diff --git a/src/HOL/Probability/Levy.thy b/src/HOL/Probability/Levy.thy
--- a/src/HOL/Probability/Levy.thy
+++ b/src/HOL/Probability/Levy.thy
@@ -1,529 +1,529 @@
 (*  Title:     HOL/Probability/Levy.thy
     Authors:   Jeremy Avigad (CMU)
 *)
 
 section \<open>The Levy inversion theorem, and the Levy continuity theorem.\<close>
 
 theory Levy
   imports Characteristic_Functions Helly_Selection Sinc_Integral
 begin
 
 subsection \<open>The Levy inversion theorem\<close>
 
 (* Actually, this is not needed for us -- but it is useful for other purposes. (See Billingsley.) *)
 lemma Levy_Inversion_aux1:
   fixes a b :: real
   assumes "a \<le> b"
   shows "((\<lambda>t. (iexp (-(t * a)) - iexp (-(t * b))) / (\<i> * t)) \<longlongrightarrow> b - a) (at 0)"
     (is "(?F \<longlongrightarrow> _) (at _)")
 proof -
   have 1: "cmod (?F t - (b - a)) \<le> a^2 / 2 * abs t + b^2 / 2 * abs t" if "t \<noteq> 0" for t
   proof -
     have "cmod (?F t - (b - a)) = cmod (
         (iexp (-(t * a)) - (1 + \<i> * -(t * a))) / (\<i> * t) -
         (iexp (-(t * b)) - (1 + \<i> * -(t * b))) / (\<i> * t))"
            (is "_ = cmod (?one / (\<i> * t) - ?two / (\<i> * t))")
       using \<open>t \<noteq> 0\<close> by (intro arg_cong[where f=norm]) (simp add: field_simps)
     also have "\<dots> \<le> cmod (?one / (\<i> * t)) + cmod (?two / (\<i> * t))"
       by (rule norm_triangle_ineq4)
     also have "cmod (?one / (\<i> * t)) = cmod ?one / abs t"
       by (simp add: norm_divide norm_mult)
     also have "cmod (?two / (\<i> * t)) = cmod ?two / abs t"
       by (simp add: norm_divide norm_mult)
     also have "cmod ?one / abs t + cmod ?two / abs t \<le>
         ((- (a * t))^2 / 2) / abs t + ((- (b * t))^2 / 2) / abs t"
       apply (rule add_mono)
       apply (rule divide_right_mono)
       using iexp_approx1 [of "-(t * a)" 1] apply (simp add: field_simps eval_nat_numeral)
       apply force
       apply (rule divide_right_mono)
       using iexp_approx1 [of "-(t * b)" 1] apply (simp add: field_simps eval_nat_numeral)
       by force
     also have "\<dots> = a^2 / 2 * abs t + b^2 / 2 * abs t"
       using \<open>t \<noteq> 0\<close> apply (case_tac "t \<ge> 0", simp add: field_simps power2_eq_square)
       using \<open>t \<noteq> 0\<close> by (subst (1 2) abs_of_neg, auto simp add: field_simps power2_eq_square)
     finally show "cmod (?F t - (b - a)) \<le> a^2 / 2 * abs t + b^2 / 2 * abs t" .
   qed
   show ?thesis
     apply (rule LIM_zero_cancel)
     apply (rule tendsto_norm_zero_cancel)
     apply (rule real_LIM_sandwich_zero [OF _ _ 1])
     apply (auto intro!: tendsto_eq_intros)
     done
 qed
 
 lemma Levy_Inversion_aux2:
   fixes a b t :: real
   assumes "a \<le> b" and "t \<noteq> 0"
   shows "cmod ((iexp (t * b) - iexp (t * a)) / (\<i> * t)) \<le> b - a" (is "?F \<le> _")
 proof -
   have "?F = cmod (iexp (t * a) * (iexp (t * (b - a)) - 1) / (\<i> * t))"
     using \<open>t \<noteq> 0\<close> by (intro arg_cong[where f=norm]) (simp add: field_simps exp_diff exp_minus)
   also have "\<dots> = cmod (iexp (t * (b - a)) - 1) / abs t"
     unfolding norm_divide norm_mult norm_exp_i_times using \<open>t \<noteq> 0\<close>
     by (simp add: complex_eq_iff norm_mult)
   also have "\<dots> \<le> abs (t * (b - a)) / abs t"
     using iexp_approx1 [of "t * (b - a)" 0]
     by (intro divide_right_mono) (auto simp add: field_simps eval_nat_numeral)
   also have "\<dots> = b - a"
     using assms by (auto simp add: abs_mult)
   finally show ?thesis .
 qed
 
