src/HOL/Probability/Measurable.thy
 author hoelzl Tue Jan 13 19:10:36 2015 +0100 (2015-01-13) changeset 59353 f0707dc3d9aa parent 59088 ff2bd4a14ddb child 59361 fd5da2434be4 permissions -rw-r--r--
measurability prover: removed app splitting, replaced by more powerful destruction rules
```     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 (
```
```    55     (Args.add >> K true || Args.del >> K false || Scan.succeed true) --
```
```    56     Scan.optional (Args.parens (
```
```    57       Scan.optional (Args.\$\$\$ "raw" >> K true) false --
```
```    58       Scan.optional (Args.\$\$\$ "generic" >> K Measurable.Generic) Measurable.Concrete))
```
```    59     (false, Measurable.Concrete) >>
```
```    60     Measurable.measurable_thm_attr)
```
```    61 *} "declaration of measurability theorems"
```
```    62
```
```    63 attribute_setup measurable_dest = Measurable.dest_thm_attr
```
```    64   "add dest rule to measurability prover"
```
```    65
```
```    66 attribute_setup measurable_cong = Measurable.cong_thm_attr
```
```    67   "add congurence rules to measurability prover"
```
```    68
```
```    69 method_setup measurable = \<open> Scan.lift (Scan.succeed (METHOD o Measurable.measurable_tac)) \<close>
```
```    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)
```
```    76 *}
```
```    77
```
```    78 declare
```
```    79   pred_sets1[measurable_dest]
```
```    80   pred_sets2[measurable_dest]
```
```    81   sets.sets_into_space[measurable_dest]
```
```    82
```
```    83 declare
```
```    84   sets.top[measurable]
```
```    85   sets.empty_sets[measurable (raw)]
```
```    86   sets.Un[measurable (raw)]
```
```    87   sets.Diff[measurable (raw)]
```
```    88
```
```    89 declare
```
```    90   measurable_count_space[measurable (raw)]
```
```    91   measurable_ident[measurable (raw)]
```
```    92   measurable_id[measurable (raw)]
```
```    93   measurable_const[measurable (raw)]
```
```    94   measurable_If[measurable (raw)]
```
```    95   measurable_comp[measurable (raw)]
```
```    96   measurable_sets[measurable (raw)]
```
```    97
```
```    98 declare measurable_cong_sets[measurable_cong]
```
```    99 declare sets_restrict_space_cong[measurable_cong]
```
```   100
```
```   101 lemma predE[measurable (raw)]:
```
```   102   "pred M P \<Longrightarrow> {x\<in>space M. P x} \<in> sets M"
```
```   103   unfolding pred_def .
```
```   104
```
```   105 lemma pred_intros_imp'[measurable (raw)]:
```
```   106   "(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<longrightarrow> P x)"
```
```   107   by (cases K) auto
```
```   108
```
```   109 lemma pred_intros_conj1'[measurable (raw)]:
```
```   110   "(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<and> P x)"
```
```   111   by (cases K) auto
```
```   112
```
```   113 lemma pred_intros_conj2'[measurable (raw)]:
```
```   114   "(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. P x \<and> K)"
```
```   115   by (cases K) auto
```
```   116
```
```   117 lemma pred_intros_disj1'[measurable (raw)]:
```
```   118   "(\<not> K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<or> P x)"
```
```   119   by (cases K) auto
```
```   120
```
```   121 lemma pred_intros_disj2'[measurable (raw)]:
```
```   122   "(\<not> K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. P x \<or> K)"
```
```   123   by (cases K) auto
```
```   124
```
```   125 lemma pred_intros_logic[measurable (raw)]:
```
```   126   "pred M (\<lambda>x. x \<in> space M)"
```
```   127   "pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. \<not> P x)"
```
```   128   "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<and> P x)"
```
```   129   "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<longrightarrow> P x)"
```
```   130   "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<or> P x)"
```
```   131   "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x = P x)"
```
```   132   "pred M (\<lambda>x. f x \<in> UNIV)"
```
```   133   "pred M (\<lambda>x. f x \<in> {})"
```
```   134   "pred M (\<lambda>x. P' (f x) x) \<Longrightarrow> pred M (\<lambda>x. f x \<in> {y. P' y x})"
```
