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
10 hide_const (open) Order_Continuity.continuous
12 subsection {* Measurability prover *}
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
23 abbreviation "pred M P \<equiv> P \<in> measurable M (count_space (UNIV::bool set))"
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
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)
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])
51 ML_file "measurable.ML"
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"
63 attribute_setup measurable_dest = Measurable.dest_thm_attr
64   "add dest rule to measurability prover"
66 attribute_setup measurable_app = Measurable.app_thm_attr
67   "add application rule to measurability prover"
69 attribute_setup measurable_cong = Measurable.cong_thm_attr
70   "add congurence rules to measurability prover"
72 method_setup measurable = \<open> Scan.lift (Scan.succeed (METHOD o Measurable.measurable_tac)) \<close>
73   "measurability prover"
75 simproc_setup measurable ("A \<in> sets M" | "f \<in> measurable M N") = {* K Measurable.simproc *}
77 setup {*
79 *}
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]
87 declare
88   sets.top[measurable]
89   sets.empty_sets[measurable (raw)]
90   sets.Un[measurable (raw)]
91   sets.Diff[measurable (raw)]
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)]
102 declare measurable_cong_sets[measurable_cong]
103 declare sets_restrict_space_cong[measurable_cong]
105 lemma predE[measurable (raw)]:
106   "pred M P \<Longrightarrow> {x\<in>space M. P x} \<in> sets M"
107   unfolding pred_def .
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
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
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
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
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
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)
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)
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)
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)
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
174 declare
175   finite_UN[measurable (raw)]
176   finite_INT[measurable (raw)]
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
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)"
205 qed
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 )
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 )
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)
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)
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)
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)
235 declare
236   sets.Int[measurable (raw)]
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
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
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
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
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
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
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
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)
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
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
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
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
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
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
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
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
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
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
352 lemma sets_UNIV [measurable (raw)]: "A \<in> sets (count_space UNIV)"
353   by simp
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
380 subsection {* Measurability for (co)inductive predicates *}
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
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
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
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
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
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
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 .
532 qed
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)
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
568 lemma measurable_bot[measurable]: "Measurable.pred M bot"
571 lemma measurable_top[measurable]: "Measurable.pred M top"
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
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
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] ..
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)
595 hide_const (open) pred
597 end