src/HOL/Probability/Measurable.thy
author haftmann
Sun Jun 23 21:16:07 2013 +0200 (2013-06-23)
changeset 52435 6646bb548c6b
parent 50530 6266e44b3396
child 53043 8cbfbeb566a4
permissions -rw-r--r--
migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
     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 = {* Measurable.attr *} "declaration of measurability theorems"
    50 attribute_setup measurable_dest = {* Measurable.dest_attr *} "add dest rule for measurability prover"
    51 attribute_setup measurable_app = {* Measurable.app_attr *} "add application rule for measurability prover"
    52 method_setup measurable = {* Measurable.method *} "measurability prover"
    53 simproc_setup measurable ("A \<in> sets M" | "f \<in> measurable M N") = {* K Measurable.simproc *}
    54 
    55 declare
    56   measurable_compose_rev[measurable_dest]
    57   pred_sets1[measurable_dest]
    58   pred_sets2[measurable_dest]
    59   sets.sets_into_space[measurable_dest]
    60 
    61 declare
    62   sets.top[measurable]
    63   sets.empty_sets[measurable (raw)]
    64   sets.Un[measurable (raw)]
    65   sets.Diff[measurable (raw)]
    66 
    67 declare
    68   measurable_count_space[measurable (raw)]
    69   measurable_ident[measurable (raw)]
    70   measurable_ident_sets[measurable (raw)]
    71   measurable_const[measurable (raw)]
    72   measurable_If[measurable (raw)]
    73   measurable_comp[measurable (raw)]
    74   measurable_sets[measurable (raw)]
    75 
    76 lemma predE[measurable (raw)]: 
    77   "pred M P \<Longrightarrow> {x\<in>space M. P x} \<in> sets M"
    78   unfolding pred_def .
    79 
    80 lemma pred_intros_imp'[measurable (raw)]:
    81   "(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<longrightarrow> P x)"
    82   by (cases K) auto
    83 
    84 lemma pred_intros_conj1'[measurable (raw)]:
    85   "(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<and> P x)"
    86   by (cases K) auto
    87 
    88 lemma pred_intros_conj2'[measurable (raw)]:
    89   "(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. P x \<and> K)"
    90   by (cases K) auto
    91 
    92 lemma pred_intros_disj1'[measurable (raw)]:
    93   "(\<not> K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<or> P x)"
    94   by (cases K) auto
    95 
    96 lemma pred_intros_disj2'[measurable (raw)]:
    97   "(\<not> K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. P x \<or> K)"
    98   by (cases K) auto
    99 
   100 lemma pred_intros_logic[measurable (raw)]:
   101   "pred M (\<lambda>x. x \<in> space M)"
   102   "pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. \<not> P x)"
   103   "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<and> P x)"
   104   "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<longrightarrow> P x)"
   105   "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<or> P x)"
   106   "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x = P x)"
   107   "pred M (\<lambda>x. f x \<in> UNIV)"
   108   "pred M (\<lambda>x. f x \<in> {})"
   109   "pred M (\<lambda>x. P' (f x) x) \<Longrightarrow> pred M (\<lambda>x. f x \<in> {y. P' y x})"
   110   "pred M (\<lambda>x. f x \<in> (B x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> - (B x))"
   111   "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))"
   112   "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))"
   113   "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))"
   114   "pred M (\<lambda>x. g x (f x) \<in> (X x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (g x) -` (X x))"
   115   by (auto simp: iff_conv_conj_imp pred_def)
   116 
   117 lemma pred_intros_countable[measurable (raw)]:
   118   fixes P :: "'a \<Rightarrow> 'i :: countable \<Rightarrow> bool"
   119   shows 
   120     "(\<And>i. pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<forall>i. P x i)"
   121     "(\<And>i. pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<exists>i. P x i)"
   122   by (auto intro!: sets.sets_Collect_countable_All sets.sets_Collect_countable_Ex simp: pred_def)
   123 
   124 lemma pred_intros_countable_bounded[measurable (raw)]:
   125   fixes X :: "'i :: countable set"
   126   shows 
   127     "(\<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))"
   128     "(\<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))"
   129     "(\<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)"
   130     "(\<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)"
   131   by (auto simp: Bex_def Ball_def)
   132 
   133 lemma pred_intros_finite[measurable (raw)]:
   134   "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))"
   135   "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))"
   136   "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)"
   137   "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)"
   138   by (auto intro!: sets.sets_Collect_finite_Ex sets.sets_Collect_finite_All simp: iff_conv_conj_imp pred_def)
   139 
   140 lemma countable_Un_Int[measurable (raw)]:
   141   "(\<And>i :: 'i :: countable. i \<in> I \<Longrightarrow> N i \<in> sets M) \<Longrightarrow> (\<Union>i\<in>I. N i) \<in> sets M"
   142   "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"
   143   by auto
   144 
   145 declare
   146   finite_UN[measurable (raw)]
   147   finite_INT[measurable (raw)]
   148 
   149 lemma sets_Int_pred[measurable (raw)]:
   150   assumes space: "A \<inter> B \<subseteq> space M" and [measurable]: "pred M (\<lambda>x. x \<in> A)" "pred M (\<lambda>x. x \<in> B)"
   151   shows "A \<inter> B \<in> sets M"
   152 proof -
   153   have "{x\<in>space M. x \<in> A \<inter> B} \<in> sets M" by auto
   154   also have "{x\<in>space M. x \<in> A \<inter> B} = A \<inter> B"
   155     using space by auto
   156   finally show ?thesis .