```   135   "pred M (\<lambda>x. f x \<in> (B x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> - (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) - (B x))"
```
```   137   "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))"
```
```   138   "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))"
```
```   139   "pred M (\<lambda>x. g x (f x) \<in> (X x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (g x) -` (X x))"
```
```   140   by (auto simp: iff_conv_conj_imp pred_def)
```
```   141
```
```   142 lemma pred_intros_countable[measurable (raw)]:
```
```   143   fixes P :: "'a \<Rightarrow> 'i :: countable \<Rightarrow> bool"
```
```   144   shows
```
```   145     "(\<And>i. pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<forall>i. P x i)"
```
```   146     "(\<And>i. pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<exists>i. P x i)"
```
```   147   by (auto intro!: sets.sets_Collect_countable_All sets.sets_Collect_countable_Ex simp: pred_def)
```
```   148
```
```   149 lemma pred_intros_countable_bounded[measurable (raw)]:
```
```   150   fixes X :: "'i :: countable set"
```
```   151   shows
```
```   152     "(\<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))"
```
```   153     "(\<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))"
```
```   154     "(\<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)"
```
```   155     "(\<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)"
```
```   156   by (auto simp: Bex_def Ball_def)
```
```   157
```
```   158 lemma pred_intros_finite[measurable (raw)]:
```
```   159   "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))"
```
```   160   "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))"
```
```   161   "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)"
```
```   162   "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)"
```
```   163   by (auto intro!: sets.sets_Collect_finite_Ex sets.sets_Collect_finite_All simp: iff_conv_conj_imp pred_def)
```
```   164
```
```   165 lemma countable_Un_Int[measurable (raw)]:
```
```   166   "(\<And>i :: 'i :: countable. i \<in> I \<Longrightarrow> N i \<in> sets M) \<Longrightarrow> (\<Union>i\<in>I. N i) \<in> sets M"
```
```   167   "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"
```
```   168   by auto
```
```   169
```
```   170 declare
```
```   171   finite_UN[measurable (raw)]
```
```   172   finite_INT[measurable (raw)]
```
```   173
```
```   174 lemma sets_Int_pred[measurable (raw)]:
```
```   175   assumes space: "A \<inter> B \<subseteq> space M" and [measurable]: "pred M (\<lambda>x. x \<in> A)" "pred M (\<lambda>x. x \<in> B)"
```
```   176   shows "A \<inter> B \<in> sets M"
```
```   177 proof -
```
```   178   have "{x\<in>space M. x \<in> A \<inter> B} \<in> sets M" by auto
```
```   179   also have "{x\<in>space M. x \<in> A \<inter> B} = A \<inter> B"
```
```   180     using space by auto
```
```   181   finally show ?thesis .
```
```   182 qed
```
```   183
```
```   184 lemma [measurable (raw generic)]:
```
```   185   assumes f: "f \<in> measurable M N" and c: "c \<in> space N \<Longrightarrow> {c} \<in> sets N"
```
```   186   shows pred_eq_const1: "pred M (\<lambda>x. f x = c)"
```
```   187     and pred_eq_const2: "pred M (\<lambda>x. c = f x)"
```
```   188 proof -
```
```   189   show "pred M (\<lambda>x. f x = c)"
```
```   190   proof cases
```
```   191     assume "c \<in> space N"
```
```   192     with measurable_sets[OF f c] show ?thesis
```
```   193       by (auto simp: Int_def conj_commute pred_def)
```
```   194   next
```
```   195     assume "c \<notin> space N"
```
```   196     with f[THEN measurable_space] have "{x \<in> space M. f x = c} = {}" by auto
```
```   197     then show ?thesis by (auto simp: pred_def cong: conj_cong)
```
```   198   qed
```
```   199   then show "pred M (\<lambda>x. c = f x)"
```
```   200     by (simp add: eq_commute)
```
```   201 qed
```
```   202
```
```   203 lemma pred_count_space_const1[measurable (raw)]:
```
```   204   "f \<in> measurable M (count_space UNIV) \<Longrightarrow> Measurable.pred M (\<lambda>x. f x = c)"
```
```   205   by (intro pred_eq_const1[where N="count_space UNIV"]) (auto )
```
```   206
```
```   207 lemma pred_count_space_const2[measurable (raw)]:
```
