src/HOL/Analysis/Measurable.thy
 author hoelzl Thu Oct 20 18:41:59 2016 +0200 (2016-10-20) changeset 64320 ba194424b895 parent 64283 979cdfdf7a79 child 66453 cc19f7ca2ed6 permissions -rw-r--r--
HOL-Probability: move stopping time from AFP/Markov_Models
```     1 (*  Title:      HOL/Analysis/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 subsection \<open>Measurability prover\<close>
```
```    11
```
```    12 lemma (in algebra) sets_Collect_finite_All:
```
```    13   assumes "\<And>i. i \<in> S \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M" "finite S"
```
```    14   shows "{x\<in>\<Omega>. \<forall>i\<in>S. P i x} \<in> M"
```
```    15 proof -
```
```    16   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})"
```
```    17     by auto
```
```    18   with assms show ?thesis by (auto intro!: sets_Collect_finite_All')
```
```    19 qed
```
```    20
```
```    21 abbreviation "pred M P \<equiv> P \<in> measurable M (count_space (UNIV::bool set))"
```
```    22
```
```    23 lemma pred_def: "pred M P \<longleftrightarrow> {x\<in>space M. P x} \<in> sets M"
```
```    24 proof
```
```    25   assume "pred M P"
```
```    26   then have "P -` {True} \<inter> space M \<in> sets M"
```
```    27     by (auto simp: measurable_count_space_eq2)
```
```    28   also have "P -` {True} \<inter> space M = {x\<in>space M. P x}" by auto
```
```    29   finally show "{x\<in>space M. P x} \<in> sets M" .
```
```    30 next
```
```    31   assume P: "{x\<in>space M. P x} \<in> sets M"
```
```    32   moreover
```
```    33   { fix X
```
```    34     have "X \<in> Pow (UNIV :: bool set)" by simp
```
```    35     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> {})}"
```
```    36       unfolding UNIV_bool Pow_insert Pow_empty by auto
```
```    37     then have "P -` X \<inter> space M \<in> sets M"
```
```    38       by (auto intro!: sets.sets_Collect_neg sets.sets_Collect_imp sets.sets_Collect_conj sets.sets_Collect_const P) }
```
```    39   then show "pred M P"
```
```    40     by (auto simp: measurable_def)
```
```    41 qed
```
```    42
```
```    43 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))"
```
```    44   by (rule measurable_compose[where f=f and N=M]) (auto simp: pred_def)
```
```    45
```
```    46 lemma pred_sets2: "A \<in> sets N \<Longrightarrow> f \<in> measurable M N \<Longrightarrow> pred M (\<lambda>x. f x \<in> A)"
```
```    47   by (rule measurable_compose[where f=f and N=N]) (auto simp: pred_def Int_def[symmetric])
```
```    48
```
```    49 ML_file "measurable.ML"
```
```    50
```
```    51 attribute_setup measurable = \<open>
```
```    52   Scan.lift (
```
```    53     (Args.add >> K true || Args.del >> K false || Scan.succeed true) --
```
```    54     Scan.optional (Args.parens (
```
```    55       Scan.optional (Args.\$\$\$ "raw" >> K true) false --
```
```    56       Scan.optional (Args.\$\$\$ "generic" >> K Measurable.Generic) Measurable.Concrete))
```
```    57     (false, Measurable.Concrete) >>
```
```    58     Measurable.measurable_thm_attr)
```
```    59 \<close> "declaration of measurability theorems"
```
```    60
```
```    61 attribute_setup measurable_dest = Measurable.dest_thm_attr
```
```    62   "add dest rule to measurability prover"
```
```    63
```
```    64 attribute_setup measurable_cong = Measurable.cong_thm_attr
```
```    65   "add congurence rules to measurability prover"
```
```    66
```
```    67 method_setup measurable = \<open> Scan.lift (Scan.succeed (METHOD o Measurable.measurable_tac)) \<close>
```
```    68   "measurability prover"
```
```    69
```
```    70 simproc_setup measurable ("A \<in> sets M" | "f \<in> measurable M N") = \<open>K Measurable.simproc\<close>
```
```    71
```
```    72 setup \<open>
```
```    73   Global_Theory.add_thms_dynamic (@{binding measurable}, Measurable.get_all)
```
```    74 \<close>
```
```    75
```
```    76 declare
```
```    77   pred_sets1[measurable_dest]
```
```    78   pred_sets2[measurable_dest]
```
```    79   sets.sets_into_space[measurable_dest]
```
```    80
```
```    81 declare
```
```    82   sets.top[measurable]
```
```    83   sets.empty_sets[measurable (raw)]
```
```    84   sets.Un[measurable (raw)]
```
```    85   sets.Diff[measurable (raw)]
```
```    86
```
```    87 declare
```
```    88   measurable_count_space[measurable (raw)]
```
```    89   measurable_ident[measurable (raw)]
```
```    90   measurable_id[measurable (raw)]
```
```    91   measurable_const[measurable (raw)]
```
```    92   measurable_If[measurable (raw)]
```
```    93   measurable_comp[measurable (raw)]
```
```    94   measurable_sets[measurable (raw)]
```
```    95
```
```    96 declare measurable_cong_sets[measurable_cong]
```
```    97 declare sets_restrict_space_cong[measurable_cong]
```
```    98 declare sets_restrict_UNIV[measurable_cong]
```
```    99
```
```   100 lemma predE[measurable (raw)]:
```
```   101   "pred M P \<Longrightarrow> {x\<in>space M. P x} \<in> sets M"
```
```   102   unfolding pred_def .
