```  1049   note YZ = finite_random_variable_pairI[OF Y Z]
```
```  1050   note XYZ = finite_random_variable_pairI[OF X YZ]
```
```  1051   note finite = Z YZ XZ XYZ
```
```  1052   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>
```
```  1053           \<Longrightarrow> joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)} = 0"
```
```  1054     unfolding joint_distribution_commute_singleton[of X]
```
```  1055     unfolding joint_distribution_assoc_singleton[symmetric]
```
```  1056     using finite_distribution_order(6)[OF Y ZX]
```
```  1057     by auto
```
```  1058
```
```  1059   note order = order
```
```  1060     finite_distribution_order(5,6)[OF X YZ]
```
```  1061     finite_distribution_order(5,6)[OF Y Z]
```
```  1062
```
```  1063   have "- conditional_mutual_information b MX MY MZ X Y Z = - (\<Sum>(x, y, z) \<in> ?M. ?dXYZ {(x, y, z)} *
```
```  1064     log b (?dXYZ {(x, y, z)} / (?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z})))"
```
```  1065     unfolding conditional_mutual_information_generic_eq[OF assms] neg_equal_iff_equal by auto
```
```  1066   also have "\<dots> \<le> log b (\<Sum>(x, y, z) \<in> ?M. ?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z})"
```
```  1067     unfolding split_beta'
```
```  1068   proof (rule log_setsum_divide)
```
```  1069     show "?M \<noteq> {}" using False by simp
```
```  1070     show "1 < b" using b_gt_1 .
```
```  1071
```
```  1072     show "finite ?M" using assms
```
```  1073       unfolding finite_sigma_algebra_def finite_sigma_algebra_axioms_def by auto
```
```  1074
```
```  1075     show "(\<Sum>x\<in>?M. ?dXYZ {(fst x, fst (snd x), snd (snd x))}) = 1"
```
```  1076       unfolding setsum_cartesian_product'
```
```  1077       unfolding setsum_commute[of _ "space MY"]
```
```  1078       unfolding setsum_commute[of _ "space MZ"]
```
```  1079       by (simp_all add: space_pair_measure
```
```  1080                         setsum_joint_distribution_singleton[OF X YZ]
```
```  1081                         setsum_joint_distribution_singleton[OF Y Z]
```
```  1082                         setsum_distribution[OF Z])
```
```  1083
```
```  1084     fix x assume "x \<in> ?M"
```
```  1085     let ?x = "(fst x, fst (snd x), snd (snd x))"
```
```  1086
```
```  1087     show "0 \<le> ?dXYZ {?x}"
```
```  1088       "0 \<le> ?dXZ {(fst x, snd (snd x))} * ?dYZ {(fst (snd x), snd (snd x))} / ?dZ {snd (snd x)}"
```
```  1089      by (simp_all add: mult_nonneg_nonneg divide_nonneg_nonneg)
```
```  1090
```
```  1091     assume *: "0 < ?dXYZ {?x}"
```
```  1092     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)}"
```
```  1093       by (cases x) (auto simp add: zero_le_mult_iff zero_le_divide_iff less_le)
```
```  1094   qed
```
```  1095   also have "(\<Sum>(x, y, z) \<in> ?M. ?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z}) = (\<Sum>z\<in>space MZ. ?dZ {z})"
```
```  1096     apply (simp add: setsum_cartesian_product')
```
```  1097     apply (subst setsum_commute)
```
```  1098     apply (subst (2) setsum_commute)
```
```  1099     by (auto simp: setsum_divide_distrib[symmetric] setsum_product[symmetric]
```
```  1100                    setsum_joint_distribution_singleton[OF X Z]
```
```  1101                    setsum_joint_distribution_singleton[OF Y Z]
```
```  1102           intro!: setsum_cong)
```
```  1103   also have "log b (\<Sum>z\<in>space MZ. ?dZ {z}) = 0"
```
```  1104     unfolding setsum_real_distribution[OF Z] by simp
```
```  1105   finally show ?thesis by simp
```
```  1106 qed
```
```  1107
```
```  1108 lemma (in information_space) conditional_mutual_information_positive:
```
```  1109   assumes "simple_function M X" and "simple_function M Y" and "simple_function M Z"
```
```  1110   shows "0 \<le> \<I>(X;Y|Z)"
```
```  1111   by (rule conditional_mutual_information_generic_positive[OF assms[THEN simple_function_imp_finite_random_variable]])
