diff --git a/src/HOL/Analysis/Change_Of_Vars.thy b/src/HOL/Analysis/Change_Of_Vars.thy
--- a/src/HOL/Analysis/Change_Of_Vars.thy
+++ b/src/HOL/Analysis/Change_Of_Vars.thy
@@ -1,3478 +1,3478 @@
 (*  Title:      HOL/Analysis/Change_Of_Vars.thy
     Authors:    LC Paulson, based on material from HOL Light
 *)
 
 section\<open>Change of Variables Theorems\<close>
 
 theory Change_Of_Vars
   imports Vitali_Covering_Theorem Determinants
 
 begin
 
 subsection \<open>Measurable Shear and Stretch\<close>
 
 proposition
   fixes a :: "real^'n"
   assumes "m \<noteq> n" and ab_ne: "cbox a b \<noteq> {}" and an: "0 \<le> a$n"
   shows measurable_shear_interval: "(\<lambda>x. \<chi> i. if i = m then x$m + x$n else x$i) ` (cbox a b) \<in> lmeasurable"
        (is  "?f ` _ \<in> _")
    and measure_shear_interval: "measure lebesgue ((\<lambda>x. \<chi> i. if i = m then x$m + x$n else x$i) ` cbox a b)
                = measure lebesgue (cbox a b)" (is "?Q")
 proof -
   have lin: "linear ?f"
     by (rule linearI) (auto simp: plus_vec_def scaleR_vec_def algebra_simps)
   show fab: "?f ` cbox a b \<in> lmeasurable"
     by (simp add: lin measurable_linear_image_interval)
   let ?c = "\<chi> i. if i = m then b$m + b$n else b$i"
   let ?mn = "axis m 1 - axis n (1::real)"
   have eq1: "measure lebesgue (cbox a ?c)
             = measure lebesgue (?f ` cbox a b)
             + measure lebesgue (cbox a ?c \<inter> {x. ?mn \<bullet> x \<le> a$m})
             + measure lebesgue (cbox a ?c \<inter> {x. ?mn \<bullet> x \<ge> b$m})"
   proof (rule measure_Un3_negligible)
     show "cbox a ?c \<inter> {x. ?mn \<bullet> x \<le> a$m} \<in> lmeasurable" "cbox a ?c \<inter> {x. ?mn \<bullet> x \<ge> b$m} \<in> lmeasurable"
       by (auto simp: convex_Int convex_halfspace_le convex_halfspace_ge bounded_Int measurable_convex)
     have "negligible {x. ?mn \<bullet> x = a$m}"
       by (metis \<open>m \<noteq> n\<close> axis_index_axis eq_iff_diff_eq_0 negligible_hyperplane)
     moreover have "?f ` cbox a b \<inter> (cbox a ?c \<inter> {x. ?mn \<bullet> x \<le> a $ m}) \<subseteq> {x. ?mn \<bullet> x = a$m}"
       using \<open>m \<noteq> n\<close> antisym_conv by (fastforce simp: algebra_simps mem_box_cart inner_axis')
     ultimately show "negligible ((?f ` cbox a b) \<inter> (cbox a ?c \<inter> {x. ?mn \<bullet> x \<le> a $ m}))"
       by (rule negligible_subset)
     have "negligible {x. ?mn \<bullet> x = b$m}"
       by (metis \<open>m \<noteq> n\<close> axis_index_axis eq_iff_diff_eq_0 negligible_hyperplane)
     moreover have "(?f ` cbox a b) \<inter> (cbox a ?c \<inter> {x. ?mn \<bullet> x \<ge> b$m}) \<subseteq> {x. ?mn \<bullet> x = b$m}"
       using \<open>m \<noteq> n\<close> antisym_conv by (fastforce simp: algebra_simps mem_box_cart inner_axis')
     ultimately show "negligible (?f ` cbox a b \<inter> (cbox a ?c \<inter> {x. ?mn \<bullet> x \<ge> b$m}))"
       by (rule negligible_subset)
     have "negligible {x. ?mn \<bullet> x = b$m}"
       by (metis \<open>m \<noteq> n\<close> axis_index_axis eq_iff_diff_eq_0 negligible_hyperplane)
     moreover have "(cbox a ?c \<inter> {x. ?mn \<bullet> x \<le> a $ m} \<inter> (cbox a ?c \<inter> {x. ?mn \<bullet> x \<ge> b$m})) \<subseteq> {x. ?mn \<bullet> x = b$m}"
       using \<open>m \<noteq> n\<close> ab_ne
       apply (auto simp: algebra_simps mem_box_cart inner_axis')
       apply (drule_tac x=m in spec)+
       apply simp
       done
     ultimately show "negligible (cbox a ?c \<inter> {x. ?mn \<bullet> x \<le> a $ m} \<inter> (cbox a ?c \<inter> {x. ?mn \<bullet> x \<ge> b$m}))"
       by (rule negligible_subset)
     show "?f ` cbox a b \<union> cbox a ?c \<inter> {x. ?mn \<bullet> x \<le> a $ m} \<union> cbox a ?c \<inter> {x. ?mn \<bullet> x \<ge> b$m} = cbox a ?c" (is "?lhs = _")
     proof
       show "?lhs \<subseteq> cbox a ?c"
         by (auto simp: mem_box_cart add_mono) (meson add_increasing2 an order_trans)
       show "cbox a ?c \<subseteq> ?lhs"
         apply (auto simp: algebra_simps image_iff inner_axis' lambda_add_Galois [OF \<open>m \<noteq> n\<close>])
         apply (auto simp: mem_box_cart split: if_split_asm)
         done
     qed
   qed (fact fab)
   let ?d = "\<chi> i. if i = m then a $ m - b $ m else 0"
   have eq2: "measure lebesgue (cbox a ?c \<inter> {x. ?mn \<bullet> x \<le> a $ m}) + measure lebesgue (cbox a ?c \<inter> {x. ?mn \<bullet> x \<ge> b$m})
            = measure lebesgue (cbox a (\<chi> i. if i = m then a $ m + b $ n else b $ i))"
   proof (rule measure_translate_add[of "cbox a ?c \<inter> {x. ?mn \<bullet> x \<le> a$m}" "cbox a ?c \<inter> {x. ?mn \<bullet> x \<ge> b$m}"
      "(\<chi> i. if i = m then a$m - b$m else 0)" "cbox a (\<chi> i. if i = m then a$m + b$n else b$i)"])
     show "(cbox a ?c \<inter> {x. ?mn \<bullet> x \<le> a$m}) \<in> lmeasurable"
       "cbox a ?c \<inter> {x. ?mn \<bullet> x \<ge> b$m} \<in> lmeasurable"
       by (auto simp: convex_Int convex_halfspace_le convex_halfspace_ge bounded_Int measurable_convex)
     have "\<And>x. \<lbrakk>x $ n + a $ m \<le> x $ m\<rbrakk>
          \<Longrightarrow> x \<in> (+) (\<chi> i. if i = m then a $ m - b $ m else 0) ` {x. x $ n + b $ m \<le> x $ m}"
       using \<open>m \<noteq> n\<close>
       by (rule_tac x="x - (\<chi> i. if i = m then a$m - b$m else 0)" in image_eqI)
          (simp_all add: mem_box_cart)
     then have imeq: "(+) ?d ` {x. b $ m \<le> ?mn \<bullet> x} = {x. a $ m \<le> ?mn \<bullet> x}"
       using \<open>m \<noteq> n\<close> by (auto simp: mem_box_cart inner_axis' algebra_simps)
     have "\<And>x. \<lbrakk>0 \<le> a $ n; x $ n + a $ m \<le> x $ m;
                 \<forall>i. i \<noteq> m \<longrightarrow> a $ i \<le> x $ i \<and> x $ i \<le> b $ i\<rbrakk>
          \<Longrightarrow> a $ m \<le> x $ m"
       using \<open>m \<noteq> n\<close>  by force
     then have "(+) ?d ` (cbox a ?c \<inter> {x. b $ m \<le> ?mn \<bullet> x})
             = cbox a (\<chi> i. if i = m then a $ m + b $ n else b $ i) \<inter> {x. a $ m \<le> ?mn \<bullet> x}"
       using an ab_ne
       apply (simp add: cbox_translation [symmetric] translation_Int interval_ne_empty_cart imeq)
       apply (auto simp: mem_box_cart inner_axis' algebra_simps if_distrib all_if_distrib)
       by (metis (full_types) add_mono mult_2_right)
     then show "cbox a ?c \<inter> {x. ?mn \<bullet> x \<le> a $ m} \<union>
           (+) ?d ` (cbox a ?c \<inter> {x. b $ m \<le> ?mn \<bullet> x}) =
           cbox a (\<chi> i. if i = m then a $ m + b $ n else b $ i)"  (is "?lhs = ?rhs")
       using an \<open>m \<noteq> n\<close>
       apply (auto simp: mem_box_cart inner_axis' algebra_simps if_distrib all_if_distrib, force)
         apply (drule_tac x=n in spec)+
       by (meson ab_ne add_mono_thms_linordered_semiring(3) dual_order.trans interval_ne_empty_cart(1))
     have "negligible{x. ?mn \<bullet> x = a$m}"
       by (metis \<open>m \<noteq> n\<close> axis_index_axis eq_iff_diff_eq_0 negligible_hyperplane)
     moreover have "(cbox a ?c \<inter> {x. ?mn \<bullet> x \<le> a $ m} \<inter>
                                  (+) ?d ` (cbox a ?c \<inter> {x. b $ m \<le> ?mn \<bullet> x})) \<subseteq> {x. ?mn \<bullet> x = a$m}"
       using \<open>m \<noteq> n\<close> antisym_conv by (fastforce simp: algebra_simps mem_box_cart inner_axis')
     ultimately show "negligible (cbox a ?c \<inter> {x. ?mn \<bullet> x \<le> a $ m} \<inter>
                                  (+) ?d ` (cbox a ?c \<inter> {x. b $ m \<le> ?mn \<bullet> x}))"
       by (rule negligible_subset)
   qed
   have ac_ne: "cbox a ?c \<noteq> {}"
     using ab_ne an
     by (clarsimp simp: interval_eq_empty_cart) (meson add_less_same_cancel1 le_less_linear less_le_trans)
   have ax_ne: "cbox a (\<chi> i. if i = m then a $ m + b $ n else b $ i) \<noteq> {}"
     using ab_ne an
     by (clarsimp simp: interval_eq_empty_cart) (meson add_less_same_cancel1 le_less_linear less_le_trans)
   have eq3: "measure lebesgue (cbox a ?c) = measure lebesgue (cbox a (\<chi> i. if i = m then a$m + b$n else b$i)) + measure lebesgue (cbox a b)"
     by (simp add: content_cbox_if_cart ab_ne ac_ne ax_ne algebra_simps prod.delta_remove
              if_distrib [of "\<lambda>u. u - z" for z] prod.remove)
   show ?Q
     using eq1 eq2 eq3
     by (simp add: algebra_simps)
 qed
 
 
 proposition
   fixes S :: "(real^'n) set"
   assumes "S \<in> lmeasurable"
   shows measurable_stretch: "((\<lambda>x. \<chi> k. m k * x$k) ` S) \<in> lmeasurable" (is  "?f ` S \<in> _")
     and measure_stretch: "measure lebesgue ((\<lambda>x. \<chi> k. m k * x$k) ` S) = \<bar>prod m UNIV\<bar> * measure lebesgue S"
     (is "?MEQ")
 proof -
   have "(?f ` S) \<in> lmeasurable \<and> ?MEQ"
   proof (cases "\<forall>k. m k \<noteq> 0")
     case True
     have m0: "0 < \<bar>prod m UNIV\<bar>"
       using True by simp
     have "(indicat_real (?f ` S) has_integral \<bar>prod m UNIV\<bar> * measure lebesgue S) UNIV"
     proof (clarsimp simp add: has_integral_alt [where i=UNIV])
       fix e :: "real"
       assume "e > 0"
       have "(indicat_real S has_integral (measure lebesgue S)) UNIV"
         using assms lmeasurable_iff_has_integral by blast
       then obtain B where "B>0"
         and B: "\<And>a b. ball 0 B \<subseteq> cbox a b \<Longrightarrow>
                         \<exists>z. (indicat_real S has_integral z) (cbox a b) \<and>
                             \<bar>z - measure lebesgue S\<bar> < e / \<bar>prod m UNIV\<bar>"
         by (simp add: has_integral_alt [where i=UNIV]) (metis (full_types) divide_pos_pos m0  m0 \<open>e > 0\<close>)
       show "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
                   (\<exists>z. (indicat_real (?f ` S) has_integral z) (cbox a b) \<and>
                        \<bar>z - \<bar>prod m UNIV\<bar> * measure lebesgue S\<bar> < e)"
       proof (intro exI conjI allI)
         let ?C = "Max (range (\<lambda>k. \<bar>m k\<bar>)) * B"
         show "?C > 0"
           using True \<open>B > 0\<close> by (simp add: Max_gr_iff)
         show "ball 0 ?C \<subseteq> cbox u v \<longrightarrow>
                   (\<exists>z. (indicat_real (?f ` S) has_integral z) (cbox u v) \<and>
                        \<bar>z - \<bar>prod m UNIV\<bar> * measure lebesgue S\<bar> < e)" for u v
         proof
           assume uv: "ball 0 ?C \<subseteq> cbox u v"
           with \<open>?C > 0\<close> have cbox_ne: "cbox u v \<noteq> {}"
             using centre_in_ball by blast
           let ?\<alpha> = "\<lambda>k. u$k / m k"
           let ?\<beta> = "\<lambda>k. v$k / m k"
           have invm0: "\<And>k. inverse (m k) \<noteq> 0"
             using True by auto
           have "ball 0 B \<subseteq> (\<lambda>x. \<chi> k. x $ k / m k) ` ball 0 ?C"
           proof clarsimp
             fix x :: "real^'n"
             assume x: "norm x < B"
             have [simp]: "\<bar>Max (range (\<lambda>k. \<bar>m k\<bar>))\<bar> = Max (range (\<lambda>k. \<bar>m k\<bar>))"
               by (meson Max_ge abs_ge_zero abs_of_nonneg finite finite_imageI order_trans rangeI)
             have "norm (\<chi> k. m k * x $ k) \<le> norm (Max (range (\<lambda>k. \<bar>m k\<bar>)) *\<^sub>R x)"
               by (rule norm_le_componentwise_cart) (auto simp: abs_mult intro: mult_right_mono)
             also have "\<dots> < ?C"
               using x by simp (metis \<open>B > 0\<close> \<open>?C > 0\<close> mult.commute real_mult_less_iff1 zero_less_mult_pos)
             finally have "norm (\<chi> k. m k * x $ k) < ?C" .
             then show "x \<in> (\<lambda>x. \<chi> k. x $ k / m k) ` ball 0 ?C"
               using stretch_Galois [of "inverse \<circ> m"] True by (auto simp: image_iff field_simps)
           qed
           then have Bsub: "ball 0 B \<subseteq> cbox (\<chi> k. min (?\<alpha> k) (?\<beta> k)) (\<chi> k. max (?\<alpha> k) (?\<beta> k))"
             using cbox_ne uv image_stretch_interval_cart [of "inverse \<circ> m" u v, symmetric]
             by (force simp: field_simps)
           obtain z where zint: "(indicat_real S has_integral z) (cbox (\<chi> k. min (?\<alpha> k) (?\<beta> k)) (\<chi> k. max (?\<alpha> k) (?\<beta> k)))"
                    and zless: "\<bar>z - measure lebesgue S\<bar> < e / \<bar>prod m UNIV\<bar>"
             using B [OF Bsub] by blast
           have ind: "indicat_real (?f ` S) = (\<lambda>x. indicator S (\<chi> k. x$k / m k))"
             using True stretch_Galois [of m] by (force simp: indicator_def)
           show "\<exists>z. (indicat_real (?f ` S) has_integral z) (cbox u v) \<and>
                        \<bar>z - \<bar>prod m UNIV\<bar> * measure lebesgue S\<bar> < e"
           proof (simp add: ind, intro conjI exI)
             have "((\<lambda>x. indicat_real S (\<chi> k. x $ k/ m k)) has_integral z *\<^sub>R \<bar>prod m UNIV\<bar>)
                 ((\<lambda>x. \<chi> k. x $ k * m k) ` cbox (\<chi> k. min (?\<alpha> k) (?\<beta> k)) (\<chi> k. max (?\<alpha> k) (?\<beta> k)))"
               using True has_integral_stretch_cart [OF zint, of "inverse \<circ> m"]
               by (simp add: field_simps prod_dividef)
             moreover have "((\<lambda>x. \<chi> k. x $ k * m k) ` cbox (\<chi> k. min (?\<alpha> k) (?\<beta> k)) (\<chi> k. max (?\<alpha> k) (?\<beta> k))) = cbox u v"
               using True image_stretch_interval_cart [of "inverse \<circ> m" u v, symmetric]
                 image_stretch_interval_cart [of "\<lambda>k. 1" u v, symmetric] \<open>cbox u v \<noteq> {}\<close>
               by (simp add: field_simps image_comp o_def)
             ultimately show "((\<lambda>x. indicat_real S (\<chi> k. x $ k/ m k)) has_integral z *\<^sub>R \<bar>prod m UNIV\<bar>) (cbox u v)"
               by simp
             have "\<bar>z *\<^sub>R \<bar>prod m UNIV\<bar> - \<bar>prod m UNIV\<bar> * measure lebesgue S\<bar>
                  = \<bar>prod m UNIV\<bar> * \<bar>z - measure lebesgue S\<bar>"
               by (metis (no_types, hide_lams) abs_abs abs_scaleR mult.commute real_scaleR_def right_diff_distrib')
             also have "\<dots> < e"
               using zless True by (simp add: field_simps)
             finally show "\<bar>z *\<^sub>R \<bar>prod m UNIV\<bar> - \<bar>prod m UNIV\<bar> * measure lebesgue S\<bar> < e" .
           qed
         qed
       qed
     qed
     then show ?thesis
       by (auto simp: has_integral_integrable integral_unique lmeasure_integral_UNIV measurable_integrable)
   next
     case False
     then obtain k where "m k = 0" and prm: "prod m UNIV = 0"
       by auto
     have nfS: "negligible (?f ` S)"
       by (rule negligible_subset [OF negligible_standard_hyperplane_cart]) (use \<open>m k = 0\<close> in auto)
     then have "(?f ` S) \<in> lmeasurable"
       by (simp add: negligible_iff_measure)
     with nfS show ?thesis
       by (simp add: prm negligible_iff_measure0)
   qed
   then show "(?f ` S) \<in> lmeasurable" ?MEQ
     by metis+
 qed
 
 
 proposition
  fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
   assumes "linear f" "S \<in> lmeasurable"
   shows measurable_linear_image: "(f ` S) \<in> lmeasurable"
     and measure_linear_image: "measure lebesgue (f ` S) = \<bar>det (matrix f)\<bar> * measure lebesgue S" (is "?Q f S")
 proof -
   have "\<forall>S \<in> lmeasurable. (f ` S) \<in> lmeasurable \<and> ?Q f S"
   proof (rule induct_linear_elementary [OF \<open>linear f\<close>]; intro ballI)
     fix f g and S :: "(real,'n) vec set"
     assume "linear f" and "linear g"
       and f [rule_format]: "\<forall>S \<in> lmeasurable. f ` S \<in> lmeasurable \<and> ?Q f S"
       and g [rule_format]: "\<forall>S \<in> lmeasurable. g ` S \<in> lmeasurable \<and> ?Q g S"
       and S: "S \<in> lmeasurable"
     then have gS: "g ` S \<in> lmeasurable"
       by blast
     show "(f \<circ> g) ` S \<in> lmeasurable \<and> ?Q (f \<circ> g) S"
       using f [OF gS] g [OF S] matrix_compose [OF \<open>linear g\<close> \<open>linear f\<close>]
       by (simp add: o_def image_comp abs_mult det_mul)
   next
     fix f :: "real^'n::_ \<Rightarrow> real^'n::_" and i and S :: "(real^'n::_) set"
     assume "linear f" and 0: "\<And>x. f x $ i = 0" and "S \<in> lmeasurable"
     then have "\<not> inj f"
       by (metis (full_types) linear_injective_imp_surjective one_neq_zero surjE vec_component)
     have detf: "det (matrix f) = 0"
       using \<open>\<not> inj f\<close> det_nz_iff_inj[OF \<open>linear f\<close>] by blast
     show "f ` S \<in> lmeasurable \<and> ?Q f S"
     proof
       show "f ` S \<in> lmeasurable"
         using lmeasurable_iff_indicator_has_integral \<open>linear f\<close> \<open>\<not> inj f\<close> negligible_UNIV negligible_linear_singular_image by blast
       have "measure lebesgue (f ` S) = 0"
         by (meson \<open>\<not> inj f\<close> \<open>linear f\<close> negligible_imp_measure0 negligible_linear_singular_image)
       also have "\<dots> = \<bar>det (matrix f)\<bar> * measure lebesgue S"
         by (simp add: detf)
       finally show "?Q f S" .
     qed
   next
     fix c and S :: "(real^'n::_) set"
     assume "S \<in> lmeasurable"
     show "(\<lambda>a. \<chi> i. c i * a $ i) ` S \<in> lmeasurable \<and> ?Q (\<lambda>a. \<chi> i. c i * a $ i) S"
     proof
       show "(\<lambda>a. \<chi> i. c i * a $ i) ` S \<in> lmeasurable"
         by (simp add: \<open>S \<in> lmeasurable\<close> measurable_stretch)
       show "?Q (\<lambda>a. \<chi> i. c i * a $ i) S"
         by (simp add: measure_stretch [OF \<open>S \<in> lmeasurable\<close>, of c] axis_def matrix_def det_diagonal)
     qed
   next
     fix m :: "'n" and n :: "'n" and S :: "(real, 'n) vec set"
     assume "m \<noteq> n" and "S \<in> lmeasurable"
     let ?h = "\<lambda>v::(real, 'n) vec. \<chi> i. v $ Fun.swap m n id i"
     have lin: "linear ?h"
       by (rule linearI) (simp_all add: plus_vec_def scaleR_vec_def)
     have meq: "measure lebesgue ((\<lambda>v::(real, 'n) vec. \<chi> i. v $ Fun.swap m n id i) ` cbox a b)
              = measure lebesgue (cbox a b)" for a b
     proof (cases "cbox a b = {}")
       case True then show ?thesis
         by simp
     next
       case False
       then have him: "?h ` (cbox a b) \<noteq> {}"
         by blast
       have eq: "?h ` (cbox a b) = cbox (?h a) (?h b)"
         by (auto simp: image_iff lambda_swap_Galois mem_box_cart) (metis swap_id_eq)+
       show ?thesis
         using him prod.permute [OF permutes_swap_id, where S=UNIV and g="\<lambda>i. (b - a)$i", symmetric]
         by (simp add: eq content_cbox_cart False)
     qed
     have "(\<chi> i j. if Fun.swap m n id i = j then 1 else 0) = (\<chi> i j. if j = Fun.swap m n id i then 1 else (0::real))"
       by (auto intro!: Cart_lambda_cong)
     then have "matrix ?h = transpose(\<chi> i j. mat 1 $ i $ Fun.swap m n id j)"
       by (auto simp: matrix_eq transpose_def axis_def mat_def matrix_def)
     then have 1: "\<bar>det (matrix ?h)\<bar> = 1"
       by (simp add: det_permute_columns permutes_swap_id sign_swap_id abs_mult)
     show "?h ` S \<in> lmeasurable \<and> ?Q ?h S"
     proof
       show "?h ` S \<in> lmeasurable" "?Q ?h S"
         using measure_linear_sufficient [OF lin \<open>S \<in> lmeasurable\<close>] meq 1 by force+
     qed
   next
     fix m n :: "'n" and S :: "(real, 'n) vec set"
     assume "m \<noteq> n" and "S \<in> lmeasurable"
     let ?h = "\<lambda>v::(real, 'n) vec. \<chi> i. if i = m then v $ m + v $ n else v $ i"
     have lin: "linear ?h"
       by (rule linearI) (auto simp: algebra_simps plus_vec_def scaleR_vec_def vec_eq_iff)
     consider "m < n" | " n < m"
       using \<open>m \<noteq> n\<close> less_linear by blast
     then have 1: "det(matrix ?h) = 1"
     proof cases
       assume "m < n"
       have *: "matrix ?h $ i $ j = (0::real)" if "j < i" for i j :: 'n
       proof -
         have "axis j 1 = (\<chi> n. if n = j then 1 else (0::real))"
           using axis_def by blast
         then have "(\<chi> p q. if p = m then axis q 1 $ m + axis q 1 $ n else axis q 1 $ p) $ i $ j = (0::real)"
           using \<open>j < i\<close> axis_def \<open>m < n\<close> by auto
         with \<open>m < n\<close> show ?thesis
           by (auto simp: matrix_def axis_def cong: if_cong)
       qed
       show ?thesis
         using \<open>m \<noteq> n\<close> by (subst det_upperdiagonal [OF *]) (auto simp: matrix_def axis_def cong: if_cong)
     next
       assume "n < m"
       have *: "matrix ?h $ i $ j = (0::real)" if "j > i" for i j :: 'n
       proof -
         have "axis j 1 = (\<chi> n. if n = j then 1 else (0::real))"
           using axis_def by blast
         then have "(\<chi> p q. if p = m then axis q 1 $ m + axis q 1 $ n else axis q 1 $ p) $ i $ j = (0::real)"
           using \<open>j > i\<close> axis_def \<open>m > n\<close> by auto
         with \<open>m > n\<close> show ?thesis
           by (auto simp: matrix_def axis_def cong: if_cong)
       qed
       show ?thesis
         using \<open>m \<noteq> n\<close>
         by (subst det_lowerdiagonal [OF *]) (auto simp: matrix_def axis_def cong: if_cong)
     qed
     have meq: "measure lebesgue (?h ` (cbox a b)) = measure lebesgue (cbox a b)" for a b
     proof (cases "cbox a b = {}")
       case True then show ?thesis by simp
     next
       case False
       then have ne: "(+) (\<chi> i. if i = n then - a $ n else 0) ` cbox a b \<noteq> {}"
         by auto
       let ?v = "\<chi> i. if i = n then - a $ n else 0"
       have "?h ` cbox a b
             = (+) (\<chi> i. if i = m \<or> i = n then a $ n else 0) ` ?h ` (+) ?v ` (cbox a b)"
         using \<open>m \<noteq> n\<close> unfolding image_comp o_def by (force simp: vec_eq_iff)
       then have "measure lebesgue (?h ` (cbox a b))
                = measure lebesgue ((\<lambda>v. \<chi> i. if i = m then v $ m + v $ n else v $ i) `
                                    (+) ?v ` cbox a b)"
         by (rule ssubst) (rule measure_translation)
       also have "\<dots> = measure lebesgue ((\<lambda>v. \<chi> i. if i = m then v $ m + v $ n else v $ i) ` cbox (?v +a) (?v + b))"
         by (metis (no_types, lifting) cbox_translation)
       also have "\<dots> = measure lebesgue ((+) (\<chi> i. if i = n then - a $ n else 0) ` cbox a b)"
         apply (subst measure_shear_interval)
         using \<open>m \<noteq> n\<close> ne apply auto
         apply (simp add: cbox_translation)
         by (metis cbox_borel cbox_translation measure_completion sets_lborel)
       also have "\<dots> = measure lebesgue (cbox a b)"
         by (rule measure_translation)
         finally show ?thesis .
       qed
     show "?h ` S \<in> lmeasurable \<and> ?Q ?h S"
       using measure_linear_sufficient [OF lin \<open>S \<in> lmeasurable\<close>] meq 1 by force
   qed
   with assms show "(f ` S) \<in> lmeasurable" "?Q f S"
     by metis+
 qed
 
 
 lemma
  fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
   assumes f: "orthogonal_transformation f" and S: "S \<in> lmeasurable"
   shows measurable_orthogonal_image: "f ` S \<in> lmeasurable"
     and measure_orthogonal_image: "measure lebesgue (f ` S) = measure lebesgue S"
 proof -
   have "linear f"
     by (simp add: f orthogonal_transformation_linear)
   then show "f ` S \<in> lmeasurable"
     by (metis S measurable_linear_image)
   show "measure lebesgue (f ` S) = measure lebesgue S"
     by (simp add: measure_linear_image \<open>linear f\<close> S f)
 qed
 
 proposition measure_semicontinuous_with_hausdist_explicit:
   assumes "bounded S" and neg: "negligible(frontier S)" and "e > 0"
   obtains d where "d > 0"
                   "\<And>T. \<lbrakk>T \<in> lmeasurable; \<And>y. y \<in> T \<Longrightarrow> \<exists>x. x \<in> S \<and> dist x y < d\<rbrakk>
                         \<Longrightarrow> measure lebesgue T < measure lebesgue S + e"
 proof (cases "S = {}")
   case True
   with that \<open>e > 0\<close> show ?thesis by force
 next
   case False
   then have frS: "frontier S \<noteq> {}"
     using \<open>bounded S\<close> frontier_eq_empty not_bounded_UNIV by blast
   have "S \<in> lmeasurable"
     by (simp add: \<open>bounded S\<close> measurable_Jordan neg)
   have null: "(frontier S) \<in> null_sets lebesgue"
     by (metis neg negligible_iff_null_sets)
   have "frontier S \<in> lmeasurable" and mS0: "measure lebesgue (frontier S) = 0"
     using neg negligible_imp_measurable negligible_iff_measure by blast+
   with \<open>e > 0\<close> sets_lebesgue_outer_open
   obtain U where "open U"
     and U: "frontier S \<subseteq> U" "U - frontier S \<in> lmeasurable" "emeasure lebesgue (U - frontier S) < e"
     by (metis fmeasurableD)
   with null have "U \<in> lmeasurable"
     by (metis borel_open measurable_Diff_null_set sets_completionI_sets sets_lborel)
   have "measure lebesgue (U - frontier S) = measure lebesgue U"
     using mS0 by (simp add: \<open>U \<in> lmeasurable\<close> fmeasurableD measure_Diff_null_set null)
   with U have mU: "measure lebesgue U < e"
     by (simp add: emeasure_eq_measure2 ennreal_less_iff)
   show ?thesis
   proof
     have "U \<noteq> UNIV"
       using \<open>U \<in> lmeasurable\<close> by auto
     then have "- U \<noteq> {}"
       by blast
     with \<open>open U\<close> \<open>frontier S \<subseteq> U\<close> show "setdist (frontier S) (- U) > 0"
       by (auto simp: \<open>bounded S\<close> open_closed compact_frontier_bounded setdist_gt_0_compact_closed frS)
     fix T
     assume "T \<in> lmeasurable"
       and T: "\<And>t. t \<in> T \<Longrightarrow> \<exists>y. y \<in> S \<and> dist y t < setdist (frontier S) (- U)"
     then have "measure lebesgue T - measure lebesgue S \<le> measure lebesgue (T - S)"
       by (simp add: \<open>S \<in> lmeasurable\<close> measure_diff_le_measure_setdiff)
     also have "\<dots>  \<le> measure lebesgue U"
     proof -
       have "T - S \<subseteq> U"
       proof clarify
         fix x
         assume "x \<in> T" and "x \<notin> S"
         then obtain y where "y \<in> S" and y: "dist y x < setdist (frontier S) (- U)"
           using T by blast
         have "closed_segment x y \<inter> frontier S \<noteq> {}"
           using connected_Int_frontier \<open>x \<notin> S\<close> \<open>y \<in> S\<close> by blast
         then obtain z where z: "z \<in> closed_segment x y" "z \<in> frontier S"
           by auto
         with y have "dist z x < setdist(frontier S) (- U)"
           by (auto simp: dist_commute dest!: dist_in_closed_segment)
         with z have False if "x \<in> -U"
           using setdist_le_dist [OF \<open>z \<in> frontier S\<close> that] by auto
         then show "x \<in> U"
           by blast
       qed
       then show ?thesis
         by (simp add: \<open>S \<in> lmeasurable\<close> \<open>T \<in> lmeasurable\<close> \<open>U \<in> lmeasurable\<close> fmeasurableD measure_mono_fmeasurable sets.Diff)
     qed
     finally have "measure lebesgue T - measure lebesgue S \<le> measure lebesgue U" .
     with mU show "measure lebesgue T < measure lebesgue S + e"
       by linarith
   qed
 qed
 
 proposition
  fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
   assumes S: "S \<in> lmeasurable"
   and deriv: "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
   and int: "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) integrable_on S"
   and bounded: "\<And>x. x \<in> S \<Longrightarrow> \<bar>det (matrix (f' x))\<bar> \<le> B"
   shows measurable_bounded_differentiable_image:
        "f ` S \<in> lmeasurable"
     and measure_bounded_differentiable_image:
        "measure lebesgue (f ` S) \<le> B * measure lebesgue S" (is "?M")
 proof -
   have "f ` S \<in> lmeasurable \<and> measure lebesgue (f ` S) \<le> B * measure lebesgue S"
   proof (cases "B < 0")
     case True
     then have "S = {}"
       by (meson abs_ge_zero bounded empty_iff equalityI less_le_trans linorder_not_less subsetI)
     then show ?thesis
       by auto
   next
     case False
     then have "B \<ge> 0"
       by arith
     let ?\<mu> = "measure lebesgue"
     have f_diff: "f differentiable_on S"
       using deriv by (auto simp: differentiable_on_def differentiable_def)
     have eps: "f ` S \<in> lmeasurable" "?\<mu> (f ` S) \<le> (B+e) * ?\<mu> S" (is "?ME")
               if "e > 0" for e
     proof -
       have eps_d: "f ` S \<in> lmeasurable"  "?\<mu> (f ` S) \<le> (B+e) * (?\<mu> S + d)" (is "?MD")
                   if "d > 0" for d
       proof -
         obtain T where T: "open T" "S \<subseteq> T" and TS: "(T-S) \<in> lmeasurable" and "emeasure lebesgue (T-S) < ennreal d"
           using S \<open>d > 0\<close> sets_lebesgue_outer_open by blast
         then have "?\<mu> (T-S) < d"
           by (metis emeasure_eq_measure2 ennreal_leI not_less)
         with S T TS have "T \<in> lmeasurable" and Tless: "?\<mu> T < ?\<mu> S + d"
           by (auto simp: measurable_measure_Diff dest!: fmeasurable_Diff_D)
         have "\<exists>r. 0 < r \<and> r < d \<and> ball x r \<subseteq> T \<and> f ` (S \<inter> ball x r) \<in> lmeasurable \<and>
                   ?\<mu> (f ` (S \<inter> ball x r)) \<le> (B + e) * ?\<mu> (ball x r)"
           if "x \<in> S" "d > 0" for x d
         proof -
           have lin: "linear (f' x)"
             and lim0: "((\<lambda>y. (f y - (f x + f' x (y - x))) /\<^sub>R norm(y - x)) \<longlongrightarrow> 0) (at x within S)"
             using deriv \<open>x \<in> S\<close> by (auto simp: has_derivative_within bounded_linear.linear field_simps)
           have bo: "bounded (f' x ` ball 0 1)"
             by (simp add: bounded_linear_image linear_linear lin)
           have neg: "negligible (frontier (f' x ` ball 0 1))"
             using deriv has_derivative_linear \<open>x \<in> S\<close>
             by (auto intro!: negligible_convex_frontier [OF convex_linear_image])
           let ?unit_vol = "content (ball (0 :: real ^ 'n :: {finite, wellorder}) 1)"
           have 0: "0 < e * ?unit_vol"
             using \<open>e > 0\<close> by (simp add: content_ball_pos)
           obtain k where "k > 0" and k:
                   "\<And>U. \<lbrakk>U \<in> lmeasurable; \<And>y. y \<in> U \<Longrightarrow> \<exists>z. z \<in> f' x ` ball 0 1 \<and> dist z y < k\<rbrakk>
                         \<Longrightarrow> ?\<mu> U < ?\<mu> (f' x ` ball 0 1) + e * ?unit_vol"
             using measure_semicontinuous_with_hausdist_explicit [OF bo neg 0] by blast
           obtain l where "l > 0" and l: "ball x l \<subseteq> T"
             using \<open>x \<in> S\<close> \<open>open T\<close> \<open>S \<subseteq> T\<close> openE by blast
           obtain \<zeta> where "0 < \<zeta>"
             and \<zeta>: "\<And>y. \<lbrakk>y \<in> S; y \<noteq> x; dist y x < \<zeta>\<rbrakk>
                         \<Longrightarrow> norm (f y - (f x + f' x (y - x))) / norm (y - x) < k"
             using lim0 \<open>k > 0\<close> by (simp add: Lim_within) (auto simp add: field_simps)
           define r where "r \<equiv> min (min l (\<zeta>/2)) (min 1 (d/2))"
           show ?thesis
           proof (intro exI conjI)
             show "r > 0" "r < d"
               using \<open>l > 0\<close> \<open>\<zeta> > 0\<close> \<open>d > 0\<close> by (auto simp: r_def)
             have "r \<le> l"
               by (auto simp: r_def)
             with l show "ball x r \<subseteq> T"
               by auto
             have ex_lessK: "\<exists>x' \<in> ball 0 1. dist (f' x x') ((f y - f x) /\<^sub>R r) < k"
               if "y \<in> S" and "dist x y < r" for y
             proof (cases "y = x")
               case True
               with lin linear_0 \<open>k > 0\<close> that show ?thesis
                 by (rule_tac x=0 in bexI) (auto simp: linear_0)
             next
               case False
               then show ?thesis
               proof (rule_tac x="(y - x) /\<^sub>R r" in bexI)
                 have "f' x ((y - x) /\<^sub>R r) = f' x (y - x) /\<^sub>R r"
                   by (simp add: lin linear_scale)
                 then have "dist (f' x ((y - x) /\<^sub>R r)) ((f y - f x) /\<^sub>R r) = norm (f' x (y - x) /\<^sub>R r - (f y - f x) /\<^sub>R r)"
                   by (simp add: dist_norm)
                 also have "\<dots> = norm (f' x (y - x) - (f y - f x)) / r"
                   using \<open>r > 0\<close> by (simp add: divide_simps scale_right_diff_distrib [symmetric])
                 also have "\<dots> \<le> norm (f y - (f x + f' x (y - x))) / norm (y - x)"
                   using that \<open>r > 0\<close> False by (simp add: field_split_simps dist_norm norm_minus_commute mult_right_mono)
                 also have "\<dots> < k"
                   using that \<open>0 < \<zeta>\<close> by (simp add: dist_commute r_def  \<zeta> [OF \<open>y \<in> S\<close> False])
                 finally show "dist (f' x ((y - x) /\<^sub>R r)) ((f y - f x) /\<^sub>R r) < k" .
                 show "(y - x) /\<^sub>R r \<in> ball 0 1"
                   using that \<open>r > 0\<close> by (simp add: dist_norm divide_simps norm_minus_commute)
               qed
             qed
             let ?rfs = "(\<lambda>x. x /\<^sub>R r) ` (+) (- f x) ` f ` (S \<inter> ball x r)"
             have rfs_mble: "?rfs \<in> lmeasurable"
             proof (rule bounded_set_imp_lmeasurable)
               have "f differentiable_on S \<inter> ball x r"
                 using f_diff by (auto simp: fmeasurableD differentiable_on_subset)
               with S show "?rfs \<in> sets lebesgue"
                 by (auto simp: sets.Int intro!: lebesgue_sets_translation differentiable_image_in_sets_lebesgue)
               let ?B = "(\<lambda>(x, y). x + y) ` (f' x ` ball 0 1 \<times> ball 0 k)"
               have "bounded ?B"
                 by (simp add: bounded_plus [OF bo])
               moreover have "?rfs \<subseteq> ?B"
                 apply (auto simp: dist_norm image_iff dest!: ex_lessK)
                 by (metis (no_types, hide_lams) add.commute diff_add_cancel dist_0_norm dist_commute dist_norm mem_ball)
               ultimately show "bounded (?rfs)"
                 by (rule bounded_subset)
             qed
             then have "(\<lambda>x. r *\<^sub>R x) ` ?rfs \<in> lmeasurable"
               by (simp add: measurable_linear_image)
             with \<open>r > 0\<close> have "(+) (- f x) ` f ` (S \<inter> ball x r) \<in> lmeasurable"
               by (simp add: image_comp o_def)
             then have "(+) (f x) ` (+) (- f x) ` f ` (S \<inter> ball x r) \<in> lmeasurable"
               using  measurable_translation by blast
             then show fsb: "f ` (S \<inter> ball x r) \<in> lmeasurable"
               by (simp add: image_comp o_def)
             have "?\<mu> (f ` (S \<inter> ball x r)) = ?\<mu> (?rfs) * r ^ CARD('n)"
               using \<open>r > 0\<close> fsb
               by (simp add: measure_linear_image measure_translation_subtract measurable_translation_subtract field_simps cong: image_cong_simp)
             also have "\<dots> \<le> (\<bar>det (matrix (f' x))\<bar> * ?unit_vol + e * ?unit_vol) * r ^ CARD('n)"
             proof -
               have "?\<mu> (?rfs) < ?\<mu> (f' x ` ball 0 1) + e * ?unit_vol"
                 using rfs_mble by (force intro: k dest!: ex_lessK)
               then have "?\<mu> (?rfs) < \<bar>det (matrix (f' x))\<bar> * ?unit_vol + e * ?unit_vol"
                 by (simp add: lin measure_linear_image [of "f' x"])
               with \<open>r > 0\<close> show ?thesis
                 by auto
             qed
             also have "\<dots> \<le> (B + e) * ?\<mu> (ball x r)"
               using bounded [OF \<open>x \<in> S\<close>] \<open>r > 0\<close>
               by (simp add: algebra_simps content_ball_conv_unit_ball[of r] content_ball_pos)
             finally show "?\<mu> (f ` (S \<inter> ball x r)) \<le> (B + e) * ?\<mu> (ball x r)" .
           qed
         qed
         then obtain r where
           r0d: "\<And>x d. \<lbrakk>x \<in> S; d > 0\<rbrakk> \<Longrightarrow> 0 < r x d \<and> r x d < d"
           and rT: "\<And>x d. \<lbrakk>x \<in> S; d > 0\<rbrakk> \<Longrightarrow> ball x (r x d) \<subseteq> T"
           and r: "\<And>x d. \<lbrakk>x \<in> S; d > 0\<rbrakk> \<Longrightarrow>
                   (f ` (S \<inter> ball x (r x d))) \<in> lmeasurable \<and>
                   ?\<mu> (f ` (S \<inter> ball x (r x d))) \<le> (B + e) * ?\<mu> (ball x (r x d))"
           by metis
         obtain C where "countable C" and Csub: "C \<subseteq> {(x,r x t) |x t. x \<in> S \<and> 0 < t}"
           and pwC: "pairwise (\<lambda>i j. disjnt (ball (fst i) (snd i)) (ball (fst j) (snd j))) C"
           and negC: "negligible(S - (\<Union>i \<in> C. ball (fst i) (snd i)))"
           apply (rule Vitali_covering_theorem_balls [of S "{(x,r x t) |x t. x \<in> S \<and> 0 < t}" fst snd])
            apply auto
           by (metis dist_eq_0_iff r0d)
         let ?UB = "(\<Union>(x,s) \<in> C. ball x s)"
         have eq: "f ` (S \<inter> ?UB) = (\<Union>(x,s) \<in> C. f ` (S \<inter> ball x s))"
           by auto
         have mle: "?\<mu> (\<Union>(x,s) \<in> K. f ` (S \<inter> ball x s)) \<le> (B + e) * (?\<mu> S + d)"  (is "?l \<le> ?r")
           if "K \<subseteq> C" and "finite K" for K
         proof -
           have gt0: "b > 0" if "(a, b) \<in> K" for a b
             using Csub that \<open>K \<subseteq> C\<close> r0d by auto
           have inj: "inj_on (\<lambda>(x, y). ball x y) K"
             by (force simp: inj_on_def ball_eq_ball_iff dest: gt0)
           have disjnt: "disjoint ((\<lambda>(x, y). ball x y) ` K)"
             using pwC that
             apply (clarsimp simp: pairwise_def case_prod_unfold ball_eq_ball_iff)
             by (metis subsetD fst_conv snd_conv)
           have "?l \<le> (\<Sum>i\<in>K. ?\<mu> (case i of (x, s) \<Rightarrow> f ` (S \<inter> ball x s)))"
           proof (rule measure_UNION_le [OF \<open>finite K\<close>], clarify)
             fix x r
             assume "(x,r) \<in> K"
             then have "x \<in> S"
               using Csub \<open>K \<subseteq> C\<close> by auto
             show "f ` (S \<inter> ball x r) \<in> sets lebesgue"
               by (meson Int_lower1 S differentiable_on_subset f_diff fmeasurableD lmeasurable_ball order_refl sets.Int differentiable_image_in_sets_lebesgue)
           qed
           also have "\<dots> \<le> (\<Sum>(x,s) \<in> K. (B + e) * ?\<mu> (ball x s))"
             apply (rule sum_mono)
             using Csub r \<open>K \<subseteq> C\<close> by auto
           also have "\<dots> = (B + e) * (\<Sum>(x,s) \<in> K. ?\<mu> (ball x s))"
             by (simp add: prod.case_distrib sum_distrib_left)
           also have "\<dots> = (B + e) * sum ?\<mu> ((\<lambda>(x, y). ball x y) ` K)"
             using \<open>B \<ge> 0\<close> \<open>e > 0\<close> by (simp add: inj sum.reindex prod.case_distrib)
           also have "\<dots> = (B + e) * ?\<mu> (\<Union>(x,s) \<in> K. ball x s)"
             using \<open>B \<ge> 0\<close> \<open>e > 0\<close> that
             by (subst measure_Union') (auto simp: disjnt measure_Union')
           also have "\<dots> \<le> (B + e) * ?\<mu> T"
             using \<open>B \<ge> 0\<close> \<open>e > 0\<close> that apply simp
             apply (rule measure_mono_fmeasurable [OF _ _ \<open>T \<in> lmeasurable\<close>])
             using Csub rT by force+
           also have "\<dots> \<le> (B + e) * (?\<mu> S + d)"
             using \<open>B \<ge> 0\<close> \<open>e > 0\<close> Tless by simp
           finally show ?thesis .
         qed
         have fSUB_mble: "(f ` (S \<inter> ?UB)) \<in> lmeasurable"
           unfolding eq using Csub r False \<open>e > 0\<close> that
           by (auto simp: intro!: fmeasurable_UN_bound [OF \<open>countable C\<close> _ mle])
         have fSUB_meas: "?\<mu> (f ` (S \<inter> ?UB)) \<le> (B + e) * (?\<mu> S + d)"  (is "?MUB")
           unfolding eq using Csub r False \<open>e > 0\<close> that
           by (auto simp: intro!: measure_UN_bound [OF \<open>countable C\<close> _ mle])
         have neg: "negligible ((f ` (S \<inter> ?UB) - f ` S) \<union> (f ` S - f ` (S \<inter> ?UB)))"
         proof (rule negligible_subset [OF negligible_differentiable_image_negligible [OF order_refl negC, where f=f]])
           show "f differentiable_on S - (\<Union>i\<in>C. ball (fst i) (snd i))"
             by (meson DiffE differentiable_on_subset subsetI f_diff)
         qed force
         show "f ` S \<in> lmeasurable"
           by (rule lmeasurable_negligible_symdiff [OF fSUB_mble neg])
         show ?MD
           using fSUB_meas measure_negligible_symdiff [OF fSUB_mble neg] by simp
       qed
       show "f ` S \<in> lmeasurable"
         using eps_d [of 1] by simp
       show ?ME
       proof (rule field_le_epsilon)
         fix \<delta> :: real
         assume "0 < \<delta>"
         then show "?\<mu> (f ` S) \<le> (B + e) * ?\<mu> S + \<delta>"
           using eps_d [of "\<delta> / (B+e)"] \<open>e > 0\<close> \<open>B \<ge> 0\<close> by (auto simp: divide_simps mult_ac)
       qed
     qed
     show ?thesis
     proof (cases "?\<mu> S = 0")
       case True
       with eps have "?\<mu> (f ` S) = 0"
         by (metis mult_zero_right not_le zero_less_measure_iff)
       then show ?thesis
         using eps [of 1] by (simp add: True)
     next
       case False
       have "?\<mu> (f ` S) \<le> B * ?\<mu> S"
       proof (rule field_le_epsilon)
         fix e :: real
         assume "e > 0"
         then show "?\<mu> (f ` S) \<le> B * ?\<mu> S + e"
           using eps [of "e / ?\<mu> S"] False by (auto simp: algebra_simps zero_less_measure_iff)
       qed
       with eps [of 1] show ?thesis by auto
     qed
   qed
   then show "f ` S \<in> lmeasurable" ?M by blast+
 qed
 
 lemma m_diff_image_weak:
  fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
   assumes S: "S \<in> lmeasurable"
     and deriv: "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
     and int: "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) integrable_on S"
   shows "f ` S \<in> lmeasurable \<and> measure lebesgue (f ` S) \<le> integral S (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)"
 proof -
   let ?\<mu> = "measure lebesgue"
   have aint_S: "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) absolutely_integrable_on S"
     using int unfolding absolutely_integrable_on_def by auto
   define m where "m \<equiv> integral S (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)"
   have *: "f ` S \<in> lmeasurable" "?\<mu> (f ` S) \<le> m + e * ?\<mu> S"
     if "e > 0" for e
   proof -
     define T where "T \<equiv> \<lambda>n. {x \<in> S. n * e \<le> \<bar>det (matrix (f' x))\<bar> \<and>
                                      \<bar>det (matrix (f' x))\<bar> < (Suc n) * e}"
     have meas_t: "T n \<in> lmeasurable" for n
     proof -
       have *: "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) \<in> borel_measurable (lebesgue_on S)"
         using aint_S by (simp add: S borel_measurable_restrict_space_iff fmeasurableD set_integrable_def)
       have [intro]: "x \<in> sets (lebesgue_on S) \<Longrightarrow> x \<in> sets lebesgue" for x
         using S sets_restrict_space_subset by blast
       have "{x \<in> S. real n * e \<le> \<bar>det (matrix (f' x))\<bar>} \<in> sets lebesgue"
         using * by (auto simp: borel_measurable_iff_halfspace_ge space_restrict_space)
       then have 1: "{x \<in> S. real n * e \<le> \<bar>det (matrix (f' x))\<bar>} \<in> lmeasurable"
         using S by (simp add: fmeasurableI2)
       have "{x \<in> S. \<bar>det (matrix (f' x))\<bar> < (1 + real n) * e} \<in> sets lebesgue"
         using * by (auto simp: borel_measurable_iff_halfspace_less space_restrict_space)
       then have 2: "{x \<in> S. \<bar>det (matrix (f' x))\<bar> < (1 + real n) * e} \<in> lmeasurable"
         using S by (simp add: fmeasurableI2)
       show ?thesis
         using fmeasurable.Int [OF 1 2] by (simp add: T_def Int_def cong: conj_cong)
     qed
     have aint_T: "\<And>k. (\<lambda>x. \<bar>det (matrix (f' x))\<bar>) absolutely_integrable_on T k"
       using set_integrable_subset [OF aint_S] meas_t T_def by blast
     have Seq: "S = (\<Union>n. T n)"
       apply (auto simp: T_def)
       apply (rule_tac x="nat(floor(abs(det(matrix(f' x))) / e))" in exI)
       using that apply auto
       using of_int_floor_le pos_le_divide_eq apply blast
       by (metis add.commute pos_divide_less_eq real_of_int_floor_add_one_gt)
     have meas_ft: "f ` T n \<in> lmeasurable" for n
     proof (rule measurable_bounded_differentiable_image)
       show "T n \<in> lmeasurable"
         by (simp add: meas_t)
     next
       fix x :: "(real,'n) vec"
       assume "x \<in> T n"
       show "(f has_derivative f' x) (at x within T n)"
-        by (metis (no_types, lifting) \<open>x \<in> T n\<close> deriv has_derivative_within_subset mem_Collect_eq subsetI T_def)
+        by (metis (no_types, lifting) \<open>x \<in> T n\<close> deriv has_derivative_subset mem_Collect_eq subsetI T_def)
       show "\<bar>det (matrix (f' x))\<bar> \<le> (Suc n) * e"
         using \<open>x \<in> T n\<close> T_def by auto
     next
       show "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) integrable_on T n"
         using aint_T absolutely_integrable_on_def by blast
     qed
     have disT: "disjoint (range T)"
       unfolding disjoint_def
     proof clarsimp
       show "T m \<inter> T n = {}" if "T m \<noteq> T n" for m n
         using that
       proof (induction m n rule: linorder_less_wlog)
         case (less m n)
         with \<open>e > 0\<close> show ?case
           unfolding T_def
           proof (clarsimp simp add: Collect_conj_eq [symmetric])
             fix x
             assume "e > 0"  "m < n"  "n * e \<le> \<bar>det (matrix (f' x))\<bar>"  "\<bar>det (matrix (f' x))\<bar> < (1 + real m) * e"
             then have "n < 1 + real m"
               by (metis (no_types, hide_lams) less_le_trans mult.commute not_le real_mult_le_cancel_iff2)
             then show "False"
               using less.hyps by linarith
           qed
       qed auto
     qed
     have injT: "inj_on T ({n. T n \<noteq> {}})"
       unfolding inj_on_def
     proof clarsimp
       show "m = n" if "T m = T n" "T n \<noteq> {}" for m n
         using that
       proof (induction m n rule: linorder_less_wlog)
         case (less m n)
         have False if "T n \<subseteq> T m" "x \<in> T n" for x
           using \<open>e > 0\<close> \<open>m < n\<close> that
           apply (auto simp: T_def  mult.commute intro: less_le_trans dest!: subsetD)
           by (metis add.commute less_le_trans nat_less_real_le not_le real_mult_le_cancel_iff2)
         then show ?case
           using less.prems by blast
       qed auto
     qed
     have sum_eq_Tim: "(\<Sum>k\<le>n. f (T k)) = sum f (T ` {..n})" if "f {} = 0" for f :: "_ \<Rightarrow> real" and n
     proof (subst sum.reindex_nontrivial)
       fix i j  assume "i \<in> {..n}" "j \<in> {..n}" "i \<noteq> j" "T i = T j"
       with that  injT [unfolded inj_on_def] show "f (T i) = 0"
         by simp metis
     qed (use atMost_atLeast0 in auto)
     let ?B = "m + e * ?\<mu> S"
     have "(\<Sum>k\<le>n. ?\<mu> (f ` T k)) \<le> ?B" for n
     proof -
       have "(\<Sum>k\<le>n. ?\<mu> (f ` T k)) \<le> (\<Sum>k\<le>n. ((k+1) * e) * ?\<mu>(T k))"
       proof (rule sum_mono [OF measure_bounded_differentiable_image])
         show "(f has_derivative f' x) (at x within T k)" if "x \<in> T k" for k x
-          using that unfolding T_def by (blast intro: deriv has_derivative_within_subset)
+          using that unfolding T_def by (blast intro: deriv has_derivative_subset)
         show "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) integrable_on T k" for k
           using absolutely_integrable_on_def aint_T by blast
         show "\<bar>det (matrix (f' x))\<bar> \<le> real (k + 1) * e" if "x \<in> T k" for k x
           using T_def that by auto
       qed (use meas_t in auto)
       also have "\<dots> \<le> (\<Sum>k\<le>n. (k * e) * ?\<mu>(T k)) + (\<Sum>k\<le>n. e * ?\<mu>(T k))"
         by (simp add: algebra_simps sum.distrib)
       also have "\<dots> \<le> ?B"
       proof (rule add_mono)
         have "(\<Sum>k\<le>n. real k * e * ?\<mu> (T k)) = (\<Sum>k\<le>n. integral (T k) (\<lambda>x. k * e))"
           by (simp add: lmeasure_integral [OF meas_t]
                    flip: integral_mult_right integral_mult_left)
         also have "\<dots> \<le> (\<Sum>k\<le>n. integral (T k) (\<lambda>x.  (abs (det (matrix (f' x))))))"
         proof (rule sum_mono)
           fix k
           assume "k \<in> {..n}"
           show "integral (T k) (\<lambda>x. k * e) \<le> integral (T k) (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)"
           proof (rule integral_le [OF integrable_on_const [OF meas_t]])
             show "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) integrable_on T k"
               using absolutely_integrable_on_def aint_T by blast
           next
             fix x assume "x \<in> T k"
             show "k * e \<le> \<bar>det (matrix (f' x))\<bar>"
               using \<open>x \<in> T k\<close> T_def by blast
           qed
         qed
         also have "\<dots> = sum (\<lambda>T. integral T (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)) (T ` {..n})"
           by (auto intro: sum_eq_Tim)
         also have "\<dots> = integral (\<Union>k\<le>n. T k) (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)"
         proof (rule integral_unique [OF has_integral_Union, symmetric])
           fix S  assume "S \<in> T ` {..n}"
           then show "((\<lambda>x. \<bar>det (matrix (f' x))\<bar>) has_integral integral S (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)) S"
           using absolutely_integrable_on_def aint_T by blast
         next
           show "pairwise (\<lambda>S S'. negligible (S \<inter> S')) (T ` {..n})"
             using disT unfolding disjnt_iff by (auto simp: pairwise_def intro!: empty_imp_negligible)
         qed auto
         also have "\<dots> \<le> m"
           unfolding m_def
         proof (rule integral_subset_le)
           have "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) absolutely_integrable_on (\<Union>k\<le>n. T k)"
             apply (rule set_integrable_subset [OF aint_S])
              apply (intro measurable meas_t fmeasurableD)
             apply (force simp: Seq)
             done
           then show "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) integrable_on (\<Union>k\<le>n. T k)"
             using absolutely_integrable_on_def by blast
         qed (use Seq int in auto)
         finally show "(\<Sum>k\<le>n. real k * e * ?\<mu> (T k)) \<le> m" .
       next
         have "(\<Sum>k\<le>n. ?\<mu> (T k)) = sum ?\<mu> (T ` {..n})"
           by (auto intro: sum_eq_Tim)
         also have "\<dots> = ?\<mu> (\<Union>k\<le>n. T k)"
           using S disT by (auto simp: pairwise_def meas_t intro: measure_Union' [symmetric])
         also have "\<dots> \<le> ?\<mu> S"
           using S by (auto simp: Seq intro: meas_t fmeasurableD measure_mono_fmeasurable)
         finally have "(\<Sum>k\<le>n. ?\<mu> (T k)) \<le> ?\<mu> S" .
         then show "(\<Sum>k\<le>n. e * ?\<mu> (T k)) \<le> e * ?\<mu> S"
           by (metis less_eq_real_def ordered_comm_semiring_class.comm_mult_left_mono sum_distrib_left that)
       qed
       finally show "(\<Sum>k\<le>n. ?\<mu> (f ` T k)) \<le> ?B" .
     qed
     moreover have "measure lebesgue (\<Union>k\<le>n. f ` T k) \<le> (\<Sum>k\<le>n. ?\<mu> (f ` T k))" for n
       by (simp add: fmeasurableD meas_ft measure_UNION_le)
     ultimately have B_ge_m: "?\<mu> (\<Union>k\<le>n. (f ` T k)) \<le> ?B" for n
       by (meson order_trans)
     have "(\<Union>n. f ` T n) \<in> lmeasurable"
       by (rule fmeasurable_countable_Union [OF meas_ft B_ge_m])
     moreover have "?\<mu> (\<Union>n. f ` T n) \<le> m + e * ?\<mu> S"
       by (rule measure_countable_Union_le [OF meas_ft B_ge_m])
     ultimately show "f ` S \<in> lmeasurable" "?\<mu> (f ` S) \<le> m + e * ?\<mu> S"
       by (auto simp: Seq image_Union)
   qed
   show ?thesis
   proof
     show "f ` S \<in> lmeasurable"
       using * linordered_field_no_ub by blast
     let ?x = "m - ?\<mu> (f ` S)"
     have False if "?\<mu> (f ` S) > integral S (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)"
     proof -
       have ml: "m < ?\<mu> (f ` S)"
         using m_def that by blast
       then have "?\<mu> S \<noteq> 0"
         using "*"(2) bgauge_existence_lemma by fastforce
       with ml have 0: "0 < - (m - ?\<mu> (f ` S))/2 / ?\<mu> S"
         using that zero_less_measure_iff by force
       then show ?thesis
         using * [OF 0] that by (auto simp: field_split_simps m_def split: if_split_asm)
     qed
     then show "?\<mu> (f ` S) \<le> integral S (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)"
       by fastforce
   qed
 qed
 
 
 theorem
  fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
   assumes S: "S \<in> sets lebesgue"
     and deriv: "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
     and int: "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) integrable_on S"
   shows measurable_differentiable_image: "f ` S \<in> lmeasurable"
     and measure_differentiable_image:
        "measure lebesgue (f ` S) \<le> integral S (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)" (is "?M")
 proof -
   let ?I = "\<lambda>n::nat. cbox (vec (-n)) (vec n) \<inter> S"
   let ?\<mu> = "measure lebesgue"
   have "x \<in> cbox (vec (- real (nat \<lceil>norm x\<rceil>))) (vec (real (nat \<lceil>norm x\<rceil>)))" for x :: "real^'n::_"
     apply (auto simp: mem_box_cart)
     apply (metis abs_le_iff component_le_norm_cart minus_le_iff of_nat_ceiling order.trans)
     by (meson abs_le_D1 norm_bound_component_le_cart real_nat_ceiling_ge)
   then have Seq: "S = (\<Union>n. ?I n)"
     by auto
   have fIn: "f ` ?I n \<in> lmeasurable"
        and mfIn: "?\<mu> (f ` ?I n) \<le> integral S (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)" (is ?MN) for n
   proof -
     have In: "?I n \<in> lmeasurable"
       by (simp add: S bounded_Int bounded_set_imp_lmeasurable sets.Int)
     moreover have "\<And>x. x \<in> ?I n \<Longrightarrow> (f has_derivative f' x) (at x within ?I n)"
-      by (meson Int_iff deriv has_derivative_within_subset subsetI)
+      by (meson Int_iff deriv has_derivative_subset subsetI)
     moreover have int_In: "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) integrable_on ?I n"
     proof -
       have "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) absolutely_integrable_on S"
         using int absolutely_integrable_integrable_bound by force
       then have "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) absolutely_integrable_on ?I n"
         by (metis (no_types) Int_lower1 In fmeasurableD inf_commute set_integrable_subset)
       then show ?thesis
         using absolutely_integrable_on_def by blast
     qed
     ultimately have "f ` ?I n \<in> lmeasurable" "?\<mu> (f ` ?I n) \<le> integral (?I n) (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)"
       using m_diff_image_weak by metis+
     moreover have "integral (?I n) (\<lambda>x. \<bar>det (matrix (f' x))\<bar>) \<le> integral S (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)"
       by (simp add: int_In int integral_subset_le)
     ultimately show "f ` ?I n \<in> lmeasurable" ?MN
       by auto
   qed
   have "?I k \<subseteq> ?I n" if "k \<le> n" for k n
     by (rule Int_mono) (use that in \<open>auto simp: subset_interval_imp_cart\<close>)
   then have "(\<Union>k\<le>n. f ` ?I k) = f ` ?I n" for n
     by (fastforce simp add:)
   with mfIn have "?\<mu> (\<Union>k\<le>n. f ` ?I k) \<le> integral S (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)" for n
     by simp
   then have "(\<Union>n. f ` ?I n) \<in> lmeasurable" "?\<mu> (\<Union>n. f ` ?I n) \<le> integral S (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)"
     by (rule fmeasurable_countable_Union [OF fIn] measure_countable_Union_le [OF fIn])+
   then show "f ` S \<in> lmeasurable" ?M
     by (metis Seq image_UN)+
 qed
 
 
 lemma borel_measurable_simple_function_limit_increasing:
   fixes f :: "'a::euclidean_space \<Rightarrow> real"
   shows "(f \<in> borel_measurable lebesgue \<and> (\<forall>x. 0 \<le> f x)) \<longleftrightarrow>
          (\<exists>g. (\<forall>n x. 0 \<le> g n x \<and> g n x \<le> f x) \<and> (\<forall>n x. g n x \<le> (g(Suc n) x)) \<and>
               (\<forall>n. g n \<in> borel_measurable lebesgue) \<and> (\<forall>n. finite(range (g n))) \<and>
               (\<forall>x. (\<lambda>n. g n x) \<longlonglongrightarrow> f x))"
          (is "?lhs = ?rhs")
 proof
   assume f: ?lhs
   have leb_f: "{x. a \<le> f x \<and> f x < b} \<in> sets lebesgue" for a b
   proof -
     have "{x. a \<le> f x \<and> f x < b} = {x. f x < b} - {x. f x < a}"
       by auto
     also have "\<dots> \<in> sets lebesgue"
       using borel_measurable_vimage_halfspace_component_lt [of f UNIV] f by auto
     finally show ?thesis .
   qed
   have "g n x \<le> f x"
         if inc_g: "\<And>n x. 0 \<le> g n x \<and> g n x \<le> g (Suc n) x"
            and meas_g: "\<And>n. g n \<in> borel_measurable lebesgue"
            and fin: "\<And>n. finite(range (g n))" and lim: "\<And>x. (\<lambda>n. g n x) \<longlonglongrightarrow> f x" for g n x
   proof -
     have "\<exists>r>0. \<forall>N. \<exists>n\<ge>N. dist (g n x) (f x) \<ge> r" if "g n x > f x"
     proof -
       have g: "g n x \<le> g (N + n) x" for N
         by (rule transitive_stepwise_le) (use inc_g in auto)
       have "\<exists>na\<ge>N. g n x - f x \<le> dist (g na x) (f x)" for N
         apply (rule_tac x="N+n" in exI)
         using g [of N] by (auto simp: dist_norm)
       with that show ?thesis
         using diff_gt_0_iff_gt by blast
     qed
     with lim show ?thesis
       apply (auto simp: lim_sequentially)
       by (meson less_le_not_le not_le_imp_less)
   qed
   moreover
   let ?\<Omega> = "\<lambda>n k. indicator {y. k/2^n \<le> f y \<and> f y < (k+1)/2^n}"
   let ?g = "\<lambda>n x. (\<Sum>k::real | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n). k/2^n * ?\<Omega> n k x)"
   have "\<exists>g. (\<forall>n x. 0 \<le> g n x \<and> g n x \<le> (g(Suc n) x)) \<and>
              (\<forall>n. g n \<in> borel_measurable lebesgue) \<and> (\<forall>n. finite(range (g n))) \<and>(\<forall>x. (\<lambda>n. g n x) \<longlonglongrightarrow> f x)"
   proof (intro exI allI conjI)
     show "0 \<le> ?g n x" for n x
     proof (clarify intro!: ordered_comm_monoid_add_class.sum_nonneg)
       fix k::real
       assume "k \<in> \<int>" and k: "\<bar>k\<bar> \<le> 2 ^ (2*n)"
       show "0 \<le> k/2^n * ?\<Omega> n k x"
         using f \<open>k \<in> \<int>\<close> apply (auto simp: indicator_def field_split_simps Ints_def)
         apply (drule spec [where x=x])
         using zero_le_power [of "2::real" n] mult_nonneg_nonneg [of "f x" "2^n"]
         by linarith
     qed
     show "?g n x \<le> ?g (Suc n) x" for n x
     proof -
       have "?g n x =
             (\<Sum>k | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n).
               k/2^n * (indicator {y. k/2^n \<le> f y \<and> f y < (k+1/2)/2^n} x +
               indicator {y. (k+1/2)/2^n \<le> f y \<and> f y < (k+1)/2^n} x))"
         by (rule sum.cong [OF refl]) (simp add: indicator_def field_split_simps)
       also have "\<dots> = (\<Sum>k | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n). k/2^n * indicator {y. k/2^n \<le> f y \<and> f y < (k+1/2)/2^n} x) +
                        (\<Sum>k | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n). k/2^n * indicator {y. (k+1/2)/2^n \<le> f y \<and> f y < (k+1)/2^n} x)"
         by (simp add:  comm_monoid_add_class.sum.distrib algebra_simps)
       also have "\<dots> = (\<Sum>k | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n). (2 * k)/2 ^ Suc n * indicator {y. (2 * k)/2 ^ Suc n \<le> f y \<and> f y < (2 * k+1)/2 ^ Suc n} x) +
                        (\<Sum>k | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n). (2 * k)/2 ^ Suc n * indicator {y. (2 * k+1)/2 ^ Suc n \<le> f y \<and> f y < ((2 * k+1) + 1)/2 ^ Suc n} x)"
         by (force simp: field_simps indicator_def intro: sum.cong)
       also have "\<dots> \<le> (\<Sum>k | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2 * Suc n). k/2 ^ Suc n * (indicator {y. k/2 ^ Suc n \<le> f y \<and> f y < (k+1)/2 ^ Suc n} x))"
                 (is "?a + _ \<le> ?b")
       proof -
         have *: "\<lbrakk>sum f I \<le> sum h I; a + sum h I \<le> b\<rbrakk> \<Longrightarrow> a + sum f I \<le> b" for I a b f and h :: "real\<Rightarrow>real"
           by linarith
         let ?h = "\<lambda>k. (2*k+1)/2 ^ Suc n *
                       (indicator {y. (2 * k+1)/2 ^ Suc n \<le> f y \<and> f y < ((2*k+1) + 1)/2 ^ Suc n} x)"
         show ?thesis
         proof (rule *)
           show "(\<Sum>k | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n).
                   2 * k/2 ^ Suc n * indicator {y. (2 * k+1)/2 ^ Suc n \<le> f y \<and> f y < (2 * k+1 + 1)/2 ^ Suc n} x)
                 \<le> sum ?h {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)}"
             by (rule sum_mono) (simp add: indicator_def field_split_simps)
         next
           have \<alpha>: "?a = (\<Sum>k \<in> (*)2 ` {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)}.
                          k/2 ^ Suc n * indicator {y. k/2 ^ Suc n \<le> f y \<and> f y < (k+1)/2 ^ Suc n} x)"
             by (auto simp: inj_on_def field_simps comm_monoid_add_class.sum.reindex)
           have \<beta>: "sum ?h {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)}
                    = (\<Sum>k \<in> (\<lambda>x. 2*x + 1) ` {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)}.
                       k/2 ^ Suc n * indicator {y. k/2 ^ Suc n \<le> f y \<and> f y < (k+1)/2 ^ Suc n} x)"
             by (auto simp: inj_on_def field_simps comm_monoid_add_class.sum.reindex)
           have 0: "(*) 2 ` {k \<in> \<int>. P k} \<inter> (\<lambda>x. 2 * x + 1) ` {k \<in> \<int>. P k} = {}" for P :: "real \<Rightarrow> bool"
           proof -
             have "2 * i \<noteq> 2 * j + 1" for i j :: int by arith
             thus ?thesis
               unfolding Ints_def by auto (use of_int_eq_iff in fastforce)
           qed
           have "?a + sum ?h {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)}
                 = (\<Sum>k \<in> (*)2 ` {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)} \<union> (\<lambda>x. 2*x + 1) ` {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)}.
                   k/2 ^ Suc n * indicator {y. k/2 ^ Suc n \<le> f y \<and> f y < (k+1)/2 ^ Suc n} x)"
             unfolding \<alpha> \<beta>
             using finite_abs_int_segment [of "2 ^ (2*n)"]
             by (subst sum_Un) (auto simp: 0)
           also have "\<dots> \<le> ?b"
           proof (rule sum_mono2)
             show "finite {k::real. k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2 * Suc n)}"
               by (rule finite_abs_int_segment)
             show "(*) 2 ` {k::real. k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2^(2*n)} \<union> (\<lambda>x. 2*x + 1) ` {k \<in> \<int>. \<bar>k\<bar> \<le> 2^(2*n)} \<subseteq> {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2 * Suc n)}"
               apply auto
               using one_le_power [of "2::real" "2*n"]  by linarith
             have *: "\<lbrakk>x \<in> (S \<union> T) - U; \<And>x. x \<in> S \<Longrightarrow> x \<in> U; \<And>x. x \<in> T \<Longrightarrow> x \<in> U\<rbrakk> \<Longrightarrow> P x" for S T U P
               by blast
             have "0 \<le> b" if "b \<in> \<int>" "f x * (2 * 2^n) < b + 1" for b
             proof -
               have "0 \<le> f x * (2 * 2^n)"
                 by (simp add: f)
               also have "\<dots> < b+1"
                 by (simp add: that)
               finally show "0 \<le> b"
                 using \<open>b \<in> \<int>\<close> by (auto simp: elim!: Ints_cases)
             qed
             then show "0 \<le> b/2 ^ Suc n * indicator {y. b/2 ^ Suc n \<le> f y \<and> f y < (b + 1)/2 ^ Suc n} x"
                   if "b \<in> {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2 * Suc n)} -
                           ((*) 2 ` {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)} \<union> (\<lambda>x. 2*x + 1) ` {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)})" for b
               using that by (simp add: indicator_def divide_simps)
           qed
           finally show "?a + sum ?h {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)} \<le> ?b" .
         qed
       qed
       finally show ?thesis .
     qed
     show "?g n \<in> borel_measurable lebesgue" for n
       apply (intro borel_measurable_indicator borel_measurable_times borel_measurable_sum)
       using leb_f sets_restrict_UNIV by auto
     show "finite (range (?g n))" for n
     proof -
       have "(\<Sum>k | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n). k/2^n * ?\<Omega> n k x)
               \<in> (\<lambda>k. k/2^n) ` {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)}" for x
       proof (cases "\<exists>k. k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n) \<and> k/2^n \<le> f x \<and> f x < (k+1)/2^n")
         case True
         then show ?thesis
           by (blast intro: indicator_sum_eq)
       next
         case False
         then have "(\<Sum>k | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n). k/2^n * ?\<Omega> n k x) = 0"
           by auto
         then show ?thesis by force
       qed
       then have "range (?g n) \<subseteq> ((\<lambda>k. (k/2^n)) ` {k. k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n)})"
         by auto
       moreover have "finite ((\<lambda>k::real. (k/2^n)) ` {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)})"
         by (intro finite_imageI finite_abs_int_segment)
       ultimately show ?thesis
         by (rule finite_subset)
     qed
     show "(\<lambda>n. ?g n x) \<longlonglongrightarrow> f x" for x
     proof (clarsimp simp add: lim_sequentially)
       fix e::real
       assume "e > 0"
       obtain N1 where N1: "2 ^ N1 > abs(f x)"
         using real_arch_pow by fastforce
       obtain N2 where N2: "(1/2) ^ N2 < e"
         using real_arch_pow_inv \<open>e > 0\<close> by fastforce
       have "dist (\<Sum>k | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n). k/2^n * ?\<Omega> n k x) (f x) < e" if "N1 + N2 \<le> n" for n
       proof -
         let ?m = "real_of_int \<lfloor>2^n * f x\<rfloor>"
         have "\<bar>?m\<bar> \<le> 2^n * 2^N1"
           using N1 apply (simp add: f)
           by (meson floor_mono le_floor_iff less_le_not_le mult_le_cancel_left_pos zero_less_numeral zero_less_power)
         also have "\<dots> \<le> 2 ^ (2*n)"
           by (metis that add_leD1 add_le_cancel_left mult.commute mult_2_right one_less_numeral_iff
                     power_add power_increasing_iff semiring_norm(76))
         finally have m_le: "\<bar>?m\<bar> \<le> 2 ^ (2*n)" .
         have "?m/2^n \<le> f x" "f x < (?m + 1)/2^n"
           by (auto simp: mult.commute pos_divide_le_eq mult_imp_less_div_pos)
         then have eq: "dist (\<Sum>k | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n). k/2^n * ?\<Omega> n k x) (f x)
                      = dist (?m/2^n) (f x)"
           by (subst indicator_sum_eq [of ?m]) (auto simp: m_le)
         have "\<bar>2^n\<bar> * \<bar>?m/2^n - f x\<bar> = \<bar>2^n * (?m/2^n - f x)\<bar>"
           by (simp add: abs_mult)
         also have "\<dots> < 2 ^ N2 * e"
           using N2 by (simp add: divide_simps mult.commute) linarith
         also have "\<dots> \<le> \<bar>2^n\<bar> * e"
           using that \<open>e > 0\<close> by auto
         finally have "dist (?m/2^n) (f x) < e"
           by (simp add: dist_norm)
         then show ?thesis
           using eq by linarith
       qed
       then show "\<exists>no. \<forall>n\<ge>no. dist (\<Sum>k | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n). k * ?\<Omega> n k x/2^n) (f x) < e"
         by force
     qed
   qed
   ultimately show ?rhs
     by metis
 next
   assume RHS: ?rhs
   with borel_measurable_simple_function_limit [of f UNIV, unfolded lebesgue_on_UNIV_eq]
   show ?lhs
     by (blast intro: order_trans)
 qed
 
 subsection\<open>Borel measurable Jacobian determinant\<close>
 
 lemma lemma_partial_derivatives0:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "linear f" and lim0: "((\<lambda>x. f x /\<^sub>R norm x) \<longlongrightarrow> 0) (at 0 within S)"
     and lb: "\<And>v. v \<noteq> 0 \<Longrightarrow> (\<exists>k>0. \<forall>e>0. \<exists>x. x \<in> S - {0} \<and> norm x < e \<and> k * norm x \<le> \<bar>v \<bullet> x\<bar>)"
   shows "f x = 0"
 proof -
   interpret linear f by fact
   have "dim {x. f x = 0} \<le> DIM('a)"
     by (rule dim_subset_UNIV)
   moreover have False if less: "dim {x. f x = 0} < DIM('a)"
   proof -
     obtain d where "d \<noteq> 0" and d: "\<And>y. f y = 0 \<Longrightarrow> d \<bullet> y = 0"
       using orthogonal_to_subspace_exists [OF less] orthogonal_def
       by (metis (mono_tags, lifting) mem_Collect_eq span_base)
     then obtain k where "k > 0"
       and k: "\<And>e. e > 0 \<Longrightarrow> \<exists>y. y \<in> S - {0} \<and> norm y < e \<and> k * norm y \<le> \<bar>d \<bullet> y\<bar>"
       using lb by blast
     have "\<exists>h. \<forall>n. ((h n \<in> S \<and> h n \<noteq> 0 \<and> k * norm (h n) \<le> \<bar>d \<bullet> h n\<bar>) \<and> norm (h n) < 1 / real (Suc n)) \<and>
                norm (h (Suc n)) < norm (h n)"
     proof (rule dependent_nat_choice)
       show "\<exists>y. (y \<in> S \<and> y \<noteq> 0 \<and> k * norm y \<le> \<bar>d \<bullet> y\<bar>) \<and> norm y < 1 / real (Suc 0)"
         by simp (metis DiffE insertCI k not_less not_one_le_zero)
     qed (use k [of "min (norm x) (1/(Suc n + 1))" for x n] in auto)
     then obtain \<alpha> where \<alpha>: "\<And>n. \<alpha> n \<in> S - {0}" and kd: "\<And>n. k * norm(\<alpha> n) \<le> \<bar>d \<bullet> \<alpha> n\<bar>"
          and norm_lt: "\<And>n. norm(\<alpha> n) < 1/(Suc n)"
       by force
     let ?\<beta> = "\<lambda>n. \<alpha> n /\<^sub>R norm (\<alpha> n)"
     have com: "\<And>g. (\<forall>n. g n \<in> sphere (0::'a) 1)
               \<Longrightarrow> \<exists>l \<in> sphere 0 1. \<exists>\<rho>::nat\<Rightarrow>nat. strict_mono \<rho> \<and> (g \<circ> \<rho>) \<longlonglongrightarrow> l"
       using compact_sphere compact_def by metis
     moreover have "\<forall>n. ?\<beta> n \<in> sphere 0 1"
       using \<alpha> by auto
     ultimately obtain l::'a and \<rho>::"nat\<Rightarrow>nat"
        where l: "l \<in> sphere 0 1" and "strict_mono \<rho>" and to_l: "(?\<beta> \<circ> \<rho>) \<longlonglongrightarrow> l"
       by meson
     moreover have "continuous (at l) (\<lambda>x. (\<bar>d \<bullet> x\<bar> - k))"
       by (intro continuous_intros)
     ultimately have lim_dl: "((\<lambda>x. (\<bar>d \<bullet> x\<bar> - k)) \<circ> (?\<beta> \<circ> \<rho>)) \<longlonglongrightarrow> (\<bar>d \<bullet> l\<bar> - k)"
       by (meson continuous_imp_tendsto)
     have "\<forall>\<^sub>F i in sequentially. 0 \<le> ((\<lambda>x. \<bar>d \<bullet> x\<bar> - k) \<circ> ((\<lambda>n. \<alpha> n /\<^sub>R norm (\<alpha> n)) \<circ> \<rho>)) i"
       using \<alpha> kd by (auto simp: field_split_simps)
     then have "k \<le> \<bar>d \<bullet> l\<bar>"
       using tendsto_lowerbound [OF lim_dl, of 0] by auto
     moreover have "d \<bullet> l = 0"
     proof (rule d)
       show "f l = 0"
       proof (rule LIMSEQ_unique [of "f \<circ> ?\<beta> \<circ> \<rho>"])
         have "isCont f l"
           using \<open>linear f\<close> linear_continuous_at linear_conv_bounded_linear by blast
         then show "(f \<circ> (\<lambda>n. \<alpha> n /\<^sub>R norm (\<alpha> n)) \<circ> \<rho>) \<longlonglongrightarrow> f l"
           unfolding comp_assoc
           using to_l continuous_imp_tendsto by blast
         have "\<alpha> \<longlonglongrightarrow> 0"
           using norm_lt LIMSEQ_norm_0 by metis
         with \<open>strict_mono \<rho>\<close> have "(\<alpha> \<circ> \<rho>) \<longlonglongrightarrow> 0"
           by (metis LIMSEQ_subseq_LIMSEQ)
         with lim0 \<alpha> have "((\<lambda>x. f x /\<^sub>R norm x) \<circ> (\<alpha> \<circ> \<rho>)) \<longlonglongrightarrow> 0"
           by (force simp: tendsto_at_iff_sequentially)
         then show "(f \<circ> (\<lambda>n. \<alpha> n /\<^sub>R norm (\<alpha> n)) \<circ> \<rho>) \<longlonglongrightarrow> 0"
           by (simp add: o_def scale)
       qed
     qed
     ultimately show False
       using \<open>k > 0\<close> by auto
   qed
   ultimately have dim: "dim {x. f x = 0} = DIM('a)"
     by force
   then show ?thesis
     using dim_eq_full
     by (metis (mono_tags, lifting) eq_0_on_span eucl.span_Basis linear_axioms linear_eq_stdbasis
         mem_Collect_eq module_hom_zero span_base span_raw_def)
 qed
 
 lemma lemma_partial_derivatives:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "linear f" and lim: "((\<lambda>x. f (x - a) /\<^sub>R norm (x - a)) \<longlongrightarrow> 0) (at a within S)"
     and lb: "\<And>v. v \<noteq> 0 \<Longrightarrow> (\<exists>k>0.  \<forall>e>0. \<exists>x \<in> S - {a}. norm(a - x) < e \<and> k * norm(a - x) \<le> \<bar>v \<bullet> (x - a)\<bar>)"
   shows "f x = 0"
 proof -
   have "((\<lambda>x. f x /\<^sub>R norm x) \<longlongrightarrow> 0) (at 0 within (\<lambda>x. x-a) ` S)"
     using lim by (simp add: Lim_within dist_norm)
   then show ?thesis
   proof (rule lemma_partial_derivatives0 [OF \<open>linear f\<close>])
     fix v :: "'a"
     assume v: "v \<noteq> 0"
     show "\<exists>k>0. \<forall>e>0. \<exists>x. x \<in> (\<lambda>x. x - a) ` S - {0} \<and> norm x < e \<and> k * norm x \<le> \<bar>v \<bullet> x\<bar>"
       using lb [OF v] by (force simp:  norm_minus_commute)
   qed
 qed
 