```
```   103
```
```   104 lemma pred_intros_imp'[measurable (raw)]:
```
```   105   "(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<longrightarrow> P x)"
```
```   106   by (cases K) auto
```
```   107
```
```   108 lemma pred_intros_conj1'[measurable (raw)]:
```
```   109   "(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<and> P x)"
```
```   110   by (cases K) auto
```
```   111
```
```   112 lemma pred_intros_conj2'[measurable (raw)]:
```
```   113   "(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. P x \<and> K)"
```
```   114   by (cases K) auto
```
```   115
```
```   116 lemma pred_intros_disj1'[measurable (raw)]:
```
```   117   "(\<not> K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<or> P x)"
```
```   118   by (cases K) auto
```
```   119
```
```   120 lemma pred_intros_disj2'[measurable (raw)]:
```
```   121   "(\<not> K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. P x \<or> K)"
```
```   122   by (cases K) auto
```
```   123
```
```   124 lemma pred_intros_logic[measurable (raw)]:
```
```   125   "pred M (\<lambda>x. x \<in> space M)"
```
```   126   "pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. \<not> P x)"
```
```   127   "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<and> P x)"
```
```   128   "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<longrightarrow> P x)"
```
```   129   "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<or> P x)"
```
```   130   "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x = P x)"
```
```   131   "pred M (\<lambda>x. f x \<in> UNIV)"
```
```   132   "pred M (\<lambda>x. f x \<in> {})"
```
```   133   "pred M (\<lambda>x. P' (f x) x) \<Longrightarrow> pred M (\<lambda>x. f x \<in> {y. P' y x})"
```
```   134   "pred M (\<lambda>x. f x \<in> (B x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> - (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) - (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) \<inter> (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) \<union> (B x))"
```
```   138   "pred M (\<lambda>x. g x (f x) \<in> (X x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (g x) -` (X x))"
```
```   139   by (auto simp: iff_conv_conj_imp pred_def)
```
```   140
```
```   141 lemma pred_intros_countable[measurable (raw)]:
```
```   142   fixes P :: "'a \<Rightarrow> 'i :: countable \<Rightarrow> bool"
```
```   143   shows
```
```   144     "(\<And>i. pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<forall>i. P x i)"
```
```   145     "(\<And>i. pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<exists>i. P x i)"
```
```   146   by (auto intro!: sets.sets_Collect_countable_All sets.sets_Collect_countable_Ex simp: pred_def)
```
```   147
```
```   148 lemma pred_intros_countable_bounded[measurable (raw)]:
```
```   149   fixes X :: "'i :: countable set"
```
```   150   shows
```
```   151     "(\<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))"
```
```   152     "(\<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))"
```
```   153     "(\<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)"
```
```   154     "(\<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)"
```
```   155   by simp_all (auto simp: Bex_def Ball_def)
```
```   156
```
```   157 lemma pred_intros_finite[measurable (raw)]:
```
```   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> (\<Inter>i\<in>I. N x i))"
```
```   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> (\<Union>i\<in>I. N x i))"
```
```   160   "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)"
```
```   161   "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)"
```
```   162   by (auto intro!: sets.sets_Collect_finite_Ex sets.sets_Collect_finite_All simp: iff_conv_conj_imp pred_def)
```
```   163
```
```   164 lemma countable_Un_Int[measurable (raw)]:
```
```   165   "(\<And>i :: 'i :: countable. i \<in> I \<Longrightarrow> N i \<in> sets M) \<Longrightarrow> (\<Union>i\<in>I. N i) \<in> sets M"
```
```   166   "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"
```
```   167   by auto
```
```   168
```
```   169 declare
```
```   170   finite_UN[measurable (raw)]
```
```   171   finite_INT[measurable (raw)]
```
```   172
```
```   173 lemma sets_Int_pred[measurable (raw)]:
```
```   174   assumes space: "A \<inter> B \<subseteq> space M" and [measurable]: "pred M (\<lambda>x. x \<in> A)" "pred M (\<lambda>x. x \<in> B)"
```
```   175   shows "A \<inter> B \<in> sets M"
```
```   176 proof -
```
```   177   have "{x\<in>space M. x \<in> A \<inter> B} \<in> sets M" by auto
```
```   178   also have "{x\<in>space M. x \<in> A \<inter> B} = A \<inter> B"
```
```   179     using space by auto
```
```   180   finally show ?thesis .
