--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Probability/Levy.thy Wed Jan 06 12:18:53 2016 +0100
@@ -0,0 +1,539 @@
+(* Theory: Levy.thy
+ Author: Jeremy Avigad
+*)
+
+section \<open>The Levy inversion theorem, and the Levy continuity theorem.\<close>
+
+theory Levy
+ imports Characteristic_Functions Helly_Selection Sinc_Integral
+begin
+
+lemma LIM_zero_cancel:
+ fixes f :: "_ \<Rightarrow> 'b::real_normed_vector"
+ shows "((\<lambda>x. f x - l) \<longlongrightarrow> 0) F \<Longrightarrow> (f \<longlongrightarrow> l) F"
+unfolding tendsto_iff dist_norm by simp
+
+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))) / (ii * 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 + ii * -(t * a))) / (ii * t) -
+ (iexp (-(t * b)) - (1 + ii * -(t * b))) / (ii * t))"
+ (is "_ = cmod (?one / (ii * t) - ?two / (ii * t))")
+ using `t \<noteq> 0` by (intro arg_cong[where f=norm]) (simp add: field_simps)
+ also have "\<dots> \<le> cmod (?one / (ii * t)) + cmod (?two / (ii * t))"
+ by (rule norm_triangle_ineq4)
+ also have "cmod (?one / (ii * t)) = cmod ?one / abs t"
+ by (simp add: norm_divide norm_mult)
+ also have "cmod (?two / (ii * 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 `t \<noteq> 0` apply (case_tac "t \<ge> 0", simp add: field_simps power2_eq_square)
+ using `t \<noteq> 0` 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)) / (ii * t)) \<le> b - a" (is "?F \<le> _")
+proof -
+ have "?F = cmod (iexp (t * a) * (iexp (t * (b - a)) - 1) / (ii * t))"
+ using `t \<noteq> 0` 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_ii_times using `t \<noteq> 0`
+ 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))) / (ii * 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))) / (ii * 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 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 `T \<ge> 0` by (simp add: interval_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))))) / (ii * t))"
+ using `T \<ge> 0` by (intro interval_integral_cong) (auto simp add: divide_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 `T \<ge> 0`
+ 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 `T \<ge> 0`])
+ 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 `T \<ge> 0` unfolding \<phi>_def char_def interval_lebesgue_integral_def
+ by (auto split: split_indicator intro!: integral_cong)
+ also have "\<dots> = (CLBINT t. (CLINT x | M. ?f' (t, x)))"
+ by (auto intro!: 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
+ 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] `a \<le> b`
+ 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 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}"] `\<mu> {a} = 0` `\<mu> {b} = 0` 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 `a \<le> b` 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
+ from M1.cdf_is_right_cont [of x] have "(cdf M1 \<longlongrightarrow> cdf M1 x) (at_right x)"
+ by (simp add: continuous_within)
+ from M2.cdf_is_right_cont [of x] have "(cdf M2 \<longlongrightarrow> cdf M2 x) (at_right x)"
+ by (simp add: continuous_within)
+
+ { fix \<epsilon> :: real
+ assume "\<epsilon> > 0"
+ from `\<epsilon> > 0` `(cdf M1 \<longlongrightarrow> 0) at_bot` `(cdf M2 \<longlongrightarrow> 0) at_bot`
+ have "eventually (\<lambda>y. \<bar>cdf M1 y\<bar> < \<epsilon> / 4 \<and> \<bar>cdf M2 y\<bar> < \<epsilon> / 4 \<and> y \<le> x) at_bot"
+ by (simp only: tendsto_iff dist_real_def diff_0_right eventually_conj eventually_le_at_bot)
+ then obtain M where "\<And>y. y \<le> M \<Longrightarrow> \<bar>cdf M1 y\<bar> < \<epsilon> / 4" "\<And>y. y \<le> M \<Longrightarrow> \<bar>cdf M2 y\<bar> < \<epsilon> / 4" "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" "\<bar>cdf M1 a\<bar> < \<epsilon> / 4" "\<bar>cdf M2 a\<bar> < \<epsilon> / 4"
+ by auto
+
+ from `\<epsilon> > 0` `(cdf M1 \<longlongrightarrow> cdf M1 x) (at_right x)` `(cdf M2 \<longlongrightarrow> cdf M2 x) (at_right x)`
+ have "eventually (\<lambda>y. \<bar>cdf M1 y - cdf M1 x\<bar> < \<epsilon> / 4 \<and> \<bar>cdf M2 y - cdf M2 x\<bar> < \<epsilon> / 4 \<and> x < y) (at_right x)"
+ 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>cdf M1 y - cdf M1 x\<bar> < \<epsilon> / 4"
+ "\<And>y. x < y \<Longrightarrow> y < N \<Longrightarrow> \<bar>cdf M2 y - cdf M2 x\<bar> < \<epsilon> / 4" "\<And>y. x < y \<Longrightarrow> y < N \<Longrightarrow> x < y"
+ 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" "\<bar>cdf M2 x - cdf M2 b\<bar> < \<epsilon> / 4" "\<bar>cdf M1 x - cdf M1 b\<bar> < \<epsilon> / 4"
+ by (auto simp: abs_minus_commute)
+ from `a \<le> x` `x < b` have "a < b" "a \<le> b" by auto
+
+ from `char M1 = char M2`
+ M1.Levy_Inversion [OF `a \<le> b` `measure M1 {a} = 0` `measure M1 {b} = 0`]
+ M2.Levy_Inversion [OF `a \<le> b` `measure M2 {a} = 0` `measure M2 {b} = 0`]
+ have "complex_of_real (measure M1 {a<..b}) = complex_of_real (measure M2 {a<..b})"
+ by (intro LIMSEQ_unique) auto
+ then have "measure M1 {a<..b} = measure M2 {a<..b}" by auto
+ then have *: "cdf M1 b - cdf M1 a = cdf M2 b - cdf M2 a"
+ unfolding M1.cdf_diff_eq [OF `a < b`] M2.cdf_diff_eq [OF `a < b`] .