 
 proposition borel_measurable_partial_derivatives:
   fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n"
   assumes S: "S \<in> sets lebesgue"
     and f: "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
   shows "(\<lambda>x. (matrix(f' x)$m$n)) \<in> borel_measurable (lebesgue_on S)"
 proof -
   have contf: "continuous_on S f"
     using continuous_on_eq_continuous_within f has_derivative_continuous by blast
   have "{x \<in> S.  (matrix (f' x)$m$n) \<le> b} \<in> sets lebesgue" for b
   proof (rule sets_negligible_symdiff)
     let ?T = "{x \<in> S. \<forall>e>0. \<exists>d>0. \<exists>A. A$m$n < b \<and> (\<forall>i j. A$i$j \<in> \<rat>) \<and>
                        (\<forall>y \<in> S. norm(y - x) < d \<longrightarrow> norm(f y - f x - A *v (y - x)) \<le> e * norm(y - x))}"
     let ?U = "S \<inter>
               (\<Inter>e \<in> {e \<in> \<rat>. e > 0}.
                 \<Union>A \<in> {A. A$m$n < b \<and> (\<forall>i j. A$i$j \<in> \<rat>)}.
                   \<Union>d \<in> {d \<in> \<rat>. 0 < d}.
                      S \<inter> (\<Inter>y \<in> S. {x \<in> S. norm(y - x) < d \<longrightarrow> norm(f y - f x - A *v (y - x)) \<le> e * norm(y - x)}))"
     have "?T = ?U"
     proof (intro set_eqI iffI)
       fix x
       assume xT: "x \<in> ?T"
       then show "x \<in> ?U"
       proof (clarsimp simp add:)
         fix q :: real
         assume "q \<in> \<rat>" "q > 0"
         then obtain d A where "d > 0" and A: "A $ m $ n < b" "\<And>i j. A $ i $ j \<in> \<rat>"
           "\<And>y. \<lbrakk>y\<in>S;  norm (y - x) < d\<rbrakk> \<Longrightarrow> norm (f y - f x - A *v (y - x)) \<le> q * norm (y - x)"
           using xT by auto
         then obtain \<delta> where "d > \<delta>" "\<delta> > 0" "\<delta> \<in> \<rat>"
           using Rats_dense_in_real by blast
         with A show "\<exists>A. A $ m $ n < b \<and> (\<forall>i j. A $ i $ j \<in> \<rat>) \<and>
                          (\<exists>s. s \<in> \<rat> \<and> 0 < s \<and> (\<forall>y\<in>S. norm (y - x) < s \<longrightarrow> norm (f y - f x - A *v (y - x)) \<le> q * norm (y - x)))"
           by force
       qed
     next
       fix x
       assume xU: "x \<in> ?U"
       then show "x \<in> ?T"
       proof clarsimp
         fix e :: "real"
         assume "e > 0"
         then obtain \<epsilon> where \<epsilon>: "e > \<epsilon>" "\<epsilon> > 0" "\<epsilon> \<in> \<rat>"
           using Rats_dense_in_real by blast
         with xU obtain A r where "x \<in> S" and Ar: "A $ m $ n < b" "\<forall>i j. A $ i $ j \<in> \<rat>" "r \<in> \<rat>" "r > 0"
           and "\<forall>y\<in>S. norm (y - x) < r \<longrightarrow> norm (f y - f x - A *v (y - x)) \<le> \<epsilon> * norm (y - x)"
           by (auto simp: split: if_split_asm)
         then have "\<forall>y\<in>S. norm (y - x) < r \<longrightarrow> norm (f y - f x - A *v (y - x)) \<le> e * norm (y - x)"
           by (meson \<open>e > \<epsilon>\<close> less_eq_real_def mult_right_mono norm_ge_zero order_trans)
         then show "\<exists>d>0. \<exists>A. A $ m $ n < b \<and> (\<forall>i j. A $ i $ j \<in> \<rat>) \<and> (\<forall>y\<in>S. norm (y - x) < d \<longrightarrow> norm (f y - f x - A *v (y - x)) \<le> e * norm (y - x))"
           using \<open>x \<in> S\<close> Ar by blast
       qed
     qed
     moreover have "?U \<in> sets lebesgue"
     proof -
       have coQ: "countable {e \<in> \<rat>. 0 < e}"
         using countable_Collect countable_rat by blast
       have ne: "{e \<in> \<rat>. (0::real) < e} \<noteq> {}"
         using zero_less_one Rats_1 by blast
       have coA: "countable {A. A $ m $ n < b \<and> (\<forall>i j. A $ i $ j \<in> \<rat>)}"
       proof (rule countable_subset)
         show "countable {A. \<forall>i j. A $ i $ j \<in> \<rat>}"
           using countable_vector [OF countable_vector, of "\<lambda>i j. \<rat>"] by (simp add: countable_rat)
       qed blast
       have *: "\<lbrakk>U \<noteq> {} \<Longrightarrow> closedin (top_of_set S) (S \<inter> \<Inter> U)\<rbrakk>
                \<Longrightarrow> closedin (top_of_set S) (S \<inter> \<Inter> U)" for U
         by fastforce
       have eq: "{x::(real,'m)vec. P x \<and> (Q x \<longrightarrow> R x)} = {x. P x \<and> \<not> Q x} \<union> {x. P x \<and> R x}" for P Q R
         by auto
       have sets: "S \<inter> (\<Inter>y\<in>S. {x \<in> S. norm (y - x) < d \<longrightarrow> norm (f y - f x - A *v (y - x)) \<le> e * norm (y - x)})
                   \<in> sets lebesgue" for e A d
       proof -
         have clo: "closedin (top_of_set S)
                      {x \<in> S. norm (y - x) < d \<longrightarrow> norm (f y - f x - A *v (y - x)) \<le> e * norm (y - x)}"
           for y
         proof -
           have cont1: "continuous_on S (\<lambda>x. norm (y - x))"
           and  cont2: "continuous_on S (\<lambda>x. e * norm (y - x) - norm (f y - f x - (A *v y - A *v x)))"
             by (force intro: contf continuous_intros)+
           have clo1: "closedin (top_of_set S) {x \<in> S. d \<le> norm (y - x)}"
             using continuous_closedin_preimage [OF cont1, of "{d..}"] by (simp add: vimage_def Int_def)
           have clo2: "closedin (top_of_set S)
                        {x \<in> S. norm (f y - f x - (A *v y - A *v x)) \<le> e * norm (y - x)}"
             using continuous_closedin_preimage [OF cont2, of "{0..}"] by (simp add: vimage_def Int_def)
           show ?thesis
             by (auto simp: eq not_less matrix_vector_mult_diff_distrib intro: clo1 clo2)
         qed
         show ?thesis
           by (rule lebesgue_closedin [of S]) (force intro: * S clo)+
       qed
       show ?thesis
         by (intro sets sets.Int S sets.countable_UN'' sets.countable_INT'' coQ coA) auto
     qed
     ultimately show "?T \<in> sets lebesgue"
       by simp
     let ?M = "(?T - {x \<in> S. matrix (f' x) $ m $ n \<le> b} \<union> ({x \<in> S. matrix (f' x) $ m $ n \<le> b} - ?T))"
     let ?\<Theta> = "\<lambda>x v. \<forall>\<xi>>0. \<exists>e>0. \<forall>y \<in> S-{x}. norm (x - y) < e \<longrightarrow> \<bar>v \<bullet> (y - x)\<bar> < \<xi> * norm (x - y)"
     have nN: "negligible {x \<in> S. \<exists>v\<noteq>0. ?\<Theta> x v}"
       unfolding negligible_eq_zero_density
     proof clarsimp
       fix x v and r e :: "real"
       assume "x \<in> S" "v \<noteq> 0" "r > 0" "e > 0"
       and Theta [rule_format]: "?\<Theta> x v"
       moreover have "(norm v * e / 2) / CARD('m) ^ CARD('m) > 0"
         by (simp add: \<open>v \<noteq> 0\<close> \<open>e > 0\<close>)
       ultimately obtain d where "d > 0"
          and dless: "\<And>y. \<lbrakk>y \<in> S - {x}; norm (x - y) < d\<rbrakk> \<Longrightarrow>
                         \<bar>v \<bullet> (y - x)\<bar> < ((norm v * e / 2) / CARD('m) ^ CARD('m)) * norm (x - y)"
         by metis
       let ?W = "ball x (min d r) \<inter> {y. \<bar>v \<bullet> (y - x)\<bar> < (norm v * e/2 * min d r) / CARD('m) ^ CARD('m)}"
       have "open {x. \<bar>v \<bullet> (x - a)\<bar> < b}" for a b
         by (intro open_Collect_less continuous_intros)
       show "\<exists>d>0. d \<le> r \<and>
             (\<exists>U. {x' \<in> S. \<exists>v\<noteq>0. ?\<Theta> x' v} \<inter> ball x d \<subseteq> U \<and>
                  U \<in> lmeasurable \<and> measure lebesgue U < e * content (ball x d))"
       proof (intro exI conjI)
         show "0 < min d r" "min d r \<le> r"
           using \<open>r > 0\<close> \<open>d > 0\<close> by auto
         show "{x' \<in> S. \<exists>v. v \<noteq> 0 \<and> (\<forall>\<xi>>0. \<exists>e>0. \<forall>z\<in>S - {x'}. norm (x' - z) < e \<longrightarrow> \<bar>v \<bullet> (z - x')\<bar> < \<xi> * norm (x' - z))} \<inter> ball x (min d r) \<subseteq> ?W"
           proof (clarsimp simp: dist_norm norm_minus_commute)
             fix y w
             assume "y \<in> S" "w \<noteq> 0"
               and less [rule_format]:
                     "\<forall>\<xi>>0. \<exists>e>0. \<forall>z\<in>S - {y}. norm (y - z) < e \<longrightarrow> \<bar>w \<bullet> (z - y)\<bar> < \<xi> * norm (y - z)"
               and d: "norm (y - x) < d" and r: "norm (y - x) < r"
             show "\<bar>v \<bullet> (y - x)\<bar> < norm v * e * min d r / (2 * real CARD('m) ^ CARD('m))"
             proof (cases "y = x")
               case True
               with \<open>r > 0\<close> \<open>d > 0\<close> \<open>e > 0\<close> \<open>v \<noteq> 0\<close> show ?thesis
                 by simp
             next
               case False
               have "\<bar>v \<bullet> (y - x)\<bar> < norm v * e / 2 / real (CARD('m) ^ CARD('m)) * norm (x - y)"
                 apply (rule dless)
                 using False \<open>y \<in> S\<close> d by (auto simp: norm_minus_commute)
               also have "\<dots> \<le> norm v * e * min d r / (2 * real CARD('m) ^ CARD('m))"
                 using d r \<open>e > 0\<close> by (simp add: field_simps norm_minus_commute mult_left_mono)
               finally show ?thesis .
             qed
           qed
           show "?W \<in> lmeasurable"
             by (simp add: fmeasurable_Int_fmeasurable borel_open)
           obtain k::'m where True
             by metis
           obtain T where T: "orthogonal_transformation T" and v: "v = T(norm v *\<^sub>R axis k (1::real))"
             using rotation_rightward_line by metis
           define b where "b \<equiv> norm v"
           have "b > 0"
             using \<open>v \<noteq> 0\<close> by (auto simp: b_def)
           obtain eqb: "inv T v = b *\<^sub>R axis k (1::real)" and "inj T" "bij T" and invT: "orthogonal_transformation (inv T)"
             by (metis UNIV_I b_def  T v bij_betw_inv_into_left orthogonal_transformation_inj orthogonal_transformation_bij orthogonal_transformation_inv)
           let ?v = "\<chi> i. min d r / CARD('m)"
           let ?v' = "\<chi> i. if i = k then (e/2 * min d r) / CARD('m) ^ CARD('m) else min d r"
           let ?x' = "inv T x"
           let ?W' = "(ball ?x' (min d r) \<inter> {y. \<bar>(y - ?x')$k\<bar> < e * min d r / (2 * CARD('m) ^ CARD('m))})"
           have abs: "x - e \<le> y \<and> y \<le> x + e \<longleftrightarrow> abs(y - x) \<le> e" for x y e::real
             by auto
           have "?W = T ` ?W'"
           proof -
             have 1: "T ` (ball (inv T x) (min d r)) = ball x (min d r)"
               by (simp add: T image_orthogonal_transformation_ball orthogonal_transformation_surj surj_f_inv_f)
             have 2: "{y. \<bar>v \<bullet> (y - x)\<bar> < b * e * min d r / (2 * real CARD('m) ^ CARD('m))} =
                       T ` {y. \<bar>y $ k - ?x' $ k\<bar> < e * min d r / (2 * real CARD('m) ^ CARD('m))}"
             proof -
               have *: "\<bar>T (b *\<^sub>R axis k 1) \<bullet> (y - x)\<bar> = b * \<bar>inv T y $ k - ?x' $ k\<bar>" for y
               proof -
                 have "\<bar>T (b *\<^sub>R axis k 1) \<bullet> (y - x)\<bar> = \<bar>(b *\<^sub>R axis k 1) \<bullet> inv T (y - x)\<bar>"
                   by (metis (no_types, hide_lams) b_def eqb invT orthogonal_transformation_def v)
                 also have "\<dots> = b * \<bar>(axis k 1) \<bullet> inv T (y - x)\<bar>"
                   using \<open>b > 0\<close> by (simp add: abs_mult)
                 also have "\<dots> = b * \<bar>inv T y $ k - ?x' $ k\<bar>"
                   using orthogonal_transformation_linear [OF invT]
                   by (simp add: inner_axis' linear_diff)
                 finally show ?thesis
                   by simp
               qed
               show ?thesis
                 using v b_def [symmetric]
                 using \<open>b > 0\<close> by (simp add: * bij_image_Collect_eq [OF \<open>bij T\<close>] mult_less_cancel_left_pos times_divide_eq_right [symmetric] del: times_divide_eq_right)
             qed
             show ?thesis
               using \<open>b > 0\<close> by (simp add: image_Int \<open>inj T\<close> 1 2 b_def [symmetric])
           qed
           moreover have "?W' \<in> lmeasurable"
             by (auto intro: fmeasurable_Int_fmeasurable)
           ultimately have "measure lebesgue ?W = measure lebesgue ?W'"
             by (metis measure_orthogonal_image T)
           also have "\<dots> \<le> measure lebesgue (cbox (?x' - ?v') (?x' + ?v'))"
           proof (rule measure_mono_fmeasurable)
             show "?W' \<subseteq> cbox (?x' - ?v') (?x' + ?v')"
               apply (clarsimp simp add: mem_box_cart abs dist_norm norm_minus_commute simp del: min_less_iff_conj min.bounded_iff)
               by (metis component_le_norm_cart less_eq_real_def le_less_trans vector_minus_component)
           qed auto
           also have "\<dots> \<le> e/2 * measure lebesgue (cbox (?x' - ?v) (?x' + ?v))"
           proof -
             have "cbox (?x' - ?v) (?x' + ?v) \<noteq> {}"
               using \<open>r > 0\<close> \<open>d > 0\<close> by (auto simp: interval_eq_empty_cart divide_less_0_iff)
             with \<open>r > 0\<close> \<open>d > 0\<close> \<open>e > 0\<close> show ?thesis
               apply (simp add: content_cbox_if_cart mem_box_cart)
               apply (auto simp: prod_nonneg)
               apply (simp add: abs if_distrib prod.delta_remove field_simps power_diff split: if_split_asm)
               done
           qed
           also have "\<dots> \<le> e/2 * measure lebesgue (cball ?x' (min d r))"
           proof (rule mult_left_mono [OF measure_mono_fmeasurable])
             have *: "norm (?x' - y) \<le> min d r"
               if y: "\<And>i. \<bar>?x' $ i - y $ i\<bar> \<le> min d r / real CARD('m)" for y
             proof -
               have "norm (?x' - y) \<le> (\<Sum>i\<in>UNIV. \<bar>(?x' - y) $ i\<bar>)"
                 by (rule norm_le_l1_cart)
               also have "\<dots> \<le> real CARD('m) * (min d r / real CARD('m))"
                 by (rule sum_bounded_above) (use y in auto)
               finally show ?thesis
                 by simp
             qed
             show "cbox (?x' - ?v) (?x' + ?v) \<subseteq> cball ?x' (min d r)"
               apply (clarsimp simp only: mem_box_cart dist_norm mem_cball intro!: *)
               by (simp add: abs_diff_le_iff abs_minus_commute)
           qed (use \<open>e > 0\<close> in auto)
           also have "\<dots> < e * content (cball ?x' (min d r))"
             using \<open>r > 0\<close> \<open>d > 0\<close> \<open>e > 0\<close> by (auto intro: content_cball_pos)
           also have "\<dots> = e * content (ball x (min d r))"
             using \<open>r > 0\<close> \<open>d > 0\<close> content_ball_conv_unit_ball[of "min d r" "inv T x"]
                   content_ball_conv_unit_ball[of "min d r" x]
             by (simp add: content_cball_conv_ball)
           finally show "measure lebesgue ?W < e * content (ball x (min d r))" .
       qed
     qed
     have *: "(\<And>x. (x \<notin> S) \<Longrightarrow> (x \<in> T \<longleftrightarrow> x \<in> U)) \<Longrightarrow> (T - U) \<union> (U - T) \<subseteq> S" for S T U :: "(real,'m) vec set"
       by blast
     have MN: "?M \<subseteq> {x \<in> S. \<exists>v\<noteq>0. ?\<Theta> x v}"
     proof (rule *)
       fix x
       assume x: "x \<notin> {x \<in> S. \<exists>v\<noteq>0. ?\<Theta> x v}"
       show "(x \<in> ?T) \<longleftrightarrow> (x \<in> {x \<in> S. matrix (f' x) $ m $ n \<le> b})"
       proof (cases "x \<in> S")
         case True
         then have x: "\<not> ?\<Theta> x v" if "v \<noteq> 0" for v
           using x that by force
         show ?thesis
         proof (rule iffI; clarsimp)
           assume b: "\<forall>e>0. \<exists>d>0. \<exists>A. A $ m $ n < b \<and> (\<forall>i j. A $ i $ j \<in> \<rat>) \<and>
                                     (\<forall>y\<in>S. norm (y - x) < d \<longrightarrow> norm (f y - f x - A *v (y - x)) \<le> e * norm (y - x))"
                      (is "\<forall>e>0. \<exists>d>0. \<exists>A. ?\<Phi> e d A")
           then have "\<forall>k. \<exists>d>0. \<exists>A. ?\<Phi> (1 / Suc k) d A"
             by (metis (no_types, hide_lams) less_Suc_eq_0_disj of_nat_0_less_iff zero_less_divide_1_iff)
           then obtain \<delta> A where \<delta>: "\<And>k. \<delta> k > 0"
                            and Ab: "\<And>k. A k $ m $ n < b"
                            and A: "\<And>k y. \<lbrakk>y \<in> S; norm (y - x) < \<delta> k\<rbrakk> \<Longrightarrow>
                                           norm (f y - f x - A k *v (y - x)) \<le> 1/(Suc k) * norm (y - x)"
             by metis
           have "\<forall>i j. \<exists>a. (\<lambda>n. A n $ i $ j) \<longlonglongrightarrow> a"
           proof (intro allI)
             fix i j
             have vax: "(A n *v axis j 1) $ i = A n $ i $ j" for n
               by (metis cart_eq_inner_axis matrix_vector_mul_component)
             let ?CA = "{x. Cauchy (\<lambda>n. (A n) *v x)}"
             have "subspace ?CA"
               unfolding subspace_def convergent_eq_Cauchy [symmetric]
                 by (force simp: algebra_simps intro: tendsto_intros)
             then have CA_eq: "?CA = span ?CA"
               by (metis span_eq_iff)
             also have "\<dots> = UNIV"
             proof -
               have "dim ?CA \<le> CARD('m)"
                 using dim_subset_UNIV[of ?CA]
                 by auto
               moreover have "False" if less: "dim ?CA < CARD('m)"
               proof -
                 obtain d where "d \<noteq> 0" and d: "\<And>y. y \<in> span ?CA \<Longrightarrow> orthogonal d y"
                   using less by (force intro: orthogonal_to_subspace_exists [of ?CA])
                 with x [OF \<open>d \<noteq> 0\<close>] obtain \<xi> where "\<xi> > 0"
                   and \<xi>: "\<And>e. e > 0 \<Longrightarrow> \<exists>y \<in> S - {x}. norm (x - y) < e \<and> \<xi> * norm (x - y) \<le> \<bar>d \<bullet> (y - x)\<bar>"
                   by (fastforce simp: not_le Bex_def)
                 obtain \<gamma> z where \<gamma>Sx: "\<And>i. \<gamma> i \<in> S - {x}"
                            and \<gamma>le:   "\<And>i. \<xi> * norm(\<gamma> i - x) \<le> \<bar>d \<bullet> (\<gamma> i - x)\<bar>"
                            and \<gamma>x:    "\<gamma> \<longlonglongrightarrow> x"
                            and z:     "(\<lambda>n. (\<gamma> n - x) /\<^sub>R norm (\<gamma> n - x)) \<longlonglongrightarrow> z"
                 proof -
                   have "\<exists>\<gamma>. (\<forall>i. (\<gamma> i \<in> S - {x} \<and>
                                   \<xi> * norm(\<gamma> i - x) \<le> \<bar>d \<bullet> (\<gamma> i - x)\<bar> \<and> norm(\<gamma> i - x) < 1/Suc i) \<and>
                                  norm(\<gamma>(Suc i) - x) < norm(\<gamma> i - x))"
                   proof (rule dependent_nat_choice)
                     show "\<exists>y. y \<in> S - {x} \<and> \<xi> * norm (y - x) \<le> \<bar>d \<bullet> (y - x)\<bar> \<and> norm (y - x) < 1 / Suc 0"
                       using \<xi> [of 1] by (auto simp: dist_norm norm_minus_commute)
                   next
                     fix y i
                     assume "y \<in> S - {x} \<and> \<xi> * norm (y - x) \<le> \<bar>d \<bullet> (y - x)\<bar> \<and> norm (y - x) < 1/Suc i"
                     then have "min (norm(y - x)) (1/((Suc i) + 1)) > 0"
                       by auto
                     then obtain y' where "y' \<in> S - {x}" and y': "norm (x - y') < min (norm (y - x)) (1/((Suc i) + 1))"
                                          "\<xi> * norm (x - y') \<le> \<bar>d \<bullet> (y' - x)\<bar>"
                       using \<xi> by metis
                     with \<xi> show "\<exists>y'. (y' \<in> S - {x} \<and> \<xi> * norm (y' - x) \<le> \<bar>d \<bullet> (y' - x)\<bar> \<and>
                               norm (y' - x) < 1/(Suc (Suc i))) \<and> norm (y' - x) < norm (y - x)"
                       by (auto simp: dist_norm norm_minus_commute)
                   qed
                   then obtain \<gamma> where
                         \<gamma>Sx: "\<And>i. \<gamma> i \<in> S - {x}"
                         and \<gamma>le: "\<And>i. \<xi> * norm(\<gamma> i - x) \<le> \<bar>d \<bullet> (\<gamma> i - x)\<bar>"
                         and \<gamma>conv: "\<And>i. norm(\<gamma> i - x) < 1/(Suc i)"
                     by blast
                   let ?f = "\<lambda>i. (\<gamma> i - x) /\<^sub>R norm (\<gamma> i - x)"
                   have "?f i \<in> sphere 0 1" for i
                     using \<gamma>Sx by auto
                   then obtain l \<rho> where "l \<in> sphere 0 1" "strict_mono \<rho>" and l: "(?f \<circ> \<rho>) \<longlonglongrightarrow> l"
                     using compact_sphere [of "0::(real,'m) vec" 1]  unfolding compact_def by meson
                   show thesis
                   proof
                     show "(\<gamma> \<circ> \<rho>) i \<in> S - {x}" "\<xi> * norm ((\<gamma> \<circ> \<rho>) i - x) \<le> \<bar>d \<bullet> ((\<gamma> \<circ> \<rho>) i - x)\<bar>" for i
                       using \<gamma>Sx \<gamma>le by auto
                     have "\<gamma> \<longlonglongrightarrow> x"
                     proof (clarsimp simp add: LIMSEQ_def dist_norm)
                       fix r :: "real"
                       assume "r > 0"
                       with real_arch_invD obtain no where "no \<noteq> 0" "real no > 1/r"
                         by (metis divide_less_0_1_iff not_less_iff_gr_or_eq of_nat_0_eq_iff reals_Archimedean2)
                       with \<gamma>conv show "\<exists>no. \<forall>n\<ge>no. norm (\<gamma> n - x) < r"
                         by (metis \<open>r > 0\<close> add.commute divide_inverse inverse_inverse_eq inverse_less_imp_less less_trans mult.left_neutral nat_le_real_less of_nat_Suc)
                     qed
                     with \<open>strict_mono \<rho>\<close> show "(\<gamma> \<circ> \<rho>) \<longlonglongrightarrow> x"
                       by (metis LIMSEQ_subseq_LIMSEQ)
                     show "(\<lambda>n. ((\<gamma> \<circ> \<rho>) n - x) /\<^sub>R norm ((\<gamma> \<circ> \<rho>) n - x)) \<longlonglongrightarrow> l"
                       using l by (auto simp: o_def)
                   qed
                 qed
                 have "isCont (\<lambda>x. (\<bar>d \<bullet> x\<bar> - \<xi>)) z"
                   by (intro continuous_intros)
                 from isCont_tendsto_compose [OF this z]
                 have lim: "(\<lambda>y. \<bar>d \<bullet> ((\<gamma> y - x) /\<^sub>R norm (\<gamma> y - x))\<bar> - \<xi>) \<longlonglongrightarrow> \<bar>d \<bullet> z\<bar> - \<xi>"
                   by auto
                 moreover have "\<forall>\<^sub>F i in sequentially. 0 \<le> \<bar>d \<bullet> ((\<gamma> i - x) /\<^sub>R norm (\<gamma> i - x))\<bar> - \<xi>"
                 proof (rule eventuallyI)
                   fix n
                   show "0 \<le> \<bar>d \<bullet> ((\<gamma> n - x) /\<^sub>R norm (\<gamma> n - x))\<bar> - \<xi>"
                   using \<gamma>le [of n] \<gamma>Sx by (auto simp: abs_mult divide_simps)
                 qed
                 ultimately have "\<xi> \<le> \<bar>d \<bullet> z\<bar>"
                   using tendsto_lowerbound [where a=0] by fastforce
                 have "Cauchy (\<lambda>n. (A n) *v z)"
                 proof (clarsimp simp add: Cauchy_def)
                   fix \<epsilon> :: "real"
                   assume "0 < \<epsilon>"
                   then obtain N::nat where "N > 0" and N: "\<epsilon>/2 > 1/N"
                     by (metis half_gt_zero inverse_eq_divide neq0_conv real_arch_inverse)
                   show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (A m *v z) (A n *v z) < \<epsilon>"
                   proof (intro exI allI impI)
                     fix i j
                     assume ij: "N \<le> i" "N \<le> j"
                     let ?V = "\<lambda>i k. A i *v ((\<gamma> k - x) /\<^sub>R norm (\<gamma> k - x))"
                     have "\<forall>\<^sub>F k in sequentially. dist (\<gamma> k) x < min (\<delta> i) (\<delta> j)"
                       using \<gamma>x [unfolded tendsto_iff] by (meson min_less_iff_conj \<delta>)
                     then have even: "\<forall>\<^sub>F k in sequentially. norm (?V i k - ?V j k) - 2 / N \<le> 0"
                     proof (rule eventually_mono, clarsimp)
                       fix p
                       assume p: "dist (\<gamma> p) x < \<delta> i" "dist (\<gamma> p) x < \<delta> j"
                       let ?C = "\<lambda>k. f (\<gamma> p) - f x - A k *v (\<gamma> p - x)"
                       have "norm ((A i - A j) *v (\<gamma> p - x)) = norm (?C j - ?C i)"
                         by (simp add: algebra_simps)
                       also have "\<dots> \<le> norm (?C j) + norm (?C i)"
                         using norm_triangle_ineq4 by blast
                       also have "\<dots> \<le> 1/(Suc j) * norm (\<gamma> p - x) + 1/(Suc i) * norm (\<gamma> p - x)"
                         by (metis A Diff_iff \<gamma>Sx dist_norm p add_mono)
                       also have "\<dots> \<le> 1/N * norm (\<gamma> p - x) + 1/N * norm (\<gamma> p - x)"
                         apply (intro add_mono mult_right_mono)
                         using ij \<open>N > 0\<close> by (auto simp: field_simps)
                       also have "\<dots> = 2 / N * norm (\<gamma> p - x)"
                         by simp
                       finally have no_le: "norm ((A i - A j) *v (\<gamma> p - x)) \<le> 2 / N * norm (\<gamma> p - x)" .
                       have "norm (?V i p - ?V j p) =
                             norm ((A i - A j) *v ((\<gamma> p - x) /\<^sub>R norm (\<gamma> p - x)))"
                         by (simp add: algebra_simps)
                       also have "\<dots> = norm ((A i - A j) *v (\<gamma> p - x)) / norm (\<gamma> p - x)"
                         by (simp add: divide_inverse matrix_vector_mult_scaleR)
                       also have "\<dots> \<le> 2 / N"
                         using no_le by (auto simp: field_split_simps)
                       finally show "norm (?V i p - ?V j p) \<le> 2 / N" .
                     qed
                     have "isCont (\<lambda>w. (norm(A i *v w - A j *v w) - 2 / N)) z"
                       by (intro continuous_intros)
                     from isCont_tendsto_compose [OF this z]
                     have lim: "(\<lambda>w. norm (A i *v ((\<gamma> w - x) /\<^sub>R norm (\<gamma> w - x)) -
                                     A j *v ((\<gamma> w - x) /\<^sub>R norm (\<gamma> w - x))) - 2 / N)
                                \<longlonglongrightarrow> norm (A i *v z - A j *v z) - 2 / N"
                       by auto
                     have "dist (A i *v z) (A j *v z) \<le> 2 / N"
                       using tendsto_upperbound [OF lim even] by (auto simp: dist_norm)
                     with N show "dist (A i *v z) (A j *v z) < \<epsilon>"
                       by linarith
                   qed
                 qed
                 then have "d \<bullet> z = 0"
                   using CA_eq d orthogonal_def by auto
                 then show False
                   using \<open>0 < \<xi>\<close> \<open>\<xi> \<le> \<bar>d \<bullet> z\<bar>\<close> by auto
               qed
               ultimately show ?thesis
                 using dim_eq_full by fastforce
             qed
             finally have "?CA = UNIV" .
             then have "Cauchy (\<lambda>n. (A n) *v axis j 1)"
               by auto
             then obtain L where "(\<lambda>n. A n *v axis j 1) \<longlonglongrightarrow> L"
               by (auto simp: Cauchy_convergent_iff convergent_def)
             then have "(\<lambda>x. (A x *v axis j 1) $ i) \<longlonglongrightarrow> L $ i"
               by (rule tendsto_vec_nth)
             then show "\<exists>a. (\<lambda>n. A n $ i $ j) \<longlonglongrightarrow> a"
               by (force simp: vax)
           qed
           then obtain B where B: "\<And>i j. (\<lambda>n. A n $ i $ j) \<longlonglongrightarrow> B $ i $ j"
             by (auto simp: lambda_skolem)
           have lin_df: "linear (f' x)"
                and lim_df: "((\<lambda>y. (1 / norm (y - x)) *\<^sub>R (f y - (f x + f' x (y - x)))) \<longlongrightarrow> 0) (at x within S)"
             using \<open>x \<in> S\<close> assms by (auto simp: has_derivative_within linear_linear)
           moreover
           interpret linear "f' x" by fact
           have "(matrix (f' x) - B) *v w = 0" for w
           proof (rule lemma_partial_derivatives [of "(*v) (matrix (f' x) - B)"])
             show "linear ((*v) (matrix (f' x) - B))"
               by (rule matrix_vector_mul_linear)
             have "((\<lambda>y. ((f x + f' x (y - x)) - f y) /\<^sub>R norm (y - x)) \<longlongrightarrow> 0) (at x within S)"
               using tendsto_minus [OF lim_df] by (simp add: field_split_simps)
             then show "((\<lambda>y. (matrix (f' x) - B) *v (y - x) /\<^sub>R norm (y - x)) \<longlongrightarrow> 0) (at x within S)"
             proof (rule Lim_transform)
               have "((\<lambda>y. ((f y + B *v x - (f x + B *v y)) /\<^sub>R norm (y - x))) \<longlongrightarrow> 0) (at x within S)"
               proof (clarsimp simp add: Lim_within dist_norm)
                 fix e :: "real"
                 assume "e > 0"
                 then obtain q::nat where "q \<noteq> 0" and qe2: "1/q < e/2"
                   by (metis divide_pos_pos inverse_eq_divide real_arch_inverse zero_less_numeral)
                 let ?g = "\<lambda>p. sum  (\<lambda>i. sum (\<lambda>j. abs((A p - B)$i$j)) UNIV) UNIV"
                 have "(\<lambda>k. onorm (\<lambda>y. (A k - B) *v y)) \<longlonglongrightarrow> 0"
                 proof (rule Lim_null_comparison)
                   show "\<forall>\<^sub>F k in sequentially. norm (onorm (\<lambda>y. (A k - B) *v y)) \<le> ?g k"
                   proof (rule eventually_sequentiallyI)
                     fix k :: "nat"
                     assume "0 \<le> k"
                     have "0 \<le> onorm ((*v) (A k - B))"
                       using matrix_vector_mul_bounded_linear
                       by (rule onorm_pos_le)
                     then show "norm (onorm ((*v) (A k - B))) \<le> (\<Sum>i\<in>UNIV. \<Sum>j\<in>UNIV. \<bar>(A k - B) $ i $ j\<bar>)"
                       by (simp add: onorm_le_matrix_component_sum del: vector_minus_component)
                   qed
                 next
                   show "?g \<longlonglongrightarrow> 0"
                     using B Lim_null tendsto_rabs_zero_iff by (fastforce intro!: tendsto_null_sum)
                 qed
                 with \<open>e > 0\<close> obtain p where "\<And>n. n \<ge> p \<Longrightarrow> \<bar>onorm ((*v) (A n - B))\<bar> < e/2"
                   unfolding lim_sequentially by (metis diff_zero dist_real_def divide_pos_pos zero_less_numeral)
                 then have pqe2: "\<bar>onorm ((*v) (A (p + q) - B))\<bar> < e/2" (*17 [`abs (onorm (\y. A (p + q) ** y - B ** y)) < e / &2`]*)
                   using le_add1 by blast
                 show "\<exists>d>0. \<forall>y\<in>S. y \<noteq> x \<and> norm (y - x) < d \<longrightarrow>
                            inverse (norm (y - x)) * norm (f y + B *v x - (f x + B *v y)) < e"
                 proof (intro exI, safe)
                   show "0 < \<delta>(p + q)"
                     by (simp add: \<delta>)
                 next
                   fix y
                   assume y: "y \<in> S" "norm (y - x) < \<delta>(p + q)" and "y \<noteq> x"
                   have *: "\<lbrakk>norm(b - c) < e - d; norm(y - x - b) \<le> d\<rbrakk> \<Longrightarrow> norm(y - x - c) < e"
                     for b c d e x and y:: "real^'n"
                     using norm_triangle_ineq2 [of "y - x - c" "y - x - b"] by simp
                   have "norm (f y - f x - B *v (y - x)) < e * norm (y - x)"
                   proof (rule *)
                     show "norm (f y - f x - A (p + q) *v (y - x)) \<le> norm (y - x) / (Suc (p + q))"
                       using A [OF y] by simp
                     have "norm (A (p + q) *v (y - x) - B *v (y - x)) \<le> onorm(\<lambda>x. (A(p + q) - B) *v x) * norm(y - x)"
                       by (metis linear_linear matrix_vector_mul_linear matrix_vector_mult_diff_rdistrib onorm)
                     also have "\<dots> < (e/2) * norm (y - x)"
                       using \<open>y \<noteq> x\<close> pqe2 by auto
                     also have "\<dots> \<le> (e - 1 / (Suc (p + q))) * norm (y - x)"
                     proof (rule mult_right_mono)
                       have "1 / Suc (p + q) \<le> 1 / q"
                         using \<open>q \<noteq> 0\<close> by (auto simp: field_split_simps)
                       also have "\<dots> < e/2"
                         using qe2 by auto
                       finally show "e / 2 \<le> e - 1 / real (Suc (p + q))"
                         by linarith
                     qed auto
                     finally show "norm (A (p + q) *v (y - x) - B *v (y - x)) < e * norm (y - x) - norm (y - x) / real (Suc (p + q))"
                       by (simp add: algebra_simps)
                   qed
                   then show "inverse (norm (y - x)) * norm (f y + B *v x - (f x + B *v y)) < e"
                     using \<open>y \<noteq> x\<close> by (simp add: field_split_simps algebra_simps)
                 qed
               qed
               then show "((\<lambda>y. (matrix (f' x) - B) *v (y - x) /\<^sub>R
                            norm (y - x) - (f x + f' x (y - x) - f y) /\<^sub>R norm (y - x)) \<longlongrightarrow> 0)
                           (at x within S)"
                 by (simp add: algebra_simps diff lin_df scalar_mult_eq_scaleR)
             qed
           qed (use x in \<open>simp; auto simp: not_less\<close>)
           ultimately have "f' x = (*v) B"
             by (force simp: algebra_simps scalar_mult_eq_scaleR)
           show "matrix (f' x) $ m $ n \<le> b"
           proof (rule tendsto_upperbound [of "\<lambda>i. (A i $ m $ n)" _ sequentially])
             show "(\<lambda>i. A i $ m $ n) \<longlonglongrightarrow> matrix (f' x) $ m $ n"
               by (simp add: B \<open>f' x = (*v) B\<close>)
             show "\<forall>\<^sub>F i in sequentially. A i $ m $ n \<le> b"
               by (simp add: Ab less_eq_real_def)
           qed auto
         next
           fix e :: "real"
           assume "x \<in> S" and b: "matrix (f' x) $ m $ n \<le> b" and "e > 0"
           then obtain d where "d>0"
             and d: "\<And>y. y\<in>S \<Longrightarrow> 0 < dist y x \<and> dist y x < d \<longrightarrow> norm (f y - f x - f' x (y - x)) / (norm (y - x))
                   < e/2"
             using f [OF \<open>x \<in> S\<close>]
             by (simp add: Deriv.has_derivative_at_within Lim_within)
               (auto simp add: field_simps dest: spec [of _ "e/2"])
           let ?A = "matrix(f' x) - (\<chi> i j. if i = m \<and> j = n then e / 4 else 0)"
           obtain B where BRats: "\<And>i j. B$i$j \<in> \<rat>" and Bo_e6: "onorm((*v) (?A - B)) < e/6"
             using matrix_rational_approximation \<open>e > 0\<close>
             by (metis zero_less_divide_iff zero_less_numeral)
           show "\<exists>d>0. \<exists>A. A $ m $ n < b \<and> (\<forall>i j. A $ i $ j \<in> \<rat>) \<and>
                 (\<forall>y\<in>S. norm (y - x) < d \<longrightarrow> norm (f y - f x - A *v (y - x)) \<le> e * norm (y - x))"
           proof (intro exI conjI ballI allI impI)
             show "d>0"
               by (rule \<open>d>0\<close>)
             show "B $ m $ n < b"
             proof -
               have "\<bar>matrix ((*v) (?A - B)) $ m $ n\<bar> \<le> onorm ((*v) (?A - B))"
                 using component_le_onorm [OF matrix_vector_mul_linear, of _ m n] by metis
               then show ?thesis
                 using b Bo_e6 by simp
             qed
             show "B $ i $ j \<in> \<rat>" for i j
               using BRats by auto
             show "norm (f y - f x - B *v (y - x)) \<le> e * norm (y - x)"
               if "y \<in> S" and y: "norm (y - x) < d" for y
             proof (cases "y = x")
               case True then show ?thesis
                 by simp
             next
               case False
               have *: "norm(d' - d) \<le> e/2 \<Longrightarrow> norm(y - (x + d')) < e/2 \<Longrightarrow> norm(y - x - d) \<le> e" for d d' e and x y::"real^'n"
                 using norm_triangle_le [of "d' - d" "y - (x + d')"] by simp
               show ?thesis
               proof (rule *)
                 have split246: "\<lbrakk>norm y \<le> e / 6; norm(x - y) \<le> e / 4\<rbrakk> \<Longrightarrow> norm x \<le> e/2" if "e > 0" for e and x y :: "real^'n"
                   using norm_triangle_le [of y "x-y" "e/2"] \<open>e > 0\<close> by simp
                 have "linear (f' x)"
                   using True f has_derivative_linear by blast
                 then have "norm (f' x (y - x) - B *v (y - x)) = norm ((matrix (f' x) - B) *v (y - x))"
                   by (simp add: matrix_vector_mult_diff_rdistrib)
                 also have "\<dots> \<le> (e * norm (y - x)) / 2"
                 proof (rule split246)
                   have "norm ((?A - B) *v (y - x)) / norm (y - x) \<le> onorm(\<lambda>x. (?A - B) *v x)"
                     by (rule le_onorm) auto
                   also have  "\<dots> < e/6"
                     by (rule Bo_e6)
                   finally have "norm ((?A - B) *v (y - x)) / norm (y - x) < e / 6" .
                   then show "norm ((?A - B) *v (y - x)) \<le> e * norm (y - x) / 6"
                     by (simp add: field_split_simps False)
                   have "norm ((matrix (f' x) - B) *v (y - x) - ((?A - B) *v (y - x))) = norm ((\<chi> i j. if i = m \<and> j = n then e / 4 else 0) *v (y - x))"
                     by (simp add: algebra_simps)
                   also have "\<dots> = norm((e/4) *\<^sub>R (y - x)$n *\<^sub>R axis m (1::real))"
                   proof -
                     have "(\<Sum>j\<in>UNIV. (if i = m \<and> j = n then e / 4 else 0) * (y $ j - x $ j)) * 4 = e * (y $ n - x $ n) * axis m 1 $ i" for i
                     proof (cases "i=m")
                       case True then show ?thesis
                         by (auto simp: if_distrib [of "\<lambda>z. z * _"] cong: if_cong)
                     next
                       case False then show ?thesis
                         by (simp add: axis_def)
                     qed
                     then have "(\<chi> i j. if i = m \<and> j = n then e / 4 else 0) *v (y - x) = (e/4) *\<^sub>R (y - x)$n *\<^sub>R axis m (1::real)"
                       by (auto simp: vec_eq_iff matrix_vector_mult_def)
                     then show ?thesis
                       by metis
                   qed
                   also have "\<dots> \<le> e * norm (y - x) / 4"
                     using \<open>e > 0\<close> apply (simp add: norm_mult abs_mult)
                     by (metis component_le_norm_cart vector_minus_component)
                   finally show "norm ((matrix (f' x) - B) *v (y - x) - ((?A - B) *v (y - x))) \<le> e * norm (y - x) / 4" .
                   show "0 < e * norm (y - x)"
                     by (simp add: False \<open>e > 0\<close>)
                 qed
                 finally show "norm (f' x (y - x) - B *v (y - x)) \<le> (e * norm (y - x)) / 2" .
                 show "norm (f y - (f x + f' x (y - x))) < (e * norm (y - x)) / 2"
                   using False d [OF \<open>y \<in> S\<close>] y by (simp add: dist_norm field_simps)
               qed
             qed
           qed
         qed
       qed auto
     qed
     show "negligible ?M"
       using negligible_subset [OF nN MN] .
   qed
   then show ?thesis
     by (simp add: borel_measurable_vimage_halfspace_component_le sets_restrict_space_iff assms)
 qed
 
 
 theorem borel_measurable_det_Jacobian:
  fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
   assumes S: "S \<in> sets lebesgue" and f: "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
   shows "(\<lambda>x. det(matrix(f' x))) \<in> borel_measurable (lebesgue_on S)"
   unfolding det_def
   by (intro measurable) (auto intro: f borel_measurable_partial_derivatives [OF S])
 
 text\<open>The localisation wrt S uses the same argument for many similar results.\<close>
 (*See HOL Light's MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_BOREL, etc.*)
 theorem borel_measurable_lebesgue_on_preimage_borel:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "S \<in> sets lebesgue"
   shows "f \<in> borel_measurable (lebesgue_on S) \<longleftrightarrow>
          (\<forall>T. T \<in> sets borel \<longrightarrow> {x \<in> S. f x \<in> T} \<in> sets lebesgue)"
 proof -
   have "{x. (if x \<in> S then f x else 0) \<in> T} \<in> sets lebesgue \<longleftrightarrow> {x \<in> S. f x \<in> T} \<in> sets lebesgue"
          if "T \<in> sets borel" for T
     proof (cases "0 \<in> T")
       case True
       then have "{x \<in> S. f x \<in> T} = {x. (if x \<in> S then f x else 0) \<in> T} \<inter> S"
                 "{x. (if x \<in> S then f x else 0) \<in> T} = {x \<in> S. f x \<in> T} \<union> -S"
         by auto
       then show ?thesis
         by (metis (no_types, lifting) Compl_in_sets_lebesgue assms sets.Int sets.Un)
     next
       case False
       then have "{x. (if x \<in> S then f x else 0) \<in> T} = {x \<in> S. f x \<in> T}"
         by auto
       then show ?thesis
         by auto
     qed
     then show ?thesis
       unfolding borel_measurable_lebesgue_preimage_borel borel_measurable_if [OF assms, symmetric]
       by blast
 qed
 
 lemma sets_lebesgue_almost_borel:
   assumes "S \<in> sets lebesgue"
   obtains B N where "B \<in> sets borel" "negligible N" "B \<union> N = S"
 proof -
   obtain T N N' where "S = T \<union> N" "N \<subseteq> N'" "N' \<in> null_sets lborel" "T \<in> sets borel"
     using sets_completionE [OF assms] by auto
   then show thesis
     by (metis negligible_iff_null_sets negligible_subset null_sets_completionI that)
 qed
 
 lemma double_lebesgue_sets:
  assumes S: "S \<in> sets lebesgue" and T: "T \<in> sets lebesgue" and fim: "f ` S \<subseteq> T"
  shows "(\<forall>U. U \<in> sets lebesgue \<and> U \<subseteq> T \<longrightarrow> {x \<in> S. f x \<in> U} \<in> sets lebesgue) \<longleftrightarrow>
           f \<in> borel_measurable (lebesgue_on S) \<and>
           (\<forall>U. negligible U \<and> U \<subseteq> T \<longrightarrow> {x \<in> S. f x \<in> U} \<in> sets lebesgue)"
          (is "?lhs \<longleftrightarrow> _ \<and> ?rhs")
   unfolding borel_measurable_lebesgue_on_preimage_borel [OF S]
 proof (intro iffI allI conjI impI, safe)
   fix V :: "'b set"
   assume *: "\<forall>U. U \<in> sets lebesgue \<and> U \<subseteq> T \<longrightarrow> {x \<in> S. f x \<in> U} \<in> sets lebesgue"
     and "V \<in> sets borel"
   then have V: "V \<in> sets lebesgue"
     by simp
   have "{x \<in> S. f x \<in> V} = {x \<in> S. f x \<in> T \<inter> V}"
     using fim by blast
   also have "{x \<in> S. f x \<in> T \<inter> V} \<in> sets lebesgue"
     using T V * le_inf_iff by blast
   finally show "{x \<in> S. f x \<in> V} \<in> sets lebesgue" .
 next
   fix U :: "'b set"
   assume "\<forall>U. U \<in> sets lebesgue \<and> U \<subseteq> T \<longrightarrow> {x \<in> S. f x \<in> U} \<in> sets lebesgue"
          "negligible U" "U \<subseteq> T"
   then show "{x \<in> S. f x \<in> U} \<in> sets lebesgue"
     using negligible_imp_sets by blast
 next
   fix U :: "'b set"
   assume 1 [rule_format]: "(\<forall>T. T \<in> sets borel \<longrightarrow> {x \<in> S. f x \<in> T} \<in> sets lebesgue)"
      and 2 [rule_format]: "\<forall>U. negligible U \<and> U \<subseteq> T \<longrightarrow> {x \<in> S. f x \<in> U} \<in> sets lebesgue"
      and "U \<in> sets lebesgue" "U \<subseteq> T"
   then obtain C N where C: "C \<in> sets borel \<and> negligible N \<and> C \<union> N = U"
     using sets_lebesgue_almost_borel
     by metis
   then have "{x \<in> S. f x \<in> C} \<in> sets lebesgue"
     by (blast intro: 1)
   moreover have "{x \<in> S. f x \<in> N} \<in> sets lebesgue"
     using C \<open>U \<subseteq> T\<close> by (blast intro: 2)
   moreover have "{x \<in> S. f x \<in> C \<union> N} = {x \<in> S. f x \<in> C} \<union> {x \<in> S. f x \<in> N}"
     by auto
   ultimately show "{x \<in> S. f x \<in> U} \<in> sets lebesgue"
     using C by auto
 qed
 
 
 subsection\<open>Simplest case of Sard's theorem (we don't need continuity of derivative)\<close>
 
 lemma Sard_lemma00:
   fixes P :: "'b::euclidean_space set"
   assumes "a \<ge> 0" and a: "a *\<^sub>R i \<noteq> 0" and i: "i \<in> Basis"
     and P: "P \<subseteq> {x. a *\<^sub>R i \<bullet> x = 0}"
     and "0 \<le> m" "0 \<le> e"
  obtains S where "S \<in> lmeasurable"
             and "{z. norm z \<le> m \<and> (\<exists>t \<in> P. norm(z - t) \<le> e)} \<subseteq> S"
             and "measure lebesgue S \<le> (2 * e) * (2 * m) ^ (DIM('b) - 1)"
 proof -
   have "a > 0"
     using assms by simp
   let ?v = "(\<Sum>j\<in>Basis. (if j = i then e else m) *\<^sub>R j)"
   show thesis
   proof
     have "- e \<le> x \<bullet> i" "x \<bullet> i \<le> e"
       if "t \<in> P" "norm (x - t) \<le> e" for x t
       using \<open>a > 0\<close> that Basis_le_norm [of i "x-t"] P i
       by (auto simp: inner_commute algebra_simps)
     moreover have "- m \<le> x \<bullet> j" "x \<bullet> j \<le> m"
       if "norm x \<le> m" "t \<in> P" "norm (x - t) \<le> e" "j \<in> Basis" and "j \<noteq> i"
       for x t j
       using that Basis_le_norm [of j x] by auto
     ultimately
     show "{z. norm z \<le> m \<and> (\<exists>t\<in>P. norm (z - t) \<le> e)} \<subseteq> cbox (-?v) ?v"
       by (auto simp: mem_box)
     have *: "\<forall>k\<in>Basis. - ?v \<bullet> k \<le> ?v \<bullet> k"
       using \<open>0 \<le> m\<close> \<open>0 \<le> e\<close> by (auto simp: inner_Basis)
     have 2: "2 ^ DIM('b) = 2 * 2 ^ (DIM('b) - Suc 0)"
       by (metis DIM_positive Suc_pred power_Suc)
     show "measure lebesgue (cbox (-?v) ?v) \<le> 2 * e * (2 * m) ^ (DIM('b) - 1)"
       using \<open>i \<in> Basis\<close>
       by (simp add: content_cbox [OF *] prod.distrib prod.If_cases Diff_eq [symmetric] 2)
   qed blast
 qed
 
 text\<open>As above, but reorienting the vector (HOL Light's @text{GEOM\_BASIS\_MULTIPLE\_TAC})\<close>
 lemma Sard_lemma0:
   fixes P :: "(real^'n::{finite,wellorder}) set"
   assumes "a \<noteq> 0"
     and P: "P \<subseteq> {x. a \<bullet> x = 0}" and "0 \<le> m" "0 \<le> e"
   obtains S where "S \<in> lmeasurable"
     and "{z. norm z \<le> m \<and> (\<exists>t \<in> P. norm(z - t) \<le> e)} \<subseteq> S"
     and "measure lebesgue S \<le> (2 * e) * (2 * m) ^ (CARD('n) - 1)"
 proof -
   obtain T and k::'n where T: "orthogonal_transformation T" and a: "a = T (norm a *\<^sub>R axis k (1::real))"
     using rotation_rightward_line by metis
   have Tinv [simp]: "T (inv T x) = x" for x
     by (simp add: T orthogonal_transformation_surj surj_f_inv_f)
   obtain S where S: "S \<in> lmeasurable"
     and subS: "{z. norm z \<le> m \<and> (\<exists>t \<in> T-`P. norm(z - t) \<le> e)} \<subseteq> S"
     and mS: "measure lebesgue S \<le> (2 * e) * (2 * m) ^ (CARD('n) - 1)"
   proof (rule Sard_lemma00 [of "norm a" "axis k (1::real)" "T-`P" m e])
     have "norm a *\<^sub>R axis k 1 \<bullet> x = 0" if "T x \<in> P" for x
     proof -
       have "a \<bullet> T x = 0"
         using P that by blast
       then show ?thesis
         by (metis (no_types, lifting) T a orthogonal_orthogonal_transformation orthogonal_def)
     qed
     then show "T -` P \<subseteq> {x. norm a *\<^sub>R axis k 1 \<bullet> x = 0}"
       by auto
   qed (use assms T in auto)
   show thesis
   proof
     show "T ` S \<in> lmeasurable"
       using S measurable_orthogonal_image T by blast
     have "{z. norm z \<le> m \<and> (\<exists>t\<in>P. norm (z - t) \<le> e)} \<subseteq> T ` {z. norm z \<le> m \<and> (\<exists>t\<in>T -` P. norm (z - t) \<le> e)}"
     proof clarsimp
       fix x t
       assume "norm x \<le> m" "t \<in> P" "norm (x - t) \<le> e"
       then have "norm (inv T x) \<le> m"
         using orthogonal_transformation_inv [OF T] by (simp add: orthogonal_transformation_norm)
       moreover have "\<exists>t\<in>T -` P. norm (inv T x - t) \<le> e"
       proof
         have "T (inv T x - inv T t) = x - t"
           using T linear_diff orthogonal_transformation_def
           by (metis (no_types, hide_lams) Tinv)
         then have "norm (inv T x - inv T t) = norm (x - t)"
           by (metis T orthogonal_transformation_norm)
         then show "norm (inv T x - inv T t) \<le> e"
           using \<open>norm (x - t) \<le> e\<close> by linarith
        next
          show "inv T t \<in> T -` P"
            using \<open>t \<in> P\<close> by force
       qed
       ultimately show "x \<in> T ` {z. norm z \<le> m \<and> (\<exists>t\<in>T -` P. norm (z - t) \<le> e)}"
         by force
     qed
     then show "{z. norm z \<le> m \<and> (\<exists>t\<in>P. norm (z - t) \<le> e)} \<subseteq> T ` S"
       using image_mono [OF subS] by (rule order_trans)
     show "measure lebesgue (T ` S) \<le> 2 * e * (2 * m) ^ (CARD('n) - 1)"
       using mS T by (simp add: S measure_orthogonal_image)
   qed
 qed
 
 text\<open>As above, but translating the sets (HOL Light's @text{GEN\_GEOM\_ORIGIN\_TAC})\<close>
 lemma Sard_lemma1:
   fixes P :: "(real^'n::{finite,wellorder}) set"
    assumes P: "dim P < CARD('n)" and "0 \<le> m" "0 \<le> e"
  obtains S where "S \<in> lmeasurable"
             and "{z. norm(z - w) \<le> m \<and> (\<exists>t \<in> P. norm(z - w - t) \<le> e)} \<subseteq> S"
             and "measure lebesgue S \<le> (2 * e) * (2 * m) ^ (CARD('n) - 1)"
 proof -
   obtain a where "a \<noteq> 0" "P \<subseteq> {x. a \<bullet> x = 0}"
     using lowdim_subset_hyperplane [of P] P span_base by auto
   then obtain S where S: "S \<in> lmeasurable"
     and subS: "{z. norm z \<le> m \<and> (\<exists>t \<in> P. norm(z - t) \<le> e)} \<subseteq> S"
     and mS: "measure lebesgue S \<le> (2 * e) * (2 * m) ^ (CARD('n) - 1)"
     by (rule Sard_lemma0 [OF _ _ \<open>0 \<le> m\<close> \<open>0 \<le> e\<close>])
   show thesis
   proof
     show "(+)w ` S \<in> lmeasurable"
       by (metis measurable_translation S)
     show "{z. norm (z - w) \<le> m \<and> (\<exists>t\<in>P. norm (z - w - t) \<le> e)} \<subseteq> (+)w ` S"
       using subS by force
     show "measure lebesgue ((+)w ` S) \<le> 2 * e * (2 * m) ^ (CARD('n) - 1)"
       by (metis measure_translation mS)
   qed
 qed
 
 lemma Sard_lemma2:
   fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n::{finite,wellorder}"
   assumes mlen: "CARD('m) \<le> CARD('n)" (is "?m \<le> ?n")
     and "B > 0" "bounded S"
     and derS: "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
     and rank: "\<And>x. x \<in> S \<Longrightarrow> rank(matrix(f' x)) < CARD('n)"
     and B: "\<And>x. x \<in> S \<Longrightarrow> onorm(f' x) \<le> B"
   shows "negligible(f ` S)"
 proof -
   have lin_f': "\<And>x. x \<in> S \<Longrightarrow> linear(f' x)"
     using derS has_derivative_linear by blast
   show ?thesis
   proof (clarsimp simp add: negligible_outer_le)
     fix e :: "real"
     assume "e > 0"
     obtain c where csub: "S \<subseteq> cbox (- (vec c)) (vec c)" and "c > 0"
     proof -
       obtain b where b: "\<And>x. x \<in> S \<Longrightarrow> norm x \<le> b"
         using \<open>bounded S\<close> by (auto simp: bounded_iff)
       show thesis
       proof
         have "- \<bar>b\<bar> - 1 \<le> x $ i \<and> x $ i \<le> \<bar>b\<bar> + 1" if "x \<in> S" for x i
           using component_le_norm_cart [of x i] b [OF that] by auto
         then show "S \<subseteq> cbox (- vec (\<bar>b\<bar> + 1)) (vec (\<bar>b\<bar> + 1))"
           by (auto simp: mem_box_cart)
       qed auto
     qed
     then have box_cc: "box (- (vec c)) (vec c) \<noteq> {}" and cbox_cc: "cbox (- (vec c)) (vec c) \<noteq> {}"
       by (auto simp: interval_eq_empty_cart)
     obtain d where "d > 0" "d \<le> B"
              and d: "(d * 2) * (4 * B) ^ (?n - 1) \<le> e / (2*c) ^ ?m / ?m ^ ?m"
       apply (rule that [of "min B (e / (2*c) ^ ?m / ?m ^ ?m / (4 * B) ^ (?n - 1) / 2)"])
       using \<open>B > 0\<close> \<open>c > 0\<close> \<open>e > 0\<close>
       by (simp_all add: divide_simps min_mult_distrib_right)
     have "\<exists>r. 0 < r \<and> r \<le> 1/2 \<and>
               (x \<in> S
                \<longrightarrow> (\<forall>y. y \<in> S \<and> norm(y - x) < r
                        \<longrightarrow> norm(f y - f x - f' x (y - x)) \<le> d * norm(y - x)))" for x
     proof (cases "x \<in> S")
       case True
       then obtain r where "r > 0"
               and "\<And>y. \<lbrakk>y \<in> S; norm (y - x) < r\<rbrakk>
                        \<Longrightarrow> norm (f y - f x - f' x (y - x)) \<le> d * norm (y - x)"
         using derS \<open>d > 0\<close> by (force simp: has_derivative_within_alt)
       then show ?thesis
         by (rule_tac x="min r (1/2)" in exI) simp
     next
       case False
       then show ?thesis
         by (rule_tac x="1/2" in exI) simp
     qed
     then obtain r where r12: "\<And>x. 0 < r x \<and> r x \<le> 1/2"
             and r: "\<And>x y. \<lbrakk>x \<in> S; y \<in> S; norm(y - x) < r x\<rbrakk>
                           \<Longrightarrow> norm(f y - f x - f' x (y - x)) \<le> d * norm(y - x)"
       by metis
     then have ga: "gauge (\<lambda>x. ball x (r x))"
       by (auto simp: gauge_def)
     obtain \<D> where \<D>: "countable \<D>" and sub_cc: "\<Union>\<D> \<subseteq> cbox (- vec c) (vec c)"
       and cbox: "\<And>K. K \<in> \<D> \<Longrightarrow> interior K \<noteq> {} \<and> (\<exists>u v. K = cbox u v)"
       and djointish: "pairwise (\<lambda>A B. interior A \<inter> interior B = {}) \<D>"
       and covered: "\<And>K. K \<in> \<D> \<Longrightarrow> \<exists>x \<in> S \<inter> K. K \<subseteq> ball x (r x)"
       and close: "\<And>u v. cbox u v \<in> \<D> \<Longrightarrow> \<exists>n. \<forall>i::'m. v $ i - u $ i = 2*c / 2^n"
       and covers: "S \<subseteq> \<Union>\<D>"
       apply (rule covering_lemma [OF csub box_cc ga])
       apply (auto simp: Basis_vec_def cart_eq_inner_axis [symmetric])
       done
     let ?\<mu> = "measure lebesgue"
     have "\<exists>T. T \<in> lmeasurable \<and> f ` (K \<inter> S) \<subseteq> T \<and> ?\<mu> T \<le> e / (2*c) ^ ?m * ?\<mu> K"
       if "K \<in> \<D>" for K
     proof -
       obtain u v where uv: "K = cbox u v"
         using cbox \<open>K \<in> \<D>\<close> by blast
       then have uv_ne: "cbox u v \<noteq> {}"
         using cbox that by fastforce
       obtain x where x: "x \<in> S \<inter> cbox u v" "cbox u v \<subseteq> ball x (r x)"
         using \<open>K \<in> \<D>\<close> covered uv by blast
       then have "dim (range (f' x)) < ?n"
         using rank_dim_range [of "matrix (f' x)"] x rank[of x]
         by (auto simp: matrix_works scalar_mult_eq_scaleR lin_f')
       then obtain T where T: "T \<in> lmeasurable"
             and subT: "{z. norm(z - f x) \<le> (2 * B) * norm(v - u) \<and> (\<exists>t \<in> range (f' x). norm(z - f x - t) \<le> d * norm(v - u))} \<subseteq> T"
             and measT: "?\<mu> T \<le> (2 * (d * norm(v - u))) * (2 * ((2 * B) * norm(v - u))) ^ (?n - 1)"
                         (is "_ \<le> ?DVU")
         apply (rule Sard_lemma1 [of "range (f' x)" "(2 * B) * norm(v - u)" "d * norm(v - u)" "f x"])
         using \<open>B > 0\<close> \<open>d > 0\<close> by simp_all
       show ?thesis
       proof (intro exI conjI)
         have "f ` (K \<inter> S) \<subseteq> {z. norm(z - f x) \<le> (2 * B) * norm(v - u) \<and> (\<exists>t \<in> range (f' x). norm(z - f x - t) \<le> d * norm(v - u))}"
           unfolding uv
         proof (clarsimp simp: mult.assoc, intro conjI)
           fix y
           assume y: "y \<in> cbox u v" and "y \<in> S"
           then have "norm (y - x) < r x"
             by (metis dist_norm mem_ball norm_minus_commute subsetCE x(2))
           then have le_dyx: "norm (f y - f x - f' x (y - x)) \<le> d * norm (y - x)"
             using r [of x y] x \<open>y \<in> S\<close> by blast
           have yx_le: "norm (y - x) \<le> norm (v - u)"
           proof (rule norm_le_componentwise_cart)
             show "norm ((y - x) $ i) \<le> norm ((v - u) $ i)" for i
               using x y by (force simp: mem_box_cart dest!: spec [where x=i])
           qed
           have *: "\<lbrakk>norm(y - x - z) \<le> d; norm z \<le> B; d \<le> B\<rbrakk> \<Longrightarrow> norm(y - x) \<le> 2 * B"
             for x y z :: "real^'n::_" and d B
             using norm_triangle_ineq2 [of "y - x" z] by auto
           show "norm (f y - f x) \<le> 2 * (B * norm (v - u))"
           proof (rule * [OF le_dyx])
             have "norm (f' x (y - x)) \<le> onorm (f' x) * norm (y - x)"
               using onorm [of "f' x" "y-x"] by (meson IntE lin_f' linear_linear x(1))
             also have "\<dots> \<le> B * norm (v - u)"
             proof (rule mult_mono)
               show "onorm (f' x) \<le> B"
                 using B x by blast
             qed (use \<open>B > 0\<close> yx_le in auto)
             finally show "norm (f' x (y - x)) \<le> B * norm (v - u)" .
             show "d * norm (y - x) \<le> B * norm (v - u)"
               using \<open>B > 0\<close> by (auto intro: mult_mono [OF \<open>d \<le> B\<close> yx_le])
           qed
           show "\<exists>t. norm (f y - f x - f' x t) \<le> d * norm (v - u)"
             apply (rule_tac x="y-x" in exI)
             using \<open>d > 0\<close> yx_le le_dyx mult_left_mono [where c=d]
             by (meson order_trans real_mult_le_cancel_iff2)
         qed
         with subT show "f ` (K \<inter> S) \<subseteq> T" by blast
         show "?\<mu> T \<le> e / (2*c) ^ ?m * ?\<mu> K"
         proof (rule order_trans [OF measT])
           have "?DVU = (d * 2 * (4 * B) ^ (?n - 1)) * norm (v - u)^?n"
             using \<open>c > 0\<close>
             apply (simp add: algebra_simps)
             by (metis Suc_pred power_Suc zero_less_card_finite)
           also have "\<dots> \<le> (e / (2*c) ^ ?m / (?m ^ ?m)) * norm(v - u) ^ ?n"
             by (rule mult_right_mono [OF d]) auto
           also have "\<dots> \<le> e / (2*c) ^ ?m * ?\<mu> K"
           proof -
             have "u \<in> ball (x) (r x)" "v \<in> ball x (r x)"
               using box_ne_empty(1) contra_subsetD [OF x(2)] mem_box(2) uv_ne by fastforce+
             moreover have "r x \<le> 1/2"
               using r12 by auto
             ultimately have "norm (v - u) \<le> 1"
               using norm_triangle_half_r [of x u 1 v]
               by (metis (no_types, hide_lams) dist_commute dist_norm less_eq_real_def less_le_trans mem_ball)
             then have "norm (v - u) ^ ?n \<le> norm (v - u) ^ ?m"
               by (simp add: power_decreasing [OF mlen])
             also have "\<dots> \<le> ?\<mu> K * real (?m ^ ?m)"
             proof -
               obtain n where n: "\<And>i. v$i - u$i = 2 * c / 2^n"
                 using close [of u v] \<open>K \<in> \<D>\<close> uv by blast
               have "norm (v - u) ^ ?m \<le> (\<Sum>i\<in>UNIV. \<bar>(v - u) $ i\<bar>) ^ ?m"
                 by (intro norm_le_l1_cart power_mono) auto
               also have "\<dots> \<le> (\<Prod>i\<in>UNIV. v $ i - u $ i) * real CARD('m) ^ CARD('m)"
                 by (simp add: n field_simps \<open>c > 0\<close> less_eq_real_def)
               also have "\<dots> = ?\<mu> K * real (?m ^ ?m)"
                 by (simp add: uv uv_ne content_cbox_cart)
               finally show ?thesis .
             qed
             finally have *: "1 / real (?m ^ ?m) * norm (v - u) ^ ?n \<le> ?\<mu> K"
               by (simp add: field_split_simps)
             show ?thesis
               using mult_left_mono [OF *, of "e / (2*c) ^ ?m"] \<open>c > 0\<close> \<open>e > 0\<close> by auto
           qed
           finally show "?DVU \<le> e / (2*c) ^ ?m * ?\<mu> K" .
         qed
       qed (use T in auto)
     qed
     then obtain g where meas_g: "\<And>K. K \<in> \<D> \<Longrightarrow> g K \<in> lmeasurable"
                     and sub_g: "\<And>K. K \<in> \<D> \<Longrightarrow> f ` (K \<inter> S) \<subseteq> g K"
                     and le_g: "\<And>K. K \<in> \<D> \<Longrightarrow> ?\<mu> (g K) \<le> e / (2*c)^?m * ?\<mu> K"
       by metis
     have le_e: "?\<mu> (\<Union>i\<in>\<F>. g i) \<le> e"
       if "\<F> \<subseteq> \<D>" "finite \<F>" for \<F>
     proof -
       have "?\<mu> (\<Union>i\<in>\<F>. g i) \<le> (\<Sum>i\<in>\<F>. ?\<mu> (g i))"
         using meas_g \<open>\<F> \<subseteq> \<D>\<close> by (auto intro: measure_UNION_le [OF \<open>finite \<F>\<close>])
       also have "\<dots> \<le> (\<Sum>K\<in>\<F>. e / (2*c) ^ ?m * ?\<mu> K)"
         using \<open>\<F> \<subseteq> \<D>\<close> sum_mono [OF le_g] by (meson le_g subsetCE sum_mono)
       also have "\<dots> = e / (2*c) ^ ?m * (\<Sum>K\<in>\<F>. ?\<mu> K)"
         by (simp add: sum_distrib_left)
       also have "\<dots> \<le> e"
       proof -
         have "\<F> division_of \<Union>\<F>"
         proof (rule division_ofI)
           show "K \<subseteq> \<Union>\<F>"  "K \<noteq> {}" "\<exists>a b. K = cbox a b" if "K \<in> \<F>" for K
             using \<open>K \<in> \<F>\<close> covered cbox \<open>\<F> \<subseteq> \<D>\<close> by (auto simp: Union_upper)
           show "interior K \<inter> interior L = {}" if "K \<in> \<F>" and "L \<in> \<F>" and "K \<noteq> L" for K L
             by (metis (mono_tags, lifting) \<open>\<F> \<subseteq> \<D>\<close> pairwiseD djointish pairwise_subset that)
         qed (use that in auto)
         then have "sum ?\<mu> \<F> \<le> ?\<mu> (\<Union>\<F>)"
           by (simp add: content_division)
         also have "\<dots> \<le> ?\<mu> (cbox (- vec c) (vec c) :: (real, 'm) vec set)"
         proof (rule measure_mono_fmeasurable)
           show "\<Union>\<F> \<subseteq> cbox (- vec c) (vec c)"
             by (meson Sup_subset_mono sub_cc order_trans \<open>\<F> \<subseteq> \<D>\<close>)
         qed (use \<open>\<F> division_of \<Union>\<F>\<close> lmeasurable_division in auto)
         also have "\<dots> = content (cbox (- vec c) (vec c) :: (real, 'm) vec set)"
           by simp
         also have "\<dots> \<le> (2 ^ ?m * c ^ ?m)"
           using \<open>c > 0\<close> by (simp add: content_cbox_if_cart)
         finally have "sum ?\<mu> \<F> \<le> (2 ^ ?m * c ^ ?m)" .
         then show ?thesis
           using \<open>e > 0\<close> \<open>c > 0\<close> by (auto simp: field_split_simps)
       qed
       finally show ?thesis .
     qed
     show "\<exists>T. f ` S \<subseteq> T \<and> T \<in> lmeasurable \<and> ?\<mu> T \<le> e"
     proof (intro exI conjI)
       show "f ` S \<subseteq> \<Union> (g ` \<D>)"
         using covers sub_g by force
       show "\<Union> (g ` \<D>) \<in> lmeasurable"
         by (rule fmeasurable_UN_bound [OF \<open>countable \<D>\<close> meas_g le_e])
       show "?\<mu> (\<Union> (g ` \<D>)) \<le> e"
         by (rule measure_UN_bound [OF \<open>countable \<D>\<close> meas_g le_e])
     qed
   qed
 qed
 
 
 theorem baby_Sard:
   fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n::{finite,wellorder}"
   assumes mlen: "CARD('m) \<le> CARD('n)"
     and der: "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
     and rank: "\<And>x. x \<in> S \<Longrightarrow> rank(matrix(f' x)) < CARD('n)"
   shows "negligible(f ` S)"
 proof -
   let ?U = "\<lambda>n. {x \<in> S. norm(x) \<le> n \<and> onorm(f' x) \<le> real n}"
   have "\<And>x. x \<in> S \<Longrightarrow> \<exists>n. norm x \<le> real n \<and> onorm (f' x) \<le> real n"
     by (meson linear order_trans real_arch_simple)
   then have eq: "S = (\<Union>n. ?U n)"
     by auto
   have "negligible (f ` ?U n)" for n
   proof (rule Sard_lemma2 [OF mlen])
     show "0 < real n + 1"
       by auto
     show "bounded (?U n)"
       using bounded_iff by blast
     show "(f has_derivative f' x) (at x within ?U n)" if "x \<in> ?U n" for x
       using der that by (force intro: has_derivative_subset)
   qed (use rank in auto)
   then show ?thesis
     by (subst eq) (simp add: image_Union negligible_Union_nat)
 qed
 
 
 subsection\<open>A one-way version of change-of-variables not assuming injectivity. \<close>
 
 lemma integral_on_image_ubound_weak:
   fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real"
   assumes S: "S \<in> sets lebesgue"
       and f: "f \<in> borel_measurable (lebesgue_on (g ` S))"
       and nonneg_fg:  "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> f(g x)"
       and der_g:   "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
       and det_int_fg: "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f(g x)) integrable_on S"
       and meas_gim: "\<And>T. \<lbrakk>T \<subseteq> g ` S; T \<in> sets lebesgue\<rbrakk> \<Longrightarrow> {x \<in> S. g x \<in> T} \<in> sets lebesgue"
     shows "f integrable_on (g ` S) \<and>
            integral (g ` S) f \<le> integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f(g x))"
          (is "_ \<and> _ \<le> ?b")
 proof -
   let ?D = "\<lambda>x. \<bar>det (matrix (g' x))\<bar>"
   have cont_g: "continuous_on S g"
     using der_g has_derivative_continuous_on by blast
   have [simp]: "space (lebesgue_on S) = S"
     by (simp add: S)
   have gS_in_sets_leb: "g ` S \<in> sets lebesgue"
     apply (rule differentiable_image_in_sets_lebesgue)
     using der_g by (auto simp: S differentiable_def differentiable_on_def)
   obtain h where nonneg_h: "\<And>n x. 0 \<le> h n x"
     and h_le_f: "\<And>n x. x \<in> S \<Longrightarrow> h n (g x) \<le> f (g x)"
     and h_inc: "\<And>n x. h n x \<le> h (Suc n) x"
     and h_meas: "\<And>n. h n \<in> borel_measurable lebesgue"
     and fin_R: "\<And>n. finite(range (h n))"
     and lim: "\<And>x. x \<in> g ` S \<Longrightarrow> (\<lambda>n. h n x) \<longlonglongrightarrow> f x"
   proof -
     let ?f = "\<lambda>x. if x \<in> g ` S then f x else 0"
     have "?f \<in> borel_measurable lebesgue \<and> (\<forall>x. 0 \<le> ?f x)"
       by (auto simp: gS_in_sets_leb f nonneg_fg measurable_restrict_space_iff [symmetric])
     then show ?thesis
       apply (clarsimp simp add: borel_measurable_simple_function_limit_increasing)
       apply (rename_tac h)
       by (rule_tac h=h in that) (auto split: if_split_asm)
   qed
   have h_lmeas: "{t. h n (g t) = y} \<inter> S \<in> sets lebesgue" for y n
   proof -
     have "space (lebesgue_on (UNIV::(real,'n) vec set)) = UNIV"
       by simp
     then have "((h n) -`{y} \<inter> g ` S) \<in> sets (lebesgue_on (g ` S))"
       by (metis Int_commute borel_measurable_vimage h_meas image_eqI inf_top.right_neutral sets_restrict_space space_borel space_completion space_lborel)
     then have "({u. h n u = y} \<inter> g ` S) \<in> sets lebesgue"
       using gS_in_sets_leb
       by (simp add: integral_indicator fmeasurableI2 sets_restrict_space_iff vimage_def)
     then have "{x \<in> S. g x \<in> ({u. h n u = y} \<inter> g ` S)} \<in> sets lebesgue"
       using meas_gim[of "({u. h n u = y} \<inter> g ` S)"] by force
     moreover have "{t. h n (g t) = y} \<inter> S = {x \<in> S. g x \<in> ({u. h n u = y} \<inter> g ` S)}"
       by blast
     ultimately show ?thesis
       by auto
   qed
   have hint: "h n integrable_on g ` S \<and> integral (g ` S) (h n) \<le> integral S (\<lambda>x. ?D x * h n (g x))"
           (is "?INT \<and> ?lhs \<le> ?rhs")  for n
   proof -
     let ?R = "range (h n)"
     have hn_eq: "h n = (\<lambda>x. \<Sum>y\<in>?R. y * indicat_real {x. h n x = y} x)"
       by (simp add: indicator_def if_distrib fin_R cong: if_cong)
     have yind: "(\<lambda>t. y * indicator{x. h n x = y} t) integrable_on (g ` S) \<and>
                 (integral (g ` S) (\<lambda>t. y * indicator {x. h n x = y} t))
                  \<le> integral S (\<lambda>t. \<bar>det (matrix (g' t))\<bar> * y * indicator {x. h n x = y} (g t))"
        if y: "y \<in> ?R" for y::real
     proof (cases "y=0")
       case True
       then show ?thesis using gS_in_sets_leb integrable_0 by force
     next
       case False
       with that have "y > 0"
         using less_eq_real_def nonneg_h by fastforce
       have "(\<lambda>x. if x \<in> {t. h n (g t) = y} then ?D x else 0) integrable_on S"
       proof (rule measurable_bounded_by_integrable_imp_integrable)
         have "(\<lambda>x. ?D x) \<in> borel_measurable (lebesgue_on ({t. h n (g t) = y} \<inter> S))"
           apply (intro borel_measurable_abs borel_measurable_det_Jacobian [OF h_lmeas, where f=g])
-          by (meson der_g IntD2 has_derivative_within_subset inf_le2)
+          by (meson der_g IntD2 has_derivative_subset inf_le2)
         then have "(\<lambda>x. if x \<in> {t. h n (g t) = y} \<inter> S then ?D x else 0) \<in> borel_measurable lebesgue"
           by (rule borel_measurable_if_I [OF _ h_lmeas])
         then show "(\<lambda>x. if x \<in> {t. h n (g t) = y} then ?D x else 0) \<in> borel_measurable (lebesgue_on S)"
           by (simp add: if_if_eq_conj Int_commute borel_measurable_if [OF S, symmetric])
         show "(\<lambda>x. ?D x *\<^sub>R f (g x) /\<^sub>R y) integrable_on S"
           by (rule integrable_cmul) (use det_int_fg in auto)
         show "norm (if x \<in> {t. h n (g t) = y} then ?D x else 0) \<le> ?D x *\<^sub>R f (g x) /\<^sub>R y"
           if "x \<in> S" for x
           using nonneg_h [of n x] \<open>y > 0\<close> nonneg_fg [of x] h_le_f [of x n] that
           by (auto simp: divide_simps mult_left_mono)
       qed (use S in auto)
       then have int_det: "(\<lambda>t. \<bar>det (matrix (g' t))\<bar>) integrable_on ({t. h n (g t) = y} \<inter> S)"
         using integrable_restrict_Int by force
       have "(g ` ({t. h n (g t) = y} \<inter> S)) \<in> lmeasurable"
         apply (rule measurable_differentiable_image [OF h_lmeas])
-         apply (blast intro: has_derivative_within_subset [OF der_g])
+         apply (blast intro: has_derivative_subset [OF der_g])
         apply (rule int_det)
         done
       moreover have "g ` ({t. h n (g t) = y} \<inter> S) = {x. h n x = y} \<inter> g ` S"
         by blast
       moreover have "measure lebesgue (g ` ({t. h n (g t) = y} \<inter> S))
                      \<le> integral ({t. h n (g t) = y} \<inter> S) (\<lambda>t. \<bar>det (matrix (g' t))\<bar>)"
         apply (rule measure_differentiable_image [OF h_lmeas _ int_det])
-        apply (blast intro: has_derivative_within_subset [OF der_g])
+        apply (blast intro: has_derivative_subset [OF der_g])
         done
       ultimately show ?thesis
         using \<open>y > 0\<close> integral_restrict_Int [of S "{t. h n (g t) = y}" "\<lambda>t. \<bar>det (matrix (g' t))\<bar> * y"]
         apply (simp add: integrable_on_indicator integral_indicator)
         apply (simp add: indicator_def if_distrib cong: if_cong)
         done
     qed
     have hn_int: "h n integrable_on g ` S"
       apply (subst hn_eq)
       using yind by (force intro: integrable_sum [OF fin_R])
     then show ?thesis
     proof
       have "?lhs = integral (g ` S) (\<lambda>x. \<Sum>y\<in>range (h n). y * indicat_real {x. h n x = y} x)"
         by (metis hn_eq)
       also have "\<dots> = (\<Sum>y\<in>range (h n). integral (g ` S) (\<lambda>x. y * indicat_real {x. h n x = y} x))"
         by (rule integral_sum [OF fin_R]) (use yind in blast)
       also have "\<dots> \<le> (\<Sum>y\<in>range (h n). integral S (\<lambda>u. \<bar>det (matrix (g' u))\<bar> * y * indicat_real {x. h n x = y} (g u)))"
         using yind by (force intro: sum_mono)
       also have "\<dots> = integral S (\<lambda>u. \<Sum>y\<in>range (h n). \<bar>det (matrix (g' u))\<bar> * y * indicat_real {x. h n x = y} (g u))"
       proof (rule integral_sum [OF fin_R, symmetric])
         fix y assume y: "y \<in> ?R"
         with nonneg_h have "y \<ge> 0"
           by auto
         show "(\<lambda>u. \<bar>det (matrix (g' u))\<bar> * y * indicat_real {x. h n x = y} (g u)) integrable_on S"
         proof (rule measurable_bounded_by_integrable_imp_integrable)
           have "(\<lambda>x. indicat_real {x. h n x = y} (g x)) \<in> borel_measurable (lebesgue_on S)"
             using h_lmeas S
             by (auto simp: indicator_vimage [symmetric] borel_measurable_indicator_iff sets_restrict_space_iff)
           then show "(\<lambda>u. \<bar>det (matrix (g' u))\<bar> * y * indicat_real {x. h n x = y} (g u)) \<in> borel_measurable (lebesgue_on S)"
             by (intro borel_measurable_times borel_measurable_abs borel_measurable_const borel_measurable_det_Jacobian [OF S der_g])
         next
           fix x
           assume "x \<in> S"
           have "y * indicat_real {x. h n x = y} (g x) \<le> f (g x)"
             by (metis (full_types) \<open>x \<in> S\<close> h_le_f indicator_def mem_Collect_eq mult.right_neutral mult_zero_right nonneg_fg)
           with \<open>y \<ge> 0\<close> show "norm (?D x * y * indicat_real {x. h n x = y} (g x)) \<le> ?D x * f(g x)"
             by (simp add: abs_mult mult.assoc mult_left_mono)
         qed (use S det_int_fg in auto)
       qed
       also have "\<dots> = integral S (\<lambda>T. \<bar>det (matrix (g' T))\<bar> *
                                         (\<Sum>y\<in>range (h n). y * indicat_real {x. h n x = y} (g T)))"
         by (simp add: sum_distrib_left mult.assoc)
       also have "\<dots> = ?rhs"
         by (metis hn_eq)
       finally show "integral (g ` S) (h n) \<le> ?rhs" .
     qed
   qed
   have le: "integral S (\<lambda>T. \<bar>det (matrix (g' T))\<bar> * h n (g T)) \<le> ?b" for n
   proof (rule integral_le)
     show "(\<lambda>T. \<bar>det (matrix (g' T))\<bar> * h n (g T)) integrable_on S"
     proof (rule measurable_bounded_by_integrable_imp_integrable)
       have "(\<lambda>T. \<bar>det (matrix (g' T))\<bar> *\<^sub>R h n (g T)) \<in> borel_measurable (lebesgue_on S)"
       proof (intro borel_measurable_scaleR borel_measurable_abs borel_measurable_det_Jacobian \<open>S \<in> sets lebesgue\<close>)
         have eq: "{x \<in> S. f x \<le> a} = (\<Union>b \<in> (f ` S) \<inter> atMost a. {x. f x = b} \<inter> S)" for f and a::real
           by auto
         have "finite ((\<lambda>x. h n (g x)) ` S \<inter> {..a})" for a
           by (force intro: finite_subset [OF _ fin_R])
         with h_lmeas [of n] show "(\<lambda>x. h n (g x)) \<in> borel_measurable (lebesgue_on S)"
           apply (simp add: borel_measurable_vimage_halfspace_component_le \<open>S \<in> sets lebesgue\<close> sets_restrict_space_iff eq)
           by (metis (mono_tags) SUP_inf sets.finite_UN)
       qed (use der_g in blast)
       then show "(\<lambda>T. \<bar>det (matrix (g' T))\<bar> * h n (g T)) \<in> borel_measurable (lebesgue_on S)"
         by simp
       show "norm (?D x * h n (g x)) \<le> ?D x *\<^sub>R f (g x)"
         if "x \<in> S" for x
         by (simp add: h_le_f mult_left_mono nonneg_h that)
     qed (use S det_int_fg in auto)
     show "?D x * h n (g x) \<le> ?D x * f (g x)" if "x \<in> S" for x
       by (simp add: \<open>x \<in> S\<close> h_le_f mult_left_mono)
     show "(\<lambda>x. ?D x * f (g x)) integrable_on S"
       using det_int_fg by blast
   qed
   have "f integrable_on g ` S \<and> (\<lambda>k. integral (g ` S) (h k)) \<longlonglongrightarrow> integral (g ` S) f"
   proof (rule monotone_convergence_increasing)
     have "\<bar>integral (g ` S) (h n)\<bar> \<le> integral S (\<lambda>x. ?D x * f (g x))" for n
     proof -
       have "\<bar>integral (g ` S) (h n)\<bar> = integral (g ` S) (h n)"
         using hint by (simp add: integral_nonneg nonneg_h)
       also have "\<dots> \<le> integral S (\<lambda>x. ?D x * f (g x))"
         using hint le by (meson order_trans)
       finally show ?thesis .
     qed
     then show "bounded (range (\<lambda>k. integral (g ` S) (h k)))"
       by (force simp: bounded_iff)
   qed (use h_inc lim hint in auto)
   moreover have "integral (g ` S) (h n) \<le> integral S (\<lambda>x. ?D x * f (g x))" for n
     using hint by (blast intro: le order_trans)
   ultimately show ?thesis
     by (auto intro: Lim_bounded)
 qed
 
 
 lemma integral_on_image_ubound_nonneg:
   fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real"
   assumes nonneg_fg: "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> f(g x)"
       and der_g:   "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
       and intS: "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f(g x)) integrable_on S"
   shows "f integrable_on (g ` S) \<and> integral (g ` S) f \<le> integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f(g x))"
          (is "_ \<and> _ \<le> ?b")
 proof -
   let ?D = "\<lambda>x. det (matrix (g' x))"
   define S' where "S' \<equiv> {x \<in> S. ?D x * f(g x) \<noteq> 0}"
   then have der_gS': "\<And>x. x \<in> S' \<Longrightarrow> (g has_derivative g' x) (at x within S')"
-    by (metis (mono_tags, lifting) der_g has_derivative_within_subset mem_Collect_eq subset_iff)
+    by (metis (mono_tags, lifting) der_g has_derivative_subset mem_Collect_eq subset_iff)
   have "(\<lambda>x. if x \<in> S then \<bar>?D x\<bar> * f (g x) else 0) integrable_on UNIV"
     by (simp add: integrable_restrict_UNIV intS)
   then have Df_borel: "(\<lambda>x. if x \<in> S then \<bar>?D x\<bar> * f (g x) else 0) \<in> borel_measurable lebesgue"
     using integrable_imp_measurable lebesgue_on_UNIV_eq by force
   have S': "S' \<in> sets lebesgue"
   proof -
     from Df_borel borel_measurable_vimage_open [of _ UNIV]
     have "{x. (if x \<in> S then \<bar>?D x\<bar> * f (g x) else 0) \<in> T} \<in> sets lebesgue"
       if "open T" for T
       using that unfolding lebesgue_on_UNIV_eq
       by (fastforce simp add: dest!: spec)
     then have "{x. (if x \<in> S then \<bar>?D x\<bar> * f (g x) else 0) \<in> -{0}} \<in> sets lebesgue"
       using open_Compl by blast
     then show ?thesis
       by (simp add: S'_def conj_ac split: if_split_asm cong: conj_cong)
   qed
   then have gS': "g ` S' \<in> sets lebesgue"
   proof (rule differentiable_image_in_sets_lebesgue)
     show "g differentiable_on S'"
       using der_g unfolding S'_def differentiable_def differentiable_on_def
-      by (blast intro: has_derivative_within_subset)
+      by (blast intro: has_derivative_subset)
   qed auto
   have f: "f \<in> borel_measurable (lebesgue_on (g ` S'))"
   proof (clarsimp simp add: borel_measurable_vimage_open)
     fix T :: "real set"
     assume "open T"
     have "{x \<in> g ` S'. f x \<in> T} = g ` {x \<in> S'. f(g x) \<in> T}"
       by blast
     moreover have "g ` {x \<in> S'. f(g x) \<in> T} \<in> sets lebesgue"
     proof (rule differentiable_image_in_sets_lebesgue)
       let ?h = "\<lambda>x. \<bar>?D x\<bar> * f (g x) /\<^sub>R \<bar>?D x\<bar>"
       have "(\<lambda>x. if x \<in> S' then \<bar>?D x\<bar> * f (g x) else 0) = (\<lambda>x. if x \<in> S then \<bar>?D x\<bar> * f (g x) else 0)"
         by (auto simp: S'_def)
       also have "\<dots> \<in> borel_measurable lebesgue"
         by (rule Df_borel)
       finally have *: "(\<lambda>x. \<bar>?D x\<bar> * f (g x)) \<in> borel_measurable (lebesgue_on S')"
         by (simp add: borel_measurable_if_D)
       have "?h \<in> borel_measurable (lebesgue_on S')"
         by (intro * S' der_gS' borel_measurable_det_Jacobian measurable) (blast intro: der_gS')
       moreover have "?h x = f(g x)" if "x \<in> S'" for x
         using that by (auto simp: S'_def)
       ultimately have "(\<lambda>x. f(g x)) \<in> borel_measurable (lebesgue_on S')"
         by (metis (no_types, lifting) measurable_lebesgue_cong)
       then show "{x \<in> S'. f (g x) \<in> T} \<in> sets lebesgue"
         by (simp add: \<open>S' \<in> sets lebesgue\<close> \<open>open T\<close> borel_measurable_vimage_open sets_restrict_space_iff)
       show "g differentiable_on {x \<in> S'. f (g x) \<in> T}"
         using der_g unfolding S'_def differentiable_def differentiable_on_def
-        by (blast intro: has_derivative_within_subset)
+        by (blast intro: has_derivative_subset)
     qed auto
     ultimately have "{x \<in> g ` S'. f x \<in> T} \<in> sets lebesgue"
       by metis
     then show "{x \<in> g ` S'. f x \<in> T} \<in> sets (lebesgue_on (g ` S'))"
       by (simp add: \<open>g ` S' \<in> sets lebesgue\<close> sets_restrict_space_iff)
   qed
   have intS': "(\<lambda>x. \<bar>?D x\<bar> * f (g x)) integrable_on S'"
     using intS
     by (rule integrable_spike_set) (auto simp: S'_def intro: empty_imp_negligible)
   have lebS': "{x \<in> S'. g x \<in> T} \<in> sets lebesgue" if "T \<subseteq> g ` S'" "T \<in> sets lebesgue" for T
   proof -
     have "g \<in> borel_measurable (lebesgue_on S')"
       using der_gS' has_derivative_continuous_on S'
       by (blast intro: continuous_imp_measurable_on_sets_lebesgue)
     moreover have "{x \<in> S'. g x \<in> U} \<in> sets lebesgue" if "negligible U" "U \<subseteq> g ` S'" for U
     proof (intro negligible_imp_sets negligible_differentiable_vimage that)
       fix x
       assume x: "x \<in> S'"
       then have "linear (g' x)"
         using der_gS' has_derivative_linear by blast
       with x show "inj (g' x)"
         by (auto simp: S'_def det_nz_iff_inj)
     qed (use der_gS' in auto)
     ultimately show ?thesis
       using double_lebesgue_sets [OF S' gS' order_refl] that by blast
   qed
   have int_gS': "f integrable_on g ` S' \<and> integral (g ` S') f \<le> integral S' (\<lambda>x. \<bar>?D x\<bar> * f(g x))"
     using integral_on_image_ubound_weak [OF S' f nonneg_fg der_gS' intS' lebS'] S'_def by blast
   have "negligible (g ` {x \<in> S. det(matrix(g' x)) = 0})"
   proof (rule baby_Sard, simp_all)
     fix x
     assume x: "x \<in> S \<and> det (matrix (g' x)) = 0"
     then show "(g has_derivative g' x) (at x within {x \<in> S. det (matrix (g' x)) = 0})"
-      by (metis (no_types, lifting) der_g has_derivative_within_subset mem_Collect_eq subsetI)
+      by (metis (no_types, lifting) der_g has_derivative_subset mem_Collect_eq subsetI)
     then show "rank (matrix (g' x)) < CARD('n)"
       using det_nz_iff_inj matrix_vector_mul_linear x
       by (fastforce simp add: less_rank_noninjective)
   qed
   then have negg: "negligible (g ` S - g ` {x \<in> S. ?D x \<noteq> 0})"
     by (rule negligible_subset) (auto simp: S'_def)
   have null: "g ` {x \<in> S. ?D x \<noteq> 0} - g ` S = {}"
     by (auto simp: S'_def)
   let ?F = "{x \<in> S. f (g x) \<noteq> 0}"
   have eq: "g ` S' = g ` ?F \<inter> g ` {x \<in> S. ?D x \<noteq> 0}"
     by (auto simp: S'_def image_iff)
   show ?thesis
   proof
     have "((\<lambda>x. if x \<in> g ` ?F then f x else 0) integrable_on g ` {x \<in> S. ?D x \<noteq> 0})"
       using int_gS' eq integrable_restrict_Int [where f=f]
       by simp
     then have "f integrable_on g ` {x \<in> S. ?D x \<noteq> 0}"
       by (auto simp: image_iff elim!: integrable_eq)
     then show "f integrable_on g ` S"
       apply (rule integrable_spike_set [OF _ empty_imp_negligible negligible_subset])
       using negg null by auto
     have "integral (g ` S) f = integral (g ` {x \<in> S. ?D x \<noteq> 0}) f"
       using negg by (auto intro: negligible_subset integral_spike_set)
     also have "\<dots> = integral (g ` {x \<in> S. ?D x \<noteq> 0}) (\<lambda>x. if x \<in> g ` ?F then f x else 0)"
       by (auto simp: image_iff intro!: integral_cong)
     also have "\<dots> = integral (g ` S') f"
       using  eq integral_restrict_Int by simp
     also have "\<dots> \<le> integral S' (\<lambda>x. \<bar>?D x\<bar> * f(g x))"
       by (metis int_gS')
     also have "\<dots> \<le> ?b"
       by (rule integral_subset_le [OF _ intS' intS]) (use nonneg_fg S'_def in auto)
     finally show "integral (g ` S) f \<le> ?b" .
   qed
 qed
 
 
 lemma absolutely_integrable_on_image_real:
   fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real" and g :: "real^'n::_ \<Rightarrow> real^'n::_"
   assumes der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
     and intS: "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f(g x)) absolutely_integrable_on S"
   shows "f absolutely_integrable_on (g ` S)"
 proof -
   let ?D = "\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f (g x)"
   let ?N = "{x \<in> S. f (g x) < 0}" and ?P = "{x \<in> S. f (g x) > 0}"
   have eq: "{x. (if x \<in> S then ?D x else 0) > 0} = {x \<in> S. ?D x > 0}"
            "{x. (if x \<in> S then ?D x else 0) < 0} = {x \<in> S. ?D x < 0}"
     by auto
   have "?D integrable_on S"
     using intS absolutely_integrable_on_def by blast
   then have "(\<lambda>x. if x \<in> S then ?D x else 0) integrable_on UNIV"
     by (simp add: integrable_restrict_UNIV)
   then have D_borel: "(\<lambda>x. if x \<in> S then ?D x else 0) \<in> borel_measurable (lebesgue_on UNIV)"
     using integrable_imp_measurable lebesgue_on_UNIV_eq by blast
   then have Dlt: "{x \<in> S. ?D x < 0} \<in> sets lebesgue"
     unfolding borel_measurable_vimage_halfspace_component_lt
     by (drule_tac x=0 in spec) (auto simp: eq)
   from D_borel have Dgt: "{x \<in> S. ?D x > 0} \<in> sets lebesgue"
     unfolding borel_measurable_vimage_halfspace_component_gt
     by (drule_tac x=0 in spec) (auto simp: eq)
 
   have dfgbm: "?D \<in> borel_measurable (lebesgue_on S)"
     using intS absolutely_integrable_on_def integrable_imp_measurable by blast
   have der_gN: "(g has_derivative g' x) (at x within ?N)" if "x \<in> ?N" for x
-      using der_g has_derivative_within_subset that by force
+      using der_g has_derivative_subset that by force
   have "(\<lambda>x. - f x) integrable_on g ` ?N \<and>
          integral (g ` ?N) (\<lambda>x. - f x) \<le> integral ?N (\<lambda>x. \<bar>det (matrix (g' x))\<bar> * - f (g x))"
   proof (rule integral_on_image_ubound_nonneg [OF _ der_gN])
     have 1: "?D integrable_on {x \<in> S. ?D x < 0}"
       using Dlt
       by (auto intro: set_lebesgue_integral_eq_integral [OF set_integrable_subset] intS)
     have "uminus \<circ> (\<lambda>x. \<bar>det (matrix (g' x))\<bar> * - f (g x)) integrable_on ?N"
       by (simp add: o_def mult_less_0_iff empty_imp_negligible integrable_spike_set [OF 1])
     then show "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> * - f (g x)) integrable_on ?N"
       by (simp add: integrable_neg_iff o_def)
   qed auto
   then have "f integrable_on g ` ?N"
     by (simp add: integrable_neg_iff)
   moreover have "g ` ?N = {y \<in> g ` S. f y < 0}"
     by auto
   ultimately have "f integrable_on {y \<in> g ` S. f y < 0}"
     by simp
   then have N: "f absolutely_integrable_on {y \<in> g ` S. f y < 0}"
     by (rule absolutely_integrable_absolutely_integrable_ubound) auto
 
   have der_gP: "(g has_derivative g' x) (at x within ?P)" if "x \<in> ?P" for x
-      using der_g has_derivative_within_subset that by force
+      using der_g has_derivative_subset that by force
   have "f integrable_on g ` ?P \<and> integral (g ` ?P) f \<le> integral ?P ?D"
   proof (rule integral_on_image_ubound_nonneg [OF _ der_gP])
     have "?D integrable_on {x \<in> S. 0 < ?D x}"
       using Dgt
       by (auto intro: set_lebesgue_integral_eq_integral [OF set_integrable_subset] intS)
     then show "?D integrable_on ?P"
       apply (rule integrable_spike_set)
       by (auto simp: zero_less_mult_iff empty_imp_negligible)
   qed auto
   then have "f integrable_on g ` ?P"
     by metis
   moreover have "g ` ?P = {y \<in> g ` S. f y > 0}"
     by auto
   ultimately have "f integrable_on {y \<in> g ` S. f y > 0}"
     by simp
   then have P: "f absolutely_integrable_on {y \<in> g ` S. f y > 0}"
     by (rule absolutely_integrable_absolutely_integrable_lbound) auto
   have "(\<lambda>x. if x \<in> g ` S \<and> f x < 0 \<or> x \<in> g ` S \<and> 0 < f x then f x else 0) = (\<lambda>x. if x \<in> g ` S then f x else 0)"
     by auto
   then show ?thesis
     using absolutely_integrable_Un [OF N P] absolutely_integrable_restrict_UNIV [symmetric, where f=f]
     by simp
 qed
 
 
 proposition absolutely_integrable_on_image:
   fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
   assumes der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
     and intS: "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S"
   shows "f absolutely_integrable_on (g ` S)"
   apply (rule absolutely_integrable_componentwise [OF absolutely_integrable_on_image_real [OF der_g]])
   using absolutely_integrable_component [OF intS]  by auto
 
 proposition integral_on_image_ubound:
   fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real" and g :: "real^'n::_ \<Rightarrow> real^'n::_"
   assumes"\<And>x. x \<in> S \<Longrightarrow> 0 \<le> f(g x)"
     and "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
     and "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f(g x)) integrable_on S"
   shows "integral (g ` S) f \<le> integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f(g x))"
   using integral_on_image_ubound_nonneg [OF assms] by simp
 
 
 subsection\<open>Change-of-variables theorem\<close>
 
 text\<open>The classic change-of-variables theorem. We have two versions with quite general hypotheses,
 the first that the transforming function has a continuous inverse, the second that the base set is
 Lebesgue measurable.\<close>
 lemma cov_invertible_nonneg_le:
   fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real" and g :: "real^'n::_ \<Rightarrow> real^'n::_"
   assumes der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
     and der_h: "\<And>y. y \<in> T \<Longrightarrow> (h has_derivative h' y) (at y within T)"
     and f0: "\<And>y. y \<in> T \<Longrightarrow> 0 \<le> f y"
     and hg: "\<And>x. x \<in> S \<Longrightarrow> g x \<in> T \<and> h(g x) = x"
     and gh: "\<And>y. y \<in> T \<Longrightarrow> h y \<in> S \<and> g(h y) = y"
     and id: "\<And>y. y \<in> T \<Longrightarrow> h' y \<circ> g'(h y) = id"
   shows "f integrable_on T \<and> (integral T f) \<le> b \<longleftrightarrow>
              (\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f(g x)) integrable_on S \<and>
              integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f(g x)) \<le> b"
         (is "?lhs = ?rhs")
 proof -
   have Teq: "T = g`S" and Seq: "S = h`T"
     using hg gh image_iff by fastforce+
   have gS: "g differentiable_on S"
     by (meson der_g differentiable_def differentiable_on_def)
   let ?D = "\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f (g x)"
   show ?thesis
   proof
     assume ?lhs
     then have fT: "f integrable_on T" and intf: "integral T f \<le> b"
       by blast+
     show ?rhs
     proof
       let ?fgh = "\<lambda>x. \<bar>det (matrix (h' x))\<bar> * (\<bar>det (matrix (g' (h x)))\<bar> * f (g (h x)))"
       have ddf: "?fgh x = f x"
         if "x \<in> T" for x
       proof -
         have "matrix (h' x) ** matrix (g' (h x)) = mat 1"
           using that id[OF that] der_g[of "h x"] gh[OF that] left_inverse_linear has_derivative_linear
           by (subst matrix_compose[symmetric]) (force simp: matrix_id_mat_1 has_derivative_linear)+
         then have "\<bar>det (matrix (h' x))\<bar> * \<bar>det (matrix (g' (h x)))\<bar> = 1"
           by (metis abs_1 abs_mult det_I det_mul)
         then show ?thesis
           by (simp add: gh that)
       qed
       have "?D integrable_on (h ` T)"
       proof (intro set_lebesgue_integral_eq_integral absolutely_integrable_on_image_real)
         show "(\<lambda>x. ?fgh x) absolutely_integrable_on T"
         proof (subst absolutely_integrable_on_iff_nonneg)
           show "(\<lambda>x. ?fgh x) integrable_on T"
             using ddf fT integrable_eq by force
         qed (simp add: zero_le_mult_iff f0 gh)
       qed (use der_h in auto)
       with Seq show "(\<lambda>x. ?D x) integrable_on S"
         by simp
       have "integral S (\<lambda>x. ?D x) \<le> integral T (\<lambda>x. ?fgh x)"
         unfolding Seq
       proof (rule integral_on_image_ubound)
         show "(\<lambda>x. ?fgh x) integrable_on T"
         using ddf fT integrable_eq by force
       qed (use f0 gh der_h in auto)
       also have "\<dots> = integral T f"
         by (force simp: ddf intro: integral_cong)
       also have "\<dots> \<le> b"
         by (rule intf)
       finally show "integral S (\<lambda>x. ?D x) \<le> b" .
     qed
   next
     assume R: ?rhs
     then have "f integrable_on g ` S"
       using der_g f0 hg integral_on_image_ubound_nonneg by blast
     moreover have "integral (g ` S) f \<le> integral S (\<lambda>x. ?D x)"
       by (rule integral_on_image_ubound [OF f0 der_g]) (use R Teq in auto)
     ultimately show ?lhs
       using R by (simp add: Teq)
   qed
 qed
 
 
 lemma cov_invertible_nonneg_eq:
   fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real" and g :: "real^'n::_ \<Rightarrow> real^'n::_"
   assumes "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
       and "\<And>y. y \<in> T \<Longrightarrow> (h has_derivative h' y) (at y within T)"
       and "\<And>y. y \<in> T \<Longrightarrow> 0 \<le> f y"
       and "\<And>x. x \<in> S \<Longrightarrow> g x \<in> T \<and> h(g x) = x"
       and "\<And>y. y \<in> T \<Longrightarrow> h y \<in> S \<and> g(h y) = y"
       and "\<And>y. y \<in> T \<Longrightarrow> h' y \<circ> g'(h y) = id"
   shows "((\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f(g x)) has_integral b) S \<longleftrightarrow> (f has_integral b) T"
   using cov_invertible_nonneg_le [OF assms]
   by (simp add: has_integral_iff) (meson eq_iff)
 
 
 lemma cov_invertible_real:
   fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real" and g :: "real^'n::_ \<Rightarrow> real^'n::_"
   assumes der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
       and der_h: "\<And>y. y \<in> T \<Longrightarrow> (h has_derivative h' y) (at y within T)"
       and hg: "\<And>x. x \<in> S \<Longrightarrow> g x \<in> T \<and> h(g x) = x"
       and gh: "\<And>y. y \<in> T \<Longrightarrow> h y \<in> S \<and> g(h y) = y"
       and id: "\<And>y. y \<in> T \<Longrightarrow> h' y \<circ> g'(h y) = id"
   shows "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f(g x)) absolutely_integrable_on S \<and>
            integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f(g x)) = b \<longleftrightarrow>
          f absolutely_integrable_on T \<and> integral T f = b"
          (is "?lhs = ?rhs")
 proof -
   have Teq: "T = g`S" and Seq: "S = h`T"
     using hg gh image_iff by fastforce+
   let ?DP = "\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f(g x)" and ?DN = "\<lambda>x. \<bar>det (matrix (g' x))\<bar> * -f(g x)"
   have "+": "(?DP has_integral b) {x \<in> S. f (g x) > 0} \<longleftrightarrow> (f has_integral b) {y \<in> T. f y > 0}" for b
   proof (rule cov_invertible_nonneg_eq)
     have *: "(\<lambda>x. f (g x)) -` Y \<inter> {x \<in> S. f (g x) > 0}
           = ((\<lambda>x. f (g x)) -` Y \<inter> S) \<inter> {x \<in> S. f (g x) > 0}" for Y
       by auto
     show "(g has_derivative g' x) (at x within {x \<in> S. f (g x) > 0})" if "x \<in> {x \<in> S. f (g x) > 0}" for x
-      using that der_g has_derivative_within_subset by fastforce
+      using that der_g has_derivative_subset by fastforce
     show "(h has_derivative h' y) (at y within {y \<in> T. f y > 0})" if "y \<in> {y \<in> T. f y > 0}" for y
-      using that der_h has_derivative_within_subset by fastforce
+      using that der_h has_derivative_subset by fastforce
   qed (use gh hg id in auto)
   have "-": "(?DN has_integral b) {x \<in> S. f (g x) < 0} \<longleftrightarrow> ((\<lambda>x. - f x) has_integral b) {y \<in> T. f y < 0}" for b
   proof (rule cov_invertible_nonneg_eq)
     have *: "(\<lambda>x. - f (g x)) -` y \<inter> {x \<in> S. f (g x) < 0}
           = ((\<lambda>x. f (g x)) -` uminus ` y \<inter> S) \<inter> {x \<in> S. f (g x) < 0}" for y
       using image_iff by fastforce
     show "(g has_derivative g' x) (at x within {x \<in> S. f (g x) < 0})" if "x \<in> {x \<in> S. f (g x) < 0}" for x
-      using that der_g has_derivative_within_subset by fastforce
+      using that der_g has_derivative_subset by fastforce
     show "(h has_derivative h' y) (at y within {y \<in> T. f y < 0})" if "y \<in> {y \<in> T. f y < 0}" for y
-      using that der_h has_derivative_within_subset by fastforce
+      using that der_h has_derivative_subset by fastforce
   qed (use gh hg id in auto)
   show ?thesis
   proof
     assume LHS: ?lhs
     have eq: "{x. (if x \<in> S then ?DP x else 0) > 0} = {x \<in> S. ?DP x > 0}"
       "{x. (if x \<in> S then ?DP x else 0) < 0} = {x \<in> S. ?DP x < 0}"
       by auto
     have "?DP integrable_on S"
       using LHS absolutely_integrable_on_def by blast
     then have "(\<lambda>x. if x \<in> S then ?DP x else 0) integrable_on UNIV"
       by (simp add: integrable_restrict_UNIV)
     then have D_borel: "(\<lambda>x. if x \<in> S then ?DP x else 0) \<in> borel_measurable (lebesgue_on UNIV)"
       using integrable_imp_measurable lebesgue_on_UNIV_eq by blast
     then have SN: "{x \<in> S. ?DP x < 0} \<in> sets lebesgue"
       unfolding borel_measurable_vimage_halfspace_component_lt
       by (drule_tac x=0 in spec) (auto simp: eq)
     from D_borel have SP: "{x \<in> S. ?DP x > 0} \<in> sets lebesgue"
       unfolding borel_measurable_vimage_halfspace_component_gt
       by (drule_tac x=0 in spec) (auto simp: eq)
     have "?DP absolutely_integrable_on {x \<in> S. ?DP x > 0}"
       using LHS by (fast intro!: set_integrable_subset [OF _, of _ S] SP)
     then have aP: "?DP absolutely_integrable_on {x \<in> S. f (g x) > 0}"
       by (rule absolutely_integrable_spike_set) (auto simp: zero_less_mult_iff empty_imp_negligible)
     have "?DP absolutely_integrable_on {x \<in> S. ?DP x < 0}"
       using LHS by (fast intro!: set_integrable_subset [OF _, of _ S] SN)
     then have aN: "?DP absolutely_integrable_on {x \<in> S. f (g x) < 0}"
       by (rule absolutely_integrable_spike_set) (auto simp: mult_less_0_iff empty_imp_negligible)
     have fN: "f integrable_on {y \<in> T. f y < 0}"
              "integral {y \<in> T. f y < 0} f = integral {x \<in> S. f (g x) < 0} ?DP"
       using "-" [of "integral {x \<in> S. f(g x) < 0} ?DN"] aN
       by (auto simp: set_lebesgue_integral_eq_integral has_integral_iff integrable_neg_iff)
     have faN: "f absolutely_integrable_on {y \<in> T. f y < 0}"
       apply (rule absolutely_integrable_integrable_bound [where g = "\<lambda>x. - f x"])
       using fN by (auto simp: integrable_neg_iff)
     have fP: "f integrable_on {y \<in> T. f y > 0}"
              "integral {y \<in> T. f y > 0} f = integral {x \<in> S. f (g x) > 0} ?DP"
       using "+" [of "integral {x \<in> S. f(g x) > 0} ?DP"] aP
       by (auto simp: set_lebesgue_integral_eq_integral has_integral_iff integrable_neg_iff)
     have faP: "f absolutely_integrable_on {y \<in> T. f y > 0}"
       apply (rule absolutely_integrable_integrable_bound [where g = f])
       using fP by auto
     have fa: "f absolutely_integrable_on ({y \<in> T. f y < 0} \<union> {y \<in> T. f y > 0})"
       by (rule absolutely_integrable_Un [OF faN faP])
     show ?rhs
     proof
       have eq: "((if x \<in> T \<and> f x < 0 \<or> x \<in> T \<and> 0 < f x then 1 else 0) * f x)
               = (if x \<in> T then 1 else 0) * f x" for x
         by auto
       show "f absolutely_integrable_on T"
         using fa by (simp add: indicator_def set_integrable_def eq)
       have [simp]: "{y \<in> T. f y < 0} \<inter> {y \<in> T. 0 < f y} = {}" for T and f :: "(real^'n::_) \<Rightarrow> real"
         by auto
       have "integral T f = integral ({y \<in> T. f y < 0} \<union> {y \<in> T. f y > 0}) f"
         by (intro empty_imp_negligible integral_spike_set) (auto simp: eq)
       also have "\<dots> = integral {y \<in> T. f y < 0} f + integral {y \<in> T. f y > 0} f"
         using fN fP by simp
       also have "\<dots> = integral {x \<in> S. f (g x) < 0} ?DP + integral {x \<in> S. 0 < f (g x)} ?DP"
         by (simp add: fN fP)
       also have "\<dots> = integral ({x \<in> S. f (g x) < 0} \<union> {x \<in> S. 0 < f (g x)}) ?DP"
         using aP aN by (simp add: set_lebesgue_integral_eq_integral)
       also have "\<dots> = integral S ?DP"
         by (intro empty_imp_negligible integral_spike_set) auto
       also have "\<dots> = b"
         using LHS by simp
       finally show "integral T f = b" .
     qed
   next
     assume RHS: ?rhs
     have eq: "{x. (if x \<in> T then f x else 0) > 0} = {x \<in> T. f x > 0}"
              "{x. (if x \<in> T then f x else 0) < 0} = {x \<in> T. f x < 0}"
       by auto
     have "f integrable_on T"
       using RHS absolutely_integrable_on_def by blast
     then have "(\<lambda>x. if x \<in> T then f x else 0) integrable_on UNIV"
       by (simp add: integrable_restrict_UNIV)
     then have D_borel: "(\<lambda>x. if x \<in> T then f x else 0) \<in> borel_measurable (lebesgue_on UNIV)"
       using integrable_imp_measurable lebesgue_on_UNIV_eq by blast
     then have TN: "{x \<in> T. f x < 0} \<in> sets lebesgue"
       unfolding borel_measurable_vimage_halfspace_component_lt
       by (drule_tac x=0 in spec) (auto simp: eq)
     from D_borel have TP: "{x \<in> T. f x > 0} \<in> sets lebesgue"
       unfolding borel_measurable_vimage_halfspace_component_gt
       by (drule_tac x=0 in spec) (auto simp: eq)
     have aint: "f absolutely_integrable_on {y. y \<in> T \<and> 0 < (f y)}"
                "f absolutely_integrable_on {y. y \<in> T \<and> (f y) < 0}"
          and intT: "integral T f = b"
       using set_integrable_subset [of _ T] TP TN RHS
       by blast+
     show ?lhs
     proof
       have fN: "f integrable_on {v \<in> T. f v < 0}"
         using absolutely_integrable_on_def aint by blast
       then have DN: "(?DN has_integral integral {y \<in> T. f y < 0} (\<lambda>x. - f x)) {x \<in> S. f (g x) < 0}"
         using "-" [of "integral {y \<in> T. f y < 0} (\<lambda>x. - f x)"]
         by (simp add: has_integral_neg_iff integrable_integral)
       have aDN: "?DP absolutely_integrable_on {x \<in> S. f (g x) < 0}"
         apply (rule absolutely_integrable_integrable_bound [where g = ?DN])
         using DN hg by (fastforce simp: abs_mult integrable_neg_iff)+
       have fP: "f integrable_on {v \<in> T. f v > 0}"
         using absolutely_integrable_on_def aint by blast
       then have DP: "(?DP has_integral integral {y \<in> T. f y > 0} f) {x \<in> S. f (g x) > 0}"
         using "+" [of "integral {y \<in> T. f y > 0} f"]
         by (simp add: has_integral_neg_iff integrable_integral)
       have aDP: "?DP absolutely_integrable_on {x \<in> S. f (g x) > 0}"
         apply (rule absolutely_integrable_integrable_bound [where g = ?DP])
         using DP hg by (fastforce simp: integrable_neg_iff)+
       have eq: "(if x \<in> S then 1 else 0) * ?DP x = (if x \<in> S \<and> f (g x) < 0 \<or> x \<in> S \<and> f (g x) > 0 then 1 else 0) * ?DP x" for x
         by force
       have "?DP absolutely_integrable_on ({x \<in> S. f (g x) < 0} \<union> {x \<in> S. f (g x) > 0})"
         by (rule absolutely_integrable_Un [OF aDN aDP])
       then show I: "?DP absolutely_integrable_on S"
         by (simp add: indicator_def eq set_integrable_def)
       have [simp]: "{y \<in> S. f y < 0} \<inter> {y \<in> S. 0 < f y} = {}" for S and f :: "(real^'n::_) \<Rightarrow> real"
         by auto
       have "integral S ?DP = integral ({x \<in> S. f (g x) < 0} \<union> {x \<in> S. f (g x) > 0}) ?DP"
         by (intro empty_imp_negligible integral_spike_set) auto
       also have "\<dots> = integral {x \<in> S. f (g x) < 0} ?DP + integral {x \<in> S. 0 < f (g x)} ?DP"
         using aDN aDP by (simp add: set_lebesgue_integral_eq_integral)
       also have "\<dots> = - integral {y \<in> T. f y < 0} (\<lambda>x. - f x) + integral {y \<in> T. f y > 0} f"
         using DN DP by (auto simp: has_integral_iff)
       also have "\<dots> = integral ({x \<in> T. f x < 0} \<union> {x \<in> T. 0 < f x}) f"
         by (simp add: fN fP)
       also have "\<dots> = integral T f"
         by (intro empty_imp_negligible integral_spike_set) auto
       also have "\<dots> = b"
         using intT by simp
       finally show "integral S ?DP = b" .
     qed
   qed
 qed
 
 
 lemma cv_inv_version3:
   fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
   assumes der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
     and der_h: "\<And>y. y \<in> T \<Longrightarrow> (h has_derivative h' y) (at y within T)"
     and hg: "\<And>x. x \<in> S \<Longrightarrow> g x \<in> T \<and> h(g x) = x"
     and gh: "\<And>y. y \<in> T \<Longrightarrow> h y \<in> S \<and> g(h y) = y"
     and id: "\<And>y. y \<in> T \<Longrightarrow> h' y \<circ> g'(h y) = id"
   shows "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S \<and>
              integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) = b
          \<longleftrightarrow> f absolutely_integrable_on T \<and> integral T f = b"
 proof -
   let ?D = "\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)"
   have "((\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f(g x) $ i) absolutely_integrable_on S \<and> integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> * (f(g x) $ i)) = b $ i) \<longleftrightarrow>
         ((\<lambda>x. f x $ i) absolutely_integrable_on T \<and> integral T (\<lambda>x. f x $ i) = b $ i)" for i
     by (rule cov_invertible_real [OF der_g der_h hg gh id])
   then have "?D absolutely_integrable_on S \<and> (?D has_integral b) S \<longleftrightarrow>
         f absolutely_integrable_on T \<and> (f has_integral b) T"
     unfolding absolutely_integrable_componentwise_iff [where f=f] has_integral_componentwise_iff [of f]
               absolutely_integrable_componentwise_iff [where f="?D"] has_integral_componentwise_iff [of ?D]
     by (auto simp: all_conj_distrib Basis_vec_def cart_eq_inner_axis [symmetric]
            has_integral_iff set_lebesgue_integral_eq_integral)
   then show ?thesis
     using absolutely_integrable_on_def by blast
 qed
 
 
 lemma cv_inv_version4:
   fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
   assumes der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S) \<and> invertible(matrix(g' x))"
     and hg: "\<And>x. x \<in> S \<Longrightarrow> continuous_on (g ` S) h \<and> h(g x) = x"
   shows "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S \<and>
              integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) = b
          \<longleftrightarrow> f absolutely_integrable_on (g ` S) \<and> integral (g ` S) f = b"
 proof -
   have "\<forall>x. \<exists>h'. x \<in> S
            \<longrightarrow> (g has_derivative g' x) (at x within S) \<and> linear h' \<and> g' x \<circ> h' = id \<and> h' \<circ> g' x = id"
     using der_g matrix_invertible has_derivative_linear by blast
   then obtain h' where h':
     "\<And>x. x \<in> S
            \<Longrightarrow> (g has_derivative g' x) (at x within S) \<and>
                linear (h' x) \<and> g' x \<circ> (h' x) = id \<and> (h' x) \<circ> g' x = id"
     by metis
   show ?thesis
   proof (rule cv_inv_version3)
     show "\<And>y. y \<in> g ` S \<Longrightarrow> (h has_derivative h' (h y)) (at y within g ` S)"
       using h' hg
       by (force simp: continuous_on_eq_continuous_within intro!: has_derivative_inverse_within)
   qed (use h' hg in auto)
 qed
 
 
 theorem has_absolute_integral_change_of_variables_invertible:
   fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
   assumes der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
       and hg: "\<And>x. x \<in> S \<Longrightarrow> h(g x) = x"
       and conth: "continuous_on (g ` S) h"
   shows "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S \<and> integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) = b \<longleftrightarrow>
          f absolutely_integrable_on (g ` S) \<and> integral (g ` S) f = b"
     (is "?lhs = ?rhs")
 proof -
   let ?S = "{x \<in> S. invertible (matrix (g' x))}" and ?D = "\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)"
   have *: "?D absolutely_integrable_on ?S \<and> integral ?S ?D = b
            \<longleftrightarrow> f absolutely_integrable_on (g ` ?S) \<and> integral (g ` ?S) f = b"
   proof (rule cv_inv_version4)
     show "(g has_derivative g' x) (at x within ?S) \<and> invertible (matrix (g' x))"
       if "x \<in> ?S" for x
-      using der_g that has_derivative_within_subset that by fastforce
+      using der_g that has_derivative_subset that by fastforce
     show "continuous_on (g ` ?S) h \<and> h (g x) = x"
       if "x \<in> ?S" for x
       using that continuous_on_subset [OF conth]  by (simp add: hg image_mono)
   qed
   have "(g has_derivative g' x) (at x within {x \<in> S. rank (matrix (g' x)) < CARD('m)})" if "x \<in> S" for x
-    by (metis (no_types, lifting) der_g has_derivative_within_subset mem_Collect_eq subsetI that)
+    by (metis (no_types, lifting) der_g has_derivative_subset mem_Collect_eq subsetI that)
   then have "negligible (g ` {x \<in> S. \<not> invertible (matrix (g' x))})"
     by (auto simp: invertible_det_nz det_eq_0_rank intro: baby_Sard)
   then have neg: "negligible {x \<in> g ` S. x \<notin> g ` ?S \<and> f x \<noteq> 0}"
     by (auto intro: negligible_subset)
   have [simp]: "{x \<in> g ` ?S. x \<notin> g ` S \<and> f x \<noteq> 0} = {}"
     by auto
   have "?D absolutely_integrable_on ?S \<and> integral ?S ?D = b
     \<longleftrightarrow> ?D absolutely_integrable_on S \<and> integral S ?D = b"
     apply (intro conj_cong absolutely_integrable_spike_set_eq)
       apply(auto simp: integral_spike_set invertible_det_nz empty_imp_negligible neg)
     done
   moreover
   have "f absolutely_integrable_on (g ` ?S) \<and> integral (g ` ?S) f = b
     \<longleftrightarrow> f absolutely_integrable_on (g ` S) \<and> integral (g ` S) f = b"
     by (auto intro!: conj_cong absolutely_integrable_spike_set_eq integral_spike_set neg)
   ultimately
   show ?thesis
     using "*" by blast
 qed
 