 (* TODO: refactor! *)
 theorem (in real_distribution) Levy_Inversion:
   fixes a b :: real
   assumes "a \<le> b"
   defines "\<mu> \<equiv> measure M" and "\<phi> \<equiv> char M"
   assumes "\<mu> {a} = 0" and "\<mu> {b} = 0"
   shows "(\<lambda>T. 1 / (2 * pi) * (CLBINT t=-T..T. (iexp (-(t * a)) - iexp (-(t * b))) / (\<i> * t) * \<phi> t))
     \<longlonglongrightarrow> \<mu> {a<..b}"
     (is "(\<lambda>T. 1 / (2 * pi) * (CLBINT t=-T..T. ?F t * \<phi> t)) \<longlonglongrightarrow> of_real (\<mu> {a<..b})")
 proof -
   interpret P: pair_sigma_finite lborel M ..
   from bounded_Si obtain B where Bprop: "\<And>T. abs (Si T) \<le> B" by auto
   from Bprop [of 0] have [simp]: "B \<ge> 0" by auto
   let ?f = "\<lambda>t x :: real. (iexp (t * (x - a)) - iexp(t * (x - b))) / (\<i> * t)"
   { fix T :: real
     assume "T \<ge> 0"
     let ?f' = "\<lambda>(t, x). indicator {-T<..<T} t *\<^sub>R ?f t x"
     { fix x
       have 1: "complex_interval_lebesgue_integrable lborel u v (\<lambda>t. ?f t x)" for u v :: real
         using Levy_Inversion_aux2[of "x - b" "x - a"]
         apply (simp add: interval_lebesgue_integrable_def set_integrable_def del: times_divide_eq_left)
         apply (intro integrableI_bounded_set_indicator[where B="b - a"] conjI impI)
         apply (auto intro!: AE_I [of _ _ "{0}"] simp: assms)
         done
       have "(CLBINT t. ?f' (t, x)) = (CLBINT t=-T..T. ?f t x)"
         using \<open>T \<ge> 0\<close> by (simp add: interval_lebesgue_integral_def set_lebesgue_integral_def)
       also have "\<dots> = (CLBINT t=-T..(0 :: real). ?f t x) + (CLBINT t=(0 :: real)..T. ?f t x)"
           (is "_ = _ + ?t")
         using 1 by (intro interval_integral_sum[symmetric]) (simp add: min_absorb1 max_absorb2 \<open>T \<ge> 0\<close>)
       also have "(CLBINT t=-T..(0 :: real). ?f t x) = (CLBINT t=(0::real)..T. ?f (-t) x)"
         by (subst interval_integral_reflect) auto
       also have "\<dots> + ?t = (CLBINT t=(0::real)..T. ?f (-t) x + ?f t x)"
         using 1
         by (intro interval_lebesgue_integral_add(2) [symmetric] interval_integrable_mirror[THEN iffD2]) simp_all
       also have "\<dots> = (CLBINT t=(0::real)..T. ((iexp(t * (x - a)) - iexp (-(t * (x - a)))) -
           (iexp(t * (x - b)) - iexp (-(t * (x - b))))) / (\<i> * t))"
         using \<open>T \<ge> 0\<close> by (intro interval_integral_cong) (auto simp add: field_split_simps)
       also have "\<dots> = (CLBINT t=(0::real)..T. complex_of_real(
           2 * (sin (t * (x - a)) / t) - 2 * (sin (t * (x - b)) / t)))"
         using \<open>T \<ge> 0\<close>
         apply (intro interval_integral_cong)
         apply (simp add: field_simps cis.ctr Im_divide Re_divide Im_exp Re_exp complex_eq_iff)
         unfolding minus_diff_eq[symmetric, of "y * x" "y * a" for y a] sin_minus cos_minus
         apply (simp add: field_simps power2_eq_square)
         done
       also have "\<dots> = complex_of_real (LBINT t=(0::real)..T.
           2 * (sin (t * (x - a)) / t) - 2 * (sin (t * (x - b)) / t))"
         by (rule interval_lebesgue_integral_of_real)
       also have "\<dots> = complex_of_real (2 * (sgn (x - a) * Si (T * abs (x - a)) -
           sgn (x - b) * Si (T * abs (x - b))))"
         apply (subst interval_lebesgue_integral_diff)
         apply (rule interval_lebesgue_integrable_mult_right, rule integrable_sinc')+
         apply (subst interval_lebesgue_integral_mult_right)+
         apply (simp add: zero_ereal_def[symmetric] LBINT_I0c_sin_scale_divide[OF \<open>T \<ge> 0\<close>])
         done
       finally have "(CLBINT t. ?f' (t, x)) =
           2 * (sgn (x - a) * Si (T * abs (x - a)) - sgn (x - b) * Si (T * abs (x - b)))" .
     } note main_eq = this
     have "(CLBINT t=-T..T. ?F t * \<phi> t) =
       (CLBINT t. (CLINT x | M. ?F t * iexp (t * x) * indicator {-T<..<T} t))"
       using \<open>T \<ge> 0\<close> unfolding \<phi>_def char_def interval_lebesgue_integral_def set_lebesgue_integral_def
       by (auto split: split_indicator intro!: Bochner_Integration.integral_cong)
     also have "\<dots> = (CLBINT t. (CLINT x | M. ?f' (t, x)))"
       by (auto intro!: Bochner_Integration.integral_cong simp: field_simps exp_diff exp_minus split: split_indicator)
     also have "\<dots> = (CLINT x | M. (CLBINT t. ?f' (t, x)))"
     proof (intro P.Fubini_integral [symmetric] integrableI_bounded_set [where B="b - a"])
       show "emeasure (lborel \<Otimes>\<^sub>M M) ({- T<..<T} \<times> space M) < \<infinity>"
         using \<open>T \<ge> 0\<close>
         by (subst emeasure_pair_measure_Times)
            (auto simp: ennreal_mult_less_top not_less top_unique)
       show "AE x\<in>{- T<..<T} \<times> space M in lborel \<Otimes>\<^sub>M M. cmod (case x of (t, x) \<Rightarrow> ?f' (t, x)) \<le> b - a"
         using Levy_Inversion_aux2[of "x - b" "x - a" for x] \<open>a \<le> b\<close>
         by (intro AE_I [of _ _ "{0} \<times> UNIV"]) (force simp: emeasure_pair_measure_Times)+
     qed (auto split: split_indicator split_indicator_asm)
     also have "\<dots> = (CLINT x | M. (complex_of_real (2 * (sgn (x - a) *
          Si (T * abs (x - a)) - sgn (x - b) * Si (T * abs (x - b))))))"
        using main_eq by (intro Bochner_Integration.integral_cong, auto)
     also have "\<dots> = complex_of_real (LINT x | M. (2 * (sgn (x - a) *
          Si (T * abs (x - a)) - sgn (x - b) * Si (T * abs (x - b)))))"
        by (rule integral_complex_of_real)
     finally have "(CLBINT t=-T..T. ?F t * \<phi> t) =
         complex_of_real (LINT x | M. (2 * (sgn (x - a) *
          Si (T * abs (x - a)) - sgn (x - b) * Si (T * abs (x - b)))))" .
   } note main_eq2 = this
 
