src/HOL/Probability/Measurable.thy
 author hoelzl Tue May 20 19:24:39 2014 +0200 (2014-05-20) changeset 57025 e7fd64f82876 parent 56993 e5366291d6aa child 58965 a62cdcc5344b permissions -rw-r--r--
```     1 (*  Title:      HOL/Probability/Measurable.thy
```
```     2     Author:     Johannes Hölzl <hoelzl@in.tum.de>
```
```     3 *)
```
```     4 theory Measurable
```
```     5   imports
```
```     6     Sigma_Algebra
```
```     7     "~~/src/HOL/Library/Order_Continuity"
```
```     8 begin
```
```     9
```
```    10 hide_const (open) Order_Continuity.continuous
```
```    11
```
```    12 subsection {* Measurability prover *}
```
```    13
```
```    14 lemma (in algebra) sets_Collect_finite_All:
```
```    15   assumes "\<And>i. i \<in> S \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M" "finite S"
```
```    16   shows "{x\<in>\<Omega>. \<forall>i\<in>S. P i x} \<in> M"
```
```    17 proof -
```
```    18   have "{x\<in>\<Omega>. \<forall>i\<in>S. P i x} = (if S = {} then \<Omega> else \<Inter>i\<in>S. {x\<in>\<Omega>. P i x})"
```
```    19     by auto
```
```    20   with assms show ?thesis by (auto intro!: sets_Collect_finite_All')
```
```    21 qed
```
```    22
```
```    23 abbreviation "pred M P \<equiv> P \<in> measurable M (count_space (UNIV::bool set))"
```
```    24
```
```    25 lemma pred_def: "pred M P \<longleftrightarrow> {x\<in>space M. P x} \<in> sets M"
```
```    26 proof
```
```    27   assume "pred M P"
```
```    28   then have "P -` {True} \<inter> space M \<in> sets M"
```
```    29     by (auto simp: measurable_count_space_eq2)
```
```    30   also have "P -` {True} \<inter> space M = {x\<in>space M. P x}" by auto
```
```    31   finally show "{x\<in>space M. P x} \<in> sets M" .
```
```    32 next
```
```    33   assume P: "{x\<in>space M. P x} \<in> sets M"
```
```    34   moreover
```
```    35   { fix X
```
```    36     have "X \<in> Pow (UNIV :: bool set)" by simp
```
```    37     then have "P -` X \<inter> space M = {x\<in>space M. ((X = {True} \<longrightarrow> P x) \<and> (X = {False} \<longrightarrow> \<not> P x) \<and> X \<noteq> {})}"
```
```    38       unfolding UNIV_bool Pow_insert Pow_empty by auto
```
```    39     then have "P -` X \<inter> space M \<in> sets M"
```
```    40       by (auto intro!: sets.sets_Collect_neg sets.sets_Collect_imp sets.sets_Collect_conj sets.sets_Collect_const P) }
```
```    41   then show "pred M P"
```
```    42     by (auto simp: measurable_def)
```
```    43 qed
```
```    44
```
```    45 lemma pred_sets1: "{x\<in>space M. P x} \<in> sets M \<Longrightarrow> f \<in> measurable N M \<Longrightarrow> pred N (\<lambda>x. P (f x))"
```
```    46   by (rule measurable_compose[where f=f and N=M]) (auto simp: pred_def)
```
```    47
```
```    48 lemma pred_sets2: "A \<in> sets N \<Longrightarrow> f \<in> measurable M N \<Longrightarrow> pred M (\<lambda>x. f x \<in> A)"
```
```    49   by (rule measurable_compose[where f=f and N=N]) (auto simp: pred_def Int_def[symmetric])
```
```    50
```
```    51 ML_file "measurable.ML"
```
```    52
```
```    53 attribute_setup measurable = {*
```
```    54   Scan.lift (Scan.optional (Args.parens (Scan.optional (Args.\$\$\$ "raw" >> K true) false --
```
```    55     Scan.optional (Args.\$\$\$ "generic" >> K Measurable.Generic) Measurable.Concrete))
```
```    56     (false, Measurable.Concrete) >> (Thm.declaration_attribute o Measurable.add_thm))
```
```    57 *} "declaration of measurability theorems"
```
```    58
```
```    59 attribute_setup measurable_dest = {*
```
```    60   Scan.lift (Scan.succeed (Thm.declaration_attribute Measurable.add_dest))
```
```    61 *} "add dest rule for measurability prover"
```
```    62
```
```    63 attribute_setup measurable_app = {*
```
```    64   Scan.lift (Scan.succeed (Thm.declaration_attribute Measurable.add_app))
```
```    65 *} "add application rule for measurability prover"
```