 
 
 theorem has_absolute_integral_change_of_variables_compact:
   fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
   assumes "compact S"
       and der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
       and inj: "inj_on g S"
   shows "((\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S \<and>
              integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) = b
       \<longleftrightarrow> f absolutely_integrable_on (g ` S) \<and> integral (g ` S) f = b)"
 proof -
   obtain h where hg: "\<And>x. x \<in> S \<Longrightarrow> h(g x) = x"
     using inj by (metis the_inv_into_f_f)
   have conth: "continuous_on (g ` S) h"
     by (metis \<open>compact S\<close> continuous_on_inv der_g has_derivative_continuous_on hg)
   show ?thesis
     by (rule has_absolute_integral_change_of_variables_invertible [OF der_g hg conth])
 qed
 
 
 lemma has_absolute_integral_change_of_variables_compact_family:
   fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
   assumes compact: "\<And>n::nat. compact (F n)"
       and der_g: "\<And>x. x \<in> (\<Union>n. F n) \<Longrightarrow> (g has_derivative g' x) (at x within (\<Union>n. F n))"
       and inj: "inj_on g (\<Union>n. F n)"
   shows "((\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) absolutely_integrable_on (\<Union>n. F n) \<and>
              integral (\<Union>n. F n) (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) = b
       \<longleftrightarrow> f absolutely_integrable_on (g ` (\<Union>n. F n)) \<and> integral (g ` (\<Union>n. F n)) f = b)"
 proof -
   let ?D = "\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f (g x)"
   let ?U = "\<lambda>n. \<Union>m\<le>n. F m"
   let ?lift = "vec::real\<Rightarrow>real^1"
   have F_leb: "F m \<in> sets lebesgue" for m
     by (simp add: compact borel_compact)
   have iff: "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f (g x)) absolutely_integrable_on (?U n) \<and>
              integral (?U n) (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f (g x)) = b
          \<longleftrightarrow> f absolutely_integrable_on (g ` (?U n)) \<and> integral (g ` (?U n)) f = b" for n b and f :: "real^'m::_ \<Rightarrow> real^'k"
   proof (rule has_absolute_integral_change_of_variables_compact)
     show "compact (?U n)"
       by (simp add: compact compact_UN)
     show "(g has_derivative g' x) (at x within (?U n))"
       if "x \<in> ?U n" for x
-      using that by (blast intro!: has_derivative_within_subset [OF der_g])
+      using that by (blast intro!: has_derivative_subset [OF der_g])
     show "inj_on g (?U n)"
       using inj by (auto simp: inj_on_def)
   qed
   show ?thesis
     unfolding image_UN
   proof safe
     assume DS: "?D absolutely_integrable_on (\<Union>n. F n)"
       and b: "b = integral (\<Union>n. F n) ?D"
     have DU: "\<And>n. ?D absolutely_integrable_on (?U n)"
              "(\<lambda>n. integral (?U n) ?D) \<longlonglongrightarrow> integral (\<Union>n. F n) ?D"
       using integral_countable_UN [OF DS F_leb] by auto
     with iff have fag: "f absolutely_integrable_on g ` (?U n)"
       and fg_int: "integral (\<Union>m\<le>n. g ` F m) f = integral (?U n) ?D" for n
       by (auto simp: image_UN)
     let ?h = "\<lambda>x. if x \<in> (\<Union>m. g ` F m) then norm(f x) else 0"
     have "(\<lambda>x. if x \<in> (\<Union>m. g ` F m) then f x else 0) absolutely_integrable_on UNIV"
     proof (rule dominated_convergence_absolutely_integrable)
       show "(\<lambda>x. if x \<in> (\<Union>m\<le>k. g ` F m) then f x else 0) absolutely_integrable_on UNIV" for k
         unfolding absolutely_integrable_restrict_UNIV
         using fag by (simp add: image_UN)
       let ?nf = "\<lambda>n x. if x \<in> (\<Union>m\<le>n. g ` F m) then norm(f x) else 0"
       show "?h integrable_on UNIV"
       proof (rule monotone_convergence_increasing [THEN conjunct1])
         show "?nf k integrable_on UNIV" for k
           using fag
           unfolding integrable_restrict_UNIV absolutely_integrable_on_def by (simp add: image_UN)
         { fix n
           have "(norm \<circ> ?D) absolutely_integrable_on ?U n"
             by (intro absolutely_integrable_norm DU)
           then have "integral (g ` ?U n) (norm \<circ> f) = integral (?U n) (norm \<circ> ?D)"
             using iff [of n "vec \<circ> norm \<circ> f" "integral (?U n) (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R (?lift \<circ> norm \<circ> f) (g x))"]
             unfolding absolutely_integrable_on_1_iff integral_on_1_eq by (auto simp: o_def)
         }
         moreover have "bounded (range (\<lambda>k. integral (?U k) (norm \<circ> ?D)))"
           unfolding bounded_iff
         proof (rule exI, clarify)
           fix k
           show "norm (integral (?U k) (norm \<circ> ?D)) \<le> integral (\<Union>n. F n) (norm \<circ> ?D)"
             unfolding integral_restrict_UNIV [of _ "norm \<circ> ?D", symmetric]
           proof (rule integral_norm_bound_integral)
             show "(\<lambda>x. if x \<in> \<Union> (F ` {..k}) then (norm \<circ> ?D) x else 0) integrable_on UNIV"
               "(\<lambda>x. if x \<in> (\<Union>n. F n) then (norm \<circ> ?D) x else 0) integrable_on UNIV"
               using DU(1) DS
               unfolding absolutely_integrable_on_def o_def integrable_restrict_UNIV by auto
           qed auto
         qed
         ultimately show "bounded (range (\<lambda>k. integral UNIV (?nf k)))"
           by (simp add: integral_restrict_UNIV image_UN [symmetric] o_def)
       next
         show "(\<lambda>k. if x \<in> (\<Union>m\<le>k. g ` F m) then norm (f x) else 0)
               \<longlonglongrightarrow> (if x \<in> (\<Union>m. g ` F m) then norm (f x) else 0)" for x
           by (force intro: tendsto_eventually eventually_sequentiallyI)
       qed auto
     next
       show "(\<lambda>k. if x \<in> (\<Union>m\<le>k. g ` F m) then f x else 0)
             \<longlonglongrightarrow> (if x \<in> (\<Union>m. g ` F m) then f x else 0)" for x
       proof clarsimp
         fix m y
         assume "y \<in> F m"
         show "(\<lambda>k. if \<exists>x\<in>{..k}. g y \<in> g ` F x then f (g y) else 0) \<longlonglongrightarrow> f (g y)"
           using \<open>y \<in> F m\<close> by (force intro: tendsto_eventually eventually_sequentiallyI [of m])
       qed
     qed auto
     then show fai: "f absolutely_integrable_on (\<Union>m. g ` F m)"
       using absolutely_integrable_restrict_UNIV by blast
     show "integral ((\<Union>x. g ` F x)) f = integral (\<Union>n. F n) ?D"
     proof (rule LIMSEQ_unique)
       show "(\<lambda>n. integral (?U n) ?D) \<longlonglongrightarrow> integral (\<Union>x. g ` F x) f"
         unfolding fg_int [symmetric]
       proof (rule integral_countable_UN [OF fai])
         show "g ` F m \<in> sets lebesgue" for m
         proof (rule differentiable_image_in_sets_lebesgue [OF F_leb])
           show "g differentiable_on F m"
             by (meson der_g differentiableI UnionI differentiable_on_def differentiable_on_subset rangeI subsetI)
         qed auto
       qed
     next
       show "(\<lambda>n. integral (?U n) ?D) \<longlonglongrightarrow> integral (\<Union>n. F n) ?D"
         by (rule DU)
     qed
   next
     assume fs: "f absolutely_integrable_on (\<Union>x. g ` F x)"
       and b: "b = integral ((\<Union>x. g ` F x)) f"
     have gF_leb: "g ` F m \<in> sets lebesgue" for m
     proof (rule differentiable_image_in_sets_lebesgue [OF F_leb])
       show "g differentiable_on F m"
         using der_g unfolding differentiable_def differentiable_on_def
-        by (meson Sup_upper UNIV_I UnionI has_derivative_within_subset image_eqI)
+        by (meson Sup_upper UNIV_I UnionI has_derivative_subset image_eqI)
     qed auto
     have fgU: "\<And>n. f absolutely_integrable_on (\<Union>m\<le>n. g ` F m)"
       "(\<lambda>n. integral (\<Union>m\<le>n. g ` F m) f) \<longlonglongrightarrow> integral (\<Union>m. g ` F m) f"
       using integral_countable_UN [OF fs gF_leb] by auto
     with iff have DUn: "?D absolutely_integrable_on ?U n"
       and D_int: "integral (?U n) ?D = integral (\<Union>m\<le>n. g ` F m) f" for n
       by (auto simp: image_UN)
     let ?h = "\<lambda>x. if x \<in> (\<Union>n. F n) then norm(?D x) else 0"
     have "(\<lambda>x. if x \<in> (\<Union>n. F n) then ?D x else 0) absolutely_integrable_on UNIV"
     proof (rule dominated_convergence_absolutely_integrable)
       show "(\<lambda>x. if x \<in> ?U k then ?D x else 0) absolutely_integrable_on UNIV" for k
         unfolding absolutely_integrable_restrict_UNIV using DUn by simp
       let ?nD = "\<lambda>n x. if x \<in> ?U n then norm(?D x) else 0"
       show "?h integrable_on UNIV"
       proof (rule monotone_convergence_increasing [THEN conjunct1])
         show "?nD k integrable_on UNIV" for k
           using DUn
           unfolding integrable_restrict_UNIV absolutely_integrable_on_def by (simp add: image_UN)
         { fix n::nat
           have "(norm \<circ> f) absolutely_integrable_on (\<Union>m\<le>n. g ` F m)"
             apply (rule absolutely_integrable_norm)
             using fgU by blast
           then have "integral (?U n) (norm \<circ> ?D) = integral (g ` ?U n) (norm \<circ> f)"
             using iff [of n "?lift \<circ> norm \<circ> f" "integral (g ` ?U n) (?lift \<circ> norm \<circ> f)"]
             unfolding absolutely_integrable_on_1_iff integral_on_1_eq image_UN by (auto simp: o_def)
         }
         moreover have "bounded (range (\<lambda>k. integral (g ` ?U k) (norm \<circ> f)))"
           unfolding bounded_iff
         proof (rule exI, clarify)
           fix k
           show "norm (integral (g ` ?U k) (norm \<circ> f)) \<le> integral (g ` (\<Union>n. F n)) (norm \<circ> f)"
             unfolding integral_restrict_UNIV [of _ "norm \<circ> f", symmetric]
           proof (rule integral_norm_bound_integral)
             show "(\<lambda>x. if x \<in> g ` ?U k then (norm \<circ> f) x else 0) integrable_on UNIV"
                  "(\<lambda>x. if x \<in> g ` (\<Union>n. F n) then (norm \<circ> f) x else 0) integrable_on UNIV"
               using fgU fs
               unfolding absolutely_integrable_on_def o_def integrable_restrict_UNIV
               by (auto simp: image_UN)
           qed auto
         qed
         ultimately show "bounded (range (\<lambda>k. integral UNIV (?nD k)))"
           unfolding integral_restrict_UNIV image_UN [symmetric] o_def by simp
       next
         show "(\<lambda>k. if x \<in> ?U k then norm (?D x) else 0) \<longlonglongrightarrow> (if x \<in> (\<Union>n. F n) then norm (?D x) else 0)" for x
           by (force intro: tendsto_eventually eventually_sequentiallyI)
       qed auto
     next
       show "(\<lambda>k. if x \<in> ?U k then ?D x else 0) \<longlonglongrightarrow> (if x \<in> (\<Union>n. F n) then ?D x else 0)" for x
       proof clarsimp
         fix n
         assume "x \<in> F n"
         show "(\<lambda>m. if \<exists>j\<in>{..m}. x \<in> F j then ?D x else 0) \<longlonglongrightarrow> ?D x"
           using \<open>x \<in> F n\<close> by (auto intro!: tendsto_eventually eventually_sequentiallyI [of n])
       qed
     qed auto
     then show Dai: "?D absolutely_integrable_on (\<Union>n. F n)"
       unfolding absolutely_integrable_restrict_UNIV by simp
     show "integral (\<Union>n. F n) ?D = integral ((\<Union>x. g ` F x)) f"
     proof (rule LIMSEQ_unique)
       show "(\<lambda>n. integral (\<Union>m\<le>n. g ` F m) f) \<longlonglongrightarrow> integral (\<Union>x. g ` F x) f"
         by (rule fgU)
       show "(\<lambda>n. integral (\<Union>m\<le>n. g ` F m) f) \<longlonglongrightarrow> integral (\<Union>n. F n) ?D"
         unfolding D_int [symmetric] by (rule integral_countable_UN [OF Dai F_leb])
     qed
   qed
 qed
 
 
 theorem has_absolute_integral_change_of_variables:
   fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
   assumes S: "S \<in> sets lebesgue"
     and der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
     and inj: "inj_on g S"
   shows "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S \<and>
            integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) = b
      \<longleftrightarrow> f absolutely_integrable_on (g ` S) \<and> integral (g ` S) f = b"
 proof -
   obtain C N where "fsigma C" and N: "N \<in> null_sets lebesgue" and CNS: "C \<union> N = S" and "disjnt C N"
     using lebesgue_set_almost_fsigma [OF S] .
   then obtain F :: "nat \<Rightarrow> (real^'m::_) set"
     where F: "range F \<subseteq> Collect compact" and Ceq: "C = Union(range F)"
     using fsigma_Union_compact by metis
   have "negligible N"
     using N by (simp add: negligible_iff_null_sets)
   let ?D = "\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f (g x)"
   have "?D absolutely_integrable_on C \<and> integral C ?D = b
     \<longleftrightarrow> f absolutely_integrable_on (g ` C) \<and> integral (g ` C) f = b"
     unfolding Ceq
   proof (rule has_absolute_integral_change_of_variables_compact_family)
     fix n x
     assume "x \<in> \<Union>(F ` UNIV)"
     then show "(g has_derivative g' x) (at x within \<Union>(F ` UNIV))"
-      using Ceq \<open>C \<union> N = S\<close> der_g has_derivative_within_subset by blast
+      using Ceq \<open>C \<union> N = S\<close> der_g has_derivative_subset by blast
   next
     have "\<Union>(F ` UNIV) \<subseteq> S"
       using Ceq \<open>C \<union> N = S\<close> by blast
     then show "inj_on g (\<Union>(F ` UNIV))"
       using inj by (meson inj_on_subset)
   qed (use F in auto)
   moreover
   have "?D absolutely_integrable_on C \<and> integral C ?D = b
     \<longleftrightarrow> ?D absolutely_integrable_on S \<and> integral S ?D = b"
   proof (rule conj_cong)
     have neg: "negligible {x \<in> C - S. ?D x \<noteq> 0}" "negligible {x \<in> S - C. ?D x \<noteq> 0}"
       using CNS by (blast intro: negligible_subset [OF \<open>negligible N\<close>])+
     then show "(?D absolutely_integrable_on C) = (?D absolutely_integrable_on S)"
       by (rule absolutely_integrable_spike_set_eq)
     show "(integral C ?D = b) \<longleftrightarrow> (integral S ?D = b)"
       using integral_spike_set [OF neg] by simp
   qed
   moreover
   have "f absolutely_integrable_on (g ` C) \<and> integral (g ` C) f = b
     \<longleftrightarrow> f absolutely_integrable_on (g ` S) \<and> integral (g ` S) f = b"
   proof (rule conj_cong)
     have "g differentiable_on N"
       by (metis CNS der_g differentiable_def differentiable_on_def differentiable_on_subset sup.cobounded2)
     with \<open>negligible N\<close>
     have neg_gN: "negligible (g ` N)"
       by (blast intro: negligible_differentiable_image_negligible)
     have neg: "negligible {x \<in> g ` C - g ` S. f x \<noteq> 0}"
               "negligible {x \<in> g ` S - g ` C. f x \<noteq> 0}"
       using CNS by (blast intro: negligible_subset [OF neg_gN])+
     then show "(f absolutely_integrable_on g ` C) = (f absolutely_integrable_on g ` S)"
       by (rule absolutely_integrable_spike_set_eq)
     show "(integral (g ` C) f = b) \<longleftrightarrow> (integral (g ` S) f = b)"
       using integral_spike_set [OF neg] by simp
   qed
   ultimately show ?thesis
     by simp
 qed
 
 
 corollary absolutely_integrable_change_of_variables:
   fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
   assumes "S \<in> sets lebesgue"
     and "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
     and "inj_on g S"
   shows "f absolutely_integrable_on (g ` S)
      \<longleftrightarrow> (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S"
   using assms has_absolute_integral_change_of_variables by blast
 
 corollary integral_change_of_variables:
   fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
   assumes S: "S \<in> sets lebesgue"
     and der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
     and inj: "inj_on g S"
     and disj: "(f absolutely_integrable_on (g ` S) \<or>
         (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S)"
   shows "integral (g ` S) f = integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x))"
   using has_absolute_integral_change_of_variables [OF S der_g inj] disj
   by blast
 
 lemma has_absolute_integral_change_of_variables_1:
   fixes f :: "real \<Rightarrow> real^'n::{finite,wellorder}" and g :: "real \<Rightarrow> real"
   assumes S: "S \<in> sets lebesgue"
     and der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_vector_derivative g' x) (at x within S)"
     and inj: "inj_on g S"
   shows "(\<lambda>x. \<bar>g' x\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S \<and>
            integral S (\<lambda>x. \<bar>g' x\<bar> *\<^sub>R f(g x)) = b
      \<longleftrightarrow> f absolutely_integrable_on (g ` S) \<and> integral (g ` S) f = b"
 proof -
   let ?lift = "vec :: real \<Rightarrow> real^1"
   let ?drop = "(\<lambda>x::real^1. x $ 1)"
   have S': "?lift ` S \<in> sets lebesgue"
     by (auto intro: differentiable_image_in_sets_lebesgue [OF S] differentiable_vec)
   have "((\<lambda>x. vec (g (x $ 1))) has_derivative (*\<^sub>R) (g' z)) (at (vec z) within ?lift ` S)"
     if "z \<in> S" for z
     using der_g [OF that]
     by (simp add: has_vector_derivative_def has_derivative_vector_1)
   then have der': "\<And>x. x \<in> ?lift ` S \<Longrightarrow>
         (?lift \<circ> g \<circ> ?drop has_derivative (*\<^sub>R) (g' (?drop x))) (at x within ?lift ` S)"
     by (auto simp: o_def)
   have inj': "inj_on (vec \<circ> g \<circ> ?drop) (vec ` S)"
     using inj by (simp add: inj_on_def)
   let ?fg = "\<lambda>x. \<bar>g' x\<bar> *\<^sub>R f(g x)"
   have "((\<lambda>x. ?fg x $ i) absolutely_integrable_on S \<and> ((\<lambda>x. ?fg x $ i) has_integral b $ i) S
     \<longleftrightarrow> (\<lambda>x. f x $ i) absolutely_integrable_on g ` S \<and> ((\<lambda>x. f x $ i) has_integral b $ i) (g ` S))" for i
     using has_absolute_integral_change_of_variables [OF S' der' inj', of "\<lambda>x. ?lift(f (?drop x) $ i)" "?lift (b$i)"]
     unfolding integrable_on_1_iff integral_on_1_eq absolutely_integrable_on_1_iff absolutely_integrable_drop absolutely_integrable_on_def
     by (auto simp: image_comp o_def integral_vec1_eq has_integral_iff)
   then have "?fg absolutely_integrable_on S \<and> (?fg has_integral b) S
          \<longleftrightarrow> f absolutely_integrable_on (g ` S) \<and> (f has_integral b) (g ` S)"
     unfolding has_integral_componentwise_iff [where y=b]
            absolutely_integrable_componentwise_iff [where f=f]
            absolutely_integrable_componentwise_iff [where f = ?fg]
     by (force simp: Basis_vec_def cart_eq_inner_axis)
   then show ?thesis
     using absolutely_integrable_on_def by blast
 qed
 
 
 corollary absolutely_integrable_change_of_variables_1:
   fixes f :: "real \<Rightarrow> real^'n::{finite,wellorder}" and g :: "real \<Rightarrow> real"
   assumes S: "S \<in> sets lebesgue"
     and der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_vector_derivative g' x) (at x within S)"
     and inj: "inj_on g S"
   shows "(f absolutely_integrable_on g ` S \<longleftrightarrow>
              (\<lambda>x. \<bar>g' x\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S)"
   using has_absolute_integral_change_of_variables_1 [OF assms] by auto
 
 
 subsection\<open>Change of variables for integrals: special case of linear function\<close>
 
 lemma has_absolute_integral_change_of_variables_linear:
   fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
   assumes "linear g"
   shows "(\<lambda>x. \<bar>det (matrix g)\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S \<and>
            integral S (\<lambda>x. \<bar>det (matrix g)\<bar> *\<^sub>R f(g x)) = b
      \<longleftrightarrow> f absolutely_integrable_on (g ` S) \<and> integral (g ` S) f = b"
 proof (cases "det(matrix g) = 0")
   case True
   then have "negligible(g ` S)"
     using assms det_nz_iff_inj negligible_linear_singular_image by blast
   with True show ?thesis
     by (auto simp: absolutely_integrable_on_def integrable_negligible integral_negligible)
 next
   case False
   then obtain h where h: "\<And>x. x \<in> S \<Longrightarrow> h (g x) = x" "linear h"
     using assms det_nz_iff_inj linear_injective_isomorphism by metis
   show ?thesis
   proof (rule has_absolute_integral_change_of_variables_invertible)
     show "(g has_derivative g) (at x within S)" for x
       by (simp add: assms linear_imp_has_derivative)
     show "continuous_on (g ` S) h"
       using continuous_on_eq_continuous_within has_derivative_continuous linear_imp_has_derivative h by blast
   qed (use h in auto)
 qed
 
 lemma absolutely_integrable_change_of_variables_linear:
   fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
   assumes "linear g"
   shows "(\<lambda>x. \<bar>det (matrix g)\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S
      \<longleftrightarrow> f absolutely_integrable_on (g ` S)"
   using assms has_absolute_integral_change_of_variables_linear by blast
 
 lemma absolutely_integrable_on_linear_image:
   fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
   assumes "linear g"
   shows "f absolutely_integrable_on (g ` S)
      \<longleftrightarrow> (f \<circ> g) absolutely_integrable_on S \<or> det(matrix g) = 0"
   unfolding assms absolutely_integrable_change_of_variables_linear [OF assms, symmetric] absolutely_integrable_on_scaleR_iff
   by (auto simp: set_integrable_def)
 
 lemma integral_change_of_variables_linear:
   fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
   assumes "linear g"
       and "f absolutely_integrable_on (g ` S) \<or> (f \<circ> g) absolutely_integrable_on S"
     shows "integral (g ` S) f = \<bar>det (matrix g)\<bar> *\<^sub>R integral S (f \<circ> g)"
 proof -
   have "((\<lambda>x. \<bar>det (matrix g)\<bar> *\<^sub>R f (g x)) absolutely_integrable_on S) \<or> (f absolutely_integrable_on g ` S)"
     using absolutely_integrable_on_linear_image assms by blast
   moreover
   have ?thesis if "((\<lambda>x. \<bar>det (matrix g)\<bar> *\<^sub>R f (g x)) absolutely_integrable_on S)" "(f absolutely_integrable_on g ` S)"
     using has_absolute_integral_change_of_variables_linear [OF \<open>linear g\<close>] that
     by (auto simp: o_def)
   ultimately show ?thesis
     using absolutely_integrable_change_of_variables_linear [OF \<open>linear g\<close>]
     by blast
 qed
 
 subsection\<open>Change of variable for measure\<close>
 
 lemma has_measure_differentiable_image:
   fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
   assumes "S \<in> sets lebesgue"
       and "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
       and "inj_on f S"
   shows "f ` S \<in> lmeasurable \<and> measure lebesgue (f ` S) = m
      \<longleftrightarrow> ((\<lambda>x. \<bar>det (matrix (f' x))\<bar>) has_integral m) S"
   using has_absolute_integral_change_of_variables [OF assms, of "\<lambda>x. (1::real^1)" "vec m"]
   unfolding absolutely_integrable_on_1_iff integral_on_1_eq integrable_on_1_iff absolutely_integrable_on_def
   by (auto simp: has_integral_iff lmeasurable_iff_integrable_on lmeasure_integral)
 
 lemma measurable_differentiable_image_eq:
   fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
   assumes "S \<in> sets lebesgue"
       and "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
       and "inj_on f S"
   shows "f ` S \<in> lmeasurable \<longleftrightarrow> (\<lambda>x. \<bar>det (matrix (f' x))\<bar>) integrable_on S"
   using has_measure_differentiable_image [OF assms]
   by blast
 
 lemma measurable_differentiable_image_alt:
   fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
   assumes "S \<in> sets lebesgue"
     and "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
     and "inj_on f S"
   shows "f ` S \<in> lmeasurable \<longleftrightarrow> (\<lambda>x. \<bar>det (matrix (f' x))\<bar>) absolutely_integrable_on S"
   using measurable_differentiable_image_eq [OF assms]
   by (simp only: absolutely_integrable_on_iff_nonneg)
 
 lemma measure_differentiable_image_eq:
   fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
   assumes S: "S \<in> sets lebesgue"
     and der_f: "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
     and inj: "inj_on f S"
     and intS: "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) integrable_on S"
   shows "measure lebesgue (f ` S) = integral S (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)"
   using measurable_differentiable_image_eq [OF S der_f inj]
         assms has_measure_differentiable_image by blast
 
 end
diff --git a/src/HOL/Analysis/Derivative.thy b/src/HOL/Analysis/Derivative.thy
--- a/src/HOL/Analysis/Derivative.thy
+++ b/src/HOL/Analysis/Derivative.thy
@@ -1,3475 +1,3475 @@
 (*  Title:      HOL/Analysis/Derivative.thy
     Author:     John Harrison
     Author:     Robert Himmelmann, TU Muenchen (translation from HOL Light); tidied by LCP
 *)
 
 section \<open>Derivative\<close>
 
 theory Derivative
   imports
     Bounded_Linear_Function
     Line_Segment
     Convex_Euclidean_Space
 begin
 
 declare bounded_linear_inner_left [intro]
 
 declare has_derivative_bounded_linear[dest]
 
 subsection \<open>Derivatives\<close>
 
 lemma has_derivative_add_const:
   "(f has_derivative f') net \<Longrightarrow> ((\<lambda>x. f x + c) has_derivative f') net"
   by (intro derivative_eq_intros) auto
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Derivative with composed bilinear function\<close>
 
 text \<open>More explicit epsilon-delta forms.\<close>
 
 proposition has_derivative_within':
   "(f has_derivative f')(at x within s) \<longleftrightarrow>
     bounded_linear f' \<and>
     (\<forall>e>0. \<exists>d>0. \<forall>x'\<in>s. 0 < norm (x' - x) \<and> norm (x' - x) < d \<longrightarrow>
       norm (f x' - f x - f'(x' - x)) / norm (x' - x) < e)"
   unfolding has_derivative_within Lim_within dist_norm
   by (simp add: diff_diff_eq)
 
 lemma has_derivative_at':
   "(f has_derivative f') (at x) 
    \<longleftrightarrow> bounded_linear f' \<and>
        (\<forall>e>0. \<exists>d>0. \<forall>x'. 0 < norm (x' - x) \<and> norm (x' - x) < d \<longrightarrow>
         norm (f x' - f x - f'(x' - x)) / norm (x' - x) < e)"
   using has_derivative_within' [of f f' x UNIV] by simp
 
 lemma has_derivative_componentwise_within:
    "(f has_derivative f') (at a within S) \<longleftrightarrow>
     (\<forall>i \<in> Basis. ((\<lambda>x. f x \<bullet> i) has_derivative (\<lambda>x. f' x \<bullet> i)) (at a within S))"
   apply (simp add: has_derivative_within)
   apply (subst tendsto_componentwise_iff)
   apply (simp add: bounded_linear_componentwise_iff [symmetric] ball_conj_distrib)
   apply (simp add: algebra_simps)
   done
 
 lemma has_derivative_at_withinI:
   "(f has_derivative f') (at x) \<Longrightarrow> (f has_derivative f') (at x within s)"
   unfolding has_derivative_within' has_derivative_at'
   by blast
 
 lemma has_derivative_right:
   fixes f :: "real \<Rightarrow> real"
     and y :: "real"
   shows "(f has_derivative ((*) y)) (at x within ({x <..} \<inter> I)) \<longleftrightarrow>
          ((\<lambda>t. (f x - f t) / (x - t)) \<longlongrightarrow> y) (at x within ({x <..} \<inter> I))"
 proof -
   have "((\<lambda>t. (f t - (f x + y * (t - x))) / \<bar>t - x\<bar>) \<longlongrightarrow> 0) (at x within ({x<..} \<inter> I)) \<longleftrightarrow>
     ((\<lambda>t. (f t - f x) / (t - x) - y) \<longlongrightarrow> 0) (at x within ({x<..} \<inter> I))"
     by (intro Lim_cong_within) (auto simp add: diff_divide_distrib add_divide_distrib)
   also have "\<dots> \<longleftrightarrow> ((\<lambda>t. (f t - f x) / (t - x)) \<longlongrightarrow> y) (at x within ({x<..} \<inter> I))"
     by (simp add: Lim_null[symmetric])
   also have "\<dots> \<longleftrightarrow> ((\<lambda>t. (f x - f t) / (x - t)) \<longlongrightarrow> y) (at x within ({x<..} \<inter> I))"
     by (intro Lim_cong_within) (simp_all add: field_simps)
   finally show ?thesis
     by (simp add: bounded_linear_mult_right has_derivative_within)
 qed
 
 subsubsection \<open>Caratheodory characterization\<close>
 
 lemma DERIV_caratheodory_within:
   "(f has_field_derivative l) (at x within S) \<longleftrightarrow>
    (\<exists>g. (\<forall>z. f z - f x = g z * (z - x)) \<and> continuous (at x within S) g \<and> g x = l)"
       (is "?lhs = ?rhs")
 proof
   assume ?lhs
   show ?rhs
   proof (intro exI conjI)
     let ?g = "(%z. if z = x then l else (f z - f x) / (z-x))"
     show "\<forall>z. f z - f x = ?g z * (z-x)" by simp
     show "continuous (at x within S) ?g" using \<open>?lhs\<close>
       by (auto simp add: continuous_within has_field_derivative_iff cong: Lim_cong_within)
     show "?g x = l" by simp
   qed
 next
   assume ?rhs
   then obtain g where
     "(\<forall>z. f z - f x = g z * (z-x))" and "continuous (at x within S) g" and "g x = l" by blast
   thus ?lhs
     by (auto simp add: continuous_within has_field_derivative_iff cong: Lim_cong_within)
 qed
 
 subsection \<open>Differentiability\<close>
 
 definition\<^marker>\<open>tag important\<close>
   differentiable_on :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a set \<Rightarrow> bool"
     (infix "differentiable'_on" 50)
   where "f differentiable_on s \<longleftrightarrow> (\<forall>x\<in>s. f differentiable (at x within s))"
 
 lemma differentiableI: "(f has_derivative f') net \<Longrightarrow> f differentiable net"
   unfolding differentiable_def
   by auto
 
 lemma differentiable_onD: "\<lbrakk>f differentiable_on S; x \<in> S\<rbrakk> \<Longrightarrow> f differentiable (at x within S)"
   using differentiable_on_def by blast
 
 lemma differentiable_at_withinI: "f differentiable (at x) \<Longrightarrow> f differentiable (at x within s)"
   unfolding differentiable_def
   using has_derivative_at_withinI
   by blast
 
 lemma differentiable_at_imp_differentiable_on:
   "(\<And>x. x \<in> s \<Longrightarrow> f differentiable at x) \<Longrightarrow> f differentiable_on s"
   by (metis differentiable_at_withinI differentiable_on_def)
 
 corollary\<^marker>\<open>tag unimportant\<close> differentiable_iff_scaleR:
   fixes f :: "real \<Rightarrow> 'a::real_normed_vector"
   shows "f differentiable F \<longleftrightarrow> (\<exists>d. (f has_derivative (\<lambda>x. x *\<^sub>R d)) F)"
   by (auto simp: differentiable_def dest: has_derivative_linear linear_imp_scaleR)
 
 lemma differentiable_on_eq_differentiable_at:
   "open s \<Longrightarrow> f differentiable_on s \<longleftrightarrow> (\<forall>x\<in>s. f differentiable at x)"
   unfolding differentiable_on_def
   by (metis at_within_interior interior_open)
 
 lemma differentiable_transform_within:
   assumes "f differentiable (at x within s)"
     and "0 < d"
     and "x \<in> s"
     and "\<And>x'. \<lbrakk>x'\<in>s; dist x' x < d\<rbrakk> \<Longrightarrow> f x' = g x'"
   shows "g differentiable (at x within s)"
    using assms has_derivative_transform_within unfolding differentiable_def
    by blast
 
 lemma differentiable_on_ident [simp, derivative_intros]: "(\<lambda>x. x) differentiable_on S"
   by (simp add: differentiable_at_imp_differentiable_on)
 
 lemma differentiable_on_id [simp, derivative_intros]: "id differentiable_on S"
   by (simp add: id_def)
 
 lemma differentiable_on_const [simp, derivative_intros]: "(\<lambda>z. c) differentiable_on S"
   by (simp add: differentiable_on_def)
 
 lemma differentiable_on_mult [simp, derivative_intros]:
   fixes f :: "'M::real_normed_vector \<Rightarrow> 'a::real_normed_algebra"
   shows "\<lbrakk>f differentiable_on S; g differentiable_on S\<rbrakk> \<Longrightarrow> (\<lambda>z. f z * g z) differentiable_on S"
   unfolding differentiable_on_def differentiable_def
   using differentiable_def differentiable_mult by blast
 
 lemma differentiable_on_compose:
    "\<lbrakk>g differentiable_on S; f differentiable_on (g ` S)\<rbrakk> \<Longrightarrow> (\<lambda>x. f (g x)) differentiable_on S"
 by (simp add: differentiable_in_compose differentiable_on_def)
 
 lemma bounded_linear_imp_differentiable_on: "bounded_linear f \<Longrightarrow> f differentiable_on S"
   by (simp add: differentiable_on_def bounded_linear_imp_differentiable)
 
 lemma linear_imp_differentiable_on:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
   shows "linear f \<Longrightarrow> f differentiable_on S"
 by (simp add: differentiable_on_def linear_imp_differentiable)
 
 lemma differentiable_on_minus [simp, derivative_intros]:
    "f differentiable_on S \<Longrightarrow> (\<lambda>z. -(f z)) differentiable_on S"
 by (simp add: differentiable_on_def)
 
 lemma differentiable_on_add [simp, derivative_intros]:
    "\<lbrakk>f differentiable_on S; g differentiable_on S\<rbrakk> \<Longrightarrow> (\<lambda>z. f z + g z) differentiable_on S"
 by (simp add: differentiable_on_def)
 
 lemma differentiable_on_diff [simp, derivative_intros]:
    "\<lbrakk>f differentiable_on S; g differentiable_on S\<rbrakk> \<Longrightarrow> (\<lambda>z. f z - g z) differentiable_on S"
 by (simp add: differentiable_on_def)
 
 lemma differentiable_on_inverse [simp, derivative_intros]:
   fixes f :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
   shows "f differentiable_on S \<Longrightarrow> (\<And>x. x \<in> S \<Longrightarrow> f x \<noteq> 0) \<Longrightarrow> (\<lambda>x. inverse (f x)) differentiable_on S"
 by (simp add: differentiable_on_def)
 
 lemma differentiable_on_scaleR [derivative_intros, simp]:
    "\<lbrakk>f differentiable_on S; g differentiable_on S\<rbrakk> \<Longrightarrow> (\<lambda>x. f x *\<^sub>R g x) differentiable_on S"
   unfolding differentiable_on_def
   by (blast intro: differentiable_scaleR)
 
 lemma has_derivative_sqnorm_at [derivative_intros, simp]:
   "((\<lambda>x. (norm x)\<^sup>2) has_derivative (\<lambda>x. 2 *\<^sub>R (a \<bullet> x))) (at a)"
   using bounded_bilinear.FDERIV  [of "(\<bullet>)" id id a _ id id]
   by (auto simp: inner_commute dot_square_norm bounded_bilinear_inner)
 
 lemma differentiable_sqnorm_at [derivative_intros, simp]:
   fixes a :: "'a :: {real_normed_vector,real_inner}"
   shows "(\<lambda>x. (norm x)\<^sup>2) differentiable (at a)"
 by (force simp add: differentiable_def intro: has_derivative_sqnorm_at)
 
 lemma differentiable_on_sqnorm [derivative_intros, simp]:
   fixes S :: "'a :: {real_normed_vector,real_inner} set"
   shows "(\<lambda>x. (norm x)\<^sup>2) differentiable_on S"
 by (simp add: differentiable_at_imp_differentiable_on)
 
 lemma differentiable_norm_at [derivative_intros, simp]:
   fixes a :: "'a :: {real_normed_vector,real_inner}"
   shows "a \<noteq> 0 \<Longrightarrow> norm differentiable (at a)"
 using differentiableI has_derivative_norm by blast
 
 lemma differentiable_on_norm [derivative_intros, simp]:
   fixes S :: "'a :: {real_normed_vector,real_inner} set"
   shows "0 \<notin> S \<Longrightarrow> norm differentiable_on S"
 by (metis differentiable_at_imp_differentiable_on differentiable_norm_at)
 
 
 subsection \<open>Frechet derivative and Jacobian matrix\<close>
 
 definition "frechet_derivative f net = (SOME f'. (f has_derivative f') net)"
 
 proposition frechet_derivative_works:
   "f differentiable net \<longleftrightarrow> (f has_derivative (frechet_derivative f net)) net"
   unfolding frechet_derivative_def differentiable_def
   unfolding some_eq_ex[of "\<lambda> f' . (f has_derivative f') net"] ..
 
 lemma linear_frechet_derivative: "f differentiable net \<Longrightarrow> linear (frechet_derivative f net)"
   unfolding frechet_derivative_works has_derivative_def
   by (auto intro: bounded_linear.linear)
 
 lemma frechet_derivative_const [simp]: "frechet_derivative (\<lambda>x. c) (at a) = (\<lambda>x. 0)"
   using differentiable_const frechet_derivative_works has_derivative_const has_derivative_unique by blast
 
 lemma frechet_derivative_id [simp]: "frechet_derivative id (at a) = id"
   using differentiable_def frechet_derivative_works has_derivative_id has_derivative_unique by blast
 
 lemma frechet_derivative_ident [simp]: "frechet_derivative (\<lambda>x. x) (at a) = (\<lambda>x. x)"
   by (metis eq_id_iff frechet_derivative_id)
 
 
 subsection \<open>Differentiability implies continuity\<close>
 
 proposition differentiable_imp_continuous_within:
   "f differentiable (at x within s) \<Longrightarrow> continuous (at x within s) f"
   by (auto simp: differentiable_def intro: has_derivative_continuous)
 
 lemma differentiable_imp_continuous_on:
   "f differentiable_on s \<Longrightarrow> continuous_on s f"
   unfolding differentiable_on_def continuous_on_eq_continuous_within
   using differentiable_imp_continuous_within by blast
 
 lemma differentiable_on_subset:
   "f differentiable_on t \<Longrightarrow> s \<subseteq> t \<Longrightarrow> f differentiable_on s"
   unfolding differentiable_on_def
   using differentiable_within_subset
   by blast
 
 lemma differentiable_on_empty: "f differentiable_on {}"
   unfolding differentiable_on_def
   by auto
 
 lemma has_derivative_continuous_on:
   "(\<And>x. x \<in> s \<Longrightarrow> (f has_derivative f' x) (at x within s)) \<Longrightarrow> continuous_on s f"
   by (auto intro!: differentiable_imp_continuous_on differentiableI simp: differentiable_on_def)
 
 text \<open>Results about neighborhoods filter.\<close>
 
 lemma eventually_nhds_metric_le:
   "eventually P (nhds a) = (\<exists>d>0. \<forall>x. dist x a \<le> d \<longrightarrow> P x)"
   unfolding eventually_nhds_metric by (safe, rule_tac x="d / 2" in exI, auto)
 
 lemma le_nhds: "F \<le> nhds a \<longleftrightarrow> (\<forall>S. open S \<and> a \<in> S \<longrightarrow> eventually (\<lambda>x. x \<in> S) F)"
   unfolding le_filter_def eventually_nhds by (fast elim: eventually_mono)
 
 lemma le_nhds_metric: "F \<le> nhds a \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist x a < e) F)"
   unfolding le_filter_def eventually_nhds_metric by (fast elim: eventually_mono)
 
 lemma le_nhds_metric_le: "F \<le> nhds a \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist x a \<le> e) F)"
   unfolding le_filter_def eventually_nhds_metric_le by (fast elim: eventually_mono)
 
 text \<open>Several results are easier using a "multiplied-out" variant.
 (I got this idea from Dieudonne's proof of the chain rule).\<close>
 
 lemma has_derivative_within_alt:
   "(f has_derivative f') (at x within s) \<longleftrightarrow> bounded_linear f' \<and>
     (\<forall>e>0. \<exists>d>0. \<forall>y\<in>s. norm(y - x) < d \<longrightarrow> norm (f y - f x - f' (y - x)) \<le> e * norm (y - x))"
   unfolding has_derivative_within filterlim_def le_nhds_metric_le eventually_filtermap
     eventually_at dist_norm diff_diff_eq
   by (force simp add: linear_0 bounded_linear.linear pos_divide_le_eq)
 
 lemma has_derivative_within_alt2:
   "(f has_derivative f') (at x within s) \<longleftrightarrow> bounded_linear f' \<and>
     (\<forall>e>0. eventually (\<lambda>y. norm (f y - f x - f' (y - x)) \<le> e * norm (y - x)) (at x within s))"
   unfolding has_derivative_within filterlim_def le_nhds_metric_le eventually_filtermap
     eventually_at dist_norm diff_diff_eq
   by (force simp add: linear_0 bounded_linear.linear pos_divide_le_eq)
 
 lemma has_derivative_at_alt:
   "(f has_derivative f') (at x) \<longleftrightarrow>
     bounded_linear f' \<and>
     (\<forall>e>0. \<exists>d>0. \<forall>y. norm(y - x) < d \<longrightarrow> norm (f y - f x - f'(y - x)) \<le> e * norm (y - x))"
   using has_derivative_within_alt[where s=UNIV]
   by simp
 
 
 subsection \<open>The chain rule\<close>
 
 proposition diff_chain_within[derivative_intros]:
   assumes "(f has_derivative f') (at x within s)"
     and "(g has_derivative g') (at (f x) within (f ` s))"
   shows "((g \<circ> f) has_derivative (g' \<circ> f'))(at x within s)"
   using has_derivative_in_compose[OF assms]
   by (simp add: comp_def)
 
 lemma diff_chain_at[derivative_intros]:
   "(f has_derivative f') (at x) \<Longrightarrow>
     (g has_derivative g') (at (f x)) \<Longrightarrow> ((g \<circ> f) has_derivative (g' \<circ> f')) (at x)"
   using has_derivative_compose[of f f' x UNIV g g']
   by (simp add: comp_def)
 
 lemma has_vector_derivative_within_open:
   "a \<in> S \<Longrightarrow> open S \<Longrightarrow>
     (f has_vector_derivative f') (at a within S) \<longleftrightarrow> (f has_vector_derivative f') (at a)"
   by (simp only: at_within_interior interior_open)
 
 lemma field_vector_diff_chain_within:
  assumes Df: "(f has_vector_derivative f') (at x within S)"
      and Dg: "(g has_field_derivative g') (at (f x) within f ` S)"
  shows "((g \<circ> f) has_vector_derivative (f' * g')) (at x within S)"
 using diff_chain_within[OF Df[unfolded has_vector_derivative_def]
                        Dg [unfolded has_field_derivative_def]]
  by (auto simp: o_def mult.commute has_vector_derivative_def)
 
 lemma vector_derivative_diff_chain_within:
   assumes Df: "(f has_vector_derivative f') (at x within S)"
      and Dg: "(g has_derivative g') (at (f x) within f`S)"
   shows "((g \<circ> f) has_vector_derivative (g' f')) (at x within S)"
 using diff_chain_within[OF Df[unfolded has_vector_derivative_def] Dg]
   linear.scaleR[OF has_derivative_linear[OF Dg]]
   unfolding has_vector_derivative_def o_def
   by (auto simp: o_def mult.commute has_vector_derivative_def)
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Composition rules stated just for differentiability\<close>
 
 lemma differentiable_chain_at:
   "f differentiable (at x) \<Longrightarrow>
     g differentiable (at (f x)) \<Longrightarrow> (g \<circ> f) differentiable (at x)"
   unfolding differentiable_def
   by (meson diff_chain_at)
 
 lemma differentiable_chain_within:
   "f differentiable (at x within S) \<Longrightarrow>
     g differentiable (at(f x) within (f ` S)) \<Longrightarrow> (g \<circ> f) differentiable (at x within S)"
   unfolding differentiable_def
   by (meson diff_chain_within)
 
 
 subsection \<open>Uniqueness of derivative\<close>
 
 
 text\<^marker>\<open>tag important\<close> \<open>
  The general result is a bit messy because we need approachability of the
  limit point from any direction. But OK for nontrivial intervals etc.
 \<close>
 
 proposition frechet_derivative_unique_within:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
   assumes 1: "(f has_derivative f') (at x within S)"
     and 2: "(f has_derivative f'') (at x within S)"
     and S: "\<And>i e. \<lbrakk>i\<in>Basis; e>0\<rbrakk> \<Longrightarrow> \<exists>d. 0 < \<bar>d\<bar> \<and> \<bar>d\<bar> < e \<and> (x + d *\<^sub>R i) \<in> S"
   shows "f' = f''"
 proof -
   note as = assms(1,2)[unfolded has_derivative_def]
   then interpret f': bounded_linear f' by auto
   from as interpret f'': bounded_linear f'' by auto
   have "x islimpt S" unfolding islimpt_approachable
   proof (intro allI impI)
     fix e :: real
     assume "e > 0"
     obtain d where "0 < \<bar>d\<bar>" and "\<bar>d\<bar> < e" and "x + d *\<^sub>R (SOME i. i \<in> Basis) \<in> S"
       using assms(3) SOME_Basis \<open>e>0\<close> by blast
     then show "\<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"
       by (rule_tac x="x + d *\<^sub>R (SOME i. i \<in> Basis)" in bexI) (auto simp: dist_norm SOME_Basis nonzero_Basis)  qed
   then have *: "netlimit (at x within S) = x"
     by (simp add: Lim_ident_at trivial_limit_within)
   show ?thesis
   proof (rule linear_eq_stdbasis)
     show "linear f'" "linear f''"
       unfolding linear_conv_bounded_linear using as by auto
   next
     fix i :: 'a
     assume i: "i \<in> Basis"
     define e where "e = norm (f' i - f'' i)"
     show "f' i = f'' i"
     proof (rule ccontr)
       assume "f' i \<noteq> f'' i"
       then have "e > 0"
         unfolding e_def by auto
       obtain d where d:
         "0 < d"
         "(\<And>y. y\<in>S \<longrightarrow> 0 < dist y x \<and> dist y x < d \<longrightarrow>
           dist ((f y - f x - f' (y - x)) /\<^sub>R norm (y - x) -
               (f y - f x - f'' (y - x)) /\<^sub>R norm (y - x)) (0 - 0) < e)"
         using tendsto_diff [OF as(1,2)[THEN conjunct2]]
         unfolding * Lim_within
         using \<open>e>0\<close> by blast
       obtain c where c: "0 < \<bar>c\<bar>" "\<bar>c\<bar> < d \<and> x + c *\<^sub>R i \<in> S"
         using assms(3) i d(1) by blast
       have *: "norm (- ((1 / \<bar>c\<bar>) *\<^sub>R f' (c *\<^sub>R i)) + (1 / \<bar>c\<bar>) *\<^sub>R f'' (c *\<^sub>R i)) =
         norm ((1 / \<bar>c\<bar>) *\<^sub>R (- (f' (c *\<^sub>R i)) + f'' (c *\<^sub>R i)))"
         unfolding scaleR_right_distrib by auto
       also have "\<dots> = norm ((1 / \<bar>c\<bar>) *\<^sub>R (c *\<^sub>R (- (f' i) + f'' i)))"
         unfolding f'.scaleR f''.scaleR
         unfolding scaleR_right_distrib scaleR_minus_right
         by auto
       also have "\<dots> = e"
         unfolding e_def
         using c(1)
         using norm_minus_cancel[of "f' i - f'' i"]
         by auto
       finally show False
         using c
         using d(2)[of "x + c *\<^sub>R i"]
         unfolding dist_norm
         unfolding f'.scaleR f''.scaleR f'.add f''.add f'.diff f''.diff
           scaleR_scaleR scaleR_right_diff_distrib scaleR_right_distrib
         using i
         by (auto simp: inverse_eq_divide)
     qed
   qed
 qed
 
 proposition frechet_derivative_unique_within_closed_interval:
   fixes f::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
   assumes ab: "\<And>i. i\<in>Basis \<Longrightarrow> a\<bullet>i < b\<bullet>i"
     and x: "x \<in> cbox a b"
     and "(f has_derivative f' ) (at x within cbox a b)"
     and "(f has_derivative f'') (at x within cbox a b)"
   shows "f' = f''"
 proof (rule frechet_derivative_unique_within)
   fix e :: real
   fix i :: 'a
   assume "e > 0" and i: "i \<in> Basis"
   then show "\<exists>d. 0 < \<bar>d\<bar> \<and> \<bar>d\<bar> < e \<and> x + d *\<^sub>R i \<in> cbox a b"
   proof (cases "x\<bullet>i = a\<bullet>i")
     case True
     with ab[of i] \<open>e>0\<close> x i show ?thesis
       by (rule_tac x="(min (b\<bullet>i - a\<bullet>i) e) / 2" in exI)
          (auto simp add: mem_box field_simps inner_simps inner_Basis)
   next
     case False
     moreover have "a \<bullet> i < x \<bullet> i"
       using False i mem_box(2) x by force
     moreover {
       have "a \<bullet> i * 2 + min (x \<bullet> i - a \<bullet> i) e \<le> a\<bullet>i *2 + x\<bullet>i - a\<bullet>i"
         by auto
       also have "\<dots> = a\<bullet>i + x\<bullet>i"
         by auto
       also have "\<dots> \<le> 2 * (x\<bullet>i)"
         using \<open>a \<bullet> i < x \<bullet> i\<close> by auto
       finally have "a \<bullet> i * 2 + min (x \<bullet> i - a \<bullet> i) e \<le> x \<bullet> i * 2"
         by auto
     }
     moreover have "min (x \<bullet> i - a \<bullet> i) e \<ge> 0"
       by (simp add: \<open>0 < e\<close> \<open>a \<bullet> i < x \<bullet> i\<close> less_eq_real_def)
     then have "x \<bullet> i * 2 \<le> b \<bullet> i * 2 + min (x \<bullet> i - a \<bullet> i) e"
       using i mem_box(2) x by force
     ultimately show ?thesis
     using ab[of i] \<open>e>0\<close> x i 
       by (rule_tac x="- (min (x\<bullet>i - a\<bullet>i) e) / 2" in exI)
          (auto simp add: mem_box field_simps inner_simps inner_Basis)
   qed
 qed (use assms in auto)
 
 lemma frechet_derivative_unique_within_open_interval:
   fixes f::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
   assumes x: "x \<in> box a b"
     and f: "(f has_derivative f' ) (at x within box a b)" "(f has_derivative f'') (at x within box a b)"
   shows "f' = f''"
 proof -
   have "at x within box a b = at x"
     by (metis x at_within_interior interior_open open_box)
   with f show "f' = f''"
     by (simp add: has_derivative_unique)
 qed
 
 lemma frechet_derivative_at:
   "(f has_derivative f') (at x) \<Longrightarrow> f' = frechet_derivative f (at x)"
   using differentiable_def frechet_derivative_works has_derivative_unique by blast
 
 lemma frechet_derivative_compose:
   "frechet_derivative (f o g) (at x) = frechet_derivative (f) (at (g x)) o frechet_derivative g (at x)"
   if "g differentiable at x" "f differentiable at (g x)"
   by (metis diff_chain_at frechet_derivative_at frechet_derivative_works that)
 
 lemma frechet_derivative_within_cbox:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
   assumes "\<And>i. i\<in>Basis \<Longrightarrow> a\<bullet>i < b\<bullet>i"
     and "x \<in> cbox a b"
     and "(f has_derivative f') (at x within cbox a b)"
   shows "frechet_derivative f (at x within cbox a b) = f'"
   using assms
   by (metis Derivative.differentiableI frechet_derivative_unique_within_closed_interval frechet_derivative_works)
 
 lemma frechet_derivative_transform_within_open:
   "frechet_derivative f (at x) = frechet_derivative g (at x)"
   if "f differentiable at x" "open X" "x \<in> X" "\<And>x. x \<in> X \<Longrightarrow> f x = g x"
   by (meson frechet_derivative_at frechet_derivative_works has_derivative_transform_within_open that)
 
 
 subsection \<open>Derivatives of local minima and maxima are zero\<close>
 
 lemma has_derivative_local_min:
   fixes f :: "'a::real_normed_vector \<Rightarrow> real"
   assumes deriv: "(f has_derivative f') (at x)"
   assumes min: "eventually (\<lambda>y. f x \<le> f y) (at x)"
   shows "f' = (\<lambda>h. 0)"
 proof
   fix h :: 'a
   interpret f': bounded_linear f'
     using deriv by (rule has_derivative_bounded_linear)
   show "f' h = 0"
   proof (cases "h = 0")
     case False
     from min obtain d where d1: "0 < d" and d2: "\<forall>y\<in>ball x d. f x \<le> f y"
       unfolding eventually_at by (force simp: dist_commute)
     have "FDERIV (\<lambda>r. x + r *\<^sub>R h) 0 :> (\<lambda>r. r *\<^sub>R h)"
       by (intro derivative_eq_intros) auto
     then have "FDERIV (\<lambda>r. f (x + r *\<^sub>R h)) 0 :> (\<lambda>k. f' (k *\<^sub>R h))"
       by (rule has_derivative_compose, simp add: deriv)
     then have "DERIV (\<lambda>r. f (x + r *\<^sub>R h)) 0 :> f' h"
       unfolding has_field_derivative_def by (simp add: f'.scaleR mult_commute_abs)
     moreover have "0 < d / norm h" using d1 and \<open>h \<noteq> 0\<close> by simp
     moreover have "\<forall>y. \<bar>0 - y\<bar> < d / norm h \<longrightarrow> f (x + 0 *\<^sub>R h) \<le> f (x + y *\<^sub>R h)"
       using \<open>h \<noteq> 0\<close> by (auto simp add: d2 dist_norm pos_less_divide_eq)
     ultimately show "f' h = 0"
       by (rule DERIV_local_min)
   qed simp
 qed
 
 lemma has_derivative_local_max:
   fixes f :: "'a::real_normed_vector \<Rightarrow> real"
   assumes "(f has_derivative f') (at x)"
   assumes "eventually (\<lambda>y. f y \<le> f x) (at x)"
   shows "f' = (\<lambda>h. 0)"
   using has_derivative_local_min [of "\<lambda>x. - f x" "\<lambda>h. - f' h" "x"]
   using assms unfolding fun_eq_iff by simp
 
 lemma differential_zero_maxmin:
   fixes f::"'a::real_normed_vector \<Rightarrow> real"
   assumes "x \<in> S"
     and "open S"
     and deriv: "(f has_derivative f') (at x)"
     and mono: "(\<forall>y\<in>S. f y \<le> f x) \<or> (\<forall>y\<in>S. f x \<le> f y)"
   shows "f' = (\<lambda>v. 0)"
   using mono
 proof
   assume "\<forall>y\<in>S. f y \<le> f x"
   with \<open>x \<in> S\<close> and \<open>open S\<close> have "eventually (\<lambda>y. f y \<le> f x) (at x)"
     unfolding eventually_at_topological by auto
   with deriv show ?thesis
     by (rule has_derivative_local_max)
 next
   assume "\<forall>y\<in>S. f x \<le> f y"
   with \<open>x \<in> S\<close> and \<open>open S\<close> have "eventually (\<lambda>y. f x \<le> f y) (at x)"
     unfolding eventually_at_topological by auto
   with deriv show ?thesis
     by (rule has_derivative_local_min)
 qed
 
 lemma differential_zero_maxmin_component:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes k: "k \<in> Basis"
     and ball: "0 < e" "(\<forall>y \<in> ball x e. (f y)\<bullet>k \<le> (f x)\<bullet>k) \<or> (\<forall>y\<in>ball x e. (f x)\<bullet>k \<le> (f y)\<bullet>k)"
     and diff: "f differentiable (at x)"
   shows "(\<Sum>j\<in>Basis. (frechet_derivative f (at x) j \<bullet> k) *\<^sub>R j) = (0::'a)" (is "?D k = 0")
 proof -
   let ?f' = "frechet_derivative f (at x)"
   have "x \<in> ball x e" using \<open>0 < e\<close> by simp
   moreover have "open (ball x e)" by simp
   moreover have "((\<lambda>x. f x \<bullet> k) has_derivative (\<lambda>h. ?f' h \<bullet> k)) (at x)"
     using bounded_linear_inner_left diff[unfolded frechet_derivative_works]
     by (rule bounded_linear.has_derivative)
   ultimately have "(\<lambda>h. frechet_derivative f (at x) h \<bullet> k) = (\<lambda>v. 0)"
     using ball(2) by (rule differential_zero_maxmin)
   then show ?thesis
     unfolding fun_eq_iff by simp
 qed
 
 subsection \<open>One-dimensional mean value theorem\<close>
 
 lemma mvt_simple:
   fixes f :: "real \<Rightarrow> real"
   assumes "a < b"
     and derf: "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> (f has_derivative f' x) (at x within {a..b})"
   shows "\<exists>x\<in>{a<..<b}. f b - f a = f' x (b - a)"
 proof (rule mvt)
   have "f differentiable_on {a..b}"
     using derf unfolding differentiable_on_def differentiable_def by force
   then show "continuous_on {a..b} f"
     by (rule differentiable_imp_continuous_on)
   show "(f has_derivative f' x) (at x)" if "a < x" "x < b" for x
     by (metis at_within_Icc_at derf leI order.asym that)
 qed (use assms in auto)
 
 lemma mvt_very_simple:
   fixes f :: "real \<Rightarrow> real"
   assumes "a \<le> b"
     and derf: "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> (f has_derivative f' x) (at x within {a..b})"
   shows "\<exists>x\<in>{a..b}. f b - f a = f' x (b - a)"
 proof (cases "a = b")
   interpret bounded_linear "f' b"
     using assms(2) assms(1) by auto
   case True
   then show ?thesis
     by force
 next
   case False
   then show ?thesis
     using mvt_simple[OF _ derf]
     by (metis \<open>a \<le> b\<close> atLeastAtMost_iff dual_order.order_iff_strict greaterThanLessThan_iff)
 qed
 
 text \<open>A nice generalization (see Havin's proof of 5.19 from Rudin's book).\<close>
 
 lemma mvt_general:
   fixes f :: "real \<Rightarrow> 'a::real_inner"
   assumes "a < b"
     and contf: "continuous_on {a..b} f"
     and derf: "\<And>x. \<lbrakk>a < x; x < b\<rbrakk> \<Longrightarrow> (f has_derivative f' x) (at x)"
   shows "\<exists>x\<in>{a<..<b}. norm (f b - f a) \<le> norm (f' x (b - a))"
 proof -
   have "\<exists>x\<in>{a<..<b}. (f b - f a) \<bullet> f b - (f b - f a) \<bullet> f a = (f b - f a) \<bullet> f' x (b - a)"
     apply (rule mvt [OF \<open>a < b\<close>, where f = "\<lambda>x. (f b - f a) \<bullet> f x"])
     apply (intro continuous_intros contf)
     using derf apply (auto intro: has_derivative_inner_right)
     done
   then obtain x where x: "x \<in> {a<..<b}"
     "(f b - f a) \<bullet> f b - (f b - f a) \<bullet> f a = (f b - f a) \<bullet> f' x (b - a)" ..
   show ?thesis
   proof (cases "f a = f b")
     case False
     have "norm (f b - f a) * norm (f b - f a) = (norm (f b - f a))\<^sup>2"
       by (simp add: power2_eq_square)
     also have "\<dots> = (f b - f a) \<bullet> (f b - f a)"
       unfolding power2_norm_eq_inner ..
     also have "\<dots> = (f b - f a) \<bullet> f' x (b - a)"
       using x(2) by (simp only: inner_diff_right)
     also have "\<dots> \<le> norm (f b - f a) * norm (f' x (b - a))"
       by (rule norm_cauchy_schwarz)
     finally show ?thesis
       using False x(1)
       by (auto simp add: mult_left_cancel)
   next
     case True
     then show ?thesis
       using \<open>a < b\<close> by (rule_tac x="(a + b) /2" in bexI) auto
   qed
 qed
 
 
 subsection \<open>More general bound theorems\<close>
 
 proposition differentiable_bound_general:
   fixes f :: "real \<Rightarrow> 'a::real_normed_vector"
   assumes "a < b"
     and f_cont: "continuous_on {a..b} f"
     and phi_cont: "continuous_on {a..b} \<phi>"
     and f': "\<And>x. a < x \<Longrightarrow> x < b \<Longrightarrow> (f has_vector_derivative f' x) (at x)"
     and phi': "\<And>x. a < x \<Longrightarrow> x < b \<Longrightarrow> (\<phi> has_vector_derivative \<phi>' x) (at x)"
     and bnd: "\<And>x. a < x \<Longrightarrow> x < b \<Longrightarrow> norm (f' x) \<le> \<phi>' x"
   shows "norm (f b - f a) \<le> \<phi> b - \<phi> a"
 proof -
   {
     fix x assume x: "a < x" "x < b"
     have "0 \<le> norm (f' x)" by simp
     also have "\<dots> \<le> \<phi>' x" using x by (auto intro!: bnd)
     finally have "0 \<le> \<phi>' x" .
   } note phi'_nonneg = this
   note f_tendsto = assms(2)[simplified continuous_on_def, rule_format]
   note phi_tendsto = assms(3)[simplified continuous_on_def, rule_format]
   {
     fix e::real assume "e > 0"
     define e2 where "e2 = e / 2"
     with \<open>e > 0\<close> have "e2 > 0" by simp
     let ?le = "\<lambda>x1. norm (f x1 - f a) \<le> \<phi> x1 - \<phi> a + e * (x1 - a) + e"
     define A where "A = {x2. a \<le> x2 \<and> x2 \<le> b \<and> (\<forall>x1\<in>{a ..< x2}. ?le x1)}"
     have A_subset: "A \<subseteq> {a..b}" by (auto simp: A_def)
     {
       fix x2
       assume a: "a \<le> x2" "x2 \<le> b" and le: "\<forall>x1\<in>{a..<x2}. ?le x1"
       have "?le x2" using \<open>e > 0\<close>
       proof cases
         assume "x2 \<noteq> a" with a have "a < x2" by simp
         have "at x2 within {a <..<x2}\<noteq> bot"
           using \<open>a < x2\<close>
           by (auto simp: trivial_limit_within islimpt_in_closure)
         moreover
         have "((\<lambda>x1. (\<phi> x1 - \<phi> a) + e * (x1 - a) + e) \<longlongrightarrow> (\<phi> x2 - \<phi> a) + e * (x2 - a) + e) (at x2 within {a <..<x2})"
           "((\<lambda>x1. norm (f x1 - f a)) \<longlongrightarrow> norm (f x2 - f a)) (at x2 within {a <..<x2})"
           using a
           by (auto intro!: tendsto_eq_intros f_tendsto phi_tendsto
             intro: tendsto_within_subset[where S="{a..b}"])
         moreover
         have "eventually (\<lambda>x. x > a) (at x2 within {a <..<x2})"
           by (auto simp: eventually_at_filter)
         hence "eventually ?le (at x2 within {a <..<x2})"
           unfolding eventually_at_filter
           by eventually_elim (insert le, auto)
         ultimately
         show ?thesis
           by (rule tendsto_le)
       qed simp
     } note le_cont = this
     have "a \<in> A"
       using assms by (auto simp: A_def)
     hence [simp]: "A \<noteq> {}" by auto
     have A_ivl: "\<And>x1 x2. x2 \<in> A \<Longrightarrow> x1 \<in> {a ..x2} \<Longrightarrow> x1 \<in> A"
       by (simp add: A_def)
     have [simp]: "bdd_above A" by (auto simp: A_def)
     define y where "y = Sup A"
     have "y \<le> b"
       unfolding y_def
       by (simp add: cSup_le_iff) (simp add: A_def)
      have leI: "\<And>x x1. a \<le> x1 \<Longrightarrow> x \<in> A \<Longrightarrow> x1 < x \<Longrightarrow> ?le x1"
        by (auto simp: A_def intro!: le_cont)
     have y_all_le: "\<forall>x1\<in>{a..<y}. ?le x1"
       by (auto simp: y_def less_cSup_iff leI)
     have "a \<le> y"
       by (metis \<open>a \<in> A\<close> \<open>bdd_above A\<close> cSup_upper y_def)
     have "y \<in> A"
       using y_all_le \<open>a \<le> y\<close> \<open>y \<le> b\<close>
       by (auto simp: A_def)
     hence "A = {a .. y}"
       using A_subset by (auto simp: subset_iff y_def cSup_upper intro: A_ivl)
     from le_cont[OF \<open>a \<le> y\<close> \<open>y \<le> b\<close> y_all_le] have le_y: "?le y" .
     have "y = b"
     proof (cases "a = y")
       case True
       with \<open>a < b\<close> have "y < b" by simp
       with \<open>a = y\<close> f_cont phi_cont \<open>e2 > 0\<close>
       have 1: "\<forall>\<^sub>F x in at y within {y..b}. dist (f x) (f y) < e2"
        and 2: "\<forall>\<^sub>F x in at y within {y..b}. dist (\<phi> x) (\<phi> y) < e2"
         by (auto simp: continuous_on_def tendsto_iff)
       have 3: "eventually (\<lambda>x. y < x) (at y within {y..b})"
         by (auto simp: eventually_at_filter)
       have 4: "eventually (\<lambda>x::real. x < b) (at y within {y..b})"
         using _ \<open>y < b\<close>
         by (rule order_tendstoD) (auto intro!: tendsto_eq_intros)
       from 1 2 3 4
       have eventually_le: "eventually (\<lambda>x. ?le x) (at y within {y .. b})"
       proof eventually_elim
         case (elim x1)
         have "norm (f x1 - f a) = norm (f x1 - f y)"
           by (simp add: \<open>a = y\<close>)
         also have "norm (f x1 - f y) \<le> e2"
           using elim \<open>a = y\<close> by (auto simp : dist_norm intro!:  less_imp_le)
         also have "\<dots> \<le> e2 + (\<phi> x1 - \<phi> a + e2 + e * (x1 - a))"
           using \<open>0 < e\<close> elim
           by (intro add_increasing2[OF add_nonneg_nonneg order.refl])
             (auto simp: \<open>a = y\<close> dist_norm intro!: mult_nonneg_nonneg)
         also have "\<dots> = \<phi> x1 - \<phi> a + e * (x1 - a) + e"
           by (simp add: e2_def)
         finally show "?le x1" .
       qed
       from this[unfolded eventually_at_topological] \<open>?le y\<close>
       obtain S where S: "open S" "y \<in> S" "\<And>x. x\<in>S \<Longrightarrow> x \<in> {y..b} \<Longrightarrow> ?le x"
         by metis
       from \<open>open S\<close> obtain d where d: "\<And>x. dist x y < d \<Longrightarrow> x \<in> S" "d > 0"
         by (force simp: dist_commute open_dist ball_def dest!: bspec[OF _ \<open>y \<in> S\<close>])
       define d' where "d' = min b (y + (d/2))"
       have "d' \<in> A"
         unfolding A_def
       proof safe
         show "a \<le> d'" using \<open>a = y\<close> \<open>0 < d\<close> \<open>y < b\<close> by (simp add: d'_def)
         show "d' \<le> b" by (simp add: d'_def)
         fix x1
         assume "x1 \<in> {a..<d'}"
         hence "x1 \<in> S" "x1 \<in> {y..b}"
           by (auto simp: \<open>a = y\<close> d'_def dist_real_def intro!: d )
         thus "?le x1"
           by (rule S)
       qed
       hence "d' \<le> y"
         unfolding y_def
         by (rule cSup_upper) simp
       then show "y = b" using \<open>d > 0\<close> \<open>y < b\<close>
         by (simp add: d'_def)
     next
       case False
       with \<open>a \<le> y\<close> have "a < y" by simp
       show "y = b"
       proof (rule ccontr)
         assume "y \<noteq> b"
         hence "y < b" using \<open>y \<le> b\<close> by simp
         let ?F = "at y within {y..<b}"
         from f' phi'
         have "(f has_vector_derivative f' y) ?F"
           and "(\<phi> has_vector_derivative \<phi>' y) ?F"
           using \<open>a < y\<close> \<open>y < b\<close>
           by (auto simp add: at_within_open[of _ "{a<..<b}"] has_vector_derivative_def
             intro!: has_derivative_subset[where s="{a<..<b}" and t="{y..<b}"])
         hence "\<forall>\<^sub>F x1 in ?F. norm (f x1 - f y - (x1 - y) *\<^sub>R f' y) \<le> e2 * \<bar>x1 - y\<bar>"
             "\<forall>\<^sub>F x1 in ?F. norm (\<phi> x1 - \<phi> y - (x1 - y) *\<^sub>R \<phi>' y) \<le> e2 * \<bar>x1 - y\<bar>"
           using \<open>e2 > 0\<close>
           by (auto simp: has_derivative_within_alt2 has_vector_derivative_def)
         moreover
         have "\<forall>\<^sub>F x1 in ?F. y \<le> x1" "\<forall>\<^sub>F x1 in ?F. x1 < b"
           by (auto simp: eventually_at_filter)
         ultimately
         have "\<forall>\<^sub>F x1 in ?F. norm (f x1 - f y) \<le> (\<phi> x1 - \<phi> y) + e * \<bar>x1 - y\<bar>"
           (is "\<forall>\<^sub>F x1 in ?F. ?le' x1")
         proof eventually_elim
           case (elim x1)
           from norm_triangle_ineq2[THEN order_trans, OF elim(1)]
           have "norm (f x1 - f y) \<le> norm (f' y) * \<bar>x1 - y\<bar> + e2 * \<bar>x1 - y\<bar>"
             by (simp add: ac_simps)
           also have "norm (f' y) \<le> \<phi>' y" using bnd \<open>a < y\<close> \<open>y < b\<close> by simp
           also have "\<phi>' y * \<bar>x1 - y\<bar> \<le> \<phi> x1 - \<phi> y + e2 * \<bar>x1 - y\<bar>"
             using elim by (simp add: ac_simps)
           finally
           have "norm (f x1 - f y) \<le> \<phi> x1 - \<phi> y + e2 * \<bar>x1 - y\<bar> + e2 * \<bar>x1 - y\<bar>"
             by (auto simp: mult_right_mono)
           thus ?case by (simp add: e2_def)
         qed
         moreover have "?le' y" by simp
         ultimately obtain S
         where S: "open S" "y \<in> S" "\<And>x. x\<in>S \<Longrightarrow> x \<in> {y..<b} \<Longrightarrow> ?le' x"
           unfolding eventually_at_topological
           by metis
         from \<open>open S\<close> obtain d where d: "\<And>x. dist x y < d \<Longrightarrow> x \<in> S" "d > 0"
           by (force simp: dist_commute open_dist ball_def dest!: bspec[OF _ \<open>y \<in> S\<close>])
         define d' where "d' = min ((y + b)/2) (y + (d/2))"
         have "d' \<in> A"
           unfolding A_def
         proof safe
           show "a \<le> d'" using \<open>a < y\<close> \<open>0 < d\<close> \<open>y < b\<close> by (simp add: d'_def)
           show "d' \<le> b" using \<open>y < b\<close> by (simp add: d'_def min_def)
           fix x1
           assume x1: "x1 \<in> {a..<d'}"
           show "?le x1"
           proof (cases "x1 < y")
             case True
             then show ?thesis
               using \<open>y \<in> A\<close> local.leI x1 by auto
           next
             case False
             hence x1': "x1 \<in> S" "x1 \<in> {y..<b}" using x1
               by (auto simp: d'_def dist_real_def intro!: d)
             have "norm (f x1 - f a) \<le> norm (f x1 - f y) + norm (f y - f a)"
               by (rule order_trans[OF _ norm_triangle_ineq]) simp
             also note S(3)[OF x1']
             also note le_y
             finally show "?le x1"
               using False by (auto simp: algebra_simps)
           qed
         qed
         hence "d' \<le> y"
           unfolding y_def by (rule cSup_upper) simp
         thus False using \<open>d > 0\<close> \<open>y < b\<close>
           by (simp add: d'_def min_def split: if_split_asm)
       qed
     qed
     with le_y have "norm (f b - f a) \<le> \<phi> b - \<phi> a + e * (b - a + 1)"
       by (simp add: algebra_simps)
   } note * = this
   show ?thesis
   proof (rule field_le_epsilon)
     fix e::real assume "e > 0"
     then show "norm (f b - f a) \<le> \<phi> b - \<phi> a + e"
       using *[of "e / (b - a + 1)"] \<open>a < b\<close> by simp
   qed
 qed
 
 lemma differentiable_bound:
   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
   assumes "convex S"
     and derf: "\<And>x. x\<in>S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
     and B: "\<And>x. x \<in> S \<Longrightarrow> onorm (f' x) \<le> B"
     and x: "x \<in> S"
     and y: "y \<in> S"
   shows "norm (f x - f y) \<le> B * norm (x - y)"
 proof -
   let ?p = "\<lambda>u. x + u *\<^sub>R (y - x)"
   let ?\<phi> = "\<lambda>h. h * B * norm (x - y)"
   have *: "x + u *\<^sub>R (y - x) \<in> S" if "u \<in> {0..1}" for u
   proof -
     have "u *\<^sub>R y = u *\<^sub>R (y - x) + u *\<^sub>R x"
       by (simp add: scale_right_diff_distrib)
     then show "x + u *\<^sub>R (y - x) \<in> S"
       using that \<open>convex S\<close> x y by (simp add: convex_alt)
         (metis pth_b(2) pth_c(1) scaleR_collapse)
   qed
   have "\<And>z. z \<in> (\<lambda>u. x + u *\<^sub>R (y - x)) ` {0..1} \<Longrightarrow>
           (f has_derivative f' z) (at z within (\<lambda>u. x + u *\<^sub>R (y - x)) ` {0..1})"
-    by (auto intro: * has_derivative_within_subset [OF derf])
+    by (auto intro: * has_derivative_subset [OF derf])
   then have "continuous_on (?p ` {0..1}) f"
     unfolding continuous_on_eq_continuous_within
     by (meson has_derivative_continuous)
   with * have 1: "continuous_on {0 .. 1} (f \<circ> ?p)"
     by (intro continuous_intros)+
   {
     fix u::real assume u: "u \<in>{0 <..< 1}"
     let ?u = "?p u"
     interpret linear "(f' ?u)"
       using u by (auto intro!: has_derivative_linear derf *)
     have "(f \<circ> ?p has_derivative (f' ?u) \<circ> (\<lambda>u. 0 + u *\<^sub>R (y - x))) (at u within box 0 1)"
-      by (intro derivative_intros has_derivative_within_subset [OF derf]) (use u * in auto)
+      by (intro derivative_intros has_derivative_subset [OF derf]) (use u * in auto)
     hence "((f \<circ> ?p) has_vector_derivative f' ?u (y - x)) (at u)"
       by (simp add: at_within_open[OF u open_greaterThanLessThan] scaleR has_vector_derivative_def o_def)
   } note 2 = this
   have 3: "continuous_on {0..1} ?\<phi>"
     by (rule continuous_intros)+
   have 4: "(?\<phi> has_vector_derivative B * norm (x - y)) (at u)" for u
     by (auto simp: has_vector_derivative_def intro!: derivative_eq_intros)
   {
     fix u::real assume u: "u \<in>{0 <..< 1}"
     let ?u = "?p u"
     interpret bounded_linear "(f' ?u)"
       using u by (auto intro!: has_derivative_bounded_linear derf *)
     have "norm (f' ?u (y - x)) \<le> onorm (f' ?u) * norm (y - x)"
       by (rule onorm) (rule bounded_linear)
     also have "onorm (f' ?u) \<le> B"
       using u by (auto intro!: assms(3)[rule_format] *)
     finally have "norm ((f' ?u) (y - x)) \<le> B * norm (x - y)"
       by (simp add: mult_right_mono norm_minus_commute)
   } note 5 = this
   have "norm (f x - f y) = norm ((f \<circ> (\<lambda>u. x + u *\<^sub>R (y - x))) 1 - (f \<circ> (\<lambda>u. x + u *\<^sub>R (y - x))) 0)"
     by (auto simp add: norm_minus_commute)
   also
   from differentiable_bound_general[OF zero_less_one 1, OF 3 2 4 5]
   have "norm ((f \<circ> ?p) 1 - (f \<circ> ?p) 0) \<le> B * norm (x - y)"
     by simp
   finally show ?thesis .
 qed
 
 lemma field_differentiable_bound:
   fixes S :: "'a::real_normed_field set"
   assumes cvs: "convex S"
       and df:  "\<And>z. z \<in> S \<Longrightarrow> (f has_field_derivative f' z) (at z within S)"
       and dn:  "\<And>z. z \<in> S \<Longrightarrow> norm (f' z) \<le> B"
       and "x \<in> S"  "y \<in> S"
     shows "norm(f x - f y) \<le> B * norm(x - y)"
   apply (rule differentiable_bound [OF cvs])
   apply (erule df [unfolded has_field_derivative_def])
   apply (rule onorm_le, simp_all add: norm_mult mult_right_mono assms)
   done
 
 lemma
   differentiable_bound_segment:
   fixes f::"'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
   assumes "\<And>t. t \<in> {0..1} \<Longrightarrow> x0 + t *\<^sub>R a \<in> G"
   assumes f': "\<And>x. x \<in> G \<Longrightarrow> (f has_derivative f' x) (at x within G)"
   assumes B: "\<And>x. x \<in> {0..1} \<Longrightarrow> onorm (f' (x0 + x *\<^sub>R a)) \<le> B"
   shows "norm (f (x0 + a) - f x0) \<le> norm a * B"
 proof -
   let ?G = "(\<lambda>x. x0 + x *\<^sub>R a) ` {0..1}"
   have "?G = (+) x0 ` (\<lambda>x. x *\<^sub>R a) ` {0..1}" by auto
   also have "convex \<dots>"
     by (intro convex_translation convex_scaled convex_real_interval)
   finally have "convex ?G" .
   moreover have "?G \<subseteq> G" "x0 \<in> ?G" "x0 + a \<in> ?G" using assms by (auto intro: image_eqI[where x=1])
   ultimately show ?thesis
     using has_derivative_subset[OF f' \<open>?G \<subseteq> G\<close>] B
       differentiable_bound[of "(\<lambda>x. x0 + x *\<^sub>R a) ` {0..1}" f f' B "x0 + a" x0]
     by (force simp: ac_simps)
 qed
 
 lemma differentiable_bound_linearization:
   fixes f::"'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
   assumes S: "\<And>t. t \<in> {0..1} \<Longrightarrow> a + t *\<^sub>R (b - a) \<in> S"
   assumes f'[derivative_intros]: "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
   assumes B: "\<And>x. x \<in> S \<Longrightarrow> onorm (f' x - f' x0) \<le> B"
   assumes "x0 \<in> S"
   shows "norm (f b - f a - f' x0 (b - a)) \<le> norm (b - a) * B"
 proof -
   define g where [abs_def]: "g x = f x - f' x0 x" for x
   have g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative (\<lambda>i. f' x i - f' x0 i)) (at x within S)"
     unfolding g_def using assms
     by (auto intro!: derivative_eq_intros
       bounded_linear.has_derivative[OF has_derivative_bounded_linear, OF f'])
   from B have "\<forall>x\<in>{0..1}. onorm (\<lambda>i. f' (a + x *\<^sub>R (b - a)) i - f' x0 i) \<le> B"
     using assms by (auto simp: fun_diff_def)
   with differentiable_bound_segment[OF S g] \<open>x0 \<in> S\<close>
   show ?thesis
     by (simp add: g_def field_simps linear_diff[OF has_derivative_linear[OF f']])
 qed
 
 lemma vector_differentiable_bound_linearization:
   fixes f::"real \<Rightarrow> 'b::real_normed_vector"
   assumes f': "\<And>x. x \<in> S \<Longrightarrow> (f has_vector_derivative f' x) (at x within S)"
   assumes "closed_segment a b \<subseteq> S"
   assumes B: "\<And>x. x \<in> S \<Longrightarrow> norm (f' x - f' x0) \<le> B"
   assumes "x0 \<in> S"
   shows "norm (f b - f a - (b - a) *\<^sub>R f' x0) \<le> norm (b - a) * B"
   using assms
   by (intro differentiable_bound_linearization[of a b S f "\<lambda>x h. h *\<^sub>R f' x" x0 B])
     (force simp: closed_segment_real_eq has_vector_derivative_def
       scaleR_diff_right[symmetric] mult.commute[of B]
       intro!: onorm_le mult_left_mono)+
 
 
 text \<open>In particular.\<close>
 
 lemma has_derivative_zero_constant:
   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
   assumes "convex s"
     and "\<And>x. x \<in> s \<Longrightarrow> (f has_derivative (\<lambda>h. 0)) (at x within s)"
   shows "\<exists>c. \<forall>x\<in>s. f x = c"
 proof -
   { fix x y assume "x \<in> s" "y \<in> s"
     then have "norm (f x - f y) \<le> 0 * norm (x - y)"
       using assms by (intro differentiable_bound[of s]) (auto simp: onorm_zero)
     then have "f x = f y"
       by simp }
   then show ?thesis
     by metis
 qed
 
 lemma has_field_derivative_zero_constant:
   assumes "convex s" "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative 0) (at x within s)"
   shows   "\<exists>c. \<forall>x\<in>s. f (x) = (c :: 'a :: real_normed_field)"
 proof (rule has_derivative_zero_constant)
   have A: "(*) 0 = (\<lambda>_. 0 :: 'a)" by (intro ext) simp
   fix x assume "x \<in> s" thus "(f has_derivative (\<lambda>h. 0)) (at x within s)"
     using assms(2)[of x] by (simp add: has_field_derivative_def A)
 qed fact
 
 lemma
   has_vector_derivative_zero_constant:
   assumes "convex s"
   assumes "\<And>x. x \<in> s \<Longrightarrow> (f has_vector_derivative 0) (at x within s)"
   obtains c where "\<And>x. x \<in> s \<Longrightarrow> f x = c"
   using has_derivative_zero_constant[of s f] assms
   by (auto simp: has_vector_derivative_def)
 
 lemma has_derivative_zero_unique:
   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
   assumes "convex s"
     and "\<And>x. x \<in> s \<Longrightarrow> (f has_derivative (\<lambda>h. 0)) (at x within s)"
     and "x \<in> s" "y \<in> s"
   shows "f x = f y"
   using has_derivative_zero_constant[OF assms(1,2)] assms(3-) by force
 
 lemma has_derivative_zero_unique_connected:
   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
   assumes "open s" "connected s"
   assumes f: "\<And>x. x \<in> s \<Longrightarrow> (f has_derivative (\<lambda>x. 0)) (at x)"
   assumes "x \<in> s" "y \<in> s"
   shows "f x = f y"
 proof (rule connected_local_const[where f=f, OF \<open>connected s\<close> \<open>x\<in>s\<close> \<open>y\<in>s\<close>])
   show "\<forall>a\<in>s. eventually (\<lambda>b. f a = f b) (at a within s)"
   proof
     fix a assume "a \<in> s"
     with \<open>open s\<close> obtain e where "0 < e" "ball a e \<subseteq> s"
       by (rule openE)
     then have "\<exists>c. \<forall>x\<in>ball a e. f x = c"
       by (intro has_derivative_zero_constant)
          (auto simp: at_within_open[OF _ open_ball] f)
     with \<open>0<e\<close> have "\<forall>x\<in>ball a e. f a = f x"
       by auto
     then show "eventually (\<lambda>b. f a = f b) (at a within s)"
       using \<open>0<e\<close> unfolding eventually_at_topological
       by (intro exI[of _ "ball a e"]) auto
   qed
 qed
 
 subsection \<open>Differentiability of inverse function (most basic form)\<close>
 
 lemma has_derivative_inverse_basic:
   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
   assumes derf: "(f has_derivative f') (at (g y))"
     and ling': "bounded_linear g'"
     and "g' \<circ> f' = id"
     and contg: "continuous (at y) g"
     and "open T"
     and "y \<in> T"
     and fg: "\<And>z. z \<in> T \<Longrightarrow> f (g z) = z"
   shows "(g has_derivative g') (at y)"
 proof -
   interpret f': bounded_linear f'
     using assms unfolding has_derivative_def by auto
   interpret g': bounded_linear g'
     using assms by auto
   obtain C where C: "0 < C" "\<And>x. norm (g' x) \<le> norm x * C"
     using bounded_linear.pos_bounded[OF assms(2)] by blast
   have lem1: "\<forall>e>0. \<exists>d>0. \<forall>z.
     norm (z - y) < d \<longrightarrow> norm (g z - g y - g'(z - y)) \<le> e * norm (g z - g y)"
   proof (intro allI impI)
     fix e :: real
     assume "e > 0"
     with C(1) have *: "e / C > 0" by auto
     obtain d0 where  "0 < d0" and d0:
         "\<And>u. norm (u - g y) < d0 \<Longrightarrow> norm (f u - f (g y) - f' (u - g y)) \<le> e / C * norm (u - g y)"
       using derf * unfolding has_derivative_at_alt by blast
     obtain d1 where "0 < d1" and d1: "\<And>x. \<lbrakk>0 < dist x y; dist x y < d1\<rbrakk> \<Longrightarrow> dist (g x) (g y) < d0"
       using contg \<open>0 < d0\<close> unfolding continuous_at Lim_at by blast
     obtain d2 where "0 < d2" and d2: "\<And>u. dist u y < d2 \<Longrightarrow> u \<in> T"
       using \<open>open T\<close> \<open>y \<in> T\<close> unfolding open_dist by blast
     obtain d where d: "0 < d" "d < d1" "d < d2"
       using field_lbound_gt_zero[OF \<open>0 < d1\<close> \<open>0 < d2\<close>] by blast
     show "\<exists>d>0. \<forall>z. norm (z - y) < d \<longrightarrow> norm (g z - g y - g' (z - y)) \<le> e * norm (g z - g y)"
     proof (intro exI allI impI conjI)
       fix z
       assume as: "norm (z - y) < d"
       then have "z \<in> T"
         using d2 d unfolding dist_norm by auto
       have "norm (g z - g y - g' (z - y)) \<le> norm (g' (f (g z) - y - f' (g z - g y)))"
         unfolding g'.diff f'.diff
         unfolding assms(3)[unfolded o_def id_def, THEN fun_cong] fg[OF \<open>z\<in>T\<close>]
         by (simp add: norm_minus_commute)
       also have "\<dots> \<le> norm (f (g z) - y - f' (g z - g y)) * C"
         by (rule C(2))
       also have "\<dots> \<le> (e / C) * norm (g z - g y) * C"
       proof -
         have "norm (g z - g y) < d0"
           by (metis as cancel_comm_monoid_add_class.diff_cancel d(2) \<open>0 < d0\<close> d1 diff_gt_0_iff_gt diff_strict_mono dist_norm dist_self zero_less_dist_iff)
         then show ?thesis
           by (metis C(1) \<open>y \<in> T\<close> d0 fg real_mult_le_cancel_iff1)
       qed
       also have "\<dots> \<le> e * norm (g z - g y)"
         using C by (auto simp add: field_simps)
       finally show "norm (g z - g y - g' (z - y)) \<le> e * norm (g z - g y)"
         by simp
     qed (use d in auto)
   qed
   have *: "(0::real) < 1 / 2"
     by auto
   obtain d where "0 < d" and d:
       "\<And>z. norm (z - y) < d \<Longrightarrow> norm (g z - g y - g' (z - y)) \<le> 1/2 * norm (g z - g y)"
     using lem1 * by blast
   define B where "B = C * 2"
   have "B > 0"
     unfolding B_def using C by auto
   have lem2: "norm (g z - g y) \<le> B * norm (z - y)" if z: "norm(z - y) < d" for z
   proof -
     have "norm (g z - g y) \<le> norm(g' (z - y)) + norm ((g z - g y) - g'(z - y))"
       by (rule norm_triangle_sub)
     also have "\<dots> \<le> norm (g' (z - y)) + 1 / 2 * norm (g z - g y)"
       by (rule add_left_mono) (use d z in auto)
     also have "\<dots> \<le> norm (z - y) * C + 1 / 2 * norm (g z - g y)"
       by (rule add_right_mono) (use C in auto)
     finally show "norm (g z - g y) \<le> B * norm (z - y)"
       unfolding B_def
       by (auto simp add: field_simps)
   qed
   show ?thesis
     unfolding has_derivative_at_alt
   proof (intro conjI assms allI impI)
     fix e :: real
     assume "e > 0"
     then have *: "e / B > 0" by (metis \<open>B > 0\<close> divide_pos_pos)
     obtain d' where "0 < d'" and d':
         "\<And>z. norm (z - y) < d' \<Longrightarrow> norm (g z - g y - g' (z - y)) \<le> e / B * norm (g z - g y)"
       using lem1 * by blast
     obtain k where k: "0 < k" "k < d" "k < d'"
       using field_lbound_gt_zero[OF \<open>0 < d\<close> \<open>0 < d'\<close>] by blast
     show "\<exists>d>0. \<forall>ya. norm (ya - y) < d \<longrightarrow> norm (g ya - g y - g' (ya - y)) \<le> e * norm (ya - y)"
     proof (intro exI allI impI conjI)
       fix z
       assume as: "norm (z - y) < k"
       then have "norm (g z - g y - g' (z - y)) \<le> e / B * norm(g z - g y)"
         using d' k by auto
       also have "\<dots> \<le> e * norm (z - y)"
         unfolding times_divide_eq_left pos_divide_le_eq[OF \<open>B>0\<close>]
         using lem2[of z] k as \<open>e > 0\<close>
         by (auto simp add: field_simps)
       finally show "norm (g z - g y - g' (z - y)) \<le> e * norm (z - y)"
         by simp
     qed (use k in auto)
   qed
 qed
 
 text\<^marker>\<open>tag unimportant\<close>\<open>Inverse function theorem for complex derivatives\<close>
 lemma has_field_derivative_inverse_basic:
   shows "DERIV f (g y) :> f' \<Longrightarrow>
         f' \<noteq> 0 \<Longrightarrow>
         continuous (at y) g \<Longrightarrow>
         open t \<Longrightarrow>
         y \<in> t \<Longrightarrow>
         (\<And>z. z \<in> t \<Longrightarrow> f (g z) = z)
         \<Longrightarrow> DERIV g y :> inverse (f')"
   unfolding has_field_derivative_def
   apply (rule has_derivative_inverse_basic)
   apply (auto simp:  bounded_linear_mult_right)
   done
 
 text \<open>Simply rewrite that based on the domain point x.\<close>
 
 lemma has_derivative_inverse_basic_x:
   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
   assumes "(f has_derivative f') (at x)"
     and "bounded_linear g'"
     and "g' \<circ> f' = id"
     and "continuous (at (f x)) g"
     and "g (f x) = x"
     and "open T"
     and "f x \<in> T"
     and "\<And>y. y \<in> T \<Longrightarrow> f (g y) = y"
   shows "(g has_derivative g') (at (f x))"
   by (rule has_derivative_inverse_basic) (use assms in auto)
 
 text \<open>This is the version in Dieudonne', assuming continuity of f and g.\<close>
 
 lemma has_derivative_inverse_dieudonne:
   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
   assumes "open S"
     and "open (f ` S)"
     and "continuous_on S f"
     and "continuous_on (f ` S) g"
     and "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x"
     and "x \<in> S"
     and "(f has_derivative f') (at x)"
     and "bounded_linear g'"
     and "g' \<circ> f' = id"
   shows "(g has_derivative g') (at (f x))"
   apply (rule has_derivative_inverse_basic_x[OF assms(7-9) _ _ assms(2)])
   using assms(3-6)
   unfolding continuous_on_eq_continuous_at[OF assms(1)] continuous_on_eq_continuous_at[OF assms(2)]
   apply auto
   done
 
 text \<open>Here's the simplest way of not assuming much about g.\<close>
 
 proposition has_derivative_inverse:
   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
   assumes "compact S"
     and "x \<in> S"
     and fx: "f x \<in> interior (f ` S)"
     and "continuous_on S f"
     and gf: "\<And>y. y \<in> S \<Longrightarrow> g (f y) = y"
     and "(f has_derivative f') (at x)"
     and "bounded_linear g'"
     and "g' \<circ> f' = id"
   shows "(g has_derivative g') (at (f x))"
 proof -
   have *: "\<And>y. y \<in> interior (f ` S) \<Longrightarrow> f (g y) = y"
     by (metis gf image_iff interior_subset subsetCE)
   show ?thesis
     apply (rule has_derivative_inverse_basic_x[OF assms(6-8), where T = "interior (f ` S)"])
     apply (rule continuous_on_interior[OF _ fx])
     apply (rule continuous_on_inv)
     apply (simp_all add: assms *)
     done
 qed
 
 
 text \<open>Invertible derivative continuous at a point implies local
 injectivity. It's only for this we need continuity of the derivative,
 except of course if we want the fact that the inverse derivative is
 also continuous. So if we know for some other reason that the inverse
 function exists, it's OK.\<close>
 
 proposition has_derivative_locally_injective:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
   assumes "a \<in> S"
       and "open S"
       and bling: "bounded_linear g'"
       and "g' \<circ> f' a = id"
       and derf: "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x)"
       and "\<And>e. e > 0 \<Longrightarrow> \<exists>d>0. \<forall>x. dist a x < d \<longrightarrow> onorm (\<lambda>v. f' x v - f' a v) < e"
   obtains r where "r > 0" "ball a r \<subseteq> S" "inj_on f (ball a r)"
 proof -
   interpret bounded_linear g'
     using assms by auto
   note f'g' = assms(4)[unfolded id_def o_def,THEN cong]
   have "g' (f' a (\<Sum>Basis)) = (\<Sum>Basis)" "(\<Sum>Basis) \<noteq> (0::'n)"
     using f'g' by auto
   then have *: "0 < onorm g'"
     unfolding onorm_pos_lt[OF assms(3)]
     by fastforce
   define k where "k = 1 / onorm g' / 2"
   have *: "k > 0"
     unfolding k_def using * by auto
   obtain d1 where d1:
       "0 < d1"
       "\<And>x. dist a x < d1 \<Longrightarrow> onorm (\<lambda>v. f' x v - f' a v) < k"
     using assms(6) * by blast
   from \<open>open S\<close> obtain d2 where "d2 > 0" "ball a d2 \<subseteq> S"
     using \<open>a\<in>S\<close> ..
   obtain d2 where d2: "0 < d2" "ball a d2 \<subseteq> S"
     using \<open>0 < d2\<close> \<open>ball a d2 \<subseteq> S\<close> by blast
   obtain d where d: "0 < d" "d < d1" "d < d2"
     using field_lbound_gt_zero[OF d1(1) d2(1)] by blast
   show ?thesis
   proof
     show "0 < d" by (fact d)
     show "ball a d \<subseteq> S"
       using \<open>d < d2\<close> \<open>ball a d2 \<subseteq> S\<close> by auto
     show "inj_on f (ball a d)"
     unfolding inj_on_def
     proof (intro strip)
       fix x y
       assume as: "x \<in> ball a d" "y \<in> ball a d" "f x = f y"
       define ph where [abs_def]: "ph w = w - g' (f w - f x)" for w
       have ph':"ph = g' \<circ> (\<lambda>w. f' a w - (f w - f x))"
         unfolding ph_def o_def  by (simp add: diff f'g')
       have "norm (ph x - ph y) \<le> (1 / 2) * norm (x - y)"
       proof (rule differentiable_bound[OF convex_ball _ _ as(1-2)])
         fix u
         assume u: "u \<in> ball a d"
         then have "u \<in> S"
           using d d2 by auto
         have *: "(\<lambda>v. v - g' (f' u v)) = g' \<circ> (\<lambda>w. f' a w - f' u w)"
           unfolding o_def and diff
           using f'g' by auto
         have blin: "bounded_linear (f' a)"
           using \<open>a \<in> S\<close> derf by blast
         show "(ph has_derivative (\<lambda>v. v - g' (f' u v))) (at u within ball a d)"
           unfolding ph' * comp_def
           by (rule \<open>u \<in> S\<close> derivative_eq_intros has_derivative_at_withinI [OF derf] bounded_linear.has_derivative [OF blin]  bounded_linear.has_derivative [OF bling] |simp)+
         have **: "bounded_linear (\<lambda>x. f' u x - f' a x)" "bounded_linear (\<lambda>x. f' a x - f' u x)"
           using \<open>u \<in> S\<close> blin bounded_linear_sub derf by auto
         then have "onorm (\<lambda>v. v - g' (f' u v)) \<le> onorm g' * onorm (\<lambda>w. f' a w - f' u w)"
           by (simp add: "*" bounded_linear_axioms onorm_compose)
         also have "\<dots> \<le> onorm g' * k"
           apply (rule mult_left_mono)
           using d1(2)[of u]
           using onorm_neg[where f="\<lambda>x. f' u x - f' a x"] d u onorm_pos_le[OF bling] apply (auto simp: algebra_simps)
           done
         also have "\<dots> \<le> 1 / 2"
           unfolding k_def by auto
         finally show "onorm (\<lambda>v. v - g' (f' u v)) \<le> 1 / 2" .
       qed
       moreover have "norm (ph y - ph x) = norm (y - x)"
         by (simp add: as(3) ph_def)
       ultimately show "x = y"
         unfolding norm_minus_commute by auto
     qed
   qed
 qed
 
 
 subsection \<open>Uniformly convergent sequence of derivatives\<close>
 
 lemma has_derivative_sequence_lipschitz_lemma:
   fixes f :: "nat \<Rightarrow> 'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
   assumes "convex S"
     and derf: "\<And>n x. x \<in> S \<Longrightarrow> ((f n) has_derivative (f' n x)) (at x within S)"
     and nle: "\<And>n x h. \<lbrakk>n\<ge>N; x \<in> S\<rbrakk> \<Longrightarrow> norm (f' n x h - g' x h) \<le> e * norm h"
     and "0 \<le> e"
   shows "\<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>S. \<forall>y\<in>S. norm ((f m x - f n x) - (f m y - f n y)) \<le> 2 * e * norm (x - y)"
 proof clarify
   fix m n x y
   assume as: "N \<le> m" "N \<le> n" "x \<in> S" "y \<in> S"
   show "norm ((f m x - f n x) - (f m y - f n y)) \<le> 2 * e * norm (x - y)"
   proof (rule differentiable_bound[where f'="\<lambda>x h. f' m x h - f' n x h", OF \<open>convex S\<close> _ _ as(3-4)])
     fix x
     assume "x \<in> S"
     show "((\<lambda>a. f m a - f n a) has_derivative (\<lambda>h. f' m x h - f' n x h)) (at x within S)"
       by (rule derivative_intros derf \<open>x\<in>S\<close>)+
     show "onorm (\<lambda>h. f' m x h - f' n x h) \<le> 2 * e"
     proof (rule onorm_bound)
       fix h
       have "norm (f' m x h - f' n x h) \<le> norm (f' m x h - g' x h) + norm (f' n x h - g' x h)"
         using norm_triangle_ineq[of "f' m x h - g' x h" "- f' n x h + g' x h"]
         by (auto simp add: algebra_simps norm_minus_commute)
       also have "\<dots> \<le> e * norm h + e * norm h"
         using nle[OF \<open>N \<le> m\<close> \<open>x \<in> S\<close>, of h] nle[OF \<open>N \<le> n\<close> \<open>x \<in> S\<close>, of h]
         by (auto simp add: field_simps)
       finally show "norm (f' m x h - f' n x h) \<le> 2 * e * norm h"
         by auto
     qed (simp add: \<open>0 \<le> e\<close>)
   qed
 qed
 
 lemma has_derivative_sequence_Lipschitz:
   fixes f :: "nat \<Rightarrow> 'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
   assumes "convex S"
     and "\<And>n x. x \<in> S \<Longrightarrow> ((f n) has_derivative (f' n x)) (at x within S)"
     and nle: "\<And>e. e > 0 \<Longrightarrow> \<forall>\<^sub>F n in sequentially. \<forall>x\<in>S. \<forall>h. norm (f' n x h - g' x h) \<le> e * norm h"
     and "e > 0"
   shows "\<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>S. \<forall>y\<in>S.
     norm ((f m x - f n x) - (f m y - f n y)) \<le> e * norm (x - y)"
 proof -
   have *: "2 * (e/2) = e"
     using \<open>e > 0\<close> by auto
   obtain N where "\<forall>n\<ge>N. \<forall>x\<in>S. \<forall>h. norm (f' n x h - g' x h) \<le> (e/2) * norm h"
     using nle \<open>e > 0\<close>
     unfolding eventually_sequentially
     by (metis less_divide_eq_numeral1(1) mult_zero_left)
   then show "\<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>S. \<forall>y\<in>S. norm (f m x - f n x - (f m y - f n y)) \<le> e * norm (x - y)"
     apply (rule_tac x=N in exI)
     apply (rule has_derivative_sequence_lipschitz_lemma[where e="e/2", unfolded *])
     using assms \<open>e > 0\<close>
     apply auto
     done
 qed
 
