(* Title: HOL/Analysis/Measurable.thy Author: Johannes Hölzl <hoelzl@in.tum.de> *) section ‹Measurability prover› theory Measurable imports Sigma_Algebra "HOL-Library.Order_Continuity" begin lemma (in algebra) sets_Collect_finite_All: assumes "⋀i. i ∈ S ⟹ {x∈Ω. P i x} ∈ M" "finite S" shows "{x∈Ω. ∀i∈S. P i x} ∈ M" proof - have "{x∈Ω. ∀i∈S. P i x} = (if S = {} then Ω else ⋂i∈S. {x∈Ω. P i x})" by auto with assms show ?thesis by (auto intro!: sets_Collect_finite_All') qed abbreviation "pred M P ≡ P ∈ measurable M (count_space (UNIV::bool set))" lemma pred_def: "pred M P ⟷ {x∈space M. P x} ∈ sets M" proof assume "pred M P" then have "P -` {True} ∩ space M ∈ sets M" by (auto simp: measurable_count_space_eq2) also have "P -` {True} ∩ space M = {x∈space M. P x}" by auto finally show "{x∈space M. P x} ∈ sets M" . next assume P: "{x∈space M. P x} ∈ sets M" moreover { fix X have "X ∈ Pow (UNIV :: bool set)" by simp then have "P -` X ∩ space M = {x∈space M. ((X = {True} ⟶ P x) ∧ (X = {False} ⟶ ¬ P x) ∧ X ≠ {})}" unfolding UNIV_bool Pow_insert Pow_empty by auto then have "P -` X ∩ space M ∈ sets M" by (auto intro!: sets.sets_Collect_neg sets.sets_Collect_imp sets.sets_Collect_conj sets.sets_Collect_const P) } then show "pred M P" by (auto simp: measurable_def) qed lemma pred_sets1: "{x∈space M. P x} ∈ sets M ⟹ f ∈ measurable N M ⟹ pred N (λx. P (f x))" by (rule measurable_compose[where f=f and N=M]) (auto simp: pred_def) lemma pred_sets2: "A ∈ sets N ⟹ f ∈ measurable M N ⟹ pred M (λx. f x ∈ A)" by (rule measurable_compose[where f=f and N=N]) (auto simp: pred_def Int_def[symmetric]) ML_file "measurable.ML" attribute_setup measurable = ‹ Scan.lift ( (Args.add >> K true || Args.del >> K false || Scan.succeed true) -- Scan.optional (Args.parens ( Scan.optional (Args.$$$ "raw" >> K true) false -- Scan.optional (Args.$$$ "generic" >> K Measurable.Generic) Measurable.Concrete)) (false, Measurable.Concrete) >> Measurable.measurable_thm_attr) › "declaration of measurability theorems" attribute_setup measurable_dest = Measurable.dest_thm_attr "add dest rule to measurability prover" attribute_setup measurable_cong = Measurable.cong_thm_attr "add congurence rules to measurability prover" method_setup%important measurable = ‹ Scan.lift (Scan.succeed (METHOD o Measurable.measurable_tac)) › "measurability prover" simproc_setup%important measurable ("A ∈ sets M" | "f ∈ measurable M N") = ‹K Measurable.simproc› setup ‹ Global_Theory.add_thms_dynamic (@{binding measurable}, Measurable.get_all) › declare pred_sets1[measurable_dest] pred_sets2[measurable_dest] sets.sets_into_space[measurable_dest] declare sets.top[measurable] sets.empty_sets[measurable (raw)] sets.Un[measurable (raw)] sets.Diff[measurable (raw)] declare measurable_count_space[measurable (raw)] measurable_ident[measurable (raw)] measurable_id[measurable (raw)] measurable_const[measurable (raw)] measurable_If[measurable (raw)] measurable_comp[measurable (raw)] measurable_sets[measurable (raw)] declare