```
```  1112
```
```  1113 subsection {* Conditional Entropy *}
```
```  1114
```
```  1115 definition (in prob_space)
```
```  1116   "conditional_entropy b S T X Y = conditional_mutual_information b S S T X X Y"
```
```  1117
```
```  1118 abbreviation (in information_space)
```
```  1119   conditional_entropy_Pow ("\<H>'(_ | _')") where
```
```  1120   "\<H>(X | Y) \<equiv> conditional_entropy b
```
```  1121     \<lparr> space = X`space M, sets = Pow (X`space M), measure = ereal\<circ>distribution X \<rparr>
```
```  1122     \<lparr> space = Y`space M, sets = Pow (Y`space M), measure = ereal\<circ>distribution Y \<rparr> X Y"
```
```  1123
```
```  1124 lemma (in information_space) conditional_entropy_positive:
```
```  1125   "simple_function M X \<Longrightarrow> simple_function M Y \<Longrightarrow> 0 \<le> \<H>(X | Y)"
```
```  1126   unfolding conditional_entropy_def by (auto intro!: conditional_mutual_information_positive)
```
```  1127
```
```  1128 lemma (in information_space) conditional_entropy_generic_eq:
```
```  1129   fixes MX :: "('c, 'd) measure_space_scheme" and MY :: "('e, 'f) measure_space_scheme"
```
```  1130   assumes MX: "finite_random_variable MX X"
```
```  1131   assumes MZ: "finite_random_variable MZ Z"
```
```  1132   shows "conditional_entropy b MX MZ X Z =
```
```  1133      - (\<Sum>(x, z)\<in>space MX \<times> space MZ.
```
```  1134          joint_distribution X Z {(x, z)} * log b (joint_distribution X Z {(x, z)} / distribution Z {z}))"
```
```  1135 proof -
```
```  1136   interpret MX: finite_sigma_algebra MX using MX by simp
```
```  1137   interpret MZ: finite_sigma_algebra MZ using MZ by simp
```
```  1138   let "?XXZ x y z" = "joint_distribution X (\<lambda>x. (X x, Z x)) {(x, y, z)}"
```
```  1139   let "?XZ x z" = "joint_distribution X Z {(x, z)}"
```
```  1140   let "?Z z" = "distribution Z {z}"
```
```  1141   let "?f x y z" = "log b (?XXZ x y z * ?Z z / (?XZ x z * ?XZ y z))"
```
```  1142   { fix x z have "?XXZ x x z = ?XZ x z"
```
```  1143       unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>']) }
```
```  1144   note this[simp]
```
```  1145   { fix x x' :: 'c and z assume "x' \<noteq> x"
```
```  1146     then have "?XXZ x x' z = 0"
```
```  1147       by (auto simp: distribution_def empty_measure'[symmetric]
```
```  1148                simp del: empty_measure' intro!: arg_cong[where f=\<mu>']) }
```
```  1149   note this[simp]
```
```  1150   { fix x x' z assume *: "x \<in> space MX" "z \<in> space MZ"
```
```  1151     then have "(\<Sum>x'\<in>space MX. ?XXZ x x' z * ?f x x' z)
```
```  1152       = (\<Sum>x'\<in>space MX. if x = x' then ?XZ x z * ?f x x z else 0)"
```
```  1153       by (auto intro!: setsum_cong)
```
```  1154     also have "\<dots> = ?XZ x z * ?f x x z"
```
```  1155       using `x \<in> space MX` by (simp add: setsum_cases[OF MX.finite_space])
```
```  1156     also have "\<dots> = ?XZ x z * log b (?Z z / ?XZ x z)" by auto
```
```  1157     also have "\<dots> = - ?XZ x z * log b (?XZ x z / ?Z z)"
```
```  1158       using finite_distribution_order(6)[OF MX MZ]
```
```  1159       by (auto simp: log_simps field_simps zero_less_mult_iff)
```
```  1160     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)" . }
```
```  1161   note * = this
```
```  1162   show ?thesis
```
```  1163     unfolding conditional_entropy_def
```
```  1164     unfolding conditional_mutual_information_generic_eq[OF MX MX MZ]
```
```  1165     by (auto simp: setsum_cartesian_product' setsum_negf[symmetric]
```
```  1166                    setsum_commute[of _ "space MZ"] *
```
```  1167              intro!: setsum_cong)
```
```  1168 qed
```
```  1169
```
```  1170 lemma (in information_space) conditional_entropy_eq:
```
```  1171   assumes "simple_function M X" "simple_function M Z"
```
```  1172   shows "\<H>(X | Z) =
```
```  1173      - (\<Sum>(x, z)\<in>X ` space M \<times> Z ` space M.
