src/HOL/Probability/Measurable.thy
 author hoelzl Mon Nov 24 12:20:14 2014 +0100 (2014-11-24) changeset 59048 7dc8ac6f0895 parent 59047 8d7cec9b861d child 59088 ff2bd4a14ddb permissions -rw-r--r--
add congruence solver to measurability prover
```     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_app = Measurable.app_thm_attr
```
```    67   "add application rule to measurability prover"
```
```    68
```
```    69 attribute_setup measurable_cong = Measurable.cong_thm_attr
```
```    70   "add congurence rules to measurability prover"
```
```    71
```
```    72 method_setup measurable = \<open> Scan.lift (Scan.succeed (METHOD o Measurable.measurable_tac)) \<close>
```
```    73   "measurability prover"
```
```    74
```
```    75 simproc_setup measurable ("A \<in> sets M" | "f \<in> measurable M N") = {* K Measurable.simproc *}
```
```    76
```
```    77 setup {*
```
```    78   Global_Theory.add_thms_dynamic (@{binding measurable}, Measurable.get_all)
```
```    79 *}
```
```    80
```
```    81 declare
```
```    82   measurable_compose_rev[measurable_dest]
```
```    83   pred_sets1[measurable_dest]
```
```    84   pred_sets2[measurable_dest]
```
```    85   sets.sets_into_space[measurable_dest]
```
```    86
```
```    87 declare
```
```    88   sets.top[measurable]
```
```    89   sets.empty_sets[measurable (raw)]
```
```    90   sets.Un[measurable (raw)]
```
```    91   sets.Diff[measurable (raw)]
```
```    92
```
```    93 declare
```
```    94   measurable_count_space[measurable (raw)]
```
```    95   measurable_ident[measurable (raw)]
```
```    96   measurable_id[measurable (raw)]
```
```    97   measurable_const[measurable (raw)]
```
```    98   measurable_If[measurable (raw)]
```
```    99   measurable_comp[measurable (raw)]
```
```   100   measurable_sets[measurable (raw)]
```
```   101
```
```   102 declare measurable_cong_sets[measurable_cong]
```
```   103 declare sets_restrict_space_cong[measurable_cong]
```
```   104
```
```   105 lemma predE[measurable (raw)]:
```
```   106   "pred M P \<Longrightarrow> {x\<in>space M. P x} \<in> sets M"
```
```   107   unfolding pred_def .
```
```   108
```
```   109 lemma pred_intros_imp'[measurable (raw)]:
```
```   110   "(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<longrightarrow> P x)"
```
```   111   by (cases K) auto
```
```   112
```
```   113 lemma pred_intros_conj1'[measurable (raw)]:
```
```   114   "(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<and> P x)"
```
```   115   by (cases K) auto
```
```   116
```
```   117 lemma pred_intros_conj2'[measurable (raw)]:
```
```   118   "(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. P x \<and> K)"
```
```   119   by (cases K) auto
```
```   120
```
```   121 lemma pred_intros_disj1'[measurable (raw)]:
```
```   122   "(\<not> K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<or> P x)"
```
```   123   by (cases K) auto
```
```   124
```
```   125 lemma pred_intros_disj2'[measurable (raw)]:
```
```   126   "(\<not> K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. P x \<or> K)"
```
```   127   by (cases K) auto
```
```   128
```
```   129 lemma pred_intros_logic[measurable (raw)]:
```
```   130   "pred M (\<lambda>x. x \<in> space M)"
```
```   131   "pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. \<not> P x)"
```
```   132   "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<and> P x)"
```
```   133   "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<longrightarrow> P x)"
```
```   134   "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<or> P x)"
```
```   135   "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x = P x)"
```
```   136   "pred M (\<lambda>x. f x \<in> UNIV)"
```
```   137   "pred M (\<lambda>x. f x \<in> {})"
```
```   138   "pred M (\<lambda>x. P' (f x) x) \<Longrightarrow> pred M (\<lambda>x. f x \<in> {y. P' y x})"
```
```   139   "pred M (\<lambda>x. f x \<in> (B x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> - (B x))"
