src/HOL/Probability/Measurable.thy
 author hoelzl Thu Nov 13 17:19:52 2014 +0100 (2014-11-13) changeset 59000 6eb0725503fc parent 58965 a62cdcc5344b child 59047 8d7cec9b861d permissions -rw-r--r--
import general theorems from AFP/Markov_Models
```     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_count_space_const1[measurable (raw)]:
```
```   202   "f \<in> measurable M (count_space UNIV) \<Longrightarrow> Measurable.pred M (\<lambda>x. f x = c)"
```
```   203   by (intro pred_eq_const1[where N="count_space UNIV"]) (auto )
```
```   204
```
```   205 lemma pred_count_space_const2[measurable (raw)]:
```
```   206   "f \<in> measurable M (count_space UNIV) \<Longrightarrow> Measurable.pred M (\<lambda>x. c = f x)"
```
```   207   by (intro pred_eq_const2[where N="count_space UNIV"]) (auto )
```
```   208
```
```   209 lemma pred_le_const[measurable (raw generic)]:
```
```   210   assumes f: "f \<in> measurable M N" and c: "{.. c} \<in> sets N" shows "pred M (\<lambda>x. f x \<le> c)"
```
```   211   using measurable_sets[OF f c]
```
```   212   by (auto simp: Int_def conj_commute eq_commute pred_def)
```
```   213
```
```   214 lemma pred_const_le[measurable (raw generic)]:
```
```   215   assumes f: "f \<in> measurable M N" and c: "{c ..} \<in> sets N" shows "pred M (\<lambda>x. c \<le> f x)"
```
```   216   using measurable_sets[OF f c]
```
```   217   by (auto simp: Int_def conj_commute eq_commute pred_def)
```
```   218
```
```   219 lemma pred_less_const[measurable (raw generic)]:
```
```   220   assumes f: "f \<in> measurable M N" and c: "{..< c} \<in> sets N" shows "pred M (\<lambda>x. f x < c)"
```
```   221   using measurable_sets[OF f c]
```
```   222   by (auto simp: Int_def conj_commute eq_commute pred_def)
```
```   223
```
```   224 lemma pred_const_less[measurable (raw generic)]:
```
```   225   assumes f: "f \<in> measurable M N" and c: "{c <..} \<in> sets N" shows "pred M (\<lambda>x. c < f x)"
```
```   226   using measurable_sets[OF f c]
```
```   227   by (auto simp: Int_def conj_commute eq_commute pred_def)
```
```   228
```
```   229 declare
```
```   230   sets.Int[measurable (raw)]
```
```   231
```
```   232 lemma pred_in_If[measurable (raw)]:
```
```   233   "(P \<Longrightarrow> pred M (\<lambda>x. x \<in> A x)) \<Longrightarrow> (\<not> P \<Longrightarrow> pred M (\<lambda>x. x \<in> B x)) \<Longrightarrow>
```
```   234     pred M (\<lambda>x. x \<in> (if P then A x else B x))"
```
```   235   by auto
```
```   236
```
```   237 lemma sets_range[measurable_dest]:
```
```   238   "A ` I \<subseteq> sets M \<Longrightarrow> i \<in> I \<Longrightarrow> A i \<in> sets M"
```
```   239   by auto
```
```   240
```
```   241 lemma pred_sets_range[measurable_dest]:
```
```   242   "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)"
```
```   243   using pred_sets2[OF sets_range] by auto
```
```   244
```
```   245 lemma sets_All[measurable_dest]:
```
```   246   "\<forall>i. A i \<in> sets (M i) \<Longrightarrow> A i \<in> sets (M i)"
```
```   247   by auto
```
```   248
```
```   249 lemma pred_sets_All[measurable_dest]:
```
```   250   "\<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)"
```
```   251   using pred_sets2[OF sets_All, of A N f] by auto
```
```   252
```
```   253 lemma sets_Ball[measurable_dest]:
```
```   254   "\<forall>i\<in>I. A i \<in> sets (M i) \<Longrightarrow> i\<in>I \<Longrightarrow> A i \<in> sets (M i)"
```
```   255   by auto
```
```   256
```
```   257 lemma pred_sets_Ball[measurable_dest]:
```
```   258   "\<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)"
```
```   259   using pred_sets2[OF sets_Ball, of _ _ _ f] by auto
```
```   260
```
```   261 lemma measurable_finite[measurable (raw)]:
```
```   262   fixes S :: "'a \<Rightarrow> nat set"
```
```   263   assumes [measurable]: "\<And>i. {x\<in>space M. i \<in> S x} \<in> sets M"
```
```   264   shows "pred M (\<lambda>x. finite (S x))"
```
```   265   unfolding finite_nat_set_iff_bounded by (simp add: Ball_def)
```
```   266
```
```   267 lemma measurable_Least[measurable]:
```