```
```  1174          joint_distribution X Z {(x, z)} *
```
```  1175          log b (joint_distribution X Z {(x, z)} / distribution Z {z}))"
```
```  1176   by (subst conditional_entropy_generic_eq[OF assms[THEN simple_function_imp_finite_random_variable]])
```
```  1177      simp
```
```  1178
```
```  1179 lemma (in information_space) conditional_entropy_eq_ce_with_hypothesis:
```
```  1180   assumes X: "simple_function M X" and Y: "simple_function M Y"
```
```  1181   shows "\<H>(X | Y) =
```
```  1182     -(\<Sum>y\<in>Y`space M. distribution Y {y} *
```
```  1183       (\<Sum>x\<in>X`space M. joint_distribution X Y {(x,y)} / distribution Y {(y)} *
```
```  1184               log b (joint_distribution X Y {(x,y)} / distribution Y {(y)})))"
```
```  1185   unfolding conditional_entropy_eq[OF assms]
```
```  1186   using finite_distribution_order(5,6)[OF assms[THEN simple_function_imp_finite_random_variable]]
```
```  1187   by (auto simp: setsum_cartesian_product'  setsum_commute[of _ "Y`space M"] setsum_right_distrib
```
```  1188            intro!: setsum_cong)
```
```  1189
```
```  1190 lemma (in information_space) conditional_entropy_eq_cartesian_product:
```
```  1191   assumes "simple_function M X" "simple_function M Y"
```
```  1192   shows "\<H>(X | Y) = -(\<Sum>x\<in>X`space M. \<Sum>y\<in>Y`space M.
```
```  1193     joint_distribution X Y {(x,y)} *
```
```  1194     log b (joint_distribution X Y {(x,y)} / distribution Y {y}))"
```
```  1195   unfolding conditional_entropy_eq[OF assms]
```
```  1196   by (auto intro!: setsum_cong simp: setsum_cartesian_product')
```
```  1197
```
```  1198 subsection {* Equalities *}
```
```  1199
```
```  1200 lemma (in information_space) mutual_information_eq_entropy_conditional_entropy:
```
```  1201   assumes X: "simple_function M X" and Z: "simple_function M Z"
```
```  1202   shows  "\<I>(X ; Z) = \<H>(X) - \<H>(X | Z)"
```
```  1203 proof -
```
```  1204   let "?XZ x z" = "joint_distribution X Z {(x, z)}"
```
```  1205   let "?Z z" = "distribution Z {z}"
```
```  1206   let "?X x" = "distribution X {x}"
```
```  1207   note fX = X[THEN simple_function_imp_finite_random_variable]
```
```  1208   note fZ = Z[THEN simple_function_imp_finite_random_variable]
```
```  1209   note finite_distribution_order[OF fX fZ, simp]
```
```  1210   { fix x z assume "x \<in> X`space M" "z \<in> Z`space M"
```
```  1211     have "?XZ x z * log b (?XZ x z / (?X x * ?Z z)) =
```
```  1212           ?XZ x z * log b (?XZ x z / ?Z z) - ?XZ x z * log b (?X x)"
```
```  1213       by (auto simp: log_simps zero_le_mult_iff field_simps less_le) }
```
```  1214   note * = this
```
```  1215   show ?thesis
```
```  1216     unfolding entropy_eq[OF X] conditional_entropy_eq[OF X Z] mutual_information_eq[OF X Z]
```
```  1217     using setsum_joint_distribution_singleton[OF fZ fX, unfolded joint_distribution_commute_singleton[of Z X]]
```
```  1218     by (simp add: * setsum_cartesian_product' setsum_subtractf setsum_left_distrib[symmetric]