measurable_cong_sets[measurable_cong] declare sets_restrict_space_cong[measurable_cong] declare sets_restrict_UNIV[measurable_cong] lemma predE[measurable (raw)]: "pred M P ⟹ {x∈space M. P x} ∈ sets M" unfolding pred_def . lemma pred_intros_imp'[measurable (raw)]: "(K ⟹ pred M (λx. P x)) ⟹ pred M (λx. K ⟶ P x)" by (cases K) auto lemma pred_intros_conj1'[measurable (raw)]: "(K ⟹ pred M (λx. P x)) ⟹ pred M (λx. K ∧ P x)" by (cases K) auto lemma pred_intros_conj2'[measurable (raw)]: "(K ⟹ pred M (λx. P x)) ⟹ pred M (λx. P x ∧ K)" by (cases K) auto lemma pred_intros_disj1'[measurable (raw)]: "(¬ K ⟹ pred M (λx. P x)) ⟹ pred M (λx. K ∨ P x)" by (cases K) auto lemma pred_intros_disj2'[measurable (raw)]: "(¬ K ⟹ pred M (λx. P x)) ⟹ pred M (λx. P x ∨ K)" by (cases K) auto lemma pred_intros_logic[measurable (raw)]: "pred M (λx. x ∈ space M)" "pred M (λx. P x) ⟹ pred M (λx. ¬ P x)" "pred M (λx. Q x) ⟹ pred M (λx. P x) ⟹ pred M (λx. Q x ∧ P x)" "pred M (λx. Q x) ⟹ pred M (λx. P x) ⟹ pred M (λx. Q x ⟶ P x)" "pred M (λx. Q x) ⟹ pred M (λx. P x) ⟹ pred M (λx. Q x ∨ P x)" "pred M (λx. Q x) ⟹ pred M (λx. P x) ⟹ pred M (λx. Q x = P x)" "pred M (λx. f x ∈ UNIV)" "pred M (λx. f x ∈ {})" "pred M (λx. P' (f x) x) ⟹ pred M (λx. f x ∈ {y. P' y x})" "pred M (λx. f x ∈ (B x)) ⟹ pred M (λx. f x ∈ - (B x))" "pred M (λx. f x ∈ (A x)) ⟹ pred M (λx. f x ∈ (B x)) ⟹ pred M (λx. f x ∈ (A x) - (B x))" "pred M (λx. f x ∈ (A x)) ⟹ pred M (λx. f x ∈ (B x)) ⟹ pred M (λx. f x ∈ (A x) ∩ (B x))" "pred M (λx. f x ∈ (A x)) ⟹ pred M (λx. f x ∈ (B x)) ⟹ pred M (λx. f x ∈ (A x) ∪ (B x))" "pred M (λx. g x (f x) ∈ (X x)) ⟹ pred M (λx. f x ∈ (g x) -` (X x))" by (auto simp: iff_conv_conj_imp pred_def) lemma pred_intros_countable[measurable (raw)]: fixes P :: "'a ⇒ 'i :: countable ⇒ bool" shows "(⋀i. pred M (λx. P x i)) ⟹ pred M (λx. ∀i. P x i)" "(⋀i. pred M (λx. P x i)) ⟹ pred M (λx. ∃i. P x i)" by (auto intro!: sets.sets_Collect_countable_All sets.sets_Collect_countable_Ex simp: pred_def) lemma pred_intros_countable_bounded[measurable (raw)]: fixes X :: "'i :: countable set" shows "(⋀i. i ∈ X ⟹ pred M (λx. x ∈ N x i)) ⟹ pred M (λx. x ∈ (⋂i∈X. N x i))" "(⋀i. i ∈ X ⟹ pred M (λx. x ∈ N x i)) ⟹ pred M (λx. x ∈ (⋃i∈X. N x i))" "(⋀i. i ∈ X ⟹ pred M (λx. P x i)) ⟹ pred M (λx. ∀i∈X. P x i)" "(⋀i. i ∈ X ⟹ pred M (λx. P x i)) ⟹ pred M (λx. ∃i∈X. P x i)" by simp_all (auto simp: Bex_def Ball_def) lemma pred_intros_finite[measurable (raw)]: "finite I ⟹ (⋀i. i ∈ I ⟹ pred M (λx. x ∈ N x i)) ⟹ pred M (λx. x ∈ (⋂i∈I. N x i))" "finite I ⟹ (⋀i. i ∈ I ⟹ pred M (λx. x ∈ N x i)) ⟹ pred M (λx. x ∈ (⋃i∈I. N x i))" "finite I ⟹ (⋀i. i ∈ I ⟹ pred M (λx. P x i)) ⟹ pred M (λx. ∀i∈I. P x i)" "finite I ⟹ (⋀i. i ∈ I ⟹ pred M (λx. P x i)) ⟹ pred M (λx. ∃i∈I. P x i)" by (auto intro!: sets.sets_Collect_finite_Ex sets.sets_Collect_finite_All simp: iff_conv_conj_imp pred_def) lemma countable_Un_Int[measurable (raw)]: "(⋀i :: 'i :: countable. i ∈ I ⟹ N i ∈ sets M) ⟹ (⋃i∈I. N i) ∈ sets M" "I ≠ {} ⟹ (⋀i :: 'i :: countable. i ∈ I ⟹ N i ∈ sets M) ⟹ (⋂i∈I. N i) ∈ sets M" by auto declare finite_UN[measurable (raw)] finite_INT[measurable (raw)] lemma sets_Int_pred[measurable (raw)]: assumes space: "A ∩ B ⊆ space M" and [measurable]: "pred M (λx. x ∈ A)" "pred M (λx. x ∈ B)" shows "A ∩ B ∈ sets M" proof - have "{x∈space M. x ∈ A ∩ B} ∈ sets M" by auto also have "{x∈space M. x ∈ A ∩ B} = A ∩ B" using space by auto finally show ?thesis . qed lemma [measurable (raw generic)]: assumes f: "f ∈ measurable M N" and c: "c ∈ space N ⟹ {c} ∈ sets N" shows pred_eq_const1: "pred M (λx. f x = c)" and pred_eq_const2: "pred M (λx. c = f x)" proof - show "pred M (λx. f x = c)" proof cases assume "c ∈ space N" with measurable_sets[OF f c] show ?thesis by (auto simp: Int_def conj_commute pred_def) next assume "c ∉ space N" with f[THEN measurable_space] have "{x ∈ space M. f x = c} = {}" by auto then show ?thesis by (auto simp: pred_def cong: conj_cong) qed then show "pred M (λx. c = f x)" by (simp add: eq_commute) qed lemma pred_count_space_const1[measurable (raw)]: "f ∈ measurable M (count_space UNIV) ⟹ Measurable.pred M (λx. f x = c)" by (intro pred_eq_const1[where N="count_space UNIV"]) (auto ) lemma pred_count_space_const2[measurable (raw)]: "f ∈ measurable M (count_space UNIV) ⟹ Measurable.pred M (λx. c = f x)" by (intro pred_eq_const2[where N="count_space UNIV"]) (auto ) lemma pred_le_const[measurable (raw generic)]: assumes f: "f ∈ measurable M N" and c: "{.. c} ∈ sets N" shows "pred M (λx. f x ≤ c)" using measurable_sets[OF f c] by (auto simp: Int_def conj_commute eq_commute pred_def) lemma pred_const_le[measurable (raw generic)]: assumes f: "f ∈ measurable M N" and c: "{c ..} ∈ sets N" shows "pred M (λx. c ≤ f x)" using measurable_sets[OF f c] by (auto simp: Int_def conj_commute eq_commute pred_def) lemma pred_less_const[measurable (raw generic)]: assumes f: "f ∈ measurable M N" and c: "{..< c} ∈ sets N" shows "pred M (λx. f x < c)" using measurable_sets[OF f c] by (auto simp: Int_def conj_commute eq_commute pred_def) lemma pred_const_less[measurable (raw generic)]: assumes f: "f ∈ measurable M N" and c: "{c <..} ∈ sets N" shows "pred M (λx. c < f x)" using measurable_sets[OF f c] by (auto simp: Int_def conj_commute eq_commute pred_def) declare sets.Int[measurable (raw)] lemma pred_in_If[measurable (raw)]: "(P ⟹ pred M (λx. x ∈ A x)) ⟹ (¬ P ⟹ pred M (λx. x ∈ B x)) ⟹ pred M (λx. x ∈ (if P then A x else B x))" by auto lemma sets_range[measurable_dest]: "A ` I ⊆ sets M ⟹ i ∈ I ⟹ A i ∈ sets M" by auto lemma pred_sets_range[measurable_dest]: "A ` I ⊆ sets N ⟹ i ∈ I ⟹ f ∈ measurable M N ⟹ pred M (λx. f x ∈ A i)" using pred_sets2[OF sets_range] by auto lemma sets_All[measurable_dest]: "∀i. A i ∈ sets (M i) ⟹ A i ∈ sets (M i)" by auto lemma pred_sets_All[measurable_dest]: "∀i. A i ∈ sets (N i) ⟹ f ∈ measurable M (N i) ⟹ pred M (λx. f x ∈ A i)" using pred_sets2[OF sets_All, of A N f] by auto lemma sets_Ball[measurable_dest]: "∀i∈I. A i ∈ sets (M i) ⟹ i∈I ⟹ A i ∈ sets (M i)" by auto lemma pred_sets_Ball[measurable_dest]: "∀i∈I. A i ∈ sets (N i) ⟹ i∈I ⟹ f ∈ measurable M (N i) ⟹ pred M (λx. f x ∈ A i)" using pred_sets2[OF sets_Ball, of _ _ _ f] by auto lemma measurable_finite[measurable (raw)]: fixes S :: "'a ⇒ nat set" assumes [measurable]: "⋀i. {x∈space M. i ∈ S x} ∈ sets M" shows "pred M (λx. finite (S x))" unfolding finite_nat_set_iff_bounded by (simp add: Ball_def) lemma measurable_Least[measurable]: assumes [measurable]: "(⋀i::nat. (λx. P i x) ∈ measurable M (count_space UNIV))" shows "(λx. LEAST i. P i x) ∈ measurable M (count_space UNIV)" unfolding measurable_def by (safe intro!: sets_Least) simp_all lemma measurable_Max_nat[measurable (raw)]: fixes P :: "nat ⇒ 'a ⇒ bool" assumes [measurable]: "⋀i. Measurable.pred M (P i)" shows "(λx. Max {i. P i x}) ∈ measurable M (count_space UNIV)" unfolding measurable_count_space_eq2_countable proof safe fix n { fix x assume "∀i. ∃n≥i. P n x" then have "infinite {i. P i x}" unfolding infinite_nat_iff_unbounded_le by auto then have "Max {i. P i x} = the None" by (rule Max.infinite) } note 1 = this { fix x i j assume "P i x" "∀n≥j. ¬ P n x" then have "finite {i. P i x}" by (auto simp: subset_eq not_le[symmetric] finite_nat_iff_bounded) with ‹P i x› have "P (Max {i. P i x}) x" "i ≤ Max {i. P i x}" "finite {i. P i x}" using Max_in[of "{i. P i x}"] by auto } note 2 = this have "(λx. Max {i. P i x}) -` {n} ∩ space M = {x∈space M. Max {i. P i x} = n}" by auto also have "… = {x∈space M. if (∀i. ∃n≥i. P n x) then the None = n else if (∃i. P i x) then P n x ∧ (∀i>n. ¬ P i x) else Max {} = n}" by (intro arg_cong[where f=Collect] ext conj_cong) (auto simp add: 1 2 not_le[symmetric] intro!: Max_eqI) also have "… ∈ sets M" by measurable finally show "(λx. Max {i. P i x}) -` {n} ∩ space M ∈ sets M" . qed simp lemma measurable_Min_nat[measurable (raw)]: fixes P :: "nat ⇒ 'a ⇒ bool" assumes [measurable]: "⋀i. Measurable.pred M (P i)" shows "(λx. Min {i. P i x}) ∈ measurable M (count_space UNIV)" unfolding measurable_count_space_eq2_countable proof safe fix n { fix x assume "∀i. ∃n≥i. P n x" then have "infinite {i. P i x}" unfolding infinite_nat_iff_unbounded_le by auto then have "Min {i. P i x} = the None" by (rule Min.infinite) } note 1 = this { fix x i j assume "P i x" "∀n≥j. ¬ P n x" then have "finite {i. P i x}" by (auto simp: subset_eq not_le[symmetric] finite_nat_iff_bounded) with ‹P i x› have "P (Min {i. P i x}) x" "Min {i. P i x} ≤ i" "finite {i. P i x}" using Min_in[of "{i. P i x}"] by auto } note 2 = this have "(λx. Min {i. P i x}) -` {n} ∩ space M = {x∈space M. Min {i. P i x} = n}" by auto also have "… = {x∈space M. if (∀i. ∃n≥i. P n x) then the None = n else if (∃i. P i x) then P n x ∧ (∀i<n. ¬ P i x) else Min {} = n}" by (intro arg_cong[where f=Collect] ext conj_cong) (auto simp add: 1 2 not_le[symmetric] intro!: Min_eqI) also have "… ∈ sets M" by measurable finally show "(λx. Min {i. P i x}) -` {n} ∩ space M ∈ sets M" . qed simp lemma measurable_count_space_insert[measurable (raw)]: "s ∈ S ⟹ A ∈ sets (count_space S) ⟹ insert