diff --git a/thys/Fourier/Fourier.thy b/thys/Fourier/Fourier.thy
--- a/thys/Fourier/Fourier.thy
+++ b/thys/Fourier/Fourier.thy
@@ -1,2740 +1,2742 @@
 section\<open>The basics of Fourier series\<close>
 
 text\<open>Ported from HOL Light; thanks to Manuel Eberl for help with the real asymp proof methods\<close>
 
 theory "Fourier"
-  imports Periodic Square_Integrable "HOL-Real_Asymp.Real_Asymp" Confine
+  imports Periodic Square_Integrable "HOL-Real_Asymp.Real_Asymp" Confine Fourier_Aux2
 begin
 
 subsection\<open>Orthonormal system of L2 functions and their Fourier coefficients\<close>
 
 definition orthonormal_system :: "'a::euclidean_space set \<Rightarrow> ('b \<Rightarrow> 'a \<Rightarrow> real) \<Rightarrow> bool"
   where "orthonormal_system S w \<equiv> \<forall>m n. l2product S (w m) (w n) = (if m = n then 1 else 0)"
 
 definition orthonormal_coeff :: "'a::euclidean_space set \<Rightarrow> (nat \<Rightarrow> 'a \<Rightarrow> real) \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real"
   where "orthonormal_coeff S w f n = l2product S (w n) f"
 
 lemma orthonormal_system_eq: "orthonormal_system S w \<Longrightarrow> l2product S (w m) (w n) = (if m = n then 1 else 0)"
   by (simp add: orthonormal_system_def)
 
 lemma orthonormal_system_l2norm:
    "orthonormal_system S w \<Longrightarrow> l2norm S (w i) = 1"
   by (simp add: l2norm_def orthonormal_system_def)
 
 lemma orthonormal_partial_sum_diff:
   assumes os: "orthonormal_system S w" and w: "\<And>i. (w i) square_integrable S"
     and f: "f square_integrable S" and "finite I"
   shows "(l2norm S (\<lambda>x. f x - (\<Sum>i\<in>I. a i * w i x)))\<^sup>2 =
         (l2norm S f)\<^sup>2 + (\<Sum>i\<in>I. (a i)\<^sup>2) -  2 * (\<Sum>i\<in>I. a i * orthonormal_coeff S w f i)"
 proof -
   have "S \<in> sets lebesgue"
     using f square_integrable_def by blast
   then have "(\<lambda>x. \<Sum>i\<in>I. a i * w i x) square_integrable S"
     by (intro square_integrable_sum square_integrable_lmult w \<open>finite I\<close>)
   with assms show ?thesis
     apply (simp add: square_integrable_diff l2norm_pow_2 orthonormal_coeff_def orthonormal_system_def)
     apply (simp add: square_integrable_diff square_integrable_lmult l2product_rdiff l2product_sym
            l2product_rsum l2product_rmult algebra_simps  power2_eq_square if_distrib [of "\<lambda>x. _ * x"])
     done
 qed
 
 lemma orthonormal_optimal_partial_sum:
   assumes "orthonormal_system S w" "\<And>i. (w i) square_integrable S"
           "f square_integrable S" "finite I"
   shows "l2norm S (\<lambda>x. f x - (\<Sum>i\<in>I. orthonormal_coeff S w f i * w i x))
        \<le> l2norm S (\<lambda>x. f x - (\<Sum>i\<in>I. a i * w i x))"
 proof -
   have "2 * (a i * l2product S f(w i)) \<le> (a i)\<^sup>2 + (l2product S f(w i))\<^sup>2" for i
     using sum_squares_bound [of "a i" "l2product S f(w i)"]
     by (force simp: algebra_simps)
   then have *: "2 * (\<Sum>i\<in>I. a i * l2product S f(w i)) \<le> (\<Sum>i\<in>I. (a i)\<^sup>2 + (l2product S f(w i))\<^sup>2)"
     by (simp add: sum_distrib_left sum_mono)
   have S: "S \<in> sets lebesgue"
     using assms square_integrable_def by blast
   with assms * show ?thesis
     apply (simp add: l2norm_le square_integrable_sum square_integrable_diff square_integrable_lmult flip: l2norm_pow_2)
     apply (simp add: orthonormal_coeff_def orthonormal_partial_sum_diff)
     apply (simp add: l2product_sym power2_eq_square sum.distrib)
     done
 qed
 
 lemma Bessel_inequality:
   assumes "orthonormal_system S w" "\<And>i. (w i) square_integrable S"
     "f square_integrable S" "finite I"
   shows "(\<Sum>i\<in>I. (orthonormal_coeff S w f i)\<^sup>2) \<le> (l2norm S f)\<^sup>2"
   using orthonormal_partial_sum_diff [OF assms, of "orthonormal_coeff S w f"]
   apply (simp add: algebra_simps flip: power2_eq_square)
   by (metis (lifting) add_diff_cancel_left' diff_ge_0_iff_ge zero_le_power2)
 
 lemma Fourier_series_square_summable:
   assumes os: "orthonormal_system S w" and w: "\<And>i. (w i) square_integrable S"
     and f: "f square_integrable S"
   shows "summable (confine (\<lambda>i. (orthonormal_coeff S w f i) ^ 2) I)"
 proof -
   have "incseq (\<lambda>n. \<Sum>i\<in>I \<inter> {..n}. (orthonormal_coeff S w f i)\<^sup>2)"
     by (auto simp: incseq_def intro: sum_mono2)
   moreover have "\<And>i. (\<Sum>i\<in>I \<inter> {..i}. (orthonormal_coeff S w f i)\<^sup>2) \<le> (l2norm S f)\<^sup>2"
     by (simp add: Bessel_inequality assms)
   ultimately obtain L where "(\<lambda>n. \<Sum>i\<in>I \<inter> {..n}. (orthonormal_coeff S w f i)\<^sup>2) \<longlonglongrightarrow> L"
     using incseq_convergent by blast
   then have "\<forall>r>0. \<exists>no. \<forall>n\<ge>no. \<bar>(\<Sum>i\<in>{..n} \<inter> I. (orthonormal_coeff S w f i)\<^sup>2) - L\<bar> < r"
     by (simp add: LIMSEQ_iff Int_commute)
   then show ?thesis
     by (auto simp: summable_def sums_confine_le LIMSEQ_iff)
 qed
 
 lemma orthonormal_Fourier_partial_sum_diff_squared:
   assumes os: "orthonormal_system S w" and w: "\<And>i. (w i) square_integrable S"
     and f: "f square_integrable S" and "finite I"
   shows "(l2norm S (\<lambda>x. f x -(\<Sum>i\<in>I. orthonormal_coeff S w f i * w i x)))\<^sup>2 =
          (l2norm S f)\<^sup>2 - (\<Sum>i\<in>I. (orthonormal_coeff S w f i)\<^sup>2)"
   using orthonormal_partial_sum_diff [OF assms, where a = "orthonormal_coeff S w f"]
   by (simp add: power2_eq_square)
 
 
 lemma Fourier_series_l2_summable:
   assumes os: "orthonormal_system S w" and w: "\<And>i. (w i) square_integrable S"
     and f: "f square_integrable S"
   obtains g where "g square_integrable S"
                   "(\<lambda>n. l2norm S (\<lambda>x. (\<Sum>i\<in>I \<inter> {..n}. orthonormal_coeff S w f i * w i x) - g x))
                    \<longlonglongrightarrow> 0"
 proof -
   have S: "S \<in> sets lebesgue"
     using f square_integrable_def by blast
   let ?\<Theta> = "\<lambda>A x. (\<Sum>i\<in>I \<inter> A. orthonormal_coeff S w f i * w i x)"
   let ?\<Psi> = "confine (\<lambda>i. (orthonormal_coeff S w f i)\<^sup>2) I"
   have si: "?\<Theta> A square_integrable S" if "finite A" for A
     by (force intro: square_integrable_lmult w square_integrable_sum S that)
   have "\<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. l2norm S (\<lambda>x. ?\<Theta> {..m} x - ?\<Theta> {..n} x) < e"
     if "e > 0" for e
   proof -
     have "summable ?\<Psi>"
       by (rule Fourier_series_square_summable [OF os w f])
     then have "Cauchy (\<lambda>n. sum ?\<Psi> {..n})"
       by (simp add: summable_def sums_def_le convergent_eq_Cauchy)
     then obtain M where M: "\<And>m n. \<lbrakk>m\<ge>M; n\<ge>M\<rbrakk> \<Longrightarrow> dist (sum ?\<Psi> {..m}) (sum ?\<Psi> {..n}) < e\<^sup>2"
       using \<open>e > 0\<close> unfolding Cauchy_def by (drule_tac x="e^2" in spec) auto
     have "\<lbrakk>m\<ge>M; n\<ge>M\<rbrakk> \<Longrightarrow> l2norm S (\<lambda>x. ?\<Theta> {..m} x - ?\<Theta> {..n} x) < e" for m n
     proof (induct m n rule: linorder_class.linorder_wlog)
       case (le m n)
       have sum_diff: "sum f(I \<inter> {..n}) - sum f(I \<inter> {..m}) = sum f(I \<inter> {Suc m..n})" for f :: "nat \<Rightarrow> real"
       proof -
         have "(I \<inter> {..n}) =  (I \<inter> {..m} \<union> I \<inter> {Suc m..n})" "(I \<inter> {..m}) \<inter> (I \<inter> {Suc m..n}) = {}"
           using le by auto
         then show ?thesis
           by (simp add: algebra_simps flip: sum.union_disjoint)
       qed
       have "(l2norm S (\<lambda>x. ?\<Theta> {..n} x - ?\<Theta> {..m} x))^2
             \<le> \<bar>(\<Sum>i\<in>I \<inter> {..n}. (orthonormal_coeff S w f i)\<^sup>2) - (\<Sum>i\<in>I \<inter> {..m}. (orthonormal_coeff S w f i)\<^sup>2)\<bar>"
       proof (simp add: sum_diff)
         have "(l2norm S (?\<Theta> {Suc m..n}))\<^sup>2
             = (\<Sum>i\<in>I \<inter> {Suc m..n}. l2product S (\<lambda>x. \<Sum>j\<in>I \<inter> {Suc m..n}. orthonormal_coeff S w f j * w j x)
                                                (\<lambda>x. orthonormal_coeff S w f i * w i x))"
           by (simp add: l2norm_pow_2 l2product_rsum si w)
         also have "\<dots> = (\<Sum>i\<in>I \<inter> {Suc m..n}. \<Sum>j\<in>I \<inter> {Suc m..n}.
                             orthonormal_coeff S w f j * orthonormal_coeff S w f i * l2product S (w j) (w i))"
           by (simp add: l2product_lsum l2product_lmult l2product_rmult si w flip: mult.assoc)
         also have "\<dots> \<le> \<bar>\<Sum>i\<in>I \<inter> {Suc m..n}. (orthonormal_coeff S w f i)\<^sup>2\<bar>"
           by (simp add: orthonormal_system_eq [OF os] power2_eq_square if_distrib [of "(*)_"] cong: if_cong)
         finally show "(l2norm S (?\<Theta> {Suc m..n}))\<^sup>2 \<le> \<bar>\<Sum>i\<in>I \<inter> {Suc m..n}. (orthonormal_coeff S w f i)\<^sup>2\<bar>" .
       qed
       also have "\<dots> < e\<^sup>2"
         using M [OF \<open>n\<ge>M\<close> \<open>m\<ge>M\<close>]
         by (simp add: dist_real_def sum_confine_eq_Int Int_commute)
       finally have "(l2norm S (\<lambda>x. ?\<Theta> {..m} x - ?\<Theta> {..n} x))^2 < e^2"
         using l2norm_diff [OF si si] by simp
       with \<open>e > 0\<close> show ?case
         by (simp add: power2_less_imp_less)
     next
       case (sym a b)
       then show ?case
         by (subst l2norm_diff) (auto simp: si)
     qed
     then show ?thesis
       by blast
   qed
   with that show thesis
     by (blast intro: si [of "{.._}"] l2_complete [of "\<lambda>n. ?\<Theta> {..n}"])
 qed
 
 lemma Fourier_series_l2_summable_strong:
   assumes os: "orthonormal_system S w" and w: "\<And>i. (w i) square_integrable S"
     and f: "f square_integrable S"
   obtains g where "g square_integrable S"
           "\<And>i. i \<in> I \<Longrightarrow> orthonormal_coeff S w (\<lambda>x. f x - g x) i = 0"
           "(\<lambda>n. l2norm S (\<lambda>x. (\<Sum>i\<in>I \<inter> {..n}. orthonormal_coeff S w f i * w i x) - g x))
            \<longlonglongrightarrow> 0"
 proof -
   have S: "S \<in> sets lebesgue"
     using f square_integrable_def by blast
   obtain g where g: "g square_integrable S"
             and teng: "(\<lambda>n. l2norm S (\<lambda>x. (\<Sum>i\<in>I \<inter> {..n}. orthonormal_coeff S w f i * w i x) - g x))
                        \<longlonglongrightarrow> 0"
     using Fourier_series_l2_summable [OF assms] .
   show thesis
   proof
     show "orthonormal_coeff S w (\<lambda>x. f x - g x) i = 0"
       if "i \<in> I" for i
     proof (rule tendsto_unique)
       let ?\<Theta> = "\<lambda>A x. (\<Sum>i\<in>I \<inter> A. orthonormal_coeff S w f i * w i x)"
       let ?f = "\<lambda>n. l2product S (w i) (\<lambda>x. (f x - ?\<Theta> {..n} x) + (?\<Theta> {..n} x - g x))"
       show "?f \<longlonglongrightarrow> orthonormal_coeff S w (\<lambda>x. f x - g x) i"
         by (simp add: orthonormal_coeff_def)
       have 1: "(\<lambda>n. l2product S (w i) (\<lambda>x. f x - ?\<Theta> {..n} x)) \<longlonglongrightarrow> 0"
         proof (rule tendsto_eventually)
           have "l2product S (w i) (\<lambda>x. f x - ?\<Theta> {..j} x) = 0"
             if "j \<ge> i" for j
             using that \<open>i \<in> I\<close>
             apply (simp add: l2product_rdiff l2product_rsum l2product_rmult orthonormal_coeff_def w f S)
             apply (simp add: orthonormal_system_eq [OF os] if_distrib [of "(*)_"] cong: if_cong)
             done
           then show "\<forall>\<^sub>F n in sequentially. l2product S (w i) (\<lambda>x. f x - ?\<Theta> {..n} x) = 0"
             using eventually_at_top_linorder by blast
         qed
         have 2: "(\<lambda>n. l2product S (w i) (\<lambda>x. ?\<Theta> {..n} x - g x)) \<longlonglongrightarrow> 0"
         proof (intro Lim_null_comparison [OF _ teng] eventuallyI)
           show "norm (l2product S (w i) (\<lambda>x. ?\<Theta> {..n} x - g x)) \<le> l2norm S (\<lambda>x. ?\<Theta> {..n} x - g x)" for n
             using Schwartz_inequality_abs [of "w i" S "(\<lambda>x. ?\<Theta> {..n} x - g x)"]
             by (simp add: S g f w orthonormal_system_l2norm [OF os])
         qed
         show "?f \<longlonglongrightarrow> 0"
           using that tendsto_add [OF 1 2]
           by (subst l2product_radd) (simp_all add: l2product_radd w f g S)
       qed auto
   qed (auto simp: g teng)
 qed
 
 
-
 subsection\<open>Actual trigonometric orthogonality relations\<close>
 
 lemma integrable_sin_cx:
   "integrable (lebesgue_on {-pi..pi}) (\<lambda>x. sin(x * c))"
   by (intro continuous_imp_integrable_real continuous_intros)
 
 lemma integrable_cos_cx:
   "integrable (lebesgue_on {-pi..pi}) (\<lambda>x. cos(x * c))"
   by (intro continuous_imp_integrable_real continuous_intros)
 
+lemma integral_cos_Z' [simp]:
+  assumes "n \<in> \<int>"
+  shows "integral\<^sup>L (lebesgue_on {-pi..pi}) (\<lambda>x. cos(n * x)) = (if n = 0 then 2 * pi else 0)"
+  by (metis assms integral_cos_Z mult_commute_abs)
+
 lemma integral_sin_and_cos:
   fixes m n::int
   shows
   "integral\<^sup>L (lebesgue_on {-pi..pi}) (\<lambda>x. cos(m * x) * cos(n * x)) = (if \<bar>m\<bar> = \<bar>n\<bar> then if n = 0 then 2 * pi else pi else 0)"
   "integral\<^sup>L (lebesgue_on {-pi..pi}) (\<lambda>x. cos(m * x) * sin(n * x)) = 0"
   "integral\<^sup>L (lebesgue_on {-pi..pi}) (\<lambda>x. sin(m * x) * cos(n * x)) = 0"
   "\<lbrakk>m \<ge> 0; n \<ge> 0\<rbrakk> \<Longrightarrow> integral\<^sup>L (lebesgue_on {-pi..pi}) (\<lambda>x. sin (m * x) * sin (n * x)) = (if m = n \<and> n \<noteq> 0 then pi else 0)"
   "\<bar>integral\<^sup>L (lebesgue_on {-pi..pi}) (\<lambda>x. sin (m * x) * sin (n * x))\<bar> = (if \<bar>m\<bar> = \<bar>n\<bar> \<and> n \<noteq> 0 then pi else 0)"
   by (simp_all add: abs_if sin_times_sin cos_times_sin sin_times_cos cos_times_cos
                     integrable_sin_cx integrable_cos_cx mult_ac
                flip: distrib_left distrib_right left_diff_distrib right_diff_distrib)
 
 lemma integral_sin_and_cos_Z [simp]:
   fixes m n::real
   assumes "m \<in> \<int>" "n \<in> \<int>"
   shows
   "integral\<^sup>L (lebesgue_on {-pi..pi}) (\<lambda>x. cos(m * x) * cos(n * x)) = (if \<bar>m\<bar> = \<bar>n\<bar> then if n = 0 then 2 * pi else pi else 0)"
   "integral\<^sup>L (lebesgue_on {-pi..pi}) (\<lambda>x. cos(m * x) * sin(n * x)) = 0"
   "integral\<^sup>L (lebesgue_on {-pi..pi}) (\<lambda>x. sin(m * x) * cos(n * x)) = 0"
   "\<bar>integral\<^sup>L (lebesgue_on {-pi..pi}) (\<lambda>x. sin (m * x) * sin (n * x))\<bar> = (if \<bar>m\<bar> = \<bar>n\<bar> \<and> n \<noteq> 0 then pi else 0)"
   using assms unfolding Ints_def
      apply safe
   unfolding integral_sin_and_cos
      apply auto
   done
 
 lemma integral_sin_and_cos_N [simp]:
   fixes m n::real
   assumes "m \<in> \<nat>" "n \<in> \<nat>"
   shows "integral\<^sup>L (lebesgue_on {-pi..pi}) (\<lambda>x. sin (m * x) * sin (n * x)) = (if m = n \<and> n \<noteq> 0 then pi else 0)"
   using assms unfolding Nats_altdef1 by (auto simp: integral_sin_and_cos)
 
 
 lemma integrable_sin_and_cos:
   fixes m n::int
   shows "integrable (lebesgue_on {a..b}) (\<lambda>x. cos(x * m) * cos(x * n))"
         "integrable (lebesgue_on {a..b}) (\<lambda>x. cos(x * m) * sin(x * n))"
         "integrable (lebesgue_on {a..b}) (\<lambda>x. sin(x * m) * cos(x * n))"
         "integrable (lebesgue_on {a..b}) (\<lambda>x. sin(x * m) * sin(x * n))"
   by (intro continuous_imp_integrable_real continuous_intros)+
 
 lemma sqrt_pi_ge1: "sqrt pi \<ge> 1"
   using pi_gt3 by auto
 
 definition trigonometric_set :: "nat \<Rightarrow> real \<Rightarrow> real"
   where "trigonometric_set n \<equiv>
     if n = 0 then \<lambda>x. 1 / sqrt(2 * pi)
     else if odd n then \<lambda>x. sin(real(Suc (n div 2)) * x) / sqrt(pi)
     else (\<lambda>x. cos((n div 2) * x) / sqrt pi)"
 
 lemma trigonometric_set:
   "trigonometric_set 0 x = 1 / sqrt(2 * pi)"
   "trigonometric_set (Suc (2 * n)) x = sin(real(Suc n) * x) / sqrt(pi)"
   "trigonometric_set (2 * n + 2) x = cos(real(Suc n) * x) / sqrt(pi)"
   "trigonometric_set (Suc (Suc (2 * n))) x = cos(real(Suc n) * x) / sqrt(pi)"
   by (simp_all add: trigonometric_set_def algebra_simps add_divide_distrib)
 
 lemma trigonometric_set_even:
    "trigonometric_set(2*k) = (if k = 0 then (\<lambda>x. 1 / sqrt(2 * pi)) else (\<lambda>x. cos(k * x) / sqrt pi))"
   by (induction k) (auto simp: trigonometric_set_def add_divide_distrib split: if_split_asm)
 
-
 lemma orthonormal_system_trigonometric_set:
     "orthonormal_system {-pi..pi} trigonometric_set"
 proof -
   have "l2product {-pi..pi} (trigonometric_set m) (trigonometric_set n) = (if m = n then 1 else 0)" for m n
   proof (induction m rule: odd_even_cases)
     case 0
     show ?case
       by (induction n rule: odd_even_cases) (auto simp: trigonometric_set l2product_def)
   next
     case (odd m)
     show ?case
       by (induction n rule: odd_even_cases) (auto simp: trigonometric_set l2product_def double_not_eq_Suc_double)
   next
     case (even m)
     show ?case
       by (induction n rule: odd_even_cases) (auto simp: trigonometric_set l2product_def Suc_double_not_eq_double)
   qed
   then show ?thesis
     unfolding orthonormal_system_def by auto
 qed
 
 
 lemma square_integrable_trigonometric_set:
    "(trigonometric_set i) square_integrable {-pi..pi}"
 proof -
   have "continuous_on {-pi..pi} (\<lambda>x. sin ((1 + real n) * x) / sqrt pi)" for n
     by (intro continuous_intros) auto
   moreover
   have "continuous_on {-pi..pi} (\<lambda>x. cos (real i * x / 2) / sqrt pi) "
     by (intro continuous_intros) auto
   ultimately show ?thesis
     by (simp add: trigonometric_set_def)
 qed
 
 subsection\<open>Weierstrass for trigonometric polynomials\<close>
 
 lemma Weierstrass_trig_1:
   fixes g :: "real \<Rightarrow> real"
   assumes contf: "continuous_on UNIV g" and periodic: "\<And>x. g(x + 2 * pi) = g x" and 1: "norm z = 1"
   shows "continuous (at z within (sphere 0 1)) (g \<circ> Im \<circ> Ln)"
 proof (cases "Re z < 0")
   let ?f = "complex_of_real \<circ> g \<circ> (\<lambda>x. x + pi) \<circ> Im \<circ> Ln \<circ> uminus"
   case True
   have "continuous (at z within (sphere 0 1)) (complex_of_real \<circ> g \<circ> Im \<circ> Ln)"
   proof (rule continuous_transform_within)
     have [simp]: "z \<notin> \<real>\<^sub>\<ge>\<^sub>0"
       using True complex_nonneg_Reals_iff by auto
     show "continuous (at z within (sphere 0 1)) ?f"
       by (intro continuous_within_Ln continuous_intros continuous_on_imp_continuous_within [OF contf]) auto
     show "?f x' =  (complex_of_real \<circ> g \<circ> Im \<circ> Ln) x'"
       if "x' \<in> (sphere 0 1)" and "dist x' z < 1" for x'
     proof -
       have "x' \<noteq> 0"
         using that by auto
       with that show ?thesis
         by (auto simp: Ln_minus add.commute periodic)
     qed
   qed (use 1 in auto)
   then have "continuous (at z within (sphere 0 1)) (Re \<circ> complex_of_real \<circ> g \<circ> Im \<circ> Ln)"
     unfolding o_def by (rule continuous_Re)
   then show ?thesis
     by (simp add: o_def)
 next
   case False
   let ?f = "complex_of_real \<circ> g \<circ> Im \<circ> Ln \<circ> uminus"
   have "z \<noteq> 0"
     using 1 by auto
   with False have "z \<notin> \<real>\<^sub>\<le>\<^sub>0"
     by (auto simp: not_less nonpos_Reals_def)
   then have "continuous (at z within (sphere 0 1)) (complex_of_real \<circ> g \<circ> Im \<circ> Ln)"
     by (intro continuous_within_Ln continuous_intros continuous_on_imp_continuous_within [OF contf]) auto
   then have "continuous (at z within (sphere 0 1)) (Re \<circ> complex_of_real \<circ> g \<circ> Im \<circ> Ln)"
     unfolding o_def by (rule continuous_Re)
   then show ?thesis
     by (simp add: o_def)
 qed
 
 inductive_set cx_poly :: "(complex \<Rightarrow> real) set" where
     Re: "Re \<in> cx_poly"
   | Im: "Im \<in> cx_poly"
   | const: "(\<lambda>x. c) \<in> cx_poly"
   | add:   "\<lbrakk>f \<in> cx_poly; g \<in> cx_poly\<rbrakk> \<Longrightarrow> (\<lambda>x. f x + g x) \<in> cx_poly"
   | mult:  "\<lbrakk>f \<in> cx_poly; g \<in> cx_poly\<rbrakk> \<Longrightarrow> (\<lambda>x. f x * g x) \<in> cx_poly"
 
 declare cx_poly.intros [intro]
 
 
 lemma Weierstrass_trig_polynomial:
   assumes contf: "continuous_on {-pi..pi} f" and fpi: "f(-pi) = f pi" and "0 < e"
   obtains n::nat and a b where
     "\<And>x::real. x \<in> {-pi..pi} \<Longrightarrow> \<bar>f x - (\<Sum>k\<le>n. a k * sin (k * x) + b k * cos (k * x))\<bar> < e"
 proof -
   interpret CR: function_ring_on "cx_poly" "sphere 0 1"
   proof
     show "continuous_on (sphere 0 1) f" if "f \<in> cx_poly" for f
       using that by induction (assumption | intro continuous_intros)+
     fix x y::complex
     assume "x \<in> sphere 0 1" and "y \<in> sphere 0 1" and "x \<noteq> y"
     then consider "Re x \<noteq> Re y" | "Im x \<noteq> Im y"
       using complex_eqI by blast
     then show "\<exists>f\<in>cx_poly. f x \<noteq> f y"
       by (metis cx_poly.Im cx_poly.Re)
   qed (auto simp: cx_poly.intros)
   have "continuous (at z within (sphere 0 1)) (f \<circ> Im \<circ> Ln)" if "norm z = 1" for z
   proof -
     obtain g where contg: "continuous_on UNIV g" and gf: "\<And>x. x \<in> {-pi..pi} \<Longrightarrow> g x = f x"
       and periodic: "\<And>x. g(x + 2*pi) = g x"
       using Tietze_periodic_interval [OF contf fpi] by auto
     let ?f = "(g \<circ> Im \<circ> Ln)"
     show ?thesis
     proof (rule continuous_transform_within)
       show "continuous (at z within (sphere 0 1)) ?f"
         using Weierstrass_trig_1 [OF contg periodic that] by (simp add: sphere_def)
       show "?f x' = (f \<circ> Im \<circ> Ln) x'"
         if "x' \<in> sphere 0 1" "dist x' z < 1" for x'
       proof -
         have "x' \<noteq> 0"
           using that by auto
         then have "Im (Ln x') \<in> {-pi..pi}"
           using Im_Ln_le_pi [of x'] mpi_less_Im_Ln [of x'] by simp
         then show ?thesis
           using gf by simp
       qed
     qed (use that in auto)
   qed
   then have "continuous_on (sphere 0 1) (f \<circ> Im \<circ> Ln)"
     using continuous_on_eq_continuous_within mem_sphere_0 by blast
   then obtain g where "g \<in> cx_poly" and g: "\<And>x. x \<in> sphere 0 1 \<Longrightarrow> \<bar>(f \<circ> Im \<circ> Ln) x - g x\<bar> < e"
     using CR.Stone_Weierstrass_basic [of "f \<circ> Im \<circ> Ln"] \<open>e > 0\<close> by meson
   define trigpoly where
     "trigpoly \<equiv> \<lambda>f. \<exists>n a b. f = (\<lambda>x. (\<Sum>k\<le>n. a k * sin(real k * x) +  b k * cos(real k * x)))"
   have tp_const: "trigpoly(\<lambda>x. c)" for c
     unfolding trigpoly_def
     by (rule_tac x=0 in exI) auto
   have tp_add: "trigpoly(\<lambda>x. f x + g x)" if "trigpoly f" "trigpoly g" for f g
   proof -
     obtain n1 a1 b1 where feq: "f = (\<lambda>x. \<Sum>k\<le>n1. a1 k * sin (real k * x) + b1 k * cos (real k * x))"
       using \<open>trigpoly f\<close> trigpoly_def by blast
     obtain n2 a2 b2 where geq: "g = (\<lambda>x. \<Sum>k\<le>n2. a2 k * sin (real k * x) + b2 k * cos (real k * x))"
       using \<open>trigpoly g\<close> trigpoly_def by blast
     let ?a = "\<lambda>n. (if n \<le> n1 then a1 n else 0) + (if n \<le> n2 then a2 n else 0)"
     let ?b = "\<lambda>n. (if n \<le> n1 then b1 n else 0) + (if n \<le> n2 then b2 n else 0)"
     have eq: "{k. k \<le> max a b \<and> k \<le> a} = {..a}" "{k. k \<le> max a b \<and> k \<le> b} = {..b}" for a b::nat
       by auto
     have "(\<lambda>x. f x + g x) = (\<lambda>x. \<Sum>k \<le> max n1 n2. ?a k * sin (real k * x) + ?b k * cos (real k * x))"
       by (simp add: feq geq algebra_simps eq sum.distrib if_distrib [of "\<lambda>u. _ * u"] cong: if_cong flip: sum.inter_filter)
     then show ?thesis
       unfolding trigpoly_def by meson
   qed
   have tp_sum: "trigpoly(\<lambda>x. \<Sum>i\<in>S. f i x)" if "finite S" "\<And>i. i \<in> S \<Longrightarrow> trigpoly(f i)" for f and S :: "nat set"
     using that
     by induction (auto simp: tp_const tp_add)
   have tp_cmul: "trigpoly(\<lambda>x. c * f x)" if "trigpoly f" for f c
   proof -
     obtain n a b where feq: "f = (\<lambda>x. \<Sum>k\<le>n. a k * sin (real k * x) + b k * cos (real k * x))"
       using \<open>trigpoly f\<close> trigpoly_def by blast
     have "(\<lambda>x. c * f x) = (\<lambda>x. \<Sum>k \<le> n. (c * a k) * sin (real k * x) + (c * b k) * cos (real k * x))"
       by (simp add: feq algebra_simps sum_distrib_left)
     then show ?thesis
       unfolding trigpoly_def by meson
   qed
   have tp_cdiv: "trigpoly(\<lambda>x. f x / c)" if "trigpoly f" for f c
     using tp_cmul [OF \<open>trigpoly f\<close>, of "inverse c"]
     by (simp add: divide_inverse_commute)
   have tp_diff: "trigpoly(\<lambda>x. f x - g x)" if "trigpoly f" "trigpoly g" for f g
     using tp_add [OF \<open>trigpoly f\<close> tp_cmul [OF \<open>trigpoly g\<close>, of "-1"]] by auto
   have tp_sin: "trigpoly(\<lambda>x. sin (n * x))" if "n \<in> \<nat>" for n
     unfolding trigpoly_def
   proof
     obtain k where "n = real k"
       using Nats_cases \<open>n \<in> \<nat>\<close> by blast
     then show "\<exists>a b. (\<lambda>x. sin (n * x)) = (\<lambda>x. \<Sum>i \<le> nat\<lfloor>n\<rfloor>. a i * sin (real i * x) + b i * cos (real i * x))"
       apply (rule_tac x="\<lambda>i. if i = k then 1 else 0" in exI)
       apply (rule_tac x="\<lambda>i. 0" in exI)
       apply (simp add: if_distrib [of "\<lambda>u. u * _"] cong: if_cong)
       done
   qed
   have tp_cos: "trigpoly(\<lambda>x. cos (n * x))" if "n \<in> \<nat>" for n
     unfolding trigpoly_def
   proof
     obtain k where "n = real k"
       using Nats_cases \<open>n \<in> \<nat>\<close> by blast
     then show "\<exists>a b. (\<lambda>x. cos (n * x)) = (\<lambda>x. \<Sum>i \<le> nat\<lfloor>n\<rfloor>. a i * sin (real i * x) + b i * cos (real i * x))"
       apply (rule_tac x="\<lambda>i. 0" in exI)
       apply (rule_tac x="\<lambda>i. if i = k then 1 else 0" in exI)
       apply (simp add: if_distrib [of "\<lambda>u. u * _"] cong: if_cong)
       done
   qed
   have tp_sincos: "trigpoly(\<lambda>x. sin(i * x) * sin(j * x)) \<and> trigpoly(\<lambda>x. sin(i * x) * cos(j * x)) \<and>
                    trigpoly(\<lambda>x. cos(i * x) * sin(j * x)) \<and> trigpoly(\<lambda>x. cos(i * x) * cos(j * x))" (is "?P i j")
     for i j::nat
   proof (rule linorder_wlog [of _ j i])
     show "?P j i" if "i \<le> j" for j i
       using that
       by (auto simp add: sin_times_sin cos_times_cos sin_times_cos cos_times_sin diff_divide_distrib
                simp flip: of_nat_diff left_diff_distrib distrib_right
                intro!: tp_add tp_diff tp_cdiv tp_cos tp_sin)
   qed (simp add: mult_ac)
   have tp_mult: "trigpoly(\<lambda>x. f x * g x)" if "trigpoly f" "trigpoly g" for f g
   proof -
     obtain n1 a1 b1 where feq: "f = (\<lambda>x. \<Sum>k\<le>n1. a1 k * sin (real k * x) + b1 k * cos (real k * x))"
       using \<open>trigpoly f\<close> trigpoly_def by blast
     obtain n2 a2 b2 where geq: "g = (\<lambda>x. \<Sum>k\<le>n2. a2 k * sin (real k * x) + b2 k * cos (real k * x))"
       using \<open>trigpoly g\<close> trigpoly_def by blast
     then show ?thesis
       unfolding feq geq
       by (simp add: algebra_simps sum_product tp_sum tp_add tp_cmul tp_sincos del: mult.commute)
   qed
   have *: "trigpoly(\<lambda>x.  f(exp(\<i> * of_real x)))" if "f \<in> cx_poly" for f
     using that
   proof induction
     case Re
     then show ?case
       using tp_cos [of 1] by (auto simp: Re_exp)
   next
     case Im
     then show ?case
       using tp_sin [of 1] by (auto simp: Im_exp)
   qed (auto simp: tp_const tp_add tp_mult)
   obtain n a b where eq: "(g (iexp x)) = (\<Sum>k\<le>n. a k * sin (real k * x) + b k * cos (real k * x))" for x
     using * [OF \<open>g \<in> cx_poly\<close>] trigpoly_def by meson
   show thesis
     proof
     show "\<bar>f \<theta> - (\<Sum>k\<le>n. a k * sin (real k * \<theta>) + b k * cos (real k * \<theta>))\<bar> < e"
       if "\<theta> \<in> {-pi..pi}" for \<theta>
     proof -
       have "f \<theta> - g (iexp \<theta>) = (f \<circ> Im \<circ> Ln) (iexp \<theta>) - g (iexp \<theta>)"
       proof (cases "\<theta> = -pi")
       case True
         then show ?thesis
           by (simp add: exp_minus fpi)
     next
       case False
       then show ?thesis
         using that by auto
     qed
     then show ?thesis
       using g [of "exp(\<i> * of_real \<theta>)"] by (simp flip: eq)
     qed
   qed
 qed
 
 
 subsection\<open>A bit of extra hacking round so that the ends of a function are OK\<close>
 
 lemma integral_tweak_ends:
   fixes a b :: real
   assumes "a < b" "e > 0"
   obtains f where "continuous_on {a..b} f" "f a = d" "f b = 0" "l2norm {a..b} f < e"
 proof -
   have cont: "continuous_on {a..b}
           (\<lambda>x. if x \<le> a+1 / real(Suc n)
                then ((Suc n) * d) * ((a+1 / (Suc n)) - x) else 0)" for n
   proof (cases "a+1/(Suc n) \<le> b")
     case True
     have *: "1 / (1 + real n) > 0"
       by auto
     have abeq: "{a..b} = {a..a+1/(Suc n)} \<union> {a+1/(Suc n)..b}"
       using \<open>a < b\<close> True
       apply auto
       using "*" by linarith
     show ?thesis
       unfolding abeq
     proof (rule continuous_on_cases)
       show "continuous_on {a..a+1 / real (Suc n)} (\<lambda>x. real (Suc n) * d * (a+1 / real (Suc n) - x))"
         by (intro continuous_intros)
     qed auto
   next
     case False
     show ?thesis
     proof (rule continuous_on_eq [where f = "\<lambda>x. ((Suc n) * d) * ((a+1/(Suc n)) - x)"])
       show " continuous_on {a..b} (\<lambda>x. (Suc n) * d * (a+1 / real (Suc n) - x))"
         by (intro continuous_intros)
     qed (use False in auto)
   qed
   let ?f = "\<lambda>k x. (if x \<le> a+1 / (Suc k) then (Suc k) * d * (a+1 / (Suc k) - x) else 0)\<^sup>2"
   let ?g = "\<lambda>x. if x = a then d\<^sup>2 else 0"
   have bmg: "?g \<in> borel_measurable (lebesgue_on {a..b})"
     apply (rule measurable_restrict_space1)
     using borel_measurable_if_I [of _ "{a}", OF borel_measurable_const] by auto
   have bmf: "?f k \<in> borel_measurable (lebesgue_on {a..b})" for k
   proof -
     have bm: "(\<lambda>x. (Suc k) * d * (a+1 / real (Suc k) - x))
               \<in> borel_measurable (lebesgue_on {..a+1 / (Suc k)})"
       by (intro measurable) (auto simp: measurable_completion measurable_restrict_space1)
     show ?thesis
       apply (intro borel_measurable_power measurable_restrict_space1)
       using borel_measurable_if_I [of _ "{.. a+1 / (Suc k)}", OF bm]  apply auto
       done
   qed
   have int_d2: "integrable (lebesgue_on {a..b}) (\<lambda>x. d\<^sup>2)"
     by (intro continuous_imp_integrable_real continuous_intros)
   have "(\<lambda>k. ?f k x) \<longlonglongrightarrow> ?g x"
     if x: "x \<in> {a..b}" for x
   proof (cases "x = a")
     case False
     then have "x > a"
       using x by auto
     with x obtain N where "N > 0" and N: "1 / real N < x-a"
       using real_arch_invD [of "x-a"]
       by (force simp: inverse_eq_divide)
     then have "x > a+1 / (1 + real k)"
       if "k \<ge> N" for k
     proof -
       have "a+1 / (1 + real k) < a+1 / real N"
         using that \<open>0 < N\<close> by (simp add: field_simps)
       also have "\<dots> < x"
         using N by linarith
       finally show ?thesis .
     qed
     then show ?thesis
       apply (intro tendsto_eventually eventually_sequentiallyI [where c=N])
       by (fastforce simp: False)
   qed auto
   then have tends: "AE x in (lebesgue_on {a..b}). (\<lambda>k. ?f k x) \<longlonglongrightarrow> ?g x"
     by force
   have le_d2: "\<And>k. AE x in (lebesgue_on {a..b}). norm (?f k x) \<le> d\<^sup>2"
   proof
     show "norm ((if x \<le> a+1 / real (Suc k) then real (Suc k) * d * (a+1 / real (Suc k) - x) else 0)\<^sup>2) \<le> d\<^sup>2"
       if "x \<in> space (lebesgue_on {a..b})" for k x
       using that
       apply (simp add: abs_mult divide_simps flip: abs_le_square_iff)
       apply (auto simp: abs_if zero_less_mult_iff mult_left_le)
       done
   qed
   have integ: "integrable (lebesgue_on {a..b}) ?g"
     using integrable_dominated_convergence [OF bmg bmf int_d2 tends le_d2] by auto
   have int: "(\<lambda>k. integral\<^sup>L (lebesgue_on {a..b}) (?f k)) \<longlonglongrightarrow> integral\<^sup>L (lebesgue_on {a..b}) ?g"
     using integral_dominated_convergence [OF bmg bmf int_d2 tends le_d2] by auto
   then obtain N where N: "\<And>k. k \<ge> N \<Longrightarrow> \<bar>integral\<^sup>L (lebesgue_on {a..b}) (?f k) - integral\<^sup>L (lebesgue_on {a..b}) ?g\<bar> < e\<^sup>2"
     apply (simp add: lim_sequentially dist_real_def)
     apply (drule_tac x="e^2" in spec)
     using \<open>e > 0\<close>
     by auto
   obtain M where "M > 0" and M: "1 / real M < b-a"
     using real_arch_invD [of "b-a"]
     by (metis \<open>a < b\<close> diff_gt_0_iff_gt inverse_eq_divide neq0_conv)
   have *: "\<bar>integral\<^sup>L (lebesgue_on {a..b}) (?f (max M N)) - integral\<^sup>L (lebesgue_on {a..b}) ?g\<bar> < e\<^sup>2"
     using N by force
   let ?\<phi> = "\<lambda>x. if x \<le> a+1/(Suc (max M N)) then ((Suc (max M N)) * d) * ((a+1/(Suc (max M N))) - x) else 0"
   show ?thesis
   proof
     show "continuous_on {a..b} ?\<phi>"
       by (rule cont)
     have "1 / (1 + real (max M N)) \<le> 1 / (real M)"
       by (simp add: \<open>0 < M\<close> frac_le)
     then have "\<not> (b \<le> a+1 / (1 + real (max M N)))"
       using M \<open>a < b\<close> \<open>M > 0\<close> max.cobounded1 [of M N]
       by linarith
     then show "?\<phi> b = 0"
       by simp
     have null_a: "{a} \<in> null_sets (lebesgue_on {a..b})"
       using \<open>a < b\<close> by (simp add: null_sets_restrict_space)
     have "LINT x|lebesgue_on {a..b}. ?g x = 0"
-      using null_AE_impI [OF null_a] by (force intro: integral_eq_zero_AE)
+      by (force intro: integral_eq_zero_AE AE_I' [OF null_a])
     then have "l2norm {a..b} ?\<phi> < sqrt (e^2)"
       unfolding l2norm_def l2product_def power2_eq_square [symmetric]
       apply (intro real_sqrt_less_mono)
       using "*" by linarith
     then show "l2norm {a..b} ?\<phi> < e"
       using \<open>e > 0\<close> by auto
   qed auto
 qed
 
 
 lemma square_integrable_approximate_continuous_ends:
   assumes f: "f square_integrable {a..b}" and "a < b" "0 < e"
   obtains g where "continuous_on {a..b} g" "g b = g a" "g square_integrable {a..b}" "l2norm {a..b} (\<lambda>x. f x - g x) < e"
 proof -
   obtain g where contg: "continuous_on UNIV g" and "g square_integrable {a..b}"
     and lg: "l2norm {a..b} (\<lambda>x. f x - g x) < e/2"
     using f \<open>e > 0\<close> square_integrable_approximate_continuous
     by (metis (full_types) box_real(2) half_gt_zero_iff lmeasurable_cbox)
   obtain h where conth: "continuous_on {a..b} h" and "h a = g b - g a" "h b = 0"
     and lh: "l2norm {a..b} h < e/2"
     using integral_tweak_ends \<open>e > 0\<close>
     by (metis \<open>a < b\<close> zero_less_divide_iff zero_less_numeral)
   have "h square_integrable {a..b}"
     using \<open>continuous_on {a..b} h\<close> continuous_imp_square_integrable by blast
   show thesis
   proof
     show "continuous_on {a..b} (\<lambda>x. g x + h x)"
       by (blast intro: continuous_on_subset [OF contg] conth continuous_intros)
     then show "(\<lambda>x. g x + h x) square_integrable {a..b}"
       using continuous_imp_square_integrable by blast
     show "g b + h b = g a + h a"
       by (simp add: \<open>h a = g b - g a\<close> \<open>h b = 0\<close>)
     have "l2norm {a..b} (\<lambda>x. (f x - g x) + - h x) < e"
     proof (rule le_less_trans [OF l2norm_triangle [of "\<lambda>x. f x - g x" "{a..b}" "\<lambda>x. - (h x)"]])
       show "(\<lambda>x. f x - g x) square_integrable {a..b}"
         using \<open>g square_integrable {a..b}\<close> f square_integrable_diff by blast
       show "(\<lambda>x. - h x) square_integrable {a..b}"
         by (simp add: \<open>h square_integrable {a..b}\<close>)
       show "l2norm {a..b} (\<lambda>x. f x - g x) + l2norm {a..b} (\<lambda>x. - h x) < e"
         using \<open>h square_integrable {a..b}\<close> l2norm_neg lg lh by auto
     qed
     then show "l2norm {a..b} (\<lambda>x. f x - (g x + h x)) < e"
       by (simp add: field_simps)
   qed
 qed
 
 
 subsection\<open>Hence the main approximation result\<close>
 
 lemma Weierstrass_l2_trig_polynomial:
   assumes f: "f square_integrable {-pi..pi}" and "0 < e"
   obtains n a b where
    "l2norm {-pi..pi} (\<lambda>x. f x - (\<Sum>k\<le>n. a k * sin(real k * x) + b k * cos(real k * x))) < e"
 proof -
   let ?\<phi> = "\<lambda>n a b x. \<Sum>k\<le>n. a k * sin(real k * x) + b k * cos(real k * x)"
   obtain g where contg: "continuous_on {-pi..pi} g" and geq: "g(-pi) = g pi"
            and g: "g square_integrable {-pi..pi}" and norm_fg: "l2norm {-pi..pi} (\<lambda>x. f x - g x) < e/2"
     using \<open>e > 0\<close> by (auto intro: square_integrable_approximate_continuous_ends [OF f, of "e/2"])
   then obtain n a b where g_phi_less: "\<And>x. x \<in> {-pi..pi} \<Longrightarrow> \<bar>g x - (?\<phi> n a b x)\<bar> < e/6"
     using \<open>e > 0\<close> Weierstrass_trig_polynomial [OF contg geq, of "e/6"]
     by (meson zero_less_divide_iff zero_less_numeral)
   show thesis
   proof
     have si: "(?\<phi> n u v) square_integrable {-pi..pi}" for n::nat and u v
     proof (intro square_integrable_sum continuous_imp_square_integrable)
       show "continuous_on {-pi..pi} (\<lambda>x. u k * sin (real k * x) + v k * cos (real k * x))"
         if "k \<in> {..n}" for k
         using that by (intro continuous_intros)
     qed auto
     have "l2norm {-pi..pi} (\<lambda>x. f x - (?\<phi> n a b x)) =  l2norm {-pi..pi} (\<lambda>x. (f x - g x) + (g x - (?\<phi> n a b x)))"
       by simp
     also have "\<dots> \<le> l2norm {-pi..pi} (\<lambda>x. f x - g x) + l2norm {-pi..pi} (\<lambda>x. g x - (?\<phi> n a b x))"
     proof (rule l2norm_triangle)
       show "(\<lambda>x. f x - g x) square_integrable {-pi..pi}"
         using f g square_integrable_diff by blast
       show "(\<lambda>x. g x - (?\<phi> n a b x)) square_integrable {-pi..pi}"
         using g si square_integrable_diff by blast
     qed
     also have "\<dots> < e"
     proof -
       have g_phi: "(\<lambda>x. g x - (?\<phi> n a b x)) square_integrable {-pi..pi}"
         using g si square_integrable_diff by blast
       have "l2norm {-pi..pi} (\<lambda>x. g x - (?\<phi> n a b x)) \<le> e/2"
         unfolding l2norm_def l2product_def power2_eq_square [symmetric]
       proof (rule real_le_lsqrt)
         show "0 \<le> LINT x|lebesgue_on {-pi..pi}. (g x - (?\<phi> n a b x))\<^sup>2"
           by (rule integral_nonneg_AE) auto
         have "LINT x|lebesgue_on {-pi..pi}. (g x - (?\<phi> n a b x))\<^sup>2
             \<le> LINT x|lebesgue_on {-pi..pi}. (e / 6) ^ 2"
         proof (rule integral_mono)
           show "integrable (lebesgue_on {-pi..pi}) (\<lambda>x. (g x - (?\<phi> n a b x))\<^sup>2)"
             using g_phi square_integrable_def by auto
           show "integrable (lebesgue_on {-pi..pi}) (\<lambda>x. (e / 6)\<^sup>2)"
             by (intro continuous_intros continuous_imp_integrable_real)
           show "(g x - (?\<phi> n a b x))\<^sup>2 \<le> (e / 6)\<^sup>2" if "x \<in> space (lebesgue_on {-pi..pi})" for x
             using \<open>e > 0\<close> g_phi_less that
             apply (simp add: abs_le_square_iff [symmetric])
             using less_eq_real_def by blast
         qed
         also have "\<dots> \<le> (e / 2)\<^sup>2"
           using \<open>e > 0\<close> pi_less_4 by (auto simp: power2_eq_square)
         finally show "LINT x|lebesgue_on {-pi..pi}. (g x - (?\<phi> n a b x))\<^sup>2 \<le> (e / 2)\<^sup>2" .
       qed (use \<open>e > 0\<close> in auto)
       with norm_fg show ?thesis
         by auto
     qed
     finally show "l2norm {-pi..pi} (\<lambda>x. f x - (?\<phi> n a b x)) < e" .
   qed
 qed
 
 
 proposition Weierstrass_l2_trigonometric_set:
   assumes f: "f square_integrable {-pi..pi}" and "0 < e"
   obtains n a where "l2norm {-pi..pi} (\<lambda>x. f x - (\<Sum>k\<le>n. a k * trigonometric_set k x)) < e"
 proof -
   obtain n a b where lee:
     "l2norm {-pi..pi} (\<lambda>x. f x - (\<Sum>k\<le>n. a k * sin(real k * x) + b k * cos(real k * x))) < e"
     using Weierstrass_l2_trig_polynomial [OF assms] .
   let ?a = "\<lambda>k. if k = 0 then sqrt(2 * pi) * b 0
                 else if even k then sqrt pi * b(k div 2)
                 else if k \<le> 2 * n then sqrt pi * a((Suc k) div 2)
                 else 0"
   show thesis
   proof
     have [simp]: "Suc (i * 2) \<le> n * 2 \<longleftrightarrow> i<n" "{..n} \<inter> {..<n} = {..<n}" for i n
       by auto
     have "(\<Sum>k\<le>n. b k * cos (real k * x)) = (\<Sum>i\<le>n. if i = 0 then b 0 else b i * cos (real i * x))" for x
       by (rule sum.cong) auto
     moreover have "(\<Sum>k\<le>n. a k * sin (real k * x)) = (\<Sum>i\<le>n. (if Suc (2 * i) \<le> 2 * n then sqrt pi * a (Suc i) * sin ((1 + real i) * x) else 0) / sqrt pi)"
                    (is "?lhs = ?rhs") for x
     proof (cases "n=0")
       case False
       then obtain n' where n': "n = Suc n'"
         using not0_implies_Suc by blast
       have "?lhs = (\<Sum>k = Suc 0..n. a k * sin (real k * x))"
         by (simp add: atMost_atLeast0 sum_shift_lb_Suc0_0)
       also have "\<dots> = (\<Sum>i<n. a (Suc i) * sin (x + x * real i))"
       proof (subst sum.reindex_bij_betw [symmetric])
         show "bij_betw Suc {..n'} {Suc 0..n}"
           by (simp add: atMost_atLeast0 n')
         show "(\<Sum>j\<le>n'. a (Suc j) * sin (real (Suc j) * x)) = (\<Sum>i<n. a (Suc i) * sin (x + x * real i))"
         unfolding n' lessThan_Suc_atMost by (simp add: algebra_simps)
       qed
       also have "\<dots> = ?rhs"
         by (simp add: field_simps if_distrib [of "\<lambda>x. x/_"] sum.inter_restrict [where B = "{..<n}", simplified, symmetric] cong: if_cong)
       finally
       show ?thesis .
     qed auto
     ultimately
     have "(\<Sum>k\<le>n. a k * sin(real k * x) + b k * cos(real k * x)) = (\<Sum>k \<le> Suc(2*n). ?a k * trigonometric_set k x)" for x
       unfolding sum.in_pairs_0 trigonometric_set_def
       by (simp add: sum.distrib if_distrib [of "\<lambda>x. x*_"] cong: if_cong)
     with lee show "l2norm {-pi..pi} (\<lambda>x. f x - (\<Sum>k \<le> Suc(2*n). ?a k * trigonometric_set k x)) < e"
       by auto
   qed
 qed
 
 subsection\<open>Convergence wrt the L2 norm of trigonometric Fourier series\<close>
 
 definition Fourier_coefficient
   where "Fourier_coefficient \<equiv> orthonormal_coeff {-pi..pi} trigonometric_set"
 
 lemma Fourier_series_l2:
   assumes "f square_integrable {-pi..pi}"
   shows "(\<lambda>n. l2norm {-pi..pi} (\<lambda>x. f x - (\<Sum>i\<le>n. Fourier_coefficient f i * trigonometric_set i x)))
          \<longlonglongrightarrow> 0"
 proof (clarsimp simp add: lim_sequentially dist_real_def Fourier_coefficient_def)
   let ?h = "\<lambda>n x. (\<Sum>i\<le>n. orthonormal_coeff {-pi..pi} trigonometric_set f i * trigonometric_set i x)"
   show "\<exists>N. \<forall>n\<ge>N. \<bar>l2norm {-pi..pi} (\<lambda>x. f x - ?h n x)\<bar> < e"
     if "0 < e" for e
   proof -
     obtain N a where lte: "l2norm {-pi..pi} (\<lambda>x. f x - (\<Sum>k\<le>N. a k * trigonometric_set k x)) < e"
       using Weierstrass_l2_trigonometric_set by (meson \<open>0 < e\<close> assms)
     show ?thesis
     proof (intro exI allI impI)
       show "\<bar>l2norm {-pi..pi} (\<lambda>x. f x - ?h m x)\<bar> < e"
         if "N \<le> m" for m
       proof -
         have "\<bar>l2norm {-pi..pi} (\<lambda>x. f x - ?h m x)\<bar> = l2norm {-pi..pi} (\<lambda>x. f x - ?h m x)"
         proof (rule abs_of_nonneg)
           show "0 \<le> l2norm {-pi..pi} (\<lambda>x. f x -  ?h m x)"
             apply (intro l2norm_pos_le square_integrable_diff square_integrable_sum square_integrable_lmult
                 square_integrable_trigonometric_set assms, auto)
             done
         qed
         also have "\<dots> \<le> l2norm {-pi..pi} (\<lambda>x. f x - (\<Sum>k\<le>N. a k * trigonometric_set k x))"
         proof -
           have "(\<Sum>i\<le>m. (if i \<le> N then a i else 0) * trigonometric_set i x) = (\<Sum>i\<le>N. a i * trigonometric_set i x)" for x
             using sum.inter_restrict [where A = "{..m}" and B = "{..N}", symmetric] that
             by (force simp: if_distrib [of "\<lambda>x. x * _"] min_absorb2 cong: if_cong)
           moreover
           have "l2norm {-pi..pi} (\<lambda>x. f x - ?h m x)
                 \<le> l2norm {-pi..pi} (\<lambda>x. f x - (\<Sum>i\<le>m. (if i \<le> N then a i else 0) * trigonometric_set i x))"
             using orthonormal_optimal_partial_sum
               [OF orthonormal_system_trigonometric_set square_integrable_trigonometric_set assms]
             by simp
           ultimately show ?thesis
             by simp
         qed
         also have "\<dots> < e"
           by (rule lte)
         finally show ?thesis .
       qed
     qed
   qed
 qed
 