   have "(\<lambda>T :: nat. LINT x | M. (2 * (sgn (x - a) *
          Si (T * abs (x - a)) - sgn (x - b) * Si (T * abs (x - b))))) \<longlonglongrightarrow>
        (LINT x | M. 2 * pi * indicator {a<..b} x)"
   proof (rule integral_dominated_convergence [where w="\<lambda>x. 4 * B"])
     show "integrable M (\<lambda>x. 4 * B)"
       by (rule integrable_const_bound [of _ "4 * B"]) auto
   next
     let ?S = "\<lambda>n::nat. \<lambda>x. sgn (x - a) * Si (n * \<bar>x - a\<bar>) - sgn (x - b) * Si (n * \<bar>x - b\<bar>)"
     { fix n x
       have "norm (?S n x) \<le> norm (sgn (x - a) * Si (n * \<bar>x - a\<bar>)) + norm (sgn (x - b) * Si (n * \<bar>x - b\<bar>))"
         by (rule norm_triangle_ineq4)
       also have "\<dots> \<le> B + B"
         using Bprop by (intro add_mono) (auto simp: abs_mult abs_sgn_eq)
       finally have "norm (2 * ?S n x) \<le> 4 * B"
         by simp }
     then show "\<And>n. AE x in M. norm (2 * ?S n x) \<le> 4 * B"
       by auto
     have "AE x in M. x \<noteq> a" "AE x in M. x \<noteq> b"
       using prob_eq_0[of "{a}"] prob_eq_0[of "{b}"] \<open>\<mu> {a} = 0\<close> \<open>\<mu> {b} = 0\<close> by (auto simp: \<mu>_def)
     then show "AE x in M. (\<lambda>n. 2 * ?S n x) \<longlonglongrightarrow> 2 * pi * indicator {a<..b} x"
     proof eventually_elim
       fix x assume x: "x \<noteq> a" "x \<noteq> b"
       then have "(\<lambda>n. 2 * (sgn (x - a) * Si (\<bar>x - a\<bar> * n) - sgn (x - b) * Si (\<bar>x - b\<bar> * n)))
           \<longlonglongrightarrow> 2 * (sgn (x - a) * (pi / 2) - sgn (x - b) * (pi / 2))"
         by (intro tendsto_intros filterlim_compose[OF Si_at_top]
             filterlim_tendsto_pos_mult_at_top[OF tendsto_const] filterlim_real_sequentially)
            auto
       also have "(\<lambda>n. 2 * (sgn (x - a) * Si (\<bar>x - a\<bar> * n) - sgn (x - b) * Si (\<bar>x - b\<bar> * n))) = (\<lambda>n. 2 * ?S n x)"
         by (auto simp: ac_simps)
       also have "2 * (sgn (x - a) * (pi / 2) - sgn (x - b) * (pi / 2)) = 2 * pi * indicator {a<..b} x"
         using x \<open>a \<le> b\<close> by (auto split: split_indicator)
       finally show "(\<lambda>n. 2 * ?S n x) \<longlonglongrightarrow> 2 * pi * indicator {a<..b} x" .
     qed
   qed simp_all
   also have "(LINT x | M. 2 * pi * indicator {a<..b} x) = 2 * pi * \<mu> {a<..b}"
     by (simp add: \<mu>_def)
   finally have "(\<lambda>T. LINT x | M. (2 * (sgn (x - a) *
          Si (T * abs (x - a)) - sgn (x - b) * Si (T * abs (x - b))))) \<longlonglongrightarrow>
        2 * pi * \<mu> {a<..b}" .
   with main_eq2 show ?thesis
     by (auto intro!: tendsto_eq_intros)
 qed
 