```
```  1219                      setsum_distribution)
```
```  1220 qed
```
```  1221
```
```  1222 lemma (in information_space) conditional_entropy_less_eq_entropy:
```
```  1223   assumes X: "simple_function M X" and Z: "simple_function M Z"
```
```  1224   shows "\<H>(X | Z) \<le> \<H>(X)"
```
```  1225 proof -
```
```  1226   have "\<I>(X ; Z) = \<H>(X) - \<H>(X | Z)" using mutual_information_eq_entropy_conditional_entropy[OF assms] .
```
```  1227   with mutual_information_positive[OF X Z] entropy_positive[OF X]
```
```  1228   show ?thesis by auto
```
```  1229 qed
```
```  1230
```
```  1231 lemma (in information_space) entropy_chain_rule:
```
```  1232   assumes X: "simple_function M X" and Y: "simple_function M Y"
```
```  1233   shows  "\<H>(\<lambda>x. (X x, Y x)) = \<H>(X) + \<H>(Y|X)"
```
```  1234 proof -
```
```  1235   let "?XY x y" = "joint_distribution X Y {(x, y)}"
```
```  1236   let "?Y y" = "distribution Y {y}"
```
```  1237   let "?X x" = "distribution X {x}"
```
```  1238   note fX = X[THEN simple_function_imp_finite_random_variable]
```
```  1239   note fY = Y[THEN simple_function_imp_finite_random_variable]
```
```  1240   note finite_distribution_order[OF fX fY, simp]
```
```  1241   { fix x y assume "x \<in> X`space M" "y \<in> Y`space M"
```
```  1242     have "?XY x y * log b (?XY x y / ?X x) =
```
```  1243           ?XY x y * log b (?XY x y) - ?XY x y * log b (?X x)"
```
```  1244       by (auto simp: log_simps zero_le_mult_iff field_simps less_le) }
```
```  1245   note * = this
```
```  1246   show ?thesis
```
```  1247     using setsum_joint_distribution_singleton[OF fY fX]
```
```  1248     unfolding entropy_eq[OF X] conditional_entropy_eq_cartesian_product[OF Y X] entropy_eq_cartesian_product[OF X Y]
```
```  1249     unfolding joint_distribution_commute_singleton[of Y X] setsum_commute[of _ "X`space M"]
```
```  1250     by (simp add: * setsum_subtractf setsum_left_distrib[symmetric])
```
```  1251 qed
```
```  1252
```
```  1253 section {* Partitioning *}
```
```  1254
```
```  1255 definition "subvimage A f g \<longleftrightarrow> (\<forall>x \<in> A. f -` {f x} \<inter> A \<subseteq> g -` {g x} \<inter> A)"
```
```  1256
```
```  1257 lemma subvimageI:
```
```  1258   assumes "\<And>x y. \<lbrakk> x \<in> A ; y \<in> A ; f x = f y \<rbrakk> \<Longrightarrow> g x = g y"
```
```  1259   shows "subvimage A f g"
```
```  1260   using assms unfolding subvimage_def by blast
```
```  1261
```
```  1262 lemma subvimageE[consumes 1]:
```
```  1263   assumes "subvimage A f g"
```
```  1264   obtains "\<And>x y. \<lbrakk> x \<in> A ; y \<in> A ; f x = f y \<rbrakk> \<Longrightarrow> g x = g y"
```
```  1265   using assms unfolding subvimage_def by blast
```
```  1266
```
```  1267 lemma subvimageD:
```
```  1268   "\<lbrakk> subvimage A f g ; x \<in> A ; y \<in> A ; f x = f y \<rbrakk> \<Longrightarrow> g x = g y"
```
```  1269   using assms unfolding subvimage_def by blast
```
```  1270
```