```   208   "f \<in> measurable M (count_space UNIV) \<Longrightarrow> Measurable.pred M (\<lambda>x. c = f x)"
```
```   209   by (intro pred_eq_const2[where N="count_space UNIV"]) (auto )
```
```   210
```
```   211 lemma pred_le_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 \<le> 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_le[measurable (raw generic)]:
```
```   217   assumes f: "f \<in> measurable M N" and c: "{c ..} \<in> sets N" shows "pred M (\<lambda>x. c \<le> f x)"
```
```   218   using measurable_sets[OF f c]
```
```   219   by (auto simp: Int_def conj_commute eq_commute pred_def)
```
```   220
```
```   221 lemma pred_less_const[measurable (raw generic)]:
```
```   222   assumes f: "f \<in> measurable M N" and c: "{..< c} \<in> sets N" shows "pred M (\<lambda>x. f x < c)"
```
```   223   using measurable_sets[OF f c]
```
```   224   by (auto simp: Int_def conj_commute eq_commute pred_def)
```
```   225
```
```   226 lemma pred_const_less[measurable (raw generic)]:
```
```   227   assumes f: "f \<in> measurable M N" and c: "{c <..} \<in> sets N" shows "pred M (\<lambda>x. c < f x)"
```
```   228   using measurable_sets[OF f c]
```
```   229   by (auto simp: Int_def conj_commute eq_commute pred_def)
```
```   230
```
```   231 declare
```
```   232   sets.Int[measurable (raw)]
```
```   233
```
```   234 lemma pred_in_If[measurable (raw)]:
```
```   235   "(P \<Longrightarrow> pred M (\<lambda>x. x \<in> A x)) \<Longrightarrow> (\<not> P \<Longrightarrow> pred M (\<lambda>x. x \<in> B x)) \<Longrightarrow>
```
```   236     pred M (\<lambda>x. x \<in> (if P then A x else B x))"
```
```   237   by auto
```
```   238
```
```   239 lemma sets_range[measurable_dest]:
```
```   240   "A ` I \<subseteq> sets M \<Longrightarrow> i \<in> I \<Longrightarrow> A i \<in> sets M"
```
```   241   by auto
```
```   242
```
```   243 lemma pred_sets_range[measurable_dest]:
```
```   244   "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)"
```
```   245   using pred_sets2[OF sets_range] by auto
```
```   246
```
```   247 lemma sets_All[measurable_dest]:
```
```   248   "\<forall>i. A i \<in> sets (M i) \<Longrightarrow> A i \<in> sets (M i)"
```
```   249   by auto
```
```   250
```
```   251 lemma pred_sets_All[measurable_dest]:
```
```   252   "\<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)"
```
```   253   using pred_sets2[OF sets_All, of A N f] by auto
```
```   254
```
```   255 lemma sets_Ball[measurable_dest]:
```
```   256   "\<forall>i\<in>I. A i \<in> sets (M i) \<Longrightarrow> i\<in>I \<Longrightarrow> A i \<in> sets (M i)"
```
```   257   by auto
```
```   258
```
```   259 lemma pred_sets_Ball[measurable_dest]:
```
```   260   "\<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)"
```
```   261   using pred_sets2[OF sets_Ball, of _ _ _ f] by auto
```
```   262
```
```   263 lemma measurable_finite[measurable (raw)]:
```
```   264   fixes S :: "'a \<Rightarrow> nat set"
```
```   265   assumes [measurable]: "\<And>i. {x\<in>space M. i \<in> S x} \<in> sets M"
```
```   266   shows "pred M (\<lambda>x. finite (S x))"
```
```   267   unfolding finite_nat_set_iff_bounded by (simp add: Ball_def)
```
```   268
```
```   269 lemma measurable_Least[measurable]:
```
```   270   assumes [measurable]: "(\<And>i::nat. (\<lambda>x. P i x) \<in> measurable M (count_space UNIV))"q
```
```   271   shows "(\<lambda>x. LEAST i. P i x) \<in> measurable M (count_space UNIV)"
```
```   272   unfolding measurable_def by (safe intro!: sets_Least) simp_all
```
```   273
```
```   274 lemma measurable_Max_nat[measurable (raw)]:
```
```   275   fixes P :: "nat \<Rightarrow> 'a \<Rightarrow> bool"
```
```   276   assumes [measurable]: "\<And>i. Measurable.pred M (P i)"
```
```   277   shows "(\<lambda>x. Max {i. P i x}) \<in> measurable M (count_space UNIV)"
```
```   278   unfolding measurable_count_space_eq2_countable
```
```   279 proof safe
```
```   280   fix n
```
```   281
```
```   282   { fix x assume "\<forall>i. \<exists>n\<ge>i. P n x"
```
```   283     then have "infinite {i. P i x}"
```
```   284       unfolding infinite_nat_iff_unbounded_le by auto
```
```   285     then have "Max {i. P i x} = the None"
```
```   286       by (rule Max.infinite) }
```