   157 qed
   158 
   159 lemma [measurable (raw generic)]:
   160   assumes f: "f \<in> measurable M N" and c: "c \<in> space N \<Longrightarrow> {c} \<in> sets N"
   161   shows pred_eq_const1: "pred M (\<lambda>x. f x = c)"
   162     and pred_eq_const2: "pred M (\<lambda>x. c = f x)"
   163 proof -
   164   show "pred M (\<lambda>x. f x = c)"
   165   proof cases
   166     assume "c \<in> space N"
   167     with measurable_sets[OF f c] show ?thesis
   168       by (auto simp: Int_def conj_commute pred_def)
   169   next
   170     assume "c \<notin> space N"
   171     with f[THEN measurable_space] have "{x \<in> space M. f x = c} = {}" by auto
   172     then show ?thesis by (auto simp: pred_def cong: conj_cong)
   173   qed
   174   then show "pred M (\<lambda>x. c = f x)"
   175     by (simp add: eq_commute)
   176 qed
   177 
   178 lemma pred_le_const[measurable (raw generic)]:
   179   assumes f: "f \<in> measurable M N" and c: "{.. c} \<in> sets N" shows "pred M (\<lambda>x. f x \<le> c)"
   180   using measurable_sets[OF f c]
   181   by (auto simp: Int_def conj_commute eq_commute pred_def)
   182 
   183 lemma pred_const_le[measurable (raw generic)]:
   184   assumes f: "f \<in> measurable M N" and c: "{c ..} \<in> sets N" shows "pred M (\<lambda>x. c \<le> f x)"
   185   using measurable_sets[OF f c]
   186   by (auto simp: Int_def conj_commute eq_commute pred_def)
   187 
   188 lemma pred_less_const[measurable (raw generic)]:
   189   assumes f: "f \<in> measurable M N" and c: "{..< c} \<in> sets N" shows "pred M (\<lambda>x. f x < c)"
   190   using measurable_sets[OF f c]
   191   by (auto simp: Int_def conj_commute eq_commute pred_def)
   192 
   193 lemma pred_const_less[measurable (raw generic)]:
   194   assumes f: "f \<in> measurable M N" and c: "{c <..} \<in> sets N" shows "pred M (\<lambda>x. c < f x)"
   195   using measurable_sets[OF f c]
   196   by (auto simp: Int_def conj_commute eq_commute pred_def)
   197 
   198 declare
   199   sets.Int[measurable (raw)]
   200 
   201 lemma pred_in_If[measurable (raw)]:
   202   "(P \<Longrightarrow> pred M (\<lambda>x. x \<in> A x)) \<Longrightarrow> (\<not> P \<Longrightarrow> pred M (\<lambda>x. x \<in> B x)) \<Longrightarrow>
   203     pred M (\<lambda>x. x \<in> (if P then A x else B x))"
   204   by auto
   205 
   206 lemma sets_range[measurable_dest]:
   207   "A ` I \<subseteq> sets M \<Longrightarrow> i \<in> I \<Longrightarrow> A i \<in> sets M"
   208   by auto
   209 
   210 lemma pred_sets_range[measurable_dest]:
   211   "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)"
   212   using pred_sets2[OF sets_range] by auto
   213 
   214 lemma sets_All[measurable_dest]:
   215   "\<forall>i. A i \<in> sets (M i) \<Longrightarrow> A i \<in> sets (M i)"
   216   by auto
   217 
   218 lemma pred_sets_All[measurable_dest]:
   219   "\<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)"
   220   using pred_sets2[OF sets_All, of A N f] by auto
   221 
   222 lemma sets_Ball[measurable_dest]:
   223   "\<forall>i\<in>I. A i \<in> sets (M i) \<Longrightarrow> i\<in>I \<Longrightarrow> A i \<in> sets (M i)"
   224   by auto
   225 
   226 lemma pred_sets_Ball[measurable_dest]:
   227   "\<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)"
   228   using pred_sets2[OF sets_Ball, of _ _ _ f] by auto
   229 
   230 lemma measurable_finite[measurable (raw)]:
   231   fixes S :: "'a \<Rightarrow> nat set"
   232   assumes [measurable]: "\<And>i. {x\<in>space M. i \<in> S x} \<in> sets M"
   233   shows "pred M (\<lambda>x. finite (S x))"
   234   unfolding finite_nat_set_iff_bounded by (simp add: Ball_def)
   235 
   236 lemma measurable_Least[measurable]:
   237   assumes [measurable]: "(\<And>i::nat. (\<lambda>x. P i x) \<in> measurable M (count_space UNIV))"q
   238   shows "(\<lambda>x. LEAST i. P i x) \<in> measurable M (count_space UNIV)"
   239   unfolding measurable_def by (safe intro!: sets_Least) simp_all
   240 
   241 lemma measurable_count_space_insert[measurable (raw)]:
   242   "s \<in> S \<Longrightarrow> A \<in> sets (count_space S) \<Longrightarrow> insert s A \<in> sets (count_space S)"
   243   by simp
   244 
   245 hide_const (open) pred
   246 
   247 end