```
```   181 qed
```
```   182
```
```   183 lemma [measurable (raw generic)]:
```
```   184   assumes f: "f \<in> measurable M N" and c: "c \<in> space N \<Longrightarrow> {c} \<in> sets N"
```
```   185   shows pred_eq_const1: "pred M (\<lambda>x. f x = c)"
```
```   186     and pred_eq_const2: "pred M (\<lambda>x. c = f x)"
```
```   187 proof -
```
```   188   show "pred M (\<lambda>x. f x = c)"
```
```   189   proof cases
```
```   190     assume "c \<in> space N"
```
```   191     with measurable_sets[OF f c] show ?thesis
```
```   192       by (auto simp: Int_def conj_commute pred_def)
```
```   193   next
```
```   194     assume "c \<notin> space N"
```
```   195     with f[THEN measurable_space] have "{x \<in> space M. f x = c} = {}" by auto
```
```   196     then show ?thesis by (auto simp: pred_def cong: conj_cong)
```
```   197   qed
```
```   198   then show "pred M (\<lambda>x. c = f x)"
```
```   199     by (simp add: eq_commute)
```
```   200 qed
```
```   201
```
```   202 lemma pred_count_space_const1[measurable (raw)]:
```
```   203   "f \<in> measurable M (count_space UNIV) \<Longrightarrow> Measurable.pred M (\<lambda>x. f x = c)"
```
```   204   by (intro pred_eq_const1[where N="count_space UNIV"]) (auto )
```
```   205
```
```   206 lemma pred_count_space_const2[measurable (raw)]:
```
```   207   "f \<in> measurable M (count_space UNIV) \<Longrightarrow> Measurable.pred M (\<lambda>x. c = f x)"
```
```   208   by (intro pred_eq_const2[where N="count_space UNIV"]) (auto )
```
```   209
```
```   210 lemma pred_le_const[measurable (raw generic)]:
```
```   211   assumes f: "f \<in> measurable M N" and c: "{.. c} \<in> sets N" shows "pred M (\<lambda>x. f x \<le> c)"
```
```   212   using measurable_sets[OF f c]
```
```   213   by (auto simp: Int_def conj_commute eq_commute pred_def)
```
```   214
```
```   215 lemma pred_const_le[measurable (raw generic)]:
```
```   216   assumes f: "f \<in> measurable M N" and c: "{c ..} \<in> sets N" shows "pred M (\<lambda>x. c \<le> f x)"
```
```   217   using measurable_sets[OF f c]
```
```   218   by (auto simp: Int_def conj_commute eq_commute pred_def)
```
```   219
```
```   220 lemma pred_less_const[measurable (raw generic)]:
```
```   221   assumes f: "f \<in> measurable M N" and c: "{..< c} \<in> sets N" shows "pred M (\<lambda>x. f x < c)"
```
```   222   using measurable_sets[OF f c]
```
```   223   by (auto simp: Int_def conj_commute eq_commute pred_def)
```
```   224
```
```   225 lemma pred_const_less[measurable (raw generic)]:
```
```   226   assumes f: "f \<in> measurable M N" and c: "{c <..} \<in> sets N" shows "pred M (\<lambda>x. c < f x)"
```
```   227   using measurable_sets[OF f c]
```
```   228   by (auto simp: Int_def conj_commute eq_commute pred_def)
```
```   229
```
```   230 declare
```
```   231   sets.Int[measurable (raw)]
```
```   232
```
```   233 lemma pred_in_If[measurable (raw)]:
```
```   234   "(P \<Longrightarrow> pred M (\<lambda>x. x \<in> A x)) \<Longrightarrow> (\<not> P \<Longrightarrow> pred M (\<lambda>x. x \<in> B x)) \<Longrightarrow>
```
```   235     pred M (\<lambda>x. x \<in> (if P then A x else B x))"
```
```   236   by auto
```
```   237
```
```   238 lemma sets_range[measurable_dest]:
```
```   239   "A ` I \<subseteq> sets M \<Longrightarrow> i \<in> I \<Longrightarrow> A i \<in> sets M"
```
```   240   by auto
```
```   241
```
```   242 lemma pred_sets_range[measurable_dest]:
```
```   243   "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)"
```
```   244   using pred_sets2[OF sets_range] by auto
```
```   245
```
```   246 lemma sets_All[measurable_dest]:
```
```   247   "\<forall>i. A i \<in> sets (M i) \<Longrightarrow> A i \<in> sets (M i)"
```
```   248   by auto
```
```   249
```
```   250 lemma pred_sets_All[measurable_dest]:
```
```   251   "\<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)"
```
```   252   using pred_sets2[OF sets_All, of A N f] by auto
```
```   253
```
```   254 lemma sets_Ball[measurable_dest]:
```
```   255   "\<forall>i\<in>I. A i \<in> sets (M i) \<Longrightarrow> i\<in>I \<Longrightarrow> A i \<in> sets (M i)"
```
```   256   by auto
```
```   257
```
```   258 lemma pred_sets_Ball[measurable_dest]:
```
```   259   "\<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)"
```
```   260   using pred_sets2[OF sets_Ball, of _ _ _ f] by auto
```
```   261
```
```   262 lemma measurable_finite[measurable (raw)]:
```