 
 
 subsection\<open>Fourier coefficients go to 0 (weak form of Riemann-Lebesgue)\<close>
 
 lemma trigonometric_set_mul_absolutely_integrable:
   assumes "f absolutely_integrable_on {-pi..pi}"
   shows "(\<lambda>x. trigonometric_set n x * f x) absolutely_integrable_on {-pi..pi}"
 proof (rule absolutely_integrable_bounded_measurable_product_real)
   show "trigonometric_set n \<in> borel_measurable (lebesgue_on {-pi..pi})"
     using square_integrable_def square_integrable_trigonometric_set by blast
   show "bounded (trigonometric_set n ` {-pi..pi})"
     unfolding bounded_iff using pi_gt3 sqrt_pi_ge1
     by (rule_tac x=1 in exI)
       (auto simp: trigonometric_set_def dist_real_def
         intro: order_trans [OF abs_sin_le_one] order_trans [OF abs_cos_le_one])
 qed (auto simp: assms)
 
 
 lemma trigonometric_set_mul_integrable:
    "f absolutely_integrable_on {-pi..pi} \<Longrightarrow> integrable (lebesgue_on {-pi..pi}) (\<lambda>x. trigonometric_set n x * f x)"
   using trigonometric_set_mul_absolutely_integrable
   by (simp add: integrable_restrict_space set_integrable_def)
 
 lemma trigonometric_set_integrable [simp]: "integrable (lebesgue_on {-pi..pi}) (trigonometric_set n)"
   using trigonometric_set_mul_integrable [where f = id]
   by simp (metis absolutely_integrable_imp_integrable fmeasurableD interval_cbox lmeasurable_cbox square_integrable_imp_absolutely_integrable square_integrable_trigonometric_set)
 
 lemma absolutely_integrable_sin_product:
   assumes "f absolutely_integrable_on {-pi..pi}"
   shows "(\<lambda>x. sin(k * x) * f x) absolutely_integrable_on {-pi..pi}"
 proof (rule absolutely_integrable_bounded_measurable_product_real)
   show "(\<lambda>x. sin (k * x)) \<in> borel_measurable (lebesgue_on {-pi..pi})"
     by (metis borel_measurable_integrable integrable_sin_cx mult_commute_abs)
   show "bounded ((\<lambda>x. sin (k * x)) ` {-pi..pi})"
     by (metis (mono_tags, lifting) abs_sin_le_one bounded_iff imageE real_norm_def)
 qed (auto simp: assms)
 
 lemma absolutely_integrable_cos_product:
   assumes "f absolutely_integrable_on {-pi..pi}"
   shows "(\<lambda>x. cos(k * x) * f x) absolutely_integrable_on {-pi..pi}"
 proof (rule absolutely_integrable_bounded_measurable_product_real)
   show "(\<lambda>x. cos (k * x)) \<in> borel_measurable (lebesgue_on {-pi..pi})"
     by (metis borel_measurable_integrable integrable_cos_cx mult_commute_abs)
   show "bounded ((\<lambda>x. cos (k * x)) ` {-pi..pi})"
     by (metis (mono_tags, lifting) abs_cos_le_one bounded_iff imageE real_norm_def)
 qed (auto simp: assms)
 
 lemma
   assumes "f absolutely_integrable_on {-pi..pi}"
   shows Fourier_products_integrable_cos: "integrable (lebesgue_on {-pi..pi}) (\<lambda>x. cos(k * x) * f x)"
   and   Fourier_products_integrable_sin: "integrable (lebesgue_on {-pi..pi}) (\<lambda>x. sin(k * x) * f x)"
   using absolutely_integrable_cos_product absolutely_integrable_sin_product assms
   by (auto simp: integrable_restrict_space set_integrable_def)
 
 
 lemma Riemann_lebesgue_square_integrable:
   assumes "orthonormal_system S w" "\<And>i. w i square_integrable S" "f square_integrable S"
   shows "orthonormal_coeff S w f \<longlonglongrightarrow> 0"
   using Fourier_series_square_summable [OF assms, of UNIV] summable_LIMSEQ_zero by force
 
 proposition Riemann_lebesgue:
   assumes "f absolutely_integrable_on {-pi..pi}"
   shows "Fourier_coefficient f \<longlonglongrightarrow> 0"
   unfolding lim_sequentially
 proof (intro allI impI)
   fix e::real
   assume "e > 0"
   then obtain g where "continuous_on UNIV g" and gabs: "g absolutely_integrable_on {-pi..pi}"
                 and fg_e2: "integral\<^sup>L (lebesgue_on {-pi..pi}) (\<lambda>x. \<bar>f x - g x\<bar>) < e/2"
     using absolutely_integrable_approximate_continuous [OF assms, of "e/2"]
     by (metis (full_types) box_real(2) half_gt_zero_iff lmeasurable_cbox)
   have "g square_integrable {-pi..pi}"
     using \<open>continuous_on UNIV g\<close> continuous_imp_square_integrable continuous_on_subset by blast
   then have "orthonormal_coeff {-pi..pi} trigonometric_set g \<longlonglongrightarrow> 0"
     using Riemann_lebesgue_square_integrable orthonormal_system_trigonometric_set square_integrable_trigonometric_set by blast
   with \<open>e > 0\<close> obtain N where N: "\<And>n. n \<ge> N \<Longrightarrow> \<bar>orthonormal_coeff {-pi..pi} trigonometric_set g n\<bar> < e/2"
     unfolding lim_sequentially by (metis half_gt_zero_iff norm_conv_dist real_norm_def)
   have "\<bar>Fourier_coefficient f n\<bar> < e"
     if "n \<ge> N" for n
   proof -
     have "\<bar>LINT x|lebesgue_on {-pi..pi}. trigonometric_set n x * g x\<bar> < e/2"
       using N [OF \<open>n \<ge> N\<close>] by (auto simp: orthonormal_coeff_def l2product_def)
 
     have "integrable (lebesgue_on {-pi..pi}) (\<lambda>x. trigonometric_set n x * g x)"
       using gabs trigonometric_set_mul_integrable by blast
     moreover have "integrable (lebesgue_on {-pi..pi}) (\<lambda>x. trigonometric_set n x * f x)"
       using assms trigonometric_set_mul_integrable by blast
     ultimately have "\<bar>(LINT x|lebesgue_on {-pi..pi}. trigonometric_set n x * g x) -
                         (LINT x|lebesgue_on {-pi..pi}. trigonometric_set n x * f x)\<bar>
                      = \<bar>(LINT x|lebesgue_on {-pi..pi}. trigonometric_set n x * (g x - f x))\<bar>"
       by (simp add:  algebra_simps flip: Bochner_Integration.integral_diff)
     also have "\<dots> \<le> LINT x|lebesgue_on {-pi..pi}. \<bar>f x - g x\<bar>"
     proof (rule integral_abs_bound_integral)
       show "integrable (lebesgue_on {-pi..pi}) (\<lambda>x. trigonometric_set n x * (g x - f x))"
         by (simp add: gabs assms trigonometric_set_mul_integrable)
       have "(\<lambda>x. f x - g x) absolutely_integrable_on {-pi..pi}"
         using gabs assms by blast
       then show "integrable (lebesgue_on {-pi..pi}) (\<lambda>x. \<bar>f x - g x\<bar>)"
         by (simp add: absolutely_integrable_imp_integrable)
       fix x
       assume "x \<in> space (lebesgue_on {-pi..pi})"
       then have "-pi \<le> x" "x \<le> pi"
         by auto
       have "\<bar>trigonometric_set n x\<bar> \<le> 1"
         using pi_ge_two
         apply (simp add: trigonometric_set_def)
         using sqrt_pi_ge1 abs_sin_le_one order_trans abs_cos_le_one by metis
       then show "\<bar>trigonometric_set n x * (g x - f x)\<bar> \<le> \<bar>f x - g x\<bar>"
         using abs_ge_zero mult_right_mono by (fastforce simp add: abs_mult abs_minus_commute)
     qed
     finally have "\<bar>(LINT x|lebesgue_on {-pi..pi}. trigonometric_set n x * g x) -
            (LINT x|lebesgue_on {-pi..pi}. trigonometric_set n x * f x)\<bar>
           \<le> LINT x|lebesgue_on {-pi..pi}. \<bar>f x - g x\<bar>" .
     then show ?thesis
       using N [OF \<open>n \<ge> N\<close>] fg_e2
       unfolding Fourier_coefficient_def orthonormal_coeff_def l2product_def
       by linarith
   qed
   then show "\<exists>no. \<forall>n\<ge>no. dist (Fourier_coefficient f n) 0 < e"
     by auto
 qed
 
 
 lemma Riemann_lebesgue_sin:
   assumes "f absolutely_integrable_on {-pi..pi}"
   shows "(\<lambda>n. integral\<^sup>L (lebesgue_on {-pi..pi}) (\<lambda>x. sin(real n * x) * f x)) \<longlonglongrightarrow> 0"
   unfolding lim_sequentially
 proof (intro allI impI)
   fix e::real
   assume "e > 0"
   then obtain N where N: "\<And>n. n \<ge> N \<Longrightarrow> \<bar>Fourier_coefficient f n\<bar> < e/4"
     using Riemann_lebesgue [OF assms]
     unfolding lim_sequentially
     by (metis norm_conv_dist real_norm_def zero_less_divide_iff zero_less_numeral)
   have "\<bar>LINT x|lebesgue_on {-pi..pi}. sin (real n * x) * f x\<bar> < e" if "n > N" for n
     using that
   proof (induction n)
     case (Suc n)
     have "\<bar>Fourier_coefficient f(Suc (2 * n))\<bar> < e/4"
       using N Suc.prems by auto
     then have "\<bar>LINT x|lebesgue_on {-pi..pi}. sin ((1 + real n) * x) * f x\<bar> < sqrt pi * e / 4"
       by (simp add: Fourier_coefficient_def orthonormal_coeff_def
           trigonometric_set_def l2product_def field_simps)
     also have "\<dots> \<le> e"
       using \<open>0 < e\<close> pi_less_4 real_sqrt_less_mono by (fastforce simp add: field_simps)
     finally show ?case
       by simp
   qed auto
   then show "\<exists>no. \<forall>n\<ge>no. dist (LINT x|lebesgue_on {-pi..pi}. sin (real n * x) * f x) 0 < e"
     by (metis (full_types) le_neq_implies_less less_add_same_cancel2 less_trans norm_conv_dist real_norm_def zero_less_one)
 qed
 
 lemma Riemann_lebesgue_cos:
   assumes "f absolutely_integrable_on {-pi..pi}"
   shows "(\<lambda>n. integral\<^sup>L (lebesgue_on {-pi..pi}) (\<lambda>x. cos(real n * x) * f x)) \<longlonglongrightarrow> 0"
   unfolding lim_sequentially
 proof (intro allI impI)
   fix e::real
   assume "e > 0"
   then obtain N where N: "\<And>n. n \<ge> N \<Longrightarrow> \<bar>Fourier_coefficient f n\<bar> < e/4"
     using Riemann_lebesgue [OF assms]
     unfolding lim_sequentially
     by (metis norm_conv_dist real_norm_def zero_less_divide_iff zero_less_numeral)
   have "\<bar>LINT x|lebesgue_on {-pi..pi}. cos (real n * x) * f x\<bar> < e" if "n > N" for n
     using that
   proof (induction n)
     case (Suc n)
     have eq: "(x * 2 + x * (real n * 2)) / 2 = x + x * (real n)" for x
       by simp
     have "\<bar>Fourier_coefficient f(2*n + 2)\<bar> < e/4"
       using N Suc.prems by auto
     then have "\<bar>LINT x|lebesgue_on {-pi..pi}. f x * cos (x + x * (real n))\<bar> < sqrt pi * e / 4"
       by (simp add: Fourier_coefficient_def orthonormal_coeff_def
           trigonometric_set_def l2product_def field_simps eq)
     also have "\<dots> \<le> e"
       using \<open>0 < e\<close> pi_less_4 real_sqrt_less_mono by (fastforce simp add: field_simps)
     finally show ?case
       by (simp add: field_simps)
   qed auto
   then show "\<exists>no. \<forall>n\<ge>no. dist (LINT x|lebesgue_on {-pi..pi}. cos (real n * x) * f x) 0 < e"
     by (metis (full_types) le_neq_implies_less less_add_same_cancel2 less_trans norm_conv_dist real_norm_def zero_less_one)
 qed
 
 
 lemma Riemann_lebesgue_sin_half:
   assumes "f absolutely_integrable_on {-pi..pi}"
   shows "(\<lambda>n. LINT x|lebesgue_on {-pi..pi}. sin ((real n + 1/2) * x) * f x) \<longlonglongrightarrow> 0"
 proof (simp add: algebra_simps sin_add)
   let ?INT = "integral\<^sup>L (lebesgue_on {-pi..pi})"
   let ?f = "(\<lambda>n. ?INT (\<lambda>x. sin(n * x) * cos(1/2 * x) * f x) +
                  ?INT (\<lambda>x. cos(n * x) * sin(1/2 * x) * f x))"
   show "(\<lambda>n. ?INT (\<lambda>x. f x * (cos (x * real n) * sin (x/2)) + f x * (sin (x * real n) * cos (x/2)))) \<longlonglongrightarrow> 0"
   proof (rule Lim_transform_eventually)
     have sin: "(\<lambda>x. sin (1/2 * x) * f x) absolutely_integrable_on {-pi..pi}"
       by (intro absolutely_integrable_sin_product assms)
     have cos: "(\<lambda>x. cos (1/2 * x) * f x) absolutely_integrable_on {-pi..pi}"
       by (intro absolutely_integrable_cos_product assms)
     show "\<forall>\<^sub>F n in sequentially. ?f n = ?INT (\<lambda>x. f x * (cos (x * real n) * sin (x/2)) + f x * (sin (x * real n) * cos (x/2)))"
       unfolding mult.assoc
       apply (rule eventuallyI)
       apply (subst Bochner_Integration.integral_add [symmetric])
         apply (safe intro!: cos absolutely_integrable_sin_product sin absolutely_integrable_cos_product absolutely_integrable_imp_integrable)
         apply (auto simp: algebra_simps)
       done
     have "?f \<longlonglongrightarrow> 0 + 0"
       unfolding mult.assoc
       by (intro tendsto_add Riemann_lebesgue_sin Riemann_lebesgue_cos sin cos)
     then show "?f \<longlonglongrightarrow> (0::real)" by simp
   qed
 qed
 
 
 lemma Fourier_sum_limit_pair:
   assumes "f absolutely_integrable_on {-pi..pi}"
   shows "(\<lambda>n. \<Sum>k\<le>2 * n. Fourier_coefficient f k * trigonometric_set k t) \<longlonglongrightarrow> l
      \<longleftrightarrow> (\<lambda>n. \<Sum>k\<le>n. Fourier_coefficient f k * trigonometric_set k t) \<longlonglongrightarrow> l"
         (is "?lhs = ?rhs")
 proof
   assume L: ?lhs
   show ?rhs
     unfolding lim_sequentially dist_real_def
   proof (intro allI impI)
     fix e::real
     assume "e > 0"
     then obtain N1 where N1: "\<And>n. n \<ge> N1 \<Longrightarrow> \<bar>Fourier_coefficient f n\<bar> < e/2"
       using Riemann_lebesgue [OF assms] unfolding lim_sequentially
       by (metis norm_conv_dist real_norm_def zero_less_divide_iff zero_less_numeral)
     obtain N2 where N2: "\<And>n. n \<ge> N2 \<Longrightarrow> \<bar>(\<Sum>k\<le>2 * n. Fourier_coefficient f k * trigonometric_set k t) - l\<bar> < e/2"
       using L unfolding lim_sequentially dist_real_def
       by (meson \<open>0 < e\<close> half_gt_zero_iff)
     show "\<exists>no. \<forall>n\<ge>no. \<bar>(\<Sum>k\<le>n. Fourier_coefficient f k * trigonometric_set k t) - l\<bar> < e"
     proof (intro exI allI impI)
       fix n
       assume n: "N1 + 2*N2 + 1 \<le> n"
       then have "n \<ge> N1" "n \<ge> N2" "n div 2 \<ge> N2"
         by linarith+
       consider "n = 2 * (n div 2)" | "n = Suc(2 * (n div 2))"
         by linarith
       then show "\<bar>(\<Sum>k\<le>n. Fourier_coefficient f k * trigonometric_set k t) - l\<bar> < e"
       proof cases
         case 1
         show ?thesis
           apply (subst 1)
           using N2 [OF \<open>n div 2 \<ge> N2\<close>] by linarith
       next
         case 2
         have "\<bar>(\<Sum>k\<le>2 * (n div 2). Fourier_coefficient f k * trigonometric_set k t) - l\<bar> < e/2"
           using N2 [OF \<open>n div 2 \<ge> N2\<close>] by linarith
         moreover have "\<bar>Fourier_coefficient f(Suc (2 * (n div 2))) * trigonometric_set (Suc (2 * (n div 2))) t\<bar> < (e/2) * 1"
         proof -
           have "\<bar>trigonometric_set (Suc (2 * (n div 2))) t\<bar> \<le> 1"
             apply (simp add: trigonometric_set)
             using abs_sin_le_one sqrt_pi_ge1 by (blast intro: order_trans)
           moreover have "\<bar>Fourier_coefficient f(Suc (2 * (n div 2)))\<bar> < e/2"
             using N1 \<open>N1 \<le> n\<close> by auto
           ultimately show ?thesis
             unfolding abs_mult
             by (meson abs_ge_zero le_less_trans mult_left_mono real_mult_less_iff1 zero_less_one)
         qed
         ultimately show ?thesis
           apply (subst 2)
           unfolding sum.atMost_Suc by linarith
       qed
     qed
   qed
 next
   assume ?rhs then show ?lhs
     unfolding lim_sequentially
     by (auto simp: elim!: imp_forward ex_forward)
 qed
 
 
 subsection\<open>Express Fourier sum in terms of the special expansion at the origin\<close>
 
 lemma Fourier_sum_0:
   "(\<Sum>k \<le> n. Fourier_coefficient f k * trigonometric_set k 0) =
      (\<Sum>k \<le> n div 2. Fourier_coefficient f(2*k) * trigonometric_set (2*k) 0)"
   (is "?lhs = ?rhs")
 proof -
   have "?lhs = (\<Sum>k = 2 * 0.. Suc (2 * (n div 2)). Fourier_coefficient f k * trigonometric_set k 0)"
   proof (rule sum.mono_neutral_left)
     show "\<forall>i\<in>{2 * 0..Suc (2 * (n div 2))} - {..n}. Fourier_coefficient f i * trigonometric_set i 0 = 0"
     proof clarsimp
       fix i
       assume "i \<le> Suc (2 * (n div 2))" "\<not> i \<le> n"
 			then have "i = Suc n" "even n"
 			  by presburger+
 			moreover
 			assume "trigonometric_set i 0 \<noteq> 0"
 			ultimately
 			show "Fourier_coefficient f i = 0"
 				by (simp add: trigonometric_set_def)
 		qed
 	qed auto
   also have "\<dots> = ?rhs"
  	  unfolding sum.in_pairs by (simp add: trigonometric_set atMost_atLeast0)
   finally show ?thesis .
 qed
 
 
 lemma Fourier_sum_0_explicit:
   "(\<Sum>k\<le>n. Fourier_coefficient f k * trigonometric_set k 0)
     = (Fourier_coefficient f 0 / sqrt 2 + (\<Sum>k = 1..n div 2. Fourier_coefficient f(2*k))) / sqrt pi"
   (is "?lhs = ?rhs")
 proof -
   have "(\<Sum>k=0..n. Fourier_coefficient f k * trigonometric_set k 0)
         = Fourier_coefficient f 0 * trigonometric_set 0 0 + (\<Sum>k = 1..n. Fourier_coefficient f k * trigonometric_set k 0)"
     by (simp add: Fourier_sum_0 sum.atLeast_Suc_atMost)
   also have "\<dots> = ?rhs"
   proof -
     have "Fourier_coefficient f 0 * trigonometric_set 0 0 = Fourier_coefficient f 0 / (sqrt 2 * sqrt pi)"
       by (simp add: real_sqrt_mult trigonometric_set(1))
     moreover have "(\<Sum>k = Suc 0..n. Fourier_coefficient f k * trigonometric_set k 0) = (\<Sum>k = Suc 0..n div 2. Fourier_coefficient f(2*k)) / sqrt pi"
     proof (induction n)
       case (Suc n)
       show ?case
         by (simp add: Suc) (auto simp: trigonometric_set_def field_simps)
     qed auto
     ultimately show ?thesis
       by (simp add: add_divide_distrib)
   qed
   finally show ?thesis
     by (simp add: atMost_atLeast0)
 qed
 
 lemma Fourier_sum_0_integrals:
   assumes "f absolutely_integrable_on {-pi..pi}"
   shows "(\<Sum>k\<le>n. Fourier_coefficient f k * trigonometric_set k 0) =
           (integral\<^sup>L (lebesgue_on {-pi..pi}) f / 2 +
            (\<Sum>k = Suc 0..n div 2. integral\<^sup>L (lebesgue_on {-pi..pi}) (\<lambda>x. cos(k * x) * f x))) / pi"
 proof -
   have "integral\<^sup>L (lebesgue_on {-pi..pi}) f / (sqrt (2 * pi) * sqrt 2 * sqrt pi) = integral\<^sup>L (lebesgue_on {-pi..pi}) f / (2 * pi)"
     by (simp add: algebra_simps real_sqrt_mult)
   moreover have "(\<Sum>k = Suc 0..n div 2. LINT x|lebesgue_on {-pi..pi}. trigonometric_set (2*k) x * f x) / sqrt pi
                = (\<Sum>k = Suc 0..n div 2. LINT x|lebesgue_on {-pi..pi}. cos (k * x) * f x) / pi"
     by (simp add: trigonometric_set_def sum_divide_distrib)
   ultimately show ?thesis
     unfolding Fourier_sum_0_explicit
     by (simp add: Fourier_coefficient_def orthonormal_coeff_def l2product_def trigonometric_set add_divide_distrib)
 qed
 
 
 lemma Fourier_sum_0_integral:
   assumes "f absolutely_integrable_on {-pi..pi}"
   shows "(\<Sum>k\<le>n. Fourier_coefficient f k * trigonometric_set k 0) =
        integral\<^sup>L (lebesgue_on {-pi..pi}) (\<lambda>x. (1/2 + (\<Sum>k = Suc 0..n div 2. cos(k * x))) * f x) / pi"
 proof -
   note * = Fourier_products_integrable_cos [OF assms]
   have "integrable (lebesgue_on {-pi..pi}) (\<lambda>x. \<Sum>n = Suc 0..n div 2. f x * cos (x * real n))"
     using * by (force simp: mult_ac)
   moreover have "integral\<^sup>L (lebesgue_on {-pi..pi}) f / 2 + (\<Sum>k = Suc 0..n div 2. LINT x|lebesgue_on {-pi..pi}. f x * cos (x * real k))
                = (LINT x|lebesgue_on {-pi..pi}. f x / 2) + (LINT x|lebesgue_on {-pi..pi}. (\<Sum>n = Suc 0..n div 2. f x * cos (x * real n)))"
   proof (subst Bochner_Integration.integral_sum)
     show "integrable (lebesgue_on {-pi..pi}) (\<lambda>x. f x * cos (x * real i))"
       if "i \<in> {Suc 0..n div 2}" for i
       using that * [of i] by (auto simp: Bochner_Integration.integral_sum mult_ac)
   qed auto
   ultimately show ?thesis
     using Fourier_products_integrable_cos [OF assms] absolutely_integrable_imp_integrable [OF assms]
     by (simp add: Fourier_sum_0_integrals sum_distrib_left assms algebra_simps)
 qed
 
 
 subsection\<open>How Fourier coefficients behave under addition etc\<close>
 
 lemma Fourier_coefficient_add:
   assumes "f absolutely_integrable_on {-pi..pi}" "g absolutely_integrable_on {-pi..pi}"
   shows "Fourier_coefficient (\<lambda>x. f x + g x) i =
                 Fourier_coefficient f i + Fourier_coefficient g i"
   using assms trigonometric_set_mul_integrable
   by (simp add: Fourier_coefficient_def orthonormal_coeff_def l2product_def distrib_left)
 
 lemma Fourier_coefficient_minus:
   assumes "f absolutely_integrable_on {-pi..pi}"
   shows "Fourier_coefficient (\<lambda>x. - f x) i = - Fourier_coefficient f i"
   using assms trigonometric_set_mul_integrable
   by (simp add: Fourier_coefficient_def orthonormal_coeff_def l2product_def)
 
 lemma Fourier_coefficient_diff:
   assumes f: "f absolutely_integrable_on {-pi..pi}" and g: "g absolutely_integrable_on {-pi..pi}"
   shows "Fourier_coefficient (\<lambda>x. f x - g x) i = Fourier_coefficient f i - Fourier_coefficient g i"
 proof -
   have mg: "(\<lambda>x. - g x) absolutely_integrable_on {-pi..pi}"
     using g by (simp add: absolutely_integrable_measurable_real)
   show ?thesis
     using Fourier_coefficient_add [OF f mg] Fourier_coefficient_minus [OF g] by simp
 qed
 
 lemma Fourier_coefficient_const:
    "Fourier_coefficient (\<lambda>x. c) i = (if i = 0 then c * sqrt(2 * pi) else 0)"
   by (auto simp: Fourier_coefficient_def orthonormal_coeff_def l2product_def trigonometric_set_def divide_simps)
 
 lemma Fourier_offset_term:
   fixes f :: "real \<Rightarrow> real"
   assumes f: "f absolutely_integrable_on {-pi..pi}" and periodic: "\<And>x. f(x + 2*pi) = f x"
   shows  "Fourier_coefficient (\<lambda>x. f(x+t)) (2 * n + 2) * trigonometric_set (2 * n + 2) 0
         = Fourier_coefficient f(2 * n+1) * trigonometric_set (2 * n+1) t
         + Fourier_coefficient f(2 * n + 2) * trigonometric_set (2 * n + 2) t"
 proof -
   have eq: "(2 + 2 * real n) * x / 2 = (1 + real n) * x" for x
     by (simp add: divide_simps)
   have 1: "integrable (lebesgue_on {-pi..pi}) (\<lambda>x. f x * (cos (x + x * n) * cos (t + t * n)))"
     using Fourier_products_integrable_cos [of f "Suc n"] f
     by (simp add: algebra_simps) (simp add: mult.assoc [symmetric])
   have 2: "integrable (lebesgue_on {-pi..pi}) (\<lambda>x. f x * (sin (x + x * n) * sin (t + t * n)))"
     using Fourier_products_integrable_sin [of f "Suc n"] f
     by (simp add: algebra_simps) (simp add: mult.assoc [symmetric])
   have "has_bochner_integral (lebesgue_on {-pi..pi}) (\<lambda>x. cos (real (Suc n) * (x + t - t)) * f(x + t))
                  (LINT x|lebesgue_on {-pi..pi}. cos (real (Suc n) * (x - t)) * f x)"
   proof (rule has_integral_periodic_offset)
     have "(\<lambda>x. cos (real (Suc n) * (x - t)) * f x) absolutely_integrable_on {-pi..pi}"
     proof (rule absolutely_integrable_bounded_measurable_product_real)
       show "(\<lambda>x. cos (real (Suc n) * (x - t))) \<in> borel_measurable (lebesgue_on {-pi..pi})"
         by (intro continuous_imp_measurable_on_sets_lebesgue continuous_intros) auto
       show "bounded ((\<lambda>x. cos (real (Suc n) * (x - t))) ` {-pi..pi})"
         by (rule boundedI [where B=1]) auto
     qed (use f in auto)
     then show "has_bochner_integral (lebesgue_on {-pi..pi}) (\<lambda>x. cos ((Suc n) * (x - t)) * f x) (LINT x|lebesgue_on {-pi..pi}. cos (real (Suc n) * (x - t)) * f x)"
       by (simp add: has_bochner_integral_integrable integrable_restrict_space set_integrable_def)
   next
     fix x
     show "cos (real (Suc n) * (x + (pi - - pi) - t)) * f(x + (pi - - pi)) = cos (real (Suc n) * (x - t)) * f x"
       using periodic cos.plus_of_nat [of "(Suc n) * (x - t)" "Suc n"] by (simp add: algebra_simps)
   qed
   then have "has_bochner_integral (lebesgue_on {-pi..pi}) (\<lambda>x. cos (real (Suc n) * x) * f(x + t))
                  (LINT x|lebesgue_on {-pi..pi}. cos (real (Suc n) * (x - t)) * f x)"
     by simp
   then have "LINT x|lebesgue_on {-pi..pi}. cos ((Suc n) * x) * f(x+t)
            = LINT x|lebesgue_on {-pi..pi}. cos ((Suc n) * (x - t)) * f x"
     using has_bochner_integral_integral_eq by blast
   also have "\<dots> = LINT x|lebesgue_on {-pi..pi}. cos ((Suc n) * x - ((Suc n) * t)) * f x"
     by (simp add: algebra_simps)
   also have "\<dots> = cos ((Suc n) * t) * (LINT x|lebesgue_on {-pi..pi}. cos ((Suc n) * x) * f x)
                  + sin ((Suc n) * t) * (LINT x|lebesgue_on {-pi..pi}. sin ((Suc n) * x) * f x)"
     by (simp add: cos_diff algebra_simps 1 2 flip: integral_mult_left_zero integral_mult_right_zero)
   finally have "LINT x|lebesgue_on {-pi..pi}. cos ((Suc n) * x) * f(x+t)
         = cos ((Suc n) * t) * (LINT x|lebesgue_on {-pi..pi}. cos ((Suc n) * x) * f x)
         + sin ((Suc n) * t) * (LINT x|lebesgue_on {-pi..pi}. sin ((Suc n) * x) * f x)" .
   then show ?thesis
     unfolding Fourier_coefficient_def orthonormal_coeff_def trigonometric_set_def
     by (simp add: eq) (simp add: divide_simps algebra_simps l2product_def)
 qed
 
 
 lemma Fourier_sum_offset:
   fixes f :: "real \<Rightarrow> real"
   assumes f: "f absolutely_integrable_on {-pi..pi}" and periodic: "\<And>x. f(x + 2*pi) = f x"
   shows  "(\<Sum>k\<le>2*n. Fourier_coefficient f k * trigonometric_set k t) =
           (\<Sum>k\<le>2*n. Fourier_coefficient (\<lambda>x. f(x+t)) k * trigonometric_set k 0)" (is "?lhs = ?rhs")
 proof -
   have "?lhs = Fourier_coefficient f 0 * trigonometric_set 0 t + (\<Sum>k = Suc 0..2*n. Fourier_coefficient f k * trigonometric_set k t)"
     by (simp add: atMost_atLeast0 sum.atLeast_Suc_atMost)
   moreover have "(\<Sum>k = Suc 0..2*n. Fourier_coefficient f k * trigonometric_set k t) =
                  (\<Sum>k = Suc 0..2*n. Fourier_coefficient (\<lambda>x. f(x+t)) k * trigonometric_set k 0)"
   proof (cases n)
     case (Suc n')
     have "(\<Sum>k = Suc 0..2 * n. Fourier_coefficient f k * trigonometric_set k t)
         = (\<Sum>k = Suc 0.. Suc (Suc (2 * n')). Fourier_coefficient f k * trigonometric_set k t)"
       by (simp add: Suc)
     also have "\<dots> = (\<Sum>k \<le> Suc (2 * n'). Fourier_coefficient f(Suc k) * trigonometric_set (Suc k) t)"
       unfolding atMost_atLeast0 using sum.shift_bounds_cl_Suc_ivl by blast
     also have "\<dots> = (\<Sum>k \<le> Suc (2 * n'). Fourier_coefficient (\<lambda>x. f(x+t)) (Suc k) * trigonometric_set (Suc k) 0)"
       unfolding sum.in_pairs_0
     proof (rule sum.cong [OF refl])
       show "Fourier_coefficient f(Suc (2 * k)) * trigonometric_set (Suc (2 * k)) t + Fourier_coefficient f(Suc (Suc (2 * k))) * trigonometric_set (Suc (Suc (2 * k))) t = Fourier_coefficient (\<lambda>x. f(x + t)) (Suc (2 * k)) * trigonometric_set (Suc (2 * k)) 0 + Fourier_coefficient (\<lambda>x. f(x + t)) (Suc (Suc (2 * k))) * trigonometric_set (Suc (Suc (2 * k))) 0"
         if "k \<in> {..n'}" for k
         using that Fourier_offset_term [OF assms, of t ] by (auto simp: trigonometric_set_def)
     qed
     also have "\<dots> = (\<Sum>k = Suc 0..Suc (Suc (2 * n')). Fourier_coefficient (\<lambda>x. f(x+t)) k * trigonometric_set k 0)"
       by (simp only: atMost_atLeast0 sum.shift_bounds_cl_Suc_ivl)
     also have "\<dots> = (\<Sum>k = Suc 0..2*n. Fourier_coefficient (\<lambda>x. f(x+t)) k * trigonometric_set k 0)"
       by (simp add: Suc)
     finally show ?thesis .
   qed auto
   moreover have "?rhs
            = Fourier_coefficient (\<lambda>x. f(x+t)) 0 * trigonometric_set 0 0 + (\<Sum>k = Suc 0..2*n. Fourier_coefficient (\<lambda>x. f(x+t)) k * trigonometric_set k 0)"
     by (simp add: atMost_atLeast0 sum.atLeast_Suc_atMost)
   moreover have "Fourier_coefficient f 0 * trigonometric_set 0 t
                = Fourier_coefficient (\<lambda>x. f(x+t)) 0 * trigonometric_set 0 0"
     by (simp add: Fourier_coefficient_def orthonormal_coeff_def trigonometric_set_def l2product_def integral_periodic_offset periodic)
   ultimately show ?thesis
     by simp
 qed
 
 
 lemma Fourier_sum_offset_unpaired:
   fixes f :: "real \<Rightarrow> real"
   assumes f: "f absolutely_integrable_on {-pi..pi}" and periodic: "\<And>x. f(x + 2*pi) = f x"
   shows  "(\<Sum>k\<le>2*n. Fourier_coefficient f k * trigonometric_set k t) =
           (\<Sum>k\<le>n. Fourier_coefficient (\<lambda>x. f(x+t)) (2*k) * trigonometric_set (2*k) 0)"
   (is "?lhs = ?rhs")
 proof -
   have "?lhs = (\<Sum>k\<le>n. Fourier_coefficient (\<lambda>x. f(x+t)) (2*k) *  trigonometric_set (2*k) 0 +
                    Fourier_coefficient (\<lambda>x. f(x+t)) (Suc (2*k)) * trigonometric_set (Suc (2*k)) 0)"
     apply (simp add: assms Fourier_sum_offset)
     apply (subst sum.in_pairs_0 [symmetric])
     by (simp add: trigonometric_set)
   also have "\<dots> = ?rhs"
     by (auto simp: trigonometric_set)
   finally show ?thesis .
 qed
 
 subsection\<open>Express partial sums using Dirichlet kernel\<close>
 
 definition Dirichlet_kernel
   where "Dirichlet_kernel \<equiv>
            \<lambda>n x. if x = 0 then real n + 1/2
                           else sin((real n + 1/2) * x) / (2 * sin(x/2))"
 
 lemma Dirichlet_kernel_0 [simp]:
    "\<bar>x\<bar> < 2 * pi \<Longrightarrow> Dirichlet_kernel 0 x = 1/2"
   by (auto simp: Dirichlet_kernel_def sin_zero_iff)
 
 lemma Dirichlet_kernel_minus [simp]: "Dirichlet_kernel n (-x) = Dirichlet_kernel n x"
   by (auto simp: Dirichlet_kernel_def)
 
 
 lemma Dirichlet_kernel_continuous_strong:
    "continuous_on {-(2 * pi)<..<2 * pi} (Dirichlet_kernel n)"
 proof -
   have "isCont (Dirichlet_kernel n) z" if "-(2 * pi) < z" "z < 2 * pi" for z
   proof (cases "z=0")
     case True
     have *: "(\<lambda>x. sin ((n + 1/2) * x) / (2 * sin (x/2))) \<midarrow>0 \<rightarrow> real n + 1/2"
       by real_asymp
     show ?thesis
       unfolding Dirichlet_kernel_def continuous_at True
       apply (rule Lim_transform_eventually [where f = "\<lambda>x. sin((n + 1/2) * x) / (2 * sin(x/2))"])
        apply (auto simp: * eventually_at_filter)
       done
   next
     case False
     have "continuous (at z) (\<lambda>x. sin((real n + 1/2) * x) / (2 * sin(x/2)))"
       using that False by (intro continuous_intros) (auto simp: sin_zero_iff)
     moreover have "\<forall>\<^sub>F x in nhds z. x \<noteq> 0"
       using False t1_space_nhds by blast
     ultimately show ?thesis
       using False
       by (force simp: Dirichlet_kernel_def continuous_at eventually_at_filter elim: Lim_transform_eventually)
   qed
   then show ?thesis
     by (simp add: continuous_on_eq_continuous_at)
 qed
 
 lemma Dirichlet_kernel_continuous: "continuous_on {-pi..pi} (Dirichlet_kernel n)"
   apply (rule continuous_on_subset [OF Dirichlet_kernel_continuous_strong], clarsimp)
   using pi_gt_zero by linarith
 
 
 lemma absolutely_integrable_mult_Dirichlet_kernel:
   assumes "f absolutely_integrable_on {-pi..pi}"
   shows "(\<lambda>x. Dirichlet_kernel n x * f x) absolutely_integrable_on {-pi..pi}"
 proof (rule absolutely_integrable_bounded_measurable_product_real)
   show "Dirichlet_kernel n \<in> borel_measurable (lebesgue_on {-pi..pi})"
     by (simp add: Dirichlet_kernel_continuous continuous_imp_measurable_on_sets_lebesgue)
   have "compact (Dirichlet_kernel n ` {-pi..pi})"
     by (auto simp: compact_continuous_image [OF Dirichlet_kernel_continuous])
   then show "bounded (Dirichlet_kernel n ` {-pi..pi})"
     using compact_imp_bounded by blast
 qed (use assms in auto)
 
 
 lemma cosine_sum_lemma:
    "(1/2 + (\<Sum>k = Suc 0..n. cos(real k * x))) * sin(x/2) =  sin((real n + 1/2) * x) / 2"
 proof -
   consider "n=0" | "n \<ge> 1" by linarith
   then show ?thesis
   proof cases
     case 2
     then have "(\<Sum>k = Suc 0..n. (sin (real k * x + x/2) * 2 - sin (real k * x - x/2) * 2) / 2)
         = sin (real n * x + x/2) - sin (x/2)"
     proof (induction n)
       case (Suc n)
       show ?case
       proof (cases "n=0")
         case False
         with Suc show ?thesis
           by (simp add: divide_simps algebra_simps)
       qed auto
     qed auto
     then show ?thesis
       by (simp add: distrib_right sum_distrib_right cos_times_sin)
   qed auto
 qed
 
 
 lemma Dirichlet_kernel_cosine_sum:
   assumes "\<bar>x\<bar> < 2 * pi"
   shows "Dirichlet_kernel n x = 1/2 + (\<Sum>k = Suc 0..n. cos(real k * x))"
 proof -
   have "sin ((real n + 1/2) * x) / (2 * sin (x/2)) = 1/2 + (\<Sum>k = Suc 0..n. cos (real k * x))"
     if "x \<noteq> 0"
   proof -
     have "sin(x/2) \<noteq> 0"
       using assms that by (auto simp: sin_zero_iff)
     then show ?thesis
      using cosine_sum_lemma [of x n] by (simp add: divide_simps)
   qed
   then show ?thesis
     by (auto simp: Dirichlet_kernel_def)
 qed
 
 lemma integrable_Dirichlet_kernel: "integrable (lebesgue_on {-pi..pi}) (Dirichlet_kernel n)"
   using Dirichlet_kernel_continuous continuous_imp_integrable_real by blast
 
 lemma integral_Dirichlet_kernel [simp]:
   "integral\<^sup>L (lebesgue_on {-pi..pi}) (Dirichlet_kernel n) = pi"
 proof -
   have "integral\<^sup>L (lebesgue_on {-pi..pi}) (Dirichlet_kernel n) = LINT x|lebesgue_on {-pi..pi}. 1/2 + (\<Sum>k = Suc 0..n. cos (k * x))"
     using pi_ge_two
     by (force intro: Bochner_Integration.integral_cong [OF refl Dirichlet_kernel_cosine_sum])
   also have "\<dots> = (LINT x|lebesgue_on {-pi..pi}. 1/2) + (LINT x|lebesgue_on {-pi..pi}. (\<Sum>k = Suc 0..n. cos (k * x)))"
   proof (rule Bochner_Integration.integral_add)
     show "integrable (lebesgue_on {-pi..pi}) (\<lambda>x. \<Sum>k = Suc 0..n. cos (real k * x))"
       by (rule Bochner_Integration.integrable_sum) (metis integrable_cos_cx mult_commute_abs)
   qed auto
   also have "\<dots> = pi"
   proof -
     have "\<And>i. \<lbrakk>Suc 0 \<le> i; i \<le> n\<rbrakk>
          \<Longrightarrow> integrable (lebesgue_on {-pi..pi}) (\<lambda>x. cos (real i * x))"
       by (metis integrable_cos_cx mult_commute_abs)
     then have "LINT x|lebesgue_on {-pi..pi}. (\<Sum>k = Suc 0..n. cos (real k * x)) = 0"
       by (simp add: Bochner_Integration.integral_sum)
     then show ?thesis
       by auto
   qed
   finally show ?thesis .
 qed
 
 lemma integral_Dirichlet_kernel_half [simp]:
   "integral\<^sup>L (lebesgue_on {0..pi}) (Dirichlet_kernel n) = pi/2"
 proof -
   have "integral\<^sup>L (lebesgue_on {- pi..0}) (Dirichlet_kernel n) +
         integral\<^sup>L (lebesgue_on {0..pi}) (Dirichlet_kernel n) = pi"
     using integral_combine [OF integrable_Dirichlet_kernel] integral_Dirichlet_kernel by simp
   moreover have "integral\<^sup>L (lebesgue_on {- pi..0}) (Dirichlet_kernel n) = integral\<^sup>L (lebesgue_on {0..pi}) (Dirichlet_kernel n)"
     using integral_reflect_real [of pi 0 "Dirichlet_kernel n"] by simp
   ultimately show ?thesis
     by simp
 qed
 
 
 lemma Fourier_sum_offset_Dirichlet_kernel:
   assumes f: "f absolutely_integrable_on {-pi..pi}" and periodic: "\<And>x. f(x + 2*pi) = f x"
   shows
    "(\<Sum>k\<le>2*n. Fourier_coefficient f k * trigonometric_set k t) =
             integral\<^sup>L (lebesgue_on {-pi..pi}) (\<lambda>x. Dirichlet_kernel n x * f(x+t)) / pi"
   (is "?lhs = ?rhs")
 proof -
   have ft: "(\<lambda>x. f(x+t)) absolutely_integrable_on {-pi..pi}"
     using absolutely_integrable_periodic_offset [OF f, of t] periodic by simp
   have "?lhs = (\<Sum>k=0..n. Fourier_coefficient (\<lambda>x. f(x+t)) (2*k) * trigonometric_set (2*k) 0)"
     using Fourier_sum_offset_unpaired assms atMost_atLeast0 by auto
   also have "\<dots> = Fourier_coefficient (\<lambda>x. f(x+t)) 0 / sqrt (2 * pi)
                  + (\<Sum>k = Suc 0..n. Fourier_coefficient (\<lambda>x. f(x+t)) (2*k) / sqrt pi)"
     by (simp add: sum.atLeast_Suc_atMost trigonometric_set_def)
   also have "\<dots> = (LINT x|lebesgue_on {-pi..pi}. f(x+t)) / (2 * pi) +
                    (\<Sum>k = Suc 0..n. (LINT x|lebesgue_on {-pi..pi}. cos (real k * x) * f(x+t)) / pi)"
     by (simp add: Fourier_coefficient_def orthonormal_coeff_def trigonometric_set_def l2product_def)
   also have "\<dots> = LINT x|lebesgue_on {-pi..pi}.
                      f(x+t) / (2 * pi) + (\<Sum>k = Suc 0..n. (cos (real k * x) * f(x+t)) / pi)"
     using Fourier_products_integrable_cos [OF ft] absolutely_integrable_imp_integrable [OF ft] by simp
   also have "\<dots> = (LINT x|lebesgue_on {-pi..pi}.
                      f(x+t) / 2 + (\<Sum>k = Suc 0..n. cos (real k * x) * f(x+t))) / pi"
     by (simp add: divide_simps sum_distrib_right mult.assoc)
   also have "\<dots> = ?rhs"
   proof -
     have "LINT x|lebesgue_on {-pi..pi}. f(x + t) / 2 + (\<Sum>k = Suc 0..n. cos (real k * x) * f(x + t))
         = LINT x|lebesgue_on {-pi..pi}. Dirichlet_kernel n x * f(x + t)"
     proof -
       have eq: "f(x+t) / 2 + (\<Sum>k = Suc 0..n. cos (real k * x) * f(x + t))
               = Dirichlet_kernel n x * f(x + t)" if "- pi \<le> x" "x \<le> pi" for x
       proof (cases "x = 0")
         case False
         then have "sin (x/2) \<noteq> 0"
           using that by (auto simp: sin_zero_iff)
         then have "f(x + t) * (1/2 + (\<Sum>k = Suc 0..n. cos(real k * x))) = f(x + t) * sin((real n + 1/2) * x) / 2 / sin(x/2)"
           using cosine_sum_lemma [of x n] by (simp add: divide_simps)
         then show ?thesis
           by (simp add: Dirichlet_kernel_def False field_simps sum_distrib_left)
       qed (simp add: Dirichlet_kernel_def algebra_simps)
       show ?thesis
         by (rule Bochner_Integration.integral_cong [OF refl]) (simp add: eq)
     qed
     then show ?thesis by simp
   qed
   finally show ?thesis .
 qed
 
 
 lemma Fourier_sum_limit_Dirichlet_kernel:
   assumes f: "f absolutely_integrable_on {-pi..pi}" and periodic: "\<And>x. f(x + 2*pi) = f x"
   shows "((\<lambda>n. (\<Sum>k\<le>n. Fourier_coefficient f k * trigonometric_set k t)) \<longlonglongrightarrow> l)
      \<longleftrightarrow> (\<lambda>n. LINT x|lebesgue_on {-pi..pi}. Dirichlet_kernel n x * f(x + t)) \<longlonglongrightarrow> pi * l"
     (is "?lhs = ?rhs")
 proof -
   have "?lhs \<longleftrightarrow> (\<lambda>n. (LINT x|lebesgue_on {-pi..pi}. Dirichlet_kernel n x * f(x + t)) / pi) \<longlonglongrightarrow> l"
     by (simp add: Fourier_sum_limit_pair [OF f, symmetric] Fourier_sum_offset_Dirichlet_kernel assms)
   also have "\<dots> \<longleftrightarrow> (\<lambda>k. (LINT x|lebesgue_on {-pi..pi}. Dirichlet_kernel k x * f(x + t)) * inverse pi)
                     \<longlonglongrightarrow> pi * l * inverse pi"
     by (auto simp: divide_simps)
   also have "\<dots> \<longleftrightarrow> ?rhs"
     using real_tendsto_mult_right_iff [of "inverse pi", symmetric] by auto
   finally show ?thesis .
 qed
 
 subsection\<open>A directly deduced sufficient condition for convergence at a point\<close>
 
 lemma simple_Fourier_convergence_periodic:
   assumes f: "f absolutely_integrable_on {-pi..pi}"
     and ft: "(\<lambda>x. (f(x+t) - f t) / sin(x/2)) absolutely_integrable_on {-pi..pi}"
     and periodic: "\<And>x. f(x + 2*pi) = f x"
   shows "(\<lambda>n. (\<Sum>k\<le>n. Fourier_coefficient f k * trigonometric_set k t)) \<longlonglongrightarrow> f t"
 proof -
   let ?\<Phi> = "\<lambda>n. \<Sum>k\<le>n. Fourier_coefficient (\<lambda>x. f x - f t) k * trigonometric_set k t"
   let ?\<Psi> = "\<lambda>n. LINT x|lebesgue_on {-pi..pi}. Dirichlet_kernel n x * (f(x + t) - f t)"
   have fft: "(\<lambda>x. f x - f t) absolutely_integrable_on {-pi..pi}"
     by (simp add: f absolutely_integrable_measurable_real)
   have fxt: "(\<lambda>x. f(x + t)) absolutely_integrable_on {-pi..pi}"
     using absolutely_integrable_periodic_offset assms by auto
   have *: "?\<Phi> \<longlonglongrightarrow> 0 \<longleftrightarrow> ?\<Psi> \<longlonglongrightarrow> 0"
     by (simp add: Fourier_sum_limit_Dirichlet_kernel fft periodic)
   moreover have "(\<lambda>n. (\<Sum>k\<le>n. Fourier_coefficient f k * trigonometric_set k t) - f t) \<longlonglongrightarrow> 0"
     if "?\<Phi> \<longlonglongrightarrow> 0"
   proof (rule Lim_transform_eventually [OF that eventually_sequentiallyI])
     show "(\<Sum>k\<le>n. Fourier_coefficient (\<lambda>x. f x - f t) k * trigonometric_set k t)
         = (\<Sum>k\<le>n. Fourier_coefficient f k * trigonometric_set k t) - f t"
       if "Suc 0 \<le> n" for n
     proof -
       have "(\<Sum>k = Suc 0..n. Fourier_coefficient (\<lambda>x. f x - f t) k * trigonometric_set k t)
           = (\<Sum>k = Suc 0..n. Fourier_coefficient f k * trigonometric_set k t)"
       proof (rule sum.cong [OF refl])
         fix k
         assume k: "k \<in> {Suc 0..n}"
         then have [simp]: "integral\<^sup>L (lebesgue_on {-pi..pi}) (trigonometric_set k) = 0"
           by (auto simp: trigonometric_set_def)
         show "Fourier_coefficient (\<lambda>x. f x - f t) k * trigonometric_set k t = Fourier_coefficient f k * trigonometric_set k t"
           using k unfolding Fourier_coefficient_def orthonormal_coeff_def l2product_def
           by (simp add: right_diff_distrib f absolutely_integrable_measurable_real trigonometric_set_mul_integrable)
       qed
       moreover have "Fourier_coefficient (\<lambda>x. f x - f t) 0 * trigonometric_set 0 t =
                      Fourier_coefficient f 0 * trigonometric_set 0 t - f t"
         using f unfolding Fourier_coefficient_def orthonormal_coeff_def l2product_def
         by (auto simp: divide_simps trigonometric_set absolutely_integrable_imp_integrable)
       ultimately show ?thesis
         by (simp add: sum.atLeast_Suc_atMost atMost_atLeast0)
     qed
   qed
   moreover
   have "?\<Psi> \<longlonglongrightarrow> 0"
   proof -
     have eq: "\<And>n. ?\<Psi> n = integral\<^sup>L (lebesgue_on {-pi..pi}) (\<lambda>x. sin((n + 1/2) * x) * ((f(x+t) - f t) / sin(x/2) / 2))"
       unfolding Dirichlet_kernel_def
       by (rule Bochner_Integration.integral_cong [OF refl]) (auto simp: divide_simps sin_zero_iff)
     show ?thesis
       unfolding eq
       by (intro ft set_integrable_divide Riemann_lebesgue_sin_half)
   qed
   ultimately show ?thesis
     by (simp add: LIM_zero_cancel)
 qed
 