```    66
```
```    67 method_setup measurable = {*
```
```    68   Scan.lift (Scan.succeed (fn ctxt => METHOD (fn facts => Measurable.measurable_tac ctxt facts)))
```
```    69 *} "measurability prover"
```
```    70
```
```    71 simproc_setup measurable ("A \<in> sets M" | "f \<in> measurable M N") = {* K Measurable.simproc *}
```
```    72
```
```    73 declare
```
```    74   measurable_compose_rev[measurable_dest]
```
```    75   pred_sets1[measurable_dest]
```
```    76   pred_sets2[measurable_dest]
```
```    77   sets.sets_into_space[measurable_dest]
```
```    78
```
```    79 declare
```
```    80   sets.top[measurable]
```
```    81   sets.empty_sets[measurable (raw)]
```
```    82   sets.Un[measurable (raw)]
```
```    83   sets.Diff[measurable (raw)]
```
```    84
```
```    85 declare
```
```    86   measurable_count_space[measurable (raw)]
```
```    87   measurable_ident[measurable (raw)]
```
```    88   measurable_ident_sets[measurable (raw)]
```
```    89   measurable_const[measurable (raw)]
```
```    90   measurable_If[measurable (raw)]
```
```    91   measurable_comp[measurable (raw)]
```
```    92   measurable_sets[measurable (raw)]
```
```    93
```
```    94 lemma predE[measurable (raw)]:
```
```    95   "pred M P \<Longrightarrow> {x\<in>space M. P x} \<in> sets M"
```
```    96   unfolding pred_def .
```
```    97
```
```    98 lemma pred_intros_imp'[measurable (raw)]:
```
```    99   "(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<longrightarrow> P x)"
```
```   100   by (cases K) auto
```
```   101
```
```   102 lemma pred_intros_conj1'[measurable (raw)]:
```
```   103   "(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<and> P x)"
```
```   104   by (cases K) auto
```
```   105
```
```   106 lemma pred_intros_conj2'[measurable (raw)]:
```
```   107   "(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. P x \<and> K)"
```
```   108   by (cases K) auto
```
```   109
```
```   110 lemma pred_intros_disj1'[measurable (raw)]:
```
```   111   "(\<not> K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<or> P x)"
```
```   112   by (cases K) auto
```
```   113
```
```   114 lemma pred_intros_disj2'[measurable (raw)]:
```
```   115   "(\<not> K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. P x \<or> K)"
```
```   116   by (cases K) auto
```
```   117
```
```   118 lemma pred_intros_logic[measurable (raw)]:
```
```   119   "pred M (\<lambda>x. x \<in> space M)"
```
```   120   "pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. \<not> P x)"
```
```   121   "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<and> P x)"
```
```   122   "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<longrightarrow> P x)"
```
```   123   "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<or> P x)"
```
```   124   "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x = P x)"
```
```   125   "pred M (\<lambda>x. f x \<in> UNIV)"
```
```   126   "pred M (\<lambda>x. f x \<in> {})"
```
```   127   "pred M (\<lambda>x. P' (f x) x) \<Longrightarrow> pred M (\<lambda>x. f x \<in> {y. P' y x})"
```
```   128   "pred M (\<lambda>x. f x \<in> (B x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> - (B x))"
```
```   129   "pred M (\<lambda>x. f x \<in> (A x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (B x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (A x) - (B x))"
```
```   130   "pred M (\<lambda>x. f x \<in> (A x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (B x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (A x) \<inter> (B x))"
```
```   131   "pred M (\<lambda>x. f x \<in> (A x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (B x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (A x) \<union> (B x))"
```
```   132   "pred M (\<lambda>x. g x (f x) \<in> (X x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (g x) -` (X x))"
```
```   133   by (auto simp: iff_conv_conj_imp pred_def)