```   263   fixes S :: "'a \<Rightarrow> nat set"
```
```   264   assumes [measurable]: "\<And>i. {x\<in>space M. i \<in> S x} \<in> sets M"
```
```   265   shows "pred M (\<lambda>x. finite (S x))"
```
```   266   unfolding finite_nat_set_iff_bounded by (simp add: Ball_def)
```
```   267
```
```   268 lemma measurable_Least[measurable]:
```
```   269   assumes [measurable]: "(\<And>i::nat. (\<lambda>x. P i x) \<in> measurable M (count_space UNIV))"
```
```   270   shows "(\<lambda>x. LEAST i. P i x) \<in> measurable M (count_space UNIV)"
```
```   271   unfolding measurable_def by (safe intro!: sets_Least) simp_all
```
```   272
```
```   273 lemma measurable_Max_nat[measurable (raw)]:
```
```   274   fixes P :: "nat \<Rightarrow> 'a \<Rightarrow> bool"
```
```   275   assumes [measurable]: "\<And>i. Measurable.pred M (P i)"
```
```   276   shows "(\<lambda>x. Max {i. P i x}) \<in> measurable M (count_space UNIV)"
```
```   277   unfolding measurable_count_space_eq2_countable
```
```   278 proof safe
```
```   279   fix n
```
```   280
```
```   281   { fix x assume "\<forall>i. \<exists>n\<ge>i. P n x"
```
```   282     then have "infinite {i. P i x}"
```
```   283       unfolding infinite_nat_iff_unbounded_le by auto
```
```   284     then have "Max {i. P i x} = the None"
```
```   285       by (rule Max.infinite) }
```
```   286   note 1 = this
```
```   287
```
```   288   { fix x i j assume "P i x" "\<forall>n\<ge>j. \<not> P n x"
```
```   289     then have "finite {i. P i x}"
```
```   290       by (auto simp: subset_eq not_le[symmetric] finite_nat_iff_bounded)
```
```   291     with \<open>P i x\<close> have "P (Max {i. P i x}) x" "i \<le> Max {i. P i x}" "finite {i. P i x}"
```
```   292       using Max_in[of "{i. P i x}"] by auto }
```
```   293   note 2 = this
```
```   294
```
```   295   have "(\<lambda>x. Max {i. P i x}) -` {n} \<inter> space M = {x\<in>space M. Max {i. P i x} = n}"
```
```   296     by auto
```
```   297   also have "\<dots> =
```
```   298     {x\<in>space M. if (\<forall>i. \<exists>n\<ge>i. P n x) then the None = n else
```
```   299       if (\<exists>i. P i x) then P n x \<and> (\<forall>i>n. \<not> P i x)
```
```   300       else Max {} = n}"
```
```   301     by (intro arg_cong[where f=Collect] ext conj_cong)
```
```   302        (auto simp add: 1 2 not_le[symmetric] intro!: Max_eqI)
```
```   303   also have "\<dots> \<in> sets M"
```
```   304     by measurable
```
```   305   finally show "(\<lambda>x. Max {i. P i x}) -` {n} \<inter> space M \<in> sets M" .
```
```   306 qed simp
```
```   307
```
```   308 lemma measurable_Min_nat[measurable (raw)]:
```
```   309   fixes P :: "nat \<Rightarrow> 'a \<Rightarrow> bool"
```
```   310   assumes [measurable]: "\<And>i. Measurable.pred M (P i)"
```
```   311   shows "(\<lambda>x. Min {i. P i x}) \<in> measurable M (count_space UNIV)"
```
```   312   unfolding measurable_count_space_eq2_countable
```
```   313 proof safe
```
```   314   fix n
```
```   315
```
```   316   { fix x assume "\<forall>i. \<exists>n\<ge>i. P n x"
```
```   317     then have "infinite {i. P i x}"
```
```   318       unfolding infinite_nat_iff_unbounded_le by auto
```
```   319     then have "Min {i. P i x} = the None"
```
```   320       by (rule Min.infinite) }
```
```   321   note 1 = this
```
```   322
```
```   323   { fix x i j assume "P i x" "\<forall>n\<ge>j. \<not> P n x"
```
```   324     then have "finite {i. P i x}"
```
```   325       by (auto simp: subset_eq not_le[symmetric] finite_nat_iff_bounded)
```
```   326     with \<open>P i x\<close> have "P (Min {i. P i x}) x" "Min {i. P i x} \<le> i" "finite {i. P i x}"
```
```   327       using Min_in[of "{i. P i x}"] by auto }
```
```   328   note 2 = this
```
```   329
```
```   330   have "(\<lambda>x. Min {i. P i x}) -` {n} \<inter> space M = {x\<in>space M. Min {i. P i x} = n}"
```
```   331     by auto
```
```   332   also have "\<dots> =
```
```   333     {x\<in>space M. if (\<forall>i. \<exists>n\<ge>i. P n x) then the None = n else
```
```   334       if (\<exists>i. P i x) then P n x \<and> (\<forall>i<n. \<not> P i x)
```
```   335       else Min {} = n}"
```
```   336     by (intro arg_cong[where f=Collect] ext conj_cong)
```
```   337        (auto simp add: 1 2 not_le[symmetric] intro!: Min_eqI)
```
```   338   also have "\<dots> \<in> sets M"
```
```   339     by measurable
```
```   340   finally show "(\<lambda>x. Min {i. P i x}) -` {n} \<inter> space M \<in> sets M" .