 
 subsection\<open>A more natural sufficient Hoelder condition at a point\<close>
 
 lemma bounded_inverse_sin_half:
   assumes "d > 0"
   obtains B where "B>0" "\<And>x. x \<in> ({-pi..pi} - {-d<..<d}) \<Longrightarrow> \<bar>inverse (sin (x/2))\<bar> \<le> B"
 proof -
   have "bounded ((\<lambda>x. inverse (sin (x/2))) ` ({-pi..pi} - {- d<..<d}))"
   proof (intro compact_imp_bounded compact_continuous_image)
     have "\<lbrakk>x \<in> {-pi..pi}; x \<noteq> 0\<rbrakk> \<Longrightarrow> sin (x/2) \<noteq> 0" for x
       using \<open>0 < d\<close> by (auto simp: sin_zero_iff)
     then show "continuous_on ({-pi..pi} - {-d<..<d}) (\<lambda>x. inverse (sin (x/2)))"
       using \<open>0 < d\<close> by (intro continuous_intros) auto
     show "compact ({-pi..pi} - {-d<..<d})"
       by (simp add: compact_diff)
   qed
   then show thesis
     using that by (auto simp: bounded_pos)
 qed
 
 proposition Hoelder_Fourier_convergence_periodic:
   assumes f: "f absolutely_integrable_on {-pi..pi}" and "d > 0" "a > 0"
     and ft: "\<And>x. \<bar>x-t\<bar> < d \<Longrightarrow> \<bar>f x - f t\<bar> \<le> M * \<bar>x-t\<bar> powr a"
     and periodic: "\<And>x. f(x + 2*pi) = f x"
   shows "(\<lambda>n. (\<Sum>k\<le>n. Fourier_coefficient f k * trigonometric_set k t)) \<longlonglongrightarrow> f t"
 proof (rule simple_Fourier_convergence_periodic [OF f])
   have half: "((\<lambda>x. sin(x/2)) has_real_derivative 1/2) (at 0)"
     by (rule derivative_eq_intros refl | force)+
   then obtain e0::real where "e0 > 0" and e0: "\<And>x. \<lbrakk>x \<noteq> 0; \<bar>x\<bar> < e0\<rbrakk> \<Longrightarrow> \<bar>sin (x/2) / x - 1/2\<bar> < 1/4"
     apply (simp add: DERIV_def Lim_at dist_real_def)
     apply (drule_tac x="1/4" in spec, auto)
     done
   obtain e where "e > 0" and e: "\<And>x. \<bar>x\<bar> < e \<Longrightarrow> \<bar>(f(x+t) - f t) / sin (x/2)\<bar> \<le> 4 * (\<bar>M\<bar> * \<bar>x\<bar> powr (a-1))"
   proof
     show "min d e0 > 0"
       using \<open>d > 0\<close> \<open>e0 > 0\<close> by auto
   next
     fix x
     assume x: "\<bar>x\<bar> < min d e0"
     show "\<bar>(f(x + t) - f t) / sin (x/2)\<bar> \<le> 4 * (\<bar>M\<bar> * \<bar>x\<bar> powr (a - 1))"
     proof (cases "sin(x/2) = 0 \<or> x=0")
       case False
       have eq: "\<bar>(f(x + t) - f t) / sin (x/2)\<bar> = \<bar>inverse (sin (x/2) / x)\<bar> * (\<bar>f(x + t) - f t\<bar> / \<bar>x\<bar>)"
         by simp
       show ?thesis
         unfolding eq
       proof (rule mult_mono)
         have "\<bar>sin (x/2) / x - 1/2\<bar> < 1/4"
           using e0 [of x] x False by force
         then have "1/4 \<le> \<bar>sin (x/2) / x\<bar>"
           by (simp add: abs_if field_simps split: if_split_asm)
         then show "\<bar>inverse (sin (x/2) / x)\<bar> \<le> 4"
           using False by (simp add: field_simps)
         have "\<bar>f(x + t) - f t\<bar> / \<bar>x\<bar> \<le> M * \<bar>x + t - t\<bar> powr (a - 1)"
           using ft [of "x+t"] x by (auto simp: divide_simps algebra_simps Transcendental.powr_mult_base)
         also have "\<dots> \<le> \<bar>M\<bar> * \<bar>x\<bar> powr (a - 1)"
           by (simp add: mult_mono)
         finally show "\<bar>f(x + t) - f t\<bar> / \<bar>x\<bar> \<le> \<bar>M\<bar> * \<bar>x\<bar> powr (a - 1)" .
       qed auto
     qed auto
   qed
   obtain B where "B>0" and B: "\<And>x. x \<in> ({-pi..pi} - {- e<..<e}) \<Longrightarrow> \<bar>inverse (sin (x/2))\<bar> \<le> B"
     using bounded_inverse_sin_half [OF \<open>e > 0\<close>] by metis
   let ?g = "\<lambda>x. max (B * \<bar>f(x + t) - f t\<bar>) ((4 * \<bar>M\<bar>) * \<bar>x\<bar> powr (a - 1))"
   show "(\<lambda>x. (f(x + t) - f t) / sin (x/2)) absolutely_integrable_on {-pi..pi}"
   proof (rule measurable_bounded_by_integrable_imp_absolutely_integrable)
     have fxt: "(\<lambda>x. f(x + t)) absolutely_integrable_on {-pi..pi}"
       using absolutely_integrable_periodic_offset assms by auto
     show "(\<lambda>x. (f(x + t) - f t) / sin (x/2)) \<in> borel_measurable (lebesgue_on {-pi..pi})"
     proof (intro measurable)
       show "(\<lambda>x. f(x + t)) \<in> borel_measurable (lebesgue_on {-pi..pi})"
         using absolutely_integrable_on_def fxt integrable_imp_measurable by blast
       show "(\<lambda>x. sin (x/2)) \<in> borel_measurable (lebesgue_on {-pi..pi})"
         by (intro continuous_imp_measurable_on_sets_lebesgue continuous_intros) auto
     qed auto
     have "(\<lambda>x. f(x + t) - f t) absolutely_integrable_on {-pi..pi}"
       by (intro set_integral_diff fxt) (simp add: set_integrable_def)
     moreover
     have "(\<lambda>x. \<bar>x\<bar> powr (a - 1)) absolutely_integrable_on {-pi..pi}"
     proof -
       have "((\<lambda>x. x powr (a - 1)) has_integral
        inverse a * pi powr a - inverse a * 0 powr a)
        {0..pi}"
       proof (rule fundamental_theorem_of_calculus_interior)
         show "continuous_on {0..pi} (\<lambda>x. inverse a * x powr a)"
           using \<open>a > 0\<close>
           by (intro continuous_on_powr' continuous_intros) auto
         show "((\<lambda>b. inverse a * b powr a) has_vector_derivative x powr (a - 1)) (at x)"
           if "x \<in> {0<..<pi}" for x
           using that \<open>a > 0\<close>
           apply (simp flip: has_field_derivative_iff_has_vector_derivative)
           apply (rule derivative_eq_intros | simp)+
           done
       qed auto
       then have int: "(\<lambda>x. x powr (a - 1)) integrable_on {0..pi}"
         by blast
       show ?thesis
       proof (rule nonnegative_absolutely_integrable_1)
         show "(\<lambda>x. \<bar>x\<bar> powr (a - 1)) integrable_on {-pi..pi}"
         proof (rule Henstock_Kurzweil_Integration.integrable_combine)
           show "(\<lambda>x. \<bar>x\<bar> powr (a - 1)) integrable_on {0..pi}"
             using int integrable_eq by fastforce
           then show "(\<lambda>x. \<bar>x\<bar> powr (a - 1)) integrable_on {- pi..0}"
             using Henstock_Kurzweil_Integration.integrable_reflect_real by fastforce
         qed auto
       qed auto
     qed
     ultimately show "?g integrable_on {-pi..pi}"
       apply (intro set_lebesgue_integral_eq_integral absolutely_integrable_max_1 absolutely_integrable_bounded_measurable_product_real set_integrable_abs measurable)
               apply (auto simp: image_constant_conv)
       done
     show "norm ((f(x + t) - f t) / sin (x/2)) \<le> ?g x" if "x \<in> {-pi..pi}" for x
     proof (cases "\<bar>x\<bar> < e")
       case True
       then show ?thesis
         using e [OF True] by (simp add: max_def)
     next
       case False
       then have "\<bar>inverse (sin (x/2))\<bar> \<le> B"
         using B that by force
       then have "\<bar>inverse (sin (x/2))\<bar> * \<bar>f(x + t) - f t\<bar> \<le> B * \<bar>f(x + t) - f t\<bar>"
         by (simp add: mult_right_mono)
       then have "\<bar>f(x + t) - f t\<bar> / \<bar>sin (x/2)\<bar> \<le> B * \<bar>f(x + t) - f t\<bar>"
         by (simp add: divide_simps split: if_split_asm)
       then show ?thesis
         by auto
     qed
   qed auto
 qed (auto simp: periodic)
 
 
 text\<open>In particular, a Lipschitz condition at the point\<close>
 corollary Lipschitz_Fourier_convergence_periodic:
   assumes f: "f absolutely_integrable_on {-pi..pi}" and "d > 0"
     and ft: "\<And>x. \<bar>x-t\<bar> < d \<Longrightarrow> \<bar>f x - f t\<bar> \<le> M * \<bar>x-t\<bar>"
     and periodic: "\<And>x. f(x + 2*pi) = f x"
   shows "(\<lambda>n. (\<Sum>k\<le>n. Fourier_coefficient f k * trigonometric_set k t)) \<longlonglongrightarrow> f t"
   using Hoelder_Fourier_convergence_periodic [OF f \<open>d > 0\<close>, of 1] assms
   by (fastforce simp add:)
 
 text\<open>In particular, if left and right derivatives both exist\<close>
 proposition bi_differentiable_Fourier_convergence_periodic:
   assumes f: "f absolutely_integrable_on {-pi..pi}"
     and f_lt: "f differentiable at_left t"
     and f_gt: "f differentiable at_right t"
     and periodic: "\<And>x. f(x + 2*pi) = f x"
   shows "(\<lambda>n. (\<Sum>k\<le>n. Fourier_coefficient f k * trigonometric_set k t)) \<longlonglongrightarrow> f t"
 proof -
   obtain D_lt where D_lt: "\<And>e. e > 0 \<Longrightarrow> \<exists>d>0. \<forall>x<t. 0 < dist x t \<and> dist x t < d \<longrightarrow> dist ((f x - f t) / (x - t)) D_lt < e"
     using f_lt unfolding has_field_derivative_iff real_differentiable_def Lim_within
     by (meson lessThan_iff)
   then obtain d_lt where "d_lt > 0"
     and d_lt: "\<And>x. \<lbrakk>x < t; 0 < \<bar>x - t\<bar>; \<bar>x - t\<bar> < d_lt\<rbrakk> \<Longrightarrow> \<bar>(f x - f t) / (x - t) - D_lt\<bar> < 1"
     unfolding dist_real_def by (metis dist_commute dist_real_def zero_less_one)
   obtain D_gt where D_gt: "\<And>e. e > 0 \<Longrightarrow> \<exists>d>0. \<forall>x>t. 0 < dist x t \<and> dist x t < d \<longrightarrow> dist ((f x - f t) / (x - t)) D_gt < e"
     using f_gt unfolding has_field_derivative_iff real_differentiable_def Lim_within
     by (meson greaterThan_iff)
   then obtain d_gt where "d_gt > 0"
     and d_gt: "\<And>x. \<lbrakk>x > t; 0 < \<bar>x - t\<bar>; \<bar>x - t\<bar> < d_gt\<rbrakk> \<Longrightarrow> \<bar>(f x - f t) / (x - t) - D_gt\<bar> < 1"
     unfolding dist_real_def by (metis dist_commute dist_real_def zero_less_one)
   show ?thesis
   proof (rule Lipschitz_Fourier_convergence_periodic [OF f])
     fix x
     assume "\<bar>x - t\<bar> < min d_lt d_gt"
     then have *: "\<bar>x - t\<bar> < d_lt" "\<bar>x - t\<bar> < d_gt"
       by auto
     consider "x<t" | "x=t" | "x>t"
       by linarith
     then show "\<bar>f x - f t\<bar> \<le> (\<bar>D_lt\<bar> + \<bar>D_gt\<bar> + 1)  * \<bar>x - t\<bar>"
     proof cases
       case 1
       then have "\<bar>(f x - f t) / (x - t) - D_lt\<bar> < 1"
         using d_lt [OF 1] * by auto
       then have "\<bar>(f x - f t) / (x - t)\<bar> - \<bar>D_lt\<bar> < 1"
         by linarith
       then have "\<bar>f x - f t\<bar> \<le> (\<bar>D_lt\<bar> + 1) * \<bar>x - t\<bar>"
         by (simp add: comm_semiring_class.distrib divide_simps split: if_split_asm)
       also have "\<dots> \<le> (\<bar>D_lt\<bar> + \<bar>D_gt\<bar> + 1) * \<bar>x - t\<bar>"
         by (simp add: mult_right_mono)
       finally show ?thesis .
     next
       case 3
       then have "\<bar>(f x - f t) / (x - t) - D_gt\<bar> < 1"
         using d_gt [OF 3] * by auto
       then have "\<bar>(f x - f t) / (x - t)\<bar> - \<bar>D_gt\<bar> < 1"
         by linarith
       then have "\<bar>f x - f t\<bar> \<le> (\<bar>D_gt\<bar> + 1) * \<bar>x - t\<bar>"
         by (simp add: comm_semiring_class.distrib divide_simps split: if_split_asm)
       also have "\<dots> \<le> (\<bar>D_lt\<bar> + \<bar>D_gt\<bar> + 1) * \<bar>x - t\<bar>"
         by (simp add: mult_right_mono)
       finally show ?thesis .
     qed auto
   qed (auto simp: \<open>0 < d_gt\<close> \<open>0 < d_lt\<close> periodic)
 qed
 
 
 text\<open>And in particular at points where the function is differentiable\<close>
 lemma differentiable_Fourier_convergence_periodic:
   assumes f: "f absolutely_integrable_on {-pi..pi}"
     and fdif: "f differentiable (at t)"
     and periodic: "\<And>x. f(x + 2*pi) = f x"
   shows "(\<lambda>n. (\<Sum>k\<le>n. Fourier_coefficient f k * trigonometric_set k t)) \<longlonglongrightarrow> f t"
   by (rule bi_differentiable_Fourier_convergence_periodic) (auto simp: differentiable_at_withinI assms)
 
 text\<open>Use reflection to halve the region of integration\<close>
 lemma absolutely_integrable_mult_Dirichlet_kernel_reflected:
   assumes f: "f absolutely_integrable_on {-pi..pi}"
     and periodic: "\<And>x. f(x + 2*pi) = f x"
   shows "(\<lambda>x. Dirichlet_kernel n x * f(t+x)) absolutely_integrable_on {-pi..pi}"
         "(\<lambda>x. Dirichlet_kernel n x * f(t-x)) absolutely_integrable_on {-pi..pi}"
         "(\<lambda>x. Dirichlet_kernel n x * c) absolutely_integrable_on {-pi..pi}"
 proof -
   show "(\<lambda>x. Dirichlet_kernel n x * f(t+x)) absolutely_integrable_on {-pi..pi}"
     apply (rule absolutely_integrable_mult_Dirichlet_kernel)
     using absolutely_integrable_periodic_offset [OF f, of t] periodic
     apply simp
     done
   then show "(\<lambda>x. Dirichlet_kernel n x * f(t-x)) absolutely_integrable_on {-pi..pi}"
     by (subst absolutely_integrable_reflect_real [symmetric]) simp
   show "(\<lambda>x. Dirichlet_kernel n x * c) absolutely_integrable_on {-pi..pi}"
     by (simp add: absolutely_integrable_measurable_real borel_measurable_integrable integrable_Dirichlet_kernel)
 qed
 
 
 lemma absolutely_integrable_mult_Dirichlet_kernel_reflected_part:
   assumes f: "f absolutely_integrable_on {-pi..pi}"
     and periodic: "\<And>x. f(x + 2*pi) = f x" and "d \<le> pi"
   shows "(\<lambda>x. Dirichlet_kernel n x * f(t+x)) absolutely_integrable_on {0..d}"
         "(\<lambda>x. Dirichlet_kernel n x * f(t-x)) absolutely_integrable_on {0..d}"
         "(\<lambda>x. Dirichlet_kernel n x * c) absolutely_integrable_on {0..d}"
   using absolutely_integrable_mult_Dirichlet_kernel_reflected [OF f periodic, of n] \<open>d \<le> pi\<close>
   by (force intro: absolutely_integrable_on_subinterval)+
 
 lemma absolutely_integrable_mult_Dirichlet_kernel_reflected_part2:
   assumes f: "f absolutely_integrable_on {-pi..pi}"
     and periodic: "\<And>x. f(x + 2*pi) = f x" and "d \<le> pi"
   shows "(\<lambda>x. Dirichlet_kernel n x * (f(t+x) + f(t-x))) absolutely_integrable_on {0..d}"
         "(\<lambda>x. Dirichlet_kernel n x * ((f(t+x) + f(t-x)) - c)) absolutely_integrable_on {0..d}"
   using absolutely_integrable_mult_Dirichlet_kernel_reflected_part assms
   by (simp_all add: distrib_left right_diff_distrib)
 
-
 lemma integral_reflect_and_add:
   fixes f :: "real \<Rightarrow> 'b::euclidean_space"
   assumes "integrable (lebesgue_on {-a..a}) f"
   shows "integral\<^sup>L (lebesgue_on {-a..a}) f = integral\<^sup>L (lebesgue_on {0..a})  (\<lambda>x. f x + f(-x))"
 proof (cases "a \<ge> 0")
   case True
   have f: "integrable (lebesgue_on {0..a}) f" "integrable (lebesgue_on {-a..0}) f"
-    using integrable_subinterval_real assms by fastforce+
+    using integrable_subinterval assms by fastforce+
   then have fm: "integrable (lebesgue_on {0..a}) (\<lambda>x. f(-x))"
     using integrable_reflect_real by fastforce
   have "integral\<^sup>L (lebesgue_on {-a..a}) f = integral\<^sup>L (lebesgue_on {-a..0}) f + integral\<^sup>L (lebesgue_on {0..a}) f"
     by (simp add: True assms integral_combine)
   also have "\<dots> = integral\<^sup>L (lebesgue_on {0..a}) (\<lambda>x. f(-x)) + integral\<^sup>L (lebesgue_on {0..a}) f"
     by (metis (no_types) add.inverse_inverse add.inverse_neutral integral_reflect_real)
   also have "\<dots> = integral\<^sup>L (lebesgue_on {0..a}) (\<lambda>x. f x + f(-x))"
     by (simp add: fm f)
   finally show ?thesis .
 qed (use assms in auto)
 
 lemma Fourier_sum_offset_Dirichlet_kernel_half:
   assumes f: "f absolutely_integrable_on {-pi..pi}"
     and periodic: "\<And>x. f(x + 2*pi) = f x"
   shows "(\<Sum>k\<le>2*n. Fourier_coefficient f k * trigonometric_set k t) - l
        = (LINT x|lebesgue_on {0..pi}. Dirichlet_kernel n x * (f(t+x) + f(t-x) - 2*l)) / pi"
 proof -
   have fxt: "(\<lambda>x. f(x + t)) absolutely_integrable_on {-pi..pi}"
     using absolutely_integrable_periodic_offset assms by auto
   have int: "integrable (lebesgue_on {0..pi}) (Dirichlet_kernel n)"
     using not_integrable_integral_eq by fastforce
   have "LINT x|lebesgue_on {-pi..pi}. Dirichlet_kernel n x * f(x + t)
       = LINT x|lebesgue_on {0..pi}. Dirichlet_kernel n x * f(x + t) + Dirichlet_kernel n (- x) * f(- x + t)"
     by (simp add: integral_reflect_and_add absolutely_integrable_imp_integrable absolutely_integrable_mult_Dirichlet_kernel fxt)
   also have "\<dots> = (LINT x|lebesgue_on {0..pi}. Dirichlet_kernel n x * (f(t+x) + f(t-x) - 2*l)) + pi * l"
     using absolutely_integrable_mult_Dirichlet_kernel_reflected_part [OF f periodic order_refl, of n t]
     apply (simp add: algebra_simps absolutely_integrable_imp_integrable int)
     done
   finally show ?thesis
     by (simp add: Fourier_sum_offset_Dirichlet_kernel assms divide_simps)
 qed
 
 lemma Fourier_sum_limit_Dirichlet_kernel_half:
   assumes f: "f absolutely_integrable_on {-pi..pi}"
     and periodic: "\<And>x. f(x + 2*pi) = f x"
   shows "(\<lambda>n. (\<Sum>k\<le>n. Fourier_coefficient f k * trigonometric_set k t)) \<longlonglongrightarrow> l
      \<longleftrightarrow> (\<lambda>n. (LINT x|lebesgue_on {0..pi}. Dirichlet_kernel n x * (f(t+x) + f(t-x) - 2*l))) \<longlonglongrightarrow> 0"
   apply (simp flip: Fourier_sum_limit_pair [OF f])
   apply (simp add: Lim_null [where l=l] Fourier_sum_offset_Dirichlet_kernel_half assms)
   done
 
 
 subsection\<open>Localization principle: convergence only depends on values "nearby"\<close>
 
 proposition Riemann_localization_integral:
   assumes f: "f absolutely_integrable_on {-pi..pi}" and g: "g absolutely_integrable_on {-pi..pi}"
     and "d > 0" and d: "\<And>x. \<bar>x\<bar> < d \<Longrightarrow> f x = g x"
   shows "(\<lambda>n. integral\<^sup>L (lebesgue_on {-pi..pi}) (\<lambda>x. Dirichlet_kernel n x * f x)
             - integral\<^sup>L (lebesgue_on {-pi..pi}) (\<lambda>x. Dirichlet_kernel n x * g x))
          \<longlonglongrightarrow> 0"  (is "?a \<longlonglongrightarrow> 0")
 proof -
   let ?f = "\<lambda>x. (if \<bar>x\<bar> < d then 0 else f x - g x) / sin(x/2) / 2"
   have eq: "?a n = integral\<^sup>L (lebesgue_on {-pi..pi}) (\<lambda>x. sin((n+1/2) * x) * ?f x)" for n
   proof -
     have "?a n = integral\<^sup>L (lebesgue_on {-pi..pi}) (\<lambda>x. Dirichlet_kernel n x * (if \<bar>x\<bar> < d then 0 else f x - g x))"
       apply (simp add: absolutely_integrable_imp_integrable absolutely_integrable_mult_Dirichlet_kernel f g flip: Bochner_Integration.integral_diff)
       apply (auto simp: d algebra_simps intro: Bochner_Integration.integral_cong)
       done
     also have "\<dots> = integral\<^sup>L (lebesgue_on {-pi..pi}) (\<lambda>x. sin((n+1/2) * x) * ?f x)"
       using \<open>d > 0\<close> by (auto simp: Dirichlet_kernel_def intro: Bochner_Integration.integral_cong)
     finally show ?thesis .
   qed
   show ?thesis
     unfolding eq
   proof (rule Riemann_lebesgue_sin_half)
     obtain B where "B>0" and B: "\<And>x. x \<in> ({-pi..pi} - {-d<..<d}) \<Longrightarrow> \<bar>inverse (sin (x/2))\<bar> \<le> B"
       using bounded_inverse_sin_half [OF \<open>d > 0\<close>] by metis
     have "(\<lambda>x. (if \<bar>x\<bar> < d then 0 else f x - g x) / sin (x/2)) absolutely_integrable_on {-pi..pi}"
     proof (rule measurable_bounded_by_integrable_imp_absolutely_integrable)
       show "(\<lambda>x. (if \<bar>x\<bar> < d then 0 else f x - g x) / sin (x/2)) \<in> borel_measurable (lebesgue_on {-pi..pi})"
       proof (intro measurable)
         show "f \<in> borel_measurable (lebesgue_on {-pi..pi})" "g \<in> borel_measurable (lebesgue_on {-pi..pi})"
           using absolutely_integrable_on_def f g integrable_imp_measurable by blast+
         show "(\<lambda>x. x) \<in> borel_measurable (lebesgue_on {-pi..pi})"
              "(\<lambda>x. sin (x/2)) \<in> borel_measurable (lebesgue_on {-pi..pi})"
           by (intro continuous_intros continuous_imp_measurable_on_sets_lebesgue | force)+
       qed auto
       have "(\<lambda>x. B * \<bar>f x - g x\<bar>) absolutely_integrable_on {-pi..pi}"
         using \<open>B > 0\<close> by (simp add: f g set_integrable_abs)
       then show "(\<lambda>x. B * \<bar>f x - g x\<bar>) integrable_on {-pi..pi}"
         using absolutely_integrable_on_def by blast
     next
       fix x
       assume x: "x \<in> {-pi..pi}"
       then have [simp]: "sin (x/2) = 0 \<longleftrightarrow> x=0"
         using \<open>0 < d\<close> by (force simp: sin_zero_iff)
       show "norm ((if \<bar>x\<bar> < d then 0 else f x - g x) / sin (x/2)) \<le> B * \<bar>f x - g x\<bar>"
       proof (cases "\<bar>x\<bar> < d")
         case False
         then have "1 \<le> B * \<bar>sin (x/2)\<bar>"
           using \<open>B > 0\<close> B [of x] x \<open>d > 0\<close>
           by (auto simp: abs_less_iff divide_simps split: if_split_asm)
         then have "1 * \<bar>f x - g x\<bar> \<le> B * \<bar>sin (x/2)\<bar> * \<bar>f x - g x\<bar>"
           by (metis (full_types) abs_ge_zero mult.commute mult_left_mono)
         then show ?thesis
           using False by (auto simp: divide_simps algebra_simps)
       qed (simp add: d)
     qed auto
     then show "(\<lambda>x. (if \<bar>x\<bar> < d then 0 else f x - g x) / sin (x/2) / 2) absolutely_integrable_on {-pi..pi}"
       using set_integrable_divide by blast
   qed
 qed
 
 lemma Riemann_localization_integral_range:
   assumes f: "f absolutely_integrable_on {-pi..pi}"
     and "0 < d" "d \<le> pi"
   shows "(\<lambda>n. integral\<^sup>L (lebesgue_on {-pi..pi}) (\<lambda>x. Dirichlet_kernel n x * f x)
             - integral\<^sup>L (lebesgue_on {-d..d}) (\<lambda>x. Dirichlet_kernel n x * f x))
              \<longlonglongrightarrow> 0"
 proof -
   have *: "(\<lambda>n. (LINT x|lebesgue_on {-pi..pi}. Dirichlet_kernel n x * f x)
               - (LINT x|lebesgue_on {-pi..pi}. Dirichlet_kernel n x * (if x \<in> {-d..d} then f x else 0)))
            \<longlonglongrightarrow> 0"
   proof (rule Riemann_localization_integral [OF f _ \<open>0 < d\<close>])
     have "f absolutely_integrable_on {-d..d}"
       using f absolutely_integrable_on_subinterval \<open>d \<le> pi\<close> by fastforce
     moreover have "(\<lambda>x. if - pi \<le> x \<and> x \<le> pi then if x \<in> {-d..d} then f x else 0  else 0)
                  = (\<lambda>x. if x \<in> {-d..d} then f x else 0)"
       using \<open>d \<le> pi\<close> by force
     ultimately
     show "(\<lambda>x. if x \<in> {-d..d} then f x else 0) absolutely_integrable_on {-pi..pi}"
       apply (subst absolutely_integrable_restrict_UNIV [symmetric])
       apply (simp flip: absolutely_integrable_restrict_UNIV [of "{-d..d}" f])
       done
   qed auto
   then show ?thesis
     using integral_restrict [of "{-d..d}" "{-pi..pi}" "\<lambda>x. Dirichlet_kernel _ x * f x"] assms
     by (simp add: if_distrib cong: if_cong)
 qed
 
 lemma Riemann_localization:
   assumes f: "f absolutely_integrable_on {-pi..pi}" and g: "g absolutely_integrable_on {-pi..pi}"
     and perf: "\<And>x. f(x + 2*pi) = f x"
     and perg: "\<And>x. g(x + 2*pi) = g x"
     and "d > 0" and d: "\<And>x. \<bar>x-t\<bar> < d \<Longrightarrow> f x = g x"
   shows "(\<lambda>n. \<Sum>k\<le>n. Fourier_coefficient f k * trigonometric_set k t) \<longlonglongrightarrow> c
      \<longleftrightarrow> (\<lambda>n. \<Sum>k\<le>n. Fourier_coefficient g k * trigonometric_set k t) \<longlonglongrightarrow> c"
 proof -
   have "(\<lambda>n. LINT x|lebesgue_on {-pi..pi}. Dirichlet_kernel n x * f(x + t)) \<longlonglongrightarrow> pi * c
     \<longleftrightarrow> (\<lambda>n. LINT x|lebesgue_on {-pi..pi}. Dirichlet_kernel n x * g (x + t)) \<longlonglongrightarrow> pi * c"
   proof (intro Lim_transform_eq Riemann_localization_integral)
     show "(\<lambda>x. f(x + t)) absolutely_integrable_on {-pi..pi}" "(\<lambda>x. g (x + t)) absolutely_integrable_on {-pi..pi}"
       using assms by (auto intro: absolutely_integrable_periodic_offset)
   qed (use assms in auto)
   then show ?thesis
     by (simp add: Fourier_sum_limit_Dirichlet_kernel assms)
 qed
 
 subsection\<open>Localize the earlier integral\<close>
 
 lemma Riemann_localization_integral_range_half:
   assumes f: "f absolutely_integrable_on {-pi..pi}"
     and "0 < d" "d \<le> pi"
   shows "(\<lambda>n. (LINT x|lebesgue_on {0..pi}. Dirichlet_kernel n x * (f x + f(-x)))
             - (LINT x|lebesgue_on {0..d}. Dirichlet_kernel n x * (f x + f(-x)))) \<longlonglongrightarrow> 0"
 proof -
   have *: "(\<lambda>x. Dirichlet_kernel n x * f x) absolutely_integrable_on {-d..d}" for n
     by (metis (full_types) absolutely_integrable_mult_Dirichlet_kernel absolutely_integrable_on_subinterval
               \<open>d \<le> pi\<close> atLeastatMost_subset_iff f neg_le_iff_le)
   show ?thesis
     using absolutely_integrable_mult_Dirichlet_kernel [OF f]
     using Riemann_localization_integral_range [OF assms]
     apply (simp add: "*" absolutely_integrable_imp_integrable integral_reflect_and_add)
     apply (simp add: algebra_simps)
     done
 qed
 
 
 lemma Fourier_sum_limit_Dirichlet_kernel_part:
   assumes f: "f absolutely_integrable_on {-pi..pi}"
     and periodic: "\<And>x. f(x + 2*pi) = f x"
     and d: "0 < d" "d \<le> pi"
   shows "(\<lambda>n. \<Sum>k\<le>n. Fourier_coefficient f k * trigonometric_set k t) \<longlonglongrightarrow> l
      \<longleftrightarrow> (\<lambda>n. (LINT x|lebesgue_on {0..d}. Dirichlet_kernel n x * ((f(t+x) + f(t-x)) - 2*l))) \<longlonglongrightarrow> 0"
 proof -
   have "(\<lambda>x. f(t+x)) absolutely_integrable_on {-pi..pi}"
     using absolutely_integrable_periodic_offset [OF f, of t] periodic by simp
   then have int: "(\<lambda>x. f(t+x) - l) absolutely_integrable_on {-pi..pi}"
     by auto
   have "(\<lambda>n. LINT x|lebesgue_on {0..pi}. Dirichlet_kernel n x * (f(t+x) + f(t-x) - 2*l)) \<longlonglongrightarrow> 0
     \<longleftrightarrow> (\<lambda>n. LINT x|lebesgue_on {0..d}. Dirichlet_kernel n x * (f(t+x) + f(t-x) - 2*l)) \<longlonglongrightarrow> 0"
     by (rule Lim_transform_eq) (use Riemann_localization_integral_range_half [OF int d] in auto)
   then show ?thesis
     by (simp add: Fourier_sum_limit_Dirichlet_kernel_half assms)
 qed
 
 subsection\<open>Make a harmless simplifying tweak to the Dirichlet kernel\<close>
 
 lemma inte_Dirichlet_kernel_mul_expand:
   assumes f: "f \<in> borel_measurable (lebesgue_on S)" and S: "S \<in> sets lebesgue"
   shows "(LINT x|lebesgue_on S. Dirichlet_kernel n x * f x
         = LINT x|lebesgue_on S. sin((n+1/2) * x) * f x / (2 * sin(x/2)))
        \<and> (integrable (lebesgue_on S) (\<lambda>x. Dirichlet_kernel n x * f x)
       \<longleftrightarrow> integrable (lebesgue_on S) (\<lambda>x. sin((n+1/2) * x) * f x / (2 * sin(x/2))))"
 proof (cases "0 \<in> S")
   case True
   have *: "{x. x = 0 \<and> x \<in> S} \<in> sets (lebesgue_on S)"
     using True  by (simp add: S sets_restrict_space_iff cong: conj_cong)
   have bm1: "(\<lambda>x. Dirichlet_kernel n x * f x) \<in> borel_measurable (lebesgue_on S)"
     unfolding Dirichlet_kernel_def
     by (force intro: * assms measurable continuous_imp_measurable_on_sets_lebesgue continuous_intros)
   have bm2: "(\<lambda>x. sin ((n + 1/2) * x) * f x / (2 * sin (x/2))) \<in> borel_measurable (lebesgue_on S)"
     by (intro assms measurable continuous_imp_measurable_on_sets_lebesgue continuous_intros) auto
   have 0: "{0} \<in> null_sets (lebesgue_on S)"
     using True by (simp add: S null_sets_restrict_space)
   show ?thesis
     apply (intro conjI integral_cong_AE integrable_cong_AE bm1 bm2 AE_I' [OF 0])
     unfolding Dirichlet_kernel_def  by auto
 next
   case False
   then show ?thesis
     unfolding Dirichlet_kernel_def by (auto intro!: Bochner_Integration.integral_cong Bochner_Integration.integrable_cong)
 qed
 
 lemma
   assumes f: "f \<in> borel_measurable (lebesgue_on S)" and S: "S \<in> sets lebesgue"
   shows integral_Dirichlet_kernel_mul_expand:
         "(LINT x|lebesgue_on S. Dirichlet_kernel n x * f x)
        = (LINT x|lebesgue_on S. sin((n+1/2) * x) * f x / (2 * sin(x/2)))" (is "?th1")
   and integrable_Dirichlet_kernel_mul_expand:
        "integrable (lebesgue_on S) (\<lambda>x. Dirichlet_kernel n x * f x)
     \<longleftrightarrow> integrable (lebesgue_on S) (\<lambda>x. sin((n+1/2) * x) * f x / (2 * sin(x/2)))" (is "?th2")
   using inte_Dirichlet_kernel_mul_expand [OF assms] by auto
 
 
 proposition Fourier_sum_limit_sine_part:
   assumes f: "f absolutely_integrable_on {-pi..pi}"
     and periodic: "\<And>x. f(x + 2*pi) = f x"
     and d: "0 < d" "d \<le> pi"
   shows "(\<lambda>n. (\<Sum>k\<le>n. Fourier_coefficient f k * trigonometric_set k t)) \<longlonglongrightarrow> l
      \<longleftrightarrow> (\<lambda>n. LINT x|lebesgue_on {0..d}. sin((n + 1/2) * x) * ((f(t+x) + f(t-x) - 2*l) / x)) \<longlonglongrightarrow> 0"
     (is "?lhs \<longleftrightarrow> ?\<Psi> \<longlonglongrightarrow> 0")
 proof -
   have ftx: "(\<lambda>x. f(t+x)) absolutely_integrable_on {-pi..pi}"
     using absolutely_integrable_periodic_offset assms by auto
   then have ftmx: "(\<lambda>x. f(t-x)) absolutely_integrable_on {-pi..pi}"
     by (simp flip: absolutely_integrable_reflect_real [where f= "(\<lambda>x. f(t+x))"])
   have fbm: "(\<lambda>x. f(t+x) + f(t-x) - 2*l) absolutely_integrable_on {-pi..pi}"
     by (force intro: ftmx ftx)
   let ?\<Phi> = "\<lambda>n. LINT x|lebesgue_on {0..d}. Dirichlet_kernel n x * ((f(t+x) + f(t-x)) - 2*l)"
   have "?lhs \<longleftrightarrow> ?\<Phi> \<longlonglongrightarrow> 0"
     by (intro Fourier_sum_limit_Dirichlet_kernel_part assms)
   moreover have "?\<Phi> \<longlonglongrightarrow> 0 \<longleftrightarrow> ?\<Psi> \<longlonglongrightarrow> 0"
   proof (rule Lim_transform_eq [OF Lim_transform_eventually])
     let ?f = "\<lambda>n. LINT x|lebesgue_on {0..d}. sin((real n + 1/2) * x) * (1 / (2 * sin(x/2)) - 1/x) * (f(t+x) + f(t-x) - 2*l)"
     have "?f n = (LINT x|lebesgue_on {-pi..pi}.
             sin ((n + 1/2) * x) * ((if x \<in> {0..d} then 1 / (2 * sin (x/2)) - 1/x else 0) * (f(t+x) + f(t-x) - 2*l)))" for n
       using d by (simp add: integral_restrict if_distrib [of "\<lambda>u. _ * (u * _)"] mult.assoc flip: atLeastAtMost_iff cong: if_cong)
     moreover have "\<dots> \<longlonglongrightarrow> 0"
     proof (intro Riemann_lebesgue_sin_half absolutely_integrable_bounded_measurable_product_real)
       have "(\<lambda>x. 1 / (2 * sin(x/2)) - 1/x) \<in> borel_measurable (lebesgue_on {0..d})"
         by (intro measurable continuous_imp_measurable_on_sets_lebesgue continuous_intros) auto
       then show "(\<lambda>x. if x \<in> {0..d} then 1 / (2 * sin(x/2)) - 1/x else 0) \<in> borel_measurable (lebesgue_on {-pi..pi})"
         using d by (simp add: borel_measurable_if_lebesgue_on flip: atLeastAtMost_iff)
 
       let ?g = "\<lambda>x::real. 1 / (2 * sin(x/2)) - 1/x"
       have limg: "(?g \<longlongrightarrow> ?g a) (at a within {0..d})"   \<comment>\<open>thanks to Manuel Eberl\<close>
         if a: "0 \<le> a" "a \<le> d" for a
       proof -
         have "(?g \<longlongrightarrow> 0) (at_right 0)" and "(?g \<longlongrightarrow> ?g d) (at_left d)"
           using d sin_gt_zero[of "d/2"] by (real_asymp simp: field_simps)+
         moreover have "(?g \<longlongrightarrow> ?g a) (at a)" if "a > 0"
           using a that d sin_gt_zero[of "a/2"] that by (real_asymp simp: field_simps)
         ultimately show ?thesis using that d
           by (cases "a = 0 \<or> a = d") (auto simp: at_within_Icc_at at_within_Icc_at_right at_within_Icc_at_left)
       qed
       have "((\<lambda>x. if x \<in> {0..d} then 1 / (2 * sin(x/2)) - 1/x else 0) ` {-pi..pi}) = ((\<lambda>x. 1 / (2 * sin(x/2)) - 1/x) ` {0..d})"
         using d  by (auto intro: image_eqI [where x=0])
       moreover have "bounded \<dots>"
         by (intro compact_imp_bounded compact_continuous_image) (auto simp: continuous_on limg)
       ultimately show "bounded ((\<lambda>x. if x \<in> {0..d} then 1 / (2 * sin(x/2)) - 1/x else 0) ` {-pi..pi})"
         by simp
     qed (auto simp: fbm)
     ultimately show "?f \<longlonglongrightarrow> (0::real)"
       by simp
     show "\<forall>\<^sub>F n in sequentially. ?f n = ?\<Phi> n - ?\<Psi> n"
     proof (rule eventually_sequentiallyI [where c = "Suc 0"])
       fix n
       assume "n \<ge> Suc 0"
       have "integrable (lebesgue_on {0..d}) (\<lambda>x. sin ((real n + 1/2) * x) * (f(t+x) + f(t-x) - 2*l) / (2 * sin(x/2)))"
         using d
         apply (subst integrable_Dirichlet_kernel_mul_expand [symmetric])
           apply (intro assms absolutely_integrable_imp_borel_measurable absolutely_integrable_on_subinterval [OF fbm]
                        absolutely_integrable_imp_integrable absolutely_integrable_mult_Dirichlet_kernel_reflected_part2 | force)+
         done
       moreover have "integrable (lebesgue_on {0..d}) (\<lambda>x. sin ((real n + 1/2) * x) * (f(t+x) + f(t-x) - 2*l) / x)"
       proof (rule integrable_cong_AE_imp)
         let ?g = "\<lambda>x. Dirichlet_kernel n x * (2 * sin(x/2) / x * (f(t+x) + f(t-x) - 2*l))"
         have *: "\<bar>2 * sin (x/2) / x\<bar> \<le> 1" for x::real
           using abs_sin_x_le_abs_x [of "x/2"]
           by (simp add: divide_simps mult.commute abs_mult)
         have "bounded ((\<lambda>x. 1 + (x/2)\<^sup>2) ` {-pi..pi})"
           by (intro compact_imp_bounded compact_continuous_image continuous_intros) auto
         then have bo: "bounded ((\<lambda>x. 2 * sin (x/2) / x) ` {-pi..pi})"
           using * unfolding bounded_real by blast
         show "integrable (lebesgue_on {0..d}) ?g"
           using d
           apply (intro absolutely_integrable_imp_integrable
               absolutely_integrable_on_subinterval [OF absolutely_integrable_mult_Dirichlet_kernel]
               absolutely_integrable_bounded_measurable_product_real bo fbm
               measurable continuous_imp_measurable_on_sets_lebesgue continuous_intros, auto)
           done
         show "(\<lambda>x. sin ((n + 1/2) * x) * (f(t+x) + f(t-x) - 2*l) / x) \<in> borel_measurable (lebesgue_on {0..d})"
           using d
           apply (intro measurable absolutely_integrable_imp_borel_measurable
                        absolutely_integrable_on_subinterval [OF ftx] absolutely_integrable_on_subinterval [OF ftmx]
                        absolutely_integrable_continuous_real continuous_intros, auto)
           done
         have "Dirichlet_kernel n x * (2 * sin(x/2)) / x = sin ((real n + 1/2) * x) / x"
           if "0 < x" "x \<le> d" for x
           using that d by (simp add: Dirichlet_kernel_def divide_simps sin_zero_pi_iff)
         then show "AE x in lebesgue_on {0..d}. ?g x = sin ((real n + 1/2) * x) * (f(t+x) + f(t-x) - 2*l) / x"
           using d by (force intro: AE_I' [where N="{0}"])
       qed
       ultimately have "?f n = (LINT x|lebesgue_on {0..d}. sin ((n + 1/2) * x) * (f(t+x) + f(t-x) - 2*l) / (2 * sin(x/2)))
                  - (LINT x|lebesgue_on {0..d}. sin ((n + 1/2) * x) * (f(t+x) + f(t-x) - 2*l) / x)"
         by (simp add: right_diff_distrib [of _ _ "1/_"] left_diff_distrib)
       also have "\<dots> = ?\<Phi> n - ?\<Psi> n"
         using d
         by (simp add: measurable_restrict_mono [OF absolutely_integrable_imp_borel_measurable [OF fbm]]
                       integral_Dirichlet_kernel_mul_expand)
       finally show "?f n = ?\<Phi> n - ?\<Psi> n" .
     qed
   qed
   ultimately show ?thesis
     by simp
 qed
 
 
 subsection\<open>Dini's test for the convergence of a Fourier series\<close>
 
 proposition Fourier_Dini_test:
   assumes f: "f absolutely_integrable_on {-pi..pi}"
     and periodic: "\<And>x. f(x + 2*pi) = f x"
     and int0d: "integrable (lebesgue_on {0..d}) (\<lambda>x. \<bar>f(t+x) + f(t-x) - 2*l\<bar> / x)"
     and "0 < d"
   shows "(\<lambda>n. (\<Sum>k\<le>n. Fourier_coefficient f k * trigonometric_set k t)) \<longlonglongrightarrow> l"
 proof -
   define \<phi> where "\<phi> \<equiv> \<lambda>n x. sin((n + 1/2) * x) * ((f(t+x) + f(t-x) - 2*l) / x)"
   have "(\<lambda>n. LINT x|lebesgue_on {0..pi}. \<phi> n x) \<longlonglongrightarrow> 0"
     unfolding lim_sequentially dist_real_def
   proof (intro allI impI)
     fix e :: real
     assume "e > 0"
     define \<theta> where "\<theta> \<equiv> \<lambda>x. LINT x|lebesgue_on {0..x}. \<bar>f(t+x) + f(t-x) - 2*l\<bar> / x"
     have [simp]: "\<theta> 0 = 0"
       by (simp add: \<theta>_def integral_eq_zero_null_sets)
     have cont: "continuous_on {0..d} \<theta>"
-      unfolding \<theta>_def using indefinite_integral_continuous_1 int0d by blast
+      unfolding \<theta>_def using indefinite_integral_continuous_real int0d by blast
     with \<open>d > 0\<close>
     have "\<forall>e>0. \<exists>da>0. \<forall>x'\<in>{0..d}. \<bar>x' - 0\<bar> < da \<longrightarrow> \<bar>\<theta> x' - \<theta> 0\<bar> < e"
       by (force simp: continuous_on_real_range dest: bspec [where x=0])
     with \<open>e > 0\<close>
     obtain k where "k > 0" and k: "\<And>x'. \<lbrakk>0 \<le> x'; x' < k; x' \<le> d\<rbrakk> \<Longrightarrow> \<bar>\<theta> x'\<bar> < e/2"
       by (drule_tac x="e/2" in spec) auto
     define \<delta> where "\<delta> \<equiv> min d (min (k/2) pi)"
     have e2: "\<bar>\<theta> \<delta>\<bar> < e/2" and "\<delta> > 0" "\<delta> \<le> pi"
       unfolding \<delta>_def using k \<open>k > 0\<close> \<open>d > 0\<close> by auto
     have ftx: "(\<lambda>x. f(t+x)) absolutely_integrable_on {-pi..pi}"
       using absolutely_integrable_periodic_offset assms by auto
     then have ftmx: "(\<lambda>x. f(t-x)) absolutely_integrable_on {-pi..pi}"
       by (simp flip: absolutely_integrable_reflect_real [where f= "(\<lambda>x. f(t+x))"])
     have 1: "\<phi> n absolutely_integrable_on {0..\<delta>}" for n::nat
       unfolding \<phi>_def
     proof (rule absolutely_integrable_bounded_measurable_product_real)
       show "(\<lambda>x. sin ((real n + 1/2) * x)) \<in> borel_measurable (lebesgue_on {0..\<delta>})"
         by (intro continuous_imp_measurable_on_sets_lebesgue continuous_intros) auto
       show "bounded ((\<lambda>x. sin ((real n + 1/2) * x)) ` {0..\<delta>})"
         by (simp add: bounded_iff) (use abs_sin_le_one in blast)
       have "(\<lambda>x. (f(t+x) + f(t-x) - 2*l) / x) \<in> borel_measurable (lebesgue_on {0..\<delta>})"
         using \<open>d > 0\<close> unfolding \<delta>_def
         by (intro measurable absolutely_integrable_imp_borel_measurable
             absolutely_integrable_on_subinterval [OF ftx] absolutely_integrable_on_subinterval [OF ftmx]
             absolutely_integrable_continuous_real continuous_intros, auto)
       moreover have "integrable (lebesgue_on {0..\<delta>}) (norm \<circ> (\<lambda>x. (f(t+x) + f(t-x) - 2*l) / x))"
-      proof (rule integrable_subinterval_real [of 0 d])
+      proof (rule integrable_subinterval [of 0 d])
         show "integrable (lebesgue_on {0..d}) (norm \<circ> (\<lambda>x. (f(t+x) + f(t-x) - 2*l) / x))"
           using int0d by (subst integrable_cong) (auto simp: o_def)
         show "{0..\<delta>} \<subseteq> {0..d}"
           using \<open>d > 0\<close> by (auto simp: \<delta>_def)
       qed
       ultimately show "(\<lambda>x. (f(t+x) + f(t-x) - 2*l) / x) absolutely_integrable_on {0..\<delta>}"
         by (auto simp: absolutely_integrable_measurable)
     qed auto
     have 2: "\<phi> n absolutely_integrable_on {\<delta>..pi}" for n::nat
       unfolding \<phi>_def
     proof (rule absolutely_integrable_bounded_measurable_product_real)
       show "(\<lambda>x. sin ((n + 1/2) * x)) \<in> borel_measurable (lebesgue_on {\<delta>..pi})"
         by (intro continuous_imp_measurable_on_sets_lebesgue continuous_intros) auto
       show "bounded ((\<lambda>x. sin ((n + 1/2) * x)) ` {\<delta>..pi})"
         by (simp add: bounded_iff) (use abs_sin_le_one in blast)
     have "(\<lambda>x. inverse x * (f(t+x) + f(t-x) - 2*l)) absolutely_integrable_on {\<delta>..pi}"
     proof (rule absolutely_integrable_bounded_measurable_product_real)
       have "(\<lambda>x. 1/x) \<in> borel_measurable (lebesgue_on {\<delta>..pi})"
         by (auto simp: measurable_completion measurable_restrict_space1)
       then show "inverse \<in> borel_measurable (lebesgue_on {\<delta>..pi})"
         by (metis (no_types, lifting) inverse_eq_divide measurable_lebesgue_cong)
       have "\<forall>x\<in>{\<delta>..pi}. inverse \<bar>x\<bar> \<le> inverse \<delta>"
         using \<open>0 < \<delta>\<close> by auto
       then show "bounded (inverse ` {\<delta>..pi})"
         by (auto simp: bounded_iff)
       show "(\<lambda>x. f(t+x) + f(t-x) - 2*l) absolutely_integrable_on {\<delta>..pi}"
       proof (rule absolutely_integrable_on_subinterval)
         show "(\<lambda>x. (f(t+x) + f(t-x) - 2*l)) absolutely_integrable_on {-pi..pi}"
           by (fast intro: ftx ftmx absolutely_integrable_on_const)
         show "{\<delta>..pi} \<subseteq> {-pi..pi}"
           using \<open>0 < \<delta>\<close> by auto
       qed
     qed auto
     then show "(\<lambda>x. (f(t+x) + f(t-x) - 2*l) / x) absolutely_integrable_on {\<delta>..pi}"
       by (metis (no_types, lifting) divide_inverse mult.commute set_integrable_cong)
   qed auto
   have "(\<lambda>x. f(t+x) - l) absolutely_integrable_on {-pi..pi}"
     using ftx by auto
   moreover have "bounded ((\<lambda>x. 0) ` {x. \<bar>x\<bar> < \<delta>})"
     using bounded_def by blast
   moreover have "bounded (inverse ` {x. \<not> \<bar>x\<bar> < \<delta>})"
     using \<open>\<delta> > 0\<close> by (auto simp: divide_simps intro: boundedI [where B = "1/\<delta>"])
   ultimately have "(\<lambda>x. (if \<bar>x\<bar> < \<delta> then 0 else inverse x) * (if x \<in> {-pi..pi} then f(t+x) - l else 0)) absolutely_integrable_on UNIV"
     apply (intro absolutely_integrable_bounded_measurable_product_real measurable set_integral_diff)
           apply (auto simp: lebesgue_on_UNIV_eq measurable_completion simp flip: absolutely_integrable_restrict_UNIV [where S = "{-pi..pi}"])
     done
   moreover have "(if x \<in> {-pi..pi} then if \<bar>x\<bar> < \<delta> then 0 else (f(t+x) - l) / x else 0)
                = (if \<bar>x\<bar> < \<delta> then 0 else inverse x) * (if x \<in> {-pi..pi} then f(t+x) - l else 0)"  for x
     by (auto simp: divide_simps)
   ultimately have *: "(\<lambda>x. if \<bar>x\<bar> < \<delta> then 0 else (f(t+x) - l) / x) absolutely_integrable_on {-pi..pi}"
     by (simp add: flip: absolutely_integrable_restrict_UNIV [where S = "{-pi..pi}"])
   have "(\<lambda>n. LINT x|lebesgue_on {0..pi}. sin ((n + 1/2) * x) * (if \<bar>x\<bar> < \<delta> then 0 else (f(t+x) - l) / x)
                                        + sin ((n + 1/2) * -x) * (if \<bar>x\<bar> < \<delta> then 0 else (f(t-x) - l) / -x))
         \<longlonglongrightarrow> 0"
     using Riemann_lebesgue_sin_half [OF *] *
     by (simp add: integral_reflect_and_add absolutely_integrable_imp_integrable absolutely_integrable_sin_product cong: if_cong)
   moreover have "sin ((n + 1/2) * x) * (if \<bar>x\<bar> < \<delta> then 0 else (f(t+x) - l) / x)
                + sin ((n + 1/2) * -x) * (if \<bar>x\<bar> < \<delta> then 0 else (f(t-x) - l) / -x)
            = (if \<not> \<bar>x\<bar> < \<delta> then \<phi> n x else 0)" for x n
     by simp (auto simp: divide_simps algebra_simps \<phi>_def)
   ultimately have "(\<lambda>n. LINT x|lebesgue_on {0..pi}. (if \<not> \<bar>x\<bar> < \<delta> then \<phi> n x else 0)) \<longlonglongrightarrow> 0"
     by simp
   moreover have "(if 0 \<le> x \<and> x \<le> pi then if \<not> \<bar>x\<bar> < \<delta> then \<phi> n x else 0 else 0)
                = (if \<delta> \<le> x \<and> x \<le> pi then \<phi> n x else 0)" for x n
     using \<open>\<delta> > 0\<close> by auto
   ultimately have \<dagger>: "(\<lambda>n. LINT x|lebesgue_on {\<delta>..pi}. \<phi> n x) \<longlonglongrightarrow> 0"
-    by (simp flip: integral_restrict_UNIV)
+    by (simp flip: Lebesgue_Measure.integral_restrict_UNIV)
   then obtain N::nat where N: "\<And>n. n\<ge>N \<Longrightarrow> \<bar>LINT x|lebesgue_on {\<delta>..pi}. \<phi> n x\<bar> < e/2"
     unfolding lim_sequentially dist_real_def
     by (metis (full_types) \<open>0 < e\<close> diff_zero half_gt_zero_iff)
   have "\<bar>integral\<^sup>L (lebesgue_on {0..pi}) (\<phi> n)\<bar> < e" if "n \<ge> N" for n::nat
   proof -
     have int: "integrable (lebesgue_on {0..pi}) (\<phi> (real n))"
       by (intro integrable_combine [of concl: 0 pi] absolutely_integrable_imp_integrable) (use  \<open>\<delta> > 0\<close>  \<open>\<delta> \<le> pi\<close> 1 2 in auto)
     then have "integral\<^sup>L (lebesgue_on {0..pi}) (\<phi> n) = integral\<^sup>L (lebesgue_on {0..\<delta>}) (\<phi> n) + integral\<^sup>L (lebesgue_on {\<delta>..pi}) (\<phi> n)"
       by (rule integral_combine) (use \<open>0 < \<delta>\<close> \<open>\<delta> \<le> pi\<close> in auto)
     moreover have "\<bar>integral\<^sup>L (lebesgue_on {0..\<delta>}) (\<phi> n)\<bar> \<le> LINT x|lebesgue_on {0..\<delta>}. \<bar>f(t + x) + f(t - x) - 2 * l\<bar> / x"
     proof (rule integral_abs_bound_integral)
       show "integrable (lebesgue_on {0..\<delta>}) (\<phi> (real n))"
-        by (meson integrable_subinterval_real \<open>\<delta> \<le> pi\<close> int atLeastatMost_subset_iff order_refl)
+        by (meson integrable_subinterval \<open>\<delta> \<le> pi\<close> int atLeastatMost_subset_iff order_refl)
       have "{0..\<delta>} \<subseteq> {0..d}"
         by (auto simp: \<delta>_def)
       then show "integrable (lebesgue_on {0..\<delta>}) (\<lambda>x. \<bar>f(t + x) + f(t - x) - 2 * l\<bar> / x)"
-        by (rule integrable_subinterval_real [OF int0d])
+        by (rule integrable_subinterval [OF int0d])
       show "\<bar>\<phi> (real n) x\<bar> \<le> \<bar>f(t + x) + f(t - x) - 2 * l\<bar> / x"
         if "x \<in> space (lebesgue_on {0..\<delta>})" for x
         using that
         apply (auto simp: \<phi>_def divide_simps abs_mult)
         by (simp add: mult.commute mult_left_le)
     qed
     ultimately have "\<bar>integral\<^sup>L (lebesgue_on {0..pi}) (\<phi> n)\<bar> < e/2 + e/2"
       using N [OF that] e2 unfolding \<theta>_def by linarith
     then show ?thesis
       by simp
   qed
   then show "\<exists>N. \<forall>n\<ge>N. \<bar>integral\<^sup>L (lebesgue_on {0..pi}) (\<phi> (real n)) - 0\<bar> < e"
     by force
 qed
   then show ?thesis
     unfolding \<phi>_def using Fourier_sum_limit_sine_part assms pi_gt_zero by blast
 qed
 
