isabelle update_cartouches -c -t;
authorwenzelm
Mon Dec 07 20:19:59 2015 +0100 (2015-12-07)
changeset 61808fc1556774cfe
parent 61807 965769fe2b63
child 61809 81d34cf268d8
child 61811 1530a0f19539
isabelle update_cartouches -c -t;
src/HOL/Multivariate_Analysis/Bounded_Continuous_Function.thy
src/HOL/Multivariate_Analysis/Brouwer_Fixpoint.thy
src/HOL/Multivariate_Analysis/Cauchy_Integral_Thm.thy
src/HOL/Multivariate_Analysis/Complex_Analysis_Basics.thy
src/HOL/Multivariate_Analysis/Complex_Transcendental.thy
src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy
src/HOL/Multivariate_Analysis/Derivative.thy
src/HOL/Multivariate_Analysis/Integration.thy
src/HOL/Multivariate_Analysis/Linear_Algebra.thy
src/HOL/Multivariate_Analysis/Operator_Norm.thy
src/HOL/Multivariate_Analysis/Ordered_Euclidean_Space.thy
src/HOL/Multivariate_Analysis/Path_Connected.thy
src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
src/HOL/Multivariate_Analysis/Uniform_Limit.thy
src/HOL/Probability/Binary_Product_Measure.thy
src/HOL/Probability/Bochner_Integration.thy
src/HOL/Probability/Borel_Space.thy
src/HOL/Probability/Caratheodory.thy
src/HOL/Probability/Complete_Measure.thy
src/HOL/Probability/Convolution.thy
src/HOL/Probability/Discrete_Topology.thy
src/HOL/Probability/Distributions.thy
src/HOL/Probability/Embed_Measure.thy
src/HOL/Probability/Fin_Map.thy
src/HOL/Probability/Finite_Product_Measure.thy
src/HOL/Probability/Giry_Monad.thy
src/HOL/Probability/Independent_Family.thy
src/HOL/Probability/Infinite_Product_Measure.thy
src/HOL/Probability/Information.thy
src/HOL/Probability/Interval_Integral.thy
src/HOL/Probability/Lebesgue_Integral_Substitution.thy
src/HOL/Probability/Lebesgue_Measure.thy
src/HOL/Probability/Measurable.thy
src/HOL/Probability/Measure_Space.thy
src/HOL/Probability/Nonnegative_Lebesgue_Integration.thy
src/HOL/Probability/Probability_Mass_Function.thy
src/HOL/Probability/Probability_Measure.thy
src/HOL/Probability/Projective_Family.thy
src/HOL/Probability/Projective_Limit.thy
src/HOL/Probability/Radon_Nikodym.thy
src/HOL/Probability/Regularity.thy
src/HOL/Probability/Set_Integral.thy
src/HOL/Probability/Sigma_Algebra.thy
src/HOL/Probability/Stream_Space.thy
src/HOL/Probability/ex/Dining_Cryptographers.thy
src/HOL/Probability/ex/Koepf_Duermuth_Countermeasure.thy
src/HOL/Probability/ex/Measure_Not_CCC.thy
     1.1 --- a/src/HOL/Multivariate_Analysis/Bounded_Continuous_Function.thy	Mon Dec 07 16:48:10 2015 +0000
     1.2 +++ b/src/HOL/Multivariate_Analysis/Bounded_Continuous_Function.thy	Mon Dec 07 20:19:59 2015 +0100
     1.3 @@ -169,7 +169,7 @@
     1.4  instance bcontfun :: (metric_space, complete_space) complete_space
     1.5  proof
     1.6    fix f :: "nat \<Rightarrow> ('a, 'b) bcontfun"
     1.7 -  assume "Cauchy f"  -- \<open>Cauchy equals uniform convergence\<close>
     1.8 +  assume "Cauchy f"  \<comment> \<open>Cauchy equals uniform convergence\<close>
     1.9    then obtain g where limit_function:
    1.10      "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x. dist (Rep_bcontfun (f n) x) (g x) < e"
    1.11      using uniformly_convergent_eq_cauchy[of "\<lambda>_. True"
    1.12 @@ -177,13 +177,13 @@
    1.13      unfolding Cauchy_def
    1.14      by (metis dist_fun_lt_imp_dist_val_lt)
    1.15  
    1.16 -  then obtain N where fg_dist:  -- \<open>for an upper bound on @{term g}\<close>
    1.17 +  then obtain N where fg_dist:  \<comment> \<open>for an upper bound on @{term g}\<close>
    1.18      "\<forall>n\<ge>N. \<forall>x. dist (g x) ( Rep_bcontfun (f n) x) < 1"
    1.19      by (force simp add: dist_commute)
    1.20    from bcontfunE'[OF Rep_bcontfun, of "f N"] obtain b where
    1.21      f_bound: "\<forall>x. dist (Rep_bcontfun (f N) x) undefined \<le> b"
    1.22      by force
    1.23 -  have "g \<in> bcontfun"  -- \<open>The limit function is bounded and continuous\<close>
    1.24 +  have "g \<in> bcontfun"  \<comment> \<open>The limit function is bounded and continuous\<close>
    1.25    proof (intro bcontfunI)
    1.26      show "continuous_on UNIV g"
    1.27        using bcontfunE[OF Rep_bcontfun] limit_function
    1.28 @@ -199,7 +199,7 @@
    1.29    qed
    1.30    show "convergent f"
    1.31    proof (rule convergentI, subst lim_sequentially, safe)
    1.32 -    -- \<open>The limit function converges according to its norm\<close>
    1.33 +    \<comment> \<open>The limit function converges according to its norm\<close>
    1.34      fix e :: real
    1.35      assume "e > 0"
    1.36      then have "e/2 > 0" by simp
     2.1 --- a/src/HOL/Multivariate_Analysis/Brouwer_Fixpoint.thy	Mon Dec 07 16:48:10 2015 +0000
     2.2 +++ b/src/HOL/Multivariate_Analysis/Brouwer_Fixpoint.thy	Mon Dec 07 20:19:59 2015 +0100
     2.3 @@ -142,7 +142,7 @@
     2.4  lemma kuhn_counting_lemma:
     2.5    fixes bnd compo compo' face S F
     2.6    defines "nF s == card {f\<in>F. face f s \<and> compo' f}"
     2.7 -  assumes [simp, intro]: "finite F" -- "faces" and [simp, intro]: "finite S" -- "simplices"
     2.8 +  assumes [simp, intro]: "finite F" \<comment> "faces" and [simp, intro]: "finite S" \<comment> "simplices"
     2.9      and "\<And>f. f \<in> F \<Longrightarrow> bnd f \<Longrightarrow> card {s\<in>S. face f s} = 1"
    2.10      and "\<And>f. f \<in> F \<Longrightarrow> \<not> bnd f \<Longrightarrow> card {s\<in>S. face f s} = 2"
    2.11      and "\<And>s. s \<in> S \<Longrightarrow> compo s \<Longrightarrow> nF s = 1"
    2.12 @@ -1932,7 +1932,7 @@
    2.13    using assms by auto
    2.14  
    2.15  text \<open>Still more general form; could derive this directly without using the
    2.16 -  rather involved @{text "HOMEOMORPHIC_CONVEX_COMPACT"} theorem, just using
    2.17 +  rather involved \<open>HOMEOMORPHIC_CONVEX_COMPACT\<close> theorem, just using
    2.18    a scaling and translation to put the set inside the unit cube.\<close>
    2.19  
    2.20  lemma brouwer:
     3.1 --- a/src/HOL/Multivariate_Analysis/Cauchy_Integral_Thm.thy	Mon Dec 07 16:48:10 2015 +0000
     3.2 +++ b/src/HOL/Multivariate_Analysis/Cauchy_Integral_Thm.thy	Mon Dec 07 20:19:59 2015 +0100
     3.3 @@ -3209,7 +3209,7 @@
     3.4        by simp
     3.5      have "f y = f x" if "y \<in> s" and ccs: "f y \<in> connected_component_set (f ` s) (f x)" for y
     3.6        apply (rule ccontr)
     3.7 -      using connected_closed [of "connected_component_set (f ` s) (f x)"] `e>0`
     3.8 +      using connected_closed [of "connected_component_set (f ` s) (f x)"] \<open>e>0\<close>
     3.9        apply (simp add: del: ex_simps)
    3.10        apply (drule spec [where x="cball (f x) (e / 2)"])
    3.11        apply (drule spec [where x="- ball(f x) e"])
    3.12 @@ -3217,7 +3217,7 @@
    3.13          apply (metis diff_self e2 ele norm_minus_commute norm_zero not_less)
    3.14         using centre_in_cball connected_component_refl_eq e2 x apply blast
    3.15        using ccs
    3.16 -      apply (force simp: cball_def dist_norm norm_minus_commute dest: ele [OF `y \<in> s`])
    3.17 +      apply (force simp: cball_def dist_norm norm_minus_commute dest: ele [OF \<open>y \<in> s\<close>])
    3.18        done
    3.19      moreover have "connected_component_set (f ` s) (f x) \<subseteq> f ` s"
    3.20        by (auto simp: connected_component_in)
    3.21 @@ -3365,7 +3365,7 @@
    3.22      using contour_integral_nearby_ends [of "UNIV - {z}" \<gamma>] assms by (auto simp: open_delete)
    3.23    then obtain h where h: "polynomial_function h \<and> pathstart h = pathstart \<gamma> \<and> pathfinish h = pathfinish \<gamma> \<and>
    3.24                            (\<forall>t \<in> {0..1}. norm(h t - \<gamma> t) < d/2)"
    3.25 -    using path_approx_polynomial_function [OF `path \<gamma>`, of "d/2"] d by auto
    3.26 +    using path_approx_polynomial_function [OF \<open>path \<gamma>\<close>, of "d/2"] d by auto
    3.27    def nn \<equiv> "1/(2* pi*ii) * contour_integral h (\<lambda>w. 1/(w - z))"
    3.28    have "\<exists>n. \<forall>e > 0. \<exists>p. valid_path p \<and> z \<notin> path_image p \<and>
    3.29                          pathstart p = pathstart \<gamma> \<and>  pathfinish p = pathfinish \<gamma> \<and>
    3.30 @@ -3377,7 +3377,7 @@
    3.31        assume e: "e>0"
    3.32        obtain p where p: "polynomial_function p \<and>
    3.33              pathstart p = pathstart \<gamma> \<and> pathfinish p = pathfinish \<gamma> \<and> (\<forall>t\<in>{0..1}. cmod (p t - \<gamma> t) < min e (d / 2))"
    3.34 -        using path_approx_polynomial_function [OF `path \<gamma>`, of "min e (d/2)"] d `0<e` by auto
    3.35 +        using path_approx_polynomial_function [OF \<open>path \<gamma>\<close>, of "min e (d/2)"] d \<open>0<e\<close> by auto
    3.36        have "(\<lambda>w. 1 / (w - z)) holomorphic_on UNIV - {z}"
    3.37          by (auto simp: intro!: holomorphic_intros)
    3.38        then show "?PP e nn"
    3.39 @@ -3389,7 +3389,7 @@
    3.40    then show ?thesis
    3.41      unfolding winding_number_def
    3.42      apply (rule someI2_ex)
    3.43 -    apply (blast intro: `0<e`)
    3.44 +    apply (blast intro: \<open>0<e\<close>)
    3.45      done
    3.46  qed
    3.47  
    3.48 @@ -3692,7 +3692,7 @@
    3.49    obtain k where fink: "finite k" and g_C1_diff: "\<gamma> C1_differentiable_on ({a..b} - k)"
    3.50      using \<gamma> by (force simp: piecewise_C1_differentiable_on_def)
    3.51    have o: "open ({a<..<b} - k)"
    3.52 -    using `finite k` by (simp add: finite_imp_closed open_Diff)
    3.53 +    using \<open>finite k\<close> by (simp add: finite_imp_closed open_Diff)
    3.54    moreover have "{a<..<b} - k \<subseteq> {a..b} - k"
    3.55      by force
    3.56    ultimately have g_diff_at: "\<And>x. \<lbrakk>x \<notin> k; x \<in> {a<..<b}\<rbrakk> \<Longrightarrow> \<gamma> differentiable at x"
    3.57 @@ -3933,31 +3933,31 @@
    3.58                      "pathstart p = pathstart \<gamma> \<and> pathfinish p = pathfinish \<gamma>"
    3.59                and pg: "\<And>t. t\<in>{0..1} \<Longrightarrow> cmod (\<gamma> t - p t) < min d e / 2"
    3.60                and pi: "contour_integral p (\<lambda>x. 1 / (x - z)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z"
    3.61 -    using winding_number [OF \<gamma> z, of "min d e / 2"] `d>0` `e>0` by auto
    3.62 +    using winding_number [OF \<gamma> z, of "min d e / 2"] \<open>d>0\<close> \<open>e>0\<close> by auto
    3.63    { fix w
    3.64      assume d2: "cmod (w - z) < d/2" and e2: "cmod (w - z) < e/2"
    3.65      then have wnotp: "w \<notin> path_image p"
    3.66 -      using cbg `d>0` `e>0`
    3.67 +      using cbg \<open>d>0\<close> \<open>e>0\<close>
    3.68        apply (simp add: path_image_def cball_def dist_norm, clarify)
    3.69        apply (frule pg)
    3.70        apply (drule_tac c="\<gamma> x" in subsetD)
    3.71        apply (auto simp: less_eq_real_def norm_minus_commute norm_triangle_half_l)
    3.72        done
    3.73      have wnotg: "w \<notin> path_image \<gamma>"
    3.74 -      using cbg e2 `e>0` by (force simp: dist_norm norm_minus_commute)
    3.75 +      using cbg e2 \<open>e>0\<close> by (force simp: dist_norm norm_minus_commute)
    3.76      { fix k::real
    3.77        assume k: "k>0"
    3.78        then obtain q where q: "valid_path q" "w \<notin> path_image q"
    3.79                               "pathstart q = pathstart \<gamma> \<and> pathfinish q = pathfinish \<gamma>"
    3.80                      and qg: "\<And>t. t \<in> {0..1} \<Longrightarrow> cmod (\<gamma> t - q t) < min k (min d e) / 2"
    3.81                      and qi: "contour_integral q (\<lambda>u. 1 / (u - w)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> w"
    3.82 -        using winding_number [OF \<gamma> wnotg, of "min k (min d e) / 2"] `d>0` `e>0` k
    3.83 +        using winding_number [OF \<gamma> wnotg, of "min k (min d e) / 2"] \<open>d>0\<close> \<open>e>0\<close> k
    3.84          by (force simp: min_divide_distrib_right)
    3.85        have "contour_integral p (\<lambda>u. 1 / (u - w)) = contour_integral q (\<lambda>u. 1 / (u - w))"
    3.86 -        apply (rule pi_eq [OF `valid_path q` `valid_path p`, THEN conjunct2, THEN conjunct2, rule_format])
    3.87 +        apply (rule pi_eq [OF \<open>valid_path q\<close> \<open>valid_path p\<close>, THEN conjunct2, THEN conjunct2, rule_format])
    3.88          apply (frule pg)
    3.89          apply (frule qg)
    3.90 -        using p q `d>0` e2
    3.91 +        using p q \<open>d>0\<close> e2
    3.92          apply (auto simp: dist_norm norm_minus_commute intro!: holomorphic_intros)
    3.93          done
    3.94        then have "contour_integral p (\<lambda>x. 1 / (x - w)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> w"
    3.95 @@ -3979,11 +3979,11 @@
    3.96        and L: "\<And>f B. \<lbrakk>f holomorphic_on - cball z (3 / 4 * pe);
    3.97                        \<forall>z \<in> - cball z (3 / 4 * pe). cmod (f z) \<le> B\<rbrakk> \<Longrightarrow>
    3.98                        cmod (contour_integral p f) \<le> L * B"
    3.99 -    using contour_integral_bound_exists [of "- cball z (3/4*pe)" p] cbp `valid_path p` by force
   3.100 +    using contour_integral_bound_exists [of "- cball z (3/4*pe)" p] cbp \<open>valid_path p\<close> by force
   3.101    { fix e::real and w::complex
   3.102      assume e: "0 < e" and w: "cmod (w - z) < pe/4" "cmod (w - z) < e * pe\<^sup>2 / (8 * L)"
   3.103      then have [simp]: "w \<notin> path_image p"
   3.104 -      using cbp p(2) `0 < pe`
   3.105 +      using cbp p(2) \<open>0 < pe\<close>
   3.106        by (force simp: dist_norm norm_minus_commute path_image_def cball_def)
   3.107      have [simp]: "contour_integral p (\<lambda>x. 1/(x - w)) - contour_integral p (\<lambda>x. 1/(x - z)) =
   3.108                    contour_integral p (\<lambda>x. 1/(x - w) - 1/(x - z))"
   3.109 @@ -4001,13 +4001,13 @@
   3.110          using pe by auto
   3.111        then have "(pe/2)^2 < cmod (w - x) ^ 2"
   3.112          apply (rule power_strict_mono)
   3.113 -        using `pe>0` by auto
   3.114 +        using \<open>pe>0\<close> by auto
   3.115        then have pe2: "pe^2 < 4 * cmod (w - x) ^ 2"
   3.116          by (simp add: power_divide)
   3.117        have "8 * L * cmod (w - z) < e * pe\<^sup>2"
   3.118 -        using w `L>0` by (simp add: field_simps)
   3.119 +        using w \<open>L>0\<close> by (simp add: field_simps)
   3.120        also have "... < e * 4 * cmod (w - x) * cmod (w - x)"
   3.121 -        using pe2 `e>0` by (simp add: power2_eq_square)
   3.122 +        using pe2 \<open>e>0\<close> by (simp add: power2_eq_square)
   3.123        also have "... < e * 4 * cmod (w - x) * (4/3 * cmod (z - x))"
   3.124          using wx
   3.125          apply (rule mult_strict_left_mono)
   3.126 @@ -4019,23 +4019,23 @@
   3.127        finally have Lwz: "L * cmod (w - z) < e * cmod (w - x) * cmod (z - x)" .
   3.128        have "L * cmod (1 / (x - w) - 1 / (x - z)) \<le> e"
   3.129          apply (cases "x=z \<or> x=w")
   3.130 -        using pe `pe>0` w `L>0`
   3.131 +        using pe \<open>pe>0\<close> w \<open>L>0\<close>
   3.132          apply (force simp: norm_minus_commute)
   3.133 -        using wx w(2) `L>0` pe pe2 Lwz
   3.134 +        using wx w(2) \<open>L>0\<close> pe pe2 Lwz
   3.135          apply (auto simp: divide_simps mult_less_0_iff norm_minus_commute norm_divide norm_mult power2_eq_square)
   3.136          done
   3.137      } note L_cmod_le = this
   3.138      have *: "cmod (contour_integral p (\<lambda>x. 1 / (x - w) - 1 / (x - z))) \<le> L * (e * pe\<^sup>2 / L / 4 * (inverse (pe / 2))\<^sup>2)"
   3.139        apply (rule L)
   3.140 -      using `pe>0` w
   3.141 +      using \<open>pe>0\<close> w
   3.142        apply (force simp: dist_norm norm_minus_commute intro!: holomorphic_intros)
   3.143 -      using `pe>0` w `L>0`
   3.144 +      using \<open>pe>0\<close> w \<open>L>0\<close>
   3.145        apply (auto simp: cball_def dist_norm field_simps L_cmod_le  simp del: less_divide_eq_numeral1 le_divide_eq_numeral1)
   3.146        done
   3.147      have "cmod (contour_integral p (\<lambda>x. 1 / (x - w)) - contour_integral p (\<lambda>x. 1 / (x - z))) < 2*e"
   3.148        apply (simp add:)
   3.149        apply (rule le_less_trans [OF *])
   3.150 -      using `L>0` e
   3.151 +      using \<open>L>0\<close> e
   3.152        apply (force simp: field_simps)
   3.153        done
   3.154      then have "cmod (winding_number p w - winding_number p z) < e"
   3.155 @@ -4044,10 +4044,10 @@
   3.156    } note cmod_wn_diff = this
   3.157    show ?thesis
   3.158      apply (rule continuous_transform_at [of "min d e / 2" _ "winding_number p"])
   3.159 -    apply (auto simp: `d>0` `e>0` dist_norm wnwn simp del: less_divide_eq_numeral1)
   3.160 +    apply (auto simp: \<open>d>0\<close> \<open>e>0\<close> dist_norm wnwn simp del: less_divide_eq_numeral1)
   3.161      apply (simp add: continuous_at_eps_delta, clarify)
   3.162      apply (rule_tac x="min (pe/4) (e/2*pe^2/L/4)" in exI)
   3.163 -    using `pe>0` `L>0`
   3.164 +    using \<open>pe>0\<close> \<open>L>0\<close>
   3.165      apply (simp add: dist_norm cmod_wn_diff)
   3.166      done
   3.167  qed
   3.168 @@ -4057,7 +4057,7 @@
   3.169    by (simp add: continuous_at_imp_continuous_on continuous_at_winding_number)
   3.170  
   3.171  
   3.172 -subsection{*The winding number is constant on a connected region*}
   3.173 +subsection\<open>The winding number is constant on a connected region\<close>
   3.174  
   3.175  lemma winding_number_constant:
   3.176    assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>" and cs: "connected s" and sg: "s \<inter> path_image \<gamma> = {}"
   3.177 @@ -4067,7 +4067,7 @@
   3.178      assume ne: "winding_number \<gamma> y \<noteq> winding_number \<gamma> z"
   3.179      assume "y \<in> s" "z \<in> s"
   3.180      then have "winding_number \<gamma> y \<in> \<int>"  "winding_number \<gamma> z \<in>  \<int>"
   3.181 -      using integer_winding_number [OF \<gamma> loop] sg `y \<in> s` by auto
   3.182 +      using integer_winding_number [OF \<gamma> loop] sg \<open>y \<in> s\<close> by auto
   3.183      with ne have "1 \<le> cmod (winding_number \<gamma> y - winding_number \<gamma> z)"
   3.184        by (auto simp: Ints_def of_int_diff [symmetric] simp del: of_int_diff)
   3.185    } note * = this
   3.186 @@ -4132,7 +4132,7 @@
   3.187        obtain p where p: "polynomial_function p" "pathstart p = pathstart \<gamma>" "pathfinish p = pathfinish \<gamma>"
   3.188                   and pg1: "(\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> cmod (p t - \<gamma> t) < 1)"
   3.189                   and pge: "(\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> cmod (p t - \<gamma> t) < e)"
   3.190 -        using path_approx_polynomial_function [OF \<gamma>, of "min 1 e"] `e>0` by force
   3.191 +        using path_approx_polynomial_function [OF \<gamma>, of "min 1 e"] \<open>e>0\<close> by force
   3.192        have pip: "path_image p \<subseteq> ball 0 (B + 1)"
   3.193          using B
   3.194          apply (clarsimp simp add: path_image_def dist_norm ball_def)
   3.195 @@ -4197,7 +4197,7 @@
   3.196      hence "x \<notin> path_image \<gamma>"
   3.197        by (meson disjoint_iff_not_equal s_disj)
   3.198      thus "x \<in> inside (path_image \<gamma>)"
   3.199 -      using `x \<in> s` by (metis (no_types) ComplI UnE cos \<gamma> loop s_disj union_with_outside winding_number_eq winding_number_zero_in_outside wn_nz z)
   3.200 +      using \<open>x \<in> s\<close> by (metis (no_types) ComplI UnE cos \<gamma> loop s_disj union_with_outside winding_number_eq winding_number_zero_in_outside wn_nz z)
   3.201  qed
   3.202    show ?thesis
   3.203      apply (rule winding_number_eq [OF \<gamma> loop w])
   3.204 @@ -4326,10 +4326,10 @@
   3.205      have *: "a \<bullet> z' \<le> b - d / 3 * cmod a"
   3.206        unfolding z'_def inner_mult_right' divide_inverse
   3.207        apply (simp add: divide_simps algebra_simps dot_square_norm power2_eq_square anz)
   3.208 -      apply (metis `0 < d` add_increasing azb less_eq_real_def mult_nonneg_nonneg mult_right_mono norm_ge_zero norm_numeral)
   3.209 +      apply (metis \<open>0 < d\<close> add_increasing azb less_eq_real_def mult_nonneg_nonneg mult_right_mono norm_ge_zero norm_numeral)
   3.210        done
   3.211      have "cmod (winding_number \<gamma> z' - winding_number \<gamma> z) < \<bar>Re (winding_number \<gamma> z)\<bar> - 1/2"
   3.212 -      using d [of z'] anz `d>0` by (simp add: dist_norm z'_def)
   3.213 +      using d [of z'] anz \<open>d>0\<close> by (simp add: dist_norm z'_def)
   3.214      then have "1/2 < \<bar>Re (winding_number \<gamma> z)\<bar> - cmod (winding_number \<gamma> z' - winding_number \<gamma> z)"
   3.215        by simp
   3.216      then have "1/2 < \<bar>Re (winding_number \<gamma> z)\<bar> - \<bar>Re (winding_number \<gamma> z') - Re (winding_number \<gamma> z)\<bar>"
   3.217 @@ -4338,7 +4338,7 @@
   3.218        by linarith
   3.219      moreover have "\<bar>Re (winding_number \<gamma> z')\<bar> < 1/2"
   3.220        apply (rule winding_number_lt_half [OF \<gamma> *])
   3.221 -      using azb `d>0` pag
   3.222 +      using azb \<open>d>0\<close> pag
   3.223        apply (auto simp: add_strict_increasing anz divide_simps algebra_simps dest!: subsetD)
   3.224        done
   3.225      ultimately have False
   3.226 @@ -4372,7 +4372,7 @@
   3.227  qed
   3.228  
   3.229  
   3.230 -subsection{* Cauchy's integral formula, again for a convex enclosing set.*}
   3.231 +subsection\<open>Cauchy's integral formula, again for a convex enclosing set.\<close>
   3.232  
   3.233  lemma Cauchy_integral_formula_weak:
   3.234      assumes s: "convex s" and "finite k" and conf: "continuous_on s f"
   3.235 @@ -4462,7 +4462,7 @@
   3.236        using contour_integral_nearby [OF \<open>open s\<close> pak pik, of atends] f by metis
   3.237      obtain d where "d>0" and d:
   3.238          "\<And>x x'. \<lbrakk>x \<in> {0..1} \<times> {0..1}; x' \<in> {0..1} \<times> {0..1}; norm (x'-x) < d\<rbrakk> \<Longrightarrow> norm (k x' - k x) < e/4"
   3.239 -      by (rule uniformly_continuous_onE [OF ucontk, of "e/4"]) (auto simp: dist_norm `e>0`)
   3.240 +      by (rule uniformly_continuous_onE [OF ucontk, of "e/4"]) (auto simp: dist_norm \<open>e>0\<close>)
   3.241      { fix t1 t2
   3.242        assume t1: "0 \<le> t1" "t1 \<le> 1" and t2: "0 \<le> t2" "t2 \<le> 1" and ltd: "\<bar>t1 - t\<bar> < d" "\<bar>t2 - t\<bar> < d"
   3.243        have no2: "\<And>g1 k1 kt. \<lbrakk>norm(g1 - k1) < e/4; norm(k1 - kt) < e/4\<rbrakk> \<Longrightarrow> norm(g1 - kt) < e"
   3.244 @@ -4544,7 +4544,7 @@
   3.245      have n'N_less: "\<bar>real (Suc n) / real N - t\<bar> < ee t"
   3.246        using t N \<open>N > 0\<close> e_le_ee [of t]
   3.247        by (simp add: dist_norm add_divide_distrib abs_diff_less_iff del: less_divide_eq_numeral1) (simp add: field_simps)
   3.248 -    have t01: "t \<in> {0..1}" using `kk \<subseteq> {0..1}` `t \<in> kk` by blast
   3.249 +    have t01: "t \<in> {0..1}" using \<open>kk \<subseteq> {0..1}\<close> \<open>t \<in> kk\<close> by blast
   3.250      obtain d1 where "d1 > 0" and d1:
   3.251          "\<And>g1 g2. \<lbrakk>valid_path g1; valid_path g2;
   3.252                     \<forall>u\<in>{0..1}. cmod (g1 u - k (n/N, u)) < d1 \<and> cmod (g2 u - k ((Suc n) / N, u)) < d1;
   3.253 @@ -4562,7 +4562,7 @@
   3.254        using N01 by auto
   3.255      then have pkn: "path (\<lambda>u. k (n/N, u))"
   3.256        by (simp add: path_def)
   3.257 -    have min12: "min d1 d2 > 0" by (simp add: `0 < d1` `0 < d2`)
   3.258 +    have min12: "min d1 d2 > 0" by (simp add: \<open>0 < d1\<close> \<open>0 < d2\<close>)
   3.259      obtain p where "polynomial_function p"
   3.260          and psf: "pathstart p = pathstart (\<lambda>u. k (n/N, u))"
   3.261                   "pathfinish p = pathfinish (\<lambda>u. k (n/N, u))"
   3.262 @@ -4573,7 +4573,7 @@
   3.263        by (metis (mono_tags, lifting) N01(1) ksf linked_paths_def pathfinish_def pathstart_def psf)
   3.264      show ?case
   3.265        apply (rule_tac x="min d1 d2" in exI)
   3.266 -      apply (simp add: `0 < d1` `0 < d2`, clarify)
   3.267 +      apply (simp add: \<open>0 < d1\<close> \<open>0 < d2\<close>, clarify)
   3.268        apply (rule_tac s="contour_integral p f" in trans)
   3.269        using pk_le N01(1) ksf pathfinish_def pathstart_def
   3.270        apply (force intro!: vpp d1 simp add: linked_paths_def psf ksf)
     4.1 --- a/src/HOL/Multivariate_Analysis/Complex_Analysis_Basics.thy	Mon Dec 07 16:48:10 2015 +0000
     4.2 +++ b/src/HOL/Multivariate_Analysis/Complex_Analysis_Basics.thy	Mon Dec 07 20:19:59 2015 +0100
     4.3 @@ -897,7 +897,7 @@
     4.4  proof -
     4.5    from assms(4) obtain g' where A: "uniform_limit s (\<lambda>n x. \<Sum>i<n. f' i x) g' sequentially"
     4.6      unfolding uniformly_convergent_on_def by blast
     4.7 -  from x and `open s` have s: "at x within s = at x" by (rule at_within_open)
     4.8 +  from x and \<open>open s\<close> have s: "at x within s = at x" by (rule at_within_open)
     4.9    have "\<exists>g. \<forall>x\<in>s. (\<lambda>n. f n x) sums g x \<and> (g has_field_derivative g' x) (at x within s)"
    4.10      by (intro has_field_derivative_series[of s f f' g' x0] assms A has_field_derivative_at_within)
    4.11    then obtain g where g: "\<And>x. x \<in> s \<Longrightarrow> (\<lambda>n. f n x) sums g x"
    4.12 @@ -906,7 +906,7 @@
    4.13    from g(2)[OF x] have g': "(g has_derivative op * (g' x)) (at x)"
    4.14      by (simp add: has_field_derivative_def s)
    4.15    have "((\<lambda>x. \<Sum>n. f n x) has_derivative op * (g' x)) (at x)"
    4.16 -    by (rule has_derivative_transform_within_open[OF `open s` x _ g'])
    4.17 +    by (rule has_derivative_transform_within_open[OF \<open>open s\<close> x _ g'])
    4.18         (insert g, auto simp: sums_iff)
    4.19    thus "(\<lambda>x. \<Sum>n. f n x) complex_differentiable (at x)" unfolding differentiable_def
    4.20      by (auto simp: summable_def complex_differentiable_def has_field_derivative_def)
    4.21 @@ -919,7 +919,7 @@
    4.22    assumes "uniformly_convergent_on s (\<lambda>n x. \<Sum>i<n. f' i x)"
    4.23    assumes "x0 \<in> s" "summable (\<lambda>n. f n x0)"
    4.24    shows   "(\<lambda>x. \<Sum>n. f n x) complex_differentiable (at x0)"
    4.25 -  using complex_differentiable_series[OF assms, of x0] `x0 \<in> s` by blast+
    4.26 +  using complex_differentiable_series[OF assms, of x0] \<open>x0 \<in> s\<close> by blast+
    4.27  
    4.28  subsection\<open>Bound theorem\<close>
    4.29  
     5.1 --- a/src/HOL/Multivariate_Analysis/Complex_Transcendental.thy	Mon Dec 07 16:48:10 2015 +0000
     5.2 +++ b/src/HOL/Multivariate_Analysis/Complex_Transcendental.thy	Mon Dec 07 20:19:59 2015 +0100
     5.3 @@ -2116,7 +2116,7 @@
     5.4  lemma sin_Arcsin [simp]: "sin(Arcsin z) = z"
     5.5  proof -
     5.6    have "\<i>*z*2 + csqrt (1 - z\<^sup>2)*2 = 0 \<longleftrightarrow> (\<i>*z)*2 + csqrt (1 - z\<^sup>2)*2 = 0"
     5.7 -    by (simp add: algebra_simps)  --\<open>Cancelling a factor of 2\<close>
     5.8 +    by (simp add: algebra_simps)  \<comment>\<open>Cancelling a factor of 2\<close>
     5.9    moreover have "... \<longleftrightarrow> (\<i>*z) + csqrt (1 - z\<^sup>2) = 0"
    5.10      by (metis Arcsin_body_lemma distrib_right no_zero_divisors zero_neq_numeral)
    5.11    ultimately show ?thesis
    5.12 @@ -2294,7 +2294,7 @@
    5.13  lemma cos_Arccos [simp]: "cos(Arccos z) = z"
    5.14  proof -
    5.15    have "z*2 + \<i> * (2 * csqrt (1 - z\<^sup>2)) = 0 \<longleftrightarrow> z*2 + \<i> * csqrt (1 - z\<^sup>2)*2 = 0"
    5.16 -    by (simp add: algebra_simps)  --\<open>Cancelling a factor of 2\<close>
    5.17 +    by (simp add: algebra_simps)  \<comment>\<open>Cancelling a factor of 2\<close>
    5.18    moreover have "... \<longleftrightarrow> z + \<i> * csqrt (1 - z\<^sup>2) = 0"
    5.19      by (metis distrib_right mult_eq_0_iff zero_neq_numeral)
    5.20    ultimately show ?thesis
     6.1 --- a/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy	Mon Dec 07 16:48:10 2015 +0000
     6.2 +++ b/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy	Mon Dec 07 20:19:59 2015 +0100
     6.3 @@ -266,7 +266,7 @@
     6.4    have [simp]: "g ` f ` S = S"
     6.5      using g by (simp add: image_comp)
     6.6    have cgf: "closed (g ` f ` S)"
     6.7 -    by (simp add: `g \<circ> f = id` S image_comp)
     6.8 +    by (simp add: \<open>g \<circ> f = id\<close> S image_comp)
     6.9    have [simp]: "{x \<in> range f. g x \<in> S} = f ` S"
    6.10      using g by (simp add: o_def id_def image_def) metis
    6.11    show ?thesis
    6.12 @@ -5695,7 +5695,7 @@
    6.13    apply auto
    6.14    done
    6.15  
    6.16 -subsection \<open>On @{text "real"}, @{text "is_interval"}, @{text "convex"} and @{text "connected"} are all equivalent.\<close>
    6.17 +subsection \<open>On \<open>real\<close>, \<open>is_interval\<close>, \<open>convex\<close> and \<open>connected\<close> are all equivalent.\<close>
    6.18  
    6.19  lemma is_interval_1:
    6.20    "is_interval (s::real set) \<longleftrightarrow> (\<forall>a\<in>s. \<forall>b\<in>s. \<forall> x. a \<le> x \<and> x \<le> b \<longrightarrow> x \<in> s)"
    6.21 @@ -9132,7 +9132,7 @@
    6.22      { fix u v x
    6.23        assume uv: "setsum u t = 1" "\<forall>x\<in>s. 0 \<le> v x" "setsum v s = 1"
    6.24                   "(\<Sum>x\<in>s. v x *\<^sub>R x) = (\<Sum>v\<in>t. u v *\<^sub>R v)" "x \<in> t"
    6.25 -      then have s: "s = (s - t) \<union> t" --\<open>split into separate cases\<close>
    6.26 +      then have s: "s = (s - t) \<union> t" \<comment>\<open>split into separate cases\<close>
    6.27          using assms by auto
    6.28        have [simp]: "(\<Sum>x\<in>t. v x *\<^sub>R x) + (\<Sum>x\<in>s - t. v x *\<^sub>R x) = (\<Sum>x\<in>t. u x *\<^sub>R x)"
    6.29                     "setsum v t + setsum v (s - t) = 1"
    6.30 @@ -9250,7 +9250,7 @@
    6.31        using assms by (simp add: aff_independent_finite)
    6.32      { fix a b and d::real
    6.33        assume ab: "a \<in> s" "b \<in> s" "a \<noteq> b"
    6.34 -      then have s: "s = (s - {a,b}) \<union> {a,b}" --\<open>split into separate cases\<close>
    6.35 +      then have s: "s = (s - {a,b}) \<union> {a,b}" \<comment>\<open>split into separate cases\<close>
    6.36          by auto
    6.37        have "(\<Sum>x\<in>s. if x = a then - d else if x = b then d else 0) = 0"
    6.38             "(\<Sum>x\<in>s. (if x = a then - d else if x = b then d else 0) *\<^sub>R x) = d *\<^sub>R b - d *\<^sub>R a"
     7.1 --- a/src/HOL/Multivariate_Analysis/Derivative.thy	Mon Dec 07 16:48:10 2015 +0000
     7.2 +++ b/src/HOL/Multivariate_Analysis/Derivative.thy	Mon Dec 07 20:19:59 2015 +0100
     7.3 @@ -1585,7 +1585,7 @@
     7.4  
     7.5  text \<open>Hence the following eccentric variant of the inverse function theorem.
     7.6    This has no continuity assumptions, but we do need the inverse function.
     7.7 -  We could put @{text "f' \<circ> g = I"} but this happens to fit with the minimal linear
     7.8 +  We could put \<open>f' \<circ> g = I\<close> but this happens to fit with the minimal linear
     7.9    algebra theory I've set up so far.\<close>
    7.10  
    7.11  (* move  before left_inverse_linear in Euclidean_Space*)
    7.12 @@ -2264,7 +2264,7 @@
    7.13  proof -
    7.14    from assms(4) obtain g' where A: "uniform_limit s (\<lambda>n x. \<Sum>i<n. f' i x) g' sequentially"
    7.15      unfolding uniformly_convergent_on_def by blast
    7.16 -  from x and `open s` have s: "at x within s = at x" by (rule at_within_open)
    7.17 +  from x and \<open>open s\<close> have s: "at x within s = at x" by (rule at_within_open)
    7.18    have "\<exists>g. \<forall>x\<in>s. (\<lambda>n. f n x) sums g x \<and> (g has_field_derivative g' x) (at x within s)"
    7.19      by (intro has_field_derivative_series[of s f f' g' x0] assms A has_field_derivative_at_within)
    7.20    then obtain g where g: "\<And>x. x \<in> s \<Longrightarrow> (\<lambda>n. f n x) sums g x"
    7.21 @@ -2273,7 +2273,7 @@
    7.22    from g(2)[OF x] have g': "(g has_derivative op * (g' x)) (at x)"
    7.23      by (simp add: has_field_derivative_def s)
    7.24    have "((\<lambda>x. \<Sum>n. f n x) has_derivative op * (g' x)) (at x)"
    7.25 -    by (rule has_derivative_transform_within_open[OF `open s` x _ g'])
    7.26 +    by (rule has_derivative_transform_within_open[OF \<open>open s\<close> x _ g'])
    7.27         (insert g, auto simp: sums_iff)
    7.28    thus "(\<lambda>x. \<Sum>n. f n x) differentiable (at x)" unfolding differentiable_def
    7.29      by (auto simp: summable_def differentiable_def has_field_derivative_def)
    7.30 @@ -2286,7 +2286,7 @@
    7.31    assumes "uniformly_convergent_on s (\<lambda>n x. \<Sum>i<n. f' i x)"
    7.32    assumes "x0 \<in> s" "summable (\<lambda>n. f n x0)"
    7.33    shows   "(\<lambda>x. \<Sum>n. f n x) differentiable (at x0)"
    7.34 -  using differentiable_series[OF assms, of x0] `x0 \<in> s` by blast+
    7.35 +  using differentiable_series[OF assms, of x0] \<open>x0 \<in> s\<close> by blast+
    7.36  
    7.37  text \<open>Considering derivative @{typ "real \<Rightarrow> 'b::real_normed_vector"} as a vector.\<close>
    7.38  
     8.1 --- a/src/HOL/Multivariate_Analysis/Integration.thy	Mon Dec 07 16:48:10 2015 +0000
     8.2 +++ b/src/HOL/Multivariate_Analysis/Integration.thy	Mon Dec 07 20:19:59 2015 +0100
     8.3 @@ -9574,7 +9574,7 @@
     8.4  subsection \<open>Geometric progression\<close>
     8.5  
     8.6  text \<open>FIXME: Should one or more of these theorems be moved to @{file
     8.7 -"~~/src/HOL/Set_Interval.thy"}, alongside @{text geometric_sum}?\<close>
     8.8 +"~~/src/HOL/Set_Interval.thy"}, alongside \<open>geometric_sum\<close>?\<close>
     8.9  
    8.10  lemma sum_gp_basic:
    8.11    fixes x :: "'a::ring_1"
     9.1 --- a/src/HOL/Multivariate_Analysis/Linear_Algebra.thy	Mon Dec 07 16:48:10 2015 +0000
     9.2 +++ b/src/HOL/Multivariate_Analysis/Linear_Algebra.thy	Mon Dec 07 20:19:59 2015 +0100
     9.3 @@ -476,7 +476,7 @@
     9.4    apply auto
     9.5    done
     9.6  
     9.7 -lemma approachable_lt_le2:  --\<open>like the above, but pushes aside an extra formula\<close>
     9.8 +lemma approachable_lt_le2:  \<comment>\<open>like the above, but pushes aside an extra formula\<close>
     9.9      "(\<exists>(d::real) > 0. \<forall>x. Q x \<longrightarrow> f x < d \<longrightarrow> P x) \<longleftrightarrow> (\<exists>d>0. \<forall>x. f x \<le> d \<longrightarrow> Q x \<longrightarrow> P x)"
    9.10    apply auto
    9.11    apply (rule_tac x="d/2" in exI, auto)
    10.1 --- a/src/HOL/Multivariate_Analysis/Operator_Norm.thy	Mon Dec 07 16:48:10 2015 +0000
    10.2 +++ b/src/HOL/Multivariate_Analysis/Operator_Norm.thy	Mon Dec 07 20:19:59 2015 +0100
    10.3 @@ -9,7 +9,7 @@
    10.4  imports Complex_Main
    10.5  begin
    10.6  
    10.7 -text \<open>This formulation yields zero if @{text 'a} is the trivial vector space.\<close>
    10.8 +text \<open>This formulation yields zero if \<open>'a\<close> is the trivial vector space.\<close>
    10.9  
   10.10  definition onorm :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> real"
   10.11    where "onorm f = (SUP x. norm (f x) / norm x)"
    11.1 --- a/src/HOL/Multivariate_Analysis/Ordered_Euclidean_Space.thy	Mon Dec 07 16:48:10 2015 +0000
    11.2 +++ b/src/HOL/Multivariate_Analysis/Ordered_Euclidean_Space.thy	Mon Dec 07 20:19:59 2015 +0100
    11.3 @@ -174,7 +174,7 @@
    11.4        inner_Basis_SUP_left Sup_prod_def less_prod_def less_eq_prod_def eucl_le[where 'a='a]
    11.5        eucl_le[where 'a='b] abs_prod_def abs_inner)
    11.6  
    11.7 -text\<open>Instantiation for intervals on @{text ordered_euclidean_space}\<close>
    11.8 +text\<open>Instantiation for intervals on \<open>ordered_euclidean_space\<close>\<close>
    11.9  
   11.10  lemma
   11.11    fixes a :: "'a::ordered_euclidean_space"
    12.1 --- a/src/HOL/Multivariate_Analysis/Path_Connected.thy	Mon Dec 07 16:48:10 2015 +0000
    12.2 +++ b/src/HOL/Multivariate_Analysis/Path_Connected.thy	Mon Dec 07 20:19:59 2015 +0100
    12.3 @@ -755,14 +755,14 @@
    12.4    then have umin': "\<And>t. \<lbrakk>0 \<le> t; t \<le> 1; t < u\<rbrakk> \<Longrightarrow>  g t \<in> S"
    12.5      using closure_def by fastforce
    12.6    { assume "u \<noteq> 0"
    12.7 -    then have "u > 0" using `0 \<le> u` by auto
    12.8 +    then have "u > 0" using \<open>0 \<le> u\<close> by auto
    12.9      { fix e::real assume "e > 0"
   12.10        obtain d where "d>0" and d: "\<And>x'. \<lbrakk>x' \<in> {0..1}; dist x' u < d\<rbrakk> \<Longrightarrow> dist (g x') (g u) < e"
   12.11 -        using continuous_onD [OF gcon _ `e > 0`] `0 \<le> _` `_ \<le> 1` atLeastAtMost_iff by auto
   12.12 +        using continuous_onD [OF gcon _ \<open>e > 0\<close>] \<open>0 \<le> _\<close> \<open>_ \<le> 1\<close> atLeastAtMost_iff by auto
   12.13        have *: "dist (max 0 (u - d / 2)) u < d"
   12.14 -        using `0 \<le> u` `u \<le> 1` `d > 0` by (simp add: dist_real_def)
   12.15 +        using \<open>0 \<le> u\<close> \<open>u \<le> 1\<close> \<open>d > 0\<close> by (simp add: dist_real_def)
   12.16        have "\<exists>y\<in>S. dist y (g u) < e"
   12.17 -        using `0 < u` `u \<le> 1` `d > 0`
   12.18 +        using \<open>0 < u\<close> \<open>u \<le> 1\<close> \<open>d > 0\<close>
   12.19          by (force intro: d [OF _ *] umin')
   12.20      }
   12.21      then have "g u \<in> closure S"
   12.22 @@ -770,8 +770,8 @@
   12.23    }
   12.24    then show ?thesis
   12.25      apply (rule_tac u=u in that)
   12.26 -    apply (auto simp: `0 \<le> u` `u \<le> 1` gu interior_closure umin)
   12.27 -    using `_ \<le> 1` interior_closure umin apply fastforce
   12.28 +    apply (auto simp: \<open>0 \<le> u\<close> \<open>u \<le> 1\<close> gu interior_closure umin)
   12.29 +    using \<open>_ \<le> 1\<close> interior_closure umin apply fastforce
   12.30      done
   12.31  qed
   12.32  
   12.33 @@ -785,9 +785,9 @@
   12.34               and gunot: "(g u \<notin> interior S)" and u0: "(u = 0 \<or> g u \<in> closure S)"
   12.35      using subpath_to_frontier_explicit [OF assms] by blast
   12.36    show ?thesis
   12.37 -    apply (rule that [OF `0 \<le> u` `u \<le> 1`])
   12.38 +    apply (rule that [OF \<open>0 \<le> u\<close> \<open>u \<le> 1\<close>])
   12.39      apply (simp add: gunot)
   12.40 -    using `0 \<le> u` u0 by (force simp: subpath_def gxin)
   12.41 +    using \<open>0 \<le> u\<close> u0 by (force simp: subpath_def gxin)
   12.42  qed
   12.43  
   12.44  lemma subpath_to_frontier:
   12.45 @@ -800,9 +800,9 @@
   12.46                          (\<forall>x. 0 \<le> x \<and> x < 1 \<longrightarrow> subpath 0 u g x \<in> interior S) \<and> g u \<in> closure S"
   12.47      using subpath_to_frontier_strong [OF g g1] by blast
   12.48    show ?thesis
   12.49 -    apply (rule that [OF `0 \<le> u` `u \<le> 1`])
   12.50 +    apply (rule that [OF \<open>0 \<le> u\<close> \<open>u \<le> 1\<close>])
   12.51      apply (metis DiffI disj frontier_def g0 notin pathstart_def)
   12.52 -    using `0 \<le> u` g0 disj
   12.53 +    using \<open>0 \<le> u\<close> g0 disj
   12.54      apply (simp add: path_image_subpath_gen)
   12.55      apply (auto simp: closed_segment_eq_real_ivl pathstart_def pathfinish_def subpath_def)
   12.56      apply (rename_tac y)
   12.57 @@ -840,7 +840,7 @@
   12.58                      "pathfinish h \<in> frontier S"
   12.59      using exists_path_subpath_to_frontier [OF g _ g1] closure_closed [OF S] g0 by auto
   12.60    show ?thesis
   12.61 -    apply (rule that [OF `path h`])
   12.62 +    apply (rule that [OF \<open>path h\<close>])
   12.63      using assms h
   12.64      apply auto
   12.65      apply (metis diff_single_insert frontier_subset_eq insert_iff interior_subset subset_iff)
   12.66 @@ -1555,9 +1555,9 @@
   12.67    case False
   12.68    then obtain a where "a \<in> s" by auto
   12.69    { fix x y assume "x \<notin> s" "y \<notin> s"
   12.70 -    then have "x \<noteq> a" "y \<noteq> a" using `a \<in> s` by auto
   12.71 +    then have "x \<noteq> a" "y \<noteq> a" using \<open>a \<in> s\<close> by auto
   12.72      then have bxy: "bounded(insert x (insert y s))"
   12.73 -      by (simp add: `bounded s`)
   12.74 +      by (simp add: \<open>bounded s\<close>)
   12.75      then obtain B::real where B: "0 < B" and Bx: "norm (a - x) < B" and By: "norm (a - y) < B"
   12.76                            and "s \<subseteq> ball a B"
   12.77        using bounded_subset_ballD [OF bxy, of a] by (auto simp: dist_norm)
   12.78 @@ -1565,7 +1565,7 @@
   12.79      { fix u
   12.80        assume u: "(1 - u) *\<^sub>R x + u *\<^sub>R (a + C *\<^sub>R (x - a)) \<in> s" and "0 \<le> u" "u \<le> 1"
   12.81        have CC: "1 \<le> 1 + (C - 1) * u"
   12.82 -        using `x \<noteq> a` `0 \<le> u`
   12.83 +        using \<open>x \<noteq> a\<close> \<open>0 \<le> u\<close>
   12.84          apply (simp add: C_def divide_simps norm_minus_commute)
   12.85          using Bx by auto
   12.86        have *: "\<And>v. (1 - u) *\<^sub>R x + u *\<^sub>R (a + v *\<^sub>R (x - a)) = a + (1 + (v - 1) * u) *\<^sub>R (x - a)"
   12.87 @@ -1583,24 +1583,24 @@
   12.88        finally have xeq: "(1 - 1 / (1 + (C - 1) * u)) *\<^sub>R a + (1 / (1 + (C - 1) * u)) *\<^sub>R (a + (1 + (C - 1) * u) *\<^sub>R (x - a)) = x"
   12.89          by (simp add: algebra_simps)
   12.90        have False
   12.91 -        using `convex s`
   12.92 +        using \<open>convex s\<close>
   12.93          apply (simp add: convex_alt)
   12.94          apply (drule_tac x=a in bspec)
   12.95 -         apply (rule  `a \<in> s`)
   12.96 +         apply (rule  \<open>a \<in> s\<close>)
   12.97          apply (drule_tac x="a + (1 + (C - 1) * u) *\<^sub>R (x - a)" in bspec)
   12.98           using u apply (simp add: *)
   12.99          apply (drule_tac x="1 / (1 + (C - 1) * u)" in spec)
  12.100 -        using `x \<noteq> a` `x \<notin> s` `0 \<le> u` CC
  12.101 +        using \<open>x \<noteq> a\<close> \<open>x \<notin> s\<close> \<open>0 \<le> u\<close> CC
  12.102          apply (auto simp: xeq)
  12.103          done
  12.104      }
  12.105      then have pcx: "path_component (- s) x (a + C *\<^sub>R (x - a))"
  12.106        by (force simp: closed_segment_def intro!: path_connected_linepath)
  12.107 -    def D == "B / norm(y - a)"  --{*massive duplication with the proof above*}
  12.108 +    def D == "B / norm(y - a)"  \<comment>\<open>massive duplication with the proof above\<close>
  12.109      { fix u
  12.110        assume u: "(1 - u) *\<^sub>R y + u *\<^sub>R (a + D *\<^sub>R (y - a)) \<in> s" and "0 \<le> u" "u \<le> 1"
  12.111        have DD: "1 \<le> 1 + (D - 1) * u"
  12.112 -        using `y \<noteq> a` `0 \<le> u`
  12.113 +        using \<open>y \<noteq> a\<close> \<open>0 \<le> u\<close>
  12.114          apply (simp add: D_def divide_simps norm_minus_commute)
  12.115          using By by auto
  12.116        have *: "\<And>v. (1 - u) *\<^sub>R y + u *\<^sub>R (a + v *\<^sub>R (y - a)) = a + (1 + (v - 1) * u) *\<^sub>R (y - a)"
  12.117 @@ -1618,14 +1618,14 @@
  12.118        finally have xeq: "(1 - 1 / (1 + (D - 1) * u)) *\<^sub>R a + (1 / (1 + (D - 1) * u)) *\<^sub>R (a + (1 + (D - 1) * u) *\<^sub>R (y - a)) = y"
  12.119          by (simp add: algebra_simps)
  12.120        have False
  12.121 -        using `convex s`
  12.122 +        using \<open>convex s\<close>
  12.123          apply (simp add: convex_alt)
  12.124          apply (drule_tac x=a in bspec)
  12.125 -         apply (rule  `a \<in> s`)
  12.126 +         apply (rule  \<open>a \<in> s\<close>)
  12.127          apply (drule_tac x="a + (1 + (D - 1) * u) *\<^sub>R (y - a)" in bspec)
  12.128           using u apply (simp add: *)
  12.129          apply (drule_tac x="1 / (1 + (D - 1) * u)" in spec)
  12.130 -        using `y \<noteq> a` `y \<notin> s` `0 \<le> u` DD
  12.131 +        using \<open>y \<noteq> a\<close> \<open>y \<notin> s\<close> \<open>0 \<le> u\<close> DD
  12.132          apply (auto simp: xeq)
  12.133          done
  12.134      }
  12.135 @@ -1633,10 +1633,10 @@
  12.136        by (force simp: closed_segment_def intro!: path_connected_linepath)
  12.137      have pyx: "path_component (- s) (a + D *\<^sub>R (y - a)) (a + C *\<^sub>R (x - a))"
  12.138        apply (rule path_component_of_subset [of "{x. norm(x - a) = B}"])
  12.139 -       using `s \<subseteq> ball a B`
  12.140 +       using \<open>s \<subseteq> ball a B\<close>
  12.141         apply (force simp: ball_def dist_norm norm_minus_commute)
  12.142        apply (rule path_connected_sphere [OF 2, of a B, simplified path_connected_component, rule_format])
  12.143 -      using `x \<noteq> a`  using `y \<noteq> a`  B apply (auto simp: C_def D_def)
  12.144 +      using \<open>x \<noteq> a\<close>  using \<open>y \<noteq> a\<close>  B apply (auto simp: C_def D_def)
  12.145        done
  12.146      have "path_component (- s) x y"
  12.147        by (metis path_component_trans path_component_sym pcx pdy pyx)
  12.148 @@ -1834,7 +1834,7 @@
  12.149  proof -
  12.150    obtain dx dy where "dx \<ge> 0" "dy \<ge> 0" "x = B + dx" "y = B + dy"
  12.151      using assms by auto (metis add.commute diff_add_cancel)
  12.152 -  with `0 \<le> u` `u \<le> 1` show ?thesis
  12.153 +  with \<open>0 \<le> u\<close> \<open>u \<le> 1\<close> show ?thesis
  12.154      by (simp add: add_increasing2 mult_left_le field_simps)
  12.155  qed
  12.156  
  12.157 @@ -2036,20 +2036,20 @@
  12.158        by (metis mem_Collect_eq)
  12.159      def C \<equiv> "((B + 1 + norm z) / norm (z-a))"
  12.160      have "C > 0"
  12.161 -      using `0 < B` zna by (simp add: C_def divide_simps add_strict_increasing)
  12.162 +      using \<open>0 < B\<close> zna by (simp add: C_def divide_simps add_strict_increasing)
  12.163      have "\<bar>norm (z + C *\<^sub>R (z-a)) - norm (C *\<^sub>R (z-a))\<bar> \<le> norm z"
  12.164        by (metis add_diff_cancel norm_triangle_ineq3)
  12.165      moreover have "norm (C *\<^sub>R (z-a)) > norm z + B"
  12.166 -      using zna `B>0` by (simp add: C_def le_max_iff_disj field_simps)
  12.167 +      using zna \<open>B>0\<close> by (simp add: C_def le_max_iff_disj field_simps)
  12.168      ultimately have C: "norm (z + C *\<^sub>R (z-a)) > B" by linarith
  12.169      { fix u::real
  12.170        assume u: "0\<le>u" "u\<le>1" and ins: "(1 - u) *\<^sub>R z + u *\<^sub>R (z + C *\<^sub>R (z - a)) \<in> s"
  12.171        then have Cpos: "1 + u * C > 0"
  12.172 -        by (meson `0 < C` add_pos_nonneg less_eq_real_def zero_le_mult_iff zero_less_one)
  12.173 +        by (meson \<open>0 < C\<close> add_pos_nonneg less_eq_real_def zero_le_mult_iff zero_less_one)
  12.174        then have *: "(1 / (1 + u * C)) *\<^sub>R z + (u * C / (1 + u * C)) *\<^sub>R z = z"
  12.175          by (simp add: scaleR_add_left [symmetric] divide_simps)
  12.176        then have False
  12.177 -        using convexD_alt [OF s `a \<in> s` ins, of "1/(u*C + 1)"] `C>0` `z \<notin> s` Cpos u
  12.178 +        using convexD_alt [OF s \<open>a \<in> s\<close> ins, of "1/(u*C + 1)"] \<open>C>0\<close> \<open>z \<notin> s\<close> Cpos u
  12.179          by (simp add: * divide_simps algebra_simps)
  12.180      } note contra = this
  12.181      have "connected_component (- s) z (z + C *\<^sub>R (z-a))"
  12.182 @@ -2250,7 +2250,7 @@
  12.183    next
  12.184      case False
  12.185        have front: "frontier t \<subseteq> - s"
  12.186 -        using `s \<subseteq> t` frontier_disjoint_eq t by auto
  12.187 +        using \<open>s \<subseteq> t\<close> frontier_disjoint_eq t by auto
  12.188        { fix \<gamma>
  12.189          assume "path \<gamma>" and pimg_sbs: "path_image \<gamma> - {pathfinish \<gamma>} \<subseteq> interior (- t)"
  12.190             and pf: "pathfinish \<gamma> \<in> frontier t" and ps: "pathstart \<gamma> = a"
  12.191 @@ -2267,20 +2267,20 @@
  12.192          have pimg_sbs_cos: "path_image \<gamma> \<subseteq> -s"
  12.193            using pimg_sbs apply (auto simp: path_image_def)
  12.194            apply (drule subsetD)
  12.195 -          using `c \<in> - s` `s \<subseteq> t` interior_subset apply (auto simp: c_def)
  12.196 +          using \<open>c \<in> - s\<close> \<open>s \<subseteq> t\<close> interior_subset apply (auto simp: c_def)
  12.197            done
  12.198          have "closed_segment c d \<le> cball c \<epsilon>"
  12.199            apply (simp add: segment_convex_hull)
  12.200            apply (rule hull_minimal)
  12.201 -          using  `\<epsilon> > 0` d apply (auto simp: dist_commute)
  12.202 +          using  \<open>\<epsilon> > 0\<close> d apply (auto simp: dist_commute)
  12.203            done
  12.204          with \<epsilon> have "closed_segment c d \<subseteq> -s" by blast
  12.205          moreover have con_gcd: "connected (path_image \<gamma> \<union> closed_segment c d)"
  12.206 -          by (rule connected_Un) (auto simp: c_def `path \<gamma>` connected_path_image)
  12.207 +          by (rule connected_Un) (auto simp: c_def \<open>path \<gamma>\<close> connected_path_image)
  12.208          ultimately have "connected_component (- s) a d"
  12.209            unfolding connected_component_def using pimg_sbs_cos ps by blast
  12.210          then have "outside s \<inter> t \<noteq> {}"
  12.211 -          using outside_same_component [OF _ a]  by (metis IntI `d \<in> t` empty_iff)
  12.212 +          using outside_same_component [OF _ a]  by (metis IntI \<open>d \<in> t\<close> empty_iff)
  12.213        } note * = this
  12.214        have pal: "pathstart (linepath a b) \<in> closure (- t)"
  12.215          by (auto simp: False closure_def)
  12.216 @@ -2328,10 +2328,10 @@
  12.217      moreover have "outside s \<inter> inside t \<noteq> {}"
  12.218        by (meson False assms(4) compact_eq_bounded_closed coms open_inside outside_compact_in_open t)
  12.219      ultimately have "inside s \<inter> t = {}"
  12.220 -      using inside_outside_intersect_connected [OF `connected t`, of s]
  12.221 +      using inside_outside_intersect_connected [OF \<open>connected t\<close>, of s]
  12.222        by (metis "2" compact_eq_bounded_closed coms connected_outside inf.commute inside_outside_intersect_connected outst)
  12.223      then show ?thesis
  12.224 -      using inside_inside [OF `s \<subseteq> inside t`] by blast
  12.225 +      using inside_inside [OF \<open>s \<subseteq> inside t\<close>] by blast
  12.226    qed
  12.227  qed
  12.228  
    13.1 --- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Mon Dec 07 16:48:10 2015 +0000
    13.2 +++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Mon Dec 07 20:19:59 2015 +0100
    13.3 @@ -728,7 +728,7 @@
    13.4                   openin (subtopology euclidean s) e2 \<and>
    13.5                   s \<subseteq> e1 \<union> e2 \<and> e1 \<inter> e2 = {} \<and> e1 \<noteq> {} \<and> e2 \<noteq> {})"
    13.6    apply (simp add: connected_def openin_open, safe)
    13.7 -  apply (simp_all, blast+)  --\<open>slow\<close>
    13.8 +  apply (simp_all, blast+)  \<comment>\<open>slow\<close>
    13.9    done
   13.10  
   13.11  lemma connected_open_in_eq:
   13.12 @@ -1898,7 +1898,7 @@
   13.13    next
   13.14      assume "\<forall>x \<in> s. connected_component_set s x = s"
   13.15      then show "connected s"
   13.16 -      by (metis `x \<in> s` connected_connected_component)
   13.17 +      by (metis \<open>x \<in> s\<close> connected_connected_component)
   13.18    qed
   13.19  qed
   13.20  
   13.21 @@ -5211,7 +5211,7 @@
   13.22  lemma Lim_trivial_limit: "trivial_limit net \<Longrightarrow> (f ---> l) net"
   13.23    by simp
   13.24  
   13.25 -lemmas continuous_on = continuous_on_def -- "legacy theorem name"
   13.26 +lemmas continuous_on = continuous_on_def \<comment> "legacy theorem name"
   13.27  
   13.28  lemma continuous_within_subset:
   13.29    "continuous (at x within s) f \<Longrightarrow> t \<subseteq> s \<Longrightarrow> continuous (at x within t) f"
    14.1 --- a/src/HOL/Multivariate_Analysis/Uniform_Limit.thy	Mon Dec 07 16:48:10 2015 +0000
    14.2 +++ b/src/HOL/Multivariate_Analysis/Uniform_Limit.thy	Mon Dec 07 20:19:59 2015 +0100
    14.3 @@ -209,7 +209,7 @@
    14.4        fix x assume x: "x \<in> X"
    14.5        with assms have "(\<lambda>n. f n x) ----> ?f x" 
    14.6          by (auto dest!: Cauchy_convergent uniformly_Cauchy_imp_Cauchy simp: convergent_LIMSEQ_iff)
    14.7 -      with `e/2 > 0` have "eventually (\<lambda>m. m \<ge> N \<and> dist (f m x) (?f x) < e/2) sequentially"
    14.8 +      with \<open>e/2 > 0\<close> have "eventually (\<lambda>m. m \<ge> N \<and> dist (f m x) (?f x) < e/2) sequentially"
    14.9          by (intro tendstoD eventually_conj eventually_ge_at_top)
   14.10        then obtain m where m: "m \<ge> N" "dist (f m x) (?f x) < e/2" 
   14.11          unfolding eventually_at_top_linorder by blast
    15.1 --- a/src/HOL/Probability/Binary_Product_Measure.thy	Mon Dec 07 16:48:10 2015 +0000
    15.2 +++ b/src/HOL/Probability/Binary_Product_Measure.thy	Mon Dec 07 20:19:59 2015 +0100
    15.3 @@ -2,7 +2,7 @@
    15.4      Author:     Johannes Hölzl, TU München
    15.5  *)
    15.6  
    15.7 -section {*Binary product measures*}
    15.8 +section \<open>Binary product measures\<close>
    15.9  
   15.10  theory Binary_Product_Measure
   15.11  imports Nonnegative_Lebesgue_Integration
   15.12 @@ -249,17 +249,17 @@
   15.13      have "(\<Sum>i. emeasure M (?C x i)) = emeasure M (\<Union>i. ?C x i)"
   15.14      proof (intro suminf_emeasure)
   15.15        show "range (?C x) \<subseteq> sets M"
   15.16 -        using F `Q \<in> sets (N \<Otimes>\<^sub>M M)` by (auto intro!: sets_Pair1)
   15.17 +        using F \<open>Q \<in> sets (N \<Otimes>\<^sub>M M)\<close> by (auto intro!: sets_Pair1)
   15.18        have "disjoint_family F" using F by auto
   15.19        show "disjoint_family (?C x)"
   15.20 -        by (rule disjoint_family_on_bisimulation[OF `disjoint_family F`]) auto
   15.21 +        by (rule disjoint_family_on_bisimulation[OF \<open>disjoint_family F\<close>]) auto
   15.22      qed
   15.23      also have "(\<Union>i. ?C x i) = Pair x -` Q"
   15.24 -      using F sets.sets_into_space[OF `Q \<in> sets (N \<Otimes>\<^sub>M M)`]
   15.25 +      using F sets.sets_into_space[OF \<open>Q \<in> sets (N \<Otimes>\<^sub>M M)\<close>]
   15.26        by (auto simp: space_pair_measure)
   15.27      finally have "emeasure M (Pair x -` Q) = (\<Sum>i. emeasure M (?C x i))"
   15.28        by simp }
   15.29 -  ultimately show ?thesis using `Q \<in> sets (N \<Otimes>\<^sub>M M)` F_sets
   15.30 +  ultimately show ?thesis using \<open>Q \<in> sets (N \<Otimes>\<^sub>M M)\<close> F_sets
   15.31      by auto
   15.32  qed
   15.33  
   15.34 @@ -320,7 +320,7 @@
   15.35      by (simp add: ac_simps)
   15.36  qed
   15.37  
   15.38 -subsection {* Binary products of $\sigma$-finite emeasure spaces *}
   15.39 +subsection \<open>Binary products of $\sigma$-finite emeasure spaces\<close>
   15.40  
   15.41  locale pair_sigma_finite = M1?: sigma_finite_measure M1 + M2?: sigma_finite_measure M2
   15.42    for M1 :: "'a measure" and M2 :: "'b measure"
   15.43 @@ -359,7 +359,7 @@
   15.44        then obtain i j where "fst x \<in> F1 i" "snd x \<in> F2 j"
   15.45          by (auto simp: space)
   15.46        then have "fst x \<in> F1 (max i j)" "snd x \<in> F2 (max j i)"
   15.47 -        using `incseq F1` `incseq F2` unfolding incseq_def
   15.48 +        using \<open>incseq F1\<close> \<open>incseq F2\<close> unfolding incseq_def
   15.49          by (force split: split_max)+
   15.50        then have "(fst x, snd x) \<in> F1 (max i j) \<times> F2 (max i j)"
   15.51          by (intro SigmaI) (auto simp add: max.commute)
   15.52 @@ -369,7 +369,7 @@
   15.53        using space by (auto simp: space)
   15.54    next
   15.55      fix i show "incseq (\<lambda>i. F1 i \<times> F2 i)"
   15.56 -      using `incseq F1` `incseq F2` unfolding incseq_Suc_iff by auto
   15.57 +      using \<open>incseq F1\<close> \<open>incseq F2\<close> unfolding incseq_Suc_iff by auto
   15.58    next
   15.59      fix i
   15.60      from F1 F2 have "F1 i \<in> sets M1" "F2 i \<in> sets M2" by auto
   15.61 @@ -453,7 +453,7 @@
   15.62      using assms unfolding eventually_ae_filter by auto
   15.63    show ?thesis
   15.64    proof (rule AE_I)
   15.65 -    from N measurable_emeasure_Pair1[OF `N \<in> sets (M1 \<Otimes>\<^sub>M M2)`]
   15.66 +    from N measurable_emeasure_Pair1[OF \<open>N \<in> sets (M1 \<Otimes>\<^sub>M M2)\<close>]
   15.67      show "emeasure M1 {x\<in>space M1. emeasure M2 (Pair x -` N) \<noteq> 0} = 0"
   15.68        by (auto simp: M2.emeasure_pair_measure_alt nn_integral_0_iff emeasure_nonneg)
   15.69      show "{x \<in> space M1. emeasure M2 (Pair x -` N) \<noteq> 0} \<in> sets M1"
   15.70 @@ -464,7 +464,7 @@
   15.71          show "emeasure M2 (Pair x -` N) = 0" by fact
   15.72          show "Pair x -` N \<in> sets M2" using N(1) by (rule sets_Pair1)
   15.73          show "{y \<in> space M2. \<not> Q (x, y)} \<subseteq> Pair x -` N"
   15.74 -          using N `x \<in> space M1` unfolding space_pair_measure by auto
   15.75 +          using N \<open>x \<in> space M1\<close> unfolding space_pair_measure by auto
   15.76        qed }
   15.77      then show "{x \<in> space M1. \<not> (AE y in M2. Q (x, y))} \<subseteq> {x \<in> space M1. emeasure M2 (Pair x -` N) \<noteq> 0}"
   15.78        by auto
   15.79 @@ -599,7 +599,7 @@
   15.80    shows "(\<integral>\<^sup>+ y. (\<integral>\<^sup>+ x. f x y \<partial>M1) \<partial>M2) = (\<integral>\<^sup>+ x. (\<integral>\<^sup>+ y. f x y \<partial>M2) \<partial>M1)"
   15.81    using Fubini[OF f] by simp
   15.82  
   15.83 -subsection {* Products on counting spaces, densities and distributions *}
   15.84 +subsection \<open>Products on counting spaces, densities and distributions\<close>
   15.85  
   15.86  lemma sigma_prod:
   15.87    assumes X_cover: "\<exists>E\<subseteq>A. countable E \<and> X = \<Union>E" and A: "A \<subseteq> Pow X"
   15.88 @@ -628,18 +628,18 @@
   15.89        fix a assume "a \<in> A"
   15.90        from Y_cover obtain E where E: "E \<subseteq> B" "countable E" and "Y = \<Union>E"
   15.91          by auto
   15.92 -      with `a \<in> A` A have eq: "fst -` a \<inter> X \<times> Y = (\<Union>e\<in>E. a \<times> e)"
   15.93 +      with \<open>a \<in> A\<close> A have eq: "fst -` a \<inter> X \<times> Y = (\<Union>e\<in>E. a \<times> e)"
   15.94          by auto
   15.95        show "fst -` a \<inter> X \<times> Y \<in> sigma_sets (X \<times> Y) {a \<times> b |a b. a \<in> A \<and> b \<in> B}"
   15.96 -        using `a \<in> A` E unfolding eq by (auto intro!: XY.countable_UN')
   15.97 +        using \<open>a \<in> A\<close> E unfolding eq by (auto intro!: XY.countable_UN')
   15.98      next
   15.99        fix b assume "b \<in> B"
  15.100        from X_cover obtain E where E: "E \<subseteq> A" "countable E" and "X = \<Union>E"
  15.101          by auto
  15.102 -      with `b \<in> B` B have eq: "snd -` b \<inter> X \<times> Y = (\<Union>e\<in>E. e \<times> b)"
  15.103 +      with \<open>b \<in> B\<close> B have eq: "snd -` b \<inter> X \<times> Y = (\<Union>e\<in>E. e \<times> b)"
  15.104          by auto
  15.105        show "snd -` b \<inter> X \<times> Y \<in> sigma_sets (X \<times> Y) {a \<times> b |a b. a \<in> A \<and> b \<in> B}"
  15.106 -        using `b \<in> B` E unfolding eq by (auto intro!: XY.countable_UN')
  15.107 +        using \<open>b \<in> B\<close> E unfolding eq by (auto intro!: XY.countable_UN')
  15.108      qed
  15.109    next
  15.110      fix Z assume "Z \<in> {a \<times> b |a b. a \<in> A \<and> b \<in> B}"
  15.111 @@ -769,9 +769,9 @@
  15.112      with A have "emeasure (?A \<Otimes>\<^sub>M ?B) C \<le> emeasure (?A \<Otimes>\<^sub>M ?B) A"
  15.113        by (intro emeasure_mono) auto
  15.114      also have "emeasure (?A \<Otimes>\<^sub>M ?B) C = emeasure (count_space UNIV) C"
  15.115 -      using `countable C` by (rule *)
  15.116 +      using \<open>countable C\<close> by (rule *)
  15.117      finally show ?thesis
  15.118 -      using `infinite C` `infinite A` by simp
  15.119 +      using \<open>infinite C\<close> \<open>infinite A\<close> by simp
  15.120    qed
  15.121  qed
  15.122  
  15.123 @@ -799,7 +799,7 @@
  15.124      by (auto simp add: nn_integral_count_space_indicator intro!: nn_integral_cong split: split_indicator)
  15.125  next
  15.126    { fix x assume "f x \<noteq> 0"
  15.127 -    with `0 \<le> f x` have "(\<exists>r. 0 < r \<and> f x = ereal r) \<or> f x = \<infinity>"
  15.128 +    with \<open>0 \<le> f x\<close> have "(\<exists>r. 0 < r \<and> f x = ereal r) \<or> f x = \<infinity>"
  15.129        by (cases "f x") (auto simp: less_le)
  15.130      then have "\<exists>n. ereal (1 / real (Suc n)) \<le> f x"
  15.131        by (auto elim!: nat_approx_posE intro!: less_imp_le) }
  15.132 @@ -814,16 +814,16 @@
  15.133      by (metis infinite_countable_subset')
  15.134  
  15.135    have [measurable]: "C \<in> sets ?P"
  15.136 -    using sets.countable[OF _ `countable C`, of ?P] by (auto simp: sets_Pair)
  15.137 +    using sets.countable[OF _ \<open>countable C\<close>, of ?P] by (auto simp: sets_Pair)
  15.138  
  15.139    have "(\<integral>\<^sup>+x. ereal (1/Suc n) * indicator C x \<partial>?P) \<le> nn_integral ?P f"
  15.140      using C by (intro nn_integral_mono) (auto split: split_indicator simp: zero_ereal_def[symmetric])
  15.141    moreover have "(\<integral>\<^sup>+x. ereal (1/Suc n) * indicator C x \<partial>?P) = \<infinity>"
  15.142 -    using `infinite C` by (simp add: nn_integral_cmult emeasure_count_space_prod_eq)
  15.143 +    using \<open>infinite C\<close> by (simp add: nn_integral_cmult emeasure_count_space_prod_eq)
  15.144    moreover have "(\<integral>\<^sup>+x. ereal (1/Suc n) * indicator C x \<partial>count_space UNIV) \<le> nn_integral (count_space UNIV) f"
  15.145      using C by (intro nn_integral_mono) (auto split: split_indicator simp: zero_ereal_def[symmetric])
  15.146    moreover have "(\<integral>\<^sup>+x. ereal (1/Suc n) * indicator C x \<partial>count_space UNIV) = \<infinity>"
  15.147 -    using `infinite C` by (simp add: nn_integral_cmult)
  15.148 +    using \<open>infinite C\<close> by (simp add: nn_integral_cmult)
  15.149    ultimately show ?thesis
  15.150      by simp
  15.151  qed
  15.152 @@ -930,11 +930,11 @@
  15.153  next
  15.154    fix X assume X: "X \<subseteq> S1 \<times> S2"
  15.155    then have "countable X"
  15.156 -    by (metis countable_subset `countable S1` `countable S2` countable_SIGMA)
  15.157 +    by (metis countable_subset \<open>countable S1\<close> \<open>countable S2\<close> countable_SIGMA)
  15.158    have "X = (\<Union>(a, b)\<in>X. {a} \<times> {b})" by auto
  15.159    also have "\<dots> \<in> sets (M \<Otimes>\<^sub>M N)"
  15.160      using X
  15.161 -    by (safe intro!: sets.countable_UN' `countable X` subsetI pair_measureI) (auto simp: M N)
  15.162 +    by (safe intro!: sets.countable_UN' \<open>countable X\<close> subsetI pair_measureI) (auto simp: M N)
  15.163    finally show "X \<in> sets (M \<Otimes>\<^sub>M N)" .