```
```   341 qed simp
```
```   342
```
```   343 lemma measurable_count_space_insert[measurable (raw)]:
```
```   344   "s \<in> S \<Longrightarrow> A \<in> sets (count_space S) \<Longrightarrow> insert s A \<in> sets (count_space S)"
```
```   345   by simp
```
```   346
```
```   347 lemma sets_UNIV [measurable (raw)]: "A \<in> sets (count_space UNIV)"
```
```   348   by simp
```
```   349
```
```   350 lemma measurable_card[measurable]:
```
```   351   fixes S :: "'a \<Rightarrow> nat set"
```
```   352   assumes [measurable]: "\<And>i. {x\<in>space M. i \<in> S x} \<in> sets M"
```
```   353   shows "(\<lambda>x. card (S x)) \<in> measurable M (count_space UNIV)"
```
```   354   unfolding measurable_count_space_eq2_countable
```
```   355 proof safe
```
```   356   fix n show "(\<lambda>x. card (S x)) -` {n} \<inter> space M \<in> sets M"
```
```   357   proof (cases n)
```
```   358     case 0
```
```   359     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)}"
```
```   360       by auto
```
```   361     also have "\<dots> \<in> sets M"
```
```   362       by measurable
```
```   363     finally show ?thesis .
```
```   364   next
```
```   365     case (Suc i)
```
```   366     then have "(\<lambda>x. card (S x)) -` {n} \<inter> space M =
```
```   367       (\<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)})"
```
```   368       unfolding set_eq_iff[symmetric] Collect_bex_eq[symmetric] by (auto intro: card_ge_0_finite)
```
```   369     also have "\<dots> \<in> sets M"
```
```   370       by (intro sets.countable_UN' countable_Collect countable_Collect_finite) auto
```
```   371     finally show ?thesis .
```
```   372   qed
```
```   373 qed rule
```
```   374
```
```   375 lemma measurable_pred_countable[measurable (raw)]:
```
```   376   assumes "countable X"
```
```   377   shows
```
```   378     "(\<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)"
```
```   379     "(\<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)"
```
```   380   unfolding pred_def
```
```   381   by (auto intro!: sets.sets_Collect_countable_All' sets.sets_Collect_countable_Ex' assms)
```
```   382
```
```   383 subsection \<open>Measurability for (co)inductive predicates\<close>
```
```   384
```
```   385 lemma measurable_bot[measurable]: "bot \<in> measurable M (count_space UNIV)"
```
```   386   by (simp add: bot_fun_def)
```
```   387
```
```   388 lemma measurable_top[measurable]: "top \<in> measurable M (count_space UNIV)"
```
```   389   by (simp add: top_fun_def)
```
```   390
```
```   391 lemma measurable_SUP[measurable]:
```
```   392   fixes F :: "'i \<Rightarrow> 'a \<Rightarrow> 'b::{complete_lattice, countable}"
```
```   393   assumes [simp]: "countable I"
```
```   394   assumes [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> measurable M (count_space UNIV)"
```
```   395   shows "(\<lambda>x. SUP i:I. F i x) \<in> measurable M (count_space UNIV)"
```
```   396   unfolding measurable_count_space_eq2_countable
```
```   397 proof (safe intro!: UNIV_I)
```
```   398   fix a
```
```   399   have "(\<lambda>x. SUP i:I. F i x) -` {a} \<inter> space M =
```
```   400     {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)}"
```
```   401     unfolding SUP_le_iff[symmetric] by auto
```
```   402   also have "\<dots> \<in> sets M"
```
```   403     by measurable
```
```   404   finally show "(\<lambda>x. SUP i:I. F i x) -` {a} \<inter> space M \<in> sets M" .
```
```   405 qed
```
```   406
```
```   407 lemma measurable_INF[measurable]:
```
```   408   fixes F :: "'i \<Rightarrow> 'a \<Rightarrow> 'b::{complete_lattice, countable}"
```
```   409   assumes [simp]: "countable I"
```
```   410   assumes [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> measurable M (count_space UNIV)"
```
```   411   shows "(\<lambda>x. INF i:I. F i x) \<in> measurable M (count_space UNIV)"
```
```   412   unfolding measurable_count_space_eq2_countable
```
```   413 proof (safe intro!: UNIV_I)
```
```   414   fix a
```
```   415   have "(\<lambda>x. INF i:I. F i x) -` {a} \<inter> space M =
```
```   416     {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)}"
```
```   417     unfolding le_INF_iff[symmetric] by auto
```
```   418   also have "\<dots> \<in> sets M"
```
```   419     by measurable
```
```   420   finally show "(\<lambda>x. INF i:I. F i x) -` {a} \<inter> space M \<in> sets M" .