 theorem Levy_uniqueness:
   fixes M1 M2 :: "real measure"
   assumes "real_distribution M1" "real_distribution M2" and
     "char M1 = char M2"
   shows "M1 = M2"
 proof -
   interpret M1: real_distribution M1 by (rule assms)
   interpret M2: real_distribution M2 by (rule assms)
   have "countable ({x. measure M1 {x} \<noteq> 0} \<union> {x. measure M2 {x} \<noteq> 0})"
     by (intro countable_Un M2.countable_support M1.countable_support)
   then have count: "countable {x. measure M1 {x} \<noteq> 0 \<or> measure M2 {x} \<noteq> 0}"
     by (simp add: Un_def)
 
   have "cdf M1 = cdf M2"
   proof (rule ext)
     fix x
     let ?D = "\<lambda>x. \<bar>cdf M1 x - cdf M2 x\<bar>"
 
     { fix \<epsilon> :: real
       assume "\<epsilon> > 0"
       have "(?D \<longlongrightarrow> 0) at_bot"
         using M1.cdf_lim_at_bot M2.cdf_lim_at_bot by (intro tendsto_eq_intros) auto
       then have "eventually (\<lambda>y. ?D y < \<epsilon> / 2 \<and> y \<le> x) at_bot"
         using \<open>\<epsilon> > 0\<close> by (simp only: tendsto_iff dist_real_def diff_0_right eventually_conj eventually_le_at_bot abs_idempotent)
       then obtain M where "\<And>y. y \<le> M \<Longrightarrow> ?D y < \<epsilon> / 2" "M \<le> x"
         unfolding eventually_at_bot_linorder by auto
       with open_minus_countable[OF count, of "{..< M}"] obtain a where
         "measure M1 {a} = 0" "measure M2 {a} = 0" "a < M" "a \<le> x" and 1: "?D a < \<epsilon> / 2"
         by auto
 
       have "(?D \<longlongrightarrow> ?D x) (at_right x)"
         using M1.cdf_is_right_cont [of x] M2.cdf_is_right_cont [of x]
         by (intro tendsto_intros) (auto simp add: continuous_within)
       then have "eventually (\<lambda>y. \<bar>?D y - ?D x\<bar> < \<epsilon> / 2) (at_right x)"
         using \<open>\<epsilon> > 0\<close> by (simp only: tendsto_iff dist_real_def eventually_conj eventually_at_right_less)
       then obtain N where "N > x" "\<And>y. x < y \<Longrightarrow> y < N \<Longrightarrow> \<bar>?D y - ?D x\<bar> < \<epsilon> / 2"
         by (auto simp add: eventually_at_right[OF less_add_one])
       with open_minus_countable[OF count, of "{x <..< N}"] obtain b where "x < b" "b < N"
           "measure M1 {b} = 0" "measure M2 {b} = 0" and 2: "\<bar>?D b - ?D x\<bar> < \<epsilon> / 2"
         by (auto simp: abs_minus_commute)
       from \<open>a \<le> x\<close> \<open>x < b\<close> have "a < b" "a \<le> b" by auto
 
       from \<open>char M1 = char M2\<close>
         M1.Levy_Inversion [OF \<open>a \<le> b\<close> \<open>measure M1 {a} = 0\<close> \<open>measure M1 {b} = 0\<close>]
         M2.Levy_Inversion [OF \<open>a \<le> b\<close> \<open>measure M2 {a} = 0\<close> \<open>measure M2 {b} = 0\<close>]
       have "complex_of_real (measure M1 {a<..b}) = complex_of_real (measure M2 {a<..b})"
         by (intro LIMSEQ_unique) auto
       then have "?D a = ?D b"
         unfolding of_real_eq_iff M1.cdf_diff_eq [OF \<open>a < b\<close>, symmetric] M2.cdf_diff_eq [OF \<open>a < b\<close>, symmetric] by simp
       then have "?D x = \<bar>(?D b - ?D x) - ?D a\<bar>"
         by simp
       also have "\<dots> \<le> \<bar>?D b - ?D x\<bar> + \<bar>?D a\<bar>"
         by (rule abs_triangle_ineq4)
       also have "\<dots> \<le> \<epsilon> / 2 + \<epsilon> / 2"
         using 1 2 by (intro add_mono) auto
       finally have "?D x \<le> \<epsilon>" by simp }
     then show "cdf M1 x = cdf M2 x"
       by (metis abs_le_zero_iff dense_ge eq_iff_diff_eq_0)
   qed
   thus ?thesis
     by (rule cdf_unique [OF \<open>real_distribution M1\<close> \<open>real_distribution M2\<close>])
 qed
 
 
 subsection \<open>The Levy continuity theorem\<close>
 
 theorem levy_continuity1:
   fixes M :: "nat \<Rightarrow> real measure" and M' :: "real measure"
   assumes "\<And>n. real_distribution (M n)" "real_distribution M'" "weak_conv_m M M'"
   shows "(\<lambda>n. char (M n) t) \<longlonglongrightarrow> char M' t"
   unfolding char_def using assms by (rule weak_conv_imp_integral_bdd_continuous_conv) auto
 