 
 subsection\<open>Cesaro summability of Fourier series using Fejér kernel\<close>
 
 definition Fejer_kernel :: "nat \<Rightarrow> real \<Rightarrow> real"
   where
   "Fejer_kernel \<equiv> \<lambda>n x. if n = 0 then 0 else (\<Sum>r<n. Dirichlet_kernel r x) / n"
 
 lemma Fejer_kernel:
      "Fejer_kernel n x =
         (if n = 0 then 0
          else if x = 0 then n/2
          else sin(n / 2 * x) ^ 2 / (2 * n * sin(x/2) ^ 2))"
 proof (cases "x=0 \<or> sin(x/2) = 0")
   case True
   have "(\<Sum>r<n. (1 + real r * 2)) = real n * real n"
     by (induction n) (auto simp: algebra_simps)
   with True show ?thesis
     by (auto simp: Fejer_kernel_def Dirichlet_kernel_def field_simps simp flip: sum_divide_distrib)
 next
   case False
   have "sin (x/2) * (\<Sum>r<n. sin ((real r * 2 + 1) * x / 2)) =
         sin (real n * x / 2) * sin (real n * x / 2)"
   proof (induction n)
   next
     case (Suc n)
     then show ?case
       apply (simp add: field_simps sin_times_sin)
       apply (simp add: add_divide_distrib)
       done
   qed auto
   then show ?thesis
     using False
     unfolding Fejer_kernel_def Dirichlet_kernel_def
     by (simp add: divide_simps power2_eq_square mult.commute flip: sum_divide_distrib)
 qed
 
 lemma Fejer_kernel_0 [simp]: "Fejer_kernel 0 x = 0"  "Fejer_kernel n 0 = n/2"
   by (auto simp: Fejer_kernel)
 
 lemma Fejer_kernel_continuous_strong:
   "continuous_on {-(2 * pi)<..<2 * pi} (Fejer_kernel n)"
 proof (cases "n=0")
   case False
   then show ?thesis
   by (simp add: Fejer_kernel_def continuous_intros Dirichlet_kernel_continuous_strong)
 qed (simp add: Fejer_kernel_def)
 
 lemma Fejer_kernel_continuous:
   "continuous_on {-pi..pi} (Fejer_kernel n)"
   apply (rule continuous_on_subset [OF Fejer_kernel_continuous_strong])
   apply (simp add: subset_iff)
   using pi_gt_zero apply linarith
   done
 
 
 lemma absolutely_integrable_mult_Fejer_kernel:
   assumes "f absolutely_integrable_on {-pi..pi}"
   shows "(\<lambda>x. Fejer_kernel n x * f x) absolutely_integrable_on {-pi..pi}"
 proof (rule absolutely_integrable_bounded_measurable_product_real)
   show "Fejer_kernel n \<in> borel_measurable (lebesgue_on {-pi..pi})"
     by (simp add: Fejer_kernel_continuous continuous_imp_measurable_on_sets_lebesgue)
   show "bounded (Fejer_kernel n ` {-pi..pi})"
     using Fejer_kernel_continuous compact_continuous_image compact_imp_bounded by blast
 qed (use assms in auto)
 
 
 lemma absolutely_integrable_mult_Fejer_kernel_reflected1:
   assumes f: "f absolutely_integrable_on {-pi..pi}"
     and periodic: "\<And>x. f(x + 2*pi) = f x"
   shows "(\<lambda>x. Fejer_kernel n x * f(t + x)) absolutely_integrable_on {-pi..pi}"
   using assms
   by (force intro: absolutely_integrable_mult_Fejer_kernel absolutely_integrable_periodic_offset)
 
 lemma absolutely_integrable_mult_Fejer_kernel_reflected2:
   assumes f: "f absolutely_integrable_on {-pi..pi}"
     and periodic: "\<And>x. f(x + 2*pi) = f x"
   shows "(\<lambda>x. Fejer_kernel n x * f(t - x)) absolutely_integrable_on {-pi..pi}"
 proof -
   have "(\<lambda>x. f(t - x)) absolutely_integrable_on {-pi..pi}"
     using assms
     apply (subst absolutely_integrable_reflect_real [symmetric])
     apply (simp add: absolutely_integrable_periodic_offset)
     done
   then show ?thesis
     by (rule absolutely_integrable_mult_Fejer_kernel)
 qed
 
 lemma absolutely_integrable_mult_Fejer_kernel_reflected3:
   shows "(\<lambda>x. Fejer_kernel n x * c) absolutely_integrable_on {-pi..pi}"
   using absolutely_integrable_on_const absolutely_integrable_mult_Fejer_kernel by blast
 
 
 lemma absolutely_integrable_mult_Fejer_kernel_reflected_part1:
   assumes f: "f absolutely_integrable_on {-pi..pi}"
     and periodic: "\<And>x. f(x + 2*pi) = f x" and "d \<le> pi"
   shows "(\<lambda>x. Fejer_kernel n x * f(t + x)) absolutely_integrable_on {0..d}"
   by (rule absolutely_integrable_on_subinterval [OF absolutely_integrable_mult_Fejer_kernel_reflected1]) (auto simp: assms)
 
 lemma absolutely_integrable_mult_Fejer_kernel_reflected_part2:
   assumes f: "f absolutely_integrable_on {-pi..pi}"
     and periodic: "\<And>x. f(x + 2*pi) = f x" and "d \<le> pi"
   shows "(\<lambda>x. Fejer_kernel n x * f(t - x)) absolutely_integrable_on {0..d}"
   by (rule absolutely_integrable_on_subinterval [OF absolutely_integrable_mult_Fejer_kernel_reflected2]) (auto simp: assms)
 
 lemma absolutely_integrable_mult_Fejer_kernel_reflected_part3:
   assumes "d \<le> pi"
   shows "(\<lambda>x. Fejer_kernel n x * c) absolutely_integrable_on {0..d}"
   by (rule absolutely_integrable_on_subinterval [OF absolutely_integrable_mult_Fejer_kernel_reflected2]) (auto simp: assms)
 
 lemma absolutely_integrable_mult_Fejer_kernel_reflected_part4:
   assumes f: "f absolutely_integrable_on {-pi..pi}"
     and periodic: "\<And>x. f(x + 2*pi) = f x" and "d \<le> pi"
   shows "(\<lambda>x. Fejer_kernel n x * (f(t + x) + f(t - x))) absolutely_integrable_on {0..d}"
   unfolding distrib_left
   by (intro set_integral_add absolutely_integrable_mult_Fejer_kernel_reflected_part1 absolutely_integrable_mult_Fejer_kernel_reflected_part2 assms)
 
 lemma absolutely_integrable_mult_Fejer_kernel_reflected_part5:
   assumes f: "f absolutely_integrable_on {-pi..pi}"
     and periodic: "\<And>x. f(x + 2*pi) = f x" and "d \<le> pi"
   shows "(\<lambda>x. Fejer_kernel n x * ((f(t + x) + f(t - x)) - c)) absolutely_integrable_on {0..d}"
   unfolding distrib_left right_diff_distrib
   by (intro set_integral_add set_integral_diff absolutely_integrable_on_const
       absolutely_integrable_mult_Fejer_kernel_reflected_part1 absolutely_integrable_mult_Fejer_kernel_reflected_part2 assms, auto)
 
 
 lemma Fourier_sum_offset_Fejer_kernel_half:
   fixes n::nat
   assumes f: "f absolutely_integrable_on {-pi..pi}"
     and periodic: "\<And>x. f(x + 2*pi) = f x" and "n > 0"
   shows "(\<Sum>r<n. \<Sum>k\<le>2*r. Fourier_coefficient f k * trigonometric_set k t) / n - l
        = (LINT x|lebesgue_on {0..pi}. Fejer_kernel n x * (f(t + x) + f(t - x) - 2 * l)) / pi"
 proof -
   have "(\<Sum>r<n. \<Sum>k\<le>2 * r. Fourier_coefficient f k * trigonometric_set k t) - real n * l
        = (\<Sum>r<n. (\<Sum>k\<le>2 * r. Fourier_coefficient f k * trigonometric_set k t) - l)"
     by (simp add: sum_subtractf)
   also have "\<dots> = (\<Sum>r<n.
                       (LINT x|lebesgue_on {0..pi}. Dirichlet_kernel r x * (f(t + x) + f(t - x) - 2 * l)) / pi)"
     by (simp add: Fourier_sum_offset_Dirichlet_kernel_half assms)
   also have "\<dots> = real n * ((LINT x|lebesgue_on {0..pi}. Fejer_kernel n x * (f(t + x) + f(t - x) - 2 * l)) / pi)"
   proof -
     have "integrable (lebesgue_on {0..pi}) (\<lambda>x. Dirichlet_kernel i x * (f(t + x) + f(t - x) - 2 * l))" for i
       using absolutely_integrable_mult_Dirichlet_kernel_reflected_part2(2) f periodic
       by (force simp: intro!: absolutely_integrable_imp_integrable)
     then show ?thesis
       using \<open>n > 0\<close>
       apply (simp add: Fejer_kernel_def flip: sum_divide_distrib)
       apply (simp add: sum_distrib_right flip: Bochner_Integration.integral_sum [symmetric])
       done
   qed
   finally have "(\<Sum>r<n. \<Sum>k\<le>2 * r. Fourier_coefficient f k * trigonometric_set k t) - real n * l = real n * ((LINT x|lebesgue_on {0..pi}. Fejer_kernel n x * (f(t + x) + f(t - x) - 2 * l)) / pi)" .
   with \<open>n > 0\<close> show ?thesis
     by (auto simp: mult.commute divide_simps)
 qed
 
 
 lemma Fourier_sum_limit_Fejer_kernel_half:
   fixes n::nat
   assumes f: "f absolutely_integrable_on {-pi..pi}"
     and periodic: "\<And>x. f(x + 2*pi) = f x"
   shows "(\<lambda>n. ((\<Sum>r<n. \<Sum>k\<le>2*r. Fourier_coefficient f k * trigonometric_set k t)) / n) \<longlonglongrightarrow> l
          \<longleftrightarrow>
          ((\<lambda>n. integral\<^sup>L (lebesgue_on {0..pi}) (\<lambda>x. Fejer_kernel n x * ((f(t + x) + f(t - x)) - 2*l)))  \<longlonglongrightarrow> 0)"
         (is "?lhs = ?rhs")
 proof -
   have "?lhs \<longleftrightarrow>
         (\<lambda>n. ((\<Sum>r<n. \<Sum>k\<le>2*r.  Fourier_coefficient f k * trigonometric_set k t)) / n - l) \<longlonglongrightarrow> 0"
     by (simp add: LIM_zero_iff)
   also have "\<dots> \<longleftrightarrow>
         (\<lambda>n. ((((\<Sum>r<n. \<Sum>k\<le>2*r.  Fourier_coefficient f k * trigonometric_set k t)) / n) - l) * pi) \<longlonglongrightarrow> 0"
     using real_tendsto_mult_right_iff [OF pi_neq_zero] by simp
   also have "\<dots> \<longleftrightarrow> ?rhs"
     apply (intro Lim_transform_eq [OF Lim_transform_eventually [of "\<lambda>n. 0"]] eventually_sequentiallyI [of 1])
     apply (simp_all add: Fourier_sum_offset_Fejer_kernel_half assms)
     done
   finally show ?thesis .
 qed
 
 
 lemma has_integral_Fejer_kernel:
   "has_bochner_integral (lebesgue_on {-pi..pi}) (Fejer_kernel n) (if n = 0 then 0 else pi)"
 proof -
   have "has_bochner_integral (lebesgue_on {-pi..pi}) (\<lambda>x. (\<Sum>r<n. Dirichlet_kernel r x) / real n) ((\<Sum>r<n. pi) / n)"
     by (simp add: has_bochner_integral_iff integrable_Dirichlet_kernel has_bochner_integral_divide has_bochner_integral_sum)
   then show ?thesis
     by (auto simp: Fejer_kernel_def)
 qed
 
 lemma has_integral_Fejer_kernel_half:
   "has_bochner_integral (lebesgue_on {0..pi}) (Fejer_kernel n) (if n = 0 then 0 else pi/2)"
 proof -
   have "has_bochner_integral (lebesgue_on {0..pi}) (\<lambda>x. (\<Sum>r<n. Dirichlet_kernel r x) / real n) ((\<Sum>r<n. pi/2) / n)"
     apply (intro has_bochner_integral_sum has_bochner_integral_divide)
     using not_integrable_integral_eq by (force simp: has_bochner_integral_iff)
   then show ?thesis
     by (auto simp: Fejer_kernel_def)
 qed
 
 lemma Fejer_kernel_pos_le [simp]: "Fejer_kernel n x \<ge> 0"
   by (simp add: Fejer_kernel)
 
 
 theorem Fourier_Fejer_Cesaro_summable:
   assumes f: "f absolutely_integrable_on {-pi..pi}"
     and periodic: "\<And>x. f(x + 2*pi) = f x"
     and fl: "(f \<longlongrightarrow> l) (at t within atMost t)"
     and fr: "(f \<longlongrightarrow> r) (at t within atLeast t)"
   shows "(\<lambda>n. (\<Sum>m<n. \<Sum>k\<le>2*m. Fourier_coefficient f k * trigonometric_set k t) / n) \<longlonglongrightarrow> (l+r) / 2"
 proof -
   define h where "h \<equiv> \<lambda>u. (f(t+u) + f(t-u)) - (l+r)"
   have "(\<lambda>n. LINT u|lebesgue_on {0..pi}. Fejer_kernel n u * h u) \<longlonglongrightarrow> 0"
   proof -
     have h0: "(h \<longlongrightarrow> 0) (at 0 within atLeast 0)"
     proof -
       have l0: "((\<lambda>u. f(t-u) - l) \<longlongrightarrow> 0) (at 0 within {0..})"
         using fl
         unfolding Lim_within
         apply (elim all_forward imp_forward ex_forward conj_forward asm_rl, clarify)
         apply (drule_tac x="t-x" in bspec)
          apply (auto simp: dist_norm)
         done
       have r0: "((\<lambda>u. f(t + u) - r) \<longlongrightarrow> 0) (at 0 within {0..})"
         using fr
         unfolding Lim_within
         apply (elim all_forward imp_forward ex_forward conj_forward asm_rl, clarify)
         apply (drule_tac x="t+x" in bspec)
          apply (auto simp: dist_norm)
         done
       show ?thesis
         using tendsto_add [OF l0 r0] by (simp add: h_def algebra_simps)
     qed
     show ?thesis
       unfolding lim_sequentially dist_real_def diff_0_right
     proof clarify
       fix e::real
       assume "e > 0"
       then obtain x' where "0 < x'" "\<And>x. \<lbrakk>0 < x; x < x'\<rbrakk> \<Longrightarrow> \<bar>h x\<bar> < e / (2 * pi)"
         using h0 unfolding Lim_within dist_real_def
         by (auto simp: dest: spec [where x="e/2/pi"])
       then obtain \<xi> where "0 < \<xi>" "\<xi> < pi" and \<xi>: "\<And>x. 0 < x \<and> x \<le> \<xi> \<Longrightarrow> \<bar>h x\<bar> < e/2/pi"
         apply (intro that [where \<xi>="min x' pi/2"], auto)
         using m2pi_less_pi by linarith
       have ftx: "(\<lambda>x. f(t+x)) absolutely_integrable_on {-pi..pi}"
         using absolutely_integrable_periodic_offset assms by auto
       then have ftmx: "(\<lambda>x. f(t-x)) absolutely_integrable_on {-pi..pi}"
         by (simp flip: absolutely_integrable_reflect_real [where f= "(\<lambda>x. f(t+x))"])
       have h_aint: "h absolutely_integrable_on {-pi..pi}"
         unfolding h_def
         by (intro absolutely_integrable_on_const set_integral_diff set_integral_add, auto simp: ftx ftmx)
       have "(\<lambda>n. LINT x|lebesgue_on {\<xi>..pi}. Fejer_kernel n x * h x) \<longlonglongrightarrow> 0"
       proof (rule Lim_null_comparison)
         define \<phi> where "\<phi> \<equiv> \<lambda>n. (LINT x|lebesgue_on {\<xi>..pi}. \<bar>h x\<bar> / (2 * sin(x/2) ^ 2)) / n"
         show "\<forall>\<^sub>F n in sequentially. norm (LINT x|lebesgue_on {\<xi>..pi}. Fejer_kernel n x * h x) \<le> \<phi> n"
         proof (rule eventually_sequentiallyI)
           fix n::nat assume "n \<ge> 1"
           have hint: "(\<lambda>x. h x / (2 * sin(x/2) ^ 2)) absolutely_integrable_on {\<xi>..pi}"
             unfolding divide_inverse mult.commute [of "h _"]
           proof (rule absolutely_integrable_bounded_measurable_product_real)
             have cont: "continuous_on {\<xi>..pi} (\<lambda>x. inverse (2 * (sin (x * inverse 2))\<^sup>2))"
               using \<open>0 < \<xi>\<close> by (intro continuous_intros) (auto simp: sin_zero_pi_iff)
             show "(\<lambda>x. inverse (2 * (sin (x * inverse 2))\<^sup>2)) \<in> borel_measurable (lebesgue_on {\<xi>..pi})"
               using \<open>0 < \<xi>\<close>
               by (intro cont continuous_imp_measurable_on_sets_lebesgue) auto
             show "bounded ((\<lambda>x. inverse (2 * (sin (x * inverse 2))\<^sup>2)) ` {\<xi>..pi})"
               using cont by (simp add: compact_continuous_image compact_imp_bounded)
             show "h absolutely_integrable_on {\<xi>..pi}"
               using \<open>0 < \<xi>\<close> \<open>\<xi> < pi\<close> by (auto intro: absolutely_integrable_on_subinterval [OF h_aint])
           qed auto
           then have *: "integrable (lebesgue_on {\<xi>..pi}) (\<lambda>x. \<bar>h x\<bar> / (2 * (sin (x/2))\<^sup>2))"
             by (simp add: absolutely_integrable_measurable o_def)
           define \<psi> where
               "\<psi> \<equiv> \<lambda>x. (if n = 0 then 0 else if x = 0 then n/2
                          else (sin (real n / 2 * x))\<^sup>2 / (2 * real n * (sin (x/2))\<^sup>2)) * h x"
           have "\<bar>LINT x|lebesgue_on {\<xi>..pi}. \<psi> x\<bar>
               \<le> (LINT x|lebesgue_on {\<xi>..pi}. \<bar>h x\<bar> / (2 * (sin (x/2))\<^sup>2) / n)"
           proof (rule integral_abs_bound_integral)
             show **: "integrable (lebesgue_on {\<xi>..pi}) (\<lambda>x. \<bar>h x\<bar> / (2 * (sin (x/2))\<^sup>2) / n)"
               using Bochner_Integration.integrable_mult_left [OF *, of "1/n"]
               by (simp add: field_simps)
             show \<dagger>: "\<bar>\<psi> x\<bar> \<le> \<bar>h x\<bar> / (2 * (sin (x/2))\<^sup>2) / real n"
               if "x \<in> space (lebesgue_on {\<xi>..pi})" for x
               using that \<open>0 < \<xi>\<close>
               apply (simp add: \<psi>_def divide_simps mult_less_0_iff abs_mult)
               apply (auto simp: square_le_1 mult_left_le_one_le)
               done
             show "integrable (lebesgue_on {\<xi>..pi}) \<psi>"
             proof (rule measurable_bounded_by_integrable_imp_lebesgue_integrable [OF _ **])
               let ?g = "\<lambda>x. \<bar>h x\<bar> / (2 * sin(x/2) ^ 2) / n"
               have ***: "integrable (lebesgue_on {\<xi>..pi}) (\<lambda>x. (sin (n/2 * x))\<^sup>2 * (inverse (2 * (sin (x/2))\<^sup>2) * h x))"
               proof (rule absolutely_integrable_imp_integrable [OF absolutely_integrable_bounded_measurable_product_real])
                 show "(\<lambda>x. (sin (real n / 2 * x))\<^sup>2) \<in> borel_measurable (lebesgue_on {\<xi>..pi})"
                   by (intro continuous_imp_measurable_on_sets_lebesgue continuous_intros) auto
                 show "bounded ((\<lambda>x. (sin (real n / 2 * x))\<^sup>2) ` {\<xi>..pi})"
                   by (force simp: square_le_1 intro: boundedI [where B=1])
                 show "(\<lambda>x. inverse (2 * (sin (x/2))\<^sup>2) * h x) absolutely_integrable_on {\<xi>..pi}"
                   using hint by (simp add: divide_simps)
               qed auto
               show "\<psi> \<in> borel_measurable (lebesgue_on {\<xi>..pi})"
                 apply (rule borel_measurable_integrable)
                 apply (rule integrable_cong [where f = "\<lambda>x. sin(n / 2 * x) ^ 2 / (2 * n * sin(x/2) ^ 2) * h x", OF refl, THEN iffD1])
                 using \<open>0 < \<xi>\<close> **
                  apply (force simp: \<psi>_def divide_simps algebra_simps mult_less_0_iff abs_mult)
                 using Bochner_Integration.integrable_mult_left [OF ***, of "1/n"]
                 by (simp add: field_simps)
               show "norm (\<psi> x) \<le> ?g x" if "x \<in> {\<xi>..pi}" for x
                 using that \<dagger> by (simp add: \<psi>_def)
             qed auto
           qed
           then show "norm (LINT x|lebesgue_on {\<xi>..pi}. Fejer_kernel n x * h x) \<le> \<phi> n"
             by (simp add: Fejer_kernel \<phi>_def \<psi>_def flip: Bochner_Integration.integral_divide [OF *])
         qed
         show "\<phi> \<longlonglongrightarrow> 0"
           unfolding \<phi>_def divide_inverse
           by (simp add: tendsto_mult_right_zero lim_inverse_n)
       qed
       then obtain N where N: "\<And>n. n \<ge> N \<Longrightarrow> \<bar>LINT x|lebesgue_on {\<xi>..pi}. Fejer_kernel n x * h x\<bar> < e/2"
         unfolding lim_sequentially by (metis half_gt_zero_iff norm_conv_dist real_norm_def \<open>e > 0\<close>)
       show "\<exists>N. \<forall>n\<ge>N. \<bar>(LINT u|lebesgue_on {0..pi}. Fejer_kernel n u * h u)\<bar> < e"
       proof (intro exI allI impI)
         fix n :: nat
         assume n: "n \<ge> max 1 N"
         with N have 1: "\<bar>LINT x|lebesgue_on {\<xi>..pi}. Fejer_kernel n x * h x\<bar> < e/2"
           by simp
         have "integral\<^sup>L (lebesgue_on {0..\<xi>}) (Fejer_kernel n) \<le> integral\<^sup>L (lebesgue_on {0..pi}) (Fejer_kernel n)"
          using \<open>\<xi> < pi\<close> has_bochner_integral_iff has_integral_Fejer_kernel_half
          by (force intro!: integral_mono_lebesgue_on_AE)
         also have "\<dots> \<le> pi"
           using has_integral_Fejer_kernel_half by (simp add: has_bochner_integral_iff)
         finally have int_le_pi: "integral\<^sup>L (lebesgue_on {0..\<xi>}) (Fejer_kernel n) \<le> pi" .
         have 2: "\<bar>LINT x|lebesgue_on {0..\<xi>}. Fejer_kernel n x * h x\<bar> \<le> (LINT x|lebesgue_on {0..\<xi>}. Fejer_kernel n x * e/2/pi)"
         proof -
           have eq: "integral\<^sup>L (lebesgue_on {0..\<xi>}) (\<lambda>x. Fejer_kernel n x * h x)
                   = integral\<^sup>L (lebesgue_on {0..\<xi>}) (\<lambda>x. Fejer_kernel n x * (if x = 0 then 0 else h x))"
           proof (rule integral_cong_AE)
             have \<dagger>: "{u. u \<noteq> 0 \<longrightarrow> P u} \<inter> {0..\<xi>} = {0} \<union> Collect P \<inter> {0..\<xi>}"
               "{u. u \<noteq> 0 \<and> P u} \<inter> {0..\<xi>} = (Collect P \<inter> {0..\<xi>}) - {0}" for P
               using \<open>0 < \<xi>\<close> by auto
             have *: "- {0} \<inter> A \<inter> {0..\<xi>} = A \<inter> {0..\<xi>} - {0}" for A
               by auto
             show "(\<lambda>x. Fejer_kernel n x * h x) \<in> borel_measurable (lebesgue_on {0..\<xi>})"
               using \<open>\<xi> < pi\<close>
               by (intro absolutely_integrable_imp_borel_measurable h_aint
                   absolutely_integrable_on_subinterval [OF absolutely_integrable_mult_Fejer_kernel], auto)
             then show "(\<lambda>x. Fejer_kernel n x * (if x = 0 then 0 else h x)) \<in> borel_measurable (lebesgue_on {0..\<xi>})"
               apply (simp add: in_borel_measurable Ball_def vimage_def Collect_conj_eq Collect_imp_eq * flip: Collect_neg_eq)
               apply (elim all_forward imp_forward asm_rl)
               using \<open>0 < \<xi>\<close>
               apply (auto simp: \<dagger> insert_sets_lebesgue_on cong: conj_cong)
               done
             have 0: "{0} \<in> null_sets (lebesgue_on {0..\<xi>})"
               using \<open>0 < \<xi>\<close> by (simp add: null_sets_restrict_space)
             then show "AE x in lebesgue_on {0..\<xi>}. Fejer_kernel n x * h x = Fejer_kernel n x * (if x = 0 then 0 else h x)"
               by (auto intro: AE_I' [OF 0])
           qed
           show ?thesis
             unfolding eq
           proof (rule integral_abs_bound_integral)
             have "(\<lambda>x. if x = 0 then 0 else h x) absolutely_integrable_on {- pi..pi}"
             proof (rule absolutely_integrable_spike [OF h_aint])
               show "negligible {0}"
                 by auto
             qed auto
             with \<open>0 < \<xi>\<close> \<open>\<xi> < pi\<close> show "integrable (lebesgue_on {0..\<xi>}) (\<lambda>x. Fejer_kernel n x * (if x = 0 then 0 else h x))"
               by (intro absolutely_integrable_imp_integrable h_aint absolutely_integrable_on_subinterval [OF absolutely_integrable_mult_Fejer_kernel]) auto
             show "integrable (lebesgue_on {0..\<xi>}) (\<lambda>x. Fejer_kernel n x * e / 2 / pi)"
               by (simp add: absolutely_integrable_imp_integrable \<open>\<xi> < pi\<close> absolutely_integrable_mult_Fejer_kernel_reflected_part3 less_eq_real_def)
             show "\<bar>Fejer_kernel n x * (if x = 0 then 0 else h x)\<bar> \<le> Fejer_kernel n x * e / 2 / pi"
               if "x \<in> space (lebesgue_on {0..\<xi>})" for x
               using that \<xi> [of x] \<open>e > 0\<close>
               by (auto simp: abs_mult eq simp flip: times_divide_eq_right intro: mult_left_mono)
           qed
         qed
         have 3: "\<dots> \<le> e/2"
           using int_le_pi \<open>0 < e\<close>
           by (simp add: divide_simps mult.commute [of e])
         have "integrable (lebesgue_on {0..pi}) (\<lambda>x. Fejer_kernel n x * h x)"
           unfolding h_def
           by (simp add: absolutely_integrable_imp_integrable absolutely_integrable_mult_Fejer_kernel_reflected_part5 assms)
         then have "LINT x|lebesgue_on {0..pi}. Fejer_kernel n x * h x
            = (LINT x|lebesgue_on {0..\<xi>}. Fejer_kernel n x * h x) + (LINT x|lebesgue_on {\<xi>..pi}. Fejer_kernel n x * h x)"
           by (rule integral_combine) (use  \<open>0 < \<xi>\<close> \<open>\<xi> < pi\<close> in auto)
         then show "\<bar>LINT u|lebesgue_on {0..pi}. Fejer_kernel n u * h u\<bar> < e"
           using 1 2 3 by linarith
       qed
     qed
   qed
   then show ?thesis
     unfolding h_def by (simp add: Fourier_sum_limit_Fejer_kernel_half assms add_divide_distrib)
 qed
 
 corollary Fourier_Fejer_Cesaro_summable_simple:
 assumes f: "continuous_on UNIV f"
     and periodic: "\<And>x. f(x + 2*pi) = f x"
   shows "(\<lambda>n. (\<Sum>m<n. \<Sum>k\<le>2*m. Fourier_coefficient f k * trigonometric_set k x) / n) \<longlonglongrightarrow> f x"
 proof -
   have "(\<lambda>n. (\<Sum>m<n. \<Sum>k\<le>2*m. Fourier_coefficient f k * trigonometric_set k x) / n) \<longlonglongrightarrow> (f x + f x) / 2"
   proof (rule Fourier_Fejer_Cesaro_summable)
     show "f absolutely_integrable_on {- pi..pi}"
       using absolutely_integrable_continuous_real continuous_on_subset f by blast
     show "(f \<longlongrightarrow> f x) (at x within {..x})"  "(f \<longlongrightarrow> f x) (at x within {x..})"
       using Lim_at_imp_Lim_at_within continuous_on_def f by blast+
   qed (auto simp: periodic Lim_at_imp_Lim_at_within continuous_on_def f)
   then show ?thesis
     by simp
 qed
 
 end
 
 
diff --git a/thys/Fourier/Fourier_Aux1.thy b/thys/Fourier/Fourier_Aux1.thy
deleted file mode 100644
--- a/thys/Fourier/Fourier_Aux1.thy
+++ /dev/null
@@ -1,577 +0,0 @@
-section\<open>Material already in development version as of September 2019\<close>
-
-theory Fourier_Aux1
-  imports "HOL-Analysis.Analysis"
-begin
-
-lemma square_powr_half [simp]:
-  fixes x::real shows "x\<^sup>2 powr (1/2) = \<bar>x\<bar>"
-  by (simp add: powr_half_sqrt)
-
-lemma frac_1_eq: "frac (x+1) = frac x"
-  by (simp add: frac_def)
-
-context comm_monoid_set
-begin
-
-lemma in_pairs: "F g {2*m..Suc(2*n)} = F (\<lambda>i. g(2*i) \<^bold>* g(Suc(2*i))) {m..n}"
-proof (induction n)
-  case 0
-  show ?case
-    by (cases "m=0") auto
-next
-  case (Suc n)
-  then show ?case
-    by (auto simp: assoc split: if_split_asm)
-qed
-
-lemma in_pairs_0: "F g {..Suc(2*n)} = F (\<lambda>i. g(2*i) \<^bold>* g(Suc(2*i))) {..n}"
-  using in_pairs [of _ 0 n] by (simp add: atLeast0AtMost)
-
-end
-
-context linordered_nonzero_semiring
-begin
-
-lemma numeral_le_iff: "numeral m \<le> numeral n \<longleftrightarrow> m \<le> n"
-proof -
-  have "of_nat (numeral m) \<le> of_nat (numeral n) \<longleftrightarrow> m \<le> n"
-    by (simp only: less_eq_num_def nat_of_num_numeral of_nat_le_iff)
-  then show ?thesis by simp
-qed
-
-lemma one_le_numeral: "1 \<le> numeral n"
-  using numeral_le_iff [of num.One n] by (simp add: numeral_One)
-
-lemma numeral_le_one_iff: "numeral n \<le> 1 \<longleftrightarrow> n \<le> num.One"
-  using numeral_le_iff [of n num.One] by (simp add: numeral_One)
-
-lemma numeral_less_iff: "numeral m < numeral n \<longleftrightarrow> m < n"
-proof -
-  have "of_nat (numeral m) < of_nat (numeral n) \<longleftrightarrow> m < n"
-    unfolding less_num_def nat_of_num_numeral of_nat_less_iff ..
-  then show ?thesis by simp
-qed
-
-lemma not_numeral_less_one: "\<not> numeral n < 1"
-  using numeral_less_iff [of n num.One] by (simp add: numeral_One)
-
-lemma one_less_numeral_iff: "1 < numeral n \<longleftrightarrow> num.One < n"
-  using numeral_less_iff [of num.One n] by (simp add: numeral_One)
-
-lemma zero_le_numeral: "0 \<le> numeral n"
-  using dual_order.trans one_le_numeral zero_le_one by blast
-
-lemma zero_less_numeral: "0 < numeral n"
-  using less_linear not_numeral_less_one order.strict_trans zero_less_one by blast
-
-lemma not_numeral_le_zero: "\<not> numeral n \<le> 0"
-  by (simp add: not_le zero_less_numeral)
-
-lemma not_numeral_less_zero: "\<not> numeral n < 0"
-  by (simp add: not_less zero_le_numeral)
-
-lemmas le_numeral_extra =
-  zero_le_one not_one_le_zero
-  order_refl [of 0] order_refl [of 1]
-
-lemmas less_numeral_extra =
-  zero_less_one not_one_less_zero
-  less_irrefl [of 0] less_irrefl [of 1]
-
-lemmas le_numeral_simps [simp] =
-  numeral_le_iff
-  one_le_numeral
-  numeral_le_one_iff
-  zero_le_numeral
-  not_numeral_le_zero
-
-lemmas less_numeral_simps [simp] =
-  numeral_less_iff
-  one_less_numeral_iff
-  not_numeral_less_one
-  zero_less_numeral
-  not_numeral_less_zero
-
-end
-
-lemma powr_numeral [simp]: "0 \<le> x \<Longrightarrow> x powr (numeral n :: real) = x ^ (numeral n)"
-  by (metis less_le power_zero_numeral powr_0 of_nat_numeral powr_realpow)
-
-context linordered_semidom
-begin
-
-lemma power_mono_iff [simp]:
-  shows "\<lbrakk>a \<ge> 0; b \<ge> 0; n>0\<rbrakk> \<Longrightarrow> a ^ n \<le> b ^ n \<longleftrightarrow> a \<le> b"
-  using power_mono [of a b] power_strict_mono [of b a] not_le by auto
-
-end
-
-lemma tendsto_sup[tendsto_intros]:
-  fixes X :: "'a \<Rightarrow> 'b::ordered_euclidean_space"
-  assumes "(X \<longlongrightarrow> x) net" "(Y \<longlongrightarrow> y) net"
-  shows "((\<lambda>i. sup (X i) (Y i)) \<longlongrightarrow> sup x y) net"
-   unfolding sup_max eucl_sup by (intro assms tendsto_intros)
-
-lemma tendsto_inf[tendsto_intros]:
-  fixes X :: "'a \<Rightarrow> 'b::ordered_euclidean_space"
-  assumes "(X \<longlongrightarrow> x) net" "(Y \<longlongrightarrow> y) net"
-  shows "((\<lambda>i. inf (X i) (Y i)) \<longlongrightarrow> inf x y) net"
-   unfolding inf_min eucl_inf by (intro assms tendsto_intros)
-
-lemma continuous_on_max [continuous_intros]:
-  fixes f g :: "'a::topological_space \<Rightarrow> 'b::linorder_topology"
-  shows "continuous_on A f \<Longrightarrow> continuous_on A g \<Longrightarrow> continuous_on A (\<lambda>x. max (f x) (g x))"
-  by (auto simp: continuous_on_def intro!: tendsto_max)
-
-lemma continuous_on_min [continuous_intros]:
-  fixes f g :: "'a::topological_space \<Rightarrow> 'b::linorder_topology"
-  shows "continuous_on A f \<Longrightarrow> continuous_on A g \<Longrightarrow> continuous_on A (\<lambda>x. min (f x) (g x))"
-  by (auto simp: continuous_on_def intro!: tendsto_min)
-
-lemma Ints_eq_abs_less1:
-  fixes x:: "'a :: linordered_idom"
-  shows "\<lbrakk>x \<in> Ints; y \<in> Ints\<rbrakk> \<Longrightarrow> x = y \<longleftrightarrow> abs (x-y) < 1"
-  using eq_iff_diff_eq_0 by (fastforce intro: Ints_nonzero_abs_less1)
-
-lemma indicator_times_eq_if:
-  fixes f :: "'a \<Rightarrow> 'b::comm_ring_1"
-  shows "indicator S x * f x = (if x \<in> S then f x else 0)" "f x * indicator S x = (if x \<in> S then f x else 0)"
-  by auto
-
-lemma indicator_scaleR_eq_if:
-  fixes f :: "'a \<Rightarrow> 'b::real_vector"
-  shows "indicator S x *\<^sub>R f x = (if x \<in> S then f x else 0)"
-  by simp
-
-lemma lebesgue_on_UNIV_eq: "lebesgue_on UNIV = lebesgue"
-proof -
-  have "measure_of UNIV (sets lebesgue) (emeasure lebesgue) = lebesgue"
-    by (metis measure_of_of_measure space_borel space_completion space_lborel)
-  then show ?thesis
-    by (auto simp: restrict_space_def)
-qed
-
-lemma mem_limsup_iff: "x \<in> limsup A \<longleftrightarrow> (\<exists>\<^sub>F n in sequentially. x \<in> A n)"
-  by (simp add: Limsup_def) (metis (mono_tags) eventually_mono not_frequently)
-
-lemma integral_restrict_Int:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes "S \<in> sets lebesgue" "T \<in> sets lebesgue"
-  shows "integral\<^sup>L (lebesgue_on T) (\<lambda>x. if x \<in> S then f x else 0) = integral\<^sup>L (lebesgue_on (S \<inter> T)) f"
-proof -
-  have "(\<lambda>x. indicat_real T x *\<^sub>R (if x \<in> S then f x else 0)) = (\<lambda>x. indicat_real (S \<inter> T) x *\<^sub>R f x)"
-    by (force simp: indicator_def)
-  then show ?thesis
-    by (simp add: assms sets.Int Bochner_Integration.integral_restrict_space)
-qed
-
-lemma space_lebesgue_on [simp]: "space (lebesgue_on S) = S"
-  by (simp add: space_restrict_space)
-
-corollary absolutely_integrable_on_const [simp]:
-  fixes c :: "'a::euclidean_space"
-  assumes "S \<in> lmeasurable"
-  shows "(\<lambda>x. c) absolutely_integrable_on S"
-  by (metis (full_types) assms absolutely_integrable_integrable_bound integrable_on_const order_refl)
-
-lemma integral_restrict_UNIV:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes "S \<in> sets lebesgue"
-  shows "integral\<^sup>L lebesgue (\<lambda>x. if x \<in> S then f x else 0) = integral\<^sup>L (lebesgue_on S) f"
-  using integral_restrict_Int [of S UNIV f] assms
-  by (simp add: lebesgue_on_UNIV_eq)
-
-lemma tendsto_eventually: "eventually (\<lambda>x. f x = c) F \<Longrightarrow> filterlim f (nhds c) F"
-  by (simp add: eventually_mono eventually_nhds_x_imp_x filterlim_iff)
-
-lemma content_cbox_plus:
-  fixes x :: "'a::euclidean_space"
-  shows "content(cbox x (x + h *\<^sub>R One)) = (if h \<ge> 0 then h ^ DIM('a) else 0)"
-  by (simp add: algebra_simps content_cbox_if box_eq_empty)
-
-lemma finite_measure_lebesgue_on: "S \<in> lmeasurable \<Longrightarrow> finite_measure (lebesgue_on S)"
-  by (auto simp: finite_measureI fmeasurable_def emeasure_restrict_space)
-
-lemma measurable_bounded_by_integrable_imp_absolutely_integrable:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes "f \<in> borel_measurable (lebesgue_on S)" "S \<in> sets lebesgue"
-    and "g integrable_on S" and "\<And>x. x \<in> S \<Longrightarrow> norm(f x) \<le> (g x)"
-  shows "f absolutely_integrable_on S"
-  using assms absolutely_integrable_integrable_bound measurable_bounded_by_integrable_imp_integrable by blast
-
-lemma absolutely_integrable_measurable:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes "S \<in> sets lebesgue"
-  shows "f absolutely_integrable_on S \<longleftrightarrow> f \<in> borel_measurable (lebesgue_on S) \<and> integrable (lebesgue_on S) (norm \<circ> f)"
-    (is "?lhs = ?rhs")
-proof
-  assume L: ?lhs
-  then have "f \<in> borel_measurable (lebesgue_on S)"
-    by (simp add: absolutely_integrable_on_def integrable_imp_measurable)
-  then show ?rhs
-    using assms set_integrable_norm [of lebesgue S f] L
-    by (simp add: integrable_restrict_space set_integrable_def)
-next
-  assume ?rhs then show ?lhs
-    using assms integrable_on_lebesgue_on
-    by (metis absolutely_integrable_integrable_bound comp_def eq_iff measurable_bounded_by_integrable_imp_integrable)
-qed
-
-lemma lim_const_over_n [tendsto_intros]:
-  fixes a :: "'a::real_normed_field"
-  shows "(\<lambda>n. a / of_nat n) \<longlonglongrightarrow> 0"
-  using tendsto_mult [OF tendsto_const [of a] lim_1_over_n] by simp
-
-lemma add_mono_ennreal: "x < ennreal y \<Longrightarrow> x' < ennreal y' \<Longrightarrow> x + x' < ennreal (y + y')"
-  by (metis (full_types) add_strict_mono ennreal_less_zero_iff ennreal_plus less_le not_less zero_le)
-
-lemma enn2real_sum:"(\<And>i. i \<in> I \<Longrightarrow> f i < top) \<Longrightarrow> enn2real (sum f I) = sum (enn2real \<circ> f) I"
-  by (induction I rule: infinite_finite_induct) (auto simp: enn2real_plus)
-
-lemma sets_lebesgue_outer_open:
-  fixes e::real
-  assumes S: "S \<in> sets lebesgue" and "e > 0"
-  obtains T where "open T" "S \<subseteq> T" "(T - S) \<in> lmeasurable" "emeasure lebesgue (T - S) < ennreal e"
-proof -
-  obtain S' where S': "S \<subseteq> S'" "S' \<in> sets borel"
-              and null: "S' - S \<in> null_sets lebesgue"
-              and em: "emeasure lebesgue S = emeasure lborel S'"
-    using completion_upper[of S lborel] S by auto
-  then have f_S': "S' \<in> sets borel"
-    using S by (auto simp: fmeasurable_def)
-  with outer_regular_lborel[OF _ \<open>0<e\<close>]
-  obtain U where U: "open U" "S' \<subseteq> U" "emeasure lborel (U - S') < e"
-    by blast
-  show thesis
-  proof
-    show "open U" "S \<subseteq> U"
-      using f_S' U S' by auto
-  have "(U - S) = (U - S') \<union> (S' - S)"
-    using S' U by auto
-  then have eq: "emeasure lebesgue (U - S) = emeasure lborel (U - S')"
-    using null  by (simp add: U(1) emeasure_Un_null_set f_S' sets.Diff)
-  have "(U - S) \<in> sets lebesgue"
-    by (simp add: S U(1) sets.Diff)
-  then show "(U - S) \<in> lmeasurable"
-    unfolding fmeasurable_def using U(3) eq less_le_trans by fastforce
-  with eq U show "emeasure lebesgue (U - S) < e"
-    by (simp add: eq)
-  qed
-qed
-
-lemma sets_lebesgue_inner_closed:
-  fixes e::real
-  assumes "S \<in> sets lebesgue" "e > 0"
-  obtains T where "closed T" "T \<subseteq> S" "S-T \<in> lmeasurable" "emeasure lebesgue (S - T) < ennreal e"
-proof -
-  have "-S \<in> sets lebesgue"
-    using assms by (simp add: Compl_in_sets_lebesgue)
-  then obtain T where "open T" "-S \<subseteq> T"
-          and T: "(T - -S) \<in> lmeasurable" "emeasure lebesgue (T - -S) < e"
-    using sets_lebesgue_outer_open assms  by blast
-  show thesis
-  proof
-    show "closed (-T)"
-      using \<open>open T\<close> by blast
-    show "-T \<subseteq> S"
-      using \<open>- S \<subseteq> T\<close> by auto
-    show "S - ( -T) \<in> lmeasurable" "emeasure lebesgue (S - (- T)) < e"
-      using T by (auto simp: Int_commute)
-  qed
-qed
-
-corollary eventually_ae_filter_negligible:
-  "eventually P (ae_filter lebesgue) \<longleftrightarrow> (\<exists>N. negligible N \<and> {x. \<not> P x} \<subseteq> N)"
-  by (auto simp: completion.AE_iff_null_sets negligible_iff_null_sets null_sets_completion_subset)
-
-
-lemma borel_measurable_affine:
-  fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
-  assumes f: "f \<in> borel_measurable lebesgue" and "c \<noteq> 0"
-  shows "(\<lambda>x. f(t + c *\<^sub>R x)) \<in> borel_measurable lebesgue"
-proof -
-  { fix a b
-    have "{x. f x \<in> cbox a b} \<in> sets lebesgue"
-      using f cbox_borel lebesgue_measurable_vimage_borel by blast
-    then have "(\<lambda>x. (x - t) /\<^sub>R c) ` {x. f x \<in> cbox a b} \<in> sets lebesgue"
-    proof (rule differentiable_image_in_sets_lebesgue)
-      show "(\<lambda>x. (x - t) /\<^sub>R c) differentiable_on {x. f x \<in> cbox a b}"
-        unfolding differentiable_on_def differentiable_def
-        by (rule \<open>c \<noteq> 0\<close> derivative_eq_intros strip exI | simp)+
-    qed auto
-    moreover
-    have "{x. f(t + c *\<^sub>R x) \<in> cbox a b} = (\<lambda>x. (x-t) /\<^sub>R c) ` {x. f x \<in> cbox a b}"
-      using \<open>c \<noteq> 0\<close> by (auto simp: image_def)
-    ultimately have "{x. f(t + c *\<^sub>R x) \<in> cbox a b} \<in> sets lebesgue"
-      by (auto simp: borel_measurable_vimage_closed_interval) }
-  then show ?thesis
-    by (subst lebesgue_on_UNIV_eq [symmetric]; auto simp: borel_measurable_vimage_closed_interval)
-qed
-
-
-corollary lebesgue_real_affine:
-  "c \<noteq> 0 \<Longrightarrow> lebesgue = density (distr lebesgue lebesgue (\<lambda>x. t + c * x)) (\<lambda>_. ennreal (abs c))"
-    using lebesgue_affine_euclidean [where c= "\<lambda>x::real. c"] by simp
-
-lemma nn_integral_real_affine_lebesgue:
-  fixes c :: real assumes f[measurable]: "f \<in> borel_measurable lebesgue" and c: "c \<noteq> 0"
-  shows "(\<integral>\<^sup>+x. f x \<partial>lebesgue) = ennreal\<bar>c\<bar> * (\<integral>\<^sup>+x. f(t + c * x) \<partial>lebesgue)"
-proof -
-  have "(\<integral>\<^sup>+x. f x \<partial>lebesgue) = (\<integral>\<^sup>+x. f x \<partial>density (distr lebesgue lebesgue (\<lambda>x. t + c * x)) (\<lambda>x. ennreal \<bar>c\<bar>))"
-    using lebesgue_real_affine c by auto
-  also have "\<dots> = \<integral>\<^sup>+ x. ennreal \<bar>c\<bar> * f x \<partial>distr lebesgue lebesgue (\<lambda>x. t + c * x)"
-    by (subst nn_integral_density) auto
-  also have "\<dots> = ennreal \<bar>c\<bar> * integral\<^sup>N (distr lebesgue lebesgue (\<lambda>x. t + c * x)) f"
-    using f measurable_distr_eq1 nn_integral_cmult by blast
-  also have "\<dots> = \<bar>c\<bar> * (\<integral>\<^sup>+x. f(t + c * x) \<partial>lebesgue)"
-    using lebesgue_affine_measurable[where c= "\<lambda>x::real. c"]
-    by (subst nn_integral_distr) (force+)
-  finally show ?thesis .
-qed
-
-lemma lebesgue_integrable_real_affine:
-  fixes f :: "real \<Rightarrow> 'a :: euclidean_space"
-  assumes f: "integrable lebesgue f" and "c \<noteq> 0"
-  shows "integrable lebesgue (\<lambda>x. f(t + c * x))"
-proof -
-  have "(\<lambda>x. norm (f x)) \<in> borel_measurable lebesgue"
-    by (simp add: borel_measurable_integrable f)
-  then show ?thesis
-    using assms borel_measurable_affine [of f c]
-    unfolding integrable_iff_bounded
-    by (subst (asm) nn_integral_real_affine_lebesgue[where c=c and t=t])  (auto simp: ennreal_mult_less_top)
-qed
-
-lemma lebesgue_integrable_real_affine_iff:
-  fixes f :: "real \<Rightarrow> 'a :: euclidean_space"
-  shows "c \<noteq> 0 \<Longrightarrow> integrable lebesgue (\<lambda>x. f(t + c * x)) \<longleftrightarrow> integrable lebesgue f"
-  using lebesgue_integrable_real_affine[of f c t]
-        lebesgue_integrable_real_affine[of "\<lambda>x. f(t + c * x)" "1/c" "-t/c"]
-  by (auto simp: field_simps)
-
-lemma\<^marker>\<open>tag important\<close> lebesgue_integral_real_affine:
-  fixes f :: "real \<Rightarrow> 'a :: euclidean_space" and c :: real
-  assumes c: "c \<noteq> 0" shows "(\<integral>x. f x \<partial> lebesgue) = \<bar>c\<bar> *\<^sub>R (\<integral>x. f(t + c * x) \<partial>lebesgue)"
-proof cases
-  have "(\<lambda>x. t + c * x) \<in> lebesgue \<rightarrow>\<^sub>M lebesgue"
-    using lebesgue_affine_measurable[where c= "\<lambda>x::real. c"] \<open>c \<noteq> 0\<close> by simp
-  moreover
-  assume "integrable lebesgue f"
-  ultimately show ?thesis
-    by (subst lebesgue_real_affine[OF c, of t]) (auto simp: integral_density integral_distr)
-next
-  assume "\<not> integrable lebesgue f" with c show ?thesis
-    by (simp add: lebesgue_integrable_real_affine_iff not_integrable_integral_eq)
-qed
-
-lemma has_bochner_integral_lebesgue_real_affine_iff:
-  fixes i :: "'a :: euclidean_space"
-  shows "c \<noteq> 0 \<Longrightarrow>
-    has_bochner_integral lebesgue f i \<longleftrightarrow>
-    has_bochner_integral lebesgue (\<lambda>x. f(t + c * x)) (i /\<^sub>R \<bar>c\<bar>)"
-  unfolding has_bochner_integral_iff lebesgue_integrable_real_affine_iff
-  by (simp_all add: lebesgue_integral_real_affine[symmetric] divideR_right cong: conj_cong)
-
-lemma absolutely_integrable_measurable_real:
-  fixes f :: "'a::euclidean_space \<Rightarrow> real"
-  assumes "S \<in> sets lebesgue"
-  shows "f absolutely_integrable_on S \<longleftrightarrow>
-         f \<in> borel_measurable (lebesgue_on S) \<and> integrable (lebesgue_on S) (\<lambda>x. \<bar>f x\<bar>)"
-  by (simp add: absolutely_integrable_measurable assms o_def)
-
-lemma continuous_imp_integrable:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes "continuous_on (cbox a b) f"
-  shows "integrable (lebesgue_on (cbox a b)) f"
-proof -
-  have "f absolutely_integrable_on cbox a b"
-    by (simp add: absolutely_integrable_continuous assms)
-  then show ?thesis
-    by (simp add: integrable_restrict_space set_integrable_def)
-qed
-
-lemma continuous_imp_integrable_real:
-  fixes f :: "real \<Rightarrow> 'b::euclidean_space"
-  assumes "continuous_on {a..b} f"
-  shows "integrable (lebesgue_on {a..b}) f"
-  by (metis assms continuous_imp_integrable interval_cbox)
-
-lemma integral_abs_bound_integral:
-  fixes f :: "'a::euclidean_space \<Rightarrow> real"
-  assumes "integrable M f" "integrable M g" "\<And>x. x \<in> space M \<Longrightarrow> \<bar>f x\<bar> \<le> g x"
-  shows "\<bar>\<integral>x. f x \<partial>M\<bar> \<le> (\<integral>x. g x \<partial>M)"
-proof -
-  have "LINT x|M. \<bar>f x\<bar> \<le> integral\<^sup>L M g"
-    by (simp add: assms integral_mono)
-  then show ?thesis
-    by (metis dual_order.trans integral_abs_bound)
-qed
-
-lemma ennreal_tendsto_0_iff: "(\<And>n. f n \<ge> 0) \<Longrightarrow> ((\<lambda>n. ennreal (f n)) \<longlonglongrightarrow> 0) \<longleftrightarrow> (f \<longlonglongrightarrow> 0)"
-  by (metis (mono_tags) ennreal_0 eventuallyI order_refl tendsto_ennreal_iff)
-
-lemma borel_measurable_continuous_onI: "continuous_on UNIV f \<Longrightarrow> f \<in> borel_measurable borel"
-  by (drule borel_measurable_continuous_on_restrict) simp
-
-lemma has_integral_integral_lebesgue_on:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes "integrable (lebesgue_on S) f" "S \<in> sets lebesgue"
-  shows "(f has_integral (integral\<^sup>L (lebesgue_on S) f)) S"
-proof -
-  let ?f = "\<lambda>x. if x \<in> S then f x else 0"
-  have "integrable lebesgue (\<lambda>x. indicat_real S x *\<^sub>R f x)"
-    using indicator_scaleR_eq_if [of S _ f] assms
-  by (metis (full_types) integrable_restrict_space sets.Int_space_eq2)
-  then have "integrable lebesgue ?f"
-    using indicator_scaleR_eq_if [of S _ f] assms by auto
-  then have "(?f has_integral (integral\<^sup>L lebesgue ?f)) UNIV"
-    by (rule has_integral_integral_lebesgue)
-  then have "(f has_integral (integral\<^sup>L lebesgue ?f)) S"
-    using has_integral_restrict_UNIV by blast
-  moreover
-  have "S \<inter> space lebesgue \<in> sets lebesgue"
-    by (simp add: assms)
-  then have "(integral\<^sup>L lebesgue ?f) = (integral\<^sup>L (lebesgue_on S) f)"
-    by (simp add: integral_restrict_space indicator_scaleR_eq_if)
-  ultimately show ?thesis
-    by auto
-qed
-
-lemma lebesgue_integral_eq_integral:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes "integrable (lebesgue_on S) f" "S \<in> sets lebesgue"
-  shows "integral\<^sup>L (lebesgue_on S) f = integral S f"
-  by (metis has_integral_integral_lebesgue_on assms integral_unique)
-
-declare isCont_Ln' [continuous_intros]
-
-lemma absolutely_integrable_imp_integrable:
-  assumes "f absolutely_integrable_on S" "S \<in> sets lebesgue"
-  shows "integrable (lebesgue_on S) f"
-  by (meson assms integrable_restrict_space set_integrable_def sets.Int sets.top)
-
-lemma insert_null_sets_iff [simp]: "insert a N \<in> null_sets lebesgue \<longleftrightarrow> N \<in> null_sets lebesgue"
-    (is "?lhs = ?rhs")
-proof
-  assume ?lhs then show ?rhs
-    by (meson completion.complete2 subset_insertI)
-next
-  assume ?rhs then show ?lhs
-  by (simp add: null_sets.insert_in_sets null_setsI)
-qed
-
-lemma
-  fixes a::real
-  shows lmeasurable_interval [iff]: "{a..b} \<in> lmeasurable" "{a<..<b} \<in> lmeasurable"
-  apply (metis box_real(2) lmeasurable_cbox)
-  by (metis box_real(1) lmeasurable_box)
-
-lemma integrable_const_ivl [iff]:
-  fixes a::"'a::ordered_euclidean_space"
-  shows "integrable (lebesgue_on {a..b}) (\<lambda>x. c)"
-  by (metis cbox_interval finite_measure.integrable_const finite_measure_lebesgue_on lmeasurable_cbox)
-
-lemma absolutely_integrable_reflect[simp]:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  shows "(\<lambda>x. f(-x)) absolutely_integrable_on cbox (-b) (-a) \<longleftrightarrow> f absolutely_integrable_on cbox a b"
-  apply (auto simp: absolutely_integrable_on_def dest: integrable_reflect [THEN iffD1])
-  unfolding integrable_on_def by auto
-
-lemma absolutely_integrable_reflect_real[simp]:
-  fixes f :: "real \<Rightarrow> 'b::euclidean_space"
-  shows "(\<lambda>x. f(-x)) absolutely_integrable_on {-b .. -a} \<longleftrightarrow> f absolutely_integrable_on {a..b::real}"
-  unfolding box_real[symmetric] by (rule absolutely_integrable_reflect)
-
-lemma absolutely_integrable_on_subcbox:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  shows "\<lbrakk>f absolutely_integrable_on S; cbox a b \<subseteq> S\<rbrakk> \<Longrightarrow> f absolutely_integrable_on cbox a b"
-  by (meson absolutely_integrable_on_def integrable_on_subcbox)
-
-lemma absolutely_integrable_on_subinterval:
-  fixes f :: "real \<Rightarrow> 'b::euclidean_space"
-  shows "\<lbrakk>f absolutely_integrable_on S; {a..b} \<subseteq> S\<rbrakk> \<Longrightarrow> f absolutely_integrable_on {a..b}"
-  using absolutely_integrable_on_subcbox by fastforce
-
-lemma integral_restrict:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes "S \<subseteq> T" "S \<in> sets lebesgue" "T \<in> sets lebesgue"
-  shows "integral\<^sup>L (lebesgue_on T) (\<lambda>x. if x \<in> S then f x else 0) = integral\<^sup>L (lebesgue_on S) f"
-  using integral_restrict_Int [of S T f] assms
-  by (simp add: Int_absorb2)
-
-lemma absolutely_integrable_continuous_real:
-  fixes f :: "real \<Rightarrow> 'b::euclidean_space"
-  shows "continuous_on {a..b} f \<Longrightarrow> f absolutely_integrable_on {a..b}"
-  by (metis absolutely_integrable_continuous box_real(2))
-
-lemma absolutely_integrable_imp_borel_measurable:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes "f absolutely_integrable_on S" "S \<in> sets lebesgue"
-  shows "f \<in> borel_measurable (lebesgue_on S)"
-  using absolutely_integrable_measurable assms by blast
-
-lemma measurable_bounded_by_integrable_imp_lebesgue_integrable:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes f: "f \<in> borel_measurable (lebesgue_on S)" and g: "integrable (lebesgue_on S) g"
-    and normf: "\<And>x. x \<in> S \<Longrightarrow> norm(f x) \<le> g x" and S: "S \<in> sets lebesgue"
-  shows "integrable (lebesgue_on S) f"
-proof -
-  have "f absolutely_integrable_on S"
-    by (metis (no_types) S absolutely_integrable_integrable_bound f g integrable_on_lebesgue_on measurable_bounded_by_integrable_imp_integrable normf)
-  then show ?thesis
-    by (simp add: S integrable_restrict_space set_integrable_def)
-qed
-
-lemma lebesgue_on_mono:
-  assumes major: "AE x in lebesgue_on S. P x" and minor: "\<And>x.\<lbrakk>P x; x \<in> S\<rbrakk> \<Longrightarrow> Q x"
-  shows "AE x in lebesgue_on S. Q x"
-proof -
-  have "AE a in lebesgue_on S. P a \<longrightarrow> Q a"
-    using minor space_restrict_space by fastforce
-  then show ?thesis
-    using major by auto
-qed
-
-lemma integral_eq_zero_null_sets:
-  assumes "S \<in> null_sets lebesgue"
-  shows "integral\<^sup>L (lebesgue_on S) f = 0"
-proof (rule integral_eq_zero_AE)
-  show "AE x in lebesgue_on S. f x = 0"
-    by (metis (no_types, lifting) assms AE_not_in lebesgue_on_mono null_setsD2 null_sets_restrict_space order_refl)
-qed
-
-lemma sin_zero_pi_iff:
-  fixes x::real
-  assumes "\<bar>x\<bar> < pi"
-  shows "sin x = 0 \<longleftrightarrow> x = 0"
-proof
-  show "x = 0" if "sin x = 0"
-    using that assms by (auto simp: sin_zero_iff)
-qed auto
-
-lemma id_borel_measurable_lebesgue [iff]: "id \<in> borel_measurable lebesgue"
-  by (simp add: measurable_completion)
-
-lemma id_borel_measurable_lebesgue_on [iff]: "id \<in> borel_measurable (lebesgue_on S)"
-  by (simp add: measurable_completion measurable_restrict_space1)
-
-lemma root_abs_power: "n > 0 \<Longrightarrow> abs (root n (y ^n)) = abs y"
-  using root_sgn_power [of n]
-  by (metis abs_ge_zero power_abs real_root_abs real_root_power_cancel)
-
-lemma power_tendsto_0_iff [simp]:
-  fixes f :: "'a \<Rightarrow> real"
-  assumes "n > 0"
-  shows "((\<lambda>x. f x ^ n) \<longlongrightarrow> 0) F \<longleftrightarrow> (f \<longlongrightarrow> 0) F"
-proof -
-  have "((\<lambda>x. \<bar>root n (f x ^ n)\<bar>) \<longlongrightarrow> 0) F \<Longrightarrow> (f \<longlongrightarrow> 0) F"
-    by (auto simp: assms root_abs_power tendsto_rabs_zero_iff)
-  then have "((\<lambda>x. f x ^ n) \<longlongrightarrow> 0) F \<Longrightarrow> (f \<longlongrightarrow> 0) F"
-    by (metis tendsto_real_root abs_0 real_root_zero tendsto_rabs)
-  with assms show ?thesis
-    by (auto simp: tendsto_null_power)
-qed
-
-end
-
diff --git a/thys/Fourier/Fourier_Aux2.thy b/thys/Fourier/Fourier_Aux2.thy
--- a/thys/Fourier/Fourier_Aux2.thy
+++ b/thys/Fourier/Fourier_Aux2.thy
@@ -1,2165 +1,139 @@
-section\<open>Lemmas destined for the development version\<close>
+section\<open>Lemmas possibly destined for future Isabelle releases\<close>
 