```
```   134
```
```   135 lemma pred_intros_countable[measurable (raw)]:
```
```   136   fixes P :: "'a \<Rightarrow> 'i :: countable \<Rightarrow> bool"
```
```   137   shows
```
```   138     "(\<And>i. pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<forall>i. P x i)"
```
```   139     "(\<And>i. pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<exists>i. P x i)"
```
```   140   by (auto intro!: sets.sets_Collect_countable_All sets.sets_Collect_countable_Ex simp: pred_def)
```
```   141
```
```   142 lemma pred_intros_countable_bounded[measurable (raw)]:
```
```   143   fixes X :: "'i :: countable set"
```
```   144   shows
```
```   145     "(\<And>i. i \<in> X \<Longrightarrow> pred M (\<lambda>x. x \<in> N x i)) \<Longrightarrow> pred M (\<lambda>x. x \<in> (\<Inter>i\<in>X. N x i))"
```
```   146     "(\<And>i. i \<in> X \<Longrightarrow> pred M (\<lambda>x. x \<in> N x i)) \<Longrightarrow> pred M (\<lambda>x. x \<in> (\<Union>i\<in>X. N x i))"
```
```   147     "(\<And>i. i \<in> X \<Longrightarrow> pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<forall>i\<in>X. P x i)"
```
```   148     "(\<And>i. i \<in> X \<Longrightarrow> pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<exists>i\<in>X. P x i)"
```
```   149   by (auto simp: Bex_def Ball_def)
```
```   150
```
```   151 lemma pred_intros_finite[measurable (raw)]:
```
```   152   "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> pred M (\<lambda>x. x \<in> N x i)) \<Longrightarrow> pred M (\<lambda>x. x \<in> (\<Inter>i\<in>I. N x i))"
```
```   153   "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> pred M (\<lambda>x. x \<in> N x i)) \<Longrightarrow> pred M (\<lambda>x. x \<in> (\<Union>i\<in>I. N x i))"
```
```   154   "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<forall>i\<in>I. P x i)"
```
```   155   "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<exists>i\<in>I. P x i)"
```
```   156   by (auto intro!: sets.sets_Collect_finite_Ex sets.sets_Collect_finite_All simp: iff_conv_conj_imp pred_def)
```
```   157
```
```   158 lemma countable_Un_Int[measurable (raw)]:
```
```   159   "(\<And>i :: 'i :: countable. i \<in> I \<Longrightarrow> N i \<in> sets M) \<Longrightarrow> (\<Union>i\<in>I. N i) \<in> sets M"
```
```   160   "I \<noteq> {} \<Longrightarrow> (\<And>i :: 'i :: countable. i \<in> I \<Longrightarrow> N i \<in> sets M) \<Longrightarrow> (\<Inter>i\<in>I. N i) \<in> sets M"
```
```   161   by auto
```
```   162
```
```   163 declare
```
```   164   finite_UN[measurable (raw)]
```
```   165   finite_INT[measurable (raw)]
```
```   166
```
```   167 lemma sets_Int_pred[measurable (raw)]:
```
```   168   assumes space: "A \<inter> B \<subseteq> space M" and [measurable]: "pred M (\<lambda>x. x \<in> A)" "pred M (\<lambda>x. x \<in> B)"
```
```   169   shows "A \<inter> B \<in> sets M"
```
```   170 proof -
```
```   171   have "{x\<in>space M. x \<in> A \<inter> B} \<in> sets M" by auto
```
```   172   also have "{x\<in>space M. x \<in> A \<inter> B} = A \<inter> B"
```
```   173     using space by auto
```
```   174   finally show ?thesis .
```
```   175 qed
```
```   176
```
```   177 lemma [measurable (raw generic)]:
```
```   178   assumes f: "f \<in> measurable M N" and c: "c \<in> space N \<Longrightarrow> {c} \<in> sets N"
```
```   179   shows pred_eq_const1: "pred M (\<lambda>x. f x = c)"
```
```   180     and pred_eq_const2: "pred M (\<lambda>x. c = f x)"
```
```   181 proof -
```
```   182   show "pred M (\<lambda>x. f x = c)"
```
```   183   proof cases
```
```   184     assume "c \<in> space N"
```
```   185     with measurable_sets[OF f c] show ?thesis
```
```   186       by (auto simp: Int_def conj_commute pred_def)
```
```   187   next
```
```   188     assume "c \<notin> space N"
```
```   189     with f[THEN measurable_space] have "{x \<in> space M. f x = c} = {}" by auto
```