```
```   421 qed
```
```   422
```
```   423 lemma measurable_lfp_coinduct[consumes 1, case_names continuity step]:
```
```   424   fixes F :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b::{complete_lattice, countable})"
```
```   425   assumes "P M"
```
```   426   assumes F: "sup_continuous F"
```
```   427   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)"
```
```   428   shows "lfp F \<in> measurable M (count_space UNIV)"
```
```   429 proof -
```
```   430   { fix i from \<open>P M\<close> have "((F ^^ i) bot) \<in> measurable M (count_space UNIV)"
```
```   431       by (induct i arbitrary: M) (auto intro!: *) }
```
```   432   then have "(\<lambda>x. SUP i. (F ^^ i) bot x) \<in> measurable M (count_space UNIV)"
```
```   433     by measurable
```
```   434   also have "(\<lambda>x. SUP i. (F ^^ i) bot x) = lfp F"
```
```   435     by (subst sup_continuous_lfp) (auto intro: F)
```
```   436   finally show ?thesis .
```
```   437 qed
```
```   438
```
```   439 lemma measurable_lfp:
```
```   440   fixes F :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b::{complete_lattice, countable})"
```
```   441   assumes F: "sup_continuous F"
```
```   442   assumes *: "\<And>A. A \<in> measurable M (count_space UNIV) \<Longrightarrow> F A \<in> measurable M (count_space UNIV)"
```
```   443   shows "lfp F \<in> measurable M (count_space UNIV)"
```
```   444   by (coinduction rule: measurable_lfp_coinduct[OF _ F]) (blast intro: *)
```
```   445
```
```   446 lemma measurable_gfp_coinduct[consumes 1, case_names continuity step]:
```
```   447   fixes F :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b::{complete_lattice, countable})"
```
```   448   assumes "P M"
```
```   449   assumes F: "inf_continuous F"
```
```   450   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)"
```
```   451   shows "gfp F \<in> measurable M (count_space UNIV)"
```
```   452 proof -
```
```   453   { fix i from \<open>P M\<close> have "((F ^^ i) top) \<in> measurable M (count_space UNIV)"
```
```   454       by (induct i arbitrary: M) (auto intro!: *) }
```
```   455   then have "(\<lambda>x. INF i. (F ^^ i) top x) \<in> measurable M (count_space UNIV)"
```
```   456     by measurable
```
```   457   also have "(\<lambda>x. INF i. (F ^^ i) top x) = gfp F"
```
```   458     by (subst inf_continuous_gfp) (auto intro: F)
```
```   459   finally show ?thesis .
```
```   460 qed
```
```   461
```
```   462 lemma measurable_gfp:
```
```   463   fixes F :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b::{complete_lattice, countable})"
```
```   464   assumes F: "inf_continuous F"
```
```   465   assumes *: "\<And>A. A \<in> measurable M (count_space UNIV) \<Longrightarrow> F A \<in> measurable M (count_space UNIV)"
```
```   466   shows "gfp F \<in> measurable M (count_space UNIV)"
```
```   467   by (coinduction rule: measurable_gfp_coinduct[OF _ F]) (blast intro: *)
```
```   468
```
```   469 lemma measurable_lfp2_coinduct[consumes 1, case_names continuity step]:
```
```   470   fixes F :: "('a \<Rightarrow> 'c \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'c \<Rightarrow> 'b::{complete_lattice, countable})"
```
```   471   assumes "P M s"
```
```   472   assumes F: "sup_continuous F"
```
```   473   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)"
```
```   474   shows "lfp F s \<in> measurable M (count_space UNIV)"
```
```   475 proof -
```
```   476   { fix i from \<open>P M s\<close> have "(\<lambda>x. (F ^^ i) bot s x) \<in> measurable M (count_space UNIV)"
```
```   477       by (induct i arbitrary: M s) (auto intro!: *) }
```
```   478   then have "(\<lambda>x. SUP i. (F ^^ i) bot s x) \<in> measurable M (count_space UNIV)"
```
```   479     by measurable
```
```   480   also have "(\<lambda>x. SUP i. (F ^^ i) bot s x) = lfp F s"
```
```   481     by (subst sup_continuous_lfp) (auto simp: F)
```
```   482   finally show ?thesis .
```
```   483 qed
```
```   484
```
```   485 lemma measurable_gfp2_coinduct[consumes 1, case_names continuity step]:
```
```   486   fixes F :: "('a \<Rightarrow> 'c \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'c \<Rightarrow> 'b::{complete_lattice, countable})"
```
```   487   assumes "P M s"
```
```   488   assumes F: "inf_continuous F"
```
```   489   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)"
```
```   490   shows "gfp F s \<in> measurable M (count_space UNIV)"
```
```   491 proof -
```
```   492   { fix i from \<open>P M s\<close> have "(\<lambda>x. (F ^^ i) top s x) \<in> measurable M (count_space UNIV)"
```
```   493       by (induct i arbitrary: M s) (auto intro!: *) }
```
```   494   then have "(\<lambda>x. INF i. (F ^^ i) top s x) \<in> measurable M (count_space UNIV)"
```
```   495     by measurable
```
```   496   also have "(\<lambda>x. INF i. (F ^^ i) top s x) = gfp F s"
```
```   497     by (subst inf_continuous_gfp) (auto simp: F)
```
```   498   finally show ?thesis .