 theory Fourier_Aux2
-  imports Fourier_Aux1 Ergodic_Theory.SG_Library_Complement
+  imports Ergodic_Theory.SG_Library_Complement
 begin
 
-lemma has_integral_divide:
-  fixes c :: "_ :: real_normed_div_algebra"
-  shows "(f has_integral y) S \<Longrightarrow> ((\<lambda>x. f x / c) has_integral (y / c)) S"
-  unfolding divide_inverse
-  by (simp add: has_integral_mult_left)
-
-lemma integral_sin [simp]:
-  fixes a::real
-  assumes "a \<le> b" shows "integral {a..b} sin = cos a - cos b"
-proof -
-  have "(sin has_integral (- cos b - - cos a)) {a..b}"
-  proof (rule fundamental_theorem_of_calculus)
-    show "((\<lambda>a. - cos a) has_vector_derivative sin x) (at x within {a..b})" for x
-      unfolding has_field_derivative_iff_has_vector_derivative [symmetric]
-      by (rule derivative_eq_intros | force)+
-  qed (use assms in auto)
-  then show ?thesis
-    by (simp add: integral_unique)
-qed
-
-lemma integral_cos [simp]:
-  fixes a::real
-  assumes "a \<le> b" shows "integral {a..b} cos = sin b - sin a"
-proof -
-  have "(cos has_integral (sin b - sin a)) {a..b}"
-  proof (rule fundamental_theorem_of_calculus)
-    show "(sin has_vector_derivative cos x) (at x within {a..b})" for x
-      unfolding has_field_derivative_iff_has_vector_derivative [symmetric]
-      by (rule derivative_eq_intros | force)+
-  qed (use assms in auto)
-  then show ?thesis
-    by (simp add: integral_unique)
-qed
-
 lemma has_integral_sin_nx: "((\<lambda>x. sin(real_of_int n * x)) has_integral 0) {-pi..pi}"
 proof (cases "n = 0")
   case False
   have "((\<lambda>x. sin (n * x)) has_integral (- cos (n * pi)/n - - cos (n * - pi)/n)) {-pi..pi}"
   proof (rule fundamental_theorem_of_calculus)
     show "((\<lambda>x. - cos (n * x) / n) has_vector_derivative sin (n * a)) (at a within {-pi..pi})"
       if "a \<in> {-pi..pi}" for a :: real
       using that False
       apply (simp only: has_vector_derivative_def)
       apply (rule derivative_eq_intros | force)+
       done
   qed auto
   then show ?thesis
     by simp
 qed auto
 
 lemma integral_sin_nx:
    "integral {-pi..pi} (\<lambda>x. sin(x * real_of_int n)) = 0"
   using has_integral_sin_nx [of n] by (force simp: mult.commute)
 
 lemma integral_sin_Z [simp]:
   assumes "n \<in> \<int>"
   shows "integral\<^sup>L (lebesgue_on {-pi..pi}) (\<lambda>x. sin(x * n)) = 0"
   proof (subst lebesgue_integral_eq_integral)
   show "integrable (lebesgue_on {-pi..pi}) (\<lambda>x. sin (x * n))"
     by (intro continuous_imp_integrable_real continuous_intros)
   show "integral {-pi..pi} (\<lambda>x. sin (x * n)) = 0"
     using assms Ints_cases integral_sin_nx by blast
 qed auto
 
 lemma integral_sin_Z' [simp]:
   assumes "n \<in> \<int>"
   shows "integral\<^sup>L (lebesgue_on {-pi..pi}) (\<lambda>x. sin(n * x)) = 0"
   by (metis assms integral_sin_Z mult_commute_abs)
 
 lemma has_integral_cos_nx:
   "((\<lambda>x. cos(real_of_int n * x)) has_integral (if n = 0 then 2 * pi else 0)) {-pi..pi}"
 proof (cases "n = 0")
   case True
   then show ?thesis
     using has_integral_const_real [of "1::real" "-pi" pi] by auto
 next
   case False
   have "((\<lambda>x. cos (n * x)) has_integral (sin (n * pi)/n - sin (n * - pi)/n)) {-pi..pi}"
   proof (rule fundamental_theorem_of_calculus)
     show "((\<lambda>x. sin (n * x) / n) has_vector_derivative cos (n * x)) (at x within {-pi..pi})"
       if "x \<in> {-pi..pi}"
       for x :: real
       using that False
       apply (simp only: has_vector_derivative_def)
       apply (rule derivative_eq_intros | force)+
       done
   qed auto
   with False show ?thesis
     by (simp add: mult.commute)
 qed
 
 lemma integral_cos_nx:
    "integral {-pi..pi} (\<lambda>x. cos(x * real_of_int n)) = (if n = 0 then 2 * pi else 0)"
   using has_integral_cos_nx [of n] by (force simp: mult.commute)
 
 lemma integral_cos_Z [simp]:
   assumes "n \<in> \<int>"
   shows "integral\<^sup>L (lebesgue_on {-pi..pi}) (\<lambda>x. cos(x * n)) = (if n = 0 then 2 * pi else 0)"
   proof (subst lebesgue_integral_eq_integral)
   show "integrable (lebesgue_on {-pi..pi}) (\<lambda>x. cos (x * n))"
     by (intro continuous_imp_integrable_real continuous_intros)
   show "integral {-pi..pi} (\<lambda>x. cos (x * n)) = (if n = 0 then 2 * pi else 0)"
     by (metis Ints_cases assms integral_cos_nx of_int_0_eq_iff)
 qed auto
 
 lemma integral_cos_Z' [simp]:
   assumes "n \<in> \<int>"
   shows "integral\<^sup>L (lebesgue_on {-pi..pi}) (\<lambda>x. cos(n * x)) = (if n = 0 then 2 * pi else 0)"
   by (metis assms integral_cos_Z mult_commute_abs)
 
-lemma real_tendsto_mult_left_iff:
-   "c \<noteq> 0 \<Longrightarrow> tendsto(\<lambda>x. c * f x) (c * l) F \<longleftrightarrow> tendsto f l F" for c :: real
-  by (auto simp: tendsto_mult_left dest: tendsto_mult_left [where c = "1/c"])
-
-lemma real_tendsto_mult_right_iff:
-   "c \<noteq> 0 \<Longrightarrow> tendsto(\<lambda>x. f x * c) (l * c) F \<longleftrightarrow> tendsto f l F" for c :: real
-  by (simp add: mult.commute real_tendsto_mult_left_iff)
-
 lemma real_tendsto_zero_mult_right_iff [simp]:
   fixes c::real assumes "c \<noteq> 0" shows "(\<lambda>n. a n * c)\<longlonglongrightarrow> 0 \<longleftrightarrow> a \<longlonglongrightarrow> 0"
   by (metis assms mult_zero_left real_tendsto_mult_right_iff)
 
 lemma real_tendsto_zero_divide_iff [simp]:
   fixes c::real assumes "c \<noteq> 0" shows "(\<lambda>n. a n / c)\<longlonglongrightarrow> 0 \<longleftrightarrow> a \<longlonglongrightarrow> 0"
   using real_tendsto_zero_mult_right_iff [of "1/c" a] assms by (simp add: field_simps)
 
-lemma measurable_restrict_mono:
-  assumes f: "f \<in> restrict_space M A \<rightarrow>\<^sub>M N" and "B \<subseteq> A"
-  shows "f \<in> restrict_space M B \<rightarrow>\<^sub>M N"
-by (rule measurable_compose[OF measurable_restrict_space3 f])
-   (insert \<open>B \<subseteq> A\<close>, auto)
-
-lemma borel_measurable_if_lebesgue_on:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes "S \<in> sets lebesgue" "T \<in> sets lebesgue" "S \<subseteq> T"
-  shows "(\<lambda>x. if x \<in> S then f x else 0) \<in> borel_measurable (lebesgue_on T) \<longleftrightarrow> f \<in> borel_measurable (lebesgue_on S)"
-  (is "?lhs = ?rhs")
-proof
-  assume ?lhs then show ?rhs
-    using measurable_restrict_mono [OF _ \<open>S \<subseteq> T\<close>]
-    by (subst measurable_lebesgue_cong [where g = "(\<lambda>x. if x \<in> S then f x else 0)"]) auto
-next
-  assume ?rhs then show ?lhs
-  by (simp add: \<open>S \<in> sets lebesgue\<close> borel_measurable_if_I measurable_restrict_space1)
-qed
-
-
 lemma insert_sets_lebesgue_on:
   assumes "A \<in> sets (lebesgue_on S)" "a \<in> S" "S \<in> sets lebesgue"
   shows "insert a A \<in> sets (lebesgue_on S)"
   by (metis assms borel_singleton insert_subset lborelD sets.Int_space_eq2 sets.empty_sets sets.insert_in_sets sets_completionI_sets sets_restrict_space_iff)
 
-lemma integrable_lebesgue_on_empty [iff]:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::{second_countable_topology,banach}"
-  shows "integrable (lebesgue_on {}) f"
-  by (simp add: integrable_restrict_space)
-
-lemma integral_lebesgue_on_empty [simp]:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::{second_countable_topology,banach}"
-  shows "integral\<^sup>L (lebesgue_on {}) f = 0"
-  by (simp add: Bochner_Integration.integral_empty)
-
 lemma odd_even_cases [case_names 0 odd even]:
   assumes "P 0" and odd: "\<And>n. P(Suc (2 * n))" and even: "\<And>n. P(2 * n + 2)"
   shows "P n"
   by (metis nat_induct2 One_nat_def Suc_1 add_Suc_right assms(1) dvdE even odd oddE)
 
-lemma null_AE_impI: "\<lbrakk>N \<in> null_sets (lebesgue_on S); S \<in> sets lebesgue\<rbrakk> \<Longrightarrow> AE x in lebesgue_on S. x \<in> N \<longrightarrow> P x"
-  by (rule AE_I' [of N]) auto
-
-lemma has_bochner_integral_restrict_space:
-  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
-  assumes \<Omega>: "\<Omega> \<inter> space M \<in> sets M"
-  shows "has_bochner_integral (restrict_space M \<Omega>) f i
-     \<longleftrightarrow> has_bochner_integral M (\<lambda>x. indicator \<Omega> x *\<^sub>R f x) i"
-  by (simp add: integrable_restrict_space [OF assms] integral_restrict_space [OF assms] has_bochner_integral_iff)
-
-lemma integrable_restrict_UNIV:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::{banach, second_countable_topology}"
-  assumes S: "S \<in> sets lebesgue"
-  shows  "integrable lebesgue (\<lambda>x. if x \<in> S then f x else 0) \<longleftrightarrow> integrable (lebesgue_on S) f"
-  using has_bochner_integral_restrict_space [of S lebesgue f] assms
-  by (simp add: integrable.simps indicator_scaleR_eq_if)
-
-lemma integral_mono_lebesgue_on_AE:
-  fixes f::"_ \<Rightarrow> real"
-  assumes f: "integrable (lebesgue_on T) f"
-    and gf: "AE x in (lebesgue_on S). g x \<le> f x"
-    and f0: "AE x in (lebesgue_on T). 0 \<le> f x"
-    and "S \<subseteq> T" and S: "S \<in> sets lebesgue" and T: "T \<in> sets lebesgue"
-  shows "(\<integral>x. g x \<partial>(lebesgue_on S)) \<le> (\<integral>x. f x \<partial>(lebesgue_on T))"
-proof -
-  have "(\<integral>x. g x \<partial>(lebesgue_on S)) = (\<integral>x. (if x \<in> S then g x else 0) \<partial>lebesgue)"
-    by (simp add: integral_restrict_UNIV S)
-  also have "\<dots> \<le> (\<integral>x. (if x \<in> T then f x else 0) \<partial>lebesgue)"
-  proof (rule Bochner_Integration.integral_mono_AE')
-    show "integrable lebesgue (\<lambda>x. if x \<in> T then f x else 0)"
-      by (simp add: integrable_restrict_UNIV T f)
-    show "AE x in lebesgue. (if x \<in> S then g x else 0) \<le> (if x \<in> T then f x else 0)"
-      using assms by (auto simp: AE_restrict_space_iff)
-    show "AE x in lebesgue. 0 \<le> (if x \<in> T then f x else 0)"
-      using f0 by (simp add: AE_restrict_space_iff T)
-  qed
-  also have "\<dots> = (\<integral>x. f x \<partial>(lebesgue_on T))"
-    using integral_restrict_UNIV T by blast
-  finally show ?thesis .
-qed
-
-
-
-lemma integrable_subinterval_real:
-  fixes f :: "real \<Rightarrow> 'a::euclidean_space"
-  assumes "integrable (lebesgue_on {a..b}) f"
-    and "{c..d} \<subseteq> {a..b}"
-  shows "integrable (lebesgue_on {c..d}) f"
-  by (metis (full_types) absolutely_integrable_on_def assms fmeasurableD integrable_restrict_space integrable_subinterval_real interval_cbox lmeasurable_cbox set_integrable_def sets.Int_space_eq2)
-
-
-lemma indefinite_integral_continuous_1:
-  fixes f :: "real \<Rightarrow> 'a::euclidean_space"
-  assumes "integrable (lebesgue_on {a..b}) f"
-  shows "continuous_on {a..b} (\<lambda>x. integral\<^sup>L (lebesgue_on {a..x}) f)"
-proof -
-  have "f integrable_on {a..b}"
-    by (simp add: assms integrable_on_lebesgue_on)
-  then have "continuous_on {a..b} (\<lambda>x. integral {a..x} f)"
-    using indefinite_integral_continuous_1 by blast
-  moreover have "integral\<^sup>L (lebesgue_on {a..x}) f = integral {a..x} f" if "a \<le> x" "x \<le> b" for x
-  proof -
-    have "{a..x} \<subseteq> {a..b}"
-      using that by auto
-    then have "integrable (lebesgue_on {a..x}) f"
-      using integrable_subinterval_real assms by blast
-    then show "integral\<^sup>L (lebesgue_on {a..x}) f = integral {a..x} f"
-      by (smt integrable_restrict_UNIV integral_restrict_UNIV Henstock_Kurzweil_Integration.integral_restrict_UNIV atLeastAtMost_borel integral_lebesgue sets_completionI_sets sets_lborel)
-  qed
-  ultimately show ?thesis
-    by (metis (no_types, lifting) atLeastAtMost_iff continuous_on_cong)
-qed
-
-
-lemma AE_null_sets_lebesgue:
-  assumes "AE x in lebesgue. P x"
-  obtains N where "N \<in> null_sets lebesgue" "\<And>x. x \<notin> N \<Longrightarrow> P x"
-  using assms unfolding completion.AE_iff_null_sets  by auto
-
-
 lemma measure_lebesgue_on_ivl [simp]: "\<lbrakk>{a..b} \<subseteq> S; S \<in> sets lebesgue\<rbrakk> \<Longrightarrow> measure (lebesgue_on S) {a..b} = content {a..b::real}"
   by (simp add: measure_restrict_space)
 