  15.164  qed
  15.165  
  15.166 @@ -977,7 +977,7 @@
  15.167    finally show ?thesis .
  15.168  next
  15.169    { fix xy assume "f xy \<noteq> 0"
  15.170 -    with `0 \<le> f xy` have "(\<exists>r. 0 < r \<and> f xy = ereal r) \<or> f xy = \<infinity>"
  15.171 +    with \<open>0 \<le> f xy\<close> have "(\<exists>r. 0 < r \<and> f xy = ereal r) \<or> f xy = \<infinity>"
  15.172        by (cases "f xy") (auto simp: less_le)
  15.173      then have "\<exists>n. ereal (1 / real (Suc n)) \<le> f xy"
  15.174        by (auto elim!: nat_approx_posE intro!: less_imp_le) }
  15.175 @@ -1060,7 +1060,7 @@
  15.176  using _ _ assms(1)
  15.177  by(rule measurable_compose_countable'[where f="\<lambda>a b. f (a, snd b)" and g=fst and I=A, simplified])simp_all
  15.178  
  15.179 -subsection {* Product of Borel spaces *}
  15.180 +subsection \<open>Product of Borel spaces\<close>
  15.181  
  15.182  lemma borel_Times:
  15.183    fixes A :: "'a::topological_space set" and B :: "'b::topological_space set"
    16.1 --- a/src/HOL/Probability/Bochner_Integration.thy	Mon Dec 07 16:48:10 2015 +0000
    16.2 +++ b/src/HOL/Probability/Bochner_Integration.thy	Mon Dec 07 20:19:59 2015 +0100
    16.3 @@ -2,18 +2,18 @@
    16.4      Author:     Johannes Hölzl, TU München
    16.5  *)
    16.6  
    16.7 -section {* Bochner Integration for Vector-Valued Functions *}
    16.8 +section \<open>Bochner Integration for Vector-Valued Functions\<close>
    16.9  
   16.10  theory Bochner_Integration
   16.11    imports Finite_Product_Measure
   16.12  begin
   16.13  
   16.14 -text {*
   16.15 +text \<open>
   16.16  
   16.17  In the following development of the Bochner integral we use second countable topologies instead
   16.18  of separable spaces. A second countable topology is also separable.
   16.19  
   16.20 -*}
   16.21 +\<close>
   16.22  
   16.23  lemma borel_measurable_implies_sequence_metric:
   16.24    fixes f :: "'a \<Rightarrow> 'b :: {metric_space, second_countable_topology}"
   16.25 @@ -28,7 +28,7 @@
   16.26    { fix n x
   16.27      obtain d where "d \<in> D" and d: "d \<in> ball x (1 / Suc n)"
   16.28        using D[of "ball x (1 / Suc n)"] by auto
   16.29 -    from `d \<in> D` D[of UNIV] `countable D` obtain i where "d = e i"
   16.30 +    from \<open>d \<in> D\<close> D[of UNIV] \<open>countable D\<close> obtain i where "d = e i"
   16.31        unfolding e_def by (auto dest: from_nat_into_surj)
   16.32      with d have "\<exists>i. dist x (e i) < 1 / Suc n"
   16.33        by auto }
   16.34 @@ -109,16 +109,16 @@
   16.35        then have "\<And>i. F i x = z"
   16.36          by (auto simp: F_def)
   16.37        then show ?thesis
   16.38 -        using `f x = z` by auto
   16.39 +        using \<open>f x = z\<close> by auto
   16.40      next
   16.41        assume "f x \<noteq> z"
   16.42  
   16.43        show ?thesis
   16.44        proof (rule tendstoI)
   16.45          fix e :: real assume "0 < e"
   16.46 -        with `f x \<noteq> z` obtain n where "1 / Suc n < e" "1 / Suc n < dist (f x) z"
   16.47 +        with \<open>f x \<noteq> z\<close> obtain n where "1 / Suc n < e" "1 / Suc n < dist (f x) z"
   16.48            by (metis dist_nz order_less_trans neq_iff nat_approx_posE)
   16.49 -        with `x\<in>space M` `f x \<noteq> z` have "x \<in> (\<Union>i. B n i)"
   16.50 +        with \<open>x\<in>space M\<close> \<open>f x \<noteq> z\<close> have "x \<in> (\<Union>i. B n i)"
   16.51            unfolding A_def B_def UN_disjointed_eq using e by auto
   16.52          then obtain i where i: "x \<in> B n i" by auto
   16.53  
   16.54 @@ -131,7 +131,7 @@
   16.55            also have "\<dots> \<le> 1 / Suc n"
   16.56              using j m_upper[OF _ _ i]
   16.57              by (auto simp: field_simps)
   16.58 -          also note `1 / Suc n < e`
   16.59 +          also note \<open>1 / Suc n < e\<close>
   16.60            finally show "dist (F j x) (f x) < e"
   16.61              by (simp add: less_imp_le dist_commute)
   16.62          qed
   16.63 @@ -292,7 +292,7 @@
   16.64      with nn show "m * indicator {x \<in> space M. 0 \<noteq> f x} x \<le> f x"
   16.65        using f by (auto split: split_indicator simp: simple_function_def m_def)
   16.66    qed
   16.67 -  also note `\<dots> < \<infinity>`
   16.68 +  also note \<open>\<dots> < \<infinity>\<close>
   16.69    finally show ?thesis
   16.70      using m by auto 
   16.71  next
   16.72 @@ -556,7 +556,7 @@
   16.73    have sbi: "simple_bochner_integrable M (indicator A::'a \<Rightarrow> real)"
   16.74    proof
   16.75      have "{y \<in> space M. (indicator A y::real) \<noteq> 0} = A"
   16.76 -      using sets.sets_into_space[OF `A\<in>sets M`] by (auto split: split_indicator)
   16.77 +      using sets.sets_into_space[OF \<open>A\<in>sets M\<close>] by (auto split: split_indicator)
   16.78      then show "emeasure M {y \<in> space M. (indicator A y::real) \<noteq> 0} \<noteq> \<infinity>"
   16.79        using A by auto
   16.80    qed (rule simple_function_indicator assms)+
   16.81 @@ -743,7 +743,7 @@
   16.82    then have "(\<integral>\<^sup>+x. norm (s i x) \<partial>M) \<le> (\<integral>\<^sup>+x. ereal m * indicator {x\<in>space M. s i x \<noteq> 0} x \<partial>M)"
   16.83      by (auto split: split_indicator intro!: Max_ge nn_integral_mono simp:)
   16.84    also have "\<dots> < \<infinity>"
   16.85 -    using s by (subst nn_integral_cmult_indicator) (auto simp: `0 \<le> m` simple_bochner_integrable.simps)
   16.86 +    using s by (subst nn_integral_cmult_indicator) (auto simp: \<open>0 \<le> m\<close> simple_bochner_integrable.simps)
   16.87    finally have s_fin: "(\<integral>\<^sup>+x. norm (s i x) \<partial>M) < \<infinity>" .
   16.88  
   16.89    have "(\<integral>\<^sup>+ x. norm (f x) \<partial>M) \<le> (\<integral>\<^sup>+ x. ereal (norm (f x - s i x)) + ereal (norm (s i x)) \<partial>M)"
   16.90 @@ -824,7 +824,7 @@
   16.91      show "eventually (\<lambda>i. norm (?s i - ?t i) \<le> ?S i + ?T i) sequentially"
   16.92        by (intro always_eventually allI simple_bochner_integral_bounded s t f)
   16.93      show "(\<lambda>i. ?S i + ?T i) ----> ereal 0"
   16.94 -      using tendsto_add_ereal[OF _ _ `?S ----> 0` `?T ----> 0`]
   16.95 +      using tendsto_add_ereal[OF _ _ \<open>?S ----> 0\<close> \<open>?T ----> 0\<close>]
   16.96        by (simp add: zero_ereal_def[symmetric])
   16.97    qed
   16.98    then have "(\<lambda>i. norm (?s i - ?t i)) ----> 0"
   16.99 @@ -1159,7 +1159,7 @@
  16.100        have "norm (?s n - ?s m) \<le> ?S n + ?S m"
  16.101          by (intro simple_bochner_integral_bounded s f)
  16.102        also have "\<dots> < ereal (e / 2) + e / 2"
  16.103 -        using ereal_add_strict_mono[OF less_imp_le[OF M[OF n]] _ `?S n \<noteq> \<infinity>` M[OF m]]
  16.104 +        using ereal_add_strict_mono[OF less_imp_le[OF M[OF n]] _ \<open>?S n \<noteq> \<infinity>\<close> M[OF m]]
  16.105          by (auto simp: nn_integral_nonneg)
  16.106        also have "\<dots> = e" by simp
  16.107        finally show "dist (?s n) (?s m) < e"
  16.108 @@ -1460,7 +1460,7 @@
  16.109      using bound[of 0] by eventually_elim (auto intro: norm_ge_zero order_trans)
  16.110    then have "(\<integral>\<^sup>+x. w x \<partial>M) = (\<integral>\<^sup>+x. norm (w x) \<partial>M)"
  16.111      by (intro nn_integral_cong_AE) auto
  16.112 -  with `integrable M w` have w: "w \<in> borel_measurable M" "(\<integral>\<^sup>+x. w x \<partial>M) < \<infinity>"
  16.113 +  with \<open>integrable M w\<close> have w: "w \<in> borel_measurable M" "(\<integral>\<^sup>+x. w x \<partial>M) < \<infinity>"
  16.114      unfolding integrable_iff_bounded by auto
  16.115  
  16.116    show int_s: "\<And>i. integrable M (s i)"
  16.117 @@ -1690,7 +1690,7 @@
  16.118     (\<And>i x. x \<in> space M \<Longrightarrow> norm (s i x) \<le> 2 * norm (f x)) \<Longrightarrow> integrable M f \<Longrightarrow> P f"
  16.119    shows "P f"
  16.120  proof -
  16.121 -  from `integrable M f` have f: "f \<in> borel_measurable M" "(\<integral>\<^sup>+x. norm (f x) \<partial>M) < \<infinity>"
  16.122 +  from \<open>integrable M f\<close> have f: "f \<in> borel_measurable M" "(\<integral>\<^sup>+x. norm (f x) \<partial>M) < \<infinity>"
  16.123      unfolding integrable_iff_bounded by auto
  16.124    from borel_measurable_implies_sequence_metric[OF f(1)]
  16.125    obtain s where s: "\<And>i. simple_function M (s i)" "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. s i x) ----> f x"
  16.126 @@ -1746,7 +1746,7 @@
  16.127      fix x assume "x \<in> space M" with s show "(\<lambda>i. s' i x) ----> f x"
  16.128        by (simp add: s'_eq_s)
  16.129      show "norm (s' i x) \<le> 2 * norm (f x)"
  16.130 -      using `x \<in> space M` s by (simp add: s'_eq_s)
  16.131 +      using \<open>x \<in> space M\<close> s by (simp add: s'_eq_s)
  16.132    qed fact
  16.133  qed
  16.134  
  16.135 @@ -1838,7 +1838,7 @@
  16.136      by (simp add: not_integrable_integral_eq)
  16.137  qed
  16.138  
  16.139 -subsection {* Restricted measure spaces *}
  16.140 +subsection \<open>Restricted measure spaces\<close>
  16.141  
  16.142  lemma integrable_restrict_space:
  16.143    fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
  16.144 @@ -1890,7 +1890,7 @@
  16.145    thus ?thesis by simp
  16.146  qed
  16.147  
  16.148 -subsection {* Measure spaces with an associated density *}
  16.149 +subsection \<open>Measure spaces with an associated density\<close>
  16.150  
  16.151  lemma integrable_density:
  16.152    fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}" and g :: "'a \<Rightarrow> real"
  16.153 @@ -1972,7 +1972,7 @@
  16.154      has_bochner_integral M (\<lambda>x. g x *\<^sub>R f x) x \<Longrightarrow> has_bochner_integral (density M g) f x"
  16.155    by (simp add: has_bochner_integral_iff integrable_density integral_density)
  16.156  
  16.157 -subsection {* Distributions *}
  16.158 +subsection \<open>Distributions\<close>
  16.159  
  16.160  lemma integrable_distr_eq:
  16.161    fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
  16.162 @@ -2044,7 +2044,7 @@
  16.163      has_bochner_integral M (\<lambda>x. f (g x)) x \<Longrightarrow> has_bochner_integral (distr M N g) f x"
  16.164    by (simp add: has_bochner_integral_iff integrable_distr_eq integral_distr)
  16.165  
  16.166 -subsection {* Lebesgue integration on @{const count_space} *}
  16.167 +subsection \<open>Lebesgue integration on @{const count_space}\<close>
  16.168  
  16.169  lemma integrable_count_space:
  16.170    fixes f :: "'a \<Rightarrow> 'b::{banach,second_countable_topology}"
  16.171 @@ -2109,7 +2109,7 @@
  16.172    shows "integrable (count_space UNIV) f \<Longrightarrow> integral\<^sup>L (count_space UNIV) f = (\<Sum>x. f x)"
  16.173    using sums_integral_count_space_nat by (rule sums_unique)
  16.174  
  16.175 -subsection {* Point measure *}
  16.176 +subsection \<open>Point measure\<close>
  16.177  
  16.178  lemma lebesgue_integral_point_measure_finite:
  16.179    fixes g :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
  16.180 @@ -2126,7 +2126,7 @@
  16.181    apply (auto simp: AE_count_space integrable_count_space)
  16.182    done
  16.183  
  16.184 -subsection {* Lebesgue integration on @{const null_measure} *}
  16.185 +subsection \<open>Lebesgue integration on @{const null_measure}\<close>
  16.186  
  16.187  lemma has_bochner_integral_null_measure_iff[iff]:
  16.188    "has_bochner_integral (null_measure M) f 0 \<longleftrightarrow> f \<in> borel_measurable M"
  16.189 @@ -2140,7 +2140,7 @@
  16.190    by (cases "integrable (null_measure M) f")
  16.191       (auto simp add: not_integrable_integral_eq has_bochner_integral_integral_eq)
  16.192  
  16.193 -subsection {* Legacy lemmas for the real-valued Lebesgue integral *}
  16.194 +subsection \<open>Legacy lemmas for the real-valued Lebesgue integral\<close>
  16.195  
  16.196  lemma real_lebesgue_integral_def:
  16.197    assumes f[measurable]: "integrable M f"
  16.198 @@ -2388,7 +2388,7 @@
  16.199        using int A by (simp add: integrable_def)
  16.200      ultimately have "emeasure M A = 0"
  16.201        using emeasure_nonneg[of M A] by simp
  16.202 -    with `(emeasure M) A \<noteq> 0` show False by auto
  16.203 +    with \<open>(emeasure M) A \<noteq> 0\<close> show False by auto
  16.204    qed
  16.205    ultimately show ?thesis by auto
  16.206  qed
  16.207 @@ -2413,7 +2413,7 @@
  16.208      show "AE x in M. (\<lambda>n. indicator {..X n} x *\<^sub>R f x) ----> f x"
  16.209      proof
  16.210        fix x
  16.211 -      from `filterlim X at_top sequentially` 
  16.212 +      from \<open>filterlim X at_top sequentially\<close> 
  16.213        have "eventually (\<lambda>n. x \<le> X n) sequentially"
  16.214          unfolding filterlim_at_top_ge[where c=x] by auto
  16.215        then show "(\<lambda>n. indicator {..X n} x *\<^sub>R f x) ----> f x"
  16.216 @@ -2455,7 +2455,7 @@
  16.217      by (auto simp: _has_bochner_integral_iff)
  16.218  qed
  16.219  
  16.220 -subsection {* Product measure *}
  16.221 +subsection \<open>Product measure\<close>
  16.222  
  16.223  lemma (in sigma_finite_measure) borel_measurable_lebesgue_integrable[measurable (raw)]:
  16.224    fixes f :: "_ \<Rightarrow> _ \<Rightarrow> _::{banach, second_countable_topology}"
  16.225 @@ -2823,7 +2823,7 @@
  16.226      have f_borel: "f \<in> borel_measurable (Pi\<^sub>M (insert i I) M)"
  16.227        using f by auto
  16.228      show "(\<lambda>y. f (x(i := y))) \<in> borel_measurable (M i)"
  16.229 -      using measurable_comp[OF measurable_component_update f_borel, OF x `i \<notin> I`]
  16.230 +      using measurable_comp[OF measurable_component_update f_borel, OF x \<open>i \<notin> I\<close>]
  16.231        unfolding comp_def .
  16.232      from x I show "(\<integral> y. f (merge I {i} (x,y)) \<partial>Pi\<^sub>M {i} M) = (\<integral> xa. f (x(i := xa i)) \<partial>Pi\<^sub>M {i} M)"
  16.233        by (auto intro!: integral_cong arg_cong[where f=f] simp: merge_def space_PiM extensional_def PiE_def)
  16.234 @@ -2867,7 +2867,7 @@
  16.235      by (intro product_integrable_setprod insert(4)) (auto intro: finite_subset)
  16.236    interpret I: finite_product_sigma_finite M I by standard fact
  16.237    have *: "\<And>x y. (\<Prod>j\<in>I. f j (if j = i then y else x j)) = (\<Prod>j\<in>I. f j (x j))"
  16.238 -    using `i \<notin> I` by (auto intro!: setprod.cong)
  16.239 +    using \<open>i \<notin> I\<close> by (auto intro!: setprod.cong)
  16.240    show ?case
  16.241      unfolding product_integral_insert[OF insert(1,2) prod[OF subset_refl]]
  16.242      by (simp add: * insert prod subset_insertI)
    17.1 --- a/src/HOL/Probability/Borel_Space.thy	Mon Dec 07 16:48:10 2015 +0000
    17.2 +++ b/src/HOL/Probability/Borel_Space.thy	Mon Dec 07 20:19:59 2015 +0100
    17.3 @@ -3,7 +3,7 @@
    17.4      Author:     Armin Heller, TU München
    17.5  *)
    17.6  
    17.7 -section {*Borel spaces*}
    17.8 +section \<open>Borel spaces\<close>
    17.9  
   17.10  theory Borel_Space
   17.11  imports
   17.12 @@ -22,7 +22,7 @@
   17.13      by (auto intro: topological_basis_prod topological_basis_trivial topological_basis_imp_subbasis)  
   17.14  qed
   17.15  
   17.16 -subsection {* Generic Borel spaces *}
   17.17 +subsection \<open>Generic Borel spaces\<close>
   17.18  
   17.19  definition borel :: "'a::topological_space measure" where
   17.20    "borel = sigma UNIV {S. open S}"
   17.21 @@ -182,7 +182,7 @@
   17.22        by metis
   17.23      def U \<equiv> "(\<Union>k\<in>K. m k)"
   17.24      with m have "countable U"
   17.25 -      by (intro countable_subset[OF _ `countable B`]) auto
   17.26 +      by (intro countable_subset[OF _ \<open>countable B\<close>]) auto
   17.27      have "\<Union>U = (\<Union>A\<in>U. A)" by simp
   17.28      also have "\<dots> = \<Union>K"
   17.29        unfolding U_def UN_simps by (simp add: m)
   17.30 @@ -195,9 +195,9 @@
   17.31      then have "(\<Union>b\<in>U. u b) \<subseteq> \<Union>K" "\<Union>U \<subseteq> (\<Union>b\<in>U. u b)"
   17.32        by auto
   17.33      then have "\<Union>K = (\<Union>b\<in>U. u b)"
   17.34 -      unfolding `\<Union>U = \<Union>K` by auto
   17.35 +      unfolding \<open>\<Union>U = \<Union>K\<close> by auto
   17.36      also have "\<dots> \<in> sigma_sets UNIV X"
   17.37 -      using u UN by (intro X.countable_UN' `countable U`) auto
   17.38 +      using u UN by (intro X.countable_UN' \<open>countable U\<close>) auto
   17.39      finally show "\<Union>K \<in> sigma_sets UNIV X" .
   17.40    qed auto
   17.41  qed (auto simp: eq intro: generate_topology.Basis)
   17.42 @@ -257,7 +257,7 @@
   17.43    fix X::"'a set" assume "open X"
   17.44    from open_countable_basisE[OF this] guess B' . note B' = this
   17.45    then show "X \<in> sigma_sets UNIV B"
   17.46 -    by (blast intro: sigma_sets_UNION `countable B` countable_subset)
   17.47 +    by (blast intro: sigma_sets_UNION \<open>countable B\<close> countable_subset)
   17.48  next
   17.49    fix b assume "b \<in> B"
   17.50    hence "open b" by (rule topological_basis_open[OF assms(2)])
   17.51 @@ -302,7 +302,7 @@
   17.52      unfolding eq by (rule borel_measurable_continuous_on[OF H]) auto
   17.53  qed
   17.54  
   17.55 -subsection {* Borel spaces on order topologies *}
   17.56 +subsection \<open>Borel spaces on order topologies\<close>
   17.57  
   17.58  
   17.59  lemma borel_Iio:
   17.60 @@ -441,7 +441,7 @@
   17.61    finally show ?thesis .
   17.62  qed
   17.63  
   17.64 -subsection {* Borel spaces on euclidean spaces *}
   17.65 +subsection \<open>Borel spaces on euclidean spaces\<close>
   17.66  
   17.67  lemma borel_measurable_inner[measurable (raw)]:
   17.68    fixes f g :: "'a \<Rightarrow> 'b::{second_countable_topology, real_inner}"
   17.69 @@ -603,7 +603,7 @@
   17.70    fix M :: "'a set" assume "M \<in> {S. open S}"
   17.71    then have "open M" by simp
   17.72    show "M \<in> ?SIGMA"
   17.73 -    apply (subst open_UNION_box[OF `open M`])
   17.74 +    apply (subst open_UNION_box[OF \<open>open M\<close>])
   17.75      apply (safe intro!: sets.countable_UN' countable_PiE countable_Collect)
   17.76      apply (auto intro: countable_rat)
   17.77      done
   17.78 @@ -746,7 +746,7 @@
   17.79    fix a :: real and i :: 'a assume "(a, i) \<in> UNIV \<times> Basis"
   17.80    then have i: "i \<in> Basis" by auto
   17.81    have "{x::'a. a \<le> x\<bullet>i} = UNIV - {x::'a. x\<bullet>i < a}" by auto
   17.82 -  also have *: "{x::'a. x\<bullet>i < a} = (\<Union>k::nat. {x. x <e (\<Sum>n\<in>Basis. (if n = i then a else real k) *\<^sub>R n)})" using `i\<in> Basis`
   17.83 +  also have *: "{x::'a. x\<bullet>i < a} = (\<Union>k::nat. {x. x <e (\<Sum>n\<in>Basis. (if n = i then a else real k) *\<^sub>R n)})" using \<open>i\<in> Basis\<close>
   17.84    proof (safe, simp_all add: eucl_less_def split: split_if_asm)
   17.85      fix x :: 'a
   17.86      from reals_Archimedean2[of "Max ((\<lambda>i. x\<bullet>i)`Basis)"]
   17.87 @@ -817,13 +817,13 @@
   17.88    fix x :: "'a set" assume "open x"
   17.89    hence "x = UNIV - (UNIV - x)" by auto
   17.90    also have "\<dots> \<in> sigma_sets UNIV (Collect closed)"
   17.91 -    by (force intro: sigma_sets.Compl simp: `open x`)
   17.92 +    by (force intro: sigma_sets.Compl simp: \<open>open x\<close>)
   17.93    finally show "x \<in> sigma_sets UNIV (Collect closed)" by simp
   17.94  next
   17.95    fix x :: "'a set" assume "closed x"
   17.96    hence "x = UNIV - (UNIV - x)" by auto
   17.97    also have "\<dots> \<in> sigma_sets UNIV (Collect open)"
   17.98 -    by (force intro: sigma_sets.Compl simp: `closed x`)
   17.99 +    by (force intro: sigma_sets.Compl simp: \<open>closed x\<close>)
  17.100    finally show "x \<in> sigma_sets UNIV (Collect open)" by simp
  17.101  qed simp_all
  17.102  
  17.103 @@ -965,12 +965,12 @@
  17.104    show "(\<lambda>x. a + b *\<^sub>R f x) -` S \<inter> space M \<in> sets M"
  17.105    proof cases
  17.106      assume "b \<noteq> 0"
  17.107 -    with `open S` have "open ((\<lambda>x. (- a + x) /\<^sub>R b) ` S)" (is "open ?S")
  17.108 +    with \<open>open S\<close> have "open ((\<lambda>x. (- a + x) /\<^sub>R b) ` S)" (is "open ?S")
  17.109        using open_affinity [of S "inverse b" "- a /\<^sub>R b"]
  17.110        by (auto simp: algebra_simps)
  17.111      hence "?S \<in> sets borel" by auto
  17.112      moreover
  17.113 -    from `b \<noteq> 0` have "(\<lambda>x. a + b *\<^sub>R f x) -` S = f -` ?S"
  17.114 +    from \<open>b \<noteq> 0\<close> have "(\<lambda>x. a + b *\<^sub>R f x) -` S = f -` ?S"
  17.115        apply auto by (rule_tac x="a + b *\<^sub>R f x" in image_eqI, simp_all)
  17.116      ultimately show ?thesis using assms unfolding in_borel_measurable_borel
  17.117        by auto
  17.118 @@ -1315,7 +1315,7 @@
  17.119    shows "(\<lambda>x. (\<Sum>i. f i x)) \<in> borel_measurable M"
  17.120    unfolding suminf_def sums_def[abs_def] lim_def[symmetric] by simp
  17.121  
  17.122 -subsection {* LIMSEQ is borel measurable *}
  17.123 +subsection \<open>LIMSEQ is borel measurable\<close>
  17.124  
  17.125  lemma borel_measurable_LIMSEQ:
  17.126    fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real"
  17.127 @@ -1352,7 +1352,7 @@
  17.128    proof cases
  17.129      assume "A \<noteq> {}"
  17.130      then have "\<And>x. infdist x A = 0 \<longleftrightarrow> x \<in> A"
  17.131 -      using `closed A` by (simp add: in_closed_iff_infdist_zero)
  17.132 +      using \<open>closed A\<close> by (simp add: in_closed_iff_infdist_zero)
  17.133      then have "g -` A \<inter> space M = {x\<in>space M. infdist (g x) A = 0}"
  17.134        by auto
  17.135      also have "\<dots> \<in> sets M"
    18.1 --- a/src/HOL/Probability/Caratheodory.thy	Mon Dec 07 16:48:10 2015 +0000
    18.2 +++ b/src/HOL/Probability/Caratheodory.thy	Mon Dec 07 20:19:59 2015 +0100
    18.3 @@ -3,15 +3,15 @@
    18.4      Author:     Johannes Hölzl, TU München
    18.5  *)
    18.6  
    18.7 -section {*Caratheodory Extension Theorem*}
    18.8 +section \<open>Caratheodory Extension Theorem\<close>
    18.9  
   18.10  theory Caratheodory
   18.11    imports Measure_Space
   18.12  begin
   18.13  
   18.14 -text {*
   18.15 +text \<open>
   18.16    Originally from the Hurd/Coble measure theory development, translated by Lawrence Paulson.
   18.17 -*}
   18.18 +\<close>
   18.19  
   18.20  lemma suminf_ereal_2dimen:
   18.21    fixes f:: "nat \<times> nat \<Rightarrow> ereal"
   18.22 @@ -45,7 +45,7 @@
   18.23                       SUP_ereal_setsum[symmetric] incseq_setsumI setsum_nonneg)
   18.24  qed
   18.25  
   18.26 -subsection {* Characterizations of Measures *}
   18.27 +subsection \<open>Characterizations of Measures\<close>
   18.28  
   18.29  definition subadditive where
   18.30    "subadditive M f \<longleftrightarrow> (\<forall>x\<in>M. \<forall>y\<in>M. x \<inter> y = {} \<longrightarrow> f (x \<union> y) \<le> f x + f y)"
   18.31 @@ -60,7 +60,7 @@
   18.32  lemma subadditiveD: "subadditive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x \<in> M \<Longrightarrow> y \<in> M \<Longrightarrow> f (x \<union> y) \<le> f x + f y"
   18.33    by (auto simp add: subadditive_def)
   18.34  
   18.35 -subsubsection {* Lambda Systems *}
   18.36 +subsubsection \<open>Lambda Systems\<close>
   18.37  
   18.38  definition lambda_system where
   18.39    "lambda_system \<Omega> M f = {l \<in> M. \<forall>x \<in> M. f (l \<inter> x) + f ((\<Omega> - l) \<inter> x) = f x}"
   18.40 @@ -393,7 +393,7 @@
   18.41    assumes posf: "positive M f" and "0 < e" and "outer_measure M f X \<noteq> \<infinity>"
   18.42    shows "\<exists>A. range A \<subseteq> M \<and> disjoint_family A \<and> X \<subseteq> (\<Union>i. A i) \<and> (\<Sum>i. f (A i)) \<le> outer_measure M f X + e"
   18.43  proof -
   18.44 -  from `outer_measure M f X \<noteq> \<infinity>` have fin: "\<bar>outer_measure M f X\<bar> \<noteq> \<infinity>"
   18.45 +  from \<open>outer_measure M f X \<noteq> \<infinity>\<close> have fin: "\<bar>outer_measure M f X\<bar> \<noteq> \<infinity>"
   18.46      using outer_measure_nonneg[OF posf, of X] by auto
   18.47    show ?thesis
   18.48      using Inf_ereal_close[OF fin[unfolded outer_measure_def INF_def], OF \<open>0 < e\<close>]
   18.49 @@ -509,7 +509,7 @@
   18.50  lemma measure_down: "measure_space \<Omega> N \<mu> \<Longrightarrow> sigma_algebra \<Omega> M \<Longrightarrow> M \<subseteq> N \<Longrightarrow> measure_space \<Omega> M \<mu>"
   18.51    by (auto simp add: measure_space_def positive_def countably_additive_def subset_eq)
   18.52  
   18.53 -subsection {* Caratheodory's theorem *}
   18.54 +subsection \<open>Caratheodory's theorem\<close>
   18.55  
   18.56  theorem (in ring_of_sets) caratheodory':
   18.57    assumes posf: "positive M f" and ca: "countably_additive M f"
   18.58 @@ -546,7 +546,7 @@
   18.59    show "\<forall>A\<in>M. f A \<noteq> \<infinity>" using fin by auto
   18.60  qed (rule cont)
   18.61  
   18.62 -subsection {* Volumes *}
   18.63 +subsection \<open>Volumes\<close>
   18.64  
   18.65  definition volume :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where
   18.66    "volume M f \<longleftrightarrow>
   18.67 @@ -575,16 +575,16 @@
   18.68  proof -
   18.69    have "A`I \<subseteq> M" "disjoint (A`I)" "finite (A`I)" "\<Union>(A`I) \<in> M"
   18.70      using A unfolding SUP_def by (auto simp: disjoint_family_on_disjoint_image)
   18.71 -  with `volume M f` have "f (\<Union>(A`I)) = (\<Sum>a\<in>A`I. f a)"
   18.72 +  with \<open>volume M f\<close> have "f (\<Union>(A`I)) = (\<Sum>a\<in>A`I. f a)"
   18.73      unfolding volume_def by blast
   18.74    also have "\<dots> = (\<Sum>i\<in>I. f (A i))"
   18.75    proof (subst setsum.reindex_nontrivial)
   18.76      fix i j assume "i \<in> I" "j \<in> I" "i \<noteq> j" "A i = A j"
   18.77 -    with `disjoint_family_on A I` have "A i = {}"
   18.78 +    with \<open>disjoint_family_on A I\<close> have "A i = {}"
   18.79        by (auto simp: disjoint_family_on_def)
   18.80      then show "f (A i) = 0"
   18.81 -      using volume_empty[OF `volume M f`] by simp
   18.82 -  qed (auto intro: `finite I`)
   18.83 +      using volume_empty[OF \<open>volume M f\<close>] by simp
   18.84 +  qed (auto intro: \<open>finite I\<close>)
   18.85    finally show "f (UNION I A) = (\<Sum>i\<in>I. f (A i))"
   18.86      by simp
   18.87  qed
   18.88 @@ -622,15 +622,15 @@
   18.89      proof (intro setsum.cong refl)
   18.90        fix d assume "d \<in> D"
   18.91        have Un_eq_d: "(\<Union>c\<in>C. c \<inter> d) = d"
   18.92 -        using `d \<in> D` `\<Union>C = \<Union>D` by auto
   18.93 +        using \<open>d \<in> D\<close> \<open>\<Union>C = \<Union>D\<close> by auto
   18.94        moreover have "\<mu> (\<Union>c\<in>C. c \<inter> d) = (\<Sum>c\<in>C. \<mu> (c \<inter> d))"
   18.95        proof (rule volume_finite_additive)
   18.96          { fix c assume "c \<in> C" then show "c \<inter> d \<in> M"
   18.97 -            using C D `d \<in> D` by auto }
   18.98 +            using C D \<open>d \<in> D\<close> by auto }
   18.99          show "(\<Union>a\<in>C. a \<inter> d) \<in> M"
  18.100 -          unfolding Un_eq_d using `d \<in> D` D by auto
  18.101 +          unfolding Un_eq_d using \<open>d \<in> D\<close> D by auto
  18.102          show "disjoint_family_on (\<lambda>a. a \<inter> d) C"
  18.103 -          using `disjoint C` by (auto simp: disjoint_family_on_def disjoint_def)
  18.104 +          using \<open>disjoint C\<close> by (auto simp: disjoint_family_on_def disjoint_def)
  18.105        qed fact+
  18.106        ultimately show "\<mu> d = (\<Sum>c\<in>C. \<mu> (c \<inter> d))" by simp
  18.107      qed }
  18.108 @@ -659,7 +659,7 @@
  18.109        by (simp add: disjoint_def)
  18.110    next
  18.111      fix a assume "a \<in> ?R" then guess Ca .. note Ca = this
  18.112 -    with \<mu>'[of Ca] `volume M \<mu>`[THEN volume_positive]
  18.113 +    with \<mu>'[of Ca] \<open>volume M \<mu>\<close>[THEN volume_positive]
  18.114      show "0 \<le> \<mu>' a"
  18.115        by (auto intro!: setsum_nonneg)
  18.116    next
  18.117 @@ -671,10 +671,10 @@
  18.118      with Ca Cb have "Ca \<inter> Cb \<subseteq> {{}}" by auto
  18.119      then have C_Int_cases: "Ca \<inter> Cb = {{}} \<or> Ca \<inter> Cb = {}" by auto
  18.120  
  18.121 -    from `a \<inter> b = {}` have "\<mu>' (\<Union>(Ca \<union> Cb)) = (\<Sum>c\<in>Ca \<union> Cb. \<mu> c)"
  18.122 +    from \<open>a \<inter> b = {}\<close> have "\<mu>' (\<Union>(Ca \<union> Cb)) = (\<Sum>c\<in>Ca \<union> Cb. \<mu> c)"
  18.123        using Ca Cb by (intro \<mu>') (auto intro!: disjoint_union)
  18.124      also have "\<dots> = (\<Sum>c\<in>Ca \<union> Cb. \<mu> c) + (\<Sum>c\<in>Ca \<inter> Cb. \<mu> c)"
  18.125 -      using C_Int_cases volume_empty[OF `volume M \<mu>`] by (elim disjE) simp_all
  18.126 +      using C_Int_cases volume_empty[OF \<open>volume M \<mu>\<close>] by (elim disjE) simp_all
  18.127      also have "\<dots> = (\<Sum>c\<in>Ca. \<mu> c) + (\<Sum>c\<in>Cb. \<mu> c)"
  18.128        using Ca Cb by (simp add: setsum.union_inter)
  18.129      also have "\<dots> = \<mu>' a + \<mu>' b"
  18.130 @@ -684,7 +684,7 @@
  18.131    qed
  18.132  qed
  18.133  
  18.134 -subsubsection {* Caratheodory on semirings *}
  18.135 +subsubsection \<open>Caratheodory on semirings\<close>
  18.136  
  18.137  theorem (in semiring_of_sets) caratheodory:
  18.138    assumes pos: "positive M \<mu>" and ca: "countably_additive M \<mu>"
  18.139 @@ -698,14 +698,14 @@
  18.140  
  18.141      fix C assume sets_C: "C \<subseteq> M" "\<Union>C \<in> M" and "disjoint C" "finite C"
  18.142      have "\<exists>F'. bij_betw F' {..<card C} C"
  18.143 -      by (rule finite_same_card_bij[OF _ `finite C`]) auto
  18.144 +      by (rule finite_same_card_bij[OF _ \<open>finite C\<close>]) auto
  18.145      then guess F' .. note F' = this
  18.146      then have F': "C = F' ` {..< card C}" "inj_on F' {..< card C}"
  18.147        by (auto simp: bij_betw_def)
  18.148      { fix i j assume *: "i < card C" "j < card C" "i \<noteq> j"
  18.149        with F' have "F' i \<in> C" "F' j \<in> C" "F' i \<noteq> F' j"
  18.150          unfolding inj_on_def by auto
  18.151 -      with `disjoint C`[THEN disjointD]
  18.152 +      with \<open>disjoint C\<close>[THEN disjointD]
  18.153        have "F' i \<inter> F' j = {}"
  18.154          by auto }
  18.155      note F'_disj = this
  18.156 @@ -733,7 +733,7 @@
  18.157      finally show "\<mu> (\<Union>C) = (\<Sum>c\<in>C. \<mu> c)" .
  18.158    next
  18.159      show "\<mu> {} = 0"
  18.160 -      using `positive M \<mu>` by (rule positiveD1)
  18.161 +      using \<open>positive M \<mu>\<close> by (rule positiveD1)
  18.162    qed
  18.163    from extend_volume[OF this] obtain \<mu>_r where
  18.164      V: "volume generated_ring \<mu>_r" "\<And>a. a \<in> M \<Longrightarrow> \<mu> a = \<mu>_r a"
  18.165 @@ -758,7 +758,7 @@
  18.166          and Un_A: "(\<Union>i. A i) \<in> generated_ring"
  18.167          using A' C'
  18.168          by (auto intro!: G.Int G.finite_Union intro: generated_ringI_Basic simp: disjoint_family_on_def)
  18.169 -      from A C' `c \<in> C'` have UN_eq: "(\<Union>i. A i) = c"
  18.170 +      from A C' \<open>c \<in> C'\<close> have UN_eq: "(\<Union>i. A i) = c"
  18.171          by (auto simp: A_def)
  18.172  
  18.173        have "\<forall>i::nat. \<exists>f::nat \<Rightarrow> 'a set. \<mu>_r (A i) = (\<Sum>j. \<mu>_r (f j)) \<and> disjoint_family f \<and> \<Union>range f = A i \<and> (\<forall>j. f j \<in> M)"
  18.174 @@ -769,7 +769,7 @@
  18.175          from generated_ringE[OF this] guess C . note C = this
  18.176  
  18.177          have "\<exists>F'. bij_betw F' {..<card C} C"
  18.178 -          by (rule finite_same_card_bij[OF _ `finite C`]) auto
  18.179 +          by (rule finite_same_card_bij[OF _ \<open>finite C\<close>]) auto
  18.180          then guess F .. note F = this
  18.181          def f \<equiv> "\<lambda>i. if i < card C then F i else {}"
  18.182          then have f: "bij_betw f {..< card C} C"
  18.183 @@ -831,7 +831,7 @@
  18.184        also have "\<dots> = (\<Sum>n. \<mu> (case_prod f (prod_decode n)))"
  18.185          using f V(2) by (auto intro!: arg_cong[where f=suminf] split: prod.split)
  18.186        also have "\<dots> = \<mu> (\<Union>i. case_prod f (prod_decode i))"
  18.187 -        using f `c \<in> C'` C'
  18.188 +        using f \<open>c \<in> C'\<close> C'
  18.189          by (intro ca[unfolded countably_additive_def, rule_format])
  18.190             (auto split: prod.split simp: UN_f_eq d UN_eq)
  18.191        finally have "(\<Sum>n. \<mu>_r (A' n \<inter> c)) = \<mu> c"
  18.192 @@ -858,7 +858,7 @@
  18.193      finally show "(\<Sum>n. \<mu>_r (A' n)) = \<mu>_r (\<Union>i. A' i)"
  18.194        using C' by simp
  18.195    qed
  18.196 -  from G.caratheodory'[OF `positive generated_ring \<mu>_r` `countably_additive generated_ring \<mu>_r`]
  18.197 +  from G.caratheodory'[OF \<open>positive generated_ring \<mu>_r\<close> \<open>countably_additive generated_ring \<mu>_r\<close>]
  18.198    guess \<mu>' ..
  18.199    with V show ?thesis
  18.200      unfolding sigma_sets_generated_ring_eq
    19.1 --- a/src/HOL/Probability/Complete_Measure.thy	Mon Dec 07 16:48:10 2015 +0000
    19.2 +++ b/src/HOL/Probability/Complete_Measure.thy	Mon Dec 07 20:19:59 2015 +0100
    19.3 @@ -256,7 +256,7 @@
    19.4        have "?F (f x) - ?N = main_part M (?F (f x)) \<union> null_part M (?F (f x)) - ?N"
    19.5          using main_part_null_part_Un[OF F] by auto
    19.6        also have "\<dots> = main_part M (?F (f x)) - ?N"
    19.7 -        using N `x \<in> space M` by auto
    19.8 +        using N \<open>x \<in> space M\<close> by auto
    19.9        finally have "?F (f x) - ?N \<in> sets M"
   19.10          using F sets by auto }
   19.11      ultimately show "?f' -` {?f' x} \<inter> space M \<in> sets M" by auto
   19.12 @@ -284,7 +284,7 @@
   19.13      proof (elim AE_mp, safe intro!: AE_I2)
   19.14        fix x assume eq: "\<forall>i. f i x = f' i x"
   19.15        moreover have "g x = (SUP i. f i x)"
   19.16 -        unfolding f using `0 \<le> g x` by (auto split: split_max)
   19.17 +        unfolding f using \<open>0 \<le> g x\<close> by (auto split: split_max)
   19.18        ultimately show "g x = ?f x" by auto
   19.19      qed
   19.20      show "?f \<in> borel_measurable M"
    20.1 --- a/src/HOL/Probability/Convolution.thy	Mon Dec 07 16:48:10 2015 +0000
    20.2 +++ b/src/HOL/Probability/Convolution.thy	Mon Dec 07 20:19:59 2015 +0100
    20.3 @@ -2,7 +2,7 @@
    20.4      Author:     Sudeep Kanav, TU München
    20.5      Author:     Johannes Hölzl, TU München *)
    20.6  
    20.7 -section {* Convolution Measure *}
    20.8 +section \<open>Convolution Measure\<close>
    20.9  
   20.10  theory Convolution
   20.11    imports Independent_Family
   20.12 @@ -160,7 +160,7 @@
   20.13        by (subst nn_integral_real_affine[where c=1 and t="-y"])
   20.14           (auto simp del: gt_0 simp add: one_ereal_def[symmetric])
   20.15      also have "\<dots> = (\<integral>\<^sup>+x. g y * (f (x - y) * indicator A x) \<partial>lborel)"
   20.16 -      using `0 \<le> g y` by (intro nn_integral_cmult[symmetric]) auto
   20.17 +      using \<open>0 \<le> g y\<close> by (intro nn_integral_cmult[symmetric]) auto
   20.18      finally show "(\<integral>\<^sup>+ x. g y * (f x * indicator A (x + y)) \<partial>lborel) =
   20.19        (\<integral>\<^sup>+ x. f (x - y) * g y * indicator A x \<partial>lborel)"
   20.20        by (simp add: ac_simps)
    21.1 --- a/src/HOL/Probability/Discrete_Topology.thy	Mon Dec 07 16:48:10 2015 +0000
    21.2 +++ b/src/HOL/Probability/Discrete_Topology.thy	Mon Dec 07 20:19:59 2015 +0100
    21.3 @@ -6,7 +6,7 @@
    21.4  imports "~~/src/HOL/Multivariate_Analysis/Multivariate_Analysis"
    21.5  begin
    21.6  
    21.7 -text {* Copy of discrete types with discrete topology. This space is polish. *}
    21.8 +text \<open>Copy of discrete types with discrete topology. This space is polish.\<close>
    21.9  
   21.10  typedef 'a discrete = "UNIV::'a set"
   21.11  morphisms of_discrete discrete
    22.1 --- a/src/HOL/Probability/Distributions.thy	Mon Dec 07 16:48:10 2015 +0000
    22.2 +++ b/src/HOL/Probability/Distributions.thy	Mon Dec 07 20:19:59 2015 +0100
    22.3 @@ -3,7 +3,7 @@
    22.4      Author:     Johannes Hölzl, TU München
    22.5      Author:     Jeremy Avigad, CMU *)
    22.6  
    22.7 -section {* Properties of Various Distributions *}
    22.8 +section \<open>Properties of Various Distributions\<close>
    22.9  
   22.10  theory Distributions
   22.11    imports Convolution Information
   22.12 @@ -69,7 +69,7 @@
   22.13    finally show ?thesis .
   22.14  qed
   22.15  
   22.16 -subsection {* Erlang *}
   22.17 +subsection \<open>Erlang\<close>
   22.18  
   22.19  lemma nn_intergal_power_times_exp_Icc:
   22.20    assumes [arith]: "0 \<le> a"
   22.21 @@ -327,7 +327,7 @@
   22.22      by simp (auto simp: power2_eq_square field_simps of_nat_Suc)
   22.23  qed
   22.24  
   22.25 -subsection {* Exponential distribution *}
   22.26 +subsection \<open>Exponential distribution\<close>
   22.27  
   22.28  abbreviation exponential_density :: "real \<Rightarrow> real \<Rightarrow> real" where
   22.29    "exponential_density \<equiv> erlang_density 0"
   22.30 @@ -353,7 +353,7 @@
   22.31        using assms by (auto simp: distributed_real_AE)
   22.32      then have "AE x in lborel. x \<le> (0::real)"
   22.33        apply eventually_elim
   22.34 -      using `l < 0`
   22.35 +      using \<open>l < 0\<close>
   22.36        apply (auto simp: exponential_density_def zero_le_mult_iff split: split_if_asm)
   22.37        done
   22.38      then show "emeasure lborel {x :: real \<in> space lborel. 0 < x} = 0"
   22.39 @@ -391,7 +391,7 @@
   22.40    shows "\<P>(x in M. a + t < X x \<bar> a < X x) = \<P>(x in M. t < X x)"
   22.41  proof -
   22.42    have "\<P>(x in M. a + t < X x \<bar> a < X x) = \<P>(x in M. a + t < X x) / \<P>(x in M. a < X x)"
   22.43 -    using `0 \<le> t` by (auto simp: cond_prob_def intro!: arg_cong[where f=prob] arg_cong2[where f="op /"])
   22.44 +    using \<open>0 \<le> t\<close> by (auto simp: cond_prob_def intro!: arg_cong[where f=prob] arg_cong2[where f="op /"])
   22.45    also have "\<dots> = exp (- (a + t) * l) / exp (- a * l)"
   22.46      using a t by (simp add: exponential_distributedD_gt[OF D])
   22.47    also have "\<dots> = exp (- t * l)"
   22.48 @@ -563,7 +563,7 @@
   22.49    assumes erlY: "distributed M lborel Y (erlang_density k\<^sub>2 l)"
   22.50    shows "distributed M lborel (\<lambda>x. X x + Y x) (erlang_density (Suc k\<^sub>1 + Suc k\<^sub>2 - 1) l)"
   22.51    using assms
   22.52 -  apply (subst convolution_erlang_density[symmetric, OF `0<l`])
   22.53 +  apply (subst convolution_erlang_density[symmetric, OF \<open>0<l\<close>])
   22.54    apply (intro distributed_convolution)
   22.55    apply auto
   22.56    done
   22.57 @@ -630,7 +630,7 @@
   22.58      by (simp add: log_def divide_simps ln_div)
   22.59  qed
   22.60  
   22.61 -subsection {* Uniform distribution *}
   22.62 +subsection \<open>Uniform distribution\<close>
   22.63  
   22.64  lemma uniform_distrI:
   22.65    assumes X: "X \<in> measurable M M'"
   22.66 @@ -679,7 +679,7 @@
   22.67      (\<integral>\<^sup>+ x. ereal (1 / measure lborel A) * indicator (A \<inter> {..a}) x \<partial>lborel)"
   22.68      by (auto intro!: nn_integral_cong split: split_indicator)
   22.69    also have "\<dots> = ereal (1 / measure lborel A) * emeasure lborel (A \<inter> {..a})"
   22.70 -    using `A \<in> sets borel`
   22.71 +    using \<open>A \<in> sets borel\<close>
   22.72      by (intro nn_integral_cmult_indicator) (auto simp: measure_nonneg)
   22.73    also have "\<dots> = ereal (measure lborel (A \<inter> {..a}) / r)"
   22.74      unfolding emeasure_eq_ereal_measure[OF fin] using A by (simp add: measure_def)
   22.75 @@ -702,27 +702,27 @@
   22.76      then have "emeasure M {x\<in>space M. X x \<le> t} \<le> emeasure M {x\<in>space M. X x \<le> a}"
   22.77        using X by (auto intro!: emeasure_mono measurable_sets)
   22.78      also have "\<dots> = 0"
   22.79 -      using distr[of a] `a < b` by (simp add: emeasure_eq_measure)
   22.80 +      using distr[of a] \<open>a < b\<close> by (simp add: emeasure_eq_measure)
   22.81      finally have "emeasure M {x\<in>space M. X x \<le> t} = 0"
   22.82        by (simp add: antisym measure_nonneg emeasure_le_0_iff)
   22.83 -    with `t < a` show ?thesis by simp
   22.84 +    with \<open>t < a\<close> show ?thesis by simp
   22.85    next
   22.86      assume bnds: "a \<le> t" "t \<le> b"
   22.87      have "{a..b} \<inter> {..t} = {a..t}"
   22.88        using bnds by auto
   22.89 -    then show ?thesis using `a \<le> t` `a < b`
   22.90 +    then show ?thesis using \<open>a \<le> t\<close> \<open>a < b\<close>
   22.91        using distr[OF bnds] by (simp add: emeasure_eq_measure)
   22.92    next
   22.93      assume "b < t"
   22.94      have "1 = emeasure M {x\<in>space M. X x \<le> b}"
   22.95 -      using distr[of b] `a < b` by (simp add: one_ereal_def emeasure_eq_measure)
   22.96 +      using distr[of b] \<open>a < b\<close> by (simp add: one_ereal_def emeasure_eq_measure)
   22.97      also have "\<dots> \<le> emeasure M {x\<in>space M. X x \<le> t}"
   22.98 -      using X `b < t` by (auto intro!: emeasure_mono measurable_sets)
   22.99 +      using X \<open>b < t\<close> by (auto intro!: emeasure_mono measurable_sets)
  22.100      finally have "emeasure M {x\<in>space M. X x \<le> t} = 1"
  22.101         by (simp add: antisym emeasure_eq_measure one_ereal_def)
  22.102 -    with `b < t` `a < b` show ?thesis by (simp add: measure_def one_ereal_def)
  22.103 +    with \<open>b < t\<close> \<open>a < b\<close> show ?thesis by (simp add: measure_def one_ereal_def)
  22.104    qed
  22.105 -qed (insert X `a < b`, auto)
  22.106 +qed (insert X \<open>a < b\<close>, auto)
  22.107  
  22.108  lemma (in prob_space) uniform_distributed_measure:
  22.109    fixes a b :: real
  22.110 @@ -734,12 +734,12 @@
  22.111      using distributed_measurable[OF D]
  22.112      by (subst emeasure_distr) (auto intro!: arg_cong2[where f=emeasure])
  22.113    also have "\<dots> = (\<integral>\<^sup>+x. ereal (1 / (b - a)) * indicator {a .. t} x \<partial>lborel)"
  22.114 -    using distributed_borel_measurable[OF D] `a \<le> t` `t \<le> b`
  22.115 +    using distributed_borel_measurable[OF D] \<open>a \<le> t\<close> \<open>t \<le> b\<close>
  22.116      unfolding distributed_distr_eq_density[OF D]
  22.117      by (subst emeasure_density)
  22.118         (auto intro!: nn_integral_cong simp: measure_def split: split_indicator)
  22.119    also have "\<dots> = ereal (1 / (b - a)) * (t - a)"
  22.120 -    using `a \<le> t` `t \<le> b`
  22.121 +    using \<open>a \<le> t\<close> \<open>t \<le> b\<close>
  22.122      by (subst nn_integral_cmult_indicator) auto
  22.123    finally show ?thesis
  22.124      by (simp add: measure_def)
  22.125 @@ -788,12 +788,12 @@
  22.126        by (auto intro!: isCont_divide)
  22.127      have *: "b\<^sup>2 / (2 * measure lborel {a..b}) - a\<^sup>2 / (2 * measure lborel {a..b}) =
  22.128        (b*b - a * a) / (2 * (b - a))"
  22.129 -      using `a < b`
  22.130 +      using \<open>a < b\<close>
  22.131        by (auto simp: measure_def power2_eq_square diff_divide_distrib[symmetric])
  22.132      show "b\<^sup>2 / (2 * measure lborel {a..b}) - a\<^sup>2 / (2 * measure lborel {a..b}) = (a + b) / 2"
  22.133 -      using `a < b`
  22.134 +      using \<open>a < b\<close>
  22.135        unfolding * square_diff_square_factored by (auto simp: field_simps)
  22.136 -  qed (insert `a < b`, simp)
  22.137 +  qed (insert \<open>a < b\<close>, simp)
  22.138    finally show "(\<integral> x. indicator {a .. b} x / measure lborel {a .. b} * x \<partial>lborel) = (a + b) / 2" .
  22.139  qed auto
  22.140  
  22.141 @@ -812,7 +812,7 @@
  22.142    finally show "(\<integral>x. x\<^sup>2 * ?D x \<partial>lborel) = (b - a)\<^sup>2 / 12" .