```
```   140   "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))"
```
```   141   "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))"
```
```   142   "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))"
```
```   143   "pred M (\<lambda>x. g x (f x) \<in> (X x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (g x) -` (X x))"
```
```   144   by (auto simp: iff_conv_conj_imp pred_def)
```
```   145
```
```   146 lemma pred_intros_countable[measurable (raw)]:
```
```   147   fixes P :: "'a \<Rightarrow> 'i :: countable \<Rightarrow> bool"
```
```   148   shows
```
```   149     "(\<And>i. pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<forall>i. P x i)"
```
```   150     "(\<And>i. pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<exists>i. P x i)"
```
```   151   by (auto intro!: sets.sets_Collect_countable_All sets.sets_Collect_countable_Ex simp: pred_def)
```
```   152
```
```   153 lemma pred_intros_countable_bounded[measurable (raw)]:
```
```   154   fixes X :: "'i :: countable set"
```
```   155   shows
```
```   156     "(\<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))"
```
```   157     "(\<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))"
```
```   158     "(\<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)"
```
```   159     "(\<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)"
```
```   160   by (auto simp: Bex_def Ball_def)
```
```   161
```
```   162 lemma pred_intros_finite[measurable (raw)]:
```
```   163   "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))"
```
```   164   "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))"
```
```   165   "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)"
```
```   166   "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)"
```
```   167   by (auto intro!: sets.sets_Collect_finite_Ex sets.sets_Collect_finite_All simp: iff_conv_conj_imp pred_def)
```
```   168
```
```   169 lemma countable_Un_Int[measurable (raw)]:
```
```   170   "(\<And>i :: 'i :: countable. i \<in> I \<Longrightarrow> N i \<in> sets M) \<Longrightarrow> (\<Union>i\<in>I. N i) \<in> sets M"
```
```   171   "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"
```
```   172   by auto
```
```   173
```
```   174 declare
```
```   175   finite_UN[measurable (raw)]
```
```   176   finite_INT[measurable (raw)]
```
```   177
```
```   178 lemma sets_Int_pred[measurable (raw)]:
```
```   179   assumes space: "A \<inter> B \<subseteq> space M" and [measurable]: "pred M (\<lambda>x. x \<in> A)" "pred M (\<lambda>x. x \<in> B)"
```
```   180   shows "A \<inter> B \<in> sets M"
```
```   181 proof -
```
```   182   have "{x\<in>space M. x \<in> A \<inter> B} \<in> sets M" by auto
```
```   183   also have "{x\<in>space M. x \<in> A \<inter> B} = A \<inter> B"
```
```   184     using space by auto
```
```   185   finally show ?thesis .
```
```   186 qed
```
```   187
```
```   188 lemma [measurable (raw generic)]:
```
```   189   assumes f: "f \<in> measurable M N" and c: "c \<in> space N \<Longrightarrow> {c} \<in> sets N"
```
```   190   shows pred_eq_const1: "pred M (\<lambda>x. f x = c)"
```
```   191     and pred_eq_const2: "pred M (\<lambda>x. c = f x)"
```
```   192 proof -
```
```   193   show "pred M (\<lambda>x. f x = c)"
```
```   194   proof cases
```
```   195     assume "c \<in> space N"
```
```   196     with measurable_sets[OF f c] show ?thesis
```
```   197       by (auto simp: Int_def conj_commute pred_def)
```
```   198   next
```
```   199     assume "c \<notin> space N"
```
```   200     with f[THEN measurable_space] have "{x \<in> space M. f x = c} = {}" by auto
```
```   201     then show ?thesis by (auto simp: pred_def cong: conj_cong)
```
```   202   qed
```
```   203   then show "pred M (\<lambda>x. c = f x)"
```
```   204     by (simp add: eq_commute)
```
```   205 qed
```
```   206
```