```   287   note 1 = this
```
```   288
```
```   289   { fix x i j assume "P i x" "\<forall>n\<ge>j. \<not> P n x"
```
```   290     then have "finite {i. P i x}"
```
```   291       by (auto simp: subset_eq not_le[symmetric] finite_nat_iff_bounded)
```
```   292     with `P i x` have "P (Max {i. P i x}) x" "i \<le> Max {i. P i x}" "finite {i. P i x}"
```
```   293       using Max_in[of "{i. P i x}"] by auto }
```
```   294   note 2 = this
```
```   295
```
```   296   have "(\<lambda>x. Max {i. P i x}) -` {n} \<inter> space M = {x\<in>space M. Max {i. P i x} = n}"
```
```   297     by auto
```
```   298   also have "\<dots> =
```
```   299     {x\<in>space M. if (\<forall>i. \<exists>n\<ge>i. P n x) then the None = n else
```
```   300       if (\<exists>i. P i x) then P n x \<and> (\<forall>i>n. \<not> P i x)
```
```   301       else Max {} = n}"
```
```   302     by (intro arg_cong[where f=Collect] ext conj_cong)
```
```   303        (auto simp add: 1 2 not_le[symmetric] intro!: Max_eqI)
```
```   304   also have "\<dots> \<in> sets M"
```
```   305     by measurable
```
```   306   finally show "(\<lambda>x. Max {i. P i x}) -` {n} \<inter> space M \<in> sets M" .
```
```   307 qed simp
```
```   308
```
```   309 lemma measurable_Min_nat[measurable (raw)]:
```
```   310   fixes P :: "nat \<Rightarrow> 'a \<Rightarrow> bool"
```
```   311   assumes [measurable]: "\<And>i. Measurable.pred M (P i)"
```
```   312   shows "(\<lambda>x. Min {i. P i x}) \<in> measurable M (count_space UNIV)"
```
```   313   unfolding measurable_count_space_eq2_countable
```
```   314 proof safe
```
```   315   fix n
```
```   316
```
```   317   { fix x assume "\<forall>i. \<exists>n\<ge>i. P n x"
```
```   318     then have "infinite {i. P i x}"
```
```   319       unfolding infinite_nat_iff_unbounded_le by auto
```
```   320     then have "Min {i. P i x} = the None"
```
```   321       by (rule Min.infinite) }
```
```   322   note 1 = this
```
```   323
```
```   324   { fix x i j assume "P i x" "\<forall>n\<ge>j. \<not> P n x"
```
```   325     then have "finite {i. P i x}"
```
```   326       by (auto simp: subset_eq not_le[symmetric] finite_nat_iff_bounded)
```
```   327     with `P i x` have "P (Min {i. P i x}) x" "Min {i. P i x} \<le> i" "finite {i. P i x}"
```
```   328       using Min_in[of "{i. P i x}"] by auto }
```
```   329   note 2 = this
```
```   330
```
```   331   have "(\<lambda>x. Min {i. P i x}) -` {n} \<inter> space M = {x\<in>space M. Min {i. P i x} = n}"
```
```   332     by auto
```
```   333   also have "\<dots> =
```
```   334     {x\<in>space M. if (\<forall>i. \<exists>n\<ge>i. P n x) then the None = n else
```
```   335       if (\<exists>i. P i x) then P n x \<and> (\<forall>i<n. \<not> P i x)
```
```   336       else Min {} = n}"
```
```   337     by (intro arg_cong[where f=Collect] ext conj_cong)
```
```   338        (auto simp add: 1 2 not_le[symmetric] intro!: Min_eqI)
```
```   339   also have "\<dots> \<in> sets M"
```
```   340     by measurable
```
```   341   finally show "(\<lambda>x. Min {i. P i x}) -` {n} \<inter> space M \<in> sets M" .
```
```   342 qed simp
```
```   343
```
```   344 lemma measurable_count_space_insert[measurable (raw)]:
```
```   345   "s \<in> S \<Longrightarrow> A \<in> sets (count_space S) \<Longrightarrow> insert s A \<in> sets (count_space S)"
```
```   346   by simp
```
```   347
```
```   348 lemma sets_UNIV [measurable (raw)]: "A \<in> sets (count_space UNIV)"
```
```   349   by simp
```
```   350
```
```   351 lemma measurable_card[measurable]:
```
```   352   fixes S :: "'a \<Rightarrow> nat set"
```
```   353   assumes [measurable]: "\<And>i. {x\<in>space M. i \<in> S x} \<in> sets M"
```
```   354   shows "(\<lambda>x. card (S x)) \<in> measurable M (count_space UNIV)"
```
```   355   unfolding measurable_count_space_eq2_countable
```
```   356 proof safe
```
```   357   fix n show "(\<lambda>x. card (S x)) -` {n} \<inter> space M \<in> sets M"
```
```   358   proof (cases n)
```
```   359     case 0
```
```   360     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)}"
```
```   361       by auto
```
```   362     also have "\<dots> \<in> sets M"
```
```   363       by measurable
```
```   364     finally show ?thesis .