  22.143  qed fact
  22.144  
  22.145 -subsection {* Normal distribution *}
  22.146 +subsection \<open>Normal distribution\<close>
  22.147  
  22.148  
  22.149  definition normal_density :: "real \<Rightarrow> real \<Rightarrow> real \<Rightarrow> real" where
  22.150 @@ -936,7 +936,7 @@
  22.151      let ?f = "\<lambda>b. \<integral>x. indicator {0..} x *\<^sub>R ?M (k + 2) x * indicator {..b} x \<partial>lborel"
  22.152      have "((\<lambda>b. (k + 1) / 2 * (\<integral>x. indicator {..b} x *\<^sub>R (indicator {0..} x *\<^sub>R ?M k x) \<partial>lborel) - ?M (k + 1) b / 2) --->
  22.153          (k + 1) / 2 * (\<integral>x. indicator {0..} x *\<^sub>R ?M k x \<partial>lborel) - 0 / 2) at_top" (is ?tendsto)
  22.154 -    proof (intro tendsto_intros `2 \<noteq> 0` tendsto_integral_at_top sets_lborel Mk[THEN integrable.intros])
  22.155 +    proof (intro tendsto_intros \<open>2 \<noteq> 0\<close> tendsto_integral_at_top sets_lborel Mk[THEN integrable.intros])
  22.156        show "(?M (k + 1) ---> 0) at_top"
  22.157        proof cases
  22.158          assume "even k"
  22.159 @@ -945,7 +945,7 @@
  22.160                     filterlim_at_top_imp_at_infinity filterlim_ident filterlim_pow_at_top filterlim_ident)
  22.161               auto
  22.162          also have "(\<lambda>x. ((x\<^sup>2)^(k div 2 + 1) / exp (x\<^sup>2)) * (1 / x) :: real) = ?M (k + 1)"
  22.163 -          using `even k` by (auto simp: fun_eq_iff exp_minus field_simps power2_eq_square power_mult elim: evenE)
  22.164 +          using \<open>even k\<close> by (auto simp: fun_eq_iff exp_minus field_simps power2_eq_square power_mult elim: evenE)
  22.165          finally show ?thesis by simp
  22.166        next
  22.167          assume "odd k"
  22.168 @@ -954,7 +954,7 @@
  22.169                      filterlim_ident filterlim_pow_at_top)
  22.170               auto
  22.171          also have "(\<lambda>x. ((x\<^sup>2)^((k - 1) div 2 + 1) / exp (x\<^sup>2)) :: real) = ?M (k + 1)"
  22.172 -          using `odd k` by (auto simp: fun_eq_iff exp_minus field_simps power2_eq_square power_mult elim: oddE)
  22.173 +          using \<open>odd k\<close> by (auto simp: fun_eq_iff exp_minus field_simps power2_eq_square power_mult elim: oddE)
  22.174          finally show ?thesis by simp
  22.175        qed
  22.176      qed
  22.177 @@ -1203,7 +1203,7 @@
  22.178      by (simp add: normal_density_def real_sqrt_mult field_simps)
  22.179         (simp add: power2_eq_square field_simps)
  22.180    show ?thesis
  22.181 -    by (rule distributed_affineI[OF _ `\<alpha> \<noteq> 0`, where t=\<beta>]) (simp_all add: eq X)
  22.182 +    by (rule distributed_affineI[OF _ \<open>\<alpha> \<noteq> 0\<close>, where t=\<beta>]) (simp_all add: eq X)
  22.183  qed
  22.184  
  22.185  lemma (in prob_space) normal_standard_normal_convert:
    23.1 --- a/src/HOL/Probability/Embed_Measure.thy	Mon Dec 07 16:48:10 2015 +0000
    23.2 +++ b/src/HOL/Probability/Embed_Measure.thy	Mon Dec 07 20:19:59 2015 +0100
    23.3 @@ -6,7 +6,7 @@
    23.4      measure on the left part of the sum type 'a + 'b)
    23.5  *)
    23.6  
    23.7 -section {* Embed Measure Spaces with a Function *}
    23.8 +section \<open>Embed Measure Spaces with a Function\<close>
    23.9  
   23.10  theory Embed_Measure
   23.11  imports Binary_Product_Measure
   23.12 @@ -216,7 +216,7 @@
   23.13    moreover {
   23.14      fix X assume "X \<in> sets A"
   23.15      from asm have "emeasure ?M (f`X) = emeasure ?N (f`X)" by simp
   23.16 -    with `X \<in> sets A` and `sets A = sets B` and assms 
   23.17 +    with \<open>X \<in> sets A\<close> and \<open>sets A = sets B\<close> and assms 
   23.18          have "emeasure A X = emeasure B X" by (simp add: emeasure_embed_measure_image)
   23.19    }
   23.20    ultimately show "A = B" by (rule measure_eqI)
   23.21 @@ -312,7 +312,7 @@
   23.22    with A have "f x \<in> f ` B" by blast
   23.23    then obtain y where "f x = f y" and "y \<in> B" by blast
   23.24    with assms and B have "x = y" by (auto dest: inj_onD)
   23.25 -  with `y \<in> B` show "x \<in> B" by simp
   23.26 +  with \<open>y \<in> B\<close> show "x \<in> B" by simp
   23.27  qed auto
   23.28    
   23.29  
    24.1 --- a/src/HOL/Probability/Fin_Map.thy	Mon Dec 07 16:48:10 2015 +0000
    24.2 +++ b/src/HOL/Probability/Fin_Map.thy	Mon Dec 07 20:19:59 2015 +0100
    24.3 @@ -2,21 +2,21 @@
    24.4      Author:     Fabian Immler, TU München
    24.5  *)
    24.6  
    24.7 -section {* Finite Maps *}
    24.8 +section \<open>Finite Maps\<close>
    24.9  
   24.10  theory Fin_Map
   24.11  imports Finite_Product_Measure
   24.12  begin
   24.13  
   24.14 -text {* Auxiliary type that is instantiated to @{class polish_space}, needed for the proof of
   24.15 +text \<open>Auxiliary type that is instantiated to @{class polish_space}, needed for the proof of
   24.16    projective limit. @{const extensional} functions are used for the representation in order to
   24.17    stay close to the developments of (finite) products @{const Pi\<^sub>E} and their sigma-algebra
   24.18 -  @{const Pi\<^sub>M}. *}
   24.19 +  @{const Pi\<^sub>M}.\<close>
   24.20  
   24.21  typedef ('i, 'a) finmap ("(_ \<Rightarrow>\<^sub>F /_)" [22, 21] 21) =
   24.22    "{(I::'i set, f::'i \<Rightarrow> 'a). finite I \<and> f \<in> extensional I}" by auto
   24.23  
   24.24 -subsection {* Domain and Application *}
   24.25 +subsection \<open>Domain and Application\<close>
   24.26  
   24.27  definition domain where "domain P = fst (Rep_finmap P)"
   24.28  
   24.29 @@ -38,7 +38,7 @@
   24.30       (auto simp add: Abs_finmap_inject extensional_def domain_def proj_def Abs_finmap_inverse
   24.31                intro: extensionalityI)
   24.32  
   24.33 -subsection {* Countable Finite Maps *}
   24.34 +subsection \<open>Countable Finite Maps\<close>
   24.35  
   24.36  instance finmap :: (countable, countable) countable
   24.37  proof
   24.38 @@ -50,15 +50,15 @@
   24.39      then have "map fst (?F f1) = map fst (?F f2)" by simp
   24.40      then have "mapper f1 = mapper f2" by (simp add: comp_def)
   24.41      then have "domain f1 = domain f2" by (simp add: mapper[symmetric])
   24.42 -    with `?F f1 = ?F f2` show "f1 = f2"
   24.43 -      unfolding `mapper f1 = mapper f2` map_eq_conv mapper
   24.44 +    with \<open>?F f1 = ?F f2\<close> show "f1 = f2"
   24.45 +      unfolding \<open>mapper f1 = mapper f2\<close> map_eq_conv mapper
   24.46        by (simp add: finmap_eq_iff)
   24.47    qed
   24.48    then show "\<exists>to_nat :: 'a \<Rightarrow>\<^sub>F 'b \<Rightarrow> nat. inj to_nat"
   24.49      by (intro exI[of _ "to_nat \<circ> ?F"] inj_comp) auto
   24.50  qed
   24.51  
   24.52 -subsection {* Constructor of Finite Maps *}
   24.53 +subsection \<open>Constructor of Finite Maps\<close>
   24.54  
   24.55  definition "finmap_of inds f = Abs_finmap (inds, restrict f inds)"
   24.56  
   24.57 @@ -93,9 +93,9 @@
   24.58    show "x = y" using assms by (simp add: extensional_restrict)
   24.59  qed
   24.60  
   24.61 -subsection {* Product set of Finite Maps *}
   24.62 +subsection \<open>Product set of Finite Maps\<close>
   24.63  
   24.64 -text {* This is @{term Pi} for Finite Maps, most of this is copied *}
   24.65 +text \<open>This is @{term Pi} for Finite Maps, most of this is copied\<close>
   24.66  
   24.67  definition Pi' :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a set) \<Rightarrow> ('i \<Rightarrow>\<^sub>F 'a) set" where
   24.68    "Pi' I A = { P. domain P = I \<and> (\<forall>i. i \<in> I \<longrightarrow> (P)\<^sub>F i \<in> A i) } "
   24.69 @@ -107,7 +107,7 @@
   24.70  translations
   24.71    "PI' x:A. B" == "CONST Pi' A (%x. B)"
   24.72  
   24.73 -subsubsection{*Basic Properties of @{term Pi'}*}
   24.74 +subsubsection\<open>Basic Properties of @{term Pi'}\<close>
   24.75  
   24.76  lemma Pi'_I[intro!]: "domain f = A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x) \<Longrightarrow> f \<in> Pi' A B"
   24.77    by (simp add: Pi'_def)
   24.78 @@ -146,7 +146,7 @@
   24.79    apply auto
   24.80    done
   24.81  
   24.82 -subsection {* Topological Space of Finite Maps *}
   24.83 +subsection \<open>Topological Space of Finite Maps\<close>
   24.84  
   24.85  instantiation finmap :: (type, topological_space) topological_space
   24.86  begin
   24.87 @@ -171,7 +171,7 @@
   24.88      fix i::"'a set"
   24.89      assume "finite i"
   24.90      hence "{m. domain m = i} = Pi' i (\<lambda>_. UNIV)" by (auto simp: Pi'_def)
   24.91 -    also have "open \<dots>" by (auto intro: open_Pi'I simp: `finite i`)
   24.92 +    also have "open \<dots>" by (auto intro: open_Pi'I simp: \<open>finite i\<close>)
   24.93      finally show "open {m. domain m = i}" .
   24.94    next
   24.95      fix i::"'a set"
   24.96 @@ -196,7 +196,7 @@
   24.97    moreover assume "\<forall>S. open S \<longrightarrow> a \<in> S \<longrightarrow> eventually (\<lambda>x. x \<in> S) F" "a i \<in> S"
   24.98    ultimately have "eventually (\<lambda>x. x \<in> ?S) F" by auto
   24.99    thus "eventually (\<lambda>x. (x)\<^sub>F i \<in> S) F"
  24.100 -    by eventually_elim (insert `a i \<in> S`, force simp: Pi'_iff split: split_if_asm)
  24.101 +    by eventually_elim (insert \<open>a i \<in> S\<close>, force simp: Pi'_iff split: split_if_asm)
  24.102  qed
  24.103  
  24.104  lemma continuous_proj:
  24.105 @@ -236,7 +236,7 @@
  24.106        case (UN B)
  24.107        then obtain b where "x \<in> b" "b \<in> B" by auto
  24.108        hence "\<exists>a\<in>?A. a \<subseteq> b" using UN by simp
  24.109 -      thus ?case using `b \<in> B` by blast
  24.110 +      thus ?case using \<open>b \<in> B\<close> by blast
  24.111      next
  24.112        case (Basis s)
  24.113        then obtain a b where xs: "x\<in> Pi' a b" "s = Pi' a b" "\<And>i. i\<in>a \<Longrightarrow> open (b i)" by auto
  24.114 @@ -254,7 +254,7 @@
  24.115    qed (insert A,auto simp: PiE_iff intro!: open_Pi'I)
  24.116  qed
  24.117  
  24.118 -subsection {* Metric Space of Finite Maps *}
  24.119 +subsection \<open>Metric Space of Finite Maps\<close>
  24.120  
  24.121  instantiation finmap :: (type, metric_space) metric_space
  24.122  begin
  24.123 @@ -342,25 +342,25 @@
  24.124          fix x assume "x \<in> s"
  24.125          hence [simp]: "finite a" and a_dom: "a = domain x" using s by (auto simp: Pi'_iff)
  24.126          obtain es where es: "\<forall>i \<in> a. es i > 0 \<and> (\<forall>y. dist y (proj x i) < es i \<longrightarrow> y \<in> b i)"
  24.127 -          using b `x \<in> s` by atomize_elim (intro bchoice, auto simp: open_dist s)
  24.128 +          using b \<open>x \<in> s\<close> by atomize_elim (intro bchoice, auto simp: open_dist s)
  24.129          hence in_b: "\<And>i y. i \<in> a \<Longrightarrow> dist y (proj x i) < es i \<Longrightarrow> y \<in> b i" by auto
  24.130          show "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> s"
  24.131          proof (cases, rule, safe)
  24.132            assume "a \<noteq> {}"
  24.133 -          show "0 < min 1 (Min (es ` a))" using es by (auto simp: `a \<noteq> {}`)
  24.134 +          show "0 < min 1 (Min (es ` a))" using es by (auto simp: \<open>a \<noteq> {}\<close>)
  24.135            fix y assume d: "dist y x < min 1 (Min (es ` a))"
  24.136            show "y \<in> s" unfolding s
  24.137            proof
  24.138 -            show "domain y = a" using d s `a \<noteq> {}` by (auto simp: dist_le_1_imp_domain_eq a_dom)
  24.139 +            show "domain y = a" using d s \<open>a \<noteq> {}\<close> by (auto simp: dist_le_1_imp_domain_eq a_dom)
  24.140              fix i assume i: "i \<in> a"
  24.141              hence "dist ((y)\<^sub>F i) ((x)\<^sub>F i) < es i" using d
  24.142 -              by (auto simp: dist_finmap_def `a \<noteq> {}` intro!: le_less_trans[OF dist_proj])
  24.143 +              by (auto simp: dist_finmap_def \<open>a \<noteq> {}\<close> intro!: le_less_trans[OF dist_proj])
  24.144              with i show "y i \<in> b i" by (rule in_b)
  24.145            qed
  24.146          next
  24.147            assume "\<not>a \<noteq> {}"
  24.148            thus "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> s"
  24.149 -            using s `x \<in> s` by (auto simp: Pi'_def dist_le_1_imp_domain_eq intro!: exI[where x=1])
  24.150 +            using s \<open>x \<in> s\<close> by (auto simp: Pi'_def dist_le_1_imp_domain_eq intro!: exI[where x=1])
  24.151          qed
  24.152        qed
  24.153      qed
  24.154 @@ -380,7 +380,7 @@
  24.155        assume "y \<in> S"
  24.156        moreover
  24.157        assume "x \<in> (\<Pi>' i\<in>domain y. ball (y i) (e y))"
  24.158 -      hence "dist x y < e y" using e_pos `y \<in> S`
  24.159 +      hence "dist x y < e y" using e_pos \<open>y \<in> S\<close>
  24.160          by (auto simp: dist_finmap_def Pi'_iff finite_proj_diag dist_commute)
  24.161        ultimately show "x \<in> S" by (rule e_in)
  24.162      qed
  24.163 @@ -415,7 +415,7 @@
  24.164  
  24.165  end
  24.166  
  24.167 -subsection {* Complete Space of Finite Maps *}
  24.168 +subsection \<open>Complete Space of Finite Maps\<close>
  24.169  
  24.170  lemma tendsto_finmap:
  24.171    fixes f::"nat \<Rightarrow> ('i \<Rightarrow>\<^sub>F ('a::metric_space))"
  24.172 @@ -430,13 +430,13 @@
  24.173      using finite_domain[of g] proj_g
  24.174    proof induct
  24.175      case (insert i G)
  24.176 -    with `0 < e` have "eventually (\<lambda>x. ?dists x i < e) sequentially" by (auto simp add: tendsto_iff)
  24.177 +    with \<open>0 < e\<close> have "eventually (\<lambda>x. ?dists x i < e) sequentially" by (auto simp add: tendsto_iff)
  24.178      moreover
  24.179      from insert have "eventually (\<lambda>x. \<forall>i\<in>G. dist ((f x)\<^sub>F i) ((g)\<^sub>F i) < e) sequentially" by simp
  24.180      ultimately show ?case by eventually_elim auto
  24.181    qed simp
  24.182    thus "eventually (\<lambda>x. dist (f x) g < e) sequentially"
  24.183 -    by eventually_elim (auto simp add: dist_finmap_def finite_proj_diag ind_f `0 < e`)
  24.184 +    by eventually_elim (auto simp add: dist_finmap_def finite_proj_diag ind_f \<open>0 < e\<close>)
  24.185  qed
  24.186  
  24.187  instance finmap :: (type, complete_space) complete_space
  24.188 @@ -457,7 +457,7 @@
  24.189      have "Cauchy (p i)" unfolding cauchy p_def
  24.190      proof safe
  24.191        fix e::real assume "0 < e"
  24.192 -      with `Cauchy P` obtain N where N: "\<And>n. n\<ge>N \<Longrightarrow> dist (P n) (P N) < min e 1"
  24.193 +      with \<open>Cauchy P\<close> obtain N where N: "\<And>n. n\<ge>N \<Longrightarrow> dist (P n) (P N) < min e 1"
  24.194          by (force simp: cauchy min_def)
  24.195        hence "\<And>n. n \<ge> N \<Longrightarrow> domain (P n) = domain (P N)" using dist_le_1_imp_domain_eq by auto
  24.196        with dim have dim: "\<And>n. n \<ge> N \<Longrightarrow> domain (P n) = d" by (metis nat_le_linear)
  24.197 @@ -465,9 +465,9 @@
  24.198        proof (safe intro!: exI[where x="N"])
  24.199          fix n assume "N \<le> n" have "N \<le> N" by simp
  24.200          have "dist ((P n) i) ((P N) i) \<le> dist (P n) (P N)"
  24.201 -          using dim[OF `N \<le> n`]  dim[OF `N \<le> N`] `i \<in> d`
  24.202 +          using dim[OF \<open>N \<le> n\<close>]  dim[OF \<open>N \<le> N\<close>] \<open>i \<in> d\<close>
  24.203            by (auto intro!: dist_proj)
  24.204 -        also have "\<dots> < e" using N[OF `N \<le> n`] by simp
  24.205 +        also have "\<dots> < e" using N[OF \<open>N \<le> n\<close>] by simp
  24.206          finally show "dist ((P n) i) ((P N) i) < e" .
  24.207        qed
  24.208      qed
  24.209 @@ -480,7 +480,7 @@
  24.210      have "\<exists>ni. \<forall>i\<in>d. \<forall>n\<ge>ni i. dist (p i n) (q i) < e"
  24.211      proof (safe intro!: bchoice)
  24.212        fix i assume "i \<in> d"
  24.213 -      from p[OF `i \<in> d`, THEN metric_LIMSEQ_D, OF `0 < e`]
  24.214 +      from p[OF \<open>i \<in> d\<close>, THEN metric_LIMSEQ_D, OF \<open>0 < e\<close>]
  24.215        show "\<exists>no. \<forall>n\<ge>no. dist (p i n) (q i) < e" .
  24.216      qed then guess ni .. note ni = this
  24.217      def N \<equiv> "max Nd (Max (ni ` d))"
  24.218 @@ -490,12 +490,12 @@
  24.219        hence dom: "domain (P n) = d" "domain Q = d" "domain (P n) = domain Q"
  24.220          using dim by (simp_all add: N_def Q_def dim_def Abs_finmap_inverse)
  24.221        show "dist (P n) Q < e"
  24.222 -      proof (rule dist_finmap_lessI[OF dom(3) `0 < e`])
  24.223 +      proof (rule dist_finmap_lessI[OF dom(3) \<open>0 < e\<close>])
  24.224          fix i
  24.225          assume "i \<in> domain (P n)"
  24.226          hence "ni i \<le> Max (ni ` d)" using dom by simp
  24.227          also have "\<dots> \<le> N" by (simp add: N_def)
  24.228 -        finally show "dist ((P n)\<^sub>F i) ((Q)\<^sub>F i) < e" using ni `i \<in> domain (P n)` `N \<le> n` dom
  24.229 +        finally show "dist ((P n)\<^sub>F i) ((Q)\<^sub>F i) < e" using ni \<open>i \<in> domain (P n)\<close> \<open>N \<le> n\<close> dom
  24.230            by (auto simp: p_def q N_def less_imp_le)
  24.231        qed
  24.232      qed
  24.233 @@ -503,7 +503,7 @@
  24.234    thus "convergent P" by (auto simp: convergent_def)
  24.235  qed
  24.236  
  24.237 -subsection {* Second Countable Space of Finite Maps *}
  24.238 +subsection \<open>Second Countable Space of Finite Maps\<close>
  24.239  
  24.240  instantiation finmap :: (countable, second_countable_topology) second_countable_topology
  24.241  begin
  24.242 @@ -582,7 +582,7 @@
  24.243    then guess B .. note B = this
  24.244    def B' \<equiv> "Pi' (domain x) (\<lambda>i. (B i)::'b set)"
  24.245    have "B' \<subseteq> Pi' (domain x) a" using B by (auto intro!: Pi'_mono simp: B'_def)
  24.246 -  also note `\<dots> \<subseteq> O'`
  24.247 +  also note \<open>\<dots> \<subseteq> O'\<close>
  24.248    finally show "\<exists>B'\<in>basis_finmap. x \<in> B' \<and> B' \<subseteq> O'" using B
  24.249      by (auto intro!: bexI[where x=B'] Pi'_mono in_basis_finmapI simp: B'_def)
  24.250  qed
  24.251 @@ -596,12 +596,12 @@
  24.252  
  24.253  end
  24.254  
  24.255 -subsection {* Polish Space of Finite Maps *}
  24.256 +subsection \<open>Polish Space of Finite Maps\<close>
  24.257  
  24.258  instance finmap :: (countable, polish_space) polish_space proof qed
  24.259  
  24.260  
  24.261 -subsection {* Product Measurable Space of Finite Maps *}
  24.262 +subsection \<open>Product Measurable Space of Finite Maps\<close>
  24.263  
  24.264  definition "PiF I M \<equiv>
  24.265    sigma (\<Union>J \<in> I. (\<Pi>' j\<in>J. space (M j))) {(\<Pi>' j\<in>J. X j) |X J. J \<in> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}"
  24.266 @@ -700,7 +700,7 @@
  24.267    proof safe
  24.268      fix x X s assume *: "x \<in> f s" "P s"
  24.269      with assms obtain l where "s = set l" using finite_list by blast
  24.270 -    with * show "x \<in> (\<Union>n. let s = set (from_nat n) in if P s then f s else {})" using `P s`
  24.271 +    with * show "x \<in> (\<Union>n. let s = set (from_nat n) in if P s then f s else {})" using \<open>P s\<close>
  24.272        by (auto intro!: exI[where x="to_nat l"])
  24.273    next
  24.274      fix x n assume "x \<in> (let s = set (from_nat n) in if P s then f s else {})"
  24.275 @@ -755,7 +755,7 @@
  24.276      apply (case_tac "set (from_nat i) \<in> I")
  24.277      apply simp_all
  24.278      apply (rule singleton_subspace_set_in_sets[OF measurable_sets[OF MN]])
  24.279 -    using assms `y \<in> sets N`
  24.280 +    using assms \<open>y \<in> sets N\<close>
  24.281      apply (auto simp: space_PiF)
  24.282      done
  24.283    finally show "A -` y \<inter> space (PiF I M) \<in> sets (PiF I M)" .
  24.284 @@ -806,7 +806,7 @@
  24.285  next
  24.286    case (Compl a)
  24.287    have "(space (PiF I M) - a) \<inter> {m. domain m \<in> J} = (space (PiF J M) - (a \<inter> {m. domain m \<in> J}))"
  24.288 -    using `J \<subseteq> I` by (auto simp: space_PiF Pi'_def)
  24.289 +    using \<open>J \<subseteq> I\<close> by (auto simp: space_PiF Pi'_def)
  24.290    also have "\<dots> \<in> sets (PiF J M)" using Compl by auto
  24.291    finally show ?case by (simp add: space_PiF)
  24.292  qed simp
  24.293 @@ -848,7 +848,7 @@
  24.294    apply (rule measurable_component_singleton)
  24.295    apply simp
  24.296    apply rule
  24.297 -  apply (rule `finite J`)
  24.298 +  apply (rule \<open>finite J\<close>)
  24.299    apply simp
  24.300    done
  24.301  
  24.302 @@ -859,9 +859,9 @@
  24.303    assume "i \<in> I"
  24.304    hence "(\<lambda>x. (x)\<^sub>F i) -` A \<inter> space (PiF {I} M) =
  24.305      Pi' I (\<lambda>x. if x = i then A else space (M x))"
  24.306 -    using sets.sets_into_space[OF ] `A \<in> sets (M i)` assms
  24.307 +    using sets.sets_into_space[OF ] \<open>A \<in> sets (M i)\<close> assms
  24.308      by (auto simp: space_PiF Pi'_def)
  24.309 -  thus ?thesis  using assms `A \<in> sets (M i)`
  24.310 +  thus ?thesis  using assms \<open>A \<in> sets (M i)\<close>
  24.311      by (intro in_sets_PiFI) auto
  24.312  next
  24.313    assume "i \<notin> I"
  24.314 @@ -874,7 +874,7 @@
  24.315    assumes "i \<in> I"
  24.316    shows "(\<lambda>x. (x)\<^sub>F i) \<in> measurable (PiF {I} M) (M i)"
  24.317    by (unfold measurable_def, intro CollectI conjI ballI proj_measurable_singleton assms)
  24.318 -     (insert `i \<in> I`, auto simp: space_PiF)
  24.319 +     (insert \<open>i \<in> I\<close>, auto simp: space_PiF)
  24.320  
  24.321  lemma measurable_proj_countable:
  24.322    fixes I::"'a::countable set set"
  24.323 @@ -889,11 +889,11 @@
  24.324      have "(\<lambda>x. if i \<in> domain x then x i else y) -` z \<inter> space (PiF {J} M) =
  24.325        (\<lambda>x. if i \<in> J then (x)\<^sub>F i else y) -` z \<inter> space (PiF {J} M)"
  24.326        by (auto simp: space_PiF Pi'_def)
  24.327 -    also have "\<dots> \<in> sets (PiF {J} M)" using `z \<in> sets (M i)` `finite J`
  24.328 +    also have "\<dots> \<in> sets (PiF {J} M)" using \<open>z \<in> sets (M i)\<close> \<open>finite J\<close>
  24.329        by (cases "i \<in> J") (auto intro!: measurable_sets[OF measurable_proj_singleton])
  24.330      finally show "(\<lambda>x. if i \<in> domain x then x i else y) -` z \<inter> space (PiF {J} M) \<in>
  24.331        sets (PiF {J} M)" .
  24.332 -  qed (insert `y \<in> space (M i)`, auto simp: space_PiF Pi'_def)
  24.333 +  qed (insert \<open>y \<in> space (M i)\<close>, auto simp: space_PiF Pi'_def)
  24.334  qed
  24.335  
  24.336  lemma measurable_restrict_proj:
  24.337 @@ -927,7 +927,7 @@
  24.338    shows "space (PiF {I} M) = (\<Pi>' i\<in>I. space (M i))"
  24.339    by (auto simp: product_def space_PiF assms)
  24.340  
  24.341 -text {* adapted from @{thm sets_PiM_single} *}
  24.342 +text \<open>adapted from @{thm sets_PiM_single}\<close>
  24.343  
  24.344  lemma sets_PiF_single:
  24.345    assumes "finite I" "I \<noteq> {}"
  24.346 @@ -942,11 +942,11 @@
  24.347    then obtain X where X: "A = Pi' I X" "X \<in> (\<Pi> j\<in>I. sets (M j))" by auto
  24.348    show "A \<in> sigma_sets ?\<Omega> ?R"
  24.349    proof -
  24.350 -    from `I \<noteq> {}` X have "A = (\<Inter>j\<in>I. {f\<in>space (PiF {I} M). f j \<in> X j})"
  24.351 +    from \<open>I \<noteq> {}\<close> X have "A = (\<Inter>j\<in>I. {f\<in>space (PiF {I} M). f j \<in> X j})"
  24.352        using sets.sets_into_space
  24.353        by (auto simp: space_PiF product_def) blast
  24.354      also have "\<dots> \<in> sigma_sets ?\<Omega> ?R"
  24.355 -      using X `I \<noteq> {}` assms by (intro R.finite_INT) (auto simp: space_PiF)
  24.356 +      using X \<open>I \<noteq> {}\<close> assms by (intro R.finite_INT) (auto simp: space_PiF)
  24.357      finally show "A \<in> sigma_sets ?\<Omega> ?R" .
  24.358    qed
  24.359  next
  24.360 @@ -965,7 +965,7 @@
  24.361    finally show "A \<in> sigma_sets ?\<Omega> {Pi' I X |X. X \<in> (\<Pi> j\<in>I. sets (M j))}" .
  24.362  qed
  24.363  
  24.364 -text {* adapted from @{thm PiE_cong} *}
  24.365 +text \<open>adapted from @{thm PiE_cong}\<close>
  24.366  
  24.367  lemma Pi'_cong:
  24.368    assumes "finite I"
  24.369 @@ -973,7 +973,7 @@
  24.370    shows "Pi' I f = Pi' I g"
  24.371  using assms by (auto simp: Pi'_def)
  24.372  
  24.373 -text {* adapted from @{thm Pi_UN} *}
  24.374 +text \<open>adapted from @{thm Pi_UN}\<close>
  24.375  
  24.376  lemma Pi'_UN:
  24.377    fixes A :: "nat \<Rightarrow> 'i \<Rightarrow> 'a set"
  24.378 @@ -982,20 +982,20 @@
  24.379    shows "(\<Union>n. Pi' I (A n)) = Pi' I (\<lambda>i. \<Union>n. A n i)"
  24.380  proof (intro set_eqI iffI)
  24.381    fix f assume "f \<in> Pi' I (\<lambda>i. \<Union>n. A n i)"
  24.382 -  then have "\<forall>i\<in>I. \<exists>n. f i \<in> A n i" "domain f = I" by (auto simp: `finite I` Pi'_def)
  24.383 +  then have "\<forall>i\<in>I. \<exists>n. f i \<in> A n i" "domain f = I" by (auto simp: \<open>finite I\<close> Pi'_def)
  24.384    from bchoice[OF this(1)] obtain n where n: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> (A (n i) i)" by auto
  24.385    obtain k where k: "\<And>i. i \<in> I \<Longrightarrow> n i \<le> k"
  24.386 -    using `finite I` finite_nat_set_iff_bounded_le[of "n`I"] by auto
  24.387 +    using \<open>finite I\<close> finite_nat_set_iff_bounded_le[of "n`I"] by auto
  24.388    have "f \<in> Pi' I (\<lambda>i. A k i)"
  24.389    proof
  24.390      fix i assume "i \<in> I"
  24.391 -    from mono[OF this, of "n i" k] k[OF this] n[OF this] `domain f = I` `i \<in> I`
  24.392 -    show "f i \<in> A k i " by (auto simp: `finite I`)
  24.393 -  qed (simp add: `domain f = I` `finite I`)
  24.394 +    from mono[OF this, of "n i" k] k[OF this] n[OF this] \<open>domain f = I\<close> \<open>i \<in> I\<close>
  24.395 +    show "f i \<in> A k i " by (auto simp: \<open>finite I\<close>)
  24.396 +  qed (simp add: \<open>domain f = I\<close> \<open>finite I\<close>)
  24.397    then show "f \<in> (\<Union>n. Pi' I (A n))" by auto
  24.398 -qed (auto simp: Pi'_def `finite I`)
  24.399 +qed (auto simp: Pi'_def \<open>finite I\<close>)
  24.400  
  24.401 -text {* adapted from @{thm sets_PiM_sigma} *}
  24.402 +text \<open>adapted from @{thm sets_PiM_sigma}\<close>
  24.403  
  24.404  lemma sigma_fprod_algebra_sigma_eq:
  24.405    fixes E :: "'i \<Rightarrow> 'a set set" and S :: "'i \<Rightarrow> nat \<Rightarrow> 'a set"
  24.406 @@ -1008,9 +1008,9 @@
  24.407    shows "sets (PiF {I} M) = sigma_sets (space (PiF {I} M)) P"
  24.408  proof
  24.409    let ?P = "sigma (space (Pi\<^sub>F {I} M)) P"
  24.410 -  from `finite I`[THEN ex_bij_betw_finite_nat] guess T ..
  24.411 +  from \<open>finite I\<close>[THEN ex_bij_betw_finite_nat] guess T ..
  24.412    then have T: "\<And>i. i \<in> I \<Longrightarrow> T i < card I" "\<And>i. i\<in>I \<Longrightarrow> the_inv_into I T (T i) = i"
  24.413 -    by (auto simp add: bij_betw_def set_eq_iff image_iff the_inv_into_f_f simp del: `finite I`)
  24.414 +    by (auto simp add: bij_betw_def set_eq_iff image_iff the_inv_into_f_f simp del: \<open>finite I\<close>)
  24.415    have P_closed: "P \<subseteq> Pow (space (Pi\<^sub>F {I} M))"
  24.416      using E_closed by (auto simp: space_PiF P_def Pi'_iff subset_eq)
  24.417    then have space_P: "space ?P = (\<Pi>' i\<in>I. space (M i))"
  24.418 @@ -1023,14 +1023,14 @@
  24.419      fix i A assume "i \<in> I" and A: "A \<in> sets (M i)"
  24.420      have "(\<lambda>x. (x)\<^sub>F i) \<in> measurable ?P (sigma (space (M i)) (E i))"
  24.421      proof (subst measurable_iff_measure_of)
  24.422 -      show "E i \<subseteq> Pow (space (M i))" using `i \<in> I` by fact
  24.423 -      from space_P `i \<in> I` show "(\<lambda>x. (x)\<^sub>F i) \<in> space ?P \<rightarrow> space (M i)"
  24.424 +      show "E i \<subseteq> Pow (space (M i))" using \<open>i \<in> I\<close> by fact
  24.425 +      from space_P \<open>i \<in> I\<close> show "(\<lambda>x. (x)\<^sub>F i) \<in> space ?P \<rightarrow> space (M i)"
  24.426          by auto
  24.427        show "\<forall>A\<in>E i. (\<lambda>x. (x)\<^sub>F i) -` A \<inter> space ?P \<in> sets ?P"
  24.428        proof
  24.429          fix A assume A: "A \<in> E i"
  24.430          then have "(\<lambda>x. (x)\<^sub>F i) -` A \<inter> space ?P = (\<Pi>' j\<in>I. if i = j then A else space (M j))"
  24.431 -          using E_closed `i \<in> I` by (auto simp: space_P Pi_iff subset_eq split: split_if_asm)
  24.432 +          using E_closed \<open>i \<in> I\<close> by (auto simp: space_P Pi_iff subset_eq split: split_if_asm)
  24.433          also have "\<dots> = (\<Pi>' j\<in>I. \<Union>n. if i = j then A else S j n)"
  24.434            by (intro Pi'_cong) (simp_all add: S_union)
  24.435          also have "\<dots> = (\<Union>xs\<in>{xs. length xs = card I}. \<Pi>' j\<in>I. if i = j then A else S j (xs ! T j))"
  24.436 @@ -1052,7 +1052,7 @@
  24.437            using P_closed by simp
  24.438        qed
  24.439      qed
  24.440 -    from measurable_sets[OF this, of A] A `i \<in> I` E_closed
  24.441 +    from measurable_sets[OF this, of A] A \<open>i \<in> I\<close> E_closed
  24.442      have "(\<lambda>x. (x)\<^sub>F i) -` A \<inter> space ?P \<in> sets ?P"
  24.443        by (simp add: E_generates)
  24.444      also have "(\<lambda>x. (x)\<^sub>F i) -` A \<inter> space ?P = {f \<in> \<Pi>' i\<in>I. space (M i). f i \<in> A}"
  24.445 @@ -1062,7 +1062,7 @@
  24.446    finally show "sets (PiF {I} M) \<subseteq> sigma_sets (space (PiF {I} M)) P"
  24.447      by (simp add: P_closed)
  24.448    show "sigma_sets (space (PiF {I} M)) P \<subseteq> sets (PiF {I} M)"
  24.449 -    using `finite I` `I \<noteq> {}`
  24.450 +    using \<open>finite I\<close> \<open>I \<noteq> {}\<close>
  24.451      by (auto intro!: sets.sigma_sets_subset product_in_sets_PiFI simp: E_generates P_def)
  24.452  qed
  24.453  
  24.454 @@ -1105,7 +1105,7 @@
  24.455    then obtain B' where B': "B'\<subseteq>basis_finmap" "a = \<Union>B'"
  24.456      using finmap_topological_basis by (force simp add: topological_basis_def)
  24.457    have "a \<in> sigma UNIV {Pi' J X |X J. finite J \<and> X \<in> J \<rightarrow> sigma_sets UNIV (Collect open)}"
  24.458 -    unfolding `a = \<Union>B'`
  24.459 +    unfolding \<open>a = \<Union>B'\<close>
  24.460    proof (rule sets.countable_Union)
  24.461      from B' countable_basis_finmap show "countable B'" by (metis countable_subset)
  24.462    next
  24.463 @@ -1134,7 +1134,7 @@
  24.464        proof cases
  24.465          assume "?b J \<noteq> {}"
  24.466          then obtain f where "f \<in> b" "domain f = {}" using ef by auto
  24.467 -        hence "?b J = {f}" using `J = {}`
  24.468 +        hence "?b J = {f}" using \<open>J = {}\<close>
  24.469            by (auto simp: finmap_eq_iff)
  24.470          also have "{f} \<in> sets borel" by simp
  24.471          finally show ?thesis .
  24.472 @@ -1143,11 +1143,11 @@
  24.473        assume "J \<noteq> ({}::'i set)"
  24.474        have "(?b J) = b \<inter> {m. domain m \<in> {J}}" by auto
  24.475        also have "\<dots> \<in> sets (PiF {J} (\<lambda>_. borel))"
  24.476 -        using b' by (rule restrict_sets_measurable) (auto simp: `finite J`)
  24.477 +        using b' by (rule restrict_sets_measurable) (auto simp: \<open>finite J\<close>)
  24.478        also have "\<dots> = sigma_sets (space (PiF {J} (\<lambda>_. borel)))
  24.479          {Pi' (J) F |F. (\<forall>j\<in>J. F j \<in> Collect open)}"
  24.480          (is "_ = sigma_sets _ ?P")
  24.481 -       by (rule product_open_generates_sets_PiF_single[OF `J \<noteq> {}` `finite J`])
  24.482 +       by (rule product_open_generates_sets_PiF_single[OF \<open>J \<noteq> {}\<close> \<open>finite J\<close>])
  24.483        also have "\<dots> \<subseteq> sigma_sets UNIV (Collect open)"
  24.484          by (intro sigma_sets_mono'') (auto intro!: open_Pi'I simp: space_PiF)
  24.485        finally have "(?b J) \<in> sets borel" by (simp add: borel_def)
  24.486 @@ -1156,7 +1156,7 @@
  24.487    finally show "b \<in> sigma_sets UNIV (Collect open)" by (simp add: borel_def)
  24.488  qed (simp add: emeasure_sigma borel_def PiF_def)
  24.489  
  24.490 -subsection {* Isomorphism between Functions and Finite Maps *}
  24.491 +subsection \<open>Isomorphism between Functions and Finite Maps\<close>
  24.492  
  24.493  lemma measurable_finmap_compose:
  24.494    shows "(\<lambda>m. compose J m f) \<in> measurable (PiM (f ` J) (\<lambda>_. M)) (PiM J (\<lambda>_. M))"
  24.495 @@ -1173,7 +1173,7 @@
  24.496    assumes inv: "i \<in> J \<Longrightarrow> f' (f i) = i"
  24.497  begin
  24.498  
  24.499 -text {* to measure finmaps *}
  24.500 +text \<open>to measure finmaps\<close>
  24.501  
  24.502  definition "fm = (finmap_of (f ` J)) o (\<lambda>g. compose (f ` J) g f')"
  24.503  
  24.504 @@ -1222,7 +1222,7 @@
  24.505    apply (auto)
  24.506    done
  24.507  
  24.508 -text {* to measure functions *}
  24.509 +text \<open>to measure functions\<close>
  24.510  
  24.511  definition "mf = (\<lambda>g. compose J g f) o proj"
  24.512  
  24.513 @@ -1284,7 +1284,7 @@
  24.514    using fm_image_measurable[OF assms]
  24.515    by (rule subspace_set_in_sets) (auto simp: finite_subset)
  24.516  
  24.517 -text {* measure on finmaps *}
  24.518 +text \<open>measure on finmaps\<close>
  24.519  
  24.520  definition "mapmeasure M N = distr M (PiF (Collect finite) N) (fm)"
  24.521  
    25.1 --- a/src/HOL/Probability/Finite_Product_Measure.thy	Mon Dec 07 16:48:10 2015 +0000
    25.2 +++ b/src/HOL/Probability/Finite_Product_Measure.thy	Mon Dec 07 20:19:59 2015 +0100
    25.3 @@ -2,7 +2,7 @@
    25.4      Author:     Johannes Hölzl, TU München
    25.5  *)
    25.6  
    25.7 -section {*Finite product measures*}
    25.8 +section \<open>Finite product measures\<close>
    25.9  
   25.10  theory Finite_Product_Measure
   25.11  imports Binary_Product_Measure
   25.12 @@ -15,7 +15,7 @@
   25.13  lemma case_prod_const: "(\<lambda>(i, j). c) = (\<lambda>_. c)"
   25.14    by auto
   25.15  
   25.16 -subsubsection {* More about Function restricted by @{const extensional}  *}
   25.17 +subsubsection \<open>More about Function restricted by @{const extensional}\<close>
   25.18  
   25.19  definition
   25.20    "merge I J = (\<lambda>(x, y) i. if i \<in> I then x i else if i \<in> J then y i else undefined)"
   25.21 @@ -92,10 +92,10 @@
   25.22    proof cases
   25.23      assume x: "x \<in> (\<Pi>\<^sub>E i\<in>J. S i)"
   25.24      have "x \<in> A \<longleftrightarrow> merge J (I - J) (x,y) \<in> (\<lambda>x. restrict x J) -` A \<inter> (\<Pi>\<^sub>E i\<in>I. S i)"
   25.25 -      using y x `J \<subseteq> I` PiE_cancel_merge[of "J" "I - J" x y S]
   25.26 +      using y x \<open>J \<subseteq> I\<close> PiE_cancel_merge[of "J" "I - J" x y S]
   25.27        by (auto simp del: PiE_cancel_merge simp add: Un_absorb1)
   25.28      then show "x \<in> A \<longleftrightarrow> x \<in> B"
   25.29 -      using y x `J \<subseteq> I` PiE_cancel_merge[of "J" "I - J" x y S]
   25.30 +      using y x \<open>J \<subseteq> I\<close> PiE_cancel_merge[of "J" "I - J" x y S]
   25.31        by (auto simp del: PiE_cancel_merge simp add: Un_absorb1 eq)
   25.32    qed (insert sets, auto)
   25.33  qed
   25.34 @@ -109,9 +109,9 @@
   25.35    "I \<inter> J = {} \<Longrightarrow> merge I J -` Pi\<^sub>E (I \<union> J) E = Pi I E \<times> Pi J E"
   25.36    by (auto simp: restrict_Pi_cancel PiE_def)
   25.37  
   25.38 -subsection {* Finite product spaces *}
   25.39 +subsection \<open>Finite product spaces\<close>
   25.40  
   25.41 -subsubsection {* Products *}
   25.42 +subsubsection \<open>Products\<close>
   25.43  
   25.44  definition prod_emb where
   25.45    "prod_emb I M K X = (\<lambda>x. restrict x K) -` X \<inter> (PIE i:I. space (M i))"
   25.46 @@ -324,7 +324,7 @@
   25.47  proof -
   25.48    have space: "\<And>i. i \<in> I \<Longrightarrow> space (M i) = space (N i)"
   25.49      using sets_eq_imp_space_eq[OF sets] by auto
   25.50 -  with sets show ?thesis unfolding `I = J`
   25.51 +  with sets show ?thesis unfolding \<open>I = J\<close>
   25.52      by (intro antisym prod_algebra_mono) auto
   25.53  qed
   25.54  
   25.55 @@ -339,7 +339,7 @@
   25.56    then have "(\<Pi>\<^sub>E i\<in>I. space (M i)) = prod_emb I M {i} (\<Pi>\<^sub>E i\<in>{i}. space (M i))"
   25.57      by (auto simp: prod_emb_def)
   25.58    also have "\<dots> \<in> prod_algebra I M"
   25.59 -    using `i \<in> I` by (intro prod_algebraI) auto
   25.60 +    using \<open>i \<in> I\<close> by (intro prod_algebraI) auto
   25.61    finally show ?thesis .
   25.62  qed
   25.63  
   25.64 @@ -370,13 +370,13 @@
   25.65    proof cases
   25.66      assume "I = {}"
   25.67      with X have "A = {\<lambda>x. undefined}" by (auto simp: prod_emb_def)
   25.68 -    with `I = {}` show ?thesis by (auto intro!: sigma_sets_top)
   25.69 +    with \<open>I = {}\<close> show ?thesis by (auto intro!: sigma_sets_top)
   25.70    next
   25.71      assume "I \<noteq> {}"
   25.72      with X have "A = (\<Inter>j\<in>J. {f\<in>(\<Pi>\<^sub>E i\<in>I. space (M i)). f j \<in> X j})"
   25.73        by (auto simp: prod_emb_def)
   25.74      also have "\<dots> \<in> sigma_sets ?\<Omega> ?R"
   25.75 -      using X `I \<noteq> {}` by (intro R.finite_INT sigma_sets.Basic) auto
   25.76 +      using X \<open>I \<noteq> {}\<close> by (intro R.finite_INT sigma_sets.Basic) auto
   25.77      finally show "A \<in> sigma_sets ?\<Omega> ?R" .
   25.78    qed
   25.79  next
   25.80 @@ -412,9 +412,9 @@
   25.81    shows "sets (\<Pi>\<^sub>M i\<in>I. sigma (\<Omega> i) (E i)) = sets (sigma (\<Pi>\<^sub>E i\<in>I. \<Omega> i) P)"
   25.82  proof cases
   25.83    assume "I = {}" 
   25.84 -  with `\<Union>J = I` have "P = {{\<lambda>_. undefined}} \<or> P = {}"
   25.85 +  with \<open>\<Union>J = I\<close> have "P = {{\<lambda>_. undefined}} \<or> P = {}"
   25.86      by (auto simp: P_def)
   25.87 -  with `I = {}` show ?thesis
   25.88 +  with \<open>I = {}\<close> show ?thesis
   25.89      by (auto simp add: sets_PiM_empty sigma_sets_empty_eq)
   25.90  next
   25.91    let ?F = "\<lambda>i. {(\<lambda>x. x i) -` A \<inter> Pi\<^sub>E I \<Omega> |A. A \<in> E i}"
   25.92 @@ -425,7 +425,7 @@
   25.93    also have "\<dots> = sets (\<Squnion>\<^sub>\<sigma> i\<in>I. sigma (Pi\<^sub>E I \<Omega>) (?F i))"
   25.94      using E by (intro SUP_sigma_cong arg_cong[where f=sets] vimage_algebra_sigma) auto
   25.95    also have "\<dots> = sets (sigma (Pi\<^sub>E I \<Omega>) (\<Union>i\<in>I. ?F i))"
   25.96 -    using `I \<noteq> {}` by (intro arg_cong[where f=sets] SUP_sigma_sigma) auto
   25.97 +    using \<open>I \<noteq> {}\<close> by (intro arg_cong[where f=sets] SUP_sigma_sigma) auto
   25.98    also have "\<dots> = sets (sigma (Pi\<^sub>E I \<Omega>) P)"
   25.99    proof (intro arg_cong[where f=sets] sigma_eqI sigma_sets_eqI)
  25.100      show "(\<Union>i\<in>I. ?F i) \<subseteq> Pow (Pi\<^sub>E I \<Omega>)" "P \<subseteq> Pow (Pi\<^sub>E I \<Omega>)"
  25.101 @@ -437,34 +437,34 @@
  25.102      fix Z assume "Z \<in> (\<Union>i\<in>I. ?F i)"
  25.103      then obtain i A where i: "i \<in> I" "A \<in> E i" and Z_def: "Z = (\<lambda>x. x i) -` A \<inter> Pi\<^sub>E I \<Omega>"
  25.104        by auto
  25.105 -    from `i \<in> I` J obtain j where j: "i \<in> j" "j \<in> J" "j \<subseteq> I" "finite j"
  25.106 +    from \<open>i \<in> I\<close> J obtain j where j: "i \<in> j" "j \<in> J" "j \<subseteq> I" "finite j"
  25.107        by auto
  25.108      obtain S where S: "\<And>i. i \<in> j \<Longrightarrow> S i \<subseteq> E i" "\<And>i. i \<in> j \<Longrightarrow> countable (S i)"
  25.109        "\<And>i. i \<in> j \<Longrightarrow> \<Omega> i = \<Union>(S i)"
  25.110 -      by (metis subset_eq \<Omega>_cover `j \<subseteq> I`)
  25.111 +      by (metis subset_eq \<Omega>_cover \<open>j \<subseteq> I\<close>)
  25.112      def A' \<equiv> "\<lambda>n. n(i := A)"
  25.113      then have A'_i: "\<And>n. A' n i = A"
  25.114        by simp
  25.115      { fix n assume "n \<in> Pi\<^sub>E (j - {i}) S"
  25.116        then have "A' n \<in> Pi j E"
  25.117 -        unfolding PiE_def Pi_def using S(1) by (auto simp: A'_def `A \<in> E i` )
  25.118 -      with `j \<in> J` have "{f \<in> Pi\<^sub>E I \<Omega>. \<forall>i\<in>j. f i \<in> A' n i} \<in> P"
  25.119 +        unfolding PiE_def Pi_def using S(1) by (auto simp: A'_def \<open>A \<in> E i\<close> )
  25.120 +      with \<open>j \<in> J\<close> have "{f \<in> Pi\<^sub>E I \<Omega>. \<forall>i\<in>j. f i \<in> A' n i} \<in> P"
  25.121          by (auto simp: P_def) }
  25.122      note A'_in_P = this
  25.123  
  25.124      { fix x assume "x i \<in> A" "x \<in> Pi\<^sub>E I \<Omega>"
  25.125 -      with S(3) `j \<subseteq> I` have "\<forall>i\<in>j. \<exists>s\<in>S i. x i \<in> s"
  25.126 +      with S(3) \<open>j \<subseteq> I\<close> have "\<forall>i\<in>j. \<exists>s\<in>S i. x i \<in> s"
  25.127          by (auto simp: PiE_def Pi_def)
  25.128        then obtain s where s: "\<And>i. i \<in> j \<Longrightarrow> s i \<in> S i" "\<And>i. i \<in> j \<Longrightarrow> x i \<in> s i"
  25.129          by metis
  25.130 -      with `x i \<in> A` have "\<exists>n\<in>PiE (j-{i}) S. \<forall>i\<in>j. x i \<in> A' n i"
  25.131 +      with \<open>x i \<in> A\<close> have "\<exists>n\<in>PiE (j-{i}) S. \<forall>i\<in>j. x i \<in> A' n i"
  25.132          by (intro bexI[of _ "restrict (s(i := A)) (j-{i})"]) (auto simp: A'_def split: if_splits) }
  25.133      then have "Z = (\<Union>n\<in>PiE (j-{i}) S. {f\<in>(\<Pi>\<^sub>E i\<in>I. \<Omega> i). \<forall>i\<in>j. f i \<in> A' n i})"
  25.134        unfolding Z_def
  25.135 -      by (auto simp add: set_eq_iff ball_conj_distrib `i\<in>j` A'_i dest: bspec[OF _ `i\<in>j`]
  25.136 +      by (auto simp add: set_eq_iff ball_conj_distrib \<open>i\<in>j\<close> A'_i dest: bspec[OF _ \<open>i\<in>j\<close>]
  25.137                 cong: conj_cong)
  25.138      also have "\<dots> \<in> sigma_sets (\<Pi>\<^sub>E i\<in>I. \<Omega> i) P"
  25.139 -      using `finite j` S(2)
  25.140 +      using \<open>finite j\<close> S(2)
  25.141        by (intro P.countable_UN' countable_PiE) (simp_all add: image_subset_iff A'_in_P)
  25.142      finally show "Z \<in> sigma_sets (\<Pi>\<^sub>E i\<in>I. \<Omega> i) P" .
  25.143    next
  25.144 @@ -487,8 +487,8 @@
  25.145          unfolding b(1)
  25.146          by (auto simp: PiE_def Pi_def)
  25.147        show ?thesis
  25.148 -        unfolding eq using `A \<in> Pi j E` `j \<in> J` J(2)
  25.149 -        by (intro F.finite_INT J `j \<in> J` `j \<noteq> {}` sigma_sets.Basic) blast
  25.150 +        unfolding eq using \<open>A \<in> Pi j E\<close> \<open>j \<in> J\<close> J(2)
  25.151 +        by (intro F.finite_INT J \<open>j \<in> J\<close> \<open>j \<noteq> {}\<close> sigma_sets.Basic) blast
  25.152      qed
  25.153    qed
  25.154    finally show "?thesis" .
  25.155 @@ -575,18 +575,18 @@
  25.156  lemma sets_PiM_I_finite[measurable]:
  25.157    assumes "finite I" and sets: "(\<And>i. i \<in> I \<Longrightarrow> E i \<in> sets (M i))"
  25.158    shows "(PIE j:I. E j) \<in> sets (PIM i:I. M i)"
  25.159 -  using sets_PiM_I[of I I E M] sets.sets_into_space[OF sets] `finite I` sets by auto
  25.160 +  using sets_PiM_I[of I I E M] sets.sets_into_space[OF sets] \<open>finite I\<close> sets by auto
  25.161  
  25.162  lemma measurable_component_singleton[measurable (raw)]:
  25.163    assumes "i \<in> I" shows "(\<lambda>x. x i) \<in> measurable (Pi\<^sub>M I M) (M i)"
  25.164  proof (unfold measurable_def, intro CollectI conjI ballI)
  25.165    fix A assume "A \<in> sets (M i)"
  25.166    then have "(\<lambda>x. x i) -` A \<inter> space (Pi\<^sub>M I M) = prod_emb I M {i} (\<Pi>\<^sub>E j\<in>{i}. A)"
  25.167 -    using sets.sets_into_space `i \<in> I`
  25.168 +    using sets.sets_into_space \<open>i \<in> I\<close>
  25.169      by (fastforce dest: Pi_mem simp: prod_emb_def space_PiM split: split_if_asm)
  25.170    then show "(\<lambda>x. x i) -` A \<inter> space (Pi\<^sub>M I M) \<in> sets (Pi\<^sub>M I M)"
  25.171 -    using `A \<in> sets (M i)` `i \<in> I` by (auto intro!: sets_PiM_I)
  25.172 -qed (insert `i \<in> I`, auto simp: space_PiM)
  25.173 +    using \<open>A \<in> sets (M i)\<close> \<open>i \<in> I\<close> by (auto intro!: sets_PiM_I)
  25.174 +qed (insert \<open>i \<in> I\<close>, auto simp: space_PiM)
  25.175  
  25.176  lemma measurable_component_singleton'[measurable_dest]:
  25.177    assumes f: "f \<in> measurable N (Pi\<^sub>M I M)"
  25.178 @@ -863,7 +863,7 @@
  25.179      show "f \<in> (\<Union>i. ?F i)" by (auto simp: incseq_def PiE_def)
  25.180    next
  25.181      fix i show "?F i \<subseteq> ?F (Suc i)"
  25.182 -      using `\<And>i. incseq (F i)`[THEN incseq_SucD] by auto
  25.183 +      using \<open>\<And>i. incseq (F i)\<close>[THEN incseq_SucD] by auto
  25.184    qed
  25.185  qed
  25.186  
  25.187 @@ -892,7 +892,7 @@
  25.188  proof (induct I arbitrary: A rule: finite_induct)
  25.189    case (insert i I)
  25.190    interpret finite_product_sigma_finite M I by standard fact
  25.191 -  have "finite (insert i I)" using `finite I` by auto
  25.192 +  have "finite (insert i I)" using \<open>finite I\<close> by auto
  25.193    interpret I': finite_product_sigma_finite M "insert i I" by standard fact
  25.194    let ?h = "(\<lambda>(f, y). f(i := y))"
  25.195  
  25.196 @@ -1065,7 +1065,7 @@
  25.197      fix x assume x: "x \<in> space (Pi\<^sub>M I M)"
  25.198      let ?f = "\<lambda>y. f (x(i := y))"
  25.199      show "?f \<in> borel_measurable (M i)"
  25.200 -      using measurable_comp[OF measurable_component_update f, OF x `i \<notin> I`]
  25.201 +      using measurable_comp[OF measurable_component_update f, OF x \<open>i \<notin> I\<close>]
  25.202        unfolding comp_def .
  25.203      show "(\<integral>\<^sup>+ y. f (merge I {i} (x, y)) \<partial>Pi\<^sub>M {i} M) = (\<integral>\<^sup>+ y. f (x(i := y i)) \<partial>Pi\<^sub>M {i} M)"
  25.204        using x
  25.205 @@ -1092,7 +1092,7 @@
  25.206    shows "(\<integral>\<^sup>+ x. (\<Prod>i\<in>I. f i (x i)) \<partial>Pi\<^sub>M I M) = (\<Prod>i\<in>I. integral\<^sup>N (M i) (f i))"
  25.207  using assms proof induct
  25.208    case (insert i I)
  25.209 -  note `finite I`[intro, simp]
  25.210 +  note \<open>finite I\<close>[intro, simp]
  25.211    interpret I: finite_product_sigma_finite M I by standard auto
  25.212    have *: "\<And>x y. (\<Prod>j\<in>I. f j (if j = i then y else x j)) = (\<Prod>j\<in>I. f j (x j))"
  25.213      using insert by (auto intro!: setprod.cong)
    26.1 --- a/src/HOL/Probability/Giry_Monad.thy	Mon Dec 07 16:48:10 2015 +0000
    26.2 +++ b/src/HOL/Probability/Giry_Monad.thy	Mon Dec 07 20:19:59 2015 +0100
    26.3 @@ -10,7 +10,7 @@
    26.4    imports Probability_Measure Lebesgue_Integral_Substitution "~~/src/HOL/Library/Monad_Syntax" 
    26.5  begin
    26.6  
    26.7 -section {* Sub-probability spaces *}
    26.8 +section \<open>Sub-probability spaces\<close>
    26.9  
   26.10  locale subprob_space = finite_measure +
   26.11    assumes emeasure_space_le_1: "emeasure M (space M) \<le> 1"
   26.12 @@ -93,7 +93,7 @@
   26.13    from mono' derivg have "\<And>x. x \<in> {a<..<b} \<Longrightarrow> g' x \<ge> 0"
   26.14      by (rule mono_on_imp_deriv_nonneg) (auto simp: interior_atLeastAtMost_real)
   26.15    from contg' this have derivg_nonneg: "\<And>x. x \<in> {a..b} \<Longrightarrow> g' x \<ge> 0"
   26.16 -    by (rule continuous_ge_on_Iii) (simp_all add: `a < b`)
   26.17 +    by (rule continuous_ge_on_Iii) (simp_all add: \<open>a < b\<close>)
   26.18  
   26.19    from derivg have contg: "continuous_on {a..b} g" by (rule has_real_derivative_imp_continuous_on)
   26.20    have A: "h -` {a..b} = {g a..g b}"
   26.21 @@ -119,13 +119,13 @@
   26.22    with assms have "(\<integral>\<^sup>+x. f x * indicator {g a..g b} x \<partial>lborel) = 
   26.23                        (\<integral>\<^sup>+x. f (g x) * g' x * indicator {a..b} x \<partial>lborel)"
   26.24      by (intro nn_integral_substitution_aux)
   26.25 -       (auto simp: derivg_nonneg A B emeasure_density mult.commute `a < b`)
   26.26 +       (auto simp: derivg_nonneg A B emeasure_density mult.commute \<open>a < b\<close>)
   26.27    also have "... = emeasure (density lborel (\<lambda>x. f (g x) * g' x)) {a..b}" 
   26.28      by (simp add: emeasure_density)
   26.29    finally show ?thesis .
   26.30  next
   26.31    assume "\<not>a < b"
   26.32 -  with `a \<le> b` have [simp]: "b = a" by (simp add: not_less del: `a \<le> b`)
   26.33 +  with \<open>a \<le> b\<close> have [simp]: "b = a" by (simp add: not_less del: \<open>a \<le> b\<close>)
   26.34    from inv and range have "h -` {a} = {g a}" by auto
   26.35    thus ?thesis by (simp_all add: emeasure_distr emeasure_density measurable_sets_borel[OF Mh])
   26.36  qed
   26.37 @@ -185,7 +185,7 @@
   26.38    using measurable_space[OF N x]
   26.39    by (auto simp: space_subprob_algebra intro!: measurable_cong_sets dest: sets_eq_imp_space_eq)
   26.40  
   26.41 -ML {*
   26.42 +ML \<open>
   26.43  
   26.44  fun subprob_cong thm ctxt = (
   26.45    let
   26.46 @@ -198,7 +198,7 @@
   26.47    end
   26.48    handle THM _ => ([], ctxt) | TERM _ => ([], ctxt))
   26.49  
   26.50 -*}
   26.51 +\<close>
   26.52  
   26.53  setup \<open>
   26.54    Context.theory_map (Measurable.add_preprocessor "subprob_cong" subprob_cong)
   26.55 @@ -460,7 +460,7 @@
   26.56    qed
   26.57  qed
   26.58  
   26.59 -section {* Properties of return *}
   26.60 +section \<open>Properties of return\<close>
   26.61  
   26.62  definition return :: "'a measure \<Rightarrow> 'a \<Rightarrow> 'a measure" where
   26.63    "return R x = measure_of (space R) (sets R) (\<lambda>A. indicator A x)"
   26.64 @@ -525,11 +525,11 @@
   26.65    assumes "g x \<ge> 0" "x \<in> space M" "g \<in> borel_measurable M"
   26.66    shows "(\<integral>\<^sup>+ a. g a \<partial>return M x) = g x"
   26.67  proof-
   26.68 -  interpret prob_space "return M x" by (rule prob_space_return[OF `x \<in> space M`])
   26.69 +  interpret prob_space "return M x" by (rule prob_space_return[OF \<open>x \<in> space M\<close>])
   26.70    have "(\<integral>\<^sup>+ a. g a \<partial>return M x) = (\<integral>\<^sup>+ a. g x \<partial>return M x)" using assms
   26.71      by (intro nn_integral_cong_AE) (auto simp: AE_return)
   26.72    also have "... = g x"
   26.73 -    using nn_integral_const[OF `g x \<ge> 0`, of "return M x"] emeasure_space_1 by simp
   26.74 +    using nn_integral_const[OF \<open>g x \<ge> 0\<close>, of "return M x"] emeasure_space_1 by simp
   26.75    finally show ?thesis .
   26.76  qed
   26.77  
   26.78 @@ -538,7 +538,7 @@
   26.79    assumes "x \<in> space M" "g \<in> borel_measurable M"
   26.80    shows "(\<integral>a. g a \<partial>return M x) = g x"
   26.81  proof-
   26.82 -  interpret prob_space "return M x" by (rule prob_space_return[OF `x \<in> space M`])
   26.83 +  interpret prob_space "return M x" by (rule prob_space_return[OF \<open>x \<in> space M\<close>])
   26.84    have "(\<integral>a. g a \<partial>return M x) = (\<integral>a. g x \<partial>return M x)" using assms
   26.85      by (intro integral_cong_AE) (auto simp: AE_return)
   26.86    then show ?thesis
   26.87 @@ -696,7 +696,7 @@
   26.88    "sets M = sets (subprob_algebra N) \<Longrightarrow> space (select_sets M) = space N"
   26.89    by (intro sets_eq_imp_space_eq sets_select_sets)
   26.90  
   26.91 -section {* Join *}
   26.92 +section \<open>Join\<close>
   26.93  
   26.94  definition join :: "'a measure measure \<Rightarrow> 'a measure" where
   26.95    "join M = measure_of (space (select_sets M)) (sets (select_sets M)) (\<lambda>B. \<integral>\<^sup>+ M'. emeasure M' B \<partial>M)"
   26.96 @@ -734,10 +734,10 @@
   26.97    proof (rule measurable_cong)
   26.98      fix M' assume "M' \<in> space (subprob_algebra (subprob_algebra N))"
   26.99      then show "emeasure (join M') A = (\<integral>\<^sup>+ M''. emeasure M'' A \<partial>M')"
  26.100 -      by (intro emeasure_join) (auto simp: space_subprob_algebra `A\<in>sets N`)
  26.101 +      by (intro emeasure_join) (auto simp: space_subprob_algebra \<open>A\<in>sets N\<close>)
  26.102    qed
  26.103    also have "(\<lambda>M'. \<integral>\<^sup>+M''. emeasure M'' A \<partial>M') \<in> ?B"
  26.104 -    using measurable_emeasure_subprob_algebra[OF `A\<in>sets N`]
  26.105 +    using measurable_emeasure_subprob_algebra[OF \<open>A\<in>sets N\<close>]
  26.106      by (rule nn_integral_measurable_subprob_algebra)
  26.107    finally show "(\<lambda>M'. emeasure (join M') A) \<in> borel_measurable (subprob_algebra (subprob_algebra N))" .