 
-
-subsection\<open>Linear functions are (uniformly) continuous on any set\<close>
-
-lemma linear_lim_0:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
-  assumes "linear f"
-  shows "(f \<longlongrightarrow> 0) (at 0)"
-proof -
-  obtain B where "B > 0" "\<And>x. norm (f x) \<le> B * norm x"
-    using assms linear_bounded_pos by blast
-  then have "\<And>e x. \<lbrakk>0 < e; x \<noteq> 0; norm x < e / B\<rbrakk> \<Longrightarrow> norm (f x) < e"
-    by (metis le_less_trans mult.commute pos_less_divide_eq)
-  then show ?thesis
-    unfolding Lim_at dist_norm
-    by (metis \<open>0 < B\<close> diff_zero divide_pos_pos zero_less_norm_iff)
-qed
-
-lemma linear_continuous_at:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
-  assumes "linear f"
-  shows "continuous (at a) f"
-  using assms has_derivative_continuous linear_imp_has_derivative by blast
-
-
-lemma linear_continuous_within:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
-  assumes "linear f"
-  shows "continuous (at x within S) f"
-  by (simp add: assms differentiable_imp_continuous_within linear_imp_differentiable)
-
-lemma linear_continuous_on:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
-  assumes "linear f"
-  shows "continuous_on S f"
-  by (simp add: continuous_at_imp_continuous_on linear_continuous_within assms)
-
-lemma Lim_linear:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space" and h :: "'b \<Rightarrow> 'c::real_normed_vector"
-  assumes "(f \<longlongrightarrow> l) F" "linear h"
-  shows "((\<lambda>x. h(f x)) \<longlongrightarrow> h l) F"
-proof -
-  obtain B where B: "B > 0" "\<And>x. norm (h x) \<le> B * norm x"
-    using linear_bounded_pos [OF \<open>linear h\<close>] by blast
-  show ?thesis
-    unfolding tendsto_iff
-  proof (intro allI impI)
-    show "\<forall>\<^sub>F x in F. dist (h (f x)) (h l) < e" if "e > 0" for e
-    proof -
-      have "\<forall>\<^sub>F x in F. dist (f x) l < e/B"
-        by (simp add: \<open>0 < B\<close> assms(1) tendstoD that)
-      then show ?thesis
-        unfolding dist_norm
-      proof (rule eventually_mono)
-        show "norm (h (f x) - h l) < e" if "norm (f x - l) < e / B" for x
-          using that B
-          apply (simp add: divide_simps)
-          by (metis \<open>linear h\<close> le_less_trans linear_diff mult.commute)
-      qed
-    qed
-  qed
-qed
-
-lemma linear_continuous_compose:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space" and g :: "'b \<Rightarrow> 'c::real_normed_vector"
-  assumes "continuous F f" "linear g"
-  shows "continuous F (\<lambda>x. g(f x))"
-  using assms unfolding continuous_def by (rule Lim_linear)
-
-lemma linear_continuous_on_compose:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space" and g :: "'b \<Rightarrow> 'c::real_normed_vector"
-  assumes "continuous_on S f" "linear g"
-  shows "continuous_on S (\<lambda>x. g(f x))"
-  using assms by (simp add: continuous_on_eq_continuous_within linear_continuous_compose)
-
-
-subsection\<open>Also bilinear functions, in composition form\<close>
-
-
-lemma bilinear_continuous_compose:
-  fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
-  assumes "continuous F f" "continuous F g" "bilinear h"
-  shows "continuous F (\<lambda>x. h (f x) (g x))"
-  using assms Lim_bilinear bilinear_conv_bounded_bilinear
-  unfolding continuous_def
-  by blast
-
-lemma bilinear_continuous_on_compose:
-  fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
-    and f :: "'d::t2_space \<Rightarrow> 'a"
-  assumes "continuous_on S f" "continuous_on S g" "bilinear h"
-  shows "continuous_on S (\<lambda>x. h (f x) (g x))"
-  using assms
-  by (simp add: continuous_on_eq_continuous_within bilinear_continuous_compose)
-
-
-
-
-
-subsection\<open>Austin's Lemma\<close>
-
-lemma Austin_Lemma:
-  fixes \<D> :: "'a::euclidean_space set set"
-  assumes "finite \<D>" and \<D>: "\<And>D. D \<in> \<D> \<Longrightarrow> \<exists>k a b. D = cbox a b \<and> (\<forall>i \<in> Basis. b\<bullet>i - a\<bullet>i = k)"
-  obtains \<C> where "\<C> \<subseteq> \<D>" "pairwise disjnt \<C>"
-                  "measure lebesgue (\<Union>\<C>) \<ge> measure lebesgue (\<Union>\<D>) / 3 ^ (DIM('a))"
-  using assms
-proof (induction "card \<D>" arbitrary: \<D> thesis rule: less_induct)
-  case less
-  show ?case
-  proof (cases "\<D> = {}")
-    case True
-    then show thesis
-      using less by auto
-  next
-    case False
-    then have "Max (Sigma_Algebra.measure lebesgue ` \<D>) \<in> Sigma_Algebra.measure lebesgue ` \<D>"
-      using Max_in finite_imageI \<open>finite \<D>\<close> by blast
-    then obtain D where "D \<in> \<D>" and "measure lebesgue D = Max (measure lebesgue ` \<D>)"
-      by auto
-    then have D: "\<And>C. C \<in> \<D> \<Longrightarrow> measure lebesgue C \<le> measure lebesgue D"
-      by (simp add: \<open>finite \<D>\<close>)
-    let ?\<E> = "{C. C \<in> \<D> - {D} \<and> disjnt C D}"
-    obtain \<D>' where \<D>'sub: "\<D>' \<subseteq> ?\<E>" and \<D>'dis: "pairwise disjnt \<D>'"
-      and \<D>'m: "measure lebesgue (\<Union>\<D>') \<ge> measure lebesgue (\<Union>?\<E>) / 3 ^ (DIM('a))"
-    proof (rule less.hyps)
-      have *: "?\<E> \<subset> \<D>"
-        using \<open>D \<in> \<D>\<close> by auto
-      then show "card ?\<E> < card \<D>" "finite ?\<E>"
-        by (auto simp: \<open>finite \<D>\<close> psubset_card_mono)
-      show "\<exists>k a b. D = cbox a b \<and> (\<forall>i\<in>Basis. b \<bullet> i - a \<bullet> i = k)" if "D \<in> ?\<E>" for D
-        using less.prems(3) that by auto
-    qed
-    then have [simp]: "\<Union>\<D>' - D = \<Union>\<D>'"
-      by (auto simp: disjnt_iff)
-    show ?thesis
-    proof (rule less.prems)
-      show "insert D \<D>' \<subseteq> \<D>"
-        using \<D>'sub \<open>D \<in> \<D>\<close> by blast
-      show "disjoint (insert D \<D>')"
-        using \<D>'dis \<D>'sub by (fastforce simp add: pairwise_def disjnt_sym)
-      obtain a3 b3 where  m3: "content (cbox a3 b3) = 3 ^ DIM('a) * measure lebesgue D"
-        and sub3: "\<And>C. \<lbrakk>C \<in> \<D>; \<not> disjnt C D\<rbrakk> \<Longrightarrow> C \<subseteq> cbox a3 b3"
-      proof -
-        obtain k a b where ab: "D = cbox a b" and k: "\<And>i. i \<in> Basis \<Longrightarrow> b\<bullet>i - a\<bullet>i = k"
-          using less.prems \<open>D \<in> \<D>\<close> by blast
-        then have eqk: "\<And>i. i \<in> Basis \<Longrightarrow> a \<bullet> i \<le> b \<bullet> i \<longleftrightarrow> k \<ge> 0"
-          by force
-        show thesis
-        proof
-          let ?a = "(a + b) /\<^sub>R 2 - (3/2) *\<^sub>R (b - a)"
-          let ?b = "(a + b) /\<^sub>R 2 + (3/2) *\<^sub>R (b - a)"
-          have eq: "(\<Prod>i\<in>Basis. b \<bullet> i * 3 - a \<bullet> i * 3) = (\<Prod>i\<in>Basis. b \<bullet> i - a \<bullet> i) * 3 ^ DIM('a)"
-            by (simp add: comm_monoid_mult_class.prod.distrib flip: left_diff_distrib inner_diff_left)
-          show "content (cbox ?a ?b) = 3 ^ DIM('a) * measure lebesgue D"
-            by (simp add: content_cbox_if box_eq_empty algebra_simps eq ab k)
-          show "C \<subseteq> cbox ?a ?b" if "C \<in> \<D>" and CD: "\<not> disjnt C D" for C
-          proof -
-            obtain k' a' b' where ab': "C = cbox a' b'" and k': "\<And>i. i \<in> Basis \<Longrightarrow> b'\<bullet>i - a'\<bullet>i = k'"
-              using less.prems \<open>C \<in> \<D>\<close> by blast
-            then have eqk': "\<And>i. i \<in> Basis \<Longrightarrow> a' \<bullet> i \<le> b' \<bullet> i \<longleftrightarrow> k' \<ge> 0"
-              by force
-            show ?thesis
-            proof (clarsimp simp add: disjoint_interval disjnt_def ab ab' not_less subset_box algebra_simps)
-              show "a \<bullet> i * 2 \<le> a' \<bullet> i + b \<bullet> i \<and> a \<bullet> i + b' \<bullet> i \<le> b \<bullet> i * 2"
-                if * [rule_format]: "\<forall>j\<in>Basis. a' \<bullet> j \<le> b' \<bullet> j" and "i \<in> Basis" for i
-              proof -
-                have "a' \<bullet> i \<le> b' \<bullet> i \<and> a \<bullet> i \<le> b \<bullet> i \<and> a \<bullet> i \<le> b' \<bullet> i \<and> a' \<bullet> i \<le> b \<bullet> i"
-                  using \<open>i \<in> Basis\<close> CD by (simp_all add: disjoint_interval disjnt_def ab ab' not_less)
-                then show ?thesis
-                  using D [OF \<open>C \<in> \<D>\<close>] \<open>i \<in> Basis\<close>
-                  apply (simp add: ab ab' k k' eqk eqk' content_cbox_cases)
-                  using k k' by fastforce
-              qed
-            qed
-          qed
-        qed
-      qed
-      have \<D>lm: "\<And>D. D \<in> \<D> \<Longrightarrow> D \<in> lmeasurable"
-        using less.prems(3) by blast
-      have "measure lebesgue (\<Union>\<D>)  \<le> measure lebesgue (cbox a3 b3 \<union> (\<Union>\<D> - cbox a3 b3))"
-      proof (rule measure_mono_fmeasurable)
-        show "\<Union>\<D> \<in> sets lebesgue"
-          using \<D>lm \<open>finite \<D>\<close> by blast
-        show "cbox a3 b3 \<union> (\<Union>\<D> - cbox a3 b3) \<in> lmeasurable"
-          by (simp add: \<D>lm fmeasurable.Un fmeasurable.finite_Union less.prems(2) subset_eq)
-      qed auto
-      also have "\<dots> = content (cbox a3 b3) + measure lebesgue (\<Union>\<D> - cbox a3 b3)"
-        by (simp add: \<D>lm fmeasurable.finite_Union less.prems(2) measure_Un2 subsetI)
-      also have "\<dots> \<le> (measure lebesgue D + measure lebesgue (\<Union>\<D>')) * 3 ^ DIM('a)"
-      proof -
-        have "(\<Union>\<D> - cbox a3 b3) \<subseteq> \<Union>?\<E>"
-          using sub3 by fastforce
-        then have "measure lebesgue (\<Union>\<D> - cbox a3 b3) \<le> measure lebesgue (\<Union>?\<E>)"
-        proof (rule measure_mono_fmeasurable)
-          show "\<Union> \<D> - cbox a3 b3 \<in> sets lebesgue"
-            by (simp add: \<D>lm fmeasurableD less.prems(2) sets.Diff sets.finite_Union subsetI)
-          show "\<Union> {C \<in> \<D> - {D}. disjnt C D} \<in> lmeasurable"
-            using \<D>lm less.prems(2) by auto
-        qed
-        then have "measure lebesgue (\<Union>\<D> - cbox a3 b3) / 3 ^ DIM('a) \<le> measure lebesgue (\<Union> \<D>')"
-          using \<D>'m by (simp add: divide_simps)
-        then show ?thesis
-          by (simp add: m3 field_simps)
-      qed
-      also have "\<dots> \<le> measure lebesgue (\<Union>(insert D \<D>')) * 3 ^ DIM('a)"
-      proof (simp add: \<D>lm \<open>D \<in> \<D>\<close>)
-        show "measure lebesgue D + measure lebesgue (\<Union>\<D>') \<le> measure lebesgue (D \<union> \<Union> \<D>')"
-        proof (subst measure_Un2)
-          show "\<Union> \<D>' \<in> lmeasurable"
-            by (meson \<D>lm \<open>insert D \<D>' \<subseteq> \<D>\<close> fmeasurable.finite_Union less.prems(2) finite_subset subset_eq subset_insertI)
-          show "measure lebesgue D + measure lebesgue (\<Union> \<D>') \<le> measure lebesgue D + measure lebesgue (\<Union> \<D>' - D)"
-            using \<open>insert D \<D>' \<subseteq> \<D>\<close> infinite_super less.prems(2) by force
-        qed (simp add: \<D>lm \<open>D \<in> \<D>\<close>)
-      qed
-      finally show "measure lebesgue (\<Union>\<D>) / 3 ^ DIM('a) \<le> measure lebesgue (\<Union>(insert D \<D>'))"
-        by (simp add: divide_simps)
-    qed
-  qed
-qed
-
-
-subsection\<open>A differentiability-like property of the indefinite integral.        \<close>
-
-proposition integrable_ccontinuous_explicit:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes "\<And>a b::'a. f integrable_on cbox a b"
-  obtains N where
-       "negligible N"
-       "\<And>x e. \<lbrakk>x \<notin> N; 0 < e\<rbrakk> \<Longrightarrow>
-               \<exists>d>0. \<forall>h. 0 < h \<and> h < d \<longrightarrow>
-                         norm(integral (cbox x (x + h *\<^sub>R One)) f /\<^sub>R h ^ DIM('a) - f x) < e"
-proof -
-  define BOX where "BOX \<equiv> \<lambda>h. \<lambda>x::'a. cbox x (x + h *\<^sub>R One)"
-  define BOX2 where "BOX2 \<equiv> \<lambda>h. \<lambda>x::'a. cbox (x - h *\<^sub>R One) (x + h *\<^sub>R One)"
-  define i where "i \<equiv> \<lambda>h x. integral (BOX h x) f /\<^sub>R h ^ DIM('a)"
-  define \<Psi> where "\<Psi> \<equiv> \<lambda>x r. \<forall>d>0. \<exists>h. 0 < h \<and> h < d \<and> r \<le> norm(i h x - f x)"
-  let ?N = "{x. \<exists>e>0. \<Psi> x e}"
-  have "\<exists>N. negligible N \<and> (\<forall>x e. x \<notin> N \<and> 0 < e \<longrightarrow> \<not> \<Psi> x e)"
-  proof (rule exI ; intro conjI allI impI)
-    let ?M =  "\<Union>n. {x. \<Psi> x (inverse(real n + 1))}"
-    have "negligible ({x. \<Psi> x \<mu>} \<inter> cbox a b)"
-      if "\<mu> > 0" for a b \<mu>
-    proof (cases "negligible(cbox a b)")
-      case True
-      then show ?thesis
-        by (simp add: negligible_Int)
-    next
-      case False
-      then have "box a b \<noteq> {}"
-        by (simp add: negligible_interval)
-      then have ab: "\<And>i. i \<in> Basis \<Longrightarrow> a\<bullet>i < b\<bullet>i"
-        by (simp add: box_ne_empty)
-      show ?thesis
-        unfolding negligible_outer_le
-      proof (intro allI impI)
-        fix e::real
-        let ?ee = "(e * \<mu>) / 2 / 6 ^ (DIM('a))"
-        assume "e > 0"
-        then have gt0: "?ee > 0"
-          using \<open>\<mu> > 0\<close> by auto
-        have f': "f integrable_on cbox (a - One) (b + One)"
-          using assms by blast
-        obtain \<gamma> where "gauge \<gamma>"
-          and \<gamma>: "\<And>p. \<lbrakk>p tagged_partial_division_of (cbox (a - One) (b + One)); \<gamma> fine p\<rbrakk>
-                    \<Longrightarrow> (\<Sum>(x, k)\<in>p. norm (content k *\<^sub>R f x - integral k f)) < ?ee"
-          using Henstock_lemma [OF f' gt0] that by auto
-        let ?E = "{x. x \<in> cbox a b \<and> \<Psi> x \<mu>}"
-        have "\<exists>h>0. BOX h x \<subseteq> \<gamma> x \<and>
-                    BOX h x \<subseteq> cbox (a - One) (b + One) \<and> \<mu> \<le> norm (i h x - f x)"
-          if "x \<in> cbox a b" "\<Psi> x \<mu>" for x
-        proof -
-          obtain d where "d > 0" and d: "ball x d \<subseteq> \<gamma> x"
-            using gaugeD [OF \<open>gauge \<gamma>\<close>, of x] openE by blast
-          then obtain h where "0 < h" "h < 1" and hless: "h < d / real DIM('a)"
-                          and mule: "\<mu> \<le> norm (i h x - f x)"
-            using \<open>\<Psi> x \<mu>\<close> [unfolded \<Psi>_def, rule_format, of "min 1 (d / DIM('a))"]
-            by auto
-          show ?thesis
-          proof (intro exI conjI)
-            show "0 < h" "\<mu> \<le> norm (i h x - f x)" by fact+
-            have "BOX h x \<subseteq> ball x d"
-            proof (clarsimp simp: BOX_def mem_box dist_norm algebra_simps)
-              fix y
-              assume "\<forall>i\<in>Basis. x \<bullet> i \<le> y \<bullet> i \<and> y \<bullet> i \<le> h + x \<bullet> i"
-              then have lt: "\<bar>(x - y) \<bullet> i\<bar> < d / real DIM('a)" if "i \<in> Basis" for i
-                using hless that by (force simp: inner_diff_left)
-              have "norm (x - y) \<le> (\<Sum>i\<in>Basis. \<bar>(x - y) \<bullet> i\<bar>)"
-                using norm_le_l1 by blast
-              also have "\<dots> < d"
-                using sum_bounded_above_strict [of Basis "\<lambda>i. \<bar>(x - y) \<bullet> i\<bar>" "d / DIM('a)", OF lt]
-                by auto
-              finally show "norm (x - y) < d" .
-            qed
-            with d show "BOX h x \<subseteq> \<gamma> x"
-              by blast
-            show "BOX h x \<subseteq> cbox (a - One) (b + One)"
-              using that \<open>h < 1\<close>
-              by (force simp: BOX_def mem_box algebra_simps intro: subset_box_imp)
-          qed
-        qed
-        then obtain \<eta> where h0: "\<And>x. x \<in> ?E \<Longrightarrow> \<eta> x > 0"
-          and BOX_\<gamma>: "\<And>x. x \<in> ?E \<Longrightarrow> BOX (\<eta> x) x \<subseteq> \<gamma> x"
-          and "\<And>x. x \<in> ?E \<Longrightarrow> BOX (\<eta> x) x \<subseteq> cbox (a - One) (b + One) \<and> \<mu> \<le> norm (i (\<eta> x) x - f x)"
-          by simp metis
-        then have BOX_cbox: "\<And>x. x \<in> ?E \<Longrightarrow> BOX (\<eta> x) x \<subseteq> cbox (a - One) (b + One)"
-             and \<mu>_le: "\<And>x. x \<in> ?E \<Longrightarrow> \<mu> \<le> norm (i (\<eta> x) x - f x)"
-          by blast+
-        define \<gamma>' where "\<gamma>' \<equiv> \<lambda>x. if x \<in> cbox a b \<and> \<Psi> x \<mu> then ball x (\<eta> x) else \<gamma> x"
-        have "gauge \<gamma>'"
-          using \<open>gauge \<gamma>\<close> by (auto simp: h0 gauge_def \<gamma>'_def)
-        obtain \<D> where "countable \<D>"
-          and \<D>: "\<Union>\<D> \<subseteq> cbox a b"
-          "\<And>K. K \<in> \<D> \<Longrightarrow> interior K \<noteq> {} \<and> (\<exists>c d. K = cbox c d)"
-          and Dcovered: "\<And>K. K \<in> \<D> \<Longrightarrow> \<exists>x. x \<in> cbox a b \<and> \<Psi> x \<mu> \<and> x \<in> K \<and> K \<subseteq> \<gamma>' x"
-          and subUD: "?E \<subseteq> \<Union>\<D>"
-          by (rule covering_lemma [of ?E a b \<gamma>']) (simp_all add: Bex_def \<open>box a b \<noteq> {}\<close> \<open>gauge \<gamma>'\<close>)
-        then have "\<D> \<subseteq> sets lebesgue"
-          by fastforce
-        show "\<exists>T. {x. \<Psi> x \<mu>} \<inter> cbox a b \<subseteq> T \<and> T \<in> lmeasurable \<and> measure lebesgue T \<le> e"
-        proof (intro exI conjI)
-          show "{x. \<Psi> x \<mu>} \<inter> cbox a b \<subseteq> \<Union>\<D>"
-            apply auto
-            using subUD by auto
-          have mUE: "measure lebesgue (\<Union> \<E>) \<le> measure lebesgue (cbox a b)"
-            if "\<E> \<subseteq> \<D>" "finite \<E>" for \<E>
-          proof (rule measure_mono_fmeasurable)
-            show "\<Union> \<E> \<subseteq> cbox a b"
-              using \<D>(1) that(1) by blast
-            show "\<Union> \<E> \<in> sets lebesgue"
-              by (metis \<D>(2) fmeasurable.finite_Union fmeasurableD lmeasurable_cbox subset_eq that)
-          qed auto
-          then show "\<Union>\<D> \<in> lmeasurable"
-            by (metis \<D>(2) \<open>countable \<D>\<close> fmeasurable_Union_bound lmeasurable_cbox)
-          then have leab: "measure lebesgue (\<Union>\<D>) \<le> measure lebesgue (cbox a b)"
-            by (meson \<D>(1) fmeasurableD lmeasurable_cbox measure_mono_fmeasurable)
-          obtain \<F> where "\<F> \<subseteq> \<D>" "finite \<F>"
-            and \<F>: "measure lebesgue (\<Union>\<D>) \<le> 2 * measure lebesgue (\<Union>\<F>)"
-          proof (cases "measure lebesgue (\<Union>\<D>) = 0")
-            case True
-            then show ?thesis
-              by (force intro: that [where \<F> = "{}"])
-          next
-            case False
-            obtain \<F> where "\<F> \<subseteq> \<D>" "finite \<F>"
-              and \<F>: "measure lebesgue (\<Union>\<D>)/2 < measure lebesgue (\<Union>\<F>)"
-            proof (rule measure_countable_Union_approachable [of \<D> "measure lebesgue (\<Union>\<D>) / 2" "content (cbox a b)"])
-              show "countable \<D>"
-                by fact
-              show "0 < measure lebesgue (\<Union> \<D>) / 2"
-                using False by (simp add: zero_less_measure_iff)
-              show Dlm: "D \<in> lmeasurable" if "D \<in> \<D>" for D
-                using \<D>(2) that by blast
-              show "measure lebesgue (\<Union> \<F>) \<le> content (cbox a b)"
-                if "\<F> \<subseteq> \<D>" "finite \<F>" for \<F>
-              proof -
-                have "measure lebesgue (\<Union> \<F>) \<le> measure lebesgue (\<Union>\<D>)"
-                proof (rule measure_mono_fmeasurable)
-                  show "\<Union> \<F> \<subseteq> \<Union> \<D>"
-                    by (simp add: Sup_subset_mono \<open>\<F> \<subseteq> \<D>\<close>)
-                  show "\<Union> \<F> \<in> sets lebesgue"
-                    by (meson Dlm fmeasurableD sets.finite_Union subset_eq that)
-                  show "\<Union> \<D> \<in> lmeasurable"
-                    by fact
-                qed
-                also have "\<dots> \<le> measure lebesgue (cbox a b)"
-                proof (rule measure_mono_fmeasurable)
-                  show "\<Union> \<D> \<in> sets lebesgue"
-                    by (simp add: \<open>\<Union> \<D> \<in> lmeasurable\<close> fmeasurableD)
-                qed (auto simp:\<D>(1))
-                finally show ?thesis
-                  by simp
-              qed
-            qed auto
-            then show ?thesis
-              using that by auto
-          qed
-          obtain tag where tag_in_E: "\<And>D. D \<in> \<D> \<Longrightarrow> tag D \<in> ?E"
-            and tag_in_self: "\<And>D. D \<in> \<D> \<Longrightarrow> tag D \<in> D"
-            and tag_sub: "\<And>D. D \<in> \<D> \<Longrightarrow> D \<subseteq> \<gamma>' (tag D)"
-            using Dcovered by simp metis
-          then have sub_ball_tag: "\<And>D. D \<in> \<D> \<Longrightarrow> D \<subseteq> ball (tag D) (\<eta> (tag D))"
-            by (simp add: \<gamma>'_def)
-          define \<Phi> where "\<Phi> \<equiv> \<lambda>D. BOX (\<eta>(tag D)) (tag D)"
-          define \<Phi>2 where "\<Phi>2 \<equiv> \<lambda>D. BOX2 (\<eta>(tag D)) (tag D)"
-          obtain \<C> where "\<C> \<subseteq> \<Phi>2 ` \<F>" "pairwise disjnt \<C>"
-            "measure lebesgue (\<Union>\<C>) \<ge> measure lebesgue (\<Union>(\<Phi>2`\<F>)) / 3 ^ (DIM('a))"
-          proof (rule Austin_Lemma)
-            show "finite (\<Phi>2`\<F>)"
-              using \<open>finite \<F>\<close> by blast
-            have "\<exists>k a b. \<Phi>2 D = cbox a b \<and> (\<forall>i\<in>Basis. b \<bullet> i - a \<bullet> i = k)" if "D \<in> \<F>" for D
-              apply (rule_tac x="2 * \<eta>(tag D)" in exI)
-              apply (rule_tac x="tag D - \<eta>(tag D) *\<^sub>R One" in exI)
-              apply (rule_tac x="tag D + \<eta>(tag D) *\<^sub>R One" in exI)
-              using that
-              apply (auto simp: \<Phi>2_def BOX2_def algebra_simps)
-              done
-            then show "\<And>D. D \<in> \<Phi>2 ` \<F> \<Longrightarrow> \<exists>k a b. D = cbox a b \<and> (\<forall>i\<in>Basis. b \<bullet> i - a \<bullet> i = k)"
-              by blast
-          qed auto
-          then obtain \<G> where "\<G> \<subseteq> \<F>" and disj: "pairwise disjnt (\<Phi>2 ` \<G>)"
-            and "measure lebesgue (\<Union>(\<Phi>2 ` \<G>)) \<ge> measure lebesgue (\<Union>(\<Phi>2`\<F>)) / 3 ^ (DIM('a))"
-            unfolding \<Phi>2_def subset_image_iff
-            by (meson empty_subsetI equals0D pairwise_imageI)
-          moreover
-          have "measure lebesgue (\<Union>(\<Phi>2 ` \<G>)) * 3 ^ DIM('a) \<le> e/2"
-          proof -
-            have "finite \<G>"
-              using \<open>finite \<F>\<close> \<open>\<G> \<subseteq> \<F>\<close> infinite_super by blast
-            have BOX2_m: "\<And>x. x \<in> tag ` \<G> \<Longrightarrow> BOX2 (\<eta> x) x \<in> lmeasurable"
-              by (auto simp: BOX2_def)
-            have BOX_m: "\<And>x. x \<in> tag ` \<G> \<Longrightarrow> BOX (\<eta> x) x \<in> lmeasurable"
-              by (auto simp: BOX_def)
-            have BOX_sub: "BOX (\<eta> x) x \<subseteq> BOX2 (\<eta> x) x" for x
-              by (auto simp: BOX_def BOX2_def subset_box algebra_simps)
-            have DISJ2: "BOX2 (\<eta> (tag X)) (tag X) \<inter> BOX2 (\<eta> (tag Y)) (tag Y) = {}"
-              if "X \<in> \<G>" "Y \<in> \<G>" "tag X \<noteq> tag Y" for X Y
-            proof -
-              obtain i where i: "i \<in> Basis" "tag X \<bullet> i \<noteq> tag Y \<bullet> i"
-                using \<open>tag X \<noteq> tag Y\<close> by (auto simp: euclidean_eq_iff [of "tag X"])
-              have XY: "X \<in> \<D>" "Y \<in> \<D>"
-                using \<open>\<F> \<subseteq> \<D>\<close> \<open>\<G> \<subseteq> \<F>\<close> that by auto
-              then have "0 \<le> \<eta> (tag X)" "0 \<le> \<eta> (tag Y)"
-                by (meson h0 le_cases not_le tag_in_E)+
-              with XY i have "BOX2 (\<eta> (tag X)) (tag X) \<noteq> BOX2 (\<eta> (tag Y)) (tag Y)"
-                unfolding eq_iff
-                by (fastforce simp add: BOX2_def subset_box algebra_simps)
-              then show ?thesis
-                using disj that by (auto simp: pairwise_def disjnt_def \<Phi>2_def)
-            qed
-            then have BOX2_disj: "pairwise (\<lambda>x y. negligible (BOX2 (\<eta> x) x \<inter> BOX2 (\<eta> y) y)) (tag ` \<G>)"
-              by (simp add: pairwise_imageI)
-            then have BOX_disj: "pairwise (\<lambda>x y. negligible (BOX (\<eta> x) x \<inter> BOX (\<eta> y) y)) (tag ` \<G>)"
-            proof (rule pairwise_mono)
-              show "negligible (BOX (\<eta> x) x \<inter> BOX (\<eta> y) y)"
-                if "negligible (BOX2 (\<eta> x) x \<inter> BOX2 (\<eta> y) y)" for x y
-                by (metis (no_types, hide_lams) that Int_mono negligible_subset BOX_sub)
-            qed auto
-
-            have eq: "\<And>box. (\<lambda>D. box (\<eta> (tag D)) (tag D)) ` \<G> = (\<lambda>t. box (\<eta> t) t) ` tag ` \<G>"
-              by (simp add: image_comp)
-            have "measure lebesgue (BOX2 (\<eta> t) t) * 3 ^ DIM('a)
-                = measure lebesgue (BOX (\<eta> t) t) * (2*3) ^ DIM('a)"
-              if "t \<in> tag ` \<G>" for t
-            proof -
-              have "content (cbox (t - \<eta> t *\<^sub>R One) (t + \<eta> t *\<^sub>R One))
-                  = content (cbox t (t + \<eta> t *\<^sub>R One)) * 2 ^ DIM('a)"
-                using that by (simp add: algebra_simps content_cbox_if box_eq_empty)
-              then show ?thesis
-                by (simp add: BOX2_def BOX_def flip: power_mult_distrib)
-            qed
-            then have "measure lebesgue (\<Union>(\<Phi>2 ` \<G>)) * 3 ^ DIM('a) = measure lebesgue (\<Union>(\<Phi> ` \<G>)) * 6 ^ DIM('a)"
-              unfolding \<Phi>_def \<Phi>2_def eq
-              by (simp add: measure_negligible_finite_Union_image
-                  \<open>finite \<G>\<close> BOX2_m BOX_m BOX2_disj BOX_disj sum_distrib_right
-                  del: UN_simps)
-            also have "\<dots> \<le> e/2"
-            proof -
-              have "\<mu> * measure lebesgue (\<Union>D\<in>\<G>. \<Phi> D) \<le> \<mu> * (\<Sum>D \<in> \<Phi>`\<G>. measure lebesgue D)"
-                using \<open>\<mu> > 0\<close> \<open>finite \<G>\<close> by (force simp: BOX_m \<Phi>_def fmeasurableD intro: measure_Union_le)
-              also have "\<dots> = (\<Sum>D \<in> \<Phi>`\<G>. measure lebesgue D * \<mu>)"
-                by (metis mult.commute sum_distrib_right)
-              also have "\<dots> \<le> (\<Sum>(x, K) \<in> (\<lambda>D. (tag D, \<Phi> D)) ` \<G>.  norm (content K *\<^sub>R f x - integral K f))"
-              proof (rule sum_le_included; clarify?)
-                fix D
-                assume "D \<in> \<G>"
-                then have "\<eta> (tag D) > 0"
-                  using \<open>\<F> \<subseteq> \<D>\<close> \<open>\<G> \<subseteq> \<F>\<close> h0 tag_in_E by auto
-                then have m_\<Phi>: "measure lebesgue (\<Phi> D) > 0"
-                  by (simp add: \<Phi>_def BOX_def algebra_simps)
-                have "\<mu> \<le> norm (i (\<eta>(tag D)) (tag D) - f(tag D))"
-                  using \<mu>_le \<open>D \<in> \<G>\<close> \<open>\<F> \<subseteq> \<D>\<close> \<open>\<G> \<subseteq> \<F>\<close> tag_in_E by auto
-                also have "\<dots> = norm ((content (\<Phi> D) *\<^sub>R f(tag D) - integral (\<Phi> D) f) /\<^sub>R measure lebesgue (\<Phi> D))"
-                  using m_\<Phi>
-                  unfolding i_def \<Phi>_def BOX_def
-                  by (simp add: algebra_simps content_cbox_plus norm_minus_commute)
-                finally have "measure lebesgue (\<Phi> D) * \<mu> \<le> norm (content (\<Phi> D) *\<^sub>R f(tag D) - integral (\<Phi> D) f)"
-                  using m_\<Phi> by (simp add: field_simps)
-                then show "\<exists>y\<in>(\<lambda>D. (tag D, \<Phi> D)) ` \<G>.
-                        snd y = \<Phi> D \<and> measure lebesgue (\<Phi> D) * \<mu> \<le> (case y of (x, k) \<Rightarrow> norm (content k *\<^sub>R f x - integral k f))"
-                  using \<open>D \<in> \<G>\<close> by auto
-              qed (use \<open>finite \<G>\<close> in auto)
-              also have "\<dots> < ?ee"
-              proof (rule \<gamma>)
-                show "(\<lambda>D. (tag D, \<Phi> D)) ` \<G> tagged_partial_division_of cbox (a - One) (b + One)"
-                  unfolding tagged_partial_division_of_def
-                proof (intro conjI allI impI ; clarify ?)
-                  show "tag D \<in> \<Phi> D"
-                    if "D \<in> \<G>" for D
-                    using that \<open>\<F> \<subseteq> \<D>\<close> \<open>\<G> \<subseteq> \<F>\<close> h0 tag_in_E
-                    by (auto simp: \<Phi>_def BOX_def mem_box algebra_simps eucl_less_le_not_le in_mono)
-                  show "y \<in> cbox (a - One) (b + One)" if "D \<in> \<G>" "y \<in> \<Phi> D" for D y
-                    using that BOX_cbox \<Phi>_def \<open>\<F> \<subseteq> \<D>\<close> \<open>\<G> \<subseteq> \<F>\<close> tag_in_E by blast
-                  show "tag D = tag E \<and> \<Phi> D = \<Phi> E"
-                    if "D \<in> \<G>" "E \<in> \<G>" and ne: "interior (\<Phi> D) \<inter> interior (\<Phi> E) \<noteq> {}" for D E
-                  proof -
-                    have "BOX2 (\<eta> (tag D)) (tag D) \<inter> BOX2 (\<eta> (tag E)) (tag E) = {} \<or> tag E = tag D"
-                      using DISJ2 \<open>D \<in> \<G>\<close> \<open>E \<in> \<G>\<close> by force
-                    then have "BOX (\<eta> (tag D)) (tag D) \<inter> BOX (\<eta> (tag E)) (tag E) = {} \<or> tag E = tag D"
-                      using BOX_sub by blast
-                    then show "tag D = tag E \<and> \<Phi> D = \<Phi> E"
-                      by (metis \<Phi>_def interior_Int interior_empty ne)
-                  qed
-                qed (use \<open>finite \<G>\<close> \<Phi>_def BOX_def in auto)
-                show "\<gamma> fine (\<lambda>D. (tag D, \<Phi> D)) ` \<G>"
-                  unfolding fine_def \<Phi>_def using BOX_\<gamma> \<open>\<F> \<subseteq> \<D>\<close> \<open>\<G> \<subseteq> \<F>\<close> tag_in_E by blast
-              qed
-              finally show ?thesis
-                using \<open>\<mu> > 0\<close> by (auto simp: divide_simps)
-          qed
-            finally show ?thesis .
-          qed
-          moreover
-          have "measure lebesgue (\<Union>\<F>) \<le> measure lebesgue (\<Union>(\<Phi>2`\<F>))"
-          proof (rule measure_mono_fmeasurable)
-            have "D \<subseteq> ball (tag D) (\<eta>(tag D))" if "D \<in> \<F>" for D
-              using \<open>\<F> \<subseteq> \<D>\<close> sub_ball_tag that by blast
-            moreover have "ball (tag D) (\<eta>(tag D)) \<subseteq> BOX2 (\<eta> (tag D)) (tag D)" if "D \<in> \<F>" for D
-            proof (clarsimp simp: \<Phi>2_def BOX2_def mem_box algebra_simps dist_norm)
-              fix x and i::'a
-              assume "norm (tag D - x) < \<eta> (tag D)" and "i \<in> Basis"
-              then have "\<bar>tag D \<bullet> i - x \<bullet> i\<bar> \<le> \<eta> (tag D)"
-                by (metis eucl_less_le_not_le inner_commute inner_diff_right norm_bound_Basis_le)
-              then show "tag D \<bullet> i \<le> x \<bullet> i + \<eta> (tag D) \<and> x \<bullet> i \<le> \<eta> (tag D) + tag D \<bullet> i"
-                by (simp add: abs_diff_le_iff)
-            qed
-            ultimately show "\<Union>\<F> \<subseteq> \<Union>(\<Phi>2`\<F>)"
-              by (force simp: \<Phi>2_def)
-            show "\<Union>\<F> \<in> sets lebesgue"
-              using \<open>finite \<F>\<close> \<open>\<D> \<subseteq> sets lebesgue\<close> \<open>\<F> \<subseteq> \<D>\<close> by blast
-            show "\<Union>(\<Phi>2`\<F>) \<in> lmeasurable"
-              unfolding \<Phi>2_def BOX2_def using \<open>finite \<F>\<close> by blast
-          qed
-          ultimately
-          have "measure lebesgue (\<Union>\<F>) \<le> e/2"
-            by (auto simp: divide_simps)
-          then show "measure lebesgue (\<Union>\<D>) \<le> e"
-            using \<F> by linarith
-        qed
-      qed
-    qed
-    then have "\<And>j. negligible {x. \<Psi> x (inverse(real j + 1))}"
-      using negligible_on_intervals
-      by (metis (full_types) inverse_positive_iff_positive le_add_same_cancel1 linorder_not_le nat_le_real_less not_add_less1 of_nat_0)
-    then have "negligible ?M"
-      by auto
-    moreover have "?N \<subseteq> ?M"
-    proof (clarsimp simp: dist_norm)
-      fix y e
-      assume "0 < e"
-        and ye [rule_format]: "\<Psi> y e"
-      then obtain k where k: "0 < k" "inverse (real k + 1) < e"
-        by (metis One_nat_def add.commute less_add_same_cancel2 less_imp_inverse_less less_trans neq0_conv of_nat_1 of_nat_Suc reals_Archimedean zero_less_one)
-      with ye show "\<exists>n. \<Psi> y (inverse (real n + 1))"
-        apply (rule_tac x=k in exI)
-        unfolding \<Psi>_def
-        by (force intro: less_le_trans)
-    qed
-    ultimately show "negligible ?N"
-      by (blast intro: negligible_subset)
-    show "\<not> \<Psi> x e" if "x \<notin> ?N \<and> 0 < e" for x e
-      using that by blast
-  qed
-  with that show ?thesis
-    unfolding i_def BOX_def \<Psi>_def by (fastforce simp add: not_le)
-qed
-
-
-
-subsection\<open>HOL Light measurability\<close>
-
-lemma measurable_on_UNIV:
-  "(\<lambda>x.  if x \<in> S then f x else 0) measurable_on UNIV \<longleftrightarrow> f measurable_on S"
-  by (auto simp: measurable_on_def)
-
-lemma measurable_on_spike_set:
-  assumes f: "f measurable_on S" and neg: "negligible ((S - T) \<union> (T - S))"
-  shows "f measurable_on T"
-proof -
-  obtain N and F
-    where N: "negligible N"
-      and conF: "\<And>n. continuous_on UNIV (F n)"
-      and tendsF: "\<And>x. x \<notin> N \<Longrightarrow> (\<lambda>n. F n x) \<longlonglongrightarrow> (if x \<in> S then f x else 0)"
-    using f by (auto simp: measurable_on_def)
-  show ?thesis
-    unfolding measurable_on_def
-  proof (intro exI conjI allI impI)
-    show "continuous_on UNIV (\<lambda>x. F n x)" for n
-      by (intro conF continuous_intros)
-    show "negligible (N \<union> (S - T) \<union> (T - S))"
-      by (metis (full_types) N neg negligible_Un_eq)
-    show "(\<lambda>n. F n x) \<longlonglongrightarrow> (if x \<in> T then f x else 0)"
-      if "x \<notin> (N \<union> (S - T) \<union> (T - S))" for x
-      using that tendsF [of x] by auto
-  qed
-qed
-
-text\<open> Various common equivalent forms of function measurability.                \<close>
-
-lemma measurable_on_0 [simp]: "(\<lambda>x. 0) measurable_on S"
-  unfolding measurable_on_def
-proof (intro exI conjI allI impI)
-  show "(\<lambda>n. 0) \<longlonglongrightarrow> (if x \<in> S then 0::'b else 0)" for x
-    by force
-qed auto
-
-lemma measurable_on_scaleR_const:
-  assumes f: "f measurable_on S"
-  shows "(\<lambda>x. c *\<^sub>R f x) measurable_on S"
-proof -
-  obtain NF and F
-    where NF: "negligible NF"
-      and conF: "\<And>n. continuous_on UNIV (F n)"
-      and tendsF: "\<And>x. x \<notin> NF \<Longrightarrow> (\<lambda>n. F n x) \<longlonglongrightarrow> (if x \<in> S then f x else 0)"
-    using f by (auto simp: measurable_on_def)
-  show ?thesis
-    unfolding measurable_on_def
-  proof (intro exI conjI allI impI)
-    show "continuous_on UNIV (\<lambda>x. c *\<^sub>R F n x)" for n
-      by (intro conF continuous_intros)
-    show "(\<lambda>n. c *\<^sub>R F n x) \<longlonglongrightarrow> (if x \<in> S then c *\<^sub>R f x else 0)"
-      if "x \<notin> NF" for x
-      using tendsto_scaleR [OF tendsto_const tendsF, of x] that by auto
-  qed (auto simp: NF)
-qed
-
-
-lemma measurable_on_cmul:
-  fixes c :: real
-  assumes "f measurable_on S"
-  shows "(\<lambda>x. c * f x) measurable_on S"
-  using measurable_on_scaleR_const [OF assms] by simp
-
-lemma measurable_on_cdivide:
-  fixes c :: real
-  assumes "f measurable_on S"
-  shows "(\<lambda>x. f x / c) measurable_on S"
-proof (cases "c=0")
-  case False
-  then show ?thesis
-    using measurable_on_cmul [of f S "1/c"]
-    by (simp add: assms)
-qed auto
-
-
-lemma measurable_on_minus:
-   "f measurable_on S \<Longrightarrow> (\<lambda>x. -(f x)) measurable_on S"
-  using measurable_on_scaleR_const [of f S "-1"] by auto
-
-
-lemma continuous_imp_measurable_on:
-   "continuous_on UNIV f \<Longrightarrow> f measurable_on UNIV"
-  unfolding measurable_on_def
-  apply (rule_tac x="{}" in exI)
-  apply (rule_tac x="\<lambda>n. f" in exI, auto)
-  done
-
-proposition integrable_subintervals_imp_measurable:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes "\<And>a b. f integrable_on cbox a b"
-  shows "f measurable_on UNIV"
-proof -
-  define BOX where "BOX \<equiv> \<lambda>h. \<lambda>x::'a. cbox x (x + h *\<^sub>R One)"
-  define i where "i \<equiv> \<lambda>h x. integral (BOX h x) f /\<^sub>R h ^ DIM('a)"
-  obtain N where "negligible N"
-    and k: "\<And>x e. \<lbrakk>x \<notin> N; 0 < e\<rbrakk>
-            \<Longrightarrow> \<exists>d>0. \<forall>h. 0 < h \<and> h < d \<longrightarrow>
-                  norm (integral (cbox x (x + h *\<^sub>R One)) f /\<^sub>R h ^ DIM('a) - f x) < e"
-    using integrable_ccontinuous_explicit assms by blast
-  show ?thesis
-    unfolding measurable_on_def
-  proof (intro exI conjI allI impI)
-    show "continuous_on UNIV ((\<lambda>n x. i (inverse(Suc n)) x) n)" for n
-    proof (clarsimp simp: continuous_on_iff)
-      show "\<exists>d>0. \<forall>x'. dist x' x < d \<longrightarrow> dist (i (inverse (1 + real n)) x') (i (inverse (1 + real n)) x) < e"
-        if "0 < e"
-        for x e
-      proof -
-        let ?e = "e / (1 + real n) ^ DIM('a)"
-        have "?e > 0"
-          using \<open>e > 0\<close> by auto
-        moreover have "x \<in> cbox (x - 2 *\<^sub>R One) (x + 2 *\<^sub>R One)"
-          by (simp add: mem_box inner_diff_left inner_left_distrib)
-        moreover have "x + One /\<^sub>R real (Suc n) \<in> cbox (x - 2 *\<^sub>R One) (x + 2 *\<^sub>R One)"
-          by (auto simp: mem_box inner_diff_left inner_left_distrib field_simps)
-        ultimately obtain \<delta> where "\<delta> > 0"
-          and \<delta>: "\<And>c' d'. \<lbrakk>c' \<in> cbox (x - 2 *\<^sub>R One) (x + 2 *\<^sub>R One);
-                           d' \<in> cbox (x - 2 *\<^sub>R One) (x + 2 *\<^sub>R One);
-                           norm(c' - x) \<le> \<delta>; norm(d' - (x + One /\<^sub>R Suc n)) \<le> \<delta>\<rbrakk>
-                          \<Longrightarrow> norm(integral(cbox c' d') f - integral(cbox x (x + One /\<^sub>R Suc n)) f) < ?e"
-          by (blast intro: indefinite_integral_continuous [of f _ _ x] assms)
-        show ?thesis
-        proof (intro exI impI conjI allI)
-          show "min \<delta> 1 > 0"
-            using \<open>\<delta> > 0\<close> by auto
-          show "dist (i (inverse (1 + real n)) y) (i (inverse (1 + real n)) x) < e"
-            if "dist y x < min \<delta> 1" for y
-          proof -
-            have no: "norm (y - x) < 1"
-              using that by (auto simp: dist_norm)
-            have le1: "inverse (1 + real n) \<le> 1"
-              by (auto simp: divide_simps)
-            have "norm (integral (cbox y (y + One /\<^sub>R real (Suc n))) f
-                - integral (cbox x (x + One /\<^sub>R real (Suc n))) f)
-                < e / (1 + real n) ^ DIM('a)"
-            proof (rule \<delta>)
-              show "y \<in> cbox (x - 2 *\<^sub>R One) (x + 2 *\<^sub>R One)"
-                using no by (auto simp: mem_box algebra_simps dest: Basis_le_norm [of _ "y-x"])
-              show "y + One /\<^sub>R real (Suc n) \<in> cbox (x - 2 *\<^sub>R One) (x + 2 *\<^sub>R One)"
-              proof (simp add: dist_norm mem_box algebra_simps, intro ballI conjI)
-                fix i::'a
-                assume "i \<in> Basis"
-                then have 1: "\<bar>y \<bullet> i - x \<bullet> i\<bar> < 1"
-                  by (metis inner_commute inner_diff_right no norm_bound_Basis_lt)
-                moreover have "\<dots> < (2 + inverse (1 + real n))" "1 \<le> 2 - inverse (1 + real n)"
-                  by (auto simp: field_simps)
-                ultimately show "x \<bullet> i \<le> y \<bullet> i + (2 + inverse (1 + real n))"
-                                "y \<bullet> i + inverse (1 + real n) \<le> x \<bullet> i + 2"
-                  by linarith+
-              qed
-              show "norm (y - x) \<le> \<delta>" "norm (y + One /\<^sub>R real (Suc n) - (x + One /\<^sub>R real (Suc n))) \<le> \<delta>"
-                using that by (auto simp: dist_norm)
-            qed
-            then show ?thesis
-              using that by (simp add: dist_norm i_def BOX_def field_simps flip: scaleR_diff_right)
-          qed
-        qed
-      qed
-    qed
-    show "negligible N"
-      by (simp add: \<open>negligible N\<close>)
-    show "(\<lambda>n. i (inverse (Suc n)) x) \<longlonglongrightarrow> (if x \<in> UNIV then f x else 0)"
-      if "x \<notin> N" for x
-      unfolding lim_sequentially
-    proof clarsimp
-      show "\<exists>no. \<forall>n\<ge>no. dist (i (inverse (1 + real n)) x) (f x) < e"
-        if "0 < e" for e
-      proof -
-        obtain d where "d > 0"
-          and d: "\<And>h. \<lbrakk>0 < h; h < d\<rbrakk> \<Longrightarrow>
-              norm (integral (cbox x (x + h *\<^sub>R One)) f /\<^sub>R h ^ DIM('a) - f x) < e"
-          using k [of x e] \<open>x \<notin> N\<close> \<open>0 < e\<close> by blast
-        then obtain M where M: "M \<noteq> 0" "0 < inverse (real M)" "inverse (real M) < d"
-          using real_arch_invD by auto
-        show ?thesis
-        proof (intro exI allI impI)
-          show "dist (i (inverse (1 + real n)) x) (f x) < e"
-            if "M \<le> n" for n
-          proof -
-            have *: "0 < inverse (1 + real n)" "inverse (1 + real n) \<le> inverse M"
-              using that \<open>M \<noteq> 0\<close> by auto
-            show ?thesis
-              using that M
-              apply (simp add: i_def BOX_def dist_norm)
-              apply (blast intro: le_less_trans * d)
-              done
-          qed
-        qed
-      qed
-    qed
-  qed
-qed
-
-subsection\<open>Composing continuous and measurable functions; a few variants\<close>
-
-lemma measurable_on_compose_continuous:
-   assumes f: "f measurable_on UNIV" and g: "continuous_on UNIV g"
-   shows "(g \<circ> f) measurable_on UNIV"
-proof -
-  obtain N and F
-    where "negligible N"
-      and conF: "\<And>n. continuous_on UNIV (F n)"
-      and tendsF: "\<And>x. x \<notin> N \<Longrightarrow> (\<lambda>n. F n x) \<longlonglongrightarrow> f x"
-    using f by (auto simp: measurable_on_def)
-  show ?thesis
-    unfolding measurable_on_def
-  proof (intro exI conjI allI impI)
-    show "negligible N"
-      by fact
-    show "continuous_on UNIV (g \<circ> (F n))" for n
-      using conF continuous_on_compose continuous_on_subset g by blast
-    show "(\<lambda>n. (g \<circ> F n) x) \<longlonglongrightarrow> (if x \<in> UNIV then (g \<circ> f) x else 0)"
-      if "x \<notin> N" for x :: 'a
-      using that g tendsF by (auto simp: continuous_on_def intro: tendsto_compose)
-  qed
-qed
-
-lemma measurable_on_compose_continuous_0:
-   assumes f: "f measurable_on S" and g: "continuous_on UNIV g" and "g 0 = 0"
-   shows "(g \<circ> f) measurable_on S"
-proof -
-  have f': "(\<lambda>x. if x \<in> S then f x else 0) measurable_on UNIV"
-    using f measurable_on_UNIV by blast
-  show ?thesis
-    using measurable_on_compose_continuous [OF f' g]
-    by (simp add: measurable_on_UNIV o_def if_distrib \<open>g 0 = 0\<close> cong: if_cong)
-qed
-
-
-lemma measurable_on_compose_continuous_box:
-  assumes fm: "f measurable_on UNIV" and fab: "\<And>x. f x \<in> box a b"
-    and contg: "continuous_on (box a b) g"
-  shows "(g \<circ> f) measurable_on UNIV"
-proof -
-  have "\<exists>\<gamma>. (\<forall>n. continuous_on UNIV (\<gamma> n)) \<and> (\<forall>x. x \<notin> N \<longrightarrow> (\<lambda>n. \<gamma> n x) \<longlonglongrightarrow> g (f x))"
-    if "negligible N"
-      and conth [rule_format]: "\<forall>n. continuous_on UNIV (\<lambda>x. h n x)"
-      and tends [rule_format]: "\<forall>x. x \<notin> N \<longrightarrow> (\<lambda>n. h n x) \<longlonglongrightarrow> f x"
-    for N and h :: "nat \<Rightarrow> 'a \<Rightarrow> 'b"
-  proof -
-    define \<theta> where "\<theta> \<equiv> \<lambda>n x. (\<Sum>i\<in>Basis. (max (a\<bullet>i + (b\<bullet>i - a\<bullet>i) / real (n+2))
-                                            (min ((h n x)\<bullet>i)
-                                                 (b\<bullet>i - (b\<bullet>i - a\<bullet>i) / real (n+2)))) *\<^sub>R i)"
-    have aibi: "\<And>i. i \<in> Basis \<Longrightarrow> a \<bullet> i < b \<bullet> i"
-      using box_ne_empty(2) fab by auto
-    then have *: "\<And>i n. i \<in> Basis \<Longrightarrow> a \<bullet> i + real n * (a \<bullet> i) < b \<bullet> i + real n * (b \<bullet> i)"
-      by (meson add_mono_thms_linordered_field(3) less_eq_real_def mult_left_mono of_nat_0_le_iff)
-    show ?thesis
-    proof (intro exI conjI allI impI)
-      show "continuous_on UNIV (g \<circ> (\<theta> n))" for n :: nat
-        unfolding \<theta>_def
-        apply (intro continuous_on_compose2 [OF contg] continuous_intros conth)
-         apply (auto simp: aibi * mem_box less_max_iff_disj min_less_iff_disj algebra_simps divide_simps)
-        done
-      show "(\<lambda>n. (g \<circ> \<theta> n) x) \<longlonglongrightarrow> g (f x)"
-        if "x \<notin> N" for x
-        unfolding o_def
-      proof (rule isCont_tendsto_compose [where g=g])
-        show "isCont g (f x)"
-          using contg fab continuous_on_eq_continuous_at by blast
-        have "(\<lambda>n. \<theta> n x) \<longlonglongrightarrow> (\<Sum>i\<in>Basis. max (a \<bullet> i) (min (f x \<bullet> i) (b \<bullet> i)) *\<^sub>R i)"
-          unfolding \<theta>_def
-        proof (intro tendsto_intros \<open>x \<notin> N\<close> tends)
-          fix i::'b
-          assume "i \<in> Basis"
-          have a: "(\<lambda>n. a \<bullet> i + (b \<bullet> i - a \<bullet> i) / real n) \<longlonglongrightarrow> a\<bullet>i + 0"
-            by (intro tendsto_add lim_const_over_n tendsto_const)
-          show "(\<lambda>n. a \<bullet> i + (b \<bullet> i - a \<bullet> i) / real (n + 2)) \<longlonglongrightarrow> a \<bullet> i"
-            using LIMSEQ_ignore_initial_segment [where k=2, OF a] by simp
-          have b: "(\<lambda>n. b\<bullet>i - (b \<bullet> i - a \<bullet> i) / (real n)) \<longlonglongrightarrow> b\<bullet>i - 0"
-            by (intro tendsto_diff lim_const_over_n tendsto_const)
-          show "(\<lambda>n. b \<bullet> i - (b \<bullet> i - a \<bullet> i) / real (n + 2)) \<longlonglongrightarrow> b \<bullet> i"
-            using LIMSEQ_ignore_initial_segment [where k=2, OF b] by simp
-        qed
-        also have "(\<Sum>i\<in>Basis. max (a \<bullet> i) (min (f x \<bullet> i) (b \<bullet> i)) *\<^sub>R i) = (\<Sum>i\<in>Basis. (f x \<bullet> i) *\<^sub>R i)"
-          apply (rule sum.cong)
-          using fab
-           apply auto
-          apply (intro order_antisym)
-           apply (auto simp: mem_box)
-          using less_imp_le apply blast
-          by (metis (full_types) linear max_less_iff_conj min.bounded_iff not_le)
-        also have "\<dots> = f x"
-          using euclidean_representation by blast
-        finally show "(\<lambda>n. \<theta> n x) \<longlonglongrightarrow> f x" .
-      qed
-    qed
-  qed
-  then show ?thesis
-    using fm by (auto simp: measurable_on_def)
-qed
-
-lemma measurable_on_Pair:
-  assumes f: "f measurable_on S" and g: "g measurable_on S"
-  shows "(\<lambda>x. (f x, g x)) measurable_on S"
-proof -
-  obtain NF and F
-    where NF: "negligible NF"
-      and conF: "\<And>n. continuous_on UNIV (F n)"
-      and tendsF: "\<And>x. x \<notin> NF \<Longrightarrow> (\<lambda>n. F n x) \<longlonglongrightarrow> (if x \<in> S then f x else 0)"
-    using f by (auto simp: measurable_on_def)
-  obtain NG and G
-    where NG: "negligible NG"
-      and conG: "\<And>n. continuous_on UNIV (G n)"
-      and tendsG: "\<And>x. x \<notin> NG \<Longrightarrow> (\<lambda>n. G n x) \<longlonglongrightarrow> (if x \<in> S then g x else 0)"
-    using g by (auto simp: measurable_on_def)
-  show ?thesis
-    unfolding measurable_on_def
-  proof (intro exI conjI allI impI)
-    show "negligible (NF \<union> NG)"
-      by (simp add: NF NG)
-    show "continuous_on UNIV (\<lambda>x. (F n x, G n x))" for n
-      using conF conG continuous_on_Pair by blast
-    show "(\<lambda>n. (F n x, G n x)) \<longlonglongrightarrow> (if x \<in> S then (f x, g x) else 0)"
-      if "x \<notin> NF \<union> NG" for x
-      using tendsto_Pair [OF tendsF tendsG, of x x] that unfolding zero_prod_def
-      by (simp add: split: if_split_asm)
-  qed
-qed
-
-lemma measurable_on_combine:
-  assumes f: "f measurable_on S" and g: "g measurable_on S"
-    and h: "continuous_on UNIV (\<lambda>x. h (fst x) (snd x))" and "h 0 0 = 0"
-  shows "(\<lambda>x. h (f x) (g x)) measurable_on S"
-proof -
-  have *: "(\<lambda>x. h (f x) (g x)) = (\<lambda>x. h (fst x) (snd x)) \<circ> (\<lambda>x. (f x, g x))"
-    by auto
-  show ?thesis
-    unfolding * by (auto simp: measurable_on_compose_continuous_0 measurable_on_Pair assms)
-qed
-
-lemma measurable_on_add:
-  assumes f: "f measurable_on S" and g: "g measurable_on S"
-  shows "(\<lambda>x. f x + g x) measurable_on S"
-  by (intro continuous_intros measurable_on_combine [OF assms]) auto
-
-lemma measurable_on_diff:
-  assumes f: "f measurable_on S" and g: "g measurable_on S"
-  shows "(\<lambda>x. f x - g x) measurable_on S"
-  by (intro continuous_intros measurable_on_combine [OF assms]) auto
-
-lemma measurable_on_scaleR:
-  assumes f: "f measurable_on S" and g: "g measurable_on S"
-  shows "(\<lambda>x. f x *\<^sub>R g x) measurable_on S"
-  by (intro continuous_intros measurable_on_combine [OF assms]) auto
-
-lemma measurable_on_sum:
-  assumes "finite I" "\<And>i. i \<in> I \<Longrightarrow> f i measurable_on S"
-  shows "(\<lambda>x. sum  (\<lambda>i. f i x) I) measurable_on S"
-  using assms by (induction I) (auto simp: measurable_on_add)
-
-lemma measurable_on_spike:
-  assumes f: "f measurable_on T" and "negligible S" and gf: "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x"
-  shows "g measurable_on T"
-proof -
-  obtain NF and F
-    where NF: "negligible NF"
-      and conF: "\<And>n. continuous_on UNIV (F n)"
-      and tendsF: "\<And>x. x \<notin> NF \<Longrightarrow> (\<lambda>n. F n x) \<longlonglongrightarrow> (if x \<in> T then f x else 0)"
-    using f by (auto simp: measurable_on_def)
-  show ?thesis
-    unfolding measurable_on_def
-  proof (intro exI conjI allI impI)
-    show "negligible (NF \<union> S)"
-      by (simp add: NF \<open>negligible S\<close>)
-    show "\<And>x. x \<notin> NF \<union> S \<Longrightarrow> (\<lambda>n. F n x) \<longlonglongrightarrow> (if x \<in> T then g x else 0)"
-      by (metis (full_types) Diff_iff Un_iff gf tendsF)
-  qed (auto simp: conF)
-qed
-
-
-lemma measurable_on_preimage_lemma0:
-  fixes f :: "'a::euclidean_space \<Rightarrow> real"
-  assumes "m \<in> \<int>" and f: "m / 2^n \<le> (f x)" "(f x) < (m+1) / 2^n" and m: "\<bar>m\<bar> \<le> 2^(2 * n)"
-  shows "(\<Sum>k\<in>{k \<in> \<int>. \<bar>k\<bar> \<le> 2^(2 * n)}.
-             (k / 2^n) * indicator {y. k / 2^n \<le> f y \<and> f y < (k+1) / 2^n} x)
-       = (m / 2^n)"  (is "?lhs = ?rhs")
-proof -
-  have "?lhs = (\<Sum>k\<in>{m}. (k / 2^n) * indicator {y. k / 2^n \<le> f y \<and> f y < (k+1) / 2^n} x)"
-  proof (intro sum.mono_neutral_right ballI)
-    show "finite {k::real. k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2^(2 * n)}"
-      using finite_abs_int_segment by blast
-    show "(i / 2^n) * indicat_real {y. i / 2^n \<le> f y \<and> f y < (i+1) / 2^n} x = 0"
-      if "i \<in> {N \<in> \<int>. \<bar>N\<bar> \<le> 2^(2 * n)} - {m}" for i
-      using f m \<open>m \<in> \<int>\<close> that Ints_eq_abs_less1 [of i m]
-      by (auto simp: indicator_def divide_simps)
-  qed (auto simp: assms)
-  also have "\<dots> = ?rhs"
-    using assms by (auto simp: indicator_def)
-  finally show ?thesis .
-qed
-
-(*see HOL Light's lebesgue_measurable BUT OUR lmeasurable IS NOT THE SAME. It's more like "sets lebesgue"
- `lebesgue_measurable s <=> (indicator s) measurable_on (:real^N)`;;
-*)
-
-proposition indicator_measurable_on:
-  assumes "S \<in> sets lebesgue"
-  shows "indicat_real S measurable_on UNIV"
-proof -
-  { fix n::nat
-    let ?\<epsilon> = "(1::real) / (2 * 2^n)"
-    have \<epsilon>: "?\<epsilon> > 0"
-      by auto
-    obtain T where "closed T" "T \<subseteq> S" "S-T \<in> lmeasurable" and ST: "emeasure lebesgue (S - T) < ?\<epsilon>"
-      by (meson \<epsilon> assms sets_lebesgue_inner_closed)
-    obtain U where "open U" "S \<subseteq> U" "(U - S) \<in> lmeasurable" and US: "emeasure lebesgue (U - S) < ?\<epsilon>"
-      by (meson \<epsilon> assms sets_lebesgue_outer_open)
-    have eq: "-T \<inter> U = (S-T) \<union> (U - S)"
-      using \<open>T \<subseteq> S\<close> \<open>S \<subseteq> U\<close> by auto
-    have "emeasure lebesgue ((S-T) \<union> (U - S)) \<le> emeasure lebesgue (S - T) + emeasure lebesgue (U - S)"
-      using \<open>S - T \<in> lmeasurable\<close> \<open>U - S \<in> lmeasurable\<close> emeasure_subadditive by blast
-    also have "\<dots> < ?\<epsilon> + ?\<epsilon>"
-      using ST US add_mono_ennreal by metis
-    finally have le: "emeasure lebesgue (-T \<inter> U) < ennreal (1 / 2^n)"
-      by (simp add: eq)
-    have 1: "continuous_on (T \<union> -U) (indicat_real S)"
-      unfolding indicator_def
-    proof (rule continuous_on_cases [OF \<open>closed T\<close>])
-      show "closed (- U)"
-        using \<open>open U\<close> by blast
-      show "continuous_on T (\<lambda>x. 1::real)" "continuous_on (- U) (\<lambda>x. 0::real)"
-        by (auto simp: continuous_on)
-      show "\<forall>x. x \<in> T \<and> x \<notin> S \<or> x \<in> - U \<and> x \<in> S \<longrightarrow> (1::real) = 0"
-        using \<open>T \<subseteq> S\<close> \<open>S \<subseteq> U\<close> by auto
-    qed
-    have 2: "closedin (top_of_set UNIV) (T \<union> -U)"
-      using \<open>closed T\<close> \<open>open U\<close> by auto
-    obtain g where "continuous_on UNIV g" "\<And>x. x \<in> T \<union> -U \<Longrightarrow> g x = indicat_real S x" "\<And>x. norm(g x) \<le> 1"
-      by (rule Tietze [OF 1 2, of 1]) auto
-    with le have "\<exists>g E. continuous_on UNIV g \<and> (\<forall>x \<in> -E. g x = indicat_real S x) \<and>
-                        (\<forall>x. norm(g x) \<le> 1) \<and> E \<in> sets lebesgue \<and> emeasure lebesgue E < ennreal (1 / 2^n)"
-      apply (rule_tac x=g in exI)
-      apply (rule_tac x="-T \<inter> U" in exI)
-      using \<open>S - T \<in> lmeasurable\<close> \<open>U - S \<in> lmeasurable\<close> eq by auto
-  }
-  then obtain g E where cont: "\<And>n. continuous_on UNIV (g n)"
-    and geq: "\<And>n x. x \<in> - E n \<Longrightarrow> g n x = indicat_real S x"
-    and ng1: "\<And>n x. norm(g n x) \<le> 1"
-    and Eset: "\<And>n. E n \<in> sets lebesgue"
-    and Em: "\<And>n. emeasure lebesgue (E n) < ennreal (1 / 2^n)"
-    by metis
-  have null: "limsup E \<in> null_sets lebesgue"
-  proof (rule borel_cantelli_limsup1 [OF Eset])
-    show "emeasure lebesgue (E n) < \<infinity>" for n
-      by (metis Em infinity_ennreal_def order.asym top.not_eq_extremum)
-    show "summable (\<lambda>n. measure lebesgue (E n))"
-    proof (rule summable_comparison_test' [OF summable_geometric, of "1/2" 0])
-      show "norm (measure lebesgue (E n)) \<le> (1/2) ^ n"  for n
-        using Em [of n] by (simp add: measure_def enn2real_leI power_one_over)
-    qed auto
-  qed
-  have tends: "(\<lambda>n. g n x) \<longlonglongrightarrow> indicat_real S x" if "x \<notin> limsup E" for x
-  proof -
-    have "\<forall>\<^sub>F n in sequentially. x \<in> - E n"
-      using that by (simp add: mem_limsup_iff not_frequently)
-    then show ?thesis
-      unfolding tendsto_iff dist_real_def
-      by (simp add: eventually_mono geq)
-  qed
-  show ?thesis
-    unfolding measurable_on_def
-  proof (intro exI conjI allI impI)
-    show "negligible (limsup E)"
-      using negligible_iff_null_sets null by blast
-    show "continuous_on UNIV (g n)" for n
-      using cont by blast
-  qed (use tends in auto)
-qed
-
-lemma measurable_on_restrict:
-  assumes f: "f measurable_on UNIV" and S: "S \<in> sets lebesgue"
-  shows "(\<lambda>x. if x \<in> S then f x else 0) measurable_on UNIV"
-proof -
-  have "indicat_real S measurable_on UNIV"
-    by (simp add: S indicator_measurable_on)
-  then show ?thesis
-    using measurable_on_scaleR [OF _ f, of "indicat_real S"]
-    by (simp add: indicator_scaleR_eq_if)
-qed
-
-lemma measurable_on_const_UNIV: "(\<lambda>x. k) measurable_on UNIV"
-  by (simp add: continuous_imp_measurable_on)
-
-lemma measurable_on_const [simp]: "S \<in> sets lebesgue \<Longrightarrow> (\<lambda>x. k) measurable_on S"
-  using measurable_on_UNIV measurable_on_const_UNIV measurable_on_restrict by blast
-
-lemma simple_function_indicator_representation_real:
-  fixes f ::"'a \<Rightarrow> real"
-  assumes f: "simple_function M f" and x: "x \<in> space M" and nn: "\<And>x. f x \<ge> 0"
-  shows "f x = (\<Sum>y \<in> f ` space M. y * indicator (f -` {y} \<inter> space M) x)"
-proof -
-  have f': "simple_function M (ennreal \<circ> f)"
-    by (simp add: f)
-  have *: "f x =
-     enn2real
-      (\<Sum>y\<in> ennreal ` f ` space M.
-         y * indicator ((ennreal \<circ> f) -` {y} \<inter> space M) x)"
-    using arg_cong [OF simple_function_indicator_representation [OF f' x], of enn2real, simplified nn o_def] nn
-    unfolding o_def image_comp
-    by (metis enn2real_ennreal)
-  have "enn2real (\<Sum>y\<in>ennreal ` f ` space M. if ennreal (f x) = y \<and> x \<in> space M then y else 0)
-      = sum (enn2real \<circ> (\<lambda>y. if ennreal (f x) = y \<and> x \<in> space M then y else 0))
-            (ennreal ` f ` space M)"
-    by (rule enn2real_sum) auto
-  also have "\<dots> = sum (enn2real \<circ> (\<lambda>y. if ennreal (f x) = y \<and> x \<in> space M then y else 0) \<circ> ennreal)
-                   (f ` space M)"
-    by (rule sum.reindex) (use nn in \<open>auto simp: inj_on_def intro: sum.cong\<close>)
-  also have "\<dots> = (\<Sum>y\<in>f ` space M. if f x = y \<and> x \<in> space M then y else 0)"
-    using nn
-    by (auto simp: inj_on_def intro: sum.cong)
-  finally show ?thesis
-    by (subst *) (simp add: enn2real_sum indicator_def if_distrib cong: if_cong)
-qed
-
-lemma\<^marker>\<open>tag important\<close> simple_function_induct_real
-    [consumes 1, case_names cong set mult add, induct set: simple_function]:
-  fixes u :: "'a \<Rightarrow> real"
-  assumes u: "simple_function M u"
-  assumes cong: "\<And>f g. simple_function M f \<Longrightarrow> simple_function M g \<Longrightarrow> (AE x in M. f x = g x) \<Longrightarrow> P f \<Longrightarrow> P g"
-  assumes set: "\<And>A. A \<in> sets M \<Longrightarrow> P (indicator A)"
-  assumes mult: "\<And>u c. P u \<Longrightarrow> P (\<lambda>x. c * u x)"
-  assumes add: "\<And>u v. P u \<Longrightarrow> P v \<Longrightarrow> P (\<lambda>x. u x + v x)"
-  and nn: "\<And>x. u x \<ge> 0"
-  shows "P u"
-proof (rule cong)
-  from AE_space show "AE x in M. (\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x) = u x"
-  proof eventually_elim
-    fix x assume x: "x \<in> space M"
-    from simple_function_indicator_representation_real[OF u x] nn
-    show "(\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x) = u x"
-      by metis
-  qed
-next
-  from u have "finite (u ` space M)"
-    unfolding simple_function_def by auto
-  then show "P (\<lambda>x. \<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x)"
-  proof induct
-    case empty
-    then show ?case
-      using set[of "{}"] by (simp add: indicator_def[abs_def])
-  next
-    case (insert a F)
-    have eq: "\<Sum> {y. u x = y \<and> (y = a \<or> y \<in> F) \<and> x \<in> space M}
-            = (if u x = a \<and> x \<in> space M then a else 0) + \<Sum> {y. u x = y \<and> y \<in> F \<and> x \<in> space M}" for x
-    proof (cases "x \<in> space M")
-      case True
-      have *: "{y. u x = y \<and> (y = a \<or> y \<in> F)} = {y. u x = a \<and> y = a} \<union> {y. u x = y \<and> y \<in> F}"
-        by auto
-      show ?thesis
-        using insert by (simp add: * True)
-    qed auto
-    have a: "P (\<lambda>x. a * indicator (u -` {a} \<inter> space M) x)"
-    proof (intro mult set)
-      show "u -` {a} \<inter> space M \<in> sets M"
-        using u by auto
-    qed
-    show ?case
-      using nn insert a
-      by (simp add: eq indicator_times_eq_if [where f = "\<lambda>x. a"] add)
-  qed
-next
-  show "simple_function M (\<lambda>x. (\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x))"
-    apply (subst simple_function_cong)
-    apply (rule simple_function_indicator_representation_real[symmetric])
-    apply (auto intro: u nn)
-    done
-qed fact
-
-proposition simple_function_measurable_on_UNIV:
-  fixes f :: "'a::euclidean_space \<Rightarrow> real"
-  assumes f: "simple_function lebesgue f" and nn: "\<And>x. f x \<ge> 0"
-  shows "f measurable_on UNIV"
-  using f
-proof (induction f)
-  case (cong f g)
-  then obtain N where "negligible N" "{x. g x \<noteq> f x} \<subseteq> N"
-    by (auto simp: eventually_ae_filter_negligible eq_commute)
-  then show ?case
-    by (blast intro: measurable_on_spike cong)
-next
-  case (set S)
-  then show ?case
-    by (simp add: indicator_measurable_on)
-next
-  case (mult u c)
-  then show ?case
-    by (simp add: measurable_on_cmul)
-  case (add u v)
-  then show ?case
-    by (simp add: measurable_on_add)
-qed (auto simp: nn)
-
-lemma simple_function_lebesgue_if:
-  fixes f :: "'a::euclidean_space \<Rightarrow> real"
-  assumes f: "simple_function lebesgue f" and S: "S \<in> sets lebesgue"
-  shows "simple_function lebesgue (\<lambda>x. if x \<in> S then f x else 0)"
-proof -
-  have ffin: "finite (range f)" and fsets: "\<forall>x. f -` {f x} \<in> sets lebesgue"
-    using f by (auto simp: simple_function_def)
-  have "finite (f ` S)"
-    by (meson finite_subset subset_image_iff ffin top_greatest)
-  moreover have "finite ((\<lambda>x. 0::real) ` T)" for T :: "'a set"
-    by (auto simp: image_def)
-  moreover have if_sets: "(\<lambda>x. if x \<in> S then f x else 0) -` {f a} \<in> sets lebesgue" for a
-  proof -
-    have *: "(\<lambda>x. if x \<in> S then f x else 0) -` {f a}
-           = (if f a = 0 then -S \<union> f -` {f a} else (f -` {f a}) \<inter> S)"
-      by (auto simp: split: if_split_asm)
-    show ?thesis
-      unfolding * by (metis Compl_in_sets_lebesgue S sets.Int sets.Un fsets)
-  qed
-  moreover have "(\<lambda>x. if x \<in> S then f x else 0) -` {0} \<in> sets lebesgue"
-  proof (cases "0 \<in> range f")
-    case True
-    then show ?thesis
-      by (metis (no_types, lifting) if_sets rangeE)
-  next
-    case False
-    then have "(\<lambda>x. if x \<in> S then f x else 0) -` {0} = -S"
-      by auto
-    then show ?thesis
-      by (simp add: Compl_in_sets_lebesgue S)
-  qed
-  ultimately show ?thesis
-    by (auto simp: simple_function_def)
-qed
-
-corollary simple_function_measurable_on:
-  fixes f :: "'a::euclidean_space \<Rightarrow> real"
-  assumes f: "simple_function lebesgue f" and nn: "\<And>x. f x \<ge> 0" and S: "S \<in> sets lebesgue"
-  shows "f measurable_on S"
-  by (simp add: measurable_on_UNIV [symmetric, of f] S f simple_function_lebesgue_if nn simple_function_measurable_on_UNIV)
-
-lemma
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::ordered_euclidean_space"
-  assumes f: "f measurable_on S" and g: "g measurable_on S"
-  shows measurable_on_sup: "(\<lambda>x. f x \<squnion> g x) measurable_on S"
-  and   measurable_on_inf: "(\<lambda>x. f x \<sqinter> g x) measurable_on S"
-proof -
-  obtain NF and F
-    where NF: "negligible NF"
-      and conF: "\<And>n. continuous_on UNIV (F n)"
-      and tendsF: "\<And>x. x \<notin> NF \<Longrightarrow> (\<lambda>n. F n x) \<longlonglongrightarrow> (if x \<in> S then f x else 0)"
-    using f by (auto simp: measurable_on_def)
-  obtain NG and G
-    where NG: "negligible NG"
-      and conG: "\<And>n. continuous_on UNIV (G n)"
-      and tendsG: "\<And>x. x \<notin> NG \<Longrightarrow> (\<lambda>n. G n x) \<longlonglongrightarrow> (if x \<in> S then g x else 0)"
-    using g by (auto simp: measurable_on_def)
-  show "(\<lambda>x. f x \<squnion> g x) measurable_on S"
-    unfolding measurable_on_def
-  proof (intro exI conjI allI impI)
-    show "continuous_on UNIV (\<lambda>x. F n x \<squnion> G n x)" for n
-      unfolding sup_max eucl_sup  by (intro conF conG continuous_intros)
-    show "(\<lambda>n. F n x \<squnion> G n x) \<longlonglongrightarrow> (if x \<in> S then f x \<squnion> g x else 0)"
-      if "x \<notin> NF \<union> NG" for x
-      using tendsto_sup [OF tendsF tendsG, of x x] that by auto
-  qed (simp add: NF NG)
-  show "(\<lambda>x. f x \<sqinter> g x) measurable_on S"
-    unfolding measurable_on_def
-  proof (intro exI conjI allI impI)
-    show "continuous_on UNIV (\<lambda>x. F n x \<sqinter> G n x)" for n
-      unfolding inf_min eucl_inf  by (intro conF conG continuous_intros)
-    show "(\<lambda>n. F n x \<sqinter> G n x) \<longlonglongrightarrow> (if x \<in> S then f x \<sqinter> g x else 0)"
-      if "x \<notin> NF \<union> NG" for x
-      using tendsto_inf [OF tendsF tendsG, of x x] that by auto
-  qed (simp add: NF NG)
-qed
-
-proposition measurable_on_componentwise_UNIV:
-  "f measurable_on UNIV \<longleftrightarrow> (\<forall>i\<in>Basis. (\<lambda>x. (f x \<bullet> i) *\<^sub>R i) measurable_on UNIV)"
-  (is "?lhs = ?rhs")
-proof
-  assume L: ?lhs
-  show ?rhs
-  proof
-    fix i::'b
-    assume "i \<in> Basis"
-    have cont: "continuous_on UNIV (\<lambda>x. (x \<bullet> i) *\<^sub>R i)"
-      by (intro continuous_intros)
-    show "(\<lambda>x. (f x \<bullet> i) *\<^sub>R i) measurable_on UNIV"
-      using measurable_on_compose_continuous [OF L cont]
-      by (simp add: o_def)
-  qed
-next
-  assume ?rhs
-  then have "\<exists>N g. negligible N \<and>
-              (\<forall>n. continuous_on UNIV (g n)) \<and>
-              (\<forall>x. x \<notin> N \<longrightarrow> (\<lambda>n. g n x) \<longlonglongrightarrow> (f x \<bullet> i) *\<^sub>R i)"
-    if "i \<in> Basis" for i
-    by (simp add: measurable_on_def that)
-  then obtain N g where N: "\<And>i. i \<in> Basis \<Longrightarrow> negligible (N i)"
-        and cont: "\<And>i n. i \<in> Basis \<Longrightarrow> continuous_on UNIV (g i n)"
-        and tends: "\<And>i x. \<lbrakk>i \<in> Basis; x \<notin> N i\<rbrakk> \<Longrightarrow> (\<lambda>n. g i n x) \<longlonglongrightarrow> (f x \<bullet> i) *\<^sub>R i"
-    by metis
-  show ?lhs
-    unfolding measurable_on_def
-  proof (intro exI conjI allI impI)
-    show "negligible (\<Union>i \<in> Basis. N i)"
-      using N eucl.finite_Basis by blast
-    show "continuous_on UNIV (\<lambda>x. (\<Sum>i\<in>Basis. g i n x))" for n
-      by (intro continuous_intros cont)
-  next
-    fix x
-    assume "x \<notin> (\<Union>i \<in> Basis. N i)"
-    then have "\<And>i. i \<in> Basis \<Longrightarrow> x \<notin> N i"
-      by auto
-    then have "(\<lambda>n. (\<Sum>i\<in>Basis. g i n x)) \<longlonglongrightarrow> (\<Sum>i\<in>Basis. (f x \<bullet> i) *\<^sub>R i)"
-      by (intro tends tendsto_intros)
-    then show "(\<lambda>n. (\<Sum>i\<in>Basis. g i n x)) \<longlonglongrightarrow> (if x \<in> UNIV then f x else 0)"
-      by (simp add: euclidean_representation)
-  qed
-qed
-
-corollary measurable_on_componentwise:
-  "f measurable_on S \<longleftrightarrow> (\<forall>i\<in>Basis. (\<lambda>x. (f x \<bullet> i) *\<^sub>R i) measurable_on S)"
-  apply (subst measurable_on_UNIV [symmetric])
-  apply (subst measurable_on_componentwise_UNIV)
-  apply (simp add: measurable_on_UNIV if_distrib [of "\<lambda>x. inner x _"] if_distrib [of "\<lambda>x. scaleR x _"] cong: if_cong)
-  done
-
-
-lemma\<^marker>\<open>tag important\<close> borel_measurable_implies_simple_function_sequence_real:
-  fixes u :: "'a \<Rightarrow> real"
-  assumes u[measurable]: "u \<in> borel_measurable M" and nn: "\<And>x. u x \<ge> 0"
-  shows "\<exists>f. incseq f \<and> (\<forall>i. simple_function M (f i)) \<and> (\<forall>x. bdd_above (range (\<lambda>i. f i x))) \<and>
-             (\<forall>i x. 0 \<le> f i x) \<and> u = (SUP i. f i)"
-proof -
-  define f where [abs_def]:
-    "f i x = real_of_int (floor ((min i (u x)) * 2^i)) / 2^i" for i x
-
-  have [simp]: "0 \<le> f i x" for i x
-    by (auto simp: f_def intro!: divide_nonneg_nonneg mult_nonneg_nonneg nn)
-
-  have *: "2^n * real_of_int x = real_of_int (2^n * x)" for n x
-    by simp
-
-  have "real_of_int \<lfloor>real i * 2 ^ i\<rfloor> = real_of_int \<lfloor>i * 2 ^ i\<rfloor>" for i
-    by (intro arg_cong[where f=real_of_int]) simp
-  then have [simp]: "real_of_int \<lfloor>real i * 2 ^ i\<rfloor> = i * 2 ^ i" for i
-    unfolding floor_of_nat by simp
-
-  have bdd: "bdd_above (range (\<lambda>i. f i x))" for x
-    by (rule bdd_aboveI [where M = "u x"]) (auto simp: f_def field_simps min_def)
-
-  have "incseq f"
-  proof (intro monoI le_funI)
-    fix m n :: nat and x assume "m \<le> n"
-    moreover
-    { fix d :: nat
-      have "\<lfloor>2^d::real\<rfloor> * \<lfloor>2^m * (min (of_nat m) (u x))\<rfloor> \<le> \<lfloor>2^d * (2^m * (min (of_nat m) (u x)))\<rfloor>"
-        by (rule le_mult_floor) (auto simp: nn)
-      also have "\<dots> \<le> \<lfloor>2^d * (2^m *  (min (of_nat d + of_nat m) (u x)))\<rfloor>"
-        by (intro floor_mono mult_mono min.mono)
-           (auto simp: nn min_less_iff_disj of_nat_less_top)
-      finally have "f m x \<le> f(m + d) x"
-        unfolding f_def
-        by (auto simp: field_simps power_add * simp del: of_int_mult) }
-    ultimately show "f m x \<le> f n x"
-      by (auto simp: le_iff_add)
-  qed
-  then have inc_f: "incseq (\<lambda>i. f i x)" for x
-    by (auto simp: incseq_def le_fun_def)
-  moreover
-  have "simple_function M (f i)" for i
-  proof (rule simple_function_borel_measurable)
-    have "\<lfloor>(min (of_nat i) (u x)) * 2 ^ i\<rfloor> \<le> \<lfloor>int i * 2 ^ i\<rfloor>" for x
-      by (auto split: split_min intro!: floor_mono)
-    then have "f i ` space M \<subseteq> (\<lambda>n. real_of_int n / 2^i) ` {0 .. of_nat i * 2^i}"
-      unfolding floor_of_int by (auto simp: f_def nn intro!: imageI)
-    then show "finite (f i ` space M)"
-      by (rule finite_subset) auto
-    show "f i \<in> borel_measurable M"
-      unfolding f_def enn2real_def by measurable
-  qed
-  moreover
-  { fix x
-    have "(SUP i. (f i x)) = u x"
-    proof -
-      obtain n where "u x \<le> of_nat n" using real_arch_simple by auto
-      then have min_eq_r: "\<forall>\<^sub>F i in sequentially. min (real i) (u x) = u x"
-        by (auto simp: eventually_sequentially intro!: exI[of _ n] split: split_min)
-      have "(\<lambda>i. real_of_int \<lfloor>min (real i) (u x) * 2^i\<rfloor> / 2^i) \<longlonglongrightarrow> u x"
-      proof (rule tendsto_sandwich)
-        show "(\<lambda>n. u x - (1/2)^n) \<longlonglongrightarrow> u x"
-          by (auto intro!: tendsto_eq_intros LIMSEQ_power_zero)
-        show "\<forall>\<^sub>F n in sequentially. real_of_int \<lfloor>min (real n) (u x) * 2 ^ n\<rfloor> / 2 ^ n \<le> u x"
-          using min_eq_r by eventually_elim (auto simp: field_simps)
-        have *: "u x * (2 ^ n * 2 ^ n) \<le> 2^n + 2^n * real_of_int \<lfloor>u x * 2 ^ n\<rfloor>" for n
-          using real_of_int_floor_ge_diff_one[of "u x * 2^n", THEN mult_left_mono, of "2^n"]
-          by (auto simp: field_simps)
-        show "\<forall>\<^sub>F n in sequentially. u x - (1/2)^n \<le> real_of_int \<lfloor>min (real n) (u x) * 2 ^ n\<rfloor> / 2 ^ n"
-          using min_eq_r by eventually_elim (insert *, auto simp: field_simps)
-      qed auto
-      then have "(\<lambda>i. (f i x)) \<longlonglongrightarrow> u x"
-        by (simp add: f_def)
-      from LIMSEQ_unique LIMSEQ_incseq_SUP [OF bdd inc_f] this
-      show ?thesis
-        by blast
-    qed }
-  ultimately show ?thesis
-    by (intro exI [of _ "\<lambda>i x. f i x"]) (auto simp: \<open>incseq f\<close> bdd image_comp)
-qed
-
-
-lemma homeomorphic_open_interval_UNIV:
-  fixes a b:: real
-  assumes "a < b"
-  shows "{a<..<b} homeomorphic (UNIV::real set)"
-proof -
-  have "{a<..<b} = ball ((b+a) / 2) ((b-a) / 2)"
-    using assms
-    by (auto simp: dist_real_def abs_if divide_simps split: if_split_asm)
-  then show ?thesis
-    by (simp add: homeomorphic_ball_UNIV assms)
-qed
-
-proposition homeomorphic_box_UNIV:
-  fixes a b:: "'a::euclidean_space"
-  assumes "box a b \<noteq> {}"
-  shows "box a b homeomorphic (UNIV::'a set)"
-proof -
-  have "{a \<bullet> i <..<b \<bullet> i} homeomorphic (UNIV::real set)" if "i \<in> Basis" for i
-    using assms box_ne_empty that by (blast intro: homeomorphic_open_interval_UNIV)
-  then have "\<exists>f g. (\<forall>x. a \<bullet> i < x \<and> x < b \<bullet> i \<longrightarrow> g (f x) = x) \<and>
-                   (\<forall>y. a \<bullet> i < g y \<and> g y < b \<bullet> i \<and> f(g y) = y) \<and>
-                   continuous_on {a \<bullet> i<..<b \<bullet> i} f \<and>
-                   continuous_on (UNIV::real set) g"
-    if "i \<in> Basis" for i
-    using that by (auto simp: homeomorphic_minimal mem_box Ball_def)
-  then obtain f g where gf: "\<And>i x. \<lbrakk>i \<in> Basis; a \<bullet> i < x; x < b \<bullet> i\<rbrakk> \<Longrightarrow> g i (f i x) = x"
-              and fg: "\<And>i y. i \<in> Basis \<Longrightarrow> a \<bullet> i < g i y \<and> g i y < b \<bullet> i \<and> f i (g i y) = y"
-              and contf: "\<And>i. i \<in> Basis \<Longrightarrow> continuous_on {a \<bullet> i<..<b \<bullet> i} (f i)"
-              and contg: "\<And>i. i \<in> Basis \<Longrightarrow> continuous_on (UNIV::real set) (g i)"
-    by metis
-  define F where "F \<equiv> \<lambda>x. \<Sum>i\<in>Basis. (f i (x \<bullet> i)) *\<^sub>R i"
-  define G where "G \<equiv> \<lambda>x. \<Sum>i\<in>Basis. (g i (x \<bullet> i)) *\<^sub>R i"
-  show ?thesis
-    unfolding homeomorphic_minimal
-  proof (intro exI conjI ballI)
-    show "G y \<in> box a b" for y
-      using fg by (simp add: G_def mem_box)
-    show "G (F x) = x" if "x \<in> box a b" for x
-      using that by (simp add: F_def G_def gf mem_box euclidean_representation)
-    show "F (G y) = y" for y
-      by (simp add: F_def G_def fg mem_box euclidean_representation)
-    show "continuous_on (box a b) F"
-      unfolding F_def
-    proof (intro continuous_intros continuous_on_compose2 [OF contf continuous_on_inner])
-      show "(\<lambda>x. x \<bullet> i) ` box a b \<subseteq> {a \<bullet> i<..<b \<bullet> i}" if "i \<in> Basis" for i
-        using that by (auto simp: mem_box)
-    qed
-    show "continuous_on UNIV G"
-      unfolding G_def
-      by (intro continuous_intros continuous_on_compose2 [OF contg continuous_on_inner]) auto
-  qed auto
-qed
-
-
-
-lemma diff_null_sets_lebesgue: "\<lbrakk>N \<in> null_sets (lebesgue_on S); X-N \<in> sets (lebesgue_on S); N \<subseteq> X\<rbrakk>
-    \<Longrightarrow> X \<in> sets (lebesgue_on S)"
-  by (metis Int_Diff_Un inf.commute inf.orderE null_setsD2 sets.Un)
-
-lemma borel_measurable_diff_null:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes N: "N \<in> null_sets (lebesgue_on S)" and S: "S \<in> sets lebesgue"
-  shows "f \<in> borel_measurable (lebesgue_on (S-N)) \<longleftrightarrow> f \<in> borel_measurable (lebesgue_on S)"
-  unfolding in_borel_measurable lebesgue_on_UNIV_eq space_lebesgue_on sets_restrict_UNIV
-proof (intro ball_cong iffI)
-  show "f -` T \<inter> S \<in> sets (lebesgue_on S)"
-    if "f -` T \<inter> (S-N) \<in> sets (lebesgue_on (S-N))" for T
-    using that  assms
-    by (smt Diff_Int_distrib completion.complete2 diff_null_sets_lebesgue inf.idem inf_le2 inf_mono lebesgue_on_UNIV_eq null_setsD2 null_sets_restrict_space sets.Diff sets_restrict_space_iff space_lebesgue_on space_restrict_space)
-  show "f -` T \<inter> (S-N) \<in> sets (lebesgue_on (S-N))"
-    if "f -` T \<inter> S \<in> sets (lebesgue_on S)" for T
-    using image_eqI inf.commute inf_top_right sets_restrict_space that
-    by (smt Int_Diff S sets.Int_space_eq2 sets_restrict_space_iff space_lebesgue_on)
-qed auto
-
-lemma lebesgue_measurable_diff_null:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes "N \<in> null_sets lebesgue"
-  shows "f \<in> borel_measurable (lebesgue_on (-N)) \<longleftrightarrow> f \<in> borel_measurable lebesgue"
-  by (simp add: Compl_eq_Diff_UNIV assms borel_measurable_diff_null lebesgue_on_UNIV_eq)
-
-
-
-proposition measurable_on_imp_borel_measurable_lebesgue_UNIV:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes "f measurable_on UNIV"
-  shows "f \<in> borel_measurable lebesgue"
-proof -
-  obtain N and F
-    where NF: "negligible N"
-      and conF: "\<And>n. continuous_on UNIV (F n)"
-      and tendsF: "\<And>x. x \<notin> N \<Longrightarrow> (\<lambda>n. F n x) \<longlonglongrightarrow> f x"
-    using assms by (auto simp: measurable_on_def)
-  obtain N where "N \<in> null_sets lebesgue" "f \<in> borel_measurable (lebesgue_on (-N))"
-  proof
-    show "f \<in> borel_measurable (lebesgue_on (- N))"
-    proof (rule borel_measurable_LIMSEQ_metric)
-      show "F i \<in> borel_measurable (lebesgue_on (- N))" for i
-        by (meson Compl_in_sets_lebesgue NF conF continuous_imp_measurable_on_sets_lebesgue continuous_on_subset negligible_imp_sets subset_UNIV)
-      show "(\<lambda>i. F i x) \<longlonglongrightarrow> f x" if "x \<in> space (lebesgue_on (- N))" for x
-        using that
-        by (simp add: tendsF)
-    qed
-    show "N \<in> null_sets lebesgue"
-      using NF negligible_iff_null_sets by blast
-  qed
-  then show ?thesis
-    using lebesgue_measurable_diff_null by blast
-qed
-
-corollary measurable_on_imp_borel_measurable_lebesgue:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes "f measurable_on S" and S: "S \<in> sets lebesgue"
-  shows "f \<in> borel_measurable (lebesgue_on S)"
-proof -
-  have "(\<lambda>x. if x \<in> S then f x else 0) measurable_on UNIV"
-    using assms(1) measurable_on_UNIV by blast
-  then show ?thesis
-    by (simp add: borel_measurable_if_D measurable_on_imp_borel_measurable_lebesgue_UNIV)
-qed
-
-
-proposition measurable_on_limit:
-  fixes f :: "nat \<Rightarrow> 'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes f: "\<And>n. f n measurable_on S" and N: "negligible N"
-    and lim: "\<And>x. x \<in> S - N \<Longrightarrow> (\<lambda>n. f n x) \<longlonglongrightarrow> g x"
-  shows "g measurable_on S"
-proof -
-  have "box (0::'b) One homeomorphic (UNIV::'b set)"
-    by (simp add: homeomorphic_box_UNIV)
-  then obtain h h':: "'b\<Rightarrow>'b" where hh': "\<And>x. x \<in> box 0 One \<Longrightarrow> h (h' x) = x"
-                  and h'im:  "h' ` box 0 One = UNIV"
-                  and conth: "continuous_on UNIV h"
-                  and conth': "continuous_on (box 0 One) h'"
-                  and h'h:   "\<And>y. h' (h y) = y"
-                  and rangeh: "range h = box 0 One"
-    by (auto simp: homeomorphic_def homeomorphism_def)
-  have "norm y \<le> DIM('b)" if y: "y \<in> box 0 One" for y::'b
-  proof -
-    have y01: "0 < y \<bullet> i" "y \<bullet> i < 1" if "i \<in> Basis" for i
-      using that y by (auto simp: mem_box)
-    have "norm y \<le> (\<Sum>i\<in>Basis. \<bar>y \<bullet> i\<bar>)"
-      using norm_le_l1 by blast
-    also have "\<dots> \<le> (\<Sum>i::'b\<in>Basis. 1)"
-    proof (rule sum_mono)
-      show "\<bar>y \<bullet> i\<bar> \<le> 1" if "i \<in> Basis" for i
-        using y01 that by fastforce
-    qed
-    also have "\<dots> \<le> DIM('b)"
-      by auto
-    finally show ?thesis .
-  qed
-  then have norm_le: "norm(h y) \<le> DIM('b)" for y
-    by (metis UNIV_I image_eqI rangeh)
-  have "(h' \<circ> (h \<circ> (\<lambda>x. if x \<in> S then g x else 0))) measurable_on UNIV"
-  proof (rule measurable_on_compose_continuous_box)
-    let ?\<chi> =  "h \<circ> (\<lambda>x. if x \<in> S then g x else 0)"
-    let ?f = "\<lambda>n. h \<circ> (\<lambda>x. if x \<in> S then f n x else 0)"
-    show "?\<chi> measurable_on UNIV"
-    proof (rule integrable_subintervals_imp_measurable)
-      show "?\<chi> integrable_on cbox a b" for a b
-      proof (rule integrable_spike_set)
-        show "?\<chi> integrable_on (cbox a b - N)"
-        proof (rule dominated_convergence_integrable)
-          show const: "(\<lambda>x. DIM('b)) integrable_on cbox a b - N"
-            by (simp add: N has_integral_iff integrable_const integrable_negligible integrable_setdiff negligible_diff)
-          show "norm ((h \<circ> (\<lambda>x. if x \<in> S then g x else 0)) x) \<le> DIM('b)" if "x \<in> cbox a b - N" for x
-            using that norm_le  by (simp add: o_def)
-          show "(\<lambda>k. ?f k x) \<longlonglongrightarrow> ?\<chi> x" if "x \<in> cbox a b - N" for x
-            using that lim [of x] conth
-            by (auto simp: continuous_on_def intro: tendsto_compose)
-          show "(?f n) absolutely_integrable_on cbox a b - N" for n
-          proof (rule measurable_bounded_by_integrable_imp_absolutely_integrable)
-            show "?f n \<in> borel_measurable (lebesgue_on (cbox a b - N))"
-            proof (rule measurable_on_imp_borel_measurable_lebesgue [OF measurable_on_spike_set])
-              show "?f n measurable_on cbox a b"
-                unfolding measurable_on_UNIV [symmetric, of _ "cbox a b"]
-              proof (rule measurable_on_restrict)
-                have f': "(\<lambda>x. if x \<in> S then f n x else 0) measurable_on UNIV"
-                  by (simp add: f measurable_on_UNIV)
-                show "?f n measurable_on UNIV"
-                  using measurable_on_compose_continuous [OF f' conth] by auto
-              qed auto
-              show "negligible (cbox a b \<Delta> (cbox a b - N))"
-                by (auto intro: negligible_subset [OF N])
-              show "cbox a b - N \<in> sets lebesgue"
-                by (simp add: N negligible_imp_sets sets.Diff)
-            qed
-            show "cbox a b - N \<in> sets lebesgue"
-              by (simp add: N negligible_imp_sets sets.Diff)
-            show "norm (?f n x) \<le> DIM('b)"
-              if "x \<in> cbox a b - N" for x
-              using that local.norm_le by simp
-          qed (auto simp: const)
-        qed
-        show "negligible {x \<in> cbox a b - N - cbox a b. ?\<chi> x \<noteq> 0}"
-          by (auto simp: empty_imp_negligible)
-        have "{x \<in> cbox a b - (cbox a b - N). ?\<chi> x \<noteq> 0} \<subseteq> N"
-          by auto
-        then show "negligible {x \<in> cbox a b - (cbox a b - N). ?\<chi> x \<noteq> 0}"
-          using N negligible_subset by blast
-      qed
-    qed
-    show "?\<chi> x \<in> box 0 One" for x
-      using rangeh by auto
-    show "continuous_on (box 0 One) h'"
-      by (rule conth')
-  qed
-  then show ?thesis
-    by (simp add: o_def h'h measurable_on_UNIV)
-qed
-
-
-lemma measurable_on_if_simple_function_limit:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  shows  "\<lbrakk>\<And>n. g n measurable_on UNIV; \<And>n. finite (range (g n)); \<And>x. (\<lambda>n. g n x) \<longlonglongrightarrow> f x\<rbrakk>
-   \<Longrightarrow> f measurable_on UNIV"
-  by (force intro: measurable_on_limit [where N="{}"])
-
-
-lemma lebesgue_measurable_imp_measurable_on_nnreal_UNIV:
-  fixes u :: "'a::euclidean_space \<Rightarrow> real"
-  assumes u: "u \<in> borel_measurable lebesgue" and nn: "\<And>x. u x \<ge> 0"
-  shows "u measurable_on UNIV"
-proof -
-  obtain f where "incseq f" and f: "\<forall>i. simple_function lebesgue (f i)"
-    and bdd: "\<And>x. bdd_above (range (\<lambda>i. f i x))"
-    and nnf: "\<And>i x. 0 \<le> f i x" and *: "u = (SUP i. f i)"
-    using borel_measurable_implies_simple_function_sequence_real nn u by metis
-  show ?thesis
-    unfolding *
-  proof (rule measurable_on_if_simple_function_limit [of concl: "\<Squnion> range f"])
-    show "(f i) measurable_on UNIV" for i
-      by (simp add: f nnf simple_function_measurable_on_UNIV)
-    show "finite (range (f i))" for i
-      by (metis f simple_function_def space_borel space_completion space_lborel)
-    show "(\<lambda>i. f i x) \<longlonglongrightarrow> (\<Squnion> range f) x" for x
-    proof -
-      have "incseq (\<lambda>i. f i x)"
-        using \<open>incseq f\<close> apply (auto simp: incseq_def)
-        by (simp add: le_funD)
-      then show ?thesis
-        by (metis SUP_apply bdd LIMSEQ_incseq_SUP)
-    qed
-  qed
-qed
-
-lemma lebesgue_measurable_imp_measurable_on_nnreal:
-  fixes u :: "'a::euclidean_space \<Rightarrow> real"
-  assumes "u \<in> borel_measurable lebesgue" "\<And>x. u x \<ge> 0""S \<in> sets lebesgue"
-  shows "u measurable_on S"
-  unfolding measurable_on_UNIV [symmetric, of u]
-  using assms
-  by (auto intro: lebesgue_measurable_imp_measurable_on_nnreal_UNIV)
-
-lemma lebesgue_measurable_imp_measurable_on_real:
-  fixes u :: "'a::euclidean_space \<Rightarrow> real"
-  assumes u: "u \<in> borel_measurable lebesgue" and S: "S \<in> sets lebesgue"
-  shows "u measurable_on S"
-proof -
-  let ?f = "\<lambda>x. \<bar>u x\<bar> + u x"
-  let ?g = "\<lambda>x. \<bar>u x\<bar> - u x"
-  have "?f measurable_on S" "?g measurable_on S"
-    using S u by (auto intro: lebesgue_measurable_imp_measurable_on_nnreal)
-  then have "(\<lambda>x. (?f x - ?g x) / 2) measurable_on S"
-    using measurable_on_cdivide measurable_on_diff by blast
-  then show ?thesis
-    by auto
-qed
-
-
-proposition lebesgue_measurable_imp_measurable_on:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes f: "f \<in> borel_measurable lebesgue" and S: "S \<in> sets lebesgue"
-  shows "f measurable_on S"
-  unfolding measurable_on_componentwise [of f]
-proof
-  fix i::'b
-  assume "i \<in> Basis"
-  have "(\<lambda>x. (f x \<bullet> i)) \<in> borel_measurable lebesgue"
-    using \<open>i \<in> Basis\<close> borel_measurable_euclidean_space f by blast
-  then have "(\<lambda>x. (f x \<bullet> i)) measurable_on S"
-    using S lebesgue_measurable_imp_measurable_on_real by blast
-  then show "(\<lambda>x. (f x \<bullet> i) *\<^sub>R i) measurable_on S"
-    by (intro measurable_on_scaleR measurable_on_const S)
-qed
-
-proposition measurable_on_iff_borel_measurable:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes "S \<in> sets lebesgue"
-  shows "f measurable_on S \<longleftrightarrow> f \<in> borel_measurable (lebesgue_on S)" (is "?lhs = ?rhs")
-proof
-  show "f \<in> borel_measurable (lebesgue_on S)"
-    if "f measurable_on S"
-    using that by (simp add: assms measurable_on_imp_borel_measurable_lebesgue)
-next
-  assume "f \<in> borel_measurable (lebesgue_on S)"
-  then have "(\<lambda>a. if a \<in> S then f a else 0) measurable_on UNIV"
-    by (simp add: assms borel_measurable_if_I lebesgue_measurable_imp_measurable_on)
-  then show "f measurable_on S"
-    using measurable_on_UNIV by blast
-qed
-
-lemma measurable_on_bilinear:
-  fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::euclidean_space"
-  assumes h: "bilinear h" and f: "f measurable_on S" and g: "g measurable_on S"
-  shows "(\<lambda>x. h (f x) (g x)) measurable_on S"
-proof (rule measurable_on_combine [where h = h])
-  show "continuous_on UNIV (\<lambda>x. h (fst x) (snd x))"
-    by (metis (mono_tags, lifting) bilinear_continuous_on_compose continuous_on_cong continuous_on_fst continuous_on_id continuous_on_snd h)
-  show "h 0 0 = 0"
-  by (simp add: bilinear_lzero h)
-qed (auto intro: assms)
-
-lemma borel_measurable_bilinear:
-  fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::euclidean_space"
-  assumes "bilinear h" "f \<in> borel_measurable (lebesgue_on S)" "g \<in> borel_measurable (lebesgue_on S)"
-    and S: "S \<in> sets lebesgue"
-  shows "(\<lambda>x. h (f x) (g x)) \<in> borel_measurable (lebesgue_on S)"
-  using assms measurable_on_bilinear [of h f S g]
-  by (simp flip: measurable_on_iff_borel_measurable)
-
-
-lemma absolutely_integrable_bounded_measurable_product:
-  fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::euclidean_space"
-  assumes "bilinear h" and f: "f \<in> borel_measurable (lebesgue_on S)" "S \<in> sets lebesgue"
-    and bou: "bounded (f ` S)" and g: "g absolutely_integrable_on S"
-  shows "(\<lambda>x. h (f x) (g x)) absolutely_integrable_on S"
-proof -
-  obtain B where "B > 0" and B: "\<And>x y. norm (h x y) \<le> B * norm x * norm y"
-    using bilinear_bounded_pos \<open>bilinear h\<close> by blast
-  obtain C where "C > 0" and C: "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> C"
-    using bounded_pos by (metis bou imageI)
-  show ?thesis
-  proof (rule measurable_bounded_by_integrable_imp_absolutely_integrable [OF _ \<open>S \<in> sets lebesgue\<close>])
-    show "norm (h (f x) (g x)) \<le> B * C * norm(g x)" if "x \<in> S" for x
-      by (meson less_le mult_left_mono mult_right_mono norm_ge_zero order_trans that \<open>B > 0\<close> B C)
-    show "(\<lambda>x. h (f x) (g x)) \<in> borel_measurable (lebesgue_on S)"
-      using \<open>bilinear h\<close> f g absolutely_integrable_measurable borel_measurable_bilinear by blast
-    show "(\<lambda>x. B * C * norm(g x)) integrable_on S"
-      using \<open>0 < B\<close> \<open>0 < C\<close> absolutely_integrable_on_def g by auto
-  qed
-qed
-
-lemma absolutely_integrable_bounded_measurable_product_real:
-  fixes f :: "real \<Rightarrow> real"
-  assumes "f \<in> borel_measurable (lebesgue_on S)" "S \<in> sets lebesgue"
-      and "bounded (f ` S)" and "g absolutely_integrable_on S"
-  shows "(\<lambda>x. f x * g x) absolutely_integrable_on S"
-  using absolutely_integrable_bounded_measurable_product bilinear_times assms by blast
-
-
-
-
-lemma set_integrable_mult_right_iff [simp]:
-  fixes a :: "'a::{real_normed_field, second_countable_topology}"
-  assumes "a \<noteq> 0"
-  shows "set_integrable M A (\<lambda>t. a * f t) \<longleftrightarrow> set_integrable M A f"
-proof
-  assume "set_integrable M A (\<lambda>t. a * f t)"
-  then have "set_integrable M A (\<lambda>t. 1/a * (a * f t))"
-    using set_integrable_mult_right by blast
-  then show "set_integrable M A f"
-    using assms by auto
-qed auto
-
-lemma set_integrable_mult_left_iff [simp]:
-  fixes a :: "'a::{real_normed_field, second_countable_topology}"
-  assumes "a \<noteq> 0"
-  shows "set_integrable M A (\<lambda>t. f t * a) \<longleftrightarrow> set_integrable M A f"
-  using assms
-  by (subst set_integrable_mult_right_iff [symmetric]) (auto simp: mult.commute)
-
-
-lemma set_integrable_mult_divide_iff [simp]:
-  fixes a :: "'a::{real_normed_field, second_countable_topology}"
-  assumes "a \<noteq> 0"
-  shows "set_integrable M A (\<lambda>t. f t / a) \<longleftrightarrow> set_integrable M A f"
-  by (simp add: divide_inverse assms)
-
-
-lemma borel_measurable_AE:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes "f \<in> borel_measurable lebesgue" and ae: "AE x in lebesgue. f x = g x"
-  shows "g \<in> borel_measurable lebesgue"
-proof -
-  obtain N where N: "N \<in> null_sets lebesgue" "\<And>x. x \<notin> N \<Longrightarrow> f x = g x"
-    using AE_null_sets_lebesgue [OF ae] by blast
-  have "f measurable_on UNIV"
-    by (simp add: assms lebesgue_measurable_imp_measurable_on)
-  then have "g measurable_on UNIV"
-    by (metis Diff_iff N measurable_on_spike negligible_iff_null_sets)
-  then show ?thesis
-    using measurable_on_imp_borel_measurable_lebesgue_UNIV by blast
-qed
-
-
-lemma has_bochner_integral_combine:
-  fixes f :: "real \<Rightarrow> 'a::euclidean_space"
-  assumes "a \<le> c" "c \<le> b"
-    and ac: "has_bochner_integral (lebesgue_on {a..c}) f i"
-    and cb: "has_bochner_integral (lebesgue_on {c..b}) f j"
-  shows "has_bochner_integral (lebesgue_on {a..b}) f(i + j)"
-proof -
-  have i: "has_bochner_integral lebesgue (\<lambda>x. indicator {a..c} x *\<^sub>R f x) i"
-   and j: "has_bochner_integral lebesgue (\<lambda>x. indicator {c..b} x *\<^sub>R f x) j"
-    using assms  by (auto simp: has_bochner_integral_restrict_space)
-  have AE: "AE x in lebesgue. indicat_real {a..c} x *\<^sub>R f x + indicat_real {c..b} x *\<^sub>R f x = indicat_real {a..b} x *\<^sub>R f x"
-  proof (rule AE_I')
-    have eq: "indicat_real {a..c} x *\<^sub>R f x + indicat_real {c..b} x *\<^sub>R f x = indicat_real {a..b} x *\<^sub>R f x" if "x \<noteq> c" for x
-      using assms that by (auto simp: indicator_def)
-    then show "{x \<in> space lebesgue. indicat_real {a..c} x *\<^sub>R f x + indicat_real {c..b} x *\<^sub>R f x \<noteq> indicat_real {a..b} x *\<^sub>R f x} \<subseteq> {c}"
-      by auto
-  qed auto
-  have "has_bochner_integral lebesgue (\<lambda>x. indicator {a..b} x *\<^sub>R f x) (i + j)"
-  proof (rule has_bochner_integralI_AE [OF has_bochner_integral_add [OF i j] _ AE])
-    have eq: "indicat_real {a..c} x *\<^sub>R f x + indicat_real {c..b} x *\<^sub>R f x = indicat_real {a..b} x *\<^sub>R f x" if "x \<noteq> c" for x
-      using assms that by (auto simp: indicator_def)
-    show "(\<lambda>x. indicat_real {a..b} x *\<^sub>R f x) \<in> borel_measurable lebesgue"
-    proof (rule borel_measurable_AE [OF borel_measurable_add AE])
-      show "(\<lambda>x. indicator {a..c} x *\<^sub>R f x) \<in> borel_measurable lebesgue"
-           "(\<lambda>x. indicator {c..b} x *\<^sub>R f x) \<in> borel_measurable lebesgue"
-        using i j by auto
-    qed
-  qed
-  then show ?thesis
-    by (simp add: has_bochner_integral_restrict_space)
-qed
-
-lemma integrable_combine:
-  fixes f :: "real \<Rightarrow> 'a::euclidean_space"
-  assumes "integrable (lebesgue_on {a..c}) f" "integrable (lebesgue_on {c..b}) f"
-    and "a \<le> c" "c \<le> b"
-  shows "integrable (lebesgue_on {a..b}) f"
-  using assms has_bochner_integral_combine has_bochner_integral_iff by blast
-
-lemma integral_combine:
-  fixes f :: "real \<Rightarrow> 'a::euclidean_space"
-  assumes f: "integrable (lebesgue_on {a..b}) f" and "a \<le> c" "c \<le> b"
-  shows "integral\<^sup>L (lebesgue_on {a..b}) f = integral\<^sup>L (lebesgue_on {a..c}) f + integral\<^sup>L (lebesgue_on {c..b}) f"
-proof -
-  have i: "has_bochner_integral (lebesgue_on {a..c}) f(integral\<^sup>L (lebesgue_on {a..c}) f)"
-    using integrable_subinterval_real \<open>c \<le> b\<close> f has_bochner_integral_iff by fastforce
-  have j: "has_bochner_integral (lebesgue_on {c..b}) f(integral\<^sup>L (lebesgue_on {c..b}) f)"
-    using integrable_subinterval_real \<open>a \<le> c\<close> f has_bochner_integral_iff by fastforce
-  show ?thesis
-    by (meson \<open>a \<le> c\<close> \<open>c \<le> b\<close> has_bochner_integral_combine has_bochner_integral_iff i j)
-qed
-
-lemma has_bochner_integral_reflect_real_lemma[intro]:
-  fixes f :: "real \<Rightarrow> 'a::euclidean_space"
-  assumes "has_bochner_integral (lebesgue_on {a..b}) f i"
-  shows "has_bochner_integral (lebesgue_on {-b..-a}) (\<lambda>x. f(-x)) i"
-proof -
-  have eq: "indicat_real {a..b} (- x) *\<^sub>R f(- x) = indicat_real {- b..- a} x *\<^sub>R f(- x)" for x
-    by (auto simp: indicator_def)
-  have i: "has_bochner_integral lebesgue (\<lambda>x. indicator {a..b} x *\<^sub>R f x) i"
-    using assms by (auto simp: has_bochner_integral_restrict_space)
-  then have "has_bochner_integral lebesgue (\<lambda>x. indicator {-b..-a} x *\<^sub>R f(-x)) i"
-    using has_bochner_integral_lebesgue_real_affine_iff [of "-1" "(\<lambda>x. indicator {a..b} x *\<^sub>R f x)" i 0]
-    by (auto simp: eq)
-  then show ?thesis
-    using assms by blast
-qed
-
-lemma has_bochner_integral_reflect_real[simp]:
-  fixes f :: "real \<Rightarrow> 'a::euclidean_space"
-  shows "has_bochner_integral (lebesgue_on {-b..-a}) (\<lambda>x. f(-x)) i \<longleftrightarrow> has_bochner_integral (lebesgue_on {a..b}) f i"
-  by (auto simp: dest: has_bochner_integral_reflect_real_lemma)
-
 lemma has_bochner_integral_null [intro]:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "N \<in> null_sets lebesgue"
   shows "has_bochner_integral (lebesgue_on N) f 0"
   unfolding has_bochner_integral_iff
 proof
   show "integrable (lebesgue_on N) f"
   proof (subst integrable_restrict_space)
     show "N \<inter> space lebesgue \<in> sets lebesgue"
       using assms by force
     show "integrable lebesgue (\<lambda>x. indicat_real N x *\<^sub>R f x)"
     proof (rule integrable_cong_AE_imp)
       show "integrable lebesgue (\<lambda>x. 0)"
         by simp
       show *: "AE x in lebesgue. 0 = indicat_real N x *\<^sub>R f x"
         using assms
         by (simp add: indicator_def completion.null_sets_iff_AE eventually_mono)
       show "(\<lambda>x. indicat_real N x *\<^sub>R f x) \<in> borel_measurable lebesgue"
         by (auto intro: borel_measurable_AE [OF _ *])
     qed
   qed
   show "integral\<^sup>L (lebesgue_on N) f = 0"
   proof (rule integral_eq_zero_AE)
     show "AE x in lebesgue_on N. f x = 0"
       by (rule AE_I' [where N=N]) (auto simp: assms null_setsD2 null_sets_restrict_space)
   qed
 qed
 