+
+ have "abs (cdf M1 x - (cdf M1 b - cdf M1 a)) = abs (cdf M1 x - cdf M1 b + cdf M1 a)"
+ by simp
+ also have "\<dots> \<le> abs (cdf M1 x - cdf M1 b) + abs (cdf M1 a)"
+ by (rule abs_triangle_ineq)
+ also have "\<dots> \<le> \<epsilon> / 4 + \<epsilon> / 4"
+ by (intro add_mono less_imp_le \<open>\<bar>cdf M1 a\<bar> < \<epsilon> / 4\<close> \<open>\<bar>cdf M1 x - cdf M1 b\<bar> < \<epsilon> / 4\<close>)
+ finally have 1: "abs (cdf M1 x - (cdf M1 b - cdf M1 a)) \<le> \<epsilon> / 2" by simp
+
+ have "abs (cdf M2 x - (cdf M2 b - cdf M2 a)) = abs (cdf M2 x - cdf M2 b + cdf M2 a)"
+ by simp
+ also have "\<dots> \<le> abs (cdf M2 x - cdf M2 b) + abs (cdf M2 a)"
+ by (rule abs_triangle_ineq)
+ also have "\<dots> \<le> \<epsilon> / 4 + \<epsilon> / 4"
+ by (intro add_mono less_imp_le \<open>\<bar>cdf M2 x - cdf M2 b\<bar> < \<epsilon> / 4\<close> \<open>\<bar>cdf M2 a\<bar> < \<epsilon> / 4\<close>)
+ finally have 2: "abs (cdf M2 x - (cdf M2 b - cdf M2 a)) \<le> \<epsilon> / 2" by simp
+
+ have "abs (cdf M1 x - cdf M2 x) = abs ((cdf M1 x - (cdf M1 b - cdf M1 a)) -
+ (cdf M2 x - (cdf M2 b - cdf M2 a)))" by (subst *, simp)
+ also have "\<dots> \<le> abs (cdf M1 x - (cdf M1 b - cdf M1 a)) +
+ abs (cdf M2 x - (cdf M2 b - cdf M2 a))" by (rule abs_triangle_ineq4)
+ also have "\<dots> \<le> \<epsilon> / 2 + \<epsilon> / 2" by (rule add_mono [OF 1 2])
+ finally have "abs (cdf M1 x - cdf M2 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 `real_distribution M1` `real_distribution M2`])
+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 `u > 0`
+ 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 `x \<noteq> 0`] 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 `0 < u`
+ by (intro integrableI_bounded_set_indicator [where B="2"])
+ (auto simp: lborel.emeasure_pair_measure_Times 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))))"
+ by (rule 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')
+ also have "\<dots> = (CLINT x | M n. (if x = 0 then 0 else 2 * (u - sin (u * x) / x)))"
+ using `u > 0` by (intro 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 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 `u > 0` by (subst 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 `u > 0`
+ by (intro integral_mono [OF _ **])
+ (auto simp: divide_simps sin_x_le_x mult.commute[of u] mult_neg_pos 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}"
+ 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 `\<epsilon> > 0` 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)"
+ by (intro integrableI_bounded_set_indicator[where B=2]) (auto simp: emeasure_lborel_Icc_eq)
+ have 3: "\<And>u v. set_integrable lborel {u..v} (\<lambda>x. 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)"
+ using integral_norm_bound[OF 2] by simp
+ also have "\<dots> \<le> LBINT t:{-d/2..d/2}. \<epsilon> / 4"
+ apply (rule integral_mono [OF 3])
+ apply (simp add: emeasure_lborel_Icc_eq)
+ apply (case_tac "x \<in> {-d/2..d/2}", auto)
+ apply (subst norm_minus_commute)
+ apply (rule less_imp_le)
+ apply (rule d1 [simplified])
+ using d0 by auto
+ also with d0 have "\<dots> = d * \<epsilon> / 4"
+ by simp
+ finally have bound: "cmod (CLBINT t:{-d/2..d/2}. 1 - char M' t) \<le> d * \<epsilon> / 4" .
+ { fix n x
+ have "cmod (1 - char (M n) x) \<le> 2"
+ by (rule order_trans [OF norm_triangle_ineq4], auto simp add: Mn.cmod_char_le_1)
+ } note bd1 = this
+ 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)"
+ using bd1
+ 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 `\<epsilon> > 0` 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 guess N ..
+ 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 `n \<ge> N`] 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 `\<epsilon> > 0` 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 `\<epsilon> > 0` 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: "subseq s"
+ 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 lim_subseq [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 * `real_distribution M'`])
+ thus "weak_conv_m (M \<circ> s) M'"
+ apply (elim subst)
+ by (rule nu)
+ qed
+qed
+
+end