  26.108  next
  26.109 @@ -1037,7 +1037,7 @@
  26.110      fix M' assume "M' \<in> space M"
  26.111      from assms have "space M = space (subprob_algebra R)"
  26.112          using sets_eq_imp_space_eq by blast
  26.113 -    with `M' \<in> space M` have [simp]: "sets M' = sets R" using space_subprob_algebra by blast
  26.114 +    with \<open>M' \<in> space M\<close> have [simp]: "sets M' = sets R" using space_subprob_algebra by blast
  26.115      show "f \<in> measurable M' N" by (simp cong: measurable_cong_sets add: assms)
  26.116      have "space M' = space R" by (rule sets_eq_imp_space_eq) simp
  26.117      thus "emeasure M' (f -` A \<inter> space R) = emeasure M' (f -` A \<inter> space M')" by simp
  26.118 @@ -1088,7 +1088,7 @@
  26.119    assume "space M \<noteq> {}"
  26.120    hence "(SOME x. x \<in> space M) \<in> space M" by (rule_tac someI_ex) blast
  26.121    with assms have "f (SOME x. x \<in> space M) = g (SOME x. x \<in> space M)" by blast
  26.122 -  with `space M \<noteq> {}` and assms show ?thesis by (simp add: bind_nonempty cong: distr_cong)
  26.123 +  with \<open>space M \<noteq> {}\<close> and assms show ?thesis by (simp add: bind_nonempty cong: distr_cong)
  26.124  qed (simp add: bind_empty)
  26.125  
  26.126  lemma bind_nonempty':
  26.127 @@ -1319,7 +1319,7 @@
  26.128    shows "(distr M X f \<guillemotright>= N) = (M \<guillemotright>= (\<lambda>x. N (f x)))"
  26.129  proof -
  26.130    have "space X \<noteq> {}" "space M \<noteq> {}"
  26.131 -    using `space M \<noteq> {}` f[THEN measurable_space] by auto
  26.132 +    using \<open>space M \<noteq> {}\<close> f[THEN measurable_space] by auto
  26.133    then show ?thesis
  26.134      by (simp add: bind_nonempty''[where N=K] distr_distr comp_def)
  26.135  qed
  26.136 @@ -1419,8 +1419,8 @@
  26.137    from assms have sets_fx[simp]: "\<And>x. x \<in> space M \<Longrightarrow> sets (f x) = sets N"
  26.138        by (simp add: sets_kernel)
  26.139    have ex_in: "\<And>A. A \<noteq> {} \<Longrightarrow> \<exists>x. x \<in> A" by blast
  26.140 -  note sets_some[simp] = sets_kernel[OF M1 someI_ex[OF ex_in[OF `space M \<noteq> {}`]]]
  26.141 -                         sets_kernel[OF M2 someI_ex[OF ex_in[OF `space N \<noteq> {}`]]]
  26.142 +  note sets_some[simp] = sets_kernel[OF M1 someI_ex[OF ex_in[OF \<open>space M \<noteq> {}\<close>]]]
  26.143 +                         sets_kernel[OF M2 someI_ex[OF ex_in[OF \<open>space N \<noteq> {}\<close>]]]
  26.144    note space_some[simp] = sets_eq_imp_space_eq[OF this(1)] sets_eq_imp_space_eq[OF this(2)]
  26.145  
  26.146    have "bind M (\<lambda>x. bind (f x) g) = 
  26.147 @@ -1504,7 +1504,7 @@
  26.148    finally show ?thesis .
  26.149  qed
  26.150  
  26.151 -section {* Measures form a $\omega$-chain complete partial order *}
  26.152 +section \<open>Measures form a $\omega$-chain complete partial order\<close>
  26.153  
  26.154  definition SUP_measure :: "(nat \<Rightarrow> 'a measure) \<Rightarrow> 'a measure" where
  26.155    "SUP_measure M = measure_of (\<Union>i. space (M i)) (\<Union>i. sets (M i)) (\<lambda>A. SUP i. emeasure (M i) A)"
    27.1 --- a/src/HOL/Probability/Independent_Family.thy	Mon Dec 07 16:48:10 2015 +0000
    27.2 +++ b/src/HOL/Probability/Independent_Family.thy	Mon Dec 07 20:19:59 2015 +0100
    27.3 @@ -3,7 +3,7 @@
    27.4      Author:     Sudeep Kanav, TU München
    27.5  *)
    27.6  
    27.7 -section {* Independent families of events, event sets, and random variables *}
    27.8 +section \<open>Independent families of events, event sets, and random variables\<close>
    27.9  
   27.10  theory Independent_Family
   27.11    imports Probability_Measure Infinite_Product_Measure
   27.12 @@ -101,7 +101,7 @@
   27.13    fix F J assume "J \<noteq> {}" "J \<subseteq> UNIV"
   27.14      and F: "\<forall>j\<in>J. F j \<in> (case j of True \<Rightarrow> A | False \<Rightarrow> B)"
   27.15    have "J \<in> Pow UNIV" by auto
   27.16 -  with F `J \<noteq> {}` indep[of "F True" "F False"]
   27.17 +  with F \<open>J \<noteq> {}\<close> indep[of "F True" "F False"]
   27.18    show "prob (\<Inter>j\<in>J. F j) = (\<Prod>j\<in>J. prob (F j))"
   27.19      unfolding UNIV_bool Pow_insert by (auto simp: ac_simps)
   27.20  qed (auto split: bool.split simp: ev)
   27.21 @@ -155,19 +155,19 @@
   27.22              next
   27.23                assume "J \<noteq> {j}"
   27.24                have "prob (\<Inter>i\<in>J. A i) = prob ((\<Inter>i\<in>J-{j}. A i) \<inter> X)"
   27.25 -                using `j \<in> J` `A j = X` by (auto intro!: arg_cong[where f=prob] split: split_if_asm)
   27.26 +                using \<open>j \<in> J\<close> \<open>A j = X\<close> by (auto intro!: arg_cong[where f=prob] split: split_if_asm)
   27.27                also have "\<dots> = prob X * (\<Prod>i\<in>J-{j}. prob (A i))"
   27.28                proof (rule indep)
   27.29                  show "J - {j} \<noteq> {}" "J - {j} \<subseteq> K" "finite (J - {j})" "j \<notin> J - {j}"
   27.30 -                  using J `J \<noteq> {j}` `j \<in> J` by auto
   27.31 +                  using J \<open>J \<noteq> {j}\<close> \<open>j \<in> J\<close> by auto
   27.32                  show "\<forall>i\<in>J - {j}. A i \<in> G i"
   27.33                    using J by auto
   27.34                qed
   27.35                also have "\<dots> = prob (A j) * (\<Prod>i\<in>J-{j}. prob (A i))"
   27.36 -                using `A j = X` by simp
   27.37 +                using \<open>A j = X\<close> by simp
   27.38                also have "\<dots> = (\<Prod>i\<in>J. prob (A i))"
   27.39 -                unfolding setprod.insert_remove[OF `finite J`, symmetric, of "\<lambda>i. prob  (A i)"]
   27.40 -                using `j \<in> J` by (simp add: insert_absorb)
   27.41 +                unfolding setprod.insert_remove[OF \<open>finite J\<close>, symmetric, of "\<lambda>i. prob  (A i)"]
   27.42 +                using \<open>j \<in> J\<close> by (simp add: insert_absorb)
   27.43                finally show ?thesis .
   27.44              qed
   27.45            next
   27.46 @@ -191,23 +191,23 @@
   27.47              using G by auto
   27.48            have "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) =
   27.49                prob ((\<Inter>j\<in>J. A j) - (\<Inter>i\<in>insert j J. (A(j := X)) i))"
   27.50 -            using A_sets sets.sets_into_space[of _ M] X `J \<noteq> {}`
   27.51 +            using A_sets sets.sets_into_space[of _ M] X \<open>J \<noteq> {}\<close>
   27.52              by (auto intro!: arg_cong[where f=prob] split: split_if_asm)
   27.53            also have "\<dots> = prob (\<Inter>j\<in>J. A j) - prob (\<Inter>i\<in>insert j J. (A(j := X)) i)"
   27.54 -            using J `J \<noteq> {}` `j \<notin> J` A_sets X sets.sets_into_space
   27.55 +            using J \<open>J \<noteq> {}\<close> \<open>j \<notin> J\<close> A_sets X sets.sets_into_space
   27.56              by (auto intro!: finite_measure_Diff sets.finite_INT split: split_if_asm)
   27.57            finally have "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) =
   27.58                prob (\<Inter>j\<in>J. A j) - prob (\<Inter>i\<in>insert j J. (A(j := X)) i)" .
   27.59            moreover {
   27.60              have "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))"
   27.61 -              using J A `finite J` by (intro indep_setsD[OF G(1)]) auto
   27.62 +              using J A \<open>finite J\<close> by (intro indep_setsD[OF G(1)]) auto
   27.63              then have "prob (\<Inter>j\<in>J. A j) = prob (space M) * (\<Prod>i\<in>J. prob (A i))"
   27.64                using prob_space by simp }
   27.65            moreover {
   27.66              have "prob (\<Inter>i\<in>insert j J. (A(j := X)) i) = (\<Prod>i\<in>insert j J. prob ((A(j := X)) i))"
   27.67 -              using J A `j \<in> K` by (intro indep_setsD[OF G']) auto
   27.68 +              using J A \<open>j \<in> K\<close> by (intro indep_setsD[OF G']) auto
   27.69              then have "prob (\<Inter>i\<in>insert j J. (A(j := X)) i) = prob X * (\<Prod>i\<in>J. prob (A i))"
   27.70 -              using `finite J` `j \<notin> J` by (auto intro!: setprod.cong) }
   27.71 +              using \<open>finite J\<close> \<open>j \<notin> J\<close> by (auto intro!: setprod.cong) }
   27.72            ultimately have "prob ((\<Inter>j\<in>J. A j) \<inter> (space M - X)) = (prob (space M) - prob X) * (\<Prod>i\<in>J. prob (A i))"
   27.73              by (simp add: field_simps)
   27.74            also have "\<dots> = prob (space M - X) * (\<Prod>i\<in>J. prob (A i))"
   27.75 @@ -223,19 +223,19 @@
   27.76            then have A_sets: "\<And>i. i\<in>J \<Longrightarrow> A i \<in> events"
   27.77              using G by auto
   27.78            have "prob ((\<Inter>j\<in>J. A j) \<inter> (\<Union>k. F k)) = prob (\<Union>k. (\<Inter>i\<in>insert j J. (A(j := F k)) i))"
   27.79 -            using `J \<noteq> {}` `j \<notin> J` `j \<in> K` by (auto intro!: arg_cong[where f=prob] split: split_if_asm)
   27.80 +            using \<open>J \<noteq> {}\<close> \<open>j \<notin> J\<close> \<open>j \<in> K\<close> by (auto intro!: arg_cong[where f=prob] split: split_if_asm)
   27.81            moreover have "(\<lambda>k. prob (\<Inter>i\<in>insert j J. (A(j := F k)) i)) sums prob (\<Union>k. (\<Inter>i\<in>insert j J. (A(j := F k)) i))"
   27.82            proof (rule finite_measure_UNION)
   27.83              show "disjoint_family (\<lambda>k. \<Inter>i\<in>insert j J. (A(j := F k)) i)"
   27.84                using disj by (rule disjoint_family_on_bisimulation) auto
   27.85              show "range (\<lambda>k. \<Inter>i\<in>insert j J. (A(j := F k)) i) \<subseteq> events"
   27.86 -              using A_sets F `finite J` `J \<noteq> {}` `j \<notin> J` by (auto intro!: sets.Int)
   27.87 +              using A_sets F \<open>finite J\<close> \<open>J \<noteq> {}\<close> \<open>j \<notin> J\<close> by (auto intro!: sets.Int)
   27.88            qed
   27.89            moreover { fix k
   27.90 -            from J A `j \<in> K` have "prob (\<Inter>i\<in>insert j J. (A(j := F k)) i) = prob (F k) * (\<Prod>i\<in>J. prob (A i))"
   27.91 +            from J A \<open>j \<in> K\<close> have "prob (\<Inter>i\<in>insert j J. (A(j := F k)) i) = prob (F k) * (\<Prod>i\<in>J. prob (A i))"
   27.92                by (subst indep_setsD[OF F(2)]) (auto intro!: setprod.cong split: split_if_asm)
   27.93              also have "\<dots> = prob (F k) * prob (\<Inter>i\<in>J. A i)"
   27.94 -              using J A `j \<in> K` by (subst indep_setsD[OF G(1)]) auto
   27.95 +              using J A \<open>j \<in> K\<close> by (subst indep_setsD[OF G(1)]) auto
   27.96              finally have "prob (\<Inter>i\<in>insert j J. (A(j := F k)) i) = prob (F k) * prob (\<Inter>i\<in>J. A i)" . }
   27.97            ultimately have "(\<lambda>k. prob (F k) * prob (\<Inter>i\<in>J. A i)) sums (prob ((\<Inter>j\<in>J. A j) \<inter> (\<Union>k. F k)))"
   27.98              by simp
   27.99 @@ -243,7 +243,7 @@
  27.100            have "(\<lambda>k. prob (F k) * prob (\<Inter>i\<in>J. A i)) sums (prob (\<Union>k. F k) * prob (\<Inter>i\<in>J. A i))"
  27.101              using disj F(1) by (intro finite_measure_UNION sums_mult2) auto
  27.102            then have "(\<lambda>k. prob (F k) * prob (\<Inter>i\<in>J. A i)) sums (prob (\<Union>k. F k) * (\<Prod>i\<in>J. prob (A i)))"
  27.103 -            using J A `j \<in> K` by (subst indep_setsD[OF G(1), symmetric]) auto
  27.104 +            using J A \<open>j \<in> K\<close> by (subst indep_setsD[OF G(1), symmetric]) auto
  27.105            ultimately
  27.106            show "prob ((\<Inter>j\<in>J. A j) \<inter> (\<Union>k. F k)) = prob (\<Union>k. F k) * (\<Prod>j\<in>J. prob (A j))"
  27.107              by (auto dest!: sums_unique)
  27.108 @@ -252,10 +252,10 @@
  27.109        then have mono: "dynkin (space M) (G j) \<subseteq> {E \<in> events. indep_sets (G(j := {E})) K}"
  27.110        proof (rule dynkin_system.dynkin_subset, safe)
  27.111          fix X assume "X \<in> G j"
  27.112 -        then show "X \<in> events" using G `j \<in> K` by auto
  27.113 -        from `indep_sets G K`
  27.114 +        then show "X \<in> events" using G \<open>j \<in> K\<close> by auto
  27.115 +        from \<open>indep_sets G K\<close>
  27.116          show "indep_sets (G(j := {X})) K"
  27.117 -          by (rule indep_sets_mono_sets) (insert `X \<in> G j`, auto)
  27.118 +          by (rule indep_sets_mono_sets) (insert \<open>X \<in> G j\<close>, auto)
  27.119        qed
  27.120        have "indep_sets (G(j:=?D)) K"
  27.121        proof (rule indep_setsI)
  27.122 @@ -279,9 +279,9 @@
  27.123        then have "indep_sets (G(j := dynkin (space M) (G j))) K"
  27.124          by (rule indep_sets_mono_sets) (insert mono, auto)
  27.125        then show ?case
  27.126 -        by (rule indep_sets_mono_sets) (insert `j \<in> K` `j \<notin> J`, auto simp: G_def)
  27.127 -    qed (insert `indep_sets F K`, simp) }
  27.128 -  from this[OF `indep_sets F J` `finite J` subset_refl]
  27.129 +        by (rule indep_sets_mono_sets) (insert \<open>j \<in> K\<close> \<open>j \<notin> J\<close>, auto simp: G_def)
  27.130 +    qed (insert \<open>indep_sets F K\<close>, simp) }
  27.131 +  from this[OF \<open>indep_sets F J\<close> \<open>finite J\<close> subset_refl]
  27.132    show "indep_sets ?F J"
  27.133      by (rule indep_sets_mono_sets) auto
  27.134  qed
  27.135 @@ -375,7 +375,7 @@
  27.136    have "indep_sets (\<lambda>i. sigma_sets (space M) (case i of True \<Rightarrow> A | False \<Rightarrow> B)) UNIV"
  27.137    proof (rule indep_sets_sigma)
  27.138      show "indep_sets (case_bool A B) UNIV"
  27.139 -      by (rule `indep_set A B`[unfolded indep_set_def])
  27.140 +      by (rule \<open>indep_set A B\<close>[unfolded indep_set_def])
  27.141      fix i show "Int_stable (case i of True \<Rightarrow> A | False \<Rightarrow> B)"
  27.142        using A B by (cases i) auto
  27.143    qed
  27.144 @@ -398,7 +398,7 @@
  27.145      then have "{{x \<in> space M. P i (X i x)}} = {X i -` {x\<in>space (N i). P i x} \<inter> space M}"
  27.146        using indep by (auto simp: indep_vars_def dest: measurable_space)
  27.147      also have "\<dots> \<subseteq> {X i -` A \<inter> space M |A. A \<in> sets (N i)}"
  27.148 -      using P[OF `i \<in> I`] by blast
  27.149 +      using P[OF \<open>i \<in> I\<close>] by blast
  27.150      finally show "{{x \<in> space M. P i (X i x)}} \<subseteq> {X i -` A \<inter> space M |A. A \<in> sets (N i)}" .
  27.151    qed
  27.152  qed                              
  27.153 @@ -457,10 +457,10 @@
  27.154          have "k = j"
  27.155          proof (rule ccontr)
  27.156            assume "k \<noteq> j"
  27.157 -          with disjoint `K \<subseteq> J` `k \<in> K` `j \<in> K` have "I k \<inter> I j = {}"
  27.158 +          with disjoint \<open>K \<subseteq> J\<close> \<open>k \<in> K\<close> \<open>j \<in> K\<close> have "I k \<inter> I j = {}"
  27.159              unfolding disjoint_family_on_def by auto
  27.160 -          with L(2,3)[OF `j \<in> K`] L(2,3)[OF `k \<in> K`]
  27.161 -          show False using `l \<in> L k` `l \<in> L j` by auto
  27.162 +          with L(2,3)[OF \<open>j \<in> K\<close>] L(2,3)[OF \<open>k \<in> K\<close>]
  27.163 +          show False using \<open>l \<in> L k\<close> \<open>l \<in> L j\<close> by auto
  27.164          qed }
  27.165        note L_inj = this
  27.166  
  27.167 @@ -494,7 +494,7 @@
  27.168        let ?A = "\<lambda>k. (if k \<in> Ka \<inter> Kb then Ea k \<inter> Eb k else if k \<in> Kb then Eb k else if k \<in> Ka then Ea k else {})"
  27.169        have "a \<inter> b = INTER (Ka \<union> Kb) ?A"
  27.170          by (simp add: a b set_eq_iff) auto
  27.171 -      with a b `j \<in> J` Int_stableD[OF Int_stable] show "a \<inter> b \<in> ?E j"
  27.172 +      with a b \<open>j \<in> J\<close> Int_stableD[OF Int_stable] show "a \<inter> b \<in> ?E j"
  27.173          by (intro CollectI exI[of _ "Ka \<union> Kb"] exI[of _ ?A]) auto
  27.174      qed
  27.175    qed
  27.176 @@ -536,10 +536,10 @@
  27.177          { interpret sigma_algebra "space M" "?UN j"
  27.178              by (rule sigma_algebra_sigma_sets) auto 
  27.179            have "\<And>A. (\<And>i. i \<in> J \<Longrightarrow> A i \<in> ?UN j) \<Longrightarrow> INTER J A \<in> ?UN j"
  27.180 -            using `finite J` `J \<noteq> {}` by (rule finite_INT) blast }
  27.181 +            using \<open>finite J\<close> \<open>J \<noteq> {}\<close> by (rule finite_INT) blast }
  27.182          note INT = this
  27.183  
  27.184 -        from `J \<noteq> {}` J K E[rule_format, THEN sets.sets_into_space] j
  27.185 +        from \<open>J \<noteq> {}\<close> J K E[rule_format, THEN sets.sets_into_space] j
  27.186          have "(\<lambda>\<omega>. \<lambda>i\<in>K j. X i \<omega>) -` prod_emb (K j) M' J (Pi\<^sub>E J E) \<inter> space M
  27.187            = (\<Inter>i\<in>J. X i -` E i \<inter> space M)"
  27.188            apply (subst prod_emb_PiE[OF _ ])
  27.189 @@ -552,7 +552,7 @@
  27.190          also have "\<dots> \<in> ?UN j"
  27.191            apply (rule INT)
  27.192            apply (rule sigma_sets.Basic)
  27.193 -          using `J \<subseteq> K j` E
  27.194 +          using \<open>J \<subseteq> K j\<close> E
  27.195            apply auto
  27.196            done
  27.197          finally show ?thesis .
  27.198 @@ -630,7 +630,7 @@
  27.199      from this[of 0] have "X \<in> sigma_sets (space M) (UNION UNIV A)" by simp
  27.200      then have "X \<subseteq> space M"
  27.201        by induct (insert A.sets_into_space, auto)
  27.202 -    with `x \<in> X` show "x \<in> space M" by auto }
  27.203 +    with \<open>x \<in> X\<close> show "x \<in> space M" by auto }
  27.204    { fix F :: "nat \<Rightarrow> 'a set" and n assume "range F \<subseteq> ?A"
  27.205      then show "(UNION UNIV F) \<in> sigma_sets (space M) (UNION {n..} A)"
  27.206        by (intro sigma_sets.Union) auto }
  27.207 @@ -661,11 +661,11 @@
  27.208        using sets.sets_into_space by auto
  27.209    next
  27.210      show "space M \<in> ?D"
  27.211 -      using prob_space `X \<subseteq> space M` by (simp add: Int_absorb2)
  27.212 +      using prob_space \<open>X \<subseteq> space M\<close> by (simp add: Int_absorb2)
  27.213    next
  27.214      fix A assume A: "A \<in> ?D"
  27.215      have "prob (X \<inter> (space M - A)) = prob (X - (X \<inter> A))"
  27.216 -      using `X \<subseteq> space M` by (auto intro!: arg_cong[where f=prob])
  27.217 +      using \<open>X \<subseteq> space M\<close> by (auto intro!: arg_cong[where f=prob])
  27.218      also have "\<dots> = prob X - prob (X \<inter> A)"
  27.219        using X_in A by (intro finite_measure_Diff) auto
  27.220      also have "\<dots> = prob X * prob (space M) - prob X * prob A"
  27.221 @@ -674,7 +674,7 @@
  27.222        using X_in A sets.sets_into_space
  27.223        by (subst finite_measure_Diff) (auto simp: field_simps)
  27.224      finally show "space M - A \<in> ?D"
  27.225 -      using A `X \<subseteq> space M` by auto
  27.226 +      using A \<open>X \<subseteq> space M\<close> by auto
  27.227    next
  27.228      fix F :: "nat \<Rightarrow> 'a set" assume dis: "disjoint_family F" and "range F \<subseteq> ?D"
  27.229      then have F: "range F \<subseteq> events" "\<And>i. prob (X \<inter> F i) = prob X * prob (F i)"
  27.230 @@ -726,7 +726,7 @@
  27.231    then have "(\<Union>n. sigma_sets (space M) (\<Union>m\<in>{..n}. A m)) \<subseteq> ?D" (is "?A \<subseteq> _")
  27.232      by auto
  27.233  
  27.234 -  note `X \<in> tail_events A`
  27.235 +  note \<open>X \<in> tail_events A\<close>
  27.236    also {
  27.237      have "\<And>n. sigma_sets (space M) (\<Union>i\<in>{n..}. A i) \<subseteq> sigma_sets (space M) ?A"
  27.238        by (intro sigma_sets_subseteq UN_mono) auto
  27.239 @@ -757,7 +757,7 @@
  27.240      qed
  27.241    qed
  27.242    also have "dynkin (space M) ?A \<subseteq> ?D"
  27.243 -    using `?A \<subseteq> ?D` by (auto intro!: D.dynkin_subset)
  27.244 +    using \<open>?A \<subseteq> ?D\<close> by (auto intro!: D.dynkin_subset)
  27.245    finally show ?thesis by auto
  27.246  qed
  27.247  
  27.248 @@ -838,8 +838,8 @@
  27.249      with indep have "prob (\<Inter>j\<in>I. ?A j) = (\<Prod>j\<in>I. prob (?A j))"
  27.250        by auto
  27.251      also have "\<dots> = (\<Prod>j\<in>J. prob (A j))"
  27.252 -      unfolding if_distrib setprod.If_cases[OF `finite I`]
  27.253 -      using prob_space `J \<subseteq> I` by (simp add: Int_absorb1 setprod.neutral_const)
  27.254 +      unfolding if_distrib setprod.If_cases[OF \<open>finite I\<close>]
  27.255 +      using prob_space \<open>J \<subseteq> I\<close> by (simp add: Int_absorb1 setprod.neutral_const)
  27.256      finally show "prob (\<Inter>j\<in>J. A j) = (\<Prod>j\<in>J. prob (A j))" ..
  27.257    qed
  27.258  qed
  27.259 @@ -858,10 +858,10 @@
  27.260      unfolding measurable_def by simp
  27.261  
  27.262    { fix i assume "i\<in>I"
  27.263 -    from closed[OF `i \<in> I`]
  27.264 +    from closed[OF \<open>i \<in> I\<close>]
  27.265      have "sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}
  27.266        = sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> E i}"
  27.267 -      unfolding sigma_sets_vimage_commute[OF X, OF `i \<in> I`, symmetric] M'[OF `i \<in> I`]
  27.268 +      unfolding sigma_sets_vimage_commute[OF X, OF \<open>i \<in> I\<close>, symmetric] M'[OF \<open>i \<in> I\<close>]
  27.269        by (subst sigma_sets_sigma_sets_eq) auto }
  27.270    note sigma_sets_X = this
  27.271  
  27.272 @@ -875,7 +875,7 @@
  27.273        then obtain B where "b = X i -` B \<inter> space M" "B \<in> E i" by auto
  27.274        moreover
  27.275        have "(X i -` A \<inter> space M) \<inter> (X i -` B \<inter> space M) = X i -` (A \<inter> B) \<inter> space M" by auto
  27.276 -      moreover note Int_stable[OF `i \<in> I`]
  27.277 +      moreover note Int_stable[OF \<open>i \<in> I\<close>]
  27.278        ultimately
  27.279        show "a \<inter> b \<in> {X i -` A \<inter> space M |A. A \<in> E i}"
  27.280          by (auto simp del: vimage_Int intro!: exI[of _ "A \<inter> B"] dest: Int_stableD)
  27.281 @@ -884,12 +884,12 @@
  27.282  
  27.283    { fix i assume "i \<in> I"
  27.284      { fix A assume "A \<in> E i"
  27.285 -      with M'[OF `i \<in> I`] have "A \<in> sets (M' i)" by auto
  27.286 +      with M'[OF \<open>i \<in> I\<close>] have "A \<in> sets (M' i)" by auto
  27.287        moreover
  27.288 -      from rv[OF `i\<in>I`] have "X i \<in> measurable M (M' i)" by auto
  27.289 +      from rv[OF \<open>i\<in>I\<close>] have "X i \<in> measurable M (M' i)" by auto
  27.290        ultimately
  27.291        have "X i -` A \<inter> space M \<in> sets M" by (auto intro: measurable_sets) }
  27.292 -    with X[OF `i\<in>I`] space[OF `i\<in>I`]
  27.293 +    with X[OF \<open>i\<in>I\<close>] space[OF \<open>i\<in>I\<close>]
  27.294      have "{X i -` A \<inter> space M |A. A \<in> E i} \<subseteq> events"
  27.295        "space M \<in> {X i -` A \<inter> space M |A. A \<in> E i}"
  27.296        by (auto intro!: exI[of _ "space (M' i)"]) }
  27.297 @@ -900,7 +900,7 @@
  27.298      (is "?L = ?R")
  27.299    proof safe
  27.300      fix A assume ?L and A: "A \<in> (\<Pi> i\<in>I. E i)"
  27.301 -    from `?L`[THEN bspec, of "\<lambda>i. X i -` A i \<inter> space M"] A `I \<noteq> {}`
  27.302 +    from \<open>?L\<close>[THEN bspec, of "\<lambda>i. X i -` A i \<inter> space M"] A \<open>I \<noteq> {}\<close>
  27.303      show "prob ((\<Inter>j\<in>I. X j -` A j) \<inter> space M) = (\<Prod>x\<in>I. prob (X x -` A x \<inter> space M))"
  27.304        by (auto simp add: Pi_iff)
  27.305    next
  27.306 @@ -908,11 +908,11 @@
  27.307      from A have "\<forall>i\<in>I. \<exists>B. A i = X i -` B \<inter> space M \<and> B \<in> E i" by auto
  27.308      from bchoice[OF this] obtain B where B: "\<forall>i\<in>I. A i = X i -` B i \<inter> space M"
  27.309        "B \<in> (\<Pi> i\<in>I. E i)" by auto
  27.310 -    from `?R`[THEN bspec, OF B(2)] B(1) `I \<noteq> {}`
  27.311 +    from \<open>?R\<close>[THEN bspec, OF B(2)] B(1) \<open>I \<noteq> {}\<close>
  27.312      show "prob (INTER I A) = (\<Prod>j\<in>I. prob (A j))"
  27.313        by simp
  27.314    qed
  27.315 -  then show ?thesis using `I \<noteq> {}`
  27.316 +  then show ?thesis using \<open>I \<noteq> {}\<close>
  27.317      by (simp add: rv indep_vars_def indep_sets_X sigma_sets_X indep_sets_finite_X cong: indep_sets_cong)
  27.318  qed
  27.319  
  27.320 @@ -922,21 +922,21 @@
  27.321    shows "indep_vars N (\<lambda>i. Y i \<circ> X i) I"
  27.322    unfolding indep_vars_def
  27.323  proof
  27.324 -  from rv `indep_vars M' X I`
  27.325 +  from rv \<open>indep_vars M' X I\<close>
  27.326    show "\<forall>i\<in>I. random_variable (N i) (Y i \<circ> X i)"
  27.327      by (auto simp: indep_vars_def)
  27.328  
  27.329    have "indep_sets (\<lambda>i. sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I"
  27.330 -    using `indep_vars M' X I` by (simp add: indep_vars_def)
  27.331 +    using \<open>indep_vars M' X I\<close> by (simp add: indep_vars_def)
  27.332    then show "indep_sets (\<lambda>i. sigma_sets (space M) {(Y i \<circ> X i) -` A \<inter> space M |A. A \<in> sets (N i)}) I"
  27.333    proof (rule indep_sets_mono_sets)
  27.334      fix i assume "i \<in> I"
  27.335 -    with `indep_vars M' X I` have X: "X i \<in> space M \<rightarrow> space (M' i)"
  27.336 +    with \<open>indep_vars M' X I\<close> have X: "X i \<in> space M \<rightarrow> space (M' i)"
  27.337        unfolding indep_vars_def measurable_def by auto
  27.338      { fix A assume "A \<in> sets (N i)"
  27.339        then have "\<exists>B. (Y i \<circ> X i) -` A \<inter> space M = X i -` B \<inter> space M \<and> B \<in> sets (M' i)"
  27.340          by (intro exI[of _ "Y i -` A \<inter> space (M' i)"])
  27.341 -           (auto simp: vimage_comp intro!: measurable_sets rv `i \<in> I` funcset_mem[OF X]) }
  27.342 +           (auto simp: vimage_comp intro!: measurable_sets rv \<open>i \<in> I\<close> funcset_mem[OF X]) }
  27.343      then show "sigma_sets (space M) {(Y i \<circ> X i) -` A \<inter> space M |A. A \<in> sets (N i)} \<subseteq>
  27.344        sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}"
  27.345        by (intro sigma_sets_subseteq) (auto simp: vimage_comp)
  27.346 @@ -1078,9 +1078,9 @@
  27.347        then have "emeasure ?D E = emeasure M (?X -` E \<inter> space M)"
  27.348          by (simp add: emeasure_distr X)
  27.349        also have "?X -` E \<inter> space M = (\<Inter>i\<in>J. X i -` Y i \<inter> space M)"
  27.350 -        using J `I \<noteq> {}` measurable_space[OF rv] by (auto simp: prod_emb_def PiE_iff split: split_if_asm)
  27.351 +        using J \<open>I \<noteq> {}\<close> measurable_space[OF rv] by (auto simp: prod_emb_def PiE_iff split: split_if_asm)
  27.352        also have "emeasure M (\<Inter>i\<in>J. X i -` Y i \<inter> space M) = (\<Prod> i\<in>J. emeasure M (X i -` Y i \<inter> space M))"
  27.353 -        using `indep_vars M' X I` J `I \<noteq> {}` using indep_varsD[of M' X I J]
  27.354 +        using \<open>indep_vars M' X I\<close> J \<open>I \<noteq> {}\<close> using indep_varsD[of M' X I J]
  27.355          by (auto simp: emeasure_eq_measure setprod_ereal)
  27.356        also have "\<dots> = (\<Prod> i\<in>J. emeasure (?D' i) (Y i))"
  27.357          using rv J by (simp add: emeasure_distr)
  27.358 @@ -1109,13 +1109,13 @@
  27.359          Y: "\<And>j. j \<in> J \<Longrightarrow> Y' j = X j -` Y j \<inter> space M" "\<And>j. j \<in> J \<Longrightarrow> Y j \<in> sets (M' j)" by auto
  27.360        let ?E = "prod_emb I M' J (Pi\<^sub>E J Y)"
  27.361        from Y have "(\<Inter>j\<in>J. Y' j) = ?X -` ?E \<inter> space M"
  27.362 -        using J `I \<noteq> {}` measurable_space[OF rv] by (auto simp: prod_emb_def PiE_iff split: split_if_asm)
  27.363 +        using J \<open>I \<noteq> {}\<close> measurable_space[OF rv] by (auto simp: prod_emb_def PiE_iff split: split_if_asm)
  27.364        then have "emeasure M (\<Inter>j\<in>J. Y' j) = emeasure M (?X -` ?E \<inter> space M)"
  27.365          by simp
  27.366        also have "\<dots> = emeasure ?D ?E"
  27.367          using Y  J by (intro emeasure_distr[symmetric] X sets_PiM_I) auto
  27.368        also have "\<dots> = emeasure ?P' ?E"
  27.369 -        using `?D = ?P'` by simp
  27.370 +        using \<open>?D = ?P'\<close> by simp
  27.371        also have "\<dots> = (\<Prod> i\<in>J. emeasure (?D' i) (Y i))"
  27.372          using P.emeasure_PiM_emb[of J Y] J Y by (simp add: prod_emb_def)
  27.373        also have "\<dots> = (\<Prod> i\<in>J. emeasure M (Y' i))"
  27.374 @@ -1191,7 +1191,7 @@
  27.375      have "emeasure ?J (A \<times> B) = emeasure M ((\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M)"
  27.376        using A B by (intro emeasure_distr[OF XY]) auto
  27.377      also have "\<dots> = emeasure M (X -` A \<inter> space M) * emeasure M (Y -` B \<inter> space M)"
  27.378 -      using indep_varD[OF `indep_var S X T Y`, of A B] A B by (simp add: emeasure_eq_measure)
  27.379 +      using indep_varD[OF \<open>indep_var S X T Y\<close>, of A B] A B by (simp add: emeasure_eq_measure)
  27.380      also have "\<dots> = emeasure ?S A * emeasure ?T B"
  27.381        using rvs A B by (simp add: emeasure_distr)
  27.382      finally show "emeasure ?S A * emeasure ?T B = emeasure ?J (A \<times> B)" by simp
  27.383 @@ -1222,15 +1222,15 @@
  27.384      show "indep_set {X -` A \<inter> space M |A. A \<in> sets S} {Y -` A \<inter> space M |A. A \<in> sets T}"
  27.385      proof (safe intro!: indep_setI)
  27.386        { fix A assume "A \<in> sets S" then show "X -` A \<inter> space M \<in> sets M"
  27.387 -        using `X \<in> measurable M S` by (auto intro: measurable_sets) }
  27.388 +        using \<open>X \<in> measurable M S\<close> by (auto intro: measurable_sets) }
  27.389        { fix A assume "A \<in> sets T" then show "Y -` A \<inter> space M \<in> sets M"
  27.390 -        using `Y \<in> measurable M T` by (auto intro: measurable_sets) }
  27.391 +        using \<open>Y \<in> measurable M T\<close> by (auto intro: measurable_sets) }
  27.392      next
  27.393        fix A B assume ab: "A \<in> sets S" "B \<in> sets T"
  27.394        then have "ereal (prob ((X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M))) = emeasure ?J (A \<times> B)"
  27.395          using XY by (auto simp add: emeasure_distr emeasure_eq_measure intro!: arg_cong[where f="prob"])
  27.396        also have "\<dots> = emeasure (?S \<Otimes>\<^sub>M ?T) (A \<times> B)"
  27.397 -        unfolding `?S \<Otimes>\<^sub>M ?T = ?J` ..
  27.398 +        unfolding \<open>?S \<Otimes>\<^sub>M ?T = ?J\<close> ..
  27.399        also have "\<dots> = emeasure ?S A * emeasure ?T B"
  27.400          using ab by (simp add: Y.emeasure_pair_measure_Times)
  27.401        finally show "prob ((X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M)) =
  27.402 @@ -1275,9 +1275,9 @@
  27.403    also have "\<dots> = (\<integral>\<^sup>+\<omega>. (\<Prod>i\<in>I. max 0 (\<omega> i)) \<partial>distr M (Pi\<^sub>M I (\<lambda>i. borel)) (\<lambda>x. \<lambda>i\<in>I. Y i x))"
  27.404      by (subst nn_integral_distr) auto
  27.405    also have "\<dots> = (\<integral>\<^sup>+\<omega>. (\<Prod>i\<in>I. max 0 (\<omega> i)) \<partial>Pi\<^sub>M I (\<lambda>i. distr M borel (Y i)))"
  27.406 -    unfolding indep_vars_iff_distr_eq_PiM[THEN iffD1, OF `I \<noteq> {}` rv_Y indep_Y] ..
  27.407 +    unfolding indep_vars_iff_distr_eq_PiM[THEN iffD1, OF \<open>I \<noteq> {}\<close> rv_Y indep_Y] ..
  27.408    also have "\<dots> = (\<Prod>i\<in>I. (\<integral>\<^sup>+\<omega>. max 0 \<omega> \<partial>distr M borel (Y i)))"
  27.409 -    by (rule product_nn_integral_setprod) (auto intro: `finite I`)
  27.410 +    by (rule product_nn_integral_setprod) (auto intro: \<open>finite I\<close>)
  27.411    also have "\<dots> = (\<Prod>i\<in>I. \<integral>\<^sup>+\<omega>. X i \<omega> \<partial>M)"
  27.412      by (intro setprod.cong nn_integral_cong)
  27.413         (auto simp: nn_integral_distr nn_integral_max_0 Y_def rv_X)
  27.414 @@ -1317,17 +1317,17 @@
  27.415    also have "\<dots> = (\<integral>\<omega>. (\<Prod>i\<in>I. \<omega> i) \<partial>distr M (Pi\<^sub>M I (\<lambda>i. borel)) (\<lambda>x. \<lambda>i\<in>I. Y i x))"
  27.416      by (subst integral_distr) auto
  27.417    also have "\<dots> = (\<integral>\<omega>. (\<Prod>i\<in>I. \<omega> i) \<partial>Pi\<^sub>M I (\<lambda>i. distr M borel (Y i)))"
  27.418 -    unfolding indep_vars_iff_distr_eq_PiM[THEN iffD1, OF `I \<noteq> {}` rv_Y indep_Y] ..
  27.419 +    unfolding indep_vars_iff_distr_eq_PiM[THEN iffD1, OF \<open>I \<noteq> {}\<close> rv_Y indep_Y] ..
  27.420    also have "\<dots> = (\<Prod>i\<in>I. (\<integral>\<omega>. \<omega> \<partial>distr M borel (Y i)))"
  27.421 -    by (rule product_integral_setprod) (auto intro: `finite I` simp: integrable_distr_eq int_Y)
  27.422 +    by (rule product_integral_setprod) (auto intro: \<open>finite I\<close> simp: integrable_distr_eq int_Y)
  27.423    also have "\<dots> = (\<Prod>i\<in>I. \<integral>\<omega>. X i \<omega> \<partial>M)"
  27.424      by (intro setprod.cong integral_cong)
  27.425         (auto simp: integral_distr Y_def rv_X)
  27.426    finally show ?eq .
  27.427  
  27.428    have "integrable (distr M (Pi\<^sub>M I (\<lambda>i. borel)) (\<lambda>x. \<lambda>i\<in>I. Y i x)) (\<lambda>\<omega>. (\<Prod>i\<in>I. \<omega> i))"
  27.429 -    unfolding indep_vars_iff_distr_eq_PiM[THEN iffD1, OF `I \<noteq> {}` rv_Y indep_Y]
  27.430 -    by (intro product_integrable_setprod[OF `finite I`])
  27.431 +    unfolding indep_vars_iff_distr_eq_PiM[THEN iffD1, OF \<open>I \<noteq> {}\<close> rv_Y indep_Y]
  27.432 +    by (intro product_integrable_setprod[OF \<open>finite I\<close>])
  27.433         (simp add: integrable_distr_eq int_Y)
  27.434    then show ?int
  27.435      by (simp add: integrable_distr_eq Y_def)
    28.1 --- a/src/HOL/Probability/Infinite_Product_Measure.thy	Mon Dec 07 16:48:10 2015 +0000
    28.2 +++ b/src/HOL/Probability/Infinite_Product_Measure.thy	Mon Dec 07 20:19:59 2015 +0100
    28.3 @@ -2,7 +2,7 @@
    28.4      Author:     Johannes Hölzl, TU München
    28.5  *)
    28.6  
    28.7 -section {*Infinite Product Measure*}
    28.8 +section \<open>Infinite Product Measure\<close>
    28.9  
   28.10  theory Infinite_Product_Measure
   28.11    imports Probability_Measure Caratheodory Projective_Family
   28.12 @@ -98,7 +98,7 @@
   28.13    moreover have "((\<lambda>\<omega>. \<omega> i) -` A \<inter> space (PiM I M)) = {x\<in>space (PiM I M). x i \<in> A}"
   28.14      by auto
   28.15    ultimately show "emeasure (M i) A = emeasure (distr (PiM I M) (M i) (\<lambda>\<omega>. \<omega> i)) A"
   28.16 -    by (auto simp: `i\<in>I` emeasure_distr measurable_component_singleton emeasure_PiM_Collect_single)
   28.17 +    by (auto simp: \<open>i\<in>I\<close> emeasure_distr measurable_component_singleton emeasure_PiM_Collect_single)
   28.18  qed simp
   28.19  
   28.20  lemma (in product_prob_space) PiM_eq:
   28.21 @@ -118,7 +118,7 @@
   28.22    apply simp_all
   28.23    done
   28.24  
   28.25 -subsection {* Sequence space *}
   28.26 +subsection \<open>Sequence space\<close>
   28.27  
   28.28  definition comb_seq :: "nat \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> (nat \<Rightarrow> 'a)" where
   28.29    "comb_seq i \<omega> \<omega>' j = (if j < i then \<omega> j else \<omega>' (j - i))"
    29.1 --- a/src/HOL/Probability/Information.thy	Mon Dec 07 16:48:10 2015 +0000
    29.2 +++ b/src/HOL/Probability/Information.thy	Mon Dec 07 20:19:59 2015 +0100
    29.3 @@ -3,7 +3,7 @@
    29.4      Author:     Armin Heller, TU München
    29.5  *)
    29.6  
    29.7 -section {*Information theory*}
    29.8 +section \<open>Information theory\<close>
    29.9  
   29.10  theory Information
   29.11  imports
   29.12 @@ -33,7 +33,7 @@
   29.13  context information_space
   29.14  begin
   29.15  
   29.16 -text {* Introduce some simplification rules for logarithm of base @{term b}. *}
   29.17 +text \<open>Introduce some simplification rules for logarithm of base @{term b}.\<close>
   29.18  
   29.19  lemma log_neg_const:
   29.20    assumes "x \<le> 0"
   29.21 @@ -69,8 +69,8 @@
   29.22  
   29.23  subsection "Kullback$-$Leibler divergence"
   29.24  
   29.25 -text {* The Kullback$-$Leibler divergence is also known as relative entropy or
   29.26 -Kullback$-$Leibler distance. *}
   29.27 +text \<open>The Kullback$-$Leibler divergence is also known as relative entropy or
   29.28 +Kullback$-$Leibler distance.\<close>
   29.29  
   29.30  definition
   29.31    "entropy_density b M N = log b \<circ> real_of_ereal \<circ> RN_deriv M N"
   29.32 @@ -118,9 +118,9 @@
   29.33      KL_divergence b (density M f) (density (density M f) (\<lambda>x. g x / f x))"
   29.34      using f g ac by (subst density_density_divide) simp_all
   29.35    also have "\<dots> = (\<integral>x. (g x / f x) * log b (g x / f x) \<partial>density M f)"
   29.36 -    using f g `1 < b` by (intro Mf.KL_density) (auto simp: AE_density)
   29.37 +    using f g \<open>1 < b\<close> by (intro Mf.KL_density) (auto simp: AE_density)
   29.38    also have "\<dots> = (\<integral>x. g x * log b (g x / f x) \<partial>M)"
   29.39 -    using ac f g `1 < b` by (subst integral_density) (auto intro!: integral_cong_AE)
   29.40 +    using ac f g \<open>1 < b\<close> by (subst integral_density) (auto intro!: integral_cong_AE)
   29.41    finally show ?thesis .
   29.42  qed
   29.43  
   29.44 @@ -135,7 +135,7 @@
   29.45    interpret N: prob_space "density M D" by fact
   29.46  
   29.47    obtain A where "A \<in> sets M" "emeasure (density M D) A \<noteq> emeasure M A"
   29.48 -    using measure_eqI[of "density M D" M] `density M D \<noteq> M` by auto
   29.49 +    using measure_eqI[of "density M D" M] \<open>density M D \<noteq> M\<close> by auto
   29.50  
   29.51    let ?D_set = "{x\<in>space M. D x \<noteq> 0}"
   29.52    have [simp, intro]: "?D_set \<in> sets M"
   29.53 @@ -157,12 +157,12 @@
   29.54    have "0 \<le> 1 - measure M ?D_set"
   29.55      using prob_le_1 by (auto simp: field_simps)
   29.56    also have "\<dots> = (\<integral> x. D x - indicator ?D_set x \<partial>M)"
   29.57 -    using `integrable M D` `integral\<^sup>L M D = 1`
   29.58 +    using \<open>integrable M D\<close> \<open>integral\<^sup>L M D = 1\<close>
   29.59      by (simp add: emeasure_eq_measure)
   29.60    also have "\<dots> < (\<integral> x. D x * (ln b * log b (D x)) \<partial>M)"
   29.61    proof (rule integral_less_AE)
   29.62      show "integrable M (\<lambda>x. D x - indicator ?D_set x)"
   29.63 -      using `integrable M D` by auto
   29.64 +      using \<open>integrable M D\<close> by auto
   29.65    next
   29.66      from integrable_mult_left(1)[OF int, of "ln b"]
   29.67      show "integrable M (\<lambda>x. D x * (ln b * log b (D x)))" 
   29.68 @@ -183,8 +183,8 @@
   29.69        then have "(\<integral>\<^sup>+x. indicator A x\<partial>M) = (\<integral>\<^sup>+x. ereal (D x) * indicator A x\<partial>M)"
   29.70          by (intro nn_integral_cong_AE) (auto simp: one_ereal_def[symmetric])
   29.71        also have "\<dots> = density M D A"
   29.72 -        using `A \<in> sets M` D by (simp add: emeasure_density)
   29.73 -      finally show False using `A \<in> sets M` `emeasure (density M D) A \<noteq> emeasure M A` by simp
   29.74 +        using \<open>A \<in> sets M\<close> D by (simp add: emeasure_density)
   29.75 +      finally show False using \<open>A \<in> sets M\<close> \<open>emeasure (density M D) A \<noteq> emeasure M A\<close> by simp
   29.76      qed
   29.77      show "{x\<in>space M. D x \<noteq> 1 \<and> D x \<noteq> 0} \<in> sets M"
   29.78        using D(1) by (auto intro: sets.sets_Collect_conj)
   29.79 @@ -200,11 +200,11 @@
   29.80          using Dt by simp
   29.81        also note eq
   29.82        also have "D t * (ln b * log b (D t)) = - D t * ln (1 / D t)"
   29.83 -        using b_gt_1 `D t \<noteq> 0` `0 \<le> D t`
   29.84 +        using b_gt_1 \<open>D t \<noteq> 0\<close> \<open>0 \<le> D t\<close>
   29.85          by (simp add: log_def ln_div less_le)
   29.86        finally have "ln (1 / D t) = 1 / D t - 1"
   29.87 -        using `D t \<noteq> 0` by (auto simp: field_simps)
   29.88 -      from ln_eq_minus_one[OF _ this] `D t \<noteq> 0` `0 \<le> D t` `D t \<noteq> 1`
   29.89 +        using \<open>D t \<noteq> 0\<close> by (auto simp: field_simps)
   29.90 +      from ln_eq_minus_one[OF _ this] \<open>D t \<noteq> 0\<close> \<open>0 \<le> D t\<close> \<open>D t \<noteq> 1\<close>
   29.91        show False by auto
   29.92      qed
   29.93  
   29.94 @@ -215,14 +215,14 @@
   29.95        show "D t - indicator ?D_set t \<le> D t * (ln b * log b (D t))"
   29.96        proof cases
   29.97          assume asm: "D t \<noteq> 0"
   29.98 -        then have "0 < D t" using `0 \<le> D t` by auto
   29.99 +        then have "0 < D t" using \<open>0 \<le> D t\<close> by auto
  29.100          then have "0 < 1 / D t" by auto
  29.101          have "D t - indicator ?D_set t \<le> - D t * (1 / D t - 1)"
  29.102 -          using asm `t \<in> space M` by (simp add: field_simps)
  29.103 +          using asm \<open>t \<in> space M\<close> by (simp add: field_simps)
  29.104          also have "- D t * (1 / D t - 1) \<le> - D t * ln (1 / D t)"
  29.105 -          using ln_le_minus_one `0 < 1 / D t` by (intro mult_left_mono_neg) auto
  29.106 +          using ln_le_minus_one \<open>0 < 1 / D t\<close> by (intro mult_left_mono_neg) auto
  29.107          also have "\<dots> = D t * (ln b * log b (D t))"
  29.108 -          using `0 < D t` b_gt_1
  29.109 +          using \<open>0 < D t\<close> b_gt_1
  29.110            by (simp_all add: log_def ln_div)
  29.111          finally show ?thesis by simp
  29.112        qed simp
  29.113 @@ -289,7 +289,7 @@
  29.114         (auto simp: N entropy_density_def)
  29.115    with D b_gt_1 have "integrable M (\<lambda>x. D x * log b (D x))"
  29.116      by (subst integrable_real_density[symmetric]) (auto simp: N[symmetric] comp_def)
  29.117 -  with `prob_space N` D show ?thesis
  29.118 +  with \<open>prob_space N\<close> D show ?thesis
  29.119      unfolding N
  29.120      by (intro KL_eq_0_iff_eq) auto
  29.121  qed
  29.122 @@ -323,7 +323,7 @@
  29.123      show "AE x in density M f. 0 \<le> g x / f x"
  29.124        using f g by (auto simp: AE_density)
  29.125      show "integrable (density M f) (\<lambda>x. g x / f x * log b (g x / f x))"
  29.126 -      using `1 < b` f g ac
  29.127 +      using \<open>1 < b\<close> f g ac
  29.128        by (subst integrable_density)
  29.129           (auto intro!: integrable_cong_AE[THEN iffD2, OF _ _ _ int] measurable_If)
  29.130    qed
  29.131 @@ -332,7 +332,7 @@
  29.132    finally show ?thesis .
  29.133  qed
  29.134  
  29.135 -subsection {* Finite Entropy *}
  29.136 +subsection \<open>Finite Entropy\<close>
  29.137  
  29.138  definition (in information_space) 
  29.139    "finite_entropy S X f \<longleftrightarrow> distributed M S X f \<and> integrable S (\<lambda>x. f x * log b (f x))"
  29.140 @@ -421,7 +421,7 @@
  29.141    using distributed_transform_integrable[of M T Y Py S X Px f "\<lambda>x. log b (Px x)"]
  29.142    by auto
  29.143  
  29.144 -subsection {* Mutual Information *}
  29.145 +subsection \<open>Mutual Information\<close>
  29.146  
  29.147  definition (in prob_space)
  29.148    "mutual_information b S T X Y =
  29.149 @@ -459,16 +459,16 @@
  29.150      have "AE x in P. 1 = RN_deriv P Q x"
  29.151      proof (rule P.RN_deriv_unique)
  29.152        show "density P (\<lambda>x. 1) = Q"
  29.153 -        unfolding `Q = P` by (intro measure_eqI) (auto simp: emeasure_density)
  29.154 +        unfolding \<open>Q = P\<close> by (intro measure_eqI) (auto simp: emeasure_density)
  29.155      qed auto
  29.156      then have ae_0: "AE x in P. entropy_density b P Q x = 0"
  29.157        by eventually_elim (auto simp: entropy_density_def)
  29.158      then have "integrable P (entropy_density b P Q) \<longleftrightarrow> integrable Q (\<lambda>x. 0::real)"
  29.159 -      using ed unfolding `Q = P` by (intro integrable_cong_AE) auto
  29.160 +      using ed unfolding \<open>Q = P\<close> by (intro integrable_cong_AE) auto
  29.161      then show "integrable Q (entropy_density b P Q)" by simp
  29.162  
  29.163      from ae_0 have "mutual_information b S T X Y = (\<integral>x. 0 \<partial>P)"
  29.164 -      unfolding mutual_information_def KL_divergence_def P_def[symmetric] Q_def[symmetric] `Q = P`
  29.165 +      unfolding mutual_information_def KL_divergence_def P_def[symmetric] Q_def[symmetric] \<open>Q = P\<close>
  29.166        by (intro integral_cong_AE) auto
  29.167      then show "mutual_information b S T X Y = 0"
  29.168        by simp }
  29.169 @@ -753,7 +753,7 @@
  29.170      Pxy (x, y) * log b (Pxy (x, y) / (Px x * Py y))) = 0" by simp
  29.171  qed
  29.172  
  29.173 -subsection {* Entropy *}
  29.174 +subsection \<open>Entropy\<close>
  29.175  
  29.176  definition (in prob_space) entropy :: "real \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> real" where
  29.177    "entropy b S X = - KL_divergence b S (distr M S X)"
  29.178 @@ -946,7 +946,7 @@
  29.179    finally show ?thesis .
  29.180  qed
  29.181  
  29.182 -subsection {* Conditional Mutual Information *}
  29.183 +subsection \<open>Conditional Mutual Information\<close>
  29.184  
  29.185  definition (in prob_space)
  29.186    "conditional_mutual_information b MX MY MZ X Y Z \<equiv>
  29.187 @@ -1173,7 +1173,7 @@
  29.188        done
  29.189    qed (auto simp: b_gt_1 minus_log_convex)
  29.190    also have "\<dots> = conditional_mutual_information b S T P X Y Z"
  29.191 -    unfolding `?eq`
  29.192 +    unfolding \<open>?eq\<close>
  29.193      apply (subst integral_real_density)
  29.194      apply simp
  29.195      apply (auto intro!: distributed_real_AE[OF Pxyz]) []
  29.196 @@ -1430,7 +1430,7 @@
  29.197        done
  29.198    qed (auto simp: b_gt_1 minus_log_convex)
  29.199    also have "\<dots> = conditional_mutual_information b S T P X Y Z"
  29.200 -    unfolding `?eq`
  29.201 +    unfolding \<open>?eq\<close>
  29.202      apply (subst integral_real_density)
  29.203      apply simp
  29.204      apply (auto intro!: distributed_real_AE[OF Pxyz]) []
  29.205 @@ -1490,7 +1490,7 @@
  29.206    have "(\<lambda>(x, y, z). ?i x y z) = (\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x, Z x)) ` space M then ?j (fst x) (fst (snd x)) (snd (snd x)) else 0)"
  29.207      by (auto intro!: ext)
  29.208    then show "(\<integral> (x, y, z). ?i x y z \<partial>?P) = (\<Sum>(x, y, z)\<in>(\<lambda>x. (X x, Y x, Z x)) ` space M. ?j x y z)"
  29.209 -    by (auto intro!: setsum.cong simp add: `?P = ?C` lebesgue_integral_count_space_finite simple_distributed_finite setsum.If_cases split_beta')
  29.210 +    by (auto intro!: setsum.cong simp add: \<open>?P = ?C\<close> lebesgue_integral_count_space_finite simple_distributed_finite setsum.If_cases split_beta')
  29.211  qed
  29.212  
  29.213  lemma (in information_space) conditional_mutual_information_nonneg:
  29.214 @@ -1514,7 +1514,7 @@
  29.215     done
  29.216  qed
  29.217  
  29.218 -subsection {* Conditional Entropy *}
  29.219 +subsection \<open>Conditional Entropy\<close>
  29.220  
  29.221  definition (in prob_space)
  29.222    "conditional_entropy b S T X Y = - (\<integral>(x, y). log b (real_of_ereal (RN_deriv (S \<Otimes>\<^sub>M T) (distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))) (x, y)) / 
  29.223 @@ -1614,7 +1614,7 @@
  29.224      by (rule conditional_entropy_eq_entropy sigma_finite_measure_count_space_finite
  29.225                   simple_functionD  X Y simple_distributed simple_distributedI[OF _ refl]
  29.226                   simple_distributed_joint simple_function_Pair integrable_count_space)+
  29.227 -       (auto simp: `?P = ?C` intro!: integrable_count_space simple_functionD  X Y)
  29.228 +       (auto simp: \<open>?P = ?C\<close> intro!: integrable_count_space simple_functionD  X Y)
  29.229  qed
  29.230  
  29.231  lemma (in information_space) conditional_entropy_eq:
  29.232 @@ -1642,7 +1642,7 @@
  29.233      by auto
  29.234    from Y show "- (\<integral> (x, y). ?f (x, y) * log b (?f (x, y) / Py y) \<partial>?P) =
  29.235      - (\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` space M. Pxy (x, y) * log b (Pxy (x, y) / Py y))"
  29.236 -    by (auto intro!: setsum.cong simp add: `?P = ?C` lebesgue_integral_count_space_finite simple_distributed_finite eq setsum.If_cases split_beta')
  29.237 +    by (auto intro!: setsum.cong simp add: \<open>?P = ?C\<close> lebesgue_integral_count_space_finite simple_distributed_finite eq setsum.If_cases split_beta')
  29.238  qed
  29.239  
  29.240  lemma (in information_space) conditional_mutual_information_eq_conditional_entropy:
  29.241 @@ -1685,7 +1685,7 @@
  29.242    using conditional_mutual_information_eq_conditional_entropy[OF X Y] conditional_mutual_information_nonneg[OF X X Y]
  29.243    by simp
  29.244  
  29.245 -subsection {* Equalities *}
  29.246 +subsection \<open>Equalities\<close>
  29.247  
  29.248  lemma (in information_space) mutual_information_eq_entropy_conditional_entropy_distr:
  29.249    fixes Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real" and Pxy :: "('b \<times> 'c) \<Rightarrow> real"
  29.250 @@ -1752,7 +1752,7 @@
  29.251      apply (simp add: field_simps)
  29.252      done
  29.253    also have "\<dots> = integral\<^sup>L (S \<Otimes>\<^sub>M T) ?g"
  29.254 -    using `AE x in _. ?f x = ?g x` by (intro integral_cong_AE) auto
  29.255 +    using \<open>AE x in _. ?f x = ?g x\<close> by (intro integral_cong_AE) auto
  29.256    also have "\<dots> = mutual_information b S T X Y"
  29.257      by (rule mutual_information_distr[OF S T Px Py Pxy, symmetric])
  29.258    finally show ?thesis ..
    30.1 --- a/src/HOL/Probability/Interval_Integral.thy	Mon Dec 07 16:48:10 2015 +0000
    30.2 +++ b/src/HOL/Probability/Interval_Integral.thy	Mon Dec 07 20:19:59 2015 +0100
    30.3 @@ -23,7 +23,7 @@
    30.4      unfolding has_vector_derivative_def has_derivative_iff_norm
    30.5      using assms by (intro conj_cong Lim_cong_within refl) auto
    30.6    then show ?thesis
    30.7 -    using has_vector_derivative_within_subset[OF f `s \<subseteq> t`] by simp
    30.8 +    using has_vector_derivative_within_subset[OF f \<open>s \<subseteq> t\<close>] by simp
    30.9  qed
   30.10  
   30.11  definition "einterval a b = {x. a < ereal x \<and> ereal x < b}"
   30.12 @@ -65,7 +65,7 @@
   30.13      "incseq X" "\<And>i. a < X i" "\<And>i. X i < b" "X ----> b"
   30.14  proof (cases b)
   30.15    case PInf
   30.16 -  with `a < b` have "a = -\<infinity> \<or> (\<exists>r. a = ereal r)"
   30.17 +  with \<open>a < b\<close> have "a = -\<infinity> \<or> (\<exists>r. a = ereal r)"
   30.18      by (cases a) auto
   30.19    moreover have "(\<lambda>x. ereal (real (Suc x))) ----> \<infinity>"
   30.20        apply (subst LIMSEQ_Suc_iff)
   30.21 @@ -82,12 +82,12 @@
   30.22  next
   30.23    case (real b')
   30.24    def d \<equiv> "b' - (if a = -\<infinity> then b' - 1 else real_of_ereal a)"
   30.25 -  with `a < b` have a': "0 < d"
   30.26 +  with \<open>a < b\<close> have a': "0 < d"
   30.27      by (cases a) (auto simp: real)
   30.28    moreover
   30.29    have "\<And>i r. r < b' \<Longrightarrow> (b' - r) * 1 < (b' - r) * real (Suc (Suc i))"
   30.30      by (intro mult_strict_left_mono) auto
   30.31 -  with `a < b` a' have "\<And>i. a < ereal (b' - d / real (Suc (Suc i)))"
   30.32 +  with \<open>a < b\<close> a' have "\<And>i. a < ereal (b' - d / real (Suc (Suc i)))"
   30.33      by (cases a) (auto simp: real d_def field_simps)
   30.34    moreover have "(\<lambda>i. b' - d / Suc (Suc i)) ----> b'"
   30.35      apply (subst filterlim_sequentially_Suc)
   30.36 @@ -99,7 +99,7 @@
   30.37    ultimately show thesis
   30.38      by (intro that[of "\<lambda>i. b' - d / Suc (Suc i)"])
   30.39         (auto simp add: real incseq_def intro!: divide_left_mono)
   30.40 -qed (insert `a < b`, auto)
   30.41 +qed (insert \<open>a < b\<close>, auto)
   30.42  
   30.43  lemma ereal_decseq_approx:
   30.44    fixes a b :: ereal
   30.45 @@ -107,7 +107,7 @@
   30.46    obtains X :: "nat \<Rightarrow> real" where 
   30.47      "decseq X" "\<And>i. a < X i" "\<And>i. X i < b" "X ----> a"
   30.48  proof -
   30.49 -  have "-b < -a" using `a < b` by simp
   30.50 +  have "-b < -a" using \<open>a < b\<close> by simp
   30.51    from ereal_incseq_approx[OF this] guess X .