```   190     then show ?thesis by (auto simp: pred_def cong: conj_cong)
```
```   191   qed
```
```   192   then show "pred M (\<lambda>x. c = f x)"
```
```   193     by (simp add: eq_commute)
```
```   194 qed
```
```   195
```
```   196 lemma pred_le_const[measurable (raw generic)]:
```
```   197   assumes f: "f \<in> measurable M N" and c: "{.. c} \<in> sets N" shows "pred M (\<lambda>x. f x \<le> c)"
```
```   198   using measurable_sets[OF f c]
```
```   199   by (auto simp: Int_def conj_commute eq_commute pred_def)
```
```   200
```
```   201 lemma pred_const_le[measurable (raw generic)]:
```
```   202   assumes f: "f \<in> measurable M N" and c: "{c ..} \<in> sets N" shows "pred M (\<lambda>x. c \<le> f x)"
```
```   203   using measurable_sets[OF f c]
```
```   204   by (auto simp: Int_def conj_commute eq_commute pred_def)
```
```   205
```
```   206 lemma pred_less_const[measurable (raw generic)]:
```
```   207   assumes f: "f \<in> measurable M N" and c: "{..< c} \<in> sets N" shows "pred M (\<lambda>x. f x < c)"
```
```   208   using measurable_sets[OF f c]
```
```   209   by (auto simp: Int_def conj_commute eq_commute pred_def)
```
```   210
```
```   211 lemma pred_const_less[measurable (raw generic)]:
```
```   212   assumes f: "f \<in> measurable M N" and c: "{c <..} \<in> sets N" shows "pred M (\<lambda>x. c < f x)"
```
```   213   using measurable_sets[OF f c]
```
```   214   by (auto simp: Int_def conj_commute eq_commute pred_def)
```
```   215
```
```   216 declare
```
```   217   sets.Int[measurable (raw)]
```
```   218
```
```   219 lemma pred_in_If[measurable (raw)]:
```
```   220   "(P \<Longrightarrow> pred M (\<lambda>x. x \<in> A x)) \<Longrightarrow> (\<not> P \<Longrightarrow> pred M (\<lambda>x. x \<in> B x)) \<Longrightarrow>
```
```   221     pred M (\<lambda>x. x \<in> (if P then A x else B x))"
```
```   222   by auto
```
```   223
```
```   224 lemma sets_range[measurable_dest]:
```
```   225   "A ` I \<subseteq> sets M \<Longrightarrow> i \<in> I \<Longrightarrow> A i \<in> sets M"
```
```   226   by auto
```
```   227
```
```   228 lemma pred_sets_range[measurable_dest]:
```
```   229   "A ` I \<subseteq> sets N \<Longrightarrow> i \<in> I \<Longrightarrow> f \<in> measurable M N \<Longrightarrow> pred M (\<lambda>x. f x \<in> A i)"
```
```   230   using pred_sets2[OF sets_range] by auto
```
```   231
```
```   232 lemma sets_All[measurable_dest]:
```
```   233   "\<forall>i. A i \<in> sets (M i) \<Longrightarrow> A i \<in> sets (M i)"
```
```   234   by auto
```
```   235
```
```   236 lemma pred_sets_All[measurable_dest]:
```
```   237   "\<forall>i. A i \<in> sets (N i) \<Longrightarrow> f \<in> measurable M (N i) \<Longrightarrow> pred M (\<lambda>x. f x \<in> A i)"
```
```   238   using pred_sets2[OF sets_All, of A N f] by auto
```
```   239
```
```   240 lemma sets_Ball[measurable_dest]:
```
```   241   "\<forall>i\<in>I. A i \<in> sets (M i) \<Longrightarrow> i\<in>I \<Longrightarrow> A i \<in> sets (M i)"
```
```   242   by auto
```
```   243
```
```   244 lemma pred_sets_Ball[measurable_dest]:
```
```   245   "\<forall>i\<in>I. A i \<in> sets (N i) \<Longrightarrow> i\<in>I \<Longrightarrow> f \<in> measurable M (N i) \<Longrightarrow> pred M (\<lambda>x. f x \<in> A i)"
```
```   246   using pred_sets2[OF sets_Ball, of _ _ _ f] by auto
```
```   247
```
```   248 lemma measurable_finite[measurable (raw)]:
```
```   249   fixes S :: "'a \<Rightarrow> nat set"
```
```   250   assumes [measurable]: "\<And>i. {x\<in>space M. i \<in> S x} \<in> sets M"
```
```   251   shows "pred M (\<lambda>x. finite (S x))"
```
```   252   unfolding finite_nat_set_iff_bounded by (simp add: Ball_def)
```
```   253
```
```   254 lemma measurable_Least[measurable]:
```
```   255   assumes [measurable]: "(\<And>i::nat. (\<lambda>x. P i x) \<in> measurable M (count_space UNIV))"q
```
```   256   shows "(\<lambda>x. LEAST i. P i x) \<in> measurable M (count_space UNIV)"