 proposition has_derivative_sequence:
   fixes f :: "nat \<Rightarrow> 'a::real_normed_vector \<Rightarrow> 'b::banach"
   assumes "convex S"
     and derf: "\<And>n x. x \<in> S \<Longrightarrow> ((f n) has_derivative (f' n x)) (at x within S)"
     and nle: "\<And>e. e > 0 \<Longrightarrow> \<forall>\<^sub>F n in sequentially. \<forall>x\<in>S. \<forall>h. norm (f' n x h - g' x h) \<le> e * norm h"
     and "x0 \<in> S"
     and lim: "((\<lambda>n. f n x0) \<longlongrightarrow> l) sequentially"
   shows "\<exists>g. \<forall>x\<in>S. (\<lambda>n. f n x) \<longlonglongrightarrow> g x \<and> (g has_derivative g'(x)) (at x within S)"
 proof -
   have lem1: "\<And>e. e > 0 \<Longrightarrow> \<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>S. \<forall>y\<in>S.
       norm ((f m x - f n x) - (f m y - f n y)) \<le> e * norm (x - y)"
     using assms(1,2,3) by (rule has_derivative_sequence_Lipschitz)
   have "\<exists>g. \<forall>x\<in>S. ((\<lambda>n. f n x) \<longlongrightarrow> g x) sequentially"
   proof (intro ballI bchoice)
     fix x
     assume "x \<in> S"
     show "\<exists>y. (\<lambda>n. f n x) \<longlonglongrightarrow> y"
     unfolding convergent_eq_Cauchy
     proof (cases "x = x0")
       case True
       then show "Cauchy (\<lambda>n. f n x)"
         using LIMSEQ_imp_Cauchy[OF lim] by auto
     next
       case False
       show "Cauchy (\<lambda>n. f n x)"
         unfolding Cauchy_def
       proof (intro allI impI)
         fix e :: real
         assume "e > 0"
         hence *: "e / 2 > 0" "e / 2 / norm (x - x0) > 0" using False by auto
         obtain M where M: "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (f m x0) (f n x0) < e / 2"
           using LIMSEQ_imp_Cauchy[OF lim] * unfolding Cauchy_def by blast
         obtain N where N:
           "\<forall>m\<ge>N. \<forall>n\<ge>N.
             \<forall>u\<in>S. \<forall>y\<in>S. norm (f m u - f n u - (f m y - f n y)) \<le>
               e / 2 / norm (x - x0) * norm (u - y)"
         using lem1 *(2) by blast
         show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (f m x) (f n x) < e"
         proof (intro exI allI impI)
           fix m n
           assume as: "max M N \<le>m" "max M N\<le>n"
           have "dist (f m x) (f n x) \<le> norm (f m x0 - f n x0) + norm (f m x - f n x - (f m x0 - f n x0))"
             unfolding dist_norm
             by (rule norm_triangle_sub)
           also have "\<dots> \<le> norm (f m x0 - f n x0) + e / 2"
             using N \<open>x\<in>S\<close> \<open>x0\<in>S\<close> as False by fastforce
           also have "\<dots> < e / 2 + e / 2"
             by (rule add_strict_right_mono) (use as M in \<open>auto simp: dist_norm\<close>)
           finally show "dist (f m x) (f n x) < e"
             by auto
         qed
       qed
     qed
   qed
   then obtain g where g: "\<forall>x\<in>S. (\<lambda>n. f n x) \<longlonglongrightarrow> g x" ..
   have lem2: "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>S. \<forall>y\<in>S. norm ((f n x - f n y) - (g x - g y)) \<le> e * norm (x - y)" if "e > 0" for e
   proof -
     obtain N where
       N: "\<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>S. \<forall>y\<in>S. norm (f m x - f n x - (f m y - f n y)) \<le> e * norm (x - y)"
       using lem1 \<open>e > 0\<close> by blast
     show "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>S. \<forall>y\<in>S. norm (f n x - f n y - (g x - g y)) \<le> e * norm (x - y)"
     proof (intro exI ballI allI impI)
       fix n x y
       assume as: "N \<le> n" "x \<in> S" "y \<in> S"
       have "((\<lambda>m. norm (f n x - f n y - (f m x - f m y))) \<longlongrightarrow> norm (f n x - f n y - (g x - g y))) sequentially"
         by (intro tendsto_intros g[rule_format] as)
       moreover have "eventually (\<lambda>m. norm (f n x - f n y - (f m x - f m y)) \<le> e * norm (x - y)) sequentially"
         unfolding eventually_sequentially
       proof (intro exI allI impI)
         fix m
         assume "N \<le> m"
         then show "norm (f n x - f n y - (f m x - f m y)) \<le> e * norm (x - y)"
           using N as by (auto simp add: algebra_simps)
       qed
       ultimately show "norm (f n x - f n y - (g x - g y)) \<le> e * norm (x - y)"
         by (simp add: tendsto_upperbound)
     qed
   qed
   have "\<forall>x\<in>S. ((\<lambda>n. f n x) \<longlongrightarrow> g x) sequentially \<and> (g has_derivative g' x) (at x within S)"
     unfolding has_derivative_within_alt2
   proof (intro ballI conjI allI impI)
     fix x
     assume "x \<in> S"
     then show "(\<lambda>n. f n x) \<longlonglongrightarrow> g x"
       by (simp add: g)
     have tog': "(\<lambda>n. f' n x u) \<longlonglongrightarrow> g' x u" for u
       unfolding filterlim_def le_nhds_metric_le eventually_filtermap dist_norm
     proof (intro allI impI)
       fix e :: real
       assume "e > 0"
       show "eventually (\<lambda>n. norm (f' n x u - g' x u) \<le> e) sequentially"
       proof (cases "u = 0")
         case True
         have "eventually (\<lambda>n. norm (f' n x u - g' x u) \<le> e * norm u) sequentially"
           using nle \<open>0 < e\<close> \<open>x \<in> S\<close> by (fast elim: eventually_mono)
         then show ?thesis
           using \<open>u = 0\<close> \<open>0 < e\<close> by (auto elim: eventually_mono)
       next
         case False
         with \<open>0 < e\<close> have "0 < e / norm u" by simp
         then have "eventually (\<lambda>n. norm (f' n x u - g' x u) \<le> e / norm u * norm u) sequentially"
           using nle \<open>x \<in> S\<close> by (fast elim: eventually_mono)
         then show ?thesis
           using \<open>u \<noteq> 0\<close> by simp
       qed
     qed
     show "bounded_linear (g' x)"
     proof
       fix x' y z :: 'a
       fix c :: real
       note lin = assms(2)[rule_format,OF \<open>x\<in>S\<close>,THEN has_derivative_bounded_linear]
       show "g' x (c *\<^sub>R x') = c *\<^sub>R g' x x'"
         apply (rule tendsto_unique[OF trivial_limit_sequentially tog'])
         unfolding lin[THEN bounded_linear.linear, THEN linear_cmul]
         apply (intro tendsto_intros tog')
         done
       show "g' x (y + z) = g' x y + g' x z"
         apply (rule tendsto_unique[OF trivial_limit_sequentially tog'])
         unfolding lin[THEN bounded_linear.linear, THEN linear_add]
         apply (rule tendsto_add)
         apply (rule tog')+
         done
       obtain N where N: "\<forall>h. norm (f' N x h - g' x h) \<le> 1 * norm h"
         using nle \<open>x \<in> S\<close> unfolding eventually_sequentially by (fast intro: zero_less_one)
       have "bounded_linear (f' N x)"
         using derf \<open>x \<in> S\<close> by fast
       from bounded_linear.bounded [OF this]
       obtain K where K: "\<forall>h. norm (f' N x h) \<le> norm h * K" ..
       {
         fix h
         have "norm (g' x h) = norm (f' N x h - (f' N x h - g' x h))"
           by simp
         also have "\<dots> \<le> norm (f' N x h) + norm (f' N x h - g' x h)"
           by (rule norm_triangle_ineq4)
         also have "\<dots> \<le> norm h * K + 1 * norm h"
           using N K by (fast intro: add_mono)
         finally have "norm (g' x h) \<le> norm h * (K + 1)"
           by (simp add: ring_distribs)
       }
       then show "\<exists>K. \<forall>h. norm (g' x h) \<le> norm h * K" by fast
     qed
     show "eventually (\<lambda>y. norm (g y - g x - g' x (y - x)) \<le> e * norm (y - x)) (at x within S)"
       if "e > 0" for e
     proof -
       have *: "e / 3 > 0"
         using that by auto
       obtain N1 where N1: "\<forall>n\<ge>N1. \<forall>x\<in>S. \<forall>h. norm (f' n x h - g' x h) \<le> e / 3 * norm h"
         using nle * unfolding eventually_sequentially by blast
       obtain N2 where
           N2[rule_format]: "\<forall>n\<ge>N2. \<forall>x\<in>S. \<forall>y\<in>S. norm (f n x - f n y - (g x - g y)) \<le> e / 3 * norm (x - y)"
         using lem2 * by blast
       let ?N = "max N1 N2"
       have "eventually (\<lambda>y. norm (f ?N y - f ?N x - f' ?N x (y - x)) \<le> e / 3 * norm (y - x)) (at x within S)"
         using derf[unfolded has_derivative_within_alt2] and \<open>x \<in> S\<close> and * by fast
       moreover have "eventually (\<lambda>y. y \<in> S) (at x within S)"
         unfolding eventually_at by (fast intro: zero_less_one)
       ultimately show "\<forall>\<^sub>F y in at x within S. norm (g y - g x - g' x (y - x)) \<le> e * norm (y - x)"
       proof (rule eventually_elim2)
         fix y
         assume "y \<in> S"
         assume "norm (f ?N y - f ?N x - f' ?N x (y - x)) \<le> e / 3 * norm (y - x)"
         moreover have "norm (g y - g x - (f ?N y - f ?N x)) \<le> e / 3 * norm (y - x)"
           using N2[OF _ \<open>y \<in> S\<close> \<open>x \<in> S\<close>]
           by (simp add: norm_minus_commute)
         ultimately have "norm (g y - g x - f' ?N x (y - x)) \<le> 2 * e / 3 * norm (y - x)"
           using norm_triangle_le[of "g y - g x - (f ?N y - f ?N x)" "f ?N y - f ?N x - f' ?N x (y - x)" "2 * e / 3 * norm (y - x)"]
           by (auto simp add: algebra_simps)
         moreover
         have " norm (f' ?N x (y - x) - g' x (y - x)) \<le> e / 3 * norm (y - x)"
           using N1 \<open>x \<in> S\<close> by auto
         ultimately show "norm (g y - g x - g' x (y - x)) \<le> e * norm (y - x)"
           using norm_triangle_le[of "g y - g x - f' (max N1 N2) x (y - x)" "f' (max N1 N2) x (y - x) - g' x (y - x)"]
           by (auto simp add: algebra_simps)
       qed
     qed
   qed
   then show ?thesis by fast
 qed
 
 text \<open>Can choose to line up antiderivatives if we want.\<close>
 
 lemma has_antiderivative_sequence:
   fixes f :: "nat \<Rightarrow> 'a::real_normed_vector \<Rightarrow> 'b::banach"
   assumes "convex S"
     and der: "\<And>n x. x \<in> S \<Longrightarrow> ((f n) has_derivative (f' n x)) (at x within S)"
     and no: "\<And>e. e > 0 \<Longrightarrow> \<forall>\<^sub>F n in sequentially.
        \<forall>x\<in>S. \<forall>h. norm (f' n x h - g' x h) \<le> e * norm h"
   shows "\<exists>g. \<forall>x\<in>S. (g has_derivative g' x) (at x within S)"
 proof (cases "S = {}")
   case False
   then obtain a where "a \<in> S"
     by auto
   have *: "\<And>P Q. \<exists>g. \<forall>x\<in>S. P g x \<and> Q g x \<Longrightarrow> \<exists>g. \<forall>x\<in>S. Q g x"
     by auto
   show ?thesis
     apply (rule *)
     apply (rule has_derivative_sequence [OF \<open>convex S\<close> _ no, of "\<lambda>n x. f n x + (f 0 a - f n a)"])
        apply (metis assms(2) has_derivative_add_const)
     using \<open>a \<in> S\<close> 
       apply auto
     done
 qed auto
 
 lemma has_antiderivative_limit:
   fixes g' :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'b::banach"
   assumes "convex S"
     and "\<And>e. e>0 \<Longrightarrow> \<exists>f f'. \<forall>x\<in>S.
            (f has_derivative (f' x)) (at x within S) \<and> (\<forall>h. norm (f' x h - g' x h) \<le> e * norm h)"
   shows "\<exists>g. \<forall>x\<in>S. (g has_derivative g' x) (at x within S)"
 proof -
   have *: "\<forall>n. \<exists>f f'. \<forall>x\<in>S.
     (f has_derivative (f' x)) (at x within S) \<and>
     (\<forall>h. norm(f' x h - g' x h) \<le> inverse (real (Suc n)) * norm h)"
     by (simp add: assms(2))
   obtain f where
     *: "\<And>x. \<exists>f'. \<forall>xa\<in>S. (f x has_derivative f' xa) (at xa within S) \<and>
         (\<forall>h. norm (f' xa h - g' xa h) \<le> inverse (real (Suc x)) * norm h)"
     using * by metis
   obtain f' where
     f': "\<And>x. \<forall>z\<in>S. (f x has_derivative f' x z) (at z within S) \<and>
             (\<forall>h. norm (f' x z h - g' z h) \<le> inverse (real (Suc x)) * norm h)"
     using * by metis
   show ?thesis
   proof (rule has_antiderivative_sequence[OF \<open>convex S\<close>, of f f'])
     fix e :: real
     assume "e > 0"
     obtain N where N: "inverse (real (Suc N)) < e"
       using reals_Archimedean[OF \<open>e>0\<close>] ..
     show "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>S.  \<forall>h. norm (f' n x h - g' x h) \<le> e * norm h"
         unfolding eventually_sequentially
     proof (intro exI allI ballI impI)
       fix n x h
       assume n: "N \<le> n" and x: "x \<in> S"
       have *: "inverse (real (Suc n)) \<le> e"
         apply (rule order_trans[OF _ N[THEN less_imp_le]])
         using n apply (auto simp add: field_simps)
         done
       show "norm (f' n x h - g' x h) \<le> e * norm h"
         by (meson "*" mult_right_mono norm_ge_zero order.trans x f')
     qed
   qed (use f' in auto)
 qed
 
 
 subsection \<open>Differentiation of a series\<close>
 
 proposition has_derivative_series:
   fixes f :: "nat \<Rightarrow> 'a::real_normed_vector \<Rightarrow> 'b::banach"
   assumes "convex S"
     and "\<And>n x. x \<in> S \<Longrightarrow> ((f n) has_derivative (f' n x)) (at x within S)"
     and "\<And>e. e>0 \<Longrightarrow> \<forall>\<^sub>F n in sequentially. \<forall>x\<in>S. \<forall>h. norm (sum (\<lambda>i. f' i x h) {..<n} - g' x h) \<le> e * norm h"
     and "x \<in> S"
     and "(\<lambda>n. f n x) sums l"
   shows "\<exists>g. \<forall>x\<in>S. (\<lambda>n. f n x) sums (g x) \<and> (g has_derivative g' x) (at x within S)"
   unfolding sums_def
   apply (rule has_derivative_sequence[OF assms(1) _ assms(3)])
   apply (metis assms(2) has_derivative_sum)
   using assms(4-5)
   unfolding sums_def
   apply auto
   done
 
 lemma has_field_derivative_series:
   fixes f :: "nat \<Rightarrow> ('a :: {real_normed_field,banach}) \<Rightarrow> 'a"
   assumes "convex S"
   assumes "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x within S)"
   assumes "uniform_limit S (\<lambda>n x. \<Sum>i<n. f' i x) g' sequentially"
   assumes "x0 \<in> S" "summable (\<lambda>n. f n x0)"
   shows   "\<exists>g. \<forall>x\<in>S. (\<lambda>n. f n x) sums g x \<and> (g has_field_derivative g' x) (at x within S)"
 unfolding has_field_derivative_def
 proof (rule has_derivative_series)
   show "\<forall>\<^sub>F n in sequentially.
        \<forall>x\<in>S. \<forall>h. norm ((\<Sum>i<n. f' i x * h) - g' x * h) \<le> e * norm h" if "e > 0" for e
     unfolding eventually_sequentially
   proof -
     from that assms(3) obtain N where N: "\<And>n x. n \<ge> N \<Longrightarrow> x \<in> S \<Longrightarrow> norm ((\<Sum>i<n. f' i x) - g' x) < e"
       unfolding uniform_limit_iff eventually_at_top_linorder dist_norm by blast
     {
       fix n :: nat and x h :: 'a assume nx: "n \<ge> N" "x \<in> S"
       have "norm ((\<Sum>i<n. f' i x * h) - g' x * h) = norm ((\<Sum>i<n. f' i x) - g' x) * norm h"
         by (simp add: norm_mult [symmetric] ring_distribs sum_distrib_right)
       also from N[OF nx] have "norm ((\<Sum>i<n. f' i x) - g' x) \<le> e" by simp
       hence "norm ((\<Sum>i<n. f' i x) - g' x) * norm h \<le> e * norm h"
         by (intro mult_right_mono) simp_all
       finally have "norm ((\<Sum>i<n. f' i x * h) - g' x * h) \<le> e * norm h" .
     }
     thus "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>S. \<forall>h. norm ((\<Sum>i<n. f' i x * h) - g' x * h) \<le> e * norm h" by blast
   qed
 qed (use assms in \<open>auto simp: has_field_derivative_def\<close>)
 
 lemma has_field_derivative_series':
   fixes f :: "nat \<Rightarrow> ('a :: {real_normed_field,banach}) \<Rightarrow> 'a"
   assumes "convex S"
   assumes "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x within S)"
   assumes "uniformly_convergent_on S (\<lambda>n x. \<Sum>i<n. f' i x)"
   assumes "x0 \<in> S" "summable (\<lambda>n. f n x0)" "x \<in> interior S"
   shows   "summable (\<lambda>n. f n x)" "((\<lambda>x. \<Sum>n. f n x) has_field_derivative (\<Sum>n. f' n x)) (at x)"
 proof -
   from \<open>x \<in> interior S\<close> have "x \<in> S" using interior_subset by blast
   define g' where [abs_def]: "g' x = (\<Sum>i. f' i x)" for x
   from assms(3) have "uniform_limit S (\<lambda>n x. \<Sum>i<n. f' i x) g' sequentially"
     by (simp add: uniformly_convergent_uniform_limit_iff suminf_eq_lim g'_def)
   from has_field_derivative_series[OF assms(1,2) this assms(4,5)] obtain g where g:
     "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>n. f n x) sums g x"
     "\<And>x. x \<in> S \<Longrightarrow> (g has_field_derivative g' x) (at x within S)" by blast
   from g(1)[OF \<open>x \<in> S\<close>] show "summable (\<lambda>n. f n x)" by (simp add: sums_iff)
   from g(2)[OF \<open>x \<in> S\<close>] \<open>x \<in> interior S\<close> have "(g has_field_derivative g' x) (at x)"
     by (simp add: at_within_interior[of x S])
   also have "(g has_field_derivative g' x) (at x) \<longleftrightarrow>
                 ((\<lambda>x. \<Sum>n. f n x) has_field_derivative g' x) (at x)"
     using eventually_nhds_in_nhd[OF \<open>x \<in> interior S\<close>] interior_subset[of S] g(1)
     by (intro DERIV_cong_ev) (auto elim!: eventually_mono simp: sums_iff)
   finally show "((\<lambda>x. \<Sum>n. f n x) has_field_derivative g' x) (at x)" .
 qed
 
 lemma differentiable_series:
   fixes f :: "nat \<Rightarrow> ('a :: {real_normed_field,banach}) \<Rightarrow> 'a"
   assumes "convex S" "open S"
   assumes "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
   assumes "uniformly_convergent_on S (\<lambda>n x. \<Sum>i<n. f' i x)"
   assumes "x0 \<in> S" "summable (\<lambda>n. f n x0)" and x: "x \<in> S"
   shows   "summable (\<lambda>n. f n x)" and "(\<lambda>x. \<Sum>n. f n x) differentiable (at x)"
 proof -
   from assms(4) obtain g' where A: "uniform_limit S (\<lambda>n x. \<Sum>i<n. f' i x) g' sequentially"
     unfolding uniformly_convergent_on_def by blast
   from x and \<open>open S\<close> have S: "at x within S = at x" by (rule at_within_open)
   have "\<exists>g. \<forall>x\<in>S. (\<lambda>n. f n x) sums g x \<and> (g has_field_derivative g' x) (at x within S)"
     by (intro has_field_derivative_series[of S f f' g' x0] assms A has_field_derivative_at_within)
   then obtain g where g: "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>n. f n x) sums g x"
     "\<And>x. x \<in> S \<Longrightarrow> (g has_field_derivative g' x) (at x within S)" by blast
   from g[OF x] show "summable (\<lambda>n. f n x)" by (auto simp: summable_def)
   from g(2)[OF x] have g': "(g has_derivative (*) (g' x)) (at x)"
     by (simp add: has_field_derivative_def S)
   have "((\<lambda>x. \<Sum>n. f n x) has_derivative (*) (g' x)) (at x)"
     by (rule has_derivative_transform_within_open[OF g' \<open>open S\<close> x])
        (insert g, auto simp: sums_iff)
   thus "(\<lambda>x. \<Sum>n. f n x) differentiable (at x)" unfolding differentiable_def
     by (auto simp: summable_def differentiable_def has_field_derivative_def)
 qed
 
 lemma differentiable_series':
   fixes f :: "nat \<Rightarrow> ('a :: {real_normed_field,banach}) \<Rightarrow> 'a"
   assumes "convex S" "open S"
   assumes "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
   assumes "uniformly_convergent_on S (\<lambda>n x. \<Sum>i<n. f' i x)"
   assumes "x0 \<in> S" "summable (\<lambda>n. f n x0)"
   shows   "(\<lambda>x. \<Sum>n. f n x) differentiable (at x0)"
   using differentiable_series[OF assms, of x0] \<open>x0 \<in> S\<close> by blast+
 
 subsection \<open>Derivative as a vector\<close>
 
 text \<open>Considering derivative \<^typ>\<open>real \<Rightarrow> 'b::real_normed_vector\<close> as a vector.\<close>
 
 definition "vector_derivative f net = (SOME f'. (f has_vector_derivative f') net)"
 
 lemma vector_derivative_unique_within:
   assumes not_bot: "at x within S \<noteq> bot"
     and f': "(f has_vector_derivative f') (at x within S)"
     and f'': "(f has_vector_derivative f'') (at x within S)"
   shows "f' = f''"
 proof -
   have "(\<lambda>x. x *\<^sub>R f') = (\<lambda>x. x *\<^sub>R f'')"
   proof (rule frechet_derivative_unique_within, simp_all)
     show "\<exists>d. d \<noteq> 0 \<and> \<bar>d\<bar> < e \<and> x + d \<in> S" if "0 < e"  for e
     proof -
       from that
       obtain x' where "x' \<in> S" "x' \<noteq> x" "\<bar>x' - x\<bar> < e"
         using islimpt_approachable_real[of x S] not_bot
         by (auto simp add: trivial_limit_within)
       then show ?thesis
         using eq_iff_diff_eq_0 by fastforce
     qed
   qed (use f' f'' in \<open>auto simp: has_vector_derivative_def\<close>)
   then show ?thesis
     unfolding fun_eq_iff by (metis scaleR_one)
 qed
 
 lemma vector_derivative_unique_at:
   "(f has_vector_derivative f') (at x) \<Longrightarrow> (f has_vector_derivative f'') (at x) \<Longrightarrow> f' = f''"
   by (rule vector_derivative_unique_within) auto
 
 lemma differentiableI_vector: "(f has_vector_derivative y) F \<Longrightarrow> f differentiable F"
   by (auto simp: differentiable_def has_vector_derivative_def)
 
 proposition vector_derivative_works:
   "f differentiable net \<longleftrightarrow> (f has_vector_derivative (vector_derivative f net)) net"
     (is "?l = ?r")
 proof
   assume ?l
   obtain f' where f': "(f has_derivative f') net"
     using \<open>?l\<close> unfolding differentiable_def ..
   then interpret bounded_linear f'
     by auto
   show ?r
     unfolding vector_derivative_def has_vector_derivative_def
     by (rule someI[of _ "f' 1"]) (simp add: scaleR[symmetric] f')
 qed (auto simp: vector_derivative_def has_vector_derivative_def differentiable_def)
 
 lemma vector_derivative_within:
   assumes not_bot: "at x within S \<noteq> bot" and y: "(f has_vector_derivative y) (at x within S)"
   shows "vector_derivative f (at x within S) = y"
   using y
   by (intro vector_derivative_unique_within[OF not_bot vector_derivative_works[THEN iffD1] y])
      (auto simp: differentiable_def has_vector_derivative_def)
 
 lemma frechet_derivative_eq_vector_derivative:
   assumes "f differentiable (at x)"
     shows  "(frechet_derivative f (at x)) = (\<lambda>r. r *\<^sub>R vector_derivative f (at x))"
 using assms
 by (auto simp: differentiable_iff_scaleR vector_derivative_def has_vector_derivative_def
          intro: someI frechet_derivative_at [symmetric])
 
 lemma has_real_derivative:
   fixes f :: "real \<Rightarrow> real"
   assumes "(f has_derivative f') F"
   obtains c where "(f has_real_derivative c) F"
 proof -
   obtain c where "f' = (\<lambda>x. x * c)"
     by (metis assms has_derivative_bounded_linear real_bounded_linear)
   then show ?thesis
     by (metis assms that has_field_derivative_def mult_commute_abs)
 qed
 
 lemma has_real_derivative_iff:
   fixes f :: "real \<Rightarrow> real"
   shows "(\<exists>c. (f has_real_derivative c) F) = (\<exists>D. (f has_derivative D) F)"
   by (metis has_field_derivative_def has_real_derivative)
 
 lemma has_vector_derivative_cong_ev:
   assumes *: "eventually (\<lambda>x. x \<in> S \<longrightarrow> f x = g x) (nhds x)" "f x = g x"
   shows "(f has_vector_derivative f') (at x within S) = (g has_vector_derivative f') (at x within S)"
   unfolding has_vector_derivative_def has_derivative_def
   using *
   apply (cases "at x within S \<noteq> bot")
   apply (intro refl conj_cong filterlim_cong)
   apply (auto simp: Lim_ident_at eventually_at_filter elim: eventually_mono)
   done
 
 lemma islimpt_closure_open:
   fixes s :: "'a::perfect_space set"
   assumes "open s" and t: "t = closure s" "x \<in> t"
   shows "x islimpt t"
 proof cases
   assume "x \<in> s"
   { fix T assume "x \<in> T" "open T"
     then have "open (s \<inter> T)"
       using \<open>open s\<close> by auto
     then have "s \<inter> T \<noteq> {x}"
       using not_open_singleton[of x] by auto
     with \<open>x \<in> T\<close> \<open>x \<in> s\<close> have "\<exists>y\<in>t. y \<in> T \<and> y \<noteq> x"
       using closure_subset[of s] by (auto simp: t) }
   then show ?thesis
     by (auto intro!: islimptI)
 next
   assume "x \<notin> s" with t show ?thesis
     unfolding t closure_def by (auto intro: islimpt_subset)
 qed
 
 lemma vector_derivative_unique_within_closed_interval:
   assumes ab: "a < b" "x \<in> cbox a b"
   assumes D: "(f has_vector_derivative f') (at x within cbox a b)" "(f has_vector_derivative f'') (at x within cbox a b)"
   shows "f' = f''"
   using ab
   by (intro vector_derivative_unique_within[OF _ D])
      (auto simp: trivial_limit_within intro!: islimpt_closure_open[where s="{a <..< b}"])
 
 lemma vector_derivative_at:
   "(f has_vector_derivative f') (at x) \<Longrightarrow> vector_derivative f (at x) = f'"
   by (intro vector_derivative_within at_neq_bot)
 
 lemma has_vector_derivative_id_at [simp]: "vector_derivative (\<lambda>x. x) (at a) = 1"
   by (simp add: vector_derivative_at)
 
 lemma vector_derivative_minus_at [simp]:
   "f differentiable at a
    \<Longrightarrow> vector_derivative (\<lambda>x. - f x) (at a) = - vector_derivative f (at a)"
   by (simp add: vector_derivative_at has_vector_derivative_minus vector_derivative_works [symmetric])
 
 lemma vector_derivative_add_at [simp]:
   "\<lbrakk>f differentiable at a; g differentiable at a\<rbrakk>
    \<Longrightarrow> vector_derivative (\<lambda>x. f x + g x) (at a) = vector_derivative f (at a) + vector_derivative g (at a)"
   by (simp add: vector_derivative_at has_vector_derivative_add vector_derivative_works [symmetric])
 
 lemma vector_derivative_diff_at [simp]:
   "\<lbrakk>f differentiable at a; g differentiable at a\<rbrakk>
    \<Longrightarrow> vector_derivative (\<lambda>x. f x - g x) (at a) = vector_derivative f (at a) - vector_derivative g (at a)"
   by (simp add: vector_derivative_at has_vector_derivative_diff vector_derivative_works [symmetric])
 
 lemma vector_derivative_mult_at [simp]:
   fixes f g :: "real \<Rightarrow> 'a :: real_normed_algebra"
   shows  "\<lbrakk>f differentiable at a; g differentiable at a\<rbrakk>
    \<Longrightarrow> vector_derivative (\<lambda>x. f x * g x) (at a) = f a * vector_derivative g (at a) + vector_derivative f (at a) * g a"
   by (simp add: vector_derivative_at has_vector_derivative_mult vector_derivative_works [symmetric])
 
 lemma vector_derivative_scaleR_at [simp]:
     "\<lbrakk>f differentiable at a; g differentiable at a\<rbrakk>
    \<Longrightarrow> vector_derivative (\<lambda>x. f x *\<^sub>R g x) (at a) = f a *\<^sub>R vector_derivative g (at a) + vector_derivative f (at a) *\<^sub>R g a"
 apply (rule vector_derivative_at)
 apply (rule has_vector_derivative_scaleR)
 apply (auto simp: vector_derivative_works has_vector_derivative_def has_field_derivative_def mult_commute_abs)
 done
 
 lemma vector_derivative_within_cbox:
   assumes ab: "a < b" "x \<in> cbox a b"
   assumes f: "(f has_vector_derivative f') (at x within cbox a b)"
   shows "vector_derivative f (at x within cbox a b) = f'"
   by (intro vector_derivative_unique_within_closed_interval[OF ab _ f]
             vector_derivative_works[THEN iffD1] differentiableI_vector)
      fact
 
 lemma vector_derivative_within_closed_interval:
   fixes f::"real \<Rightarrow> 'a::euclidean_space"
   assumes "a < b" and "x \<in> {a..b}"
   assumes "(f has_vector_derivative f') (at x within {a..b})"
   shows "vector_derivative f (at x within {a..b}) = f'"
   using assms vector_derivative_within_cbox
   by fastforce
 
 lemma has_vector_derivative_within_subset:
   "(f has_vector_derivative f') (at x within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f has_vector_derivative f') (at x within T)"
-  by (auto simp: has_vector_derivative_def intro: has_derivative_within_subset)
+  by (auto simp: has_vector_derivative_def intro: has_derivative_subset)
 
 lemma has_vector_derivative_at_within:
   "(f has_vector_derivative f') (at x) \<Longrightarrow> (f has_vector_derivative f') (at x within S)"
   unfolding has_vector_derivative_def
   by (rule has_derivative_at_withinI)
 
 lemma has_vector_derivative_weaken:
   fixes x D and f g S T
   assumes f: "(f has_vector_derivative D) (at x within T)"
     and "x \<in> S" "S \<subseteq> T"
     and "\<And>x. x \<in> S \<Longrightarrow> f x = g x"
   shows "(g has_vector_derivative D) (at x within S)"
 proof -
   have "(f has_vector_derivative D) (at x within S) \<longleftrightarrow> (g has_vector_derivative D) (at x within S)"
     unfolding has_vector_derivative_def has_derivative_iff_norm
     using assms by (intro conj_cong Lim_cong_within refl) auto
   then show ?thesis
     using has_vector_derivative_within_subset[OF f \<open>S \<subseteq> T\<close>] by simp
 qed
 
 lemma has_vector_derivative_transform_within:
   assumes "(f has_vector_derivative f') (at x within S)"
     and "0 < d"
     and "x \<in> S"
     and "\<And>x'. \<lbrakk>x'\<in>S; dist x' x < d\<rbrakk> \<Longrightarrow> f x' = g x'"
     shows "(g has_vector_derivative f') (at x within S)"
   using assms
   unfolding has_vector_derivative_def
   by (rule has_derivative_transform_within)
 
 lemma has_vector_derivative_transform_within_open:
   assumes "(f has_vector_derivative f') (at x)"
     and "open S"
     and "x \<in> S"
     and "\<And>y. y\<in>S \<Longrightarrow> f y = g y"
   shows "(g has_vector_derivative f') (at x)"
   using assms
   unfolding has_vector_derivative_def
   by (rule has_derivative_transform_within_open)
 
 lemma has_vector_derivative_transform:
   assumes "x \<in> S" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
   assumes f': "(f has_vector_derivative f') (at x within S)"
   shows "(g has_vector_derivative f') (at x within S)"
   using assms
   unfolding has_vector_derivative_def
   by (rule has_derivative_transform)
 
 lemma vector_diff_chain_at:
   assumes "(f has_vector_derivative f') (at x)"
     and "(g has_vector_derivative g') (at (f x))"
   shows "((g \<circ> f) has_vector_derivative (f' *\<^sub>R g')) (at x)"
   using assms has_vector_derivative_at_within has_vector_derivative_def vector_derivative_diff_chain_within by blast
 
 lemma vector_diff_chain_within:
   assumes "(f has_vector_derivative f') (at x within s)"
     and "(g has_vector_derivative g') (at (f x) within f ` s)"
   shows "((g \<circ> f) has_vector_derivative (f' *\<^sub>R g')) (at x within s)"
   using assms has_vector_derivative_def vector_derivative_diff_chain_within by blast
 
 lemma vector_derivative_const_at [simp]: "vector_derivative (\<lambda>x. c) (at a) = 0"
   by (simp add: vector_derivative_at)
 
 lemma vector_derivative_at_within_ivl:
   "(f has_vector_derivative f') (at x) \<Longrightarrow>
     a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> a<b \<Longrightarrow> vector_derivative f (at x within {a..b}) = f'"
   using has_vector_derivative_at_within vector_derivative_within_cbox by fastforce
 
 lemma vector_derivative_chain_at:
   assumes "f differentiable at x" "(g differentiable at (f x))"
   shows "vector_derivative (g \<circ> f) (at x) =
          vector_derivative f (at x) *\<^sub>R vector_derivative g (at (f x))"
 by (metis vector_diff_chain_at vector_derivative_at vector_derivative_works assms)
 
 lemma field_vector_diff_chain_at:  (*thanks to Wenda Li*)
  assumes Df: "(f has_vector_derivative f') (at x)"
      and Dg: "(g has_field_derivative g') (at (f x))"
  shows "((g \<circ> f) has_vector_derivative (f' * g')) (at x)"
 using diff_chain_at[OF Df[unfolded has_vector_derivative_def]
                        Dg [unfolded has_field_derivative_def]]
  by (auto simp: o_def mult.commute has_vector_derivative_def)
 
 lemma vector_derivative_chain_within: 
   assumes "at x within S \<noteq> bot" "f differentiable (at x within S)" 
     "(g has_derivative g') (at (f x) within f ` S)" 
   shows "vector_derivative (g \<circ> f) (at x within S) =
         g' (vector_derivative f (at x within S)) "
   apply (rule vector_derivative_within [OF \<open>at x within S \<noteq> bot\<close>])
   apply (rule vector_derivative_diff_chain_within)
   using assms(2-3) vector_derivative_works
   by auto
 
 subsection \<open>Field differentiability\<close>
 
 definition\<^marker>\<open>tag important\<close> field_differentiable :: "['a \<Rightarrow> 'a::real_normed_field, 'a filter] \<Rightarrow> bool"
            (infixr "(field'_differentiable)" 50)
   where "f field_differentiable F \<equiv> \<exists>f'. (f has_field_derivative f') F"
 
 lemma field_differentiable_imp_differentiable:
   "f field_differentiable F \<Longrightarrow> f differentiable F"
   unfolding field_differentiable_def differentiable_def 
   using has_field_derivative_imp_has_derivative by auto
 
 lemma field_differentiable_imp_continuous_at:
     "f field_differentiable (at x within S) \<Longrightarrow> continuous (at x within S) f"
   by (metis DERIV_continuous field_differentiable_def)
 
 lemma field_differentiable_within_subset:
     "\<lbrakk>f field_differentiable (at x within S); T \<subseteq> S\<rbrakk> \<Longrightarrow> f field_differentiable (at x within T)"
   by (metis DERIV_subset field_differentiable_def)
 
 lemma field_differentiable_at_within:
     "\<lbrakk>f field_differentiable (at x)\<rbrakk>
      \<Longrightarrow> f field_differentiable (at x within S)"
   unfolding field_differentiable_def
   by (metis DERIV_subset top_greatest)
 
 lemma field_differentiable_linear [simp,derivative_intros]: "((*) c) field_differentiable F"
   unfolding field_differentiable_def has_field_derivative_def mult_commute_abs
   by (force intro: has_derivative_mult_right)
 
 lemma field_differentiable_const [simp,derivative_intros]: "(\<lambda>z. c) field_differentiable F"
   unfolding field_differentiable_def has_field_derivative_def
   using DERIV_const has_field_derivative_imp_has_derivative by blast
 
 lemma field_differentiable_ident [simp,derivative_intros]: "(\<lambda>z. z) field_differentiable F"
   unfolding field_differentiable_def has_field_derivative_def
   using DERIV_ident has_field_derivative_def by blast
 
 lemma field_differentiable_id [simp,derivative_intros]: "id field_differentiable F"
   unfolding id_def by (rule field_differentiable_ident)
 
 lemma field_differentiable_minus [derivative_intros]:
   "f field_differentiable F \<Longrightarrow> (\<lambda>z. - (f z)) field_differentiable F"
   unfolding field_differentiable_def
   by (metis field_differentiable_minus)
 
 lemma field_differentiable_add [derivative_intros]:
   assumes "f field_differentiable F" "g field_differentiable F"
     shows "(\<lambda>z. f z + g z) field_differentiable F"
   using assms unfolding field_differentiable_def
   by (metis field_differentiable_add)
 
 lemma field_differentiable_add_const [simp,derivative_intros]:
      "(+) c field_differentiable F"
   by (simp add: field_differentiable_add)
 
 lemma field_differentiable_sum [derivative_intros]:
   "(\<And>i. i \<in> I \<Longrightarrow> (f i) field_differentiable F) \<Longrightarrow> (\<lambda>z. \<Sum>i\<in>I. f i z) field_differentiable F"
   by (induct I rule: infinite_finite_induct)
      (auto intro: field_differentiable_add field_differentiable_const)
 
 lemma field_differentiable_diff [derivative_intros]:
   assumes "f field_differentiable F" "g field_differentiable F"
     shows "(\<lambda>z. f z - g z) field_differentiable F"
   using assms unfolding field_differentiable_def
   by (metis field_differentiable_diff)
 
 lemma field_differentiable_inverse [derivative_intros]:
   assumes "f field_differentiable (at a within S)" "f a \<noteq> 0"
   shows "(\<lambda>z. inverse (f z)) field_differentiable (at a within S)"
   using assms unfolding field_differentiable_def
   by (metis DERIV_inverse_fun)
 
 lemma field_differentiable_mult [derivative_intros]:
   assumes "f field_differentiable (at a within S)"
           "g field_differentiable (at a within S)"
     shows "(\<lambda>z. f z * g z) field_differentiable (at a within S)"
   using assms unfolding field_differentiable_def
   by (metis DERIV_mult [of f _ a S g])
 
 lemma field_differentiable_divide [derivative_intros]:
   assumes "f field_differentiable (at a within S)"
           "g field_differentiable (at a within S)"
           "g a \<noteq> 0"
     shows "(\<lambda>z. f z / g z) field_differentiable (at a within S)"
   using assms unfolding field_differentiable_def
   by (metis DERIV_divide [of f _ a S g])
 
 lemma field_differentiable_power [derivative_intros]:
   assumes "f field_differentiable (at a within S)"
     shows "(\<lambda>z. f z ^ n) field_differentiable (at a within S)"
   using assms unfolding field_differentiable_def
   by (metis DERIV_power)
 
 lemma field_differentiable_transform_within:
   "0 < d \<Longrightarrow>
         x \<in> S \<Longrightarrow>
         (\<And>x'. x' \<in> S \<Longrightarrow> dist x' x < d \<Longrightarrow> f x' = g x') \<Longrightarrow>
         f field_differentiable (at x within S)
         \<Longrightarrow> g field_differentiable (at x within S)"
   unfolding field_differentiable_def has_field_derivative_def
   by (blast intro: has_derivative_transform_within)
 
 lemma field_differentiable_compose_within:
   assumes "f field_differentiable (at a within S)"
           "g field_differentiable (at (f a) within f`S)"
     shows "(g o f) field_differentiable (at a within S)"
   using assms unfolding field_differentiable_def
   by (metis DERIV_image_chain)
 
 lemma field_differentiable_compose:
   "f field_differentiable at z \<Longrightarrow> g field_differentiable at (f z)
           \<Longrightarrow> (g o f) field_differentiable at z"
 by (metis field_differentiable_at_within field_differentiable_compose_within)
 
 lemma field_differentiable_within_open:
      "\<lbrakk>a \<in> S; open S\<rbrakk> \<Longrightarrow> f field_differentiable at a within S \<longleftrightarrow>
                           f field_differentiable at a"
   unfolding field_differentiable_def
   by (metis at_within_open)
 
 lemma exp_scaleR_has_vector_derivative_right:
   "((\<lambda>t. exp (t *\<^sub>R A)) has_vector_derivative exp (t *\<^sub>R A) * A) (at t within T)"
   unfolding has_vector_derivative_def
 proof (rule has_derivativeI)
   let ?F = "at t within (T \<inter> {t - 1 <..< t + 1})"
   have *: "at t within T = ?F"
     by (rule at_within_nhd[where S="{t - 1 <..< t + 1}"]) auto
   let ?e = "\<lambda>i x. (inverse (1 + real i) * inverse (fact i) * (x - t) ^ i) *\<^sub>R (A * A ^ i)"
   have "\<forall>\<^sub>F n in sequentially.
       \<forall>x\<in>T \<inter> {t - 1<..<t + 1}. norm (?e n x) \<le> norm (A ^ (n + 1) /\<^sub>R fact (n + 1))"
     apply (auto simp: algebra_split_simps intro!: eventuallyI)
     apply (rule mult_left_mono)
      apply (auto simp add: field_simps power_abs intro!: divide_right_mono power_le_one)
     done
   then have "uniform_limit (T \<inter> {t - 1<..<t + 1}) (\<lambda>n x. \<Sum>i<n. ?e i x) (\<lambda>x. \<Sum>i. ?e i x) sequentially"
     by (rule Weierstrass_m_test_ev) (intro summable_ignore_initial_segment summable_norm_exp)
   moreover
   have "\<forall>\<^sub>F x in sequentially. x > 0"
     by (metis eventually_gt_at_top)
   then have
     "\<forall>\<^sub>F n in sequentially. ((\<lambda>x. \<Sum>i<n. ?e i x) \<longlongrightarrow> A) ?F"
     by eventually_elim
       (auto intro!: tendsto_eq_intros
         simp: power_0_left if_distrib if_distribR
         cong: if_cong)
   ultimately
   have [tendsto_intros]: "((\<lambda>x. \<Sum>i. ?e i x) \<longlongrightarrow> A) ?F"
     by (auto intro!: swap_uniform_limit[where f="\<lambda>n x. \<Sum>i < n. ?e i x" and F = sequentially])
   have [tendsto_intros]: "((\<lambda>x. if x = t then 0 else 1) \<longlongrightarrow> 1) ?F"
     by (rule tendsto_eventually) (simp add: eventually_at_filter)
   have "((\<lambda>y. ((y - t) / abs (y - t)) *\<^sub>R ((\<Sum>n. ?e n y) - A)) \<longlongrightarrow> 0) (at t within T)"
     unfolding *
     by (rule tendsto_norm_zero_cancel) (auto intro!: tendsto_eq_intros)
 
   moreover have "\<forall>\<^sub>F x in at t within T. x \<noteq> t"
     by (simp add: eventually_at_filter)
   then have "\<forall>\<^sub>F x in at t within T. ((x - t) / \<bar>x - t\<bar>) *\<^sub>R ((\<Sum>n. ?e n x) - A) =
     (exp ((x - t) *\<^sub>R A) - 1 - (x - t) *\<^sub>R A) /\<^sub>R norm (x - t)"
   proof eventually_elim
     case (elim x)
     have "(exp ((x - t) *\<^sub>R A) - 1 - (x - t) *\<^sub>R A) /\<^sub>R norm (x - t) =
       ((\<Sum>n. (x - t) *\<^sub>R ?e n x) - (x - t) *\<^sub>R A) /\<^sub>R norm (x - t)"
       unfolding exp_first_term
       by (simp add: ac_simps)
     also
     have "summable (\<lambda>n. ?e n x)"
     proof -
       from elim have "?e n x = (((x - t) *\<^sub>R A) ^ (n + 1)) /\<^sub>R fact (n + 1) /\<^sub>R (x - t)" for n
         by simp
       then show ?thesis
         by (auto simp only:
           intro!: summable_scaleR_right summable_ignore_initial_segment summable_exp_generic)
     qed
     then have "(\<Sum>n. (x - t) *\<^sub>R ?e n x) = (x - t) *\<^sub>R (\<Sum>n. ?e n x)"
       by (rule suminf_scaleR_right[symmetric])
     also have "(\<dots> - (x - t) *\<^sub>R A) /\<^sub>R norm (x - t) = (x - t) *\<^sub>R ((\<Sum>n. ?e n x) - A) /\<^sub>R norm (x - t)"
       by (simp add: algebra_simps)
     finally show ?case
       by simp (simp add: field_simps)
   qed
 
   ultimately have "((\<lambda>y. (exp ((y - t) *\<^sub>R A) - 1 - (y - t) *\<^sub>R A) /\<^sub>R norm (y - t)) \<longlongrightarrow> 0) (at t within T)"
     by (rule Lim_transform_eventually)
   from tendsto_mult_right_zero[OF this, where c="exp (t *\<^sub>R A)"]
   show "((\<lambda>y. (exp (y *\<^sub>R A) - exp (t *\<^sub>R A) - (y - t) *\<^sub>R (exp (t *\<^sub>R A) * A)) /\<^sub>R norm (y - t)) \<longlongrightarrow> 0)
       (at t within T)"
     by (rule Lim_transform_eventually)
       (auto simp: field_split_simps exp_add_commuting[symmetric])
 qed (rule bounded_linear_scaleR_left)
 
 lemma exp_times_scaleR_commute: "exp (t *\<^sub>R A) * A = A * exp (t *\<^sub>R A)"
   using exp_times_arg_commute[symmetric, of "t *\<^sub>R A"]
   by (auto simp: algebra_simps)
 
 lemma exp_scaleR_has_vector_derivative_left: "((\<lambda>t. exp (t *\<^sub>R A)) has_vector_derivative A * exp (t *\<^sub>R A)) (at t)"
   using exp_scaleR_has_vector_derivative_right[of A t]
   by (simp add: exp_times_scaleR_commute)
 
 lemma field_differentiable_series:
   fixes f :: "nat \<Rightarrow> 'a::{real_normed_field,banach} \<Rightarrow> 'a"
   assumes "convex S" "open S"
   assumes "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
   assumes "uniformly_convergent_on S (\<lambda>n x. \<Sum>i<n. f' i x)"
   assumes "x0 \<in> S" "summable (\<lambda>n. f n x0)" and x: "x \<in> S"
   shows  "(\<lambda>x. \<Sum>n. f n x) field_differentiable (at x)"
 proof -
   from assms(4) obtain g' where A: "uniform_limit S (\<lambda>n x. \<Sum>i<n. f' i x) g' sequentially"
     unfolding uniformly_convergent_on_def by blast
   from x and \<open>open S\<close> have S: "at x within S = at x" by (rule at_within_open)
   have "\<exists>g. \<forall>x\<in>S. (\<lambda>n. f n x) sums g x \<and> (g has_field_derivative g' x) (at x within S)"
     by (intro has_field_derivative_series[of S f f' g' x0] assms A has_field_derivative_at_within)
   then obtain g where g: "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>n. f n x) sums g x"
     "\<And>x. x \<in> S \<Longrightarrow> (g has_field_derivative g' x) (at x within S)" by blast
   from g(2)[OF x] have g': "(g has_derivative (*) (g' x)) (at x)"
     by (simp add: has_field_derivative_def S)
   have "((\<lambda>x. \<Sum>n. f n x) has_derivative (*) (g' x)) (at x)"
     by (rule has_derivative_transform_within_open[OF g' \<open>open S\<close> x])
        (insert g, auto simp: sums_iff)
   thus "(\<lambda>x. \<Sum>n. f n x) field_differentiable (at x)" unfolding differentiable_def
     by (auto simp: summable_def field_differentiable_def has_field_derivative_def)
 qed
 
 subsubsection\<^marker>\<open>tag unimportant\<close>\<open>Caratheodory characterization\<close>
 
 lemma field_differentiable_caratheodory_at:
   "f field_differentiable (at z) \<longleftrightarrow>
          (\<exists>g. (\<forall>w. f(w) - f(z) = g(w) * (w - z)) \<and> continuous (at z) g)"
   using CARAT_DERIV [of f]
   by (simp add: field_differentiable_def has_field_derivative_def)
 
 lemma field_differentiable_caratheodory_within:
   "f field_differentiable (at z within s) \<longleftrightarrow>
          (\<exists>g. (\<forall>w. f(w) - f(z) = g(w) * (w - z)) \<and> continuous (at z within s) g)"
   using DERIV_caratheodory_within [of f]
   by (simp add: field_differentiable_def has_field_derivative_def)
 
 
 subsection \<open>Field derivative\<close>
 
 definition\<^marker>\<open>tag important\<close> deriv :: "('a \<Rightarrow> 'a::real_normed_field) \<Rightarrow> 'a \<Rightarrow> 'a" where
   "deriv f x \<equiv> SOME D. DERIV f x :> D"
 
 lemma DERIV_imp_deriv: "DERIV f x :> f' \<Longrightarrow> deriv f x = f'"
   unfolding deriv_def by (metis some_equality DERIV_unique)
 
 lemma DERIV_deriv_iff_has_field_derivative:
   "DERIV f x :> deriv f x \<longleftrightarrow> (\<exists>f'. (f has_field_derivative f') (at x))"
   by (auto simp: has_field_derivative_def DERIV_imp_deriv)
 
 lemma DERIV_deriv_iff_real_differentiable:
   fixes x :: real
   shows "DERIV f x :> deriv f x \<longleftrightarrow> f differentiable at x"
   unfolding differentiable_def by (metis DERIV_imp_deriv has_real_derivative_iff)
 
 lemma deriv_cong_ev:
   assumes "eventually (\<lambda>x. f x = g x) (nhds x)" "x = y"
   shows   "deriv f x = deriv g y"
 proof -
   have "(\<lambda>D. (f has_field_derivative D) (at x)) = (\<lambda>D. (g has_field_derivative D) (at y))"
     by (intro ext DERIV_cong_ev refl assms)
   thus ?thesis by (simp add: deriv_def assms)
 qed
 
 lemma higher_deriv_cong_ev:
   assumes "eventually (\<lambda>x. f x = g x) (nhds x)" "x = y"
   shows   "(deriv ^^ n) f x = (deriv ^^ n) g y"
 proof -
   from assms(1) have "eventually (\<lambda>x. (deriv ^^ n) f x = (deriv ^^ n) g x) (nhds x)"
   proof (induction n arbitrary: f g)
     case (Suc n)
     from Suc.prems have "eventually (\<lambda>y. eventually (\<lambda>z. f z = g z) (nhds y)) (nhds x)"
       by (simp add: eventually_eventually)
     hence "eventually (\<lambda>x. deriv f x = deriv g x) (nhds x)"
       by eventually_elim (rule deriv_cong_ev, simp_all)
     thus ?case by (auto intro!: deriv_cong_ev Suc simp: funpow_Suc_right simp del: funpow.simps)
   qed auto
   from eventually_nhds_x_imp_x[OF this] assms(2) show ?thesis by simp
 qed
 
 lemma real_derivative_chain:
   fixes x :: real
   shows "f differentiable at x \<Longrightarrow> g differentiable at (f x)
     \<Longrightarrow> deriv (g o f) x = deriv g (f x) * deriv f x"
   by (metis DERIV_deriv_iff_real_differentiable DERIV_chain DERIV_imp_deriv)
 lemma field_derivative_eq_vector_derivative:
    "(deriv f x) = vector_derivative f (at x)"
 by (simp add: mult.commute deriv_def vector_derivative_def has_vector_derivative_def has_field_derivative_def)
 
 proposition field_differentiable_derivI:
     "f field_differentiable (at x) \<Longrightarrow> (f has_field_derivative deriv f x) (at x)"
 by (simp add: field_differentiable_def DERIV_deriv_iff_has_field_derivative)
 
 lemma vector_derivative_chain_at_general:
   assumes "f differentiable at x" "g field_differentiable at (f x)"
   shows "vector_derivative (g \<circ> f) (at x) = vector_derivative f (at x) * deriv g (f x)"
   apply (rule vector_derivative_at [OF field_vector_diff_chain_at])
   using assms vector_derivative_works by (auto simp: field_differentiable_derivI)
 
 lemma DERIV_deriv_iff_field_differentiable:
   "DERIV f x :> deriv f x \<longleftrightarrow> f field_differentiable at x"
   unfolding field_differentiable_def by (metis DERIV_imp_deriv)
 
 lemma deriv_chain:
   "f field_differentiable at x \<Longrightarrow> g field_differentiable at (f x)
     \<Longrightarrow> deriv (g o f) x = deriv g (f x) * deriv f x"
   by (metis DERIV_deriv_iff_field_differentiable DERIV_chain DERIV_imp_deriv)
 
 lemma deriv_linear [simp]: "deriv (\<lambda>w. c * w) = (\<lambda>z. c)"
   by (metis DERIV_imp_deriv DERIV_cmult_Id)
 
 lemma deriv_uminus [simp]: "deriv (\<lambda>w. -w) = (\<lambda>z. -1)"
   using deriv_linear[of "-1"] by (simp del: deriv_linear)
 
 lemma deriv_ident [simp]: "deriv (\<lambda>w. w) = (\<lambda>z. 1)"
   by (metis DERIV_imp_deriv DERIV_ident)
 
 lemma deriv_id [simp]: "deriv id = (\<lambda>z. 1)"
   by (simp add: id_def)
 
 lemma deriv_const [simp]: "deriv (\<lambda>w. c) = (\<lambda>z. 0)"
   by (metis DERIV_imp_deriv DERIV_const)
 
 lemma deriv_add [simp]:
   "\<lbrakk>f field_differentiable at z; g field_differentiable at z\<rbrakk>
    \<Longrightarrow> deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g z"
   unfolding DERIV_deriv_iff_field_differentiable[symmetric]
   by (auto intro!: DERIV_imp_deriv derivative_intros)
 
 lemma deriv_diff [simp]:
   "\<lbrakk>f field_differentiable at z; g field_differentiable at z\<rbrakk>
    \<Longrightarrow> deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z"
   unfolding DERIV_deriv_iff_field_differentiable[symmetric]
   by (auto intro!: DERIV_imp_deriv derivative_intros)
 
 lemma deriv_mult [simp]:
   "\<lbrakk>f field_differentiable at z; g field_differentiable at z\<rbrakk>
    \<Longrightarrow> deriv (\<lambda>w. f w * g w) z = f z * deriv g z + deriv f z * g z"
   unfolding DERIV_deriv_iff_field_differentiable[symmetric]
   by (auto intro!: DERIV_imp_deriv derivative_eq_intros)
 
 lemma deriv_cmult:
   "f field_differentiable at z \<Longrightarrow> deriv (\<lambda>w. c * f w) z = c * deriv f z"
   by simp
 
 lemma deriv_cmult_right:
   "f field_differentiable at z \<Longrightarrow> deriv (\<lambda>w. f w * c) z = deriv f z * c"
   by simp
 
 lemma deriv_inverse [simp]:
   "\<lbrakk>f field_differentiable at z; f z \<noteq> 0\<rbrakk>
    \<Longrightarrow> deriv (\<lambda>w. inverse (f w)) z = - deriv f z / f z ^ 2"
   unfolding DERIV_deriv_iff_field_differentiable[symmetric]
   by (safe intro!: DERIV_imp_deriv derivative_eq_intros) (auto simp: field_split_simps power2_eq_square)
 
 lemma deriv_divide [simp]:
   "\<lbrakk>f field_differentiable at z; g field_differentiable at z; g z \<noteq> 0\<rbrakk>
    \<Longrightarrow> deriv (\<lambda>w. f w / g w) z = (deriv f z * g z - f z * deriv g z) / g z ^ 2"
   by (simp add: field_class.field_divide_inverse field_differentiable_inverse)
      (simp add: field_split_simps power2_eq_square)
 
 lemma deriv_cdivide_right:
   "f field_differentiable at z \<Longrightarrow> deriv (\<lambda>w. f w / c) z = deriv f z / c"
   by (simp add: field_class.field_divide_inverse)
 
 lemma deriv_compose_linear:
   "f field_differentiable at (c * z) \<Longrightarrow> deriv (\<lambda>w. f (c * w)) z = c * deriv f (c * z)"
 apply (rule DERIV_imp_deriv)
   unfolding DERIV_deriv_iff_field_differentiable [symmetric]
   by (metis (full_types) DERIV_chain2 DERIV_cmult_Id mult.commute)
 
 
 lemma nonzero_deriv_nonconstant:
   assumes df: "DERIV f \<xi> :> df" and S: "open S" "\<xi> \<in> S" and "df \<noteq> 0"
     shows "\<not> f constant_on S"
 unfolding constant_on_def
 by (metis \<open>df \<noteq> 0\<close> has_field_derivative_transform_within_open [OF df S] DERIV_const DERIV_unique)
 
 
 subsection \<open>Relation between convexity and derivative\<close>
 
 (* TODO: Generalise to real vector spaces? *)
 proposition convex_on_imp_above_tangent:
   assumes convex: "convex_on A f" and connected: "connected A"
   assumes c: "c \<in> interior A" and x : "x \<in> A"
   assumes deriv: "(f has_field_derivative f') (at c within A)"
   shows   "f x - f c \<ge> f' * (x - c)"
 proof (cases x c rule: linorder_cases)
   assume xc: "x > c"
   let ?A' = "interior A \<inter> {c<..}"
   from c have "c \<in> interior A \<inter> closure {c<..}" by auto
   also have "\<dots> \<subseteq> closure (interior A \<inter> {c<..})" by (intro open_Int_closure_subset) auto
   finally have "at c within ?A' \<noteq> bot" by (subst at_within_eq_bot_iff) auto
   moreover from deriv have "((\<lambda>y. (f y - f c) / (y - c)) \<longlongrightarrow> f') (at c within ?A')"
     unfolding has_field_derivative_iff using interior_subset[of A] by (blast intro: tendsto_mono at_le)
   moreover from eventually_at_right_real[OF xc]
     have "eventually (\<lambda>y. (f y - f c) / (y - c) \<le> (f x - f c) / (x - c)) (at_right c)"
   proof eventually_elim
     fix y assume y: "y \<in> {c<..<x}"
     with convex connected x c have "f y \<le> (f x - f c) / (x - c) * (y - c) + f c"
       using interior_subset[of A]
       by (intro convex_onD_Icc' convex_on_subset[OF convex] connected_contains_Icc) auto
     hence "f y - f c \<le> (f x - f c) / (x - c) * (y - c)" by simp
     thus "(f y - f c) / (y - c) \<le> (f x - f c) / (x - c)" using y xc by (simp add: field_split_simps)
   qed
   hence "eventually (\<lambda>y. (f y - f c) / (y - c) \<le> (f x - f c) / (x - c)) (at c within ?A')"
     by (blast intro: filter_leD at_le)
   ultimately have "f' \<le> (f x - f c) / (x - c)" by (simp add: tendsto_upperbound)
   thus ?thesis using xc by (simp add: field_simps)
 next
   assume xc: "x < c"
   let ?A' = "interior A \<inter> {..<c}"
   from c have "c \<in> interior A \<inter> closure {..<c}" by auto
   also have "\<dots> \<subseteq> closure (interior A \<inter> {..<c})" by (intro open_Int_closure_subset) auto
   finally have "at c within ?A' \<noteq> bot" by (subst at_within_eq_bot_iff) auto
   moreover from deriv have "((\<lambda>y. (f y - f c) / (y - c)) \<longlongrightarrow> f') (at c within ?A')"
     unfolding has_field_derivative_iff using interior_subset[of A] by (blast intro: tendsto_mono at_le)
   moreover from eventually_at_left_real[OF xc]
     have "eventually (\<lambda>y. (f y - f c) / (y - c) \<ge> (f x - f c) / (x - c)) (at_left c)"
   proof eventually_elim
     fix y assume y: "y \<in> {x<..<c}"
     with convex connected x c have "f y \<le> (f x - f c) / (c - x) * (c - y) + f c"
       using interior_subset[of A]
       by (intro convex_onD_Icc'' convex_on_subset[OF convex] connected_contains_Icc) auto
     hence "f y - f c \<le> (f x - f c) * ((c - y) / (c - x))" by simp
     also have "(c - y) / (c - x) = (y - c) / (x - c)" using y xc by (simp add: field_simps)
     finally show "(f y - f c) / (y - c) \<ge> (f x - f c) / (x - c)" using y xc
       by (simp add: field_split_simps)
   qed
   hence "eventually (\<lambda>y. (f y - f c) / (y - c) \<ge> (f x - f c) / (x - c)) (at c within ?A')"
     by (blast intro: filter_leD at_le)
   ultimately have "f' \<ge> (f x - f c) / (x - c)" by (simp add: tendsto_lowerbound)
   thus ?thesis using xc by (simp add: field_simps)
 qed simp_all
 
 
 subsection \<open>Partial derivatives\<close>
 
 lemma eventually_at_Pair_within_TimesI1:
   fixes x::"'a::metric_space"
   assumes "\<forall>\<^sub>F x' in at x within X. P x'"
   assumes "P x"
   shows "\<forall>\<^sub>F (x', y') in at (x, y) within X \<times> Y. P x'"
 proof -
   from assms[unfolded eventually_at_topological]
   obtain S where S: "open S" "x \<in> S" "\<And>x'. x' \<in> X \<Longrightarrow> x' \<in> S \<Longrightarrow> P x'"
     by metis
   show "\<forall>\<^sub>F (x', y') in at (x, y) within X \<times> Y. P x'"
     unfolding eventually_at_topological
     by (auto intro!: exI[where x="S \<times> UNIV"] S open_Times)
 qed
 
 lemma eventually_at_Pair_within_TimesI2:
   fixes x::"'a::metric_space"
   assumes "\<forall>\<^sub>F y' in at y within Y. P y'" "P y"
   shows "\<forall>\<^sub>F (x', y') in at (x, y) within X \<times> Y. P y'"
 proof -
   from assms[unfolded eventually_at_topological]
   obtain S where S: "open S" "y \<in> S" "\<And>y'. y' \<in> Y \<Longrightarrow> y' \<in> S \<Longrightarrow> P y'"
     by metis
   show "\<forall>\<^sub>F (x', y') in at (x, y) within X \<times> Y. P y'"
     unfolding eventually_at_topological
     by (auto intro!: exI[where x="UNIV \<times> S"] S open_Times)
 qed
 
 proposition has_derivative_partialsI:
   fixes f::"'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector \<Rightarrow> 'c::real_normed_vector"
   assumes fx: "((\<lambda>x. f x y) has_derivative fx) (at x within X)"
   assumes fy: "\<And>x y. x \<in> X \<Longrightarrow> y \<in> Y \<Longrightarrow> ((\<lambda>y. f x y) has_derivative blinfun_apply (fy x y)) (at y within Y)"
   assumes fy_cont[unfolded continuous_within]: "continuous (at (x, y) within X \<times> Y) (\<lambda>(x, y). fy x y)"
   assumes "y \<in> Y" "convex Y"
   shows "((\<lambda>(x, y). f x y) has_derivative (\<lambda>(tx, ty). fx tx + fy x y ty)) (at (x, y) within X \<times> Y)"
 proof (safe intro!: has_derivativeI tendstoI, goal_cases)
   case (2 e')
   interpret fx: bounded_linear "fx" using fx by (rule has_derivative_bounded_linear)
   define e where "e = e' / 9"
   have "e > 0" using \<open>e' > 0\<close> by (simp add: e_def)
 
   from fy_cont[THEN tendstoD, OF \<open>e > 0\<close>]
   have "\<forall>\<^sub>F (x', y') in at (x, y) within X \<times> Y. dist (fy x' y') (fy x y) < e"
     by (auto simp: split_beta')
   from this[unfolded eventually_at] obtain d' where
     "d' > 0"
     "\<And>x' y'. x' \<in> X \<Longrightarrow> y' \<in> Y \<Longrightarrow> (x', y') \<noteq> (x, y) \<Longrightarrow> dist (x', y') (x, y) < d' \<Longrightarrow>
       dist (fy x' y') (fy x y) < e"
     by auto
   then
   have d': "x' \<in> X \<Longrightarrow> y' \<in> Y \<Longrightarrow> dist (x', y') (x, y) < d' \<Longrightarrow> dist (fy x' y') (fy x y) < e"
     for x' y'
     using \<open>0 < e\<close>
     by (cases "(x', y') = (x, y)") auto
   define d where "d = d' / sqrt 2"
   have "d > 0" using \<open>0 < d'\<close> by (simp add: d_def)
   have d: "x' \<in> X \<Longrightarrow> y' \<in> Y \<Longrightarrow> dist x' x < d \<Longrightarrow> dist y' y < d \<Longrightarrow> dist (fy x' y') (fy x y) < e"
     for x' y'
     by (auto simp: dist_prod_def d_def intro!: d' real_sqrt_sum_squares_less)
 
   let ?S = "ball y d \<inter> Y"
   have "convex ?S"
     by (auto intro!: convex_Int \<open>convex Y\<close>)
   {
     fix x'::'a and y'::'b
     assume x': "x' \<in> X" and y': "y' \<in> Y"
     assume dx': "dist x' x < d" and dy': "dist y' y < d"
     have "norm (fy x' y' - fy x' y) \<le> dist (fy x' y') (fy x y) + dist (fy x' y) (fy x y)"
       by norm
     also have "dist (fy x' y') (fy x y) < e"
       by (rule d; fact)
     also have "dist (fy x' y) (fy x y) < e"
       by (auto intro!: d simp: dist_prod_def x' \<open>d > 0\<close> \<open>y \<in> Y\<close> dx')
     finally
     have "norm (fy x' y' - fy x' y) < e + e"
       by arith
     then have "onorm (blinfun_apply (fy x' y') - blinfun_apply (fy x' y)) < e + e"
       by (auto simp: norm_blinfun.rep_eq blinfun.diff_left[abs_def] fun_diff_def)
   } note onorm = this
 
   have ev_mem: "\<forall>\<^sub>F (x', y') in at (x, y) within X \<times> Y. (x', y') \<in> X \<times> Y"
     using \<open>y \<in> Y\<close>
     by (auto simp: eventually_at intro!: zero_less_one)
   moreover
   have ev_dist: "\<forall>\<^sub>F xy in at (x, y) within X \<times> Y. dist xy (x, y) < d" if "d > 0" for d
     using eventually_at_ball[OF that]
     by (rule eventually_elim2) (auto simp: dist_commute intro!: eventually_True)
   note ev_dist[OF \<open>0 < d\<close>]
   ultimately
   have "\<forall>\<^sub>F (x', y') in at (x, y) within X \<times> Y.
     norm (f x' y' - f x' y - (fy x' y) (y' - y)) \<le> norm (y' - y) * (e + e)"
   proof (eventually_elim, safe)
     fix x' y'
     assume "x' \<in> X" and y': "y' \<in> Y"
     assume dist: "dist (x', y') (x, y) < d"
     then have dx: "dist x' x < d" and dy: "dist y' y < d"
       unfolding dist_prod_def fst_conv snd_conv atomize_conj
       by (metis le_less_trans real_sqrt_sum_squares_ge1 real_sqrt_sum_squares_ge2)
     {
       fix t::real
       assume "t \<in> {0 .. 1}"
       then have "y + t *\<^sub>R (y' - y) \<in> closed_segment y y'"
         by (auto simp: closed_segment_def algebra_simps intro!: exI[where x=t])
       also
       have "\<dots> \<subseteq> ball y d \<inter> Y"
         using \<open>y \<in> Y\<close> \<open>0 < d\<close> dy y'
         by (intro \<open>convex ?S\<close>[unfolded convex_contains_segment, rule_format, of y y'])
           (auto simp: dist_commute)
       finally have "y + t *\<^sub>R (y' - y) \<in> ?S" .
     } note seg = this
 
     have "\<And>x. x \<in> ball y d \<inter> Y \<Longrightarrow> onorm (blinfun_apply (fy x' x) - blinfun_apply (fy x' y)) \<le> e + e"
       by (safe intro!: onorm less_imp_le \<open>x' \<in> X\<close> dx) (auto simp: dist_commute \<open>0 < d\<close> \<open>y \<in> Y\<close>)
-    with seg has_derivative_within_subset[OF assms(2)[OF \<open>x' \<in> X\<close>]]
+    with seg has_derivative_subset[OF assms(2)[OF \<open>x' \<in> X\<close>]]
     show "norm (f x' y' - f x' y - (fy x' y) (y' - y)) \<le> norm (y' - y) * (e + e)"
       by (rule differentiable_bound_linearization[where S="?S"])
         (auto intro!: \<open>0 < d\<close> \<open>y \<in> Y\<close>)
   qed
   moreover
   let ?le = "\<lambda>x'. norm (f x' y - f x y - (fx) (x' - x)) \<le> norm (x' - x) * e"
   from fx[unfolded has_derivative_within, THEN conjunct2, THEN tendstoD, OF \<open>0 < e\<close>]
   have "\<forall>\<^sub>F x' in at x within X. ?le x'"
     by eventually_elim (simp, 
       simp add: dist_norm field_split_simps split: if_split_asm)
   then have "\<forall>\<^sub>F (x', y') in at (x, y) within X \<times> Y. ?le x'"
     by (rule eventually_at_Pair_within_TimesI1)
        (simp add: blinfun.bilinear_simps)
   moreover have "\<forall>\<^sub>F (x', y') in at (x, y) within X \<times> Y. norm ((x', y') - (x, y)) \<noteq> 0"
     unfolding norm_eq_zero right_minus_eq
     by (auto simp: eventually_at intro!: zero_less_one)
   moreover
   from fy_cont[THEN tendstoD, OF \<open>0 < e\<close>]
   have "\<forall>\<^sub>F x' in at x within X. norm (fy x' y - fy x y) < e"
     unfolding eventually_at
     using \<open>y \<in> Y\<close>
     by (auto simp: dist_prod_def dist_norm)
   then have "\<forall>\<^sub>F (x', y') in at (x, y) within X \<times> Y. norm (fy x' y - fy x y) < e"
     by (rule eventually_at_Pair_within_TimesI1)
        (simp add: blinfun.bilinear_simps \<open>0 < e\<close>)
   ultimately
   have "\<forall>\<^sub>F (x', y') in at (x, y) within X \<times> Y.
             norm ((f x' y' - f x y - (fx (x' - x) + fy x y (y' - y))) /\<^sub>R
               norm ((x', y') - (x, y)))
             < e'"
     apply eventually_elim
   proof safe
     fix x' y'
     have "norm (f x' y' - f x y - (fx (x' - x) + fy x y (y' - y))) \<le>
         norm (f x' y' - f x' y - fy x' y (y' - y)) +
         norm (fy x y (y' - y) - fy x' y (y' - y)) +
         norm (f x' y - f x y - fx (x' - x))"
       by norm
     also
     assume nz: "norm ((x', y') - (x, y)) \<noteq> 0"
       and nfy: "norm (fy x' y - fy x y) < e"
     assume "norm (f x' y' - f x' y - blinfun_apply (fy x' y) (y' - y)) \<le> norm (y' - y) * (e + e)"
     also assume "norm (f x' y - f x y - (fx) (x' - x)) \<le> norm (x' - x) * e"
     also
     have "norm ((fy x y) (y' - y) - (fy x' y) (y' - y)) \<le> norm ((fy x y) - (fy x' y)) * norm (y' - y)"
       by (auto simp: blinfun.bilinear_simps[symmetric] intro!: norm_blinfun)
     also have "\<dots> \<le> (e + e) * norm (y' - y)"
       using \<open>e > 0\<close> nfy
       by (auto simp: norm_minus_commute intro!: mult_right_mono)
     also have "norm (x' - x) * e \<le> norm (x' - x) * (e + e)"
       using \<open>0 < e\<close> by simp
     also have "norm (y' - y) * (e + e) + (e + e) * norm (y' - y) + norm (x' - x) * (e + e) \<le>
         (norm (y' - y) + norm (x' - x)) * (4 * e)"
       using \<open>e > 0\<close>
       by (simp add: algebra_simps)
     also have "\<dots> \<le> 2 * norm ((x', y') - (x, y)) * (4 * e)"
       using \<open>0 < e\<close> real_sqrt_sum_squares_ge1[of "norm (x' - x)" "norm (y' - y)"]
         real_sqrt_sum_squares_ge2[of "norm (y' - y)" "norm (x' - x)"]
       by (auto intro!: mult_right_mono simp: norm_prod_def
         simp del: real_sqrt_sum_squares_ge1 real_sqrt_sum_squares_ge2)
     also have "\<dots> \<le> norm ((x', y') - (x, y)) * (8 * e)"
       by simp
     also have "\<dots> < norm ((x', y') - (x, y)) * e'"
       using \<open>0 < e'\<close> nz
       by (auto simp: e_def)
     finally show "norm ((f x' y' - f x y - (fx (x' - x) + fy x y (y' - y))) /\<^sub>R norm ((x', y') - (x, y))) < e'"
       by (simp add: dist_norm) (auto simp add: field_split_simps)
   qed
   then show ?case
     by eventually_elim (auto simp: dist_norm field_simps)
 next
   from has_derivative_bounded_linear[OF fx]
   obtain fxb where "fx = blinfun_apply fxb"
     by (metis bounded_linear_Blinfun_apply)
   then show "bounded_linear (\<lambda>(tx, ty). fx tx + blinfun_apply (fy x y) ty)"
     by (auto intro!: bounded_linear_intros simp: split_beta')
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Differentiable case distinction\<close>
 
 lemma has_derivative_within_If_eq:
   "((\<lambda>x. if P x then f x else g x) has_derivative f') (at x within S) =
     (bounded_linear f' \<and>
      ((\<lambda>y.(if P y then (f y - ((if P x then f x else g x) + f' (y - x)))/\<^sub>R norm (y - x)
            else (g y - ((if P x then f x else g x) + f' (y - x)))/\<^sub>R norm (y - x)))
       \<longlongrightarrow> 0) (at x within S))"
   (is "_ = (_ \<and> (?if \<longlongrightarrow> 0) _)")
 proof -
   have "(\<lambda>y. (1 / norm (y - x)) *\<^sub>R
            ((if P y then f y else g y) -
             ((if P x then f x else g x) + f' (y - x)))) = ?if"
     by (auto simp: inverse_eq_divide)
   thus ?thesis by (auto simp: has_derivative_within)
 qed
 
 lemma has_derivative_If_within_closures:
   assumes f': "x \<in> S \<union> (closure S \<inter> closure T) \<Longrightarrow>
     (f has_derivative f' x) (at x within S \<union> (closure S \<inter> closure T))"
   assumes g': "x \<in> T \<union> (closure S \<inter> closure T) \<Longrightarrow>
     (g has_derivative g' x) (at x within T \<union> (closure S \<inter> closure T))"
   assumes connect: "x \<in> closure S \<Longrightarrow> x \<in> closure T \<Longrightarrow> f x = g x"
   assumes connect': "x \<in> closure S \<Longrightarrow> x \<in> closure T \<Longrightarrow> f' x = g' x"
   assumes x_in: "x \<in> S \<union> T"
   shows "((\<lambda>x. if x \<in> S then f x else g x) has_derivative
       (if x \<in> S then f' x else g' x)) (at x within (S \<union> T))"
 proof -
   from f' x_in interpret f': bounded_linear "if x \<in> S then f' x else (\<lambda>x. 0)"
     by (auto simp add: has_derivative_within)
   from g' interpret g': bounded_linear "if x \<in> T then g' x else (\<lambda>x. 0)"
     by (auto simp add: has_derivative_within)
   have bl: "bounded_linear (if x \<in> S then f' x else g' x)"
     using f'.scaleR f'.bounded f'.add g'.scaleR g'.bounded g'.add x_in
     by (unfold_locales; force)
   show ?thesis
     using f' g' closure_subset[of T] closure_subset[of S]
     unfolding has_derivative_within_If_eq
     by (intro conjI bl tendsto_If_within_closures x_in)
       (auto simp: has_derivative_within inverse_eq_divide connect connect' subsetD)
 qed
 
 lemma has_vector_derivative_If_within_closures:
   assumes x_in: "x \<in> S \<union> T"
   assumes "u = S \<union> T"
   assumes f': "x \<in> S \<union> (closure S \<inter> closure T) \<Longrightarrow>
     (f has_vector_derivative f' x) (at x within S \<union> (closure S \<inter> closure T))"
   assumes g': "x \<in> T \<union> (closure S \<inter> closure T) \<Longrightarrow>
     (g has_vector_derivative g' x) (at x within T \<union> (closure S \<inter> closure T))"
   assumes connect: "x \<in> closure S \<Longrightarrow> x \<in> closure T \<Longrightarrow> f x = g x"
   assumes connect': "x \<in> closure S \<Longrightarrow> x \<in> closure T \<Longrightarrow> f' x = g' x"
   shows "((\<lambda>x. if x \<in> S then f x else g x) has_vector_derivative
     (if x \<in> S then f' x else g' x)) (at x within u)"
   unfolding has_vector_derivative_def assms
   using x_in
   apply (intro has_derivative_If_within_closures[where ?f' = "\<lambda>x a. a *\<^sub>R f' x" and ?g' = "\<lambda>x a. a *\<^sub>R g' x",
         THEN has_derivative_eq_rhs])
   subgoal by (rule f'[unfolded has_vector_derivative_def]; assumption)
   subgoal by (rule g'[unfolded has_vector_derivative_def]; assumption)
   by (auto simp: assms)
 
 subsection\<^marker>\<open>tag important\<close>\<open>The Inverse Function Theorem\<close>
 
 lemma linear_injective_contraction:
   assumes "linear f" "c < 1" and le: "\<And>x. norm (f x - x) \<le> c * norm x"
   shows "inj f"
   unfolding linear_injective_0[OF \<open>linear f\<close>]
 proof safe
   fix x
   assume "f x = 0"
   with le [of x] have "norm x \<le> c * norm x"
     by simp
   then show "x = 0"
     using \<open>c < 1\<close> by (simp add: mult_le_cancel_right1)
 qed
 
 text\<open>From an online proof by J. Michael Boardman, Department of Mathematics, Johns Hopkins University\<close>
 lemma inverse_function_theorem_scaled:
   fixes f::"'a::euclidean_space \<Rightarrow> 'a"
     and f'::"'a \<Rightarrow> ('a \<Rightarrow>\<^sub>L 'a)"
   assumes "open U"
     and derf: "\<And>x. x \<in> U \<Longrightarrow> (f has_derivative blinfun_apply (f' x)) (at x)"
     and contf: "continuous_on U f'"
     and "0 \<in> U" and [simp]: "f 0 = 0"
     and id: "f' 0 = id_blinfun"
   obtains U' V g g' where "open U'" "U' \<subseteq> U" "0 \<in> U'" "open V" "0 \<in> V" "homeomorphism U' V f g"
                 "\<And>y. y \<in> V \<Longrightarrow> (g has_derivative (g' y)) (at y)"
                 "\<And>y. y \<in> V \<Longrightarrow> g' y = inv (blinfun_apply (f'(g y)))"
                 "\<And>y. y \<in> V \<Longrightarrow> bij (blinfun_apply (f'(g y)))"
 proof -
   obtain d1 where "cball 0 d1 \<subseteq> U" "d1 > 0"
     using \<open>open U\<close> \<open>0 \<in> U\<close> open_contains_cball by blast
   obtain d2 where d2: "\<And>x. \<lbrakk>x \<in> U; dist x 0 \<le> d2\<rbrakk> \<Longrightarrow> dist (f' x) (f' 0) < 1/2" "0 < d2"
     using continuous_onE [OF contf, of 0 "1/2"] by (metis \<open>0 \<in> U\<close> half_gt_zero_iff zero_less_one)
   obtain \<delta> where le: "\<And>x. norm x \<le> \<delta> \<Longrightarrow> dist (f' x) id_blinfun \<le> 1/2" and "0 < \<delta>"
     and subU: "cball 0 \<delta> \<subseteq> U"
   proof
     show "min d1 d2 > 0"
       by (simp add: \<open>0 < d1\<close> \<open>0 < d2\<close>)
     show "cball 0 (min d1 d2) \<subseteq> U"
       using \<open>cball 0 d1 \<subseteq> U\<close> by auto
     show "dist (f' x) id_blinfun \<le> 1/2" if "norm x \<le> min d1 d2" for x
       using \<open>cball 0 d1 \<subseteq> U\<close> d2 that id by fastforce
   qed
   let ?D = "cball 0 \<delta>"
   define V:: "'a set" where "V \<equiv> ball 0 (\<delta>/2)"
   have 4: "norm (f (x + h) - f x - h) \<le> 1/2 * norm h"
     if "x \<in> ?D" "x+h \<in> ?D" for x h
   proof -
     let ?w = "\<lambda>x. f x - x"
     have B: "\<And>x. x \<in> ?D \<Longrightarrow> onorm (blinfun_apply (f' x - id_blinfun)) \<le> 1/2"
       by (metis dist_norm le mem_cball_0 norm_blinfun.rep_eq)
     have "\<And>x. x \<in> ?D \<Longrightarrow> (?w has_derivative (blinfun_apply (f' x - id_blinfun))) (at x)"
       by (rule derivative_eq_intros derf subsetD [OF subU] | force simp: blinfun.diff_left)+
     then have Dw: "\<And>x. x \<in> ?D \<Longrightarrow> (?w has_derivative (blinfun_apply (f' x - id_blinfun))) (at x within ?D)"
       using has_derivative_at_withinI by blast
     have "norm (?w (x+h) - ?w x) \<le> (1/2) * norm h"
       using differentiable_bound [OF convex_cball Dw B] that by fastforce
     then show ?thesis
       by (auto simp: algebra_simps)
   qed
   have for_g: "\<exists>!x. norm x < \<delta> \<and> f x = y" if y: "norm y < \<delta>/2" for y
   proof -
     let ?u = "\<lambda>x. x + (y - f x)"
     have *: "norm (?u x) < \<delta>" if "x \<in> ?D" for x
     proof -
       have fxx: "norm (f x - x) \<le> \<delta>/2"
         using 4 [of 0 x] \<open>0 < \<delta>\<close> \<open>f 0 = 0\<close> that by auto
       have "norm (?u x) \<le> norm y + norm (f x - x)"
         by (metis add.commute add_diff_eq norm_minus_commute norm_triangle_ineq)
       also have "\<dots> < \<delta>/2 + \<delta>/2"
         using fxx y by auto
       finally show ?thesis
         by simp
     qed
     have "\<exists>!x \<in> ?D. ?u x = x"
     proof (rule banach_fix)
       show "cball 0 \<delta> \<noteq> {}"
         using \<open>0 < \<delta>\<close> by auto
       show "(\<lambda>x. x + (y - f x)) ` cball 0 \<delta> \<subseteq> cball 0 \<delta>"
         using * by force
       have "dist (x + (y - f x)) (xh + (y - f xh)) * 2 \<le> dist x xh"
         if "norm x \<le> \<delta>" and "norm xh \<le> \<delta>" for x xh
         using that 4 [of x "xh-x"] by (auto simp: dist_norm norm_minus_commute algebra_simps)
       then show "\<forall>x\<in>cball 0 \<delta>. \<forall>ya\<in>cball 0 \<delta>. dist (x + (y - f x)) (ya + (y - f ya)) \<le> (1/2) * dist x ya"
         by auto
     qed (auto simp: complete_eq_closed)
     then show ?thesis
       by (metis "*" add_cancel_right_right eq_iff_diff_eq_0 le_less mem_cball_0)
   qed
   define g where "g \<equiv> \<lambda>y. THE x. norm x < \<delta> \<and> f x = y"
   have g: "norm (g y) < \<delta> \<and> f (g y) = y" if "norm y < \<delta>/2" for y
     unfolding g_def using that theI' [OF for_g] by meson
   then have fg[simp]: "f (g y) = y" if "y \<in> V" for y
     using that by (auto simp: V_def)
   have 5: "norm (g y' - g y) \<le> 2 * norm (y' - y)" if "y \<in> V" "y' \<in> V" for y y'
   proof -
     have no: "norm (g y) \<le> \<delta>" "norm (g y') \<le> \<delta>" and [simp]: "f (g y) = y"
       using that g unfolding V_def by force+
     have "norm (g y' - g y) \<le> norm (g y' - g y - (y' - y)) + norm (y' - y)"
       by (simp add: add.commute norm_triangle_sub)
     also have "\<dots> \<le> (1/2) * norm (g y' - g y) + norm (y' - y)"
       using 4 [of "g y" "g y' - g y"] that no by (simp add: g norm_minus_commute V_def)
     finally show ?thesis
       by auto
   qed
   have contg: "continuous_on V g"
   proof
     fix y::'a and e::real
     assume "0 < e" and y: "y \<in> V"
     show "\<exists>d>0. \<forall>x'\<in>V. dist x' y < d \<longrightarrow> dist (g x') (g y) \<le> e"
     proof (intro exI conjI ballI impI)
       show "0 < e/2"
         by (simp add: \<open>0 < e\<close>)
     qed (use 5 y in \<open>force simp: dist_norm\<close>)
   qed
   show thesis
   proof
     define U' where "U' \<equiv> (f -` V) \<inter> ball 0 \<delta>"
     have contf: "continuous_on U f"
       using derf has_derivative_at_withinI by (fast intro: has_derivative_continuous_on)
     then have "continuous_on (ball 0 \<delta>) f"
       by (meson ball_subset_cball continuous_on_subset subU)
     then show "open U'"
       by (simp add: U'_def V_def Int_commute continuous_open_preimage)
     show "0 \<in> U'" "U' \<subseteq> U" "open V" "0 \<in> V"
       using \<open>0 < \<delta>\<close> subU by (auto simp: U'_def V_def)
     show hom: "homeomorphism U' V f g"
     proof
       show "continuous_on U' f"
         using \<open>U' \<subseteq> U\<close> contf continuous_on_subset by blast
       show "continuous_on V g"
         using contg by blast
       show "f ` U' \<subseteq> V"
         using U'_def by blast
       show "g ` V \<subseteq> U'"
         by (simp add: U'_def V_def g image_subset_iff)
       show "g (f x) = x" if "x \<in> U'" for x
         by (metis that fg Int_iff U'_def V_def for_g g mem_ball_0 vimage_eq)
       show "f (g y) = y" if "y \<in> V" for y
         using that by (simp add: g V_def)
     qed
     show bij: "bij (blinfun_apply (f'(g y)))" if "y \<in> V" for y
     proof -
       have inj: "inj (blinfun_apply (f' (g y)))"
       proof (rule linear_injective_contraction)
         show "linear (blinfun_apply (f' (g y)))"
           using blinfun.bounded_linear_right bounded_linear_def by blast
       next
         fix x
         have "norm (blinfun_apply (f' (g y)) x - x) = norm (blinfun_apply (f' (g y) - id_blinfun) x)"
           by (simp add: blinfun.diff_left)
         also have "\<dots> \<le> norm (f' (g y) - id_blinfun) * norm x"
           by (rule norm_blinfun)
         also have "\<dots> \<le> (1/2) * norm x"
         proof (rule mult_right_mono)
           show "norm (f' (g y) - id_blinfun) \<le> 1/2"
             using that g [of y] le by (auto simp: V_def dist_norm)
         qed auto
         finally show "norm (blinfun_apply (f' (g y)) x - x) \<le> (1/2) * norm x" .
       qed auto
       moreover
       have "surj (blinfun_apply (f' (g y)))"
         using blinfun.bounded_linear_right bounded_linear_def
         by (blast intro!: linear_inj_imp_surj [OF _ inj])
       ultimately show ?thesis
         using bijI by blast
     qed
     define g' where "g' \<equiv> \<lambda>y. inv (blinfun_apply (f'(g y)))"
     show "(g has_derivative g' y) (at y)" if "y \<in> V" for y
     proof -
       have gy: "g y \<in> U"
         using g subU that unfolding V_def by fastforce
       obtain e where e: "\<And>h. f (g y + h) = y + blinfun_apply (f' (g y)) h + e h"
         and e0: "(\<lambda>h. norm (e h) / norm h) \<midarrow>0\<rightarrow> 0"
         using iffD1 [OF has_derivative_iff_Ex derf [OF gy]] \<open>y \<in> V\<close> by auto
       have [simp]: "e 0 = 0"
         using e [of 0] that by simp
       let ?INV = "inv (blinfun_apply (f' (g y)))"
       have inj: "inj (blinfun_apply (f' (g y)))"
         using bij bij_betw_def that by blast
       have "(g has_derivative g' y) (at y within V)"
         unfolding has_derivative_at_within_iff_Ex [OF \<open>y \<in> V\<close> \<open>open V\<close>]
       proof
         show blinv: "bounded_linear (g' y)"
           unfolding g'_def using derf gy inj inj_linear_imp_inv_bounded_linear by blast
         define eg where "eg \<equiv> \<lambda>k. - ?INV (e (g (y+k) - g y))"
         have "g (y+k) = g y + g' y k + eg k" if "y + k \<in> V" for k
         proof -
           have "?INV k = ?INV (blinfun_apply (f' (g y)) (g (y+k) - g y) + e (g (y+k) - g y))"
             using e [of "g(y+k) - g y"] that by simp
           then have "g (y+k) = g y + ?INV k - ?INV (e (g (y+k) - g y))"
             using inj blinv by (simp add: linear_simps g'_def)
           then show ?thesis
             by (auto simp: eg_def g'_def)
         qed
         moreover have "(\<lambda>k. norm (eg k) / norm k) \<midarrow>0\<rightarrow> 0"
         proof (rule Lim_null_comparison)
           let ?g = "\<lambda>k. 2 * onorm ?INV * norm (e (g (y+k) - g y)) / norm (g (y+k) - g y)"
           show "\<forall>\<^sub>F k in at 0. norm (norm (eg k) / norm k) \<le> ?g k"
             unfolding eventually_at_topological
           proof (intro exI conjI ballI impI)
             show "open ((+)(-y) ` V)"
               using \<open>open V\<close> open_translation by blast
             show "0 \<in> (+)(-y) ` V"
               by (simp add: that)
             show "norm (norm (eg k) / norm k) \<le> 2 * onorm (inv (blinfun_apply (f' (g y)))) * norm (e (g (y+k) - g y)) / norm (g (y+k) - g y)"
               if "k \<in> (+)(-y) ` V" "k \<noteq> 0" for k
             proof -
               have "y+k \<in> V"
                 using that by auto
               have "norm (norm (eg k) / norm k) \<le> onorm ?INV * norm (e (g (y+k) - g y)) / norm k"
                 using blinv g'_def onorm by (force simp: eg_def divide_simps)
               also have "\<dots> = (norm (g (y+k) - g y) / norm k) * (onorm ?INV * (norm (e (g (y+k) - g y)) / norm (g (y+k) - g y)))"
                 by (simp add: divide_simps)
               also have "\<dots> \<le> 2 * (onorm ?INV * (norm (e (g (y+k) - g y)) / norm (g (y+k) - g y)))"
                 apply (rule mult_right_mono)
                 using 5 [of y "y+k"] \<open>y \<in> V\<close> \<open>y + k \<in> V\<close>  onorm_pos_le [OF blinv]
                  apply (auto simp: divide_simps zero_le_mult_iff zero_le_divide_iff g'_def)
                 done
               finally show "norm (norm (eg k) / norm k) \<le> 2 * onorm ?INV * norm (e (g (y+k) - g y)) / norm (g (y+k) - g y)"
                 by simp
             qed
           qed
           have 1: "(\<lambda>h. norm (e h) / norm h) \<midarrow>0\<rightarrow> (norm (e 0) / norm 0)"
             using e0 by auto
           have 2: "(\<lambda>k. g (y+k) - g y) \<midarrow>0\<rightarrow> 0"
             using contg \<open>open V\<close> \<open>y \<in> V\<close> LIM_offset_zero_iff LIM_zero_iff at_within_open continuous_on_def by fastforce
           from tendsto_compose [OF 1 2, simplified]
           have "(\<lambda>k. norm (e (g (y+k) - g y)) / norm (g (y+k) - g y)) \<midarrow>0\<rightarrow> 0" .
           from tendsto_mult_left [OF this] show "?g \<midarrow>0\<rightarrow> 0" by auto
         qed
         ultimately show "\<exists>e. (\<forall>k. y + k \<in> V \<longrightarrow> g (y+k) = g y + g' y k + e k) \<and> (\<lambda>k. norm (e k) / norm k) \<midarrow>0\<rightarrow> 0"
           by blast
       qed
       then show ?thesis
         by (metis \<open>open V\<close> at_within_open that)
     qed
     show "g' y = inv (blinfun_apply (f' (g y)))"
       if "y \<in> V" for y
       by (simp add: g'_def)
   qed
 qed
 
 
 text\<open>We need all this to justify the scaling and translations.\<close>
 theorem inverse_function_theorem:
   fixes f::"'a::euclidean_space \<Rightarrow> 'a"
     and f'::"'a \<Rightarrow> ('a \<Rightarrow>\<^sub>L 'a)"
   assumes "open U"
     and derf: "\<And>x. x \<in> U \<Longrightarrow> (f has_derivative (blinfun_apply (f' x))) (at x)"
     and contf:  "continuous_on U f'"
     and "x0 \<in> U"
     and invf: "invf o\<^sub>L f' x0 = id_blinfun"
   obtains U' V g g' where "open U'" "U' \<subseteq> U" "x0 \<in> U'" "open V" "f x0 \<in> V" "homeomorphism U' V f g"
     "\<And>y. y \<in> V \<Longrightarrow> (g has_derivative (g' y)) (at y)"
     "\<And>y. y \<in> V \<Longrightarrow> g' y = inv (blinfun_apply (f'(g y)))"
     "\<And>y. y \<in> V \<Longrightarrow> bij (blinfun_apply (f'(g y)))"
 proof -
   have apply1 [simp]: "\<And>i. blinfun_apply invf (blinfun_apply (f' x0) i) = i"
     by (metis blinfun_apply_blinfun_compose blinfun_apply_id_blinfun invf)
   have apply2 [simp]: "\<And>i. blinfun_apply (f' x0) (blinfun_apply invf i) = i"
     by (metis apply1 bij_inv_eq_iff blinfun_bij1 invf)
   have [simp]: "(range (blinfun_apply invf)) = UNIV"
     using apply1 surjI by blast
   let ?f = "invf \<circ> (\<lambda>x. (f \<circ> (+)x0)x - f x0)"
   let ?f' = "\<lambda>x. invf o\<^sub>L (f' (x + x0))"
   obtain U' V g g' where "open U'" and U': "U' \<subseteq> (+)(-x0) ` U" "0 \<in> U'"
     and "open V" "0 \<in> V" and hom: "homeomorphism U' V ?f g"
     and derg: "\<And>y. y \<in> V \<Longrightarrow> (g has_derivative (g' y)) (at y)"
     and g': "\<And>y. y \<in> V \<Longrightarrow> g' y = inv (?f'(g y))"
     and bij: "\<And>y. y \<in> V \<Longrightarrow> bij (?f'(g y))"
   proof (rule inverse_function_theorem_scaled [of "(+)(-x0) ` U" ?f "?f'"])
     show ope: "open ((+) (- x0) ` U)"
       using \<open>open U\<close> open_translation by blast
     show "(?f has_derivative blinfun_apply (?f' x)) (at x)"
       if "x \<in> (+) (- x0) ` U" for x
       using that
       apply clarify
       apply (rule derf derivative_eq_intros | simp add: blinfun_compose.rep_eq)+
       done
     have YY: "(\<lambda>x. f' (x + x0)) \<midarrow>u-x0\<rightarrow> f' u"
       if "f' \<midarrow>u\<rightarrow> f' u" "u \<in> U" for u
       using that LIM_offset [where k = x0] by (auto simp: algebra_simps)
     then have "continuous_on ((+) (- x0) ` U) (\<lambda>x. f' (x + x0))"
       using contf \<open>open U\<close> Lim_at_imp_Lim_at_within
       by (fastforce simp: continuous_on_def at_within_open_NO_MATCH ope)
     then show "continuous_on ((+) (- x0) ` U) ?f'"
       by (intro continuous_intros) simp
   qed (auto simp: invf \<open>x0 \<in> U\<close>)
   show thesis
   proof
     let ?U' = "(+)x0 ` U'"
     let ?V = "((+)(f x0) \<circ> f' x0) ` V"
     let ?g = "(+)x0 \<circ> g \<circ> invf \<circ> (+)(- f x0)"
     let ?g' = "\<lambda>y. inv (blinfun_apply (f' (?g y)))"
     show oU': "open ?U'"
       by (simp add: \<open>open U'\<close> open_translation)
     show subU: "?U' \<subseteq> U"
       using ComplI \<open>U' \<subseteq> (+) (- x0) ` U\<close> by auto
     show "x0 \<in> ?U'"
       by (simp add: \<open>0 \<in> U'\<close>)
     show "open ?V"
       using blinfun_bij2 [OF invf]
       by (metis \<open>open V\<close> bij_is_surj blinfun.bounded_linear_right bounded_linear_def image_comp open_surjective_linear_image open_translation)
     show "f x0 \<in> ?V"
       using \<open>0 \<in> V\<close> image_iff by fastforce
     show "homeomorphism ?U' ?V f ?g"
     proof
       show "continuous_on ?U' f"
         by (meson subU continuous_on_eq_continuous_at derf has_derivative_continuous oU' subsetD)
       have "?f ` U' \<subseteq> V"
         using hom homeomorphism_image1 by blast
       then show "f ` ?U' \<subseteq> ?V"
         unfolding image_subset_iff
         by (clarsimp simp: image_def) (metis apply2 add.commute diff_add_cancel)
       show "?g ` ?V \<subseteq> ?U'"
         using hom invf by (auto simp: image_def homeomorphism_def)
       show "?g (f x) = x"
         if "x \<in> ?U'" for x
         using that hom homeomorphism_apply1 by fastforce
       have "continuous_on V g"
         using hom homeomorphism_def by blast
       then show "continuous_on ?V ?g"
         by (intro continuous_intros) (auto elim!: continuous_on_subset)
       have fg: "?f (g x) = x" if "x \<in> V" for x
         using hom homeomorphism_apply2 that by blast
       show "f (?g y) = y"
         if "y \<in> ?V" for y
         using that fg by (simp add: image_iff) (metis apply2 add.commute diff_add_cancel)
     qed
     show "(?g has_derivative ?g' y) (at y)" "bij (blinfun_apply (f' (?g y)))"
       if "y \<in> ?V" for y
     proof -
       have 1: "bij (blinfun_apply invf)"
         using blinfun_bij1 invf by blast
       then have 2: "bij (blinfun_apply (f' (x0 + g x)))" if "x \<in> V" for x
         by (metis add.commute bij bij_betw_comp_iff2 blinfun_compose.rep_eq that top_greatest)
       then show "bij (blinfun_apply (f' (?g y)))"
         using that by auto
       have "g' x \<circ> blinfun_apply invf = inv (blinfun_apply (f' (x0 + g x)))"
         if "x \<in> V" for x
         using that
         by (simp add: g' o_inv_distrib blinfun_compose.rep_eq 1 2 add.commute bij_is_inj flip: o_assoc)
       then show "(?g has_derivative ?g' y) (at y)"
         using that invf
         by clarsimp (rule derg derivative_eq_intros | simp flip: id_def)+
     qed
   qed auto
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Piecewise differentiable functions\<close>
 
 definition piecewise_differentiable_on
            (infixr "piecewise'_differentiable'_on" 50)
   where "f piecewise_differentiable_on i  \<equiv>
            continuous_on i f \<and>
            (\<exists>S. finite S \<and> (\<forall>x \<in> i - S. f differentiable (at x within i)))"
 
 lemma piecewise_differentiable_on_imp_continuous_on:
     "f piecewise_differentiable_on S \<Longrightarrow> continuous_on S f"
 by (simp add: piecewise_differentiable_on_def)
 
 lemma piecewise_differentiable_on_subset:
     "f piecewise_differentiable_on S \<Longrightarrow> T \<le> S \<Longrightarrow> f piecewise_differentiable_on T"
   using continuous_on_subset
   unfolding piecewise_differentiable_on_def
   apply safe
   apply (blast elim: continuous_on_subset)
   by (meson Diff_iff differentiable_within_subset subsetCE)
 
 lemma differentiable_on_imp_piecewise_differentiable:
   fixes a:: "'a::{linorder_topology,real_normed_vector}"
   shows "f differentiable_on {a..b} \<Longrightarrow> f piecewise_differentiable_on {a..b}"
   apply (simp add: piecewise_differentiable_on_def differentiable_imp_continuous_on)
   apply (rule_tac x="{a,b}" in exI, simp add: differentiable_on_def)
   done
 
 lemma differentiable_imp_piecewise_differentiable:
     "(\<And>x. x \<in> S \<Longrightarrow> f differentiable (at x within S))
          \<Longrightarrow> f piecewise_differentiable_on S"
 by (auto simp: piecewise_differentiable_on_def differentiable_imp_continuous_on differentiable_on_def
          intro: differentiable_within_subset)
 
 lemma piecewise_differentiable_const [iff]: "(\<lambda>x. z) piecewise_differentiable_on S"
   by (simp add: differentiable_imp_piecewise_differentiable)
 
 lemma piecewise_differentiable_compose:
     "\<lbrakk>f piecewise_differentiable_on S; g piecewise_differentiable_on (f ` S);
       \<And>x. finite (S \<inter> f-`{x})\<rbrakk>
       \<Longrightarrow> (g \<circ> f) piecewise_differentiable_on S"
   apply (simp add: piecewise_differentiable_on_def, safe)
   apply (blast intro: continuous_on_compose2)
   apply (rename_tac A B)
   apply (rule_tac x="A \<union> (\<Union>x\<in>B. S \<inter> f-`{x})" in exI)
   apply (blast intro!: differentiable_chain_within)
   done
 
 lemma piecewise_differentiable_affine:
   fixes m::real
   assumes "f piecewise_differentiable_on ((\<lambda>x. m *\<^sub>R x + c) ` S)"
   shows "(f \<circ> (\<lambda>x. m *\<^sub>R x + c)) piecewise_differentiable_on S"
 proof (cases "m = 0")
   case True
   then show ?thesis
     unfolding o_def
     by (force intro: differentiable_imp_piecewise_differentiable differentiable_const)
 next
   case False
   show ?thesis
     apply (rule piecewise_differentiable_compose [OF differentiable_imp_piecewise_differentiable])
     apply (rule assms derivative_intros | simp add: False vimage_def real_vector_affinity_eq)+
     done
 qed
 
 lemma piecewise_differentiable_cases:
   fixes c::real
   assumes "f piecewise_differentiable_on {a..c}"
           "g piecewise_differentiable_on {c..b}"
            "a \<le> c" "c \<le> b" "f c = g c"
   shows "(\<lambda>x. if x \<le> c then f x else g x) piecewise_differentiable_on {a..b}"
 proof -
   obtain S T where st: "finite S" "finite T"
                and fd: "\<And>x. x \<in> {a..c} - S \<Longrightarrow> f differentiable at x within {a..c}"
                and gd: "\<And>x. x \<in> {c..b} - T \<Longrightarrow> g differentiable at x within {c..b}"
     using assms
     by (auto simp: piecewise_differentiable_on_def)
   have finabc: "finite ({a,b,c} \<union> (S \<union> T))"
     by (metis \<open>finite S\<close> \<open>finite T\<close> finite_Un finite_insert finite.emptyI)
   have "continuous_on {a..c} f" "continuous_on {c..b} g"
     using assms piecewise_differentiable_on_def by auto
   then have "continuous_on {a..b} (\<lambda>x. if x \<le> c then f x else g x)"
     using continuous_on_cases [OF closed_real_atLeastAtMost [of a c],
                                OF closed_real_atLeastAtMost [of c b],
                                of f g "\<lambda>x. x\<le>c"]  assms
     by (force simp: ivl_disj_un_two_touch)
   moreover
   { fix x
     assume x: "x \<in> {a..b} - ({a,b,c} \<union> (S \<union> T))"
     have "(\<lambda>x. if x \<le> c then f x else g x) differentiable at x within {a..b}" (is "?diff_fg")
     proof (cases x c rule: le_cases)
       case le show ?diff_fg
       proof (rule differentiable_transform_within [where d = "dist x c"])
         have "f differentiable at x"
           using x le fd [of x] at_within_interior [of x "{a..c}"] by simp
         then show "f differentiable at x within {a..b}"
           by (simp add: differentiable_at_withinI)
       qed (use x le st dist_real_def in auto)
     next
       case ge show ?diff_fg
       proof (rule differentiable_transform_within [where d = "dist x c"])
         have "g differentiable at x"
           using x ge gd [of x] at_within_interior [of x "{c..b}"] by simp
         then show "g differentiable at x within {a..b}"
           by (simp add: differentiable_at_withinI)
       qed (use x ge st dist_real_def in auto)
     qed
   }
   then have "\<exists>S. finite S \<and>
                  (\<forall>x\<in>{a..b} - S. (\<lambda>x. if x \<le> c then f x else g x) differentiable at x within {a..b})"
     by (meson finabc)
   ultimately show ?thesis
     by (simp add: piecewise_differentiable_on_def)
 qed
 
 lemma piecewise_differentiable_neg:
     "f piecewise_differentiable_on S \<Longrightarrow> (\<lambda>x. -(f x)) piecewise_differentiable_on S"
   by (auto simp: piecewise_differentiable_on_def continuous_on_minus)
 
 lemma piecewise_differentiable_add:
   assumes "f piecewise_differentiable_on i"
           "g piecewise_differentiable_on i"
     shows "(\<lambda>x. f x + g x) piecewise_differentiable_on i"
 proof -
   obtain S T where st: "finite S" "finite T"
                        "\<forall>x\<in>i - S. f differentiable at x within i"
                        "\<forall>x\<in>i - T. g differentiable at x within i"
     using assms by (auto simp: piecewise_differentiable_on_def)
   then have "finite (S \<union> T) \<and> (\<forall>x\<in>i - (S \<union> T). (\<lambda>x. f x + g x) differentiable at x within i)"
     by auto
   moreover have "continuous_on i f" "continuous_on i g"
     using assms piecewise_differentiable_on_def by auto
   ultimately show ?thesis
     by (auto simp: piecewise_differentiable_on_def continuous_on_add)
 qed
 
 lemma piecewise_differentiable_diff:
     "\<lbrakk>f piecewise_differentiable_on S;  g piecewise_differentiable_on S\<rbrakk>
      \<Longrightarrow> (\<lambda>x. f x - g x) piecewise_differentiable_on S"
   unfolding diff_conv_add_uminus
   by (metis piecewise_differentiable_add piecewise_differentiable_neg)
 
 
 subsection\<open>The concept of continuously differentiable\<close>
 
 text \<open>
 John Harrison writes as follows:
 
 ``The usual assumption in complex analysis texts is that a path \<open>\<gamma>\<close> should be piecewise
 continuously differentiable, which ensures that the path integral exists at least for any continuous
 f, since all piecewise continuous functions are integrable. However, our notion of validity is
 weaker, just piecewise differentiability\ldots{} [namely] continuity plus differentiability except on a
 finite set\ldots{} [Our] underlying theory of integration is the Kurzweil-Henstock theory. In contrast to
 the Riemann or Lebesgue theory (but in common with a simple notion based on antiderivatives), this
 can integrate all derivatives.''
 