   30.52    then show thesis
   30.53      apply (intro that[of "\<lambda>i. - X i"])
   30.54 @@ -125,25 +125,25 @@
   30.55      "incseq u" "decseq l" "\<And>i. l i < u i" "\<And>i. a < l i" "\<And>i. u i < b"
   30.56      "l ----> a" "u ----> b"
   30.57  proof -
   30.58 -  from dense[OF `a < b`] obtain c where "a < c" "c < b" by safe
   30.59 -  from ereal_incseq_approx[OF `c < b`] guess u . note u = this
   30.60 -  from ereal_decseq_approx[OF `a < c`] guess l . note l = this
   30.61 -  { fix i from less_trans[OF `l i < c` `c < u i`] have "l i < u i" by simp }
   30.62 +  from dense[OF \<open>a < b\<close>] obtain c where "a < c" "c < b" by safe
   30.63 +  from ereal_incseq_approx[OF \<open>c < b\<close>] guess u . note u = this
   30.64 +  from ereal_decseq_approx[OF \<open>a < c\<close>] guess l . note l = this
   30.65 +  { fix i from less_trans[OF \<open>l i < c\<close> \<open>c < u i\<close>] have "l i < u i" by simp }
   30.66    have "einterval a b = (\<Union>i. {l i .. u i})"
   30.67    proof (auto simp: einterval_iff)
   30.68      fix x assume "a < ereal x" "ereal x < b"
   30.69      have "eventually (\<lambda>i. ereal (l i) < ereal x) sequentially"
   30.70 -      using l(4) `a < ereal x` by (rule order_tendstoD)
   30.71 +      using l(4) \<open>a < ereal x\<close> by (rule order_tendstoD)
   30.72      moreover 
   30.73      have "eventually (\<lambda>i. ereal x < ereal (u i)) sequentially"
   30.74 -      using u(4) `ereal x< b` by (rule order_tendstoD)
   30.75 +      using u(4) \<open>ereal x< b\<close> by (rule order_tendstoD)
   30.76      ultimately have "eventually (\<lambda>i. l i < x \<and> x < u i) sequentially"
   30.77        by eventually_elim auto
   30.78      then show "\<exists>i. l i \<le> x \<and> x \<le> u i"
   30.79        by (auto intro: less_imp_le simp: eventually_sequentially)
   30.80    next
   30.81      fix x i assume "l i \<le> x" "x \<le> u i" 
   30.82 -    with `a < ereal (l i)` `ereal (u i) < b`
   30.83 +    with \<open>a < ereal (l i)\<close> \<open>ereal (u i) < b\<close>
   30.84      show "a < ereal x" "ereal x < b"
   30.85        by (auto simp del: ereal_less_eq(3) simp add: ereal_less_eq(3)[symmetric])
   30.86    qed
   30.87 @@ -553,15 +553,15 @@
   30.88      proof (intro AE_I2 tendsto_intros Lim_eventually)
   30.89        fix x
   30.90        { fix i assume "l i \<le> x" "x \<le> u i" 
   30.91 -        with `incseq u`[THEN incseqD, of i] `decseq l`[THEN decseqD, of i]
   30.92 +        with \<open>incseq u\<close>[THEN incseqD, of i] \<open>decseq l\<close>[THEN decseqD, of i]
   30.93          have "eventually (\<lambda>i. l i \<le> x \<and> x \<le> u i) sequentially"
   30.94            by (auto simp: eventually_sequentially decseq_def incseq_def intro: order_trans) }
   30.95        then show "eventually (\<lambda>xa. indicator {l xa..u xa} x = (indicator (einterval a b) x::real)) sequentially"
   30.96 -        using approx order_tendstoD(2)[OF `l ----> a`, of x] order_tendstoD(1)[OF `u ----> b`, of x]
   30.97 +        using approx order_tendstoD(2)[OF \<open>l ----> a\<close>, of x] order_tendstoD(1)[OF \<open>u ----> b\<close>, of x]
   30.98          by (auto split: split_indicator)
   30.99      qed
  30.100    qed
  30.101 -  with `a < b` `\<And>i. l i < u i` show ?thesis
  30.102 +  with \<open>a < b\<close> \<open>\<And>i. l i < u i\<close> show ?thesis
  30.103      by (simp add: interval_lebesgue_integral_le_eq[symmetric] interval_integral_Icc less_imp_le)
  30.104  qed
  30.105  
  30.106 @@ -615,7 +615,7 @@
  30.107      "set_integrable lborel (einterval a b) f" 
  30.108      "(LBINT x=a..b. f x) = B - A"
  30.109  proof -
  30.110 -  from einterval_Icc_approximation[OF `a < b`] guess u l . note approx = this
  30.111 +  from einterval_Icc_approximation[OF \<open>a < b\<close>] guess u l . note approx = this
  30.112    have [simp]: "\<And>x i. l i \<le> x \<Longrightarrow> a < ereal x"
  30.113      by (rule order_less_le_trans, rule approx, force)
  30.114    have [simp]: "\<And>x i. x \<le> u i \<Longrightarrow> ereal x < b"
  30.115 @@ -629,7 +629,7 @@
  30.116    have 1: "\<And>i. set_integrable lborel {l i..u i} f"
  30.117    proof -
  30.118      fix i show "set_integrable lborel {l i .. u i} f"
  30.119 -      using `a < l i` `u i < b`
  30.120 +      using \<open>a < l i\<close> \<open>u i < b\<close>
  30.121        by (intro borel_integrable_compact f continuous_at_imp_continuous_on compact_Icc ballI)
  30.122           (auto simp del: ereal_less_eq simp add: ereal_less_eq(3)[symmetric])
  30.123    qed
  30.124 @@ -645,9 +645,9 @@
  30.125      using A approx unfolding tendsto_at_iff_sequentially comp_def
  30.126      by (elim allE[of _ "\<lambda>i. ereal (l i)"], auto)
  30.127    show "(LBINT x=a..b. f x) = B - A"
  30.128 -    by (rule interval_integral_Icc_approx_nonneg [OF `a < b` approx 1 f_nonneg 2 3])
  30.129 +    by (rule interval_integral_Icc_approx_nonneg [OF \<open>a < b\<close> approx 1 f_nonneg 2 3])
  30.130    show "set_integrable lborel (einterval a b) f" 
  30.131 -    by (rule interval_integral_Icc_approx_nonneg [OF `a < b` approx 1 f_nonneg 2 3])
  30.132 +    by (rule interval_integral_Icc_approx_nonneg [OF \<open>a < b\<close> approx 1 f_nonneg 2 3])
  30.133  qed
  30.134  
  30.135  lemma interval_integral_FTC_integrable:
  30.136 @@ -660,7 +660,7 @@
  30.137    assumes B: "((F \<circ> real_of_ereal) ---> B) (at_left b)"
  30.138    shows "(LBINT x=a..b. f x) = B - A"
  30.139  proof -
  30.140 -  from einterval_Icc_approximation[OF `a < b`] guess u l . note approx = this
  30.141 +  from einterval_Icc_approximation[OF \<open>a < b\<close>] guess u l . note approx = this
  30.142    have [simp]: "\<And>x i. l i \<le> x \<Longrightarrow> a < ereal x"
  30.143      by (rule order_less_le_trans, rule approx, force)
  30.144    have [simp]: "\<And>x i. x \<le> u i \<Longrightarrow> ereal x < b"
  30.145 @@ -678,7 +678,7 @@
  30.146      using A approx unfolding tendsto_at_iff_sequentially comp_def
  30.147      by (elim allE[of _ "\<lambda>i. ereal (l i)"], auto)
  30.148    moreover have "(\<lambda>i. LBINT x=l i..u i. f x) ----> (LBINT x=a..b. f x)"
  30.149 -    by (rule interval_integral_Icc_approx_integrable [OF `a < b` approx f_integrable])
  30.150 +    by (rule interval_integral_Icc_approx_integrable [OF \<open>a < b\<close> approx f_integrable])
  30.151    ultimately show ?thesis
  30.152      by (elim LIMSEQ_unique)
  30.153  qed
  30.154 @@ -701,7 +701,7 @@
  30.155      by (rule borel_integrable_atLeastAtMost', rule contf)
  30.156    have "((\<lambda>u. integral {a..u} f) has_vector_derivative f x) (at x within {a..b})"
  30.157      apply (intro integral_has_vector_derivative)
  30.158 -    using `a \<le> x` `x \<le> b` by (intro continuous_on_subset [OF contf], auto)
  30.159 +    using \<open>a \<le> x\<close> \<open>x \<le> b\<close> by (intro continuous_on_subset [OF contf], auto)
  30.160    then have "((\<lambda>u. integral {a..u} f) has_vector_derivative (f x)) (at x within {a..b})"
  30.161      by simp
  30.162    then have "(?F has_vector_derivative (f x)) (at x within {a..b})"
  30.163 @@ -725,7 +725,7 @@
  30.164    assumes "a < b" and contf: "\<And>x :: real. a < x \<Longrightarrow> x < b \<Longrightarrow> isCont f x"
  30.165    shows "\<exists>F. \<forall>x :: real. a < x \<longrightarrow> x < b \<longrightarrow> (F has_vector_derivative f x) (at x)"
  30.166  proof -
  30.167 -  from einterval_nonempty [OF `a < b`] obtain c :: real where [simp]: "a < c" "c < b" 
  30.168 +  from einterval_nonempty [OF \<open>a < b\<close>] obtain c :: real where [simp]: "a < c" "c < b" 
  30.169      by (auto simp add: einterval_def)
  30.170    let ?F = "(\<lambda>u. LBINT y=c..u. f y)"
  30.171    show ?thesis
  30.172 @@ -747,9 +747,9 @@
  30.173        apply (rule interval_integral_FTC2, auto simp add: less_imp_le)
  30.174        apply (rule continuous_at_imp_continuous_on)
  30.175        apply (auto intro!: contf)
  30.176 -      apply (rule order_less_le_trans, rule `a < d`, auto)
  30.177 +      apply (rule order_less_le_trans, rule \<open>a < d\<close>, auto)
  30.178        apply (rule order_le_less_trans) prefer 2
  30.179 -      by (rule `e < b`, auto)
  30.180 +      by (rule \<open>e < b\<close>, auto)
  30.181    qed
  30.182  qed
  30.183  
  30.184 @@ -778,13 +778,13 @@
  30.185      apply (auto simp add: min_def max_def less_imp_le)
  30.186      apply (frule (1) IVT' [of g], auto simp add: assms)
  30.187      by (frule (1) IVT2' [of g], auto simp add: assms)
  30.188 -  from contg `a \<le> b` have "\<exists>c d. g ` {a..b} = {c..d} \<and> c \<le> d"
  30.189 +  from contg \<open>a \<le> b\<close> have "\<exists>c d. g ` {a..b} = {c..d} \<and> c \<le> d"
  30.190      by (elim continuous_image_closed_interval)
  30.191    then obtain c d where g_im: "g ` {a..b} = {c..d}" and "c \<le> d" by auto
  30.192    have "\<exists>F. \<forall>x\<in>{a..b}. (F has_vector_derivative (f (g x))) (at (g x) within (g ` {a..b}))"
  30.193      apply (rule exI, auto, subst g_im)
  30.194      apply (rule interval_integral_FTC2 [of c c d])
  30.195 -    using `c \<le> d` apply auto
  30.196 +    using \<open>c \<le> d\<close> apply auto
  30.197      apply (rule continuous_on_subset [OF contf])
  30.198      using g_im by auto
  30.199    then guess F ..
  30.200 @@ -798,7 +798,7 @@
  30.201      by (blast intro: continuous_on_compose2 contf contg)
  30.202    have "LBINT x=a..b. g' x *\<^sub>R f (g x) = F (g b) - F (g a)"
  30.203      apply (subst interval_integral_Icc, simp add: assms)
  30.204 -    apply (rule integral_FTC_atLeastAtMost[of a b "\<lambda>x. F (g x)", OF `a \<le> b`])
  30.205 +    apply (rule integral_FTC_atLeastAtMost[of a b "\<lambda>x. F (g x)", OF \<open>a \<le> b\<close>])
  30.206      apply (rule vector_diff_chain_within[OF v_derivg derivF, unfolded comp_def])
  30.207      apply (auto intro!: continuous_on_scaleR contg' contfg)
  30.208      done
  30.209 @@ -827,7 +827,7 @@
  30.210    and integrable2: "set_integrable lborel (einterval A B) (\<lambda>x. f x)"
  30.211    shows "(LBINT x=A..B. f x) = (LBINT x=a..b. g' x *\<^sub>R f (g x))"
  30.212  proof -
  30.213 -  from einterval_Icc_approximation[OF `a < b`] guess u l . note approx [simp] = this
  30.214 +  from einterval_Icc_approximation[OF \<open>a < b\<close>] guess u l . note approx [simp] = this
  30.215    note less_imp_le [simp]
  30.216    have [simp]: "\<And>x i. l i \<le> x \<Longrightarrow> a < ereal x"
  30.217      by (rule order_less_le_trans, rule approx, force)
  30.218 @@ -891,7 +891,7 @@
  30.219          done
  30.220    } note eq1 = this
  30.221    have "(\<lambda>i. LBINT x=l i..u i. g' x *\<^sub>R f (g x)) ----> (LBINT x=a..b. g' x *\<^sub>R f (g x))"
  30.222 -    apply (rule interval_integral_Icc_approx_integrable [OF `a < b` approx])
  30.223 +    apply (rule interval_integral_Icc_approx_integrable [OF \<open>a < b\<close> approx])
  30.224      by (rule assms)
  30.225    hence 2: "(\<lambda>i. (LBINT y=g (l i)..g (u i). f y)) ----> (LBINT x=a..b. g' x *\<^sub>R f (g x))"
  30.226      by (simp add: eq1)
  30.227 @@ -902,7 +902,7 @@
  30.228      by (erule order_less_le_trans, rule g_nondec, auto)
  30.229    have "(\<lambda>i. (LBINT y=g (l i)..g (u i). f y)) ----> (LBINT x = A..B. f x)"
  30.230      apply (subst interval_lebesgue_integral_le_eq, auto simp del: real_scaleR_def)
  30.231 -    apply (subst interval_lebesgue_integral_le_eq, rule `A \<le> B`)
  30.232 +    apply (subst interval_lebesgue_integral_le_eq, rule \<open>A \<le> B\<close>)
  30.233      apply (subst un, rule set_integral_cont_up, auto simp del: real_scaleR_def)
  30.234      apply (rule incseq)
  30.235      apply (subst un [symmetric])
  30.236 @@ -929,7 +929,7 @@
  30.237      "set_integrable lborel (einterval A B) f"
  30.238      "(LBINT x=A..B. f x) = (LBINT x=a..b. (f (g x) * g' x))"
  30.239  proof -
  30.240 -  from einterval_Icc_approximation[OF `a < b`] guess u l . note approx [simp] = this
  30.241 +  from einterval_Icc_approximation[OF \<open>a < b\<close>] guess u l . note approx [simp] = this
  30.242    note less_imp_le [simp]
  30.243    have [simp]: "\<And>x i. l i \<le> x \<Longrightarrow> a < ereal x"
  30.244      by (rule order_less_le_trans, rule approx, force)
  30.245 @@ -994,7 +994,7 @@
  30.246    } note eq1 = this
  30.247    have "(\<lambda>i. LBINT x=l i..u i. f (g x) * g' x)
  30.248        ----> (LBINT x=a..b. f (g x) * g' x)"
  30.249 -    apply (rule interval_integral_Icc_approx_integrable [OF `a < b` approx])
  30.250 +    apply (rule interval_integral_Icc_approx_integrable [OF \<open>a < b\<close> approx])
  30.251      by (rule assms)
  30.252    hence 2: "(\<lambda>i. (LBINT y=g (l i)..g (u i). f y)) ----> (LBINT x=a..b. f (g x) * g' x)"
  30.253      by (simp add: eq1)
    31.1 --- a/src/HOL/Probability/Lebesgue_Integral_Substitution.thy	Mon Dec 07 16:48:10 2015 +0000
    31.2 +++ b/src/HOL/Probability/Lebesgue_Integral_Substitution.thy	Mon Dec 07 20:19:59 2015 +0100
    31.3 @@ -6,7 +6,7 @@
    31.4      This could probably be weakened somehow.
    31.5  *)
    31.6  
    31.7 -section {* Integration by Substition *}
    31.8 +section \<open>Integration by Substition\<close>
    31.9  
   31.10  theory Lebesgue_Integral_Substitution
   31.11  imports Interval_Integral
   31.12 @@ -36,7 +36,7 @@
   31.13    also from assms(1) have "closed (g -` {a..} \<inter> {c..d})"
   31.14      by (auto simp: continuous_on_closed_vimage)
   31.15    hence "closure (g -` {a..} \<inter> {c..d}) = g -` {a..} \<inter> {c..d}" by simp
   31.16 -  finally show ?thesis using `x \<in> {c..d}` by auto 
   31.17 +  finally show ?thesis using \<open>x \<in> {c..d}\<close> by auto 
   31.18  qed 
   31.19  
   31.20  lemma interior_real_semiline':
   31.21 @@ -103,7 +103,7 @@
   31.22      using assms by (subst borel_measurable_restrict_space_iff) auto
   31.23    then have "f -` B \<inter> space (restrict_space M X) \<in> sets (restrict_space M X)"
   31.24      by (rule measurable_sets) fact
   31.25 -  with `X \<in> sets M` show ?thesis
   31.26 +  with \<open>X \<in> sets M\<close> show ?thesis
   31.27      by (subst (asm) sets_restrict_space_iff) (auto simp: space_restrict_space)
   31.28  qed
   31.29  
   31.30 @@ -171,8 +171,8 @@
   31.31    shows "strict_mono g"
   31.32  proof
   31.33    fix x y :: 'b assume "x < y"
   31.34 -  from `surj f` obtain x' y' where [simp]: "x = f x'" "y = f y'" by blast
   31.35 -  with `x < y` and `strict_mono f` have "x' < y'" by (simp add: strict_mono_less)
   31.36 +  from \<open>surj f\<close> obtain x' y' where [simp]: "x = f x'" "y = f y'" by blast
   31.37 +  with \<open>x < y\<close> and \<open>strict_mono f\<close> have "x' < y'" by (simp add: strict_mono_less)
   31.38    with inv show "g x < g y" by simp
   31.39  qed
   31.40  
   31.41 @@ -218,11 +218,11 @@
   31.42      also have "(op + (-x) ` interior A) = ?A'" by auto
   31.43      finally show "open ?A'" .
   31.44    next
   31.45 -    from `x \<in> interior A` show "0 \<in> ?A'" by auto
   31.46 +    from \<open>x \<in> interior A\<close> show "0 \<in> ?A'" by auto
   31.47    next
   31.48      fix h assume "h \<in> ?A'"
   31.49      hence "x + h \<in> interior A" by auto
   31.50 -    with mono' and `x \<in> interior A` show "(f (x + h) - f x) / h \<ge> 0"
   31.51 +    with mono' and \<open>x \<in> interior A\<close> show "(f (x + h) - f x) / h \<ge> 0"
   31.52        by (cases h rule: linorder_cases[of _ 0])
   31.53           (simp_all add: divide_nonpos_neg divide_nonneg_pos mono_onD field_simps)
   31.54    qed
   31.55 @@ -267,7 +267,7 @@
   31.56  proof (cases "a < b")
   31.57    assume "a < b"
   31.58    from deriv have "\<forall>x. x \<ge> a \<and> x \<le> b \<longrightarrow> (g has_real_derivative g' x) (at x)" by simp
   31.59 -  from MVT2[OF `a < b` this] and deriv 
   31.60 +  from MVT2[OF \<open>a < b\<close> this] and deriv 
   31.61      obtain \<xi> where \<xi>_ab: "\<xi> > a" "\<xi> < b" and g_ab: "g b - g a = (b - a) * g' \<xi>" by blast
   31.62    from \<xi>_ab ab nonneg have "(b - a) * g' \<xi> \<ge> 0" by simp
   31.63    with g_ab show ?thesis by simp
   31.64 @@ -279,9 +279,9 @@
   31.65    obtains c' d' where "{a..b} \<inter> g -` {c..d} = {c'..d'}" "c' \<le> d'" "g c' = c" "g d' = d"
   31.66  proof-
   31.67      let ?A = "{a..b} \<inter> g -` {c..d}"
   31.68 -    from IVT'[of g a c b, OF _ _ `a \<le> b` assms(1)] assms(4,5) 
   31.69 +    from IVT'[of g a c b, OF _ _ \<open>a \<le> b\<close> assms(1)] assms(4,5) 
   31.70           obtain c'' where c'': "c'' \<in> ?A" "g c'' = c" by auto
   31.71 -    from IVT'[of g a d b, OF _ _ `a \<le> b` assms(1)] assms(4,5) 
   31.72 +    from IVT'[of g a d b, OF _ _ \<open>a \<le> b\<close> assms(1)] assms(4,5) 
   31.73           obtain d'' where d'': "d'' \<in> ?A" "g d'' = d" by auto
   31.74      hence [simp]: "?A \<noteq> {}" by blast
   31.75  
   31.76 @@ -319,7 +319,7 @@
   31.77    shows "(\<integral>\<^sup>+x. f x * indicator {g a..g b} x \<partial>lborel) = 
   31.78               (\<integral>\<^sup>+x. f (g x) * g' x * indicator {a..b} x \<partial>lborel)"
   31.79  proof-
   31.80 -  from `a < b` have [simp]: "a \<le> b" by simp
   31.81 +  from \<open>a < b\<close> have [simp]: "a \<le> b" by simp
   31.82    from derivg have contg: "continuous_on {a..b} g" by (rule has_real_derivative_imp_continuous_on)
   31.83    from this and contg' have Mg: "set_borel_measurable borel {a..b} g" and 
   31.84                               Mg': "set_borel_measurable borel {a..b} g'" 
   31.85 @@ -364,7 +364,7 @@
   31.86              by (simp only: u'v' max_absorb2 min_absorb1) 
   31.87                 (intro continuous_on_subset[OF contg'], insert u'v', auto)
   31.88          have "\<And>x. x \<in> {u'..v'} \<Longrightarrow> (g has_real_derivative g' x) (at x within {u'..v'})"
   31.89 -           using asm by (intro has_field_derivative_subset[OF derivg] set_mp[OF `{u'..v'} \<subseteq> {a..b}`]) auto
   31.90 +           using asm by (intro has_field_derivative_subset[OF derivg] set_mp[OF \<open>{u'..v'} \<subseteq> {a..b}\<close>]) auto
   31.91          hence B: "\<And>x. min u' v' \<le> x \<Longrightarrow> x \<le> max u' v' \<Longrightarrow> 
   31.92                        (g has_vector_derivative g' x) (at x within {min u' v'..max u' v'})" 
   31.93              by (simp only: u'v' max_absorb2 min_absorb1) 
   31.94 @@ -377,7 +377,7 @@
   31.95               (auto intro: measurable_sets Mg simp: derivg_nonneg mult.commute split: split_indicator)
   31.96          also from interval_integral_FTC_finite[OF A B]
   31.97              have "LBINT x:{a..b} \<inter> g-`{u..v}. g' x = v - u"
   31.98 -                by (simp add: u'v' interval_integral_Icc `u \<le> v`)
   31.99 +                by (simp add: u'v' interval_integral_Icc \<open>u \<le> v\<close>)
  31.100          finally have "(\<integral>\<^sup>+ x. ereal (g' x) * indicator ({a..b} \<inter> g -` {u..v}) x \<partial>lborel) =
  31.101                             ereal (v - u)" .
  31.102        } note A = this
  31.103 @@ -386,11 +386,11 @@
  31.104                 (\<integral>\<^sup>+ x. ereal (g' x) * indicator ({a..b} \<inter> g -` {c..d}) x \<partial>lborel)" 
  31.105          by (intro nn_integral_cong) (simp split: split_indicator)
  31.106        also have "{a..b} \<inter> g-`{c..d} = {a..b} \<inter> g-`{max (g a) c..min (g b) d}" 
  31.107 -        using `a \<le> b` `c \<le> d`
  31.108 +        using \<open>a \<le> b\<close> \<open>c \<le> d\<close>
  31.109          by (auto intro!: monog intro: order.trans)
  31.110        also have "(\<integral>\<^sup>+ x. ereal (g' x) * indicator ... x \<partial>lborel) =
  31.111          (if max (g a) c \<le> min (g b) d then min (g b) d - max (g a) c else 0)"
  31.112 -         using `c \<le> d` by (simp add: A)
  31.113 +         using \<open>c \<le> d\<close> by (simp add: A)
  31.114        also have "... = (\<integral>\<^sup>+ x. indicator ({g a..g b} \<inter> {c..d}) x \<partial>lborel)"
  31.115          by (subst nn_integral_indicator) (auto intro!: measurable_sets Mg simp:)
  31.116        also have "... = (\<integral>\<^sup>+ x. indicator {c..d} x * indicator {g a..g b} x \<partial>lborel)"
  31.117 @@ -400,7 +400,7 @@
  31.118        next
  31.119  
  31.120        case (compl A)
  31.121 -      note `A \<in> sets borel`[measurable]
  31.122 +      note \<open>A \<in> sets borel\<close>[measurable]
  31.123        from emeasure_mono[of "A \<inter> {g a..g b}" "{g a..g b}" lborel]
  31.124            have [simp]: "emeasure lborel (A \<inter> {g a..g b}) \<noteq> \<infinity>" by auto
  31.125        have [simp]: "g -` A \<inter> {a..b} \<in> sets borel"
  31.126 @@ -415,10 +415,10 @@
  31.127          by (simp add: vimage_Compl diff_eq Int_commute[of "-A"])
  31.128        also have "{g a..g b} - A = {g a..g b} - A \<inter> {g a..g b}" by blast
  31.129        also have "emeasure lborel ... = g b - g a - emeasure lborel (A \<inter> {g a..g b})"
  31.130 -             using `A \<in> sets borel` by (subst emeasure_Diff) (auto simp: real_of_ereal_minus)
  31.131 +             using \<open>A \<in> sets borel\<close> by (subst emeasure_Diff) (auto simp: real_of_ereal_minus)
  31.132       also have "emeasure lborel (A \<inter> {g a..g b}) = 
  31.133                      \<integral>\<^sup>+x. indicator A x * indicator {g a..g b} x \<partial>lborel" 
  31.134 -       using `A \<in> sets borel`
  31.135 +       using \<open>A \<in> sets borel\<close>
  31.136         by (subst nn_integral_indicator[symmetric], simp, intro nn_integral_cong)
  31.137            (simp split: split_indicator)
  31.138        also have "... = \<integral>\<^sup>+ x. indicator (g-`A \<inter> {a..b}) x * ereal (g' x * indicator {a..b} x) \<partial>lborel" (is "_ = ?I")
  31.139 @@ -500,7 +500,7 @@
  31.140  
  31.141  next
  31.142    case (mult f c)
  31.143 -    note Mf[measurable] = `f \<in> borel_measurable borel`
  31.144 +    note Mf[measurable] = \<open>f \<in> borel_measurable borel\<close>
  31.145      let ?I = "indicator {a..b}"
  31.146      have "(\<lambda>x. f (g x * ?I x) * ereal (g' x * ?I x)) \<in> borel_measurable borel" using Mg Mg'
  31.147        by (intro borel_measurable_ereal_times measurable_compose[OF _ Mf])
  31.148 @@ -522,7 +522,7 @@
  31.149        also have "(\<lambda>x. f (g x * ?I x) * ereal (g' x * ?I x)) = (\<lambda>x. f (g x) * ereal (g' x) * ?I x)"
  31.150          by (intro ext) (simp split: split_indicator)
  31.151        finally have "(\<lambda>x. f (g x) * ereal (g' x) * ?I x) \<in> borel_measurable borel" .
  31.152 -    } note Mf' = this[OF `f1 \<in> borel_measurable borel`] this[OF `f2 \<in> borel_measurable borel`]
  31.153 +    } note Mf' = this[OF \<open>f1 \<in> borel_measurable borel\<close>] this[OF \<open>f2 \<in> borel_measurable borel\<close>]
  31.154      from add have not_neginf: "\<And>x. f1 x \<noteq> -\<infinity>" "\<And>x. f2 x \<noteq> -\<infinity>" 
  31.155        by (metis Infty_neq_0(1) ereal_0_le_uminus_iff ereal_infty_less_eq(1))+
  31.156  
  31.157 @@ -583,7 +583,7 @@
  31.158               (\<integral>\<^sup>+x. f (g x) * g' x * indicator {a..b} x \<partial>lborel)"
  31.159  proof (cases "a = b")
  31.160    assume "a \<noteq> b"
  31.161 -  with `a \<le> b` have "a < b" by auto
  31.162 +  with \<open>a \<le> b\<close> have "a < b" by auto
  31.163    let ?f' = "\<lambda>x. max 0 (f x * indicator {g a..g b} x)"
  31.164  
  31.165    from derivg derivg_nonneg have monog: "\<And>x y. a \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<le> b \<Longrightarrow> g x \<le> g y"
  31.166 @@ -602,7 +602,7 @@
  31.167      by (subst nn_integral_max_0[symmetric], intro nn_integral_cong) 
  31.168         (auto split: split_indicator simp: zero_ereal_def)
  31.169    also have "... = \<integral>\<^sup>+ x. ?f' (g x) * ereal (g' x) * indicator {a..b} x \<partial>lborel" using Mf
  31.170 -    by (subst nn_integral_substitution_aux[OF _ _ derivg contg' derivg_nonneg `a < b`]) 
  31.171 +    by (subst nn_integral_substitution_aux[OF _ _ derivg contg' derivg_nonneg \<open>a < b\<close>]) 
  31.172         (auto simp add: zero_ereal_def mult.commute)
  31.173    also have "... = \<integral>\<^sup>+ x. max 0 (f (g x)) * ereal (g' x) * indicator {a..b} x \<partial>lborel"
  31.174      by (intro nn_integral_cong) 
  31.175 @@ -653,14 +653,14 @@
  31.176      by (intro nn_integral_cong) (simp split: split_indicator)
  31.177    also with M1 have A: "(\<integral>\<^sup>+ x. ereal (f x * indicator {g a..g b} x) \<partial>lborel) =
  31.178                              (\<integral>\<^sup>+ x. ereal (f (g x) * g' x * indicator {a..b} x) \<partial>lborel)"
  31.179 -    by (subst nn_integral_substitution[OF _ derivg contg' derivg_nonneg `a \<le> b`]) 
  31.180 +    by (subst nn_integral_substitution[OF _ derivg contg' derivg_nonneg \<open>a \<le> b\<close>]) 
  31.181         (auto simp: nn_integral_set_ereal mult.commute)
  31.182    also have "(\<integral>\<^sup>+ x. ereal (- (f x)) * indicator {g a..g b} x \<partial>lborel) =
  31.183                 (\<integral>\<^sup>+ x. ereal (- (f x) * indicator {g a..g b} x) \<partial>lborel)"
  31.184      by (intro nn_integral_cong) (simp split: split_indicator)
  31.185    also with M2 have B: "(\<integral>\<^sup>+ x. ereal (- (f x) * indicator {g a..g b} x) \<partial>lborel) =
  31.186                              (\<integral>\<^sup>+ x. ereal (- (f (g x)) * g' x * indicator {a..b} x) \<partial>lborel)"
  31.187 -    by (subst nn_integral_substitution[OF _ derivg contg' derivg_nonneg `a \<le> b`])
  31.188 +    by (subst nn_integral_substitution[OF _ derivg contg' derivg_nonneg \<open>a \<le> b\<close>])
  31.189         (auto simp: nn_integral_set_ereal mult.commute)
  31.190  
  31.191    also {
    32.1 --- a/src/HOL/Probability/Lebesgue_Measure.thy	Mon Dec 07 16:48:10 2015 +0000
    32.2 +++ b/src/HOL/Probability/Lebesgue_Measure.thy	Mon Dec 07 20:19:59 2015 +0100
    32.3 @@ -5,13 +5,13 @@
    32.4      Author:     Luke Serafin
    32.5  *)
    32.6  
    32.7 -section {* Lebesgue measure *}
    32.8 +section \<open>Lebesgue measure\<close>
    32.9  
   32.10  theory Lebesgue_Measure
   32.11    imports Finite_Product_Measure Bochner_Integration Caratheodory
   32.12  begin
   32.13  
   32.14 -subsection {* Every right continuous and nondecreasing function gives rise to a measure *}
   32.15 +subsection \<open>Every right continuous and nondecreasing function gives rise to a measure\<close>
   32.16  
   32.17  definition interval_measure :: "(real \<Rightarrow> real) \<Rightarrow> real measure" where
   32.18    "interval_measure F = extend_measure UNIV {(a, b). a \<le> b} (\<lambda>(a, b). {a <.. b}) (\<lambda>(a, b). ereal (F b - F a))"
   32.19 @@ -21,7 +21,7 @@
   32.20    assumes mono_F: "\<And>x y. x \<le> y \<Longrightarrow> F x \<le> F y"
   32.21    assumes right_cont_F : "\<And>a. continuous (at_right a) F"
   32.22    shows "emeasure (interval_measure F) {a <.. b} = F b - F a"
   32.23 -proof (rule extend_measure_caratheodory_pair[OF interval_measure_def `a \<le> b`])
   32.24 +proof (rule extend_measure_caratheodory_pair[OF interval_measure_def \<open>a \<le> b\<close>])
   32.25    show "semiring_of_sets UNIV {{a<..b} |a b :: real. a \<le> b}"
   32.26    proof (unfold_locales, safe)
   32.27      fix a b c d :: real assume *: "a \<le> b" "c \<le> d"
   32.28 @@ -50,7 +50,7 @@
   32.29      by (auto intro!: l_r mono_F)
   32.30  
   32.31    { fix S :: "nat set" assume "finite S"
   32.32 -    moreover note `a \<le> b`
   32.33 +    moreover note \<open>a \<le> b\<close>
   32.34      moreover have "\<And>i. i \<in> S \<Longrightarrow> {l i <.. r i} \<subseteq> {a <.. b}"
   32.35        unfolding lr_eq_ab[symmetric] by auto
   32.36      ultimately have "(\<Sum>i\<in>S. F (r i) - F (l i)) \<le> F b - F a"
   32.37 @@ -59,7 +59,7 @@
   32.38        show ?case
   32.39        proof cases
   32.40          assume "\<exists>i\<in>S. l i < r i"
   32.41 -        with `finite S` have "Min (l ` {i\<in>S. l i < r i}) \<in> l ` {i\<in>S. l i < r i}"
   32.42 +        with \<open>finite S\<close> have "Min (l ` {i\<in>S. l i < r i}) \<in> l ` {i\<in>S. l i < r i}"
   32.43            by (intro Min_in) auto
   32.44          then obtain m where m: "m \<in> S" "l m < r m" "l m = Min (l ` {i\<in>S. l i < r i})"
   32.45            by fastforce
   32.46 @@ -69,14 +69,14 @@
   32.47          also have "(\<Sum>i\<in>S - {m}. F (r i) - F (l i)) \<le> F b - F (r m)"
   32.48          proof (intro psubset.IH)
   32.49            show "S - {m} \<subset> S"
   32.50 -            using `m\<in>S` by auto
   32.51 +            using \<open>m\<in>S\<close> by auto
   32.52            show "r m \<le> b"
   32.53 -            using psubset.prems(2)[OF `m\<in>S`] `l m < r m` by auto
   32.54 +            using psubset.prems(2)[OF \<open>m\<in>S\<close>] \<open>l m < r m\<close> by auto
   32.55          next
   32.56            fix i assume "i \<in> S - {m}"
   32.57            then have i: "i \<in> S" "i \<noteq> m" by auto
   32.58            { assume i': "l i < r i" "l i < r m"
   32.59 -            moreover with `finite S` i m have "l m \<le> l i"
   32.60 +            moreover with \<open>finite S\<close> i m have "l m \<le> l i"
   32.61                by auto
   32.62              ultimately have "{l i <.. r i} \<inter> {l m <.. r m} \<noteq> {}"
   32.63                by auto
   32.64 @@ -85,14 +85,14 @@
   32.65            then have "l i \<noteq> r i \<Longrightarrow> r m \<le> l i"
   32.66              unfolding not_less[symmetric] using l_r[of i] by auto
   32.67            then show "{l i <.. r i} \<subseteq> {r m <.. b}"
   32.68 -            using psubset.prems(2)[OF `i\<in>S`] by auto
   32.69 +            using psubset.prems(2)[OF \<open>i\<in>S\<close>] by auto
   32.70          qed
   32.71          also have "F (r m) - F (l m) \<le> F (r m) - F a"
   32.72 -          using psubset.prems(2)[OF `m \<in> S`] `l m < r m`
   32.73 +          using psubset.prems(2)[OF \<open>m \<in> S\<close>] \<open>l m < r m\<close>
   32.74            by (auto simp add: Ioc_subset_iff intro!: mono_F)
   32.75          finally show ?case
   32.76            by (auto intro: add_mono)
   32.77 -      qed (auto simp add: `a \<le> b` less_le)
   32.78 +      qed (auto simp add: \<open>a \<le> b\<close> less_le)
   32.79      qed }
   32.80    note claim1 = this
   32.81  
   32.82 @@ -117,13 +117,13 @@
   32.83          show ?case
   32.84          proof cases
   32.85            assume "?R"
   32.86 -          with `j \<in> S` psubset.prems have "{u..v} \<subseteq> (\<Union>i\<in>S-{j}. {l i<..< r i})"
   32.87 +          with \<open>j \<in> S\<close> psubset.prems have "{u..v} \<subseteq> (\<Union>i\<in>S-{j}. {l i<..< r i})"
   32.88              apply (auto simp: subset_eq Ball_def)
   32.89              apply (metis Diff_iff less_le_trans leD linear singletonD)
   32.90              apply (metis Diff_iff less_le_trans leD linear singletonD)
   32.91              apply (metis order_trans less_le_not_le linear)
   32.92              done
   32.93 -          with `j \<in> S` have "F v - F u \<le> (\<Sum>i\<in>S - {j}. F (r i) - F (l i))"
   32.94 +          with \<open>j \<in> S\<close> have "F v - F u \<le> (\<Sum>i\<in>S - {j}. F (r i) - F (l i))"
   32.95              by (intro psubset) auto
   32.96            also have "\<dots> \<le> (\<Sum>i\<in>S. F (r i) - F (l i))"
   32.97              using psubset.prems
   32.98 @@ -137,7 +137,7 @@
   32.99            let ?S2 = "{i \<in> S. r i > r j}"
  32.100  
  32.101            have "(\<Sum>i\<in>S. F (r i) - F (l i)) \<ge> (\<Sum>i\<in>?S1 \<union> ?S2 \<union> {j}. F (r i) - F (l i))"
  32.102 -            using `j \<in> S` `finite S` psubset.prems j
  32.103 +            using \<open>j \<in> S\<close> \<open>finite S\<close> psubset.prems j
  32.104              by (intro setsum_mono2) (auto intro: less_imp_le)
  32.105            also have "(\<Sum>i\<in>?S1 \<union> ?S2 \<union> {j}. F (r i) - F (l i)) =
  32.106              (\<Sum>i\<in>?S1. F (r i) - F (l i)) + (\<Sum>i\<in>?S2 . F (r i) - F (l i)) + (F (r j) - F (l j))"
  32.107 @@ -149,13 +149,13 @@
  32.108              apply (metis less_le_not_le)
  32.109              done
  32.110            also (xtrans) have "(\<Sum>i\<in>?S1. F (r i) - F (l i)) \<ge> F (l j) - F u"
  32.111 -            using `j \<in> S` `finite S` psubset.prems j
  32.112 +            using \<open>j \<in> S\<close> \<open>finite S\<close> psubset.prems j
  32.113              apply (intro psubset.IH psubset)
  32.114              apply (auto simp: subset_eq Ball_def)
  32.115              apply (metis less_le_trans not_le)
  32.116              done
  32.117            also (xtrans) have "(\<Sum>i\<in>?S2. F (r i) - F (l i)) \<ge> F v - F (r j)"
  32.118 -            using `j \<in> S` `finite S` psubset.prems j
  32.119 +            using \<open>j \<in> S\<close> \<open>finite S\<close> psubset.prems j
  32.120              apply (intro psubset.IH psubset)
  32.121              apply (auto simp: subset_eq Ball_def)
  32.122              apply (metis le_less_trans not_le)
  32.123 @@ -326,7 +326,7 @@
  32.124    proof (rule tendsto_at_left_sequentially)
  32.125      show "a - 1 < a" by simp
  32.126      fix X assume "\<And>n. X n < a" "incseq X" "X ----> a"
  32.127 -    with `a \<le> b` have "(\<lambda>n. emeasure ?F {X n<..b}) ----> emeasure ?F (\<Inter>n. {X n <..b})"
  32.128 +    with \<open>a \<le> b\<close> have "(\<lambda>n. emeasure ?F {X n<..b}) ----> emeasure ?F (\<Inter>n. {X n <..b})"
  32.129        apply (intro Lim_emeasure_decseq)
  32.130        apply (auto simp: decseq_def incseq_def emeasure_interval_measure_Ioc *)
  32.131        apply force
  32.132 @@ -334,14 +334,14 @@
  32.133        apply (auto intro: less_le_trans less_imp_le)
  32.134        done
  32.135      also have "(\<Inter>n. {X n <..b}) = {a..b}"
  32.136 -      using `\<And>n. X n < a`
  32.137 +      using \<open>\<And>n. X n < a\<close>
  32.138        apply auto
  32.139 -      apply (rule LIMSEQ_le_const2[OF `X ----> a`])
  32.140 +      apply (rule LIMSEQ_le_const2[OF \<open>X ----> a\<close>])
  32.141        apply (auto intro: less_imp_le)
  32.142        apply (auto intro: less_le_trans)
  32.143        done
  32.144      also have "(\<lambda>n. emeasure ?F {X n<..b}) = (\<lambda>n. F b - F (X n))"
  32.145 -      using `\<And>n. X n < a` `a \<le> b` by (subst *) (auto intro: less_imp_le less_le_trans)
  32.146 +      using \<open>\<And>n. X n < a\<close> \<open>a \<le> b\<close> by (subst *) (auto intro: less_imp_le less_le_trans)
  32.147      finally show "(\<lambda>n. F b - F (X n)) ----> emeasure ?F {a..b}" .
  32.148    qed
  32.149    show "((\<lambda>a. ereal (F b - F a)) ---> F b - F a) (at_left a)"
  32.150 @@ -359,7 +359,7 @@
  32.151    apply (auto intro!: Rats_no_top_le Rats_no_bot_less countable_rat simp: emeasure_interval_measure_Ioc_eq[OF assms])
  32.152    done
  32.153  
  32.154 -subsection {* Lebesgue-Borel measure *}
  32.155 +subsection \<open>Lebesgue-Borel measure\<close>
  32.156  
  32.157  definition lborel :: "('a :: euclidean_space) measure" where
  32.158    "lborel = distr (\<Pi>\<^sub>M b\<in>Basis. interval_measure (\<lambda>x. x)) borel (\<lambda>f. \<Sum>b\<in>Basis. f b *\<^sub>R b)"
  32.159 @@ -557,7 +557,7 @@
  32.160    ultimately show False by contradiction
  32.161  qed
  32.162  
  32.163 -subsection {* Affine transformation on the Lebesgue-Borel *}
  32.164 +subsection \<open>Affine transformation on the Lebesgue-Borel\<close>
  32.165  
  32.166  lemma lborel_eqI:
  32.167    fixes M :: "'a::euclidean_space measure"
  32.168 @@ -595,13 +595,13 @@
  32.169      assume "0 < c"
  32.170      then have "(\<lambda>x. t + c *\<^sub>R x) -` box l u = box ((l - t) /\<^sub>R c) ((u - t) /\<^sub>R c)"
  32.171        by (auto simp: field_simps box_def inner_simps)
  32.172 -    with `0 < c` show ?thesis
  32.173 +    with \<open>0 < c\<close> show ?thesis
  32.174        using le
  32.175        by (auto simp: field_simps emeasure_density nn_integral_distr nn_integral_cmult
  32.176                       emeasure_lborel_box_eq inner_simps setprod_dividef setprod_constant
  32.177                       borel_measurable_indicator' emeasure_distr)
  32.178    next
  32.179 -    assume "\<not> 0 < c" with `c \<noteq> 0` have "c < 0" by auto
  32.180 +    assume "\<not> 0 < c" with \<open>c \<noteq> 0\<close> have "c < 0" by auto
  32.181      then have "box ((u - t) /\<^sub>R c) ((l - t) /\<^sub>R c) = (\<lambda>x. t + c *\<^sub>R x) -` box l u"
  32.182        by (auto simp: field_simps box_def inner_simps)
  32.183      then have "\<And>x. indicator (box l u) (t + c *\<^sub>R x) = (indicator (box ((u - t) /\<^sub>R c) ((l - t) /\<^sub>R c)) x :: ereal)"
  32.184 @@ -615,7 +615,7 @@
  32.185        finally have "(- c) ^ card ?B * (\<Prod>x\<in>?B. l \<bullet> x - u \<bullet> x) = c ^ card ?B * (\<Prod>b\<in>?B. u \<bullet> b - l \<bullet> b)"
  32.186          by simp }
  32.187      ultimately show ?thesis
  32.188 -      using `c < 0` le
  32.189 +      using \<open>c < 0\<close> le
  32.190        by (auto simp: field_simps emeasure_density nn_integral_distr nn_integral_cmult
  32.191                       emeasure_lborel_box_eq inner_simps setprod_dividef setprod_constant
  32.192                       borel_measurable_indicator' emeasure_distr)
  32.193 @@ -736,7 +736,7 @@
  32.194  lemma lborelD_Collect[measurable (raw)]: "{x\<in>space borel. P x} \<in> sets borel \<Longrightarrow> {x\<in>space lborel. P x} \<in> sets lborel" by simp
  32.195  lemma lborelD[measurable (raw)]: "A \<in> sets borel \<Longrightarrow> A \<in> sets lborel" by simp
  32.196  
  32.197 -subsection {* Equivalence Lebesgue integral on @{const lborel} and HK-integral *}
  32.198 +subsection \<open>Equivalence Lebesgue integral on @{const lborel} and HK-integral\<close>
  32.199  
  32.200  lemma has_integral_measure_lborel:
  32.201    fixes A :: "'a::euclidean_space set"
  32.202 @@ -915,7 +915,7 @@
  32.203    have *: "f integrable_on UNIV \<and> (\<lambda>k. integral UNIV (U k)) ----> integral UNIV f"
  32.204    proof (rule monotone_convergence_increasing)
  32.205      show "\<forall>k. U k integrable_on UNIV" using U_int by auto
  32.206 -    show "\<forall>k. \<forall>x\<in>UNIV. U k x \<le> U (Suc k) x" using `incseq U` by (auto simp: incseq_def le_fun_def)
  32.207 +    show "\<forall>k. \<forall>x\<in>UNIV. U k x \<le> U (Suc k) x" using \<open>incseq U\<close> by (auto simp: incseq_def le_fun_def)
  32.208      then show "bounded {integral UNIV (U k) |k. True}"
  32.209        using bnd int_eq by (auto simp: bounded_real intro!: exI[of _ r])
  32.210      show "\<forall>x\<in>UNIV. (\<lambda>k. U k x) ----> f x"
  32.211 @@ -1067,7 +1067,7 @@
  32.212    proof (rule has_integral_dominated_convergence)
  32.213      show "\<And>i. (s i has_integral integral\<^sup>L lborel (s i)) UNIV" by fact
  32.214      show "(\<lambda>x. norm (2 * f x)) integrable_on UNIV"
  32.215 -      using `integrable lborel f`
  32.216 +      using \<open>integrable lborel f\<close>
  32.217        by (intro nn_integral_integrable_on)
  32.218           (auto simp: integrable_iff_bounded abs_mult times_ereal.simps(1)[symmetric] nn_integral_cmult
  32.219                 simp del: times_ereal.simps)
  32.220 @@ -1106,12 +1106,12 @@
  32.221  
  32.222  end
  32.223  
  32.224 -subsection {* Fundamental Theorem of Calculus for the Lebesgue integral *}
  32.225 +subsection \<open>Fundamental Theorem of Calculus for the Lebesgue integral\<close>
  32.226  
  32.227  lemma emeasure_bounded_finite:
  32.228    assumes "bounded A" shows "emeasure lborel A < \<infinity>"
  32.229  proof -
  32.230 -  from bounded_subset_cbox[OF `bounded A`] obtain a b where "A \<subseteq> cbox a b"
  32.231 +  from bounded_subset_cbox[OF \<open>bounded A\<close>] obtain a b where "A \<subseteq> cbox a b"
  32.232      by auto
  32.233    then have "emeasure lborel A \<le> emeasure lborel (cbox a b)"
  32.234      by (intro emeasure_mono) auto
  32.235 @@ -1130,7 +1130,7 @@
  32.236    assume "S \<noteq> {}"
  32.237    have "continuous_on S (\<lambda>x. norm (f x))"
  32.238      using assms by (intro continuous_intros)
  32.239 -  from continuous_attains_sup[OF `compact S` `S \<noteq> {}` this]
  32.240 +  from continuous_attains_sup[OF \<open>compact S\<close> \<open>S \<noteq> {}\<close> this]
  32.241    obtain M where M: "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> M"
  32.242      by auto
  32.243  
  32.244 @@ -1159,11 +1159,11 @@
  32.245      by (auto simp: mult.commute)
  32.246  qed
  32.247  
  32.248 -text {*
  32.249 +text \<open>
  32.250  
  32.251  For the positive integral we replace continuity with Borel-measurability.
  32.252  
  32.253 -*}
  32.254 +\<close>
  32.255  
  32.256  lemma
  32.257    fixes f :: "real \<Rightarrow> real"
  32.258 @@ -1181,7 +1181,7 @@
  32.259    have "(f has_integral F b - F a) {a..b}"
  32.260      by (intro fundamental_theorem_of_calculus)
  32.261         (auto simp: has_field_derivative_iff_has_vector_derivative[symmetric]
  32.262 -             intro: has_field_derivative_subset[OF f(1)] `a \<le> b`)
  32.263 +             intro: has_field_derivative_subset[OF f(1)] \<open>a \<le> b\<close>)
  32.264    then have i: "((\<lambda>x. f x * indicator {a .. b} x) has_integral F b - F a) UNIV"
  32.265      unfolding indicator_def if_distrib[where f="\<lambda>x. a * x" for a]
  32.266      by (simp cong del: if_cong del: atLeastAtMost_iff)
  32.267 @@ -1235,7 +1235,7 @@
  32.268    have 2: "continuous_on {a .. b} f"
  32.269      using cont by (intro continuous_at_imp_continuous_on) auto
  32.270    show ?has ?eq
  32.271 -    using has_bochner_integral_FTC_Icc[OF `a \<le> b` 1 2] integral_FTC_Icc[OF `a \<le> b` 1 2]
  32.272 +    using has_bochner_integral_FTC_Icc[OF \<open>a \<le> b\<close> 1 2] integral_FTC_Icc[OF \<open>a \<le> b\<close> 1 2]
  32.273      by (auto simp: mult.commute)
  32.274  qed
  32.275  
  32.276 @@ -1300,7 +1300,7 @@
  32.277      by (intro derivative_eq_intros) auto
  32.278  qed (auto simp: field_simps simp del: of_nat_Suc)
  32.279  
  32.280 -subsection {* Integration by parts *}
  32.281 +subsection \<open>Integration by parts\<close>
  32.282  
  32.283  lemma integral_by_parts_integrable:
  32.284    fixes f g F G::"real \<Rightarrow> real"
    33.1 --- a/src/HOL/Probability/Measurable.thy	Mon Dec 07 16:48:10 2015 +0000
    33.2 +++ b/src/HOL/Probability/Measurable.thy	Mon Dec 07 20:19:59 2015 +0100
    33.3 @@ -7,7 +7,7 @@
    33.4      "~~/src/HOL/Library/Order_Continuity"
    33.5  begin
    33.6  
    33.7 -subsection {* Measurability prover *}
    33.8 +subsection \<open>Measurability prover\<close>
    33.9  
   33.10  lemma (in algebra) sets_Collect_finite_All:
   33.11    assumes "\<And>i. i \<in> S \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M" "finite S"
   33.12 @@ -48,7 +48,7 @@
   33.13  
   33.14  ML_file "measurable.ML"
   33.15  
   33.16 -attribute_setup measurable = {*
   33.17 +attribute_setup measurable = \<open>
   33.18    Scan.lift (
   33.19      (Args.add >> K true || Args.del >> K false || Scan.succeed true) --
   33.20      Scan.optional (Args.parens (
   33.21 @@ -56,7 +56,7 @@
   33.22        Scan.optional (Args.$$$ "generic" >> K Measurable.Generic) Measurable.Concrete))
   33.23      (false, Measurable.Concrete) >>
   33.24      Measurable.measurable_thm_attr)
   33.25 -*} "declaration of measurability theorems"
   33.26 +\<close> "declaration of measurability theorems"
   33.27  
   33.28  attribute_setup measurable_dest = Measurable.dest_thm_attr
   33.29    "add dest rule to measurability prover"
   33.30 @@ -67,11 +67,11 @@
   33.31  method_setup measurable = \<open> Scan.lift (Scan.succeed (METHOD o Measurable.measurable_tac)) \<close>
   33.32    "measurability prover"
   33.33  
   33.34 -simproc_setup measurable ("A \<in> sets M" | "f \<in> measurable M N") = {* K Measurable.simproc *}
   33.35 +simproc_setup measurable ("A \<in> sets M" | "f \<in> measurable M N") = \<open>K Measurable.simproc\<close>
   33.36  
   33.37 -setup {*
   33.38 +setup \<open>
   33.39    Global_Theory.add_thms_dynamic (@{binding measurable}, Measurable.get_all)
   33.40 -*}
   33.41 +\<close>
   33.42    
   33.43  declare
   33.44    pred_sets1[measurable_dest]
   33.45 @@ -288,7 +288,7 @@
   33.46    { fix x i j assume "P i x" "\<forall>n\<ge>j. \<not> P n x"
   33.47      then have "finite {i. P i x}"
   33.48        by (auto simp: subset_eq not_le[symmetric] finite_nat_iff_bounded)
   33.49 -    with `P i x` have "P (Max {i. P i x}) x" "i \<le> Max {i. P i x}" "finite {i. P i x}"
   33.50 +    with \<open>P i x\<close> have "P (Max {i. P i x}) x" "i \<le> Max {i. P i x}" "finite {i. P i x}"
   33.51        using Max_in[of "{i. P i x}"] by auto }
   33.52    note 2 = this
   33.53  
   33.54 @@ -323,7 +323,7 @@
   33.55    { fix x i j assume "P i x" "\<forall>n\<ge>j. \<not> P n x"
   33.56      then have "finite {i. P i x}"
   33.57        by (auto simp: subset_eq not_le[symmetric] finite_nat_iff_bounded)
   33.58 -    with `P i x` have "P (Min {i. P i x}) x" "Min {i. P i x} \<le> i" "finite {i. P i x}"
   33.59 +    with \<open>P i x\<close> have "P (Min {i. P i x}) x" "Min {i. P i x} \<le> i" "finite {i. P i x}"
   33.60        using Min_in[of "{i. P i x}"] by auto }
   33.61    note 2 = this
   33.62  
   33.63 @@ -380,7 +380,7 @@
   33.64    unfolding pred_def
   33.65    by (auto intro!: sets.sets_Collect_countable_All' sets.sets_Collect_countable_Ex' assms)
   33.66  
   33.67 -subsection {* Measurability for (co)inductive predicates *}
   33.68 +subsection \<open>Measurability for (co)inductive predicates\<close>
   33.69  
   33.70  lemma measurable_bot[measurable]: "bot \<in> measurable M (count_space UNIV)"
   33.71    by (simp add: bot_fun_def)
   33.72 @@ -427,7 +427,7 @@
   33.73    assumes *: "\<And>M A. P M \<Longrightarrow> (\<And>N. P N \<Longrightarrow> A \<in> measurable N (count_space UNIV)) \<Longrightarrow> F A \<in> measurable M (count_space UNIV)"
   33.74    shows "lfp F \<in> measurable M (count_space UNIV)"
   33.75  proof -
   33.76 -  { fix i from `P M` have "((F ^^ i) bot) \<in> measurable M (count_space UNIV)"
   33.77 +  { fix i from \<open>P M\<close> have "((F ^^ i) bot) \<in> measurable M (count_space UNIV)"
   33.78        by (induct i arbitrary: M) (auto intro!: *) }
   33.79    then have "(\<lambda>x. SUP i. (F ^^ i) bot x) \<in> measurable M (count_space UNIV)"
   33.80      by measurable
   33.81 @@ -450,7 +450,7 @@
   33.82    assumes *: "\<And>M A. P M \<Longrightarrow> (\<And>N. P N \<Longrightarrow> A \<in> measurable N (count_space UNIV)) \<Longrightarrow> F A \<in> measurable M (count_space UNIV)"
   33.83    shows "gfp F \<in> measurable M (count_space UNIV)"
   33.84  proof -
   33.85 -  { fix i from `P M` have "((F ^^ i) top) \<in> measurable M (count_space UNIV)"
   33.86 +  { fix i from \<open>P M\<close> have "((F ^^ i) top) \<in> measurable M (count_space UNIV)"
   33.87        by (induct i arbitrary: M) (auto intro!: *) }
   33.88    then have "(\<lambda>x. INF i. (F ^^ i) top x) \<in> measurable M (count_space UNIV)"
   33.89      by measurable
   33.90 @@ -473,7 +473,7 @@
   33.91    assumes *: "\<And>M A s. P M s \<Longrightarrow> (\<And>N t. P N t \<Longrightarrow> A t \<in> measurable N (count_space UNIV)) \<Longrightarrow> F A s \<in> measurable M (count_space UNIV)"
   33.92    shows "lfp F s \<in> measurable M (count_space UNIV)"
   33.93  proof -
   33.94 -  { fix i from `P M s` have "(\<lambda>x. (F ^^ i) bot s x) \<in> measurable M (count_space UNIV)"
   33.95 +  { fix i from \<open>P M s\<close> have "(\<lambda>x. (F ^^ i) bot s x) \<in> measurable M (count_space UNIV)"
   33.96        by (induct i arbitrary: M s) (auto intro!: *) }
   33.97    then have "(\<lambda>x. SUP i. (F ^^ i) bot s x) \<in> measurable M (count_space UNIV)"
   33.98      by measurable
   33.99 @@ -489,7 +489,7 @@
  33.100    assumes *: "\<And>M A s. P M s \<Longrightarrow> (\<And>N t. P N t \<Longrightarrow> A t \<in> measurable N (count_space UNIV)) \<Longrightarrow> F A s \<in> measurable M (count_space UNIV)"
  33.101    shows "gfp F s \<in> measurable M (count_space UNIV)"
  33.102  proof -
  33.103 -  { fix i from `P M s` have "(\<lambda>x. (F ^^ i) top s x) \<in> measurable M (count_space UNIV)"
  33.104 +  { fix i from \<open>P M s\<close> have "(\<lambda>x. (F ^^ i) top s x) \<in> measurable M (count_space UNIV)"
  33.105        by (induct i arbitrary: M s) (auto intro!: *) }
  33.106    then have "(\<lambda>x. INF i. (F ^^ i) top s x) \<in> measurable M (count_space UNIV)"
  33.107      by measurable
  33.108 @@ -511,7 +511,7 @@
  33.109    have "f -` {a} \<inter> space M = {x\<in>space M. f x = a}"
  33.110      by auto
  33.111    { fix i :: nat
  33.112 -    from `R f` have "Measurable.pred M (\<lambda>x. f x = enat i)"
  33.113 +    from \<open>R f\<close> have "Measurable.pred M (\<lambda>x. f x = enat i)"
  33.114      proof (induction i arbitrary: f)
  33.115        case 0
  33.116        from *[OF this] obtain g h i P
  33.117 @@ -533,7 +533,7 @@
  33.118          (\<lambda>x. (P x \<longrightarrow> h x = enat (Suc n)) \<and> (\<not> P x \<longrightarrow> g (i x) = enat n))"
  33.119          by (auto simp: f zero_enat_def[symmetric] eSuc_enat[symmetric])
  33.120        also have "Measurable.pred M \<dots>"
  33.121 -        by (intro pred_intros_logic measurable_compose[OF M(2)] Suc `R g`) measurable
  33.122 +        by (intro pred_intros_logic measurable_compose[OF M(2)] Suc \<open>R g\<close>) measurable
  33.123        finally show ?case .
  33.124      qed
  33.125      then have "f -` {enat i} \<inter> space M \<in> sets M"
    34.1 --- a/src/HOL/Probability/Measure_Space.thy	Mon Dec 07 16:48:10 2015 +0000
    34.2 +++ b/src/HOL/Probability/Measure_Space.thy	Mon Dec 07 20:19:59 2015 +0100
    34.3 @@ -4,7 +4,7 @@
    34.4      Author:     Armin Heller, TU München
    34.5  *)
    34.6  
    34.7 -section {* Measure spaces and their properties *}
    34.8 +section \<open>Measure spaces and their properties\<close>
    34.9  
   34.10  theory Measure_Space
   34.11  imports
   34.12 @@ -19,7 +19,7 @@
   34.13    shows "(\<Sum>n. f n * indicator (A n) x) = f i"
   34.14  proof -
   34.15    have **: "\<And>n. f n * indicator (A n) x = (if n = i then f n else 0 :: ereal)"
   34.16 -    using `x \<in> A i` assms unfolding disjoint_family_on_def indicator_def by auto
   34.17 +    using \<open>x \<in> A i\<close> assms unfolding disjoint_family_on_def indicator_def by auto
   34.18    then have "\<And>n. (\<Sum>j<n. f j * indicator (A j) x) = (if i < n then f i else 0 :: ereal)"
   34.19      by (auto simp: setsum.If_cases)
   34.20    moreover have "(SUP n. if i < n then f i else 0) = (f i :: ereal)"
   34.21 @@ -37,7 +37,7 @@
   34.22  proof cases
   34.23    assume *: "x \<in> (\<Union>i. A i)"
   34.24    then obtain i where "x \<in> A i" by auto
   34.25 -  from suminf_cmult_indicator[OF assms(1), OF `x \<in> A i`, of "\<lambda>k. 1"]
   34.26 +  from suminf_cmult_indicator[OF assms(1), OF \<open>x \<in> A i\<close>, of "\<lambda>k. 1"]
   34.27    show ?thesis using * by simp
   34.28  qed simp
   34.29  
   34.30 @@ -47,17 +47,17 @@
   34.31    shows "(\<Sum>i\<in>P. f i * indicator (A i) x) = f j"
   34.32  proof -
   34.33    have "P \<inter> {i. x \<in> A i} = {j}"
   34.34 -    using d `x \<in> A j` `j \<in> P` unfolding disjoint_family_on_def
   34.35 +    using d \<open>x \<in> A j\<close> \<open>j \<in> P\<close> unfolding disjoint_family_on_def
   34.36      by auto
   34.37    thus ?thesis
   34.38      unfolding indicator_def
   34.39 -    by (simp add: if_distrib setsum.If_cases[OF `finite P`])
   34.40 +    by (simp add: if_distrib setsum.If_cases[OF \<open>finite P\<close>])
   34.41  qed
   34.42  
   34.43 -text {*
   34.44 +text \<open>
   34.45    The type for emeasure spaces is already defined in @{theory Sigma_Algebra}, as it is also used to
   34.46    represent sigma algebras (with an arbitrary emeasure).
   34.47 -*}
   34.48 +\<close>
   34.49  
   34.50  subsection "Extend binary sets"
   34.51  
   34.52 @@ -91,12 +91,12 @@
   34.53    shows "f {} = 0 \<Longrightarrow> (\<Sum>n. f (binaryset A B n)) = f A + f B"
   34.54    by (metis binaryset_sums sums_unique)
   34.55  
   34.56 -subsection {* Properties of a premeasure @{term \<mu>} *}
   34.57 +subsection \<open>Properties of a premeasure @{term \<mu>}\<close>
   34.58  
   34.59 -text {*
   34.60 +text \<open>
   34.61    The definitions for @{const positive} and @{const countably_additive} should be here, by they are
   34.62    necessary to define @{typ "'a measure"} in @{theory Sigma_Algebra}.
   34.63 -*}
   34.64 +\<close>
   34.65  
   34.66  definition additive where
   34.67    "additive M \<mu> \<longleftrightarrow> (\<forall>x\<in>M. \<forall>y\<in>M. x \<inter> y = {} \<longrightarrow> \<mu> (x \<union> y) = \<mu> x + \<mu> y)"
   34.68 @@ -134,7 +134,7 @@
   34.69    also have "\<dots> = f (A n \<union> disjointed A (Suc n))"
   34.70      using A by (subst f(2)[THEN additiveD]) (auto simp: disjointed_mono)
   34.71    also have "A n \<union> disjointed A (Suc n) = A (Suc n)"
   34.72 -    using `incseq A` by (auto dest: incseq_SucD simp: disjointed_mono)
   34.73 +    using \<open>incseq A\<close> by (auto dest: incseq_SucD simp: disjointed_mono)
   34.74    finally show ?case .
   34.75  qed simp
   34.76  
   34.77 @@ -144,7 +144,7 @@
   34.78        and A: "A`S \<subseteq> M"
   34.79        and disj: "disjoint_family_on A S"
   34.80    shows  "(\<Sum>i\<in>S. f (A i)) = f (\<Union>i\<in>S. A i)"
   34.81 -  using `finite S` disj A
   34.82 +  using \<open>finite S\<close> disj A
   34.83  proof induct
   34.84    case empty show ?case using f by (simp add: positive_def)
   34.85  next
   34.86 @@ -154,7 +154,7 @@
   34.87    moreover
   34.88    have "A s \<in> M" using insert by blast
   34.89    moreover have "(\<Union>i\<in>S. A i) \<in> M"
   34.90 -    using insert `finite S` by auto
   34.91 +    using insert \<open>finite S\<close> by auto
   34.92    ultimately have "f (A s \<union> (\<Union>i\<in>S. A i)) = f (A s) + f(\<Union>i\<in>S. A i)"
   34.93      using ad UNION_in_sets A by (auto simp add: additive_def)
   34.94    with insert show ?case using ad disjoint_family_on_mono[of S "insert s S" A]
   34.95 @@ -254,7 +254,7 @@
   34.96      by (metis F(2) assms(1) infinite_super sets_into_space)
   34.97  
   34.98    have F_subset: "{i. \<mu> (F i) \<noteq> 0} \<subseteq> {i. F i \<noteq> {}}"
   34.99 -    by (auto simp: positiveD_empty[OF `positive M \<mu>`])
  34.100 +    by (auto simp: positiveD_empty[OF \<open>positive M \<mu>\<close>])
  34.101    moreover have fin_not_empty: "finite {i. F i \<noteq> {}}"
  34.102    proof (rule finite_imageD)
  34.103      from f have "f`{i. F i \<noteq> {}} \<subseteq> (\<Union>i. F i)" by auto
  34.104 @@ -272,7 +272,7 @@
  34.105    also have "\<dots> = (\<Sum>i | F i \<noteq> {}. \<mu> (F i))"
  34.106      using fin_not_empty F_subset by (rule setsum.mono_neutral_left) auto
  34.107    also have "\<dots> = \<mu> (\<Union>i\<in>{i. F i \<noteq> {}}. F i)"
  34.108 -    using `positive M \<mu>` `additive M \<mu>` fin_not_empty disj_not_empty F by (intro additive_sum) auto
  34.109 +    using \<open>positive M \<mu>\<close> \<open>additive M \<mu>\<close> fin_not_empty disj_not_empty F by (intro additive_sum) auto
  34.110    also have "\<dots> = \<mu> (\<Union>i. F i)"
  34.111      by (rule arg_cong[where f=\<mu>]) auto
  34.112    finally show "(\<Sum>i. \<mu> (F i)) = \<mu> (\<Union>i. F i)" .
  34.113 @@ -327,7 +327,7 @@
  34.114    assume cont: "(\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) \<in> M \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) ----> f (\<Inter>i. A i))"
  34.115    fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" "decseq A" "(\<Inter>i. A i) = {}" "\<forall>i. f (A i) \<noteq> \<infinity>"
  34.116    with cont[THEN spec, of A] show "(\<lambda>i. f (A i)) ----> 0"
  34.117 -    using `positive M f`[unfolded positive_def] by auto
  34.118 +    using \<open>positive M f\<close>[unfolded positive_def] by auto
  34.119  next
  34.120    assume cont: "\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) ----> 0"
  34.121    fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" "decseq A" "(\<Inter>i. A i) \<in> M" "\<forall>i. f (A i) \<noteq> \<infinity>"
  34.122 @@ -415,7 +415,7 @@
  34.123    using empty_continuous_imp_continuous_from_below[OF f fin] cont
  34.124    by blast
  34.125  
  34.126 -subsection {* Properties of @{const emeasure} *}
  34.127 +subsection \<open>Properties of @{const emeasure}\<close>
  34.128  
  34.129  lemma emeasure_positive: "positive (sets M) (emeasure M)"
  34.130    by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)
  34.131 @@ -483,7 +483,7 @@
  34.132      by (rule plus_emeasure[symmetric]) (auto simp add: s)
  34.133    finally have "emeasure M (space M) = emeasure M s + emeasure M (space M - s)" .
