src/HOL/Probability/Measurable.thy
 author Andreas Lochbihler Tue Nov 11 08:57:46 2014 +0100 (2014-11-11) changeset 58965 a62cdcc5344b parent 57025 e7fd64f82876 child 59000 6eb0725503fc permissions -rw-r--r--
make measurability rules available as dynamic theorem;
```     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.\$\$\$ "del" >> K false) true --
```
```    55     Scan.optional (Args.parens (Scan.optional (Args.\$\$\$ "raw" >> K true) false --
```
```    56       Scan.optional (Args.\$\$\$ "generic" >> K Measurable.Generic) Measurable.Concrete))
```
```    57     (false, Measurable.Concrete) >> (Thm.declaration_attribute o uncurry Measurable.add_del_thm))
```
```    58 *} "declaration of measurability theorems"
```
```    59
```
```    60 attribute_setup measurable_dest = {*
```
```    61   Scan.lift (Scan.succeed (Thm.declaration_attribute Measurable.add_dest))
```
```    62 *} "add dest rule for measurability prover"
```
```    63
```
```    64 attribute_setup measurable_app = {*
```
```    65   Scan.lift (Scan.succeed (Thm.declaration_attribute Measurable.add_app))
```
```    66 *} "add application rule for measurability prover"
```
```    67
```
```    68 method_setup measurable = {*
```
```    69   Scan.lift (Scan.succeed (fn ctxt => METHOD (fn facts => Measurable.measurable_tac ctxt facts)))
```
```    70 *} "measurability prover"
```
```    71
```
```    72 simproc_setup measurable ("A \<in> sets M" | "f \<in> measurable M N") = {* K Measurable.simproc *}
```
```    73
```
```    74 setup {*
```
```    75   Global_Theory.add_thms_dynamic (@{binding measurable}, Measurable.get_all o Context.proof_of)
```
```    76 *}
```
```    77
```
```    78 declare
```
```    79   measurable_compose_rev[measurable_dest]
```
```    80   pred_sets1[measurable_dest]
```
```    81   pred_sets2[measurable_dest]
```
```    82   sets.sets_into_space[measurable_dest]
```
```    83
```
```    84 declare
```
```    85   sets.top[measurable]
```
```    86   sets.empty_sets[measurable (raw)]
```
```    87   sets.Un[measurable (raw)]
```
```    88   sets.Diff[measurable (raw)]
```
```    89
```
```    90 declare
```
```    91   measurable_count_space[measurable (raw)]
```
```    92   measurable_ident[measurable (raw)]
```
```    93   measurable_ident_sets[measurable (raw)]
```
```    94   measurable_const[measurable (raw)]
```
```    95   measurable_If[measurable (raw)]
```
```    96   measurable_comp[measurable (raw)]
```
```    97   measurable_sets[measurable (raw)]
```
```    98
```
```    99 lemma predE[measurable (raw)]:
```
```   100   "pred M P \<Longrightarrow> {x\<in>space M. P x} \<in> sets M"
```
```   101   unfolding pred_def .