```  1271 lemma subvimage_subset:
```
```  1272   "\<lbrakk> subvimage B f g ; A \<subseteq> B \<rbrakk> \<Longrightarrow> subvimage A f g"
```
```  1273   unfolding subvimage_def by auto
```
```  1274
```
```  1275 lemma subvimage_idem[intro]: "subvimage A g g"
```
```  1276   by (safe intro!: subvimageI)
```
```  1277
```
```  1278 lemma subvimage_comp_finer[intro]:
```
```  1279   assumes svi: "subvimage A g h"
```
```  1280   shows "subvimage A g (f \<circ> h)"
```
```  1281 proof (rule subvimageI, simp)
```
```  1282   fix x y assume "x \<in> A" "y \<in> A" "g x = g y"
```
```  1283   from svi[THEN subvimageD, OF this]
```
```  1284   show "f (h x) = f (h y)" by simp
```
```  1285 qed
```
```  1286
```
```  1287 lemma subvimage_comp_gran:
```
```  1288   assumes svi: "subvimage A g h"
```
```  1289   assumes inj: "inj_on f (g ` A)"
```
```  1290   shows "subvimage A (f \<circ> g) h"
```
```  1291   by (rule subvimageI) (auto intro!: subvimageD[OF svi] simp: inj_on_iff[OF inj])
```
```  1292
```
```  1293 lemma subvimage_comp:
```
```  1294   assumes svi: "subvimage (f ` A) g h"
```
```  1295   shows "subvimage A (g \<circ> f) (h \<circ> f)"
```
```  1296   by (rule subvimageI) (auto intro!: svi[THEN subvimageD])
```
```  1297
```
```  1298 lemma subvimage_trans:
```
```  1299   assumes fg: "subvimage A f g"
```
```  1300   assumes gh: "subvimage A g h"
```
```  1301   shows "subvimage A f h"
```
```  1302   by (rule subvimageI) (auto intro!: fg[THEN subvimageD] gh[THEN subvimageD])
```
```  1303
```
```  1304 lemma subvimage_translator:
```
```  1305   assumes svi: "subvimage A f g"
```
```  1306   shows "\<exists>h. \<forall>x \<in> A. h (f x)  = g x"
```
```  1307 proof (safe intro!: exI[of _ "\<lambda>x. (THE z. z \<in> (g ` (f -` {x} \<inter> A)))"])
```
```  1308   fix x assume "x \<in> A"
```
```  1309   show "(THE x'. x' \<in> (g ` (f -` {f x} \<inter> A))) = g x"
```
```  1310     by (rule theI2[of _ "g x"])
```
```  1311       (insert `x \<in> A`, auto intro!: svi[THEN subvimageD])
```
```  1312 qed
```
```  1313
```
```  1314 lemma subvimage_translator_image:
```
```  1315   assumes svi: "subvimage A f g"
```
```  1316   shows "\<exists>h. h ` f ` A = g ` A"
```
```  1317 proof -
```
```  1318   from subvimage_translator[OF svi]
```
```  1319   obtain h where "\<And>x. x \<in> A \<Longrightarrow> h (f x) = g x" by auto
```
```  1320   thus ?thesis
```
```  1321     by (auto intro!: exI[of _ h]
```
```  1322       simp: image_compose[symmetric] comp_def cong: image_cong)
```
```  1323 qed
```
```  1324
```
```  1325 lemma subvimage_finite:
```
```  1326   assumes svi: "subvimage A f g" and fin: "finite (f`A)"
```
```  1327   shows "finite (g`A)"
```
```  1328 proof -
```
```  1329   from subvimage_translator_image[OF svi]
```
```  1330   obtain h where "g`A = h`f`A" by fastsimp
```
```  1331   with fin show "finite (g`A)" by simp
```
```  1332 qed
```
```  1333
```
```  1334 lemma subvimage_disj:
```
```  1335   assumes svi: "subvimage A f g"