  34.134    then show ?thesis
  34.135 -    using fin `0 \<le> emeasure M s`
  34.136 +    using fin \<open>0 \<le> emeasure M s\<close>
  34.137      unfolding ereal_eq_minus_iff by (auto simp: ac_simps)
  34.138  qed
  34.139  
  34.140 @@ -493,13 +493,13 @@
  34.141    shows "emeasure M (A - B) = emeasure M A - emeasure M B"
  34.142  proof -
  34.143    have "0 \<le> emeasure M B" using assms by auto
  34.144 -  have "(A - B) \<union> B = A" using `B \<subseteq> A` by auto
  34.145 +  have "(A - B) \<union> B = A" using \<open>B \<subseteq> A\<close> by auto
  34.146    then have "emeasure M A = emeasure M ((A - B) \<union> B)" by simp
  34.147    also have "\<dots> = emeasure M (A - B) + emeasure M B"
  34.148      by (subst plus_emeasure[symmetric]) auto
  34.149    finally show "emeasure M (A - B) = emeasure M A - emeasure M B"
  34.150      unfolding ereal_eq_minus_iff
  34.151 -    using finite `0 \<le> emeasure M B` by auto
  34.152 +    using finite \<open>0 \<le> emeasure M B\<close> by auto
  34.153  qed
  34.154  
  34.155  lemma Lim_emeasure_incseq:
  34.156 @@ -541,13 +541,13 @@
  34.157      unfolding minus_ereal_def using A0 assms
  34.158      by (subst SUP_ereal_add) (auto simp add: decseq_emeasure)
  34.159    also have "\<dots> = (SUP n. emeasure M (A 0 - A n))"
  34.160 -    using A finite `decseq A`[unfolded decseq_def] by (subst emeasure_Diff) auto
  34.161 +    using A finite \<open>decseq A\<close>[unfolded decseq_def] by (subst emeasure_Diff) auto
  34.162    also have "\<dots> = emeasure M (\<Union>i. A 0 - A i)"
  34.163    proof (rule SUP_emeasure_incseq)
  34.164      show "range (\<lambda>n. A 0 - A n) \<subseteq> sets M"
  34.165        using A by auto
  34.166      show "incseq (\<lambda>n. A 0 - A n)"
  34.167 -      using `decseq A` by (auto simp add: incseq_def decseq_def)
  34.168 +      using \<open>decseq A\<close> by (auto simp add: incseq_def decseq_def)
  34.169    qed
  34.170    also have "\<dots> = emeasure M (A 0) - emeasure M (\<Inter>i. A i)"
  34.171      using A finite * by (simp, subst emeasure_Diff) auto
  34.172 @@ -616,7 +616,7 @@
  34.173  proof -
  34.174    have "emeasure M {x\<in>space M. lfp F x} = emeasure M (\<Union>i. {x\<in>space M. (F ^^ i) (\<lambda>x. False) x})"
  34.175      using sup_continuous_lfp[OF cont] by (auto simp add: bot_fun_def intro!: arg_cong2[where f=emeasure])
  34.176 -  moreover { fix i from `P M` have "{x\<in>space M. (F ^^ i) (\<lambda>x. False) x} \<in> sets M"
  34.177 +  moreover { fix i from \<open>P M\<close> have "{x\<in>space M. (F ^^ i) (\<lambda>x. False) x} \<in> sets M"
  34.178      by (induct i arbitrary: M) (auto simp add: pred_def[symmetric] intro: *) }
  34.179    moreover have "incseq (\<lambda>i. {x\<in>space M. (F ^^ i) (\<lambda>x. False) x})"
  34.180    proof (rule incseq_SucI)
  34.181 @@ -694,7 +694,7 @@
  34.182    assumes sets: "{x} \<in> sets M" "A \<in> sets M" and "x \<notin> A"
  34.183    shows "emeasure M (insert x A) = emeasure M {x} + emeasure M A"
  34.184  proof -
  34.185 -  have "{x} \<inter> A = {}" using `x \<notin> A` by auto
  34.186 +  have "{x} \<inter> A = {}" using \<open>x \<notin> A\<close> by auto
  34.187    from plus_emeasure[OF sets this] show ?thesis by simp
  34.188  qed
  34.189  
  34.190 @@ -717,7 +717,7 @@
  34.191    have "(\<Sum>i\<in>S. emeasure M (A \<inter> (B i))) = emeasure M (\<Union>i\<in>S. A \<inter> (B i))"
  34.192    proof (rule setsum_emeasure)
  34.193      show "disjoint_family_on (\<lambda>i. A \<inter> B i) S"
  34.194 -      using `disjoint_family_on B S`
  34.195 +      using \<open>disjoint_family_on B S\<close>
  34.196        unfolding disjoint_family_on_def by auto
  34.197    qed (insert assms, auto)
  34.198    also have "(\<Union>i\<in>S. A \<inter> (B i)) = A"
  34.199 @@ -747,11 +747,11 @@
  34.200    fix X assume "X \<in> sets M"
  34.201    then have X: "X \<subseteq> A" by auto
  34.202    then have "emeasure M X = (\<Sum>a\<in>X. emeasure M {a})"
  34.203 -    using `finite A` by (subst emeasure_eq_setsum_singleton) (auto dest: finite_subset)
  34.204 +    using \<open>finite A\<close> by (subst emeasure_eq_setsum_singleton) (auto dest: finite_subset)
  34.205    also have "\<dots> = (\<Sum>a\<in>X. emeasure N {a})"
  34.206      using X eq by (auto intro!: setsum.cong)
  34.207    also have "\<dots> = emeasure N X"
  34.208 -    using X `finite A` by (subst emeasure_eq_setsum_singleton) (auto dest: finite_subset)
  34.209 +    using X \<open>finite A\<close> by (subst emeasure_eq_setsum_singleton) (auto dest: finite_subset)
  34.210    finally show "emeasure M X = emeasure N X" .
  34.211  qed simp
  34.212  
  34.213 @@ -767,18 +767,18 @@
  34.214    let ?\<mu>  = "emeasure M" and ?\<nu> = "emeasure N"
  34.215    interpret S: sigma_algebra \<Omega> "sigma_sets \<Omega> E" by (rule sigma_algebra_sigma_sets) fact
  34.216    have "space M = \<Omega>"
  34.217 -    using sets.top[of M] sets.space_closed[of M] S.top S.space_closed `sets M = sigma_sets \<Omega> E`
  34.218 +    using sets.top[of M] sets.space_closed[of M] S.top S.space_closed \<open>sets M = sigma_sets \<Omega> E\<close>
  34.219      by blast
  34.220  
  34.221    { fix F D assume "F \<in> E" and "?\<mu> F \<noteq> \<infinity>"
  34.222      then have [intro]: "F \<in> sigma_sets \<Omega> E" by auto
  34.223 -    have "?\<nu> F \<noteq> \<infinity>" using `?\<mu> F \<noteq> \<infinity>` `F \<in> E` eq by simp
  34.224 +    have "?\<nu> F \<noteq> \<infinity>" using \<open>?\<mu> F \<noteq> \<infinity>\<close> \<open>F \<in> E\<close> eq by simp
  34.225      assume "D \<in> sets M"
  34.226 -    with `Int_stable E` `E \<subseteq> Pow \<Omega>` have "emeasure M (F \<inter> D) = emeasure N (F \<inter> D)"
  34.227 +    with \<open>Int_stable E\<close> \<open>E \<subseteq> Pow \<Omega>\<close> have "emeasure M (F \<inter> D) = emeasure N (F \<inter> D)"
  34.228        unfolding M
  34.229      proof (induct rule: sigma_sets_induct_disjoint)
  34.230        case (basic A)
  34.231 -      then have "F \<inter> A \<in> E" using `Int_stable E` `F \<in> E` by (auto simp: Int_stable_def)
  34.232 +      then have "F \<inter> A \<in> E" using \<open>Int_stable E\<close> \<open>F \<in> E\<close> by (auto simp: Int_stable_def)
  34.233        then show ?case using eq by auto
  34.234      next
  34.235        case empty then show ?case by simp
  34.236 @@ -786,19 +786,19 @@
  34.237        case (compl A)
  34.238        then have **: "F \<inter> (\<Omega> - A) = F - (F \<inter> A)"
  34.239          and [intro]: "F \<inter> A \<in> sigma_sets \<Omega> E"
  34.240 -        using `F \<in> E` S.sets_into_space by (auto simp: M)
  34.241 +        using \<open>F \<in> E\<close> S.sets_into_space by (auto simp: M)
  34.242        have "?\<nu> (F \<inter> A) \<le> ?\<nu> F" by (auto intro!: emeasure_mono simp: M N)
  34.243 -      then have "?\<nu> (F \<inter> A) \<noteq> \<infinity>" using `?\<nu> F \<noteq> \<infinity>` by auto
  34.244 +      then have "?\<nu> (F \<inter> A) \<noteq> \<infinity>" using \<open>?\<nu> F \<noteq> \<infinity>\<close> by auto
  34.245        have "?\<mu> (F \<inter> A) \<le> ?\<mu> F" by (auto intro!: emeasure_mono simp: M N)
  34.246 -      then have "?\<mu> (F \<inter> A) \<noteq> \<infinity>" using `?\<mu> F \<noteq> \<infinity>` by auto
  34.247 +      then have "?\<mu> (F \<inter> A) \<noteq> \<infinity>" using \<open>?\<mu> F \<noteq> \<infinity>\<close> by auto
  34.248        then have "?\<mu> (F \<inter> (\<Omega> - A)) = ?\<mu> F - ?\<mu> (F \<inter> A)" unfolding **
  34.249 -        using `F \<inter> A \<in> sigma_sets \<Omega> E` by (auto intro!: emeasure_Diff simp: M N)
  34.250 -      also have "\<dots> = ?\<nu> F - ?\<nu> (F \<inter> A)" using eq `F \<in> E` compl by simp
  34.251 +        using \<open>F \<inter> A \<in> sigma_sets \<Omega> E\<close> by (auto intro!: emeasure_Diff simp: M N)
  34.252 +      also have "\<dots> = ?\<nu> F - ?\<nu> (F \<inter> A)" using eq \<open>F \<in> E\<close> compl by simp
  34.253        also have "\<dots> = ?\<nu> (F \<inter> (\<Omega> - A))" unfolding **
  34.254 -        using `F \<inter> A \<in> sigma_sets \<Omega> E` `?\<nu> (F \<inter> A) \<noteq> \<infinity>`
  34.255 +        using \<open>F \<inter> A \<in> sigma_sets \<Omega> E\<close> \<open>?\<nu> (F \<inter> A) \<noteq> \<infinity>\<close>
  34.256          by (auto intro!: emeasure_Diff[symmetric] simp: M N)
  34.257        finally show ?case
  34.258 -        using `space M = \<Omega>` by auto
  34.259 +        using \<open>space M = \<Omega>\<close> by auto
  34.260      next
  34.261        case (union A)
  34.262        then have "?\<mu> (\<Union>x. F \<inter> A x) = ?\<nu> (\<Union>x. F \<inter> A x)"
  34.263 @@ -815,10 +815,10 @@
  34.264        using A(1) by (auto simp: subset_eq M)
  34.265      fix F assume "F \<in> sets M"
  34.266      let ?D = "disjointed (\<lambda>i. F \<inter> A i)"
  34.267 -    from `space M = \<Omega>` have F_eq: "F = (\<Union>i. ?D i)"
  34.268 -      using `F \<in> sets M`[THEN sets.sets_into_space] A(2)[symmetric] by (auto simp: UN_disjointed_eq)
  34.269 +    from \<open>space M = \<Omega>\<close> have F_eq: "F = (\<Union>i. ?D i)"
  34.270 +      using \<open>F \<in> sets M\<close>[THEN sets.sets_into_space] A(2)[symmetric] by (auto simp: UN_disjointed_eq)
  34.271      have [simp, intro]: "\<And>i. ?D i \<in> sets M"
  34.272 -      using sets.range_disjointed_sets[of "\<lambda>i. F \<inter> A i" M] `F \<in> sets M`
  34.273 +      using sets.range_disjointed_sets[of "\<lambda>i. F \<inter> A i" M] \<open>F \<in> sets M\<close>
  34.274        by (auto simp: subset_eq)
  34.275      have "disjoint_family ?D"
  34.276        by (auto simp: disjoint_family_disjointed)
  34.277 @@ -832,7 +832,7 @@
  34.278          using *[of "A i" "?D i", OF _ A(3)] A(1) by auto
  34.279      qed
  34.280      ultimately show "emeasure M F = emeasure N F"
  34.281 -      by (simp add: image_subset_iff `sets M = sets N`[symmetric] F_eq[symmetric] suminf_emeasure)
  34.282 +      by (simp add: image_subset_iff \<open>sets M = sets N\<close>[symmetric] F_eq[symmetric] suminf_emeasure)
  34.283    qed
  34.284  qed
  34.285  
  34.286 @@ -845,7 +845,7 @@
  34.287      by (simp add: emeasure_countably_additive)
  34.288  qed simp_all
  34.289  
  34.290 -subsection {* @{text \<mu>}-null sets *}
  34.291 +subsection \<open>\<open>\<mu>\<close>-null sets\<close>
  34.292  
  34.293  definition null_sets :: "'a measure \<Rightarrow> 'a set set" where
  34.294    "null_sets M = {N\<in>sets M. emeasure M N = 0}"
  34.295 @@ -901,10 +901,10 @@
  34.296      show "(\<Union>i\<in>I. N i) \<in> sets M"
  34.297        using assms by (intro sets.countable_UN') auto
  34.298      have "emeasure M (\<Union>i\<in>I. N i) \<le> (\<Sum>n. emeasure M (N (from_nat_into I n)))"
  34.299 -      unfolding UN_from_nat_into[OF `countable I` `I \<noteq> {}`]
  34.300 -      using assms `I \<noteq> {}` by (intro emeasure_subadditive_countably) (auto intro: from_nat_into)
  34.301 +      unfolding UN_from_nat_into[OF \<open>countable I\<close> \<open>I \<noteq> {}\<close>]
  34.302 +      using assms \<open>I \<noteq> {}\<close> by (intro emeasure_subadditive_countably) (auto intro: from_nat_into)
  34.303      also have "(\<lambda>n. emeasure M (N (from_nat_into I n))) = (\<lambda>_. 0)"
  34.304 -      using assms `I \<noteq> {}` by (auto intro: from_nat_into)
  34.305 +      using assms \<open>I \<noteq> {}\<close> by (auto intro: from_nat_into)
  34.306      finally show "emeasure M (\<Union>i\<in>I. N i) = 0"
  34.307        by (intro antisym emeasure_nonneg) simp
  34.308    qed
  34.309 @@ -953,7 +953,7 @@
  34.310      by (subst plus_emeasure[symmetric]) auto
  34.311  qed
  34.312  
  34.313 -subsection {* The almost everywhere filter (i.e.\ quantifier) *}
  34.314 +subsection \<open>The almost everywhere filter (i.e.\ quantifier)\<close>
  34.315  
  34.316  definition ae_filter :: "'a measure \<Rightarrow> 'a filter" where
  34.317    "ae_filter M = (INF N:null_sets M. principal (space M - N))"
  34.318 @@ -983,7 +983,7 @@
  34.319    have "0 \<le> emeasure M ?P" by auto
  34.320    moreover have "emeasure M ?P \<le> emeasure M N"
  34.321      using assms N(1,2) by (auto intro: emeasure_mono)
  34.322 -  ultimately have "emeasure M ?P = 0" unfolding `emeasure M N = 0` by auto
  34.323 +  ultimately have "emeasure M ?P = 0" unfolding \<open>emeasure M N = 0\<close> by auto
  34.324    then show "?P \<in> null_sets M" using assms by auto
  34.325  next
  34.326    assume "?P \<in> null_sets M" with assms show "AE x in M. P x" by (auto intro: AE_I')
  34.327 @@ -1138,7 +1138,7 @@
  34.328  lemma AE_finite_allI:
  34.329    assumes "finite S"
  34.330    shows "(\<And>s. s \<in> S \<Longrightarrow> AE x in M. Q s x) \<Longrightarrow> AE x in M. \<forall>s\<in>S. Q s x"
  34.331 -  using AE_finite_all[OF `finite S`] by auto
  34.332 +  using AE_finite_all[OF \<open>finite S\<close>] by auto
  34.333  
  34.334  lemma emeasure_mono_AE:
  34.335    assumes imp: "AE x in M. x \<in> A \<longrightarrow> x \<in> B"
  34.336 @@ -1187,7 +1187,7 @@
  34.337    finally show ?thesis .
  34.338  qed
  34.339  
  34.340 -subsection {* @{text \<sigma>}-finite Measures *}
  34.341 +subsection \<open>\<open>\<sigma>\<close>-finite Measures\<close>
  34.342  
  34.343  locale sigma_finite_measure =
  34.344    fixes M :: "'a measure"
  34.345 @@ -1204,19 +1204,19 @@
  34.346      using sigma_finite_countable by metis
  34.347    show thesis
  34.348    proof cases
  34.349 -    assume "A = {}" with `(\<Union>A) = space M` show thesis
  34.350 +    assume "A = {}" with \<open>(\<Union>A) = space M\<close> show thesis
  34.351        by (intro that[of "\<lambda>_. {}"]) auto
  34.352    next
  34.353      assume "A \<noteq> {}"
  34.354      show thesis
  34.355      proof
  34.356        show "range (from_nat_into A) \<subseteq> sets M"
  34.357 -        using `A \<noteq> {}` A by auto
  34.358 +        using \<open>A \<noteq> {}\<close> A by auto
  34.359        have "(\<Union>i. from_nat_into A i) = \<Union>A"
  34.360 -        using range_from_nat_into[OF `A \<noteq> {}` `countable A`] by auto
  34.361 +        using range_from_nat_into[OF \<open>A \<noteq> {}\<close> \<open>countable A\<close>] by auto
  34.362        with A show "(\<Union>i. from_nat_into A i) = space M"
  34.363          by auto
  34.364 -    qed (intro A from_nat_into `A \<noteq> {}`)
  34.365 +    qed (intro A from_nat_into \<open>A \<noteq> {}\<close>)
  34.366    qed
  34.367  qed
  34.368  
  34.369 @@ -1275,7 +1275,7 @@
  34.370    qed
  34.371  qed
  34.372  
  34.373 -subsection {* Measure space induced by distribution of @{const measurable}-functions *}
  34.374 +subsection \<open>Measure space induced by distribution of @{const measurable}-functions\<close>
  34.375  
  34.376  definition distr :: "'a measure \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b measure" where
  34.377    "distr M N f = measure_of (space N) (sets N) (\<lambda>A. emeasure M (f -` A \<inter> space M))"
  34.378 @@ -1312,7 +1312,7 @@
  34.379      moreover have "(\<Union>i. f -`  A i \<inter> space M) \<in> sets M"
  34.380        using * by blast
  34.381      moreover have **: "disjoint_family (\<lambda>i. f -` A i \<inter> space M)"
  34.382 -      using `disjoint_family A` by (auto simp: disjoint_family_on_def)
  34.383 +      using \<open>disjoint_family A\<close> by (auto simp: disjoint_family_on_def)
  34.384      ultimately show "(\<Sum>i. ?\<mu> (A i)) = ?\<mu> (\<Union>i. A i)"
  34.385        using suminf_emeasure[OF _ **] A f
  34.386        by (auto simp: comp_def vimage_UN)
  34.387 @@ -1334,21 +1334,21 @@
  34.388    shows "emeasure M' {x\<in>space M'. lfp F (f x)} = (SUP i. emeasure M' {x\<in>space M'. (F ^^ i) (\<lambda>x. False) (f x)})"
  34.389  proof (subst (1 2) emeasure_Collect_distr[symmetric, where X=f])
  34.390    show "f \<in> measurable M' M"  "f \<in> measurable M' M"
  34.391 -    using f[OF `P M`] by auto
  34.392 +    using f[OF \<open>P M\<close>] by auto
  34.393    { fix i show "Measurable.pred M ((F ^^ i) (\<lambda>x. False))"
  34.394 -    using `P M` by (induction i arbitrary: M) (auto intro!: *) }
  34.395 +    using \<open>P M\<close> by (induction i arbitrary: M) (auto intro!: *) }
  34.396    show "Measurable.pred M (lfp F)"
  34.397 -    using `P M` cont * by (rule measurable_lfp_coinduct[of P])
  34.398 +    using \<open>P M\<close> cont * by (rule measurable_lfp_coinduct[of P])
  34.399  
  34.400    have "emeasure (distr M' M f) {x \<in> space (distr M' M f). lfp F x} =
  34.401      (SUP i. emeasure (distr M' M f) {x \<in> space (distr M' M f). (F ^^ i) (\<lambda>x. False) x})"
  34.402 -    using `P M`
  34.403 +    using \<open>P M\<close>
  34.404    proof (coinduction arbitrary: M rule: emeasure_lfp')
  34.405      case (measurable A N) then have "\<And>N. P N \<Longrightarrow> Measurable.pred (distr M' N f) A"
  34.406        by metis
  34.407      then have "\<And>N. P N \<Longrightarrow> Measurable.pred N A"
  34.408        by simp
  34.409 -    with `P N`[THEN *] show ?case
  34.410 +    with \<open>P N\<close>[THEN *] show ?case
  34.411        by auto
  34.412    qed fact
  34.413    then show "emeasure (distr M' M f) {x \<in> space M. lfp F x} =
  34.414 @@ -1405,7 +1405,7 @@
  34.415    by (auto simp add: emeasure_distr measurable_space
  34.416             intro!: arg_cong[where f="emeasure M"] measure_eqI)
  34.417  
  34.418 -subsection {* Real measure values *}
  34.419 +subsection \<open>Real measure values\<close>
  34.420  
  34.421  lemma measure_nonneg: "0 \<le> measure M A"
  34.422    using emeasure_nonneg[of M A] unfolding measure_def by (auto intro: real_of_ereal_pos)
  34.423 @@ -1449,7 +1449,7 @@
  34.424      using measurable by (auto intro!: emeasure_mono)
  34.425    hence "measure M ((A - B) \<union> B) = measure M (A - B) + measure M B"
  34.426      using measurable finite by (rule_tac measure_Union) auto
  34.427 -  thus ?thesis using `B \<subseteq> A` by (auto simp: Un_absorb2)
  34.428 +  thus ?thesis using \<open>B \<subseteq> A\<close> by (auto simp: Un_absorb2)
  34.429  qed
  34.430  
  34.431  lemma measure_UNION:
  34.432 @@ -1548,7 +1548,7 @@
  34.433      by (intro lim_real_of_ereal) simp
  34.434  qed
  34.435  
  34.436 -subsection {* Measure spaces with @{term "emeasure M (space M) < \<infinity>"} *}
  34.437 +subsection \<open>Measure spaces with @{term "emeasure M (space M) < \<infinity>"}\<close>
  34.438  
  34.439  locale finite_measure = sigma_finite_measure M for M +
  34.440    assumes finite_emeasure_space: "emeasure M (space M) \<noteq> \<infinity>"
  34.441 @@ -1606,7 +1606,7 @@
  34.442    assumes A: "range A \<subseteq> sets M" and sum: "summable (\<lambda>i. measure M (A i))"
  34.443    shows "measure M (\<Union>i. A i) \<le> (\<Sum>i. measure M (A i))"
  34.444  proof -
  34.445 -  from `summable (\<lambda>i. measure M (A i))`
  34.446 +  from \<open>summable (\<lambda>i. measure M (A i))\<close>
  34.447    have "(\<lambda>i. ereal (measure M (A i))) sums ereal (\<Sum>i. measure M (A i))"
  34.448      by (simp add: sums_ereal) (rule summable_sums)
  34.449    from sums_unique[OF this, symmetric]
  34.450 @@ -1729,7 +1729,7 @@
  34.451    shows "measure M e = (\<Sum> x \<in> s. measure M (e \<inter> f x))"
  34.452  proof -
  34.453    have e: "e = (\<Union>i \<in> s. e \<inter> f i)"
  34.454 -    using `e \<in> sets M` sets.sets_into_space upper by blast
  34.455 +    using \<open>e \<in> sets M\<close> sets.sets_into_space upper by blast
  34.456    hence "measure M e = measure M (\<Union>i \<in> s. e \<inter> f i)" by simp
  34.457    also have "\<dots> = (\<Sum> x \<in> s. measure M (e \<inter> f x))"
  34.458    proof (rule finite_measure_finite_Union)
  34.459 @@ -1774,7 +1774,7 @@
  34.460      using bound[of x] sets_eq_imp_space_eq[OF sets] by (simp add: iter)
  34.461  qed (auto simp add: iter le_fun_def INF_apply[abs_def] intro!: meas cont)
  34.462  
  34.463 -subsection {* Counting space *}
  34.464 +subsection \<open>Counting space\<close>
  34.465  
  34.466  lemma strict_monoI_Suc:
  34.467    assumes ord [simp]: "(\<And>n. f n < f (Suc n))" shows "strict_mono f"
  34.468 @@ -1789,7 +1789,7 @@
  34.469      (is "_ = ?M X")
  34.470    unfolding count_space_def
  34.471  proof (rule emeasure_measure_of_sigma)
  34.472 -  show "X \<in> Pow A" using `X \<subseteq> A` by auto
  34.473 +  show "X \<in> Pow A" using \<open>X \<subseteq> A\<close> by auto
  34.474    show "sigma_algebra A (Pow A)" by (rule sigma_algebra_Pow)
  34.475    show positive: "positive (Pow A) ?M"
  34.476      by (auto simp: positive_def)
  34.477 @@ -1806,7 +1806,7 @@
  34.478      proof cases
  34.479        assume "\<exists>i. \<forall>j\<ge>i. F i = F j"
  34.480        then guess i .. note i = this
  34.481 -      { fix j from i `incseq F` have "F j \<subseteq> F i"
  34.482 +      { fix j from i \<open>incseq F\<close> have "F j \<subseteq> F i"
  34.483            by (cases "i \<le> j") (auto simp: incseq_def) }
  34.484        then have eq: "(\<Union>i. F i) = F i"
  34.485          by auto
  34.486 @@ -1815,11 +1815,11 @@
  34.487      next
  34.488        assume "\<not> (\<exists>i. \<forall>j\<ge>i. F i = F j)"
  34.489        then obtain f where f: "\<And>i. i \<le> f i" "\<And>i. F i \<noteq> F (f i)" by metis
  34.490 -      then have "\<And>i. F i \<subseteq> F (f i)" using `incseq F` by (auto simp: incseq_def)
  34.491 +      then have "\<And>i. F i \<subseteq> F (f i)" using \<open>incseq F\<close> by (auto simp: incseq_def)
  34.492        with f have *: "\<And>i. F i \<subset> F (f i)" by auto
  34.493  
  34.494        have "incseq (\<lambda>i. ?M (F i))"
  34.495 -        using `incseq F` unfolding incseq_def by (auto simp: card_mono dest: finite_subset)
  34.496 +        using \<open>incseq F\<close> unfolding incseq_def by (auto simp: card_mono dest: finite_subset)
  34.497        then have "(\<lambda>i. ?M (F i)) ----> (SUP n. ?M (F n))"
  34.498          by (rule LIMSEQ_SUP)
  34.499  
  34.500 @@ -1830,9 +1830,9 @@
  34.501            case (Suc n)
  34.502            then guess k .. note k = this
  34.503            moreover have "finite (F k) \<Longrightarrow> finite (F (f k)) \<Longrightarrow> card (F k) < card (F (f k))"
  34.504 -            using `F k \<subset> F (f k)` by (simp add: psubset_card_mono)
  34.505 +            using \<open>F k \<subset> F (f k)\<close> by (simp add: psubset_card_mono)
  34.506            moreover have "finite (F (f k)) \<Longrightarrow> finite (F k)"
  34.507 -            using `k \<le> f k` `incseq F` by (auto simp: incseq_def dest: finite_subset)
  34.508 +            using \<open>k \<le> f k\<close> \<open>incseq F\<close> by (auto simp: incseq_def dest: finite_subset)
  34.509            ultimately show ?case
  34.510              by (auto intro!: exI[of _ "f k"])
  34.511          qed auto
  34.512 @@ -1926,7 +1926,7 @@
  34.513    show "sigma_finite_measure (count_space A)" ..
  34.514  qed
  34.515  
  34.516 -subsection {* Measure restricted to space *}
  34.517 +subsection \<open>Measure restricted to space\<close>
  34.518  
  34.519  lemma emeasure_restrict_space:
  34.520    assumes "\<Omega> \<inter> space M \<in> sets M" "A \<subseteq> \<Omega>"
  34.521 @@ -1936,7 +1936,7 @@
  34.522    show ?thesis
  34.523    proof (rule emeasure_measure_of[OF restrict_space_def])
  34.524      show "op \<inter> \<Omega> ` sets M \<subseteq> Pow (\<Omega> \<inter> space M)" "A \<in> sets (restrict_space M \<Omega>)"
  34.525 -      using `A \<subseteq> \<Omega>` `A \<in> sets M` sets.space_closed by (auto simp: sets_restrict_space)
  34.526 +      using \<open>A \<subseteq> \<Omega>\<close> \<open>A \<in> sets M\<close> sets.space_closed by (auto simp: sets_restrict_space)
  34.527      show "positive (sets (restrict_space M \<Omega>)) (emeasure M)"
  34.528        by (auto simp: positive_def emeasure_nonneg)
  34.529      show "countably_additive (sets (restrict_space M \<Omega>)) (emeasure M)"
  34.530 @@ -2085,7 +2085,7 @@
  34.531    finally show "emeasure M X = emeasure N X" .
  34.532  qed fact
  34.533  
  34.534 -subsection {* Null measure *}
  34.535 +subsection \<open>Null measure\<close>
  34.536  
  34.537  definition "null_measure M = sigma (space M) (sets M)"
  34.538  
    35.1 --- a/src/HOL/Probability/Nonnegative_Lebesgue_Integration.thy	Mon Dec 07 16:48:10 2015 +0000
    35.2 +++ b/src/HOL/Probability/Nonnegative_Lebesgue_Integration.thy	Mon Dec 07 20:19:59 2015 +0100
    35.3 @@ -3,7 +3,7 @@
    35.4      Author:     Armin Heller, TU München
    35.5  *)
    35.6  
    35.7 -section {* Lebesgue Integration for Nonnegative Functions *}
    35.8 +section \<open>Lebesgue Integration for Nonnegative Functions\<close>
    35.9  
   35.10  theory Nonnegative_Lebesgue_Integration
   35.11    imports Measure_Space Borel_Space
   35.12 @@ -23,13 +23,13 @@
   35.13  
   35.14  subsection "Simple function"
   35.15  
   35.16 -text {*
   35.17 +text \<open>
   35.18  
   35.19  Our simple functions are not restricted to nonnegative real numbers. Instead
   35.20  they are just functions with a finite range and are measurable when singleton
   35.21  sets are measurable.
   35.22  
   35.23 -*}
   35.24 +\<close>
   35.25  
   35.26  definition "simple_function M g \<longleftrightarrow>
   35.27      finite (g ` space M) \<and>
   35.28 @@ -170,7 +170,7 @@
   35.29    have "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M =
   35.30        (f -` {f x} \<inter> space M) \<inter> (g -` {g x} \<inter> space M)"
   35.31      by auto
   35.32 -  with `x \<in> space M` show "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M \<in> sets M"
   35.33 +  with \<open>x \<in> space M\<close> show "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M \<in> sets M"
   35.34      using assms unfolding simple_function_def by auto
   35.35  qed
   35.36  
   35.37 @@ -316,7 +316,7 @@
   35.38            ultimately show False by auto
   35.39          qed
   35.40          then show "max 0 (u x) \<le> y" using real ux by simp
   35.41 -      qed (insert `0 \<le> y`, auto)
   35.42 +      qed (insert \<open>0 \<le> y\<close>, auto)
   35.43      qed
   35.44    qed auto
   35.45  qed
   35.46 @@ -425,7 +425,7 @@
   35.47      unfolding u_eq
   35.48    proof (rule seq)
   35.49      fix i show "P (U i)"
   35.50 -      using `simple_function M (U i)` nn[of i] not_inf[of _ i]
   35.51 +      using \<open>simple_function M (U i)\<close> nn[of i] not_inf[of _ i]
   35.52      proof (induct rule: simple_function_induct_nn)
   35.53        case (mult u c)
   35.54        show ?case
   35.55 @@ -441,7 +441,7 @@
   35.56            by auto
   35.57          with mult have "P u"
   35.58            by auto
   35.59 -        from x mult(5)[OF `x \<in> space M`] mult(1) mult(3)[of x] have "c < \<infinity>"
   35.60 +        from x mult(5)[OF \<open>x \<in> space M\<close>] mult(1) mult(3)[of x] have "c < \<infinity>"
   35.61            by auto
   35.62          with u_fin mult
   35.63          show ?thesis
   35.64 @@ -715,7 +715,7 @@
   35.65    shows "(\<integral>\<^sup>Sx. u x * indicator N x \<partial>M) = 0"
   35.66  proof -
   35.67    have "AE x in M. indicator N x = (0 :: ereal)"
   35.68 -    using `N \<in> null_sets M` by (auto simp: indicator_def intro!: AE_I[of _ _ N])
   35.69 +    using \<open>N \<in> null_sets M\<close> by (auto simp: indicator_def intro!: AE_I[of _ _ N])
   35.70    then have "(\<integral>\<^sup>Sx. u x * indicator N x \<partial>M) = (\<integral>\<^sup>Sx. 0 \<partial>M)"
   35.71      using assms apply (intro simple_integral_cong_AE) by auto
   35.72    then show ?thesis by simp
   35.73 @@ -741,7 +741,7 @@
   35.74    then show ?thesis by simp
   35.75  qed
   35.76  
   35.77 -subsection {* Integral on nonnegative functions *}
   35.78 +subsection \<open>Integral on nonnegative functions\<close>
   35.79  
   35.80  definition nn_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> ereal" ("integral\<^sup>N") where
   35.81    "integral\<^sup>N M f = (SUP g : {g. simple_function M g \<and> g \<le> max 0 \<circ> f}. integral\<^sup>S M g)"
   35.82 @@ -798,12 +798,12 @@
   35.83        have "real n \<le> ?y * (emeasure M) ?G"
   35.84          using \<mu>_G \<mu>_G_pos by (cases "(emeasure M) ?G") (auto simp: field_simps)
   35.85        also have "\<dots> = (\<integral>\<^sup>Sx. ?y * indicator ?G x \<partial>M)"
   35.86 -        using `0 \<le> ?y` `?g ?y \<in> ?A` gM
   35.87 +        using \<open>0 \<le> ?y\<close> \<open>?g ?y \<in> ?A\<close> gM
   35.88          by (subst simple_integral_cmult_indicator) auto
   35.89 -      also have "\<dots> \<le> integral\<^sup>S M (?g ?y)" using `?g ?y \<in> ?A` gM
   35.90 +      also have "\<dots> \<le> integral\<^sup>S M (?g ?y)" using \<open>?g ?y \<in> ?A\<close> gM
   35.91          by (intro simple_integral_mono) auto
   35.92        finally show "\<exists>i\<in>?A. real n \<le> integral\<^sup>S M i"
   35.93 -        using `?g ?y \<in> ?A` by blast
   35.94 +        using \<open>?g ?y \<in> ?A\<close> by blast
   35.95      qed
   35.96      then show ?thesis by simp
   35.97    qed
   35.98 @@ -898,7 +898,7 @@
   35.99    hence "a \<noteq> 0" by auto
  35.100    let ?B = "\<lambda>i. {x \<in> space M. a * u x \<le> f i x}"
  35.101    have B: "\<And>i. ?B i \<in> sets M"
  35.102 -    using f `simple_function M u`[THEN borel_measurable_simple_function] by auto
  35.103 +    using f \<open>simple_function M u\<close>[THEN borel_measurable_simple_function] by auto
  35.104  
  35.105    let ?uB = "\<lambda>i x. u x * indicator (?B i) x"
  35.106  
  35.107 @@ -906,7 +906,7 @@
  35.108      proof safe
  35.109        fix i x assume "a * u x \<le> f i x"
  35.110        also have "\<dots> \<le> f (Suc i) x"
  35.111 -        using `incseq f`[THEN incseq_SucD] unfolding le_fun_def by auto
  35.112 +        using \<open>incseq f\<close>[THEN incseq_SucD] unfolding le_fun_def by auto
  35.113        finally show "a * u x \<le> f (Suc i) x" .
  35.114      qed }
  35.115    note B_mono = this
  35.116 @@ -924,24 +924,24 @@
  35.117        fix x i assume x: "x \<in> space M"
  35.118        show "x \<in> (\<Union>i. ?B' (u x) i)"
  35.119        proof cases
  35.120 -        assume "u x = 0" thus ?thesis using `x \<in> space M` f(3) by simp
  35.121 +        assume "u x = 0" thus ?thesis using \<open>x \<in> space M\<close> f(3) by simp
  35.122        next
  35.123          assume "u x \<noteq> 0"
  35.124 -        with `a < 1` u_range[OF `x \<in> space M`]
  35.125 +        with \<open>a < 1\<close> u_range[OF \<open>x \<in> space M\<close>]
  35.126          have "a * u x < 1 * u x"
  35.127            by (intro ereal_mult_strict_right_mono) (auto simp: image_iff)
  35.128          also have "\<dots> \<le> (SUP i. f i x)" using u(2) by (auto simp: le_fun_def)
  35.129          finally obtain i where "a * u x < f i x" unfolding SUP_def
  35.130            by (auto simp add: less_SUP_iff)
  35.131          hence "a * u x \<le> f i x" by auto
  35.132 -        thus ?thesis using `x \<in> space M` by auto
  35.133 +        thus ?thesis using \<open>x \<in> space M\<close> by auto
  35.134        qed
  35.135      qed
  35.136      then show "?thesis i" using SUP_emeasure_incseq[OF 1 2] by simp
  35.137    qed
  35.138  
  35.139    have "integral\<^sup>S M u = (SUP i. integral\<^sup>S M (?uB i))"
  35.140 -    unfolding simple_integral_indicator[OF B `simple_function M u`]
  35.141 +    unfolding simple_integral_indicator[OF B \<open>simple_function M u\<close>]
  35.142    proof (subst SUP_ereal_setsum, safe)
  35.143      fix x n assume "x \<in> space M"
  35.144      with u_range show "incseq (\<lambda>i. u x * (emeasure M) (?B' (u x) i))" "\<And>i. 0 \<le> u x * (emeasure M) (?B' (u x) i)"
  35.145 @@ -957,21 +957,21 @@
  35.146    proof (safe intro!: SUP_mono bexI)
  35.147      fix i
  35.148      have "a * integral\<^sup>S M (?uB i) = (\<integral>\<^sup>Sx. a * ?uB i x \<partial>M)"
  35.149 -      using B `simple_function M u` u_range
  35.150 +      using B \<open>simple_function M u\<close> u_range
  35.151        by (subst simple_integral_mult) (auto split: split_indicator)
  35.152      also have "\<dots> \<le> integral\<^sup>N M (f i)"
  35.153      proof -
  35.154 -      have *: "simple_function M (\<lambda>x. a * ?uB i x)" using B `0 < a` u(1) by auto
  35.155 -      show ?thesis using f(3) * u_range `0 < a`
  35.156 +      have *: "simple_function M (\<lambda>x. a * ?uB i x)" using B \<open>0 < a\<close> u(1) by auto
  35.157 +      show ?thesis using f(3) * u_range \<open>0 < a\<close>
  35.158          by (subst nn_integral_eq_simple_integral[symmetric])
  35.159             (auto intro!: nn_integral_mono split: split_indicator)
  35.160      qed
  35.161      finally show "a * integral\<^sup>S M (?uB i) \<le> integral\<^sup>N M (f i)"
  35.162        by auto
  35.163    next
  35.164 -    fix i show "0 \<le> \<integral>\<^sup>S x. ?uB i x \<partial>M" using B `0 < a` u(1) u_range
  35.165 +    fix i show "0 \<le> \<integral>\<^sup>S x. ?uB i x \<partial>M" using B \<open>0 < a\<close> u(1) u_range
  35.166        by (intro simple_integral_nonneg) (auto split: split_indicator)
  35.167 -  qed (insert `0 < a`, auto)
  35.168 +  qed (insert \<open>0 < a\<close>, auto)
  35.169    ultimately show "a * integral\<^sup>S M u \<le> ?S" by simp
  35.170  qed
  35.171  
  35.172 @@ -987,7 +987,7 @@
  35.173  lemma nn_integral_max_0: "(\<integral>\<^sup>+x. max 0 (f x) \<partial>M) = integral\<^sup>N M f"
  35.174    by (simp add: le_fun_def nn_integral_def)
  35.175  
  35.176 -text {* Beppo-Levi monotone convergence theorem *}
  35.177 +text \<open>Beppo-Levi monotone convergence theorem\<close>
  35.178  lemma nn_integral_monotone_convergence_SUP:
  35.179    assumes f: "incseq f" "\<And>i. f i \<in> borel_measurable M"
  35.180    shows "(\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^sup>N M (f i))"
  35.181 @@ -1104,11 +1104,11 @@
  35.182    note l = nn_integral_monotone_convergence_simple[OF this(2,5,1)] this
  35.183  
  35.184    have inc: "incseq (\<lambda>i. a * integral\<^sup>S M (u i))" "incseq (\<lambda>i. integral\<^sup>S M (v i))"
  35.185 -    using u v `0 \<le> a`
  35.186 +    using u v \<open>0 \<le> a\<close>
  35.187      by (auto simp: incseq_Suc_iff le_fun_def
  35.188               intro!: add_mono ereal_mult_left_mono simple_integral_mono)
  35.189    have pos: "\<And>i. 0 \<le> integral\<^sup>S M (u i)" "\<And>i. 0 \<le> integral\<^sup>S M (v i)" "\<And>i. 0 \<le> a * integral\<^sup>S M (u i)"
  35.190 -    using u v `0 \<le> a` by (auto simp: simple_integral_nonneg)
  35.191 +    using u v \<open>0 \<le> a\<close> by (auto simp: simple_integral_nonneg)
  35.192    { fix i from pos[of i] have "a * integral\<^sup>S M (u i) \<noteq> -\<infinity>" "integral\<^sup>S M (v i) \<noteq> -\<infinity>"
  35.193        by (auto split: split_if_asm) }
  35.194    note not_MInf = this
  35.195 @@ -1116,26 +1116,26 @@
  35.196    have l': "(SUP i. integral\<^sup>S M (l i)) = (SUP i. integral\<^sup>S M (?L' i))"
  35.197    proof (rule SUP_simple_integral_sequences[OF l(3,6,2)])
  35.198      show "incseq ?L'" "\<And>i x. 0 \<le> ?L' i x" "\<And>i. simple_function M (?L' i)"
  35.199 -      using u v  `0 \<le> a` unfolding incseq_Suc_iff le_fun_def
  35.200 +      using u v  \<open>0 \<le> a\<close> unfolding incseq_Suc_iff le_fun_def
  35.201        by (auto intro!: add_mono ereal_mult_left_mono)
  35.202      { fix x
  35.203 -      { fix i have "a * u i x \<noteq> -\<infinity>" "v i x \<noteq> -\<infinity>" "u i x \<noteq> -\<infinity>" using `0 \<le> a` u(6)[of i x] v(6)[of i x]
  35.204 +      { fix i have "a * u i x \<noteq> -\<infinity>" "v i x \<noteq> -\<infinity>" "u i x \<noteq> -\<infinity>" using \<open>0 \<le> a\<close> u(6)[of i x] v(6)[of i x]
  35.205            by auto }
  35.206        then have "(SUP i. a * u i x + v i x) = a * (SUP i. u i x) + (SUP i. v i x)"
  35.207 -        using `0 \<le> a` u(3) v(3) u(6)[of _ x] v(6)[of _ x]
  35.208 -        by (subst SUP_ereal_mult_left [symmetric, OF _ u(6) `0 \<le> a`])
  35.209 +        using \<open>0 \<le> a\<close> u(3) v(3) u(6)[of _ x] v(6)[of _ x]
  35.210 +        by (subst SUP_ereal_mult_left [symmetric, OF _ u(6) \<open>0 \<le> a\<close>])
  35.211             (auto intro!: SUP_ereal_add
  35.212                   simp: incseq_Suc_iff le_fun_def add_mono ereal_mult_left_mono) }
  35.213      then show "AE x in M. (SUP i. l i x) = (SUP i. ?L' i x)"
  35.214 -      unfolding l(5) using `0 \<le> a` u(5) v(5) l(5) f(2) g(2)
  35.215 +      unfolding l(5) using \<open>0 \<le> a\<close> u(5) v(5) l(5) f(2) g(2)
  35.216        by (intro AE_I2) (auto split: split_max)
  35.217    qed
  35.218    also have "\<dots> = (SUP i. a * integral\<^sup>S M (u i) + integral\<^sup>S M (v i))"
  35.219 -    using u(2, 6) v(2, 6) `0 \<le> a` by (auto intro!: arg_cong[where f="SUPREMUM UNIV"] ext)
  35.220 +    using u(2, 6) v(2, 6) \<open>0 \<le> a\<close> by (auto intro!: arg_cong[where f="SUPREMUM UNIV"] ext)
  35.221    finally have "(\<integral>\<^sup>+ x. max 0 (a * f x + g x) \<partial>M) = a * (\<integral>\<^sup>+x. max 0 (f x) \<partial>M) + (\<integral>\<^sup>+x. max 0 (g x) \<partial>M)"
  35.222      unfolding l(5)[symmetric] u(5)[symmetric] v(5)[symmetric]
  35.223      unfolding l(1)[symmetric] u(1)[symmetric] v(1)[symmetric]
  35.224 -    apply (subst SUP_ereal_mult_left [symmetric, OF _ pos(1) `0 \<le> a`])
  35.225 +    apply (subst SUP_ereal_mult_left [symmetric, OF _ pos(1) \<open>0 \<le> a\<close>])
  35.226      apply simp
  35.227      apply (subst SUP_ereal_add [symmetric, OF inc not_MInf])
  35.228      .
  35.229 @@ -1146,12 +1146,12 @@
  35.230    assumes f: "f \<in> borel_measurable M" "0 \<le> c"
  35.231    shows "(\<integral>\<^sup>+ x. c * f x \<partial>M) = c * integral\<^sup>N M f"
  35.232  proof -
  35.233 -  have [simp]: "\<And>x. c * max 0 (f x) = max 0 (c * f x)" using `0 \<le> c`
  35.234 +  have [simp]: "\<And>x. c * max 0 (f x) = max 0 (c * f x)" using \<open>0 \<le> c\<close>
  35.235      by (auto split: split_max simp: ereal_zero_le_0_iff)
  35.236    have "(\<integral>\<^sup>+ x. c * f x \<partial>M) = (\<integral>\<^sup>+ x. c * max 0 (f x) \<partial>M)"
  35.237      by (simp add: nn_integral_max_0)
  35.238    then show ?thesis
  35.239 -    using nn_integral_linear[OF _ _ `0 \<le> c`, of "\<lambda>x. max 0 (f x)" _ "\<lambda>x. 0"] f
  35.240 +    using nn_integral_linear[OF _ _ \<open>0 \<le> c\<close>, of "\<lambda>x. max 0 (f x)" _ "\<lambda>x. 0"] f
  35.241      by (auto simp: nn_integral_max_0)
  35.242  qed
  35.243  
  35.244 @@ -1252,7 +1252,7 @@
  35.245      (is "(emeasure M) ?A \<le> _ * ?PI")
  35.246  proof -
  35.247    have "?A \<in> sets M"
  35.248 -    using `A \<in> sets M` u by auto
  35.249 +    using \<open>A \<in> sets M\<close> u by auto
  35.250    hence "(emeasure M) ?A = (\<integral>\<^sup>+ x. indicator ?A x \<partial>M)"
  35.251      using nn_integral_indicator by simp
  35.252    also have "\<dots> \<le> (\<integral>\<^sup>+ x. c * (u x * indicator A x) \<partial>M)" using u c
  35.253 @@ -1279,7 +1279,7 @@
  35.254      using g by (subst nn_integral_cmult_indicator) auto
  35.255    also have "\<dots> \<le> integral\<^sup>N M g"
  35.256      using assms by (auto intro!: nn_integral_mono_AE simp: indicator_def)
  35.257 -  finally show False using `integral\<^sup>N M g \<noteq> \<infinity>` by auto
  35.258 +  finally show False using \<open>integral\<^sup>N M g \<noteq> \<infinity>\<close> by auto
  35.259  qed
  35.260  
  35.261  lemma nn_integral_PInf:
  35.262 @@ -1371,7 +1371,7 @@
  35.263    finally show ?thesis .
  35.264  qed
  35.265  
  35.266 -text {* Fatou's lemma: convergence theorem on limes inferior *}
  35.267 +text \<open>Fatou's lemma: convergence theorem on limes inferior\<close>
  35.268  
  35.269  lemma nn_integral_liminf:
  35.270    fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"
  35.271 @@ -1624,7 +1624,7 @@
  35.272          fix n :: nat and x
  35.273          assume *: "1 \<le> real n * u x"
  35.274          also from gt_1[OF *] have "real n * u x \<le> real (Suc n) * u x"
  35.275 -          using `0 \<le> u x` by (auto intro!: ereal_mult_right_mono)
  35.276 +          using \<open>0 \<le> u x\<close> by (auto intro!: ereal_mult_right_mono)
  35.277          finally show "1 \<le> real (Suc n) * u x" by auto
  35.278        qed
  35.279      qed
  35.280 @@ -1633,12 +1633,12 @@
  35.281        fix x assume "0 < u x" and [simp, intro]: "x \<in> space M"
  35.282        show "x \<in> (\<Union>n. ?M n \<inter> ?A)"
  35.283        proof (cases "u x")
  35.284 -        case (real r) with `0 < u x` have "0 < r" by auto
  35.285 +        case (real r) with \<open>0 < u x\<close> have "0 < r" by auto
  35.286          obtain j :: nat where "1 / r \<le> real j" using real_arch_simple ..
  35.287 -        hence "1 / r * r \<le> real j * r" unfolding mult_le_cancel_right using `0 < r` by auto
  35.288 -        hence "1 \<le> real j * r" using real `0 < r` by auto
  35.289 -        thus ?thesis using `0 < r` real by (auto simp: one_ereal_def)
  35.290 -      qed (insert `0 < u x`, auto)
  35.291 +        hence "1 / r * r \<le> real j * r" unfolding mult_le_cancel_right using \<open>0 < r\<close> by auto
  35.292 +        hence "1 \<le> real j * r" using real \<open>0 < r\<close> by auto
  35.293 +        thus ?thesis using \<open>0 < r\<close> real by (auto simp: one_ereal_def)
  35.294 +      qed (insert \<open>0 < u x\<close>, auto)
  35.295      qed auto
  35.296      finally have "(emeasure M) {x\<in>space M. 0 < u x} = 0" by simp
  35.297      moreover
  35.298 @@ -1729,7 +1729,7 @@
  35.299      then have "(\<lambda>i. ereal(if i < f x then 1 else 0)) sums (ereal(of_nat(f x)))"
  35.300        unfolding sums_ereal .
  35.301      moreover have "\<And>i. ereal (if i < f x then 1 else 0) = indicator (F i) x"
  35.302 -      using `x \<in> space M` by (simp add: one_ereal_def F_def)
  35.303 +      using \<open>x \<in> space M\<close> by (simp add: one_ereal_def F_def)
  35.304      ultimately have "ereal(of_nat(f x)) = (\<Sum>i. indicator (F i) x)"
  35.305        by (simp add: sums_iff) }
  35.306    then have "(\<integral>\<^sup>+x. ereal (of_nat (f x)) \<partial>M) = (\<integral>\<^sup>+x. (\<Sum>i. indicator (F i) x) \<partial>M)"
  35.307 @@ -1793,7 +1793,7 @@
  35.308      by (subst step) auto
  35.309  qed (insert bound, auto simp add: le_fun_def INF_apply[abs_def] top_fun_def intro!: meas f g)
  35.310  
  35.311 -subsection {* Integral under concrete measures *}
  35.312 +subsection \<open>Integral under concrete measures\<close>
  35.313  
  35.314  lemma nn_integral_empty:
  35.315    assumes "space M = {}"
  35.316 @@ -1804,7 +1804,7 @@
  35.317    thus ?thesis by simp
  35.318  qed
  35.319  
  35.320 -subsubsection {* Distributions *}
  35.321 +subsubsection \<open>Distributions\<close>
  35.322  
  35.323  lemma nn_integral_distr':
  35.324    assumes T: "T \<in> measurable M M'"
  35.325 @@ -1835,7 +1835,7 @@
  35.326    by (subst (1 2) nn_integral_max_0[symmetric])
  35.327       (simp add: nn_integral_distr')
  35.328  
  35.329 -subsubsection {* Counting space *}
  35.330 +subsubsection \<open>Counting space\<close>
  35.331  
  35.332  lemma simple_function_count_space[simp]:
  35.333    "simple_function (count_space A) f \<longleftrightarrow> finite (f ` A)"
  35.334 @@ -1868,7 +1868,7 @@
  35.335  proof -
  35.336    have "(\<integral>\<^sup>+x. f x \<partial>count_space B) = (\<Sum>a | a \<in> B \<and> 0 < f a. f a)"
  35.337      using assms(2,3)
  35.338 -    by (intro nn_integral_count_space finite_subset[OF _ `finite A`]) (auto simp: less_le)
  35.339 +    by (intro nn_integral_count_space finite_subset[OF _ \<open>finite A\<close>]) (auto simp: less_le)
  35.340    also have "\<dots> = (\<Sum>a\<in>A. f a)"
  35.341      using assms by (intro setsum.mono_neutral_cong_left) (auto simp: less_le)
  35.342    finally show ?thesis .
  35.343 @@ -1927,7 +1927,7 @@
  35.344    assume "infinite I"
  35.345    then have [simp]: "I \<noteq> {}"
  35.346      by auto
  35.347 -  note * = bij_betw_from_nat_into[OF `countable I` `infinite I`]
  35.348 +  note * = bij_betw_from_nat_into[OF \<open>countable I\<close> \<open>infinite I\<close>]
  35.349    have **: "\<And>f. (\<And>i. 0 \<le> f i) \<Longrightarrow> (\<integral>\<^sup>+i. f i \<partial>count_space I) = (\<Sum>n. f (from_nat_into I n))"
  35.350      by (simp add: nn_integral_bij_count_space[symmetric, OF *] nn_integral_count_space_nat)
  35.351    show ?thesis
  35.352 @@ -2147,7 +2147,7 @@
  35.353    finally show ?thesis .
  35.354  qed
  35.355  
  35.356 -subsubsection {* Measures with Restricted Space *}
  35.357 +subsubsection \<open>Measures with Restricted Space\<close>
  35.358  
  35.359  lemma simple_function_iff_borel_measurable:
  35.360    fixes f :: "'a \<Rightarrow> 'x::{t2_space}"
  35.361 @@ -2271,7 +2271,7 @@
  35.362    finally show ?thesis .
  35.363  qed
  35.364  
  35.365 -subsubsection {* Measure spaces with an associated density *}
  35.366 +subsubsection \<open>Measure spaces with an associated density\<close>
  35.367  
  35.368  definition density :: "'a measure \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> 'a measure" where
  35.369    "density M f = measure_of (space M) (sets M) (\<lambda>A. \<integral>\<^sup>+ x. f x * indicator A x \<partial>M)"
  35.370 @@ -2351,10 +2351,10 @@
  35.371      have "(\<integral>\<^sup>+x. f x * indicator A x \<partial>M) = 0 \<longleftrightarrow>
  35.372        emeasure M {x \<in> space M. max 0 (f x) * indicator A x \<noteq> 0} = 0"
  35.373        unfolding eq
  35.374 -      using f `A \<in> sets M`
  35.375 +      using f \<open>A \<in> sets M\<close>
  35.376        by (intro nn_integral_0_iff) auto
  35.377      also have "\<dots> \<longleftrightarrow> (AE x in M. max 0 (f x) * indicator A x = 0)"
  35.378 -      using f `A \<in> sets M`
  35.379 +      using f \<open>A \<in> sets M\<close>
  35.380        by (intro AE_iff_measurable[OF _ refl, symmetric]) auto
  35.381      also have "(AE x in M. max 0 (f x) * indicator A x = 0) \<longleftrightarrow> (AE x in M. x \<in> A \<longrightarrow> f x \<le> 0)"
  35.382        by (auto simp add: indicator_def max_def split: split_if_asm)
  35.383 @@ -2517,7 +2517,7 @@
  35.384    apply (intro nn_integral_cong, simp split: split_indicator)
  35.385    done
  35.386  
  35.387 -subsubsection {* Point measure *}
  35.388 +subsubsection \<open>Point measure\<close>
  35.389  
  35.390  definition point_measure :: "'a set \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> 'a measure" where
  35.391    "point_measure A f = density (count_space A) f"
  35.392 @@ -2549,7 +2549,7 @@
  35.393    shows "emeasure (point_measure A f) X = (\<Sum>a|a\<in>X \<and> 0 < f a. f a)"
  35.394  proof -
  35.395    have "{a. (a \<in> X \<longrightarrow> a \<in> A \<and> 0 < f a) \<and> a \<in> X} = {a\<in>X. 0 < f a}"
  35.396 -    using `X \<subseteq> A` by auto
  35.397 +    using \<open>X \<subseteq> A\<close> by auto
  35.398    with A show ?thesis
  35.399      by (simp add: emeasure_density nn_integral_count_space ereal_zero_le_0_iff
  35.400                    point_measure_def indicator_def)
  35.401 @@ -2593,7 +2593,7 @@
  35.402      integral\<^sup>N (point_measure A f) g = (\<Sum>a\<in>A. f a * g a)"
  35.403    by (subst nn_integral_point_measure) (auto intro!: setsum.mono_neutral_left simp: less_le)
  35.404  
  35.405 -subsubsection {* Uniform measure *}
  35.406 +subsubsection \<open>Uniform measure\<close>
  35.407  
  35.408  definition "uniform_measure M A = density M (\<lambda>x. indicator A x / emeasure M A)"
  35.409  
  35.410 @@ -2666,7 +2666,7 @@
  35.411    shows "(AE x in uniform_measure M A. P x) \<longleftrightarrow> (AE x in M. x \<in> A \<longrightarrow> P x)"
  35.412  proof -
  35.413    have "A \<in> sets M"
  35.414 -    using `emeasure M A \<noteq> 0` by (metis emeasure_notin_sets)
  35.415 +    using \<open>emeasure M A \<noteq> 0\<close> by (metis emeasure_notin_sets)
  35.416    moreover have "\<And>x. 0 < indicator A x / emeasure M A \<longleftrightarrow> x \<in> A"
  35.417      using emeasure_nonneg[of M A] assms
  35.418      by (cases "emeasure M A") (auto split: split_indicator simp: one_ereal_def)
  35.419 @@ -2674,7 +2674,7 @@
  35.420      unfolding uniform_measure_def by (simp add: AE_density)
  35.421  qed
  35.422  
  35.423 -subsubsection {* Null measure *}
  35.424 +subsubsection \<open>Null measure\<close>
  35.425  
  35.426  lemma null_measure_eq_density: "null_measure M = density M (\<lambda>_. 0)"
  35.427    by (intro measure_eqI) (simp_all add: emeasure_density)
  35.428 @@ -2689,7 +2689,7 @@
  35.429      by (simp add: density_def) (simp only: null_measure_def[symmetric] emeasure_null_measure)
  35.430  qed simp
  35.431  
  35.432 -subsubsection {* Uniform count measure *}
  35.433 +subsubsection \<open>Uniform count measure\<close>
  35.434  
  35.435  definition "uniform_count_measure A = point_measure A (\<lambda>x. 1 / card A)"
  35.436  
    36.1 --- a/src/HOL/Probability/Probability_Mass_Function.thy	Mon Dec 07 16:48:10 2015 +0000
    36.2 +++ b/src/HOL/Probability/Probability_Mass_Function.thy	Mon Dec 07 20:19:59 2015 +0100
    36.3 @@ -54,16 +54,16 @@
    36.4      from this(3) have *: "\<And>x. x \<in> X \<Longrightarrow> ?M / Suc n \<le> ?m x"
    36.5        by auto
    36.6      { fix x assume "x \<in> X"
    36.7 -      from `?M \<noteq> 0` *[OF this] have "?m x \<noteq> 0" by (auto simp: field_simps measure_le_0_iff)
    36.8 +      from \<open>?M \<noteq> 0\<close> *[OF this] have "?m x \<noteq> 0" by (auto simp: field_simps measure_le_0_iff)
    36.9        then have "{x} \<in> sets M" by (auto dest: measure_notin_sets) }
   36.10      note singleton_sets = this
   36.11      have "?M < (\<Sum>x\<in>X. ?M / Suc n)"
   36.12 -      using `?M \<noteq> 0`
   36.13 -      by (simp add: `card X = Suc (Suc n)` of_nat_Suc field_simps less_le measure_nonneg)
   36.14 +      using \<open>?M \<noteq> 0\<close>
   36.15 +      by (simp add: \<open>card X = Suc (Suc n)\<close> of_nat_Suc field_simps less_le measure_nonneg)
   36.16      also have "\<dots> \<le> (\<Sum>x\<in>X. ?m x)"
   36.17        by (rule setsum_mono) fact
   36.18      also have "\<dots> = measure M (\<Union>x\<in>X. {x})"
   36.19 -      using singleton_sets `finite X`
   36.20 +      using singleton_sets \<open>finite X\<close>
   36.21        by (intro finite_measure_finite_Union[symmetric]) (auto simp: disjoint_family_on_def)
   36.22      finally have "?M < measure M (\<Union>x\<in>X. {x})" .