```
```   102
```
```   103 lemma pred_intros_imp'[measurable (raw)]:
```
```   104   "(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<longrightarrow> P x)"
```
```   105   by (cases K) auto
```
```   106
```
```   107 lemma pred_intros_conj1'[measurable (raw)]:
```
```   108   "(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<and> P x)"
```
```   109   by (cases K) auto
```
```   110
```
```   111 lemma pred_intros_conj2'[measurable (raw)]:
```
```   112   "(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. P x \<and> K)"
```
```   113   by (cases K) auto
```
```   114
```
```   115 lemma pred_intros_disj1'[measurable (raw)]:
```
```   116   "(\<not> K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<or> P x)"
```
```   117   by (cases K) auto
```
```   118
```
```   119 lemma pred_intros_disj2'[measurable (raw)]:
```
```   120   "(\<not> K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. P x \<or> K)"
```
```   121   by (cases K) auto
```
```   122
```
```   123 lemma pred_intros_logic[measurable (raw)]:
```
```   124   "pred M (\<lambda>x. x \<in> space M)"
```
```   125   "pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. \<not> P x)"
```
```   126   "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<and> P x)"
```
```   127   "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<longrightarrow> P x)"
```
```   128   "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<or> P x)"
```
```   129   "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x = P x)"
```
```   130   "pred M (\<lambda>x. f x \<in> UNIV)"
```
```   131   "pred M (\<lambda>x. f x \<in> {})"
```
```   132   "pred M (\<lambda>x. P' (f x) x) \<Longrightarrow> pred M (\<lambda>x. f x \<in> {y. P' y x})"
```
```   133   "pred M (\<lambda>x. f x \<in> (B x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> - (B x))"
```
```   134   "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))"
```
```   135   "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))"
```
```   136   "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))"
```
```   137   "pred M (\<lambda>x. g x (f x) \<in> (X x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (g x) -` (X x))"
```
```   138   by (auto simp: iff_conv_conj_imp pred_def)
```
```   139
```
```   140 lemma pred_intros_countable[measurable (raw)]:
```
```   141   fixes P :: "'a \<Rightarrow> 'i :: countable \<Rightarrow> bool"
```
```   142   shows
```
```   143     "(\<And>i. pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<forall>i. P x i)"
```
```   144     "(\<And>i. pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<exists>i. P x i)"
```
```   145   by (auto intro!: sets.sets_Collect_countable_All sets.sets_Collect_countable_Ex simp: pred_def)
```
```   146
```
```   147 lemma pred_intros_countable_bounded[measurable (raw)]:
```
```   148   fixes X :: "'i :: countable set"
```
```   149   shows
```
```   150     "(\<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))"
```
```   151     "(\<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))"
```
```   152     "(\<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)"
```
```   153     "(\<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)"
```
```   154   by (auto simp: Bex_def Ball_def)
```
```   155
```
```   156 lemma pred_intros_finite[measurable (raw)]:
```
```   157   "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))"
```
```   158   "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))"
```
```   159   "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)"
```
```   160   "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)"
```
```   161   by (auto intro!: sets.sets_Collect_finite_Ex sets.sets_Collect_finite_All simp: iff_conv_conj_imp pred_def)
```
```   162
```
```   163 lemma countable_Un_Int[measurable (raw)]:
```
```   164   "(\<And>i :: 'i :: countable. i \<in> I \<Longrightarrow> N i \<in> sets M) \<Longrightarrow> (\<Union>i\<in>I. N i) \<in> sets M"
```
```   165   "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"
```
```   166   by auto
```
```   167
```
```   168 declare
```
```   169   finite_UN[measurable (raw)]
```
```   170   finite_INT[measurable (raw)]
```
```   171
```
```   172 lemma sets_Int_pred[measurable (raw)]:
```
```   173   assumes space: "A \<inter> B \<subseteq> space M" and [measurable]: "pred M (\<lambda>x. x \<in> A)" "pred M (\<lambda>x. x \<in> B)"
```
```   174   shows "A \<inter> B \<in> sets M"
```
```   175 proof -
```
```   176   have "{x\<in>space M. x \<in> A \<inter> B} \<in> sets M" by auto
```
```   177   also have "{x\<in>space M. x \<in> A \<inter> B} = A \<inter> B"
```
```   178     using space by auto
```
```   179   finally show ?thesis .