```   207 lemma pred_count_space_const1[measurable (raw)]:
```
```   208   "f \<in> measurable M (count_space UNIV) \<Longrightarrow> Measurable.pred M (\<lambda>x. f x = c)"
```
```   209   by (intro pred_eq_const1[where N="count_space UNIV"]) (auto )
```
```   210
```
```   211 lemma pred_count_space_const2[measurable (raw)]:
```
```   212   "f \<in> measurable M (count_space UNIV) \<Longrightarrow> Measurable.pred M (\<lambda>x. c = f x)"
```
```   213   by (intro pred_eq_const2[where N="count_space UNIV"]) (auto )
```
```   214
```
```   215 lemma pred_le_const[measurable (raw generic)]:
```
```   216   assumes f: "f \<in> measurable M N" and c: "{.. c} \<in> sets N" shows "pred M (\<lambda>x. f x \<le> c)"
```
```   217   using measurable_sets[OF f c]
```
```   218   by (auto simp: Int_def conj_commute eq_commute pred_def)
```
```   219
```
```   220 lemma pred_const_le[measurable (raw generic)]:
```
```   221   assumes f: "f \<in> measurable M N" and c: "{c ..} \<in> sets N" shows "pred M (\<lambda>x. c \<le> f x)"
```
```   222   using measurable_sets[OF f c]
```
```   223   by (auto simp: Int_def conj_commute eq_commute pred_def)
```
```   224
```
```   225 lemma pred_less_const[measurable (raw generic)]:
```
```   226   assumes f: "f \<in> measurable M N" and c: "{..< c} \<in> sets N" shows "pred M (\<lambda>x. f x < c)"
```
```   227   using measurable_sets[OF f c]
```
```   228   by (auto simp: Int_def conj_commute eq_commute pred_def)
```
```   229
```
```   230 lemma pred_const_less[measurable (raw generic)]:
```
```   231   assumes f: "f \<in> measurable M N" and c: "{c <..} \<in> sets N" shows "pred M (\<lambda>x. c < f x)"
```
```   232   using measurable_sets[OF f c]
```
```   233   by (auto simp: Int_def conj_commute eq_commute pred_def)
```
```   234
```
```   235 declare
```
```   236   sets.Int[measurable (raw)]
```
```   237
```
```   238 lemma pred_in_If[measurable (raw)]:
```
```   239   "(P \<Longrightarrow> pred M (\<lambda>x. x \<in> A x)) \<Longrightarrow> (\<not> P \<Longrightarrow> pred M (\<lambda>x. x \<in> B x)) \<Longrightarrow>
```
```   240     pred M (\<lambda>x. x \<in> (if P then A x else B x))"
```
```   241   by auto
```
```   242
```
```   243 lemma sets_range[measurable_dest]:
```
```   244   "A ` I \<subseteq> sets M \<Longrightarrow> i \<in> I \<Longrightarrow> A i \<in> sets M"
```
```   245   by auto
```
```   246
```
```   247 lemma pred_sets_range[measurable_dest]:
```
```   248   "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)"
```
```   249   using pred_sets2[OF sets_range] by auto
```
```   250
```
```   251 lemma sets_All[measurable_dest]:
```
```   252   "\<forall>i. A i \<in> sets (M i) \<Longrightarrow> A i \<in> sets (M i)"
```
```   253   by auto
```
```   254
```
```   255 lemma pred_sets_All[measurable_dest]:
```
```   256   "\<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)"
```
```   257   using pred_sets2[OF sets_All, of A N f] by auto
```
```   258
```
```   259 lemma sets_Ball[measurable_dest]:
```
```   260   "\<forall>i\<in>I. A i \<in> sets (M i) \<Longrightarrow> i\<in>I \<Longrightarrow> A i \<in> sets (M i)"
```
```   261   by auto
```
```   262
```
```   263 lemma pred_sets_Ball[measurable_dest]:
```
```   264   "\<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)"
```
```   265   using pred_sets2[OF sets_Ball, of _ _ _ f] by auto
```
```   266
```
```   267 lemma measurable_finite[measurable (raw)]:
```
```   268   fixes S :: "'a \<Rightarrow> nat set"
```
```   269   assumes [measurable]: "\<And>i. {x\<in>space M. i \<in> S x} \<in> sets M"
```
```   270   shows "pred M (\<lambda>x. finite (S x))"
```
```   271   unfolding finite_nat_set_iff_bounded by (simp add: Ball_def)
```
```   272
```
```   273 lemma measurable_Least[measurable]:
```
```   274   assumes [measurable]: "(\<And>i::nat. (\<lambda>x. P i x) \<in> measurable M (count_space UNIV))"q
```
```   275   shows "(\<lambda>x. LEAST i. P i x) \<in> measurable M (count_space UNIV)"