 "Formalizing basic complex analysis." From Insight to Proof: Festschrift in Honour of Andrzej Trybulec.
 Studies in Logic, Grammar and Rhetoric 10.23 (2007): 151-165.
 
 And indeed he does not assume that his derivatives are continuous, but the penalty is unreasonably
 difficult proofs concerning winding numbers. We need a self-contained and straightforward theorem
 asserting that all derivatives can be integrated before we can adopt Harrison's choice.\<close>
 
 definition\<^marker>\<open>tag important\<close> C1_differentiable_on :: "(real \<Rightarrow> 'a::real_normed_vector) \<Rightarrow> real set \<Rightarrow> bool"
            (infix "C1'_differentiable'_on" 50)
   where
   "f C1_differentiable_on S \<longleftrightarrow>
    (\<exists>D. (\<forall>x \<in> S. (f has_vector_derivative (D x)) (at x)) \<and> continuous_on S D)"
 
 lemma C1_differentiable_on_eq:
     "f C1_differentiable_on S \<longleftrightarrow>
      (\<forall>x \<in> S. f differentiable at x) \<and> continuous_on S (\<lambda>x. vector_derivative f (at x))"
      (is "?lhs = ?rhs")
 proof
   assume ?lhs
   then show ?rhs
     unfolding C1_differentiable_on_def
     by (metis (no_types, lifting) continuous_on_eq  differentiableI_vector vector_derivative_at)
 next
   assume ?rhs
   then show ?lhs
     using C1_differentiable_on_def vector_derivative_works by fastforce
 qed
 
 lemma C1_differentiable_on_subset:
   "f C1_differentiable_on T \<Longrightarrow> S \<subseteq> T \<Longrightarrow> f C1_differentiable_on S"
   unfolding C1_differentiable_on_def  continuous_on_eq_continuous_within
   by (blast intro:  continuous_within_subset)
 
 lemma C1_differentiable_compose:
   assumes fg: "f C1_differentiable_on S" "g C1_differentiable_on (f ` S)" and fin: "\<And>x. finite (S \<inter> f-`{x})"
   shows "(g \<circ> f) C1_differentiable_on S"
 proof -
   have "\<And>x. x \<in> S \<Longrightarrow> g \<circ> f differentiable at x"
     by (meson C1_differentiable_on_eq assms differentiable_chain_at imageI)
   moreover have "continuous_on S (\<lambda>x. vector_derivative (g \<circ> f) (at x))"
   proof (rule continuous_on_eq [of _ "\<lambda>x. vector_derivative f (at x) *\<^sub>R vector_derivative g (at (f x))"])
     show "continuous_on S (\<lambda>x. vector_derivative f (at x) *\<^sub>R vector_derivative g (at (f x)))"
       using fg
       apply (clarsimp simp add: C1_differentiable_on_eq)
       apply (rule Limits.continuous_on_scaleR, assumption)
       by (metis (mono_tags, lifting) continuous_at_imp_continuous_on continuous_on_compose continuous_on_cong differentiable_imp_continuous_within o_def)
     show "\<And>x. x \<in> S \<Longrightarrow> vector_derivative f (at x) *\<^sub>R vector_derivative g (at (f x)) = vector_derivative (g \<circ> f) (at x)"
       by (metis (mono_tags, hide_lams) C1_differentiable_on_eq fg imageI vector_derivative_chain_at)
   qed
   ultimately show ?thesis
     by (simp add: C1_differentiable_on_eq)
 qed
 
 lemma C1_diff_imp_diff: "f C1_differentiable_on S \<Longrightarrow> f differentiable_on S"
   by (simp add: C1_differentiable_on_eq differentiable_at_imp_differentiable_on)
 
 lemma C1_differentiable_on_ident [simp, derivative_intros]: "(\<lambda>x. x) C1_differentiable_on S"
   by (auto simp: C1_differentiable_on_eq)
 
 lemma C1_differentiable_on_const [simp, derivative_intros]: "(\<lambda>z. a) C1_differentiable_on S"
   by (auto simp: C1_differentiable_on_eq)
 
 lemma C1_differentiable_on_add [simp, derivative_intros]:
   "f C1_differentiable_on S \<Longrightarrow> g C1_differentiable_on S \<Longrightarrow> (\<lambda>x. f x + g x) C1_differentiable_on S"
   unfolding C1_differentiable_on_eq  by (auto intro: continuous_intros)
 
 lemma C1_differentiable_on_minus [simp, derivative_intros]:
   "f C1_differentiable_on S \<Longrightarrow> (\<lambda>x. - f x) C1_differentiable_on S"
   unfolding C1_differentiable_on_eq  by (auto intro: continuous_intros)
 
 lemma C1_differentiable_on_diff [simp, derivative_intros]:
   "f C1_differentiable_on S \<Longrightarrow> g C1_differentiable_on S \<Longrightarrow> (\<lambda>x. f x - g x) C1_differentiable_on S"
   unfolding C1_differentiable_on_eq  by (auto intro: continuous_intros)
 
 lemma C1_differentiable_on_mult [simp, derivative_intros]:
   fixes f g :: "real \<Rightarrow> 'a :: real_normed_algebra"
   shows "f C1_differentiable_on S \<Longrightarrow> g C1_differentiable_on S \<Longrightarrow> (\<lambda>x. f x * g x) C1_differentiable_on S"
   unfolding C1_differentiable_on_eq
   by (auto simp: continuous_on_add continuous_on_mult continuous_at_imp_continuous_on differentiable_imp_continuous_within)
 
 lemma C1_differentiable_on_scaleR [simp, derivative_intros]:
   "f C1_differentiable_on S \<Longrightarrow> g C1_differentiable_on S \<Longrightarrow> (\<lambda>x. f x *\<^sub>R g x) C1_differentiable_on S"
   unfolding C1_differentiable_on_eq
   by (rule continuous_intros | simp add: continuous_at_imp_continuous_on differentiable_imp_continuous_within)+
 
 
 definition\<^marker>\<open>tag important\<close> piecewise_C1_differentiable_on
            (infixr "piecewise'_C1'_differentiable'_on" 50)
   where "f piecewise_C1_differentiable_on i  \<equiv>
            continuous_on i f \<and>
            (\<exists>S. finite S \<and> (f C1_differentiable_on (i - S)))"
 
 lemma C1_differentiable_imp_piecewise:
     "f C1_differentiable_on S \<Longrightarrow> f piecewise_C1_differentiable_on S"
   by (auto simp: piecewise_C1_differentiable_on_def C1_differentiable_on_eq continuous_at_imp_continuous_on differentiable_imp_continuous_within)
 
 lemma piecewise_C1_imp_differentiable:
     "f piecewise_C1_differentiable_on i \<Longrightarrow> f piecewise_differentiable_on i"
   by (auto simp: piecewise_C1_differentiable_on_def piecewise_differentiable_on_def
            C1_differentiable_on_def differentiable_def has_vector_derivative_def
            intro: has_derivative_at_withinI)
 
 lemma piecewise_C1_differentiable_compose:
   assumes fg: "f piecewise_C1_differentiable_on S" "g piecewise_C1_differentiable_on (f ` S)" and fin: "\<And>x. finite (S \<inter> f-`{x})"
   shows "(g \<circ> f) piecewise_C1_differentiable_on S"
 proof -
   have "continuous_on S (\<lambda>x. g (f x))"
     by (metis continuous_on_compose2 fg order_refl piecewise_C1_differentiable_on_def)
   moreover have "\<exists>T. finite T \<and> g \<circ> f C1_differentiable_on S - T"
   proof -
     obtain F where "finite F" and F: "f C1_differentiable_on S - F" and f: "f piecewise_C1_differentiable_on S"
       using fg by (auto simp: piecewise_C1_differentiable_on_def)
     obtain G where "finite G" and G: "g C1_differentiable_on f ` S - G" and g: "g piecewise_C1_differentiable_on f ` S"
       using fg by (auto simp: piecewise_C1_differentiable_on_def)
     show ?thesis
     proof (intro exI conjI)
       show "finite (F \<union> (\<Union>x\<in>G. S \<inter> f-`{x}))"
         using fin by (auto simp only: Int_Union \<open>finite F\<close> \<open>finite G\<close> finite_UN finite_imageI)
       show "g \<circ> f C1_differentiable_on S - (F \<union> (\<Union>x\<in>G. S \<inter> f -` {x}))"
         apply (rule C1_differentiable_compose)
           apply (blast intro: C1_differentiable_on_subset [OF F])
           apply (blast intro: C1_differentiable_on_subset [OF G])
         by (simp add:  C1_differentiable_on_subset G Diff_Int_distrib2 fin)
     qed
   qed
   ultimately show ?thesis
     by (simp add: piecewise_C1_differentiable_on_def)
 qed
 
 lemma piecewise_C1_differentiable_on_subset:
     "f piecewise_C1_differentiable_on S \<Longrightarrow> T \<le> S \<Longrightarrow> f piecewise_C1_differentiable_on T"
   by (auto simp: piecewise_C1_differentiable_on_def elim!: continuous_on_subset C1_differentiable_on_subset)
 
 lemma C1_differentiable_imp_continuous_on:
   "f C1_differentiable_on S \<Longrightarrow> continuous_on S f"
   unfolding C1_differentiable_on_eq continuous_on_eq_continuous_within
   using differentiable_at_withinI differentiable_imp_continuous_within by blast
 
 lemma C1_differentiable_on_empty [iff]: "f C1_differentiable_on {}"
   unfolding C1_differentiable_on_def
   by auto
 
 lemma piecewise_C1_differentiable_affine:
   fixes m::real
   assumes "f piecewise_C1_differentiable_on ((\<lambda>x. m * x + c) ` S)"
   shows "(f \<circ> (\<lambda>x. m *\<^sub>R x + c)) piecewise_C1_differentiable_on S"
 proof (cases "m = 0")
   case True
   then show ?thesis
     unfolding o_def by (auto simp: piecewise_C1_differentiable_on_def)
 next
   case False
   have *: "\<And>x. finite (S \<inter> {y. m * y + c = x})"
     using False not_finite_existsD by fastforce
   show ?thesis
     apply (rule piecewise_C1_differentiable_compose [OF C1_differentiable_imp_piecewise])
     apply (rule * assms derivative_intros | simp add: False vimage_def)+
     done
 qed
 
 lemma piecewise_C1_differentiable_cases:
   fixes c::real
   assumes "f piecewise_C1_differentiable_on {a..c}"
           "g piecewise_C1_differentiable_on {c..b}"
            "a \<le> c" "c \<le> b" "f c = g c"
   shows "(\<lambda>x. if x \<le> c then f x else g x) piecewise_C1_differentiable_on {a..b}"
 proof -
   obtain S T where st: "f C1_differentiable_on ({a..c} - S)"
                        "g C1_differentiable_on ({c..b} - T)"
                        "finite S" "finite T"
     using assms
     by (force simp: piecewise_C1_differentiable_on_def)
   then have f_diff: "f differentiable_on {a..<c} - S"
         and g_diff: "g differentiable_on {c<..b} - T"
     by (simp_all add: C1_differentiable_on_eq differentiable_at_withinI differentiable_on_def)
   have "continuous_on {a..c} f" "continuous_on {c..b} g"
     using assms piecewise_C1_differentiable_on_def by auto
   then have cab: "continuous_on {a..b} (\<lambda>x. if x \<le> c then f x else g x)"
     using continuous_on_cases [OF closed_real_atLeastAtMost [of a c],
                                OF closed_real_atLeastAtMost [of c b],
                                of f g "\<lambda>x. x\<le>c"]  assms
     by (force simp: ivl_disj_un_two_touch)
   { fix x
     assume x: "x \<in> {a..b} - insert c (S \<union> T)"
     have "(\<lambda>x. if x \<le> c then f x else g x) differentiable at x" (is "?diff_fg")
     proof (cases x c rule: le_cases)
       case le show ?diff_fg
         apply (rule differentiable_transform_within [where f=f and d = "dist x c"])
         using x dist_real_def le st by (auto simp: C1_differentiable_on_eq)
     next
       case ge show ?diff_fg
         apply (rule differentiable_transform_within [where f=g and d = "dist x c"])
         using dist_nz x dist_real_def ge st x by (auto simp: C1_differentiable_on_eq)
     qed
   }
   then have "(\<forall>x \<in> {a..b} - insert c (S \<union> T). (\<lambda>x. if x \<le> c then f x else g x) differentiable at x)"
     by auto
   moreover
   { assume fcon: "continuous_on ({a<..<c} - S) (\<lambda>x. vector_derivative f (at x))"
        and gcon: "continuous_on ({c<..<b} - T) (\<lambda>x. vector_derivative g (at x))"
     have "open ({a<..<c} - S)"  "open ({c<..<b} - T)"
       using st by (simp_all add: open_Diff finite_imp_closed)
     moreover have "continuous_on ({a<..<c} - S) (\<lambda>x. vector_derivative (\<lambda>x. if x \<le> c then f x else g x) (at x))"
     proof -
       have "((\<lambda>x. if x \<le> c then f x else g x) has_vector_derivative vector_derivative f (at x))            (at x)"
         if "a < x" "x < c" "x \<notin> S" for x
       proof -
         have f: "f differentiable at x"
           by (meson C1_differentiable_on_eq Diff_iff atLeastAtMost_iff less_eq_real_def st(1) that)
         show ?thesis
           using that
           apply (rule_tac f=f and d="dist x c" in has_vector_derivative_transform_within)
              apply (auto simp: dist_norm vector_derivative_works [symmetric] f)
           done
       qed
       then show ?thesis
         by (metis (no_types, lifting) continuous_on_eq [OF fcon] DiffE greaterThanLessThan_iff vector_derivative_at)
     qed
     moreover have "continuous_on ({c<..<b} - T) (\<lambda>x. vector_derivative (\<lambda>x. if x \<le> c then f x else g x) (at x))"
     proof -
       have "((\<lambda>x. if x \<le> c then f x else g x) has_vector_derivative vector_derivative g (at x))            (at x)"
         if "c < x" "x < b" "x \<notin> T" for x
       proof -
         have g: "g differentiable at x"
           by (metis C1_differentiable_on_eq DiffD1 DiffI atLeastAtMost_diff_ends greaterThanLessThan_iff st(2) that)
         show ?thesis
           using that
           apply (rule_tac f=g and d="dist x c" in has_vector_derivative_transform_within)
              apply (auto simp: dist_norm vector_derivative_works [symmetric] g)
           done
       qed
       then show ?thesis
         by (metis (no_types, lifting) continuous_on_eq [OF gcon] DiffE greaterThanLessThan_iff vector_derivative_at)
     qed
     ultimately have "continuous_on ({a<..<b} - insert c (S \<union> T))
         (\<lambda>x. vector_derivative (\<lambda>x. if x \<le> c then f x else g x) (at x))"
       by (rule continuous_on_subset [OF continuous_on_open_Un], auto)
   } note * = this
   have "continuous_on ({a<..<b} - insert c (S \<union> T)) (\<lambda>x. vector_derivative (\<lambda>x. if x \<le> c then f x else g x) (at x))"
     using st
     by (auto simp: C1_differentiable_on_eq elim!: continuous_on_subset intro: *)
   ultimately have "\<exists>S. finite S \<and> ((\<lambda>x. if x \<le> c then f x else g x) C1_differentiable_on {a..b} - S)"
     apply (rule_tac x="{a,b,c} \<union> S \<union> T" in exI)
     using st  by (auto simp: C1_differentiable_on_eq elim!: continuous_on_subset)
   with cab show ?thesis
     by (simp add: piecewise_C1_differentiable_on_def)
 qed
 
 lemma piecewise_C1_differentiable_neg:
     "f piecewise_C1_differentiable_on S \<Longrightarrow> (\<lambda>x. -(f x)) piecewise_C1_differentiable_on S"
   unfolding piecewise_C1_differentiable_on_def
   by (auto intro!: continuous_on_minus C1_differentiable_on_minus)
 
 lemma piecewise_C1_differentiable_add:
   assumes "f piecewise_C1_differentiable_on i"
           "g piecewise_C1_differentiable_on i"
     shows "(\<lambda>x. f x + g x) piecewise_C1_differentiable_on i"
 proof -
   obtain S t where st: "finite S" "finite t"
                        "f C1_differentiable_on (i-S)"
                        "g C1_differentiable_on (i-t)"
     using assms by (auto simp: piecewise_C1_differentiable_on_def)
   then have "finite (S \<union> t) \<and> (\<lambda>x. f x + g x) C1_differentiable_on i - (S \<union> t)"
     by (auto intro: C1_differentiable_on_add elim!: C1_differentiable_on_subset)
   moreover have "continuous_on i f" "continuous_on i g"
     using assms piecewise_C1_differentiable_on_def by auto
   ultimately show ?thesis
     by (auto simp: piecewise_C1_differentiable_on_def continuous_on_add)
 qed
 
 lemma piecewise_C1_differentiable_diff:
     "\<lbrakk>f piecewise_C1_differentiable_on S;  g piecewise_C1_differentiable_on S\<rbrakk>
      \<Longrightarrow> (\<lambda>x. f x - g x) piecewise_C1_differentiable_on S"
   unfolding diff_conv_add_uminus
   by (metis piecewise_C1_differentiable_add piecewise_C1_differentiable_neg)
 
 end
diff --git a/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy b/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy
--- a/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy
+++ b/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy
@@ -1,7664 +1,7591 @@
 (*  Author:     John Harrison
     Author:     Robert Himmelmann, TU Muenchen (Translation from HOL light)
                 Huge cleanup by LCP
 *)
 
 section \<open>Henstock-Kurzweil Gauge Integration in Many Dimensions\<close>
 
 theory Henstock_Kurzweil_Integration
 imports
   Lebesgue_Measure Tagged_Division
 begin
 
 lemma norm_diff2: "\<lbrakk>y = y1 + y2; x = x1 + x2; e = e1 + e2; norm(y1 - x1) \<le> e1; norm(y2 - x2) \<le> e2\<rbrakk>
   \<Longrightarrow> norm(y-x) \<le> e"
   using norm_triangle_mono [of "y1 - x1" "e1" "y2 - x2" "e2"]
   by (simp add: add_diff_add)
 
 lemma setcomp_dot1: "{z. P (z \<bullet> (i,0))} = {(x,y). P(x \<bullet> i)}"
   by auto
 
 lemma setcomp_dot2: "{z. P (z \<bullet> (0,i))} = {(x,y). P(y \<bullet> i)}"
   by auto
 
 lemma Sigma_Int_Paircomp1: "(Sigma A B) \<inter> {(x, y). P x} = Sigma (A \<inter> {x. P x}) B"
   by blast
 
 lemma Sigma_Int_Paircomp2: "(Sigma A B) \<inter> {(x, y). P y} = Sigma A (\<lambda>z. B z \<inter> {y. P y})"
   by blast
 (* END MOVE *)
 
 subsection \<open>Content (length, area, volume...) of an interval\<close>
 
 abbreviation content :: "'a::euclidean_space set \<Rightarrow> real"
   where "content s \<equiv> measure lborel s"
 
 lemma content_cbox_cases:
   "content (cbox a b) = (if \<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i then prod (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis else 0)"
   by (simp add: measure_lborel_cbox_eq inner_diff)
 
 lemma content_cbox: "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow> content (cbox a b) = (\<Prod>i\<in>Basis. b\<bullet>i - a\<bullet>i)"
   unfolding content_cbox_cases by simp
 
 lemma content_cbox': "cbox a b \<noteq> {} \<Longrightarrow> content (cbox a b) = (\<Prod>i\<in>Basis. b\<bullet>i - a\<bullet>i)"
   by (simp add: box_ne_empty inner_diff)
 
 lemma content_cbox_if: "content (cbox a b) = (if cbox a b = {} then 0 else \<Prod>i\<in>Basis. b\<bullet>i - a\<bullet>i)"
   by (simp add: content_cbox')
 
 lemma content_cbox_cart:
    "cbox a b \<noteq> {} \<Longrightarrow> content(cbox a b) = prod (\<lambda>i. b$i - a$i) UNIV"
   by (simp add: content_cbox_if Basis_vec_def cart_eq_inner_axis axis_eq_axis prod.UNION_disjoint)
 
 lemma content_cbox_if_cart:
    "content(cbox a b) = (if cbox a b = {} then 0 else prod (\<lambda>i. b$i - a$i) UNIV)"
   by (simp add: content_cbox_cart)
 
 lemma content_division_of:
   assumes "K \<in> \<D>" "\<D> division_of S"
   shows "content K = (\<Prod>i \<in> Basis. interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)"
 proof -
   obtain a b where "K = cbox a b"
     using cbox_division_memE assms by metis
   then show ?thesis
     using assms by (force simp: division_of_def content_cbox')
 qed
 
 lemma content_real: "a \<le> b \<Longrightarrow> content {a..b} = b - a"
   by simp
 
 lemma abs_eq_content: "\<bar>y - x\<bar> = (if x\<le>y then content {x..y} else content {y..x})"
   by (auto simp: content_real)
 
 lemma content_singleton: "content {a} = 0"
   by simp
 
 lemma content_unit[iff]: "content (cbox 0 (One::'a::euclidean_space)) = 1"
   by simp
 
 lemma content_pos_le [iff]: "0 \<le> content X"
   by simp
 
 corollary\<^marker>\<open>tag unimportant\<close> content_nonneg [simp]: "\<not> content (cbox a b) < 0"
   using not_le by blast
 
 lemma content_pos_lt: "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i \<Longrightarrow> 0 < content (cbox a b)"
   by (auto simp: less_imp_le inner_diff box_eq_empty intro!: prod_pos)
 
 lemma content_eq_0: "content (cbox a b) = 0 \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i \<le> a\<bullet>i)"
   by (auto simp: content_cbox_cases not_le intro: less_imp_le antisym eq_refl)
 
 lemma content_eq_0_interior: "content (cbox a b) = 0 \<longleftrightarrow> interior(cbox a b) = {}"
   unfolding content_eq_0 interior_cbox box_eq_empty by auto
 
 lemma content_pos_lt_eq: "0 < content (cbox a (b::'a::euclidean_space)) \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i)"
   by (auto simp add: content_cbox_cases less_le prod_nonneg)
 
 lemma content_empty [simp]: "content {} = 0"
   by simp
 
 lemma content_real_if [simp]: "content {a..b} = (if a \<le> b then b - a else 0)"
   by (simp add: content_real)
 
 lemma content_subset: "cbox a b \<subseteq> cbox c d \<Longrightarrow> content (cbox a b) \<le> content (cbox c d)"
   unfolding measure_def
   by (intro enn2real_mono emeasure_mono) (auto simp: emeasure_lborel_cbox_eq)
 
 lemma content_lt_nz: "0 < content (cbox a b) \<longleftrightarrow> content (cbox a b) \<noteq> 0"
   unfolding content_pos_lt_eq content_eq_0 unfolding not_ex not_le by fastforce
 
 lemma content_Pair: "content (cbox (a,c) (b,d)) = content (cbox a b) * content (cbox c d)"
   unfolding measure_lborel_cbox_eq Basis_prod_def
   apply (subst prod.union_disjoint)
   apply (auto simp: bex_Un ball_Un)
   apply (subst (1 2) prod.reindex_nontrivial)
   apply auto
   done
 
 lemma content_cbox_pair_eq0_D:
    "content (cbox (a,c) (b,d)) = 0 \<Longrightarrow> content (cbox a b) = 0 \<or> content (cbox c d) = 0"
   by (simp add: content_Pair)
 
 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 content_0_subset: "content(cbox a b) = 0 \<Longrightarrow> s \<subseteq> cbox a b \<Longrightarrow> content s = 0"
   using emeasure_mono[of s "cbox a b" lborel]
   by (auto simp: measure_def enn2real_eq_0_iff emeasure_lborel_cbox_eq)
 
 lemma content_ball_pos:
   assumes "r > 0"
   shows   "content (ball c r) > 0"
 proof -
   from rational_boxes[OF assms, of c] obtain a b where ab: "c \<in> box a b" "box a b \<subseteq> ball c r"
     by auto
   from ab have "0 < content (box a b)"
     by (subst measure_lborel_box_eq) (auto intro!: prod_pos simp: algebra_simps box_def)
   have "emeasure lborel (box a b) \<le> emeasure lborel (ball c r)"
     using ab by (intro emeasure_mono) auto
   also have "emeasure lborel (box a b) = ennreal (content (box a b))"
     using emeasure_lborel_box_finite[of a b] by (intro emeasure_eq_ennreal_measure) auto
   also have "emeasure lborel (ball c r) = ennreal (content (ball c r))"
     using emeasure_lborel_ball_finite[of c r] by (intro emeasure_eq_ennreal_measure) auto
   finally show ?thesis
     using \<open>content (box a b) > 0\<close> by simp
 qed
 
 lemma content_cball_pos:
   assumes "r > 0"
   shows   "content (cball c r) > 0"
 proof -
   from rational_boxes[OF assms, of c] obtain a b where ab: "c \<in> box a b" "box a b \<subseteq> ball c r"
     by auto
   from ab have "0 < content (box a b)"
     by (subst measure_lborel_box_eq) (auto intro!: prod_pos simp: algebra_simps box_def)
   have "emeasure lborel (box a b) \<le> emeasure lborel (ball c r)"
     using ab by (intro emeasure_mono) auto
   also have "\<dots> \<le> emeasure lborel (cball c r)"
     by (intro emeasure_mono) auto
   also have "emeasure lborel (box a b) = ennreal (content (box a b))"
     using emeasure_lborel_box_finite[of a b] by (intro emeasure_eq_ennreal_measure) auto
   also have "emeasure lborel (cball c r) = ennreal (content (cball c r))"
     using emeasure_lborel_cball_finite[of c r] by (intro emeasure_eq_ennreal_measure) auto
   finally show ?thesis
     using \<open>content (box a b) > 0\<close> by simp
 qed
 
 lemma content_split:
   fixes a :: "'a::euclidean_space"
   assumes "k \<in> Basis"
   shows "content (cbox a b) = content(cbox a b \<inter> {x. x\<bullet>k \<le> c}) + content(cbox a b \<inter> {x. x\<bullet>k \<ge> c})"
   \<comment> \<open>Prove using measure theory\<close>
 proof (cases "\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i")
   case True
   have 1: "\<And>X Y Z. (\<Prod>i\<in>Basis. Z i (if i = k then X else Y i)) = Z k X * (\<Prod>i\<in>Basis-{k}. Z i (Y i))"
     by (simp add: if_distrib prod.delta_remove assms)
   note simps = interval_split[OF assms] content_cbox_cases
   have 2: "(\<Prod>i\<in>Basis. b\<bullet>i - a\<bullet>i) = (\<Prod>i\<in>Basis-{k}. b\<bullet>i - a\<bullet>i) * (b\<bullet>k - a\<bullet>k)"
     by (metis (no_types, lifting) assms finite_Basis mult.commute prod.remove)
   have "\<And>x. min (b \<bullet> k) c = max (a \<bullet> k) c \<Longrightarrow>
     x * (b\<bullet>k - a\<bullet>k) = x * (max (a \<bullet> k) c - a \<bullet> k) + x * (b \<bullet> k - max (a \<bullet> k) c)"
     by  (auto simp add: field_simps)
   moreover
   have **: "(\<Prod>i\<in>Basis. ((\<Sum>i\<in>Basis. (if i = k then min (b \<bullet> k) c else b \<bullet> i) *\<^sub>R i) \<bullet> i - a \<bullet> i)) =
       (\<Prod>i\<in>Basis. (if i = k then min (b \<bullet> k) c else b \<bullet> i) - a \<bullet> i)"
     "(\<Prod>i\<in>Basis. b \<bullet> i - ((\<Sum>i\<in>Basis. (if i = k then max (a \<bullet> k) c else a \<bullet> i) *\<^sub>R i) \<bullet> i)) =
       (\<Prod>i\<in>Basis. b \<bullet> i - (if i = k then max (a \<bullet> k) c else a \<bullet> i))"
     by (auto intro!: prod.cong)
   have "\<not> a \<bullet> k \<le> c \<Longrightarrow> \<not> c \<le> b \<bullet> k \<Longrightarrow> False"
     unfolding not_le using True assms by auto
   ultimately show ?thesis
     using assms unfolding simps ** 1[of "\<lambda>i x. b\<bullet>i - x"] 1[of "\<lambda>i x. x - a\<bullet>i"] 2
     by auto
 next
   case False
   then have "cbox a b = {}"
     unfolding box_eq_empty by (auto simp: not_le)
   then show ?thesis
     by (auto simp: not_le)
 qed
 
 lemma division_of_content_0:
   assumes "content (cbox a b) = 0" "d division_of (cbox a b)" "K \<in> d"
   shows "content K = 0"
   unfolding forall_in_division[OF assms(2)]
   by (meson assms content_0_subset division_of_def)
 
 lemma sum_content_null:
   assumes "content (cbox a b) = 0"
     and "p tagged_division_of (cbox a b)"
   shows "(\<Sum>(x,K)\<in>p. content K *\<^sub>R f x) = (0::'a::real_normed_vector)"
 proof (rule sum.neutral, rule)
   fix y
   assume y: "y \<in> p"
   obtain x K where xk: "y = (x, K)"
     using surj_pair[of y] by blast
   then obtain c d where k: "K = cbox c d" "K \<subseteq> cbox a b"
     by (metis assms(2) tagged_division_ofD(3) tagged_division_ofD(4) y)
   have "(\<lambda>(x',K'). content K' *\<^sub>R f x') y = content K *\<^sub>R f x"
     unfolding xk by auto
   also have "\<dots> = 0"
     using assms(1) content_0_subset k(2) by auto
   finally show "(\<lambda>(x, k). content k *\<^sub>R f x) y = 0" .
 qed
 
 global_interpretation sum_content: operative plus 0 content
   rewrites "comm_monoid_set.F plus 0 = sum"
 proof -
   interpret operative plus 0 content
     by standard (auto simp add: content_split [symmetric] content_eq_0_interior)
   show "operative plus 0 content"
     by standard
   show "comm_monoid_set.F plus 0 = sum"
     by (simp add: sum_def)
 qed
 
 lemma additive_content_division: "d division_of (cbox a b) \<Longrightarrow> sum content d = content (cbox a b)"
   by (fact sum_content.division)
 
 lemma additive_content_tagged_division:
   "d tagged_division_of (cbox a b) \<Longrightarrow> sum (\<lambda>(x,l). content l) d = content (cbox a b)"
   by (fact sum_content.tagged_division)
 
 lemma subadditive_content_division:
   assumes "\<D> division_of S" "S \<subseteq> cbox a b"
   shows "sum content \<D> \<le> content(cbox a b)"
 proof -
   have "\<D> division_of \<Union>\<D>" "\<Union>\<D> \<subseteq> cbox a b"
     using assms by auto
   then obtain \<D>' where "\<D> \<subseteq> \<D>'" "\<D>' division_of cbox a b"
     using partial_division_extend_interval by metis
   then have "sum content \<D> \<le> sum content \<D>'"
     using sum_mono2 by blast
   also have "... \<le> content(cbox a b)"
     by (simp add: \<open>\<D>' division_of cbox a b\<close> additive_content_division less_eq_real_def)
   finally show ?thesis .
 qed
 
 lemma content_real_eq_0: "content {a..b::real} = 0 \<longleftrightarrow> a \<ge> b"
   by (metis atLeastatMost_empty_iff2 content_empty content_real diff_self eq_iff le_cases le_iff_diff_le_0)
 
 lemma property_empty_interval: "\<forall>a b. content (cbox a b) = 0 \<longrightarrow> P (cbox a b) \<Longrightarrow> P {}"
   using content_empty unfolding empty_as_interval by auto
 
 lemma interval_bounds_nz_content [simp]:
   assumes "content (cbox a b) \<noteq> 0"
   shows "interval_upperbound (cbox a b) = b"
     and "interval_lowerbound (cbox a b) = a"
   by (metis assms content_empty interval_bounds')+
 
 subsection \<open>Gauge integral\<close>
 
 text \<open>Case distinction to define it first on compact intervals first, then use a limit. This is only
 much later unified. In Fremlin: Measure Theory, Volume 4I this is generalized using residual sets.\<close>
 
 definition has_integral :: "('n::euclidean_space \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'b \<Rightarrow> 'n set \<Rightarrow> bool"
   (infixr "has'_integral" 46)
   where "(f has_integral I) s \<longleftrightarrow>
     (if \<exists>a b. s = cbox a b
       then ((\<lambda>p. \<Sum>(x,k)\<in>p. content k *\<^sub>R f x) \<longlongrightarrow> I) (division_filter s)
       else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
         (\<exists>z. ((\<lambda>p. \<Sum>(x,k)\<in>p. content k *\<^sub>R (if x \<in> s then f x else 0)) \<longlongrightarrow> z) (division_filter (cbox a b)) \<and>
           norm (z - I) < e)))"
 
 lemma has_integral_cbox:
   "(f has_integral I) (cbox a b) \<longleftrightarrow> ((\<lambda>p. \<Sum>(x,k)\<in>p. content k *\<^sub>R f x) \<longlongrightarrow> I) (division_filter (cbox a b))"
   by (auto simp add: has_integral_def)
 
 lemma has_integral:
   "(f has_integral y) (cbox a b) \<longleftrightarrow>
     (\<forall>e>0. \<exists>\<gamma>. gauge \<gamma> \<and>
       (\<forall>\<D>. \<D> tagged_division_of (cbox a b) \<and> \<gamma> fine \<D> \<longrightarrow>
         norm (sum (\<lambda>(x,k). content(k) *\<^sub>R f x) \<D> - y) < e))"
   by (auto simp: dist_norm eventually_division_filter has_integral_def tendsto_iff)
 
 lemma has_integral_real:
   "(f has_integral y) {a..b::real} \<longleftrightarrow>
     (\<forall>e>0. \<exists>\<gamma>. gauge \<gamma> \<and>
       (\<forall>\<D>. \<D> tagged_division_of {a..b} \<and> \<gamma> fine \<D> \<longrightarrow>
         norm (sum (\<lambda>(x,k). content(k) *\<^sub>R f x) \<D> - y) < e))"
   unfolding box_real[symmetric] by (rule has_integral)
 
 lemma has_integralD[dest]:
   assumes "(f has_integral y) (cbox a b)"
     and "e > 0"
   obtains \<gamma>
     where "gauge \<gamma>"
       and "\<And>\<D>. \<D> tagged_division_of (cbox a b) \<Longrightarrow> \<gamma> fine \<D> \<Longrightarrow>
         norm ((\<Sum>(x,k)\<in>\<D>. content k *\<^sub>R f x) - y) < e"
   using assms unfolding has_integral by auto
 
 lemma has_integral_alt:
   "(f has_integral y) i \<longleftrightarrow>
     (if \<exists>a b. i = cbox a b
      then (f has_integral y) i
      else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
       (\<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral z) (cbox a b) \<and> norm (z - y) < e)))"
   by (subst has_integral_def) (auto simp add: has_integral_cbox)
 
 lemma has_integral_altD:
   assumes "(f has_integral y) i"
     and "\<not> (\<exists>a b. i = cbox a b)"
     and "e>0"
   obtains B where "B > 0"
     and "\<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
       (\<exists>z. ((\<lambda>x. if x \<in> i then f(x) else 0) has_integral z) (cbox a b) \<and> norm(z - y) < e)"
   using assms has_integral_alt[of f y i] by auto
 
 definition integrable_on (infixr "integrable'_on" 46)
   where "f integrable_on i \<longleftrightarrow> (\<exists>y. (f has_integral y) i)"
 
 definition "integral i f = (SOME y. (f has_integral y) i \<or> \<not> f integrable_on i \<and> y=0)"
 
 lemma integrable_integral[intro]: "f integrable_on i \<Longrightarrow> (f has_integral (integral i f)) i"
   unfolding integrable_on_def integral_def by (metis (mono_tags, lifting) someI_ex)
 
 lemma not_integrable_integral: "\<not> f integrable_on i \<Longrightarrow> integral i f = 0"
   unfolding integrable_on_def integral_def by blast
 
 lemma has_integral_integrable[dest]: "(f has_integral i) s \<Longrightarrow> f integrable_on s"
   unfolding integrable_on_def by auto
 
 lemma has_integral_integral: "f integrable_on s \<longleftrightarrow> (f has_integral (integral s f)) s"
   by auto
 
 subsection \<open>Basic theorems about integrals\<close>
 
 lemma has_integral_eq_rhs: "(f has_integral j) S \<Longrightarrow> i = j \<Longrightarrow> (f has_integral i) S"
   by (rule forw_subst)
 
 lemma has_integral_unique_cbox:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
   shows "(f has_integral k1) (cbox a b) \<Longrightarrow> (f has_integral k2) (cbox a b) \<Longrightarrow> k1 = k2"
     by (auto simp: has_integral_cbox intro: tendsto_unique[OF division_filter_not_empty])    
 
 lemma has_integral_unique:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
   assumes "(f has_integral k1) i" "(f has_integral k2) i"
   shows "k1 = k2"
 proof (rule ccontr)
   let ?e = "norm (k1 - k2)/2"
   let ?F = "(\<lambda>x. if x \<in> i then f x else 0)"
   assume "k1 \<noteq> k2"
   then have e: "?e > 0"
     by auto
   have nonbox: "\<not> (\<exists>a b. i = cbox a b)"
     using \<open>k1 \<noteq> k2\<close> assms has_integral_unique_cbox by blast
   obtain B1 where B1:
       "0 < B1"
       "\<And>a b. ball 0 B1 \<subseteq> cbox a b \<Longrightarrow>
         \<exists>z. (?F has_integral z) (cbox a b) \<and> norm (z - k1) < norm (k1 - k2)/2"
     by (rule has_integral_altD[OF assms(1) nonbox,OF e]) blast
   obtain B2 where B2:
       "0 < B2"
       "\<And>a b. ball 0 B2 \<subseteq> cbox a b \<Longrightarrow>
         \<exists>z. (?F has_integral z) (cbox a b) \<and> norm (z - k2) < norm (k1 - k2)/2"
     by (rule has_integral_altD[OF assms(2) nonbox,OF e]) blast
   obtain a b :: 'n where ab: "ball 0 B1 \<subseteq> cbox a b" "ball 0 B2 \<subseteq> cbox a b"
     by (metis Un_subset_iff bounded_Un bounded_ball bounded_subset_cbox_symmetric)
   obtain w where w: "(?F has_integral w) (cbox a b)" "norm (w - k1) < norm (k1 - k2)/2"
     using B1(2)[OF ab(1)] by blast
   obtain z where z: "(?F has_integral z) (cbox a b)" "norm (z - k2) < norm (k1 - k2)/2"
     using B2(2)[OF ab(2)] by blast
   have "z = w"
     using has_integral_unique_cbox[OF w(1) z(1)] by auto
   then have "norm (k1 - k2) \<le> norm (z - k2) + norm (w - k1)"
     using norm_triangle_ineq4 [of "k1 - w" "k2 - z"]
     by (auto simp add: norm_minus_commute)
   also have "\<dots> < norm (k1 - k2)/2 + norm (k1 - k2)/2"
     by (metis add_strict_mono z(2) w(2))
   finally show False by auto
 qed
 
 lemma integral_unique [intro]: "(f has_integral y) k \<Longrightarrow> integral k f = y"
   unfolding integral_def
   by (rule some_equality) (auto intro: has_integral_unique)
 
 lemma has_integral_iff: "(f has_integral i) S \<longleftrightarrow> (f integrable_on S \<and> integral S f = i)"
   by blast
 
 lemma eq_integralD: "integral k f = y \<Longrightarrow> (f has_integral y) k \<or> \<not> f integrable_on k \<and> y=0"
   unfolding integral_def integrable_on_def
   apply (erule subst)
   apply (rule someI_ex)
   by blast
 
 lemma has_integral_const [intro]:
   fixes a b :: "'a::euclidean_space"
   shows "((\<lambda>x. c) has_integral (content (cbox a b) *\<^sub>R c)) (cbox a b)"
   using eventually_division_filter_tagged_division[of "cbox a b"]
      additive_content_tagged_division[of _ a b]
   by (auto simp: has_integral_cbox split_beta' scaleR_sum_left[symmetric]
            elim!: eventually_mono intro!: tendsto_cong[THEN iffD1, OF _ tendsto_const])
 
 lemma has_integral_const_real [intro]:
   fixes a b :: real
   shows "((\<lambda>x. c) has_integral (content {a..b} *\<^sub>R c)) {a..b}"
   by (metis box_real(2) has_integral_const)
 
 lemma has_integral_integrable_integral: "(f has_integral i) s \<longleftrightarrow> f integrable_on s \<and> integral s f = i"
   by blast
 
 lemma integral_const [simp]:
   fixes a b :: "'a::euclidean_space"
   shows "integral (cbox a b) (\<lambda>x. c) = content (cbox a b) *\<^sub>R c"
   by (rule integral_unique) (rule has_integral_const)
 
 lemma integral_const_real [simp]:
   fixes a b :: real
   shows "integral {a..b} (\<lambda>x. c) = content {a..b} *\<^sub>R c"
   by (metis box_real(2) integral_const)
 
 lemma has_integral_is_0_cbox:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
   assumes "\<And>x. x \<in> cbox a b \<Longrightarrow> f x = 0"
   shows "(f has_integral 0) (cbox a b)"
     unfolding has_integral_cbox
     using eventually_division_filter_tagged_division[of "cbox a b"] assms
     by (subst tendsto_cong[where g="\<lambda>_. 0"])
        (auto elim!: eventually_mono intro!: sum.neutral simp: tag_in_interval)
 
 lemma has_integral_is_0:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
   assumes "\<And>x. x \<in> S \<Longrightarrow> f x = 0"
   shows "(f has_integral 0) S"
 proof (cases "(\<exists>a b. S = cbox a b)")
   case True with assms has_integral_is_0_cbox show ?thesis
     by blast
 next
   case False
   have *: "(\<lambda>x. if x \<in> S then f x else 0) = (\<lambda>x. 0)"
     by (auto simp add: assms)
   show ?thesis
     using has_integral_is_0_cbox False
     by (subst has_integral_alt) (force simp add: *)
 qed
 
 lemma has_integral_0[simp]: "((\<lambda>x::'n::euclidean_space. 0) has_integral 0) S"
   by (rule has_integral_is_0) auto
 
 lemma has_integral_0_eq[simp]: "((\<lambda>x. 0) has_integral i) S \<longleftrightarrow> i = 0"
   using has_integral_unique[OF has_integral_0] by auto
 
 lemma has_integral_linear_cbox:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
   assumes f: "(f has_integral y) (cbox a b)"
     and h: "bounded_linear h"
   shows "((h \<circ> f) has_integral (h y)) (cbox a b)"
 proof -
   interpret bounded_linear h using h .
   show ?thesis
     unfolding has_integral_cbox using tendsto [OF f [unfolded has_integral_cbox]]
     by (simp add: sum scaleR split_beta')
 qed
 
 lemma has_integral_linear:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
   assumes f: "(f has_integral y) S"
     and h: "bounded_linear h"
   shows "((h \<circ> f) has_integral (h y)) S"
 proof (cases "(\<exists>a b. S = cbox a b)")
   case True with f h has_integral_linear_cbox show ?thesis 
     by blast
 next
   case False
   interpret bounded_linear h using h .
   from pos_bounded obtain B where B: "0 < B" "\<And>x. norm (h x) \<le> norm x * B"
     by blast
   let ?S = "\<lambda>f x. if x \<in> S then f x else 0"
   show ?thesis
   proof (subst has_integral_alt, clarsimp simp: False)
     fix e :: real
     assume e: "e > 0"
     have *: "0 < e/B" using e B(1) by simp
     obtain M where M:
       "M > 0"
       "\<And>a b. ball 0 M \<subseteq> cbox a b \<Longrightarrow>
         \<exists>z. (?S f has_integral z) (cbox a b) \<and> norm (z - y) < e/B"
       using has_integral_altD[OF f False *] by blast
     show "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
       (\<exists>z. (?S(h \<circ> f) has_integral z) (cbox a b) \<and> norm (z - h y) < e)"
     proof (rule exI, intro allI conjI impI)
       show "M > 0" using M by metis
     next
       fix a b::'n
       assume sb: "ball 0 M \<subseteq> cbox a b"
       obtain z where z: "(?S f has_integral z) (cbox a b)" "norm (z - y) < e/B"
         using M(2)[OF sb] by blast
       have *: "?S(h \<circ> f) = h \<circ> ?S f"
         using zero by auto
       show "\<exists>z. (?S(h \<circ> f) has_integral z) (cbox a b) \<and> norm (z - h y) < e"
-        apply (rule_tac x="h z" in exI)
-        apply (simp add: * has_integral_linear_cbox[OF z(1) h])
-        apply (metis B diff le_less_trans pos_less_divide_eq z(2))
-        done
+      proof (intro exI conjI)
+        show "(?S(h \<circ> f) has_integral h z) (cbox a b)"
+          by (simp add: * has_integral_linear_cbox[OF z(1) h])
+        show "norm (h z - h y) < e"
+          by (metis B diff le_less_trans pos_less_divide_eq z(2))
+      qed
     qed
   qed
 qed
 
 lemma has_integral_scaleR_left:
   "(f has_integral y) S \<Longrightarrow> ((\<lambda>x. f x *\<^sub>R c) has_integral (y *\<^sub>R c)) S"
   using has_integral_linear[OF _ bounded_linear_scaleR_left] by (simp add: comp_def)
 
 lemma integrable_on_scaleR_left:
   assumes "f integrable_on A"
   shows "(\<lambda>x. f x *\<^sub>R y) integrable_on A"
   using assms has_integral_scaleR_left unfolding integrable_on_def by blast
 
 lemma has_integral_mult_left:
   fixes c :: "_ :: real_normed_algebra"
   shows "(f has_integral y) S \<Longrightarrow> ((\<lambda>x. f x * c) has_integral (y * c)) S"
   using has_integral_linear[OF _ bounded_linear_mult_left] by (simp add: comp_def)
 
 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)
 
 text\<open>The case analysis eliminates the condition \<^term>\<open>f integrable_on S\<close> at the cost
      of the type class constraint \<open>division_ring\<close>\<close>
 corollary integral_mult_left [simp]:
   fixes c:: "'a::{real_normed_algebra,division_ring}"
   shows "integral S (\<lambda>x. f x * c) = integral S f * c"
 proof (cases "f integrable_on S \<or> c = 0")
   case True then show ?thesis
     by (force intro: has_integral_mult_left)
 next
   case False then have "\<not> (\<lambda>x. f x * c) integrable_on S"
     using has_integral_mult_left [of "(\<lambda>x. f x * c)" _ S "inverse c"]
     by (auto simp add: mult.assoc)
   with False show ?thesis by (simp add: not_integrable_integral)
 qed
 
 corollary integral_mult_right [simp]:
   fixes c:: "'a::{real_normed_field}"
   shows "integral S (\<lambda>x. c * f x) = c * integral S f"
 by (simp add: mult.commute [of c])
 
 corollary integral_divide [simp]:
   fixes z :: "'a::real_normed_field"
   shows "integral S (\<lambda>x. f x / z) = integral S (\<lambda>x. f x) / z"
 using integral_mult_left [of S f "inverse z"]
   by (simp add: divide_inverse_commute)
 
 lemma has_integral_mult_right:
   fixes c :: "'a :: real_normed_algebra"
   shows "(f has_integral y) i \<Longrightarrow> ((\<lambda>x. c * f x) has_integral (c * y)) i"
   using has_integral_linear[OF _ bounded_linear_mult_right] by (simp add: comp_def)
 
 lemma has_integral_cmul: "(f has_integral k) S \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_integral (c *\<^sub>R k)) S"
   unfolding o_def[symmetric]
   by (metis has_integral_linear bounded_linear_scaleR_right)
 
 lemma has_integral_cmult_real:
   fixes c :: real
   assumes "c \<noteq> 0 \<Longrightarrow> (f has_integral x) A"
   shows "((\<lambda>x. c * f x) has_integral c * x) A"
 proof (cases "c = 0")
   case True
   then show ?thesis by simp
 next
   case False
   from has_integral_cmul[OF assms[OF this], of c] show ?thesis
     unfolding real_scaleR_def .
 qed
 
 lemma has_integral_neg: "(f has_integral k) S \<Longrightarrow> ((\<lambda>x. -(f x)) has_integral -k) S"
   by (drule_tac c="-1" in has_integral_cmul) auto
 
 lemma has_integral_neg_iff: "((\<lambda>x. - f x) has_integral k) S \<longleftrightarrow> (f has_integral - k) S"
   using has_integral_neg[of f "- k"] has_integral_neg[of "\<lambda>x. - f x" k] by auto
 
 lemma has_integral_add_cbox:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
   assumes "(f has_integral k) (cbox a b)" "(g has_integral l) (cbox a b)"
   shows "((\<lambda>x. f x + g x) has_integral (k + l)) (cbox a b)"
   using assms
     unfolding has_integral_cbox
     by (simp add: split_beta' scaleR_add_right sum.distrib[abs_def] tendsto_add)
 
 lemma has_integral_add:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
   assumes f: "(f has_integral k) S" and g: "(g has_integral l) S"
   shows "((\<lambda>x. f x + g x) has_integral (k + l)) S"
 proof (cases "\<exists>a b. S = cbox a b")
   case True with has_integral_add_cbox assms show ?thesis
     by blast 
 next
   let ?S = "\<lambda>f x. if x \<in> S then f x else 0"
   case False
   then show ?thesis
   proof (subst has_integral_alt, clarsimp, goal_cases)
     case (1 e)
     then have e2: "e/2 > 0"
       by auto
     obtain Bf where "0 < Bf"
       and Bf: "\<And>a b. ball 0 Bf \<subseteq> cbox a b \<Longrightarrow>
                      \<exists>z. (?S f has_integral z) (cbox a b) \<and> norm (z - k) < e/2"
       using has_integral_altD[OF f False e2] by blast
     obtain Bg where "0 < Bg"
       and Bg: "\<And>a b. ball 0 Bg \<subseteq> (cbox a b) \<Longrightarrow>
                      \<exists>z. (?S g has_integral z) (cbox a b) \<and> norm (z - l) < e/2"
       using has_integral_altD[OF g False e2] by blast
     show ?case
     proof (rule_tac x="max Bf Bg" in exI, clarsimp simp add: max.strict_coboundedI1 \<open>0 < Bf\<close>)
       fix a b
       assume "ball 0 (max Bf Bg) \<subseteq> cbox a (b::'n)"
       then have fs: "ball 0 Bf \<subseteq> cbox a (b::'n)" and gs: "ball 0 Bg \<subseteq> cbox a (b::'n)"
         by auto
       obtain w where w: "(?S f has_integral w) (cbox a b)" "norm (w - k) < e/2"
         using Bf[OF fs] by blast
       obtain z where z: "(?S g has_integral z) (cbox a b)" "norm (z - l) < e/2"
         using Bg[OF gs] by blast
       have *: "\<And>x. (if x \<in> S then f x + g x else 0) = (?S f x) + (?S g x)"
         by auto
       show "\<exists>z. (?S(\<lambda>x. f x + g x) has_integral z) (cbox a b) \<and> norm (z - (k + l)) < e"
-        apply (rule_tac x="w + z" in exI)
-        apply (simp add: has_integral_add_cbox[OF w(1) z(1), unfolded *[symmetric]])
-        using norm_triangle_ineq[of "w - k" "z - l"] w(2) z(2)
-        apply (auto simp add: field_simps)
-        done
+      proof (intro exI conjI)
+        show "(?S(\<lambda>x. f x + g x) has_integral (w + z)) (cbox a b)"
+          by (simp add: has_integral_add_cbox[OF w(1) z(1), unfolded *[symmetric]])
+        show "norm (w + z - (k + l)) < e"
+          by (metis dist_norm dist_triangle_add_half w(2) z(2))
+      qed
     qed
   qed
 qed
 
 lemma has_integral_diff:
   "(f has_integral k) S \<Longrightarrow> (g has_integral l) S \<Longrightarrow>
     ((\<lambda>x. f x - g x) has_integral (k - l)) S"
   using has_integral_add[OF _ has_integral_neg, of f k S g l]
   by (auto simp: algebra_simps)
 
 lemma integral_0 [simp]:
   "integral S (\<lambda>x::'n::euclidean_space. 0::'m::real_normed_vector) = 0"
   by (rule integral_unique has_integral_0)+
 
 lemma integral_add: "f integrable_on S \<Longrightarrow> g integrable_on S \<Longrightarrow>
     integral S (\<lambda>x. f x + g x) = integral S f + integral S g"
   by (rule integral_unique) (metis integrable_integral has_integral_add)
 
 lemma integral_cmul [simp]: "integral S (\<lambda>x. c *\<^sub>R f x) = c *\<^sub>R integral S f"
 proof (cases "f integrable_on S \<or> c = 0")
   case True with has_integral_cmul integrable_integral show ?thesis
     by fastforce
 next
   case False then have "\<not> (\<lambda>x. c *\<^sub>R f x) integrable_on S"
     using has_integral_cmul [of "(\<lambda>x. c *\<^sub>R f x)" _ S "inverse c"] by auto
   with False show ?thesis by (simp add: not_integrable_integral)
 qed
 
 lemma integral_mult:
   fixes K::real
   shows "f integrable_on X \<Longrightarrow> K * integral X f = integral X (\<lambda>x. K * f x)"
   unfolding real_scaleR_def[symmetric] integral_cmul ..
 
 lemma integral_neg [simp]: "integral S (\<lambda>x. - f x) = - integral S f"
 proof (cases "f integrable_on S")
   case True then show ?thesis
     by (simp add: has_integral_neg integrable_integral integral_unique)
 next
   case False then have "\<not> (\<lambda>x. - f x) integrable_on S"
     using has_integral_neg [of "(\<lambda>x. - f x)" _ S ] by auto
   with False show ?thesis by (simp add: not_integrable_integral)
 qed
 
 lemma integral_diff: "f integrable_on S \<Longrightarrow> g integrable_on S \<Longrightarrow>
     integral S (\<lambda>x. f x - g x) = integral S f - integral S g"
   by (rule integral_unique) (metis integrable_integral has_integral_diff)
 
 lemma integrable_0: "(\<lambda>x. 0) integrable_on S"
   unfolding integrable_on_def using has_integral_0 by auto
 
 lemma integrable_add: "f integrable_on S \<Longrightarrow> g integrable_on S \<Longrightarrow> (\<lambda>x. f x + g x) integrable_on S"
   unfolding integrable_on_def by(auto intro: has_integral_add)
 
 lemma integrable_cmul: "f integrable_on S \<Longrightarrow> (\<lambda>x. c *\<^sub>R f(x)) integrable_on S"
   unfolding integrable_on_def by(auto intro: has_integral_cmul)
 
 lemma integrable_on_scaleR_iff [simp]:
   fixes c :: real
   assumes "c \<noteq> 0"
   shows "(\<lambda>x. c *\<^sub>R f x) integrable_on S \<longleftrightarrow> f integrable_on S"
   using integrable_cmul[of "\<lambda>x. c *\<^sub>R f x" S "1 / c"] integrable_cmul[of f S c] \<open>c \<noteq> 0\<close>
   by auto
 
 lemma integrable_on_cmult_iff [simp]:
   fixes c :: real
   assumes "c \<noteq> 0"
   shows "(\<lambda>x. c * f x) integrable_on S \<longleftrightarrow> f integrable_on S"
   using integrable_on_scaleR_iff [of c f] assms by simp
 
 lemma integrable_on_cmult_left:
   assumes "f integrable_on S"
   shows "(\<lambda>x. of_real c * f x) integrable_on S"
     using integrable_cmul[of f S "of_real c"] assms
     by (simp add: scaleR_conv_of_real)
 
 lemma integrable_neg: "f integrable_on S \<Longrightarrow> (\<lambda>x. -f(x)) integrable_on S"
   unfolding integrable_on_def by(auto intro: has_integral_neg)
 
 lemma integrable_neg_iff: "(\<lambda>x. -f(x)) integrable_on S \<longleftrightarrow> f integrable_on S"
   using integrable_neg by fastforce
 
 lemma integrable_diff:
   "f integrable_on S \<Longrightarrow> g integrable_on S \<Longrightarrow> (\<lambda>x. f x - g x) integrable_on S"
   unfolding integrable_on_def by(auto intro: has_integral_diff)
 
 lemma integrable_linear:
   "f integrable_on S \<Longrightarrow> bounded_linear h \<Longrightarrow> (h \<circ> f) integrable_on S"
   unfolding integrable_on_def by(auto intro: has_integral_linear)
 
 lemma integral_linear:
   "f integrable_on S \<Longrightarrow> bounded_linear h \<Longrightarrow> integral S (h \<circ> f) = h (integral S f)"
-  apply (rule has_integral_unique [where i=S and f = "h \<circ> f"])
-  apply (simp_all add: integrable_integral integrable_linear has_integral_linear )
-  done
+  by (meson has_integral_iff has_integral_linear)
 
 lemma integrable_on_cnj_iff:
   "(\<lambda>x. cnj (f x)) integrable_on A \<longleftrightarrow> f integrable_on A"
   using integrable_linear[OF _ bounded_linear_cnj, of f A]
         integrable_linear[OF _ bounded_linear_cnj, of "cnj \<circ> f" A]
   by (auto simp: o_def)
 
 lemma integral_cnj: "cnj (integral A f) = integral A (\<lambda>x. cnj (f x))"
   by (cases "f integrable_on A")
      (simp_all add: integral_linear[OF _ bounded_linear_cnj, symmetric]
                     o_def integrable_on_cnj_iff not_integrable_integral)
 
 lemma integral_component_eq[simp]:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
   assumes "f integrable_on S"
   shows "integral S (\<lambda>x. f x \<bullet> k) = integral S f \<bullet> k"
   unfolding integral_linear[OF assms(1) bounded_linear_inner_left,unfolded o_def] ..
 
 lemma has_integral_sum:
   assumes "finite T"
     and "\<And>a. a \<in> T \<Longrightarrow> ((f a) has_integral (i a)) S"
   shows "((\<lambda>x. sum (\<lambda>a. f a x) T) has_integral (sum i T)) S"
   using assms(1) subset_refl[of T]
 proof (induct rule: finite_subset_induct)
   case empty
   then show ?case by auto
 next
   case (insert x F)
   with assms show ?case
     by (simp add: has_integral_add)
 qed
 
 lemma integral_sum:
   "\<lbrakk>finite I;  \<And>a. a \<in> I \<Longrightarrow> f a integrable_on S\<rbrakk> \<Longrightarrow>
    integral S (\<lambda>x. \<Sum>a\<in>I. f a x) = (\<Sum>a\<in>I. integral S (f a))"
   by (simp add: has_integral_sum integrable_integral integral_unique)
 
 lemma integrable_sum:
   "\<lbrakk>finite I;  \<And>a. a \<in> I \<Longrightarrow> f a integrable_on S\<rbrakk> \<Longrightarrow> (\<lambda>x. \<Sum>a\<in>I. f a x) integrable_on S"
   unfolding integrable_on_def using has_integral_sum[of I] by metis
 
 lemma has_integral_eq:
   assumes "\<And>x. x \<in> s \<Longrightarrow> f x = g x"
     and "(f has_integral k) s"
   shows "(g has_integral k) s"
   using has_integral_diff[OF assms(2), of "\<lambda>x. f x - g x" 0]
   using has_integral_is_0[of s "\<lambda>x. f x - g x"]
   using assms(1)
   by auto
 
 lemma integrable_eq: "\<lbrakk>f integrable_on s; \<And>x. x \<in> s \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> g integrable_on s"
   unfolding integrable_on_def
   using has_integral_eq[of s f g] has_integral_eq by blast
 
 lemma has_integral_cong:
   assumes "\<And>x. x \<in> s \<Longrightarrow> f x = g x"
   shows "(f has_integral i) s = (g has_integral i) s"
   using has_integral_eq[of s f g] has_integral_eq[of s g f] assms
   by auto
 
 lemma integral_cong:
   assumes "\<And>x. x \<in> s \<Longrightarrow> f x = g x"
   shows "integral s f = integral s g"
   unfolding integral_def
 by (metis (full_types, hide_lams) assms has_integral_cong integrable_eq)
 
 lemma integrable_on_cmult_left_iff [simp]:
   assumes "c \<noteq> 0"
   shows "(\<lambda>x. of_real c * f x) integrable_on s \<longleftrightarrow> f integrable_on s"
         (is "?lhs = ?rhs")
 proof
   assume ?lhs
   then have "(\<lambda>x. of_real (1 / c) * (of_real c * f x)) integrable_on s"
     using integrable_cmul[of "\<lambda>x. of_real c * f x" s "1 / of_real c"]
     by (simp add: scaleR_conv_of_real)
   then have "(\<lambda>x. (of_real (1 / c) * of_real c * f x)) integrable_on s"
     by (simp add: algebra_simps)
   with \<open>c \<noteq> 0\<close> show ?rhs
     by (metis (no_types, lifting) integrable_eq mult.left_neutral nonzero_divide_eq_eq of_real_1 of_real_mult)
 qed (blast intro: integrable_on_cmult_left)
 
 lemma integrable_on_cmult_right:
   fixes f :: "_ \<Rightarrow> 'b :: {comm_ring,real_algebra_1,real_normed_vector}"
   assumes "f integrable_on s"
   shows "(\<lambda>x. f x * of_real c) integrable_on s"
 using integrable_on_cmult_left [OF assms] by (simp add: mult.commute)
 
 lemma integrable_on_cmult_right_iff [simp]:
   fixes f :: "_ \<Rightarrow> 'b :: {comm_ring,real_algebra_1,real_normed_vector}"
   assumes "c \<noteq> 0"
   shows "(\<lambda>x. f x * of_real c) integrable_on s \<longleftrightarrow> f integrable_on s"
 using integrable_on_cmult_left_iff [OF assms] by (simp add: mult.commute)
 
 lemma integrable_on_cdivide:
   fixes f :: "_ \<Rightarrow> 'b :: real_normed_field"
   assumes "f integrable_on s"
   shows "(\<lambda>x. f x / of_real c) integrable_on s"
 by (simp add: integrable_on_cmult_right divide_inverse assms flip: of_real_inverse)
 
 lemma integrable_on_cdivide_iff [simp]:
   fixes f :: "_ \<Rightarrow> 'b :: real_normed_field"
   assumes "c \<noteq> 0"
   shows "(\<lambda>x. f x / of_real c) integrable_on s \<longleftrightarrow> f integrable_on s"
 by (simp add: divide_inverse assms flip: of_real_inverse)
 
 lemma has_integral_null [intro]: "content(cbox a b) = 0 \<Longrightarrow> (f has_integral 0) (cbox a b)"
   unfolding has_integral_cbox
   using eventually_division_filter_tagged_division[of "cbox a b"]
   by (subst tendsto_cong[where g="\<lambda>_. 0"]) (auto elim: eventually_mono intro: sum_content_null)
 
 lemma has_integral_null_real [intro]: "content {a..b::real} = 0 \<Longrightarrow> (f has_integral 0) {a..b}"
   by (metis box_real(2) has_integral_null)
 
 lemma has_integral_null_eq[simp]: "content (cbox a b) = 0 \<Longrightarrow> (f has_integral i) (cbox a b) \<longleftrightarrow> i = 0"
   by (auto simp add: has_integral_null dest!: integral_unique)
 
 lemma integral_null [simp]: "content (cbox a b) = 0 \<Longrightarrow> integral (cbox a b) f = 0"
   by (metis has_integral_null integral_unique)
 
 lemma integrable_on_null [intro]: "content (cbox a b) = 0 \<Longrightarrow> f integrable_on (cbox a b)"
   by (simp add: has_integral_integrable)
 
 lemma has_integral_empty[intro]: "(f has_integral 0) {}"
   by (meson ex_in_conv has_integral_is_0)
 
 lemma has_integral_empty_eq[simp]: "(f has_integral i) {} \<longleftrightarrow> i = 0"
   by (auto simp add: has_integral_empty has_integral_unique)
 
 lemma integrable_on_empty[intro]: "f integrable_on {}"
   unfolding integrable_on_def by auto
 
 lemma integral_empty[simp]: "integral {} f = 0"
   by (rule integral_unique) (rule has_integral_empty)
 
 lemma has_integral_refl[intro]:
   fixes a :: "'a::euclidean_space"
   shows "(f has_integral 0) (cbox a a)"
     and "(f has_integral 0) {a}"
 proof -
   show "(f has_integral 0) (cbox a a)"
      by (rule has_integral_null) simp
   then show "(f has_integral 0) {a}"
     by simp
 qed
 
 lemma integrable_on_refl[intro]: "f integrable_on cbox a a"
   unfolding integrable_on_def by auto
 
 lemma integral_refl [simp]: "integral (cbox a a) f = 0"
   by (rule integral_unique) auto
 
 lemma integral_singleton [simp]: "integral {a} f = 0"
   by auto
 
 lemma integral_blinfun_apply:
   assumes "f integrable_on s"
   shows "integral s (\<lambda>x. blinfun_apply h (f x)) = blinfun_apply h (integral s f)"
   by (subst integral_linear[symmetric, OF assms blinfun.bounded_linear_right]) (simp add: o_def)
 
 lemma blinfun_apply_integral:
   assumes "f integrable_on s"
   shows "blinfun_apply (integral s f) x = integral s (\<lambda>y. blinfun_apply (f y) x)"
   by (metis (no_types, lifting) assms blinfun.prod_left.rep_eq integral_blinfun_apply integral_cong)
 
 lemma has_integral_componentwise_iff:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
   shows "(f has_integral y) A \<longleftrightarrow> (\<forall>b\<in>Basis. ((\<lambda>x. f x \<bullet> b) has_integral (y \<bullet> b)) A)"
 proof safe
   fix b :: 'b assume "(f has_integral y) A"
   from has_integral_linear[OF this(1) bounded_linear_inner_left, of b]
     show "((\<lambda>x. f x \<bullet> b) has_integral (y \<bullet> b)) A" by (simp add: o_def)
 next
   assume "(\<forall>b\<in>Basis. ((\<lambda>x. f x \<bullet> b) has_integral (y \<bullet> b)) A)"
   hence "\<forall>b\<in>Basis. (((\<lambda>x. x *\<^sub>R b) \<circ> (\<lambda>x. f x \<bullet> b)) has_integral ((y \<bullet> b) *\<^sub>R b)) A"
     by (intro ballI has_integral_linear) (simp_all add: bounded_linear_scaleR_left)
   hence "((\<lambda>x. \<Sum>b\<in>Basis. (f x \<bullet> b) *\<^sub>R b) has_integral (\<Sum>b\<in>Basis. (y \<bullet> b) *\<^sub>R b)) A"
     by (intro has_integral_sum) (simp_all add: o_def)
   thus "(f has_integral y) A" by (simp add: euclidean_representation)
 qed
 
 lemma has_integral_componentwise:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
   shows "(\<And>b. b \<in> Basis \<Longrightarrow> ((\<lambda>x. f x \<bullet> b) has_integral (y \<bullet> b)) A) \<Longrightarrow> (f has_integral y) A"
   by (subst has_integral_componentwise_iff) blast
 
 lemma integrable_componentwise_iff:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
   shows "f integrable_on A \<longleftrightarrow> (\<forall>b\<in>Basis. (\<lambda>x. f x \<bullet> b) integrable_on A)"
 proof
   assume "f integrable_on A"
   then obtain y where "(f has_integral y) A" by (auto simp: integrable_on_def)
   hence "(\<forall>b\<in>Basis. ((\<lambda>x. f x \<bullet> b) has_integral (y \<bullet> b)) A)"
     by (subst (asm) has_integral_componentwise_iff)
   thus "(\<forall>b\<in>Basis. (\<lambda>x. f x \<bullet> b) integrable_on A)" by (auto simp: integrable_on_def)
 next
   assume "(\<forall>b\<in>Basis. (\<lambda>x. f x \<bullet> b) integrable_on A)"
   then obtain y where "\<forall>b\<in>Basis. ((\<lambda>x. f x \<bullet> b) has_integral y b) A"
     unfolding integrable_on_def by (subst (asm) bchoice_iff) blast
   hence "\<forall>b\<in>Basis. (((\<lambda>x. x *\<^sub>R b) \<circ> (\<lambda>x. f x \<bullet> b)) has_integral (y b *\<^sub>R b)) A"
     by (intro ballI has_integral_linear) (simp_all add: bounded_linear_scaleR_left)
   hence "((\<lambda>x. \<Sum>b\<in>Basis. (f x \<bullet> b) *\<^sub>R b) has_integral (\<Sum>b\<in>Basis. y b *\<^sub>R b)) A"
     by (intro has_integral_sum) (simp_all add: o_def)
   thus "f integrable_on A" by (auto simp: integrable_on_def o_def euclidean_representation)
 qed
 
 lemma integrable_componentwise:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
   shows "(\<And>b. b \<in> Basis \<Longrightarrow> (\<lambda>x. f x \<bullet> b) integrable_on A) \<Longrightarrow> f integrable_on A"
   by (subst integrable_componentwise_iff) blast
 
 lemma integral_componentwise:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
   assumes "f integrable_on A"
   shows "integral A f = (\<Sum>b\<in>Basis. integral A (\<lambda>x. (f x \<bullet> b) *\<^sub>R b))"
 proof -
   from assms have integrable: "\<forall>b\<in>Basis. (\<lambda>x. x *\<^sub>R b) \<circ> (\<lambda>x. (f x \<bullet> b)) integrable_on A"
     by (subst (asm) integrable_componentwise_iff, intro integrable_linear ballI)
        (simp_all add: bounded_linear_scaleR_left)
   have "integral A f = integral A (\<lambda>x. \<Sum>b\<in>Basis. (f x \<bullet> b) *\<^sub>R b)"
     by (simp add: euclidean_representation)
   also from integrable have "\<dots> = (\<Sum>a\<in>Basis. integral A (\<lambda>x. (f x \<bullet> a) *\<^sub>R a))"
     by (subst integral_sum) (simp_all add: o_def)
   finally show ?thesis .
 qed
 
 lemma integrable_component:
   "f integrable_on A \<Longrightarrow> (\<lambda>x. f x \<bullet> (y :: 'b :: euclidean_space)) integrable_on A"
   by (drule integrable_linear[OF _ bounded_linear_inner_left[of y]]) (simp add: o_def)
 
 
 
 subsection \<open>Cauchy-type criterion for integrability\<close>
 
 proposition integrable_Cauchy:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::{real_normed_vector,complete_space}"
   shows "f integrable_on cbox a b \<longleftrightarrow>
         (\<forall>e>0. \<exists>\<gamma>. gauge \<gamma> \<and>
           (\<forall>\<D>1 \<D>2. \<D>1 tagged_division_of (cbox a b) \<and> \<gamma> fine \<D>1 \<and>
             \<D>2 tagged_division_of (cbox a b) \<and> \<gamma> fine \<D>2 \<longrightarrow>
             norm ((\<Sum>(x,K)\<in>\<D>1. content K *\<^sub>R f x) - (\<Sum>(x,K)\<in>\<D>2. content K *\<^sub>R f x)) < e))"
   (is "?l = (\<forall>e>0. \<exists>\<gamma>. ?P e \<gamma>)")
 proof (intro iffI allI impI)
   assume ?l
   then obtain y
     where y: "\<And>e. e > 0 \<Longrightarrow>
         \<exists>\<gamma>. gauge \<gamma> \<and>
             (\<forall>\<D>. \<D> tagged_division_of cbox a b \<and> \<gamma> fine \<D> \<longrightarrow>
                  norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x) - y) < e)"
     by (auto simp: integrable_on_def has_integral)
   show "\<exists>\<gamma>. ?P e \<gamma>" if "e > 0" for e
   proof -
     have "e/2 > 0" using that by auto
     with y obtain \<gamma> where "gauge \<gamma>"
       and \<gamma>: "\<And>\<D>. \<D> tagged_division_of cbox a b \<and> \<gamma> fine \<D> \<Longrightarrow>
                   norm ((\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f x) - y) < e/2"
       by meson
     show ?thesis
     apply (rule_tac x=\<gamma> in exI, clarsimp simp: \<open>gauge \<gamma>\<close>)
         by (blast intro!: \<gamma> dist_triangle_half_l[where y=y,unfolded dist_norm])
     qed
 next
   assume "\<forall>e>0. \<exists>\<gamma>. ?P e \<gamma>"
   then have "\<forall>n::nat. \<exists>\<gamma>. ?P (1 / (n + 1)) \<gamma>"
     by auto
   then obtain \<gamma> :: "nat \<Rightarrow> 'n \<Rightarrow> 'n set" where \<gamma>:
     "\<And>m. gauge (\<gamma> m)"
     "\<And>m \<D>1 \<D>2. \<lbrakk>\<D>1 tagged_division_of cbox a b;
               \<gamma> m fine \<D>1; \<D>2 tagged_division_of cbox a b; \<gamma> m fine \<D>2\<rbrakk>
               \<Longrightarrow> norm ((\<Sum>(x,K) \<in> \<D>1. content K *\<^sub>R f x) - (\<Sum>(x,K) \<in> \<D>2. content K *\<^sub>R f x))
                   < 1 / (m + 1)"
     by metis
-  have "\<And>n. gauge (\<lambda>x. \<Inter>{\<gamma> i x |i. i \<in> {0..n}})"
-    apply (rule gauge_Inter)
-    using \<gamma> by auto
+  have "gauge (\<lambda>x. \<Inter>{\<gamma> i x |i. i \<in> {0..n}})" for n
+    using \<gamma> by (intro gauge_Inter) auto
   then have "\<forall>n. \<exists>p. p tagged_division_of (cbox a b) \<and> (\<lambda>x. \<Inter>{\<gamma> i x |i. i \<in> {0..n}}) fine p"
     by (meson fine_division_exists)
   then obtain p where p: "\<And>z. p z tagged_division_of cbox a b"
                          "\<And>z. (\<lambda>x. \<Inter>{\<gamma> i x |i. i \<in> {0..z}}) fine p z"
     by meson
   have dp: "\<And>i n. i\<le>n \<Longrightarrow> \<gamma> i fine p n"
     using p unfolding fine_Inter
     using atLeastAtMost_iff by blast
   have "Cauchy (\<lambda>n. sum (\<lambda>(x,K). content K *\<^sub>R (f x)) (p n))"
   proof (rule CauchyI)
     fix e::real
     assume "0 < e"
     then obtain N where "N \<noteq> 0" and N: "inverse (real N) < e"
       using real_arch_inverse[of e] by blast
     show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm ((\<Sum>(x,K) \<in> p m. content K *\<^sub>R f x) - (\<Sum>(x,K) \<in> p n. content K *\<^sub>R f x)) < e"
     proof (intro exI allI impI)
       fix m n
       assume mn: "N \<le> m" "N \<le> n"
       have "norm ((\<Sum>(x,K) \<in> p m. content K *\<^sub>R f x) - (\<Sum>(x,K) \<in> p n. content K *\<^sub>R f x)) < 1 / (real N + 1)"
         by (simp add: p(1) dp mn \<gamma>)
       also have "... < e"
         using  N \<open>N \<noteq> 0\<close> \<open>0 < e\<close> by (auto simp: field_simps)
       finally show "norm ((\<Sum>(x,K) \<in> p m. content K *\<^sub>R f x) - (\<Sum>(x,K) \<in> p n. content K *\<^sub>R f x)) < e" .
     qed
   qed
   then obtain y where y: "\<exists>no. \<forall>n\<ge>no. norm ((\<Sum>(x,K) \<in> p n. content K *\<^sub>R f x) - y) < r" if "r > 0" for r
     by (auto simp: convergent_eq_Cauchy[symmetric] dest: LIMSEQ_D)
   show ?l
     unfolding integrable_on_def has_integral
   proof (rule_tac x=y in exI, clarify)
     fix e :: real
     assume "e>0"
     then have e2: "e/2 > 0" by auto
     then obtain N1::nat where N1: "N1 \<noteq> 0" "inverse (real N1) < e/2"
       using real_arch_inverse by blast
     obtain N2::nat where N2: "\<And>n. n \<ge> N2 \<Longrightarrow> norm ((\<Sum>(x,K) \<in> p n. content K *\<^sub>R f x) - y) < e/2"
       using y[OF e2] by metis
     show "\<exists>\<gamma>. gauge \<gamma> \<and>
               (\<forall>\<D>. \<D> tagged_division_of (cbox a b) \<and> \<gamma> fine \<D> \<longrightarrow>
                 norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x) - y) < e)"
     proof (intro exI conjI allI impI)
       show "gauge (\<gamma> (N1+N2))"
         using \<gamma> by auto
       show "norm ((\<Sum>(x,K) \<in> q. content K *\<^sub>R f x) - y) < e"
            if "q tagged_division_of cbox a b \<and> \<gamma> (N1+N2) fine q" for q
       proof (rule norm_triangle_half_r)
         have "norm ((\<Sum>(x,K) \<in> p (N1+N2). content K *\<^sub>R f x) - (\<Sum>(x,K) \<in> q. content K *\<^sub>R f x))
                < 1 / (real (N1+N2) + 1)"
           by (rule \<gamma>; simp add: dp p that)
         also have "... < e/2"
           using N1 \<open>0 < e\<close> by (auto simp: field_simps intro: less_le_trans)
         finally show "norm ((\<Sum>(x,K) \<in> p (N1+N2). content K *\<^sub>R f x) - (\<Sum>(x,K) \<in> q. content K *\<^sub>R f x)) < e/2" .
         show "norm ((\<Sum>(x,K) \<in> p (N1+N2). content K *\<^sub>R f x) - y) < e/2"
           using N2 le_add_same_cancel2 by blast
       qed
     qed
   qed
 qed
 
 
 subsection \<open>Additivity of integral on abutting intervals\<close>
 
 lemma tagged_division_split_left_inj_content:
   assumes \<D>: "\<D> tagged_division_of S"
     and "(x1, K1) \<in> \<D>" "(x2, K2) \<in> \<D>" "K1 \<noteq> K2" "K1 \<inter> {x. x\<bullet>k \<le> c} = K2 \<inter> {x. x\<bullet>k \<le> c}" "k \<in> Basis"
   shows "content (K1 \<inter> {x. x\<bullet>k \<le> c}) = 0"
 proof -
   from tagged_division_ofD(4)[OF \<D> \<open>(x1, K1) \<in> \<D>\<close>] obtain a b where K1: "K1 = cbox a b"
     by auto
   then have "interior (K1 \<inter> {x. x \<bullet> k \<le> c}) = {}"
     by (metis tagged_division_split_left_inj assms)
   then show ?thesis
     unfolding K1 interval_split[OF \<open>k \<in> Basis\<close>] by (auto simp: content_eq_0_interior)
 qed
 
 lemma tagged_division_split_right_inj_content:
   assumes \<D>: "\<D> tagged_division_of S"
     and "(x1, K1) \<in> \<D>" "(x2, K2) \<in> \<D>" "K1 \<noteq> K2" "K1 \<inter> {x. x\<bullet>k \<ge> c} = K2 \<inter> {x. x\<bullet>k \<ge> c}" "k \<in> Basis"
   shows "content (K1 \<inter> {x. x\<bullet>k \<ge> c}) = 0"
 proof -
   from tagged_division_ofD(4)[OF \<D> \<open>(x1, K1) \<in> \<D>\<close>] obtain a b where K1: "K1 = cbox a b"
     by auto
   then have "interior (K1 \<inter> {x. c \<le> x \<bullet> k}) = {}"
     by (metis tagged_division_split_right_inj assms)
   then show ?thesis
     unfolding K1 interval_split[OF \<open>k \<in> Basis\<close>]
     by (auto simp: content_eq_0_interior)
 qed
 
 
 proposition has_integral_split:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
   assumes fi: "(f has_integral i) (cbox a b \<inter> {x. x\<bullet>k \<le> c})"
       and fj: "(f has_integral j) (cbox a b \<inter> {x. x\<bullet>k \<ge> c})"
       and k: "k \<in> Basis"
 shows "(f has_integral (i + j)) (cbox a b)"
   unfolding has_integral
 proof clarify
   fix e::real
   assume "0 < e"
   then have e: "e/2 > 0"
     by auto
     obtain \<gamma>1 where \<gamma>1: "gauge \<gamma>1"
       and \<gamma>1norm:
         "\<And>\<D>. \<lbrakk>\<D> tagged_division_of cbox a b \<inter> {x. x \<bullet> k \<le> c}; \<gamma>1 fine \<D>\<rbrakk>
              \<Longrightarrow> norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x) - i) < e/2"
        apply (rule has_integralD[OF fi[unfolded interval_split[OF k]] e])
        apply (simp add: interval_split[symmetric] k)
       done
     obtain \<gamma>2 where \<gamma>2: "gauge \<gamma>2"
       and \<gamma>2norm:
         "\<And>\<D>. \<lbrakk>\<D> tagged_division_of cbox a b \<inter> {x. c \<le> x \<bullet> k}; \<gamma>2 fine \<D>\<rbrakk>
              \<Longrightarrow> norm ((\<Sum>(x, k) \<in> \<D>. content k *\<^sub>R f x) - j) < e/2"
        apply (rule has_integralD[OF fj[unfolded interval_split[OF k]] e])
        apply (simp add: interval_split[symmetric] k)
        done
   let ?\<gamma> = "\<lambda>x. if x\<bullet>k = c then (\<gamma>1 x \<inter> \<gamma>2 x) else ball x \<bar>x\<bullet>k - c\<bar> \<inter> \<gamma>1 x \<inter> \<gamma>2 x"
   have "gauge ?\<gamma>"
     using \<gamma>1 \<gamma>2 unfolding gauge_def by auto
   then show "\<exists>\<gamma>. gauge \<gamma> \<and>
                  (\<forall>\<D>. \<D> tagged_division_of cbox a b \<and> \<gamma> fine \<D> \<longrightarrow>
                       norm ((\<Sum>(x, k)\<in>\<D>. content k *\<^sub>R f x) - (i + j)) < e)"
   proof (rule_tac x="?\<gamma>" in exI, safe)
     fix p
     assume p: "p tagged_division_of (cbox a b)" and "?\<gamma> fine p"
     have ab_eqp: "cbox a b = \<Union>{K. \<exists>x. (x, K) \<in> p}"
       using p by blast
     have xk_le_c: "x\<bullet>k \<le> c" if as: "(x,K) \<in> p" and K: "K \<inter> {x. x\<bullet>k \<le> c} \<noteq> {}" for x K
     proof (rule ccontr)
       assume **: "\<not> x \<bullet> k \<le> c"
       then have "K \<subseteq> ball x \<bar>x \<bullet> k - c\<bar>"
         using \<open>?\<gamma> fine p\<close> as by (fastforce simp: not_le algebra_simps)
       with K obtain y where y: "y \<in> ball x \<bar>x \<bullet> k - c\<bar>" "y\<bullet>k \<le> c"
         by blast
       then have "\<bar>x \<bullet> k - y \<bullet> k\<bar> < \<bar>x \<bullet> k - c\<bar>"
         using Basis_le_norm[OF k, of "x - y"]
         by (auto simp add: dist_norm inner_diff_left intro: le_less_trans)
       with y show False
         using ** by (auto simp add: field_simps)
     qed
     have xk_ge_c: "x\<bullet>k \<ge> c" if as: "(x,K) \<in> p" and K: "K \<inter> {x. x\<bullet>k \<ge> c} \<noteq> {}" for x K
     proof (rule ccontr)
       assume **: "\<not> x \<bullet> k \<ge> c"
       then have "K \<subseteq> ball x \<bar>x \<bullet> k - c\<bar>"
         using \<open>?\<gamma> fine p\<close> as by (fastforce simp: not_le algebra_simps)
       with K obtain y where y: "y \<in> ball x \<bar>x \<bullet> k - c\<bar>" "y\<bullet>k \<ge> c"
         by blast
       then have "\<bar>x \<bullet> k - y \<bullet> k\<bar> < \<bar>x \<bullet> k - c\<bar>"
         using Basis_le_norm[OF k, of "x - y"]
         by (auto simp add: dist_norm inner_diff_left intro: le_less_trans)
       with y show False
         using ** by (auto simp add: field_simps)
     qed
     have fin_finite: "finite {(x,f K) | x K. (x,K) \<in> s \<and> P x K}"
       if "finite s" for s and f :: "'a set \<Rightarrow> 'a set" and P :: "'a \<Rightarrow> 'a set \<Rightarrow> bool"
     proof -
       from that have "finite ((\<lambda>(x,K). (x, f K)) ` s)"
         by auto
       then show ?thesis
         by (rule rev_finite_subset) auto
     qed
     { fix \<G> :: "'a set \<Rightarrow> 'a set"
       fix i :: "'a \<times> 'a set"
       assume "i \<in> (\<lambda>(x, k). (x, \<G> k)) ` p - {(x, \<G> k) |x k. (x, k) \<in> p \<and> \<G> k \<noteq> {}}"
       then obtain x K where xk: "i = (x, \<G> K)"  "(x,K) \<in> p"
                                  "(x, \<G> K) \<notin> {(x, \<G> K) |x K. (x,K) \<in> p \<and> \<G> K \<noteq> {}}"
         by auto
       have "content (\<G> K) = 0"
         using xk using content_empty by auto
       then have "(\<lambda>(x,K). content K *\<^sub>R f x) i = 0"
         unfolding xk split_conv by auto
     } note [simp] = this
     have "finite p"
       using p by blast
     let ?M1 = "{(x, K \<inter> {x. x\<bullet>k \<le> c}) |x K. (x,K) \<in> p \<and> K \<inter> {x. x\<bullet>k \<le> c} \<noteq> {}}"
     have \<gamma>1_fine: "\<gamma>1 fine ?M1"
       using \<open>?\<gamma> fine p\<close> by (fastforce simp: fine_def split: if_split_asm)
     have "norm ((\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) - i) < e/2"
     proof (rule \<gamma>1norm [OF tagged_division_ofI \<gamma>1_fine])
       show "finite ?M1"
         by (rule fin_finite) (use p in blast)
       show "\<Union>{k. \<exists>x. (x, k) \<in> ?M1} = cbox a b \<inter> {x. x\<bullet>k \<le> c}"
         by (auto simp: ab_eqp)
 
       fix x L
       assume xL: "(x, L) \<in> ?M1"
       then obtain x' L' where xL': "x = x'" "L = L' \<inter> {x. x \<bullet> k \<le> c}"
                                    "(x', L') \<in> p" "L' \<inter> {x. x \<bullet> k \<le> c} \<noteq> {}"
         by blast
       then obtain a' b' where ab': "L' = cbox a' b'"
         using p by blast
       show "x \<in> L" "L \<subseteq> cbox a b \<inter> {x. x \<bullet> k \<le> c}"
         using p xk_le_c xL' by auto
       show "\<exists>a b. L = cbox a b"
         using p xL' ab' by (auto simp add: interval_split[OF k,where c=c])
 
       fix y R
       assume yR: "(y, R) \<in> ?M1"
       then obtain y' R' where yR': "y = y'" "R = R' \<inter> {x. x \<bullet> k \<le> c}"
                                    "(y', R') \<in> p" "R' \<inter> {x. x \<bullet> k \<le> c} \<noteq> {}"
         by blast
       assume as: "(x, L) \<noteq> (y, R)"
       show "interior L \<inter> interior R = {}"
       proof (cases "L' = R' \<longrightarrow> x' = y'")
         case False
         have "interior R' = {}"
           by (metis (no_types) False Pair_inject inf.idem tagged_division_ofD(5) [OF p] xL'(3) yR'(3))
         then show ?thesis
           using yR' by simp
       next
         case True
         then have "L' \<noteq> R'"
           using as unfolding xL' yR' by auto
         have "interior L' \<inter> interior R' = {}"
           by (metis (no_types) Pair_inject \<open>L' \<noteq> R'\<close> p tagged_division_ofD(5) xL'(3) yR'(3))
         then show ?thesis
           using xL'(2) yR'(2) by auto
       qed
     qed
     moreover
     let ?M2 = "{(x,K \<inter> {x. x\<bullet>k \<ge> c}) |x K. (x,K) \<in> p \<and> K \<inter> {x. x\<bullet>k \<ge> c} \<noteq> {}}"
     have \<gamma>2_fine: "\<gamma>2 fine ?M2"
       using \<open>?\<gamma> fine p\<close> by (fastforce simp: fine_def split: if_split_asm)
     have "norm ((\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - j) < e/2"
     proof (rule \<gamma>2norm [OF tagged_division_ofI \<gamma>2_fine])
       show "finite ?M2"
         by (rule fin_finite) (use p in blast)
       show "\<Union>{k. \<exists>x. (x, k) \<in> ?M2} = cbox a b \<inter> {x. x\<bullet>k \<ge> c}"
         by (auto simp: ab_eqp)
 
       fix x L
       assume xL: "(x, L) \<in> ?M2"
       then obtain x' L' where xL': "x = x'" "L = L' \<inter> {x. x \<bullet> k \<ge> c}"
                                    "(x', L') \<in> p" "L' \<inter> {x. x \<bullet> k \<ge> c} \<noteq> {}"
         by blast
       then obtain a' b' where ab': "L' = cbox a' b'"
         using p by blast
       show "x \<in> L" "L \<subseteq> cbox a b \<inter> {x. x \<bullet> k \<ge> c}"
         using p xk_ge_c xL' by auto
       show "\<exists>a b. L = cbox a b"
         using p xL' ab' by (auto simp add: interval_split[OF k,where c=c])
 
       fix y R
       assume yR: "(y, R) \<in> ?M2"
       then obtain y' R' where yR': "y = y'" "R = R' \<inter> {x. x \<bullet> k \<ge> c}"
                                    "(y', R') \<in> p" "R' \<inter> {x. x \<bullet> k \<ge> c} \<noteq> {}"
         by blast
       assume as: "(x, L) \<noteq> (y, R)"
       show "interior L \<inter> interior R = {}"
       proof (cases "L' = R' \<longrightarrow> x' = y'")
         case False
         have "interior R' = {}"
           by (metis (no_types) False Pair_inject inf.idem tagged_division_ofD(5) [OF p] xL'(3) yR'(3))
         then show ?thesis
           using yR' by simp
       next
         case True
         then have "L' \<noteq> R'"
           using as unfolding xL' yR' by auto
         have "interior L' \<inter> interior R' = {}"
           by (metis (no_types) Pair_inject \<open>L' \<noteq> R'\<close> p tagged_division_ofD(5) xL'(3) yR'(3))
         then show ?thesis
           using xL'(2) yR'(2) by auto
       qed
     qed
     ultimately
     have "norm (((\<Sum>(x,K) \<in> ?M1. content K *\<^sub>R f x) - i) + ((\<Sum>(x,K) \<in> ?M2. content K *\<^sub>R f x) - j)) < e/2 + e/2"
       using norm_add_less by blast
     moreover have "((\<Sum>(x,K) \<in> ?M1. content K *\<^sub>R f x) - i) +
                    ((\<Sum>(x,K) \<in> ?M2. content K *\<^sub>R f x) - j) =
                    (\<Sum>(x, ka)\<in>p. content ka *\<^sub>R f x) - (i + j)"
     proof -
       have eq0: "\<And>x y. x = (0::real) \<Longrightarrow> x *\<^sub>R (y::'b) = 0"
          by auto
       have cont_eq: "\<And>g. (\<lambda>(x,l). content l *\<^sub>R f x) \<circ> (\<lambda>(x,l). (x,g l)) = (\<lambda>(x,l). content (g l) *\<^sub>R f x)"
         by auto
       have *: "\<And>\<G> :: 'a set \<Rightarrow> 'a set.
                   (\<Sum>(x,K)\<in>{(x, \<G> K) |x K. (x,K) \<in> p \<and> \<G> K \<noteq> {}}. content K *\<^sub>R f x) =
                   (\<Sum>(x,K)\<in>(\<lambda>(x,K). (x, \<G> K)) ` p. content K *\<^sub>R f x)"
         by (rule sum.mono_neutral_left) (auto simp: \<open>finite p\<close>)
       have "((\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) - i) + ((\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - j) =
         (\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - (i + j)"
         by auto
       moreover have "\<dots> = (\<Sum>(x,K) \<in> p. content (K \<inter> {x. x \<bullet> k \<le> c}) *\<^sub>R f x) +
         (\<Sum>(x,K) \<in> p. content (K \<inter> {x. c \<le> x \<bullet> k}) *\<^sub>R f x) - (i + j)"
         unfolding *
         apply (subst (1 2) sum.reindex_nontrivial)
            apply (auto intro!: k p eq0 tagged_division_split_left_inj_content tagged_division_split_right_inj_content
                        simp: cont_eq \<open>finite p\<close>)
         done
       moreover have "\<And>x. x \<in> p \<Longrightarrow> (\<lambda>(a,B). content (B \<inter> {a. a \<bullet> k \<le> c}) *\<^sub>R f a) x +
                                 (\<lambda>(a,B). content (B \<inter> {a. c \<le> a \<bullet> k}) *\<^sub>R f a) x =
                                 (\<lambda>(a,B). content B *\<^sub>R f a) x"
       proof clarify
         fix a B
         assume "(a, B) \<in> p"
         with p obtain u v where uv: "B = cbox u v" by blast
         then show "content (B \<inter> {x. x \<bullet> k \<le> c}) *\<^sub>R f a + content (B \<inter> {x. c \<le> x \<bullet> k}) *\<^sub>R f a = content B *\<^sub>R f a"
           by (auto simp: scaleR_left_distrib uv content_split[OF k,of u v c])
       qed
       ultimately show ?thesis
         by (auto simp: sum.distrib[symmetric])
       qed
     ultimately show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - (i + j)) < e"
       by auto
   qed
 qed
 
 
 subsection \<open>A sort of converse, integrability on subintervals\<close>
 
 lemma has_integral_separate_sides:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
   assumes f: "(f has_integral i) (cbox a b)"
     and "e > 0"
     and k: "k \<in> Basis"
   obtains d where "gauge d"
     "\<forall>p1 p2. p1 tagged_division_of (cbox a b \<inter> {x. x\<bullet>k \<le> c}) \<and> d fine p1 \<and>
         p2 tagged_division_of (cbox a b \<inter> {x. x\<bullet>k \<ge> c}) \<and> d fine p2 \<longrightarrow>
         norm ((sum (\<lambda>(x,k). content k *\<^sub>R f x) p1 + sum (\<lambda>(x,k). content k *\<^sub>R f x) p2) - i) < e"
 proof -
   obtain \<gamma> where d: "gauge \<gamma>"
       "\<And>p. \<lbrakk>p tagged_division_of cbox a b; \<gamma> fine p\<rbrakk>
             \<Longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - i) < e"
     using has_integralD[OF f \<open>e > 0\<close>] by metis
   { fix p1 p2
     assume tdiv1: "p1 tagged_division_of (cbox a b) \<inter> {x. x \<bullet> k \<le> c}" and "\<gamma> fine p1"
     note p1=tagged_division_ofD[OF this(1)] 
     assume tdiv2: "p2 tagged_division_of (cbox a b) \<inter> {x. c \<le> x \<bullet> k}" and "\<gamma> fine p2"
     note p2=tagged_division_ofD[OF this(1)] 
     note tagged_division_Un_interval[OF tdiv1 tdiv2] 
     note p12 = tagged_division_ofD[OF this] this
     { fix a b
       assume ab: "(a, b) \<in> p1 \<inter> p2"
       have "(a, b) \<in> p1"
         using ab by auto
       obtain u v where uv: "b = cbox u v"
         using \<open>(a, b) \<in> p1\<close> p1(4) by moura
       have "b \<subseteq> {x. x\<bullet>k = c}"
         using ab p1(3)[of a b] p2(3)[of a b] by fastforce
       moreover
       have "interior {x::'a. x \<bullet> k = c} = {}"
       proof (rule ccontr)
         assume "\<not> ?thesis"
         then obtain x where x: "x \<in> interior {x::'a. x\<bullet>k = c}"
           by auto
         then obtain \<epsilon> where "0 < \<epsilon>" and \<epsilon>: "ball x \<epsilon> \<subseteq> {x. x \<bullet> k = c}"
           using mem_interior by metis
         have x: "x\<bullet>k = c"
           using x interior_subset by fastforce
         have *: "\<And>i. i \<in> Basis \<Longrightarrow> \<bar>(x - (x + (\<epsilon>/2) *\<^sub>R k)) \<bullet> i\<bar> = (if i = k then \<epsilon>/2 else 0)"
           using \<open>0 < \<epsilon>\<close> k by (auto simp: inner_simps inner_not_same_Basis)
         have "(\<Sum>i\<in>Basis. \<bar>(x - (x + (\<epsilon>/2 ) *\<^sub>R k)) \<bullet> i\<bar>) =
               (\<Sum>i\<in>Basis. (if i = k then \<epsilon>/2 else 0))"
           using "*" by (blast intro: sum.cong)
         also have "\<dots> < \<epsilon>"
           by (subst sum.delta) (use \<open>0 < \<epsilon>\<close> in auto)
         finally have "x + (\<epsilon>/2) *\<^sub>R k \<in> ball x \<epsilon>"
           unfolding mem_ball dist_norm by(rule le_less_trans[OF norm_le_l1])
         then have "x + (\<epsilon>/2) *\<^sub>R k \<in> {x. x\<bullet>k = c}"
           using \<epsilon> by auto
         then show False
           using \<open>0 < \<epsilon>\<close> x k by (auto simp: inner_simps)
       qed
       ultimately have "content b = 0"
         unfolding uv content_eq_0_interior
         using interior_mono by blast
       then have "content b *\<^sub>R f a = 0"
         by auto
     }
     then have "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - i) =
                norm ((\<Sum>(x, k)\<in>p1 \<union> p2. content k *\<^sub>R f x) - i)"
       by (subst sum.union_inter_neutral) (auto simp: p1 p2)
     also have "\<dots> < e"
       using d(2) p12 by (simp add: fine_Un k \<open>\<gamma> fine p1\<close> \<open>\<gamma> fine p2\<close>)
     finally have "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - i) < e" .
    }
   then show ?thesis
     using d(1) that by auto
 qed
 