```
```   180 qed
```
```   181
```
```   182 lemma [measurable (raw generic)]:
```
```   183   assumes f: "f \<in> measurable M N" and c: "c \<in> space N \<Longrightarrow> {c} \<in> sets N"
```
```   184   shows pred_eq_const1: "pred M (\<lambda>x. f x = c)"
```
```   185     and pred_eq_const2: "pred M (\<lambda>x. c = f x)"
```
```   186 proof -
```
```   187   show "pred M (\<lambda>x. f x = c)"
```
```   188   proof cases
```
```   189     assume "c \<in> space N"
```
```   190     with measurable_sets[OF f c] show ?thesis
```
```   191       by (auto simp: Int_def conj_commute pred_def)
```
```   192   next
```
```   193     assume "c \<notin> space N"
```
```   194     with f[THEN measurable_space] have "{x \<in> space M. f x = c} = {}" by auto
```
```   195     then show ?thesis by (auto simp: pred_def cong: conj_cong)
```
```   196   qed
```
```   197   then show "pred M (\<lambda>x. c = f x)"
```
```   198     by (simp add: eq_commute)
```
```   199 qed
```
```   200
```
```   201 lemma pred_le_const[measurable (raw generic)]:
```
```   202   assumes f: "f \<in> measurable M N" and c: "{.. c} \<in> sets N" shows "pred M (\<lambda>x. f x \<le> c)"
```
```   203   using measurable_sets[OF f c]
```
```   204   by (auto simp: Int_def conj_commute eq_commute pred_def)
```
```   205
```
```   206 lemma pred_const_le[measurable (raw generic)]:
```
```   207   assumes f: "f \<in> measurable M N" and c: "{c ..} \<in> sets N" shows "pred M (\<lambda>x. c \<le> f x)"
```
```   208   using measurable_sets[OF f c]
```
```   209   by (auto simp: Int_def conj_commute eq_commute pred_def)
```
```   210
```
```   211 lemma pred_less_const[measurable (raw generic)]:
```
```   212   assumes f: "f \<in> measurable M N" and c: "{..< c} \<in> sets N" shows "pred M (\<lambda>x. f x < c)"
```
```   213   using measurable_sets[OF f c]
```
```   214   by (auto simp: Int_def conj_commute eq_commute pred_def)
```
```   215
```
```   216 lemma pred_const_less[measurable (raw generic)]:
```
```   217   assumes f: "f \<in> measurable M N" and c: "{c <..} \<in> sets N" shows "pred M (\<lambda>x. c < f x)"
```
```   218   using measurable_sets[OF f c]
```
```   219   by (auto simp: Int_def conj_commute eq_commute pred_def)
```
```   220
```
```   221 declare
```
```   222   sets.Int[measurable (raw)]
```
```   223
```
```   224 lemma pred_in_If[measurable (raw)]:
```
```   225   "(P \<Longrightarrow> pred M (\<lambda>x. x \<in> A x)) \<Longrightarrow> (\<not> P \<Longrightarrow> pred M (\<lambda>x. x \<in> B x)) \<Longrightarrow>
```
```   226     pred M (\<lambda>x. x \<in> (if P then A x else B x))"
```
```   227   by auto
```
```   228
```
```   229 lemma sets_range[measurable_dest]:
```
```   230   "A ` I \<subseteq> sets M \<Longrightarrow> i \<in> I \<Longrightarrow> A i \<in> sets M"
```
```   231   by auto
```
```   232
```
```   233 lemma pred_sets_range[measurable_dest]:
```
```   234   "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)"
```
```   235   using pred_sets2[OF sets_range] by auto
```
```   236
```
```   237 lemma sets_All[measurable_dest]:
```
```   238   "\<forall>i. A i \<in> sets (M i) \<Longrightarrow> A i \<in> sets (M i)"
```
```   239   by auto
```
```   240
```
```   241 lemma pred_sets_All[measurable_dest]:
```
```   242   "\<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)"
```
```   243   using pred_sets2[OF sets_All, of A N f] by auto
```
```   244
```
```   245 lemma sets_Ball[measurable_dest]:
```
```   246   "\<forall>i\<in>I. A i \<in> sets (M i) \<Longrightarrow> i\<in>I \<Longrightarrow> A i \<in> sets (M i)"
```
```   247   by auto
```
```   248
```
```   249 lemma pred_sets_Ball[measurable_dest]:
```
```   250   "\<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)"
```
```   251   using pred_sets2[OF sets_Ball, of _ _ _ f] by auto
```
```   252
```
```   253 lemma measurable_finite[measurable (raw)]:
```
```   254   fixes S :: "'a \<Rightarrow> nat set"
```
```   255   assumes [measurable]: "\<And>i. {x\<in>space M. i \<in> S x} \<in> sets M"
```
```   256   shows "pred M (\<lambda>x. finite (S x))"
```
```   257   unfolding finite_nat_set_iff_bounded by (simp add: Ball_def)