```
```   365   next
```
```   366     case (Suc i)
```
```   367     then have "(\<lambda>x. card (S x)) -` {n} \<inter> space M =
```
```   368       (\<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)})"
```
```   369       unfolding set_eq_iff[symmetric] Collect_bex_eq[symmetric] by (auto intro: card_ge_0_finite)
```
```   370     also have "\<dots> \<in> sets M"
```
```   371       by (intro sets.countable_UN' countable_Collect countable_Collect_finite) auto
```
```   372     finally show ?thesis .
```
```   373   qed
```
```   374 qed rule
```
```   375
```
```   376 lemma measurable_pred_countable[measurable (raw)]:
```
```   377   assumes "countable X"
```
```   378   shows
```
```   379     "(\<And>i. i \<in> X \<Longrightarrow> Measurable.pred M (\<lambda>x. P x i)) \<Longrightarrow> Measurable.pred M (\<lambda>x. \<forall>i\<in>X. P x i)"
```
```   380     "(\<And>i. i \<in> X \<Longrightarrow> Measurable.pred M (\<lambda>x. P x i)) \<Longrightarrow> Measurable.pred M (\<lambda>x. \<exists>i\<in>X. P x i)"
```
```   381   unfolding pred_def
```
```   382   by (auto intro!: sets.sets_Collect_countable_All' sets.sets_Collect_countable_Ex' assms)
```
```   383
```
```   384 subsection {* Measurability for (co)inductive predicates *}
```
```   385
```
```   386 lemma measurable_bot[measurable]: "bot \<in> measurable M (count_space UNIV)"
```
```   387   by (simp add: bot_fun_def)
```
```   388
```
```   389 lemma measurable_top[measurable]: "top \<in> measurable M (count_space UNIV)"
```
```   390   by (simp add: top_fun_def)
```
```   391
```
```   392 lemma measurable_SUP[measurable]:
```
```   393   fixes F :: "'i \<Rightarrow> 'a \<Rightarrow> 'b::{complete_lattice, countable}"
```
```   394   assumes [simp]: "countable I"
```
```   395   assumes [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> measurable M (count_space UNIV)"
```
```   396   shows "(\<lambda>x. SUP i:I. F i x) \<in> measurable M (count_space UNIV)"
```
```   397   unfolding measurable_count_space_eq2_countable
```
```   398 proof (safe intro!: UNIV_I)
```
```   399   fix a
```
```   400   have "(\<lambda>x. SUP i:I. F i x) -` {a} \<inter> space M =
```
```   401     {x\<in>space M. (\<forall>i\<in>I. F i x \<le> a) \<and> (\<forall>b. (\<forall>i\<in>I. F i x \<le> b) \<longrightarrow> a \<le> b)}"
```
```   402     unfolding SUP_le_iff[symmetric] by auto
```
```   403   also have "\<dots> \<in> sets M"
```
```   404     by measurable
```
```   405   finally show "(\<lambda>x. SUP i:I. F i x) -` {a} \<inter> space M \<in> sets M" .
```
```   406 qed
```
```   407
```
```   408 lemma measurable_INF[measurable]:
```
```   409   fixes F :: "'i \<Rightarrow> 'a \<Rightarrow> 'b::{complete_lattice, countable}"
```
```   410   assumes [simp]: "countable I"
```
```   411   assumes [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> measurable M (count_space UNIV)"
```
```   412   shows "(\<lambda>x. INF i:I. F i x) \<in> measurable M (count_space UNIV)"
```
```   413   unfolding measurable_count_space_eq2_countable
```
```   414 proof (safe intro!: UNIV_I)
```
```   415   fix a
```
```   416   have "(\<lambda>x. INF i:I. F i x) -` {a} \<inter> space M =
```
```   417     {x\<in>space M. (\<forall>i\<in>I. a \<le> F i x) \<and> (\<forall>b. (\<forall>i\<in>I. b \<le> F i x) \<longrightarrow> b \<le> a)}"
```
```   418     unfolding le_INF_iff[symmetric] by auto
```
```   419   also have "\<dots> \<in> sets M"
```
```   420     by measurable
```
```   421   finally show "(\<lambda>x. INF i:I. F i x) -` {a} \<inter> space M \<in> sets M" .
