src/HOL/Probability/Measurable.thy
author wenzelm
Fri Aug 16 21:33:36 2013 +0200 (2013-08-16)
changeset 53043 8cbfbeb566a4
parent 50530 6266e44b3396
child 56021 e0c9d76c2a6d
permissions -rw-r--r--
more standard attribute_setup / method_setup -- export key ML operations instead of parsers;
     1 (*  Title:      HOL/Probability/Measurable.thy
     2     Author:     Johannes Hölzl <hoelzl@in.tum.de>
     3 *)
     4 theory Measurable
     5   imports Sigma_Algebra
     6 begin
     7 
     8 subsection {* Measurability prover *}
     9 
    10 lemma (in algebra) sets_Collect_finite_All:
    11   assumes "\<And>i. i \<in> S \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M" "finite S"
    12   shows "{x\<in>\<Omega>. \<forall>i\<in>S. P i x} \<in> M"
    13 proof -
    14   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})"
    15     by auto
    16   with assms show ?thesis by (auto intro!: sets_Collect_finite_All')
    17 qed
    18 
    19 abbreviation "pred M P \<equiv> P \<in> measurable M (count_space (UNIV::bool set))"
    20 
    21 lemma pred_def: "pred M P \<longleftrightarrow> {x\<in>space M. P x} \<in> sets M"
    22 proof
    23   assume "pred M P"
    24   then have "P -` {True} \<inter> space M \<in> sets M"
    25     by (auto simp: measurable_count_space_eq2)
    26   also have "P -` {True} \<inter> space M = {x\<in>space M. P x}" by auto
    27   finally show "{x\<in>space M. P x} \<in> sets M" .
    28 next
    29   assume P: "{x\<in>space M. P x} \<in> sets M"
    30   moreover
    31   { fix X
    32     have "X \<in> Pow (UNIV :: bool set)" by simp
    33     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> {})}"
    34       unfolding UNIV_bool Pow_insert Pow_empty by auto
    35     then have "P -` X \<inter> space M \<in> sets M"
    36       by (auto intro!: sets.sets_Collect_neg sets.sets_Collect_imp sets.sets_Collect_conj sets.sets_Collect_const P) }
    37   then show "pred M P"
    38     by (auto simp: measurable_def)
    39 qed
    40 
    41 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))"
    42   by (rule measurable_compose[where f=f and N=M]) (auto simp: pred_def)
    43 
    44 lemma pred_sets2: "A \<in> sets N \<Longrightarrow> f \<in> measurable M N \<Longrightarrow> pred M (\<lambda>x. f x \<in> A)"
    45   by (rule measurable_compose[where f=f and N=N]) (auto simp: pred_def Int_def[symmetric])
    46 
    47 ML_file "measurable.ML"
    48 
    49 attribute_setup measurable = {*
    50   Scan.lift (Scan.optional (Args.parens (Scan.optional (Args.$$$ "raw" >> K true) false --
    51     Scan.optional (Args.$$$ "generic" >> K Measurable.Generic) Measurable.Concrete))
    52     (false, Measurable.Concrete) >> (Thm.declaration_attribute o Measurable.add_thm))
    53 *} "declaration of measurability theorems"
    54 
    55 attribute_setup measurable_dest = {*
    56   Scan.lift (Scan.succeed (Thm.declaration_attribute Measurable.add_dest))
    57 *} "add dest rule for measurability prover"
    58 
    59 attribute_setup measurable_app = {*
    60   Scan.lift (Scan.succeed (Thm.declaration_attribute Measurable.add_app))
    61 *} "add application rule for measurability prover"
    62 
    63 method_setup measurable = {*
    64   Scan.lift (Scan.succeed (fn ctxt => METHOD (fn facts => Measurable.measurable_tac ctxt facts)))
    65 *} "measurability prover"
    66 
    67 simproc_setup measurable ("A \<in> sets M" | "f \<in> measurable M N") = {* K Measurable.simproc *}
    68 
    69 declare
    70   measurable_compose_rev[measurable_dest]
    71   pred_sets1[measurable_dest]
    72   pred_sets2[measurable_dest]
    73   sets.sets_into_space[measurable_dest]
    74 
    75 declare
    76   sets.top[measurable]
    77   sets.empty_sets[measurable (raw)]
    78   sets.Un[measurable (raw)]
    79   sets.Diff[measurable (raw)]
    80 
    81 declare
    82   measurable_count_space[measurable (raw)]
    83   measurable_ident[measurable (raw)]
    84   measurable_ident_sets[measurable (raw)]
    85   measurable_const[measurable (raw)]
    86   measurable_If[measurable (raw)]
    87   measurable_comp[measurable (raw)]
    88   measurable_sets[measurable (raw)]
    89 
    90 lemma predE[measurable (raw)]: 
    91   "pred M P \<Longrightarrow> {x\<in>space M. P x} \<in> sets M"
    92   unfolding pred_def .