```
```   276   unfolding measurable_def by (safe intro!: sets_Least) simp_all
```
```   277
```
```   278 lemma measurable_Max_nat[measurable (raw)]:
```
```   279   fixes P :: "nat \<Rightarrow> 'a \<Rightarrow> bool"
```
```   280   assumes [measurable]: "\<And>i. Measurable.pred M (P i)"
```
```   281   shows "(\<lambda>x. Max {i. P i x}) \<in> measurable M (count_space UNIV)"
```
```   282   unfolding measurable_count_space_eq2_countable
```
```   283 proof safe
```
```   284   fix n
```
```   285
```
```   286   { fix x assume "\<forall>i. \<exists>n\<ge>i. P n x"
```
```   287     then have "infinite {i. P i x}"
```
```   288       unfolding infinite_nat_iff_unbounded_le by auto
```
```   289     then have "Max {i. P i x} = the None"
```
```   290       by (rule Max.infinite) }
```
```   291   note 1 = this
```
```   292
```
```   293   { fix x i j assume "P i x" "\<forall>n\<ge>j. \<not> P n x"
```
```   294     then have "finite {i. P i x}"
```
```   295       by (auto simp: subset_eq not_le[symmetric] finite_nat_iff_bounded)
```
```   296     with `P i x` have "P (Max {i. P i x}) x" "i \<le> Max {i. P i x}" "finite {i. P i x}"
```
```   297       using Max_in[of "{i. P i x}"] by auto }
```
```   298   note 2 = this
```
```   299
```
```   300   have "(\<lambda>x. Max {i. P i x}) -` {n} \<inter> space M = {x\<in>space M. Max {i. P i x} = n}"
```
```   301     by auto
```
```   302   also have "\<dots> =
```
```   303     {x\<in>space M. if (\<forall>i. \<exists>n\<ge>i. P n x) then the None = n else
```
```   304       if (\<exists>i. P i x) then P n x \<and> (\<forall>i>n. \<not> P i x)
```
```   305       else Max {} = n}"
```
```   306     by (intro arg_cong[where f=Collect] ext conj_cong)
```
```   307        (auto simp add: 1 2 not_le[symmetric] intro!: Max_eqI)
```
```   308   also have "\<dots> \<in> sets M"
```
```   309     by measurable
```
```   310   finally show "(\<lambda>x. Max {i. P i x}) -` {n} \<inter> space M \<in> sets M" .
```
```   311 qed simp
```
```   312
```
```   313 lemma measurable_Min_nat[measurable (raw)]:
```
```   314   fixes P :: "nat \<Rightarrow> 'a \<Rightarrow> bool"
```
```   315   assumes [measurable]: "\<And>i. Measurable.pred M (P i)"
```
```   316   shows "(\<lambda>x. Min {i. P i x}) \<in> measurable M (count_space UNIV)"
```
```   317   unfolding measurable_count_space_eq2_countable
```
```   318 proof safe
```
```   319   fix n
```
```   320
```
```   321   { fix x assume "\<forall>i. \<exists>n\<ge>i. P n x"
```
```   322     then have "infinite {i. P i x}"
```
```   323       unfolding infinite_nat_iff_unbounded_le by auto
```
```   324     then have "Min {i. P i x} = the None"
```
```   325       by (rule Min.infinite) }
```
```   326   note 1 = this
```
```   327
```
```   328   { fix x i j assume "P i x" "\<forall>n\<ge>j. \<not> P n x"
```
```   329     then have "finite {i. P i x}"
```
```   330       by (auto simp: subset_eq not_le[symmetric] finite_nat_iff_bounded)
```
```   331     with `P i x` have "P (Min {i. P i x}) x" "Min {i. P i x} \<le> i" "finite {i. P i x}"
```
```   332       using Min_in[of "{i. P i x}"] by auto }
```
```   333   note 2 = this
```
```   334
```
```   335   have "(\<lambda>x. Min {i. P i x}) -` {n} \<inter> space M = {x\<in>space M. Min {i. P i x} = n}"
```
```   336     by auto
```
```   337   also have "\<dots> =
```
```   338     {x\<in>space M. if (\<forall>i. \<exists>n\<ge>i. P n x) then the None = n else
```
```   339       if (\<exists>i. P i x) then P n x \<and> (\<forall>i<n. \<not> P i x)
```
```   340       else Min {} = n}"
```
```   341     by (intro arg_cong[where f=Collect] ext conj_cong)
```
```   342        (auto simp add: 1 2 not_le[symmetric] intro!: Min_eqI)
```
```   343   also have "\<dots> \<in> sets M"
```
```   344     by measurable
```
```   345   finally show "(\<lambda>x. Min {i. P i x}) -` {n} \<inter> space M \<in> sets M" .