```
```   422 qed
```
```   423
```
```   424 lemma measurable_lfp_coinduct[consumes 1, case_names continuity step]:
```
```   425   fixes F :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b::{complete_lattice, countable})"
```
```   426   assumes "P M"
```
```   427   assumes F: "Order_Continuity.continuous F"
```
```   428   assumes *: "\<And>M A. P M \<Longrightarrow> (\<And>N. P N \<Longrightarrow> A \<in> measurable N (count_space UNIV)) \<Longrightarrow> F A \<in> measurable M (count_space UNIV)"
```
```   429   shows "lfp F \<in> measurable M (count_space UNIV)"
```
```   430 proof -
```
```   431   { fix i from `P M` have "((F ^^ i) bot) \<in> measurable M (count_space UNIV)"
```
```   432       by (induct i arbitrary: M) (auto intro!: *) }
```
```   433   then have "(\<lambda>x. SUP i. (F ^^ i) bot x) \<in> measurable M (count_space UNIV)"
```
```   434     by measurable
```
```   435   also have "(\<lambda>x. SUP i. (F ^^ i) bot x) = lfp F"
```
```   436     by (subst continuous_lfp) (auto intro: F)
```
```   437   finally show ?thesis .
```
```   438 qed
```
```   439
```
```   440 lemma measurable_lfp:
```
```   441   fixes F :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b::{complete_lattice, countable})"
```
```   442   assumes F: "Order_Continuity.continuous F"
```
```   443   assumes *: "\<And>A. A \<in> measurable M (count_space UNIV) \<Longrightarrow> F A \<in> measurable M (count_space UNIV)"
```
```   444   shows "lfp F \<in> measurable M (count_space UNIV)"
```
```   445   by (coinduction rule: measurable_lfp_coinduct[OF _ F]) (blast intro: *)
```
```   446
```
```   447 lemma measurable_gfp_coinduct[consumes 1, case_names continuity step]:
```
```   448   fixes F :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b::{complete_lattice, countable})"
```
```   449   assumes "P M"
```
```   450   assumes F: "Order_Continuity.down_continuous F"
```
```   451   assumes *: "\<And>M A. P M \<Longrightarrow> (\<And>N. P N \<Longrightarrow> A \<in> measurable N (count_space UNIV)) \<Longrightarrow> F A \<in> measurable M (count_space UNIV)"
```
```   452   shows "gfp F \<in> measurable M (count_space UNIV)"
```
```   453 proof -
```
```   454   { fix i from `P M` have "((F ^^ i) top) \<in> measurable M (count_space UNIV)"
```
```   455       by (induct i arbitrary: M) (auto intro!: *) }
```
```   456   then have "(\<lambda>x. INF i. (F ^^ i) top x) \<in> measurable M (count_space UNIV)"
```
```   457     by measurable
```
```   458   also have "(\<lambda>x. INF i. (F ^^ i) top x) = gfp F"
```
```   459     by (subst down_continuous_gfp) (auto intro: F)
```
```   460   finally show ?thesis .
```
```   461 qed
```
```   462
```
```   463 lemma measurable_gfp:
```
```   464   fixes F :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b::{complete_lattice, countable})"
```
```   465   assumes F: "Order_Continuity.down_continuous F"
```
```   466   assumes *: "\<And>A. A \<in> measurable M (count_space UNIV) \<Longrightarrow> F A \<in> measurable M (count_space UNIV)"
```
```   467   shows "gfp F \<in> measurable M (count_space UNIV)"
```
```   468   by (coinduction rule: measurable_gfp_coinduct[OF _ F]) (blast intro: *)
```
```   469
```
```   470 lemma measurable_lfp2_coinduct[consumes 1, case_names continuity step]:
```
```   471   fixes F :: "('a \<Rightarrow> 'c \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'c \<Rightarrow> 'b::{complete_lattice, countable})"
```
```   472   assumes "P M s"
```
```   473   assumes F: "Order_Continuity.continuous F"
```
```   474   assumes *: "\<And>M A s. P M s \<Longrightarrow> (\<And>N t. P N t \<Longrightarrow> A t \<in> measurable N (count_space UNIV)) \<Longrightarrow> F A s \<in> measurable M (count_space UNIV)"
```
```   475   shows "lfp F s \<in> measurable M (count_space UNIV)"
```
```   476 proof -
```
```   477   { fix i from `P M s` have "(\<lambda>x. (F ^^ i) bot s x) \<in> measurable M (count_space UNIV)"
```
```   478       by (induct i arbitrary: M s) (auto intro!: *) }
```
```   479   then have "(\<lambda>x. SUP i. (F ^^ i) bot s x) \<in> measurable M (count_space UNIV)"
```
```   480     by measurable
```
```   481   also have "(\<lambda>x. SUP i. (F ^^ i) bot s x) = lfp F s"
```
```   482     by (subst continuous_lfp) (auto simp: F)
```
```   483   finally show ?thesis .