    93 
    94 lemma pred_intros_imp'[measurable (raw)]:
    95   "(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<longrightarrow> P x)"
    96   by (cases K) auto
    97 
    98 lemma pred_intros_conj1'[measurable (raw)]:
    99   "(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<and> P x)"
   100   by (cases K) auto
   101 
   102 lemma pred_intros_conj2'[measurable (raw)]:
   103   "(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. P x \<and> K)"
   104   by (cases K) auto
   105 
   106 lemma pred_intros_disj1'[measurable (raw)]:
   107   "(\<not> K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<or> P x)"
   108   by (cases K) auto
   109 
   110 lemma pred_intros_disj2'[measurable (raw)]:
   111   "(\<not> K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. P x \<or> K)"
   112   by (cases K) auto
   113 
   114 lemma pred_intros_logic[measurable (raw)]:
   115   "pred M (\<lambda>x. x \<in> space M)"
   116   "pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. \<not> P x)"
   117   "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<and> P x)"
   118   "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<longrightarrow> P x)"
   119   "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<or> P x)"
   120   "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x = P x)"
   121   "pred M (\<lambda>x. f x \<in> UNIV)"
   122   "pred M (\<lambda>x. f x \<in> {})"
   123   "pred M (\<lambda>x. P' (f x) x) \<Longrightarrow> pred M (\<lambda>x. f x \<in> {y. P' y x})"
   124   "pred M (\<lambda>x. f x \<in> (B x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> - (B x))"
   125   "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))"
   126   "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))"
   127   "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))"
   128   "pred M (\<lambda>x. g x (f x) \<in> (X x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (g x) -` (X x))"
   129   by (auto simp: iff_conv_conj_imp pred_def)
   130 
   131 lemma pred_intros_countable[measurable (raw)]:
   132   fixes P :: "'a \<Rightarrow> 'i :: countable \<Rightarrow> bool"
   133   shows 
   134     "(\<And>i. pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<forall>i. P x i)"
   135     "(\<And>i. pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<exists>i. P x i)"
   136   by (auto intro!: sets.sets_Collect_countable_All sets.sets_Collect_countable_Ex simp: pred_def)
   137 
   138 lemma pred_intros_countable_bounded[measurable (raw)]:
   139   fixes X :: "'i :: countable set"
   140   shows 
   141     "(\<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))"
   142     "(\<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))"
   143     "(\<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)"
   144     "(\<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)"
   145   by (auto simp: Bex_def Ball_def)
   146 
   147 lemma pred_intros_finite[measurable (raw)]:
   148   "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))"
   149   "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))"
   150   "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)"
   151   "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)"
   152   by (auto intro!: sets.sets_Collect_finite_Ex sets.sets_Collect_finite_All simp: iff_conv_conj_imp pred_def)
   153 
   154 lemma countable_Un_Int[measurable (raw)]:
   155   "(\<And>i :: 'i :: countable. i \<in> I \<Longrightarrow> N i \<in> sets M) \<Longrightarrow> (\<Union>i\<in>I. N i) \<in> sets M"
   156   "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"
   157   by auto
   158 
   159 declare
   160   finite_UN[measurable (raw)]
   161   finite_INT[measurable (raw)]
   162 
   163 lemma sets_Int_pred[measurable (raw)]:
   164   assumes space: "A \<inter> B \<subseteq> space M" and [measurable]: "pred M (\<lambda>x. x \<in> A)" "pred M (\<lambda>x. x \<in> B)"
   165   shows "A \<inter> B \<in> sets M"
   166 proof -
   167   have "{x\<in>space M. x \<in> A \<inter> B} \<in> sets M" by auto
   168   also have "{x\<in>space M. x \<in> A \<inter> B} = A \<inter> B"
   169     using space by auto
   170   finally show ?thesis .