s A ∈ sets (count_space S)" by simp lemma sets_UNIV [measurable (raw)]: "A ∈ sets (count_space UNIV)" by simp lemma measurable_card[measurable]: fixes S :: "'a ⇒ nat set" assumes [measurable]: "⋀i. {x∈space M. i ∈ S x} ∈ sets M" shows "(λx. card (S x)) ∈ measurable M (count_space UNIV)" unfolding measurable_count_space_eq2_countable proof safe fix n show "(λx. card (S x)) -` {n} ∩ space M ∈ sets M" proof (cases n) case 0 then have "(λx. card (S x)) -` {n} ∩ space M = {x∈space M. infinite (S x) ∨ (∀i. i ∉ S x)}" by auto also have "… ∈ sets M" by measurable finally show ?thesis . next case (Suc i) then have "(λx. card (S x)) -` {n} ∩ space M = (⋃F∈{A∈{A. finite A}. card A = n}. {x∈space M. (∀i. i ∈ S x ⟷ i ∈ F)})" unfolding set_eq_iff[symmetric] Collect_bex_eq[symmetric] by (auto intro: card_ge_0_finite) also have "… ∈ sets M" by (intro sets.countable_UN' countable_Collect countable_Collect_finite) auto finally show ?thesis . qed qed rule lemma measurable_pred_countable[measurable (raw)]: assumes "countable X" shows "(⋀i. i ∈ X ⟹ Measurable.pred M (λx. P x i)) ⟹ Measurable.pred M (λx. ∀i∈X. P x i)" "(⋀i. i ∈ X ⟹ Measurable.pred M (λx. P x i)) ⟹ Measurable.pred M (λx. ∃i∈X. P x i)" unfolding pred_def by (auto intro!: sets.sets_Collect_countable_All' sets.sets_Collect_countable_Ex' assms) subsection%unimportant ‹Measurability for (co)inductive predicates› lemma measurable_bot[measurable]: "bot ∈ measurable M (count_space UNIV)" by (simp add: bot_fun_def) lemma measurable_top[measurable]: "top ∈ measurable M (count_space UNIV)" by (simp add: top_fun_def) lemma measurable_SUP[measurable]: fixes F :: "'i ⇒ 'a ⇒ 'b::{complete_lattice, countable}" assumes [simp]: "countable I" assumes [measurable]: "⋀i. i ∈ I ⟹ F i ∈ measurable M (count_space UNIV)" shows "(λx. SUP i:I. F i x) ∈ measurable M (count_space UNIV)" unfolding measurable_count_space_eq2_countable proof (safe intro!: UNIV_I) fix a have "(λx. SUP i:I. F i x) -` {a} ∩ space M = {x∈space M. (∀i∈I. F i x ≤ a) ∧ (∀b. (∀i∈I. F i x ≤ b) ⟶ a ≤ b)}" unfolding SUP_le_iff[symmetric] by auto also have "… ∈ sets M" by measurable finally show "(λx. SUP i:I. F i x) -` {a} ∩ space M ∈ sets M" . qed lemma measurable_INF[measurable]: fixes F :: "'i ⇒ 'a ⇒ 'b::{complete_lattice, countable}" assumes [simp]: "countable I" assumes [measurable]: "⋀i. i ∈ I ⟹ F i ∈ measurable M (count_space UNIV)" shows "(λx. INF i:I. F i x) ∈ measurable M (count_space UNIV)" unfolding measurable_count_space_eq2_countable proof (safe intro!: UNIV_I) fix a have "(λx. INF i:I. F i x) -` {a} ∩ space M = {x∈space M. (∀i∈I. a ≤ F i x) ∧ (∀b. (∀i∈I. b ≤ F i x) ⟶ b ≤ a)}" unfolding le_INF_iff[symmetric] by auto also have "… ∈ sets M" by measurable finally show "(λx. INF i:I. F i x) -` {a} ∩ space M ∈ sets M" . qed lemma measurable_lfp_coinduct[consumes 1, case_names continuity step]: fixes F :: "('a ⇒ 'b) ⇒ ('a ⇒ 'b::{complete_lattice, countable})" assumes "P M" assumes F: "sup_continuous F" assumes *: "⋀M A. P M ⟹ (⋀N. P N ⟹ A ∈ measurable N (count_space UNIV)) ⟹ F A ∈ measurable M (count_space UNIV)" shows "lfp F ∈ measurable M (count_space UNIV)" proof - { fix i from ‹P M› have "((F ^^ i) bot) ∈ measurable M (count_space UNIV)" by (induct i arbitrary: M) (auto intro!: *) } then have "(λx. SUP i. (F ^^ i) bot x) ∈ measurable M (count_space UNIV)" by measurable also have "(λx. SUP i. (F ^^ i) bot x) = lfp F" by (subst sup_continuous_lfp) (auto intro: F) finally show ?thesis . qed lemma measurable_lfp: fixes F :: "('a ⇒ 'b) ⇒ ('a ⇒ 'b::{complete_lattice, countable})" assumes F: "sup_continuous F" assumes *: "⋀A. A ∈ measurable M (count_space UNIV) ⟹ F A ∈ measurable M (count_space UNIV)" shows "lfp F ∈ measurable M (count_space UNIV)" by (coinduction rule: measurable_lfp_coinduct[OF _ F]) (blast intro: *) lemma measurable_gfp_coinduct[consumes 1, case_names continuity step]: fixes F :: "('a ⇒ 'b) ⇒ ('a ⇒ 'b::{complete_lattice, countable})" assumes "P M" assumes F: "inf_continuous F" assumes *: "⋀M A. P M ⟹ (⋀N. P N ⟹ A ∈ measurable N (count_space UNIV)) ⟹ F A ∈ measurable M (count_space UNIV)" shows "gfp F ∈ measurable M (count_space UNIV)" proof - { fix i from ‹P M› have "((F ^^ i) top) ∈ measurable M (count_space UNIV)" by (induct i arbitrary: M) (auto intro!: *) } then have "(λx. INF i. (F ^^ i) top x) ∈ measurable M (count_space UNIV)" by measurable also have "(λx. INF i. (F ^^ i) top x) = gfp F" by (subst inf_continuous_gfp) (auto intro: F) finally show ?thesis . qed lemma measurable_gfp: fixes F :: "('a ⇒ 'b) ⇒ ('a ⇒ 'b::{complete_lattice, countable})" assumes F: "inf_continuous F" assumes *: "⋀A. A ∈ measurable M (count_space UNIV) ⟹ F A ∈ measurable M (count_space UNIV)" shows "gfp F ∈ measurable M (count_space UNIV)" by (coinduction rule: measurable_gfp_coinduct[OF _ F]) (blast intro: *) lemma measurable_lfp2_coinduct[consumes 1, case_names continuity step]: fixes F :: "('a ⇒ 'c ⇒ 'b) ⇒ ('a ⇒ 'c ⇒ 'b::{complete_lattice, countable})" assumes "P M s" assumes F: "sup_continuous F" assumes *: "⋀M A s. P M s ⟹ (⋀N t. P N t ⟹ A t ∈ measurable N (count_space UNIV)) ⟹ F A s ∈ measurable M (count_space UNIV)" shows "lfp F s ∈ measurable M (count_space UNIV)" proof - { fix i from ‹P M s› have "(λx. (F ^^ i) bot s x) ∈ measurable M (count_space UNIV)" by (induct i arbitrary: M s) (auto intro!: *) } then have "(λx. SUP i. (F ^^ i) bot s x) ∈ measurable M (count_space UNIV)" by measurable also have "(λx. SUP i. (F ^^ i) bot s x) = lfp F s" by (subst sup_continuous_lfp) (auto simp: F) finally show ?thesis . qed lemma measurable_gfp2_coinduct[consumes 1, case_names continuity step]: fixes F :: "('a ⇒ 'c ⇒ 'b) ⇒ ('a ⇒ 'c ⇒ 'b::{complete_lattice, countable})" assumes "P M s" assumes F: "inf_continuous F" assumes *: "⋀M A s. P M s ⟹ (⋀N t. P N t ⟹ A t ∈ measurable N (count_space UNIV)) ⟹ F A s ∈ measurable M (count_space UNIV)" shows "gfp F s ∈ measurable M (count_space UNIV)" proof - { fix i from ‹P M s› have "(λx. (F ^^ i) top s x) ∈ measurable M (count_space UNIV)" by (induct i arbitrary: M