```
```   484 qed
```
```   485
```
```   486 lemma measurable_gfp2_coinduct[consumes 1, case_names continuity step]:
```
```   487   fixes F :: "('a \<Rightarrow> 'c \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'c \<Rightarrow> 'b::{complete_lattice, countable})"
```
```   488   assumes "P M s"
```
```   489   assumes F: "Order_Continuity.down_continuous F"
```
```   490   assumes *: "\<And>M A s. P M s \<Longrightarrow> (\<And>N t. P N t \<Longrightarrow> A t \<in> measurable N (count_space UNIV)) \<Longrightarrow> F A s \<in> measurable M (count_space UNIV)"
```
```   491   shows "gfp F s \<in> measurable M (count_space UNIV)"
```
```   492 proof -
```
```   493   { fix i from `P M s` have "(\<lambda>x. (F ^^ i) top s x) \<in> measurable M (count_space UNIV)"
```
```   494       by (induct i arbitrary: M s) (auto intro!: *) }
```
```   495   then have "(\<lambda>x. INF i. (F ^^ i) top s x) \<in> measurable M (count_space UNIV)"
```
```   496     by measurable
```
```   497   also have "(\<lambda>x. INF i. (F ^^ i) top s x) = gfp F s"
```
```   498     by (subst down_continuous_gfp) (auto simp: F)
```
```   499   finally show ?thesis .
```
```   500 qed
```
```   501
```
```   502 lemma measurable_enat_coinduct:
```
```   503   fixes f :: "'a \<Rightarrow> enat"
```
```   504   assumes "R f"
```
```   505   assumes *: "\<And>f. R f \<Longrightarrow> \<exists>g h i P. R g \<and> f = (\<lambda>x. if P x then h x else eSuc (g (i x))) \<and>
```
```   506     Measurable.pred M P \<and>
```
```   507     i \<in> measurable M M \<and>
```
```   508     h \<in> measurable M (count_space UNIV)"
```
```   509   shows "f \<in> measurable M (count_space UNIV)"
```
```   510 proof (simp add: measurable_count_space_eq2_countable, rule )
```
```   511   fix a :: enat
```
```   512   have "f -` {a} \<inter> space M = {x\<in>space M. f x = a}"
```
```   513     by auto
```
```   514   { fix i :: nat
```
```   515     from `R f` have "Measurable.pred M (\<lambda>x. f x = enat i)"
```
```   516     proof (induction i arbitrary: f)
```
```   517       case 0
```
```   518       from *[OF this] obtain g h i P
```
```   519         where f: "f = (\<lambda>x. if P x then h x else eSuc (g (i x)))" and
```
```   520           [measurable]: "Measurable.pred M P" "i \<in> measurable M M" "h \<in> measurable M (count_space UNIV)"
```
```   521         by auto
```
```   522       have "Measurable.pred M (\<lambda>x. P x \<and> h x = 0)"
```
```   523         by measurable
```
```   524       also have "(\<lambda>x. P x \<and> h x = 0) = (\<lambda>x. f x = enat 0)"
```
```   525         by (auto simp: f zero_enat_def[symmetric])
```
```   526       finally show ?case .
```
```   527     next
```
```   528       case (Suc n)
```
```   529       from *[OF Suc.prems] obtain g h i P
```
```   530         where f: "f = (\<lambda>x. if P x then h x else eSuc (g (i x)))" and "R g" and
```
```   531           M[measurable]: "Measurable.pred M P" "i \<in> measurable M M" "h \<in> measurable M (count_space UNIV)"
```
```   532         by auto
```
```   533       have "(\<lambda>x. f x = enat (Suc n)) =
```
```   534         (\<lambda>x. (P x \<longrightarrow> h x = enat (Suc n)) \<and> (\<not> P x \<longrightarrow> g (i x) = enat n))"
```
```   535         by (auto simp: f zero_enat_def[symmetric] eSuc_enat[symmetric])
```
```   536       also have "Measurable.pred M \<dots>"
```
```   537         by (intro pred_intros_logic measurable_compose[OF M(2)] Suc `R g`) measurable
```
```   538       finally show ?case .