```
```   257   unfolding measurable_def by (safe intro!: sets_Least) simp_all
```
```   258
```
```   259 lemma measurable_Max_nat[measurable (raw)]:
```
```   260   fixes P :: "nat \<Rightarrow> 'a \<Rightarrow> bool"
```
```   261   assumes [measurable]: "\<And>i. Measurable.pred M (P i)"
```
```   262   shows "(\<lambda>x. Max {i. P i x}) \<in> measurable M (count_space UNIV)"
```
```   263   unfolding measurable_count_space_eq2_countable
```
```   264 proof safe
```
```   265   fix n
```
```   266
```
```   267   { fix x assume "\<forall>i. \<exists>n\<ge>i. P n x"
```
```   268     then have "infinite {i. P i x}"
```
```   269       unfolding infinite_nat_iff_unbounded_le by auto
```
```   270     then have "Max {i. P i x} = the None"
```
```   271       by (rule Max.infinite) }
```
```   272   note 1 = this
```
```   273
```
```   274   { fix x i j assume "P i x" "\<forall>n\<ge>j. \<not> P n x"
```
```   275     then have "finite {i. P i x}"
```
```   276       by (auto simp: subset_eq not_le[symmetric] finite_nat_iff_bounded)
```
```   277     with `P i x` have "P (Max {i. P i x}) x" "i \<le> Max {i. P i x}" "finite {i. P i x}"
```
```   278       using Max_in[of "{i. P i x}"] by auto }
```
```   279   note 2 = this
```
```   280
```
```   281   have "(\<lambda>x. Max {i. P i x}) -` {n} \<inter> space M = {x\<in>space M. Max {i. P i x} = n}"
```
```   282     by auto
```
```   283   also have "\<dots> =
```
```   284     {x\<in>space M. if (\<forall>i. \<exists>n\<ge>i. P n x) then the None = n else
```
```   285       if (\<exists>i. P i x) then P n x \<and> (\<forall>i>n. \<not> P i x)
```
```   286       else Max {} = n}"
```
```   287     by (intro arg_cong[where f=Collect] ext conj_cong)
```
```   288        (auto simp add: 1 2 not_le[symmetric] intro!: Max_eqI)
```
```   289   also have "\<dots> \<in> sets M"
```
```   290     by measurable
```
```   291   finally show "(\<lambda>x. Max {i. P i x}) -` {n} \<inter> space M \<in> sets M" .
```
```   292 qed simp
```
```   293
```
```   294 lemma measurable_Min_nat[measurable (raw)]:
```
```   295   fixes P :: "nat \<Rightarrow> 'a \<Rightarrow> bool"
```
```   296   assumes [measurable]: "\<And>i. Measurable.pred M (P i)"
```
```   297   shows "(\<lambda>x. Min {i. P i x}) \<in> measurable M (count_space UNIV)"
```
```   298   unfolding measurable_count_space_eq2_countable
```
```   299 proof safe
```
```   300   fix n
```
```   301
```
```   302   { fix x assume "\<forall>i. \<exists>n\<ge>i. P n x"
```
```   303     then have "infinite {i. P i x}"
```
```   304       unfolding infinite_nat_iff_unbounded_le by auto
```
```   305     then have "Min {i. P i x} = the None"
```
```   306       by (rule Min.infinite) }
```
```   307   note 1 = this
```
```   308
```
```   309   { fix x i j assume "P i x" "\<forall>n\<ge>j. \<not> P n x"
```
```   310     then have "finite {i. P i x}"
```
```   311       by (auto simp: subset_eq not_le[symmetric] finite_nat_iff_bounded)
```
```   312     with `P i x` have "P (Min {i. P i x}) x" "Min {i. P i x} \<le> i" "finite {i. P i x}"
```
```   313       using Min_in[of "{i. P i x}"] by auto }
```
```   314   note 2 = this
```
```   315
```
```   316   have "(\<lambda>x. Min {i. P i x}) -` {n} \<inter> space M = {x\<in>space M. Min {i. P i x} = n}"
```
```   317     by auto
```
```   318   also have "\<dots> =
```
```   319     {x\<in>space M. if (\<forall>i. \<exists>n\<ge>i. P n x) then the None = n else
```
```   320       if (\<exists>i. P i x) then P n x \<and> (\<forall>i<n. \<not> P i x)
```
```   321       else Min {} = n}"
```
```   322     by (intro arg_cong[where f=Collect] ext conj_cong)
```
```   323        (auto simp add: 1 2 not_le[symmetric] intro!: Min_eqI)
```
```   324   also have "\<dots> \<in> sets M"
```
```   325     by measurable
```
```   326   finally show "(\<lambda>x. Min {i. P i x}) -` {n} \<inter> space M \<in> sets M" .