 lemma integrable_split [intro]:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::{real_normed_vector,complete_space}"
   assumes f: "f integrable_on cbox a b"
       and k: "k \<in> Basis"
     shows "f integrable_on (cbox a b \<inter> {x. x\<bullet>k \<le> c})"   (is ?thesis1)
     and   "f integrable_on (cbox a b \<inter> {x. x\<bullet>k \<ge> c})"   (is ?thesis2)
 proof -
   obtain y where y: "(f has_integral y) (cbox a b)"
     using f by blast
   define a' where "a' = (\<Sum>i\<in>Basis. (if i = k then max (a\<bullet>k) c else a\<bullet>i)*\<^sub>R i)"
   define b' where "b' = (\<Sum>i\<in>Basis. (if i = k then min (b\<bullet>k) c else b\<bullet>i)*\<^sub>R i)"
   have "\<exists>d. gauge d \<and>
             (\<forall>p1 p2. p1 tagged_division_of cbox a b \<inter> {x. x \<bullet> k \<le> c} \<and> d fine p1 \<and>
                      p2 tagged_division_of cbox a b \<inter> {x. x \<bullet> k \<le> c} \<and> d fine p2 \<longrightarrow>
                      norm ((\<Sum>(x,K) \<in> p1. content K *\<^sub>R f x) - (\<Sum>(x,K) \<in> p2. content K *\<^sub>R f x)) < e)"
     if "e > 0" for e
   proof -
     have "e/2 > 0" using that by auto
   with has_integral_separate_sides[OF y this k, of c]
   obtain d
     where "gauge d"
          and d: "\<And>p1 p2. \<lbrakk>p1 tagged_division_of cbox a b \<inter> {x. x \<bullet> k \<le> c}; d fine p1;
                           p2 tagged_division_of cbox a b \<inter> {x. c \<le> x \<bullet> k}; d fine p2\<rbrakk>
                   \<Longrightarrow> norm ((\<Sum>(x,K)\<in>p1. content K *\<^sub>R f x) + (\<Sum>(x,K)\<in>p2. content K *\<^sub>R f x) - y) < e/2"
     by metis
   show ?thesis
     proof (rule_tac x=d in exI, clarsimp simp add: \<open>gauge d\<close>)
       fix p1 p2
       assume as: "p1 tagged_division_of (cbox a b) \<inter> {x. x \<bullet> k \<le> c}" "d fine p1"
                  "p2 tagged_division_of (cbox a b) \<inter> {x. x \<bullet> k \<le> c}" "d fine p2"
       show "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e"
       proof (rule fine_division_exists[OF \<open>gauge d\<close>, of a' b])
         fix p
         assume "p tagged_division_of cbox a' b" "d fine p"
         then show ?thesis
           using as norm_triangle_half_l[OF d[of p1 p] d[of p2 p]]
           unfolding interval_split[OF k] b'_def[symmetric] a'_def[symmetric]
           by (auto simp add: algebra_simps)
       qed
     qed
   qed
   with f show ?thesis1
     by (simp add: interval_split[OF k] integrable_Cauchy)
   have "\<exists>d. gauge d \<and>
             (\<forall>p1 p2. p1 tagged_division_of cbox a b \<inter> {x. x \<bullet> k \<ge> c} \<and> d fine p1 \<and>
                      p2 tagged_division_of cbox a b \<inter> {x. x \<bullet> k \<ge> c} \<and> d fine p2 \<longrightarrow>
                      norm ((\<Sum>(x,K) \<in> p1. content K *\<^sub>R f x) - (\<Sum>(x,K) \<in> p2. content K *\<^sub>R f x)) < e)"
     if "e > 0" for e
   proof -
     have "e/2 > 0" using that by auto
   with has_integral_separate_sides[OF y this k, of c]
   obtain d
     where "gauge d"
          and d: "\<And>p1 p2. \<lbrakk>p1 tagged_division_of cbox a b \<inter> {x. x \<bullet> k \<le> c}; d fine p1;
                           p2 tagged_division_of cbox a b \<inter> {x. c \<le> x \<bullet> k}; d fine p2\<rbrakk>
                   \<Longrightarrow> norm ((\<Sum>(x,K)\<in>p1. content K *\<^sub>R f x) + (\<Sum>(x,K)\<in>p2. content K *\<^sub>R f x) - y) < e/2"
     by metis
   show ?thesis
     proof (rule_tac x=d in exI, clarsimp simp add: \<open>gauge d\<close>)
       fix p1 p2
       assume as: "p1 tagged_division_of (cbox a b) \<inter> {x. x \<bullet> k \<ge> c}" "d fine p1"
                  "p2 tagged_division_of (cbox a b) \<inter> {x. x \<bullet> k \<ge> c}" "d fine p2"
       show "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e"
       proof (rule fine_division_exists[OF \<open>gauge d\<close>, of a b'])
         fix p
         assume "p tagged_division_of cbox a b'" "d fine p"
         then show ?thesis
           using as norm_triangle_half_l[OF d[of p p1] d[of p p2]]
           unfolding interval_split[OF k] b'_def[symmetric] a'_def[symmetric]
           by (auto simp add: algebra_simps)
       qed
     qed
   qed
   with f show ?thesis2
     by (simp add: interval_split[OF k] integrable_Cauchy)
 qed
 
 lemma operative_integralI:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
   shows "operative (lift_option (+)) (Some 0)
     (\<lambda>i. if f integrable_on i then Some (integral i f) else None)"
 proof -
   interpret comm_monoid "lift_option plus" "Some (0::'b)"
     by (rule comm_monoid_lift_option)
       (rule add.comm_monoid_axioms)
   show ?thesis
   proof
     fix a b c
     fix k :: 'a
     assume k: "k \<in> Basis"
     show "(if f integrable_on cbox a b then Some (integral (cbox a b) f) else None) =
           lift_option (+) (if f integrable_on cbox a b \<inter> {x. x \<bullet> k \<le> c} then Some (integral (cbox a b \<inter> {x. x \<bullet> k \<le> c}) f) else None)
           (if f integrable_on cbox a b \<inter> {x. c \<le> x \<bullet> k} then Some (integral (cbox a b \<inter> {x. c \<le> x \<bullet> k}) f) else None)"
     proof (cases "f integrable_on cbox a b")
       case True
       with k show ?thesis
-        apply (simp add: integrable_split)
-        apply (rule integral_unique [OF has_integral_split[OF _ _ k]])
-        apply (auto intro: integrable_integral)
-        done
+        by (auto simp: integrable_split intro: integral_unique [OF has_integral_split[OF _ _ k]])
     next
     case False
       have "\<not> (f integrable_on cbox a b \<inter> {x. x \<bullet> k \<le> c}) \<or> \<not> ( f integrable_on cbox a b \<inter> {x. c \<le> x \<bullet> k})"
       proof (rule ccontr)
         assume "\<not> ?thesis"
         then have "f integrable_on cbox a b"
           unfolding integrable_on_def
           apply (rule_tac x="integral (cbox a b \<inter> {x. x \<bullet> k \<le> c}) f + integral (cbox a b \<inter> {x. x \<bullet> k \<ge> c}) f" in exI)
-          apply (rule has_integral_split[OF _ _ k])
-          apply (auto intro: integrable_integral)
+          apply (auto intro: has_integral_split[OF _ _ k])
           done
         then show False
           using False by auto
       qed
       then show ?thesis
         using False by auto
     qed
   next
     fix a b :: 'a
     assume "box a b = {}"
     then show "(if f integrable_on cbox a b then Some (integral (cbox a b) f) else None) = Some 0"
       using has_integral_null_eq
       by (auto simp: integrable_on_null content_eq_0_interior)
   qed
 qed
 
 subsection \<open>Bounds on the norm of Riemann sums and the integral itself\<close>
 
 lemma dsum_bound:
-  assumes "p division_of (cbox a b)"
+  assumes p: "p division_of (cbox a b)"
     and "norm c \<le> e"
   shows "norm (sum (\<lambda>l. content l *\<^sub>R c) p) \<le> e * content(cbox a b)"
 proof -
   have sumeq: "(\<Sum>i\<in>p. \<bar>content i\<bar>) = sum content p"
-    apply (rule sum.cong)
-    using assms
-    apply simp
-    apply (metis abs_of_nonneg content_pos_le)
-    done
+    by simp
   have e: "0 \<le> e"
     using assms(2) norm_ge_zero order_trans by blast
   have "norm (sum (\<lambda>l. content l *\<^sub>R c) p) \<le> (\<Sum>i\<in>p. norm (content i *\<^sub>R c))"
     using norm_sum by blast
   also have "...  \<le> e * (\<Sum>i\<in>p. \<bar>content i\<bar>)"
     by (simp add: sum_distrib_left[symmetric] mult.commute assms(2) mult_right_mono sum_nonneg)
   also have "... \<le> e * content (cbox a b)"
-    apply (rule mult_left_mono [OF _ e])
-    apply (simp add: sumeq)
-    using additive_content_division assms(1) eq_iff apply blast
-    done
+    by (metis additive_content_division p eq_iff sumeq)
   finally show ?thesis .
 qed
 
 lemma rsum_bound:
   assumes p: "p tagged_division_of (cbox a b)"
       and "\<forall>x\<in>cbox a b. norm (f x) \<le> e"
     shows "norm (sum (\<lambda>(x,k). content k *\<^sub>R f x) p) \<le> e * content (cbox a b)"
 proof (cases "cbox a b = {}")
   case True show ?thesis
     using p unfolding True tagged_division_of_trivial by auto
 next
   case False
   then have e: "e \<ge> 0"
     by (meson ex_in_conv assms(2) norm_ge_zero order_trans)
   have sum_le: "sum (content \<circ> snd) p \<le> content (cbox a b)"
     unfolding additive_content_tagged_division[OF p, symmetric] split_def
     by (auto intro: eq_refl)
   have con: "\<And>xk. xk \<in> p \<Longrightarrow> 0 \<le> content (snd xk)"
     using tagged_division_ofD(4) [OF p] content_pos_le
     by force
-  have norm: "\<And>xk. xk \<in> p \<Longrightarrow> norm (f (fst xk)) \<le> e"
-    unfolding fst_conv using tagged_division_ofD(2,3)[OF p] assms
-    by (metis prod.collapse subset_eq)
   have "norm (sum (\<lambda>(x,k). content k *\<^sub>R f x) p) \<le> (\<Sum>i\<in>p. norm (case i of (x, k) \<Rightarrow> content k *\<^sub>R f x))"
     by (rule norm_sum)
   also have "...  \<le> e * content (cbox a b)"
-    unfolding split_def norm_scaleR
-    apply (rule order_trans[OF sum_mono])
-    apply (rule mult_left_mono[OF _ abs_ge_zero, of _ e])
-    apply (metis norm)
-    unfolding sum_distrib_right[symmetric]
-    using con sum_le
-    apply (auto simp: mult.commute intro: mult_left_mono [OF _ e])
-    done
+  proof -
+    have "\<And>xk. xk \<in> p \<Longrightarrow> norm (f (fst xk)) \<le> e"
+      using assms(2) p tag_in_interval by force
+    moreover have "(\<Sum>i\<in>p. \<bar>content (snd i)\<bar> * e) \<le> e * content (cbox a b)"
+      unfolding sum_distrib_right[symmetric]
+      using con sum_le by (auto simp: mult.commute intro: mult_left_mono [OF _ e])
+    ultimately show ?thesis
+      unfolding split_def norm_scaleR
+      by (metis (no_types, lifting) mult_left_mono[OF _ abs_ge_zero]   order_trans[OF sum_mono])
+  qed
   finally show ?thesis .
 qed
 
 lemma rsum_diff_bound:
   assumes "p tagged_division_of (cbox a b)"
     and "\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e"
   shows "norm (sum (\<lambda>(x,k). content k *\<^sub>R f x) p - sum (\<lambda>(x,k). content k *\<^sub>R g x) p) \<le>
          e * content (cbox a b)"
-  apply (rule order_trans[OF _ rsum_bound[OF assms]])
-  apply (simp add: split_def scaleR_diff_right sum_subtractf eq_refl)
-  done
+  using order_trans[OF _ rsum_bound[OF assms]]
+  by (simp add: split_def scaleR_diff_right sum_subtractf eq_refl)
 
 lemma has_integral_bound:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
   assumes "0 \<le> B"
       and f: "(f has_integral i) (cbox a b)"
       and "\<And>x. x\<in>cbox a b \<Longrightarrow> norm (f x) \<le> B"
     shows "norm i \<le> B * content (cbox a b)"
 proof (rule ccontr)
   assume "\<not> ?thesis"
   then have "norm i - B * content (cbox a b) > 0"
     by auto
   with f[unfolded has_integral]
   obtain \<gamma> where "gauge \<gamma>" and \<gamma>:
     "\<And>p. \<lbrakk>p tagged_division_of cbox a b; \<gamma> fine p\<rbrakk>
           \<Longrightarrow> norm ((\<Sum>(x, K)\<in>p. content K *\<^sub>R f x) - i) < norm i - B * content (cbox a b)"
     by metis
   then obtain p where p: "p tagged_division_of cbox a b" and "\<gamma> fine p"
     using fine_division_exists by blast
   have "\<And>s B. norm s \<le> B \<Longrightarrow> \<not> norm (s - i) < norm i - B"
     unfolding not_less
     by (metis diff_left_mono dist_commute dist_norm norm_triangle_ineq2 order_trans)
   then show False
     using \<gamma> [OF p \<open>\<gamma> fine p\<close>] rsum_bound[OF p] assms by metis
 qed
 
 corollary integrable_bound:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
   assumes "0 \<le> B"
       and "f integrable_on (cbox a b)"
       and "\<And>x. x\<in>cbox a b \<Longrightarrow> norm (f x) \<le> B"
     shows "norm (integral (cbox a b) f) \<le> B * content (cbox a b)"
 by (metis integrable_integral has_integral_bound assms)
 
 
 subsection \<open>Similar theorems about relationship among components\<close>
 
 lemma rsum_component_le:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes p: "p tagged_division_of (cbox a b)"
       and "\<And>x. x \<in> cbox a b \<Longrightarrow> (f x)\<bullet>i \<le> (g x)\<bullet>i"
     shows "(\<Sum>(x, K)\<in>p. content K *\<^sub>R f x) \<bullet> i \<le> (\<Sum>(x, K)\<in>p. content K *\<^sub>R g x) \<bullet> i"
 unfolding inner_sum_left
 proof (rule sum_mono, clarify)
   fix x K
   assume ab: "(x, K) \<in> p"
   with p obtain u v where K: "K = cbox u v"
     by blast
   then show "(content K *\<^sub>R f x) \<bullet> i \<le> (content K *\<^sub>R g x) \<bullet> i"
     by (metis ab assms inner_scaleR_left measure_nonneg mult_left_mono tag_in_interval)
 qed
 
 lemma has_integral_component_le:
   fixes f g :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes k: "k \<in> Basis"
   assumes "(f has_integral i) S" "(g has_integral j) S"
     and f_le_g: "\<And>x. x \<in> S \<Longrightarrow> (f x)\<bullet>k \<le> (g x)\<bullet>k"
   shows "i\<bullet>k \<le> j\<bullet>k"
 proof -
   have ik_le_jk: "i\<bullet>k \<le> j\<bullet>k"
     if f_i: "(f has_integral i) (cbox a b)"
     and g_j: "(g has_integral j) (cbox a b)"
     and le: "\<forall>x\<in>cbox a b. (f x)\<bullet>k \<le> (g x)\<bullet>k"
     for a b i and j :: 'b and f g :: "'a \<Rightarrow> 'b"
   proof (rule ccontr)
     assume "\<not> ?thesis"
     then have *: "0 < (i\<bullet>k - j\<bullet>k) / 3"
       by auto
     obtain \<gamma>1 where "gauge \<gamma>1" 
       and \<gamma>1: "\<And>p. \<lbrakk>p tagged_division_of cbox a b; \<gamma>1 fine p\<rbrakk>
                 \<Longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - i) < (i \<bullet> k - j \<bullet> k) / 3"
       using f_i[unfolded has_integral,rule_format,OF *] by fastforce 
     obtain \<gamma>2 where "gauge \<gamma>2" 
       and \<gamma>2: "\<And>p. \<lbrakk>p tagged_division_of cbox a b; \<gamma>2 fine p\<rbrakk>
                 \<Longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R g x) - j) < (i \<bullet> k - j \<bullet> k) / 3"
       using g_j[unfolded has_integral,rule_format,OF *] by fastforce 
     obtain p where p: "p tagged_division_of cbox a b" and "\<gamma>1 fine p" "\<gamma>2 fine p"
        using fine_division_exists[OF gauge_Int[OF \<open>gauge \<gamma>1\<close> \<open>gauge \<gamma>2\<close>], of a b] unfolding fine_Int
        by metis
     then have "\<bar>((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - i) \<bullet> k\<bar> < (i \<bullet> k - j \<bullet> k) / 3"
          "\<bar>((\<Sum>(x, k)\<in>p. content k *\<^sub>R g x) - j) \<bullet> k\<bar> < (i \<bullet> k - j \<bullet> k) / 3"
       using le_less_trans[OF Basis_le_norm[OF k]] k \<gamma>1 \<gamma>2 by metis+ 
     then show False
       unfolding inner_simps
       using rsum_component_le[OF p] le
       by (fastforce simp add: abs_real_def split: if_split_asm)
   qed
   show ?thesis
   proof (cases "\<exists>a b. S = cbox a b")
     case True
     with ik_le_jk assms show ?thesis
       by auto
   next
     case False
     show ?thesis
     proof (rule ccontr)
       assume "\<not> i\<bullet>k \<le> j\<bullet>k"
       then have ij: "(i\<bullet>k - j\<bullet>k) / 3 > 0"
         by auto
       obtain B1 where "0 < B1" 
            and B1: "\<And>a b. ball 0 B1 \<subseteq> cbox a b \<Longrightarrow>
                     \<exists>z. ((\<lambda>x. if x \<in> S then f x else 0) has_integral z) (cbox a b) \<and>
                         norm (z - i) < (i \<bullet> k - j \<bullet> k) / 3"
         using has_integral_altD[OF _ False ij] assms by blast
       obtain B2 where "0 < B2" 
            and B2: "\<And>a b. ball 0 B2 \<subseteq> cbox a b \<Longrightarrow>
                     \<exists>z. ((\<lambda>x. if x \<in> S then g x else 0) has_integral z) (cbox a b) \<and>
                         norm (z - j) < (i \<bullet> k - j \<bullet> k) / 3"
         using has_integral_altD[OF _ False ij] assms by blast
       have "bounded (ball 0 B1 \<union> ball (0::'a) B2)"
         unfolding bounded_Un by(rule conjI bounded_ball)+
       from bounded_subset_cbox_symmetric[OF this] 
       obtain a b::'a where ab: "ball 0 B1 \<subseteq> cbox a b" "ball 0 B2 \<subseteq> cbox a b"
         by (meson Un_subset_iff)
       then obtain w1 w2 where int_w1: "((\<lambda>x. if x \<in> S then f x else 0) has_integral w1) (cbox a b)"
                         and norm_w1:  "norm (w1 - i) < (i \<bullet> k - j \<bullet> k) / 3"
                         and int_w2: "((\<lambda>x. if x \<in> S then g x else 0) has_integral w2) (cbox a b)"
                         and norm_w2: "norm (w2 - j) < (i \<bullet> k - j \<bullet> k) / 3"
         using B1 B2 by blast
       have *: "\<And>w1 w2 j i::real .\<bar>w1 - i\<bar> < (i - j) / 3 \<Longrightarrow> \<bar>w2 - j\<bar> < (i - j) / 3 \<Longrightarrow> w1 \<le> w2 \<Longrightarrow> False"
         by (simp add: abs_real_def split: if_split_asm)
       have "\<bar>(w1 - i) \<bullet> k\<bar> < (i \<bullet> k - j \<bullet> k) / 3"
            "\<bar>(w2 - j) \<bullet> k\<bar> < (i \<bullet> k - j \<bullet> k) / 3"
         using Basis_le_norm k le_less_trans norm_w1 norm_w2 by blast+
       moreover
       have "w1\<bullet>k \<le> w2\<bullet>k"
         using ik_le_jk int_w1 int_w2 f_le_g by auto
       ultimately show False
         unfolding inner_simps by(rule *)
     qed
   qed
 qed
 
 lemma integral_component_le:
   fixes g f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "k \<in> Basis"
     and "f integrable_on S" "g integrable_on S"
     and "\<And>x. x \<in> S \<Longrightarrow> (f x)\<bullet>k \<le> (g x)\<bullet>k"
   shows "(integral S f)\<bullet>k \<le> (integral S g)\<bullet>k"
-  apply (rule has_integral_component_le)
-  using integrable_integral assms
-  apply auto
-  done
+  using has_integral_component_le assms by blast
 
 lemma has_integral_component_nonneg:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "k \<in> Basis"
     and "(f has_integral i) S"
     and "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> (f x)\<bullet>k"
   shows "0 \<le> i\<bullet>k"
   using has_integral_component_le[OF assms(1) has_integral_0 assms(2)]
   using assms(3-)
   by auto
 
 lemma integral_component_nonneg:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "k \<in> Basis"
     and  "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> (f x)\<bullet>k"
   shows "0 \<le> (integral S f)\<bullet>k"
 proof (cases "f integrable_on S")
   case True show ?thesis
-    apply (rule has_integral_component_nonneg)
-    using assms True
-    apply auto
-    done
+    using True assms has_integral_component_nonneg by blast
 next
   case False then show ?thesis by (simp add: not_integrable_integral)
 qed
 
 lemma has_integral_component_neg:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "k \<in> Basis"
     and "(f has_integral i) S"
     and "\<And>x. x \<in> S \<Longrightarrow> (f x)\<bullet>k \<le> 0"
   shows "i\<bullet>k \<le> 0"
   using has_integral_component_le[OF assms(1,2) has_integral_0] assms(2-)
   by auto
 
 lemma has_integral_component_lbound:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "(f has_integral i) (cbox a b)"
     and "\<forall>x\<in>cbox a b. B \<le> f(x)\<bullet>k"
     and "k \<in> Basis"
   shows "B * content (cbox a b) \<le> i\<bullet>k"
   using has_integral_component_le[OF assms(3) has_integral_const assms(1),of "(\<Sum>i\<in>Basis. B *\<^sub>R i)::'b"] assms(2-)
   by (auto simp add: field_simps)
 
 lemma has_integral_component_ubound:
   fixes f::"'a::euclidean_space => 'b::euclidean_space"
   assumes "(f has_integral i) (cbox a b)"
     and "\<forall>x\<in>cbox a b. f x\<bullet>k \<le> B"
     and "k \<in> Basis"
   shows "i\<bullet>k \<le> B * content (cbox a b)"
   using has_integral_component_le[OF assms(3,1) has_integral_const, of "\<Sum>i\<in>Basis. B *\<^sub>R i"] assms(2-)
   by (auto simp add: field_simps)
 
 lemma integral_component_lbound:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "f integrable_on cbox a b"
     and "\<forall>x\<in>cbox a b. B \<le> f(x)\<bullet>k"
     and "k \<in> Basis"
   shows "B * content (cbox a b) \<le> (integral(cbox a b) f)\<bullet>k"
-  apply (rule has_integral_component_lbound)
-  using assms
-  unfolding has_integral_integral
-  apply auto
-  done
+  using assms has_integral_component_lbound by blast
 
 lemma integral_component_lbound_real:
   assumes "f integrable_on {a ::real..b}"
     and "\<forall>x\<in>{a..b}. B \<le> f(x)\<bullet>k"
     and "k \<in> Basis"
   shows "B * content {a..b} \<le> (integral {a..b} f)\<bullet>k"
   using assms
   by (metis box_real(2) integral_component_lbound)
 
 lemma integral_component_ubound:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "f integrable_on cbox a b"
     and "\<forall>x\<in>cbox a b. f x\<bullet>k \<le> B"
     and "k \<in> Basis"
   shows "(integral (cbox a b) f)\<bullet>k \<le> B * content (cbox a b)"
-  apply (rule has_integral_component_ubound)
-  using assms
-  unfolding has_integral_integral
-  apply auto
-  done
+  using assms has_integral_component_ubound by blast
 
 lemma integral_component_ubound_real:
   fixes f :: "real \<Rightarrow> 'a::euclidean_space"
   assumes "f integrable_on {a..b}"
     and "\<forall>x\<in>{a..b}. f x\<bullet>k \<le> B"
     and "k \<in> Basis"
   shows "(integral {a..b} f)\<bullet>k \<le> B * content {a..b}"
   using assms
   by (metis box_real(2) integral_component_ubound)
 
 subsection \<open>Uniform limit of integrable functions is integrable\<close>
 
 lemma real_arch_invD:
   "0 < (e::real) \<Longrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
   by (subst(asm) real_arch_inverse)
 
 
 lemma integrable_uniform_limit:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
   assumes "\<And>e. e > 0 \<Longrightarrow> \<exists>g. (\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e) \<and> g integrable_on cbox a b"
   shows "f integrable_on cbox a b"
 proof (cases "content (cbox a b) > 0")
   case False then show ?thesis
     using has_integral_null by (simp add: content_lt_nz integrable_on_def)
 next
   case True
   have "1 / (real n + 1) > 0" for n
     by auto
   then have "\<exists>g. (\<forall>x\<in>cbox a b. norm (f x - g x) \<le> 1 / (real n + 1)) \<and> g integrable_on cbox a b" for n
     using assms by blast
   then obtain g where g_near_f: "\<And>n x. x \<in> cbox a b \<Longrightarrow> norm (f x - g n x) \<le> 1 / (real n + 1)"
                   and int_g: "\<And>n. g n integrable_on cbox a b"
     by metis
   then obtain h where h: "\<And>n. (g n has_integral h n) (cbox a b)"
     unfolding integrable_on_def by metis
   have "Cauchy h"
     unfolding Cauchy_def
   proof clarify
     fix e :: real
     assume "e>0"
     then have "e/4 / content (cbox a b) > 0"
       using True by (auto simp: field_simps)
     then obtain M where "M \<noteq> 0" and M: "1 / (real M) < e/4 / content (cbox a b)"
       by (metis inverse_eq_divide real_arch_inverse)
     show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (h m) (h n) < e"
     proof (rule exI [where x=M], clarify)
       fix m n
       assume m: "M \<le> m" and n: "M \<le> n"
       have "e/4>0" using \<open>e>0\<close> by auto
       then obtain gm gn where "gauge gm" "gauge gn"
               and gm: "\<And>\<D>. \<D> tagged_division_of cbox a b \<and> gm fine \<D> 
                             \<Longrightarrow> norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R g m x) - h m) < e/4"
               and gn: "\<And>\<D>. \<D> tagged_division_of cbox a b \<and> gn fine \<D> \<Longrightarrow>
                       norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R g n x) - h n) < e/4"
         using h[unfolded has_integral] by meson
       then obtain \<D> where \<D>: "\<D> tagged_division_of cbox a b" "(\<lambda>x. gm x \<inter> gn x) fine \<D>"
         by (metis (full_types) fine_division_exists gauge_Int)
       have triangle3: "norm (i1 - i2) < e"
         if no: "norm(s2 - s1) \<le> e/2" "norm (s1 - i1) < e/4" "norm (s2 - i2) < e/4" for s1 s2 i1 and i2::'b
       proof -
         have "norm (i1 - i2) \<le> norm (i1 - s1) + norm (s1 - s2) + norm (s2 - i2)"
           using norm_triangle_ineq[of "i1 - s1" "s1 - i2"]
           using norm_triangle_ineq[of "s1 - s2" "s2 - i2"]
           by (auto simp: algebra_simps)
         also have "\<dots> < e"
           using no by (auto simp: algebra_simps norm_minus_commute)
         finally show ?thesis .
       qed
       have finep: "gm fine \<D>" "gn fine \<D>"
         using fine_Int \<D>  by auto
       have norm_le: "norm (g n x - g m x) \<le> 2 / real M" if x: "x \<in> cbox a b" for x
       proof -
         have "norm (f x - g n x) + norm (f x - g m x) \<le> 1 / (real n + 1) + 1 / (real m + 1)"          
           using g_near_f[OF x, of n] g_near_f[OF x, of m] by simp
         also have "\<dots> \<le> 1 / (real M) + 1 / (real M)"
-          apply (rule add_mono)
-          using \<open>M \<noteq> 0\<close> m n by (auto simp: field_split_simps)
+          using \<open>M \<noteq> 0\<close> m n by (intro add_mono; force simp: field_split_simps)
         also have "\<dots> = 2 / real M"
           by auto
         finally show "norm (g n x - g m x) \<le> 2 / real M"
           using norm_triangle_le[of "g n x - f x" "f x - g m x" "2 / real M"]
           by (auto simp: algebra_simps simp add: norm_minus_commute)
       qed
       have "norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R g n x) - (\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R g m x)) \<le> 2 / real M * content (cbox a b)"
         by (blast intro: norm_le rsum_diff_bound[OF \<D>(1), where e="2 / real M"])
       also have "... \<le> e/2"
         using M True
         by (auto simp: field_simps)
       finally have le_e2: "norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R g n x) - (\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R g m x)) \<le> e/2" .
       then show "dist (h m) (h n) < e"
         unfolding dist_norm using gm gn \<D> finep by (auto intro!: triangle3)
     qed
   qed
   then obtain s where s: "h \<longlonglongrightarrow> s"
     using convergent_eq_Cauchy[symmetric] by blast
   show ?thesis
     unfolding integrable_on_def has_integral
   proof (rule_tac x=s in exI, clarify)
     fix e::real
     assume e: "0 < e"
     then have "e/3 > 0" by auto
     then obtain N1 where N1: "\<forall>n\<ge>N1. norm (h n - s) < e/3"
       using LIMSEQ_D [OF s] by metis
     from e True have "e/3 / content (cbox a b) > 0"
       by (auto simp: field_simps)
     then obtain N2 :: nat
          where "N2 \<noteq> 0" and N2: "1 / (real N2) < e/3 / content (cbox a b)"
       by (metis inverse_eq_divide real_arch_inverse)
     obtain g' where "gauge g'"
             and g': "\<And>\<D>. \<D> tagged_division_of cbox a b \<and> g' fine \<D> \<Longrightarrow>
                     norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R g (N1 + N2) x) - h (N1 + N2)) < e/3"
       by (metis h has_integral \<open>e/3 > 0\<close>)
     have *: "norm (sf - s) < e" 
         if no: "norm (sf - sg) \<le> e/3" "norm(h - s) < e/3" "norm (sg - h) < e/3" for sf sg h
     proof -
       have "norm (sf - s) \<le> norm (sf - sg) + norm (sg - h) + norm (h - s)"
         using norm_triangle_ineq[of "sf - sg" "sg - s"]
         using norm_triangle_ineq[of "sg -  h" " h - s"]
         by (auto simp: algebra_simps)
       also have "\<dots> < e"
         using no by (auto simp: algebra_simps norm_minus_commute)
       finally show ?thesis .
     qed
     { fix \<D>
       assume ptag: "\<D> tagged_division_of (cbox a b)" and "g' fine \<D>"
       then have norm_less: "norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R g (N1 + N2) x) - h (N1 + N2)) < e/3"
         using g' by blast
       have "content (cbox a b) < e/3 * (of_nat N2)"
         using \<open>N2 \<noteq> 0\<close> N2 using True by (auto simp: field_split_simps)
       moreover have "e/3 * of_nat N2 \<le> e/3 * (of_nat (N1 + N2) + 1)"
         using \<open>e>0\<close> by auto
       ultimately have "content (cbox a b) < e/3 * (of_nat (N1 + N2) + 1)"
         by linarith
       then have le_e3: "1 / (real (N1 + N2) + 1) * content (cbox a b) \<le> e/3"
         unfolding inverse_eq_divide
         by (auto simp: field_simps)
       have ne3: "norm (h (N1 + N2) - s) < e/3"
         using N1 by auto
       have "norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x) - (\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R g (N1 + N2) x))
             \<le> 1 / (real (N1 + N2) + 1) * content (cbox a b)"
         by (blast intro: g_near_f rsum_diff_bound[OF ptag])
       then have "norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x) - s) < e"
         by (rule *[OF order_trans [OF _ le_e3] ne3 norm_less])
     }
     then show "\<exists>d. gauge d \<and>
              (\<forall>\<D>. \<D> tagged_division_of cbox a b \<and> d fine \<D> \<longrightarrow> norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x) - s) < e)"
       by (blast intro: g' \<open>gauge g'\<close>)
   qed
 qed
 
 lemmas integrable_uniform_limit_real = integrable_uniform_limit [where 'a=real, simplified]
 
 
 subsection \<open>Negligible sets\<close>
 
 definition "negligible (s:: 'a::euclidean_space set) \<longleftrightarrow>
   (\<forall>a b. ((indicator s :: 'a\<Rightarrow>real) has_integral 0) (cbox a b))"
 
 
 subsubsection \<open>Negligibility of hyperplane\<close>
 
 lemma content_doublesplit:
   fixes a :: "'a::euclidean_space"
   assumes "0 < e"
     and k: "k \<in> Basis"
   obtains d where "0 < d" and "content (cbox a b \<inter> {x. \<bar>x\<bullet>k - c\<bar> \<le> d}) < e"
 proof cases
   assume *: "a \<bullet> k \<le> c \<and> c \<le> b \<bullet> k \<and> (\<forall>j\<in>Basis. a \<bullet> j \<le> b \<bullet> j)"
   define a' where "a' d = (\<Sum>j\<in>Basis. (if j = k then max (a\<bullet>j) (c - d) else a\<bullet>j) *\<^sub>R j)" for d
   define b' where "b' d = (\<Sum>j\<in>Basis. (if j = k then min (b\<bullet>j) (c + d) else b\<bullet>j) *\<^sub>R j)" for d
 
   have "((\<lambda>d. \<Prod>j\<in>Basis. (b' d - a' d) \<bullet> j) \<longlongrightarrow> (\<Prod>j\<in>Basis. (b' 0 - a' 0) \<bullet> j)) (at_right 0)"
     by (auto simp: b'_def a'_def intro!: tendsto_min tendsto_max tendsto_eq_intros)
   also have "(\<Prod>j\<in>Basis. (b' 0 - a' 0) \<bullet> j) = 0"
     using k *
     by (intro prod_zero bexI[OF _ k])
        (auto simp: b'_def a'_def inner_diff inner_sum_left inner_not_same_Basis intro!: sum.cong)
   also have "((\<lambda>d. \<Prod>j\<in>Basis. (b' d - a' d) \<bullet> j) \<longlongrightarrow> 0) (at_right 0) =
     ((\<lambda>d. content (cbox a b \<inter> {x. \<bar>x\<bullet>k - c\<bar> \<le> d})) \<longlongrightarrow> 0) (at_right 0)"
   proof (intro tendsto_cong eventually_at_rightI)
     fix d :: real assume d: "d \<in> {0<..<1}"
     have "cbox a b \<inter> {x. \<bar>x\<bullet>k - c\<bar> \<le> d} = cbox (a' d) (b' d)" for d
       using * d k by (auto simp add: cbox_def set_eq_iff Int_def ball_conj_distrib abs_diff_le_iff a'_def b'_def)
     moreover have "j \<in> Basis \<Longrightarrow> a' d \<bullet> j \<le> b' d \<bullet> j" for j
       using * d k by (auto simp: a'_def b'_def)
     ultimately show "(\<Prod>j\<in>Basis. (b' d - a' d) \<bullet> j) = content (cbox a b \<inter> {x. \<bar>x\<bullet>k - c\<bar> \<le> d})"
       by simp
   qed simp
   finally have "((\<lambda>d. content (cbox a b \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})) \<longlongrightarrow> 0) (at_right 0)" .
   from order_tendstoD(2)[OF this \<open>0<e\<close>]
   obtain d' where "0 < d'" and d': "\<And>y. y > 0 \<Longrightarrow> y < d' \<Longrightarrow> content (cbox a b \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> y}) < e"
     by (subst (asm) eventually_at_right[of _ 1]) auto
   show ?thesis
     by (rule that[of "d'/2"], insert \<open>0<d'\<close> d'[of "d'/2"], auto)
 next
   assume *: "\<not> (a \<bullet> k \<le> c \<and> c \<le> b \<bullet> k \<and> (\<forall>j\<in>Basis. a \<bullet> j \<le> b \<bullet> j))"
   then have "(\<exists>j\<in>Basis. b \<bullet> j < a \<bullet> j) \<or> (c < a \<bullet> k \<or> b \<bullet> k < c)"
     by (auto simp: not_le)
   show thesis
   proof cases
     assume "\<exists>j\<in>Basis. b \<bullet> j < a \<bullet> j"
     then have [simp]: "cbox a b = {}"
       using box_ne_empty(1)[of a b] by auto
     show ?thesis
       by (rule that[of 1]) (simp_all add: \<open>0<e\<close>)
   next
     assume "\<not> (\<exists>j\<in>Basis. b \<bullet> j < a \<bullet> j)"
     with * have "c < a \<bullet> k \<or> b \<bullet> k < c"
       by auto
     then show thesis
     proof
       assume c: "c < a \<bullet> k"
       moreover have "x \<in> cbox a b \<Longrightarrow> c \<le> x \<bullet> k" for x
         using k c by (auto simp: cbox_def)
       ultimately have "cbox a b \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> (a \<bullet> k - c)/2} = {}"
         using k by (auto simp: cbox_def)
       with \<open>0<e\<close> c that[of "(a \<bullet> k - c)/2"] show ?thesis
         by auto
     next
       assume c: "b \<bullet> k < c"
       moreover have "x \<in> cbox a b \<Longrightarrow> x \<bullet> k \<le> c" for x
         using k c by (auto simp: cbox_def)
       ultimately have "cbox a b \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> (c - b \<bullet> k)/2} = {}"
         using k by (auto simp: cbox_def)
       with \<open>0<e\<close> c that[of "(c - b \<bullet> k)/2"] show ?thesis
         by auto
     qed
   qed
 qed
 
 
 proposition negligible_standard_hyperplane[intro]:
   fixes k :: "'a::euclidean_space"
   assumes k: "k \<in> Basis"
   shows "negligible {x. x\<bullet>k = c}"
   unfolding negligible_def has_integral
 proof clarsimp
   fix a b and e::real assume "e > 0"
   with k obtain d where "0 < d" and d: "content (cbox a b \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) < e"
     by (metis content_doublesplit)
   let ?i = "indicator {x::'a. x\<bullet>k = c} :: 'a\<Rightarrow>real"
   show "\<exists>\<gamma>. gauge \<gamma> \<and>
            (\<forall>\<D>. \<D> tagged_division_of cbox a b \<and> \<gamma> fine \<D> \<longrightarrow>
                  \<bar>\<Sum>(x,K) \<in> \<D>. content K * ?i x\<bar> < e)"
   proof (intro exI, safe)
     show "gauge (\<lambda>x. ball x d)"
       using \<open>0 < d\<close> by blast
   next
     fix \<D>
     assume p: "\<D> tagged_division_of (cbox a b)" "(\<lambda>x. ball x d) fine \<D>"
     have "content L = content (L \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})" 
       if "(x, L) \<in> \<D>" "?i x \<noteq> 0" for x L
     proof -
       have xk: "x\<bullet>k = c"
         using that by (simp add: indicator_def split: if_split_asm)
       have "L \<subseteq> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}"
       proof 
         fix y
         assume y: "y \<in> L"
         have "L \<subseteq> ball x d"
           using p(2) that(1) by auto
         then have "norm (x - y) < d"
           by (simp add: dist_norm subset_iff y)
         then have "\<bar>(x - y) \<bullet> k\<bar> < d"
           using k norm_bound_Basis_lt by blast
         then show "y \<in> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}"
           unfolding inner_simps xk by auto
       qed 
       then show "content L = content (L \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})"
         by (metis inf.orderE)
     qed
     then have *: "(\<Sum>(x,K)\<in>\<D>. content K * ?i x) = (\<Sum>(x,K)\<in>\<D>. content (K \<inter> {x. \<bar>x\<bullet>k - c\<bar> \<le> d}) *\<^sub>R ?i x)"
       by (force simp add: split_paired_all intro!: sum.cong [OF refl])
     note p'= tagged_division_ofD[OF p(1)] and p''=division_of_tagged_division[OF p(1)]
     have "(\<Sum>(x,K)\<in>\<D>. content (K \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) * indicator {x. x \<bullet> k = c} x) < e"
     proof -
       have "(\<Sum>(x,K)\<in>\<D>. content (K \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) * ?i x) \<le> (\<Sum>(x,K)\<in>\<D>. content (K \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}))"
         by (force simp add: indicator_def intro!: sum_mono)
       also have "\<dots> < e"
       proof (subst sum.over_tagged_division_lemma[OF p(1)])
         fix u v::'a
         assume "box u v = {}"
         then have *: "content (cbox u v) = 0"
           unfolding content_eq_0_interior by simp
         have "cbox u v \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} \<subseteq> cbox u v"
           by auto
         then have "content (cbox u v \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) \<le> content (cbox u v)"
           unfolding interval_doublesplit[OF k] by (rule content_subset)
         then show "content (cbox u v \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) = 0"
           unfolding * interval_doublesplit[OF k]
           by (blast intro: antisym)
       next
         have "(\<Sum>l\<in>snd ` \<D>. content (l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})) =
           sum content ((\<lambda>l. l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})`{l\<in>snd ` \<D>. l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} \<noteq> {}})"
         proof (subst (2) sum.reindex_nontrivial)
           fix x y assume "x \<in> {l \<in> snd ` \<D>. l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} \<noteq> {}}" "y \<in> {l \<in> snd ` \<D>. l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} \<noteq> {}}"
             "x \<noteq> y" and eq: "x \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} = y \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}"
           then obtain x' y' where "(x', x) \<in> \<D>" "x \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} \<noteq> {}" "(y', y) \<in> \<D>" "y \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} \<noteq> {}"
             by (auto)
           from p'(5)[OF \<open>(x', x) \<in> \<D>\<close> \<open>(y', y) \<in> \<D>\<close>] \<open>x \<noteq> y\<close> have "interior (x \<inter> y) = {}"
             by auto
           moreover have "interior ((x \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) \<inter> (y \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})) \<subseteq> interior (x \<inter> y)"
             by (auto intro: interior_mono)
           ultimately have "interior (x \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) = {}"
             by (auto simp: eq)
           then show "content (x \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) = 0"
             using p'(4)[OF \<open>(x', x) \<in> \<D>\<close>] by (auto simp: interval_doublesplit[OF k] content_eq_0_interior simp del: interior_Int)
         qed (insert p'(1), auto intro!: sum.mono_neutral_right)
         also have "\<dots> \<le> norm (\<Sum>l\<in>(\<lambda>l. l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})`{l\<in>snd ` \<D>. l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} \<noteq> {}}. content l *\<^sub>R 1::real)"
           by simp
         also have "\<dots> \<le> 1 * content (cbox a b \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})"
           using division_doublesplit[OF p'' k, unfolded interval_doublesplit[OF k]]
           unfolding interval_doublesplit[OF k] by (intro dsum_bound) auto
         also have "\<dots> < e"
           using d by simp
         finally show "(\<Sum>K\<in>snd ` \<D>. content (K \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})) < e" .
       qed
       finally show "(\<Sum>(x, K)\<in>\<D>. content (K \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) * ?i x) < e" .
     qed
     then show "\<bar>\<Sum>(x, K)\<in>\<D>. content K * ?i x\<bar> < e"
-      unfolding * 
-      apply (subst abs_of_nonneg)
-      using measure_nonneg       
-      by (force simp add: indicator_def intro: sum_nonneg)+
+      unfolding *  by (simp add: sum_nonneg split: prod.split)
   qed
 qed
 
 corollary negligible_standard_hyperplane_cart:
   fixes k :: "'a::finite"
   shows "negligible {x. x$k = (0::real)}"
   by (simp add: cart_eq_inner_axis negligible_standard_hyperplane)
 
 
 subsubsection \<open>Hence the main theorem about negligible sets\<close>
 
 
 lemma has_integral_negligible_cbox:
   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::real_normed_vector"
   assumes negs: "negligible S"
     and 0: "\<And>x. x \<notin> S \<Longrightarrow> f x = 0"
   shows "(f has_integral 0) (cbox a b)"
   unfolding has_integral
 proof clarify
   fix e::real
   assume "e > 0"
   then have nn_gt0: "e/2 / ((real n+1) * (2 ^ n)) > 0" for n
     by simp
   then have "\<exists>\<gamma>. gauge \<gamma> \<and>
-                   (\<forall>\<D>. \<D> tagged_division_of cbox a b \<and> \<gamma> fine \<D> \<longrightarrow>
-                        \<bar>\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R indicator S x\<bar>
-                        < e/2 / ((real n + 1) * 2 ^ n))" for n
+                 (\<forall>\<D>. \<D> tagged_division_of cbox a b \<and> \<gamma> fine \<D> \<longrightarrow>
+                      \<bar>\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R indicator S x\<bar>
+                      < e/2 / ((real n + 1) * 2 ^ n))" for n
     using negs [unfolded negligible_def has_integral] by auto
   then obtain \<gamma> where 
     gd: "\<And>n. gauge (\<gamma> n)"
     and \<gamma>: "\<And>n \<D>. \<lbrakk>\<D> tagged_division_of cbox a b; \<gamma> n fine \<D>\<rbrakk>
                   \<Longrightarrow> \<bar>\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R indicator S x\<bar> < e/2 / ((real n + 1) * 2 ^ n)"
     by metis
   show "\<exists>\<gamma>. gauge \<gamma> \<and>
              (\<forall>\<D>. \<D> tagged_division_of cbox a b \<and> \<gamma> fine \<D> \<longrightarrow>
                   norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x) - 0) < e)"
   proof (intro exI, safe)
     show "gauge (\<lambda>x. \<gamma> (nat \<lfloor>norm (f x)\<rfloor>) x)"
       using gd by (auto simp: gauge_def)
 
     show "norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x) - 0) < e"
       if "\<D> tagged_division_of (cbox a b)" "(\<lambda>x. \<gamma> (nat \<lfloor>norm (f x)\<rfloor>) x) fine \<D>" for \<D>
     proof (cases "\<D> = {}")
       case True with \<open>0 < e\<close> show ?thesis by simp
     next
       case False
       obtain N where "Max ((\<lambda>(x, K). norm (f x)) ` \<D>) \<le> real N"
         using real_arch_simple by blast
       then have N: "\<And>x. x \<in> (\<lambda>(x, K). norm (f x)) ` \<D> \<Longrightarrow> x \<le> real N"
         by (meson Max_ge that(1) dual_order.trans finite_imageI tagged_division_of_finite)
       have "\<forall>i. \<exists>q. q tagged_division_of (cbox a b) \<and> (\<gamma> i) fine q \<and> (\<forall>(x,K) \<in> \<D>. K \<subseteq> (\<gamma> i) x \<longrightarrow> (x, K) \<in> q)"
         by (auto intro: tagged_division_finer[OF that(1) gd])
       from choice[OF this] 
       obtain q where q: "\<And>n. q n tagged_division_of cbox a b"
                         "\<And>n. \<gamma> n fine q n"
                         "\<And>n x K. \<lbrakk>(x, K) \<in> \<D>; K \<subseteq> \<gamma> n x\<rbrakk> \<Longrightarrow> (x, K) \<in> q n"
         by fastforce
       have "finite \<D>"
         using that(1) by blast
       then have sum_le_inc: "\<lbrakk>finite T; \<And>x y. (x,y) \<in> T \<Longrightarrow> (0::real) \<le> g(x,y);
                       \<And>y. y\<in>\<D> \<Longrightarrow> \<exists>x. (x,y) \<in> T \<and> f(y) \<le> g(x,y)\<rbrakk> \<Longrightarrow> sum f \<D> \<le> sum g T" for f g T
         by (rule sum_le_included[of \<D> T g snd f]; force)
       have "norm (\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x) \<le> (\<Sum>(x,K) \<in> \<D>. norm (content K *\<^sub>R f x))"
         unfolding split_def by (rule norm_sum)
       also have "... \<le> (\<Sum>(i, j) \<in> Sigma {..N + 1} q.
                           (real i + 1) * (case j of (x, K) \<Rightarrow> content K *\<^sub>R indicator S x))"
       proof (rule sum_le_inc, safe)
         show "finite (Sigma {..N+1} q)"
           by (meson finite_SigmaI finite_atMost tagged_division_of_finite q(1)) 
       next
         fix x K
         assume xk: "(x, K) \<in> \<D>"
         define n where "n = nat \<lfloor>norm (f x)\<rfloor>"
         have *: "norm (f x) \<in> (\<lambda>(x, K). norm (f x)) ` \<D>"
           using xk by auto
         have nfx: "real n \<le> norm (f x)" "norm (f x) \<le> real n + 1"
           unfolding n_def by auto
         then have "n \<in> {0..N + 1}"
           using N[OF *] by auto
         moreover have "K \<subseteq> \<gamma> (nat \<lfloor>norm (f x)\<rfloor>) x"
           using that(2) xk by auto
         moreover then have "(x, K) \<in> q (nat \<lfloor>norm (f x)\<rfloor>)"
           by (simp add: q(3) xk)
         moreover then have "(x, K) \<in> q n"
           using n_def by blast
         moreover
         have "norm (content K *\<^sub>R f x) \<le> (real n + 1) * (content K * indicator S x)"
         proof (cases "x \<in> S")
           case False
           then show ?thesis by (simp add: 0)
         next
           case True
           have *: "content K \<ge> 0"
             using tagged_division_ofD(4)[OF that(1) xk] by auto
           moreover have "content K * norm (f x) \<le> content K * (real n + 1)"
             by (simp add: mult_left_mono nfx(2))
           ultimately show ?thesis
             using nfx True by (auto simp: field_simps)
         qed
         ultimately show "\<exists>y. (y, x, K) \<in> (Sigma {..N + 1} q) \<and> norm (content K *\<^sub>R f x) \<le>
           (real y + 1) * (content K *\<^sub>R indicator S x)"
           by force
       qed auto
       also have "... = (\<Sum>i\<le>N + 1. \<Sum>j\<in>q i. (real i + 1) * (case j of (x, K) \<Rightarrow> content K *\<^sub>R indicator S x))"
-        apply (rule sum_Sigma_product [symmetric])
-        using q(1) apply auto
-        done
+        using q(1) by (intro sum_Sigma_product [symmetric]) auto
       also have "... \<le> (\<Sum>i\<le>N + 1. (real i + 1) * \<bar>\<Sum>(x,K) \<in> q i. content K *\<^sub>R indicator S x\<bar>)"
         by (rule sum_mono) (simp add: sum_distrib_left [symmetric])
       also have "... \<le> (\<Sum>i\<le>N + 1. e/2/2 ^ i)"
       proof (rule sum_mono)
         show "(real i + 1) * \<bar>\<Sum>(x,K) \<in> q i. content K *\<^sub>R indicator S x\<bar> \<le> e/2/2 ^ i"
           if "i \<in> {..N + 1}" for i
           using \<gamma>[of "q i" i] q by (simp add: divide_simps mult.left_commute)
       qed
       also have "... = e/2 * (\<Sum>i\<le>N + 1. (1/2) ^ i)"
         unfolding sum_distrib_left by (metis divide_inverse inverse_eq_divide power_one_over)
       also have "\<dots> < e/2 * 2"
       proof (rule mult_strict_left_mono)
         have "sum (power (1/2)) {..N + 1} = sum (power (1/2::real)) {..<N + 2}"
           using lessThan_Suc_atMost by auto
         also have "... < 2"
           by (auto simp: geometric_sum)
         finally show "sum (power (1/2::real)) {..N + 1} < 2" .
       qed (use \<open>0 < e\<close> in auto)
       finally  show ?thesis by auto
     qed
   qed
 qed
 
 
 proposition has_integral_negligible:
   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::real_normed_vector"
   assumes negs: "negligible S"
     and "\<And>x. x \<in> (T - S) \<Longrightarrow> f x = 0"
   shows "(f has_integral 0) T"
 proof (cases "\<exists>a b. T = cbox a b")
   case True
   then have "((\<lambda>x. if x \<in> T then f x else 0) has_integral 0) T"
     using assms by (auto intro!: has_integral_negligible_cbox)
   then show ?thesis
     by (rule has_integral_eq [rotated]) auto
 next
   case False
   let ?f = "(\<lambda>x. if x \<in> T then f x else 0)"
   have "((\<lambda>x. if x \<in> T then f x else 0) has_integral 0) T"
     apply (auto simp: False has_integral_alt [of ?f])
     apply (rule_tac x=1 in exI, auto)
     apply (rule_tac x=0 in exI, simp add: has_integral_negligible_cbox [OF negs] assms)
     done
   then show ?thesis
     by (rule_tac f="?f" in has_integral_eq) auto
 qed
 
 lemma
   assumes "negligible S"
   shows integrable_negligible: "f integrable_on S" and integral_negligible: "integral S f = 0"
   using has_integral_negligible [OF assms]
   by (auto simp: has_integral_iff)
 
 lemma has_integral_spike:
   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::real_normed_vector"
   assumes "negligible S"
     and gf: "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x"
     and fint: "(f has_integral y) T"
   shows "(g has_integral y) T"
 proof -
   have *: "(g has_integral y) (cbox a b)"
        if "(f has_integral y) (cbox a b)" "\<forall>x \<in> cbox a b - S. g x = f x" for a b f and g:: "'b \<Rightarrow> 'a" and y
   proof -
     have "((\<lambda>x. f x + (g x - f x)) has_integral (y + 0)) (cbox a b)"
       using that by (intro has_integral_add has_integral_negligible) (auto intro!: \<open>negligible S\<close>)
     then show ?thesis
       by auto
   qed
+  have \<section>: "\<And>a b z. \<lbrakk>\<And>x. x \<in> T \<and> x \<notin> S \<Longrightarrow> g x = f x;
+                     ((\<lambda>x. if x \<in> T then f x else 0) has_integral z) (cbox a b)\<rbrakk>
+                    \<Longrightarrow> ((\<lambda>x. if x \<in> T then g x else 0) has_integral z) (cbox a b)"
+      by (auto dest!: *[where f="\<lambda>x. if x\<in>T then f x else 0" and g="\<lambda>x. if x \<in> T then g x else 0"])
   show ?thesis
     using fint gf
     apply (subst has_integral_alt)
     apply (subst (asm) has_integral_alt)
-    apply (simp split: if_split_asm)
+    apply (auto split: if_split_asm)
      apply (blast dest: *)
-      apply (erule_tac V = "\<forall>a b. T \<noteq> cbox a b" in thin_rl)
-    apply (elim all_forward imp_forward ex_forward all_forward conj_forward asm_rl)
-     apply (auto dest!: *[where f="\<lambda>x. if x\<in>T then f x else 0" and g="\<lambda>x. if x \<in> T then g x else 0"])
-    done
+    using \<section> by meson
 qed
 
 lemma has_integral_spike_eq:
   assumes "negligible S"
     and gf: "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x"
   shows "(f has_integral y) T \<longleftrightarrow> (g has_integral y) T"
     using has_integral_spike [OF \<open>negligible S\<close>] gf
     by metis
 
 lemma integrable_spike:
   assumes "f integrable_on T" "negligible S" "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x"
     shows "g integrable_on T"
   using assms unfolding integrable_on_def by (blast intro: has_integral_spike)
 
 lemma integral_spike:
   assumes "negligible S"
     and "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x"
   shows "integral T f = integral T g"
   using has_integral_spike_eq[OF assms]
     by (auto simp: integral_def integrable_on_def)
 
 
 subsection \<open>Some other trivialities about negligible sets\<close>
 
 lemma negligible_subset:
   assumes "negligible s" "t \<subseteq> s"
   shows "negligible t"
   unfolding negligible_def
     by (metis (no_types) Diff_iff assms contra_subsetD has_integral_negligible indicator_simps(2))
 
 lemma negligible_diff[intro?]:
   assumes "negligible s"
   shows "negligible (s - t)"
   using assms by (meson Diff_subset negligible_subset)
 
 lemma negligible_Int:
   assumes "negligible s \<or> negligible t"
   shows "negligible (s \<inter> t)"
   using assms negligible_subset by force
 
 lemma negligible_Un:
   assumes "negligible S" and T: "negligible T"
   shows "negligible (S \<union> T)"
 proof -
   have "(indicat_real (S \<union> T) has_integral 0) (cbox a b)"
     if S0: "(indicat_real S has_integral 0) (cbox a b)" 
       and  "(indicat_real T has_integral 0) (cbox a b)" for a b
   proof (subst has_integral_spike_eq[OF T])
     show "indicat_real S x = indicat_real (S \<union> T) x" if "x \<in> cbox a b - T" for x
       by (metis Diff_iff Un_iff indicator_def that)
     show "(indicat_real S has_integral 0) (cbox a b)"
       by (simp add: S0)
   qed
   with assms show ?thesis
     unfolding negligible_def by blast
 qed
 
 lemma negligible_Un_eq[simp]: "negligible (s \<union> t) \<longleftrightarrow> negligible s \<and> negligible t"
   using negligible_Un negligible_subset by blast
 
 lemma negligible_sing[intro]: "negligible {a::'a::euclidean_space}"
   using negligible_standard_hyperplane[OF SOME_Basis, of "a \<bullet> (SOME i. i \<in> Basis)"] negligible_subset by blast
 
 lemma negligible_insert[simp]: "negligible (insert a s) \<longleftrightarrow> negligible s"
-  apply (subst insert_is_Un)
-  unfolding negligible_Un_eq
-  apply auto
-  done
+  by (metis insert_is_Un negligible_Un_eq negligible_sing)
 
 lemma negligible_empty[iff]: "negligible {}"
   using negligible_insert by blast
 
 text\<open>Useful in this form for backchaining\<close>
 lemma empty_imp_negligible: "S = {} \<Longrightarrow> negligible S"
   by simp
 
 lemma negligible_finite[intro]:
   assumes "finite s"
   shows "negligible s"
   using assms by (induct s) auto
 
 lemma negligible_Union[intro]:
   assumes "finite \<T>"
     and "\<And>t. t \<in> \<T> \<Longrightarrow> negligible t"
   shows "negligible(\<Union>\<T>)"
   using assms by induct auto
 
 lemma negligible: "negligible S \<longleftrightarrow> (\<forall>T. (indicat_real S has_integral 0) T)"
 proof (intro iffI allI)
   fix T
   assume "negligible S"
   then show "(indicator S has_integral 0) T"
     by (meson Diff_iff has_integral_negligible indicator_simps(2))
 qed (simp add: negligible_def)
 
 subsection \<open>Finite case of the spike theorem is quite commonly needed\<close>
 
 lemma has_integral_spike_finite:
   assumes "finite S"
     and "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x"
     and "(f has_integral y) T"
   shows "(g has_integral y) T"
   using assms has_integral_spike negligible_finite by blast
 
 lemma has_integral_spike_finite_eq:
   assumes "finite S"
     and "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x"
   shows "((f has_integral y) T \<longleftrightarrow> (g has_integral y) T)"
   by (metis assms has_integral_spike_finite)
 
 lemma integrable_spike_finite:
   assumes "finite S"
     and "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x"
     and "f integrable_on T"
   shows "g integrable_on T"
   using assms has_integral_spike_finite by blast
 
 lemma has_integral_bound_spike_finite:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
   assumes "0 \<le> B" "finite S"
       and f: "(f has_integral i) (cbox a b)"
       and leB: "\<And>x. x \<in> cbox a b - S \<Longrightarrow> norm (f x) \<le> B"
     shows "norm i \<le> B * content (cbox a b)"
 proof -
   define g where "g \<equiv> (\<lambda>x. if x \<in> S then 0 else f x)"
   then have "\<And>x. x \<in> cbox a b - S \<Longrightarrow> norm (g x) \<le> B"
     using leB by simp
   moreover have "(g has_integral i) (cbox a b)"
     using has_integral_spike_finite [OF \<open>finite S\<close> _ f]
     by (simp add: g_def)
   ultimately show ?thesis
     by (simp add: \<open>0 \<le> B\<close> g_def has_integral_bound)
 qed
 
 corollary has_integral_bound_real:
   fixes f :: "real \<Rightarrow> 'b::real_normed_vector"
   assumes "0 \<le> B" "finite S"
       and "(f has_integral i) {a..b}"
       and "\<And>x. x \<in> {a..b} - S \<Longrightarrow> norm (f x) \<le> B"
     shows "norm i \<le> B * content {a..b}"
   by (metis assms box_real(2) has_integral_bound_spike_finite)
 
 
 subsection \<open>In particular, the boundary of an interval is negligible\<close>
 
 lemma negligible_frontier_interval: "negligible(cbox (a::'a::euclidean_space) b - box a b)"
 proof -
   let ?A = "\<Union>((\<lambda>k. {x. x\<bullet>k = a\<bullet>k} \<union> {x::'a. x\<bullet>k = b\<bullet>k}) ` Basis)"
   have "negligible ?A"
     by (force simp add: negligible_Union[OF finite_imageI])
   moreover have "cbox a b - box a b \<subseteq> ?A"
     by (force simp add: mem_box)
   ultimately show ?thesis
     by (rule negligible_subset)
 qed
 
 lemma has_integral_spike_interior:
   assumes f: "(f has_integral y) (cbox a b)" and gf: "\<And>x. x \<in> box a b \<Longrightarrow> g x = f x"
   shows "(g has_integral y) (cbox a b)"
-  apply (rule has_integral_spike[OF negligible_frontier_interval _ f])
-  using gf by auto
-
+  by (meson Diff_iff gf has_integral_spike[OF negligible_frontier_interval _ f])
+  
 lemma has_integral_spike_interior_eq:
   assumes "\<And>x. x \<in> box a b \<Longrightarrow> g x = f x"
   shows "(f has_integral y) (cbox a b) \<longleftrightarrow> (g has_integral y) (cbox a b)"
   by (metis assms has_integral_spike_interior)
 
 lemma integrable_spike_interior:
   assumes "\<And>x. x \<in> box a b \<Longrightarrow> g x = f x"
     and "f integrable_on cbox a b"
   shows "g integrable_on cbox a b"
   using assms has_integral_spike_interior_eq by blast
 
 
 subsection \<open>Integrability of continuous functions\<close>
 
 lemma operative_approximableI:
   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::banach"
   assumes "0 \<le> e"
   shows "operative conj True (\<lambda>i. \<exists>g. (\<forall>x\<in>i. norm (f x - g (x::'b)) \<le> e) \<and> g integrable_on i)"
 proof -
   interpret comm_monoid conj True
     by standard auto
   show ?thesis
   proof (standard, safe)
     fix a b :: 'b
     show "\<exists>g. (\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e) \<and> g integrable_on cbox a b"
       if "box a b = {}" for a b
-      apply (rule_tac x=f in exI)
-      using assms that by (auto simp: content_eq_0_interior)
+      using assms that
+      by (metis content_eq_0_interior integrable_on_null interior_cbox norm_zero right_minus_eq)
     {
       fix c g and k :: 'b
       assume fg: "\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e" and g: "g integrable_on cbox a b"
       assume k: "k \<in> Basis"
       show "\<exists>g. (\<forall>x\<in>cbox a b \<inter> {x. x \<bullet> k \<le> c}. norm (f x - g x) \<le> e) \<and> g integrable_on cbox a b \<inter> {x. x \<bullet> k \<le> c}"
            "\<exists>g. (\<forall>x\<in>cbox a b \<inter> {x. c \<le> x \<bullet> k}. norm (f x - g x) \<le> e) \<and> g integrable_on cbox a b \<inter> {x. c \<le> x \<bullet> k}"
-         apply (rule_tac[!] x=g in exI)
-        using fg integrable_split[OF g k] by auto
+        using fg g k by auto
     }
     show "\<exists>g. (\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e) \<and> g integrable_on cbox a b"
       if fg1: "\<forall>x\<in>cbox a b \<inter> {x. x \<bullet> k \<le> c}. norm (f x - g1 x) \<le> e" 
         and g1: "g1 integrable_on cbox a b \<inter> {x. x \<bullet> k \<le> c}"
         and fg2: "\<forall>x\<in>cbox a b \<inter> {x. c \<le> x \<bullet> k}. norm (f x - g2 x) \<le> e" 
         and g2: "g2 integrable_on cbox a b \<inter> {x. c \<le> x \<bullet> k}" 
         and k: "k \<in> Basis"
       for c k g1 g2
     proof -
       let ?g = "\<lambda>x. if x\<bullet>k = c then f x else if x\<bullet>k \<le> c then g1 x else g2 x"
       show "\<exists>g. (\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e) \<and> g integrable_on cbox a b"
       proof (intro exI conjI ballI)
         show "norm (f x - ?g x) \<le> e" if "x \<in> cbox a b" for x
           by (auto simp: that assms fg1 fg2)
         show "?g integrable_on cbox a b"
         proof -
           have "?g integrable_on cbox a b \<inter> {x. x \<bullet> k \<le> c}" "?g integrable_on cbox a b \<inter> {x. x \<bullet> k \<ge> c}"
             by(rule integrable_spike[OF _ negligible_standard_hyperplane[of k c]], use k g1 g2 in auto)+
           with has_integral_split[OF _ _ k] show ?thesis
             unfolding integrable_on_def by blast
         qed
       qed
     qed
   qed
 qed
 
 lemma comm_monoid_set_F_and: "comm_monoid_set.F (\<and>) True f s \<longleftrightarrow> (finite s \<longrightarrow> (\<forall>x\<in>s. f x))"
 proof -
   interpret bool: comm_monoid_set \<open>(\<and>)\<close> True ..
   show ?thesis
     by (induction s rule: infinite_finite_induct) auto
 qed
 
 lemma approximable_on_division:
   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::banach"
   assumes "0 \<le> e"
     and d: "d division_of (cbox a b)"
     and f: "\<forall>i\<in>d. \<exists>g. (\<forall>x\<in>i. norm (f x - g x) \<le> e) \<and> g integrable_on i"
   obtains g where "\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e" "g integrable_on cbox a b"
 proof -
   interpret operative conj True "\<lambda>i. \<exists>g. (\<forall>x\<in>i. norm (f x - g (x::'b)) \<le> e) \<and> g integrable_on i"
     using \<open>0 \<le> e\<close> by (rule operative_approximableI)
   from f local.division [OF d] that show thesis
     by auto
 qed
 
 lemma integrable_continuous:
   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::banach"
   assumes "continuous_on (cbox a b) f"
   shows "f integrable_on cbox a b"
 proof (rule integrable_uniform_limit)
   fix e :: real
   assume e: "e > 0"
   then obtain d where "0 < d" and d: "\<And>x x'. \<lbrakk>x \<in> cbox a b; x' \<in> cbox a b; dist x' x < d\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e"
     using compact_uniformly_continuous[OF assms compact_cbox] unfolding uniformly_continuous_on_def by metis
   obtain p where ptag: "p tagged_division_of cbox a b" and finep: "(\<lambda>x. ball x d) fine p"
     using fine_division_exists[OF gauge_ball[OF \<open>0 < d\<close>], of a b] .
   have *: "\<forall>i\<in>snd ` p. \<exists>g. (\<forall>x\<in>i. norm (f x - g x) \<le> e) \<and> g integrable_on i"
   proof (safe, unfold snd_conv)
     fix x l
     assume as: "(x, l) \<in> p"
     obtain a b where l: "l = cbox a b"
       using as ptag by blast
     then have x: "x \<in> cbox a b"
       using as ptag by auto
     show "\<exists>g. (\<forall>x\<in>l. norm (f x - g x) \<le> e) \<and> g integrable_on l"
-      apply (rule_tac x="\<lambda>y. f x" in exI)
-    proof safe
+    proof (intro exI conjI strip)
       show "(\<lambda>y. f x) integrable_on l"
         unfolding integrable_on_def l by blast
     next
       fix y
       assume y: "y \<in> l"
       then have "y \<in> ball x d"
         using as finep by fastforce
       then show "norm (f y - f x) \<le> e"
         using d x y as l
         by (metis dist_commute dist_norm less_imp_le mem_ball ptag subsetCE tagged_division_ofD(3))
     qed
   qed
   from e have "e \<ge> 0"
     by auto
   from approximable_on_division[OF this division_of_tagged_division[OF ptag] *]
   show "\<exists>g. (\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e) \<and> g integrable_on cbox a b"
     by metis
 qed
 
 lemma integrable_continuous_interval:
   fixes f :: "'b::ordered_euclidean_space \<Rightarrow> 'a::banach"
   assumes "continuous_on {a..b} f"
   shows "f integrable_on {a..b}"
   by (metis assms integrable_continuous interval_cbox)
 
 lemmas integrable_continuous_real = integrable_continuous_interval[where 'b=real]
 
 lemma integrable_continuous_closed_segment:
   fixes f :: "real \<Rightarrow> 'a::banach"
   assumes "continuous_on (closed_segment a b) f"
   shows "f integrable_on (closed_segment a b)"
   using assms
   by (auto intro!: integrable_continuous_interval simp: closed_segment_eq_real_ivl)
 
 
 subsection \<open>Specialization of additivity to one dimension\<close>
 
 
 subsection \<open>A useful lemma allowing us to factor out the content size\<close>
 
 lemma has_integral_factor_content:
   "(f has_integral i) (cbox a b) \<longleftrightarrow>
     (\<forall>e>0. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow>
       norm (sum (\<lambda>(x,k). content k *\<^sub>R f x) p - i) \<le> e * content (cbox a b)))"
 proof (cases "content (cbox a b) = 0")
   case True
   have "\<And>e p. p tagged_division_of cbox a b \<Longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)) \<le> e * content (cbox a b)"
     unfolding sum_content_null[OF True] True by force
   moreover have "i = 0" 
     if "\<And>e. e > 0 \<Longrightarrow> \<exists>d. gauge d \<and>
               (\<forall>p. p tagged_division_of cbox a b \<and>
                    d fine p \<longrightarrow>
                    norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - i) \<le> e * content (cbox a b))"
     using that [of 1]
     by (force simp add: True sum_content_null[OF True] intro: fine_division_exists[of _ a b])
   ultimately show ?thesis
     unfolding has_integral_null_eq[OF True]
     by (force simp add: )
 next
   case False
   then have F: "0 < content (cbox a b)"
     using zero_less_measure_iff by blast
   let ?P = "\<lambda>e opp. \<exists>d. gauge d \<and>
     (\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow> opp (norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - i)) e)"
   show ?thesis
-    apply (subst has_integral)
-  proof safe
+  proof (subst has_integral, safe)
     fix e :: real
     assume e: "e > 0"
-    { assume "\<forall>e>0. ?P e (<)"
-      then show "?P (e * content (cbox a b)) (\<le>)"
-        apply (rule allE [where x="e * content (cbox a b)"])
-        apply (elim impE ex_forward conj_forward)
-        using F e apply (auto simp add: algebra_simps)
-        done }
-    { assume "\<forall>e>0. ?P (e * content (cbox a b)) (\<le>)"
-      then show "?P e (<)"
-        apply (rule allE [where x="e/2 / content (cbox a b)"])
-        apply (elim impE ex_forward conj_forward)
-        using F e apply (auto simp add: algebra_simps)
-        done }
+    show "?P (e * content (cbox a b)) (\<le>)" if \<section>[rule_format]: "\<forall>\<epsilon>>0. ?P \<epsilon> (<)"
+      using \<section> [of "e * content (cbox a b)"]
+      by (meson F e less_imp_le mult_pos_pos)
+    show "?P e (<)" if \<section>[rule_format]:  "\<forall>\<epsilon>>0. ?P (\<epsilon> * content (cbox a b)) (\<le>)"
+      using \<section> [of "e/2 / content (cbox a b)"]
+        using F e by (force simp add: algebra_simps)
   qed
 qed
 
 lemma has_integral_factor_content_real:
   "(f has_integral i) {a..b::real} \<longleftrightarrow>
     (\<forall>e>0. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b}  \<and> d fine p \<longrightarrow>
       norm (sum (\<lambda>(x,k). content k *\<^sub>R f x) p - i) \<le> e * content {a..b} ))"
   unfolding box_real[symmetric]
   by (rule has_integral_factor_content)
 
 
 subsection \<open>Fundamental theorem of calculus\<close>
 
 lemma interval_bounds_real:
   fixes q b :: real
   assumes "a \<le> b"
   shows "Sup {a..b} = b"
     and "Inf {a..b} = a"
   using assms by auto
 
 theorem fundamental_theorem_of_calculus:
   fixes f :: "real \<Rightarrow> 'a::banach"
   assumes "a \<le> b" 
       and vecd: "\<And>x. x \<in> {a..b} \<Longrightarrow> (f has_vector_derivative f' x) (at x within {a..b})"
   shows "(f' has_integral (f b - f a)) {a..b}"
   unfolding has_integral_factor_content box_real[symmetric]
 proof safe
   fix e :: real
   assume "e > 0"
   then have "\<forall>x. \<exists>d>0. x \<in> {a..b} \<longrightarrow>
          (\<forall>y\<in>{a..b}. norm (y-x) < d \<longrightarrow> norm (f y - f x - (y-x) *\<^sub>R f' x) \<le> e * norm (y-x))"
     using vecd[unfolded has_vector_derivative_def has_derivative_within_alt] by blast
   then obtain d where d: "\<And>x. 0 < d x"
                  "\<And>x y. \<lbrakk>x \<in> {a..b}; y \<in> {a..b}; norm (y-x) < d x\<rbrakk>
                         \<Longrightarrow> norm (f y - f x - (y-x) *\<^sub>R f' x) \<le> e * norm (y-x)"
     by metis  
   show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow>
     norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f' x) - (f b - f a)) \<le> e * content (cbox a b))"
   proof (rule exI, safe)
     show "gauge (\<lambda>x. ball x (d x))"
       using d(1) gauge_ball_dependent by blast
   next
     fix p
     assume ptag: "p tagged_division_of cbox a b" and finep: "(\<lambda>x. ball x (d x)) fine p"
     have ba: "b - a = (\<Sum>(x,K)\<in>p. Sup K - Inf K)" "f b - f a = (\<Sum>(x,K)\<in>p. f(Sup K) - f(Inf K))"
       using additive_tagged_division_1[where f= "\<lambda>x. x"] additive_tagged_division_1[where f= f]
             \<open>a \<le> b\<close> ptag by auto
     have "norm (\<Sum>(x, K) \<in> p. (content K *\<^sub>R f' x) - (f (Sup K) - f (Inf K)))
           \<le> (\<Sum>n\<in>p. e * (case n of (x, k) \<Rightarrow> Sup k - Inf k))"
     proof (rule sum_norm_le,safe)
       fix x K
       assume "(x, K) \<in> p"
       then have "x \<in> K" and kab: "K \<subseteq> cbox a b"
         using ptag by blast+
       then obtain u v where k: "K = cbox u v" and "x \<in> K" and kab: "K \<subseteq> cbox a b"
         using ptag \<open>(x, K) \<in> p\<close> by auto 
       have "u \<le> v"
         using \<open>x \<in> K\<close> unfolding k by auto
       have ball: "\<forall>y\<in>K. y \<in> ball x (d x)"
         using finep \<open>(x, K) \<in> p\<close> by blast
       have "u \<in> K" "v \<in> K"
         by (simp_all add: \<open>u \<le> v\<close> k)
       have "norm ((v - u) *\<^sub>R f' x - (f v - f u)) = norm (f u - f x - (u - x) *\<^sub>R f' x - (f v - f x - (v - x) *\<^sub>R f' x))"
         by (auto simp add: algebra_simps)
       also have "... \<le> norm (f u - f x - (u - x) *\<^sub>R f' x) + norm (f v - f x - (v - x) *\<^sub>R f' x)"
         by (rule norm_triangle_ineq4)
       finally have "norm ((v - u) *\<^sub>R f' x - (f v - f u)) \<le>
         norm (f u - f x - (u - x) *\<^sub>R f' x) + norm (f v - f x - (v - x) *\<^sub>R f' x)" .
       also have "\<dots> \<le> e * norm (u - x) + e * norm (v - x)"
       proof (rule add_mono)
         show "norm (f u - f x - (u - x) *\<^sub>R f' x) \<le> e * norm (u - x)"
-          apply (rule d(2)[of x u])
-          using \<open>x \<in> K\<close> kab \<open>u \<in> K\<close> ball dist_real_def by (auto simp add:dist_real_def)
+        proof (rule d)
+          show "norm (u - x) < d x"
+            using \<open>u \<in> K\<close> ball by (auto simp add: dist_real_def)
+        qed (use \<open>x \<in> K\<close> \<open>u \<in> K\<close> kab in auto)
         show "norm (f v - f x - (v - x) *\<^sub>R f' x) \<le> e * norm (v - x)"
-          apply (rule d(2)[of x v])
-          using \<open>x \<in> K\<close> kab \<open>v \<in> K\<close> ball dist_real_def by (auto simp add:dist_real_def)
+        proof (rule d)
+          show "norm (v - x) < d x"
+            using \<open>v \<in> K\<close> ball by (auto simp add: dist_real_def)
+        qed (use \<open>x \<in> K\<close> \<open>v \<in> K\<close> kab in auto)
       qed
       also have "\<dots> \<le> e * (Sup K - Inf K)"
         using \<open>x \<in> K\<close> by (auto simp: k interval_bounds_real[OF \<open>u \<le> v\<close>] field_simps)
       finally show "norm (content K *\<^sub>R f' x - (f (Sup K) - f (Inf K))) \<le> e * (Sup K - Inf K)"
         using \<open>u \<le> v\<close> by (simp add: k)
     qed
     with \<open>a \<le> b\<close> show "norm ((\<Sum>(x, K)\<in>p. content K *\<^sub>R f' x) - (f b - f a)) \<le> e * content (cbox a b)"
       by (auto simp: ba split_def sum_subtractf [symmetric] sum_distrib_left)
   qed
 qed
 
 lemma ident_has_integral:
   fixes a::real
   assumes "a \<le> b"
   shows "((\<lambda>x. x) has_integral (b\<^sup>2 - a\<^sup>2)/2) {a..b}"
 proof -
   have "((\<lambda>x. x) has_integral inverse 2 * b\<^sup>2 - inverse 2 * a\<^sup>2) {a..b}"
-    apply (rule fundamental_theorem_of_calculus [OF assms])
     unfolding power2_eq_square
-    by (rule derivative_eq_intros | simp)+
+    by (rule fundamental_theorem_of_calculus [OF assms] derivative_eq_intros | simp)+
   then show ?thesis
     by (simp add: field_simps)
 qed
 
 lemma integral_ident [simp]:
   fixes a::real
   assumes "a \<le> b"
   shows "integral {a..b} (\<lambda>x. x) = (if a \<le> b then (b\<^sup>2 - a\<^sup>2)/2 else 0)"
   by (metis assms ident_has_integral integral_unique)
 
 lemma ident_integrable_on:
   fixes a::real
   shows "(\<lambda>x. x) integrable_on {a..b}"
 by (metis atLeastatMost_empty_iff integrable_on_def has_integral_empty ident_has_integral)
 
 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
+      unfolding has_vector_derivative_def
+      by (intro derivative_eq_intros | force)+
   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 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
+      unfolding has_vector_derivative_def
+      by (intro derivative_eq_intros | force)+
   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)
 
 
 subsection \<open>Taylor series expansion\<close>
 
 lemma mvt_integral:
   fixes f::"'a::real_normed_vector\<Rightarrow>'b::banach"
   assumes f'[derivative_intros]:
     "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
   assumes line_in: "\<And>t. t \<in> {0..1} \<Longrightarrow> x + t *\<^sub>R y \<in> S"
   shows "f (x + y) - f x = integral {0..1} (\<lambda>t. f' (x + t *\<^sub>R y) y)" (is ?th1)
 proof -
   from assms have subset: "(\<lambda>xa. x + xa *\<^sub>R y) ` {0..1} \<subseteq> S" by auto
   note [derivative_intros] =
     has_derivative_subset[OF _ subset]
     has_derivative_in_compose[where f="(\<lambda>xa. x + xa *\<^sub>R y)" and g = f]
   note [continuous_intros] =
     continuous_on_compose2[where f="(\<lambda>xa. x + xa *\<^sub>R y)"]
     continuous_on_subset[OF _ subset]
   have "\<And>t. t \<in> {0..1} \<Longrightarrow>
     ((\<lambda>t. f (x + t *\<^sub>R y)) has_vector_derivative f' (x + t *\<^sub>R y) y)
     (at t within {0..1})"
     using assms
     by (auto simp: has_vector_derivative_def
         linear_cmul[OF has_derivative_linear[OF f'], symmetric]
       intro!: derivative_eq_intros)
   from fundamental_theorem_of_calculus[rule_format, OF _ this]
   show ?th1
     by (auto intro!: integral_unique[symmetric])
 qed
 
 lemma (in bounded_bilinear) sum_prod_derivatives_has_vector_derivative:
   assumes "p>0"
   and f0: "Df 0 = f"
   and Df: "\<And>m t. m < p \<Longrightarrow> a \<le> t \<Longrightarrow> t \<le> b \<Longrightarrow>
     (Df m has_vector_derivative Df (Suc m) t) (at t within {a..b})"
   and g0: "Dg 0 = g"
   and Dg: "\<And>m t. m < p \<Longrightarrow> a \<le> t \<Longrightarrow> t \<le> b \<Longrightarrow>
     (Dg m has_vector_derivative Dg (Suc m) t) (at t within {a..b})"
   and ivl: "a \<le> t" "t \<le> b"
   shows "((\<lambda>t. \<Sum>i<p. (-1)^i *\<^sub>R prod (Df i t) (Dg (p - Suc i) t))
     has_vector_derivative
       prod (f t) (Dg p t) - (-1)^p *\<^sub>R prod (Df p t) (g t))
     (at t within {a..b})"
   using assms
 proof cases
   assume p: "p \<noteq> 1"
   define p' where "p' = p - 2"
   from assms p have p': "{..<p} = {..Suc p'}" "p = Suc (Suc p')"
     by (auto simp: p'_def)
   have *: "\<And>i. i \<le> p' \<Longrightarrow> Suc (Suc p' - i) = (Suc (Suc p') - i)"
     by auto
   let ?f = "\<lambda>i. (-1) ^ i *\<^sub>R (prod (Df i t) (Dg ((p - i)) t))"
   have "(\<Sum>i<p. (-1) ^ i *\<^sub>R (prod (Df i t) (Dg (Suc (p - Suc i)) t) +
     prod (Df (Suc i) t) (Dg (p - Suc i) t))) =
     (\<Sum>i\<le>(Suc p'). ?f i - ?f (Suc i))"
     by (auto simp: algebra_simps p'(2) numeral_2_eq_2 * lessThan_Suc_atMost)
   also note sum_telescope
   finally
   have "(\<Sum>i<p. (-1) ^ i *\<^sub>R (prod (Df i t) (Dg (Suc (p - Suc i)) t) +
     prod (Df (Suc i) t) (Dg (p - Suc i) t)))
     = prod (f t) (Dg p t) - (- 1) ^ p *\<^sub>R prod (Df p t) (g t)"
     unfolding p'[symmetric]
     by (simp add: assms)
   thus ?thesis
     using assms
     by (auto intro!: derivative_eq_intros has_vector_derivative)
 qed (auto intro!: derivative_eq_intros has_vector_derivative)
 
 lemma
   fixes f::"real\<Rightarrow>'a::banach"
   assumes "p>0"
   and f0: "Df 0 = f"
   and Df: "\<And>m t. m < p \<Longrightarrow> a \<le> t \<Longrightarrow> t \<le> b \<Longrightarrow>
     (Df m has_vector_derivative Df (Suc m) t) (at t within {a..b})"
   and ivl: "a \<le> b"
   defines "i \<equiv> \<lambda>x. ((b - x) ^ (p - 1) / fact (p - 1)) *\<^sub>R Df p x"
   shows Taylor_has_integral:
     "(i has_integral f b - (\<Sum>i<p. ((b-a) ^ i / fact i) *\<^sub>R Df i a)) {a..b}"
   and Taylor_integral:
     "f b = (\<Sum>i<p. ((b-a) ^ i / fact i) *\<^sub>R Df i a) + integral {a..b} i"
   and Taylor_integrable:
     "i integrable_on {a..b}"
 proof goal_cases
   case 1
   interpret bounded_bilinear "scaleR::real\<Rightarrow>'a\<Rightarrow>'a"
     by (rule bounded_bilinear_scaleR)
   define g where "g s = (b - s)^(p - 1)/fact (p - 1)" for s
   define Dg where [abs_def]:
     "Dg n s = (if n < p then (-1)^n * (b - s)^(p - 1 - n) / fact (p - 1 - n) else 0)" for n s
   have g0: "Dg 0 = g"
     using \<open>p > 0\<close>
     by (auto simp add: Dg_def field_split_simps g_def split: if_split_asm)
   {
     fix m
     assume "p > Suc m"
     hence "p - Suc m = Suc (p - Suc (Suc m))"
       by auto
     hence "real (p - Suc m) * fact (p - Suc (Suc m)) = fact (p - Suc m)"
       by auto
   } note fact_eq = this
   have Dg: "\<And>m t. m < p \<Longrightarrow> a \<le> t \<Longrightarrow> t \<le> b \<Longrightarrow>
     (Dg m has_vector_derivative Dg (Suc m) t) (at t within {a..b})"
     unfolding Dg_def
     by (auto intro!: derivative_eq_intros simp: has_vector_derivative_def fact_eq field_split_simps)
   let ?sum = "\<lambda>t. \<Sum>i<p. (- 1) ^ i *\<^sub>R Dg i t *\<^sub>R Df (p - Suc i) t"
   from sum_prod_derivatives_has_vector_derivative[of _ Dg _ _ _ Df,
       OF \<open>p > 0\<close> g0 Dg f0 Df]
   have deriv: "\<And>t. a \<le> t \<Longrightarrow> t \<le> b \<Longrightarrow>
     (?sum has_vector_derivative
       g t *\<^sub>R Df p t - (- 1) ^ p *\<^sub>R Dg p t *\<^sub>R f t) (at t within {a..b})"
     by auto
   from fundamental_theorem_of_calculus[rule_format, OF \<open>a \<le> b\<close> deriv]
   have "(i has_integral ?sum b - ?sum a) {a..b}"
     using atLeastatMost_empty'[simp del]
     by (simp add: i_def g_def Dg_def)
   also
   have one: "(- 1) ^ p' * (- 1) ^ p' = (1::real)"
     and "{..<p} \<inter> {i. p = Suc i} = {p - 1}"
     for p'
     using \<open>p > 0\<close>
     by (auto simp: power_mult_distrib[symmetric])
   then have "?sum b = f b"
     using Suc_pred'[OF \<open>p > 0\<close>]
     by (simp add: diff_eq_eq Dg_def power_0_left le_Suc_eq if_distrib
         if_distribR sum.If_cases f0)
   also
   have "{..<p} = (\<lambda>x. p - x - 1) ` {..<p}"
   proof safe
     fix x
     assume "x < p"
     thus "x \<in> (\<lambda>x. p - x - 1) ` {..<p}"
       by (auto intro!: image_eqI[where x = "p - x - 1"])
   qed simp
   from _ this
   have "?sum a = (\<Sum>i<p. ((b-a) ^ i / fact i) *\<^sub>R Df i a)"
     by (rule sum.reindex_cong) (auto simp add: inj_on_def Dg_def one)
   finally show c: ?case .
   case 2 show ?case using c integral_unique
     by (metis (lifting) add.commute diff_eq_eq integral_unique)
   case 3 show ?case using c by force
 qed
 
 
 subsection \<open>Only need trivial subintervals if the interval itself is trivial\<close>
 
 proposition division_of_nontrivial:
   fixes \<D> :: "'a::euclidean_space set set"
   assumes sdiv: "\<D> division_of (cbox a b)"
      and cont0: "content (cbox a b) \<noteq> 0"
   shows "{k. k \<in> \<D> \<and> content k \<noteq> 0} division_of (cbox a b)"
   using sdiv
 proof (induction "card \<D>" arbitrary: \<D> rule: less_induct)
   case less
   note \<D> = division_ofD[OF less.prems]
   {
     presume *: "{k \<in> \<D>. content k \<noteq> 0} \<noteq> \<D> \<Longrightarrow> ?case"
     then show ?case
       using less.prems by fastforce
   }
   assume noteq: "{k \<in> \<D>. content k \<noteq> 0} \<noteq> \<D>"
   then obtain K c d where "K \<in> \<D>" and contk: "content K = 0" and keq: "K = cbox c d"
     using \<D>(4) by blast 
   then have "card \<D> > 0"
     unfolding card_gt_0_iff using less by auto
   then have card: "card (\<D> - {K}) < card \<D>"
     using less \<open>K \<in> \<D>\<close> by (simp add: \<D>(1))
   have closed: "closed (\<Union>(\<D> - {K}))"
     using less.prems by auto
   have "x islimpt \<Union>(\<D> - {K})" if "x \<in> K" for x 
     unfolding islimpt_approachable
   proof (intro allI impI)
     fix e::real
     assume "e > 0"
     obtain i where i: "c\<bullet>i = d\<bullet>i" "i\<in>Basis"
       using contk \<D>(3) [OF \<open>K \<in> \<D>\<close>] unfolding box_ne_empty keq
       by (meson content_eq_0 dual_order.antisym)
     then have xi: "x\<bullet>i = d\<bullet>i"
       using \<open>x \<in> K\<close> unfolding keq mem_box by (metis antisym)
     define y where "y = (\<Sum>j\<in>Basis. (if j = i then if c\<bullet>i \<le> (a\<bullet>i + b\<bullet>i)/2 then c\<bullet>i +
       min e (b\<bullet>i - c\<bullet>i)/2 else c\<bullet>i - min e (c\<bullet>i - a\<bullet>i)/2 else x\<bullet>j) *\<^sub>R j)"
     show "\<exists>x'\<in>\<Union>(\<D> - {K}). x' \<noteq> x \<and> dist x' x < e"
     proof (intro bexI conjI)
       have "d \<in> cbox c d"
         using \<D>(3)[OF \<open>K \<in> \<D>\<close>] by (simp add: box_ne_empty(1) keq mem_box(2))
       then have "d \<in> cbox a b"
         using \<D>(2)[OF \<open>K \<in> \<D>\<close>] by (auto simp: keq)
       then have di: "a \<bullet> i \<le> d \<bullet> i \<and> d \<bullet> i \<le> b \<bullet> i"
         using \<open>i \<in> Basis\<close> mem_box(2) by blast
       then have xyi: "y\<bullet>i \<noteq> x\<bullet>i"
         unfolding y_def i xi using \<open>e > 0\<close> cont0 \<open>i \<in> Basis\<close>
         by (auto simp: content_eq_0 elim!: ballE[of _ _ i])
       then show "y \<noteq> x"
         unfolding euclidean_eq_iff[where 'a='a] using i by auto
       have "norm (y-x) \<le> (\<Sum>b\<in>Basis. \<bar>(y - x) \<bullet> b\<bar>)"
         by (rule norm_le_l1)
       also have "... = \<bar>(y - x) \<bullet> i\<bar> + (\<Sum>b \<in> Basis - {i}. \<bar>(y - x) \<bullet> b\<bar>)"
         by (meson finite_Basis i(2) sum.remove)
       also have "... <  e + sum (\<lambda>i. 0) Basis"
       proof (rule add_less_le_mono)
         show "\<bar>(y-x) \<bullet> i\<bar> < e"
           using di \<open>e > 0\<close> y_def i xi by (auto simp: inner_simps)
         show "(\<Sum>i\<in>Basis - {i}. \<bar>(y-x) \<bullet> i\<bar>) \<le> (\<Sum>i\<in>Basis. 0)"
           unfolding y_def by (auto simp: inner_simps)
       qed 
       finally have "norm (y-x) < e + sum (\<lambda>i. 0) Basis" .
       then show "dist y x < e"
         unfolding dist_norm by auto
       have "y \<notin> K"
         unfolding keq mem_box using i(1) i(2) xi xyi by fastforce
       moreover have "y \<in> \<Union>\<D>"
         using subsetD[OF \<D>(2)[OF \<open>K \<in> \<D>\<close>] \<open>x \<in> K\<close>] \<open>e > 0\<close> di i
         by (auto simp: \<D> mem_box y_def field_simps elim!: ballE[of _ _ i])
       ultimately show "y \<in> \<Union>(\<D> - {K})" by auto
     qed
   qed
   then have "K \<subseteq> \<Union>(\<D> - {K})"
     using closed closed_limpt by blast
   then have "\<Union>(\<D> - {K}) = cbox a b"
     unfolding \<D>(6)[symmetric] by auto
   then have "\<D> - {K} division_of cbox a b"
     by (metis Diff_subset less.prems division_of_subset \<D>(6))
   then have "{ka \<in> \<D> - {K}. content ka \<noteq> 0} division_of (cbox a b)"
     using card less.hyps by blast
   moreover have "{ka \<in> \<D> - {K}. content ka \<noteq> 0} = {K \<in> \<D>. content K \<noteq> 0}"
     using contk by auto
   ultimately show ?case by auto
 qed
 
 
 subsection \<open>Integrability on subintervals\<close>
 
 lemma operative_integrableI:
   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::banach"
   assumes "0 \<le> e"
   shows "operative conj True (\<lambda>i. f integrable_on i)"
 proof -
   interpret comm_monoid conj True
-    by standard auto
-  have 1: "\<And>a b. box a b = {} \<Longrightarrow> f integrable_on cbox a b"
-    by (simp add: content_eq_0_interior integrable_on_null)
-  have 2: "\<And>a b c k.
-       \<lbrakk>k \<in> Basis;
-        f integrable_on cbox a b \<inter> {x. x \<bullet> k \<le> c};
-        f integrable_on cbox a b \<inter> {x. c \<le> x \<bullet> k}\<rbrakk>
-       \<Longrightarrow> f integrable_on cbox a b"
-    unfolding integrable_on_def by (auto intro!: has_integral_split)
+  proof qed
   show ?thesis
-    apply standard
-    using 1 2 by blast+
+  proof
+    show "\<And>a b. box a b = {} \<Longrightarrow> (f integrable_on cbox a b) = True"
+      by (simp add: content_eq_0_interior integrable_on_null)
+    show "\<And>a b c k.
+             k \<in> Basis \<Longrightarrow>
+             (f integrable_on cbox a b) \<longleftrightarrow>
+             (f integrable_on cbox a b \<inter> {x. x \<bullet> k \<le> c} \<and> f integrable_on cbox a b \<inter> {x. c \<le> x \<bullet> k})"
+      unfolding integrable_on_def by (auto intro!: has_integral_split)
+  qed
 qed
 
 lemma integrable_subinterval:
   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::banach"
   assumes f: "f integrable_on cbox a b"
     and cd: "cbox c d \<subseteq> cbox a b"
   shows "f integrable_on cbox c d"
 proof -
   interpret operative conj True "\<lambda>i. f integrable_on i"
     using order_refl by (rule operative_integrableI)
   show ?thesis
   proof (cases "cbox c d = {}")
     case True
     then show ?thesis
       using division [symmetric] f by (auto simp: comm_monoid_set_F_and)
   next
     case False
     then show ?thesis
       by (metis cd comm_monoid_set_F_and division division_of_finite f partial_division_extend_1)
   qed
 qed
 
 lemma integrable_subinterval_real:
   fixes f :: "real \<Rightarrow> 'a::banach"
   assumes "f integrable_on {a..b}"
     and "{c..d} \<subseteq> {a..b}"
   shows "f integrable_on {c..d}"
   by (metis assms box_real(2) integrable_subinterval)
 
 subsection \<open>Combining adjacent intervals in 1 dimension\<close>
 
 lemma has_integral_combine:
   fixes a b c :: real and j :: "'a::banach"
   assumes "a \<le> c"
       and "c \<le> b"
       and ac: "(f has_integral i) {a..c}"
       and cb: "(f has_integral j) {c..b}"
   shows "(f has_integral (i + j)) {a..b}"
 proof -
   interpret operative_real "lift_option plus" "Some 0"
     "\<lambda>i. if f integrable_on i then Some (integral i f) else None"
     using operative_integralI by (rule operative_realI)
   from \<open>a \<le> c\<close> \<open>c \<le> b\<close> ac cb coalesce_less_eq
   have *: "lift_option (+)
              (if f integrable_on {a..c} then Some (integral {a..c} f) else None)
              (if f integrable_on {c..b} then Some (integral {c..b} f) else None) =
             (if f integrable_on {a..b} then Some (integral {a..b} f) else None)"
     by (auto simp: split: if_split_asm)
   then have "f integrable_on cbox a b"
     using ac cb by (auto split: if_split_asm)
   with *
   show ?thesis
     using ac cb by (auto simp add: integrable_on_def integral_unique split: if_split_asm)
 qed
 
 lemma integral_combine:
   fixes f :: "real \<Rightarrow> 'a::banach"
   assumes "a \<le> c"
     and "c \<le> b"
     and ab: "f integrable_on {a..b}"
   shows "integral {a..c} f + integral {c..b} f = integral {a..b} f"
 proof -
   have "(f has_integral integral {a..c} f) {a..c}"
     using ab \<open>c \<le> b\<close> integrable_subinterval_real by fastforce
   moreover
   have "(f has_integral integral {c..b} f) {c..b}"
     using ab \<open>a \<le> c\<close> integrable_subinterval_real by fastforce
   ultimately have "(f has_integral integral {a..c} f + integral {c..b} f) {a..b}"
     using \<open>a \<le> c\<close> \<open>c \<le> b\<close> has_integral_combine by blast
   then show ?thesis
     by (simp add: has_integral_integrable_integral)
 qed
 
 lemma integrable_combine:
   fixes f :: "real \<Rightarrow> 'a::banach"
   assumes "a \<le> c"
     and "c \<le> b"
     and "f integrable_on {a..c}"
     and "f integrable_on {c..b}"
   shows "f integrable_on {a..b}"
   using assms
   unfolding integrable_on_def
   by (auto intro!: has_integral_combine)
 
 lemma integral_minus_sets:
   fixes f::"real \<Rightarrow> 'a::banach"
   shows "c \<le> a \<Longrightarrow> c \<le> b \<Longrightarrow> f integrable_on {c .. max a b} \<Longrightarrow>
     integral {c .. a} f - integral {c .. b} f =
     (if a \<le> b then - integral {a .. b} f else integral {b .. a} f)"
   using integral_combine[of c a b f]  integral_combine[of c b a f]
   by (auto simp: algebra_simps max_def)
 
 lemma integral_minus_sets':
   fixes f::"real \<Rightarrow> 'a::banach"
   shows "c \<ge> a \<Longrightarrow> c \<ge> b \<Longrightarrow> f integrable_on {min a b .. c} \<Longrightarrow>
     integral {a .. c} f - integral {b .. c} f =
     (if a \<le> b then integral {a .. b} f else - integral {b .. a} f)"
   using integral_combine[of b a c f] integral_combine[of a b c f]
   by (auto simp: algebra_simps min_def)
 
 subsection \<open>Reduce integrability to "local" integrability\<close>
 
 lemma integrable_on_little_subintervals:
   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::banach"
   assumes "\<forall>x\<in>cbox a b. \<exists>d>0. \<forall>u v. x \<in> cbox u v \<and> cbox u v \<subseteq> ball x d \<and> cbox u v \<subseteq> cbox a b \<longrightarrow>
     f integrable_on cbox u v"
   shows "f integrable_on cbox a b"
 proof -
   interpret operative conj True "\<lambda>i. f integrable_on i"
     using order_refl by (rule operative_integrableI)
   have "\<forall>x. \<exists>d>0. x\<in>cbox a b \<longrightarrow> (\<forall>u v. x \<in> cbox u v \<and> cbox u v \<subseteq> ball x d \<and> cbox u v \<subseteq> cbox a b \<longrightarrow>
     f integrable_on cbox u v)"
     using assms by (metis zero_less_one)
   then obtain d where d: "\<And>x. 0 < d x"
      "\<And>x u v. \<lbrakk>x \<in> cbox a b; x \<in> cbox u v; cbox u v \<subseteq> ball x (d x); cbox u v \<subseteq> cbox a b\<rbrakk> 
                \<Longrightarrow> f integrable_on cbox u v"
     by metis
   obtain p where p: "p tagged_division_of cbox a b" "(\<lambda>x. ball x (d x)) fine p"
     using fine_division_exists[OF gauge_ball_dependent,of d a b] d(1) by blast 
   then have sndp: "snd ` p division_of cbox a b"
     by (metis division_of_tagged_division)
   have "f integrable_on k" if "(x, k) \<in> p" for x k
     using tagged_division_ofD(2-4)[OF p(1) that] fineD[OF p(2) that] d[of x] by auto
   then show ?thesis
     unfolding division [symmetric, OF sndp] comm_monoid_set_F_and
     by auto
 qed
 
 
 subsection \<open>Second FTC or existence of antiderivative\<close>
 
 lemma integrable_const[intro]: "(\<lambda>x. c) integrable_on cbox a b"
   unfolding integrable_on_def by blast
 
 lemma integral_has_vector_derivative_continuous_at:
   fixes f :: "real \<Rightarrow> 'a::banach"
   assumes f: "f integrable_on {a..b}"
      and x: "x \<in> {a..b} - S"
      and "finite S"
      and fx: "continuous (at x within ({a..b} - S)) f"
  shows "((\<lambda>u. integral {a..u} f) has_vector_derivative f x) (at x within ({a..b} - S))"
 proof -
   let ?I = "\<lambda>a b. integral {a..b} f"
   { fix e::real
     assume "e > 0"
     obtain d where "d>0" and d: "\<And>x'. \<lbrakk>x' \<in> {a..b} - S; \<bar>x' - x\<bar> < d\<rbrakk> \<Longrightarrow> norm(f x' - f x) \<le> e"
       using \<open>e>0\<close> fx by (auto simp: continuous_within_eps_delta dist_norm less_imp_le)
-    have "norm (integral {a..y} f - integral {a..x} f - (y-x) *\<^sub>R f x) \<le> e * \<bar>y - x\<bar>"
+    have "norm (integral {a..y} f - integral {a..x} f - (y-x) *\<^sub>R f x) \<le> e * \<bar>y - x\<bar>" (is "?lhs \<le> ?rhs")
            if y: "y \<in> {a..b} - S" and yx: "\<bar>y - x\<bar> < d" for y
     proof (cases "y < x")
       case False
       have "f integrable_on {a..y}"
         using f y by (simp add: integrable_subinterval_real)
       then have Idiff: "?I a y - ?I a x = ?I x y"
         using False x by (simp add: algebra_simps integral_combine)
       have fux_int: "((\<lambda>u. f u - f x) has_integral integral {x..y} f - (y-x) *\<^sub>R f x) {x..y}"
-        apply (rule has_integral_diff)
-        using x y apply (auto intro: integrable_integral [OF integrable_subinterval_real [OF f]])
-        using has_integral_const_real [of "f x" x y] False
-        apply simp
-        done
-      have "\<And>xa. y - x < d \<Longrightarrow> (\<And>x'. a \<le> x' \<and> x' \<le> b \<and> x' \<notin> S \<Longrightarrow> \<bar>x' - x\<bar> < d \<Longrightarrow> norm (f x' - f x) \<le> e) \<Longrightarrow> 0 < e \<Longrightarrow> xa \<notin> S \<Longrightarrow> a \<le> x \<Longrightarrow> x \<notin> S \<Longrightarrow> y \<le> b \<Longrightarrow> y \<notin> S \<Longrightarrow> x \<le> xa \<Longrightarrow> xa \<le> y \<Longrightarrow> norm (f xa - f x) \<le> e"
-        using assms by auto
-      show ?thesis
-        using False
-        apply (simp add: abs_eq_content del: content_real_if measure_lborel_Icc)
-        apply (rule has_integral_bound_real[where f="(\<lambda>u. f u - f x)"])
-        using yx False d x y \<open>e>0\<close> assms by (auto simp: Idiff fux_int)
+      proof (rule has_integral_diff)
+        show "(f has_integral integral {x..y} f) {x..y}"
+          using x y by (auto intro: integrable_integral [OF integrable_subinterval_real [OF f]])
+        show "((\<lambda>u. f x) has_integral (y - x) *\<^sub>R f x) {x..y}"
+          using has_integral_const_real [of "f x" x y] False by simp
+      qed
+      have "?lhs \<le> e * content {x..y}"
+        using yx False d x y \<open>e>0\<close> assms 
+        by (intro has_integral_bound_real[where f="(\<lambda>u. f u - f x)"]) (auto simp: Idiff fux_int)
+      also have "... \<le> ?rhs"
+        using False by auto
+      finally show ?thesis .
     next
       case True
       have "f integrable_on {a..x}"
         using f x by (simp add: integrable_subinterval_real)
       then have Idiff: "?I a x - ?I a y = ?I y x"
         using True x y by (simp add: algebra_simps integral_combine)
       have fux_int: "((\<lambda>u. f u - f x) has_integral integral {y..x} f - (x - y) *\<^sub>R f x) {y..x}"
-        apply (rule has_integral_diff)
-        using x y apply (auto intro: integrable_integral [OF integrable_subinterval_real [OF f]])
-        using has_integral_const_real [of "f x" y x] True
-        apply simp
-        done
-      have "norm (integral {a..x} f - integral {a..y} f - (x - y) *\<^sub>R f x) \<le> e * \<bar>y - x\<bar>"
-        using True
-        apply (simp add: abs_eq_content del: content_real_if measure_lborel_Icc)
-        apply (rule has_integral_bound_real[where f="(\<lambda>u. f u - f x)"])
-        using yx True d x y \<open>e>0\<close> assms by (auto simp: Idiff fux_int)
+      proof (rule has_integral_diff)
+        show "(f has_integral integral {y..x} f) {y..x}"
+          using x y by (auto intro: integrable_integral [OF integrable_subinterval_real [OF f]])
+        show "((\<lambda>u. f x) has_integral (x - y) *\<^sub>R f x) {y..x}"
+          using has_integral_const_real [of "f x" y x] True by simp
+      qed
+      have "norm (integral {a..x} f - integral {a..y} f - (x - y) *\<^sub>R f x) \<le> e * content {y..x}"
+        using yx True d x y \<open>e>0\<close> assms 
+        by (intro has_integral_bound_real[where f="(\<lambda>u. f u - f x)"]) (auto simp: Idiff fux_int)
+      also have "... \<le> e * \<bar>y - x\<bar>"
+        using True by auto
+      finally have "norm (integral {a..x} f - integral {a..y} f - (x - y) *\<^sub>R f x) \<le> e * \<bar>y - x\<bar>" .
       then show ?thesis
         by (simp add: algebra_simps norm_minus_commute)
     qed
     then have "\<exists>d>0. \<forall>y\<in>{a..b} - S. \<bar>y - x\<bar> < d \<longrightarrow> norm (integral {a..y} f - integral {a..x} f - (y-x) *\<^sub>R f x) \<le> e * \<bar>y - x\<bar>"
       using \<open>d>0\<close> by blast
   }
   then show ?thesis
     by (simp add: has_vector_derivative_def has_derivative_within_alt bounded_linear_scaleR_left)
 qed
 