```
```   346 qed simp
```
```   347
```
```   348 lemma measurable_count_space_insert[measurable (raw)]:
```
```   349   "s \<in> S \<Longrightarrow> A \<in> sets (count_space S) \<Longrightarrow> insert s A \<in> sets (count_space S)"
```
```   350   by simp
```
```   351
```
```   352 lemma sets_UNIV [measurable (raw)]: "A \<in> sets (count_space UNIV)"
```
```   353   by simp
```
```   354
```
```   355 lemma measurable_card[measurable]:
```
```   356   fixes S :: "'a \<Rightarrow> nat set"
```
```   357   assumes [measurable]: "\<And>i. {x\<in>space M. i \<in> S x} \<in> sets M"
```
```   358   shows "(\<lambda>x. card (S x)) \<in> measurable M (count_space UNIV)"
```
```   359   unfolding measurable_count_space_eq2_countable
```
```   360 proof safe
```
```   361   fix n show "(\<lambda>x. card (S x)) -` {n} \<inter> space M \<in> sets M"
```
```   362   proof (cases n)
```
```   363     case 0
```
```   364     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)}"
```
```   365       by auto
```
```   366     also have "\<dots> \<in> sets M"
```
```   367       by measurable
```
```   368     finally show ?thesis .
```
```   369   next
```
```   370     case (Suc i)
```
```   371     then have "(\<lambda>x. card (S x)) -` {n} \<inter> space M =
```
```   372       (\<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)})"
```
```   373       unfolding set_eq_iff[symmetric] Collect_bex_eq[symmetric] by (auto intro: card_ge_0_finite)
```
```   374     also have "\<dots> \<in> sets M"
```
```   375       by (intro sets.countable_UN' countable_Collect countable_Collect_finite) auto
```
```   376     finally show ?thesis .
```
```   377   qed
```
```   378 qed rule
```
```   379
```
```   380 subsection {* Measurability for (co)inductive predicates *}
```
```   381
```
```   382 lemma measurable_lfp:
```
```   383   assumes "Order_Continuity.continuous F"
```
```   384   assumes *: "\<And>A. pred M A \<Longrightarrow> pred M (F A)"
```
```   385   shows "pred M (lfp F)"
```
```   386 proof -
```
```   387   { fix i have "Measurable.pred M (\<lambda>x. (F ^^ i) (\<lambda>x. False) x)"
```
```   388       by (induct i) (auto intro!: *) }
```
```   389   then have "Measurable.pred M (\<lambda>x. \<exists>i. (F ^^ i) (\<lambda>x. False) x)"
```
```   390     by measurable
```
```   391   also have "(\<lambda>x. \<exists>i. (F ^^ i) (\<lambda>x. False) x) = (SUP i. (F ^^ i) bot)"
```
```   392     by (auto simp add: bot_fun_def)
```
```   393   also have "\<dots> = lfp F"
```
```   394     by (rule continuous_lfp[symmetric]) fact
```
```   395   finally show ?thesis .
```
```   396 qed
```
```   397
```
```   398 lemma measurable_gfp:
```
```   399   assumes "Order_Continuity.down_continuous F"
```
```   400   assumes *: "\<And>A. pred M A \<Longrightarrow> pred M (F A)"
```
```   401   shows "pred M (gfp F)"
```
```   402 proof -
```
```   403   { fix i have "Measurable.pred M (\<lambda>x. (F ^^ i) (\<lambda>x. True) x)"
```
```   404       by (induct i) (auto intro!: *) }
```
```   405   then have "Measurable.pred M (\<lambda>x. \<forall>i. (F ^^ i) (\<lambda>x. True) x)"
```
```   406     by measurable
```
```   407   also have "(\<lambda>x. \<forall>i. (F ^^ i) (\<lambda>x. True) x) = (INF i. (F ^^ i) top)"
```
```   408     by (auto simp add: top_fun_def)
```
```   409   also have "\<dots> = gfp F"
```
```   410     by (rule down_continuous_gfp[symmetric]) fact
```
```   411   finally show ?thesis .