   36.23      moreover have "measure M (\<Union>x\<in>X. {x}) \<le> ?M"
   36.24 @@ -399,7 +399,7 @@
   36.25  lemma bind_pmf_cong:
   36.26    assumes "p = q"
   36.27    shows "(\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x) \<Longrightarrow> bind_pmf p f = bind_pmf q g"
   36.28 -  unfolding `p = q`[symmetric] measure_pmf_inject[symmetric] bind_pmf.rep_eq
   36.29 +  unfolding \<open>p = q\<close>[symmetric] measure_pmf_inject[symmetric] bind_pmf.rep_eq
   36.30    by (auto simp: AE_measure_pmf_iff Pi_iff space_subprob_algebra subprob_space_measure_pmf
   36.31                   sets_bind[where N="count_space UNIV"] emeasure_bind[where N="count_space UNIV"]
   36.32             intro!: nn_integral_cong_AE measure_eqI)
   36.33 @@ -736,7 +736,7 @@
   36.34  lemma set_pmf_transfer[transfer_rule]:
   36.35    assumes "bi_total A"
   36.36    shows "rel_fun (pcr_pmf A) (rel_set A) (\<lambda>f. {x. f x \<noteq> 0}) set_pmf"
   36.37 -  using `bi_total A`
   36.38 +  using \<open>bi_total A\<close>
   36.39    by (auto simp: pcr_pmf_def cr_pmf_def rel_fun_def rel_set_def bi_total_def Bex_def set_pmf_iff)
   36.40       metis+
   36.41  
   36.42 @@ -1079,9 +1079,9 @@
   36.43      with in_set eq have "measure p {x. R x y} = measure q {y. R x y}"
   36.44        by auto
   36.45      moreover have "{y. R x y} = C"
   36.46 -      using assms `x \<in> C` C quotientD[of UNIV ?R C x] by (simp add: equivp_equiv)
   36.47 +      using assms \<open>x \<in> C\<close> C quotientD[of UNIV ?R C x] by (simp add: equivp_equiv)
   36.48      moreover have "{x. R x y} = C"
   36.49 -      using assms `y \<in> C` C quotientD[of UNIV "?R" C y] sympD[of R]
   36.50 +      using assms \<open>y \<in> C\<close> C quotientD[of UNIV "?R" C y] sympD[of R]
   36.51        by (auto simp add: equivp_equiv elim: equivpE)
   36.52      ultimately show ?thesis
   36.53        by auto
   36.54 @@ -1114,7 +1114,7 @@
   36.55  
   36.56    fix x y assume "x \<in> set_pmf p" "y \<in> set_pmf q" "R x y"
   36.57    have "{y. R x y} \<in> UNIV // ?R" "{x. R x y} = {y. R x y}"
   36.58 -    using assms `R x y` by (auto simp: quotient_def dest: equivp_symp equivp_transp)
   36.59 +    using assms \<open>R x y\<close> by (auto simp: quotient_def dest: equivp_symp equivp_transp)
   36.60    with eq show "measure p {x. R x y} = measure q {y. R x y}"
   36.61      by auto
   36.62  qed
   36.63 @@ -1198,7 +1198,7 @@
   36.64      by (force elim: rel_pmf.cases)
   36.65    moreover have "set_pmf (return_pmf x) = {x}"
   36.66      by simp
   36.67 -  with `a \<in> M` have "(x, a) \<in> pq"
   36.68 +  with \<open>a \<in> M\<close> have "(x, a) \<in> pq"
   36.69      by (force simp: eq)
   36.70    with * show "R x a"
   36.71      by auto
   36.72 @@ -1366,12 +1366,12 @@
   36.73    by (rule rel_pmf_joinI)
   36.74       (auto simp add: pmf.rel_map intro: pmf.rel_mono[THEN le_funD, THEN le_funD, THEN le_boolD, THEN mp, OF _ pq] fg)
   36.75  
   36.76 -text {*
   36.77 +text \<open>
   36.78    Proof that @{const rel_pmf} preserves orders.
   36.79    Antisymmetry proof follows Thm. 1 in N. Saheb-Djahromi, Cpo's of measures for nondeterminism,
   36.80    Theoretical Computer Science 12(1):19--37, 1980,
   36.81    @{url "http://dx.doi.org/10.1016/0304-3975(80)90003-1"}
   36.82 -*}
   36.83 +\<close>
   36.84  
   36.85  lemma
   36.86    assumes *: "rel_pmf R p q"
    37.1 --- a/src/HOL/Probability/Probability_Measure.thy	Mon Dec 07 16:48:10 2015 +0000
    37.2 +++ b/src/HOL/Probability/Probability_Measure.thy	Mon Dec 07 20:19:59 2015 +0100
    37.3 @@ -3,7 +3,7 @@
    37.4      Author:     Armin Heller, TU München
    37.5  *)
    37.6  
    37.7 -section {*Probability measure*}
    37.8 +section \<open>Probability measure\<close>
    37.9  
   37.10  theory Probability_Measure
   37.11    imports Lebesgue_Measure Radon_Nikodym
   37.12 @@ -88,21 +88,21 @@
   37.13  proof
   37.14    assume ae: "AE x in M. x \<in> A"
   37.15    have "{x \<in> space M. x \<in> A} = A" "{x \<in> space M. x \<notin> A} = space M - A"
   37.16 -    using `A \<in> events`[THEN sets.sets_into_space] by auto
   37.17 -  with AE_E2[OF ae] `A \<in> events` have "1 - emeasure M A = 0"
   37.18 +    using \<open>A \<in> events\<close>[THEN sets.sets_into_space] by auto
   37.19 +  with AE_E2[OF ae] \<open>A \<in> events\<close> have "1 - emeasure M A = 0"
   37.20      by (simp add: emeasure_compl emeasure_space_1)
   37.21    then show "prob A = 1"
   37.22 -    using `A \<in> events` by (simp add: emeasure_eq_measure one_ereal_def)
   37.23 +    using \<open>A \<in> events\<close> by (simp add: emeasure_eq_measure one_ereal_def)
   37.24  next
   37.25    assume prob: "prob A = 1"
   37.26    show "AE x in M. x \<in> A"
   37.27    proof (rule AE_I)
   37.28      show "{x \<in> space M. x \<notin> A} \<subseteq> space M - A" by auto
   37.29      show "emeasure M (space M - A) = 0"
   37.30 -      using `A \<in> events` prob
   37.31 +      using \<open>A \<in> events\<close> prob
   37.32        by (simp add: prob_compl emeasure_space_1 emeasure_eq_measure one_ereal_def)
   37.33      show "space M - A \<in> events"
   37.34 -      using `A \<in> events` by auto
   37.35 +      using \<open>A \<in> events\<close> by auto
   37.36    qed
   37.37  qed
   37.38  
   37.39 @@ -117,7 +117,7 @@
   37.40  lemma (in prob_space) AE_prob_1:
   37.41    assumes "prob A = 1" shows "AE x in M. x \<in> A"
   37.42  proof -
   37.43 -  from `prob A = 1` have "A \<in> events"
   37.44 +  from \<open>prob A = 1\<close> have "A \<in> events"
   37.45      by (metis measure_notin_sets zero_neq_one)
   37.46    with AE_in_set_eq_1 assms show ?thesis by simp
   37.47  qed
   37.48 @@ -204,21 +204,21 @@
   37.49      by (elim disjE)  (auto simp: subset_eq)
   37.50    moreover
   37.51    { fix y assume y: "y \<in> I"
   37.52 -    with q(2) `open I` have "Sup ((\<lambda>x. q x + ?F x * (y - x)) ` I) = q y"
   37.53 +    with q(2) \<open>open I\<close> have "Sup ((\<lambda>x. q x + ?F x * (y - x)) ` I) = q y"
   37.54        by (auto intro!: cSup_eq_maximum convex_le_Inf_differential image_eqI [OF _ y] simp: interior_open simp del: Sup_image_eq Inf_image_eq) }
   37.55    ultimately have "q (expectation X) = Sup ((\<lambda>x. q x + ?F x * (expectation X - x)) ` I)"
   37.56      by simp
   37.57    also have "\<dots> \<le> expectation (\<lambda>w. q (X w))"
   37.58    proof (rule cSup_least)
   37.59      show "(\<lambda>x. q x + ?F x * (expectation X - x)) ` I \<noteq> {}"
   37.60 -      using `I \<noteq> {}` by auto
   37.61 +      using \<open>I \<noteq> {}\<close> by auto
   37.62    next
   37.63      fix k assume "k \<in> (\<lambda>x. q x + ?F x * (expectation X - x)) ` I"
   37.64      then guess x .. note x = this
   37.65      have "q x + ?F x * (expectation X  - x) = expectation (\<lambda>w. q x + ?F x * (X w - x))"
   37.66        using prob_space by (simp add: X)
   37.67      also have "\<dots> \<le> expectation (\<lambda>w. q (X w))"
   37.68 -      using `x \<in> I` `open I` X(2)
   37.69 +      using \<open>x \<in> I\<close> \<open>open I\<close> X(2)
   37.70        apply (intro integral_mono_AE integrable_add integrable_mult_right integrable_diff
   37.71                  integrable_const X q)
   37.72        apply (elim eventually_elim1)
   37.73 @@ -230,7 +230,7 @@
   37.74    finally show "q (expectation X) \<le> expectation (\<lambda>x. q (X x))" .
   37.75  qed
   37.76  
   37.77 -subsection  {* Introduce binder for probability *}
   37.78 +subsection  \<open>Introduce binder for probability\<close>
   37.79  
   37.80  syntax
   37.81    "_prob" :: "pttrn \<Rightarrow> logic \<Rightarrow> logic \<Rightarrow> logic" ("('\<P>'((/_ in _./ _)'))")
   37.82 @@ -238,7 +238,7 @@
   37.83  translations
   37.84    "\<P>(x in M. P)" => "CONST measure M {x \<in> CONST space M. P}"
   37.85  
   37.86 -print_translation {*
   37.87 +print_translation \<open>
   37.88    let
   37.89      fun to_pattern (Const (@{const_syntax Pair}, _) $ l $ r) =
   37.90        Syntax.const @{const_syntax Pair} :: to_pattern l @ to_pattern r
   37.91 @@ -299,7 +299,7 @@
   37.92    in
   37.93      [(@{const_syntax Sigma_Algebra.measure}, K tr')]
   37.94    end
   37.95 -*}
   37.96 +\<close>
   37.97  
   37.98  definition
   37.99    "cond_prob M P Q = \<P>(\<omega> in M. P \<omega> \<and> Q \<omega>) / \<P>(\<omega> in M. Q \<omega>)"
  37.100 @@ -495,7 +495,7 @@
  37.101    finally show ?thesis by simp
  37.102  qed
  37.103  
  37.104 -subsection {* Distributions *}
  37.105 +subsection \<open>Distributions\<close>
  37.106  
  37.107  definition "distributed M N X f \<longleftrightarrow> distr M N X = density N f \<and> 
  37.108    f \<in> borel_measurable N \<and> (AE x in N. 0 \<le> f x) \<and> X \<in> measurable M N"
    38.1 --- a/src/HOL/Probability/Projective_Family.thy	Mon Dec 07 16:48:10 2015 +0000
    38.2 +++ b/src/HOL/Probability/Projective_Family.thy	Mon Dec 07 20:19:59 2015 +0100
    38.3 @@ -3,7 +3,7 @@
    38.4      Author:     Johannes Hölzl, TU München
    38.5  *)
    38.6  
    38.7 -section {*Projective Family*}
    38.8 +section \<open>Projective Family\<close>
    38.9  
   38.10  theory Projective_Family
   38.11  imports Finite_Product_Measure Giry_Monad
   38.12 @@ -22,11 +22,11 @@
   38.13    proof cases
   38.14      assume x: "x \<in> (\<Pi>\<^sub>E i\<in>J. S i)"
   38.15      have "merge J (I - J) (x,y) \<in> (\<lambda>x. restrict x J) -` A \<inter> (\<Pi>\<^sub>E i\<in>I. S i)"
   38.16 -      using y x `J \<subseteq> I` PiE_cancel_merge[of "J" "I - J" x y S] \<open>x\<in>A\<close>
   38.17 +      using y x \<open>J \<subseteq> I\<close> PiE_cancel_merge[of "J" "I - J" x y S] \<open>x\<in>A\<close>
   38.18        by (auto simp del: PiE_cancel_merge simp add: Un_absorb1)
   38.19      also have "\<dots> \<subseteq> (\<lambda>x. restrict x J) -` B \<inter> (\<Pi>\<^sub>E i\<in>I. S i)" by fact
   38.20      finally show "x \<in> B"
   38.21 -      using y x `J \<subseteq> I` PiE_cancel_merge[of "J" "I - J" x y S] \<open>x\<in>A\<close> eq
   38.22 +      using y x \<open>J \<subseteq> I\<close> PiE_cancel_merge[of "J" "I - J" x y S] \<open>x\<in>A\<close> eq
   38.23        by (auto simp del: PiE_cancel_merge simp add: Un_absorb1)
   38.24    qed (insert \<open>x\<in>A\<close> sets, auto)
   38.25  qed
   38.26 @@ -62,7 +62,7 @@
   38.27    show "(\<Pi>\<^sub>E i\<in>L. space (M i)) \<noteq> {}"
   38.28      using \<open>L \<subseteq> I\<close> by (auto simp add: not_empty_M space_PiM[symmetric])
   38.29    show "(\<lambda>x. restrict x J) -` X \<inter> (\<Pi>\<^sub>E i\<in>L. space (M i)) \<subseteq> (\<lambda>x. restrict x J) -` Y \<inter> (\<Pi>\<^sub>E i\<in>L. space (M i))"
   38.30 -    using `prod_emb L M J X \<subseteq> prod_emb L M J Y` by (simp add: prod_emb_def)
   38.31 +    using \<open>prod_emb L M J X \<subseteq> prod_emb L M J Y\<close> by (simp add: prod_emb_def)
   38.32  qed fact
   38.33  
   38.34  lemma emb_injective:
    39.1 --- a/src/HOL/Probability/Projective_Limit.thy	Mon Dec 07 16:48:10 2015 +0000
    39.2 +++ b/src/HOL/Probability/Projective_Limit.thy	Mon Dec 07 20:19:59 2015 +0100
    39.3 @@ -2,7 +2,7 @@
    39.4      Author:     Fabian Immler, TU München
    39.5  *)
    39.6  
    39.7 -section {* Projective Limit *}
    39.8 +section \<open>Projective Limit\<close>
    39.9  
   39.10  theory Projective_Limit
   39.11    imports
   39.12 @@ -14,7 +14,7 @@
   39.13      "~~/src/HOL/Library/Diagonal_Subsequence"
   39.14  begin
   39.15  
   39.16 -subsection {* Sequences of Finite Maps in Compact Sets *}
   39.17 +subsection \<open>Sequences of Finite Maps in Compact Sets\<close>
   39.18  
   39.19  locale finmap_seqs_into_compact =
   39.20    fixes K::"nat \<Rightarrow> (nat \<Rightarrow>\<^sub>F 'a::metric_space) set" and f::"nat \<Rightarrow> (nat \<Rightarrow>\<^sub>F 'a)" and M
   39.21 @@ -31,8 +31,8 @@
   39.22    obtain k where "k \<in> K (Suc 0)" using f_in_K by auto
   39.23    assume "\<forall>n. t \<notin> domain (f n)"
   39.24    thus ?thesis
   39.25 -    by (auto intro!: exI[where x=1] image_eqI[OF _ `k \<in> K (Suc 0)`]
   39.26 -      simp: domain_K[OF `k \<in> K (Suc 0)`])
   39.27 +    by (auto intro!: exI[where x=1] image_eqI[OF _ \<open>k \<in> K (Suc 0)\<close>]
   39.28 +      simp: domain_K[OF \<open>k \<in> K (Suc 0)\<close>])
   39.29  qed blast
   39.30  
   39.31  lemma proj_in_KE:
   39.32 @@ -52,9 +52,9 @@
   39.33  proof atomize_elim
   39.34    have "subseq (op + m)" by (simp add: subseq_def)
   39.35    have "\<forall>n. (f o (\<lambda>i. m + i)) n \<in> S" using assms by auto
   39.36 -  from seq_compactE[OF `compact S`[unfolded compact_eq_seq_compact_metric] this] guess l r .
   39.37 +  from seq_compactE[OF \<open>compact S\<close>[unfolded compact_eq_seq_compact_metric] this] guess l r .
   39.38    hence "l \<in> S" "subseq ((\<lambda>i. m + i) o r) \<and> (f \<circ> ((\<lambda>i. m + i) o r)) ----> l"
   39.39 -    using subseq_o[OF `subseq (op + m)` `subseq r`] by (auto simp: o_def)
   39.40 +    using subseq_o[OF \<open>subseq (op + m)\<close> \<open>subseq r\<close>] by (auto simp: o_def)
   39.41    thus "\<exists>l r. l \<in> S \<and> subseq r \<and> (f \<circ> r) ----> l" by blast
   39.42  qed
   39.43  
   39.44 @@ -84,7 +84,7 @@
   39.45      assume "\<exists>l. (\<lambda>i. ((f \<circ> s) i)\<^sub>F n) ----> l"
   39.46      then obtain l where "((\<lambda>i. (f i)\<^sub>F n) o s) ----> l"
   39.47        by (auto simp: o_def)
   39.48 -    hence "((\<lambda>i. (f i)\<^sub>F n) o s o r) ----> l" using `subseq r`
   39.49 +    hence "((\<lambda>i. (f i)\<^sub>F n) o s o r) ----> l" using \<open>subseq r\<close>
   39.50        by (rule LIMSEQ_subseq_LIMSEQ)
   39.51      thus "\<exists>l. (\<lambda>i. ((f \<circ> (s \<circ> r)) i)\<^sub>F n) ----> l" by (auto simp add: o_def)
   39.52    qed
   39.53 @@ -93,9 +93,9 @@
   39.54    thus ?thesis ..
   39.55  qed
   39.56  
   39.57 -subsection {* Daniell-Kolmogorov Theorem *}
   39.58 +subsection \<open>Daniell-Kolmogorov Theorem\<close>
   39.59  
   39.60 -text {* Existence of Projective Limit *}
   39.61 +text \<open>Existence of Projective Limit\<close>
   39.62  
   39.63  locale polish_projective = projective_family I P "\<lambda>_. borel::'a::polish_space measure"
   39.64    for I::"'i set" and P
   39.65 @@ -175,15 +175,15 @@
   39.66          hence "?SUP n - emeasure (P' n) K > 2 powr (-n) * ?a"
   39.67            unfolding not_less[symmetric] by simp
   39.68          hence "?SUP n - 2 powr (-n) * ?a > emeasure (P' n) K"
   39.69 -          using `0 < ?a` by (auto simp add: ereal_less_minus_iff ac_simps)
   39.70 +          using \<open>0 < ?a\<close> by (auto simp add: ereal_less_minus_iff ac_simps)
   39.71          thus "?SUP n - 2 powr (-n) * ?a \<ge> emeasure (P' n) K" by simp
   39.72        qed
   39.73 -      hence "?SUP n + 0 \<le> ?SUP n - (2 powr (-n) * ?a)" using `0 < ?a` by simp
   39.74 +      hence "?SUP n + 0 \<le> ?SUP n - (2 powr (-n) * ?a)" using \<open>0 < ?a\<close> by simp
   39.75        hence "?SUP n + 0 \<le> ?SUP n + - (2 powr (-n) * ?a)" unfolding minus_ereal_def .
   39.76        hence "0 \<le> - (2 powr (-n) * ?a)"
   39.77 -        using `?SUP n \<noteq> \<infinity>` `?SUP n \<noteq> - \<infinity>`
   39.78 +        using \<open>?SUP n \<noteq> \<infinity>\<close> \<open>?SUP n \<noteq> - \<infinity>\<close>
   39.79          by (subst (asm) ereal_add_le_add_iff) (auto simp:)
   39.80 -      moreover have "ereal (2 powr - real n) * ?a > 0" using `0 < ?a`
   39.81 +      moreover have "ereal (2 powr - real n) * ?a > 0" using \<open>0 < ?a\<close>
   39.82          by (auto simp: ereal_zero_less_0_iff)
   39.83        ultimately show False by simp
   39.84      qed
   39.85 @@ -195,7 +195,7 @@
   39.86    def K \<equiv> "\<lambda>n. fm n -` K' n \<inter> space (Pi\<^sub>M (J n) (\<lambda>_. borel))"
   39.87    have K_sets: "\<And>n. K n \<in> sets (Pi\<^sub>M (J n) (\<lambda>_. borel))"
   39.88      unfolding K_def
   39.89 -    using compact_imp_closed[OF `compact (K' _)`]
   39.90 +    using compact_imp_closed[OF \<open>compact (K' _)\<close>]
   39.91      by (intro measurable_sets[OF fm_measurable, of _ "Collect finite"])
   39.92         (auto simp: borel_eq_PiF_borel[symmetric])
   39.93    have K_B: "\<And>n. K n \<subseteq> B n"
   39.94 @@ -204,7 +204,7 @@
   39.95      then have fm_in: "fm n x \<in> fm n ` B n"
   39.96        using K' by (force simp: K_def)
   39.97      show "x \<in> B n"
   39.98 -      using `x \<in> K n` K_sets sets.sets_into_space J(1,2,3)[of n] inj_on_image_mem_iff[OF inj_on_fm]
   39.99 +      using \<open>x \<in> K n\<close> K_sets sets.sets_into_space J(1,2,3)[of n] inj_on_image_mem_iff[OF inj_on_fm]
  39.100      by (metis (no_types) Int_iff K_def fm_in space_borel)
  39.101    qed
  39.102    def Z' \<equiv> "\<lambda>n. emb I (J n) (K n)"
  39.103 @@ -224,7 +224,7 @@
  39.104          by (auto simp add: space_P sets_P)
  39.105        assume "fm n x = fm n y"
  39.106        note inj_onD[OF inj_on_fm[OF space_borel],
  39.107 -        OF `fm n x = fm n y` `x \<in> space _` `y \<in> space _`]
  39.108 +        OF \<open>fm n x = fm n y\<close> \<open>x \<in> space _\<close> \<open>y \<in> space _\<close>]
  39.109        with y show "x \<in> B n" by simp
  39.110      qed
  39.111    qed
  39.112 @@ -243,39 +243,39 @@
  39.113      have "Y n = (\<Inter>i\<in>{1..n}. emb I (J n) (emb (J n) (J i) (K i)))" using J J_mono
  39.114        by (auto simp: Y_def Z'_def)
  39.115      also have "\<dots> = prod_emb I (\<lambda>_. borel) (J n) (\<Inter>i\<in>{1..n}. emb (J n) (J i) (K i))"
  39.116 -      using `n \<ge> 1`
  39.117 +      using \<open>n \<ge> 1\<close>
  39.118        by (subst prod_emb_INT) auto
  39.119      finally
  39.120      have Y_emb:
  39.121        "Y n = prod_emb I (\<lambda>_. borel) (J n) (\<Inter>i\<in>{1..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i))" .
  39.122 -    hence "Y n \<in> generator" using J J_mono K_sets `n \<ge> 1`
  39.123 +    hence "Y n \<in> generator" using J J_mono K_sets \<open>n \<ge> 1\<close>
  39.124        by (auto simp del: prod_emb_INT intro!: generator.intros)
  39.125      have *: "\<mu>G (Z n) = P (J n) (B n)"
  39.126        unfolding Z_def using J by (intro mu_G_spec) auto
  39.127      then have "\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>" by auto
  39.128      note *
  39.129      moreover have *: "\<mu>G (Y n) = P (J n) (\<Inter>i\<in>{Suc 0..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i))"
  39.130 -      unfolding Y_emb using J J_mono K_sets `n \<ge> 1` by (subst mu_G_spec) auto
  39.131 +      unfolding Y_emb using J J_mono K_sets \<open>n \<ge> 1\<close> by (subst mu_G_spec) auto
  39.132      then have "\<bar>\<mu>G (Y n)\<bar> \<noteq> \<infinity>" by auto
  39.133      note *
  39.134      moreover have "\<mu>G (Z n - Y n) =
  39.135          P (J n) (B n - (\<Inter>i\<in>{Suc 0..n}. prod_emb (J n) (\<lambda>_. borel) (J i) (K i)))"
  39.136 -      unfolding Z_def Y_emb prod_emb_Diff[symmetric] using J J_mono K_sets `n \<ge> 1`
  39.137 +      unfolding Z_def Y_emb prod_emb_Diff[symmetric] using J J_mono K_sets \<open>n \<ge> 1\<close>
  39.138        by (subst mu_G_spec) (auto intro!: sets.Diff)
  39.139      ultimately
  39.140      have "\<mu>G (Z n) - \<mu>G (Y n) = \<mu>G (Z n - Y n)"
  39.141 -      using J J_mono K_sets `n \<ge> 1`
  39.142 +      using J J_mono K_sets \<open>n \<ge> 1\<close>
  39.143        by (simp only: emeasure_eq_measure Z_def)
  39.144          (auto dest!: bspec[where x=n] intro!: measure_Diff[symmetric] set_mp[OF K_B]
  39.145            simp: extensional_restrict emeasure_eq_measure prod_emb_iff sets_P space_P)
  39.146      also have subs: "Z n - Y n \<subseteq> (\<Union>i\<in>{1..n}. (Z i - Z' i))"
  39.147 -      using `n \<ge> 1` unfolding Y_def UN_extend_simps(7) by (intro UN_mono Diff_mono Z_mono order_refl) auto
  39.148 +      using \<open>n \<ge> 1\<close> unfolding Y_def UN_extend_simps(7) by (intro UN_mono Diff_mono Z_mono order_refl) auto
  39.149      have "Z n - Y n \<in> generator" "(\<Union>i\<in>{1..n}. (Z i - Z' i)) \<in> generator"
  39.150 -      using `Z' _ \<in> generator` `Z _ \<in> generator` `Y _ \<in> generator` by auto
  39.151 +      using \<open>Z' _ \<in> generator\<close> \<open>Z _ \<in> generator\<close> \<open>Y _ \<in> generator\<close> by auto
  39.152      hence "\<mu>G (Z n - Y n) \<le> \<mu>G (\<Union>i\<in>{1..n}. (Z i - Z' i))"
  39.153        using subs generator.additive_increasing[OF positive_mu_G additive_mu_G]
  39.154        unfolding increasing_def by auto
  39.155 -    also have "\<dots> \<le> (\<Sum> i\<in>{1..n}. \<mu>G (Z i - Z' i))" using `Z _ \<in> generator` `Z' _ \<in> generator`
  39.156 +    also have "\<dots> \<le> (\<Sum> i\<in>{1..n}. \<mu>G (Z i - Z' i))" using \<open>Z _ \<in> generator\<close> \<open>Z' _ \<in> generator\<close>
  39.157        by (intro generator.subadditive[OF positive_mu_G additive_mu_G]) auto
  39.158      also have "\<dots> \<le> (\<Sum> i\<in>{1..n}. 2 powr -real i * ?a)"
  39.159      proof (rule setsum_mono)
  39.160 @@ -285,11 +285,11 @@
  39.161        also have "\<dots> = P (J i) (B i - K i)"
  39.162          using J K_sets by (subst mu_G_spec) auto
  39.163        also have "\<dots> = P (J i) (B i) - P (J i) (K i)"
  39.164 -        using K_sets J `K _ \<subseteq> B _` by (simp add: emeasure_Diff)
  39.165 +        using K_sets J \<open>K _ \<subseteq> B _\<close> by (simp add: emeasure_Diff)
  39.166        also have "\<dots> = P (J i) (B i) - P' i (K' i)"
  39.167          unfolding K_def P'_def
  39.168          by (auto simp: mapmeasure_PiF borel_eq_PiF_borel[symmetric]
  39.169 -          compact_imp_closed[OF `compact (K' _)`] space_PiM PiE_def)
  39.170 +          compact_imp_closed[OF \<open>compact (K' _)\<close>] space_PiM PiE_def)
  39.171        also have "\<dots> \<le> ereal (2 powr - real i) * ?a" using K'(1)[of i] .
  39.172        finally show "\<mu>G (Z i - Z' i) \<le> (2 powr - real i) * ?a" .
  39.173      qed
  39.174 @@ -310,12 +310,12 @@
  39.175        using J by (auto intro: INF_lower simp: Z_def mu_G_spec)
  39.176      finally have "\<mu>G (Z n) - \<mu>G (Y n) < \<mu>G (Z n)" .
  39.177      hence R: "\<mu>G (Z n) < \<mu>G (Z n) + \<mu>G (Y n)"
  39.178 -      using `\<bar>\<mu>G (Y n)\<bar> \<noteq> \<infinity>` by (simp add: ereal_minus_less)
  39.179 -    have "0 \<le> (- \<mu>G (Z n)) + \<mu>G (Z n)" using `\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>` by auto
  39.180 +      using \<open>\<bar>\<mu>G (Y n)\<bar> \<noteq> \<infinity>\<close> by (simp add: ereal_minus_less)
  39.181 +    have "0 \<le> (- \<mu>G (Z n)) + \<mu>G (Z n)" using \<open>\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>\<close> by auto
  39.182      also have "\<dots> < (- \<mu>G (Z n)) + (\<mu>G (Z n) + \<mu>G (Y n))"
  39.183 -      apply (rule ereal_less_add[OF _ R]) using `\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>` by auto
  39.184 +      apply (rule ereal_less_add[OF _ R]) using \<open>\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>\<close> by auto
  39.185      finally have "\<mu>G (Y n) > 0"
  39.186 -      using `\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>` by (auto simp: ac_simps zero_ereal_def[symmetric])
  39.187 +      using \<open>\<bar>\<mu>G (Z n)\<bar> \<noteq> \<infinity>\<close> by (auto simp: ac_simps zero_ereal_def[symmetric])
  39.188      thus "Y n \<noteq> {}" using positive_mu_G by (auto simp add: positive_def)
  39.189    qed
  39.190    hence "\<forall>n\<in>{1..}. \<exists>y. y \<in> Y n" by auto
  39.191 @@ -323,8 +323,8 @@
  39.192    {
  39.193      fix t and n m::nat
  39.194      assume "1 \<le> n" "n \<le> m" hence "1 \<le> m" by simp
  39.195 -    from Y_mono[OF `m \<ge> n`] y[OF `1 \<le> m`] have "y m \<in> Y n" by auto
  39.196 -    also have "\<dots> \<subseteq> Z' n" using Y_Z'[OF `1 \<le> n`] .
  39.197 +    from Y_mono[OF \<open>m \<ge> n\<close>] y[OF \<open>1 \<le> m\<close>] have "y m \<in> Y n" by auto
  39.198 +    also have "\<dots> \<subseteq> Z' n" using Y_Z'[OF \<open>1 \<le> n\<close>] .
  39.199      finally
  39.200      have "fm n (restrict (y m) (J n)) \<in> K' n"
  39.201        unfolding Z'_def K_def prod_emb_iff by (simp add: Z'_def K_def prod_emb_iff)
  39.202 @@ -354,12 +354,12 @@
  39.203      assume "n \<le> m" hence "Suc n \<le> Suc m" by simp
  39.204      assume "t \<in> domain (fm (Suc n) (y (Suc n)))"
  39.205      then obtain j where j: "t = Utn j" "j \<in> J (Suc n)" by auto
  39.206 -    hence "j \<in> J (Suc m)" using J_mono[OF `Suc n \<le> Suc m`] by auto
  39.207 -    have img: "fm (Suc n) (y (Suc m)) \<in> K' (Suc n)" using `n \<le> m`
  39.208 +    hence "j \<in> J (Suc m)" using J_mono[OF \<open>Suc n \<le> Suc m\<close>] by auto
  39.209 +    have img: "fm (Suc n) (y (Suc m)) \<in> K' (Suc n)" using \<open>n \<le> m\<close>
  39.210        by (intro fm_in_K') simp_all
  39.211      show "(fm (Suc m) (y (Suc m)))\<^sub>F t \<in> (\<lambda>k. (k)\<^sub>F t) ` K' (Suc n)"
  39.212        apply (rule image_eqI[OF _ img])
  39.213 -      using `j \<in> J (Suc n)` `j \<in> J (Suc m)`
  39.214 +      using \<open>j \<in> J (Suc n)\<close> \<open>j \<in> J (Suc m)\<close>
  39.215        unfolding j by (subst proj_fm, auto)+
  39.216    qed
  39.217    have "\<forall>t. \<exists>z. (\<lambda>i. (fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>F t) ----> z"
  39.218 @@ -383,7 +383,7 @@
  39.219          fix e :: real assume "0 < e"
  39.220          { fix i and x :: "'i \<Rightarrow> 'a" assume i: "i \<ge> n"
  39.221            assume "t \<in> domain (fm n x)"
  39.222 -          hence "t \<in> domain (fm i x)" using J_mono[OF `i \<ge> n`] by auto
  39.223 +          hence "t \<in> domain (fm i x)" using J_mono[OF \<open>i \<ge> n\<close>] by auto
  39.224            with i have "(fm i x)\<^sub>F t = (fm n x)\<^sub>F t"
  39.225              using j by (auto simp: proj_fm dest!: inj_onD[OF inj_on_Utn])
  39.226          } note index_shift = this
  39.227 @@ -394,7 +394,7 @@
  39.228            done
  39.229          from z
  39.230          have "\<exists>N. \<forall>i\<ge>N. dist ((fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>F t) (z t) < e"
  39.231 -          unfolding tendsto_iff eventually_sequentially using `0 < e` by auto
  39.232 +          unfolding tendsto_iff eventually_sequentially using \<open>0 < e\<close> by auto
  39.233          then obtain N where N: "\<And>i. i \<ge> N \<Longrightarrow>
  39.234            dist ((fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>F t) (z t) < e" by auto
  39.235          show "\<exists>N. \<forall>na\<ge>N. dist ((fm n (y (Suc (diagseq na))))\<^sub>F t) (z t) < e "
  39.236 @@ -403,7 +403,7 @@
  39.237            hence  "dist ((fm n (y (Suc (diagseq na))))\<^sub>F t) (z t) =
  39.238                    dist ((fm (Suc (diagseq na)) (y (Suc (diagseq na))))\<^sub>F t) (z t)" using t
  39.239              by (subst index_shift[OF I]) auto
  39.240 -          also have "\<dots> < e" using `max N n \<le> na` by (intro N) simp
  39.241 +          also have "\<dots> < e" using \<open>max N n \<le> na\<close> by (intro N) simp
  39.242            finally show "dist ((fm n (y (Suc (diagseq na))))\<^sub>F t) (z t) < e" .
  39.243          qed
  39.244        qed
  39.245 @@ -416,14 +416,14 @@
  39.246        by (intro lim_subseq) (simp add: subseq_def)
  39.247      moreover
  39.248      have "(\<forall>i. ((\<lambda>i. fm n (y (Suc (diagseq i)))) o (\<lambda>i. i + n)) i \<in> K' n)"
  39.249 -      apply (auto simp add: o_def intro!: fm_in_K' `1 \<le> n` le_SucI)
  39.250 +      apply (auto simp add: o_def intro!: fm_in_K' \<open>1 \<le> n\<close> le_SucI)
  39.251        apply (rule le_trans)
  39.252        apply (rule le_add2)
  39.253        using seq_suble[OF subseq_diagseq]
  39.254        apply auto
  39.255        done
  39.256      moreover
  39.257 -    from `compact (K' n)` have "closed (K' n)" by (rule compact_imp_closed)
  39.258 +    from \<open>compact (K' n)\<close> have "closed (K' n)" by (rule compact_imp_closed)
  39.259      ultimately
  39.260      have "finmap_of (Utn ` J n) z \<in> K' n"
  39.261        unfolding closed_sequential_limits by blast
    40.1 --- a/src/HOL/Probability/Radon_Nikodym.thy	Mon Dec 07 16:48:10 2015 +0000
    40.2 +++ b/src/HOL/Probability/Radon_Nikodym.thy	Mon Dec 07 20:19:59 2015 +0100
    40.3 @@ -2,7 +2,7 @@
    40.4      Author:     Johannes Hölzl, TU München
    40.5  *)
    40.6  
    40.7 -section {*Radon-Nikod{\'y}m derivative*}
    40.8 +section \<open>Radon-Nikod{\'y}m derivative\<close>
    40.9  
   40.10  theory Radon_Nikodym
   40.11  imports Bochner_Integration
   40.12 @@ -80,7 +80,7 @@
   40.13          by (cases rule: ereal2_cases[of "n N" "emeasure M (A N)"])
   40.14             (simp_all add: inverse_eq_divide power_divide one_ereal_def ereal_power_divide)
   40.15        finally show "n N * emeasure M (A N) \<le> (1 / 2) ^ Suc N" .
   40.16 -      show "0 \<le> n N * emeasure M (A N)" using n[of N] `A N \<in> sets M` by (simp add: emeasure_nonneg)
   40.17 +      show "0 \<le> n N * emeasure M (A N)" using n[of N] \<open>A N \<in> sets M\<close> by (simp add: emeasure_nonneg)
   40.18      qed
   40.19      finally show "integral\<^sup>N M ?h \<noteq> \<infinity>" unfolding suminf_half_series_ereal by auto
   40.20    next
   40.21 @@ -121,12 +121,12 @@
   40.22      and "absolutely_continuous M M'" "AE x in M. P x"
   40.23     shows "AE x in M'. P x"
   40.24  proof -
   40.25 -  from `AE x in M. P x` obtain N where N: "N \<in> null_sets M" "{x\<in>space M. \<not> P x} \<subseteq> N"
   40.26 +  from \<open>AE x in M. P x\<close> obtain N where N: "N \<in> null_sets M" "{x\<in>space M. \<not> P x} \<subseteq> N"
   40.27      unfolding eventually_ae_filter by auto
   40.28    show "AE x in M'. P x"
   40.29    proof (rule AE_I')
   40.30      show "{x\<in>space M'. \<not> P x} \<subseteq> N" using sets_eq_imp_space_eq[OF sets_eq] N(2) by simp
   40.31 -    from `absolutely_continuous M M'` show "N \<in> null_sets M'"
   40.32 +    from \<open>absolutely_continuous M M'\<close> show "N \<in> null_sets M'"
   40.33        using N unfolding absolutely_continuous_def sets_eq null_sets_def by auto
   40.34    qed
   40.35  qed
   40.36 @@ -167,14 +167,14 @@
   40.37        fix B assume "B \<in> sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> - e"
   40.38        hence "A n \<inter> B = {}" "B \<in> sets M" and dB: "?d B \<le> -e" by auto
   40.39        hence "?d (A n \<union> B) = ?d (A n) + ?d B"
   40.40 -        using `A n \<in> sets M` finite_measure_Union M'.finite_measure_Union by (simp add: sets_eq)
   40.41 +        using \<open>A n \<in> sets M\<close> finite_measure_Union M'.finite_measure_Union by (simp add: sets_eq)
   40.42        also have "\<dots> \<le> ?d (A n) - e" using dB by simp
   40.43        finally show "?d (A n \<union> B) \<le> ?d (A n) - e" .
   40.44      qed }
   40.45    note dA_epsilon = this
   40.46    { fix n have "?d (A (Suc n)) \<le> ?d (A n)"
   40.47      proof (cases "\<exists>B. B\<in>sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> - e")
   40.48 -      case True from dA_epsilon[OF this] show ?thesis using `0 < e` by simp
   40.49 +      case True from dA_epsilon[OF this] show ?thesis using \<open>0 < e\<close> by simp
   40.50      next
   40.51        case False
   40.52        hence "\<forall>B\<in>sets M. B \<subseteq> space M - A n \<longrightarrow> -e < ?d B" by (auto simp: not_le)
   40.53 @@ -214,13 +214,13 @@
   40.54        fix n show "- ?d (A n) \<le> - ?d (A (Suc n))" using dA_mono[of n] by auto
   40.55      qed
   40.56      have A: "incseq A" by (auto intro!: incseq_SucI)
   40.57 -    from finite_Lim_measure_incseq[OF _ A] `range A \<subseteq> sets M`
   40.58 +    from finite_Lim_measure_incseq[OF _ A] \<open>range A \<subseteq> sets M\<close>
   40.59        M'.finite_Lim_measure_incseq[OF _ A]
   40.60      have convergent: "(\<lambda>i. ?d (A i)) ----> ?d (\<Union>i. A i)"
   40.61        by (auto intro!: tendsto_diff simp: sets_eq)
   40.62      obtain n :: nat where "- ?d (\<Union>i. A i) / e < real n" using reals_Archimedean2 by auto
   40.63      moreover from order_trans[OF decseq_le[OF decseq convergent] dA_less]
   40.64 -    have "real n \<le> - ?d (\<Union>i. A i) / e" using `0<e` by (simp add: field_simps)
   40.65 +    have "real n \<le> - ?d (\<Union>i. A i) / e" using \<open>0<e\<close> by (simp add: field_simps)
   40.66      ultimately show ?thesis by auto
   40.67    qed
   40.68  qed
   40.69 @@ -258,7 +258,7 @@
   40.70      by (auto simp add: mono_iff_le_Suc)
   40.71    show ?thesis
   40.72    proof (safe intro!: bexI[of _ "\<Inter>i. A i"])
   40.73 -    show "(\<Inter>i. A i) \<in> sets M" using `\<And>n. A n \<in> sets M` by auto
   40.74 +    show "(\<Inter>i. A i) \<in> sets M" using \<open>\<And>n. A n \<in> sets M\<close> by auto
   40.75      have "decseq A" using A by (auto intro!: decseq_SucI)
   40.76      from A(1) finite_Lim_measure_decseq[OF _ this] N.finite_Lim_measure_decseq[OF _ this]
   40.77      have "(\<lambda>i. ?d (A i)) ----> ?d (\<Inter>i. A i)" by (auto intro!: tendsto_diff simp: sets_eq)
   40.78 @@ -299,10 +299,10 @@
   40.79        let ?A = "{x \<in> space M. f x \<le> g x}"
   40.80        have "?A \<in> sets M" using f g unfolding G_def by auto
   40.81        fix A assume "A \<in> sets M"
   40.82 -      hence sets: "?A \<inter> A \<in> sets M" "(space M - ?A) \<inter> A \<in> sets M" using `?A \<in> sets M` by auto
   40.83 +      hence sets: "?A \<inter> A \<in> sets M" "(space M - ?A) \<inter> A \<in> sets M" using \<open>?A \<in> sets M\<close> by auto
   40.84        hence sets': "?A \<inter> A \<in> sets N" "(space M - ?A) \<inter> A \<in> sets N" by (auto simp: sets_eq)
   40.85        have union: "((?A \<inter> A) \<union> ((space M - ?A) \<inter> A)) = A"
   40.86 -        using sets.sets_into_space[OF `A \<in> sets M`] by auto
   40.87 +        using sets.sets_into_space[OF \<open>A \<in> sets M\<close>] by auto
   40.88        have "\<And>x. x \<in> space M \<Longrightarrow> max (g x) (f x) * indicator A x =
   40.89          g x * indicator (?A \<inter> A) x + f x * indicator ((space M - ?A) \<inter> A) x"
   40.90          by (auto simp: indicator_def max_def)
   40.91 @@ -333,11 +333,11 @@
   40.92          (\<integral>\<^sup>+x. (SUP i. f i x * indicator A x) \<partial>M)"
   40.93          by (intro nn_integral_cong) (simp split: split_indicator)
   40.94        also have "\<dots> = (SUP i. (\<integral>\<^sup>+x. f i x * indicator A x \<partial>M))"
   40.95 -        using `incseq f` f `A \<in> sets M`
   40.96 +        using \<open>incseq f\<close> f \<open>A \<in> sets M\<close>
   40.97          by (intro nn_integral_monotone_convergence_SUP)
   40.98             (auto simp: G_def incseq_Suc_iff le_fun_def split: split_indicator)
   40.99        finally show "(\<integral>\<^sup>+x. (SUP i. f i x) * indicator A x \<partial>M) \<le> N A"
  40.100 -        using f `A \<in> sets M` by (auto intro!: SUP_least simp: G_def)
  40.101 +        using f \<open>A \<in> sets M\<close> by (auto intro!: SUP_least simp: G_def)
  40.102      qed }
  40.103    note SUP_in_G = this
  40.104    let ?y = "SUP g : G. integral\<^sup>N M g"
  40.105 @@ -347,7 +347,7 @@
  40.106      from this[THEN bspec, OF sets.top] show "integral\<^sup>N M g \<le> N (space M)"
  40.107        by (simp cong: nn_integral_cong)
  40.108    qed
  40.109 -  from SUP_countable_SUP [OF `G \<noteq> {}`, of "integral\<^sup>N M"] guess ys .. note ys = this
  40.110 +  from SUP_countable_SUP [OF \<open>G \<noteq> {}\<close>, of "integral\<^sup>N M"] guess ys .. note ys = this
  40.111    then have "\<forall>n. \<exists>g. g\<in>G \<and> integral\<^sup>N M g = ys n"
  40.112    proof safe
  40.113      fix n assume "range ys \<subseteq> integral\<^sup>N M ` G"
  40.114 @@ -365,7 +365,7 @@
  40.115        case 0 thus ?case by simp fact
  40.116      next
  40.117        case (Suc i)
  40.118 -      with Suc gs_not_empty `gs (Suc i) \<in> G` show ?case
  40.119 +      with Suc gs_not_empty \<open>gs (Suc i) \<in> G\<close> show ?case
  40.120          by (auto simp add: atMost_Suc intro!: max_in_G)
  40.121      qed }
  40.122    note g_in_G = this
  40.123 @@ -374,7 +374,7 @@
  40.124    from SUP_in_G[OF this g_in_G] have [measurable]: "f \<in> G" unfolding f_def .
  40.125    then have [simp, intro]: "f \<in> borel_measurable M" unfolding G_def by auto
  40.126    have "integral\<^sup>N M f = (SUP i. integral\<^sup>N M (?g i))" unfolding f_def
  40.127 -    using g_in_G `incseq ?g`
  40.128 +    using g_in_G \<open>incseq ?g\<close>
  40.129      by (auto intro!: nn_integral_monotone_convergence_SUP simp: G_def)
  40.130    also have "\<dots> = ?y"
  40.131    proof (rule antisym)
  40.132 @@ -385,12 +385,12 @@
  40.133    qed
  40.134    finally have int_f_eq_y: "integral\<^sup>N M f = ?y" .
  40.135    have "\<And>x. 0 \<le> f x"
  40.136 -    unfolding f_def using `\<And>i. gs i \<in> G`
  40.137 +    unfolding f_def using \<open>\<And>i. gs i \<in> G\<close>
  40.138      by (auto intro!: SUP_upper2 Max_ge_iff[THEN iffD2] simp: G_def)
  40.139    let ?t = "\<lambda>A. N A - (\<integral>\<^sup>+x. ?F A x \<partial>M)"
  40.140    let ?M = "diff_measure N (density M f)"
  40.141    have f_le_N: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>\<^sup>+x. ?F A x \<partial>M) \<le> N A"
  40.142 -    using `f \<in> G` unfolding G_def by auto
  40.143 +    using \<open>f \<in> G\<close> unfolding G_def by auto
  40.144    have emeasure_M: "\<And>A. A \<in> sets M \<Longrightarrow> emeasure ?M A = ?t A"
  40.145    proof (subst emeasure_diff_measure)
  40.146      from f_le_N[of "space M"] show "finite_measure N" "finite_measure (density M f)"
  40.147 @@ -406,9 +406,9 @@
  40.148    have ac: "absolutely_continuous M ?M" unfolding absolutely_continuous_def
  40.149    proof
  40.150      fix A assume A_M: "A \<in> null_sets M"
  40.151 -    with `absolutely_continuous M N` have A_N: "A \<in> null_sets N"
  40.152 +    with \<open>absolutely_continuous M N\<close> have A_N: "A \<in> null_sets N"
  40.153        unfolding absolutely_continuous_def by auto
  40.154 -    moreover from A_M A_N have "(\<integral>\<^sup>+ x. ?F A x \<partial>M) \<le> N A" using `f \<in> G` by (auto simp: G_def)
  40.155 +    moreover from A_M A_N have "(\<integral>\<^sup>+ x. ?F A x \<partial>M) \<le> N A" using \<open>f \<in> G\<close> by (auto simp: G_def)
  40.156      ultimately have "N A - (\<integral>\<^sup>+ x. ?F A x \<partial>M) = 0"
  40.157        using nn_integral_nonneg[of M] by (auto intro!: antisym)
  40.158      then show "A \<in> null_sets ?M"
  40.159 @@ -430,7 +430,7 @@
  40.160        using emeasure_nonneg[of M "space M"] by (simp add: le_less)
  40.161      moreover
  40.162      have "(\<integral>\<^sup>+x. f x * indicator (space M) x \<partial>M) \<le> N (space M)"
  40.163 -      using `f \<in> G` unfolding G_def by auto
  40.164 +      using \<open>f \<in> G\<close> unfolding G_def by auto
  40.165      hence "(\<integral>\<^sup>+x. f x * indicator (space M) x \<partial>M) \<noteq> \<infinity>"
  40.166        using M'.finite_emeasure_space by auto
  40.167      moreover
  40.168 @@ -452,31 +452,31 @@
  40.169      note bM_le_t = this
  40.170      let ?f0 = "\<lambda>x. f x + b * indicator A0 x"
  40.171      { fix A assume A: "A \<in> sets M"
  40.172 -      hence "A \<inter> A0 \<in> sets M" using `A0 \<in> sets M` by auto
  40.173 +      hence "A \<inter> A0 \<in> sets M" using \<open>A0 \<in> sets M\<close> by auto
  40.174        have "(\<integral>\<^sup>+x. ?f0 x  * indicator A x \<partial>M) =
  40.175          (\<integral>\<^sup>+x. f x * indicator A x + b * indicator (A \<inter> A0) x \<partial>M)"
  40.176          by (auto intro!: nn_integral_cong split: split_indicator)
  40.177        hence "(\<integral>\<^sup>+x. ?f0 x * indicator A x \<partial>M) =
  40.178            (\<integral>\<^sup>+x. f x * indicator A x \<partial>M) + b * emeasure M (A \<inter> A0)"
  40.179 -        using `A0 \<in> sets M` `A \<inter> A0 \<in> sets M` A b `f \<in> G`
  40.180 +        using \<open>A0 \<in> sets M\<close> \<open>A \<inter> A0 \<in> sets M\<close> A b \<open>f \<in> G\<close>
  40.181          by (simp add: nn_integral_add nn_integral_cmult_indicator G_def) }
  40.182      note f0_eq = this
  40.183      { fix A assume A: "A \<in> sets M"
  40.184 -      hence "A \<inter> A0 \<in> sets M" using `A0 \<in> sets M` by auto
  40.185 -      have f_le_v: "(\<integral>\<^sup>+x. ?F A x \<partial>M) \<le> N A" using `f \<in> G` A unfolding G_def by auto
  40.186 +      hence "A \<inter> A0 \<in> sets M" using \<open>A0 \<in> sets M\<close> by auto
  40.187 +      have f_le_v: "(\<integral>\<^sup>+x. ?F A x \<partial>M) \<le> N A" using \<open>f \<in> G\<close> A unfolding G_def by auto
  40.188        note f0_eq[OF A]
  40.189        also have "(\<integral>\<^sup>+x. ?F A x \<partial>M) + b * emeasure M (A \<inter> A0) \<le> (\<integral>\<^sup>+x. ?F A x \<partial>M) + ?M (A \<inter> A0)"
  40.190 -        using bM_le_t[OF `A \<inter> A0 \<in> sets M`] `A \<in> sets M` `A0 \<in> sets M`
  40.191 +        using bM_le_t[OF \<open>A \<inter> A0 \<in> sets M\<close>] \<open>A \<in> sets M\<close> \<open>A0 \<in> sets M\<close>
  40.192          by (auto intro!: add_left_mono)
  40.193        also have "\<dots> \<le> (\<integral>\<^sup>+x. f x * indicator A x \<partial>M) + ?M A"
  40.194 -        using emeasure_mono[of "A \<inter> A0" A ?M] `A \<in> sets M` `A0 \<in> sets M`
  40.195 +        using emeasure_mono[of "A \<inter> A0" A ?M] \<open>A \<in> sets M\<close> \<open>A0 \<in> sets M\<close>
  40.196          by (auto intro!: add_left_mono simp: sets_eq)
  40.197        also have "\<dots> \<le> N A"
  40.198 -        unfolding emeasure_M[OF `A \<in> sets M`]
  40.199 +        unfolding emeasure_M[OF \<open>A \<in> sets M\<close>]
  40.200          using f_le_v N.emeasure_eq_measure[of A] nn_integral_nonneg[of M "?F A"]
  40.201          by (cases "\<integral>\<^sup>+x. ?F A x \<partial>M", cases "N A") auto
  40.202        finally have "(\<integral>\<^sup>+x. ?f0 x * indicator A x \<partial>M) \<le> N A" . }
  40.203 -    hence "?f0 \<in> G" using `A0 \<in> sets M` b `f \<in> G` by (auto simp: G_def)
  40.204 +    hence "?f0 \<in> G" using \<open>A0 \<in> sets M\<close> b \<open>f \<in> G\<close> by (auto simp: G_def)
  40.205      have int_f_finite: "integral\<^sup>N M f \<noteq> \<infinity>"
  40.206        by (metis N.emeasure_finite ereal_infty_less_eq2(1) int_f_eq_y y_le)
  40.207      have  "0 < ?M (space M) - emeasure ?Mb (space M)"
  40.208 @@ -484,25 +484,25 @@
  40.209        by (simp add: b emeasure_density_const)
  40.210           (simp add: M'.emeasure_eq_measure emeasure_eq_measure pos_M b_def)
  40.211      also have "\<dots> \<le> ?M A0 - b * emeasure M A0"
  40.212 -      using space_less_A0 `A0 \<in> sets M` b
  40.213 +      using space_less_A0 \<open>A0 \<in> sets M\<close> b
  40.214        by (cases b) (auto simp add: b emeasure_density_const sets_eq M'.emeasure_eq_measure emeasure_eq_measure)
  40.215      finally have 1: "b * emeasure M A0 < ?M A0"
  40.216 -      by (metis M'.emeasure_real `A0 \<in> sets M` bM_le_t diff_self ereal_less(1) ereal_minus(1)
  40.217 +      by (metis M'.emeasure_real \<open>A0 \<in> sets M\<close> bM_le_t diff_self ereal_less(1) ereal_minus(1)
  40.218                  less_eq_ereal_def mult_zero_left not_square_less_zero subset_refl zero_ereal_def)
  40.219      with b have "0 < ?M A0"
  40.220        by (metis M'.emeasure_real MInfty_neq_PInfty(1) emeasure_real ereal_less_eq(5) ereal_zero_times
  40.221                 ereal_mult_eq_MInfty ereal_mult_eq_PInfty ereal_zero_less_0_iff less_eq_ereal_def)
  40.222 -    then have "emeasure M A0 \<noteq> 0" using ac `A0 \<in> sets M`
  40.223 +    then have "emeasure M A0 \<noteq> 0" using ac \<open>A0 \<in> sets M\<close>
  40.224        by (auto simp: absolutely_continuous_def null_sets_def)
  40.225      then have "0 < emeasure M A0" using emeasure_nonneg[of M A0] by auto
  40.226      hence "0 < b * emeasure M A0" using b by (auto simp: ereal_zero_less_0_iff)
  40.227      with int_f_finite have "?y + 0 < integral\<^sup>N M f + b * emeasure M A0" unfolding int_f_eq_y
  40.228 -      using `f \<in> G`
  40.229 +      using \<open>f \<in> G\<close>
  40.230        by (intro ereal_add_strict_mono) (auto intro!: SUP_upper2 nn_integral_nonneg)
  40.231 -    also have "\<dots> = integral\<^sup>N M ?f0" using f0_eq[OF sets.top] `A0 \<in> sets M` sets.sets_into_space
  40.232 +    also have "\<dots> = integral\<^sup>N M ?f0" using f0_eq[OF sets.top] \<open>A0 \<in> sets M\<close> sets.sets_into_space
  40.233        by (simp cong: nn_integral_cong)
  40.234      finally have "?y < integral\<^sup>N M ?f0" by simp
  40.235 -    moreover from `?f0 \<in> G` have "integral\<^sup>N M ?f0 \<le> ?y" by (auto intro!: SUP_upper)
  40.236 +    moreover from \<open>?f0 \<in> G\<close> have "integral\<^sup>N M ?f0 \<le> ?y" by (auto intro!: SUP_upper)
  40.237      ultimately show False by auto
  40.238    qed
  40.239    let ?f = "\<lambda>x. max 0 (f x)"
  40.240 @@ -512,7 +512,7 @@
  40.241        by (simp add: sets_eq)
  40.242      fix A assume A: "A\<in>sets (density M ?f)"
  40.243      then show "emeasure (density M ?f) A = emeasure N A"
  40.244 -      using `f \<in> G` A upper_bound[THEN bspec, of A] N.emeasure_eq_measure[of A]
  40.245 +      using \<open>f \<in> G\<close> A upper_bound[THEN bspec, of A] N.emeasure_eq_measure[of A]
  40.246        by (cases "integral\<^sup>N M (?F A)")
  40.247           (auto intro!: antisym simp add: emeasure_density G_def emeasure_M density_max_0[symmetric])
  40.248    qed auto
  40.249 @@ -599,7 +599,7 @@
  40.250            also have "\<dots> = (SUP i. emeasure M (?O i \<union> A))"
  40.251            proof (rule SUP_emeasure_incseq[of "\<lambda>i. ?O i \<union> A", symmetric, simplified])
  40.252              show "range (\<lambda>i. ?O i \<union> A) \<subseteq> sets M"
  40.253 -              using `N A \<noteq> \<infinity>` O_sets A by auto
  40.254 +              using \<open>N A \<noteq> \<infinity>\<close> O_sets A by auto
  40.255            qed (fastforce intro!: incseq_SucI)
  40.256            also have "\<dots> \<le> ?a"
  40.257            proof (safe intro!: SUP_least)
  40.258 @@ -609,13 +609,13 @@
  40.259                from O_in_G[of i] have "N (?O i \<union> A) \<le> N (?O i) + N A"
  40.260                  using emeasure_subadditive[of "?O i" N A] A O_sets by (auto simp: sets_eq)
  40.261                with O_in_G[of i] show "N (?O i \<union> A) \<noteq> \<infinity>"
  40.262 -                using `N A \<noteq> \<infinity>` by auto
  40.263 +                using \<open>N A \<noteq> \<infinity>\<close> by auto
  40.264              qed
  40.265              then show "emeasure M (?O i \<union> A) \<le> ?a" by (rule SUP_upper)
  40.266            qed
  40.267            finally have "emeasure M A = 0"
  40.268              unfolding a_eq using measure_nonneg[of M A] by (simp add: emeasure_eq_measure)
  40.269 -          with `emeasure M A \<noteq> 0` show ?thesis by auto
  40.270 +          with \<open>emeasure M A \<noteq> 0\<close> show ?thesis by auto
  40.271          qed
  40.272        qed }
  40.273      { fix i show "N (Q i) \<noteq> \<infinity>"
  40.274 @@ -624,7 +624,7 @@
  40.275            unfolding Q_def using Q'[of 0] by simp
  40.276        next
  40.277          case (Suc n)
  40.278 -        with `?O n \<in> ?Q` `?O (Suc n) \<in> ?Q`
  40.279 +        with \<open>?O n \<in> ?Q\<close> \<open>?O (Suc n) \<in> ?Q\<close>
  40.280              emeasure_Diff[OF _ _ _ O_mono, of N n] emeasure_nonneg[of N "(\<Union>x\<le>n. Q' x)"]
  40.281          show ?thesis
  40.282            by (auto simp: sets_eq ereal_minus_eq_PInfty_iff Q_def)
  40.283 @@ -671,7 +671,7 @@
  40.284      show "sets (?N i) = sets (?M i)" by (simp add: sets_eq)
  40.285      have [measurable]: "\<And>A. A \<in> sets M \<Longrightarrow> A \<in> sets N" by (simp add: sets_eq)
  40.286      show "absolutely_continuous (?M i) (?N i)"
  40.287 -      using `absolutely_continuous M N` `Q i \<in> sets M`
  40.288 +      using \<open>absolutely_continuous M N\<close> \<open>Q i \<in> sets M\<close>
  40.289        by (auto simp: absolutely_continuous_def null_sets_density_iff sets_eq
  40.290                 intro!: absolutely_continuous_AE[OF sets_eq])
  40.291    qed
  40.292 @@ -700,31 +700,31 @@
  40.293        have Qi: "\<And>i. Q i \<in> sets M" using Q by auto
  40.294        have [intro,simp]: "\<And>i. (\<lambda>x. f i x * indicator (Q i \<inter> A) x) \<in> borel_measurable M"
  40.295          "\<And>i. AE x in M. 0 \<le> f i x * indicator (Q i \<inter> A) x"
  40.296 -        using borel Qi Q0(1) `A \<in> sets M` by (auto intro!: borel_measurable_ereal_times)
  40.297 +        using borel Qi Q0(1) \<open>A \<in> sets M\<close> by (auto intro!: borel_measurable_ereal_times)
  40.298        have "(\<integral>\<^sup>+x. ?f x * indicator A x \<partial>M) = (\<integral>\<^sup>+x. (\<Sum>i. f i x * indicator (Q i \<inter> A) x) + \<infinity> * indicator (Q0 \<inter> A) x \<partial>M)"
  40.299          using borel by (intro nn_integral_cong) (auto simp: indicator_def)
  40.300        also have "\<dots> = (\<integral>\<^sup>+x. (\<Sum>i. f i x * indicator (Q i \<inter> A) x) \<partial>M) + \<infinity> * emeasure M (Q0 \<inter> A)"
  40.301 -        using borel Qi Q0(1) `A \<in> sets M`
  40.302 +        using borel Qi Q0(1) \<open>A \<in> sets M\<close>
  40.303          by (subst nn_integral_add) (auto simp del: ereal_infty_mult
  40.304              simp add: nn_integral_cmult_indicator sets.Int intro!: suminf_0_le)
  40.305        also have "\<dots> = (\<Sum>i. N (Q i \<inter> A)) + \<infinity> * emeasure M (Q0 \<inter> A)"
  40.306 -        by (subst integral_eq[OF `A \<in> sets M`], subst nn_integral_suminf) auto
  40.307 +        by (subst integral_eq[OF \<open>A \<in> sets M\<close>], subst nn_integral_suminf) auto
  40.308        finally have "(\<integral>\<^sup>+x. ?f x * indicator A x \<partial>M) = (\<Sum>i. N (Q i \<inter> A)) + \<infinity> * emeasure M (Q0 \<inter> A)" .
  40.309        moreover have "(\<Sum>i. N (Q i \<inter> A)) = N ((\<Union>i. Q i) \<inter> A)"
  40.310 -        using Q Q_sets `A \<in> sets M`
  40.311 +        using Q Q_sets \<open>A \<in> sets M\<close>
  40.312          by (subst suminf_emeasure) (auto simp: disjoint_family_on_def sets_eq)
  40.313        moreover have "\<infinity> * emeasure M (Q0 \<inter> A) = N (Q0 \<inter> A)"
  40.314        proof -
  40.315 -        have "Q0 \<inter> A \<in> sets M" using Q0(1) `A \<in> sets M` by blast
  40.316 +        have "Q0 \<inter> A \<in> sets M" using Q0(1) \<open>A \<in> sets M\<close> by blast
  40.317          from in_Q0[OF this] show ?thesis by auto
  40.318        qed
  40.319        moreover have "Q0 \<inter> A \<in> sets M" "((\<Union>i. Q i) \<inter> A) \<in> sets M"
  40.320 -        using Q_sets `A \<in> sets M` Q0(1) by auto
  40.321 +        using Q_sets \<open>A \<in> sets M\<close> Q0(1) by auto
  40.322        moreover have "((\<Union>i. Q i) \<inter> A) \<union> (Q0 \<inter> A) = A" "((\<Union>i. Q i) \<inter> A) \<inter> (Q0 \<inter> A) = {}"
  40.323 -        using `A \<in> sets M` sets.sets_into_space Q0 by auto
  40.324 +        using \<open>A \<in> sets M\<close> sets.sets_into_space Q0 by auto
  40.325        ultimately have "N A = (\<integral>\<^sup>+x. ?f x * indicator A x \<partial>M)"
  40.326          using plus_emeasure[of "(\<Union>i. Q i) \<inter> A" N "Q0 \<inter> A"] by (simp add: sets_eq)
  40.327 -      with `A \<in> sets M` borel Q Q0(1) show "emeasure (density M ?f) A = N A"
  40.328 +      with \<open>A \<in> sets M\<close> borel Q Q0(1) show "emeasure (density M ?f) A = N A"
  40.329          by (auto simp: subset_eq emeasure_density)
  40.330      qed (simp add: sets_eq)
  40.331    qed
  40.332 @@ -752,7 +752,7 @@
  40.333      with pos sets.sets_into_space have "AE x in M. x \<notin> A"
  40.334        by (elim eventually_elim1) (auto simp: not_le[symmetric])
  40.335      then have "A \<in> null_sets M"
  40.336 -      using `A \<in> sets M` by (simp add: AE_iff_null_sets)
  40.337 +      using \<open>A \<in> sets M\<close> by (simp add: AE_iff_null_sets)
  40.338      with ac show "A \<in> null_sets N"
  40.339        by (auto simp: absolutely_continuous_def)
  40.340    qed (auto simp add: sets_eq)
  40.341 @@ -762,7 +762,7 @@
  40.342      by (auto intro!: bexI[of _ "\<lambda>x. h x * f x"] simp: density_density_eq)
  40.343  qed
  40.344  
  40.345 -subsection {* Uniqueness of densities *}
  40.346 +subsection \<open>Uniqueness of densities\<close>
  40.347  
  40.348  lemma finite_density_unique:
  40.349    assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
  40.350 @@ -848,13 +848,13 @@
  40.351          fix i ::nat have "?A i \<in> sets M"
  40.352            using borel Q0(1) by auto
  40.353          have "?N (?A i) \<le> (\<integral>\<^sup>+x. (i::ereal) * indicator (?A i) x \<partial>M)"
  40.354 -          unfolding eq[OF `?A i \<in> sets M`]
  40.355 +          unfolding eq[OF \<open>?A i \<in> sets M\<close>]
  40.356            by (auto intro!: nn_integral_mono simp: indicator_def)
  40.357          also have "\<dots> = i * emeasure M (?A i)"
  40.358 -          using `?A i \<in> sets M` by (auto intro!: nn_integral_cmult_indicator)
  40.359 +          using \<open>?A i \<in> sets M\<close> by (auto intro!: nn_integral_cmult_indicator)
  40.360          also have "\<dots> < \<infinity>" using emeasure_real[of "?A i"] by simp
  40.361          finally have "?N (?A i) \<noteq> \<infinity>" by simp
  40.362 -        then show "?A i \<in> null_sets M" using in_Q0[OF `?A i \<in> sets M`] `?A i \<in> sets M` by auto
  40.363 +        then show "?A i \<in> null_sets M" using in_Q0[OF \<open>?A i \<in> sets M\<close>] \<open>?A i \<in> sets M\<close> by auto
  40.364        qed
  40.365        also have "(\<Union>i. ?A i) = Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>}"
  40.366          by (auto simp: less_PInf_Ex_of_nat)
  40.367 @@ -894,21 +894,21 @@
  40.368      then have "{x \<in> space M. h x * indicator A x \<noteq> 0} = A"
  40.369        using pos(1) sets.sets_into_space by (force simp: indicator_def)
  40.370      then have "(\<integral>\<^sup>+x. h x * indicator A x \<partial>M) = 0 \<longleftrightarrow> A \<in> null_sets M"
  40.371 -      using h_borel `A \<in> sets M` h_nn by (subst nn_integral_0_iff) auto }
  40.372 +      using h_borel \<open>A \<in> sets M\<close> h_nn by (subst nn_integral_0_iff) auto }
  40.373    note h_null_sets = this
  40.374    { fix A assume "A \<in> sets M"
  40.375      have "(\<integral>\<^sup>+x. f x * (h x * indicator A x) \<partial>M) = (\<integral>\<^sup>+x. h x * indicator A x \<partial>?fM)"
  40.376 -      using `A \<in> sets M` h_borel h_nn f f'
  40.377 +      using \<open>A \<in> sets M\<close> h_borel h_nn f f'
  40.378        by (intro nn_integral_density[symmetric]) auto
  40.379      also have "\<dots> = (\<integral>\<^sup>+x. h x * indicator A x \<partial>?f'M)"
  40.380        by (simp_all add: density_eq)
  40.381      also have "\<dots> = (\<integral>\<^sup>+x. f' x * (h x * indicator A x) \<partial>M)"
  40.382 -      using `A \<in> sets M` h_borel h_nn f f'
  40.383 +      using \<open>A \<in> sets M\<close> h_borel h_nn f f'
  40.384        by (intro nn_integral_density) auto
  40.385      finally have "(\<integral>\<^sup>+x. h x * (f x * indicator A x) \<partial>M) = (\<integral>\<^sup>+x. h x * (f' x * indicator A x) \<partial>M)"
  40.386        by (simp add: ac_simps)
  40.387      then have "(\<integral>\<^sup>+x. (f x * indicator A x) \<partial>?H) = (\<integral>\<^sup>+x. (f' x * indicator A x) \<partial>?H)"
  40.388 -      using `A \<in> sets M` h_borel h_nn f f'
  40.389 +      using \<open>A \<in> sets M\<close> h_borel h_nn f f'
  40.390        by (subst (asm) (1 2) nn_integral_density[symmetric]) auto }
  40.391    then have "AE x in ?H. f x = f' x" using h_borel h_nn f f'
  40.392      by (intro h.density_unique_finite_measure absolutely_continuous_AE[of M])
  40.393 @@ -1025,7 +1025,7 @@
  40.394      proof (cases i)
  40.395        case 0
  40.396        have "AE x in M. f x * indicator (A i \<inter> Q j) x = 0"
  40.397 -        using AE by (auto simp: A_def `i = 0`)
  40.398 +        using AE by (auto simp: A_def \<open>i = 0\<close>)
  40.399        from nn_integral_cong_AE[OF this] show ?thesis by simp
  40.400      next
  40.401        case (Suc n)
  40.402 @@ -1050,7 +1050,7 @@
  40.403    apply (auto simp: max_def intro!: measurable_If)
  40.404    done
  40.405  
  40.406 -subsection {* Radon-Nikodym derivative *}
  40.407 +subsection \<open>Radon-Nikodym derivative\<close>
  40.408  
  40.409  definition RN_deriv :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> 'a \<Rightarrow> ereal" where
  40.410    "RN_deriv M N =
  40.411 @@ -1164,11 +1164,11 @@
  40.412      next
  40.413        fix X assume "X \<in> (\<lambda>A. T' -` A \<inter> space ?M')`F"
  40.414        then obtain A where [simp]: "X = T' -` A \<inter> space ?M'" and "A \<in> F" by auto
  40.415 -      have "X \<in> sets M'" using F T' `A\<in>F` by auto
  40.416 +      have "X \<in> sets M'" using F T' \<open>A\<in>F\<close> by auto
  40.417        moreover
  40.418 -      have Fi: "A \<in> sets M" using F `A\<in>F` by auto
  40.419 +      have Fi: "A \<in> sets M" using F \<open>A\<in>F\<close> by auto
  40.420        ultimately show "emeasure ?M' X \<noteq> \<infinity>"
  40.421 -        using F T T' `A\<in>F` by (simp add: emeasure_distr)
  40.422 +        using F T T' \<open>A\<in>F\<close> by (simp add: emeasure_distr)
  40.423      qed (insert F, auto)
  40.424    qed
  40.425    have "(RN_deriv ?M' ?N') \<circ> T \<in> borel_measurable M"
  40.426 @@ -1291,7 +1291,7 @@
  40.427    and x: "{x} \<in> sets M"
  40.428    shows "N {x} = RN_deriv M N x * emeasure M {x}"
  40.429  proof -
  40.430 -  from `{x} \<in> sets M`
  40.431 +  from \<open>{x} \<in> sets M\<close>
  40.432    have "density M (RN_deriv M N) {x} = (\<integral>\<^sup>+w. RN_deriv M N x * indicator {x} w \<partial>M)"
  40.433      by (auto simp: indicator_def emeasure_density intro!: nn_integral_cong)
  40.434    with x density_RN_deriv[OF ac] RN_deriv_nonneg[of M N] show ?thesis
    41.1 --- a/src/HOL/Probability/Regularity.thy	Mon Dec 07 16:48:10 2015 +0000
    41.2 +++ b/src/HOL/Probability/Regularity.thy	Mon Dec 07 20:19:59 2015 +0100
    41.3 @@ -2,7 +2,7 @@
    41.4      Author:     Fabian Immler, TU München
    41.5  *)
    41.6  
    41.7 -section {* Regularity of Measures *}
    41.8 +section \<open>Regularity of Measures\<close>
    41.9  
   41.10  theory Regularity
   41.11  imports Measure_Space Borel_Space
   41.12 @@ -24,12 +24,12 @@
   41.13    show "x \<le> y"
   41.14    proof (rule ccontr)
   41.15      assume "\<not> x \<le> y" hence "x > y" by simp
   41.16 -    hence y_fin: "\<bar>y\<bar> \<noteq> \<infinity>" using `y \<ge> 0` by auto
   41.17 -    have x_fin: "\<bar>x\<bar> \<noteq> \<infinity>" using `x > y` f_fin approx[where e = 1] by auto
   41.18 +    hence y_fin: "\<bar>y\<bar> \<noteq> \<infinity>" using \<open>y \<ge> 0\<close> by auto
   41.19 +    have x_fin: "\<bar>x\<bar> \<noteq> \<infinity>" using \<open>x > y\<close> f_fin approx[where e = 1] by auto
   41.20      def e \<equiv> "real_of_ereal ((x - y) / 2)"
   41.21 -    have e: "x > y + e" "e > 0" using `x > y` y_fin x_fin by (auto simp: e_def field_simps)
   41.22 +    have e: "x > y + e" "e > 0" using \<open>x > y\<close> y_fin x_fin by (auto simp: e_def field_simps)
   41.23      note e(1)
   41.24 -    also from approx[OF `e > 0`] obtain i where i: "i \<in> A" "x \<le> f i + e" by blast
   41.25 +    also from approx[OF \<open>e > 0\<close>] obtain i where i: "i \<in> A" "x \<le> f i + e" by blast
   41.26      note i(2)
   41.27      finally have "y < f i" using y_fin f_fin by (metis add_right_mono linorder_not_le)
   41.28      moreover have "f i \<le> y" by (rule f_le_y) fact
   41.29 @@ -53,12 +53,12 @@
   41.30    show "y \<le> x"
   41.31    proof (rule ccontr)
   41.32      assume "\<not> y \<le> x" hence "y > x" by simp hence "y \<noteq> - \<infinity>" by auto
   41.33 -    hence y_fin: "\<bar>y\<bar> \<noteq> \<infinity>" using `y \<noteq> \<infinity>` by auto
   41.34 -    have x_fin: "\<bar>x\<bar> \<noteq> \<infinity>" using `y > x` f_fin f_nonneg approx[where e = 1] A_notempty
   41.35 +    hence y_fin: "\<bar>y\<bar> \<noteq> \<infinity>" using \<open>y \<noteq> \<infinity>\<close> by auto
   41.36 +    have x_fin: "\<bar>x\<bar> \<noteq> \<infinity>" using \<open>y > x\<close> f_fin f_nonneg approx[where e = 1] A_notempty
   41.37        by auto
   41.38      def e \<equiv> "real_of_ereal ((y - x) / 2)"
   41.39 -    have e: "y > x + e" "e > 0" using `y > x` y_fin x_fin by (auto simp: e_def field_simps)
   41.40 -    from approx[OF `e > 0`] obtain i where i: "i \<in> A" "x + e \<ge> f i" by blast
   41.41 +    have e: "y > x + e" "e > 0" using \<open>y > x\<close> y_fin x_fin by (auto simp: e_def field_simps)
   41.42 +    from approx[OF \<open>e > 0\<close>] obtain i where i: "i \<in> A" "x + e \<ge> f i" by blast
   41.43      note i(2)
   41.44      also note e(1)
   41.45      finally have "y > f i" .