```
```   539     qed
```
```   540     then have "f -` {enat i} \<inter> space M \<in> sets M"
```
```   541       by (simp add: pred_def Int_def conj_commute) }
```
```   542   note fin = this
```
```   543   show "f -` {a} \<inter> space M \<in> sets M"
```
```   544   proof (cases a)
```
```   545     case infinity
```
```   546     then have "f -` {a} \<inter> space M = space M - (\<Union>n. f -` {enat n} \<inter> space M)"
```
```   547       by auto
```
```   548     also have "\<dots> \<in> sets M"
```
```   549       by (intro sets.Diff sets.top sets.Un sets.countable_UN) (auto intro!: fin)
```
```   550     finally show ?thesis .
```
```   551   qed (simp add: fin)
```
```   552 qed
```
```   553
```
```   554 lemma measurable_THE:
```
```   555   fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"
```
```   556   assumes [measurable]: "\<And>i. Measurable.pred M (P i)"
```
```   557   assumes I[simp]: "countable I" "\<And>i x. x \<in> space M \<Longrightarrow> P i x \<Longrightarrow> i \<in> I"
```
```   558   assumes unique: "\<And>x i j. x \<in> space M \<Longrightarrow> P i x \<Longrightarrow> P j x \<Longrightarrow> i = j"
```
```   559   shows "(\<lambda>x. THE i. P i x) \<in> measurable M (count_space UNIV)"
```
```   560   unfolding measurable_def
```
```   561 proof safe
```
```   562   fix X
```
```   563   def f \<equiv> "\<lambda>x. THE i. P i x" def undef \<equiv> "THE i::'a. False"
```
```   564   { fix i x assume "x \<in> space M" "P i x" then have "f x = i"
```
```   565       unfolding f_def using unique by auto }
```
```   566   note f_eq = this
```
```   567   { fix x assume "x \<in> space M" "\<forall>i\<in>I. \<not> P i x"
```
```   568     then have "\<And>i. \<not> P i x"
```
```   569       using I(2)[of x] by auto
```
```   570     then have "f x = undef"
```
```   571       by (auto simp: undef_def f_def) }
```
```   572   then have "f -` X \<inter> space M = (\<Union>i\<in>I \<inter> X. {x\<in>space M. P i x}) \<union>
```
```   573      (if undef \<in> X then space M - (\<Union>i\<in>I. {x\<in>space M. P i x}) else {})"
```
```   574     by (auto dest: f_eq)
```
```   575   also have "\<dots> \<in> sets M"
```
```   576     by (auto intro!: sets.Diff sets.countable_UN')
```
```   577   finally show "f -` X \<inter> space M \<in> sets M" .
```
```   578 qed simp
```
```   579
```
```   580 lemma measurable_Ex1[measurable (raw)]:
```
```   581   assumes [simp]: "countable I" and [measurable]: "\<And>i. i \<in> I \<Longrightarrow> Measurable.pred M (P i)"
```
```   582   shows "Measurable.pred M (\<lambda>x. \<exists>!i\<in>I. P i x)"
```
```   583   unfolding bex1_def by measurable
```
```   584
```
```   585 lemma measurable_split_if[measurable (raw)]:
```
```   586   "(c \<Longrightarrow> Measurable.pred M f) \<Longrightarrow> (\<not> c \<Longrightarrow> Measurable.pred M g) \<Longrightarrow>
```
```   587    Measurable.pred M (if c then f else g)"
```
```   588   by simp
```
```   589
```
```   590 lemma pred_restrict_space:
```
```   591   assumes "S \<in> sets M"
```
```   592   shows "Measurable.pred (restrict_space M S) P \<longleftrightarrow> Measurable.pred M (\<lambda>x. x \<in> S \<and> P x)"
```
```   593   unfolding pred_def sets_Collect_restrict_space_iff[OF assms] ..
```
```   594
```
```   595 lemma measurable_predpow[measurable]:
```
```   596   assumes "Measurable.pred M T"
```
```   597   assumes "\<And>Q. Measurable.pred M Q \<Longrightarrow> Measurable.pred M (R Q)"
```
```   598   shows "Measurable.pred M ((R ^^ n) T)"
```
```   599   by (induct n) (auto intro: assms)
```
```   600
```
```   601 hide_const (open) pred
```
```   602
```
```   603 end
```
```   604
```