```
```   258
```
```   259 lemma measurable_Least[measurable]:
```
```   260   assumes [measurable]: "(\<And>i::nat. (\<lambda>x. P i x) \<in> measurable M (count_space UNIV))"q
```
```   261   shows "(\<lambda>x. LEAST i. P i x) \<in> measurable M (count_space UNIV)"
```
```   262   unfolding measurable_def by (safe intro!: sets_Least) simp_all
```
```   263
```
```   264 lemma measurable_Max_nat[measurable (raw)]:
```
```   265   fixes P :: "nat \<Rightarrow> 'a \<Rightarrow> bool"
```
```   266   assumes [measurable]: "\<And>i. Measurable.pred M (P i)"
```
```   267   shows "(\<lambda>x. Max {i. P i x}) \<in> measurable M (count_space UNIV)"
```
```   268   unfolding measurable_count_space_eq2_countable
```
```   269 proof safe
```
```   270   fix n
```
```   271
```
```   272   { fix x assume "\<forall>i. \<exists>n\<ge>i. P n x"
```
```   273     then have "infinite {i. P i x}"
```
```   274       unfolding infinite_nat_iff_unbounded_le by auto
```
```   275     then have "Max {i. P i x} = the None"
```
```   276       by (rule Max.infinite) }
```
```   277   note 1 = this
```
```   278
```
```   279   { fix x i j assume "P i x" "\<forall>n\<ge>j. \<not> P n x"
```
```   280     then have "finite {i. P i x}"
```
```   281       by (auto simp: subset_eq not_le[symmetric] finite_nat_iff_bounded)
```
```   282     with `P i x` have "P (Max {i. P i x}) x" "i \<le> Max {i. P i x}" "finite {i. P i x}"
```
```   283       using Max_in[of "{i. P i x}"] by auto }
```
```   284   note 2 = this
```
```   285
```
```   286   have "(\<lambda>x. Max {i. P i x}) -` {n} \<inter> space M = {x\<in>space M. Max {i. P i x} = n}"
```
```   287     by auto
```
```   288   also have "\<dots> =
```
```   289     {x\<in>space M. if (\<forall>i. \<exists>n\<ge>i. P n x) then the None = n else
```
```   290       if (\<exists>i. P i x) then P n x \<and> (\<forall>i>n. \<not> P i x)
```
```   291       else Max {} = n}"
```
```   292     by (intro arg_cong[where f=Collect] ext conj_cong)
```
```   293        (auto simp add: 1 2 not_le[symmetric] intro!: Max_eqI)
```
```   294   also have "\<dots> \<in> sets M"
```
```   295     by measurable
```
```   296   finally show "(\<lambda>x. Max {i. P i x}) -` {n} \<inter> space M \<in> sets M" .
```
```   297 qed simp
```
```   298
```
```   299 lemma measurable_Min_nat[measurable (raw)]:
```
```   300   fixes P :: "nat \<Rightarrow> 'a \<Rightarrow> bool"
```
```   301   assumes [measurable]: "\<And>i. Measurable.pred M (P i)"
```
```   302   shows "(\<lambda>x. Min {i. P i x}) \<in> measurable M (count_space UNIV)"
```
```   303   unfolding measurable_count_space_eq2_countable
```
```   304 proof safe
```
```   305   fix n
```
```   306
```
```   307   { fix x assume "\<forall>i. \<exists>n\<ge>i. P n x"
```
```   308     then have "infinite {i. P i x}"
```
```   309       unfolding infinite_nat_iff_unbounded_le by auto
```
```   310     then have "Min {i. P i x} = the None"
```
```   311       by (rule Min.infinite) }
```
```   312   note 1 = this
```
```   313
```
```   314   { fix x i j assume "P i x" "\<forall>n\<ge>j. \<not> P n x"
```
```   315     then have "finite {i. P i x}"
```
```   316       by (auto simp: subset_eq not_le[symmetric] finite_nat_iff_bounded)
```
```   317     with `P i x` have "P (Min {i. P i x}) x" "Min {i. P i x} \<le> i" "finite {i. P i x}"
```
```   318       using Min_in[of "{i. P i x}"] by auto }
```
```   319   note 2 = this
```
```   320
```
```   321   have "(\<lambda>x. Min {i. P i x}) -` {n} \<inter> space M = {x\<in>space M. Min {i. P i x} = n}"
```
```   322     by auto
```
```   323   also have "\<dots> =
```
```   324     {x\<in>space M. if (\<forall>i. \<exists>n\<ge>i. P n x) then the None = n else
```
```   325       if (\<exists>i. P i x) then P n x \<and> (\<forall>i<n. \<not> P i x)
```
```   326       else Min {} = n}"
```
```   327     by (intro arg_cong[where f=Collect] ext conj_cong)
```
```   328        (auto simp add: 1 2 not_le[symmetric] intro!: Min_eqI)
```
```   329   also have "\<dots> \<in> sets M"
```
```   330     by measurable
```
```   331   finally show "(\<lambda>x. Min {i. P i x}) -` {n} \<inter> space M \<in> sets M" .