   171 qed
   172 
   173 lemma [measurable (raw generic)]:
   174   assumes f: "f \<in> measurable M N" and c: "c \<in> space N \<Longrightarrow> {c} \<in> sets N"
   175   shows pred_eq_const1: "pred M (\<lambda>x. f x = c)"
   176     and pred_eq_const2: "pred M (\<lambda>x. c = f x)"
   177 proof -
   178   show "pred M (\<lambda>x. f x = c)"
   179   proof cases
   180     assume "c \<in> space N"
   181     with measurable_sets[OF f c] show ?thesis
   182       by (auto simp: Int_def conj_commute pred_def)
   183   next
   184     assume "c \<notin> space N"
   185     with f[THEN measurable_space] have "{x \<in> space M. f x = c} = {}" by auto
   186     then show ?thesis by (auto simp: pred_def cong: conj_cong)
   187   qed
   188   then show "pred M (\<lambda>x. c = f x)"
   189     by (simp add: eq_commute)
   190 qed
   191 
   192 lemma pred_le_const[measurable (raw generic)]:
   193   assumes f: "f \<in> measurable M N" and c: "{.. c} \<in> sets N" shows "pred M (\<lambda>x. f x \<le> c)"
   194   using measurable_sets[OF f c]
   195   by (auto simp: Int_def conj_commute eq_commute pred_def)
   196 
   197 lemma pred_const_le[measurable (raw generic)]:
   198   assumes f: "f \<in> measurable M N" and c: "{c ..} \<in> sets N" shows "pred M (\<lambda>x. c \<le> f x)"
   199   using measurable_sets[OF f c]
   200   by (auto simp: Int_def conj_commute eq_commute pred_def)
   201 
   202 lemma pred_less_const[measurable (raw generic)]:
   203   assumes f: "f \<in> measurable M N" and c: "{..< c} \<in> sets N" shows "pred M (\<lambda>x. f x < c)"
   204   using measurable_sets[OF f c]
   205   by (auto simp: Int_def conj_commute eq_commute pred_def)
   206 
   207 lemma pred_const_less[measurable (raw generic)]:
   208   assumes f: "f \<in> measurable M N" and c: "{c <..} \<in> sets N" shows "pred M (\<lambda>x. c < f x)"
   209   using measurable_sets[OF f c]
   210   by (auto simp: Int_def conj_commute eq_commute pred_def)
   211 
   212 declare
   213   sets.Int[measurable (raw)]
   214 
   215 lemma pred_in_If[measurable (raw)]:
   216   "(P \<Longrightarrow> pred M (\<lambda>x. x \<in> A x)) \<Longrightarrow> (\<not> P \<Longrightarrow> pred M (\<lambda>x. x \<in> B x)) \<Longrightarrow>
   217     pred M (\<lambda>x. x \<in> (if P then A x else B x))"
   218   by auto
   219 
   220 lemma sets_range[measurable_dest]:
   221   "A ` I \<subseteq> sets M \<Longrightarrow> i \<in> I \<Longrightarrow> A i \<in> sets M"
   222   by auto
   223 
   224 lemma pred_sets_range[measurable_dest]:
   225   "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)"
   226   using pred_sets2[OF sets_range] by auto
   227 
   228 lemma sets_All[measurable_dest]:
   229   "\<forall>i. A i \<in> sets (M i) \<Longrightarrow> A i \<in> sets (M i)"
   230   by auto
   231 
   232 lemma pred_sets_All[measurable_dest]:
   233   "\<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)"
   234   using pred_sets2[OF sets_All, of A N f] by auto
   235 
   236 lemma sets_Ball[measurable_dest]:
   237   "\<forall>i\<in>I. A i \<in> sets (M i) \<Longrightarrow> i\<in>I \<Longrightarrow> A i \<in> sets (M i)"
   238   by auto
   239 
   240 lemma pred_sets_Ball[measurable_dest]:
   241   "\<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)"
   242   using pred_sets2[OF sets_Ball, of _ _ _ f] by auto
   243 
   244 lemma measurable_finite[measurable (raw)]:
   245   fixes S :: "'a \<Rightarrow> nat set"
   246   assumes [measurable]: "\<And>i. {x\<in>space M. i \<in> S x} \<in> sets M"
   247   shows "pred M (\<lambda>x. finite (S x))"
   248   unfolding finite_nat_set_iff_bounded by (simp add: Ball_def)
   249 
   250 lemma measurable_Least[measurable]:
   251   assumes [measurable]: "(\<And>i::nat. (\<lambda>x. P i x) \<in> measurable M (count_space UNIV))"q
   252   shows "(\<lambda>x. LEAST i. P i x) \<in> measurable M (count_space UNIV)"
   253   unfolding measurable_def by (safe intro!: sets_Least) simp_all
   254 
   255 lemma measurable_count_space_insert[measurable (raw)]:
   256   "s \<in> S \<Longrightarrow> A \<in> sets (count_space S) \<Longrightarrow> insert s A \<in> sets (count_space S)"
   257   by simp
   258 
   259 hide_const (open) pred
   260 
   261 end