```   268   assumes [measurable]: "(\<And>i::nat. (\<lambda>x. P i x) \<in> measurable M (count_space UNIV))"q
```
```   269   shows "(\<lambda>x. LEAST i. P i x) \<in> measurable M (count_space UNIV)"
```
```   270   unfolding measurable_def by (safe intro!: sets_Least) simp_all
```
```   271
```
```   272 lemma measurable_Max_nat[measurable (raw)]:
```
```   273   fixes P :: "nat \<Rightarrow> 'a \<Rightarrow> bool"
```
```   274   assumes [measurable]: "\<And>i. Measurable.pred M (P i)"
```
```   275   shows "(\<lambda>x. Max {i. P i x}) \<in> measurable M (count_space UNIV)"
```
```   276   unfolding measurable_count_space_eq2_countable
```
```   277 proof safe
```
```   278   fix n
```
```   279
```
```   280   { fix x assume "\<forall>i. \<exists>n\<ge>i. P n x"
```
```   281     then have "infinite {i. P i x}"
```
```   282       unfolding infinite_nat_iff_unbounded_le by auto
```
```   283     then have "Max {i. P i x} = the None"
```
```   284       by (rule Max.infinite) }
```
```   285   note 1 = this
```
```   286
```
```   287   { fix x i j assume "P i x" "\<forall>n\<ge>j. \<not> P n x"
```
```   288     then have "finite {i. P i x}"
```
```   289       by (auto simp: subset_eq not_le[symmetric] finite_nat_iff_bounded)
```
```   290     with `P i x` have "P (Max {i. P i x}) x" "i \<le> Max {i. P i x}" "finite {i. P i x}"
```
```   291       using Max_in[of "{i. P i x}"] by auto }
```
```   292   note 2 = this
```
```   293
```
```   294   have "(\<lambda>x. Max {i. P i x}) -` {n} \<inter> space M = {x\<in>space M. Max {i. P i x} = n}"
```
```   295     by auto
```
```   296   also have "\<dots> =
```
```   297     {x\<in>space M. if (\<forall>i. \<exists>n\<ge>i. P n x) then the None = n else
```
```   298       if (\<exists>i. P i x) then P n x \<and> (\<forall>i>n. \<not> P i x)
```
```   299       else Max {} = n}"
```
```   300     by (intro arg_cong[where f=Collect] ext conj_cong)
```
```   301        (auto simp add: 1 2 not_le[symmetric] intro!: Max_eqI)
```
```   302   also have "\<dots> \<in> sets M"
```
```   303     by measurable
```
```   304   finally show "(\<lambda>x. Max {i. P i x}) -` {n} \<inter> space M \<in> sets M" .
```
```   305 qed simp
```
```   306
```
```   307 lemma measurable_Min_nat[measurable (raw)]:
```
```   308   fixes P :: "nat \<Rightarrow> 'a \<Rightarrow> bool"
```
```   309   assumes [measurable]: "\<And>i. Measurable.pred M (P i)"
```
```   310   shows "(\<lambda>x. Min {i. P i x}) \<in> measurable M (count_space UNIV)"
```
```   311   unfolding measurable_count_space_eq2_countable
```
```   312 proof safe
```
```   313   fix n
```
```   314
```
```   315   { fix x assume "\<forall>i. \<exists>n\<ge>i. P n x"
```
```   316     then have "infinite {i. P i x}"
```
```   317       unfolding infinite_nat_iff_unbounded_le by auto
```
```   318     then have "Min {i. P i x} = the None"
```
```   319       by (rule Min.infinite) }
```
```   320   note 1 = this
```
```   321
```
```   322   { fix x i j assume "P i x" "\<forall>n\<ge>j. \<not> P n x"
```
```   323     then have "finite {i. P i x}"
```
```   324       by (auto simp: subset_eq not_le[symmetric] finite_nat_iff_bounded)
```
```   325     with `P i x` have "P (Min {i. P i x}) x" "Min {i. P i x} \<le> i" "finite {i. P i x}"
```
```   326       using Min_in[of "{i. P i x}"] by auto }
```
```   327   note 2 = this
```
```   328
```
```   329   have "(\<lambda>x. Min {i. P i x}) -` {n} \<inter> space M = {x\<in>space M. Min {i. P i x} = n}"
```
```   330     by auto
```
```   331   also have "\<dots> =
```
```   332     {x\<in>space M. if (\<forall>i. \<exists>n\<ge>i. P n x) then the None = n else
```
```   333       if (\<exists>i. P i x) then P n x \<and> (\<forall>i<n. \<not> P i x)
```
```   334       else Min {} = n}"
```
```   335     by (intro arg_cong[where f=Collect] ext conj_cong)
```
```   336        (auto simp add: 1 2 not_le[symmetric] intro!: Min_eqI)
```
```   337   also have "\<dots> \<in> sets M"
```
```   338     by measurable
```
```   339   finally show "(\<lambda>x. Min {i. P i x}) -` {n} \<inter> space M \<in> sets M" .