-
 lemma has_bochner_integral_null_eq[simp]:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "N \<in> null_sets lebesgue"
   shows "has_bochner_integral (lebesgue_on N) f i \<longleftrightarrow> i = 0"
   using assms has_bochner_integral_eq by blast
 
-lemma integrable_reflect_real[simp]:
-  fixes f :: "real \<Rightarrow> 'a::euclidean_space"
-  shows "integrable (lebesgue_on {-b..-a}) (\<lambda>x. f(-x)) \<longleftrightarrow> integrable (lebesgue_on {a..b}) f"
-  by (metis has_bochner_integral_iff has_bochner_integral_reflect_real)
-
-lemma integral_reflect_real[simp]:
-  fixes f :: "real \<Rightarrow> 'a::euclidean_space"
-  shows "integral\<^sup>L (lebesgue_on {-b .. -a}) (\<lambda>x. f(-x)) = integral\<^sup>L (lebesgue_on {a..b::real}) f"
-  using has_bochner_integral_reflect_real [of b a f]
-  by (metis has_bochner_integral_iff not_integrable_integral_eq)
-
-lemma monoseq_convergent:
-  fixes X :: "nat \<Rightarrow> real"
-  assumes X: "monoseq X" and B: "\<And>i. \<bar>X i\<bar> \<le> B"
-  obtains L where "X \<longlonglongrightarrow> L"
-  using X unfolding monoseq_iff
-proof
-  assume "incseq X"
-  show thesis
-    using abs_le_D1 [OF B] incseq_convergent [OF \<open>incseq X\<close>] that by meson
-next
-  assume "decseq X"
-  show thesis
-    using decseq_convergent [OF \<open>decseq X\<close>] that
-    by (metis B abs_le_iff add.inverse_inverse neg_le_iff_le)
-qed
-
 end
diff --git a/thys/Fourier/Lspace.thy b/thys/Fourier/Lspace.thy
--- a/thys/Fourier/Lspace.thy
+++ b/thys/Fourier/Lspace.thy
@@ -1,150 +1,150 @@
 section\<open>Lspace as it is in HOL Light\<close>
 
 text\<open>Mainly a repackaging of existing material from Lp\<close>
 
 theory Lspace
-  imports Lp.Lp Fourier_Aux1
+  imports Lp.Lp 
 begin
 
 abbreviation lspace :: "'a measure \<Rightarrow> ennreal \<Rightarrow> ('a \<Rightarrow> real) set"
   where "lspace M p \<equiv> space\<^sub>N (\<LL> p M)"
 
 lemma lspace_1:
   assumes "S \<in> sets lebesgue"
   shows "f \<in> lspace (lebesgue_on S) 1 \<longleftrightarrow> f absolutely_integrable_on S"
   using assms by (simp add: L1_space integrable_restrict_space set_integrable_def)
 
 lemma lspace_ennreal_iff:
   assumes "p > 0"
   shows "lspace (lebesgue_on S) (ennreal p) = {f \<in> borel_measurable (lebesgue_on S). integrable (lebesgue_on S) (\<lambda>x. (norm(f x) powr p))}"
   using assms  by (fastforce simp: Lp_measurable Lp_D intro: Lp_I)
 
 lemma lspace_iff:
   assumes "\<infinity> > p" "p > 0"
   shows "lspace (lebesgue_on S) p = {f \<in> borel_measurable (lebesgue_on S). integrable (lebesgue_on S) (\<lambda>x. (norm(f x) powr (enn2real p)))}"
 proof -
   obtain q::real where "p = enn2real q"
     using Lp_cases assms by auto
   then show ?thesis
     using assms lspace_ennreal_iff by auto
 qed
 
 lemma lspace_iff':
   assumes p: "\<infinity> > p" "p > 0" and S: "S \<in> sets lebesgue"
   shows "lspace (lebesgue_on S) p = {f \<in> borel_measurable (lebesgue_on S). (\<lambda>x. (norm(f x) powr (enn2real p))) integrable_on S}"
     (is "?lhs = ?rhs")
 proof
   show "?lhs \<subseteq> ?rhs"
     using assms integrable_on_lebesgue_on by (auto simp: lspace_iff)
 next
   show "?rhs \<subseteq> ?lhs"
   proof (clarsimp simp: lspace_iff [OF p])
     show "integrable (lebesgue_on S) (\<lambda>xa. \<bar>f xa\<bar> powr enn2real p)"
       if "f \<in> borel_measurable (lebesgue_on S)" and "(\<lambda>x. \<bar>f x\<bar> powr enn2real p) integrable_on S" for f
     proof -
       have "(\<lambda>x. \<bar>f x\<bar> powr enn2real p) absolutely_integrable_on S"
         by (simp add: absolutely_integrable_on_iff_nonneg that(2))
       then show ?thesis
         using L1_space S lspace_1 by blast
     qed
   qed
 qed
 
 lemma lspace_mono:
   assumes "f \<in> lspace (lebesgue_on S) q" and S: "S \<in> lmeasurable" and "p > 0" "p \<le> q" "q < \<infinity>"
   shows "f \<in> lspace (lebesgue_on S) p"
 proof -
   have "p < \<infinity>"
     using assms by (simp add: top.not_eq_extremum)
   with assms show ?thesis
   proof (clarsimp simp add: lspace_iff')
     show "(\<lambda>x. \<bar>f x\<bar> powr enn2real p) integrable_on S"
       if "f \<in> borel_measurable (lebesgue_on S)"
         and "(\<lambda>x. \<bar>f x\<bar> powr enn2real q) integrable_on S"
     proof (rule measurable_bounded_by_integrable_imp_integrable)
       show "(\<lambda>x. \<bar>f x\<bar> powr enn2real p) \<in> borel_measurable (lebesgue_on S)"
         using measurable_abs_powr that(1) by blast
       let ?g = "\<lambda>x. max 1 (norm(f x) powr enn2real q)"
       have "?g absolutely_integrable_on S"
       proof (rule absolutely_integrable_max_1)
         show "(\<lambda>x. norm (f x) powr enn2real q) absolutely_integrable_on S"
           by (simp add: nonnegative_absolutely_integrable_1 that(2))
       qed (simp add: S)
       then show "?g integrable_on S"
         using absolutely_integrable_abs_iff by blast
       show "norm (\<bar>f x\<bar> powr enn2real p) \<le> ?g x" if "x \<in> S" for x
       proof -
         have "\<bar>f x\<bar> powr enn2real p \<le> \<bar>f x\<bar> powr enn2real q" if "1 \<le> \<bar>f x\<bar>"
           using assms enn2real_mono powr_mono that by auto
         then show ?thesis
           using powr_le1 by (fastforce simp add: le_max_iff_disj)
       qed
       show "S \<in> sets lebesgue"
         by (simp add: S fmeasurableD)
     qed
   qed
 qed
 
 lemma lspace_inclusion:
   assumes "S \<in> lmeasurable" and "p > 0" "p \<le> q" "q < \<infinity>"
   shows "lspace (lebesgue_on S) q \<subseteq> lspace (lebesgue_on S) p"
   using assms lspace_mono by auto
 
 lemma lspace_const:
   fixes p::real
   assumes "p > 0""S \<in> lmeasurable"
   shows "(\<lambda>x. c) \<in> lspace (lebesgue_on S) p"
   by (simp add: Lp_space assms finite_measure.integrable_const finite_measure_lebesgue_on)
 
 lemma lspace_max:
   fixes p::real
   assumes "f \<in> lspace (lebesgue_on S) p" "g \<in> lspace (lebesgue_on S) p" "p > 0"
   shows "(\<lambda>x. max (f x) (g x)) \<in> lspace (lebesgue_on S) p"
 proof -
   have "integrable (lebesgue_on S) (\<lambda>x. \<bar>max (f x) (g x)\<bar> powr p)"
     if f: "f \<in> borel_measurable (lebesgue_on S)" "integrable (lebesgue_on S) (\<lambda>x. \<bar>f x\<bar> powr p)"
       and g: "g \<in> borel_measurable (lebesgue_on S)" "integrable (lebesgue_on S) (\<lambda>x. \<bar>g x\<bar> powr p)"
   proof -
     have "integrable (lebesgue_on S) (\<lambda>x. \<bar>\<bar>f x\<bar> powr p\<bar>)" "integrable (lebesgue_on S) (\<lambda>x. \<bar>\<bar>g x\<bar> powr p\<bar>)"
       using integrable_abs that by blast+
     then have "integrable (lebesgue_on S) (\<lambda>x. max (\<bar>\<bar>f x\<bar> powr p\<bar>) (\<bar>\<bar>g x\<bar> powr p\<bar>))"
       using integrable_max by blast
     then show ?thesis
     proof (rule Bochner_Integration.integrable_bound)
       show "(\<lambda>x. \<bar>max (f x) (g x)\<bar> powr p) \<in> borel_measurable (lebesgue_on S)"
         using borel_measurable_max measurable_abs_powr that by blast
     qed auto
   qed
   then show ?thesis
     using assms by (auto simp: Lp_space borel_measurable_max)
 qed
 
 lemma lspace_min:
   fixes p::real
   assumes "f \<in> lspace (lebesgue_on S) p" "g \<in> lspace (lebesgue_on S) p" "p > 0"
   shows "(\<lambda>x. min (f x) (g x)) \<in> lspace (lebesgue_on S) p"
 proof -
   have "integrable (lebesgue_on S) (\<lambda>x. \<bar>min (f x) (g x)\<bar> powr p)"
     if f: "f \<in> borel_measurable (lebesgue_on S)" "integrable (lebesgue_on S) (\<lambda>x. \<bar>f x\<bar> powr p)"
       and g: "g \<in> borel_measurable (lebesgue_on S)" "integrable (lebesgue_on S) (\<lambda>x. \<bar>g x\<bar> powr p)"
   proof -
     have "integrable (lebesgue_on S) (\<lambda>x. \<bar>\<bar>f x\<bar> powr p\<bar>)" "integrable (lebesgue_on S) (\<lambda>x. \<bar>\<bar>g x\<bar> powr p\<bar>)"
       using integrable_abs that by blast+
     then have "integrable (lebesgue_on S) (\<lambda>x. max (\<bar>\<bar>f x\<bar> powr p\<bar>) (\<bar>\<bar>g x\<bar> powr p\<bar>))"
       using integrable_max by blast
     then show ?thesis
     proof (rule Bochner_Integration.integrable_bound)
       show "(\<lambda>x. \<bar>min (f x) (g x)\<bar> powr p) \<in> borel_measurable (lebesgue_on S)"
         using borel_measurable_min measurable_abs_powr that by blast
     qed auto
   qed
   then show ?thesis
     using assms by (auto simp: Lp_space borel_measurable_min)
 qed
 
 lemma Lp_space_numeral:
   assumes "numeral n > (0::int)"
   shows "space\<^sub>N (\<LL> (numeral n) M) = {f \<in> borel_measurable M. integrable M (\<lambda>x. \<bar>f x\<bar> ^ numeral n)}"
   using Lp_space [of "numeral n" M] assms by simp
 
 end
diff --git a/thys/Fourier/Periodic.thy b/thys/Fourier/Periodic.thy
--- a/thys/Fourier/Periodic.thy
+++ b/thys/Fourier/Periodic.thy
@@ -1,187 +1,221 @@
 section\<open>Shifting the origin for integration of periodic functions\<close>
 
 theory Periodic
-  imports "HOL-Analysis.Analysis" Fourier_Aux2
+  imports "HOL-Analysis.Analysis" 
 begin
 