 theorem levy_continuity:
   fixes M :: "nat \<Rightarrow> real measure" and M' :: "real measure"
   assumes real_distr_M : "\<And>n. real_distribution (M n)"
     and real_distr_M': "real_distribution M'"
     and char_conv: "\<And>t. (\<lambda>n. char (M n) t) \<longlonglongrightarrow> char M' t"
   shows "weak_conv_m M M'"
 proof -
   interpret Mn: real_distribution "M n" for n by fact
   interpret M': real_distribution M' by fact
 
   have *: "\<And>u x. u > 0 \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> (CLBINT t:{-u..u}. 1 - iexp (t * x)) =
       2 * (u  - sin (u * x) / x)"
   proof -
     fix u :: real and x :: real
     assume "u > 0" and "x \<noteq> 0"
     hence "(CLBINT t:{-u..u}. 1 - iexp (t * x)) = (CLBINT t=-u..u. 1 - iexp (t * x))"
       by (subst interval_integral_Icc, auto)
     also have "\<dots> = (CLBINT t=-u..0. 1 - iexp (t * x)) + (CLBINT t=0..u. 1 - iexp (t * x))"
       using \<open>u > 0\<close>
       apply (subst interval_integral_sum)
       apply (simp add: min_absorb1 min_absorb2 max_absorb1 max_absorb2)
       apply (rule interval_integrable_isCont)
       apply auto
       done
     also have "\<dots> = (CLBINT t=ereal 0..u. 1 - iexp (t * -x)) + (CLBINT t=ereal 0..u. 1 - iexp (t * x))"
       apply (subgoal_tac "0 = ereal 0", erule ssubst)
       by (subst interval_integral_reflect, auto)
     also have "\<dots> = (LBINT t=ereal 0..u. 2 - 2 * cos (t * x))"
       apply (subst interval_lebesgue_integral_add (2) [symmetric])
       apply ((rule interval_integrable_isCont, auto)+) [2]
       unfolding exp_Euler cos_of_real
       apply (simp add: of_real_mult interval_lebesgue_integral_of_real[symmetric])
       done
     also have "\<dots> = 2 * u - 2 * sin (u * x) / x"
       by (subst interval_lebesgue_integral_diff)
          (auto intro!: interval_integrable_isCont
                simp: interval_lebesgue_integral_of_real integral_cos [OF \<open>x \<noteq> 0\<close>] mult.commute[of _ x])
     finally show "(CLBINT t:{-u..u}. 1 - iexp (t * x)) = 2 * (u  - sin (u * x) / x)"
       by (simp add: field_simps)
   qed
   have main_bound: "\<And>u n. u > 0 \<Longrightarrow> Re (CLBINT t:{-u..u}. 1 - char (M n) t) \<ge>
     u * measure (M n) {x. abs x \<ge> 2 / u}"
   proof -
     fix u :: real and n
     assume "u > 0"
     interpret P: pair_sigma_finite "M n" lborel ..
     (* TODO: put this in the real_distribution locale as a simp rule? *)
     have Mn1 [simp]: "measure (M n) UNIV = 1" by (metis Mn.prob_space Mn.space_eq_univ)
     (* TODO: make this automatic somehow? *)
     have Mn2 [simp]: "\<And>x. complex_integrable (M n) (\<lambda>t. exp (\<i> * complex_of_real (x * t)))"
       by (rule Mn.integrable_const_bound [where B = 1], auto)
     have Mn3: "set_integrable (M n \<Otimes>\<^sub>M lborel) (UNIV \<times> {- u..u}) (\<lambda>a. 1 - exp (\<i> * complex_of_real (snd a * fst a)))"
       using \<open>0 < u\<close>
       unfolding set_integrable_def
       by (intro integrableI_bounded_set_indicator [where B="2"])
          (auto simp: lborel.emeasure_pair_measure_Times ennreal_mult_less_top not_less top_unique