```
```   340 qed simp
```
```   341
```
```   342 lemma measurable_count_space_insert[measurable (raw)]:
```
```   343   "s \<in> S \<Longrightarrow> A \<in> sets (count_space S) \<Longrightarrow> insert s A \<in> sets (count_space S)"
```
```   344   by simp
```
```   345
```
```   346 lemma sets_UNIV [measurable (raw)]: "A \<in> sets (count_space UNIV)"
```
```   347   by simp
```
```   348
```
```   349 lemma measurable_card[measurable]:
```
```   350   fixes S :: "'a \<Rightarrow> nat set"
```
```   351   assumes [measurable]: "\<And>i. {x\<in>space M. i \<in> S x} \<in> sets M"
```
```   352   shows "(\<lambda>x. card (S x)) \<in> measurable M (count_space UNIV)"
```
```   353   unfolding measurable_count_space_eq2_countable
```
```   354 proof safe
```
```   355   fix n show "(\<lambda>x. card (S x)) -` {n} \<inter> space M \<in> sets M"
```
```   356   proof (cases n)
```
```   357     case 0
```
```   358     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)}"
```
```   359       by auto
```
```   360     also have "\<dots> \<in> sets M"
```
```   361       by measurable
```
```   362     finally show ?thesis .
```
```   363   next
```
```   364     case (Suc i)
```
```   365     then have "(\<lambda>x. card (S x)) -` {n} \<inter> space M =
```
```   366       (\<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)})"
```
```   367       unfolding set_eq_iff[symmetric] Collect_bex_eq[symmetric] by (auto intro: card_ge_0_finite)
```
```   368     also have "\<dots> \<in> sets M"
```
```   369       by (intro sets.countable_UN' countable_Collect countable_Collect_finite) auto
```
```   370     finally show ?thesis .
```
```   371   qed
```
```   372 qed rule
```
```   373
```
```   374 subsection {* Measurability for (co)inductive predicates *}
```
```   375
```
```   376 lemma measurable_lfp:
```
```   377   assumes "Order_Continuity.continuous F"
```
```   378   assumes *: "\<And>A. pred M A \<Longrightarrow> pred M (F A)"
```
```   379   shows "pred M (lfp F)"
```
```   380 proof -
```
```   381   { fix i have "Measurable.pred M (\<lambda>x. (F ^^ i) (\<lambda>x. False) x)"
```
```   382       by (induct i) (auto intro!: *) }
```
```   383   then have "Measurable.pred M (\<lambda>x. \<exists>i. (F ^^ i) (\<lambda>x. False) x)"
```
```   384     by measurable
```
```   385   also have "(\<lambda>x. \<exists>i. (F ^^ i) (\<lambda>x. False) x) = (SUP i. (F ^^ i) bot)"
```
```   386     by (auto simp add: bot_fun_def)
```
```   387   also have "\<dots> = lfp F"
```
```   388     by (rule continuous_lfp[symmetric]) fact
```
```   389   finally show ?thesis .
```
```   390 qed
```
```   391
```
```   392 lemma measurable_gfp:
```
```   393   assumes "Order_Continuity.down_continuous F"
```
```   394   assumes *: "\<And>A. pred M A \<Longrightarrow> pred M (F A)"
```
```   395   shows "pred M (gfp F)"
```
```   396 proof -
```
```   397   { fix i have "Measurable.pred M (\<lambda>x. (F ^^ i) (\<lambda>x. True) x)"
```
```   398       by (induct i) (auto intro!: *) }
```
```   399   then have "Measurable.pred M (\<lambda>x. \<forall>i. (F ^^ i) (\<lambda>x. True) x)"
```
```   400     by measurable
```
```   401   also have "(\<lambda>x. \<forall>i. (F ^^ i) (\<lambda>x. True) x) = (INF i. (F ^^ i) top)"
```
```   402     by (auto simp add: top_fun_def)
```
```   403   also have "\<dots> = gfp F"
```
```   404     by (rule down_continuous_gfp[symmetric]) fact
```
```   405   finally show ?thesis .