 
 lemma integral_has_vector_derivative:
   fixes f :: "real \<Rightarrow> 'a::banach"
   assumes "continuous_on {a..b} f"
     and "x \<in> {a..b}"
   shows "((\<lambda>u. integral {a..u} f) has_vector_derivative f(x)) (at x within {a..b})"
 using assms integral_has_vector_derivative_continuous_at [OF integrable_continuous_real]
   by (fastforce simp: continuous_on_eq_continuous_within)
 
 lemma integral_has_real_derivative:
   assumes "continuous_on {a..b} g"
   assumes "t \<in> {a..b}"
   shows "((\<lambda>x. integral {a..x} g) has_real_derivative g t) (at t within {a..b})"
   using integral_has_vector_derivative[of a b g t] assms
   by (auto simp: has_field_derivative_iff_has_vector_derivative)
 
 lemma antiderivative_continuous:
   fixes q b :: real
   assumes "continuous_on {a..b} f"
   obtains g where "\<And>x. x \<in> {a..b} \<Longrightarrow> (g has_vector_derivative (f x::_::banach)) (at x within {a..b})"
   using integral_has_vector_derivative[OF assms] by auto
 
 subsection \<open>Combined fundamental theorem of calculus\<close>
 
 lemma antiderivative_integral_continuous:
   fixes f :: "real \<Rightarrow> 'a::banach"
   assumes "continuous_on {a..b} f"
   obtains g where "\<forall>u\<in>{a..b}. \<forall>v \<in> {a..b}. u \<le> v \<longrightarrow> (f has_integral (g v - g u)) {u..v}"
 proof -
   obtain g 
     where g: "\<And>x. x \<in> {a..b} \<Longrightarrow> (g has_vector_derivative f x) (at x within {a..b})" 
     using  antiderivative_continuous[OF assms] by metis
   have "(f has_integral g v - g u) {u..v}" if "u \<in> {a..b}" "v \<in> {a..b}" "u \<le> v" for u v
   proof -
     have "\<And>x. x \<in> cbox u v \<Longrightarrow> (g has_vector_derivative f x) (at x within cbox u v)"
       by (metis atLeastAtMost_iff atLeastatMost_subset_iff box_real(2) g
           has_vector_derivative_within_subset subsetCE that(1,2))
     then show ?thesis
       by (metis box_real(2) that(3) fundamental_theorem_of_calculus)
   qed
   then show ?thesis
     using that by blast
 qed
 
 
 subsection \<open>General "twiddling" for interval-to-interval function image\<close>
 
 lemma has_integral_twiddle:
   assumes "0 < r"
     and hg: "\<And>x. h(g x) = x"
     and gh: "\<And>x. g(h x) = x"
     and contg: "\<And>x. continuous (at x) g"
     and g: "\<And>u v. \<exists>w z. g ` cbox u v = cbox w z"
     and h: "\<And>u v. \<exists>w z. h ` cbox u v = cbox w z"
     and r: "\<And>u v. content(g ` cbox u v) = r * content (cbox u v)"
     and intfi: "(f has_integral i) (cbox a b)"
   shows "((\<lambda>x. f(g x)) has_integral (1 / r) *\<^sub>R i) (h ` cbox a b)"
 proof (cases "cbox a b = {}")
   case True
   then show ?thesis 
     using intfi by auto
 next
   case False
   obtain w z where wz: "h ` cbox a b = cbox w z"
     using h by blast
   have inj: "inj g" "inj h"
     using hg gh injI by metis+
   from h obtain ha hb where h_eq: "h ` cbox a b = cbox ha hb" by blast
   have "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of h ` cbox a b \<and> d fine p 
               \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f (g x)) - (1 / r) *\<^sub>R i) < e)"
     if "e > 0" for e
   proof -
     have "e * r > 0" using that \<open>0 < r\<close> by simp
     with intfi[unfolded has_integral]
     obtain d where "gauge d"
                and d: "\<And>p. p tagged_division_of cbox a b \<and> d fine p 
                         \<Longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - i) < e * r" 
       by metis
     define d' where "d' x = g -` d (g x)" for x
     show ?thesis
     proof (rule_tac x=d' in exI, safe)
       show "gauge d'"
         using \<open>gauge d\<close> continuous_open_vimage[OF _ contg] by (auto simp: gauge_def d'_def)
     next
       fix p
       assume ptag: "p tagged_division_of h ` cbox a b" and finep: "d' fine p"
       note p = tagged_division_ofD[OF ptag]
       have gab: "g y \<in> cbox a b" if "y \<in> K" "(x, K) \<in> p" for x y K
         by (metis hg inj(2) inj_image_mem_iff p(3) subsetCE that that)
       have gimp: "(\<lambda>(x,K). (g x, g ` K)) ` p tagged_division_of (cbox a b) \<and> 
                   d fine (\<lambda>(x, k). (g x, g ` k)) ` p"
         unfolding tagged_division_of
       proof safe
         show "finite ((\<lambda>(x, k). (g x, g ` k)) ` p)"
           using ptag by auto
         show "d fine (\<lambda>(x, k). (g x, g ` k)) ` p"
           using finep unfolding fine_def d'_def by auto
       next
         fix x k
         assume xk: "(x, k) \<in> p"
         show "g x \<in> g ` k"
           using p(2)[OF xk] by auto
         show "\<exists>u v. g ` k = cbox u v"
           using p(4)[OF xk] using assms(5-6) by auto
         fix x' K' u
         assume xk': "(x', K') \<in> p" and u: "u \<in> interior (g ` k)" "u \<in> interior (g ` K')"
         have "interior k \<inter> interior K' \<noteq> {}"
         proof 
           assume "interior k \<inter> interior K' = {}"
           moreover have "u \<in> g ` (interior k \<inter> interior K')"
             using interior_image_subset[OF \<open>inj g\<close> contg] u
             unfolding image_Int[OF inj(1)] by blast
           ultimately show False by blast
         qed
         then have same: "(x, k) = (x', K')"
           using ptag xk' xk by blast
         then show "g x = g x'"
           by auto
         show "g u \<in> g ` K'"if "u \<in> k" for u
           using that same by auto
         show "g u \<in> g ` k"if "u \<in> K'" for u
           using that same by auto
       next
         fix x
         assume "x \<in> cbox a b"
         then have "h x \<in>  \<Union>{k. \<exists>x. (x, k) \<in> p}"
           using p(6) by auto
         then obtain X y where "h x \<in> X" "(y, X) \<in> p" by blast
         then show "x \<in> \<Union>{k. \<exists>x. (x, k) \<in> (\<lambda>(x, k). (g x, g ` k)) ` p}"
-          apply clarsimp
-          by (metis (no_types, lifting) assms(3) image_eqI pair_imageI)
+          by clarsimp (metis (no_types, lifting) gh image_eqI pair_imageI)
       qed (use gab in auto)
       have *: "inj_on (\<lambda>(x, k). (g x, g ` k)) p"
         using inj(1) unfolding inj_on_def by fastforce
-      have "(\<Sum>(x, k)\<in>(\<lambda>(x, k). (g x, g ` k)) ` p. content k *\<^sub>R f x) - i = r *\<^sub>R (\<Sum>(x, k)\<in>p. content k *\<^sub>R f (g x)) - i" (is "?l = _")
-        using r
-        apply (simp only: algebra_simps add_left_cancel scaleR_right.sum)
-        apply (subst sum.reindex_bij_betw[symmetric, where h="\<lambda>(x, k). (g x, g ` k)" and S=p])
-         apply (auto intro!: * sum.cong simp: bij_betw_def dest!: p(4))
-        done
-      also have "\<dots> = r *\<^sub>R ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f (g x)) - (1 / r) *\<^sub>R i)" (is "_ = ?r")
+      have "(\<Sum>(x,K)\<in>(\<lambda>(y,L). (g y, g ` L)) ` p. content K *\<^sub>R f x) 
+          = (\<Sum>u\<in>p. case case u of (x,K) \<Rightarrow> (g x, g ` K) of (y,L) \<Rightarrow> content L *\<^sub>R f y)"
+        by (metis (mono_tags, lifting) "*" sum.reindex_cong)
+      also have "... = (\<Sum>(x,K)\<in>p. r *\<^sub>R content K *\<^sub>R f (g x))"
+        using r by (auto intro!: * sum.cong simp: bij_betw_def dest!: p(4))
+      finally
+      have "(\<Sum>(x, K)\<in>(\<lambda>(x,K). (g x, g ` K)) ` p. content K *\<^sub>R f x) - i = r *\<^sub>R (\<Sum>(x,K)\<in>p. content K *\<^sub>R f (g x)) - i" 
+        by (simp add: scaleR_right.sum split_def)
+      also have "\<dots> = r *\<^sub>R ((\<Sum>(x,K)\<in>p. content K *\<^sub>R f (g x)) - (1 / r) *\<^sub>R i)" 
         using \<open>0 < r\<close> by (auto simp: scaleR_diff_right)
-      finally have eq: "?l = ?r" .
-      show "norm ((\<Sum>(x,K)\<in>p. content K *\<^sub>R f (g x)) - (1 / r) *\<^sub>R i) < e"
-        using d[OF gimp] \<open>0 < r\<close> by (auto simp add: eq)
+      finally show "norm ((\<Sum>(x,K)\<in>p. content K *\<^sub>R f (g x)) - (1 / r) *\<^sub>R i) < e"
+        using d[OF gimp] \<open>0 < r\<close> by auto
     qed
   qed
   then show ?thesis
     by (auto simp: h_eq has_integral)
 qed
 
 
 subsection \<open>Special case of a basic affine transformation\<close>
 
 lemma AE_lborel_inner_neq:
   assumes k: "k \<in> Basis"
   shows "AE x in lborel. x \<bullet> k \<noteq> c"
 proof -
   interpret finite_product_sigma_finite "\<lambda>_. lborel" Basis
     proof qed simp
-
-  have "emeasure lborel {x\<in>space lborel. x \<bullet> k = c} = emeasure (\<Pi>\<^sub>M j::'a\<in>Basis. lborel) (\<Pi>\<^sub>E j\<in>Basis. if j = k then {c} else UNIV)"
+    have "emeasure lborel {x\<in>space lborel. x \<bullet> k = c} 
+        = emeasure (\<Pi>\<^sub>M j::'a\<in>Basis. lborel) (\<Pi>\<^sub>E j\<in>Basis. if j = k then {c} else UNIV)"
     using k
     by (auto simp add: lborel_eq[where 'a='a] emeasure_distr intro!: arg_cong2[where f=emeasure])
        (auto simp: space_PiM PiE_iff extensional_def split: if_split_asm)
   also have "\<dots> = (\<Prod>j\<in>Basis. emeasure lborel (if j = k then {c} else UNIV))"
     by (intro measure_times) auto
   also have "\<dots> = 0"
     by (intro prod_zero bexI[OF _ k]) auto
   finally show ?thesis
     by (subst AE_iff_measurable[OF _ refl]) auto
 qed
 
 lemma content_image_stretch_interval:
   fixes m :: "'a::euclidean_space \<Rightarrow> real"
   defines "s f x \<equiv> (\<Sum>k::'a\<in>Basis. (f k * (x\<bullet>k)) *\<^sub>R k)"
   shows "content (s m ` cbox a b) = \<bar>\<Prod>k\<in>Basis. m k\<bar> * content (cbox a b)"
 proof cases
   have s[measurable]: "s f \<in> borel \<rightarrow>\<^sub>M borel" for f
     by (auto simp: s_def[abs_def])
   assume m: "\<forall>k\<in>Basis. m k \<noteq> 0"
   then have s_comp_s: "s (\<lambda>k. 1 / m k) \<circ> s m = id" "s m \<circ> s (\<lambda>k. 1 / m k) = id"
     by (auto simp: s_def[abs_def] fun_eq_iff euclidean_representation)
   then have "inv (s (\<lambda>k. 1 / m k)) = s m" "bij (s (\<lambda>k. 1 / m k))"
     by (auto intro: inv_unique_comp o_bij)
   then have eq: "s m ` cbox a b = s (\<lambda>k. 1 / m k) -` cbox a b"
     using bij_vimage_eq_inv_image[OF \<open>bij (s (\<lambda>k. 1 / m k))\<close>, of "cbox a b"] by auto
   show ?thesis
     using m unfolding eq measure_def
     by (subst lborel_affine_euclidean[where c=m and t=0])
        (simp_all add: emeasure_density measurable_sets_borel[OF s] abs_prod nn_integral_cmult
                       s_def[symmetric] emeasure_distr vimage_comp s_comp_s enn2real_mult prod_nonneg)
 next
   assume "\<not> (\<forall>k\<in>Basis. m k \<noteq> 0)"
   then obtain k where k: "k \<in> Basis" "m k = 0" by auto
   then have [simp]: "(\<Prod>k\<in>Basis. m k) = 0"
     by (intro prod_zero) auto
   have "emeasure lborel {x\<in>space lborel. x \<in> s m ` cbox a b} = 0"
   proof (rule emeasure_eq_0_AE)
     show "AE x in lborel. x \<notin> s m ` cbox a b"
       using AE_lborel_inner_neq[OF \<open>k\<in>Basis\<close>]
     proof eventually_elim
       show "x \<bullet> k \<noteq> 0 \<Longrightarrow> x \<notin> s m ` cbox a b " for x
         using k by (auto simp: s_def[abs_def] cbox_def)
     qed
   qed
   then show ?thesis
     by (simp add: measure_def)
 qed
 
 lemma interval_image_affinity_interval:
   "\<exists>u v. (\<lambda>x. m *\<^sub>R (x::'a::euclidean_space) + c) ` cbox a b = cbox u v"
   unfolding image_affinity_cbox
   by auto
 
 lemma content_image_affinity_cbox:
   "content((\<lambda>x::'a::euclidean_space. m *\<^sub>R x + c) ` cbox a b) =
     \<bar>m\<bar> ^ DIM('a) * content (cbox a b)" (is "?l = ?r")
 proof (cases "cbox a b = {}")
   case True then show ?thesis by simp
 next
   case False
   show ?thesis
   proof (cases "m \<ge> 0")
     case True
     with \<open>cbox a b \<noteq> {}\<close> have "cbox (m *\<^sub>R a + c) (m *\<^sub>R b + c) \<noteq> {}"
-      unfolding box_ne_empty
-      apply (intro ballI)
-      apply (erule_tac x=i in ballE)
-      apply (auto simp: inner_simps mult_left_mono)
-      done
+      by (simp add: box_ne_empty inner_left_distrib mult_left_mono)
     moreover from True have *: "\<And>i. (m *\<^sub>R b + c) \<bullet> i - (m *\<^sub>R a + c) \<bullet> i = m *\<^sub>R (b-a) \<bullet> i"
       by (simp add: inner_simps field_simps)
     ultimately show ?thesis
       by (simp add: image_affinity_cbox True content_cbox'
         prod.distrib inner_diff_left)
   next
     case False
     with \<open>cbox a b \<noteq> {}\<close> have "cbox (m *\<^sub>R b + c) (m *\<^sub>R a + c) \<noteq> {}"
-      unfolding box_ne_empty
-      apply (intro ballI)
-      apply (erule_tac x=i in ballE)
-      apply (auto simp: inner_simps mult_left_mono)
-      done
+      by (simp add: box_ne_empty inner_left_distrib mult_left_mono)
     moreover from False have *: "\<And>i. (m *\<^sub>R a + c) \<bullet> i - (m *\<^sub>R b + c) \<bullet> i = (-m) *\<^sub>R (b-a) \<bullet> i"
       by (simp add: inner_simps field_simps)
     ultimately show ?thesis using False
       by (simp add: image_affinity_cbox content_cbox'
         prod.distrib[symmetric] inner_diff_left flip: prod_constant)
   qed
 qed
 
 lemma has_integral_affinity:
   fixes a :: "'a::euclidean_space"
   assumes "(f has_integral i) (cbox a b)"
       and "m \<noteq> 0"
-  shows "((\<lambda>x. f(m *\<^sub>R x + c)) has_integral ((1 / (\<bar>m\<bar> ^ DIM('a))) *\<^sub>R i)) ((\<lambda>x. (1 / m) *\<^sub>R x + -((1 / m) *\<^sub>R c)) ` cbox a b)"
-  apply (rule has_integral_twiddle)
-  using assms
-  apply (safe intro!: interval_image_affinity_interval content_image_affinity_cbox)
-  apply (rule zero_less_power)
-  unfolding scaleR_right_distrib
-  apply auto
-  done
+  shows "((\<lambda>x. f(m *\<^sub>R x + c)) has_integral (1 / (\<bar>m\<bar> ^ DIM('a))) *\<^sub>R i) ((\<lambda>x. (1 / m) *\<^sub>R x + -((1 / m) *\<^sub>R c)) ` cbox a b)"
+proof (rule has_integral_twiddle)
+  show "\<exists>w z. (\<lambda>x. (1 / m) *\<^sub>R x + - ((1 / m) *\<^sub>R c)) ` cbox u v = cbox w z" 
+       "\<exists>w z. (\<lambda>x. m *\<^sub>R x + c) ` cbox u v = cbox w z" for u v
+    using interval_image_affinity_interval by blast+
+  show "content ((\<lambda>x. m *\<^sub>R x + c) ` cbox u v) = \<bar>m\<bar> ^ DIM('a) * content (cbox u v)" for u v
+    using content_image_affinity_cbox by blast
+qed (use assms zero_less_power in \<open>auto simp: field_simps\<close>)
 
 lemma integrable_affinity:
   assumes "f integrable_on cbox a b"
     and "m \<noteq> 0"
   shows "(\<lambda>x. f(m *\<^sub>R x + c)) integrable_on ((\<lambda>x. (1 / m) *\<^sub>R x + -((1/m) *\<^sub>R c)) ` cbox a b)"
-  using assms
-  unfolding integrable_on_def
-  apply safe
-  apply (drule has_integral_affinity)
-  apply auto
-  done
+  using has_integral_affinity assms
+  unfolding integrable_on_def by blast
 
 lemmas has_integral_affinity01 = has_integral_affinity [of _ _ 0 "1::real", simplified]
 
 lemma integrable_on_affinity:
   assumes "m \<noteq> 0" "f integrable_on (cbox a b)"
   shows   "(\<lambda>x. f (m *\<^sub>R x + c)) integrable_on ((\<lambda>x. (1 / m) *\<^sub>R x - ((1 / m) *\<^sub>R c)) ` cbox a b)"
 proof -
   from assms obtain I where "(f has_integral I) (cbox a b)"
     by (auto simp: integrable_on_def)
   from has_integral_affinity[OF this assms(1), of c] show ?thesis
     by (auto simp: integrable_on_def)
 qed
 
 lemma has_integral_cmul_iff:
   assumes "c \<noteq> 0"
   shows   "((\<lambda>x. c *\<^sub>R f x) has_integral (c *\<^sub>R I)) A \<longleftrightarrow> (f has_integral I) A"
   using assms has_integral_cmul[of f I A c]
         has_integral_cmul[of "\<lambda>x. c *\<^sub>R f x" "c *\<^sub>R I" A "inverse c"] by (auto simp: field_simps)
 
 lemma has_integral_affinity':
   fixes a :: "'a::euclidean_space"
   assumes "(f has_integral i) (cbox a b)" and "m > 0"
   shows "((\<lambda>x. f(m *\<^sub>R x + c)) has_integral (i /\<^sub>R m ^ DIM('a)))
            (cbox ((a - c) /\<^sub>R m) ((b - c) /\<^sub>R m))"
 proof (cases "cbox a b = {}")
   case True
   hence "(cbox ((a - c) /\<^sub>R m) ((b - c) /\<^sub>R m)) = {}"
     using \<open>m > 0\<close> unfolding box_eq_empty by (auto simp: algebra_simps)
   with True and assms show ?thesis by simp
 next
   case False
   have "((\<lambda>x. f (m *\<^sub>R x + c)) has_integral (1 / \<bar>m\<bar> ^ DIM('a)) *\<^sub>R i)
           ((\<lambda>x. (1 / m) *\<^sub>R x + - ((1 / m) *\<^sub>R c)) ` cbox a b)"
     using assms by (intro has_integral_affinity) auto
   also have "((\<lambda>x. (1 / m) *\<^sub>R x + - ((1 / m) *\<^sub>R c)) ` cbox a b) =
                ((\<lambda>x.  - ((1 / m) *\<^sub>R c) + x) ` (\<lambda>x. (1 / m) *\<^sub>R x) ` cbox a b)"
     by (simp add: image_image algebra_simps)
   also have "(\<lambda>x. (1 / m) *\<^sub>R x) ` cbox a b = cbox ((1 / m) *\<^sub>R a) ((1 / m) *\<^sub>R b)" using \<open>m > 0\<close> False
     by (subst image_smult_cbox) simp_all
   also have "(\<lambda>x. - ((1 / m) *\<^sub>R c) + x) ` \<dots> = cbox ((a - c) /\<^sub>R m) ((b - c) /\<^sub>R m)"
     by (subst cbox_translation [symmetric]) (simp add: field_simps vector_add_divide_simps)
   finally show ?thesis using \<open>m > 0\<close> by (simp add: field_simps)
 qed
 
 lemma has_integral_affinity_iff:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: real_normed_vector"
   assumes "m > 0"
   shows   "((\<lambda>x. f (m *\<^sub>R x + c)) has_integral (I /\<^sub>R m ^ DIM('a)))
                (cbox ((a - c) /\<^sub>R m) ((b - c) /\<^sub>R m)) \<longleftrightarrow>
            (f has_integral I) (cbox a b)" (is "?lhs = ?rhs")
 proof
   assume ?lhs
   from has_integral_affinity'[OF this, of "1 / m" "-c /\<^sub>R m"] and \<open>m > 0\<close>
     show ?rhs by (simp add: vector_add_divide_simps) (simp add: field_simps)
 next
   assume ?rhs
   from has_integral_affinity'[OF this, of m c] and \<open>m > 0\<close>
   show ?lhs by simp
 qed
 
 
 subsection \<open>Special case of stretching coordinate axes separately\<close>
 
 lemma has_integral_stretch:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
   assumes "(f has_integral i) (cbox a b)"
     and "\<forall>k\<in>Basis. m k \<noteq> 0"
   shows "((\<lambda>x. f (\<Sum>k\<in>Basis. (m k * (x\<bullet>k))*\<^sub>R k)) has_integral
          ((1/ \<bar>prod m Basis\<bar>) *\<^sub>R i)) ((\<lambda>x. (\<Sum>k\<in>Basis. (1 / m k * (x\<bullet>k))*\<^sub>R k)) ` cbox a b)"
   apply (rule has_integral_twiddle[where f=f])
   unfolding zero_less_abs_iff content_image_stretch_interval
   unfolding image_stretch_interval empty_as_interval euclidean_eq_iff[where 'a='a]
   using assms
   by auto
 
 lemma has_integral_stretch_real:
   fixes f :: "real \<Rightarrow> 'b::real_normed_vector"
   assumes "(f has_integral i) {a..b}" and "m \<noteq> 0"
   shows "((\<lambda>x. f (m * x)) has_integral (1 / \<bar>m\<bar>) *\<^sub>R i) ((\<lambda>x. x / m) ` {a..b})"
   using has_integral_stretch [of f i a b "\<lambda>b. m"] assms by simp
 
 lemma integrable_stretch:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
   assumes "f integrable_on cbox a b"
     and "\<forall>k\<in>Basis. m k \<noteq> 0"
   shows "(\<lambda>x::'a. f (\<Sum>k\<in>Basis. (m k * (x\<bullet>k))*\<^sub>R k)) integrable_on
     ((\<lambda>x. \<Sum>k\<in>Basis. (1 / m k * (x\<bullet>k))*\<^sub>R k) ` cbox a b)"
   using assms unfolding integrable_on_def
   by (force dest: has_integral_stretch)
 
 lemma vec_lambda_eq_sum:
-  shows "(\<chi> k. f k (x $ k)) = (\<Sum>k\<in>Basis. (f (axis_index k) (x \<bullet> k)) *\<^sub>R k)"
-    apply (simp add: Basis_vec_def cart_eq_inner_axis UNION_singleton_eq_range sum.reindex axis_eq_axis inj_on_def)
-    apply (simp add: vec_eq_iff axis_def if_distrib cong: if_cong)
-    done
+     "(\<chi> k. f k (x $ k)) = (\<Sum>k\<in>Basis. (f (axis_index k) (x \<bullet> k)) *\<^sub>R k)" (is "?lhs = ?rhs")
+proof -
+  have "?lhs = (\<chi> k. f k (x \<bullet> axis k 1))"
+    by (simp add: cart_eq_inner_axis)
+  also have "... = (\<Sum>u\<in>UNIV. f u (x \<bullet> axis u 1) *\<^sub>R axis u 1)"
+    by (simp add: vec_eq_iff axis_def if_distrib cong: if_cong)
+  also have "... = ?rhs"
+    by (simp add: Basis_vec_def UNION_singleton_eq_range sum.reindex axis_eq_axis inj_on_def)
+  finally show ?thesis .
+qed
 
 lemma has_integral_stretch_cart:
   fixes m :: "'n::finite \<Rightarrow> real"
   assumes f: "(f has_integral i) (cbox a b)" and m: "\<And>k. m k \<noteq> 0"
   shows "((\<lambda>x. f(\<chi> k. m k * x$k)) has_integral i /\<^sub>R \<bar>prod m UNIV\<bar>)
             ((\<lambda>x. \<chi> k. x$k / m k) ` (cbox a b))"
 proof -
   have *: "\<forall>k:: real^'n \<in> Basis. m (axis_index k) \<noteq> 0"
     using axis_index by (simp add: m)
   have eqp: "(\<Prod>k:: real^'n \<in> Basis. m (axis_index k)) = prod m UNIV"
     by (simp add: Basis_vec_def UNION_singleton_eq_range prod.reindex axis_eq_axis inj_on_def)
   show ?thesis
     using has_integral_stretch [OF f *] vec_lambda_eq_sum [where f="\<lambda>i x. m i * x"] vec_lambda_eq_sum [where f="\<lambda>i x. x / m i"]
     by (simp add: field_simps eqp)
 qed
 
 lemma image_stretch_interval_cart:
   fixes m :: "'n::finite \<Rightarrow> real"
   shows "(\<lambda>x. \<chi> k. m k * x$k) ` cbox a b =
             (if cbox a b = {} then {}
             else cbox (\<chi> k. min (m k * a$k) (m k * b$k)) (\<chi> k. max (m k * a$k) (m k * b$k)))"
 proof -
   have *: "(\<Sum>k\<in>Basis. min (m (axis_index k) * (a \<bullet> k)) (m (axis_index k) * (b \<bullet> k)) *\<^sub>R k)
            = (\<chi> k. min (m k * a $ k) (m k * b $ k))"
           "(\<Sum>k\<in>Basis. max (m (axis_index k) * (a \<bullet> k)) (m (axis_index k) * (b \<bullet> k)) *\<^sub>R k)
            = (\<chi> k. max (m k * a $ k) (m k * b $ k))"
     apply (simp_all add: Basis_vec_def cart_eq_inner_axis UNION_singleton_eq_range sum.reindex axis_eq_axis inj_on_def)
     apply (simp_all add: vec_eq_iff axis_def if_distrib cong: if_cong)
     done
   show ?thesis
     by (simp add: vec_lambda_eq_sum [where f="\<lambda>i x. m i * x"] image_stretch_interval eq_cbox *)
 qed
 
 
 subsection \<open>even more special cases\<close>
 
 lemma uminus_interval_vector[simp]:
   fixes a b :: "'a::euclidean_space"
   shows "uminus ` cbox a b = cbox (-b) (-a)"
-  apply safe
-   apply (simp add: mem_box(2))
-  apply (rule_tac x="-x" in image_eqI)
-   apply (auto simp add: mem_box)
-  done
+proof -
+  have "x \<in> uminus ` cbox a b" if "x \<in> cbox (- b) (- a)" for x
+  proof -
+    have "-x \<in> cbox a b"
+      using that by (auto simp: mem_box)
+    then show ?thesis
+      by force
+  qed
+  then show ?thesis
+    by (auto simp: mem_box)
+qed
 
 lemma has_integral_reflect_lemma[intro]:
   assumes "(f has_integral i) (cbox a b)"
   shows "((\<lambda>x. f(-x)) has_integral i) (cbox (-b) (-a))"
   using has_integral_affinity[OF assms, of "-1" 0]
   by auto
 
 lemma has_integral_reflect_lemma_real[intro]:
   assumes "(f has_integral i) {a..b::real}"
   shows "((\<lambda>x. f(-x)) has_integral i) {-b .. -a}"
   using assms
   unfolding box_real[symmetric]
   by (rule has_integral_reflect_lemma)
 
 lemma has_integral_reflect[simp]:
   "((\<lambda>x. f (-x)) has_integral i) (cbox (-b) (-a)) \<longleftrightarrow> (f has_integral i) (cbox a b)"
   by (auto dest: has_integral_reflect_lemma)
 
 lemma has_integral_reflect_real[simp]:
   fixes a b::real
   shows "((\<lambda>x. f (-x)) has_integral i) {-b..-a} \<longleftrightarrow> (f has_integral i) {a..b}"
   by (metis has_integral_reflect interval_cbox)
 
 lemma integrable_reflect[simp]: "(\<lambda>x. f(-x)) integrable_on cbox (-b) (-a) \<longleftrightarrow> f integrable_on cbox a b"
   unfolding integrable_on_def by auto
 
 lemma integrable_reflect_real[simp]: "(\<lambda>x. f(-x)) integrable_on {-b .. -a} \<longleftrightarrow> f integrable_on {a..b::real}"
   unfolding box_real[symmetric]
   by (rule integrable_reflect)
 
 lemma integral_reflect[simp]: "integral (cbox (-b) (-a)) (\<lambda>x. f (-x)) = integral (cbox a b) f"
   unfolding integral_def by auto
 
 lemma integral_reflect_real[simp]: "integral {-b .. -a} (\<lambda>x. f (-x)) = integral {a..b::real} f"
   unfolding box_real[symmetric]
   by (rule integral_reflect)
 
 subsection \<open>Stronger form of FCT; quite a tedious proof\<close>
 
 lemma split_minus[simp]: "(\<lambda>(x, k). f x k) x - (\<lambda>(x, k). g x k) x = (\<lambda>(x, k). f x k - g x k) x"
   by (simp add: split_def)
 
 theorem fundamental_theorem_of_calculus_interior:
   fixes f :: "real \<Rightarrow> 'a::real_normed_vector"
   assumes "a \<le> b"
     and contf: "continuous_on {a..b} f"
     and derf: "\<And>x. x \<in> {a <..< b} \<Longrightarrow> (f has_vector_derivative f' x) (at x)"
   shows "(f' has_integral (f b - f a)) {a..b}"
 proof (cases "a = b")
   case True
   then have *: "cbox a b = {b}" "f b - f a = 0"
     by (auto simp add:  order_antisym)
   with True show ?thesis by auto
 next
   case False
   with \<open>a \<le> b\<close> have ab: "a < b" by arith
   show ?thesis
     unfolding has_integral_factor_content_real
   proof (intro allI impI)
     fix e :: real
     assume e: "e > 0"
     then have eba8: "(e * (b-a)) / 8 > 0"
       using ab by (auto simp add: field_simps)
-    note derf_exp = derf[unfolded has_vector_derivative_def has_derivative_at_alt]
+    note derf_exp = derf[unfolded has_vector_derivative_def has_derivative_at_alt, THEN conjunct2, rule_format]
+    thm derf_exp
     have bounded: "\<And>x. x \<in> {a<..<b} \<Longrightarrow> bounded_linear (\<lambda>u. u *\<^sub>R f' x)"
-      using derf_exp by simp
+      by (simp add: bounded_linear_scaleR_left)
     have "\<forall>x \<in> box a b. \<exists>d>0. \<forall>y. norm (y-x) < d \<longrightarrow> norm (f y - f x - (y-x) *\<^sub>R f' x) \<le> e/2 * norm (y-x)"
-      (is "\<forall>x \<in> box a b. ?Q x")
+      (is "\<forall>x \<in> box a b. ?Q x") \<comment>\<open>The explicit quantifier is required by the following step\<close>
     proof
-      fix x assume x: "x \<in> box a b"
-      show "?Q x"
-        apply (rule allE [where x="e/2", OF derf_exp [THEN conjunct2, of x]])
-        using x e by auto
+      fix x assume "x \<in> box a b"
+      with e show "?Q x"
+        using derf_exp [of x "e/2"] by auto
     qed
-    from this [unfolded bgauge_existence_lemma]
-    obtain d where d: "\<And>x. 0 < d x"
-      "\<And>x y. \<lbrakk>x \<in> box a b; norm (y-x) < d x\<rbrakk>
-               \<Longrightarrow> norm (f y - f x - (y-x) *\<^sub>R f' x) \<le> e/2 * norm (y-x)"
-      by metis
+    then obtain d where d: "\<And>x. 0 < d x"
+      "\<And>x y. \<lbrakk>x \<in> box a b; norm (y-x) < d x\<rbrakk> \<Longrightarrow> norm (f y - f x - (y-x) *\<^sub>R f' x) \<le> e/2 * norm (y-x)"
+      unfolding bgauge_existence_lemma by metis
     have "bounded (f ` cbox a b)"
       using compact_cbox assms by (auto simp: compact_imp_bounded compact_continuous_image)
     then obtain B 
       where "0 < B" and B: "\<And>x. x \<in> f ` cbox a b \<Longrightarrow> norm x \<le> B"
       unfolding bounded_pos by metis
     obtain da where "0 < da"
       and da: "\<And>c. \<lbrakk>a \<le> c; {a..c} \<subseteq> {a..b}; {a..c} \<subseteq> ball a da\<rbrakk>
                           \<Longrightarrow> norm (content {a..c} *\<^sub>R f' a - (f c - f a)) \<le> (e * (b-a)) / 4"
     proof -
       have "continuous (at a within {a..b}) f"
         using contf continuous_on_eq_continuous_within by force
       with eba8 obtain k where "0 < k"
         and k: "\<And>x. \<lbrakk>x \<in> {a..b}; 0 < norm (x-a); norm (x-a) < k\<rbrakk> \<Longrightarrow> norm (f x - f a) < e * (b-a) / 8"
         unfolding continuous_within Lim_within dist_norm by metis
       obtain l where l: "0 < l" "norm (l *\<^sub>R f' a) \<le> e * (b-a) / 8" 
       proof (cases "f' a = 0")
         case True with ab e that show ?thesis by auto
       next
         case False
-        then show ?thesis
-          apply (rule_tac l="(e * (b-a)) / 8 / norm (f' a)" in that)
-          using ab e apply (auto simp add: field_simps)
-          done
+        show ?thesis
+        proof
+          show "norm ((e * (b - a) / 8 / norm (f' a)) *\<^sub>R f' a) \<le> e * (b - a) / 8"
+               "0 < e * (b - a) / 8 / norm (f' a)"
+            using False ab e by (auto simp add: field_simps)
+        qed 
       qed
       have "norm (content {a..c} *\<^sub>R f' a - (f c - f a)) \<le> e * (b-a) / 4"
         if "a \<le> c" "{a..c} \<subseteq> {a..b}" and bmin: "{a..c} \<subseteq> ball a (min k l)" for c
       proof -
         have minkl: "\<bar>a - x\<bar> < min k l" if "x \<in> {a..c}" for x
           using bmin dist_real_def that by auto
         then have lel: "\<bar>c - a\<bar> \<le> \<bar>l\<bar>"
           using that by force
         have "norm ((c - a) *\<^sub>R f' a - (f c - f a)) \<le> norm ((c - a) *\<^sub>R f' a) + norm (f c - f a)"
           by (rule norm_triangle_ineq4)
         also have "\<dots> \<le> e * (b-a) / 8 + e * (b-a) / 8"
         proof (rule add_mono)
           have "norm ((c - a) *\<^sub>R f' a) \<le> norm (l *\<^sub>R f' a)"
             by (auto intro: mult_right_mono [OF lel])
           also have "... \<le> e * (b-a) / 8"
             by (rule l)
           finally show "norm ((c - a) *\<^sub>R f' a) \<le> e * (b-a) / 8" .
         next
           have "norm (f c - f a) < e * (b-a) / 8"
           proof (cases "a = c")
             case True then show ?thesis
               using eba8 by auto
           next
             case False show ?thesis
               by (rule k) (use minkl \<open>a \<le> c\<close> that False in auto)
           qed
           then show "norm (f c - f a) \<le> e * (b-a) / 8" by simp
         qed
         finally show "norm (content {a..c} *\<^sub>R f' a - (f c - f a)) \<le> e * (b-a) / 4"
           unfolding content_real[OF \<open>a \<le> c\<close>] by auto
       qed
       then show ?thesis
         by (rule_tac da="min k l" in that) (auto simp: l \<open>0 < k\<close>)
     qed
     obtain db where "0 < db"
       and db: "\<And>c. \<lbrakk>c \<le> b; {c..b} \<subseteq> {a..b}; {c..b} \<subseteq> ball b db\<rbrakk>
                           \<Longrightarrow> norm (content {c..b} *\<^sub>R f' b - (f b - f c)) \<le> (e * (b-a)) / 4"
     proof -
       have "continuous (at b within {a..b}) f"
         using contf continuous_on_eq_continuous_within by force
       with eba8 obtain k
         where "0 < k"
           and k: "\<And>x. \<lbrakk>x \<in> {a..b}; 0 < norm(x-b); norm(x-b) < k\<rbrakk>
                      \<Longrightarrow> norm (f b - f x) < e * (b-a) / 8"
         unfolding continuous_within Lim_within dist_norm norm_minus_commute by metis
       obtain l where l: "0 < l" "norm (l *\<^sub>R f' b) \<le> (e * (b-a)) / 8"
       proof (cases "f' b = 0")
         case True thus ?thesis 
           using ab e that by auto
       next
-        case False then show ?thesis
-          apply (rule_tac l="(e * (b-a)) / 8 / norm (f' b)" in that)
-          using ab e by (auto simp add: field_simps)
+        case False show ?thesis
+        proof
+          show "norm ((e * (b - a) / 8 / norm (f' b)) *\<^sub>R f' b) \<le> e * (b - a) / 8" 
+               "0 < e * (b - a) / 8 / norm (f' b)"
+            using False ab e by (auto simp add: field_simps)
+        qed
       qed
       have "norm (content {c..b} *\<^sub>R f' b - (f b - f c)) \<le> e * (b-a) / 4" 
         if "c \<le> b" "{c..b} \<subseteq> {a..b}" and bmin: "{c..b} \<subseteq> ball b (min k l)" for c
       proof -
         have minkl: "\<bar>b - x\<bar> < min k l" if "x \<in> {c..b}" for x
           using bmin dist_real_def that by auto
         then have lel: "\<bar>b - c\<bar> \<le> \<bar>l\<bar>"
           using that by force
         have "norm ((b - c) *\<^sub>R f' b - (f b - f c)) \<le> norm ((b - c) *\<^sub>R f' b) + norm (f b - f c)"
           by (rule norm_triangle_ineq4)
         also have "\<dots> \<le> e * (b-a) / 8 + e * (b-a) / 8"
         proof (rule add_mono)
           have "norm ((b - c) *\<^sub>R f' b) \<le> norm (l *\<^sub>R f' b)"
             by (auto intro: mult_right_mono [OF lel])
           also have "... \<le> e * (b-a) / 8"
             by (rule l)
           finally show "norm ((b - c) *\<^sub>R f' b) \<le> e * (b-a) / 8" .
         next
           have "norm (f b - f c) < e * (b-a) / 8"
           proof (cases "b = c")
             case True with eba8 show ?thesis
               by auto
           next
             case False show ?thesis
               by (rule k) (use minkl \<open>c \<le> b\<close> that False in auto)
           qed
           then show "norm (f b - f c) \<le> e * (b-a) / 8" by simp
         qed
         finally show "norm (content {c..b} *\<^sub>R f' b - (f b - f c)) \<le> e * (b-a) / 4"
           unfolding content_real[OF \<open>c \<le> b\<close>] by auto
       qed
       then show ?thesis
         by (rule_tac db="min k l" in that) (auto simp: l \<open>0 < k\<close>)
     qed
     let ?d = "(\<lambda>x. ball x (if x=a then da else if x=b then db else d x))"
     show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow>
               norm ((\<Sum>(x,K)\<in>p. content K *\<^sub>R f' x) - (f b - f a)) \<le> e * content {a..b})"
     proof (rule exI, safe)
       show "gauge ?d"
         using ab \<open>db > 0\<close> \<open>da > 0\<close> d(1) by (auto intro: gauge_ball_dependent)
     next
       fix p
       assume ptag: "p tagged_division_of {a..b}" and fine: "?d fine p"
       let ?A = "{t. fst t \<in> {a, b}}"
       note p = tagged_division_ofD[OF ptag]
       have pA: "p = (p \<inter> ?A) \<union> (p - ?A)" "finite (p \<inter> ?A)" "finite (p - ?A)" "(p \<inter> ?A) \<inter> (p - ?A) = {}"
         using ptag fine by auto
       have le_xz: "\<And>w x y z::real. y \<le> z/2 \<Longrightarrow> w - x \<le> z/2 \<Longrightarrow> w + y \<le> x + z"
         by arith
       have non: False if xk: "(x,K) \<in> p" and "x \<noteq> a" "x \<noteq> b"
         and less: "e * (Sup K - Inf K)/2 < norm (content K *\<^sub>R f' x - (f (Sup K) - f (Inf K)))"
       for x K
       proof -
         obtain u v where k: "K = cbox u v"
           using p(4) xk by blast
         then have "u \<le> v" and uv: "{u, v} \<subseteq> cbox u v"
           using p(2)[OF xk] by auto
         then have result: "e * (v - u)/2 < norm ((v - u) *\<^sub>R f' x - (f v - f u))"
           using less[unfolded k box_real interval_bounds_real content_real] by auto
         then have "x \<in> box a b"
           using p(2) p(3) \<open>x \<noteq> a\<close> \<open>x \<noteq> b\<close> xk by fastforce
         with d have *: "\<And>y. norm (y-x) < d x 
                 \<Longrightarrow> norm (f y - f x - (y-x) *\<^sub>R f' x) \<le> e/2 * norm (y-x)"
           by metis
         have xd: "norm (u - x) < d x" "norm (v - x) < d x"
           using fineD[OF fine xk] \<open>x \<noteq> a\<close> \<open>x \<noteq> b\<close> uv 
           by (auto simp add: k subset_eq dist_commute dist_real_def)
         have "norm ((v - u) *\<^sub>R f' x - (f v - f u)) =
               norm ((f u - f x - (u - x) *\<^sub>R f' x) - (f v - f x - (v - x) *\<^sub>R f' x))"
           by (rule arg_cong[where f=norm]) (auto simp: scaleR_left.diff)
         also have "\<dots> \<le> e/2 * norm (u - x) + e/2 * norm (v - x)"
           by (metis norm_triangle_le_diff add_mono * xd)
         also have "\<dots> \<le> e/2 * norm (v - u)"
           using p(2)[OF xk] by (auto simp add: field_simps k)
         also have "\<dots> < norm ((v - u) *\<^sub>R f' x - (f v - f u))"
           using result by (simp add: \<open>u \<le> v\<close>)
         finally have "e * (v - u)/2 < e * (v - u)/2"
           using uv by auto
         then show False by auto
       qed
       have "norm (\<Sum>(x, K)\<in>p - ?A. content K *\<^sub>R f' x - (f (Sup K) - f (Inf K)))
           \<le> (\<Sum>(x, K)\<in>p - ?A. norm (content K *\<^sub>R f' x - (f (Sup K) - f (Inf K))))"
         by (auto intro: sum_norm_le)
       also have "... \<le> (\<Sum>n\<in>p - ?A. e * (case n of (x, k) \<Rightarrow> Sup k - Inf k)/2)"
         using non by (fastforce intro: sum_mono)
       finally have I: "norm (\<Sum>(x, k)\<in>p - ?A.
                   content k *\<^sub>R f' x - (f (Sup k) - f (Inf k)))
              \<le> (\<Sum>n\<in>p - ?A. e * (case n of (x, k) \<Rightarrow> Sup k - Inf k))/2"
         by (simp add: sum_divide_distrib)
       have II: "norm (\<Sum>(x, k)\<in>p \<inter> ?A. content k *\<^sub>R f' x - (f (Sup k) - f (Inf k))) -
              (\<Sum>n\<in>p \<inter> ?A. e * (case n of (x, k) \<Rightarrow> Sup k - Inf k))
              \<le> (\<Sum>n\<in>p - ?A. e * (case n of (x, k) \<Rightarrow> Sup k - Inf k))/2"
       proof -
         have ge0: "0 \<le> e * (Sup k - Inf k)" if xkp: "(x, k) \<in> p \<inter> ?A" for x k
         proof -
           obtain u v where uv: "k = cbox u v"
             by (meson Int_iff xkp p(4))
           with p(2) that uv have "cbox u v \<noteq> {}"
             by blast
           then show "0 \<le> e * ((Sup k) - (Inf k))"
             unfolding uv using e by (auto simp add: field_simps)
         qed
         let ?B = "\<lambda>x. {t \<in> p. fst t = x \<and> content (snd t) \<noteq> 0}"
         let ?C = "{t \<in> p. fst t \<in> {a, b} \<and> content (snd t) \<noteq> 0}"
         have "norm (\<Sum>(x, k)\<in>p \<inter> {t. fst t \<in> {a, b}}. content k *\<^sub>R f' x - (f (Sup k) - f (Inf k))) \<le> e * (b-a)/2"
         proof -
           have *: "\<And>S f e. sum f S = sum f (p \<inter> ?C) \<Longrightarrow> norm (sum f (p \<inter> ?C)) \<le> e \<Longrightarrow> norm (sum f S) \<le> e"
             by auto
           have 1: "content K *\<^sub>R (f' x) - (f ((Sup K)) - f ((Inf K))) = 0"
             if "(x,K) \<in> p \<inter> {t. fst t \<in> {a, b}} - p \<inter> ?C" for x K
           proof -
             have xk: "(x,K) \<in> p" and k0: "content K = 0"
               using that by auto
             then obtain u v where uv: "K = cbox u v"
               using p(4) by blast
             then have "u = v"
               using xk k0 p(2) by force
             then show "content K *\<^sub>R (f' x) - (f ((Sup K)) - f ((Inf K))) = 0"
               using xk unfolding uv by auto
           qed
           have 2: "norm(\<Sum>(x,K)\<in>p \<inter> ?C. content K *\<^sub>R f' x - (f (Sup K) - f (Inf K)))  \<le> e * (b-a)/2"
           proof -
             have norm_le: "norm (sum f S) \<le> e"
               if "\<And>x y. \<lbrakk>x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> x = y" "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> e" "e > 0"
               for S f and e :: real
             proof (cases "S = {}")
               case True
               with that show ?thesis by auto
             next
               case False then obtain x where "x \<in> S"
                 by auto
               then have "S = {x}"
                 using that(1) by auto
               then show ?thesis
                 using \<open>x \<in> S\<close> that(2) by auto
             qed
             have *: "p \<inter> ?C = ?B a \<union> ?B b"
               by blast
             then have "norm (\<Sum>(x,K)\<in>p \<inter> ?C. content K *\<^sub>R f' x - (f (Sup K) - f (Inf K))) =
                        norm (\<Sum>(x,K) \<in> ?B a \<union> ?B b. content K *\<^sub>R f' x - (f (Sup K) - f (Inf K)))"
               by simp
             also have "... = norm ((\<Sum>(x,K) \<in> ?B a. content K *\<^sub>R f' x - (f (Sup K) - f (Inf K))) + 
                                    (\<Sum>(x,K) \<in> ?B b. content K *\<^sub>R f' x - (f (Sup K) - f (Inf K))))"
-              apply (subst sum.union_disjoint)
-              using p(1) ab e by auto
+              using p(1) ab e by (subst sum.union_disjoint) auto
             also have "... \<le> e * (b - a) / 4 + e * (b - a) / 4"
             proof (rule norm_triangle_le [OF add_mono])
               have pa: "\<exists>v. k = cbox a v \<and> a \<le> v" if "(a, k) \<in> p" for k
                 using p(2) p(3) p(4) that by fastforce
               show "norm (\<Sum>(x,K) \<in> ?B a. content K *\<^sub>R f' x - (f (Sup K) - f (Inf K))) \<le> e * (b - a) / 4"
               proof (intro norm_le; clarsimp)
                 fix K K'
                 assume K: "(a, K) \<in> p" "(a, K') \<in> p" and ne0: "content K \<noteq> 0" "content K' \<noteq> 0"
                 with pa obtain v v' where v: "K = cbox a v" "a \<le> v" and v': "K' = cbox a v'" "a \<le> v'"
                   by blast
                 let ?v = "min v v'"
                 have "box a ?v \<subseteq> K \<inter> K'"
                   unfolding v v' by (auto simp add: mem_box)
                 then have "interior (box a (min v v')) \<subseteq> interior K \<inter> interior K'"
                   using interior_Int interior_mono by blast
                 moreover have "(a + ?v)/2 \<in> box a ?v"
                   using ne0  unfolding v v' content_eq_0 not_le
                   by (auto simp add: mem_box)
                 ultimately have "(a + ?v)/2 \<in> interior K \<inter> interior K'"
                   unfolding interior_open[OF open_box] by auto
                 then show "K = K'"
                   using p(5)[OF K] by auto
               next
                 fix K 
                 assume K: "(a, K) \<in> p" and ne0: "content K \<noteq> 0"
                 show "norm (content c *\<^sub>R f' a - (f (Sup c) - f (Inf c))) * 4 \<le> e * (b-a)"
                   if "(a, c) \<in> p" and ne0: "content c \<noteq> 0" for c
                 proof -
                   obtain v where v: "c = cbox a v" and "a \<le> v"
                     using pa[OF \<open>(a, c) \<in> p\<close>] by metis 
                   then have "a \<in> {a..v}" "v \<le> b"
                     using p(3)[OF \<open>(a, c) \<in> p\<close>] by auto
                   moreover have "{a..v} \<subseteq> ball a da"
                     using fineD[OF \<open>?d fine p\<close> \<open>(a, c) \<in> p\<close>] by (simp add: v split: if_split_asm)
                   ultimately show ?thesis
                     unfolding v interval_bounds_real[OF \<open>a \<le> v\<close>] box_real
                     using da \<open>a \<le> v\<close> by auto
                 qed
               qed (use ab e in auto)
             next
               have pb: "\<exists>v. k = cbox v b \<and> b \<ge> v" if "(b, k) \<in> p" for k
                 using p(2) p(3) p(4) that by fastforce
               show "norm (\<Sum>(x,K) \<in> ?B b. content K *\<^sub>R f' x - (f (Sup K) - f (Inf K))) \<le> e * (b - a) / 4"
               proof (intro norm_le; clarsimp)
                 fix K K'
                 assume K: "(b, K) \<in> p" "(b, K') \<in> p" and ne0: "content K \<noteq> 0" "content K' \<noteq> 0"
                 with pb obtain v v' where v: "K = cbox v b" "v \<le> b" and v': "K' = cbox v' b" "v' \<le> b"
                   by blast
                 let ?v = "max v v'"
                 have "box ?v b \<subseteq> K \<inter> K'"
                   unfolding v v' by (auto simp: mem_box)
                 then have "interior (box (max v v') b) \<subseteq> interior K \<inter> interior K'"
                   using interior_Int interior_mono by blast
                 moreover have " ((b + ?v)/2) \<in> box ?v b"
                   using ne0 unfolding v v' content_eq_0 not_le by (auto simp: mem_box)
                 ultimately have "((b + ?v)/2) \<in> interior K \<inter> interior K'"
                   unfolding interior_open[OF open_box] by auto
                 then show "K = K'"
                   using p(5)[OF K] by auto
               next
                 fix K 
                 assume K: "(b, K) \<in> p" and ne0: "content K \<noteq> 0"
                 show "norm (content c *\<^sub>R f' b - (f (Sup c) - f (Inf c))) * 4 \<le> e * (b-a)"
                   if "(b, c) \<in> p" and ne0: "content c \<noteq> 0" for c
                 proof -
                 obtain v where v: "c = cbox v b" and "v \<le> b"
                   using \<open>(b, c) \<in> p\<close> pb by blast
                 then have "v \<ge> a""b \<in> {v.. b}"  
                   using p(3)[OF \<open>(b, c) \<in> p\<close>] by auto
                 moreover have "{v..b} \<subseteq> ball b db"
                   using fineD[OF \<open>?d fine p\<close> \<open>(b, c) \<in> p\<close>] box_real(2) v False by force
                 ultimately show ?thesis
                   using db v by auto
                 qed
               qed (use ab e in auto)
             qed
             also have "... = e * (b-a)/2"
               by simp
             finally show "norm (\<Sum>(x,k)\<in>p \<inter> ?C.
                         content k *\<^sub>R f' x - (f (Sup k) - f (Inf k))) \<le> e * (b-a)/2" .
           qed
           show "norm (\<Sum>(x, k)\<in>p \<inter> ?A. content k *\<^sub>R f' x - (f ((Sup k)) - f ((Inf k)))) \<le> e * (b-a)/2"
             apply (rule * [OF sum.mono_neutral_right[OF pA(2)]])
             using 1 2 by (auto simp: split_paired_all)
         qed
         also have "... = (\<Sum>n\<in>p. e * (case n of (x, k) \<Rightarrow> Sup k - Inf k))/2"
           unfolding sum_distrib_left[symmetric]
-          apply (subst additive_tagged_division_1[OF \<open>a \<le> b\<close> ptag])
-          by auto
+          by (subst additive_tagged_division_1[OF \<open>a \<le> b\<close> ptag]) auto
         finally have norm_le: "norm (\<Sum>(x,K)\<in>p \<inter> {t. fst t \<in> {a, b}}. content K *\<^sub>R f' x - (f (Sup K) - f (Inf K)))
                 \<le> (\<Sum>n\<in>p. e * (case n of (x, K) \<Rightarrow> Sup K - Inf K))/2" .
         have le2: "\<And>x s1 s2::real. 0 \<le> s1 \<Longrightarrow> x \<le> (s1 + s2)/2 \<Longrightarrow> x - s1 \<le> s2/2"
           by auto
         show ?thesis
           apply (rule le2 [OF sum_nonneg])
           using ge0 apply force
-          unfolding sum.union_disjoint[OF pA(2-), symmetric] pA(1)[symmetric]
-          by (metis norm_le)
+          by (metis (no_types, lifting) Diff_Diff_Int Diff_subset norm_le p(1) sum.subset_diff)
       qed
       note * = additive_tagged_division_1[OF assms(1) ptag, symmetric]
       have "norm (\<Sum>(x,K)\<in>p \<inter> ?A \<union> (p - ?A). content K *\<^sub>R f' x - (f (Sup K) - f (Inf K)))
                \<le> e * (\<Sum>(x,K)\<in>p \<inter> ?A \<union> (p - ?A). Sup K - Inf K)"
         unfolding sum_distrib_left
         unfolding sum.union_disjoint[OF pA(2-)]
         using le_xz norm_triangle_le I II by blast
       then
       show "norm ((\<Sum>(x,K)\<in>p. content K *\<^sub>R f' x) - (f b - f a)) \<le> e * content {a..b}"
         by (simp only: content_real[OF \<open>a \<le> b\<close>] *[of "\<lambda>x. x"] *[of f] sum_subtractf[symmetric] split_minus pA(1) [symmetric])
     qed
   qed
 qed
 
 
 subsection \<open>Stronger form with finite number of exceptional points\<close>
 
 lemma fundamental_theorem_of_calculus_interior_strong:
  fixes f :: "real \<Rightarrow> 'a::banach"
  assumes "finite S"
    and "a \<le> b" "\<And>x. x \<in> {a <..< b} - S \<Longrightarrow> (f has_vector_derivative f'(x)) (at x)"
    and "continuous_on {a .. b} f"
  shows "(f' has_integral (f b - f a)) {a .. b}"
   using assms
 proof (induction arbitrary: a b)
 case empty
   then show ?case
     using fundamental_theorem_of_calculus_interior by force
 next
 case (insert x S)
   show ?case
   proof (cases "x \<in> {a<..<b}")
     case False then show ?thesis
       using insert by blast
   next
     case True then have "a < x" "x < b"
       by auto
     have "(f' has_integral f x - f a) {a..x}" "(f' has_integral f b - f x) {x..b}"
       using \<open>continuous_on {a..b} f\<close> \<open>a < x\<close> \<open>x < b\<close> continuous_on_subset by (force simp: intro!: insert)+
     then have "(f' has_integral f x - f a + (f b - f x)) {a..b}"
       using \<open>a < x\<close> \<open>x < b\<close> has_integral_combine less_imp_le by blast
     then show ?thesis
       by simp
   qed
 qed
 
 corollary fundamental_theorem_of_calculus_strong:
   fixes f :: "real \<Rightarrow> 'a::banach"
   assumes "finite S"
     and "a \<le> b"
     and vec: "\<And>x. x \<in> {a..b} - S \<Longrightarrow> (f has_vector_derivative f'(x)) (at x)"
     and "continuous_on {a..b} f"
   shows "(f' has_integral (f b - f a)) {a..b}"
   by (rule fundamental_theorem_of_calculus_interior_strong [OF \<open>finite S\<close>]) (force simp: assms)+
 
 proposition indefinite_integral_continuous_left:
   fixes f:: "real \<Rightarrow> 'a::banach"
   assumes intf: "f integrable_on {a..b}" and "a < c" "c \<le> b" "e > 0"
   obtains d where "d > 0"
     and "\<forall>t. c - d < t \<and> t \<le> c \<longrightarrow> norm (integral {a..c} f - integral {a..t} f) < e"
 proof -
   obtain w where "w > 0" and w: "\<And>t. \<lbrakk>c - w < t; t < c\<rbrakk> \<Longrightarrow> norm (f c) * norm(c - t) < e/3"
   proof (cases "f c = 0")
     case False
     hence e3: "0 < e/3 / norm (f c)" using \<open>e>0\<close> by simp
     moreover have "norm (f c) * norm (c - t) < e/3" 
       if "t < c" and "c - e/3 / norm (f c) < t" for t
     proof -
       have "norm (c - t) < e/3 / norm (f c)"
         using that by auto
       then show "norm (f c) * norm (c - t) < e/3"
         by (metis e3 mult.commute norm_not_less_zero pos_less_divide_eq zero_less_divide_iff)
     qed
     ultimately show ?thesis
       using that by auto
   next
     case True then show ?thesis
       using \<open>e > 0\<close> that by auto
   qed
 
   let ?SUM = "\<lambda>p. (\<Sum>(x,K) \<in> p. content K *\<^sub>R f x)"
   have e3: "e/3 > 0"
     using \<open>e > 0\<close> by auto
   have "f integrable_on {a..c}"
-    apply (rule integrable_subinterval_real[OF intf])
-    using \<open>a < c\<close> \<open>c \<le> b\<close> by auto
+    using \<open>a < c\<close> \<open>c \<le> b\<close> by (auto intro: integrable_subinterval_real[OF intf])
   then obtain d1 where "gauge d1" and d1:
     "\<And>p. \<lbrakk>p tagged_division_of {a..c}; d1 fine p\<rbrakk> \<Longrightarrow> norm (?SUM p - integral {a..c} f) < e/3"
     using integrable_integral has_integral_real e3 by metis
   define d where [abs_def]: "d x = ball x w \<inter> d1 x" for x
   have "gauge d"
     unfolding d_def using \<open>w > 0\<close> \<open>gauge d1\<close> by auto
   then obtain k where "0 < k" and k: "ball c k \<subseteq> d c"
     by (meson gauge_def open_contains_ball)
 
   let ?d = "min k (c - a)/2"
   show thesis
   proof (intro that[of ?d] allI impI, safe)
     show "?d > 0"
       using \<open>0 < k\<close> \<open>a < c\<close> by auto
   next
     fix t
     assume t: "c - ?d < t" "t \<le> c"
     show "norm (integral ({a..c}) f - integral ({a..t}) f) < e"
     proof (cases "t < c")
       case False with \<open>t \<le> c\<close> show ?thesis
         by (simp add: \<open>e > 0\<close>)
     next
       case True
       have "f integrable_on {a..t}"
-        apply (rule integrable_subinterval_real[OF intf])
-        using \<open>t < c\<close> \<open>c \<le> b\<close> by auto
+        using \<open>t < c\<close> \<open>c \<le> b\<close> by (auto intro: integrable_subinterval_real[OF intf])
       then obtain d2 where d2: "gauge d2"
         "\<And>p. p tagged_division_of {a..t} \<and> d2 fine p \<Longrightarrow> norm (?SUM p - integral {a..t} f) < e/3"
         using integrable_integral has_integral_real e3 by metis
       define d3 where "d3 x = (if x \<le> t then d1 x \<inter> d2 x else d1 x)" for x
       have "gauge d3"
         using \<open>gauge d1\<close> \<open>gauge d2\<close> unfolding d3_def gauge_def by auto
       then obtain p where ptag: "p tagged_division_of {a..t}" and pfine: "d3 fine p"
         by (metis box_real(2) fine_division_exists)
       note p' = tagged_division_ofD[OF ptag]
       have pt: "(x,K)\<in>p \<Longrightarrow> x \<le> t" for x K
         by (meson atLeastAtMost_iff p'(2) p'(3) subsetCE)
       with pfine have "d2 fine p"
         unfolding fine_def d3_def by fastforce
       then have d2_fin: "norm (?SUM p - integral {a..t} f) < e/3"
         using d2(2) ptag by auto
       have eqs: "{a..c} \<inter> {x. x \<le> t} = {a..t}" "{a..c} \<inter> {x. x \<ge> t} = {t..c}"
         using t by (auto simp add: field_simps)
       have "p \<union> {(c, {t..c})} tagged_division_of {a..c}"
-        apply (rule tagged_division_Un_interval_real[of _ _ _ 1 "t"])
-        using  \<open>t \<le> c\<close> by (auto simp: eqs ptag tagged_division_of_self_real)
+      proof (intro tagged_division_Un_interval_real)
+        show "{(c, {t..c})} tagged_division_of {a..c} \<inter> {x. t \<le> x \<bullet> 1}"
+          using \<open>t \<le> c\<close> by (auto simp: eqs tagged_division_of_self_real)
+      qed (auto simp: eqs ptag)
       moreover
       have "d1 fine p \<union> {(c, {t..c})}"
         unfolding fine_def
       proof safe
         fix x K y
         assume "(x,K) \<in> p" and "y \<in> K" then show "y \<in> d1 x"
           by (metis Int_iff d3_def subsetD fineD pfine)
       next
         fix x assume "x \<in> {t..c}"
         then have "dist c x < k"
           using t(1) by (auto simp add: field_simps dist_real_def)
         with k show "x \<in> d1 c"
           unfolding d_def by auto
       qed  
       ultimately have d1_fin: "norm (?SUM(p \<union> {(c, {t..c})}) - integral {a..c} f) < e/3"
         using d1 by metis
       have SUMEQ: "?SUM(p \<union> {(c, {t..c})}) = (c - t) *\<^sub>R f c + ?SUM p"
       proof -
         have "?SUM(p \<union> {(c, {t..c})}) = (content{t..c} *\<^sub>R f c) + ?SUM p"
         proof (subst sum.union_disjoint)
           show "p \<inter> {(c, {t..c})} = {}"
             using \<open>t < c\<close> pt by force
         qed (use p'(1) in auto)
         also have "... = (c - t) *\<^sub>R f c + ?SUM p"
           using \<open>t \<le> c\<close> by auto
         finally show ?thesis .
       qed
       have "c - k < t"
         using \<open>k>0\<close> t(1) by (auto simp add: field_simps)
       moreover have "k \<le> w"
       proof (rule ccontr)
         assume "\<not> k \<le> w"
         then have "c + (k + w) / 2 \<notin> d c"
           by (auto simp add: field_simps not_le not_less dist_real_def d_def)
         then have "c + (k + w) / 2 \<notin> ball c k"
           using k by blast
         then show False
           using \<open>0 < w\<close> \<open>\<not> k \<le> w\<close> dist_real_def by auto
       qed
       ultimately have cwt: "c - w < t"
         by (auto simp add: field_simps)
       have eq: "integral {a..c} f - integral {a..t} f = -(((c - t) *\<^sub>R f c + ?SUM p) -
              integral {a..c} f) + (?SUM p - integral {a..t} f) + (c - t) *\<^sub>R f c"
         by auto
       have "norm (integral {a..c} f - integral {a..t} f) < e/3 + e/3 + e/3"
         unfolding eq
       proof (intro norm_triangle_lt add_strict_mono)
         show "norm (- ((c - t) *\<^sub>R f c + ?SUM p - integral {a..c} f)) < e/3"
           by (metis SUMEQ d1_fin norm_minus_cancel)
         show "norm (?SUM p - integral {a..t} f) < e/3"
           using d2_fin by blast
         show "norm ((c - t) *\<^sub>R f c) < e/3"
           using w cwt \<open>t < c\<close> by simp (simp add: field_simps)
       qed
       then show ?thesis by simp
     qed
   qed
 qed
 
 lemma indefinite_integral_continuous_right:
   fixes f :: "real \<Rightarrow> 'a::banach"
   assumes "f integrable_on {a..b}"
     and "a \<le> c"
     and "c < b"
     and "e > 0"
   obtains d where "0 < d"
     and "\<forall>t. c \<le> t \<and> t < c + d \<longrightarrow> norm (integral {a..c} f - integral {a..t} f) < e"
 proof -
   have intm: "(\<lambda>x. f (- x)) integrable_on {-b .. -a}" "- b < - c" "- c \<le> - a"
     using assms by auto
   from indefinite_integral_continuous_left[OF intm \<open>e>0\<close>]
   obtain d where "0 < d"
     and d: "\<And>t. \<lbrakk>- c - d < t; t \<le> -c\<rbrakk> 
              \<Longrightarrow> norm (integral {- b..- c} (\<lambda>x. f (-x)) - integral {- b..t} (\<lambda>x. f (-x))) < e"
     by metis
   let ?d = "min d (b - c)" 
   show ?thesis
   proof (intro that[of "?d"] allI impI)
     show "0 < ?d"
       using \<open>0 < d\<close> \<open>c < b\<close> by auto
     fix t :: real
     assume t: "c \<le> t \<and> t < c + ?d"
     have *: "integral {a..c} f = integral {a..b} f - integral {c..b} f"
             "integral {a..t} f = integral {a..b} f - integral {t..b} f"
-      apply (simp_all only: algebra_simps)
-      using assms t by (auto simp: integral_combine)
+      using assms t by (auto simp: algebra_simps integral_combine)
     have "(- c) - d < (- t)" "- t \<le> - c"
       using t by auto 
     from d[OF this] show "norm (integral {a..c} f - integral {a..t} f) < e"
       by (auto simp add: algebra_simps norm_minus_commute *)
   qed
 qed
 
 lemma indefinite_integral_continuous_1:
   fixes f :: "real \<Rightarrow> 'a::banach"
   assumes "f integrable_on {a..b}"
   shows "continuous_on {a..b} (\<lambda>x. integral {a..x} f)"
 proof -
   have "\<exists>d>0. \<forall>x'\<in>{a..b}. dist x' x < d \<longrightarrow> dist (integral {a..x'} f) (integral {a..x} f) < e" 
        if x: "x \<in> {a..b}" and "e > 0" for x e :: real
   proof (cases "a = b")
     case True
     with that show ?thesis by force
   next
     case False
     with x have "a < b" by force
     with x consider "x = a" | "x = b" | "a < x" "x < b"
       by force
     then show ?thesis 
     proof cases
-      case 1 show ?thesis
-        apply (rule indefinite_integral_continuous_right [OF assms _ \<open>a < b\<close> \<open>e > 0\<close>], force)
-        using \<open>x = a\<close> apply (force simp: dist_norm algebra_simps)
-        done
+      case 1 then show ?thesis
+        by (force simp: dist_norm algebra_simps intro: indefinite_integral_continuous_right [OF assms _ \<open>a < b\<close> \<open>e > 0\<close>])
     next
-      case 2 show ?thesis 
-        apply (rule indefinite_integral_continuous_left [OF assms \<open>a < b\<close> _ \<open>e > 0\<close>], force)
-        using \<open>x = b\<close> apply (force simp: dist_norm norm_minus_commute algebra_simps)
-        done
+      case 2 then show ?thesis
+        by (force simp: dist_norm norm_minus_commute algebra_simps intro: indefinite_integral_continuous_left [OF assms \<open>a < b\<close> _ \<open>e > 0\<close>])
     next
       case 3
       obtain d1 where "0 < d1" 
         and d1: "\<And>t. \<lbrakk>x - d1 < t; t \<le> x\<rbrakk> \<Longrightarrow> norm (integral {a..x} f - integral {a..t} f) < e"
         using 3 by (auto intro: indefinite_integral_continuous_left [OF assms \<open>a < x\<close> _ \<open>e > 0\<close>])
       obtain d2 where "0 < d2" 
         and d2: "\<And>t. \<lbrakk>x \<le> t; t < x + d2\<rbrakk> \<Longrightarrow> norm (integral {a..x} f - integral {a..t} f) < e"
         using 3 by (auto intro: indefinite_integral_continuous_right [OF assms _ \<open>x < b\<close> \<open>e > 0\<close>])
       show ?thesis
       proof (intro exI ballI conjI impI)
         show "0 < min d1 d2"
           using \<open>0 < d1\<close> \<open>0 < d2\<close> by simp
         show "dist (integral {a..y} f) (integral {a..x} f) < e"
              if "y \<in> {a..b}" "dist y x < min d1 d2" for y
         proof (cases "y < x")
           case True
           with that d1 show ?thesis by (auto simp: dist_commute dist_norm)
         next
           case False
           with that d2 show ?thesis
             by (auto simp: dist_commute dist_norm)
         qed
       qed
     qed
   qed
   then show ?thesis
     by (auto simp: continuous_on_iff)
 qed
 
 lemma indefinite_integral_continuous_1':
   fixes f::"real \<Rightarrow> 'a::banach"
   assumes "f integrable_on {a..b}"
   shows "continuous_on {a..b} (\<lambda>x. integral {x..b} f)"
 proof -
   have "integral {a..b} f - integral {a..x} f = integral {x..b} f" if "x \<in> {a..b}" for x
     using integral_combine[OF _ _ assms, of x] that
     by (auto simp: algebra_simps)
   with _ show ?thesis
     by (rule continuous_on_eq) (auto intro!: continuous_intros indefinite_integral_continuous_1 assms)
 qed
 
 theorem integral_has_vector_derivative':
   fixes f :: "real \<Rightarrow> 'b::banach"
   assumes "continuous_on {a..b} f"
     and "x \<in> {a..b}"
   shows "((\<lambda>u. integral {u..b} f) has_vector_derivative - f x) (at x within {a..b})"
 proof -
   have *: "integral {x..b} f = integral {a .. b} f - integral {a .. x} f" if "a \<le> x" "x \<le> b" for x
     using integral_combine[of a x b for x, OF that integrable_continuous_real[OF assms(1)]]
     by (simp add: algebra_simps)
   show ?thesis
     using \<open>x \<in> _\<close> *
     by (rule has_vector_derivative_transform)
       (auto intro!: derivative_eq_intros assms integral_has_vector_derivative)
 qed
 
 lemma integral_has_real_derivative':
   assumes "continuous_on {a..b} g"
   assumes "t \<in> {a..b}"
   shows "((\<lambda>x. integral {x..b} g) has_real_derivative -g t) (at t within {a..b})"
   using integral_has_vector_derivative'[OF assms]
   by (auto simp: has_field_derivative_iff_has_vector_derivative)
 
 
 subsection \<open>This doesn't directly involve integration, but that gives an easy proof\<close>
 
 lemma has_derivative_zero_unique_strong_interval:
   fixes f :: "real \<Rightarrow> 'a::banach"
   assumes "finite k"
     and contf: "continuous_on {a..b} f"
     and "f a = y"
     and fder: "\<And>x. x \<in> {a..b} - k \<Longrightarrow> (f has_derivative (\<lambda>h. 0)) (at x within {a..b})"
     and x: "x \<in> {a..b}"
   shows "f x = y"
 proof -
   have "a \<le> b" "a \<le> x"
     using assms by auto
   have "((\<lambda>x. 0::'a) has_integral f x - f a) {a..x}"
   proof (rule fundamental_theorem_of_calculus_interior_strong[OF \<open>finite k\<close> \<open>a \<le> x\<close>]; clarify?)
     have "{a..x} \<subseteq> {a..b}"
       using x by auto
     then show "continuous_on {a..x} f"
       by (rule continuous_on_subset[OF contf])
     show "(f has_vector_derivative 0) (at z)" if z: "z \<in> {a<..<x}" and notin: "z \<notin> k" for z
       unfolding has_vector_derivative_def
     proof (simp add: at_within_open[OF z, symmetric])
       show "(f has_derivative (\<lambda>x. 0)) (at z within {a<..<x})"
-        by (rule has_derivative_within_subset [OF fder]) (use x z notin in auto)
+        by (rule has_derivative_subset [OF fder]) (use x z notin in auto)
     qed
   qed
   from has_integral_unique[OF has_integral_0 this]
   show ?thesis
     unfolding assms by auto
 qed
 
 
 subsection \<open>Generalize a bit to any convex set\<close>
 
 lemma has_derivative_zero_unique_strong_convex:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
   assumes "convex S" "finite K"
     and contf: "continuous_on S f"
     and "c \<in> S" "f c = y"
     and derf: "\<And>x. x \<in> S - K \<Longrightarrow> (f has_derivative (\<lambda>h. 0)) (at x within S)"
     and "x \<in> S"
   shows "f x = y"
 proof (cases "x = c")
   case True with \<open>f c = y\<close> show ?thesis
     by blast
 next
   case False
   let ?\<phi> = "\<lambda>u. (1 - u) *\<^sub>R c + u *\<^sub>R x"
   have contf': "continuous_on {0 ..1} (f \<circ> ?\<phi>)"
   proof (rule continuous_intros continuous_on_subset[OF contf])+
     show "(\<lambda>u. (1 - u) *\<^sub>R c + u *\<^sub>R x) ` {0..1} \<subseteq> S"
       using \<open>convex S\<close> \<open>x \<in> S\<close> \<open>c \<in> S\<close> by (auto simp add: convex_alt algebra_simps)
   qed
   have "t = u" if "?\<phi> t = ?\<phi> u" for t u
   proof -
     from that have "(t - u) *\<^sub>R x = (t - u) *\<^sub>R c"
       by (auto simp add: algebra_simps)
     then show ?thesis
       using \<open>x \<noteq> c\<close> by auto
   qed
   then have eq: "(SOME t. ?\<phi> t = ?\<phi> u) = u" for u
     by blast
   then have "(?\<phi> -` K) \<subseteq> (\<lambda>z. SOME t. ?\<phi> t = z) ` K"
     by (clarsimp simp: image_iff) (metis (no_types) eq)
   then have fin: "finite (?\<phi> -` K)"
     by (rule finite_surj[OF \<open>finite K\<close>])
 
   have derf': "((\<lambda>u. f (?\<phi> u)) has_derivative (\<lambda>h. 0)) (at t within {0..1})"
                if "t \<in> {0..1} - {t. ?\<phi> t \<in> K}" for t
   proof -
     have df: "(f has_derivative (\<lambda>h. 0)) (at (?\<phi> t) within ?\<phi> ` {0..1})"
-      apply (rule has_derivative_within_subset [OF derf])
-      using \<open>convex S\<close> \<open>x \<in> S\<close> \<open>c \<in> S\<close> that by (auto simp add: convex_alt algebra_simps)
+      using \<open>convex S\<close> \<open>x \<in> S\<close> \<open>c \<in> S\<close> that 
+      by (auto simp add: convex_alt algebra_simps intro: has_derivative_subset [OF derf])
     have "(f \<circ> ?\<phi> has_derivative (\<lambda>x. 0) \<circ> (\<lambda>z. (0 - z *\<^sub>R c) + z *\<^sub>R x)) (at t within {0..1})"
       by (rule derivative_eq_intros df | simp)+
     then show ?thesis
       unfolding o_def .
   qed
   have "(f \<circ> ?\<phi>) 1 = y"
     apply (rule has_derivative_zero_unique_strong_interval[OF fin contf'])
     unfolding o_def using \<open>f c = y\<close> derf' by auto
   then show ?thesis
     by auto
 qed
 
 
 text \<open>Also to any open connected set with finite set of exceptions. Could
  generalize to locally convex set with limpt-free set of exceptions.\<close>
 
 lemma has_derivative_zero_unique_strong_connected:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
   assumes "connected S"
     and "open S"
     and "finite K"
     and contf: "continuous_on S f"
     and "c \<in> S"
     and "f c = y"
     and derf: "\<And>x. x \<in> S - K \<Longrightarrow> (f has_derivative (\<lambda>h. 0)) (at x within S)"
     and "x \<in> S"
   shows "f x = y"
 proof -
   have "\<exists>e>0. ball x e \<subseteq> (S \<inter> f -` {f x})" if "x \<in> S" for x
   proof -
     obtain e where "0 < e" and e: "ball x e \<subseteq> S"
       using \<open>x \<in> S\<close> \<open>open S\<close> open_contains_ball by blast
     have "ball x e \<subseteq> {u \<in> S. f u \<in> {f x}}"
     proof safe
       fix y
       assume y: "y \<in> ball x e"
       then show "y \<in> S"
         using e by auto
       show "f y = f x"
       proof (rule has_derivative_zero_unique_strong_convex[OF convex_ball \<open>finite K\<close>])
         show "continuous_on (ball x e) f"
           using contf continuous_on_subset e by blast
         show "(f has_derivative (\<lambda>h. 0)) (at u within ball x e)"
              if "u \<in> ball x e - K" for u
-          by (metis Diff_iff contra_subsetD derf e has_derivative_within_subset that)
+          by (metis Diff_iff contra_subsetD derf e has_derivative_subset that)
       qed (use y e \<open>0 < e\<close> in auto)
     qed
     then show "\<exists>e>0. ball x e \<subseteq> (S \<inter> f -` {f x})"
       using \<open>0 < e\<close> by blast
   qed
   then have "openin (top_of_set S) (S \<inter> f -` {y})"
     by (auto intro!: open_openin_trans[OF \<open>open S\<close>] simp: open_contains_ball)
   moreover have "closedin (top_of_set S) (S \<inter> f -` {y})"
     by (force intro!: continuous_closedin_preimage [OF contf])
   ultimately have "(S \<inter> f -` {y}) = {} \<or> (S \<inter> f -` {y}) = S"
     using \<open>connected S\<close> by (simp add: connected_clopen)
   then show ?thesis
     using \<open>x \<in> S\<close> \<open>f c = y\<close> \<open>c \<in> S\<close> by auto
 qed
 
 lemma has_derivative_zero_connected_constant:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
   assumes "connected S"
       and "open S"
       and "finite k"
       and "continuous_on S f"
       and "\<forall>x\<in>(S - k). (f has_derivative (\<lambda>h. 0)) (at x within S)"
     obtains c where "\<And>x. x \<in> S \<Longrightarrow> f(x) = c"
 proof (cases "S = {}")
   case True
   then show ?thesis
     by (metis empty_iff that)
 next
   case False
   then obtain c where "c \<in> S"
     by (metis equals0I)
   then show ?thesis
     by (metis has_derivative_zero_unique_strong_connected assms that)
 qed
 
 lemma DERIV_zero_connected_constant:
   fixes f :: "'a::{real_normed_field,euclidean_space} \<Rightarrow> 'a"
   assumes "connected S"
       and "open S"
       and "finite K"
       and "continuous_on S f"
       and "\<forall>x\<in>(S - K). DERIV f x :> 0"
     obtains c where "\<And>x. x \<in> S \<Longrightarrow> f(x) = c"
   using has_derivative_zero_connected_constant [OF assms(1-4)] assms
   by (metis DERIV_const has_derivative_const Diff_iff at_within_open frechet_derivative_at has_field_derivative_def)
 
 
 subsection \<open>Integrating characteristic function of an interval\<close>
 
 lemma has_integral_restrict_open_subinterval:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
   assumes intf: "(f has_integral i) (cbox c d)"
     and cb: "cbox c d \<subseteq> cbox a b"
   shows "((\<lambda>x. if x \<in> box c d then f x else 0) has_integral i) (cbox a b)"
 proof (cases "cbox c d = {}")
   case True
   then have "box c d = {}"
     by (metis bot.extremum_uniqueI box_subset_cbox)
   then show ?thesis
     using True intf by auto
 next
   case False
   then obtain p where pdiv: "p division_of cbox a b" and inp: "cbox c d \<in> p"
     using cb partial_division_extend_1 by blast
   define g where [abs_def]: "g x = (if x \<in>box c d then f x else 0)" for x
   interpret operative "lift_option plus" "Some (0 :: 'b)"
     "\<lambda>i. if g integrable_on i then Some (integral i g) else None"
     by (fact operative_integralI)
   note operat = division [OF pdiv, symmetric]
   show ?thesis
   proof (cases "(g has_integral i) (cbox a b)")
     case True then show ?thesis
       by (simp add: g_def)
   next
     case False
     have iterate:"F (\<lambda>i. if g integrable_on i then Some (integral i g) else None) (p - {cbox c d}) = Some 0"
     proof (intro neutral ballI)
       fix x
       assume x: "x \<in> p - {cbox c d}"
       then have "x \<in> p"
         by auto
       then obtain u v where uv: "x = cbox u v"
         using pdiv by blast
       have "interior x \<inter> interior (cbox c d) = {}"
         using pdiv inp x by blast 
       then have "(g has_integral 0) x"
         unfolding uv using has_integral_spike_interior[where f="\<lambda>x. 0"]
         by (metis (no_types, lifting) disjoint_iff_not_equal g_def has_integral_0_eq interior_cbox)
       then show "(if g integrable_on x then Some (integral x g) else None) = Some 0"
         by auto
     qed
     interpret comm_monoid_set "lift_option plus" "Some (0 :: 'b)"
       by (intro comm_monoid_set.intro comm_monoid_lift_option add.comm_monoid_axioms)
     have intg: "g integrable_on cbox c d"
       using integrable_spike_interior[where f=f]
       by (meson g_def has_integral_integrable intf)
     moreover have "integral (cbox c d) g = i"
     proof (rule has_integral_unique[OF has_integral_spike_interior intf])
       show "\<And>x. x \<in> box c d \<Longrightarrow> f x = g x"
         by (auto simp: g_def)
       show "(g has_integral integral (cbox c d) g) (cbox c d)"
         by (rule integrable_integral[OF intg])
     qed
     ultimately have "F (\<lambda>A. if g integrable_on A then Some (integral A g) else None) p = Some i"
       by (metis (full_types, lifting) division_of_finite inp iterate pdiv remove right_neutral)
     then
     have "(g has_integral i) (cbox a b)"
       by (metis integrable_on_def integral_unique operat option.inject option.simps(3))
     with False show ?thesis
       by blast
   qed
 qed
 
 
 lemma has_integral_restrict_closed_subinterval:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
   assumes "(f has_integral i) (cbox c d)"
     and "cbox c d \<subseteq> cbox a b"
   shows "((\<lambda>x. if x \<in> cbox c d then f x else 0) has_integral i) (cbox a b)"
 proof -
   note has_integral_restrict_open_subinterval[OF assms]
   note * = has_integral_spike[OF negligible_frontier_interval _ this]
   show ?thesis
     by (rule *[of c d]) (use box_subset_cbox[of c d] in auto)
 qed
 
 lemma has_integral_restrict_closed_subintervals_eq:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
   assumes "cbox c d \<subseteq> cbox a b"
   shows "((\<lambda>x. if x \<in> cbox c d then f x else 0) has_integral i) (cbox a b) \<longleftrightarrow> (f has_integral i) (cbox c d)"
   (is "?l = ?r")
 proof (cases "cbox c d = {}")
   case False
   let ?g = "\<lambda>x. if x \<in> cbox c d then f x else 0"
   show ?thesis
   proof 
     assume ?l
     then have "?g integrable_on cbox c d"
       using assms has_integral_integrable integrable_subinterval by blast
     then have "f integrable_on cbox c d"
       by (rule integrable_eq) auto
     moreover then have "i = integral (cbox c d) f"
       by (meson \<open>((\<lambda>x. if x \<in> cbox c d then f x else 0) has_integral i) (cbox a b)\<close> assms has_integral_restrict_closed_subinterval has_integral_unique integrable_integral)
     ultimately show ?r by auto
   next
     assume ?r then show ?l
       by (rule has_integral_restrict_closed_subinterval[OF _ assms])
   qed
 qed auto
 
 
 text \<open>Hence we can apply the limit process uniformly to all integrals.\<close>
 
 lemma has_integral':
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   shows "(f has_integral i) S \<longleftrightarrow>
     (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
       (\<exists>z. ((\<lambda>x. if x \<in> S then f(x) else 0) has_integral z) (cbox a b) \<and> norm(z - i) < e))"
   (is "?l \<longleftrightarrow> (\<forall>e>0. ?r e)")
 proof (cases "\<exists>a b. S = cbox a b")
   case False then show ?thesis 
     by (simp add: has_integral_alt)
 next
   case True
   then obtain a b where S: "S = cbox a b"
     by blast 
   obtain B where " 0 < B" and B: "\<And>x. x \<in> cbox a b \<Longrightarrow> norm x \<le> B"
     using bounded_cbox[unfolded bounded_pos] by blast
   show ?thesis
   proof safe
     fix e :: real
     assume ?l and "e > 0"
     have "((\<lambda>x. if x \<in> S then f x else 0) has_integral i) (cbox c d)"
       if "ball 0 (B+1) \<subseteq> cbox c d" for c d
         unfolding S using B that
         by (force intro: \<open>?l\<close>[unfolded S] has_integral_restrict_closed_subinterval)
     then show "?r e"
-      apply (rule_tac x="B+1" in exI)
-      using \<open>B>0\<close> \<open>e>0\<close> by force
+      by (meson \<open>0 < B\<close> \<open>0 < e\<close> add_pos_pos le_less_trans zero_less_one norm_pths(2))
   next
     assume as: "\<forall>e>0. ?r e"
     then obtain C 
       where C: "\<And>a b. ball 0 C \<subseteq> cbox a b \<Longrightarrow>
                 \<exists>z. ((\<lambda>x. if x \<in> S then f x else 0) has_integral z) (cbox a b)"
       by (meson zero_less_one)
     define c :: 'n where "c = (\<Sum>i\<in>Basis. (- max B C) *\<^sub>R i)"
     define d :: 'n where "d = (\<Sum>i\<in>Basis. max B C *\<^sub>R i)"
     have "c \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> d \<bullet> i" if "norm x \<le> B" "i \<in> Basis" for x i
       using that and Basis_le_norm[OF \<open>i\<in>Basis\<close>, of x]
       by (auto simp add: field_simps sum_negf c_def d_def)
     then have c_d: "cbox a b \<subseteq> cbox c d"
       by (meson B mem_box(2) subsetI)
     have "c \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> d \<bullet> i"
       if x: "norm (0 - x) < C" and i: "i \<in> Basis" for x i
         using Basis_le_norm[OF i, of x] x i by (auto simp: sum_negf c_def d_def)
       then have "ball 0 C \<subseteq> cbox c d"
         by (auto simp: mem_box dist_norm)
     with C obtain y where y: "(f has_integral y) (cbox a b)"
       using c_d has_integral_restrict_closed_subintervals_eq S by blast
     have "y = i"
     proof (rule ccontr)
       assume "y \<noteq> i"
       then have "0 < norm (y - i)"
         by auto
       from as[rule_format,OF this]
       obtain C where C: "\<And>a b. ball 0 C \<subseteq> cbox a b \<Longrightarrow> 
            \<exists>z. ((\<lambda>x. if x \<in> S then f x else 0) has_integral z) (cbox a b) \<and> norm (z-i) < norm (y-i)"
         by auto
       define c :: 'n where "c = (\<Sum>i\<in>Basis. (- max B C) *\<^sub>R i)"
       define d :: 'n where "d = (\<Sum>i\<in>Basis. max B C *\<^sub>R i)"
       have "c \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> d \<bullet> i"
         if "norm x \<le> B" and "i \<in> Basis" for x i
           using that Basis_le_norm[of i x] by (auto simp add: field_simps sum_negf c_def d_def)
         then have c_d: "cbox a b \<subseteq> cbox c d"
         by (simp add: B mem_box(2) subset_eq)
       have "c \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> d \<bullet> i" if "norm (0 - x) < C" and "i \<in> Basis" for x i
         using Basis_le_norm[of i x] that by (auto simp: sum_negf c_def d_def)
       then have "ball 0 C \<subseteq> cbox c d"
         by (auto simp: mem_box dist_norm)
       with C obtain z where z: "(f has_integral z) (cbox a b)" "norm (z-i) < norm (y-i)"
         using has_integral_restrict_closed_subintervals_eq[OF c_d] S by blast
       moreover then have "z = y" 
         by (blast intro: has_integral_unique[OF _ y])
       ultimately show False
         by auto
     qed
     then show ?l
       using y by (auto simp: S)
   qed
 qed
 
 lemma has_integral_le:
   fixes f :: "'n::euclidean_space \<Rightarrow> real"
   assumes fg: "(f has_integral i) S" "(g has_integral j) S"
     and le: "\<And>x. x \<in> S \<Longrightarrow> f x \<le> g x"
   shows "i \<le> j"
   using has_integral_component_le[OF _ fg, of 1] le  by auto
 
 lemma integral_le:
   fixes f :: "'n::euclidean_space \<Rightarrow> real"
   assumes "f integrable_on S"
     and "g integrable_on S"
     and "\<And>x. x \<in> S \<Longrightarrow> f x \<le> g x"
   shows "integral S f \<le> integral S g"
   by (rule has_integral_le[OF assms(1,2)[unfolded has_integral_integral] assms(3)])
 
 lemma has_integral_nonneg:
   fixes f :: "'n::euclidean_space \<Rightarrow> real"
   assumes "(f has_integral i) S"
     and "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> f x"
   shows "0 \<le> i"
   using has_integral_component_nonneg[of 1 f i S]
   unfolding o_def
   using assms
   by auto
 
 lemma integral_nonneg:
   fixes f :: "'n::euclidean_space \<Rightarrow> real"
   assumes f: "f integrable_on S" and 0: "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> f x"
   shows "0 \<le> integral S f"
   by (rule has_integral_nonneg[OF f[unfolded has_integral_integral] 0])
 
 
 text \<open>Hence a general restriction property.\<close>
 
 lemma has_integral_restrict [simp]:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: banach"
   assumes "S \<subseteq> T"
   shows "((\<lambda>x. if x \<in> S then f x else 0) has_integral i) T \<longleftrightarrow> (f has_integral i) S"
 proof -
   have *: "\<And>x. (if x \<in> T then if x \<in> S then f x else 0 else 0) =  (if x\<in>S then f x else 0)"
     using assms by auto
   show ?thesis
     apply (subst(2) has_integral')
     apply (subst has_integral')
       apply (simp add: *)
     done
 qed
 
 corollary has_integral_restrict_UNIV:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   shows "((\<lambda>x. if x \<in> s then f x else 0) has_integral i) UNIV \<longleftrightarrow> (f has_integral i) s"
   by auto
 
 lemma has_integral_restrict_Int:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: banach"
   shows "((\<lambda>x. if x \<in> S then f x else 0) has_integral i) T \<longleftrightarrow> (f has_integral i) (S \<inter> T)"
 proof -
   have "((\<lambda>x. if x \<in> T then if x \<in> S then f x else 0 else 0) has_integral i) UNIV =
         ((\<lambda>x. if x \<in> S \<inter> T then f x else 0) has_integral i) UNIV"
     by (rule has_integral_cong) auto
   then show ?thesis
     using has_integral_restrict_UNIV by fastforce
 qed
 
 lemma integral_restrict_Int:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: banach"
   shows "integral T (\<lambda>x. if x \<in> S then f x else 0) = integral (S \<inter> T) f"
   by (metis (no_types, lifting) has_integral_cong has_integral_restrict_Int integrable_integral integral_unique not_integrable_integral)
 
 lemma integrable_restrict_Int:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: banach"
   shows "(\<lambda>x. if x \<in> S then f x else 0) integrable_on T \<longleftrightarrow> f integrable_on (S \<inter> T)"
   using has_integral_restrict_Int by fastforce
 
 lemma has_integral_on_superset:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes f: "(f has_integral i) S"
       and "\<And>x. x \<notin> S \<Longrightarrow> f x = 0"
       and "S \<subseteq> T"
     shows "(f has_integral i) T"
 proof -
   have "(\<lambda>x. if x \<in> S then f x else 0) = (\<lambda>x. if x \<in> T then f x else 0)"
     using assms by fastforce
   with f show ?thesis
     by (simp only: has_integral_restrict_UNIV [symmetric, of f])
 qed
 
 lemma integrable_on_superset:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes "f integrable_on S"
     and "\<And>x. x \<notin> S \<Longrightarrow> f x = 0"
     and "S \<subseteq> t"
   shows "f integrable_on t"
   using assms
   unfolding integrable_on_def
   by (auto intro:has_integral_on_superset)
 
 lemma integral_restrict_UNIV:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   shows "integral UNIV (\<lambda>x. if x \<in> S then f x else 0) = integral S f"
   by (simp add: integral_restrict_Int)
 
 lemma integrable_restrict_UNIV:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   shows "(\<lambda>x. if x \<in> s then f x else 0) integrable_on UNIV \<longleftrightarrow> f integrable_on s"
   unfolding integrable_on_def
   by auto
 
 lemma has_integral_subset_component_le:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
   assumes k: "k \<in> Basis"
     and as: "S \<subseteq> T" "(f has_integral i) S" "(f has_integral j) T" "\<And>x. x\<in>T \<Longrightarrow> 0 \<le> f(x)\<bullet>k"
   shows "i\<bullet>k \<le> j\<bullet>k"
 proof -
-  have "((\<lambda>x. if x \<in> S then f x else 0) has_integral i) UNIV"
-        "((\<lambda>x. if x \<in> T then f x else 0) has_integral j) UNIV"
+  have \<section>: "((\<lambda>x. if x \<in> S then f x else 0) has_integral i) UNIV"
+          "((\<lambda>x. if x \<in> T then f x else 0) has_integral j) UNIV"
     by (simp_all add: assms)
-  then show ?thesis
-    apply (rule has_integral_component_le[OF k])
-    using as by auto
+  show ?thesis
+    using as by (force intro!: has_integral_component_le[OF k \<section>])
 qed
 
 subsection\<open>Integrals on set differences\<close>
 
 lemma has_integral_setdiff:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
   assumes S: "(f has_integral i) S" and T: "(f has_integral j) T"
     and neg: "negligible (T - S)"
   shows "(f has_integral (i - j)) (S - T)"
 proof -
   show ?thesis
     unfolding has_integral_restrict_UNIV [symmetric, of f]
   proof (rule has_integral_spike [OF neg])
     have eq: "(\<lambda>x. (if x \<in> S then f x else 0) - (if x \<in> T then f x else 0)) =
             (\<lambda>x. if x \<in> T - S then - f x else if x \<in> S - T then f x else 0)"
       by (force simp add: )
     have "((\<lambda>x. if x \<in> S then f x else 0) has_integral i) UNIV"
       "((\<lambda>x. if x \<in> T then f x else 0) has_integral j) UNIV"
       using S T has_integral_restrict_UNIV by auto
     from has_integral_diff [OF this]
     show "((\<lambda>x. if x \<in> T - S then - f x else if x \<in> S - T then f x else 0)
                    has_integral i-j) UNIV"
       by (simp add: eq)
   qed force
 qed
 
 lemma integral_setdiff:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
   assumes "f integrable_on S" "f integrable_on T" "negligible(T - S)"
  shows "integral (S - T) f = integral S f - integral T f"
   by (rule integral_unique) (simp add: assms has_integral_setdiff integrable_integral)
 
 lemma integrable_setdiff:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
   assumes "(f has_integral i) S" "(f has_integral j) T" "negligible (T - S)"
   shows "f integrable_on (S - T)"
   using has_integral_setdiff [OF assms]
   by (simp add: has_integral_iff)
 
 lemma negligible_setdiff [simp]: "T \<subseteq> S \<Longrightarrow> negligible (T - S)"
   by (metis Diff_eq_empty_iff negligible_empty)
 
 lemma negligible_on_intervals: "negligible s \<longleftrightarrow> (\<forall>a b. negligible(s \<inter> cbox a b))" (is "?l \<longleftrightarrow> ?r")
 proof
-  assume ?r
+  assume R: ?r
   show ?l
     unfolding negligible_def
   proof safe
     fix a b
-    show "(indicator s has_integral 0) (cbox a b)"
-      apply (rule has_integral_negligible[OF \<open>?r\<close>[rule_format,of a b]])
-      unfolding indicator_def
-      apply auto
-      done
+    have "negligible (s \<inter> cbox a b)"
+      by (simp add: R)
+    then show "(indicator s has_integral 0) (cbox a b)"
+      by (meson Diff_iff Int_iff has_integral_negligible indicator_simps(2))
   qed
 qed (simp add: negligible_Int)
 
 lemma negligible_translation:
   assumes "negligible S"
     shows "negligible ((+) c ` S)"
 proof -
   have inj: "inj ((+) c)"
     by simp
   show ?thesis
   using assms
   proof (clarsimp simp: negligible_def)
     fix a b
     assume "\<forall>x y. (indicator S has_integral 0) (cbox x y)"
     then have *: "(indicator S has_integral 0) (cbox (a-c) (b-c))"
       by (meson Diff_iff assms has_integral_negligible indicator_simps(2))
     have eq: "indicator ((+) c ` S) = (\<lambda>x. indicator S (x - c))"
       by (force simp add: indicator_def)
     show "(indicator ((+) c ` S) has_integral 0) (cbox a b)"
       using has_integral_affinity [OF *, of 1 "-c"]
             cbox_translation [of "c" "-c+a" "-c+b"]
       by (simp add: eq) (simp add: ac_simps)
   qed
 qed
 
 lemma negligible_translation_rev:
   assumes "negligible ((+) c ` S)"
     shows "negligible S"
 by (metis negligible_translation [OF assms, of "-c"] translation_galois)
 
 lemma has_integral_spike_set_eq:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes "negligible {x \<in> S - T. f x \<noteq> 0}" "negligible {x \<in> T - S. f x \<noteq> 0}"
   shows "(f has_integral y) S \<longleftrightarrow> (f has_integral y) T"
 proof -
   have "((\<lambda>x. if x \<in> S then f x else 0) has_integral y) UNIV =
         ((\<lambda>x. if x \<in> T then f x else 0) has_integral y) UNIV"
   proof (rule has_integral_spike_eq)
     show "negligible ({x \<in> S - T. f x \<noteq> 0} \<union> {x \<in> T - S. f x \<noteq> 0})"
       by (rule negligible_Un [OF assms])
   qed auto
   then show ?thesis
     by (simp add: has_integral_restrict_UNIV)
 qed
 
 corollary integral_spike_set:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes "negligible {x \<in> S - T. f x \<noteq> 0}" "negligible {x \<in> T - S. f x \<noteq> 0}"
   shows "integral S f = integral T f"
   using has_integral_spike_set_eq [OF assms]
   by (metis eq_integralD integral_unique)
 
 lemma integrable_spike_set:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes f: "f integrable_on S" and neg: "negligible {x \<in> S - T. f x \<noteq> 0}" "negligible {x \<in> T - S. f x \<noteq> 0}"
   shows "f integrable_on T"
   using has_integral_spike_set_eq [OF neg] f by blast
 
 lemma integrable_spike_set_eq:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes "negligible ((S - T) \<union> (T - S))"
   shows "f integrable_on S \<longleftrightarrow> f integrable_on T"
   by (blast intro: integrable_spike_set assms negligible_subset)
 
 lemma integrable_on_insert_iff: "f integrable_on (insert x X) \<longleftrightarrow> f integrable_on X"
   for f::"_ \<Rightarrow> 'a::banach"
   by (rule integrable_spike_set_eq) (auto simp: insert_Diff_if)
 
 lemma has_integral_interior:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: banach"
   shows "negligible(frontier S) \<Longrightarrow> (f has_integral y) (interior S) \<longleftrightarrow> (f has_integral y) S"
   by (rule has_integral_spike_set_eq [OF empty_imp_negligible negligible_subset])
      (use interior_subset in \<open>auto simp: frontier_def closure_def\<close>)
 
 lemma has_integral_closure:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: banach"
   shows "negligible(frontier S) \<Longrightarrow> (f has_integral y) (closure S) \<longleftrightarrow> (f has_integral y) S"
   by (rule has_integral_spike_set_eq [OF negligible_subset empty_imp_negligible]) (auto simp: closure_Un_frontier )
 
 lemma has_integral_open_interval:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: banach"
   shows "(f has_integral y) (box a b) \<longleftrightarrow> (f has_integral y) (cbox a b)"
   unfolding interior_cbox [symmetric]
   by (metis frontier_cbox has_integral_interior negligible_frontier_interval)
 
 lemma integrable_on_open_interval:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: banach"
   shows "f integrable_on box a b \<longleftrightarrow> f integrable_on cbox a b"
   by (simp add: has_integral_open_interval integrable_on_def)
 