```
```   412 qed
```
```   413
```
```   414 lemma measurable_lfp_coinduct[consumes 1, case_names continuity step]:
```
```   415   assumes "P M"
```
```   416   assumes "Order_Continuity.continuous F"
```
```   417   assumes *: "\<And>M A. P M \<Longrightarrow> (\<And>N. P N \<Longrightarrow> Measurable.pred N A) \<Longrightarrow> Measurable.pred M (F A)"
```
```   418   shows "Measurable.pred M (lfp F)"
```
```   419 proof -
```
```   420   { fix i from `P M` have "Measurable.pred M (\<lambda>x. (F ^^ i) (\<lambda>x. False) x)"
```
```   421       by (induct i arbitrary: M) (auto intro!: *) }
```
```   422   then have "Measurable.pred M (\<lambda>x. \<exists>i. (F ^^ i) (\<lambda>x. False) x)"
```
```   423     by measurable
```
```   424   also have "(\<lambda>x. \<exists>i. (F ^^ i) (\<lambda>x. False) x) = (SUP i. (F ^^ i) bot)"
```
```   425     by (auto simp add: bot_fun_def)
```
```   426   also have "\<dots> = lfp F"
```
```   427     by (rule continuous_lfp[symmetric]) fact
```
```   428   finally show ?thesis .
```
```   429 qed
```
```   430
```
```   431 lemma measurable_gfp_coinduct[consumes 1, case_names continuity step]:
```
```   432   assumes "P M"
```
```   433   assumes "Order_Continuity.down_continuous F"
```
```   434   assumes *: "\<And>M A. P M \<Longrightarrow> (\<And>N. P N \<Longrightarrow> Measurable.pred N A) \<Longrightarrow> Measurable.pred M (F A)"
```
```   435   shows "Measurable.pred M (gfp F)"
```
```   436 proof -
```
```   437   { fix i from `P M` have "Measurable.pred M (\<lambda>x. (F ^^ i) (\<lambda>x. True) x)"
```
```   438       by (induct i arbitrary: M) (auto intro!: *) }
```
```   439   then have "Measurable.pred M (\<lambda>x. \<forall>i. (F ^^ i) (\<lambda>x. True) x)"
```
```   440     by measurable
```
```   441   also have "(\<lambda>x. \<forall>i. (F ^^ i) (\<lambda>x. True) x) = (INF i. (F ^^ i) top)"
```
```   442     by (auto simp add: top_fun_def)
```
```   443   also have "\<dots> = gfp F"
```
```   444     by (rule down_continuous_gfp[symmetric]) fact
```
```   445   finally show ?thesis .
```
```   446 qed
```
```   447
```
```   448 lemma measurable_lfp2_coinduct[consumes 1, case_names continuity step]:
```
```   449   assumes "P M s"
```
```   450   assumes "Order_Continuity.continuous F"
```
```   451   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)"
```
```   452   shows "Measurable.pred M (lfp F s)"
```
```   453 proof -
```
```   454   { fix i from `P M s` have "Measurable.pred M (\<lambda>x. (F ^^ i) (\<lambda>t x. False) s x)"
```
```   455       by (induct i arbitrary: M s) (auto intro!: *) }
```
```   456   then have "Measurable.pred M (\<lambda>x. \<exists>i. (F ^^ i) (\<lambda>t x. False) s x)"
```
```   457     by measurable
```
```   458   also have "(\<lambda>x. \<exists>i. (F ^^ i) (\<lambda>t x. False) s x) = (SUP i. (F ^^ i) bot) s"
```
```   459     by (auto simp add: bot_fun_def)
```
```   460   also have "(SUP i. (F ^^ i) bot) = lfp F"
```
```   461     by (rule continuous_lfp[symmetric]) fact
```
```   462   finally show ?thesis .