```
```   406 qed
```
```   407
```
```   408 lemma measurable_lfp_coinduct[consumes 1, case_names continuity step]:
```
```   409   assumes "P M"
```
```   410   assumes "Order_Continuity.continuous F"
```
```   411   assumes *: "\<And>M A. P M \<Longrightarrow> (\<And>N. P N \<Longrightarrow> Measurable.pred N A) \<Longrightarrow> Measurable.pred M (F A)"
```
```   412   shows "Measurable.pred M (lfp F)"
```
```   413 proof -
```
```   414   { fix i from `P M` have "Measurable.pred M (\<lambda>x. (F ^^ i) (\<lambda>x. False) x)"
```
```   415       by (induct i arbitrary: M) (auto intro!: *) }
```
```   416   then have "Measurable.pred M (\<lambda>x. \<exists>i. (F ^^ i) (\<lambda>x. False) x)"
```
```   417     by measurable
```
```   418   also have "(\<lambda>x. \<exists>i. (F ^^ i) (\<lambda>x. False) x) = (SUP i. (F ^^ i) bot)"
```
```   419     by (auto simp add: bot_fun_def)
```
```   420   also have "\<dots> = lfp F"
```
```   421     by (rule continuous_lfp[symmetric]) fact
```
```   422   finally show ?thesis .
```
```   423 qed
```
```   424
```
```   425 lemma measurable_gfp_coinduct[consumes 1, case_names continuity step]:
```
```   426   assumes "P M"
```
```   427   assumes "Order_Continuity.down_continuous F"
```
```   428   assumes *: "\<And>M A. P M \<Longrightarrow> (\<And>N. P N \<Longrightarrow> Measurable.pred N A) \<Longrightarrow> Measurable.pred M (F A)"
```
```   429   shows "Measurable.pred M (gfp F)"
```
```   430 proof -
```
```   431   { fix i from `P M` have "Measurable.pred M (\<lambda>x. (F ^^ i) (\<lambda>x. True) x)"
```
```   432       by (induct i arbitrary: M) (auto intro!: *) }
```
```   433   then have "Measurable.pred M (\<lambda>x. \<forall>i. (F ^^ i) (\<lambda>x. True) x)"
```
```   434     by measurable
```
```   435   also have "(\<lambda>x. \<forall>i. (F ^^ i) (\<lambda>x. True) x) = (INF i. (F ^^ i) top)"
```
```   436     by (auto simp add: top_fun_def)
```
```   437   also have "\<dots> = gfp F"
```
```   438     by (rule down_continuous_gfp[symmetric]) fact
```
```   439   finally show ?thesis .
```
```   440 qed
```
```   441
```
```   442 lemma measurable_lfp2_coinduct[consumes 1, case_names continuity step]:
```
```   443   assumes "P M s"
```
```   444   assumes "Order_Continuity.continuous F"
```
```   445   assumes *: "\<And>M A s. P M s \<Longrightarrow> (\<And>N t. P N t \<Longrightarrow> Measurable.pred N (A t)) \<Longrightarrow> Measurable.pred M (F A s)"
```
```   446   shows "Measurable.pred M (lfp F s)"
```
```   447 proof -
```
```   448   { fix i from `P M s` have "Measurable.pred M (\<lambda>x. (F ^^ i) (\<lambda>t x. False) s x)"
```
```   449       by (induct i arbitrary: M s) (auto intro!: *) }
```
```   450   then have "Measurable.pred M (\<lambda>x. \<exists>i. (F ^^ i) (\<lambda>t x. False) s x)"
```
```   451     by measurable
```
```   452   also have "(\<lambda>x. \<exists>i. (F ^^ i) (\<lambda>t x. False) s x) = (SUP i. (F ^^ i) bot) s"
```
```   453     by (auto simp add: bot_fun_def)
```
```   454   also have "(SUP i. (F ^^ i) bot) = lfp F"
```
```   455     by (rule continuous_lfp[symmetric]) fact
```
```   456   finally show ?thesis .