```
```  1336   shows "f -` {x} \<inter> A \<subseteq> g -` {y} \<inter> A \<or>
```
```  1337       f -` {x} \<inter> g -` {y} \<inter> A = {}" (is "?sub \<or> ?dist")
```
```  1338 proof (rule disjCI)
```
```  1339   assume "\<not> ?dist"
```
```  1340   then obtain z where "z \<in> A" and "x = f z" and "y = g z" by auto
```
```  1341   thus "?sub" using svi unfolding subvimage_def by auto
```
```  1342 qed
```
```  1343
```
```  1344 lemma setsum_image_split:
```
```  1345   assumes svi: "subvimage A f g" and fin: "finite (f ` A)"
```
```  1346   shows "(\<Sum>x\<in>f`A. h x) = (\<Sum>y\<in>g`A. \<Sum>x\<in>f`(g -` {y} \<inter> A). h x)"
```
```  1347     (is "?lhs = ?rhs")
```
```  1348 proof -
```
```  1349   have "f ` A =
```
```  1350       snd ` (SIGMA x : g ` A. f ` (g -` {x} \<inter> A))"
```
```  1351       (is "_ = snd ` ?SIGMA")
```
```  1352     unfolding image_split_eq_Sigma[symmetric]
```
```  1353     by (simp add: image_compose[symmetric] comp_def)
```
```  1354   moreover
```
```  1355   have snd_inj: "inj_on snd ?SIGMA"
```
```  1356     unfolding image_split_eq_Sigma[symmetric]
```
```  1357     by (auto intro!: inj_onI subvimageD[OF svi])
```
```  1358   ultimately
```
```  1359   have "(\<Sum>x\<in>f`A. h x) = (\<Sum>(x,y)\<in>?SIGMA. h y)"
```
```  1360     by (auto simp: setsum_reindex intro: setsum_cong)
```
```  1361   also have "... = ?rhs"
```
```  1362     using subvimage_finite[OF svi fin] fin
```
```  1363     apply (subst setsum_Sigma[symmetric])
```
```  1364     by (auto intro!: finite_subset[of _ "f`A"])
```
```  1365   finally show ?thesis .
```
```  1366 qed
```
```  1367
```
```  1368 lemma (in information_space) entropy_partition:
```
```  1369   assumes sf: "simple_function M X" "simple_function M P"
```
```  1370   assumes svi: "subvimage (space M) X P"
```
```  1371   shows "\<H>(X) = \<H>(P) + \<H>(X|P)"
```
```  1372 proof -
```
```  1373   let "?XP x p" = "joint_distribution X P {(x, p)}"
```
```  1374   let "?X x" = "distribution X {x}"
```
```  1375   let "?P p" = "distribution P {p}"
```
```  1376   note fX = sf(1)[THEN simple_function_imp_finite_random_variable]
```
```  1377   note fP = sf(2)[THEN simple_function_imp_finite_random_variable]
```
```  1378   note finite_distribution_order[OF fX fP, simp]
```
```  1379   have "(\<Sum>x\<in>X ` space M. ?X x * log b (?X x)) =
```
```  1380     (\<Sum>y\<in>P `space M. \<Sum>x\<in>X ` space M. ?XP x y * log b (?XP x y))"
```
```  1381   proof (subst setsum_image_split[OF svi],
```
```  1382       safe intro!: setsum_mono_zero_cong_left imageI)
```
```  1383     show "finite (X ` space M)" "finite (X ` space M)" "finite (P ` space M)"
```
```  1384       using sf unfolding simple_function_def by auto
```
```  1385   next
```
```  1386     fix p x assume in_space: "p \<in> space M" "x \<in> space M"
```
```  1387     assume "?XP (X x) (P p) * log b (?XP (X x) (P p)) \<noteq> 0"
```
```  1388     hence "(\<lambda>x. (X x, P x)) -` {(X x, P p)} \<inter> space M \<noteq> {}" by (auto simp: distribution_def)