   41.46 @@ -78,7 +78,7 @@
   41.47    moreover
   41.48    from INF have "\<And>y. (\<And>i. i \<in> A \<Longrightarrow> y \<le> f i) \<Longrightarrow> y \<le> x" by (auto intro: INF_greatest)
   41.49    ultimately
   41.50 -  have "(INF i : A. f i) = x + e" using `e > 0`
   41.51 +  have "(INF i : A. f i) = x + e" using \<open>e > 0\<close>
   41.52      by (intro INF_eqI)
   41.53        (force, metis add.comm_neutral add_left_mono ereal_less(1)
   41.54          linorder_not_le not_less_iff_gr_or_eq)
   41.55 @@ -96,7 +96,7 @@
   41.56    moreover
   41.57    from SUP have "\<And>y. (\<And>i. i \<in> A \<Longrightarrow> f i \<le> y) \<Longrightarrow> y \<ge> x" by (auto intro: SUP_least)
   41.58    ultimately
   41.59 -  have "(SUP i : A. f i) = x - e" using `e > 0` `\<bar>x\<bar> \<noteq> \<infinity>`
   41.60 +  have "(SUP i : A. f i) = x - e" using \<open>e > 0\<close> \<open>\<bar>x\<bar> \<noteq> \<infinity>\<close>
   41.61      by (intro SUP_eqI)
   41.62         (metis PInfty_neq_ereal(2) abs_ereal.simps(1) ereal_minus_le linorder_linear,
   41.63          metis ereal_between(1) ereal_less(2) less_eq_ereal_def order_trans)
   41.64 @@ -136,7 +136,7 @@
   41.65          (auto intro!: borel_closed bexI simp: closed_cball incseq_def Us sb)
   41.66      also have "?U = space M"
   41.67      proof safe
   41.68 -      fix x from X(2)[OF open_ball[of x r]] `r > 0` obtain d where d: "d\<in>X" "d \<in> ball x r" by auto
   41.69 +      fix x from X(2)[OF open_ball[of x r]] \<open>r > 0\<close> obtain d where d: "d\<in>X" "d \<in> ball x r" by auto
   41.70        show "x \<in> ?U"
   41.71          using X(1) d by (auto intro!: exI[where x="to_nat_on X d"] simp: dist_commute Bex_def)
   41.72      qed (simp add: sU)
   41.73 @@ -145,10 +145,10 @@
   41.74    {
   41.75      fix e ::real and n :: nat assume "e > 0" "n > 0"
   41.76      hence "1/n > 0" "e * 2 powr - n > 0" by (auto)
   41.77 -    from M_space[OF `1/n>0`]
   41.78 +    from M_space[OF \<open>1/n>0\<close>]
   41.79      have "(\<lambda>k. measure M (\<Union>i\<in>{0..k}. cball (from_nat_into X i) (1/real n))) ----> measure M (space M)"
   41.80        unfolding emeasure_eq_measure by simp
   41.81 -    from metric_LIMSEQ_D[OF this `0 < e * 2 powr -n`]
   41.82 +    from metric_LIMSEQ_D[OF this \<open>0 < e * 2 powr -n\<close>]
   41.83      obtain k where "dist (measure M (\<Union>i\<in>{0..k}. cball (from_nat_into X i) (1/real n))) (measure M (space M)) <
   41.84        e * 2 powr -n"
   41.85        by auto
   41.86 @@ -176,13 +176,13 @@
   41.87      def B \<equiv> "\<lambda>n. \<Union>i\<in>{0..k e (Suc n)}. cball (from_nat_into X i) (1 / Suc n)"
   41.88      have "\<And>n. closed (B n)" by (auto simp: B_def closed_cball)
   41.89      hence [simp]: "\<And>n. B n \<in> sets M" by (simp add: sb)
   41.90 -    from k[OF `e > 0` zero_less_Suc]
   41.91 +    from k[OF \<open>e > 0\<close> zero_less_Suc]
   41.92      have "\<And>n. measure M (space M) - measure M (B n) \<le> e * 2 powr - real (Suc n)"
   41.93        by (simp add: algebra_simps B_def finite_measure_compl)
   41.94      hence B_compl_le: "\<And>n::nat. measure M (space M - B n) \<le> e * 2 powr - real (Suc n)"
   41.95        by (simp add: finite_measure_compl)
   41.96      def K \<equiv> "\<Inter>n. B n"
   41.97 -    from `closed (B _)` have "closed K" by (auto simp: K_def)
   41.98 +    from \<open>closed (B _)\<close> have "closed K" by (auto simp: K_def)
   41.99      hence [simp]: "K \<in> sets M" by (simp add: sb)
  41.100      have "measure M (space M) - measure M K = measure M (space M - K)"
  41.101        by (simp add: finite_measure_compl)
  41.102 @@ -197,14 +197,14 @@
  41.103        unfolding times_ereal.simps[symmetric] ereal_power[symmetric] one_ereal_def numeral_eq_ereal
  41.104        by simp
  41.105      also have "\<dots> = ereal e * (\<Sum>n. ((1 / 2) ^ Suc n))"
  41.106 -      by (rule suminf_cmult_ereal) (auto simp: `0 < e` less_imp_le)
  41.107 +      by (rule suminf_cmult_ereal) (auto simp: \<open>0 < e\<close> less_imp_le)
  41.108      also have "\<dots> = e" unfolding suminf_half_series_ereal by simp
  41.109      finally have "measure M (space M) \<le> measure M K + e" by simp
  41.110      hence "emeasure M (space M) \<le> emeasure M K + e" by (simp add: emeasure_eq_measure)
  41.111      moreover have "compact K"
  41.112        unfolding compact_eq_totally_bounded
  41.113      proof safe
  41.114 -      show "complete K" using `closed K` by (simp add: complete_eq_closed)
  41.115 +      show "complete K" using \<open>closed K\<close> by (simp add: complete_eq_closed)
  41.116        fix e'::real assume "0 < e'"
  41.117        from nat_approx_posE[OF this] guess n . note n = this
  41.118        let ?k = "from_nat_into X ` {0..k e (Suc n)}"
  41.119 @@ -236,7 +236,7 @@
  41.120        also have "\<dots> \<le> e" using K by (simp add: emeasure_eq_measure)
  41.121        finally have "emeasure M A \<le> emeasure M (A \<inter> K) + ereal e"
  41.122          by (simp add: emeasure_eq_measure algebra_simps)
  41.123 -      moreover have "A \<inter> K \<subseteq> A" "compact (A \<inter> K)" using `closed A` `compact K` by auto
  41.124 +      moreover have "A \<inter> K \<subseteq> A" "compact (A \<inter> K)" using \<open>closed A\<close> \<open>compact K\<close> by auto
  41.125        ultimately show "\<exists>K \<subseteq> A. compact K \<and> emeasure M A \<le> emeasure M K + ereal e"
  41.126          by blast
  41.127      qed simp
  41.128 @@ -251,7 +251,7 @@
  41.129            by (intro continuous_open_vimage) (auto intro!: continuous_infdist continuous_at_id)
  41.130          finally have "open (?G d)" .
  41.131        } note open_G = this
  41.132 -      from in_closed_iff_infdist_zero[OF `closed A` `A \<noteq> {}`]
  41.133 +      from in_closed_iff_infdist_zero[OF \<open>closed A\<close> \<open>A \<noteq> {}\<close>]
  41.134        have "A = {x. infdist x A = 0}" by auto
  41.135        also have "\<dots> = (\<Inter>i. ?G (1/real (Suc i)))"
  41.136        proof (auto simp del: of_nat_Suc, rule ccontr)
  41.137 @@ -291,9 +291,9 @@
  41.138          by (rule INF_greatest) (auto intro!: emeasure_mono simp: sb)
  41.139        ultimately show ?thesis by simp
  41.140      qed (auto intro!: INF_eqI)
  41.141 -    note `?inner A` `?outer A` }
  41.142 +    note \<open>?inner A\<close> \<open>?outer A\<close> }
  41.143    note closed_in_D = this
  41.144 -  from `B \<in> sets borel`
  41.145 +  from \<open>B \<in> sets borel\<close>
  41.146    have "Int_stable (Collect closed)" "Collect closed \<subseteq> Pow UNIV" "B \<in> sigma_sets UNIV (Collect closed)"
  41.147      by (auto simp: Int_stable_def borel_eq_closed)
  41.148    then show "?inner B" "?outer B"
  41.149 @@ -340,10 +340,10 @@
  41.150      also have "\<dots> = (SUP K:{K. K \<subseteq> space M - B \<and> compact K}. emeasure M K)"
  41.151      proof (safe intro!: antisym SUP_least)
  41.152        fix K assume "closed K" "K \<subseteq> space M - B"
  41.153 -      from closed_in_D[OF `closed K`]
  41.154 +      from closed_in_D[OF \<open>closed K\<close>]
  41.155        have K_inner: "emeasure M K = (SUP K:{Ka. Ka \<subseteq> K \<and> compact Ka}. emeasure M K)" by simp
  41.156        show "emeasure M K \<le> (SUP K:{K. K \<subseteq> space M - B \<and> compact K}. emeasure M K)"
  41.157 -        unfolding K_inner using `K \<subseteq> space M - B`
  41.158 +        unfolding K_inner using \<open>K \<subseteq> space M - B\<close>
  41.159          by (auto intro!: SUP_upper SUP_least)
  41.160      qed (fastforce intro!: SUP_least SUP_upper simp: compact_imp_closed)
  41.161      finally show ?case by (auto intro!: antisym simp: sets_eq_imp_space_eq[OF sb])
  41.162 @@ -355,7 +355,7 @@
  41.163        by (intro summable_LIMSEQ summable_ereal_pos emeasure_nonneg)
  41.164      finally have measure_LIMSEQ: "(\<lambda>n. \<Sum>i<n. measure M (D i)) ----> measure M (\<Union>i. D i)"
  41.165        by (simp add: emeasure_eq_measure)
  41.166 -    have "(\<Union>i. D i) \<in> sets M" using `range D \<subseteq> sets M` by auto
  41.167 +    have "(\<Union>i. D i) \<in> sets M" using \<open>range D \<subseteq> sets M\<close> by auto
  41.168  
  41.169      case 1
  41.170      show ?case
  41.171 @@ -377,10 +377,10 @@
  41.172        have "\<forall>i. \<exists>K. K \<subseteq> D i \<and> compact K \<and> emeasure M (D i) \<le> emeasure M K + e/(2*Suc n0)"
  41.173        proof
  41.174          fix i
  41.175 -        from `0 < e` have "0 < e/(2*Suc n0)" by simp
  41.176 +        from \<open>0 < e\<close> have "0 < e/(2*Suc n0)" by simp
  41.177          have "emeasure M (D i) = (SUP K:{K. K \<subseteq> (D i) \<and> compact K}. emeasure M K)"
  41.178            using union by blast
  41.179 -        from SUP_approx_ereal[OF `0 < e/(2*Suc n0)` this]
  41.180 +        from SUP_approx_ereal[OF \<open>0 < e/(2*Suc n0)\<close> this]
  41.181          show "\<exists>K. K \<subseteq> D i \<and> compact K \<and> emeasure M (D i) \<le> emeasure M K + e/(2*Suc n0)"
  41.182            by (auto simp: emeasure_eq_measure)
  41.183        qed
  41.184 @@ -388,7 +388,7 @@
  41.185          "\<And>i. emeasure M (D i) \<le> emeasure M (K i) + e/(2*Suc n0)"
  41.186          unfolding choice_iff by blast
  41.187        let ?K = "\<Union>i\<in>{..<n0}. K i"
  41.188 -      have "disjoint_family_on K {..<n0}" using K `disjoint_family D`
  41.189 +      have "disjoint_family_on K {..<n0}" using K \<open>disjoint_family D\<close>
  41.190          unfolding disjoint_family_on_def by blast
  41.191        hence mK: "measure M ?K = (\<Sum>i<n0. measure M (K i))" using K
  41.192          by (intro finite_measure_finite_Union) (auto simp: sb compact_imp_closed)
  41.193 @@ -397,7 +397,7 @@
  41.194          using K by (auto intro: setsum_mono simp: emeasure_eq_measure)
  41.195        also have "\<dots> = (\<Sum>i<n0. measure M (K i)) + (\<Sum>i<n0. e/(2*Suc n0))"
  41.196          by (simp add: setsum.distrib)
  41.197 -      also have "\<dots> \<le> (\<Sum>i<n0. measure M (K i)) +  e / 2" using `0 < e`
  41.198 +      also have "\<dots> \<le> (\<Sum>i<n0. measure M (K i)) +  e / 2" using \<open>0 < e\<close>
  41.199          by (auto simp: field_simps intro!: mult_left_mono)
  41.200        finally
  41.201        have "measure M (\<Union>i. D i) < (\<Sum>i<n0. measure M (K i)) + e / 2 + e / 2"
  41.202 @@ -413,15 +413,15 @@
  41.203      qed fact
  41.204      case 2
  41.205      show ?case
  41.206 -    proof (rule approx_outer[OF `(\<Union>i. D i) \<in> sets M`])
  41.207 +    proof (rule approx_outer[OF \<open>(\<Union>i. D i) \<in> sets M\<close>])
  41.208        fix e::real assume "e > 0"
  41.209        have "\<forall>i::nat. \<exists>U. D i \<subseteq> U \<and> open U \<and> e/(2 powr Suc i) > emeasure M U - emeasure M (D i)"
  41.210        proof
  41.211          fix i::nat
  41.212 -        from `0 < e` have "0 < e/(2 powr Suc i)" by simp
  41.213 +        from \<open>0 < e\<close> have "0 < e/(2 powr Suc i)" by simp
  41.214          have "emeasure M (D i) = (INF U:{U. (D i) \<subseteq> U \<and> open U}. emeasure M U)"
  41.215            using union by blast
  41.216 -        from INF_approx_ereal[OF `0 < e/(2 powr Suc i)` this]
  41.217 +        from INF_approx_ereal[OF \<open>0 < e/(2 powr Suc i)\<close> this]
  41.218          show "\<exists>U. D i \<subseteq> U \<and> open U \<and> e/(2 powr Suc i) > emeasure M U - emeasure M (D i)"
  41.219            by (auto simp: emeasure_eq_measure)
  41.220        qed
  41.221 @@ -429,13 +429,13 @@
  41.222          "\<And>i. e/(2 powr Suc i) > emeasure M (U i) - emeasure M (D i)"
  41.223          unfolding choice_iff by blast
  41.224        let ?U = "\<Union>i. U i"
  41.225 -      have "M ?U - M (\<Union>i. D i) = M (?U - (\<Union>i. D i))" using U  `(\<Union>i. D i) \<in> sets M`
  41.226 +      have "M ?U - M (\<Union>i. D i) = M (?U - (\<Union>i. D i))" using U  \<open>(\<Union>i. D i) \<in> sets M\<close>
  41.227          by (subst emeasure_Diff) (auto simp: sb)
  41.228 -      also have "\<dots> \<le> M (\<Union>i. U i - D i)" using U  `range D \<subseteq> sets M`
  41.229 +      also have "\<dots> \<le> M (\<Union>i. U i - D i)" using U  \<open>range D \<subseteq> sets M\<close>
  41.230          by (intro emeasure_mono) (auto simp: sb intro!: sets.countable_nat_UN sets.Diff)
  41.231 -      also have "\<dots> \<le> (\<Sum>i. M (U i - D i))" using U  `range D \<subseteq> sets M`
  41.232 +      also have "\<dots> \<le> (\<Sum>i. M (U i - D i))" using U  \<open>range D \<subseteq> sets M\<close>
  41.233          by (intro emeasure_subadditive_countably) (auto intro!: sets.Diff simp: sb)
  41.234 -      also have "\<dots> \<le> (\<Sum>i. ereal e/(2 powr Suc i))" using U `range D \<subseteq> sets M`
  41.235 +      also have "\<dots> \<le> (\<Sum>i. ereal e/(2 powr Suc i))" using U \<open>range D \<subseteq> sets M\<close>
  41.236          by (intro suminf_le_pos, subst emeasure_Diff)
  41.237             (auto simp: emeasure_Diff emeasure_eq_measure sb measure_nonneg intro: less_imp_le)
  41.238        also have "\<dots> \<le> (\<Sum>n. ereal (e * (1 / 2) ^ Suc n))"
  41.239 @@ -444,7 +444,7 @@
  41.240          unfolding times_ereal.simps[symmetric] ereal_power[symmetric] one_ereal_def numeral_eq_ereal
  41.241          by simp
  41.242        also have "\<dots> = ereal e * (\<Sum>n. ((1 / 2) ^ Suc n))"
  41.243 -        by (rule suminf_cmult_ereal) (auto simp: `0 < e` less_imp_le)
  41.244 +        by (rule suminf_cmult_ereal) (auto simp: \<open>0 < e\<close> less_imp_le)
  41.245        also have "\<dots> = e" unfolding suminf_half_series_ereal by simp
  41.246        finally
  41.247        have "emeasure M ?U \<le> emeasure M (\<Union>i. D i) + ereal e" by (simp add: emeasure_eq_measure)
    42.1 --- a/src/HOL/Probability/Set_Integral.thy	Mon Dec 07 16:48:10 2015 +0000
    42.2 +++ b/src/HOL/Probability/Set_Integral.thy	Mon Dec 07 20:19:59 2015 +0100
    42.3 @@ -104,7 +104,7 @@
    42.4  proof -
    42.5    have "set_integrable M B (\<lambda>x. indicator A x *\<^sub>R f x)"
    42.6      by (rule integrable_mult_indicator) fact+
    42.7 -  with `B \<subseteq> A` show ?thesis
    42.8 +  with \<open>B \<subseteq> A\<close> show ?thesis
    42.9      by (simp add: indicator_inter_arith[symmetric] Int_absorb2)
   42.10  qed
   42.11  
   42.12 @@ -287,7 +287,7 @@
   42.13    have "set_borel_measurable M B (\<lambda>x. indicator A x *\<^sub>R f x)"
   42.14      by measurable
   42.15    also have "(\<lambda>x. indicator B x *\<^sub>R indicator A x *\<^sub>R f x) = (\<lambda>x. indicator B x *\<^sub>R f x)"
   42.16 -    using `B \<subseteq> A` by (auto simp: fun_eq_iff split: split_indicator)
   42.17 +    using \<open>B \<subseteq> A\<close> by (auto simp: fun_eq_iff split: split_indicator)
   42.18    finally show ?thesis .
   42.19  qed
   42.20  
   42.21 @@ -340,7 +340,7 @@
   42.22    apply (rule intgbl)
   42.23    prefer 3 apply (rule lim)
   42.24    apply (rule AE_I2)
   42.25 -  using `mono A` apply (auto simp: mono_def nneg split: split_indicator) []
   42.26 +  using \<open>mono A\<close> apply (auto simp: mono_def nneg split: split_indicator) []
   42.27  proof (rule AE_I2)
   42.28    { fix x assume "x \<in> space M"
   42.29      show "(\<lambda>i. indicator (A i) x *\<^sub>R f x) ----> indicator (\<Union>i. A i) x *\<^sub>R f x"
   42.30 @@ -348,7 +348,7 @@
   42.31        assume "\<exists>i. x \<in> A i"
   42.32        then guess i ..
   42.33        then have *: "eventually (\<lambda>i. x \<in> A i) sequentially"
   42.34 -        using `x \<in> A i` `mono A` by (auto simp: eventually_sequentially mono_def)
   42.35 +        using \<open>x \<in> A i\<close> \<open>mono A\<close> by (auto simp: eventually_sequentially mono_def)
   42.36        show ?thesis
   42.37          apply (intro Lim_eventually)
   42.38          using *
    43.1 --- a/src/HOL/Probability/Sigma_Algebra.thy	Mon Dec 07 16:48:10 2015 +0000
    43.2 +++ b/src/HOL/Probability/Sigma_Algebra.thy	Mon Dec 07 20:19:59 2015 +0100
    43.3 @@ -5,7 +5,7 @@
    43.4      translated by Lawrence Paulson.
    43.5  *)
    43.6  
    43.7 -section {* Describing measurable sets *}
    43.8 +section \<open>Describing measurable sets\<close>
    43.9  
   43.10  theory Sigma_Algebra
   43.11  imports
   43.12 @@ -17,15 +17,15 @@
   43.13    "~~/src/HOL/Library/Disjoint_Sets"
   43.14  begin
   43.15  
   43.16 -text {* Sigma algebras are an elementary concept in measure
   43.17 +text \<open>Sigma algebras are an elementary concept in measure
   43.18    theory. To measure --- that is to integrate --- functions, we first have
   43.19    to measure sets. Unfortunately, when dealing with a large universe,
   43.20    it is often not possible to consistently assign a measure to every
   43.21    subset. Therefore it is necessary to define the set of measurable
   43.22    subsets of the universe. A sigma algebra is such a set that has
   43.23 -  three very natural and desirable properties. *}
   43.24 +  three very natural and desirable properties.\<close>
   43.25  
   43.26 -subsection {* Families of sets *}
   43.27 +subsection \<open>Families of sets\<close>
   43.28  
   43.29  locale subset_class =
   43.30    fixes \<Omega> :: "'a set" and M :: "'a set set"
   43.31 @@ -34,7 +34,7 @@
   43.32  lemma (in subset_class) sets_into_space: "x \<in> M \<Longrightarrow> x \<subseteq> \<Omega>"
   43.33    by (metis PowD contra_subsetD space_closed)
   43.34  
   43.35 -subsubsection {* Semiring of sets *}
   43.36 +subsubsection \<open>Semiring of sets\<close>
   43.37  
   43.38  locale semiring_of_sets = subset_class +
   43.39    assumes empty_sets[iff]: "{} \<in> M"
   43.40 @@ -67,7 +67,7 @@
   43.41    shows "{x\<in>\<Omega>. \<forall>i\<in>S. P i x} \<in> M"
   43.42  proof -
   43.43    have "{x\<in>\<Omega>. \<forall>i\<in>S. P i x} = (\<Inter>i\<in>S. {x\<in>\<Omega>. P i x})"
   43.44 -    using `S \<noteq> {}` by auto
   43.45 +    using \<open>S \<noteq> {}\<close> by auto
   43.46    with assms show ?thesis by auto
   43.47  qed
   43.48  
   43.49 @@ -158,13 +158,13 @@
   43.50    interpret ring_of_sets \<Omega> M
   43.51    proof (rule ring_of_setsI)
   43.52      show \<Omega>: "M \<subseteq> Pow \<Omega>" "{} \<in> M"
   43.53 -      using `?Un` by auto
   43.54 +      using \<open>?Un\<close> by auto
   43.55      fix a b assume a: "a \<in> M" and b: "b \<in> M"
   43.56 -    then show "a \<union> b \<in> M" using `?Un` by auto
   43.57 +    then show "a \<union> b \<in> M" using \<open>?Un\<close> by auto
   43.58      have "a - b = \<Omega> - ((\<Omega> - a) \<union> b)"
   43.59        using \<Omega> a b by auto
   43.60      then show "a - b \<in> M"
   43.61 -      using a b  `?Un` by auto
   43.62 +      using a b  \<open>?Un\<close> by auto
   43.63    qed
   43.64    show "algebra \<Omega> M" proof qed fact
   43.65  qed
   43.66 @@ -183,13 +183,13 @@
   43.67    show "algebra \<Omega> M"
   43.68    proof (unfold algebra_iff_Un, intro conjI ballI)
   43.69      show \<Omega>: "M \<subseteq> Pow \<Omega>" "{} \<in> M"
   43.70 -      using `?Int` by auto
   43.71 -    from `?Int` show "\<And>a. a \<in> M \<Longrightarrow> \<Omega> - a \<in> M" by auto
   43.72 +      using \<open>?Int\<close> by auto
   43.73 +    from \<open>?Int\<close> show "\<And>a. a \<in> M \<Longrightarrow> \<Omega> - a \<in> M" by auto
   43.74      fix a b assume M: "a \<in> M" "b \<in> M"
   43.75      hence "a \<union> b = \<Omega> - ((\<Omega> - a) \<inter> (\<Omega> - b))"
   43.76        using \<Omega> by blast
   43.77      also have "... \<in> M"
   43.78 -      using M `?Int` by auto
   43.79 +      using M \<open>?Int\<close> by auto
   43.80      finally show "a \<union> b \<in> M" .
   43.81    qed
   43.82  qed
   43.83 @@ -214,7 +214,7 @@
   43.84    "X \<subseteq> S \<Longrightarrow> algebra S { {}, X, S - X, S }"
   43.85    by (auto simp: algebra_iff_Int)
   43.86  
   43.87 -subsubsection {* Restricted algebras *}
   43.88 +subsubsection \<open>Restricted algebras\<close>
   43.89  
   43.90  abbreviation (in algebra)
   43.91    "restricted_space A \<equiv> (op \<inter> A) ` M"
   43.92 @@ -223,7 +223,7 @@
   43.93    assumes "A \<in> M" shows "algebra A (restricted_space A)"
   43.94    using assms by (auto simp: algebra_iff_Int)
   43.95  
   43.96 -subsubsection {* Sigma Algebras *}
   43.97 +subsubsection \<open>Sigma Algebras\<close>
   43.98  
   43.99  locale sigma_algebra = algebra +
  43.100    assumes countable_nat_UN [intro]: "\<And>A. range A \<subseteq> M \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
  43.101 @@ -236,7 +236,7 @@
  43.102    then have "(\<Union>i. A i) = (\<Union>s\<in>M \<inter> range A. s)"
  43.103      by auto
  43.104    also have "(\<Union>s\<in>M \<inter> range A. s) \<in> M"
  43.105 -    using `finite M` by auto
  43.106 +    using \<open>finite M\<close> by auto
  43.107    finally show "(\<Union>i. A i) \<in> M" .
  43.108  qed
  43.109  
  43.110 @@ -267,7 +267,7 @@
  43.111    hence "\<Union>X = (\<Union>n. from_nat_into X n)"
  43.112      using assms by (auto intro: from_nat_into) (metis from_nat_into_surj)
  43.113    also have "\<dots> \<in> M" using assms
  43.114 -    by (auto intro!: countable_nat_UN) (metis `X \<noteq> {}` from_nat_into set_mp)
  43.115 +    by (auto intro!: countable_nat_UN) (metis \<open>X \<noteq> {}\<close> from_nat_into set_mp)
  43.116    finally show ?thesis .
  43.117  qed simp
  43.118  
  43.119 @@ -421,9 +421,9 @@
  43.120  lemma sigma_algebra_single_set:
  43.121    assumes "X \<subseteq> S"
  43.122    shows "sigma_algebra S { {}, X, S - X, S }"
  43.123 -  using algebra.is_sigma_algebra[OF algebra_single_set[OF `X \<subseteq> S`]] by simp
  43.124 +  using algebra.is_sigma_algebra[OF algebra_single_set[OF \<open>X \<subseteq> S\<close>]] by simp
  43.125  
  43.126 -subsubsection {* Binary Unions *}
  43.127 +subsubsection \<open>Binary Unions\<close>
  43.128  
  43.129  definition binary :: "'a \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"
  43.130    where "binary a b =  (\<lambda>x. b)(0 := a)"
  43.131 @@ -445,10 +445,10 @@
  43.132    by (auto simp add: range_binary_eq sigma_algebra_def sigma_algebra_axioms_def
  43.133           algebra_iff_Un Un_range_binary)
  43.134  
  43.135 -subsubsection {* Initial Sigma Algebra *}
  43.136 +subsubsection \<open>Initial Sigma Algebra\<close>
  43.137  
  43.138 -text {*Sigma algebras can naturally be created as the closure of any set of
  43.139 -  M with regard to the properties just postulated.  *}
  43.140 +text \<open>Sigma algebras can naturally be created as the closure of any set of
  43.141 +  M with regard to the properties just postulated.\<close>
  43.142  
  43.143  inductive_set sigma_sets :: "'a set \<Rightarrow> 'a set set \<Rightarrow> 'a set set"
  43.144    for sp :: "'a set" and A :: "'a set set"
  43.145 @@ -482,7 +482,7 @@
  43.146  proof safe
  43.147    fix B X assume "A \<subseteq> B" and sa: "sigma_algebra S B"
  43.148      and X: "X \<in> sigma_sets S A"
  43.149 -  from sigma_algebra.sigma_sets_subset[OF sa, simplified, OF `A \<subseteq> B`] X
  43.150 +  from sigma_algebra.sigma_sets_subset[OF sa, simplified, OF \<open>A \<subseteq> B\<close>] X
  43.151    show "X \<in> B" by auto
  43.152  next
  43.153    fix X assume "X \<in> \<Inter>{B. A \<subseteq> B \<and> sigma_algebra S B}"
  43.154 @@ -569,19 +569,19 @@
  43.155  lemma sigma_sets_subseteq: assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
  43.156  proof
  43.157    fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
  43.158 -    by induct (insert `A \<subseteq> B`, auto intro: sigma_sets.intros(2-))
  43.159 +    by induct (insert \<open>A \<subseteq> B\<close>, auto intro: sigma_sets.intros(2-))
  43.160  qed
  43.161  
  43.162  lemma sigma_sets_mono: assumes "A \<subseteq> sigma_sets X B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
  43.163  proof
  43.164    fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
  43.165 -    by induct (insert `A \<subseteq> sigma_sets X B`, auto intro: sigma_sets.intros(2-))
  43.166 +    by induct (insert \<open>A \<subseteq> sigma_sets X B\<close>, auto intro: sigma_sets.intros(2-))
  43.167  qed
  43.168  
  43.169  lemma sigma_sets_mono': assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
  43.170  proof
  43.171    fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
  43.172 -    by induct (insert `A \<subseteq> B`, auto intro: sigma_sets.intros(2-))
  43.173 +    by induct (insert \<open>A \<subseteq> B\<close>, auto intro: sigma_sets.intros(2-))
  43.174  qed
  43.175  
  43.176  lemma sigma_sets_superset_generator: "A \<subseteq> sigma_sets X A"
  43.177 @@ -595,7 +595,7 @@
  43.178  proof -
  43.179    { fix i have "A i \<in> ?r" using * by auto
  43.180      hence "\<exists>B. A i = B \<inter> S \<and> B \<in> M" by auto
  43.181 -    hence "A i \<subseteq> S" "A i \<in> M" using `S \<in> M` by auto }
  43.182 +    hence "A i \<subseteq> S" "A i \<in> M" using \<open>S \<in> M\<close> by auto }
  43.183    thus "range A \<subseteq> M" "(\<Union>i. A i) \<in> (\<lambda>A. S \<inter> A) ` M"
  43.184      by (auto intro!: image_eqI[of _ _ "(\<Union>i. A i)"])
  43.185  qed
  43.186 @@ -630,14 +630,14 @@
  43.187                 simp add: UN_extend_simps simp del: UN_simps)
  43.188    qed (auto intro!: sigma_sets.intros(2-))
  43.189    then show "x \<in> sigma_sets A (op \<inter> A ` st)"
  43.190 -    using `A \<subseteq> sp` by (simp add: Int_absorb2)
  43.191 +    using \<open>A \<subseteq> sp\<close> by (simp add: Int_absorb2)
  43.192  next
  43.193    fix x assume "x \<in> sigma_sets A (op \<inter> A ` st)"
  43.194    then show "x \<in> op \<inter> A ` sigma_sets sp st"
  43.195    proof induct
  43.196      case (Compl a)
  43.197      then obtain x where "a = A \<inter> x" "x \<in> sigma_sets sp st" by auto
  43.198 -    then show ?case using `A \<subseteq> sp`
  43.199 +    then show ?case using \<open>A \<subseteq> sp\<close>
  43.200        by (force simp add: image_iff intro!: bexI[of _ "sp - x"] sigma_sets.Compl)
  43.201    next
  43.202      case (Union a)
  43.203 @@ -793,7 +793,7 @@
  43.204    thus "(\<Union>i::nat. A i) \<in> M" by (simp add: UN_disjointed_eq)
  43.205  qed
  43.206  
  43.207 -subsubsection {* Ring generated by a semiring *}
  43.208 +subsubsection \<open>Ring generated by a semiring\<close>
  43.209  
  43.210  definition (in semiring_of_sets)
  43.211    "generated_ring = { \<Union>C | C. C \<subseteq> M \<and> finite C \<and> disjoint C }"
  43.212 @@ -822,7 +822,7 @@
  43.213    show ?thesis
  43.214    proof
  43.215      show "disjoint (Ca \<union> Cb)"
  43.216 -      using `a \<inter> b = {}` Ca Cb by (auto intro!: disjoint_union)
  43.217 +      using \<open>a \<inter> b = {}\<close> Ca Cb by (auto intro!: disjoint_union)
  43.218    qed (insert Ca Cb, auto)
  43.219  qed
  43.220  
  43.221 @@ -888,7 +888,7 @@
  43.222    
  43.223      show "a - b \<in> ?R"
  43.224      proof cases
  43.225 -      assume "Cb = {}" with Cb `a \<in> ?R` show ?thesis
  43.226 +      assume "Cb = {}" with Cb \<open>a \<in> ?R\<close> show ?thesis
  43.227          by simp
  43.228      next
  43.229        assume "Cb \<noteq> {}"
  43.230 @@ -900,7 +900,7 @@
  43.231            by (auto simp add: generated_ring_def)
  43.232        next
  43.233          show "disjoint ((\<lambda>a'. \<Inter>b'\<in>Cb. a' - b')`Ca)"
  43.234 -          using Ca by (auto simp add: disjoint_def `Cb \<noteq> {}`)
  43.235 +          using Ca by (auto simp add: disjoint_def \<open>Cb \<noteq> {}\<close>)
  43.236        next
  43.237          show "finite Ca" "finite Cb" "Cb \<noteq> {}" by fact+
  43.238        qed
  43.239 @@ -923,7 +923,7 @@
  43.240      by (blast intro!: sigma_sets_mono elim: generated_ringE)
  43.241  qed (auto intro!: generated_ringI_Basic sigma_sets_mono)
  43.242  
  43.243 -subsubsection {* A Two-Element Series *}
  43.244 +subsubsection \<open>A Two-Element Series\<close>
  43.245  
  43.246  definition binaryset :: "'a set \<Rightarrow> 'a set \<Rightarrow> nat \<Rightarrow> 'a set "
  43.247    where "binaryset A B = (\<lambda>x. {})(0 := A, Suc 0 := B)"
  43.248 @@ -937,7 +937,7 @@
  43.249  lemma UN_binaryset_eq: "(\<Union>i. binaryset A B i) = A \<union> B"
  43.250    by (simp add: SUP_def range_binaryset_eq)
  43.251  
  43.252 -subsubsection {* Closed CDI *}
  43.253 +subsubsection \<open>Closed CDI\<close>
  43.254  
  43.255  definition closed_cdi where
  43.256    "closed_cdi \<Omega> M \<longleftrightarrow>
  43.257 @@ -1171,7 +1171,7 @@
  43.258      by blast
  43.259  qed
  43.260  
  43.261 -subsubsection {* Dynkin systems *}
  43.262 +subsubsection \<open>Dynkin systems\<close>
  43.263  
  43.264  locale dynkin_system = subset_class +
  43.265    assumes space: "\<Omega> \<in> M"
  43.266 @@ -1193,7 +1193,7 @@
  43.267      by (auto simp: image_iff split: split_if_asm)
  43.268    moreover
  43.269    have "disjoint_family ?f" unfolding disjoint_family_on_def
  43.270 -    using `D \<in> M`[THEN sets_into_space] `D \<subseteq> E` by auto
  43.271 +    using \<open>D \<in> M\<close>[THEN sets_into_space] \<open>D \<subseteq> E\<close> by auto
  43.272    ultimately have "\<Omega> - (D \<union> (\<Omega> - E)) \<in> M"
  43.273      using sets by auto
  43.274    also have "\<Omega> - (D \<union> (\<Omega> - E)) = E - D"
  43.275 @@ -1265,7 +1265,7 @@
  43.276                "\<Omega> - A \<in> M" "\<Omega> - B \<in> M"
  43.277        using sets_into_space by auto
  43.278      then show "A \<union> B \<in> M"
  43.279 -      using `Int_stable M` unfolding Int_stable_def by auto
  43.280 +      using \<open>Int_stable M\<close> unfolding Int_stable_def by auto
  43.281    qed auto
  43.282  qed
  43.283  
  43.284 @@ -1314,15 +1314,15 @@
  43.285    shows "dynkin_system \<Omega> {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> D \<in> M}"
  43.286  proof (rule dynkin_systemI, simp_all)
  43.287    have "\<Omega> \<inter> D = D"
  43.288 -    using `D \<in> M` sets_into_space by auto
  43.289 +    using \<open>D \<in> M\<close> sets_into_space by auto
  43.290    then show "\<Omega> \<inter> D \<in> M"
  43.291 -    using `D \<in> M` by auto
  43.292 +    using \<open>D \<in> M\<close> by auto
  43.293  next
  43.294    fix A assume "A \<subseteq> \<Omega> \<and> A \<inter> D \<in> M"
  43.295    moreover have "(\<Omega> - A) \<inter> D = (\<Omega> - (A \<inter> D)) - (\<Omega> - D)"
  43.296      by auto
  43.297    ultimately show "\<Omega> - A \<subseteq> \<Omega> \<and> (\<Omega> - A) \<inter> D \<in> M"
  43.298 -    using  `D \<in> M` by (auto intro: diff)
  43.299 +    using  \<open>D \<in> M\<close> by (auto intro: diff)
  43.300  next
  43.301    fix A :: "nat \<Rightarrow> 'a set"
  43.302    assume "disjoint_family A" "range A \<subseteq> {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> D \<in> M}"
  43.303 @@ -1340,7 +1340,7 @@
  43.304    have "dynkin_system \<Omega> M" ..
  43.305    then have "dynkin_system \<Omega> M"
  43.306      using assms unfolding dynkin_system_def dynkin_system_axioms_def subset_class_def by simp
  43.307 -  with `N \<subseteq> M` show ?thesis by (auto simp add: dynkin_def)
  43.308 +  with \<open>N \<subseteq> M\<close> show ?thesis by (auto simp add: dynkin_def)
  43.309  qed
  43.310  
  43.311  lemma sigma_eq_dynkin:
  43.312 @@ -1363,22 +1363,22 @@
  43.313      proof
  43.314        fix E assume "E \<in> M"
  43.315        then have "M \<subseteq> ?D E" "E \<in> dynkin \<Omega> M"
  43.316 -        using sets_into_space `Int_stable M` by (auto simp: Int_stable_def)
  43.317 +        using sets_into_space \<open>Int_stable M\<close> by (auto simp: Int_stable_def)
  43.318        then have "dynkin \<Omega> M \<subseteq> ?D E"
  43.319 -        using restricted_dynkin_system `E \<in> dynkin \<Omega> M`
  43.320 +        using restricted_dynkin_system \<open>E \<in> dynkin \<Omega> M\<close>
  43.321          by (intro dynkin_system.dynkin_subset) simp_all
  43.322        then have "B \<in> ?D E"
  43.323 -        using `B \<in> dynkin \<Omega> M` by auto
  43.324 +        using \<open>B \<in> dynkin \<Omega> M\<close> by auto
  43.325        then have "E \<inter> B \<in> dynkin \<Omega> M"
  43.326          by (subst Int_commute) simp
  43.327        then show "E \<in> ?D B"
  43.328 -        using sets `E \<in> M` by auto
  43.329 +        using sets \<open>E \<in> M\<close> by auto
  43.330      qed
  43.331      then have "dynkin \<Omega> M \<subseteq> ?D B"
  43.332 -      using restricted_dynkin_system `B \<in> dynkin \<Omega> M`
  43.333 +      using restricted_dynkin_system \<open>B \<in> dynkin \<Omega> M\<close>
  43.334        by (intro dynkin_system.dynkin_subset) simp_all
  43.335      then show "A \<inter> B \<in> dynkin \<Omega> M"
  43.336 -      using `A \<in> dynkin \<Omega> M` sets_into_space by auto
  43.337 +      using \<open>A \<in> dynkin \<Omega> M\<close> sets_into_space by auto
  43.338    qed
  43.339    from sigma_algebra.sigma_sets_subset[OF this, of "M"]
  43.340    have "sigma_sets (\<Omega>) (M) \<subseteq> dynkin \<Omega> M" by auto
  43.341 @@ -1409,17 +1409,17 @@
  43.342    have "E \<subseteq> Pow \<Omega>"
  43.343      using E sets_into_space by force
  43.344    then have *: "sigma_sets \<Omega> E = dynkin \<Omega> E"
  43.345 -    using `Int_stable E` by (rule sigma_eq_dynkin)
  43.346 +    using \<open>Int_stable E\<close> by (rule sigma_eq_dynkin)
  43.347    then have "dynkin \<Omega> E = M"
  43.348      using assms dynkin_subset[OF E(1)] by simp
  43.349    with * show ?thesis
  43.350      using assms by (auto simp: dynkin_def)
  43.351  qed
  43.352  
  43.353 -subsubsection {* Induction rule for intersection-stable generators *}
  43.354 +subsubsection \<open>Induction rule for intersection-stable generators\<close>
  43.355  
  43.356 -text {* The reason to introduce Dynkin-systems is the following induction rules for $\sigma$-algebras
  43.357 -generated by a generator closed under intersection. *}
  43.358 +text \<open>The reason to introduce Dynkin-systems is the following induction rules for $\sigma$-algebras
  43.359 +generated by a generator closed under intersection.\<close>
  43.360  
  43.361  lemma sigma_sets_induct_disjoint[consumes 3, case_names basic empty compl union]:
  43.362    assumes "Int_stable G"
  43.363 @@ -1438,11 +1438,11 @@
  43.364    interpret dynkin_system \<Omega> ?D
  43.365      by standard (auto dest: sets_into_space intro!: space compl union)
  43.366    have "sigma_sets \<Omega> G = ?D"
  43.367 -    by (rule dynkin_lemma) (auto simp: basic `Int_stable G`)
  43.368 +    by (rule dynkin_lemma) (auto simp: basic \<open>Int_stable G\<close>)
  43.369    with A show ?thesis by auto
  43.370  qed
  43.371  
  43.372 -subsection {* Measure type *}
  43.373 +subsection \<open>Measure type\<close>
  43.374  
  43.375  definition positive :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where
  43.376    "positive M \<mu> \<longleftrightarrow> \<mu> {} = 0 \<and> (\<forall>A\<in>M. 0 \<le> \<mu> A)"
  43.377 @@ -1554,7 +1554,7 @@
  43.378      hence "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A)
  43.379        (\<lambda>a. if a \<in> sigma_sets \<Omega> A then \<mu> a else 0)"
  43.380        by(rule measure_space_eq) auto
  43.381 -    with True `A \<subseteq> Pow \<Omega>` show ?thesis
  43.382 +    with True \<open>A \<subseteq> Pow \<Omega>\<close> show ?thesis
  43.383        by(simp add: measure_of_def space_def sets_def emeasure_def Abs_measure_inverse)
  43.384    next
  43.385      case False thus ?thesis
  43.386 @@ -1599,10 +1599,10 @@
  43.387    next
  43.388      case Empty show ?case by (rule sigma_sets.Empty)
  43.389    next
  43.390 -    from assms have "A \<in> sets (sigma C D)" by (subst sets_measure_of[OF `D \<subseteq> Pow C`])
  43.391 -    moreover case (Compl a) hence "a \<in> sets (sigma C D)" by (subst sets_measure_of[OF `D \<subseteq> Pow C`])
  43.392 +    from assms have "A \<in> sets (sigma C D)" by (subst sets_measure_of[OF \<open>D \<subseteq> Pow C\<close>])
  43.393 +    moreover case (Compl a) hence "a \<in> sets (sigma C D)" by (subst sets_measure_of[OF \<open>D \<subseteq> Pow C\<close>])
  43.394      ultimately have "A - a \<in> sets (sigma C D)" ..
  43.395 -    thus ?case by (subst (asm) sets_measure_of[OF `D \<subseteq> Pow C`])
  43.396 +    thus ?case by (subst (asm) sets_measure_of[OF \<open>D \<subseteq> Pow C\<close>])
  43.397    next
  43.398      case (Union a)
  43.399      thus ?case by (intro sigma_sets.Union)
  43.400 @@ -1616,7 +1616,7 @@
  43.401    by (metis Pow_empty Sup_bot_conv(1) cSup_singleton empty_iff
  43.402              sets.sigma_sets_eq sets.space_closed sigma_sets_top subset_singletonD)
  43.403  
  43.404 -subsubsection {* Constructing simple @{typ "'a measure"} *}
  43.405 +subsubsection \<open>Constructing simple @{typ "'a measure"}\<close>
  43.406  
  43.407  lemma emeasure_measure_of:
  43.408    assumes M: "M = measure_of \<Omega> A \<mu>"
  43.409 @@ -1671,17 +1671,17 @@
  43.410    interpret N: sigma_algebra \<Omega>' A' using measure_measure by (auto simp: measure_space_def)
  43.411    have "A = sets M" "A' = sets N"
  43.412      using measure_measure by (simp_all add: sets_def Abs_measure_inverse)
  43.413 -  with `sets M = sets N` have AA': "A = A'" by simp
  43.414 +  with \<open>sets M = sets N\<close> have AA': "A = A'" by simp
  43.415    moreover from M.top N.top M.space_closed N.space_closed AA' have "\<Omega> = \<Omega>'" by auto
  43.416    moreover { fix B have "\<mu> B = \<mu>' B"
  43.417      proof cases
  43.418        assume "B \<in> A"
  43.419 -      with eq `A = sets M` have "emeasure M B = emeasure N B" by simp
  43.420 +      with eq \<open>A = sets M\<close> have "emeasure M B = emeasure N B" by simp
  43.421        with measure_measure show "\<mu> B = \<mu>' B"
  43.422          by (simp add: emeasure_def Abs_measure_inverse)
  43.423      next
  43.424        assume "B \<notin> A"
  43.425 -      with `A = sets M` `A' = sets N` `A = A'` have "B \<notin> sets M" "B \<notin> sets N"
  43.426 +      with \<open>A = sets M\<close> \<open>A' = sets N\<close> \<open>A = A'\<close> have "B \<notin> sets M" "B \<notin> sets N"
  43.427          by auto
  43.428        then have "emeasure M B = 0" "emeasure N B = 0"
  43.429          by (simp_all add: emeasure_notin_sets)
  43.430 @@ -1698,7 +1698,7 @@
  43.431    shows "sigma \<Omega> M = sigma \<Omega> N"
  43.432    by (rule measure_eqI) (simp_all add: emeasure_sigma)
  43.433  
  43.434 -subsubsection {* Measurable functions *}
  43.435 +subsubsection \<open>Measurable functions\<close>
  43.436  
  43.437  definition measurable :: "'a measure \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) set" where
  43.438    "measurable A B = {f \<in> space A \<rightarrow> space B. \<forall>y \<in> sets B. f -` y \<inter> space A \<in> sets A}"
  43.439 @@ -1860,7 +1860,7 @@
  43.440      measurable (measure_of \<Omega> M \<mu>) N \<subseteq> measurable (measure_of \<Omega> M' \<mu>') N"
  43.441    using measure_of_subset[of M' \<Omega> M] by (auto simp add: measurable_def)
  43.442  
  43.443 -subsubsection {* Counting space *}
  43.444 +subsubsection \<open>Counting space\<close>
  43.445  
  43.446  definition count_space :: "'a set \<Rightarrow> 'a measure" where
  43.447    "count_space \<Omega> = measure_of \<Omega> (Pow \<Omega>) (\<lambda>A. if finite A then ereal (card A) else \<infinity>)"
  43.448 @@ -1898,11 +1898,11 @@
  43.449    shows "f \<in> measurable M (count_space A) \<longleftrightarrow> (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M))"
  43.450  proof -
  43.451    { fix X assume "X \<subseteq> A" "f \<in> space M \<rightarrow> A"
  43.452 -    with `countable A` have "f -` X \<inter> space M = (\<Union>a\<in>X. f -` {a} \<inter> space M)" "countable X"
  43.453 +    with \<open>countable A\<close> have "f -` X \<inter> space M = (\<Union>a\<in>X. f -` {a} \<inter> space M)" "countable X"
  43.454        by (auto dest: countable_subset)
  43.455      moreover assume "\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M"
  43.456      ultimately have "f -` X \<inter> space M \<in> sets M"
  43.457 -      using `X \<subseteq> A` by (auto intro!: sets.countable_UN' simp del: UN_simps) }
  43.458 +      using \<open>X \<subseteq> A\<close> by (auto intro!: sets.countable_UN' simp del: UN_simps) }
  43.459    then show ?thesis
  43.460      unfolding measurable_def by auto
  43.461  qed
  43.462 @@ -1938,7 +1938,7 @@
  43.463    "space N = {} \<Longrightarrow> f \<in> measurable M N \<longleftrightarrow> space M = {}"
  43.464    by (auto simp add: measurable_def Pi_iff)
  43.465  
  43.466 -subsubsection {* Extend measure *}
  43.467 +subsubsection \<open>Extend measure\<close>
  43.468  
  43.469  definition "extend_measure \<Omega> I G \<mu> =
  43.470    (if (\<exists>\<mu>'. (\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G`I)) \<mu>') \<and> \<not> (\<forall>i\<in>I. \<mu> i = 0)
  43.471 @@ -1961,10 +1961,10 @@
  43.472    assume *: "(\<forall>i\<in>I. \<mu> i = 0)"
  43.473    with M have M_eq: "M = measure_of \<Omega> (G`I) (\<lambda>_. 0)"
  43.474     by (simp add: extend_measure_def)
  43.475 -  from measure_space_0[OF ms(1)] ms `i\<in>I`
  43.476 +  from measure_space_0[OF ms(1)] ms \<open>i\<in>I\<close>
  43.477    have "emeasure M (G i) = 0"
  43.478      by (intro emeasure_measure_of[OF M_eq]) (auto simp add: M measure_space_def sets_extend_measure)
  43.479 -  with `i\<in>I` * show ?thesis
  43.480 +  with \<open>i\<in>I\<close> * show ?thesis
  43.481      by simp
  43.482  next
  43.483    def P \<equiv> "\<lambda>\<mu>'. (\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G`I)) \<mu>'"
  43.484 @@ -1978,14 +1978,14 @@
  43.485    ultimately have M_eq: "M = measure_of \<Omega> (G`I) (Eps P)"
  43.486      by (simp add: M extend_measure_def P_def[symmetric])
  43.487  
  43.488 -  from `\<exists>\<mu>'. P \<mu>'` have P: "P (Eps P)" by (rule someI_ex)
  43.489 +  from \<open>\<exists>\<mu>'. P \<mu>'\<close> have P: "P (Eps P)" by (rule someI_ex)
  43.490    show "emeasure M (G i) = \<mu> i"
  43.491    proof (subst emeasure_measure_of[OF M_eq])
  43.492      have sets_M: "sets M = sigma_sets \<Omega> (G`I)"
  43.493        using M_eq ms by (auto simp: sets_extend_measure)
  43.494 -    then show "G i \<in> sets M" using `i \<in> I` by auto
  43.495 +    then show "G i \<in> sets M" using \<open>i \<in> I\<close> by auto
  43.496      show "positive (sets M) (Eps P)" "countably_additive (sets M) (Eps P)" "Eps P (G i) = \<mu> i"
  43.497 -      using P `i\<in>I` by (auto simp add: sets_M measure_space_def P_def)
  43.498 +      using P \<open>i\<in>I\<close> by (auto simp add: sets_M measure_space_def P_def)
  43.499    qed fact
  43.500  qed
  43.501  
  43.502 @@ -1995,10 +1995,10 @@
  43.503      and ms: "\<And>i j. I i j \<Longrightarrow> G i j \<in> Pow \<Omega>" "positive (sets M) \<mu>'" "countably_additive (sets M) \<mu>'"
  43.504      and "I i j"
  43.505    shows "emeasure M (G i j) = \<mu> i j"
  43.506 -  using emeasure_extend_measure[OF M _ _ ms(2,3), of "(i,j)"] eq ms(1) `I i j`
  43.507 +  using emeasure_extend_measure[OF M _ _ ms(2,3), of "(i,j)"] eq ms(1) \<open>I i j\<close>
  43.508    by (auto simp: subset_eq)
  43.509  
  43.510 -subsubsection {* Supremum of a set of $\sigma$-algebras *}
  43.511 +subsubsection \<open>Supremum of a set of $\sigma$-algebras\<close>
  43.512  
  43.513  definition "Sup_sigma M = sigma (\<Union>x\<in>M. space x) (\<Union>x\<in>M. sets x)"
  43.514  
  43.515 @@ -2078,7 +2078,7 @@
  43.516      by (simp add: image_image)
  43.517  qed
  43.518  
  43.519 -subsection {* The smallest $\sigma$-algebra regarding a function *}
  43.520 +subsection \<open>The smallest $\sigma$-algebra regarding a function\<close>
  43.521  
  43.522  definition
  43.523    "vimage_algebra X f M = sigma X {f -` A \<inter> X | A. A \<in> sets M}"
  43.524 @@ -2178,7 +2178,7 @@
  43.525      using assms by (rule sets_vimage_Sup_eq)
  43.526  qed (simp add: vimage_algebra_def Sup_sigma_def emeasure_sigma)
  43.527  
  43.528 -subsubsection {* Restricted Space Sigma Algebra *}
  43.529 +subsubsection \<open>Restricted Space Sigma Algebra\<close>
  43.530  
  43.531  definition restrict_space where
  43.532    "restrict_space M \<Omega> = measure_of (\<Omega> \<inter> space M) ((op \<inter> \<Omega>) ` sets M) (emeasure M)"
  43.533 @@ -2263,7 +2263,7 @@
  43.534      by (auto simp: space_restrict_space)
  43.535    also have "\<dots> \<in> sets (restrict_space M \<Omega>)"
  43.536      unfolding sets_restrict_space
  43.537 -    using measurable_sets[OF f `A \<in> sets N`] by blast
  43.538 +    using measurable_sets[OF f \<open>A \<in> sets N\<close>] by blast
  43.539    finally show "f -` A \<inter> space (restrict_space M \<Omega>) \<in> sets (restrict_space M \<Omega>)" .
  43.540  qed
  43.541  
  43.542 @@ -2324,7 +2324,7 @@
  43.543    shows "(\<lambda>x. if x \<in> A then f x else g x) \<in> measurable M M'"
  43.544  proof (rule measurable_If[OF measure])
  43.545    have "{x \<in> space M. x \<in> A} = A \<inter> space M" by auto
  43.546 -  thus "{x \<in> space M. x \<in> A} \<in> sets M" using `A \<inter> space M \<in> sets M` by auto
  43.547 +  thus "{x \<in> space M. x \<in> A} \<in> sets M" using \<open>A \<inter> space M \<in> sets M\<close> by auto
  43.548  qed
  43.549  
  43.550  lemma measurable_restrict_space_iff:
    44.1 --- a/src/HOL/Probability/Stream_Space.thy	Mon Dec 07 16:48:10 2015 +0000
    44.2 +++ b/src/HOL/Probability/Stream_Space.thy	Mon Dec 07 20:19:59 2015 +0100
    44.3 @@ -76,7 +76,7 @@
    44.4    shows "f \<in> measurable N (stream_space M)"
    44.5  proof (rule measurable_stream_space2)
    44.6    fix n show "(\<lambda>x. f x !! n) \<in> measurable N M"
    44.7 -    using `F f` by (induction n arbitrary: f) (auto intro: h t)
    44.8 +    using \<open>F f\<close> by (induction n arbitrary: f) (auto intro: h t)
    44.9  qed
   44.10  
   44.11  lemma measurable_sdrop[measurable]: "sdrop n \<in> measurable (stream_space M) (stream_space M)"
   44.12 @@ -355,11 +355,11 @@
   44.13          case (Suc i) from this[of "stl x"] show ?case
   44.14            by (simp add: length_Suc_conv Bex_def ex_simps[symmetric] del: ex_simps)
   44.15               (metis stream.collapse streams_Stream)
   44.16 -      qed (insert `a \<in> S`, auto intro: streams_stl in_streams) }
   44.17 +      qed (insert \<open>a \<in> S\<close>, auto intro: streams_stl in_streams) }
   44.18      then have "(\<lambda>x. x !! i) -` {a} \<inter> streams S = (\<Union>xs\<in>{xs\<in>lists S. length xs = i}. sstart S (xs @ [a]))"
   44.19        by (auto simp add: set_eq_iff)
   44.20      also have "\<dots> \<in> sets ?S"
   44.21 -      using `a\<in>S` by (intro sets.countable_UN') (auto intro!: sigma_sets.Basic image_eqI)
   44.22 +      using \<open>a\<in>S\<close> by (intro sets.countable_UN') (auto intro!: sigma_sets.Basic image_eqI)
   44.23      finally have " (\<lambda>x. x !! i) -` {a} \<inter> streams S \<in> sets ?S" . }
   44.24    then show "sets (stream_space (count_space S)) \<subseteq> sets (sigma (streams S) (sstart S`lists S \<union> {{}}))"
   44.25      by (intro sets_stream_space_in_sets) (auto simp: measurable_count_space_eq_countable snth_in)
    45.1 --- a/src/HOL/Probability/ex/Dining_Cryptographers.thy	Mon Dec 07 16:48:10 2015 +0000
    45.2 +++ b/src/HOL/Probability/ex/Dining_Cryptographers.thy	Mon Dec 07 20:19:59 2015 +0100
    45.3 @@ -8,9 +8,9 @@
    45.4  lemma Ex1_eq: "\<exists>!x. P x \<Longrightarrow> P x \<Longrightarrow> P y \<Longrightarrow> x = y"
    45.5    by auto
    45.6  
    45.7 -subsection {* Define the state space *}
    45.8 +subsection \<open>Define the state space\<close>
    45.9  
   45.10 -text {*
   45.11 +text \<open>
   45.12  
   45.13  We introduce the state space on which the algorithm operates.
   45.14  
   45.15 @@ -35,7 +35,7 @@
   45.16  
   45.17  The observables are the \emph{inversions}
   45.18  
   45.19 -*}
   45.20 +\<close>
   45.21  
   45.22  locale dining_cryptographers_space =
   45.23    fixes n :: nat
   45.24 @@ -64,11 +64,11 @@
   45.25    have foldl_coin:
   45.26      "\<not> ?XOR (\<lambda>c. coin dc c \<noteq> coin dc (c + 1)) n"
   45.27    proof -
   45.28 -    def n' \<equiv> n -- "Need to hide n, as it is hidden in coin"
   45.29 +    def n' \<equiv> n \<comment> "Need to hide n, as it is hidden in coin"
   45.30      have "?XOR (\<lambda>c. coin dc c \<noteq> coin dc (c + 1)) n'
   45.31          = (coin dc 0 \<noteq> coin dc n')"
   45.32        by (induct n') auto
   45.33 -    thus ?thesis using `n' \<equiv> n` by simp
   45.34 +    thus ?thesis using \<open>n' \<equiv> n\<close> by simp
   45.35    qed
   45.36  
   45.37    from assms have "payer dc = None \<or> (\<exists>k<n. payer dc = Some k)"
   45.38 @@ -81,22 +81,22 @@
   45.39    next
   45.40      assume "\<exists>k<n. payer dc = Some k"
   45.41      then obtain k where "k < n" and "payer dc = Some k" by auto
   45.42 -    def l \<equiv> n -- "Need to hide n, as it is hidden in coin, payer etc."
   45.43 +    def l \<equiv> n \<comment> "Need to hide n, as it is hidden in coin, payer etc."
   45.44      have "?XOR (\<lambda>c. (payer dc = Some c) \<noteq> (coin dc c \<noteq> coin dc (c + 1))) l =
   45.45          ((k < l) \<noteq> ?XOR (\<lambda>c. (coin dc c \<noteq> coin dc (c + 1))) l)"
   45.46 -      using `payer dc = Some k` by (induct l) auto
   45.47 +      using \<open>payer dc = Some k\<close> by (induct