```
```   457 qed
```
```   458
```
```   459 lemma measurable_gfp2_coinduct[consumes 1, case_names continuity step]:
```
```   460   assumes "P M s"
```
```   461   assumes "Order_Continuity.down_continuous F"
```
```   462   assumes *: "\<And>M A s. P M s \<Longrightarrow> (\<And>N t. P N t \<Longrightarrow> Measurable.pred N (A t)) \<Longrightarrow> Measurable.pred M (F A s)"
```
```   463   shows "Measurable.pred M (gfp F s)"
```
```   464 proof -
```
```   465   { fix i from `P M s` have "Measurable.pred M (\<lambda>x. (F ^^ i) (\<lambda>t x. True) s x)"
```
```   466       by (induct i arbitrary: M s) (auto intro!: *) }
```
```   467   then have "Measurable.pred M (\<lambda>x. \<forall>i. (F ^^ i) (\<lambda>t x. True) s x)"
```
```   468     by measurable
```
```   469   also have "(\<lambda>x. \<forall>i. (F ^^ i) (\<lambda>t x. True) s x) = (INF i. (F ^^ i) top) s"
```
```   470     by (auto simp add: top_fun_def)
```
```   471   also have "(INF i. (F ^^ i) top) = gfp F"
```
```   472     by (rule down_continuous_gfp[symmetric]) fact
```
```   473   finally show ?thesis .
```
```   474 qed
```
```   475
```
```   476 lemma measurable_enat_coinduct:
```
```   477   fixes f :: "'a \<Rightarrow> enat"
```
```   478   assumes "R f"
```
```   479   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>
```
```   480     Measurable.pred M P \<and>
```
```   481     i \<in> measurable M M \<and>
```
```   482     h \<in> measurable M (count_space UNIV)"
```
```   483   shows "f \<in> measurable M (count_space UNIV)"
```
```   484 proof (simp add: measurable_count_space_eq2_countable, rule )
```
```   485   fix a :: enat
```
```   486   have "f -` {a} \<inter> space M = {x\<in>space M. f x = a}"
```
```   487     by auto
```
```   488   { fix i :: nat
```
```   489     from `R f` have "Measurable.pred M (\<lambda>x. f x = enat i)"
```
```   490     proof (induction i arbitrary: f)
```
```   491       case 0
```
```   492       from *[OF this] obtain g h i P
```
```   493         where f: "f = (\<lambda>x. if P x then h x else eSuc (g (i x)))" and
```
```   494           [measurable]: "Measurable.pred M P" "i \<in> measurable M M" "h \<in> measurable M (count_space UNIV)"
```
```   495         by auto
```
```   496       have "Measurable.pred M (\<lambda>x. P x \<and> h x = 0)"
```
```   497         by measurable
```
```   498       also have "(\<lambda>x. P x \<and> h x = 0) = (\<lambda>x. f x = enat 0)"
```
```   499         by (auto simp: f zero_enat_def[symmetric])
```
```   500       finally show ?case .
```
```   501     next
```
```   502       case (Suc n)
```
```   503       from *[OF Suc.prems] obtain g h i P
```
```   504         where f: "f = (\<lambda>x. if P x then h x else eSuc (g (i x)))" and "R g" and
```
```   505           M[measurable]: "Measurable.pred M P" "i \<in> measurable M M" "h \<in> measurable M (count_space UNIV)"
```
```   506         by auto
```
```   507       have "(\<lambda>x. f x = enat (Suc n)) =
```
```   508         (\<lambda>x. (P x \<longrightarrow> h x = enat (Suc n)) \<and> (\<not> P x \<longrightarrow> g (i x) = enat n))"
```
```   509         by (auto simp: f zero_enat_def[symmetric] eSuc_enat[symmetric])
```
```   510       also have "Measurable.pred M \<dots>"
```
```   511         by (intro pred_intros_logic measurable_compose[OF M(2)] Suc `R g`) measurable
```
```   512       finally show ?case .
```
```   513     qed
```
```   514     then have "f -` {enat i} \<inter> space M \<in> sets M"
```
```   515       by (simp add: pred_def Int_def conj_commute) }
```
```   516   note fin = this
```
```   517   show "f -` {a} \<inter> space M \<in> sets M"
```
```   518   proof (cases a)
```
```   519     case infinity
```
```   520     then have "f -` {a} \<inter> space M = space M - (\<Union>n. f -` {enat n} \<inter> space M)"
```
```   521       by auto
```
```   522     also have "\<dots> \<in> sets M"
```
```   523       by (intro sets.Diff sets.top sets.Un sets.countable_UN) (auto intro!: fin)
```
```   524     finally show ?thesis .