```
```   463 qed
```
```   464
```
```   465 lemma measurable_gfp2_coinduct[consumes 1, case_names continuity step]:
```
```   466   assumes "P M s"
```
```   467   assumes "Order_Continuity.down_continuous F"
```
```   468   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)"
```
```   469   shows "Measurable.pred M (gfp F s)"
```
```   470 proof -
```
```   471   { fix i from `P M s` have "Measurable.pred M (\<lambda>x. (F ^^ i) (\<lambda>t x. True) s x)"
```
```   472       by (induct i arbitrary: M s) (auto intro!: *) }
```
```   473   then have "Measurable.pred M (\<lambda>x. \<forall>i. (F ^^ i) (\<lambda>t x. True) s x)"
```
```   474     by measurable
```
```   475   also have "(\<lambda>x. \<forall>i. (F ^^ i) (\<lambda>t x. True) s x) = (INF i. (F ^^ i) top) s"
```
```   476     by (auto simp add: top_fun_def)
```
```   477   also have "(INF i. (F ^^ i) top) = gfp F"
```
```   478     by (rule down_continuous_gfp[symmetric]) fact
```
```   479   finally show ?thesis .
```
```   480 qed
```
```   481
```
```   482 lemma measurable_enat_coinduct:
```
```   483   fixes f :: "'a \<Rightarrow> enat"
```
```   484   assumes "R f"
```
```   485   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>
```
```   486     Measurable.pred M P \<and>
```
```   487     i \<in> measurable M M \<and>
```
```   488     h \<in> measurable M (count_space UNIV)"
```
```   489   shows "f \<in> measurable M (count_space UNIV)"
```
```   490 proof (simp add: measurable_count_space_eq2_countable, rule )
```
```   491   fix a :: enat
```
```   492   have "f -` {a} \<inter> space M = {x\<in>space M. f x = a}"
```
```   493     by auto
```
```   494   { fix i :: nat
```
```   495     from `R f` have "Measurable.pred M (\<lambda>x. f x = enat i)"
```
```   496     proof (induction i arbitrary: f)
```
```   497       case 0
```
```   498       from *[OF this] obtain g h i P
```
```   499         where f: "f = (\<lambda>x. if P x then h x else eSuc (g (i x)))" and
```
```   500           [measurable]: "Measurable.pred M P" "i \<in> measurable M M" "h \<in> measurable M (count_space UNIV)"
```
```   501         by auto
```
```   502       have "Measurable.pred M (\<lambda>x. P x \<and> h x = 0)"
```
```   503         by measurable
```
```   504       also have "(\<lambda>x. P x \<and> h x = 0) = (\<lambda>x. f x = enat 0)"
```
```   505         by (auto simp: f zero_enat_def[symmetric])
```
```   506       finally show ?case .
```
```   507     next
```
```   508       case (Suc n)
```
```   509       from *[OF Suc.prems] obtain g h i P
```
```   510         where f: "f = (\<lambda>x. if P x then h x else eSuc (g (i x)))" and "R g" and
```
```   511           M[measurable]: "Measurable.pred M P" "i \<in> measurable M M" "h \<in> measurable M (count_space UNIV)"
```
```   512         by auto
```
```   513       have "(\<lambda>x. f x = enat (Suc n)) =
```
```   514         (\<lambda>x. (P x \<longrightarrow> h x = enat (Suc n)) \<and> (\<not> P x \<longrightarrow> g (i x) = enat n))"
```
```   515         by (auto simp: f zero_enat_def[symmetric] eSuc_enat[symmetric])
```
```   516       also have "Measurable.pred M \<dots>"
```
```   517         by (intro pred_intros_logic measurable_compose[OF M(2)] Suc `R g`) measurable
```
```   518       finally show ?case .
```
```   519     qed
```
```   520     then have "f -` {enat i} \<inter> space M \<in> sets M"
```
```   521       by (simp add: pred_def Int_def conj_commute) }
```
```   522   note fin = this
```
```   523   show "f -` {a} \<inter> space M \<in> sets M"
```
```   524   proof (cases a)
```
```   525     case infinity
```
```   526     then have "f -` {a} \<inter> space M = space M - (\<Union>n. f -` {enat n} \<inter> space M)"
```
```   527       by auto
```
```   528     also have "\<dots> \<in> sets M"
```
```   529       by (intro sets.Diff sets.top sets.Un sets.countable_UN) (auto intro!: fin)
```
```   530     finally show ?thesis .