```
```   327 qed simp
```
```   328
```
```   329 lemma measurable_count_space_insert[measurable (raw)]:
```
```   330   "s \<in> S \<Longrightarrow> A \<in> sets (count_space S) \<Longrightarrow> insert s A \<in> sets (count_space S)"
```
```   331   by simp
```
```   332
```
```   333 lemma measurable_card[measurable]:
```
```   334   fixes S :: "'a \<Rightarrow> nat set"
```
```   335   assumes [measurable]: "\<And>i. {x\<in>space M. i \<in> S x} \<in> sets M"
```
```   336   shows "(\<lambda>x. card (S x)) \<in> measurable M (count_space UNIV)"
```
```   337   unfolding measurable_count_space_eq2_countable
```
```   338 proof safe
```
```   339   fix n show "(\<lambda>x. card (S x)) -` {n} \<inter> space M \<in> sets M"
```
```   340   proof (cases n)
```
```   341     case 0
```
```   342     then have "(\<lambda>x. card (S x)) -` {n} \<inter> space M = {x\<in>space M. infinite (S x) \<or> (\<forall>i. i \<notin> S x)}"
```
```   343       by auto
```
```   344     also have "\<dots> \<in> sets M"
```
```   345       by measurable
```
```   346     finally show ?thesis .
```
```   347   next
```
```   348     case (Suc i)
```
```   349     then have "(\<lambda>x. card (S x)) -` {n} \<inter> space M =
```
```   350       (\<Union>F\<in>{A\<in>{A. finite A}. card A = n}. {x\<in>space M. (\<forall>i. i \<in> S x \<longleftrightarrow> i \<in> F)})"
```
```   351       unfolding set_eq_iff[symmetric] Collect_bex_eq[symmetric] by (auto intro: card_ge_0_finite)
```
```   352     also have "\<dots> \<in> sets M"
```
```   353       by (intro sets.countable_UN' countable_Collect countable_Collect_finite) auto
```
```   354     finally show ?thesis .
```
```   355   qed
```
```   356 qed rule
```
```   357
```
```   358 subsection {* Measurability for (co)inductive predicates *}
```
```   359
```
```   360 lemma measurable_lfp:
```
```   361   assumes "Order_Continuity.continuous F"
```
```   362   assumes *: "\<And>A. pred M A \<Longrightarrow> pred M (F A)"
```
```   363   shows "pred M (lfp F)"
```
```   364 proof -
```
```   365   { fix i have "Measurable.pred M (\<lambda>x. (F ^^ i) (\<lambda>x. False) x)"
```
```   366       by (induct i) (auto intro!: *) }
```
```   367   then have "Measurable.pred M (\<lambda>x. \<exists>i. (F ^^ i) (\<lambda>x. False) x)"
```
```   368     by measurable
```
```   369   also have "(\<lambda>x. \<exists>i. (F ^^ i) (\<lambda>x. False) x) = (SUP i. (F ^^ i) bot)"
```
```   370     by (auto simp add: bot_fun_def)
```
```   371   also have "\<dots> = lfp F"
```
```   372     by (rule continuous_lfp[symmetric]) fact
```
```   373   finally show ?thesis .
```
```   374 qed
```
```   375
```
```   376 lemma measurable_gfp:
```
```   377   assumes "Order_Continuity.down_continuous F"
```
```   378   assumes *: "\<And>A. pred M A \<Longrightarrow> pred M (F A)"
```
```   379   shows "pred M (gfp F)"
```
```   380 proof -
```
```   381   { fix i have "Measurable.pred M (\<lambda>x. (F ^^ i) (\<lambda>x. True) x)"
```
```   382       by (induct i) (auto intro!: *) }
```
```   383   then have "Measurable.pred M (\<lambda>x. \<forall>i. (F ^^ i) (\<lambda>x. True) x)"
```
```   384     by measurable
```
```   385   also have "(\<lambda>x. \<forall>i. (F ^^ i) (\<lambda>x. True) x) = (INF i. (F ^^ i) top)"
```
```   386     by (auto simp add: top_fun_def)
```
```   387   also have "\<dots> = gfp F"
```
```   388     by (rule down_continuous_gfp[symmetric]) fact
```
```   389   finally show ?thesis .
```
```   390 qed
```
```   391
```
```   392 hide_const (open) pred
```
```   393
```
```   394 end
```