```
```   332 qed simp
```
```   333
```
```   334 lemma measurable_count_space_insert[measurable (raw)]:
```
```   335   "s \<in> S \<Longrightarrow> A \<in> sets (count_space S) \<Longrightarrow> insert s A \<in> sets (count_space S)"
```
```   336   by simp
```
```   337
```
```   338 lemma measurable_card[measurable]:
```
```   339   fixes S :: "'a \<Rightarrow> nat set"
```
```   340   assumes [measurable]: "\<And>i. {x\<in>space M. i \<in> S x} \<in> sets M"
```
```   341   shows "(\<lambda>x. card (S x)) \<in> measurable M (count_space UNIV)"
```
```   342   unfolding measurable_count_space_eq2_countable
```
```   343 proof safe
```
```   344   fix n show "(\<lambda>x. card (S x)) -` {n} \<inter> space M \<in> sets M"
```
```   345   proof (cases n)
```
```   346     case 0
```
```   347     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)}"
```
```   348       by auto
```
```   349     also have "\<dots> \<in> sets M"
```
```   350       by measurable
```
```   351     finally show ?thesis .
```
```   352   next
```
```   353     case (Suc i)
```
```   354     then have "(\<lambda>x. card (S x)) -` {n} \<inter> space M =
```
```   355       (\<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)})"
```
```   356       unfolding set_eq_iff[symmetric] Collect_bex_eq[symmetric] by (auto intro: card_ge_0_finite)
```
```   357     also have "\<dots> \<in> sets M"
```
```   358       by (intro sets.countable_UN' countable_Collect countable_Collect_finite) auto
```
```   359     finally show ?thesis .
```
```   360   qed
```
```   361 qed rule
```
```   362
```
```   363 subsection {* Measurability for (co)inductive predicates *}
```
```   364
```
```   365 lemma measurable_lfp:
```
```   366   assumes "Order_Continuity.continuous F"
```
```   367   assumes *: "\<And>A. pred M A \<Longrightarrow> pred M (F A)"
```
```   368   shows "pred M (lfp F)"
```
```   369 proof -
```
```   370   { fix i have "Measurable.pred M (\<lambda>x. (F ^^ i) (\<lambda>x. False) x)"
```
```   371       by (induct i) (auto intro!: *) }
```
```   372   then have "Measurable.pred M (\<lambda>x. \<exists>i. (F ^^ i) (\<lambda>x. False) x)"
```
```   373     by measurable
```
```   374   also have "(\<lambda>x. \<exists>i. (F ^^ i) (\<lambda>x. False) x) = (SUP i. (F ^^ i) bot)"
```
```   375     by (auto simp add: bot_fun_def)
```
```   376   also have "\<dots> = lfp F"
```
```   377     by (rule continuous_lfp[symmetric]) fact
```
```   378   finally show ?thesis .
```
```   379 qed
```
```   380
```
```   381 lemma measurable_gfp:
```
```   382   assumes "Order_Continuity.down_continuous F"
```
```   383   assumes *: "\<And>A. pred M A \<Longrightarrow> pred M (F A)"
```
```   384   shows "pred M (gfp F)"
```
```   385 proof -
```
```   386   { fix i have "Measurable.pred M (\<lambda>x. (F ^^ i) (\<lambda>x. True) x)"
```
```   387       by (induct i) (auto intro!: *) }
```
```   388   then have "Measurable.pred M (\<lambda>x. \<forall>i. (F ^^ i) (\<lambda>x. True) x)"
```
```   389     by measurable
```
```   390   also have "(\<lambda>x. \<forall>i. (F ^^ i) (\<lambda>x. True) x) = (INF i. (F ^^ i) top)"
```
```   391     by (auto simp add: top_fun_def)
```
```   392   also have "\<dots> = gfp F"
```
```   393     by (rule down_continuous_gfp[symmetric]) fact
```
```   394   finally show ?thesis .
```
```   395 qed
```
```   396
```
```   397 hide_const (open) pred
```
```   398
```
```   399 end
```