```
```   525   qed (simp add: fin)
```
```   526 qed
```
```   527
```
```   528 lemma measurable_pred_countable[measurable (raw)]:
```
```   529   assumes "countable X"
```
```   530   shows
```
```   531     "(\<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)"
```
```   532     "(\<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)"
```
```   533   unfolding pred_def
```
```   534   by (auto intro!: sets.sets_Collect_countable_All' sets.sets_Collect_countable_Ex' assms)
```
```   535
```
```   536 lemma measurable_THE:
```
```   537   fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"
```
```   538   assumes [measurable]: "\<And>i. Measurable.pred M (P i)"
```
```   539   assumes I[simp]: "countable I" "\<And>i x. x \<in> space M \<Longrightarrow> P i x \<Longrightarrow> i \<in> I"
```
```   540   assumes unique: "\<And>x i j. x \<in> space M \<Longrightarrow> P i x \<Longrightarrow> P j x \<Longrightarrow> i = j"
```
```   541   shows "(\<lambda>x. THE i. P i x) \<in> measurable M (count_space UNIV)"
```
```   542   unfolding measurable_def
```
```   543 proof safe
```
```   544   fix X
```
```   545   def f \<equiv> "\<lambda>x. THE i. P i x" def undef \<equiv> "THE i::'a. False"
```
```   546   { fix i x assume "x \<in> space M" "P i x" then have "f x = i"
```
```   547       unfolding f_def using unique by auto }
```
```   548   note f_eq = this
```
```   549   { fix x assume "x \<in> space M" "\<forall>i\<in>I. \<not> P i x"
```
```   550     then have "\<And>i. \<not> P i x"
```
```   551       using I(2)[of x] by auto
```
```   552     then have "f x = undef"
```
```   553       by (auto simp: undef_def f_def) }
```
```   554   then have "f -` X \<inter> space M = (\<Union>i\<in>I \<inter> X. {x\<in>space M. P i x}) \<union>
```
```   555      (if undef \<in> X then space M - (\<Union>i\<in>I. {x\<in>space M. P i x}) else {})"
```
```   556     by (auto dest: f_eq)
```
```   557   also have "\<dots> \<in> sets M"
```
```   558     by (auto intro!: sets.Diff sets.countable_UN')
```
```   559   finally show "f -` X \<inter> space M \<in> sets M" .
```
```   560 qed simp
```
```   561
```
```   562 lemma measurable_bot[measurable]: "Measurable.pred M bot"
```
```   563   by (simp add: bot_fun_def)
```
```   564
```
```   565 lemma measurable_top[measurable]: "Measurable.pred M top"
```
```   566   by (simp add: top_fun_def)
```
```   567
```
```   568 lemma measurable_Ex1[measurable (raw)]:
```
```   569   assumes [simp]: "countable I" and [measurable]: "\<And>i. i \<in> I \<Longrightarrow> Measurable.pred M (P i)"
```
```   570   shows "Measurable.pred M (\<lambda>x. \<exists>!i\<in>I. P i x)"
```
```   571   unfolding bex1_def by measurable
```
```   572
```
```   573 lemma measurable_split_if[measurable (raw)]:
```
```   574   "(c \<Longrightarrow> Measurable.pred M f) \<Longrightarrow> (\<not> c \<Longrightarrow> Measurable.pred M g) \<Longrightarrow>
```
```   575    Measurable.pred M (if c then f else g)"
```
```   576   by simp
```
```   577
```
```   578 lemma pred_restrict_space:
```
```   579   assumes "S \<in> sets M"
```
```   580   shows "Measurable.pred (restrict_space M S) P \<longleftrightarrow> Measurable.pred M (\<lambda>x. x \<in> S \<and> P x)"
```
```   581   unfolding pred_def sets_Collect_restrict_space_iff[OF assms] ..
```
```   582
```
```   583 lemma measurable_predpow[measurable]:
```
```   584   assumes "Measurable.pred M T"
```
```   585   assumes "\<And>Q. Measurable.pred M Q \<Longrightarrow> Measurable.pred M (R Q)"
```
```   586   shows "Measurable.pred M ((R ^^ n) T)"
```
```   587   by (induct n) (auto intro: assms)
```
```   588
```
```   589 hide_const (open) pred
```
```   590
```
```   591 end
```