```
```   531   qed (simp add: fin)
```
```   532 qed
```
```   533
```
```   534 lemma measurable_pred_countable[measurable (raw)]:
```
```   535   assumes "countable X"
```
```   536   shows
```
```   537     "(\<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)"
```
```   538     "(\<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)"
```
```   539   unfolding pred_def
```
```   540   by (auto intro!: sets.sets_Collect_countable_All' sets.sets_Collect_countable_Ex' assms)
```
```   541
```
```   542 lemma measurable_THE:
```
```   543   fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"
```
```   544   assumes [measurable]: "\<And>i. Measurable.pred M (P i)"
```
```   545   assumes I[simp]: "countable I" "\<And>i x. x \<in> space M \<Longrightarrow> P i x \<Longrightarrow> i \<in> I"
```
```   546   assumes unique: "\<And>x i j. x \<in> space M \<Longrightarrow> P i x \<Longrightarrow> P j x \<Longrightarrow> i = j"
```
```   547   shows "(\<lambda>x. THE i. P i x) \<in> measurable M (count_space UNIV)"
```
```   548   unfolding measurable_def
```
```   549 proof safe
```
```   550   fix X
```
```   551   def f \<equiv> "\<lambda>x. THE i. P i x" def undef \<equiv> "THE i::'a. False"
```
```   552   { fix i x assume "x \<in> space M" "P i x" then have "f x = i"
```
```   553       unfolding f_def using unique by auto }
```
```   554   note f_eq = this
```
```   555   { fix x assume "x \<in> space M" "\<forall>i\<in>I. \<not> P i x"
```
```   556     then have "\<And>i. \<not> P i x"
```
```   557       using I(2)[of x] by auto
```
```   558     then have "f x = undef"
```
```   559       by (auto simp: undef_def f_def) }
```
```   560   then have "f -` X \<inter> space M = (\<Union>i\<in>I \<inter> X. {x\<in>space M. P i x}) \<union>
```
```   561      (if undef \<in> X then space M - (\<Union>i\<in>I. {x\<in>space M. P i x}) else {})"
```
```   562     by (auto dest: f_eq)
```
```   563   also have "\<dots> \<in> sets M"
```
```   564     by (auto intro!: sets.Diff sets.countable_UN')
```
```   565   finally show "f -` X \<inter> space M \<in> sets M" .
```
```   566 qed simp
```
```   567
```
```   568 lemma measurable_bot[measurable]: "Measurable.pred M bot"
```
```   569   by (simp add: bot_fun_def)
```
```   570
```
```   571 lemma measurable_top[measurable]: "Measurable.pred M top"
```
```   572   by (simp add: top_fun_def)
```
```   573
```
```   574 lemma measurable_Ex1[measurable (raw)]:
```
```   575   assumes [simp]: "countable I" and [measurable]: "\<And>i. i \<in> I \<Longrightarrow> Measurable.pred M (P i)"
```
```   576   shows "Measurable.pred M (\<lambda>x. \<exists>!i\<in>I. P i x)"
```
```   577   unfolding bex1_def by measurable
```
```   578
```
```   579 lemma measurable_split_if[measurable (raw)]:
```
```   580   "(c \<Longrightarrow> Measurable.pred M f) \<Longrightarrow> (\<not> c \<Longrightarrow> Measurable.pred M g) \<Longrightarrow>
```
```   581    Measurable.pred M (if c then f else g)"
```
```   582   by simp
```
```   583
```
```   584 lemma pred_restrict_space:
```
```   585   assumes "S \<in> sets M"
```
```   586   shows "Measurable.pred (restrict_space M S) P \<longleftrightarrow> Measurable.pred M (\<lambda>x. x \<in> S \<and> P x)"
```
```   587   unfolding pred_def sets_Collect_restrict_space_iff[OF assms] ..
```
```   588
```
```   589 lemma measurable_predpow[measurable]:
```
```   590   assumes "Measurable.pred M T"
```
```   591   assumes "\<And>Q. Measurable.pred M Q \<Longrightarrow> Measurable.pred M (R Q)"
```
```   592   shows "Measurable.pred M ((R ^^ n) T)"
```
```   593   by (induct n) (auto intro: assms)
```
```   594
```
```   595 hide_const (open) pred
```
```   596
```
```   597 end
```
```   598
```