```
```  1389     with svi[unfolded subvimage_def, rule_format, OF `x \<in> space M`]
```
```  1390     show "x \<in> P -` {P p}" by auto
```
```  1391   next
```
```  1392     fix p x assume in_space: "p \<in> space M" "x \<in> space M"
```
```  1393     assume "P x = P p"
```
```  1394     from this[symmetric] svi[unfolded subvimage_def, rule_format, OF `x \<in> space M`]
```
```  1395     have "X -` {X x} \<inter> space M \<subseteq> P -` {P p} \<inter> space M"
```
```  1396       by auto
```
```  1397     hence "(\<lambda>x. (X x, P x)) -` {(X x, P p)} \<inter> space M = X -` {X x} \<inter> space M"
```
```  1398       by auto
```
```  1399     thus "?X (X x) * log b (?X (X x)) = ?XP (X x) (P p) * log b (?XP (X x) (P p))"
```
```  1400       by (auto simp: distribution_def)
```
```  1401   qed
```
```  1402   moreover have "\<And>x y. ?XP x y * log b (?XP x y / ?P y) =
```
```  1403       ?XP x y * log b (?XP x y) - ?XP x y * log b (?P y)"
```
```  1404     by (auto simp add: log_simps zero_less_mult_iff field_simps)
```
```  1405   ultimately show ?thesis
```
```  1406     unfolding sf[THEN entropy_eq] conditional_entropy_eq[OF sf]
```
```  1407     using setsum_joint_distribution_singleton[OF fX fP]
```
```  1408     by (simp add: setsum_cartesian_product' setsum_subtractf setsum_distribution
```
```  1409       setsum_left_distrib[symmetric] setsum_commute[where B="P`space M"])
```
```  1410 qed
```
```  1411
```
```  1412 corollary (in information_space) entropy_data_processing:
```
```  1413   assumes X: "simple_function M X" shows "\<H>(f \<circ> X) \<le> \<H>(X)"
```
```  1414 proof -
```
```  1415   note X
```
```  1416   moreover have fX: "simple_function M (f \<circ> X)" using X by auto
```
```  1417   moreover have "subvimage (space M) X (f \<circ> X)" by auto
```
```  1418   ultimately have "\<H>(X) = \<H>(f\<circ>X) + \<H>(X|f\<circ>X)" by (rule entropy_partition)
```
```  1419   then show "\<H>(f \<circ> X) \<le> \<H>(X)"
```
```  1420     by (auto intro: conditional_entropy_positive[OF X fX])
```
```  1421 qed
```
```  1422
```
```  1423 corollary (in information_space) entropy_of_inj:
```
```  1424   assumes X: "simple_function M X" and inj: "inj_on f (X`space M)"
```
```  1425   shows "\<H>(f \<circ> X) = \<H>(X)"
```
```  1426 proof (rule antisym)
```
```  1427   show "\<H>(f \<circ> X) \<le> \<H>(X)" using entropy_data_processing[OF X] .
```
```  1428 next
```
```  1429   have sf: "simple_function M (f \<circ> X)"
```
```  1430     using X by auto
```
```  1431   have "\<H>(X) = \<H>(the_inv_into (X`space M) f \<circ> (f \<circ> X))"
```
```  1432     by (auto intro!: mutual_information_cong simp: entropy_def the_inv_into_f_f[OF inj])
```
```  1433   also have "... \<le> \<H>(f \<circ> X)"
```
```  1434     using entropy_data_processing[OF sf] .
```
```  1435   finally show "\<H>(X) \<le> \<H>(f \<circ> X)" .
```
```  1436 qed
```
```  1437
```
```  1438 end
```