                split: split_indicator
                intro!: order_trans [OF norm_triangle_ineq4])
     have "(CLBINT t:{-u..u}. 1 - char (M n) t) =
         (CLBINT t:{-u..u}. (CLINT x | M n. 1 - iexp (t * x)))"
       unfolding char_def by (rule set_lebesgue_integral_cong, auto simp del: of_real_mult)
     also have "\<dots> = (CLBINT t. (CLINT x | M n. indicator {-u..u} t *\<^sub>R (1 - iexp (t * x))))"
       unfolding set_lebesgue_integral_def
       by (rule Bochner_Integration.integral_cong) (auto split: split_indicator)
     also have "\<dots> = (CLINT x | M n. (CLBINT t:{-u..u}. 1 - iexp (t * x)))"
       using Mn3 by (subst P.Fubini_integral) (auto simp: indicator_times split_beta' set_integrable_def set_lebesgue_integral_def)
     also have "\<dots> = (CLINT x | M n. (if x = 0 then 0 else 2 * (u  - sin (u * x) / x)))"
       using \<open>u > 0\<close> by (intro Bochner_Integration.integral_cong, auto simp add: * simp del: of_real_mult)
     also have "\<dots> = (LINT x | M n. (if x = 0 then 0 else 2 * (u  - sin (u * x) / x)))"
       by (rule integral_complex_of_real)
     finally have "Re (CLBINT t:{-u..u}. 1 - char (M n) t) =
        (LINT x | M n. (if x = 0 then 0 else 2 * (u  - sin (u * x) / x)))" by simp
     also have "\<dots> \<ge> (LINT x : {x. abs x \<ge> 2 / u} | M n. u)"
     proof -
       have "complex_integrable (M n) (\<lambda>x. CLBINT t:{-u..u}. 1 - iexp (snd (x, t) * fst (x, t)))"
         using Mn3 unfolding set_integrable_def set_lebesgue_integral_def
         by (intro P.integrable_fst) (simp add: indicator_times split_beta')
       hence "complex_integrable (M n) (\<lambda>x. if x = 0 then 0 else 2 * (u  - sin (u * x) / x))"
         using \<open>u > 0\<close>
         unfolding set_integrable_def
-        by (subst integrable_cong) (auto simp add: * simp del: of_real_mult)
+        by (subst Bochner_Integration.integrable_cong) (auto simp add: * simp del: of_real_mult)
       hence **: "integrable (M n) (\<lambda>x. if x = 0 then 0 else 2 * (u  - sin (u * x) / x))"
         unfolding complex_of_real_integrable_eq .
       have "2 * sin x \<le> x" if "2 \<le> x" for x :: real
         by (rule order_trans[OF _ \<open>2 \<le> x\<close>]) auto
       moreover have "x \<le> 2 * sin x" if "x \<le> - 2" for x :: real
         by (rule order_trans[OF \<open>x \<le> - 2\<close>]) auto
       moreover have "x < 0 \<Longrightarrow> x \<le> sin x" for x :: real
         using sin_x_le_x[of "-x"] by simp
       ultimately show ?thesis
         using \<open>u > 0\<close> unfolding set_lebesgue_integral_def
         by (intro integral_mono [OF _ **])
            (auto simp: divide_simps sin_x_le_x mult.commute[of u] mult_neg_pos top_unique less_top[symmetric]
                  split: split_indicator)
     qed
     also (xtrans) have "(LINT x : {x. abs x \<ge> 2 / u} | M n. u) = u * measure (M n) {x. abs x \<ge> 2 / u}"
       unfolding set_lebesgue_integral_def
       by (simp add: Mn.emeasure_eq_measure)
     finally show "Re (CLBINT t:{-u..u}. 1 - char (M n) t) \<ge> u * measure (M n) {x. abs x \<ge> 2 / u}" .
   qed
 