s) (auto intro!: *) } then have "(λx. INF i. (F ^^ i) top s x) ∈ measurable M (count_space UNIV)" by measurable also have "(λx. INF i. (F ^^ i) top s x) = gfp F s" by (subst inf_continuous_gfp) (auto simp: F) finally show ?thesis . qed lemma measurable_enat_coinduct: fixes f :: "'a ⇒ enat" assumes "R f" assumes *: "⋀f. R f ⟹ ∃g h i P. R g ∧ f = (λx. if P x then h x else eSuc (g (i x))) ∧ Measurable.pred M P ∧ i ∈ measurable M M ∧ h ∈ measurable M (count_space UNIV)" shows "f ∈ measurable M (count_space UNIV)" proof (simp add: measurable_count_space_eq2_countable, rule ) fix a :: enat have "f -` {a} ∩ space M = {x∈space M. f x = a}" by auto { fix i :: nat from ‹R f› have "Measurable.pred M (λx. f x = enat i)" proof (induction i arbitrary: f) case 0 from *[OF this] obtain g h i P where f: "f = (λx. if P x then h x else eSuc (g (i x)))" and [measurable]: "Measurable.pred M P" "i ∈ measurable M M" "h ∈ measurable M (count_space UNIV)" by auto have "Measurable.pred M (λx. P x ∧ h x = 0)" by measurable also have "(λx. P x ∧ h x = 0) = (λx. f x = enat 0)" by (auto simp: f zero_enat_def[symmetric]) finally show ?case . next case (Suc n) from *[OF Suc.prems] obtain g h i P where f: "f = (λx. if P x then h x else eSuc (g (i x)))" and "R g" and M[measurable]: "Measurable.pred M P" "i ∈ measurable M M" "h ∈ measurable M (count_space UNIV)" by auto have "(λx. f x = enat (Suc n)) = (λx. (P x ⟶ h x = enat (Suc n)) ∧ (¬ P x ⟶ g (i x) = enat n))" by (auto simp: f zero_enat_def[symmetric] eSuc_enat[symmetric]) also have "Measurable.pred M …" by (intro pred_intros_logic measurable_compose[OF M(2)] Suc ‹R g›) measurable finally show ?case . qed then have "f -` {enat i} ∩ space M ∈ sets M" by (simp add: pred_def Int_def conj_commute) } note fin = this show "f -` {a} ∩ space M ∈ sets M" proof (cases a) case infinity then have "f -` {a} ∩ space M = space M - (⋃n. f -` {enat n} ∩ space M)" by auto also have "… ∈ sets M" by (intro sets.Diff sets.top sets.Un sets.countable_UN) (auto intro!: fin) finally show ?thesis . qed (simp add: fin) qed lemma measurable_THE: fixes P :: "'a ⇒ 'b ⇒ bool" assumes [measurable]: "⋀i. Measurable.pred M (P i)" assumes I[simp]: "countable I" "⋀i x. x ∈ space M ⟹ P i x ⟹ i ∈ I" assumes unique: "⋀x i j. x ∈ space M ⟹ P i x ⟹ P j x ⟹ i = j" shows "(λx. THE i. P i x) ∈ measurable M (count_space UNIV)" unfolding measurable_def proof safe fix X define f where "f x = (THE i. P i x)" for x define undef where "undef = (THE i::'a. False)" { fix i x assume "x ∈ space M" "P i x" then have "f x = i" unfolding f_def using unique by auto } note f_eq = this { fix x assume "x ∈ space M" "∀i∈I. ¬ P i x" then have "⋀i. ¬ P i x" using I(2)[of x] by auto then have "f x = undef" by (auto simp: undef_def f_def) } then have "f -` X ∩ space M = (⋃i∈I ∩ X. {x∈space M. P i x}) ∪ (if undef ∈ X then space M - (⋃i∈I. {x∈space M. P i x}) else {})" by (auto dest: f_eq) also have "… ∈ sets M" by (auto intro!: sets.Diff sets.countable_UN') finally show "f -` X ∩ space M ∈ sets