```
```   499 qed
```
```   500
```
```   501 lemma measurable_enat_coinduct:
```
```   502   fixes f :: "'a \<Rightarrow> enat"
```
```   503   assumes "R f"
```
```   504   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>
```
```   505     Measurable.pred M P \<and>
```
```   506     i \<in> measurable M M \<and>
```
```   507     h \<in> measurable M (count_space UNIV)"
```
```   508   shows "f \<in> measurable M (count_space UNIV)"
```
```   509 proof (simp add: measurable_count_space_eq2_countable, rule )
```
```   510   fix a :: enat
```
```   511   have "f -` {a} \<inter> space M = {x\<in>space M. f x = a}"
```
```   512     by auto
```
```   513   { fix i :: nat
```
```   514     from \<open>R f\<close> have "Measurable.pred M (\<lambda>x. f x = enat i)"
```
```   515     proof (induction i arbitrary: f)
```
```   516       case 0
```
```   517       from *[OF this] obtain g h i P
```
```   518         where f: "f = (\<lambda>x. if P x then h x else eSuc (g (i x)))" and
```
```   519           [measurable]: "Measurable.pred M P" "i \<in> measurable M M" "h \<in> measurable M (count_space UNIV)"
```
```   520         by auto
```
```   521       have "Measurable.pred M (\<lambda>x. P x \<and> h x = 0)"
```
```   522         by measurable
```
```   523       also have "(\<lambda>x. P x \<and> h x = 0) = (\<lambda>x. f x = enat 0)"
```
```   524         by (auto simp: f zero_enat_def[symmetric])
```
```   525       finally show ?case .
```
```   526     next
```
```   527       case (Suc n)
```
```   528       from *[OF Suc.prems] obtain g h i P
```
```   529         where f: "f = (\<lambda>x. if P x then h x else eSuc (g (i x)))" and "R g" and
```
```   530           M[measurable]: "Measurable.pred M P" "i \<in> measurable M M" "h \<in> measurable M (count_space UNIV)"
```
```   531         by auto
```
```   532       have "(\<lambda>x. f x = enat (Suc n)) =
```
```   533         (\<lambda>x. (P x \<longrightarrow> h x = enat (Suc n)) \<and> (\<not> P x \<longrightarrow> g (i x) = enat n))"
```
```   534         by (auto simp: f zero_enat_def[symmetric] eSuc_enat[symmetric])
```
```   535       also have "Measurable.pred M \<dots>"
```
```   536         by (intro pred_intros_logic measurable_compose[OF M(2)] Suc \<open>R g\<close>) measurable
```
```   537       finally show ?case .
```
```   538     qed
```
```   539     then have "f -` {enat i} \<inter> space M \<in> sets M"
```
```   540       by (simp add: pred_def Int_def conj_commute) }
```
```   541   note fin = this
```
```   542   show "f -` {a} \<inter> space M \<in> sets M"
```
```   543   proof (cases a)
```
```   544     case infinity
```
```   545     then have "f -` {a} \<inter> space M = space M - (\<Union>n. f -` {enat n} \<inter> space M)"
```
```   546       by auto
```
```   547     also have "\<dots> \<in> sets M"
```
```   548       by (intro sets.Diff sets.top sets.Un sets.countable_UN) (auto intro!: fin)
```
```   549     finally show ?thesis .