   have tight_aux: "\<And>\<epsilon>. \<epsilon> > 0 \<Longrightarrow> \<exists>a b. a < b \<and> (\<forall>n. 1 - \<epsilon> < measure (M n) {a<..b})"
   proof -
     fix \<epsilon> :: real
     assume "\<epsilon> > 0"
     note M'.isCont_char [of 0]
     hence "\<exists>d>0. \<forall>t. abs t < d \<longrightarrow> cmod (char M' t - 1) < \<epsilon> / 4"
       apply (subst (asm) continuous_at_eps_delta)
       apply (drule_tac x = "\<epsilon> / 4" in spec)
       using \<open>\<epsilon> > 0\<close> by (auto simp add: dist_real_def dist_complex_def M'.char_zero)
     then obtain d where "d > 0 \<and> (\<forall>t. (abs t < d \<longrightarrow> cmod (char M' t - 1) < \<epsilon> / 4))" ..
     hence d0: "d > 0" and d1: "\<And>t. abs t < d \<Longrightarrow> cmod (char M' t - 1) < \<epsilon> / 4" by auto
     have 1: "\<And>x. cmod (1 - char M' x) \<le> 2"
       by (rule order_trans [OF norm_triangle_ineq4], auto simp add: M'.cmod_char_le_1)
     then have 2: "\<And>u v. complex_set_integrable lborel {u..v} (\<lambda>x. 1 - char M' x)"
       unfolding set_integrable_def
       by (intro integrableI_bounded_set_indicator[where B=2]) (auto simp: emeasure_lborel_Icc_eq)
     have 3: "\<And>u v. integrable lborel (\<lambda>x. indicat_real {u..v} x *\<^sub>R cmod (1 - char M' x))"
       by (intro borel_integrable_compact[OF compact_Icc] continuous_at_imp_continuous_on
                 continuous_intros ballI M'.isCont_char continuous_intros)
     have "cmod (CLBINT t:{-d/2..d/2}. 1 - char M' t) \<le> LBINT t:{-d/2..d/2}. cmod (1 - char M' t)"
       unfolding set_lebesgue_integral_def
       using integral_norm_bound[of _ "\<lambda>x. indicator {u..v} x *\<^sub>R (1 - char M' x)" for u v] by simp
     also have 4: "\<dots> \<le> LBINT t:{-d/2..d/2}. \<epsilon> / 4"
       unfolding set_lebesgue_integral_def
       apply (rule integral_mono [OF 3])
        apply (simp add: emeasure_lborel_Icc_eq)
       apply (case_tac "x \<in> {-d/2..d/2}")
        apply auto
       apply (subst norm_minus_commute)
       apply (rule less_imp_le)
       apply (rule d1 [simplified])
       using d0 apply auto
       done
     also from d0 4 have "\<dots> = d * \<epsilon> / 4"
       unfolding set_lebesgue_integral_def by simp
     finally have bound: "cmod (CLBINT t:{-d/2..d/2}. 1 - char M' t) \<le> d * \<epsilon> / 4" .
     have "cmod (1 - char (M n) x) \<le> 2" for n x
       by (rule order_trans [OF norm_triangle_ineq4], auto simp add: Mn.cmod_char_le_1)
     then have "(\<lambda>n. CLBINT t:{-d/2..d/2}. 1 - char (M n) t) \<longlonglongrightarrow> (CLBINT t:{-d/2..d/2}. 1 - char M' t)"
       unfolding set_lebesgue_integral_def
       apply (intro integral_dominated_convergence[where w="\<lambda>x. indicator {-d/2..d/2} x *\<^sub>R 2"])
       apply (auto intro!: char_conv tendsto_intros
                   simp: emeasure_lborel_Icc_eq
                   split: split_indicator)
       done
     hence "eventually (\<lambda>n. cmod ((CLBINT t:{-d/2..d/2}. 1 - char (M n) t) -
         (CLBINT t:{-d/2..d/2}. 1 - char M' t)) < d * \<epsilon> / 4) sequentially"
       using d0 \<open>\<epsilon> > 0\<close> apply (subst (asm) tendsto_iff)
       by (subst (asm) dist_complex_def, drule spec, erule mp, auto)
     hence "\<exists>N. \<forall>n \<ge> N. cmod ((CLBINT t:{-d/2..d/2}. 1 - char (M n) t) -
         (CLBINT t:{-d/2..d/2}. 1 - char M' t)) < d * \<epsilon> / 4" by (simp add: eventually_sequentially)
     then obtain N
       where "\<forall>n\<ge>N. cmod ((CLBINT t:{- d / 2..d / 2}. 1 - char (M n) t) -
         (CLBINT t:{- d / 2..d / 2}. 1 - char M' t)) < d * \<epsilon> / 4" ..
     hence N: "\<And>n. n \<ge> N \<Longrightarrow> cmod ((CLBINT t:{-d/2..d/2}. 1 - char (M n) t) -
         (CLBINT t:{-d/2..d/2}. 1 - char M' t)) < d * \<epsilon> / 4" by auto
     { fix n
       assume "n \<ge> N"
       have "cmod (CLBINT t:{-d/2..d/2}. 1 - char (M n) t) =
         cmod ((CLBINT t:{-d/2..d/2}. 1 - char (M n) t) - (CLBINT t:{-d/2..d/2}. 1 - char M' t)
           + (CLBINT t:{-d/2..d/2}. 1 - char M' t))" by simp
       also have "\<dots> \<le> cmod ((CLBINT t:{-d/2..d/2}. 1 - char (M n) t) -
           (CLBINT t:{-d/2..d/2}. 1 - char M' t)) + cmod(CLBINT t:{-d/2..d/2}. 1 - char M' t)"
         by (rule norm_triangle_ineq)
       also have "\<dots> < d * \<epsilon> / 4 + d * \<epsilon> / 4"
         by (rule add_less_le_mono [OF N [OF \<open>n \<ge> N\<close>] bound])
       also have "\<dots> = d * \<epsilon> / 2" by auto
       finally have "cmod (CLBINT t:{-d/2..d/2}. 1 - char (M n) t) < d * \<epsilon> / 2" .
       hence "d * \<epsilon> / 2 > Re (CLBINT t:{-d/2..d/2}. 1 - char (M n) t)"
         by (rule order_le_less_trans [OF complex_Re_le_cmod])
       hence "d * \<epsilon> / 2 > Re (CLBINT t:{-(d/2)..d/2}. 1 - char (M n) t)" (is "_ > ?lhs") by simp