M" . qed simp lemma measurable_Ex1[measurable (raw)]: assumes [simp]: "countable I" and [measurable]: "⋀i. i ∈ I ⟹ Measurable.pred M (P i)" shows "Measurable.pred M (λx. ∃!i∈I. P i x)" unfolding bex1_def by measurable lemma measurable_Sup_nat[measurable (raw)]: fixes F :: "'a ⇒ nat set" assumes [measurable]: "⋀i. Measurable.pred M (λx. i ∈ F x)" shows "(λx. Sup (F x)) ∈ M →⇩_{M}count_space UNIV" proof (clarsimp simp add: measurable_count_space_eq2_countable) fix a have F_empty_iff: "F x = {} ⟷ (∀i. i ∉ F x)" for x by auto have "Measurable.pred M (λx. if finite (F x) then if F x = {} then a = Max {} else a ∈ F x ∧ (∀j. j ∈ F x ⟶ j ≤ a) else a = the None)" unfolding finite_nat_set_iff_bounded Ball_def F_empty_iff by measurable moreover have "(λx. Sup (F x)) -` {a} ∩ space M = {x∈space M. if finite (F x) then if F x = {} then a = Max {} else a ∈ F x ∧ (∀j. j ∈ F x ⟶ j ≤ a) else a = the None}" by (intro set_eqI) (auto simp: Sup_nat_def Max.infinite intro!: Max_in Max_eqI) ultimately show "(λx. Sup (F x)) -` {a} ∩ space M ∈ sets M" by auto qed lemma measurable_if_split[measurable (raw)]: "(c ⟹ Measurable.pred M f) ⟹ (¬ c ⟹ Measurable.pred M g) ⟹ Measurable.pred M (if c then f else g)" by simp lemma pred_restrict_space: assumes "S ∈ sets M" shows "Measurable.pred (restrict_space M S) P ⟷ Measurable.pred M (λx. x ∈ S ∧ P x)" unfolding pred_def sets_Collect_restrict_space_iff[OF assms] .. lemma measurable_predpow[measurable]: assumes "Measurable.pred M T" assumes "⋀Q. Measurable.pred M Q ⟹ Measurable.pred M (R Q)" shows "Measurable.pred M ((R ^^ n) T)" by (induct n) (auto intro: assms) lemma measurable_compose_countable_restrict: assumes P: "countable {i. P i}" and f: "f ∈ M →⇩_{M}count_space UNIV" and Q: "⋀i. P i ⟹ pred M (Q i)" shows "pred M (λx. P (f x) ∧ Q (f x) x)" proof - have P_f: "{x ∈ space M. P (f x)} ∈ sets M" unfolding pred_def[symmetric] by (rule measurable_compose[OF f]) simp have "pred (restrict_space M {x∈space M. P (f x)}) (λx. Q (f x) x)" proof (rule measurable_compose_countable'[where g=f, OF _ _ P]) show "f ∈ restrict_space M {x∈space M. P (f x)} →⇩_{M}count_space {i. P i}" by (rule measurable_count_space_extend[OF subset_UNIV]) (auto simp: space_restrict_space intro!: measurable_restrict_space1 f) qed (auto intro!: measurable_restrict_space1 Q) then show ?thesis unfolding pred_restrict_space[OF P_f] by (simp cong: measurable_cong) qed lemma measurable_limsup [measurable (raw)]: assumes [measurable]: "⋀n. A n ∈ sets M" shows "limsup A ∈ sets M" by (subst limsup_INF_SUP, auto) lemma measurable_liminf [measurable (raw)]: assumes [measurable]: "⋀n. A n ∈ sets M" shows "liminf A ∈ sets M" by (subst liminf_SUP_INF, auto) lemma measurable_case_enat[measurable (raw)]: assumes f: "f ∈ M →⇩_{M}count_space UNIV" and g: "⋀i. g i ∈ M →⇩_{M}N" and h: "h ∈ M →⇩_{M}N" shows "(λx. case f x of enat i ⇒ g i x | ∞ ⇒ h x) ∈ M →⇩_{M}N" apply (rule measurable_compose_countable[OF _ f]) subgoal for i by (cases i) (auto intro: g h) done hide_const (open) pred end