+lemma has_bochner_integral_null [intro]:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  assumes "N \<in> null_sets lebesgue"
+  shows "has_bochner_integral (lebesgue_on N) f 0"
+  unfolding has_bochner_integral_iff
+proof
+  show "integrable (lebesgue_on N) f"
+  proof (subst integrable_restrict_space)
+    show "N \<inter> space lebesgue \<in> sets lebesgue"
+      using assms by force
+    show "integrable lebesgue (\<lambda>x. indicat_real N x *\<^sub>R f x)"
+    proof (rule integrable_cong_AE_imp)
+      show "integrable lebesgue (\<lambda>x. 0)"
+        by simp
+      show *: "AE x in lebesgue. 0 = indicat_real N x *\<^sub>R f x"
+        using assms
+        by (simp add: indicator_def completion.null_sets_iff_AE eventually_mono)
+      show "(\<lambda>x. indicat_real N x *\<^sub>R f x) \<in> borel_measurable lebesgue"
+        by (auto intro: borel_measurable_AE [OF _ *])
+    qed
+  qed
+  show "integral\<^sup>L (lebesgue_on N) f = 0"
+  proof (rule integral_eq_zero_AE)
+    show "AE x in lebesgue_on N. f x = 0"
+      by (rule AE_I' [where N=N]) (auto simp: assms null_setsD2 null_sets_restrict_space)
+  qed
+qed
+
+lemma has_bochner_integral_null_eq[simp]:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  assumes "N \<in> null_sets lebesgue"
+  shows "has_bochner_integral (lebesgue_on N) f i \<longleftrightarrow> i = 0"
+  using assms has_bochner_integral_eq by blast
+
 lemma periodic_integer_multiple:
    "(\<forall>x. f(x + a) = f x) \<longleftrightarrow> (\<forall>x. \<forall>n \<in> \<int>. f(x + n * a) = f x)" (is "?lhs = ?rhs")
 proof
   assume L [rule_format]: ?lhs
   have nat: "f(x + of_nat n * a) = f x" for x n
   proof (induction n)
     case (Suc n)
     with L [of "x + of_nat n * a"] show ?case
       by (simp add: algebra_simps)
   qed auto
   have "f(x + of_int z * a) = f x" for x z
   proof (cases "z \<ge> 0")
     case True
     then show ?thesis
       by (metis nat of_nat_nat)
   next
     case False
     then show ?thesis
       using nat [of "x + of_int z * a" "nat (-z)"] by auto
   qed
   then show ?rhs
     by (auto simp: Ints_def)
 qed (use Ints_1 in fastforce)
 
 lemma has_integral_offset:
   fixes f :: "real \<Rightarrow> 'a::euclidean_space"
   assumes "has_bochner_integral (lebesgue_on {a..b}) f i"
   shows "has_bochner_integral (lebesgue_on {a-c..b-c}) (\<lambda>x. f(x + c)) i"
 proof -
   have eq: "indicat_real {a..b} (x + c) = indicat_real {a-c..b-c} x" for x
     by (auto simp: indicator_def)
   have "has_bochner_integral lebesgue (\<lambda>x. indicator {a..b} x *\<^sub>R f x) i"
     using assms  by (auto simp: has_bochner_integral_restrict_space)
   then have "has_bochner_integral lebesgue (\<lambda>x. indicat_real {a-c..b-c} x *\<^sub>R f(x + c)) i"
     using has_bochner_integral_lebesgue_real_affine_iff [of 1 "(\<lambda>x. indicator {a..b} x *\<^sub>R f x)" i c]
     by (simp add: add_ac eq)
   then show ?thesis
     using assms by (auto simp: has_bochner_integral_restrict_space)
 qed
 
 
 lemma has_integral_periodic_offset_lemma:
   fixes f :: "real \<Rightarrow> 'a::euclidean_space"
   assumes periodic: "\<And>x. f(x + (b-a)) = f x" and f: "has_bochner_integral (lebesgue_on {a..a+c}) f i"
   shows "has_bochner_integral (lebesgue_on {b..b+c}) f i"
 proof -
   have "f(x + a-b) = f x" for x
     using periodic [of "x + (a-b)"] by (simp add: algebra_simps)
   then show ?thesis
     using has_integral_offset [OF f, of "a-b"]
     by (auto simp: algebra_simps)
 qed
 
 
 lemma has_integral_periodic_offset_pos:
   fixes f :: "real \<Rightarrow> real"
   assumes f: "has_bochner_integral (lebesgue_on {a..b}) f i" and periodic: "\<And>x. f(x + (b-a)) = f x"
       and c: "c \<ge> 0" "a + c \<le> b"
   shows "has_bochner_integral (lebesgue_on {a..b}) (\<lambda>x. f(x + c)) i"
 proof -
   have "{a..a + c} \<subseteq> {a..b}"
     by (simp add: assms(4))
   then have 1: "has_bochner_integral (lebesgue_on {a..a+c}) f(integral\<^sup>L (lebesgue_on {a..a+c}) f)"
-    by (meson integrable_subinterval_real f has_bochner_integral_iff)
+    by (meson integrable_subinterval f has_bochner_integral_iff)
   then have 3: "has_bochner_integral (lebesgue_on {b..b+c}) f(integral\<^sup>L (lebesgue_on {a..a+c}) f)"
     using has_integral_periodic_offset_lemma periodic by blast
   have "{a + c..b} \<subseteq> {a..b}"
     by (simp add: c)
   then have 2: "has_bochner_integral (lebesgue_on {a+c..b}) f(integral\<^sup>L (lebesgue_on {a+c..b}) f)"
-    by (meson integrable_subinterval_real f has_bochner_integral_integrable integrable.intros)
+    by (meson integrable_subinterval f has_bochner_integral_integrable integrable.intros)
   have "integral\<^sup>L (lebesgue_on {a + c..b}) f + integral\<^sup>L (lebesgue_on {a..a + c}) f = i"
     by (metis integral_combine add.commute c f has_bochner_integral_iff le_add_same_cancel1)
   then have "has_bochner_integral (lebesgue_on {a+c..b+c}) f i"
     using has_bochner_integral_combine [OF _ _ 2 3] 1 2 by (simp add: c)
   then show ?thesis
     by (metis add_diff_cancel_right' has_integral_offset)
 qed
 
 
 lemma has_integral_periodic_offset_weak:
   fixes f :: "real \<Rightarrow> real"
   assumes f: "has_bochner_integral (lebesgue_on {a..b}) f i" and periodic: "\<And>x. f(x + (b-a)) = f x" and c: "\<bar>c\<bar> \<le> b-a"
   shows "has_bochner_integral (lebesgue_on {a..b}) (\<lambda>x. f(x + c)) i"
 proof (cases "c \<ge> 0")
   case True
   then show ?thesis
     using c f has_integral_periodic_offset_pos periodic by auto
 next
   case False
   have per': "f(- (x + (- a - - b))) = f(- x)" for x
     using periodic [of "(a-b)-x"] by simp
   have f': "has_bochner_integral (lebesgue_on {- b..- a}) (\<lambda>x. f(- x)) i"
     using f by blast
   show ?thesis
     using has_integral_periodic_offset_pos [of "-b" "-a" "\<lambda>x. f(-x)" i "-c", OF f' per'] c  False
     by (simp flip: has_bochner_integral_reflect_real [of b a])
 qed
 
 lemma has_integral_periodic_offset:
   fixes f :: "real \<Rightarrow> real"
   assumes f: "has_bochner_integral (lebesgue_on {a..b}) f i" and periodic: "\<And>x. f(x + (b-a)) = f x"
   shows "has_bochner_integral (lebesgue_on {a..b}) (\<lambda>x. f(x + c)) i"
 proof -
   consider "b \<le> a" | "a < b" by linarith
   then show ?thesis
   proof cases
     case 1
     then have "{a..b} \<in> null_sets lebesgue"
       using less_eq_real_def by auto
     with f show ?thesis
       by auto
   next
     case 2
     define fba where "fba \<equiv> \<lambda>x. f(x + (b-a) * frac(c / (b-a)))"
     have eq: "fba x = f(x + c)"
       if "x \<in> {a..b}" for x
     proof -
       have "f(x + n * (b-a)) = f x" if "n \<in> \<int>" for n x
         using periodic periodic_integer_multiple that by blast
       then have "f((x + c) + - floor (c / (b - a)) * (b - a)) = f(x + c)"
         using Ints_of_int by blast
       moreover have "((x + c) + -floor (c / (b - a)) * (b - a)) = (x + (b - a) * frac (c / (b - a)))"
         using 2 by (simp add: field_simps frac_def)
       ultimately show ?thesis
         unfolding fba_def by metis
     qed
     have *: "has_bochner_integral (lebesgue_on {a..b}) fba i"
       unfolding fba_def
     proof (rule has_integral_periodic_offset_weak [OF f])
       show "f(x + (b - a)) = f x" for x
         by fact
       have "\<bar>frac (c / (b - a))\<bar> \<le> 1"
         using frac_unique_iff less_eq_real_def by auto
       then show "\<bar>(b - a) * frac (c / (b - a))\<bar> \<le> b - a"
         using "2" by auto
     qed
     then show ?thesis
     proof (rule has_bochner_integralI_AE [OF _ _ AE_I2])
       have "fba \<in> borel_measurable (lebesgue_on {a..b})"
         using "*" borel_measurable_has_bochner_integral by blast
       then show "(\<lambda>x. f(x + c)) \<in> borel_measurable (lebesgue_on {a..b})"
         by (subst measurable_lebesgue_cong [OF eq, symmetric])
     qed (auto simp: eq)
   qed
 qed
 
 lemma integrable_periodic_offset:
   fixes f :: "real \<Rightarrow> real"
   assumes f: "integrable (lebesgue_on {a..b}) f" and periodic: "\<And>x. f(x + (b-a)) = f x"
   shows "integrable (lebesgue_on {a..b}) (\<lambda>x. f(x + c))"
   using f has_integral_periodic_offset periodic
   by (simp add: has_bochner_integral_iff)
 
 lemma absolutely_integrable_periodic_offset:
   fixes f :: "real \<Rightarrow> real"
   assumes f: "f absolutely_integrable_on {a..b}" and periodic: "\<And>x. f(x + (b-a)) = f x"
   shows "(\<lambda>x. f(x + c)) absolutely_integrable_on {a..b}" "(\<lambda>x. f(c + x)) absolutely_integrable_on {a..b}"
   using assms integrable_periodic_offset [of a b "f"]
   by (auto simp: integrable_restrict_space set_integrable_def add.commute [of c])
 
 lemma integral_periodic_offset:
   fixes f :: "real \<Rightarrow> real"
   assumes periodic: "\<And>x. f(x + (b-a)) = f x"
   shows "integral\<^sup>L (lebesgue_on {a..b}) (\<lambda>x. f(x + c)) = integral\<^sup>L (lebesgue_on {a..b}) f"
 proof (cases "integrable (lebesgue_on {a..b}) f")
   case True
   then show ?thesis
     using has_integral_periodic_offset periodic
     by (simp add: has_bochner_integral_iff)
 next
   case False
   then have "\<not> integrable (lebesgue_on {a..b}) (\<lambda>x. f(x + c))"
     using periodic[of "_+c"]
     by (auto simp: algebra_simps intro: dest: integrable_periodic_offset [where c = "-c"])
   then have "integral\<^sup>L (lebesgue_on {a..b}) f = 0" "integral\<^sup>L (lebesgue_on {a..b}) (\<lambda>x. f(x + c)) = 0"
     using False not_integrable_integral_eq by blast+
   then show ?thesis
     by simp
 qed
 
 end
diff --git a/thys/Fourier/Square_Integrable.thy b/thys/Fourier/Square_Integrable.thy
--- a/thys/Fourier/Square_Integrable.thy
+++ b/thys/Fourier/Square_Integrable.thy
@@ -1,849 +1,824 @@
 section\<open>Square integrable functions over the reals\<close>
 
 theory Square_Integrable
-  imports Lspace Fourier_Aux2
+  imports Lspace 
 begin
 
 subsection\<open>Basic definitions\<close>
 
 definition square_integrable:: "(real \<Rightarrow> real) \<Rightarrow> real set \<Rightarrow> bool" (infixr "square'_integrable" 46)
   where "f square_integrable S \<equiv> S \<in> sets lebesgue \<and> f \<in> borel_measurable (lebesgue_on S) \<and> integrable (lebesgue_on S) (\<lambda>x. f x ^ 2)"
 
 lemma square_integrable_imp_measurable:
    "f square_integrable S \<Longrightarrow> f \<in> borel_measurable (lebesgue_on S)"
   by (simp add: square_integrable_def)
 
 lemma square_integrable_imp_lebesgue:
    "f square_integrable S \<Longrightarrow> S \<in> sets lebesgue"
   by (simp add: square_integrable_def)
 
 lemma square_integrable_imp_lspace:
   assumes "f square_integrable S" shows "f \<in> lspace (lebesgue_on S) 2"
 proof -
   have "(\<lambda>x. (f x)\<^sup>2) absolutely_integrable_on S"
     by (metis assms integrable_on_lebesgue_on nonnegative_absolutely_integrable_1 square_integrable_def zero_le_power2)
   moreover have "S \<in> sets lebesgue"
     using assms square_integrable_def by blast
   ultimately show ?thesis
     by (simp add: assms Lp_space_numeral integrable_restrict_space set_integrable_def square_integrable_imp_measurable)
 qed
 
 lemma square_integrable_iff_lspace:
   assumes "S \<in> sets lebesgue"
   shows "f square_integrable S \<longleftrightarrow> f \<in> lspace (lebesgue_on S) 2" (is "?lhs = ?rhs")
 proof
   assume L: ?lhs
   then show ?rhs
     using square_integrable_imp_lspace by blast
 next
   assume ?rhs then show ?lhs
   using assms by (auto simp: Lp_space_numeral square_integrable_def integrable_on_lebesgue_on)
 qed
 
 lemma square_integrable_0 [simp]:
    "S \<in> sets lebesgue \<Longrightarrow> (\<lambda>x. 0) square_integrable S"
   by (simp add: square_integrable_def power2_eq_square integrable_0)
 
 lemma square_integrable_neg_eq [simp]:
    "(\<lambda>x. -(f x)) square_integrable S \<longleftrightarrow> f square_integrable S"
   by (auto simp: square_integrable_def)
 
 lemma square_integrable_lmult [simp]:
   assumes "f square_integrable S"
   shows "(\<lambda>x. c * f x) square_integrable S"
 proof (simp add: square_integrable_def, intro conjI)
   have f: "f \<in> borel_measurable (lebesgue_on S)" "integrable (lebesgue_on S) (\<lambda>x. f x ^ 2)"
     using assms by (simp_all add: square_integrable_def)
   then show "(\<lambda>x. c * f x) \<in> borel_measurable (lebesgue_on S)"
     using borel_measurable_scaleR [of "\<lambda>x. c" "lebesgue_on S" f]  by simp
   have "integrable (lebesgue_on S) (\<lambda>x. c\<^sup>2 * (f x)\<^sup>2)"
   by (cases "c=0") (auto simp: f integrable_0)
   then show "integrable (lebesgue_on S) (\<lambda>x. (c * f x)\<^sup>2)"
     by (simp add: power2_eq_square mult_ac)
   show "S \<in> sets lebesgue"
     using assms square_integrable_def by blast
 qed
 
 lemma square_integrable_rmult [simp]:
    "f square_integrable S \<Longrightarrow> (\<lambda>x. f x * c) square_integrable S"
   using square_integrable_lmult [of f S c] by (simp add: mult.commute)
 
 lemma square_integrable_imp_absolutely_integrable_product:
   assumes f: "f square_integrable S" and g: "g square_integrable S"
   shows "(\<lambda>x. f x * g x) absolutely_integrable_on S"
 proof -
   have fS: "integrable (lebesgue_on S) (\<lambda>r. (f r)\<^sup>2)" "integrable (lebesgue_on S) (\<lambda>r. (g r)\<^sup>2)"
     using assms square_integrable_def by blast+
   have "integrable (lebesgue_on S) (\<lambda>x. \<bar>f x * g x\<bar>)"
   proof (intro integrable_abs Holder_inequality [of 2 2])
     show "f \<in> borel_measurable (lebesgue_on S)" "g \<in> borel_measurable (lebesgue_on S)"
       using f g square_integrable_def by blast+
     show "integrable (lebesgue_on S) (\<lambda>x. \<bar>f x\<bar> powr 2)" "integrable (lebesgue_on S) (\<lambda>x. \<bar>g x\<bar> powr 2)"
       using nonnegative_absolutely_integrable_1 [of "(\<lambda>x. (f x)\<^sup>2)"] nonnegative_absolutely_integrable_1 [of "(\<lambda>x. (g x)\<^sup>2)"]
       by (simp_all add: fS integrable_restrict_space set_integrable_def)
   qed auto
   then show ?thesis
     using assms
     by (simp add: absolutely_integrable_measurable_real borel_measurable_times square_integrable_def)
 qed
 
 lemma square_integrable_imp_integrable_product:
   assumes "f square_integrable S" "g square_integrable S"
   shows  "integrable (lebesgue_on S) (\<lambda>x. f x * g x)"
   using absolutely_integrable_measurable assms integrable_abs_iff
   by (metis (full_types) absolutely_integrable_measurable_real square_integrable_def square_integrable_imp_absolutely_integrable_product)
 
 lemma square_integrable_add [simp]:
   assumes f: "f square_integrable S" and g: "g square_integrable S"
   shows "(\<lambda>x. f x + g x) square_integrable S"
   unfolding square_integrable_def
 proof (intro conjI)
   show "S \<in> sets lebesgue"
     using assms square_integrable_def by blast
   show "(\<lambda>x. f x + g x) \<in> borel_measurable (lebesgue_on S)"
     by (simp add: f g borel_measurable_add square_integrable_imp_measurable)
   show "integrable (lebesgue_on S) (\<lambda>x. (f x + g x)\<^sup>2)"
     unfolding power2_eq_square distrib_right distrib_left
   proof (intro Bochner_Integration.integrable_add)
     show "integrable (lebesgue_on S) (\<lambda>x. f x * f x)" "integrable (lebesgue_on S) (\<lambda>x. g x * g x)"
       using f g square_integrable_imp_integrable_product by blast+
     show "integrable (lebesgue_on S) (\<lambda>x. f x * g x)" "integrable (lebesgue_on S) (\<lambda>x. g x * f x)"
       using f g square_integrable_imp_integrable_product by blast+
   qed
 qed
 
 lemma square_integrable_diff [simp]:
    "\<lbrakk>f square_integrable S; g square_integrable S\<rbrakk> \<Longrightarrow> (\<lambda>x. f x - g x) square_integrable S"
   using square_integrable_neg_eq square_integrable_add [of f S "\<lambda>x. - (g x)"] by auto
 
 lemma square_integrable_abs [simp]:
    "f square_integrable S \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) square_integrable S"
   by (simp add: square_integrable_def borel_measurable_abs)
 
 lemma square_integrable_sum [simp]:
   assumes I: "finite I" "\<And>i. i \<in> I \<Longrightarrow> f i square_integrable S" and S: "S \<in> sets lebesgue"
   shows "(\<lambda>x. \<Sum>i\<in>I. f i x) square_integrable S"
   using I by induction (simp_all add: S)
 
 lemma continuous_imp_square_integrable [simp]:
    "continuous_on {a..b} f \<Longrightarrow> f square_integrable {a..b}"
   using continuous_imp_integrable [of a b "(\<lambda>x. (f x)\<^sup>2)"]
   by (simp add: square_integrable_def continuous_on_power continuous_imp_measurable_on_sets_lebesgue)
 
 lemma square_integrable_imp_absolutely_integrable:
   assumes f: "f square_integrable S" and S: "S \<in> lmeasurable"
   shows "f absolutely_integrable_on S"
 proof -
   have "f \<in> lspace (lebesgue_on S) 2"
     using f S square_integrable_iff_lspace by blast
   then have "f \<in> lspace (lebesgue_on S) 1"
     by (rule lspace_mono) (use S in auto)
   then show ?thesis
     using S by (simp flip: lspace_1)
 qed
 
 lemma square_integrable_imp_integrable:
   assumes f: "f square_integrable S" and S: "S \<in> lmeasurable"
   shows "integrable (lebesgue_on S) f"
   by (meson S absolutely_integrable_measurable_real f fmeasurableD integrable_abs_iff square_integrable_imp_absolutely_integrable)
 
 subsection\<open> The norm and inner product in L2\<close>
 
 definition l2product :: "'a::euclidean_space set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> real"
   where "l2product S f g \<equiv> (\<integral>x. f x * g x \<partial>(lebesgue_on S))"
 
 definition l2norm :: "['a::euclidean_space set, 'a \<Rightarrow> real] \<Rightarrow> real"
   where "l2norm S f \<equiv> sqrt (l2product S f f)"
 
 definition lnorm :: "['a measure, real, 'a \<Rightarrow> real] \<Rightarrow> real"
   where "lnorm M p f \<equiv> (\<integral>x. \<bar>f x\<bar> powr p \<partial>M) powr (1/p)"
 
 corollary Holder_inequality_lnorm:
   assumes "p > (0::real)" "q > 0" "1/p+1/q = 1"
       and "f \<in> borel_measurable M" "g \<in> borel_measurable M"
           "integrable M (\<lambda>x. \<bar>f x\<bar> powr p)"
           "integrable M (\<lambda>x. \<bar>g x\<bar> powr q)"
   shows "(\<integral>x. \<bar>f x * g x\<bar> \<partial>M) \<le> lnorm M p f * lnorm M q g"
         "\<bar>\<integral>x. f x * g x \<partial>M \<bar> \<le> lnorm M p f * lnorm M q g"
   by (simp_all add: Holder_inequality assms lnorm_def)
 
 lemma l2norm_lnorm: "l2norm S f = lnorm (lebesgue_on S) 2 f"
 proof -
   have "(LINT x|lebesgue_on S. (f x)\<^sup>2) \<ge> 0"
     by simp
   then show ?thesis
     by (auto simp: lnorm_def l2norm_def l2product_def power2_eq_square powr_half_sqrt)
 qed
 
 lemma lnorm_nonneg: "lnorm M p f \<ge> 0"
   by (simp add: lnorm_def)
 
 lemma lnorm_minus_commute: "lnorm M p (g - f) = lnorm M p (f - g)"
   by (simp add: lnorm_def abs_minus_commute)
 
 
 text\<open> Extending a continuous function in a periodic way\<close>
 
-lemma isCont_id [continuous_intros,iff]:
-  fixes x :: "'a::t2_space" shows "isCont id x"
-  unfolding id_def by (intro continuous_intros)
-
-lemma continuous_floor:
-  fixes x::real
-  shows "x \<notin> \<int> \<Longrightarrow> continuous (at x) (real_of_int \<circ> floor)"
-  using floor_has_real_derivative [where f=id]
-  by (auto simp: o_def has_field_derivative_def intro: has_derivative_continuous)
-
-lemma continuous_frac:
-  fixes x::real
-  assumes "x \<notin> \<int>"
-  shows "continuous (at x) frac"
-proof -
-  have "isCont (\<lambda>x. real_of_int \<lfloor>x\<rfloor>) x"
-    using continuous_floor [OF assms] by (simp add: o_def)
-  then have *: "continuous (at x) (\<lambda>x. x - real_of_int \<lfloor>x\<rfloor>)"
-    by (intro continuous_intros)
-  moreover have "\<forall>\<^sub>F x in nhds x. frac x = x - real_of_int \<lfloor>x\<rfloor>"
-    by (simp add: frac_def)
-  ultimately show ?thesis
-    by (simp add: LIM_imp_LIM frac_def isCont_def)
-qed
-
 proposition continuous_on_compose_frac:
   fixes f:: "real \<Rightarrow> real"
   assumes contf: "continuous_on {0..1} f" and f10: "f 1 = f 0"
   shows "continuous_on UNIV (f \<circ> frac)"
 proof -
   have *: "isCont (f \<circ> frac) x"
     if caf: "\<And>x. \<lbrakk>0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> continuous (at x within {0..1}) f" for x
   proof (cases "x \<in> \<int>")
     case True
     then have [simp]: "frac x = 0"
       by simp
     show ?thesis
     proof (clarsimp simp add: continuous_at_eps_delta dist_real_def)
       have f0: "continuous (at 0 within {0..1}) f" and f1: "continuous (at 1 within {0..1}) f"
         by (auto intro: caf)
       show "\<exists>d>0. \<forall>x'. \<bar>x'-x\<bar> < d \<longrightarrow> \<bar>f(frac x') - f 0\<bar> < e"
         if "0 < e" for e
       proof -
         obtain d0 where "d0 > 0" and d0: "\<And>x'. \<lbrakk>x'\<in>{0..1}; \<bar>x'\<bar> < d0\<rbrakk> \<Longrightarrow> \<bar>f x' - f 0\<bar> < e"
           using \<open>e > 0\<close> caf [of 0] dist_not_less_zero
           by (auto simp: continuous_within_eps_delta dist_real_def)
         obtain d1 where "d1 > 0" and d1: "\<And>x'. \<lbrakk>x'\<in>{0..1}; \<bar>x' - 1\<bar> < d1\<rbrakk> \<Longrightarrow> \<bar>f x' - f 0\<bar> < e"
           using \<open>e > 0\<close> caf [of 1] dist_not_less_zero f10
           by (auto simp: continuous_within_eps_delta dist_real_def)
         show ?thesis
         proof (intro exI conjI allI impI)
           show "0 < min 1 (min d0 d1)"
             by (auto simp: \<open>d0 > 0\<close> \<open>d1 > 0\<close>)
           show "\<bar>f(frac x') - f 0\<bar> < e"
             if "\<bar>x'-x\<bar> < min 1 (min d0 d1)" for x'
           proof (cases "x \<le> x'")
             case True
             with \<open>x \<in> \<int>\<close> that have "frac x' = x' - x"
               by (simp add: frac_unique_iff)
             then show ?thesis
               using True d0 that by auto
           next
             case False
             then have [simp]: "frac x' = 1 - (x - x')"
               using that \<open>x \<in> \<int>\<close> by (simp add: not_le frac_unique_iff)
             show ?thesis
               using False d1 that by auto
           qed
         qed
       qed
     qed
   next
     case False
     show ?thesis
     proof (rule continuous_at_compose)
       show "isCont frac x"
         by (simp add: False continuous_frac)
       have "frac x \<in> {0<..<1}"
         by (simp add: False frac_lt_1)
       then show "isCont f(frac x)"
         by (metis at_within_Icc_at greaterThanLessThan_iff le_cases not_le that)
     qed
   qed
   then show ?thesis
     using contf by (simp add: o_def continuous_on_eq_continuous_within)
 qed
 
 
 proposition Tietze_periodic_interval:
   fixes f:: "real \<Rightarrow> real"
   assumes contf: "continuous_on {a..b} f" and fab: "f a = f b"
   obtains g where "continuous_on UNIV g" "\<And>x. x \<in> {a..b} \<Longrightarrow> g x = f x"
                   "\<And>x. g(x + (b-a)) = g x"
 proof (cases "a < b")
   case True
   let ?g = "f \<circ> (\<lambda>y. a + (b-a) * y) \<circ> frac \<circ>
                 (\<lambda>x. (x - a) / (b-a))"
   show ?thesis
   proof
     have "a + (b - a) * y \<le> b" if "a < b" "0 \<le> y" "y \<le> 1" for y
       using that affine_ineq by (force simp: field_simps)
     then have *: "continuous_on (range (\<lambda>x. (x - a) / (b - a))) (f \<circ> (\<lambda>y. a + (b - a) * y) \<circ> frac)"
       apply (intro continuous_on_subset [OF continuous_on_compose_frac] continuous_on_subset [OF contf]
           continuous_intros)
       using \<open>a < b\<close>
       by (auto simp: fab)
     show "continuous_on UNIV ?g"
       by (intro * continuous_on_compose continuous_intros) (use True in auto)
     show "?g x = f x" if "x \<in> {a..b}" for x :: real
     proof (cases "x=b")
       case True
       then show ?thesis
         by (auto simp: frac_def intro: fab)
     next
       case False
       with \<open>a < b\<close> that have "frac ((x - a) / (b - a)) = (x - a) / (b - a)"
         by (subst frac_eq) (auto simp: divide_simps)
       with \<open>a < b\<close> show ?thesis
         by auto
     qed
     have "a + (b-a) * frac ((x + b - 2 * a) / (b-a)) = a + (b-a) * frac ((x - a) / (b-a))" for x
       using True frac_1_eq [of "(x - a) / (b-a)"] by (auto simp: divide_simps)
     then show "?g (x + (b-a)) = (?g x::real)" for x
       by force
   qed
 next
   case False
   show ?thesis
   proof
     show "f a = f x" if "x \<in> {a..b}" for x
       using that False order_trans by fastforce
   qed auto
 qed
 
 
 subsection\<open>Lspace stuff\<close>
 
 lemma eNorm_triangle_eps:
   assumes "eNorm N (x' - x) < a" "defect N = 1"
   obtains e where "e > 0" "\<And>y. eNorm N (y - x') < e \<Longrightarrow> eNorm N (y - x) < a"
 proof -
   let ?d = "a - Norm N (x' - x)"
   have nt: "eNorm N (x' - x) < \<top>"
     using assms top.not_eq_extremum by fastforce
   with assms have d: "?d > 0"
     by (simp add: Norm_def diff_gr0_ennreal)
   have [simp]: "ennreal (1 - Norm N (x' - x)) = 1 - eNorm N (x' - x)"
     using that nt  unfolding Norm_def  by (metis enn2real_nonneg ennreal_1 ennreal_enn2real ennreal_minus)
   show ?thesis
   proof
     show "(0::ennreal) < ?d"
       using d ennreal_less_zero_iff by blast
     show "eNorm N (y - x) < a"
       if "eNorm N (y - x') < ?d" for y
       using that assms eNorm_triangular_ineq [of N "y - x'" "x' - x"] le_less_trans less_diff_eq_ennreal
       by (simp add: Norm_def nt)
   qed
 qed
 
 lemma topspace_topology\<^sub>N [simp]:
   assumes "defect N = 1" shows "topspace (topology\<^sub>N N) = UNIV"
 proof -
   have "x \<in> topspace (topology\<^sub>N N)" for x
   proof -
     have "\<exists>e>0. \<forall>y. eNorm N (y - x') < e \<longrightarrow> eNorm N (y - x) < 1"
       if "eNorm N (x' - x) < 1" for x'
       using eNorm_triangle_eps
       by (metis assms that)
     then show ?thesis
       unfolding topspace_def
       by (rule_tac X="{y. eNorm N (y - x) < 1}" in UnionI) (auto intro: openin_topology\<^sub>N_I)
   qed
   then show ?thesis
     by auto
 qed
 
 lemma tendsto_ine\<^sub>N_iff_limitin:
   assumes "defect N = 1"
   shows "tendsto_ine\<^sub>N N u x = limitin (topology\<^sub>N N) u x sequentially"
 proof -
   have "\<forall>\<^sub>F x in sequentially. u x \<in> U"
     if 0: "(\<lambda>n. eNorm N (u n - x)) \<longlonglongrightarrow> 0" and U: "openin (topology\<^sub>N N) U" "x \<in> U" for U
   proof -
     obtain e where "e > 0" and e: "\<And>y. eNorm N (y-x) < e \<Longrightarrow> y \<in> U"
       using openin_topology\<^sub>N_D U by metis
     then show ?thesis
       using eventually_mono order_tendstoD(2)[OF 0] by force
   qed
   moreover have "(\<lambda>n. eNorm N (u n - x)) \<longlonglongrightarrow> 0"
     if x: "x \<in> topspace (topology\<^sub>N N)"
       and *: "\<And>U. \<lbrakk>openin (topology\<^sub>N N) U; x \<in> U\<rbrakk> \<Longrightarrow> (\<forall>\<^sub>F x in sequentially. u x \<in> U)"
   proof (rule order_tendstoI)
     show "\<forall>\<^sub>F n in sequentially. eNorm N (u n - x) < a" if "a > 0" for a
       apply (rule * [OF openin_topology\<^sub>N_I, of "{v. eNorm N (v - x) < a}", simplified])
       using assms eNorm_triangle_eps that apply blast+
       done
   qed simp
   ultimately show ?thesis
     by (auto simp: tendsto_ine\<^sub>N_def limitin_def assms)
 qed
 
 corollary tendsto_ine\<^sub>N_iff_limitin_ge1:
   fixes p :: ennreal
   assumes "p \<ge> 1"
   shows "tendsto_ine\<^sub>N (\<LL> p M) u x = limitin (topology\<^sub>N (\<LL> p M)) u x sequentially"
 proof (rule tendsto_ine\<^sub>N_iff_limitin)
   show "defect (\<LL> p M) = 1"
     by (metis (full_types) L_infinity(2) L_zero(2) Lp(2) Lp_cases assms ennreal_ge_1)
 qed
 
 corollary tendsto_in\<^sub>N_iff_limitin:
   assumes "defect N = 1" "x \<in> space\<^sub>N N" "\<And>n. u n \<in> space\<^sub>N N"
   shows "tendsto_in\<^sub>N N u x = limitin (topology\<^sub>N N) u x sequentially"
   using assms tendsto_ine\<^sub>N_iff_limitin tendsto_ine_in by blast
 
 corollary tendsto_in\<^sub>N_iff_limitin_ge1:
   fixes p :: ennreal
   assumes "p \<ge> 1" "x \<in> lspace M p" "\<And>n. u n \<in> lspace M p"
   shows "tendsto_in\<^sub>N (\<LL> p M) u x = limitin (topology\<^sub>N (\<LL> p M)) u x sequentially"
 proof (rule tendsto_in\<^sub>N_iff_limitin)
   show "defect (\<LL> p M) = 1"
     by (metis (full_types) L_infinity(2) L_zero(2) Lp(2) Lp_cases \<open>p \<ge> 1\<close> ennreal_ge_1)
 qed (auto simp: assms)
 
 
 lemma l2product_sym: "l2product S f g = l2product S g f"
   by (simp add: l2product_def mult.commute)
 
 lemma l2product_pos_le:
    "f square_integrable S \<Longrightarrow> 0 \<le> l2product S f f"
   by (simp add: square_integrable_def l2product_def flip: power2_eq_square)
 
 lemma l2norm_pow_2:
    "f square_integrable S \<Longrightarrow> (l2norm S f) ^ 2 = l2product S f f"
   by (simp add: l2norm_def l2product_pos_le)
 
 lemma l2norm_pos_le:
    "f square_integrable S \<Longrightarrow> 0 \<le> l2norm S f"
   by (simp add: l2norm_def l2product_pos_le)
 
 lemma l2norm_le: "(l2norm S f \<le> l2norm S g \<longleftrightarrow> l2product S f f \<le> l2product S g g)"
   by (simp add: l2norm_def)
 
 lemma l2norm_eq:
    "(l2norm S f = l2norm S g \<longleftrightarrow> l2product S f f = l2product S g g)"
   by (simp add: l2norm_def)
 
 lemma Schwartz_inequality_strong:
   assumes "f square_integrable S" "g square_integrable S"
   shows "l2product S (\<lambda>x. \<bar>f x\<bar>) (\<lambda>x. \<bar>g x\<bar>) \<le> l2norm S f * l2norm S g"
   using Holder_inequality_lnorm [of 2 2 f "lebesgue_on S" g] assms
   by (simp add: square_integrable_def l2product_def abs_mult flip: l2norm_lnorm)
 
 lemma Schwartz_inequality_abs:
   assumes "f square_integrable S" "g square_integrable S"
   shows "\<bar>l2product S f g\<bar> \<le> l2norm S f * l2norm S g"
 proof -
   have "\<bar>l2product S f g\<bar> \<le> l2product S (\<lambda>x. \<bar>f x\<bar>) (\<lambda>x. \<bar>g x\<bar>)"
     unfolding l2product_def
   proof (rule integral_abs_bound_integral)
     show "integrable (lebesgue_on S) (\<lambda>x. f x * g x)" "integrable (lebesgue_on S) (\<lambda>x. \<bar>f x\<bar> * \<bar>g x\<bar>)"
       by (simp_all add: assms square_integrable_imp_integrable_product)
   qed (simp add: abs_mult)
   also have "\<dots> \<le> l2norm S f * l2norm S g"
     by (simp add: Schwartz_inequality_strong assms)
   finally show ?thesis .
 qed
 
 lemma Schwartz_inequality:
   assumes "f square_integrable S" "g square_integrable S"
   shows "l2product S f g \<le> l2norm S f * l2norm S g"
   using Schwartz_inequality_abs assms by fastforce
 
 
 lemma lnorm_triangle:
   assumes f: "f \<in> lspace M p" and g: "g \<in> lspace M p" and "p \<ge> 1"
   shows "lnorm M p (\<lambda>x. f x + g x) \<le> lnorm M p f + lnorm M p g"
 proof -
   have "p > 0"
     using assms by linarith
   then have "integrable M (\<lambda>x. \<bar>f x\<bar> powr p)" "integrable M (\<lambda>x. \<bar>g x\<bar> powr p)"
     by (simp_all add: Lp_D(2) assms)
   moreover have "f \<in> borel_measurable M" "g \<in> borel_measurable M"
     using Lp_measurable f g by blast+
   ultimately show ?thesis
     unfolding lnorm_def using Minkowski_inequality(2) \<open>p \<ge> 1\<close> by blast
 qed
 
 lemma lnorm_triangle_fun:
   assumes f: "f \<in> lspace M p" and g: "g \<in> lspace M p" and "p \<ge> 1"
   shows "lnorm M p (f + g) \<le> lnorm M p f + lnorm M p g"
   using lnorm_triangle [OF assms] by (simp add: plus_fun_def)
 
 lemma l2norm_triangle:
   assumes "f square_integrable S" "g square_integrable S"
   shows "l2norm S (\<lambda>x. f x + g x) \<le> l2norm S f + l2norm S g"
 proof -
   have "f \<in> lspace (lebesgue_on S) 2" "g \<in> lspace (lebesgue_on S) 2"
     using assms by (simp_all add: square_integrable_imp_lspace)
   then show ?thesis
     using lnorm_triangle [of f 2 "lebesgue_on S"]
     by (simp add: l2norm_lnorm)
 qed
 
 
 lemma l2product_ladd:
    "\<lbrakk>f square_integrable S; g square_integrable S; h square_integrable S\<rbrakk>
     \<Longrightarrow> l2product S (\<lambda>x. f x + g x) h = l2product S f h + l2product S g h"
   by (simp add: l2product_def algebra_simps square_integrable_imp_integrable_product)
 
 lemma l2product_radd:
    "\<lbrakk>f square_integrable S; g square_integrable S; h square_integrable S\<rbrakk>
     \<Longrightarrow> l2product S f(\<lambda>x. g x + h x) = l2product S f g + l2product S f h"
   by (simp add: l2product_def algebra_simps square_integrable_imp_integrable_product)
 
 lemma l2product_ldiff:
    "\<lbrakk>f square_integrable S; g square_integrable S; h square_integrable S\<rbrakk>
     \<Longrightarrow> l2product S (\<lambda>x. f x - g x) h = l2product S f h - l2product S g h"
   by (simp add: l2product_def algebra_simps square_integrable_imp_integrable_product)
 
 lemma l2product_rdiff:
    "\<lbrakk>f square_integrable S; g square_integrable S; h square_integrable S\<rbrakk>
     \<Longrightarrow> l2product S f(\<lambda>x. g x - h x) = l2product S f g - l2product S f h"
   by (simp add: l2product_def algebra_simps square_integrable_imp_integrable_product)
 
 lemma l2product_lmult:
    "\<lbrakk>f square_integrable S; g square_integrable S\<rbrakk>
     \<Longrightarrow> l2product S (\<lambda>x. c * f x) g = c * l2product S f g"
   by (simp add: l2product_def algebra_simps)
 
 lemma l2product_rmult:
    "\<lbrakk>f square_integrable S; g square_integrable S\<rbrakk>
     \<Longrightarrow> l2product S f(\<lambda>x. c * g x) = c * l2product S f g"
   by (simp add: l2product_def algebra_simps)
 
 lemma l2product_lzero [simp]: "l2product S (\<lambda>x. 0) f = 0"
   by (simp add: l2product_def)
 
 lemma l2product_rzero [simp]: "l2product S f(\<lambda>x. 0) = 0"
   by (simp add: l2product_def)
 
 lemma l2product_lsum:
   assumes I: "finite I" "\<And>i. i \<in> I \<Longrightarrow> (f i) square_integrable S" and S: "g square_integrable S"
   shows "l2product S (\<lambda>x. \<Sum>i\<in>I. f i x) g = (\<Sum>i\<in>I. l2product S (f i) g)"
   using I
 proof induction
   case (insert i I)
   with S show ?case
     by (simp add: l2product_ladd square_integrable_imp_lebesgue)
 qed auto
 
 lemma l2product_rsum:
   assumes I: "finite I" "\<And>i. i \<in> I \<Longrightarrow> (f i) square_integrable S" and S: "g square_integrable S"
   shows "l2product S g (\<lambda>x. \<Sum>i\<in>I. f i x) = (\<Sum>i\<in>I. l2product S g (f i))"
   using l2product_lsum [OF assms] by (simp add: l2product_sym)
 
 lemma l2norm_lmult:
    "f square_integrable S \<Longrightarrow> l2norm S (\<lambda>x. c * f x) = \<bar>c\<bar> * l2norm S f"
   by (simp add: l2norm_def l2product_rmult l2product_sym real_sqrt_mult)
 
 lemma l2norm_rmult:
    "f square_integrable S \<Longrightarrow> l2norm S (\<lambda>x. f x * c) = l2norm S f * \<bar>c\<bar>"
   using l2norm_lmult by (simp add: mult.commute)
 
 lemma l2norm_neg:
    "f square_integrable S \<Longrightarrow> l2norm S (\<lambda>x. - f x) = l2norm S f"
   using l2norm_lmult [of f S "-1"] by simp
 
 lemma l2norm_diff:
   assumes "f square_integrable S" "g square_integrable S"
   shows "l2norm S (\<lambda>x. f x - g x) = l2norm S (\<lambda>x. g x - f x)"
 proof -
   have "(\<lambda>x. f x - g x) square_integrable S"
     using assms square_integrable_diff by blast
   then show ?thesis
     using l2norm_neg [of "\<lambda>x. f x - g x" S] by (simp add: algebra_simps)
 qed
 
 
 subsection\<open>Completeness (Riesz-Fischer)\<close>
 
 lemma eNorm_eq_lnorm: "\<lbrakk>f \<in> lspace M p; p > 0\<rbrakk> \<Longrightarrow> eNorm (\<LL> (ennreal p) M) f = ennreal (lnorm M p f)"
   by (simp add: Lp_D(4) lnorm_def)
 
 lemma Norm_eq_lnorm: "\<lbrakk>f \<in> lspace M p; p > 0\<rbrakk> \<Longrightarrow> Norm (\<LL> (ennreal p) M) f = lnorm M p f"
   by (simp add: Lp_D(3) lnorm_def)
 
 
 lemma eNorm_ge1_triangular_ineq:
   assumes "p \<ge> (1::real)"
   shows "eNorm (\<LL> p M) (x + y) \<le> eNorm (\<LL> p M) x + eNorm (\<LL> p M) y"
   using eNorm_triangular_ineq [of "(\<LL> p M)"] assms
   by (simp add: Lp(2))
 
 text\<open>A mere repackaging of the theorem @{thm Lp_complete}, but nearly as much work again.\<close>
 proposition l2_complete:
   assumes f: "\<And>i::nat. f i square_integrable S"
     and cauchy: "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. l2norm S (\<lambda>x. f m x - f n x) < e"
   obtains g where "g square_integrable S" "((\<lambda>n. l2norm S (\<lambda>x. f n x - g x)) \<longlonglongrightarrow> 0)"
 proof -
   have finite: "eNorm (\<LL> 2 (lebesgue_on S)) (f n - f m) < \<top>" for m n
     by (metis f infinity_ennreal_def spaceN_diff spaceN_iff square_integrable_imp_lspace)
   have *: "cauchy_ine\<^sub>N (\<LL> 2 (lebesgue_on S)) f"
   proof (clarsimp simp: cauchy_ine\<^sub>N_def)
     show "\<exists>M. \<forall>n\<ge>M. \<forall>m\<ge>M. eNorm (\<LL> 2 (lebesgue_on S)) (f n - f m) < e"
       if "e > 0" for e
     proof (cases e)
       case (real r)
       then have "r > 0"
         using that by auto
       with cauchy obtain N::nat where N: "\<And>m n. \<lbrakk>m \<ge> N; n \<ge> N\<rbrakk> \<Longrightarrow> l2norm S (\<lambda>x. f n x - f m x) < r"
         by blast
       show ?thesis
       proof (intro exI allI impI)
         show "eNorm (\<LL> 2 (lebesgue_on S)) (f n - f m) < e"
           if "N \<le> m" "N \<le> n" for m n
         proof -
           have fnm: "(f n - f m) \<in> borel_measurable (lebesgue_on S)"
             using f unfolding square_integrable_def by (blast intro: borel_measurable_diff')
           have "l2norm S (\<lambda>x. f n x - f m x) = lnorm (lebesgue_on S) 2 (\<lambda>x. f n x - f m x)"
             by (metis l2norm_lnorm)
           also have "\<dots> = Norm (\<LL> 2 (lebesgue_on S)) (f n - f m)"
             using Lp_Norm [OF _ fnm, of 2] by (simp add: lnorm_def)
           finally show ?thesis
             using N [OF that] real finite
             by (simp add: Norm_def)
         qed
       qed
     qed (simp add: finite)
   qed
   then obtain g where g: "tendsto_ine\<^sub>N (\<LL> 2 (lebesgue_on S)) f g"
     using Lp_complete complete\<^sub>N_def by blast
   show ?thesis
   proof
     have fng_to_0: "(\<lambda>n. eNorm (\<LL> 2 (lebesgue_on S)) (\<lambda>x. f n x - g x)) \<longlonglongrightarrow> 0"
       using g Lp_D(4) [of 2 _ "lebesgue_on S"]
       by (simp add: tendsto_ine\<^sub>N_def minus_fun_def)
     then obtain M where "\<And>n . n \<ge> M \<Longrightarrow> eNorm (\<LL> 2 (lebesgue_on S)) (\<lambda>x. f n x - g x) < \<top>"
       apply (simp add: lim_explicit)
       by (metis (full_types) open_lessThan diff_self eNorm_zero lessThan_iff local.finite)
     then have "eNorm (\<LL> 2 (lebesgue_on S)) (\<lambda>x. g x - f M x) < \<top>"
       using eNorm_uminus [of _ "\<lambda>x. g x - f _ x"] by (simp add: uminus_fun_def)
     moreover have "eNorm (\<LL> 2 (lebesgue_on S)) (\<lambda>x. f M x) < \<top>"
       using f square_integrable_imp_lspace by (simp add: spaceN_iff)
     ultimately have "eNorm (\<LL> 2 (lebesgue_on S)) g < \<top>"
       using eNorm_ge1_triangular_ineq [of 2 "lebesgue_on S" "g - f M" "f M", simplified] not_le top.not_eq_extremum
       by (fastforce simp add: minus_fun_def)
     then have g_space: "g \<in> space\<^sub>N (\<LL> 2 (lebesgue_on S))"
       by (simp add: spaceN_iff)
     show "g square_integrable S"
       unfolding square_integrable_def
     proof (intro conjI)
       show "g \<in> borel_measurable (lebesgue_on S)"
         using Lp_measurable g_space by blast
       show "S \<in> sets lebesgue"
         using f square_integrable_def by blast
       then show "integrable (lebesgue_on S) (\<lambda>x. (g x)\<^sup>2)"
         using g_space square_integrable_def square_integrable_iff_lspace by blast
     qed
     then have "f n - g \<in> lspace (lebesgue_on S) 2" for n
       using f spaceN_diff square_integrable_imp_lspace by blast
     with fng_to_0 have "(\<lambda>n. ennreal (lnorm (lebesgue_on S) 2 (\<lambda>x. f n x - g x))) \<longlonglongrightarrow> 0"
       by (simp add: minus_fun_def flip: eNorm_eq_lnorm)
     then have "(\<lambda>n. lnorm (lebesgue_on S) 2 (\<lambda>x. f n x - g x)) \<longlonglongrightarrow> 0"
       by (simp add: ennreal_tendsto_0_iff lnorm_def)
     then show "(\<lambda>n. l2norm S (\<lambda>x. f n x - g x)) \<longlonglongrightarrow> 0"
       using g by (simp add:  l2norm_lnorm lnorm_def)
   qed
 qed
 
 subsection\<open>Approximation of functions in Lp by bounded and continuous ones\<close>
 
 lemma lspace_bounded_measurable:
   fixes p::real
   assumes f: "f \<in> borel_measurable (lebesgue_on S)" and g: "g \<in> lspace (lebesgue_on S) p" and "p > 0"
     and le: " AE x in lebesgue_on S. norm (\<bar>f x\<bar> powr p) \<le> norm (\<bar>g x\<bar> powr p)"
   shows "f \<in> lspace (lebesgue_on S) p"
   using assms by (auto simp: lspace_ennreal_iff intro: Bochner_Integration.integrable_bound)
 
 lemma lspace_approximate_bounded:
   assumes f: "f \<in> lspace (lebesgue_on S) p" and S: "S \<in> lmeasurable" and "p > 0" "e > 0"
   obtains g where "g \<in> lspace (lebesgue_on S) p" "bounded (g ` S)"
     "lnorm (lebesgue_on S) p (f - g) < e"
 proof -
   have f_bm: "f \<in> borel_measurable (lebesgue_on S)"
     using Lp_measurable f by blast
   let ?f = "\<lambda>n::nat. \<lambda>x. max (- n) (min n (f x))"
   have "tendsto_in\<^sub>N (\<LL> p (lebesgue_on S)) ?f f"
   proof (rule Lp_domination_limit)
     show "\<And>n::nat. ?f n \<in> borel_measurable (lebesgue_on S)"
       by (intro f_bm borel_measurable_max borel_measurable_min borel_measurable_const)
     show "abs \<circ> f \<in> lspace (lebesgue_on S) p"
       using Lp_Banach_lattice [OF f] by (simp add: o_def)
     have *: "\<forall>\<^sub>F n in sequentially. dist (?f n x) (f x) < e"
       if x: "x \<in> space (lebesgue_on S)" and "e > 0" for x e
     proof
       show "dist (?f n x) (f x) < e"
         if "nat \<lceil>\<bar>f x\<bar>\<rceil> \<le> n" for n :: nat
         using that \<open>0 < e\<close> by (simp add: dist_real_def max_def min_def abs_if split: if_split_asm)
     qed
     then show "AE x in lebesgue_on S. (\<lambda>n::nat. max (- n) (min n (f x))) \<longlonglongrightarrow> f x"
       by (blast intro: tendstoI)
   qed (auto simp: f_bm)
   moreover
   have lspace: "?f n \<in> lspace (lebesgue_on S) p" for n::nat
     by (intro f lspace_const lspace_min lspace_max \<open>p > 0\<close> S)
   ultimately have "(\<lambda>n. lnorm (lebesgue_on S) p (?f n - f)) \<longlonglongrightarrow> 0"
     by (simp add: tendsto_in\<^sub>N_def Norm_eq_lnorm \<open>p > 0\<close> f)
   with \<open>e > 0\<close> obtain N where N: "\<bar>lnorm (lebesgue_on S) p (?f N - f)\<bar> < e"
     by (auto simp: LIMSEQ_iff)
   show ?thesis
   proof
     have "\<forall>x\<in>S. \<bar>max (- real N) (min (real N) (f x))\<bar> \<le> N"
       by auto
     then show "bounded (?f N ` S::real set)"
       by (force simp: bounded_iff)
     show "lnorm (lebesgue_on S) p (f - ?f N) < e"
       using N by (simp add: lnorm_minus_commute)
   qed (auto simp: lspace)
 qed
 
 lemma borel_measurable_imp_continuous_limit:
   fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes h: "h \<in> borel_measurable (lebesgue_on S)" and S: "S \<in> sets lebesgue"
   obtains g where "\<And>n. continuous_on UNIV (g n)" "AE x in lebesgue_on S. (\<lambda>n::nat. g n x) \<longlonglongrightarrow> h x"
 proof -
   have "h measurable_on S"
     using S h measurable_on_iff_borel_measurable by blast
   then obtain N g where N: "N \<in> null_sets lebesgue" and g: "\<And>n. continuous_on UNIV (g n)"
     and tends: "\<And>x. x \<notin> N \<Longrightarrow> (\<lambda>n. g n x) \<longlonglongrightarrow> (if x \<in> S then h x else 0)"
     by (auto simp: measurable_on_def negligible_iff_null_sets)
   moreover have "AE x in lebesgue_on S. (\<lambda>n::nat. g n x) \<longlonglongrightarrow> h x"
   proof (rule AE_I')
     show "N \<inter> S \<in> null_sets (lebesgue_on S)"
       by (simp add: S N null_set_Int2 null_sets_restrict_space)
     show "{x \<in> space (lebesgue_on S). \<not> (\<lambda>n. g n x) \<longlonglongrightarrow> h x} \<subseteq> N \<inter> S"
       using tends by force
   qed
   ultimately show thesis
     using that by blast
 qed
 
 
 proposition lspace_approximate_continuous:
   assumes f: "f \<in> lspace (lebesgue_on S) p" and S: "S \<in> lmeasurable" and "1 \<le> p" "e > 0"
   obtains g where "continuous_on UNIV g" "g \<in> lspace (lebesgue_on S) p" "lnorm (lebesgue_on S) p (f - g) < e"
 proof -
   have "p > 0"
     using assms by simp
   obtain h where h: "h \<in> lspace (lebesgue_on S) p" and "bounded (h ` S)"
     and lesse2: "lnorm (lebesgue_on S) p (f - h) < e/2"
     by (rule lspace_approximate_bounded [of f p S "e/2"]) (use assms in auto)
   then obtain B where "B > 0" and B: "\<And>x. x \<in> S \<Longrightarrow> \<bar>h x\<bar> \<le> B"
     by (auto simp: bounded_pos)
   have bmh: "h \<in> borel_measurable (lebesgue_on S)"
     using h lspace_ennreal_iff [of p] \<open>p \<ge> 1\<close> by auto
   obtain g where contg: "\<And>n. continuous_on UNIV (g n)"
     and gle: "\<And>n x. x \<in> S \<Longrightarrow> \<bar>g n x\<bar> \<le> B"
     and tends: "AE x in lebesgue_on S. (\<lambda>n::nat. g n x) \<longlonglongrightarrow> h x"
   proof -
     obtain \<gamma> where cont: "\<And>n. continuous_on UNIV (\<gamma> n)"
       and tends: "AE x in lebesgue_on S. (\<lambda>n::nat. \<gamma> n x) \<longlonglongrightarrow> h x"
       using borel_measurable_imp_continuous_limit S bmh by blast
     let ?g = "\<lambda>n::nat. \<lambda>x. max (- B) (min B (\<gamma> n x))"
     show thesis
     proof
       show "continuous_on UNIV (?g n)" for n
         by (intro continuous_intros cont)
       show "\<bar>?g n x\<bar> \<le> B" if "x \<in> S"  for n x
         using that \<open>B > 0\<close> by (auto simp: max_def min_def)
       have "(\<lambda>n. max (- B) (min B (\<gamma> n x))) \<longlonglongrightarrow> h x"
         if "(\<lambda>n. \<gamma> n x) \<longlonglongrightarrow> h x" "x \<in> S" for x
         using that \<open>B > 0\<close> B [OF \<open>x \<in> S\<close>]
         unfolding LIMSEQ_def by (fastforce simp: min_def max_def dist_real_def)
       then show "AE x in lebesgue_on S. (\<lambda>n. ?g n x) \<longlonglongrightarrow> h x"
         using tends by auto
     qed
   qed
   have lspace_B: "(\<lambda>x. B) \<in> lspace (lebesgue_on S) p"
     by (simp add: S \<open>0 < p\<close> lspace_const)
   have lspace_g: "g n \<in> lspace (lebesgue_on S) p" for n
   proof (rule lspace_bounded_measurable)
     show "g n \<in> borel_measurable (lebesgue_on S)"
       by (simp add: borel_measurable_continuous_onI contg measurable_completion measurable_restrict_space1)
     show "AE x in lebesgue_on S. norm (\<bar>g n x\<bar> powr p) \<le> norm (\<bar>B\<bar> powr p)"
       using \<open>B > 0\<close> gle S \<open>0 < p\<close> powr_mono2 by auto
   qed (use \<open>p > 0\<close> lspace_B in auto)
   have "tendsto_in\<^sub>N (\<LL> p (lebesgue_on S)) g h"
   proof (rule Lp_domination_limit [OF bmh _ lspace_B tends])
     show "\<And>n::nat. g n \<in> borel_measurable (lebesgue_on S)"
       using Lp_measurable lspace_g by blast
     show "\<And>n. AE x in lebesgue_on S. \<bar>g n x\<bar> \<le> B"
       using S gle by auto
   qed
   then have 0: "(\<lambda>n. Norm (\<LL> p (lebesgue_on S)) (g n - h)) \<longlonglongrightarrow> 0"
     by (simp add: tendsto_in\<^sub>N_def)
   have "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. lnorm (lebesgue_on S) p (g n - h) < e"
     using LIMSEQ_D [OF 0] \<open>e > 0\<close>
     by (force simp: Norm_eq_lnorm \<open>0 < p\<close> h lspace_g)
   then obtain N where N: "lnorm (lebesgue_on S) p (g N - h) < e/2"
     unfolding minus_fun_def by (meson \<open>e>0\<close> half_gt_zero order_refl)
   show ?thesis
   proof
     show "continuous_on UNIV (g N)"
       by (simp add: contg)
     show "g N \<in> lspace (lebesgue_on S) (ennreal p)"
       by (simp add: lspace_g)
     have "lnorm (lebesgue_on S) p (f - h + - (g N - h)) \<le> lnorm (lebesgue_on S) p (f - h) + lnorm (lebesgue_on S) p (- (g N - h))"
       by (rule lnorm_triangle_fun) (auto simp: lspace_g h assms)
     also have "\<dots>  < e/2 + e/2"
       using lesse2 N by (simp add: lnorm_minus_commute)
     finally show "lnorm (lebesgue_on S) p (f - g N) < e"
       by simp
   qed
 qed
 
 proposition square_integrable_approximate_continuous:
   assumes f: "f square_integrable S" and S: "S \<in> lmeasurable" and "e > 0"
   obtains g where "continuous_on UNIV g" "g square_integrable S" "l2norm S (\<lambda>x. f x - g x) < e"
 proof -
   have f2: "f \<in> lspace (lebesgue_on S) 2"
     by (simp add: f square_integrable_imp_lspace)
   then obtain g where contg: "continuous_on UNIV g"
              and g2: "g \<in> lspace (lebesgue_on S) 2"
              and less_e: "lnorm (lebesgue_on S) 2 (\<lambda>x. f x - g x) < e"
     using lspace_approximate_continuous [of f 2 S e] S \<open>0 < e\<close> by (auto simp: minus_fun_def)
   show thesis
   proof
     show "g square_integrable S"
       using g2 by (simp add: S fmeasurableD square_integrable_iff_lspace)
     show "l2norm S (\<lambda>x. f x - g x) < e"
       using less_e by (simp add: l2norm_lnorm)
   qed (simp add: contg)
 qed
 
 lemma absolutely_integrable_approximate_continuous:
   fixes f :: "real \<Rightarrow> real"
   assumes f: "f absolutely_integrable_on S" and S: "S \<in> lmeasurable" and "0 < e"
   obtains g where "continuous_on UNIV g" "g absolutely_integrable_on S" "integral\<^sup>L (lebesgue_on S) (\<lambda>x. \<bar>f x - g x\<bar>) < e"
 proof -
   obtain g where "continuous_on UNIV g" "g \<in> lspace (lebesgue_on S) 1"
               and lnorm: "lnorm (lebesgue_on S) 1 (f - g) < e"
   proof (rule lspace_approximate_continuous)
     show "f \<in> lspace (lebesgue_on S) (ennreal 1)"
       by (simp add: S f fmeasurableD lspace_1)
   qed (auto simp: assms)
   show thesis
   proof
     show "continuous_on UNIV g"
       by fact
     show "g absolutely_integrable_on S"
       using S \<open>g \<in> lspace (lebesgue_on S) 1\<close> lspace_1 by blast
     have *: "(\<lambda>x. f x - g x) absolutely_integrable_on S"
       by (simp add: \<open>g absolutely_integrable_on S\<close> f)
     moreover have "integrable (lebesgue_on S) (\<lambda>x. \<bar>f x - g x\<bar>)"
       by (simp add: L1_D(2) S * fmeasurableD lspace_1)
     ultimately show "integral\<^sup>L (lebesgue_on S)  (\<lambda>x. \<bar>f x - g x\<bar>) < e"
       using lnorm S unfolding lnorm_def absolutely_integrable_on_def
       by simp
   qed
 qed
 
 end