src/HOL/Probability/Measurable.thy
 author hoelzl Wed Dec 05 15:59:08 2012 +0100 (2012-12-05) changeset 50387 3d8863c41fe8 child 50530 6266e44b3396 permissions -rw-r--r--
Move the measurability prover to its own file
```     1 (*  Title:      HOL/Probability/measurable.ML
```
```     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
```