       also have "?lhs \<ge> (d / 2) * measure (M n) {x. abs x \<ge> 2 / (d / 2)}"
         using d0 by (intro main_bound, simp)
       finally (xtrans) have "d * \<epsilon> / 2 > (d / 2) * measure (M n) {x. abs x \<ge> 2 / (d / 2)}" .
       with d0 \<open>\<epsilon> > 0\<close> have "\<epsilon> > measure (M n) {x. abs x \<ge> 2 / (d / 2)}" by (simp add: field_simps)
       hence "\<epsilon> > 1 - measure (M n) (UNIV - {x. abs x \<ge> 2 / (d / 2)})"
         apply (subst Mn.borel_UNIV [symmetric])
         by (subst Mn.prob_compl, auto)
       also have "UNIV - {x. abs x \<ge> 2 / (d / 2)} = {x. -(4 / d) < x \<and> x < (4 / d)}"
         using d0 apply (auto simp add: field_simps)
         (* very annoying -- this should be automatic *)
         apply (case_tac "x \<ge> 0", auto simp add: field_simps)
         apply (subgoal_tac "0 \<le> x * d", arith, rule mult_nonneg_nonneg, auto)
         apply (case_tac "x \<ge> 0", auto simp add: field_simps)
         apply (subgoal_tac "x * d \<le> 0", arith)
         apply (rule mult_nonpos_nonneg, auto)
         by (case_tac "x \<ge> 0", auto simp add: field_simps)
       finally have "measure (M n) {x. -(4 / d) < x \<and> x < (4 / d)} > 1 - \<epsilon>"
         by auto
     } note 6 = this
     { fix n :: nat
       have *: "(UN (k :: nat). {- real k<..real k}) = UNIV"
         by (auto, metis leI le_less_trans less_imp_le minus_less_iff reals_Archimedean2)
       have "(\<lambda>k. measure (M n) {- real k<..real k}) \<longlonglongrightarrow>
           measure (M n) (UN (k :: nat). {- real k<..real k})"
         by (rule Mn.finite_Lim_measure_incseq, auto simp add: incseq_def)
       hence "(\<lambda>k. measure (M n) {- real k<..real k}) \<longlonglongrightarrow> 1"
         using Mn.prob_space unfolding * Mn.borel_UNIV by simp
       hence "eventually (\<lambda>k. measure (M n) {- real k<..real k} > 1 - \<epsilon>) sequentially"
         apply (elim order_tendstoD (1))
         using \<open>\<epsilon> > 0\<close> by auto
     } note 7 = this
     { fix n :: nat
       have "eventually (\<lambda>k. \<forall>m < n. measure (M m) {- real k<..real k} > 1 - \<epsilon>) sequentially"
         (is "?P n")
       proof (induct n)
         case (Suc n) with 7[of n] show ?case
           by eventually_elim (auto simp add: less_Suc_eq)
       qed simp
     } note 8 = this
     from 8 [of N] have "\<exists>K :: nat. \<forall>k \<ge> K. \<forall>m<N. 1 - \<epsilon> <
         Sigma_Algebra.measure (M m) {- real k<..real k}"
       by (auto simp add: eventually_sequentially)
     hence "\<exists>K :: nat. \<forall>m<N. 1 - \<epsilon> < Sigma_Algebra.measure (M m) {- real K<..real K}" by auto
     then obtain K :: nat where
       "\<forall>m<N. 1 - \<epsilon> < Sigma_Algebra.measure (M m) {- real K<..real K}" ..
     hence K: "\<And>m. m < N \<Longrightarrow> 1 - \<epsilon> < Sigma_Algebra.measure (M m) {- real K<..real K}"
       by auto
     let ?K' = "max K (4 / d)"
     have "-?K' < ?K' \<and> (\<forall>n. 1 - \<epsilon> < measure (M n) {-?K'<..?K'})"
       using d0 apply auto
       apply (rule max.strict_coboundedI2, auto)
     proof -
       fix n
       show " 1 - \<epsilon> < measure (M n) {- max (real K) (4 / d)<..max (real K) (4 / d)}"
         apply (case_tac "n < N")
         apply (rule order_less_le_trans)
         apply (erule K)
         apply (rule Mn.finite_measure_mono, auto)
         apply (rule order_less_le_trans)
         apply (rule 6, erule leI)
         by (rule Mn.finite_measure_mono, auto)
     qed
     thus "\<exists>a b. a < b \<and> (\<forall>n. 1 - \<epsilon> < measure (M n) {a<..b})" by (intro exI)
   qed
   have tight: "tight M"
     by (auto simp: tight_def intro: assms tight_aux)
   show ?thesis
   proof (rule tight_subseq_weak_converge [OF real_distr_M real_distr_M' tight])
     fix s \<nu>
     assume s: "strict_mono (s :: nat \<Rightarrow> nat)"
     assume nu: "weak_conv_m (M \<circ> s) \<nu>"
     assume *: "real_distribution \<nu>"
     have 2: "\<And>n. real_distribution ((M \<circ> s) n)" unfolding comp_def by (rule assms)
     have 3: "\<And>t. (\<lambda>n. char ((M \<circ> s) n) t) \<longlonglongrightarrow> char \<nu> t" by (intro levy_continuity1 [OF 2 * nu])
     have 4: "\<And>t. (\<lambda>n. char ((M \<circ> s) n) t) = ((\<lambda>n. char (M n) t) \<circ> s)" by (rule ext, simp)
     have 5: "\<And>t. (\<lambda>n. char ((M \<circ> s) n) t) \<longlonglongrightarrow> char M' t"
       by (subst 4, rule LIMSEQ_subseq_LIMSEQ [OF _ s], rule assms)
     hence "char \<nu> = char M'" by (intro ext, intro LIMSEQ_unique [OF 3 5])
     hence "\<nu> = M'" by (rule Levy_uniqueness [OF * \<open>real_distribution M'\<close>])
     thus "weak_conv_m (M \<circ> s) M'"
       by (elim subst) (rule nu)
   qed
 qed
 
 end