```
```   550   qed (simp add: fin)
```
```   551 qed
```
```   552
```
```   553 lemma measurable_THE:
```
```   554   fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"
```
```   555   assumes [measurable]: "\<And>i. Measurable.pred M (P i)"
```
```   556   assumes I[simp]: "countable I" "\<And>i x. x \<in> space M \<Longrightarrow> P i x \<Longrightarrow> i \<in> I"
```
```   557   assumes unique: "\<And>x i j. x \<in> space M \<Longrightarrow> P i x \<Longrightarrow> P j x \<Longrightarrow> i = j"
```
```   558   shows "(\<lambda>x. THE i. P i x) \<in> measurable M (count_space UNIV)"
```
```   559   unfolding measurable_def
```
```   560 proof safe
```
```   561   fix X
```
```   562   define f where "f x = (THE i. P i x)" for x
```
```   563   define undef where "undef = (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_Sup_nat[measurable (raw)]:
```
```   586   fixes F :: "'a \<Rightarrow> nat set"
```
```   587   assumes [measurable]: "\<And>i. Measurable.pred M (\<lambda>x. i \<in> F x)"
```
```   588   shows "(\<lambda>x. Sup (F x)) \<in> M \<rightarrow>\<^sub>M count_space UNIV"
```
```   589 proof (clarsimp simp add: measurable_count_space_eq2_countable)
```
```   590   fix a
```
```   591   have F_empty_iff: "F x = {} \<longleftrightarrow> (\<forall>i. i \<notin> F x)" for x
```
```   592     by auto
```
```   593   have "Measurable.pred M (\<lambda>x. if finite (F x) then if F x = {} then a = Max {}
```
```   594     else a \<in> F x \<and> (\<forall>j. j \<in> F x \<longrightarrow> j \<le> a) else a = the None)"
```
```   595     unfolding finite_nat_set_iff_bounded Ball_def F_empty_iff by measurable
```
```   596   moreover have "(\<lambda>x. Sup (F x)) -` {a} \<inter> space M =
```
```   597     {x\<in>space M. if finite (F x) then if F x = {} then a = Max {}
```
```   598       else a \<in> F x \<and> (\<forall>j. j \<in> F x \<longrightarrow> j \<le> a) else a = the None}"
```
```   599     by (intro set_eqI)
```
```   600        (auto simp: Sup_nat_def Max.infinite intro!: Max_in Max_eqI)
```
```   601   ultimately show "(\<lambda>x. Sup (F x)) -` {a} \<inter> space M \<in> sets M"
```
```   602     by auto
```
```   603 qed
```
```   604
```
```   605 lemma measurable_if_split[measurable (raw)]:
```
```   606   "(c \<Longrightarrow> Measurable.pred M f) \<Longrightarrow> (\<not> c \<Longrightarrow> Measurable.pred M g) \<Longrightarrow>
```
```   607    Measurable.pred M (if c then f else g)"
```
```   608   by simp
```
```   609
```
```   610 lemma pred_restrict_space:
```
```   611   assumes "S \<in> sets M"
```
```   612   shows "Measurable.pred (restrict_space M S) P \<longleftrightarrow> Measurable.pred M (\<lambda>x. x \<in> S \<and> P x)"
```
```   613   unfolding pred_def sets_Collect_restrict_space_iff[OF assms] ..
```
```   614
```
```   615 lemma measurable_predpow[measurable]:
```
```   616   assumes "Measurable.pred M T"
```
```   617   assumes "\<And>Q. Measurable.pred M Q \<Longrightarrow> Measurable.pred M (R Q)"
```
```   618   shows "Measurable.pred M ((R ^^ n) T)"
```
```   619   by (induct n) (auto intro: assms)
```
```   620
```
```   621 lemma measurable_compose_countable_restrict:
```
```   622   assumes P: "countable {i. P i}"
```
```   623     and f: "f \<in> M \<rightarrow>\<^sub>M count_space UNIV"
```
```   624     and Q: "\<And>i. P i \<Longrightarrow> pred M (Q i)"
```
```   625   shows "pred M (\<lambda>x. P (f x) \<and> Q (f x) x)"
```
```   626 proof -
```
```   627   have P_f: "{x \<in> space M. P (f x)} \<in> sets M"
```
```   628     unfolding pred_def[symmetric] by (rule measurable_compose[OF f]) simp
```
```   629   have "pred (restrict_space M {x\<in>space M. P (f x)}) (\<lambda>x. Q (f x) x)"
```
```   630   proof (rule measurable_compose_countable'[where g=f, OF _ _ P])
```
```   631     show "f \<in> restrict_space M {x\<in>space M. P (f x)} \<rightarrow>\<^sub>M count_space {i. P i}"
```
```   632       by (rule measurable_count_space_extend[OF subset_UNIV])
```
```   633          (auto simp: space_restrict_space intro!: measurable_restrict_space1 f)
```
```   634   qed (auto intro!: measurable_restrict_space1 Q)
```
```   635   then show ?thesis
```
```   636     unfolding pred_restrict_space[OF P_f] by (simp cong: measurable_cong)
```
```   637 qed
```
```   638
```
```   639 lemma measurable_limsup [measurable (raw)]:
```
```   640   assumes [measurable]: "\<And>n. A n \<in> sets M"
```
```   641   shows "limsup A \<in> sets M"
```
```   642 by (subst limsup_INF_SUP, auto)
```
```   643
```
```   644 lemma measurable_liminf [measurable (raw)]:
```
```   645   assumes [measurable]: "\<And>n. A n \<in> sets M"
```
```   646   shows "liminf A \<in> sets M"
```
```   647 by (subst liminf_SUP_INF, auto)
```
```   648
```
```   649 lemma measurable_case_enat[measurable (raw)]:
```
```   650   assumes f: "f \<in> M \<rightarrow>\<^sub>M count_space UNIV" and g: "\<And>i. g i \<in> M \<rightarrow>\<^sub>M N" and h: "h \<in> M \<rightarrow>\<^sub>M N"
```
```   651   shows "(\<lambda>x. case f x of enat i \<Rightarrow> g i x | \<infinity> \<Rightarrow> h x) \<in> M \<rightarrow>\<^sub>M N"
```
```   652   apply (rule measurable_compose_countable[OF _ f])
```
```   653   subgoal for i
```
```   654     by (cases i) (auto intro: g h)
```
```   655   done
```
```   656
```
```   657 hide_const (open) pred
```
```   658
```
```   659 end
```
```   660
```