 lemma integral_open_interval:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: banach"
   shows "integral(box a b) f = integral(cbox a b) f"
   by (metis has_integral_integrable_integral has_integral_open_interval not_integrable_integral)
 
 
 subsection \<open>More lemmas that are useful later\<close>
 
 lemma has_integral_subset_le:
   fixes f :: "'n::euclidean_space \<Rightarrow> real"
   assumes "s \<subseteq> t"
     and "(f has_integral i) s"
     and "(f has_integral j) t"
     and "\<forall>x\<in>t. 0 \<le> f x"
   shows "i \<le> j"
   using has_integral_subset_component_le[OF _ assms(1), of 1 f i j]
   using assms
   by auto
 
 lemma integral_subset_component_le:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
   assumes "k \<in> Basis"
     and "s \<subseteq> t"
     and "f integrable_on s"
     and "f integrable_on t"
     and "\<forall>x \<in> t. 0 \<le> f x \<bullet> k"
   shows "(integral s f)\<bullet>k \<le> (integral t f)\<bullet>k"
-  apply (rule has_integral_subset_component_le)
-  using assms
-  apply auto
-  done
+  by (meson assms has_integral_subset_component_le integrable_integral)
 
 lemma integral_subset_le:
   fixes f :: "'n::euclidean_space \<Rightarrow> real"
   assumes "s \<subseteq> t"
     and "f integrable_on s"
     and "f integrable_on t"
     and "\<forall>x \<in> t. 0 \<le> f x"
   shows "integral s f \<le> integral t f"
-  apply (rule has_integral_subset_le)
-  using assms
-  apply auto
-  done
+  using assms has_integral_subset_le by blast
 
 lemma has_integral_alt':
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   shows "(f has_integral i) s \<longleftrightarrow> 
           (\<forall>a b. (\<lambda>x. if x \<in> s then f x else 0) integrable_on cbox a b) \<and>
           (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
             norm (integral (cbox a b) (\<lambda>x. if x \<in> s then f x else 0) - i) < e)"
   (is "?l = ?r")
 proof
   assume rhs: ?r
   show ?l
   proof (subst has_integral', intro allI impI)
     fix e::real
     assume "e > 0"
     from rhs[THEN conjunct2,rule_format,OF this] 
     show "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
                    (\<exists>z. ((\<lambda>x. if x \<in> s then f x else 0) has_integral z)
                          (cbox a b) \<and> norm (z - i) < e)"
-      apply (rule ex_forward)
-      using rhs by blast
+      by (simp add: has_integral_iff rhs)
   qed
 next
   let ?\<Phi> = "\<lambda>e a b. \<exists>z. ((\<lambda>x. if x \<in> s then f x else 0) has_integral z) (cbox a b) \<and> norm (z - i) < e"
   assume ?l 
   then have lhs: "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow> ?\<Phi> e a b" if "e > 0" for e
     using that has_integral'[of f] by auto
   let ?f = "\<lambda>x. if x \<in> s then f x else 0"
   show ?r
   proof (intro conjI allI impI)
     fix a b :: 'n
     from lhs[OF zero_less_one]
     obtain B where "0 < B" and B: "\<And>a b. ball 0 B \<subseteq> cbox a b \<Longrightarrow> ?\<Phi> 1 a b"
       by blast
     let ?a = "\<Sum>i\<in>Basis. min (a\<bullet>i) (-B) *\<^sub>R i::'n"
     let ?b = "\<Sum>i\<in>Basis. max (b\<bullet>i) B *\<^sub>R i::'n"
     show "?f integrable_on cbox a b"
     proof (rule integrable_subinterval[of _ ?a ?b])
       have "?a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> ?b \<bullet> i" if "norm (0 - x) < B" "i \<in> Basis" for x i
         using Basis_le_norm[of i x] that by (auto simp add:field_simps)
       then have "ball 0 B \<subseteq> cbox ?a ?b"
         by (auto simp: mem_box dist_norm)
       then show "?f integrable_on cbox ?a ?b"
         unfolding integrable_on_def using B by blast
       show "cbox a b \<subseteq> cbox ?a ?b"
         by (force simp: mem_box)
     qed
   
     fix e :: real
     assume "e > 0"
     with lhs show "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
       norm (integral (cbox a b) (\<lambda>x. if x \<in> s then f x else 0) - i) < e"
       by (metis (no_types, lifting) has_integral_integrable_integral)
   qed
 qed
 
 
 subsection \<open>Continuity of the integral (for a 1-dimensional interval)\<close>
 
 lemma integrable_alt:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   shows "f integrable_on s \<longleftrightarrow>
     (\<forall>a b. (\<lambda>x. if x \<in> s then f x else 0) integrable_on cbox a b) \<and>
     (\<forall>e>0. \<exists>B>0. \<forall>a b c d. ball 0 B \<subseteq> cbox a b \<and> ball 0 B \<subseteq> cbox c d \<longrightarrow>
     norm (integral (cbox a b) (\<lambda>x. if x \<in> s then f x else 0) -
       integral (cbox c d)  (\<lambda>x. if x \<in> s then f x else 0)) < e)"
   (is "?l = ?r")
 proof
   let ?F = "\<lambda>x. if x \<in> s then f x else 0"
   assume ?l
   then obtain y where intF: "\<And>a b. ?F integrable_on cbox a b"
           and y: "\<And>e. 0 < e \<Longrightarrow>
               \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow> norm (integral (cbox a b) ?F - y) < e"
     unfolding integrable_on_def has_integral_alt'[of f] by auto
   show ?r
   proof (intro conjI allI impI intF)
     fix e::real
     assume "e > 0"
     then have "e/2 > 0"
       by auto
     obtain B where "0 < B" 
        and B: "\<And>a b. ball 0 B \<subseteq> cbox a b \<Longrightarrow> norm (integral (cbox a b) ?F - y) < e/2"
       using \<open>0 < e/2\<close> y by blast
     show "\<exists>B>0. \<forall>a b c d. ball 0 B \<subseteq> cbox a b \<and> ball 0 B \<subseteq> cbox c d \<longrightarrow>
                   norm (integral (cbox a b) ?F - integral (cbox c d) ?F) < e"
     proof (intro conjI exI impI allI, rule \<open>0 < B\<close>)
       fix a b c d::'n
       assume sub: "ball 0 B \<subseteq> cbox a b \<and> ball 0 B \<subseteq> cbox c d"
       show "norm (integral (cbox a b) ?F - integral (cbox c d) ?F) < e"
         using sub by (auto intro: norm_triangle_half_l dest: B)
     qed
   qed
 next
   let ?F = "\<lambda>x. if x \<in> s then f x else 0"
   assume rhs: ?r
   let ?cube = "\<lambda>n. cbox (\<Sum>i\<in>Basis. - real n *\<^sub>R i::'n) (\<Sum>i\<in>Basis. real n *\<^sub>R i)"
   have "Cauchy (\<lambda>n. integral (?cube n) ?F)"
     unfolding Cauchy_def
   proof (intro allI impI)
     fix e::real
     assume "e > 0"
     with rhs obtain B where "0 < B" 
       and B: "\<And>a b c d. ball 0 B \<subseteq> cbox a b \<and> ball 0 B \<subseteq> cbox c d 
                         \<Longrightarrow> norm (integral (cbox a b) ?F - integral (cbox c d) ?F) < e"
       by blast
     obtain N where N: "B \<le> real N"
       using real_arch_simple by blast
     have "ball 0 B \<subseteq> ?cube n" if n: "n \<ge> N" for n
     proof -
       have "sum ((*\<^sub>R) (- real n)) Basis \<bullet> i \<le> x \<bullet> i \<and>
             x \<bullet> i \<le> sum ((*\<^sub>R) (real n)) Basis \<bullet> i"
         if "norm x < B" "i \<in> Basis" for x i::'n
           using Basis_le_norm[of i x] n N that by (auto simp add: field_simps sum_negf)
       then show ?thesis
         by (auto simp: mem_box dist_norm)
     qed
     then show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (integral (?cube m) ?F) (integral (?cube n) ?F) < e"
       by (fastforce simp add: dist_norm intro!: B)
   qed
   then obtain i where i: "(\<lambda>n. integral (?cube n) ?F) \<longlonglongrightarrow> i"
     using convergent_eq_Cauchy by blast
   have "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow> norm (integral (cbox a b) ?F - i) < e"
     if "e > 0" for e
   proof -
     have *: "e/2 > 0" using that by auto
     then obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> norm (i - integral (?cube n) ?F) < e/2"
       using i[THEN LIMSEQ_D, simplified norm_minus_commute] by meson
     obtain B where "0 < B" 
       and B: "\<And>a b c d. \<lbrakk>ball 0 B \<subseteq> cbox a b; ball 0 B \<subseteq> cbox c d\<rbrakk> \<Longrightarrow>
                   norm (integral (cbox a b) ?F - integral (cbox c d) ?F) < e/2"
       using rhs * by meson
     let ?B = "max (real N) B"
     show ?thesis
     proof (intro exI conjI allI impI)
       show "0 < ?B"
         using \<open>B > 0\<close> by auto
       fix a b :: 'n
       assume "ball 0 ?B \<subseteq> cbox a b"
       moreover obtain n where n: "max (real N) B \<le> real n"
         using real_arch_simple by blast
       moreover have "ball 0 B \<subseteq> ?cube n"
       proof 
         fix x :: 'n
         assume x: "x \<in> ball 0 B"
         have "\<lbrakk>norm (0 - x) < B; i \<in> Basis\<rbrakk>
               \<Longrightarrow> sum ((*\<^sub>R) (-n)) Basis \<bullet> i\<le> x \<bullet> i \<and> x \<bullet> i \<le> sum ((*\<^sub>R) n) Basis \<bullet> i" for i
           using Basis_le_norm[of i x] n by (auto simp add: field_simps sum_negf)
         then show "x \<in> ?cube n"
           using x by (auto simp: mem_box dist_norm)
       qed 
       ultimately show "norm (integral (cbox a b) ?F - i) < e"
         using norm_triangle_half_l [OF B N] by force
     qed
   qed
   then show ?l unfolding integrable_on_def has_integral_alt'[of f]
     using rhs by blast
 qed
 
 lemma integrable_altD:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes "f integrable_on s"
   shows "\<And>a b. (\<lambda>x. if x \<in> s then f x else 0) integrable_on cbox a b"
     and "\<And>e. e > 0 \<Longrightarrow> \<exists>B>0. \<forall>a b c d. ball 0 B \<subseteq> cbox a b \<and> ball 0 B \<subseteq> cbox c d \<longrightarrow>
       norm (integral (cbox a b) (\<lambda>x. if x \<in> s then f x else 0) - integral (cbox c d)  (\<lambda>x. if x \<in> s then f x else 0)) < e"
   using assms[unfolded integrable_alt[of f]] by auto
 
 lemma integrable_alt_subset:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
   shows
      "f integrable_on S \<longleftrightarrow>
       (\<forall>a b. (\<lambda>x. if x \<in> S then f x else 0) integrable_on cbox a b) \<and>
       (\<forall>e>0. \<exists>B>0. \<forall>a b c d.
                       ball 0 B \<subseteq> cbox a b \<and> cbox a b \<subseteq> cbox c d
                       \<longrightarrow> norm(integral (cbox a b) (\<lambda>x. if x \<in> S then f x else 0) -
                                integral (cbox c d) (\<lambda>x. if x \<in> S then f x else 0)) < e)"
       (is "_ = ?rhs")
 proof -
   let ?g = "\<lambda>x. if x \<in> S then f x else 0"
   have "f integrable_on S \<longleftrightarrow>
         (\<forall>a b. ?g integrable_on cbox a b) \<and>
         (\<forall>e>0. \<exists>B>0. \<forall>a b c d. ball 0 B \<subseteq> cbox a b \<and> ball 0 B \<subseteq> cbox c d \<longrightarrow>
            norm (integral (cbox a b) ?g - integral (cbox c d)  ?g) < e)"
     by (rule integrable_alt)
   also have "\<dots> = ?rhs"
   proof -
     { fix e :: "real"
       assume e: "\<And>e. e>0 \<Longrightarrow> \<exists>B>0. \<forall>a b c d. ball 0 B \<subseteq> cbox a b \<and> cbox a b \<subseteq> cbox c d \<longrightarrow>
                                    norm (integral (cbox a b) ?g - integral (cbox c d) ?g) < e"
         and "e > 0"
       obtain B where "B > 0"
         and B: "\<And>a b c d. \<lbrakk>ball 0 B \<subseteq> cbox a b; cbox a b \<subseteq> cbox c d\<rbrakk> \<Longrightarrow>
                            norm (integral (cbox a b) ?g - integral (cbox c d) ?g) < e/2"
         using \<open>e > 0\<close> e [of "e/2"] by force
       have "\<exists>B>0. \<forall>a b c d.
                ball 0 B \<subseteq> cbox a b \<and> ball 0 B \<subseteq> cbox c d \<longrightarrow>
                norm (integral (cbox a b) ?g - integral (cbox c d) ?g) < e"
       proof (intro exI allI conjI impI)
         fix a b c d :: "'a"
         let ?\<alpha> = "\<Sum>i\<in>Basis. max (a \<bullet> i) (c \<bullet> i) *\<^sub>R i"
         let ?\<beta> = "\<Sum>i\<in>Basis. min (b \<bullet> i) (d \<bullet> i) *\<^sub>R i"
         show "norm (integral (cbox a b) ?g - integral (cbox c d) ?g) < e"
           if ball: "ball 0 B \<subseteq> cbox a b \<and> ball 0 B \<subseteq> cbox c d"
         proof -
           have B': "norm (integral (cbox a b \<inter> cbox c d) ?g - integral (cbox x y) ?g) < e/2"
             if "cbox a b \<inter> cbox c d \<subseteq> cbox x y" for x y
             using B [of ?\<alpha> ?\<beta> x y] ball that by (simp add: Int_interval [symmetric])
           show ?thesis
             using B' [of a b] B' [of c d] norm_triangle_half_r by blast
         qed
       qed (use \<open>B > 0\<close> in auto)}
     then show ?thesis
       by force
   qed
   finally show ?thesis .
 qed
 
 lemma integrable_on_subcbox:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes intf: "f integrable_on S"
     and sub: "cbox a b \<subseteq> S"
   shows "f integrable_on cbox a b"
 proof -
   have "(\<lambda>x. if x \<in> S then f x else 0) integrable_on cbox a b"
     by (simp add: intf integrable_altD(1))
 then show ?thesis
   by (metis (mono_tags) sub integrable_restrict_Int le_inf_iff order_refl subset_antisym)
 qed
 
 
 subsection \<open>A straddling criterion for integrability\<close>
 
 lemma integrable_straddle_interval:
   fixes f :: "'n::euclidean_space \<Rightarrow> real"
   assumes "\<And>e. e>0 \<Longrightarrow> \<exists>g h i j. (g has_integral i) (cbox a b) \<and> (h has_integral j) (cbox a b) \<and>
                             \<bar>i - j\<bar> < e \<and> (\<forall>x\<in>cbox a b. (g x) \<le> f x \<and> f x \<le> h x)"
   shows "f integrable_on cbox a b"
 proof -
   have "\<exists>d. gauge d \<and>
             (\<forall>p1 p2. p1 tagged_division_of cbox a b \<and> d fine p1 \<and>
                      p2 tagged_division_of cbox a b \<and> d fine p2 \<longrightarrow>
                      \<bar>(\<Sum>(x,K)\<in>p1. content K *\<^sub>R f x) - (\<Sum>(x,K)\<in>p2. content K *\<^sub>R f x)\<bar> < e)"
     if "e > 0" for e
   proof -
     have e: "e/3 > 0"
       using that by auto
     then obtain g h i j where ij: "\<bar>i - j\<bar> < e/3"
             and "(g has_integral i) (cbox a b)"
             and "(h has_integral j) (cbox a b)"
             and fgh: "\<And>x. x \<in> cbox a b \<Longrightarrow> g x \<le> f x \<and> f x \<le> h x"
       using assms real_norm_def by metis
     then obtain d1 d2 where "gauge d1" "gauge d2"
             and d1: "\<And>p. \<lbrakk>p tagged_division_of cbox a b; d1 fine p\<rbrakk> \<Longrightarrow>
                           \<bar>(\<Sum>(x,K)\<in>p. content K *\<^sub>R g x) - i\<bar> < e/3"
             and d2: "\<And>p. \<lbrakk>p tagged_division_of cbox a b; d2 fine p\<rbrakk> \<Longrightarrow>
                           \<bar>(\<Sum>(x,K) \<in> p. content K *\<^sub>R h x) - j\<bar> < e/3"
       by (metis e has_integral real_norm_def)
     have "\<bar>(\<Sum>(x,K) \<in> p1. content K *\<^sub>R f x) - (\<Sum>(x,K) \<in> p2. content K *\<^sub>R f x)\<bar> < e"
       if p1: "p1 tagged_division_of cbox a b" and 11: "d1 fine p1" and 21: "d2 fine p1"
        and p2: "p2 tagged_division_of cbox a b" and 12: "d1 fine p2" and 22: "d2 fine p2" for p1 p2
     proof -
       have *: "\<And>g1 g2 h1 h2 f1 f2.
                   \<lbrakk>\<bar>g2 - i\<bar> < e/3; \<bar>g1 - i\<bar> < e/3; \<bar>h2 - j\<bar> < e/3; \<bar>h1 - j\<bar> < e/3;
                    g1 - h2 \<le> f1 - f2; f1 - f2 \<le> h1 - g2\<rbrakk>
                   \<Longrightarrow> \<bar>f1 - f2\<bar> < e"
         using \<open>e > 0\<close> ij by arith
       have 0: "(\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p1. content k *\<^sub>R g x) \<ge> 0"
               "0 \<le> (\<Sum>(x, k)\<in>p2. content k *\<^sub>R h x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)"
               "(\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R g x) \<ge> 0"
               "0 \<le> (\<Sum>(x, k)\<in>p1. content k *\<^sub>R h x) - (\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x)"
         unfolding sum_subtractf[symmetric]
            apply (auto intro!: sum_nonneg)
            apply (meson fgh measure_nonneg mult_left_mono tag_in_interval that sum_nonneg)+
         done
       show ?thesis
       proof (rule *)
         show "\<bar>(\<Sum>(x,K) \<in> p2. content K *\<^sub>R g x) - i\<bar> < e/3"
           by (rule d1[OF p2 12])
         show "\<bar>(\<Sum>(x,K) \<in> p1. content K *\<^sub>R g x) - i\<bar> < e/3"
           by (rule d1[OF p1 11])
         show "\<bar>(\<Sum>(x,K) \<in> p2. content K *\<^sub>R h x) - j\<bar> < e/3"
           by (rule d2[OF p2 22])
         show "\<bar>(\<Sum>(x,K) \<in> p1. content K *\<^sub>R h x) - j\<bar> < e/3"
           by (rule d2[OF p1 21])
       qed (use 0 in auto)
     qed
     then show ?thesis
       by (rule_tac x="\<lambda>x. d1 x \<inter> d2 x" in exI)
         (auto simp: fine_Int intro: \<open>gauge d1\<close> \<open>gauge d2\<close> d1 d2)
   qed
   then show ?thesis
     by (simp add: integrable_Cauchy)
 qed
 
 lemma integrable_straddle:
   fixes f :: "'n::euclidean_space \<Rightarrow> real"
   assumes "\<And>e. e>0 \<Longrightarrow> \<exists>g h i j. (g has_integral i) s \<and> (h has_integral j) s \<and>
                      \<bar>i - j\<bar> < e \<and> (\<forall>x\<in>s. g x \<le> f x \<and> f x \<le> h x)"
   shows "f integrable_on s"
 proof -
   let ?fs = "(\<lambda>x. if x \<in> s then f x else 0)"
   have "?fs integrable_on cbox a b" for a b
   proof (rule integrable_straddle_interval)
     fix e::real
     assume "e > 0"
     then have *: "e/4 > 0"
       by auto
     with assms obtain g h i j where g: "(g has_integral i) s" and h: "(h has_integral j) s"
                  and ij: "\<bar>i - j\<bar> < e/4"
                  and fgh: "\<And>x. x \<in> s \<Longrightarrow> g x \<le> f x \<and> f x \<le> h x"
       by metis
     let ?gs = "(\<lambda>x. if x \<in> s then g x else 0)"
     let ?hs = "(\<lambda>x. if x \<in> s then h x else 0)"
     obtain Bg where Bg: "\<And>a b. ball 0 Bg \<subseteq> cbox a b \<Longrightarrow> \<bar>integral (cbox a b) ?gs - i\<bar> < e/4"
               and int_g: "\<And>a b. ?gs integrable_on cbox a b"
       using g * unfolding has_integral_alt' real_norm_def by meson
     obtain Bh where
           Bh: "\<And>a b. ball 0 Bh \<subseteq> cbox a b \<Longrightarrow> \<bar>integral (cbox a b) ?hs - j\<bar> < e/4"
          and int_h: "\<And>a b. ?hs integrable_on cbox a b"
       using h * unfolding has_integral_alt' real_norm_def by meson
     define c where "c = (\<Sum>i\<in>Basis. min (a\<bullet>i) (- (max Bg Bh)) *\<^sub>R i)"
     define d where "d = (\<Sum>i\<in>Basis. max (b\<bullet>i) (max Bg Bh) *\<^sub>R i)"
     have "\<lbrakk>norm (0 - x) < Bg; i \<in> Basis\<rbrakk> \<Longrightarrow> c \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> d \<bullet> i" for x i
       using Basis_le_norm[of i x] unfolding c_def d_def by auto
     then have ballBg: "ball 0 Bg \<subseteq> cbox c d"
       by (auto simp: mem_box dist_norm)
     have "\<lbrakk>norm (0 - x) < Bh; i \<in> Basis\<rbrakk> \<Longrightarrow> c \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> d \<bullet> i" for x i
       using Basis_le_norm[of i x] unfolding c_def d_def by auto
     then have ballBh: "ball 0 Bh \<subseteq> cbox c d"
       by (auto simp: mem_box dist_norm)
     have ab_cd: "cbox a b \<subseteq> cbox c d"
       by (auto simp: c_def d_def subset_box_imp)
     have **: "\<And>ch cg ag ah::real. \<lbrakk>\<bar>ah - ag\<bar> \<le> \<bar>ch - cg\<bar>; \<bar>cg - i\<bar> < e/4; \<bar>ch - j\<bar> < e/4\<rbrakk>
        \<Longrightarrow> \<bar>ag - ah\<bar> < e"
       using ij by arith
     show "\<exists>g h i j. (g has_integral i) (cbox a b) \<and> (h has_integral j) (cbox a b) \<and> \<bar>i - j\<bar> < e \<and>
           (\<forall>x\<in>cbox a b. g x \<le> (if x \<in> s then f x else 0) \<and>
                         (if x \<in> s then f x else 0) \<le> h x)"
     proof (intro exI ballI conjI)
       have eq: "\<And>x f g. (if x \<in> s then f x else 0) - (if x \<in> s then g x else 0) =
                        (if x \<in> s then f x - g x else (0::real))"
         by auto
       have int_hg: "(\<lambda>x. if x \<in> s then h x - g x else 0) integrable_on cbox a b"
                    "(\<lambda>x. if x \<in> s then h x - g x else 0) integrable_on cbox c d"
         by (metis (no_types) integrable_diff g h has_integral_integrable integrable_altD(1))+
       show "(?gs has_integral integral (cbox a b) ?gs) (cbox a b)"
            "(?hs has_integral integral (cbox a b) ?hs) (cbox a b)"
         by (intro integrable_integral int_g int_h)+
       then have "integral (cbox a b) ?gs \<le> integral (cbox a b) ?hs"
-        apply (rule has_integral_le)
-        using fgh by force
+        using fgh by (force intro: has_integral_le)
       then have "0 \<le> integral (cbox a b) ?hs - integral (cbox a b) ?gs"
         by simp
       then have "\<bar>integral (cbox a b) ?hs - integral (cbox a b) ?gs\<bar>
               \<le> \<bar>integral (cbox c d) ?hs - integral (cbox c d) ?gs\<bar>"
         apply (simp add: integral_diff [symmetric] int_g int_h)
         apply (subst abs_of_nonneg[OF integral_nonneg[OF integrable_diff, OF int_h int_g]])
         using fgh apply (force simp: eq intro!: integral_subset_le [OF ab_cd int_hg])+
         done
       then show "\<bar>integral (cbox a b) ?gs - integral (cbox a b) ?hs\<bar> < e"
-        apply (rule **)
-         apply (rule Bg ballBg Bh ballBh)+
-        done
+        using ** Bg ballBg Bh ballBh by blast
       show "\<And>x. x \<in> cbox a b \<Longrightarrow> ?gs x \<le> ?fs x" "\<And>x. x \<in> cbox a b \<Longrightarrow> ?fs x \<le> ?hs x"
         using fgh by auto
     qed
   qed
   then have int_f: "?fs integrable_on cbox a b" for a b
     by simp
   have "\<exists>B>0. \<forall>a b c d.
                   ball 0 B \<subseteq> cbox a b \<and> ball 0 B \<subseteq> cbox c d \<longrightarrow>
                   abs (integral (cbox a b) ?fs - integral (cbox c d) ?fs) < e"
       if "0 < e" for e
   proof -
     have *: "e/3 > 0"
       using that by auto
     with assms obtain g h i j where g: "(g has_integral i) s" and h: "(h has_integral j) s"
                  and ij: "\<bar>i - j\<bar> < e/3"
                  and fgh: "\<And>x. x \<in> s \<Longrightarrow> g x \<le> f x \<and> f x \<le> h x"
       by metis
     let ?gs = "(\<lambda>x. if x \<in> s then g x else 0)"
     let ?hs = "(\<lambda>x. if x \<in> s then h x else 0)"
     obtain Bg where "Bg > 0"
               and Bg: "\<And>a b. ball 0 Bg \<subseteq> cbox a b \<Longrightarrow> \<bar>integral (cbox a b) ?gs - i\<bar> < e/3"
               and int_g: "\<And>a b. ?gs integrable_on cbox a b"
       using g * unfolding has_integral_alt' real_norm_def by meson
     obtain Bh where "Bh > 0"
               and Bh: "\<And>a b. ball 0 Bh \<subseteq> cbox a b \<Longrightarrow> \<bar>integral (cbox a b) ?hs - j\<bar> < e/3"
               and int_h: "\<And>a b. ?hs integrable_on cbox a b"
       using h * unfolding has_integral_alt' real_norm_def by meson
     { fix a b c d :: 'n
       assume as: "ball 0 (max Bg Bh) \<subseteq> cbox a b" "ball 0 (max Bg Bh) \<subseteq> cbox c d"
       have **: "ball 0 Bg \<subseteq> ball (0::'n) (max Bg Bh)" "ball 0 Bh \<subseteq> ball (0::'n) (max Bg Bh)"
         by auto
       have *: "\<And>ga gc ha hc fa fc. \<lbrakk>\<bar>ga - i\<bar> < e/3; \<bar>gc - i\<bar> < e/3; \<bar>ha - j\<bar> < e/3;
                      \<bar>hc - j\<bar> < e/3; ga \<le> fa; fa \<le> ha; gc \<le> fc; fc \<le> hc\<rbrakk> \<Longrightarrow>
         \<bar>fa - fc\<bar> < e"
         using ij by arith
       have "abs (integral (cbox a b) (\<lambda>x. if x \<in> s then f x else 0) - integral (cbox c d)
         (\<lambda>x. if x \<in> s then f x else 0)) < e"
       proof (rule *)
         show "\<bar>integral (cbox a b) ?gs - i\<bar> < e/3"
           using "**" Bg as by blast
         show "\<bar>integral (cbox c d) ?gs - i\<bar> < e/3"
           using "**" Bg as by blast
         show "\<bar>integral (cbox a b) ?hs - j\<bar> < e/3"
           using "**" Bh as by blast
         show "\<bar>integral (cbox c d) ?hs - j\<bar> < e/3"
           using "**" Bh as by blast
       qed (use int_f int_g int_h fgh in \<open>simp_all add: integral_le\<close>)
     }
     then show ?thesis
       apply (rule_tac x="max Bg Bh" in exI)
         using \<open>Bg > 0\<close> by auto
   qed
   then show ?thesis
     unfolding integrable_alt[of f] real_norm_def by (blast intro: int_f)
 qed
 
 
 subsection \<open>Adding integrals over several sets\<close>
 
 lemma has_integral_Un:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes f: "(f has_integral i) S" "(f has_integral j) T"
     and neg: "negligible (S \<inter> T)"
   shows "(f has_integral (i + j)) (S \<union> T)"
   unfolding has_integral_restrict_UNIV[symmetric, of f]
 proof (rule has_integral_spike[OF neg])
   let ?f = "\<lambda>x. (if x \<in> S then f x else 0) + (if x \<in> T then f x else 0)"
   show "(?f has_integral i + j) UNIV"
     by (simp add: f has_integral_add)
 qed auto
 
 lemma integral_Un [simp]:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes "f integrable_on S" "f integrable_on T" "negligible (S \<inter> T)"
   shows "integral (S \<union> T) f = integral S f + integral T f"
   by (simp add: has_integral_Un assms integrable_integral integral_unique)
 
 lemma integrable_Un:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b :: banach"
   assumes "negligible (A \<inter> B)" "f integrable_on A" "f integrable_on B"
   shows   "f integrable_on (A \<union> B)"
 proof -
   from assms obtain y z where "(f has_integral y) A" "(f has_integral z) B"
      by (auto simp: integrable_on_def)
   from has_integral_Un[OF this assms(1)] show ?thesis by (auto simp: integrable_on_def)
 qed
 
 lemma integrable_Un':
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b :: banach"
   assumes "f integrable_on A" "f integrable_on B" "negligible (A \<inter> B)" "C = A \<union> B"
   shows   "f integrable_on C"
   using integrable_Un[of A B f] assms by simp
 
 lemma has_integral_Union:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes \<T>: "finite \<T>"
     and int: "\<And>S. S \<in> \<T> \<Longrightarrow> (f has_integral (i S)) S"
     and neg: "pairwise (\<lambda>S S'. negligible (S \<inter> S')) \<T>"
   shows "(f has_integral (sum i \<T>)) (\<Union>\<T>)"
 proof -
   let ?\<U> = "((\<lambda>(a,b). a \<inter> b) ` {(a,b). a \<in> \<T> \<and> b \<in> {y. y \<in> \<T> \<and> a \<noteq> y}})"
   have "((\<lambda>x. if x \<in> \<Union>\<T> then f x else 0) has_integral sum i \<T>) UNIV"
   proof (rule has_integral_spike)
     show "negligible (\<Union>?\<U>)"
     proof (rule negligible_Union)
       have "finite (\<T> \<times> \<T>)"
         by (simp add: \<T>)
       moreover have "{(a, b). a \<in> \<T> \<and> b \<in> {y \<in> \<T>. a \<noteq> y}} \<subseteq> \<T> \<times> \<T>"
         by auto
       ultimately show "finite ?\<U>"
         by (blast intro: finite_subset[of _ "\<T> \<times> \<T>"])
       show "\<And>t. t \<in> ?\<U> \<Longrightarrow> negligible t"
         using neg unfolding pairwise_def by auto
     qed
   next
     show "(if x \<in> \<Union>\<T> then f x else 0) = (\<Sum>A\<in>\<T>. if x \<in> A then f x else 0)"
       if "x \<in> UNIV - (\<Union>?\<U>)" for x
     proof clarsimp
       fix S assume "S \<in> \<T>" "x \<in> S"
       moreover then have "\<forall>b\<in>\<T>. x \<in> b \<longleftrightarrow> b = S"
         using that by blast
       ultimately show "f x = (\<Sum>A\<in>\<T>. if x \<in> A then f x else 0)"
         by (simp add: sum.delta[OF \<T>])
     qed
   next
     show "((\<lambda>x. \<Sum>A\<in>\<T>. if x \<in> A then f x else 0) has_integral (\<Sum>A\<in>\<T>. i A)) UNIV"
-      apply (rule has_integral_sum [OF \<T>])
-      using int by (simp add: has_integral_restrict_UNIV)
+      using int by (simp add: has_integral_restrict_UNIV has_integral_sum [OF \<T>])
   qed
   then show ?thesis
     using has_integral_restrict_UNIV by blast
 qed
 
 
 text \<open>In particular adding integrals over a division, maybe not of an interval.\<close>
 
 lemma has_integral_combine_division:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes "\<D> division_of S"
     and "\<And>k. k \<in> \<D> \<Longrightarrow> (f has_integral (i k)) k"
   shows "(f has_integral (sum i \<D>)) S"
 proof -
   note \<D> = division_ofD[OF assms(1)]
   have neg: "negligible (S \<inter> s')" if "S \<in> \<D>" "s' \<in> \<D>" "S \<noteq> s'" for S s'
   proof -
     obtain a c b \<D> where obt: "S = cbox a b" "s' = cbox c \<D>"
       by (meson \<open>S \<in> \<D>\<close> \<open>s' \<in> \<D>\<close> \<D>(4))
     from \<D>(5)[OF that] show ?thesis
       unfolding obt interior_cbox
       by (metis (no_types, lifting) Diff_empty Int_interval box_Int_box negligible_frontier_interval)
   qed
   show ?thesis
     unfolding \<D>(6)[symmetric]
     by (auto intro: \<D> neg assms has_integral_Union pairwiseI)
 qed
 
 lemma integral_combine_division_bottomup:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes "\<D> division_of S" "\<And>k. k \<in> \<D> \<Longrightarrow> f integrable_on k"
   shows "integral S f = sum (\<lambda>i. integral i f) \<D>"
-  apply (rule integral_unique)
-  by (meson assms has_integral_combine_division has_integral_integrable_integral)
+  by (meson assms integral_unique has_integral_combine_division has_integral_integrable_integral)
 
 lemma has_integral_combine_division_topdown:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes f: "f integrable_on S"
     and \<D>: "\<D> division_of K"
     and "K \<subseteq> S"
   shows "(f has_integral (sum (\<lambda>i. integral i f) \<D>)) K"
 proof -
   have "f integrable_on L" if "L \<in> \<D>" for L
   proof -
     have "L \<subseteq> S"
       using \<open>K \<subseteq> S\<close> \<D> that by blast
     then show "f integrable_on L"
       using that by (metis (no_types) f \<D> division_ofD(4) integrable_on_subcbox)
   qed
   then show ?thesis
     by (meson \<D> has_integral_combine_division has_integral_integrable_integral)
 qed
 
 lemma integral_combine_division_topdown:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes "f integrable_on S"
     and "\<D> division_of S"
   shows "integral S f = sum (\<lambda>i. integral i f) \<D>"
-  apply (rule integral_unique)
-  apply (rule has_integral_combine_division_topdown)
-  using assms
-  apply auto
-  done
+  using assms has_integral_combine_division_topdown by blast
 
 lemma integrable_combine_division:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes \<D>: "\<D> division_of S"
     and f: "\<And>i. i \<in> \<D> \<Longrightarrow> f integrable_on i"
   shows "f integrable_on S"
   using f unfolding integrable_on_def by (metis has_integral_combine_division[OF \<D>])
 
 lemma integrable_on_subdivision:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes \<D>: "\<D> division_of i"
     and f: "f integrable_on S"
     and "i \<subseteq> S"
   shows "f integrable_on i"
 proof -
   have "f integrable_on i" if "i \<in> \<D>" for i
 proof -
   have "i \<subseteq> S"
     using assms that by auto
   then show "f integrable_on i"
     using that by (metis (no_types) \<D> f division_ofD(4) integrable_on_subcbox)
 qed
   then show ?thesis
     using \<D> integrable_combine_division by blast
 qed
 
 
 subsection \<open>Also tagged divisions\<close>
 
 lemma has_integral_combine_tagged_division:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes "p tagged_division_of S"
-    and "\<forall>(x,k) \<in> p. (f has_integral (i k)) k"
+    and "\<And>x k. (x,k) \<in> p \<Longrightarrow> (f has_integral (i k)) k"
   shows "(f has_integral (\<Sum>(x,k)\<in>p. i k)) S"
 proof -
   have *: "(f has_integral (\<Sum>k\<in>snd`p. integral k f)) S"
-    using assms(2)
-    apply (intro has_integral_combine_division)
-    apply (auto simp: has_integral_integral[symmetric] intro: division_of_tagged_division[OF assms(1)])
-    apply auto
-    done
+  proof -
+    have "snd ` p division_of S"
+      by (simp add: assms(1) division_of_tagged_division)
+    with assms show ?thesis
+      by (metis (mono_tags, lifting) has_integral_combine_division has_integral_integrable_integral imageE prod.collapse)
+  qed
   also have "(\<Sum>k\<in>snd`p. integral k f) = (\<Sum>(x, k)\<in>p. integral k f)"
     by (intro sum.over_tagged_division_lemma[OF assms(1), symmetric] integral_null)
        (simp add: content_eq_0_interior)
   finally show ?thesis
     using assms by (auto simp add: has_integral_iff intro!: sum.cong)
 qed
 
 lemma integral_combine_tagged_division_bottomup:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
-  assumes "p tagged_division_of (cbox a b)"
-    and "\<forall>(x,k)\<in>p. f integrable_on k"
+  assumes p: "p tagged_division_of (cbox a b)"
+    and f: "\<And>x k. (x,k)\<in>p \<Longrightarrow> f integrable_on k"
   shows "integral (cbox a b) f = sum (\<lambda>(x,k). integral k f) p"
-  apply (rule integral_unique)
-  apply (rule has_integral_combine_tagged_division[OF assms(1)])
-  using assms(2)
-  apply auto
-  done
+  by (simp add: has_integral_combine_tagged_division[OF p] integral_unique f integrable_integral)
 
 lemma has_integral_combine_tagged_division_topdown:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes f: "f integrable_on cbox a b"
     and p: "p tagged_division_of (cbox a b)"
   shows "(f has_integral (sum (\<lambda>(x,K). integral K f) p)) (cbox a b)"
 proof -
   have "(f has_integral integral K f) K" if "(x,K) \<in> p" for x K
     by (metis assms integrable_integral integrable_on_subcbox tagged_division_ofD(3,4) that)
   then show ?thesis
-    by (metis assms case_prodI2 has_integral_integrable_integral integral_combine_tagged_division_bottomup)
+    by (simp add: has_integral_combine_tagged_division p)
 qed
 
 lemma integral_combine_tagged_division_topdown:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes "f integrable_on cbox a b"
     and "p tagged_division_of (cbox a b)"
   shows "integral (cbox a b) f = sum (\<lambda>(x,k). integral k f) p"
-  apply (rule integral_unique [OF has_integral_combine_tagged_division_topdown])
-  using assms apply auto
-  done
+  using assms by (auto intro: integral_unique [OF has_integral_combine_tagged_division_topdown])
 
 
 subsection \<open>Henstock's lemma\<close>
 
 lemma Henstock_lemma_part1:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes intf: "f integrable_on cbox a b"
     and "e > 0"
     and "gauge d"
     and less_e: "\<And>p. \<lbrakk>p tagged_division_of (cbox a b); d fine p\<rbrakk> \<Longrightarrow>
                      norm (sum (\<lambda>(x,K). content K *\<^sub>R f x) p - integral(cbox a b) f) < e"
     and p: "p tagged_partial_division_of (cbox a b)" "d fine p"
   shows "norm (sum (\<lambda>(x,K). content K *\<^sub>R f x - integral K f) p) \<le> e" (is "?lhs \<le> e")
 proof (rule field_le_epsilon)
   fix k :: real
   assume "k > 0"
   let ?SUM = "\<lambda>p. (\<Sum>(x,K) \<in> p. content K *\<^sub>R f x)"
   note p' = tagged_partial_division_ofD[OF p(1)]
   have "\<Union>(snd ` p) \<subseteq> cbox a b"
     using p'(3) by fastforce
   then obtain q where q: "snd ` p \<subseteq> q" and qdiv: "q division_of cbox a b"
     by (meson p(1) partial_division_extend_interval partial_division_of_tagged_division)
   note q' = division_ofD[OF qdiv]
   define r where "r = q - snd ` p"
   have "snd ` p \<inter> r = {}"
     unfolding r_def by auto
   have "finite r"
     using q' unfolding r_def by auto
   have "\<exists>p. p tagged_division_of i \<and> d fine p \<and>
         norm (?SUM p - integral i f) < k / (real (card r) + 1)"
     if "i\<in>r" for i
   proof -
     have gt0: "k / (real (card r) + 1) > 0" using \<open>k > 0\<close> by simp
     have i: "i \<in> q"
       using that unfolding r_def by auto
     then obtain u v where uv: "i = cbox u v"
       using q'(4) by blast
     then have "cbox u v \<subseteq> cbox a b"
       using i q'(2) by auto  
     then have "f integrable_on cbox u v"
       by (rule integrable_subinterval[OF intf])
     with integrable_integral[OF this, unfolded has_integral[of f]]
     obtain dd where "gauge dd" and dd:
       "\<And>\<D>. \<lbrakk>\<D> tagged_division_of cbox u v; dd fine \<D>\<rbrakk> \<Longrightarrow>
     norm (?SUM \<D> - integral (cbox u v) f) < k / (real (card r) + 1)"
       using gt0 by auto
     with gauge_Int[OF \<open>gauge d\<close> \<open>gauge dd\<close>]
     obtain qq where qq: "qq tagged_division_of cbox u v" "(\<lambda>x. d x \<inter> dd x) fine qq"
       using fine_division_exists by blast
     with dd[of qq]  show ?thesis
       by (auto simp: fine_Int uv)
   qed
   then obtain qq where qq: "\<And>i. i \<in> r \<Longrightarrow> qq i tagged_division_of i \<and>
       d fine qq i \<and> norm (?SUM (qq i) - integral i f) < k / (real (card r) + 1)"
     by metis
 
   let ?p = "p \<union> \<Union>(qq ` r)"
   have "norm (?SUM ?p - integral (cbox a b) f) < e"
   proof (rule less_e)
     show "d fine ?p"
       by (metis (mono_tags, hide_lams) qq fine_Un fine_Union imageE p(2))
     note ptag = tagged_partial_division_of_Union_self[OF p(1)]
     have "p \<union> \<Union>(qq ` r) tagged_division_of \<Union>(snd ` p) \<union> \<Union>r"
     proof (rule tagged_division_Un[OF ptag tagged_division_Union [OF \<open>finite r\<close>]])
       show "\<And>i. i \<in> r \<Longrightarrow> qq i tagged_division_of i"
         using qq by auto
       show "\<And>i1 i2. \<lbrakk>i1 \<in> r; i2 \<in> r; i1 \<noteq> i2\<rbrakk> \<Longrightarrow> interior i1 \<inter> interior i2 = {}"
         by (simp add: q'(5) r_def)
       show "interior (\<Union>(snd ` p)) \<inter> interior (\<Union>r) = {}"
       proof (rule Int_interior_Union_intervals [OF \<open>finite r\<close>])
         show "open (interior (\<Union>(snd ` p)))"
           by blast
         show "\<And>T. T \<in> r \<Longrightarrow> \<exists>a b. T = cbox a b"
           by (simp add: q'(4) r_def)
-        have "finite (snd ` p)"
-          by (simp add: p'(1))
+        have "interior T \<inter> interior (\<Union>(snd ` p)) = {}" if "T \<in> r" for T
+        proof (rule Int_interior_Union_intervals)
+          show "\<And>U. U \<in> snd ` p \<Longrightarrow> \<exists>a b. U = cbox a b"
+            using q q'(4) by blast
+          show "\<And>U. U \<in> snd ` p \<Longrightarrow> interior T \<inter> interior U = {}"
+            by (metis DiffE q q'(5) r_def subsetD that)
+        qed (use p' in auto)
         then show "\<And>T. T \<in> r \<Longrightarrow> interior (\<Union>(snd ` p)) \<inter> interior T = {}"
-          apply (subst Int_commute)
-          apply (rule Int_interior_Union_intervals)
-          using r_def q'(5) q(1) apply auto
-          by (simp add: p'(4))
+          by (metis Int_commute)
       qed
     qed
     moreover have "\<Union>(snd ` p) \<union> \<Union>r = cbox a b" and "{qq i |i. i \<in> r} = qq ` r"
       using qdiv q unfolding Union_Un_distrib[symmetric] r_def by auto
     ultimately show "?p tagged_division_of (cbox a b)"
       by fastforce
   qed
   then have "norm (?SUM p + (?SUM (\<Union>(qq ` r))) - integral (cbox a b) f) < e"
   proof (subst sum.union_inter_neutral[symmetric, OF \<open>finite p\<close>], safe)
     show "content L *\<^sub>R f x = 0" if "(x, L) \<in> p" "(x, L) \<in> qq K" "K \<in> r" for x K L 
     proof -
       obtain u v where uv: "L = cbox u v"
         using \<open>(x,L) \<in> p\<close> p'(4) by blast
       have "L \<subseteq> K"
         using  qq[OF that(3)] tagged_division_ofD(3) \<open>(x,L) \<in> qq K\<close> by metis
       have "L \<in> snd ` p" 
         using \<open>(x,L) \<in> p\<close> image_iff by fastforce 
       then have "L \<in> q" "K \<in> q" "L \<noteq> K"
         using that(1,3) q(1) unfolding r_def by auto
       with q'(5) have "interior L = {}"
         using interior_mono[OF \<open>L \<subseteq> K\<close>] by blast
       then show "content L *\<^sub>R f x = 0"
         unfolding uv content_eq_0_interior[symmetric] by auto
     qed
     show "finite (\<Union>(qq ` r))"
       by (meson finite_UN qq \<open>finite r\<close> tagged_division_of_finite)
   qed
   moreover have "content M *\<^sub>R f x = 0" 
       if x: "(x,M) \<in> qq K" "(x,M) \<in> qq L" and KL: "qq K \<noteq> qq L" and r: "K \<in> r" "L \<in> r"
     for x M K L
   proof -
     note kl = tagged_division_ofD(3,4)[OF qq[THEN conjunct1]]
     obtain u v where uv: "M = cbox u v"
       using \<open>(x, M) \<in> qq L\<close> \<open>L \<in> r\<close> kl(2) by blast
     have empty: "interior (K \<inter> L) = {}"
       by (metis DiffD1 interior_Int q'(5) r_def KL r)
     have "interior M = {}"
       by (metis (no_types, lifting) Int_assoc empty inf.absorb_iff2 interior_Int kl(1) subset_empty x r)
     then show "content M *\<^sub>R f x = 0"
       unfolding uv content_eq_0_interior[symmetric]
       by auto
   qed 
   ultimately have "norm (?SUM p + sum ?SUM (qq ` r) - integral (cbox a b) f) < e"
     apply (subst (asm) sum.Union_comp)
     using qq by (force simp: split_paired_all)+
   moreover have "content M *\<^sub>R f x = 0" 
        if "K \<in> r" "L \<in> r" "K \<noteq> L" "qq K = qq L" "(x, M) \<in> qq K" for K L x M
     using tagged_division_ofD(6) qq that by (metis (no_types, lifting)) 
   ultimately have less_e: "norm (?SUM p + sum (?SUM \<circ> qq) r - integral (cbox a b) f) < e"
-    apply (subst (asm) sum.reindex_nontrivial [OF \<open>finite r\<close>])
-     apply (auto simp: split_paired_all sum.neutral)
-    done
+  proof (subst (asm) sum.reindex_nontrivial [OF \<open>finite r\<close>])
+    qed (auto simp: split_paired_all sum.neutral)
   have norm_le: "norm (cp - ip) \<le> e + k"
                   if "norm ((cp + cr) - i) < e" "norm (cr - ir) < k" "ip + ir = i"
                   for ir ip i cr cp::'a
   proof -
     from that show ?thesis
       using norm_triangle_le[of "cp + cr - i" "- (cr - ir)"]
       unfolding that(3)[symmetric] norm_minus_cancel
       by (auto simp add: algebra_simps)
   qed
 
   have "?lhs =  norm (?SUM p - (\<Sum>(x, k)\<in>p. integral k f))"
     unfolding split_def sum_subtractf ..
   also have "\<dots> \<le> e + k"
   proof (rule norm_le[OF less_e])
     have lessk: "k * real (card r) / (1 + real (card r)) < k"
       using \<open>k>0\<close> by (auto simp add: field_simps)
     have "norm (sum (?SUM \<circ> qq) r - (\<Sum>k\<in>r. integral k f)) \<le> (\<Sum>x\<in>r. k / (real (card r) + 1))"
       unfolding sum_subtractf[symmetric] by (force dest: qq intro!: sum_norm_le)
     also have "... < k"
       by (simp add: lessk add.commute mult.commute)
     finally show "norm (sum (?SUM \<circ> qq) r - (\<Sum>k\<in>r. integral k f)) < k" .
   next
     from q(1) have [simp]: "snd ` p \<union> q = q" by auto
     have "integral l f = 0"
       if inp: "(x, l) \<in> p" "(y, m) \<in> p" and ne: "(x, l) \<noteq> (y, m)" and "l = m" for x l y m
     proof -
       obtain u v where uv: "l = cbox u v"
         using inp p'(4) by blast
       have "content (cbox u v) = 0"
         unfolding content_eq_0_interior using that p(1) uv by auto
       then show ?thesis
         using uv by blast
     qed
-    then have "(\<Sum>(x, k)\<in>p. integral k f) = (\<Sum>k\<in>snd ` p. integral k f)"
+    then have "(\<Sum>(x, K)\<in>p. integral K f) = (\<Sum>K\<in>snd ` p. integral K f)"
       apply (subst sum.reindex_nontrivial [OF \<open>finite p\<close>])
       unfolding split_paired_all split_def by auto
     then show "(\<Sum>(x, k)\<in>p. integral k f) + (\<Sum>k\<in>r. integral k f) = integral (cbox a b) f"
       unfolding integral_combine_division_topdown[OF intf qdiv] r_def
       using q'(1) p'(1) sum.union_disjoint [of "snd ` p" "q - snd ` p", symmetric]
         by simp
   qed
   finally show "?lhs \<le> e + k" .
 qed
 
 lemma Henstock_lemma_part2:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes fed: "f integrable_on cbox a b" "e > 0" "gauge d"
     and less_e: "\<And>\<D>. \<lbrakk>\<D> tagged_division_of (cbox a b); d fine \<D>\<rbrakk> \<Longrightarrow>
                      norm (sum (\<lambda>(x,k). content k *\<^sub>R f x) \<D> - integral (cbox a b) f) < e"
     and tag: "p tagged_partial_division_of (cbox a b)"
     and "d fine p"
   shows "sum (\<lambda>(x,k). norm (content k *\<^sub>R f x - integral k f)) p \<le> 2 * real (DIM('n)) * e"
 proof -
   have "finite p"
     using tag by blast
   then show ?thesis
     unfolding split_def
   proof (rule sum_norm_allsubsets_bound)
     fix Q
     assume Q: "Q \<subseteq> p"
     then have fine: "d fine Q"
       by (simp add: \<open>d fine p\<close> fine_subset)
     show "norm (\<Sum>x\<in>Q. content (snd x) *\<^sub>R f (fst x) - integral (snd x) f) \<le> e"
       apply (rule Henstock_lemma_part1[OF fed less_e, unfolded split_def])
-      using Q tag tagged_partial_division_subset apply (force simp add: fine)+
-      done
+      using Q tag tagged_partial_division_subset by (force simp add: fine)+
   qed
 qed
 
 lemma Henstock_lemma:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes intf: "f integrable_on cbox a b"
     and "e > 0"
   obtains \<gamma> where "gauge \<gamma>"
     and "\<And>p. \<lbrakk>p tagged_partial_division_of (cbox a b); \<gamma> fine p\<rbrakk> \<Longrightarrow>
              sum (\<lambda>(x,k). norm(content k *\<^sub>R f x - integral k f)) p < e"
 proof -
   have *: "e/(2 * (real DIM('n) + 1)) > 0" using \<open>e > 0\<close> by simp
   with integrable_integral[OF intf, unfolded has_integral]
   obtain \<gamma> where "gauge \<gamma>"
     and \<gamma>: "\<And>\<D>. \<lbrakk>\<D> tagged_division_of cbox a b; \<gamma> fine \<D>\<rbrakk> \<Longrightarrow>
          norm ((\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f x) - integral (cbox a b) f)
          < e/(2 * (real DIM('n) + 1))"
     by metis
   show thesis
   proof (rule that [OF \<open>gauge \<gamma>\<close>])
     fix p
     assume p: "p tagged_partial_division_of cbox a b" "\<gamma> fine p"
     have "(\<Sum>(x,K)\<in>p. norm (content K *\<^sub>R f x - integral K f)) 
           \<le> 2 * real DIM('n) * (e/(2 * (real DIM('n) + 1)))"
       using Henstock_lemma_part2[OF intf * \<open>gauge \<gamma>\<close> \<gamma> p] by metis
     also have "... < e"
       using \<open>e > 0\<close> by (auto simp add: field_simps)
     finally
     show "(\<Sum>(x,K)\<in>p. norm (content K *\<^sub>R f x - integral K f)) < e" .
   qed
 qed
 
 
 subsection \<open>Monotone convergence (bounded interval first)\<close>
 
 lemma bounded_increasing_convergent:
   fixes f :: "nat \<Rightarrow> real"
   shows "\<lbrakk>bounded (range f); \<And>n. f n \<le> f (Suc n)\<rbrakk> \<Longrightarrow> \<exists>l. f \<longlonglongrightarrow> l"
   using Bseq_mono_convergent[of f] incseq_Suc_iff[of f]
   by (auto simp: image_def Bseq_eq_bounded convergent_def incseq_def)
 
 lemma monotone_convergence_interval:
   fixes f :: "nat \<Rightarrow> 'n::euclidean_space \<Rightarrow> real"
   assumes intf: "\<And>k. (f k) integrable_on cbox a b"
     and le: "\<And>k x. x \<in> cbox a b \<Longrightarrow> (f k x) \<le> f (Suc k) x"
     and fg: "\<And>x. x \<in> cbox a b \<Longrightarrow> ((\<lambda>k. f k x) \<longlongrightarrow> g x) sequentially"
     and bou: "bounded (range (\<lambda>k. integral (cbox a b) (f k)))"
   shows "g integrable_on cbox a b \<and> ((\<lambda>k. integral (cbox a b) (f k)) \<longlongrightarrow> integral (cbox a b) g) sequentially"
 proof (cases "content (cbox a b) = 0")
   case True then show ?thesis
     by auto
 next
   case False
   have fg1: "(f k x) \<le> (g x)" if x: "x \<in> cbox a b" for x k
   proof -
     have "\<forall>\<^sub>F j in sequentially. f k x \<le> f j x"
-      apply (rule eventually_sequentiallyI [of k])
-      using le x apply (force intro: transitive_stepwise_le)
-      done
+    proof (rule eventually_sequentiallyI [of k])
+      show "\<And>j. k \<le> j \<Longrightarrow> f k x \<le> f j x"
+        using le x by (force intro: transitive_stepwise_le)
+    qed
     then show "f k x \<le> g x"
       using tendsto_lowerbound [OF fg] x trivial_limit_sequentially by blast
   qed
   have int_inc: "\<And>n. integral (cbox a b) (f n) \<le> integral (cbox a b) (f (Suc n))"
     by (metis integral_le intf le)
   then obtain i where i: "(\<lambda>k. integral (cbox a b) (f k)) \<longlonglongrightarrow> i"
     using bounded_increasing_convergent bou by blast
   have "\<And>k. \<forall>\<^sub>F x in sequentially. integral (cbox a b) (f k) \<le> integral (cbox a b) (f x)"
     unfolding eventually_sequentially
     by (force intro: transitive_stepwise_le int_inc)
   then have i': "\<And>k. (integral(cbox a b) (f k)) \<le> i"
     using tendsto_le [OF trivial_limit_sequentially i] by blast
   have "(g has_integral i) (cbox a b)"
     unfolding has_integral real_norm_def
   proof clarify
     fix e::real
     assume e: "e > 0"
     have "\<And>k. (\<exists>\<gamma>. gauge \<gamma> \<and> (\<forall>\<D>. \<D> tagged_division_of (cbox a b) \<and> \<gamma> fine \<D> \<longrightarrow>
       abs ((\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f k x) - integral (cbox a b) (f k)) < e/2 ^ (k + 2)))"
       using intf e by (auto simp: has_integral_integral has_integral)
     then obtain c where c: "\<And>x. gauge (c x)"
           "\<And>x \<D>. \<lbrakk>\<D> tagged_division_of cbox a b; c x fine \<D>\<rbrakk> \<Longrightarrow>
               abs ((\<Sum>(u,K)\<in>\<D>. content K *\<^sub>R f x u) - integral (cbox a b) (f x))
               < e/2 ^ (x + 2)"
       by metis
 
     have "\<exists>r. \<forall>k\<ge>r. 0 \<le> i - (integral (cbox a b) (f k)) \<and> i - (integral (cbox a b) (f k)) < e/4"
     proof -
       have "e/4 > 0"
         using e by auto
       show ?thesis
         using LIMSEQ_D [OF i \<open>e/4 > 0\<close>] i' by auto
     qed
     then obtain r where r: "\<And>k. r \<le> k \<Longrightarrow> 0 \<le> i - integral (cbox a b) (f k)"
                        "\<And>k. r \<le> k \<Longrightarrow> i - integral (cbox a b) (f k) < e/4" 
       by metis
     have "\<exists>n\<ge>r. \<forall>k\<ge>n. 0 \<le> (g x) - (f k x) \<and> (g x) - (f k x) < e/(4 * content(cbox a b))"
       if "x \<in> cbox a b" for x
     proof -
       have "e/(4 * content (cbox a b)) > 0"
         by (simp add: False content_lt_nz e)
       with fg that LIMSEQ_D
       obtain N where "\<forall>n\<ge>N. norm (f n x - g x) < e/(4 * content (cbox a b))"
         by metis
-      then show "\<exists>n\<ge>r.
-            \<forall>k\<ge>n.
-               0 \<le> g x - f k x \<and>
-               g x - f k x
-               < e/(4 * content (cbox a b))"
+      then show "\<exists>n\<ge>r. \<forall>k\<ge>n. 0 \<le> g x - f k x \<and> g x - f k x < e/(4 * content (cbox a b))"
         apply (rule_tac x="N + r" in exI)
-        using fg1[OF that] apply (auto simp add: field_simps)
-        done
+        using fg1[OF that] by (auto simp add: field_simps)
     qed
     then obtain m where r_le_m: "\<And>x. x \<in> cbox a b \<Longrightarrow> r \<le> m x"
        and m: "\<And>x k. \<lbrakk>x \<in> cbox a b; m x \<le> k\<rbrakk>
                      \<Longrightarrow> 0 \<le> g x - f k x \<and> g x - f k x < e/(4 * content (cbox a b))"
       by metis
     define d where "d x = c (m x) x" for x
     show "\<exists>\<gamma>. gauge \<gamma> \<and>
              (\<forall>\<D>. \<D> tagged_division_of cbox a b \<and>
                   \<gamma> fine \<D> \<longrightarrow> abs ((\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R g x) - i) < e)"
     proof (rule exI, safe)
       show "gauge d"
         using c(1) unfolding gauge_def d_def by auto
     next
       fix \<D>
       assume ptag: "\<D> tagged_division_of (cbox a b)" and "d fine \<D>"
       note p'=tagged_division_ofD[OF ptag]
       obtain s where s: "\<And>x. x \<in> \<D> \<Longrightarrow> m (fst x) \<le> s"
         by (metis finite_imageI finite_nat_set_iff_bounded_le p'(1) rev_image_eqI)
       have *: "\<bar>a - d\<bar> < e" if "\<bar>a - b\<bar> \<le> e/4" "\<bar>b - c\<bar> < e/2" "\<bar>c - d\<bar> < e/4" for a b c d
         using that norm_triangle_lt[of "a - b" "b - c" "3* e/4"]
           norm_triangle_lt[of "a - b + (b - c)" "c - d" e]
         by (auto simp add: algebra_simps)
       show "\<bar>(\<Sum>(x, k)\<in>\<D>. content k *\<^sub>R g x) - i\<bar> < e"
       proof (rule *)
         have "\<bar>(\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R g x) - (\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f (m x) x)\<bar> 
               \<le> (\<Sum>i\<in>\<D>. \<bar>(case i of (x, K) \<Rightarrow> content K *\<^sub>R g x) - (case i of (x, K) \<Rightarrow> content K *\<^sub>R f (m x) x)\<bar>)"
           by (metis (mono_tags) sum_subtractf sum_abs) 
         also have "... \<le> (\<Sum>(x, k)\<in>\<D>. content k * (e/(4 * content (cbox a b))))"
         proof (rule sum_mono, simp add: split_paired_all)
           fix x K
           assume xk: "(x,K) \<in> \<D>"
           with ptag have x: "x \<in> cbox a b"
             by blast
           then have "abs (content K * (g x - f (m x) x)) \<le> content K * (e/(4 * content (cbox a b)))"
             by (metis m[OF x] mult_nonneg_nonneg abs_of_nonneg less_eq_real_def measure_nonneg mult_left_mono order_refl)
           then show "\<bar>content K * g x - content K * f (m x) x\<bar> \<le> content K * e/(4 * content (cbox a b))"
             by (simp add: algebra_simps)
         qed
         also have "... = (e/(4 * content (cbox a b))) * (\<Sum>(x, k)\<in>\<D>. content k)"
           by (simp add: sum_distrib_left sum_divide_distrib split_def mult.commute)
         also have "... \<le> e/4"
           by (metis False additive_content_tagged_division [OF ptag] nonzero_mult_divide_mult_cancel_right order_refl times_divide_eq_left)
         finally show "\<bar>(\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R g x) - (\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f (m x) x)\<bar> \<le> e/4" .
 
       next
         have "norm ((\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f (m x) x) - (\<Sum>(x,K)\<in>\<D>. integral K (f (m x))))
               \<le> norm (\<Sum>j = 0..s. \<Sum>(x,K)\<in>{xk \<in> \<D>. m (fst xk) = j}. content K *\<^sub>R f (m x) x - integral K (f (m x)))"
           apply (subst sum.group)
           using s by (auto simp: sum_subtractf split_def p'(1))
         also have "\<dots> < e/2"
         proof -
           have "norm (\<Sum>j = 0..s. \<Sum>(x, k)\<in>{xk \<in> \<D>. m (fst xk) = j}. content k *\<^sub>R f (m x) x - integral k (f (m x)))
                 \<le> (\<Sum>i = 0..s. e/2 ^ (i + 2))"
           proof (rule sum_norm_le)
             fix t
             assume "t \<in> {0..s}"
             have "norm (\<Sum>(x,k)\<in>{xk \<in> \<D>. m (fst xk) = t}. content k *\<^sub>R f (m x) x - integral k (f (m x))) =
                   norm (\<Sum>(x,k)\<in>{xk \<in> \<D>. m (fst xk) = t}. content k *\<^sub>R f t x - integral k (f t))"
               by (force intro!: sum.cong arg_cong[where f=norm])
             also have "... \<le> e/2 ^ (t + 2)"
             proof (rule Henstock_lemma_part1 [OF intf])
               show "{xk \<in> \<D>. m (fst xk) = t} tagged_partial_division_of cbox a b"
-                apply (rule tagged_partial_division_subset[of \<D>])
-                using ptag by (auto simp: tagged_division_of_def)
+              proof (rule tagged_partial_division_subset[of \<D>])
+                show "\<D> tagged_partial_division_of cbox a b"
+                  using ptag tagged_division_of_def by blast
+              qed auto
               show "c t fine {xk \<in> \<D>. m (fst xk) = t}"
                 using \<open>d fine \<D>\<close> by (auto simp: fine_def d_def)
             qed (use c e in auto)
             finally show "norm (\<Sum>(x,K)\<in>{xk \<in> \<D>. m (fst xk) = t}. content K *\<^sub>R f (m x) x -
                                 integral K (f (m x))) \<le> e/2 ^ (t + 2)" .
           qed
           also have "... = (e/2/2) * (\<Sum>i = 0..s. (1/2) ^ i)"
             by (simp add: sum_distrib_left field_simps)
           also have "\<dots> < e/2"
             by (simp add: sum_gp mult_strict_left_mono[OF _ e])
           finally show "norm (\<Sum>j = 0..s. \<Sum>(x, k)\<in>{xk \<in> \<D>.
             m (fst xk) = j}. content k *\<^sub>R f (m x) x - integral k (f (m x))) < e/2" .
         qed 
         finally show "\<bar>(\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f (m x) x) - (\<Sum>(x,K)\<in>\<D>. integral K (f (m x)))\<bar> < e/2"
           by simp
       next
         have comb: "integral (cbox a b) (f y) = (\<Sum>(x, k)\<in>\<D>. integral k (f y))" for y
           using integral_combine_tagged_division_topdown[OF intf ptag] by metis
         have f_le: "\<And>y m n. \<lbrakk>y \<in> cbox a b; n\<ge>m\<rbrakk> \<Longrightarrow> f m y \<le> f n y"
           using le by (auto intro: transitive_stepwise_le)        
         have "(\<Sum>(x, k)\<in>\<D>. integral k (f r)) \<le> (\<Sum>(x, K)\<in>\<D>. integral K (f (m x)))"
         proof (rule sum_mono, simp add: split_paired_all)
           fix x K
           assume xK: "(x, K) \<in> \<D>"
           show "integral K (f r) \<le> integral K (f (m x))"
           proof (rule integral_le)
             show "f r integrable_on K"
               by (metis integrable_on_subcbox intf p'(3) p'(4) xK)
             show "f (m x) integrable_on K"
               by (metis elementary_interval integrable_on_subdivision intf p'(3) p'(4) xK)
             show "f r y \<le> f (m x) y" if "y \<in> K" for y
               using that r_le_m[of x] p'(2-3)[OF xK] f_le by auto
           qed
         qed
         moreover have "(\<Sum>(x, K)\<in>\<D>. integral K (f (m x))) \<le> (\<Sum>(x, k)\<in>\<D>. integral k (f s))"
         proof (rule sum_mono, simp add: split_paired_all)
           fix x K
           assume xK: "(x, K) \<in> \<D>"
           show "integral K (f (m x)) \<le> integral K (f s)"
           proof (rule integral_le)
             show "f (m x) integrable_on K"
               by (metis elementary_interval integrable_on_subdivision intf p'(3) p'(4) xK)
             show "f s integrable_on K"
               by (metis integrable_on_subcbox intf p'(3) p'(4) xK)
             show "f (m x) y \<le> f s y" if "y \<in> K" for y
               using that s xK f_le p'(3) by fastforce
           qed
         qed
         moreover have "0 \<le> i - integral (cbox a b) (f r)" "i - integral (cbox a b) (f r) < e/4"
           using r by auto
         ultimately show "\<bar>(\<Sum>(x,K)\<in>\<D>. integral K (f (m x))) - i\<bar> < e/4"
           using comb i'[of s] by auto
       qed
     qed
   qed 
   with i integral_unique show ?thesis
     by blast
 qed
 
 lemma monotone_convergence_increasing:
   fixes f :: "nat \<Rightarrow> 'n::euclidean_space \<Rightarrow> real"
   assumes int_f: "\<And>k. (f k) integrable_on S"
     and "\<And>k x. x \<in> S \<Longrightarrow> (f k x) \<le> (f (Suc k) x)"
     and fg: "\<And>x. x \<in> S \<Longrightarrow> ((\<lambda>k. f k x) \<longlongrightarrow> g x) sequentially"
     and bou: "bounded (range (\<lambda>k. integral S (f k)))"
   shows "g integrable_on S \<and> ((\<lambda>k. integral S (f k)) \<longlongrightarrow> integral S g) sequentially"
 proof -
   have lem: "g integrable_on S \<and> ((\<lambda>k. integral S (f k)) \<longlongrightarrow> integral S g) sequentially"
     if f0: "\<And>k x. x \<in> S \<Longrightarrow> 0 \<le> f k x"
     and int_f: "\<And>k. (f k) integrable_on S"
     and le: "\<And>k x. x \<in> S \<Longrightarrow> f k x \<le> f (Suc k) x"
     and lim: "\<And>x. x \<in> S \<Longrightarrow> ((\<lambda>k. f k x) \<longlongrightarrow> g x) sequentially"
     and bou: "bounded (range(\<lambda>k. integral S (f k)))"
     for f :: "nat \<Rightarrow> 'n::euclidean_space \<Rightarrow> real" and g S
   proof -
     have fg: "(f k x) \<le> (g x)" if "x \<in> S" for x k
-      apply (rule tendsto_lowerbound [OF lim [OF that]])
-      apply (rule eventually_sequentiallyI [of k])
-      using le  by (force intro: transitive_stepwise_le that)+
+    proof -
+      have "\<And>xa. k \<le> xa \<Longrightarrow> f k x \<le> f xa x"
+        using le  by (force intro: transitive_stepwise_le that)
+      then show ?thesis
+        using tendsto_lowerbound [OF lim [OF that]] eventually_sequentiallyI by force
+    qed
     obtain i where i: "(\<lambda>k. integral S (f k)) \<longlonglongrightarrow> i"
       using bounded_increasing_convergent [OF bou] le int_f integral_le by blast
     have i': "(integral S (f k)) \<le> i" for k
     proof -
       have "\<And>k. \<And>x. x \<in> S \<Longrightarrow> \<forall>n\<ge>k. f k x \<le> f n x"
         using le  by (force intro: transitive_stepwise_le)
       then show ?thesis
         using tendsto_lowerbound [OF i eventually_sequentiallyI trivial_limit_sequentially]
         by (meson int_f integral_le)
     qed
-
     let ?f = "(\<lambda>k x. if x \<in> S then f k x else 0)"
     let ?g = "(\<lambda>x. if x \<in> S then g x else 0)"
     have int: "?f k integrable_on cbox a b" for a b k
       by (simp add: int_f integrable_altD(1))
     have int': "\<And>k a b. f k integrable_on cbox a b \<inter> S"
       using int by (simp add: Int_commute integrable_restrict_Int) 
     have g: "?g integrable_on cbox a b \<and>
              (\<lambda>k. integral (cbox a b) (?f k)) \<longlonglongrightarrow> integral (cbox a b) ?g" for a b
     proof (rule monotone_convergence_interval)
       have "norm (integral (cbox a b) (?f k)) \<le> norm (integral S (f k))" for k
       proof -
         have "0 \<le> integral (cbox a b) (?f k)"
           by (metis (no_types) integral_nonneg Int_iff f0 inf_commute integral_restrict_Int int')
         moreover have "0 \<le> integral S (f k)"
           by (simp add: integral_nonneg f0 int_f)
         moreover have "integral (S \<inter> cbox a b) (f k) \<le> integral S (f k)"
           by (metis f0 inf_commute int' int_f integral_subset_le le_inf_iff order_refl)
         ultimately show ?thesis
           by (simp add: integral_restrict_Int)
       qed
       moreover obtain B where "\<And>x. x \<in> range (\<lambda>k. integral S (f k)) \<Longrightarrow> norm x \<le> B"
         using bou unfolding bounded_iff by blast
       ultimately show "bounded (range (\<lambda>k. integral (cbox a b) (?f k)))"
         unfolding bounded_iff by (blast intro: order_trans)
     qed (use int le lim in auto)
     moreover have "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow> norm (integral (cbox a b) ?g - i) < e"
       if "0 < e" for e
     proof -
       have "e/4>0"
         using that by auto
       with LIMSEQ_D [OF i] obtain N where N: "\<And>n. n \<ge> N \<Longrightarrow> norm (integral S (f n) - i) < e/4"
         by metis
       with int_f[of N, unfolded has_integral_integral has_integral_alt'[of "f N"]] 
       obtain B where "0 < B" and B: 
         "\<And>a b. ball 0 B \<subseteq> cbox a b \<Longrightarrow> norm (integral (cbox a b) (?f N) - integral S (f N)) < e/4"
         by (meson \<open>0 < e/4\<close>)
       have "norm (integral (cbox a b) ?g - i) < e" if  ab: "ball 0 B \<subseteq> cbox a b" for a b
       proof -
         obtain M where M: "\<And>n. n \<ge> M \<Longrightarrow> abs (integral (cbox a b) (?f n) - integral (cbox a b) ?g) < e/2"
           using \<open>e > 0\<close> g by (fastforce simp add: dest!: LIMSEQ_D [where r = "e/2"])
         have *: "\<And>\<alpha> \<beta> g. \<lbrakk>\<bar>\<alpha> - i\<bar> < e/2; \<bar>\<beta> - g\<bar> < e/2; \<alpha> \<le> \<beta>; \<beta> \<le> i\<rbrakk> \<Longrightarrow> \<bar>g - i\<bar> < e"
           unfolding real_inner_1_right by arith
         show "norm (integral (cbox a b) ?g - i) < e"
           unfolding real_norm_def
         proof (rule *)
           show "\<bar>integral (cbox a b) (?f N) - i\<bar> < e/2"
           proof (rule abs_triangle_half_l)
             show "\<bar>integral (cbox a b) (?f N) - integral S (f N)\<bar> < e/2/2"
               using B[OF ab] by simp
             show "abs (i - integral S (f N)) < e/2/2"
               using N by (simp add: abs_minus_commute)
           qed
           show "\<bar>integral (cbox a b) (?f (M + N)) - integral (cbox a b) ?g\<bar> < e/2"
             by (metis le_add1 M[of "M + N"])
           show "integral (cbox a b) (?f N) \<le> integral (cbox a b) (?f (M + N))"
           proof (intro ballI integral_le[OF int int])
             fix x assume "x \<in> cbox a b"
             have "(f m x) \<le> (f n x)" if "x \<in> S" "n \<ge> m" for m n
-              apply (rule transitive_stepwise_le [OF \<open>n \<ge> m\<close> order_refl])
-              using dual_order.trans apply blast
-              by (simp add: le \<open>x \<in> S\<close>)
+            proof (rule transitive_stepwise_le [OF \<open>n \<ge> m\<close> order_refl])
+              show "\<And>u y z. \<lbrakk>f u x \<le> f y x; f y x \<le> f z x\<rbrakk> \<Longrightarrow> f u x \<le> f z x"
+                using dual_order.trans by blast
+            qed (simp add: le \<open>x \<in> S\<close>)
             then show "(?f N)x \<le> (?f (M+N))x"
               by auto
           qed
           have "integral (cbox a b \<inter> S) (f (M + N)) \<le> integral S (f (M + N))"
             by (metis Int_lower1 f0 inf_commute int' int_f integral_subset_le)
           then have "integral (cbox a b) (?f (M + N)) \<le> integral S (f (M + N))"
             by (metis (no_types) inf_commute integral_restrict_Int)
           also have "... \<le> i"
             using i'[of "M + N"] by auto
           finally show "integral (cbox a b) (?f (M + N)) \<le> i" .
         qed
       qed
       then show ?thesis
         using \<open>0 < B\<close> by blast
     qed
     ultimately have "(g has_integral i) S"
       unfolding has_integral_alt' by auto
     then show ?thesis
       using has_integral_integrable_integral i integral_unique by metis
   qed
 
   have sub: "\<And>k. integral S (\<lambda>x. f k x - f 0 x) = integral S (f k) - integral S (f 0)"
     by (simp add: integral_diff int_f)
   have *: "\<And>x m n. x \<in> S \<Longrightarrow> n\<ge>m \<Longrightarrow> f m x \<le> f n x"
     using assms(2) by (force intro: transitive_stepwise_le)
   have gf: "(\<lambda>x. g x - f 0 x) integrable_on S \<and> ((\<lambda>k. integral S (\<lambda>x. f (Suc k) x - f 0 x)) \<longlongrightarrow>
     integral S (\<lambda>x. g x - f 0 x)) sequentially"
   proof (rule lem)
     show "\<And>k. (\<lambda>x. f (Suc k) x - f 0 x) integrable_on S"
       by (simp add: integrable_diff int_f)
     show "(\<lambda>k. f (Suc k) x - f 0 x) \<longlonglongrightarrow> g x - f 0 x" if "x \<in> S" for x
     proof -
       have "(\<lambda>n. f (Suc n) x) \<longlonglongrightarrow> g x"
         using LIMSEQ_ignore_initial_segment[OF fg[OF \<open>x \<in> S\<close>], of 1] by simp
       then show ?thesis
         by (simp add: tendsto_diff)
     qed
     show "bounded (range (\<lambda>k. integral S (\<lambda>x. f (Suc k) x - f 0 x)))"
     proof -
       obtain B where B: "\<And>k. norm (integral S (f k)) \<le> B"
         using  bou by (auto simp: bounded_iff)
       then have "norm (integral S (\<lambda>x. f (Suc k) x - f 0 x))
               \<le> B + norm (integral S (f 0))" for k
         unfolding sub by (meson add_le_cancel_right norm_triangle_le_diff)
       then show ?thesis
         unfolding bounded_iff by blast
     qed
   qed (use * in auto)
   then have "(\<lambda>x. integral S (\<lambda>xa. f (Suc x) xa - f 0 xa) + integral S (f 0))
              \<longlonglongrightarrow> integral S (\<lambda>x. g x - f 0 x) + integral S (f 0)"
     by (auto simp add: tendsto_add)
   moreover have "(\<lambda>x. g x - f 0 x + f 0 x) integrable_on S"
     using gf integrable_add int_f [of 0] by metis
   ultimately show ?thesis
     by (simp add: integral_diff int_f LIMSEQ_imp_Suc sub)
 qed
 
 lemma has_integral_monotone_convergence_increasing:
   fixes f :: "nat \<Rightarrow> 'a::euclidean_space \<Rightarrow> real"
   assumes f: "\<And>k. (f k has_integral x k) s"
   assumes "\<And>k x. x \<in> s \<Longrightarrow> f k x \<le> f (Suc k) x"
   assumes "\<And>x. x \<in> s \<Longrightarrow> (\<lambda>k. f k x) \<longlonglongrightarrow> g x"
   assumes "x \<longlonglongrightarrow> x'"
   shows "(g has_integral x') s"
 proof -
   have x_eq: "x = (\<lambda>i. integral s (f i))"
     by (simp add: integral_unique[OF f])
   then have x: "range(\<lambda>k. integral s (f k)) = range x"
     by auto
   have *: "g integrable_on s \<and> (\<lambda>k. integral s (f k)) \<longlonglongrightarrow> integral s g"
   proof (intro monotone_convergence_increasing allI ballI assms)
     show "bounded (range(\<lambda>k. integral s (f k)))"
       using x convergent_imp_bounded assms by metis
   qed (use f in auto)
   then have "integral s g = x'"
     by (intro LIMSEQ_unique[OF _ \<open>x \<longlonglongrightarrow> x'\<close>]) (simp add: x_eq)
   with * show ?thesis
     by (simp add: has_integral_integral)
 qed
 
 lemma monotone_convergence_decreasing:
   fixes f :: "nat \<Rightarrow> 'n::euclidean_space \<Rightarrow> real"
   assumes intf: "\<And>k. (f k) integrable_on S"
     and le: "\<And>k x. x \<in> S \<Longrightarrow> f (Suc k) x \<le> f k x"
     and fg: "\<And>x. x \<in> S \<Longrightarrow> ((\<lambda>k. f k x) \<longlongrightarrow> g x) sequentially"
     and bou: "bounded (range(\<lambda>k. integral S (f k)))"
   shows "g integrable_on S \<and> (\<lambda>k. integral S (f k)) \<longlonglongrightarrow> integral S g"
 proof -
   have *: "range(\<lambda>k. integral S (\<lambda>x. - f k x)) = (*\<^sub>R) (- 1) ` (range(\<lambda>k. integral S (f k)))"
     by force
   have "(\<lambda>x. - g x) integrable_on S \<and> (\<lambda>k. integral S (\<lambda>x. - f k x)) \<longlonglongrightarrow> integral S (\<lambda>x. - g x)"
   proof (rule monotone_convergence_increasing)
     show "\<And>k. (\<lambda>x. - f k x) integrable_on S"
       by (blast intro: integrable_neg intf)
     show "\<And>k x. x \<in> S \<Longrightarrow> - f k x \<le> - f (Suc k) x"
       by (simp add: le)
     show "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>k. - f k x) \<longlonglongrightarrow> - g x"
       by (simp add: fg tendsto_minus)
     show "bounded (range(\<lambda>k. integral S (\<lambda>x. - f k x)))"
       using "*" bou bounded_scaling by auto
   qed
   then show ?thesis
     by (force dest: integrable_neg tendsto_minus)
 qed
 
 lemma integral_norm_bound_integral:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes int_f: "f integrable_on S"
     and int_g: "g integrable_on S"
     and le_g: "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> g x"
   shows "norm (integral S f) \<le> integral S g"
 proof -
   have norm: "norm \<eta> \<le> y + e"
     if "norm \<zeta> \<le> x" and "\<bar>x - y\<bar> < e/2" and "norm (\<zeta> - \<eta>) < e/2"
     for e x y and \<zeta> \<eta> :: 'a
   proof -
     have "norm (\<eta> - \<zeta>) < e/2"
       by (metis norm_minus_commute that(3))
     moreover have "x \<le> y + e/2"
       using that(2) by linarith
     ultimately show ?thesis
       using that(1) le_less_trans[OF norm_triangle_sub[of \<eta> \<zeta>]] by (auto simp: less_imp_le)
   qed
   have lem: "norm (integral(cbox a b) f) \<le> integral (cbox a b) g"
     if f: "f integrable_on cbox a b"
     and g: "g integrable_on cbox a b"
     and nle: "\<And>x. x \<in> cbox a b \<Longrightarrow> norm (f x) \<le> g x"
     for f :: "'n \<Rightarrow> 'a" and g a b
   proof (rule field_le_epsilon)
     fix e :: real
     assume "e > 0"
     then have e: "e/2 > 0"
       by auto
     with integrable_integral[OF f,unfolded has_integral[of f]]
     obtain \<gamma> where \<gamma>: "gauge \<gamma>"
               "\<And>\<D>. \<D> tagged_division_of cbox a b \<and> \<gamma> fine \<D> 
            \<Longrightarrow> norm ((\<Sum>(x, k)\<in>\<D>. content k *\<^sub>R f x) - integral (cbox a b) f) < e/2"
       by meson 
     moreover
     from integrable_integral[OF g,unfolded has_integral[of g]] e
     obtain \<delta> where \<delta>: "gauge \<delta>"
               "\<And>\<D>. \<D> tagged_division_of cbox a b \<and> \<delta> fine \<D> 
            \<Longrightarrow> norm ((\<Sum>(x, k)\<in>\<D>. content k *\<^sub>R g x) - integral (cbox a b) g) < e/2"
       by meson
     ultimately have "gauge (\<lambda>x. \<gamma> x \<inter> \<delta> x)"
       using gauge_Int by blast
     with fine_division_exists obtain \<D> 
       where p: "\<D> tagged_division_of cbox a b" "(\<lambda>x. \<gamma> x \<inter> \<delta> x) fine \<D>" 
       by metis
     have "\<gamma> fine \<D>" "\<delta> fine \<D>"
       using fine_Int p(2) by blast+
     show "norm (integral (cbox a b) f) \<le> integral (cbox a b) g + e"
     proof (rule norm)
       have "norm (content K *\<^sub>R f x) \<le> content K *\<^sub>R g x" if  "(x, K) \<in> \<D>" for x K
       proof-
         have K: "x \<in> K" "K \<subseteq> cbox a b"
           using \<open>(x, K) \<in> \<D>\<close> p(1) by blast+
         obtain u v where  "K = cbox u v"
           using \<open>(x, K) \<in> \<D>\<close> p(1) by blast
         moreover have "content K * norm (f x) \<le> content K * g x"
           by (metis K subsetD dual_order.antisym measure_nonneg mult_zero_left nle not_le real_mult_le_cancel_iff2)
         then show ?thesis
           by simp
       qed
       then show "norm (\<Sum>(x, k)\<in>\<D>. content k *\<^sub>R f x) \<le> (\<Sum>(x, k)\<in>\<D>. content k *\<^sub>R g x)"
         by (simp add: sum_norm_le split_def)
       show "norm ((\<Sum>(x, k)\<in>\<D>. content k *\<^sub>R f x) - integral (cbox a b) f) < e/2"
         using \<open>\<gamma> fine \<D>\<close> \<gamma> p(1) by simp
       show "\<bar>(\<Sum>(x, k)\<in>\<D>. content k *\<^sub>R g x) - integral (cbox a b) g\<bar> < e/2"
         using \<open>\<delta> fine \<D>\<close> \<delta> p(1) by simp
     qed
   qed
   show ?thesis
   proof (rule field_le_epsilon)
     fix e :: real
     assume "e > 0"
     then have e: "e/2 > 0"
       by auto
     let ?f = "(\<lambda>x. if x \<in> S then f x else 0)"
     let ?g = "(\<lambda>x. if x \<in> S then g x else 0)"
     have f: "?f integrable_on cbox a b" and g: "?g integrable_on cbox a b" for a b
       using int_f int_g integrable_altD by auto
     obtain Bf where "0 < Bf"
       and Bf: "\<And>a b. ball 0 Bf \<subseteq> cbox a b \<Longrightarrow>
         \<exists>z. (?f has_integral z) (cbox a b) \<and> norm (z - integral S f) < e/2"
       using integrable_integral [OF int_f,unfolded has_integral'[of f]] e that by blast
     obtain Bg where "0 < Bg"
       and Bg: "\<And>a b. ball 0 Bg \<subseteq> cbox a b \<Longrightarrow>
         \<exists>z. (?g has_integral z) (cbox a b) \<and> norm (z - integral S g) < e/2"
       using integrable_integral [OF int_g,unfolded has_integral'[of g]] e that by blast
     obtain a b::'n where ab: "ball 0 Bf \<union> ball 0 Bg \<subseteq> cbox a b"
       using ball_max_Un  by (metis bounded_ball bounded_subset_cbox_symmetric)
     have "ball 0 Bf \<subseteq> cbox a b"
       using ab by auto
     with Bf obtain z where int_fz: "(?f has_integral z) (cbox a b)" and z: "norm (z - integral S f) < e/2"
       by meson
     have "ball 0 Bg \<subseteq> cbox a b"
       using ab by auto
     with Bg obtain w where int_gw: "(?g has_integral w) (cbox a b)" and w: "norm (w - integral S g) < e/2"
       by meson
     show "norm (integral S f) \<le> integral S g + e"
     proof (rule norm)
       show "norm (integral (cbox a b) ?f) \<le> integral (cbox a b) ?g"
         by (simp add: le_g lem[OF f g, of a b])
       show "\<bar>integral (cbox a b) ?g - integral S g\<bar> < e/2"
         using int_gw integral_unique w by auto
       show "norm (integral (cbox a b) ?f - integral S f) < e/2"
         using int_fz integral_unique z by blast
     qed
   qed
 qed
 
 lemma continuous_on_imp_absolutely_integrable_on:
   fixes f::"real \<Rightarrow> 'a::banach"
   shows "continuous_on {a..b} f \<Longrightarrow>
     norm (integral {a..b} f) \<le> integral {a..b} (\<lambda>x. norm (f x))"
   by (intro integral_norm_bound_integral integrable_continuous_real continuous_on_norm) auto
 
 lemma integral_bound:
   fixes f::"real \<Rightarrow> 'a::banach"
   assumes "a \<le> b"
   assumes "continuous_on {a .. b} f"
   assumes "\<And>t. t \<in> {a .. b} \<Longrightarrow> norm (f t) \<le> B"
   shows "norm (integral {a .. b} f) \<le> B * (b - a)"
 proof -
   note continuous_on_imp_absolutely_integrable_on[OF assms(2)]
   also have "integral {a..b} (\<lambda>x. norm (f x)) \<le> integral {a..b} (\<lambda>_. B)"
     by (rule integral_le)
       (auto intro!: integrable_continuous_real continuous_intros assms)
   also have "\<dots> = B * (b - a)" using assms by simp
   finally show ?thesis .
 qed
 
 lemma integral_norm_bound_integral_component:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   fixes g :: "'n \<Rightarrow> 'b::euclidean_space"
   assumes f: "f integrable_on S" and g: "g integrable_on S"
     and fg: "\<And>x. x \<in> S \<Longrightarrow> norm(f x) \<le> (g x)\<bullet>k"
   shows "norm (integral S f) \<le> (integral S g)\<bullet>k"
 proof -
   have "norm (integral S f) \<le> integral S ((\<lambda>x. x \<bullet> k) \<circ> g)"
-    apply (rule integral_norm_bound_integral[OF f integrable_linear[OF g]])
-    apply (simp add: bounded_linear_inner_left)
-    apply (metis fg o_def)
-    done
+    using integral_norm_bound_integral[OF f integrable_linear[OF g]]
+    by (simp add: bounded_linear_inner_left fg)
   then show ?thesis
     unfolding o_def integral_component_eq[OF g] .
 qed
 
 lemma has_integral_norm_bound_integral_component:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   fixes g :: "'n \<Rightarrow> 'b::euclidean_space"
   assumes f: "(f has_integral i) S"
     and g: "(g has_integral j) S"
     and "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> (g x)\<bullet>k"
   shows "norm i \<le> j\<bullet>k"
   using integral_norm_bound_integral_component[of f S g k] 
   unfolding integral_unique[OF f] integral_unique[OF g]
   using assms
   by auto
 
 
 lemma uniformly_convergent_improper_integral:
   fixes f :: "'b \<Rightarrow> real \<Rightarrow> 'a :: {banach}"
   assumes deriv: "\<And>x. x \<ge> a \<Longrightarrow> (G has_field_derivative g x) (at x within {a..})"
   assumes integrable: "\<And>a' b x. x \<in> A \<Longrightarrow> a' \<ge> a \<Longrightarrow> b \<ge> a' \<Longrightarrow> f x integrable_on {a'..b}"
   assumes G: "convergent G"
   assumes le: "\<And>y x. y \<in> A \<Longrightarrow> x \<ge> a \<Longrightarrow> norm (f y x) \<le> g x"
   shows   "uniformly_convergent_on A (\<lambda>b x. integral {a..b} (f x))"
 proof (intro Cauchy_uniformly_convergent uniformly_Cauchy_onI', goal_cases)
   case (1 \<epsilon>)
   from G have "Cauchy G"
     by (auto intro!: convergent_Cauchy)
   with 1 obtain M where M: "dist (G (real m)) (G (real n)) < \<epsilon>" if "m \<ge> M" "n \<ge> M" for m n
     by (force simp: Cauchy_def)
   define M' where "M' = max (nat \<lceil>a\<rceil>) M"
 
   show ?case
   proof (rule exI[of _ M'], safe, goal_cases)
     case (1 x m n)
     have M': "M' \<ge> a" "M' \<ge> M" unfolding M'_def by linarith+
     have int_g: "(g has_integral (G (real n) - G (real m))) {real m..real n}"
       using 1 M' by (intro fundamental_theorem_of_calculus) 
                     (auto simp: has_field_derivative_iff_has_vector_derivative [symmetric] 
                           intro!: DERIV_subset[OF deriv])
     have int_f: "f x integrable_on {a'..real n}" if "a' \<ge> a" for a'
       using that 1 by (cases "a' \<le> real n") (auto intro: integrable)
 
     have "dist (integral {a..real m} (f x)) (integral {a..real n} (f x)) =
             norm (integral {a..real n} (f x) - integral {a..real m} (f x))"
       by (simp add: dist_norm norm_minus_commute)
     also have "integral {a..real m} (f x) + integral {real m..real n} (f x) = 
                  integral {a..real n} (f x)"
       using M' and 1 by (intro integral_combine int_f) auto
     hence "integral {a..real n} (f x) - integral {a..real m} (f x) = 
              integral {real m..real n} (f x)"
       by (simp add: algebra_simps)
     also have "norm \<dots> \<le> integral {real m..real n} g"
       using le 1 M' int_f int_g by (intro integral_norm_bound_integral) auto 
     also from int_g have "integral {real m..real n} g = G (real n) - G (real m)"
       by (simp add: has_integral_iff)
     also have "\<dots> \<le> dist (G m) (G n)" 
       by (simp add: dist_norm)
     also from 1 and M' have "\<dots> < \<epsilon>"
       by (intro M) auto
     finally show ?case .
   qed
 qed
 
 lemma uniformly_convergent_improper_integral':
   fixes f :: "'b \<Rightarrow> real \<Rightarrow> 'a :: {banach, real_normed_algebra}"
   assumes deriv: "\<And>x. x \<ge> a \<Longrightarrow> (G has_field_derivative g x) (at x within {a..})"
   assumes integrable: "\<And>a' b x. x \<in> A \<Longrightarrow> a' \<ge> a \<Longrightarrow> b \<ge> a' \<Longrightarrow> f x integrable_on {a'..b}"
   assumes G: "convergent G"
   assumes le: "eventually (\<lambda>x. \<forall>y\<in>A. norm (f y x) \<le> g x) at_top"
   shows   "uniformly_convergent_on A (\<lambda>b x. integral {a..b} (f x))"
 proof -
   from le obtain a'' where le: "\<And>y x. y \<in> A \<Longrightarrow> x \<ge> a'' \<Longrightarrow> norm (f y x) \<le> g x"
     by (auto simp: eventually_at_top_linorder)
   define a' where "a' = max a a''"
 
   have "uniformly_convergent_on A (\<lambda>b x. integral {a'..real b} (f x))"
   proof (rule uniformly_convergent_improper_integral)
     fix t assume t: "t \<ge> a'"
     hence "(G has_field_derivative g t) (at t within {a..})"
       by (intro deriv) (auto simp: a'_def)
     moreover have "{a'..} \<subseteq> {a..}" unfolding a'_def by auto
     ultimately show "(G has_field_derivative g t) (at t within {a'..})"
       by (rule DERIV_subset)
   qed (insert le, auto simp: a'_def intro: integrable G)
   hence "uniformly_convergent_on A (\<lambda>b x. integral {a..a'} (f x) + integral {a'..real b} (f x))" 
     (is ?P) by (intro uniformly_convergent_add) auto
   also have "eventually (\<lambda>x. \<forall>y\<in>A. integral {a..a'} (f y) + integral {a'..x} (f y) =
                    integral {a..x} (f y)) sequentially"
     by (intro eventually_mono [OF eventually_ge_at_top[of "nat \<lceil>a'\<rceil>"]] ballI integral_combine)
        (auto simp: a'_def intro: integrable)
   hence "?P \<longleftrightarrow> ?thesis"
     by (intro uniformly_convergent_cong) simp_all
   finally show ?thesis .
 qed
 
 subsection \<open>differentiation under the integral sign\<close>
 
 lemma integral_continuous_on_param:
   fixes f::"'a::topological_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::banach"
   assumes cont_fx: "continuous_on (U \<times> cbox a b) (\<lambda>(x, t). f x t)"
   shows "continuous_on U (\<lambda>x. integral (cbox a b) (f x))"
 proof cases
   assume "content (cbox a b) \<noteq> 0"
   then have ne: "cbox a b \<noteq> {}" by auto
 
   note [continuous_intros] =
     continuous_on_compose2[OF cont_fx, where f="\<lambda>y. Pair x y" for x,
       unfolded split_beta fst_conv snd_conv]
 
   show ?thesis
     unfolding continuous_on_def
   proof (safe intro!: tendstoI)
     fix e'::real and x
     assume "e' > 0"
     define e where "e = e' / (content (cbox a b) + 1)"
     have "e > 0" using \<open>e' > 0\<close> by (auto simp: e_def intro!: divide_pos_pos add_nonneg_pos)
     assume "x \<in> U"
     from continuous_on_prod_compactE[OF cont_fx compact_cbox \<open>x \<in> U\<close> \<open>0 < e\<close>]
     obtain X0 where X0: "x \<in> X0" "open X0"
       and fx_bound: "\<And>y t. y \<in> X0 \<inter> U \<Longrightarrow> t \<in> cbox a b \<Longrightarrow> norm (f y t - f x t) \<le> e"
       unfolding split_beta fst_conv snd_conv dist_norm
       by metis
     have "\<forall>\<^sub>F y in at x within U. y \<in> X0 \<inter> U"
       using X0(1) X0(2) eventually_at_topological by auto
     then show "\<forall>\<^sub>F y in at x within U. dist (integral (cbox a b) (f y)) (integral (cbox a b) (f x)) < e'"
     proof eventually_elim
       case (elim y)
       have "dist (integral (cbox a b) (f y)) (integral (cbox a b) (f x)) =
         norm (integral (cbox a b) (\<lambda>t. f y t - f x t))"
         using elim \<open>x \<in> U\<close>
         unfolding dist_norm
         by (subst integral_diff)
            (auto intro!: integrable_continuous continuous_intros)
       also have "\<dots> \<le> e * content (cbox a b)"
         using elim \<open>x \<in> U\<close>
         by (intro integrable_bound)
            (auto intro!: fx_bound \<open>x \<in> U \<close> less_imp_le[OF \<open>0 < e\<close>]
               integrable_continuous continuous_intros)
       also have "\<dots> < e'"
         using \<open>0 < e'\<close> \<open>e > 0\<close>
         by (auto simp: e_def field_split_simps)
       finally show "dist (integral (cbox a b) (f y)) (integral (cbox a b) (f x)) < e'" .
     qed
   qed
 qed (auto intro!: continuous_on_const)
 
 lemma leibniz_rule:
   fixes f::"'a::banach \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::banach"
   assumes fx: "\<And>x t. x \<in> U \<Longrightarrow> t \<in> cbox a b \<Longrightarrow>
     ((\<lambda>x. f x t) has_derivative blinfun_apply (fx x t)) (at x within U)"
   assumes integrable_f2: "\<And>x. x \<in> U \<Longrightarrow> f x integrable_on cbox a b"
   assumes cont_fx: "continuous_on (U \<times> (cbox a b)) (\<lambda>(x, t). fx x t)"
   assumes [intro]: "x0 \<in> U"
   assumes "convex U"
   shows
     "((\<lambda>x. integral (cbox a b) (f x)) has_derivative integral (cbox a b) (fx x0)) (at x0 within U)"
     (is "(?F has_derivative ?dF) _")
 proof cases
   assume "content (cbox a b) \<noteq> 0"
   then have ne: "cbox a b \<noteq> {}" by auto
   note [continuous_intros] =
     continuous_on_compose2[OF cont_fx, where f="\<lambda>y. Pair x y" for x,
       unfolded split_beta fst_conv snd_conv]
   show ?thesis
   proof (intro has_derivativeI bounded_linear_scaleR_left tendstoI, fold norm_conv_dist)
     have cont_f1: "\<And>t. t \<in> cbox a b \<Longrightarrow> continuous_on U (\<lambda>x. f x t)"
       by (auto simp: continuous_on_eq_continuous_within intro!: has_derivative_continuous fx)
     note [continuous_intros] = continuous_on_compose2[OF cont_f1]
     fix e'::real
     assume "e' > 0"
     define e where "e = e' / (content (cbox a b) + 1)"
     have "e > 0" using \<open>e' > 0\<close> by (auto simp: e_def intro!: divide_pos_pos add_nonneg_pos)
     from continuous_on_prod_compactE[OF cont_fx compact_cbox \<open>x0 \<in> U\<close> \<open>e > 0\<close>]
     obtain X0 where X0: "x0 \<in> X0" "open X0"
       and fx_bound: "\<And>x t. x \<in> X0 \<inter> U \<Longrightarrow> t \<in> cbox a b \<Longrightarrow> norm (fx x t - fx x0 t) \<le> e"
       unfolding split_beta fst_conv snd_conv
       by (metis dist_norm)
 
     note eventually_closed_segment[OF \<open>open X0\<close> \<open>x0 \<in> X0\<close>, of U]
     moreover
     have "\<forall>\<^sub>F x in at x0 within U. x \<in> X0"
       using \<open>open X0\<close> \<open>x0 \<in> X0\<close> eventually_at_topological by blast
     moreover have "\<forall>\<^sub>F x in at x0 within U. x \<noteq> x0"
       by (auto simp: eventually_at_filter)
     moreover have "\<forall>\<^sub>F x in at x0 within U. x \<in> U"
       by (auto simp: eventually_at_filter)
     ultimately
     show "\<forall>\<^sub>F x in at x0 within U. norm ((?F x - ?F x0 - ?dF (x - x0)) /\<^sub>R norm (x - x0)) < e'"
     proof eventually_elim
       case (elim x)
       from elim have "0 < norm (x - x0)" by simp
       have "closed_segment x0 x \<subseteq> U"
         by (rule \<open>convex U\<close>[unfolded convex_contains_segment, rule_format, OF \<open>x0 \<in> U\<close> \<open>x \<in> U\<close>])
       from elim have [intro]: "x \<in> U" by auto
       have "?F x - ?F x0 - ?dF (x - x0) =
         integral (cbox a b) (\<lambda>y. f x y - f x0 y - fx x0 y (x - x0))"
         (is "_ = ?id")
         using \<open>x \<noteq> x0\<close>
         by (subst blinfun_apply_integral integral_diff,
             auto intro!: integrable_diff integrable_f2 continuous_intros
               intro: integrable_continuous)+
       also
       {
         fix t assume t: "t \<in> (cbox a b)"
         have seg: "\<And>t. t \<in> {0..1} \<Longrightarrow> x0 + t *\<^sub>R (x - x0) \<in> X0 \<inter> U"
           using \<open>closed_segment x0 x \<subseteq> U\<close>
             \<open>closed_segment x0 x \<subseteq> X0\<close>
           by (force simp: closed_segment_def algebra_simps)
         from t have deriv:
           "((\<lambda>x. f x t) has_derivative (fx y t)) (at y within X0 \<inter> U)"
           if "y \<in> X0 \<inter> U" for y
           unfolding has_vector_derivative_def[symmetric]
           using that \<open>x \<in> X0\<close>
-          by (intro has_derivative_within_subset[OF fx]) auto
+          by (intro has_derivative_subset[OF fx]) auto
         have "\<And>x. x \<in> X0 \<inter> U \<Longrightarrow> onorm (blinfun_apply (fx x t) - (fx x0 t)) \<le> e"
           using fx_bound t
           by (auto simp add: norm_blinfun_def fun_diff_def blinfun.bilinear_simps[symmetric])
         from differentiable_bound_linearization[OF seg deriv this] X0
         have "norm (f x t - f x0 t - fx x0 t (x - x0)) \<le> e * norm (x - x0)"
           by (auto simp add: ac_simps)
       }
       then have "norm ?id \<le> integral (cbox a b) (\<lambda>_. e * norm (x - x0))"
         by (intro integral_norm_bound_integral)
           (auto intro!: continuous_intros integrable_diff integrable_f2
             intro: integrable_continuous)
       also have "\<dots> = content (cbox a b) * e * norm (x - x0)"
         by simp
       also have "\<dots> < e' * norm (x - x0)"
-        using \<open>e' > 0\<close>
-        apply (intro mult_strict_right_mono[OF _ \<open>0 < norm (x - x0)\<close>])
-        apply (simp add: divide_simps e_def not_less)
-        done
+      proof (intro mult_strict_right_mono[OF _ \<open>0 < norm (x - x0)\<close>])
+        show "content (cbox a b) * e < e'"
+          using \<open>e' > 0\<close> by (simp add: divide_simps e_def not_less)
+      qed
       finally have "norm (?F x - ?F x0 - ?dF (x - x0)) < e' * norm (x - x0)" .
       then show ?case
         by (auto simp: divide_simps)
     qed
   qed (rule blinfun.bounded_linear_right)
 qed (auto intro!: derivative_eq_intros simp: blinfun.bilinear_simps)
 
 lemma has_vector_derivative_eq_has_derivative_blinfun:
   "(f has_vector_derivative f') (at x within U) \<longleftrightarrow>
     (f has_derivative blinfun_scaleR_left f') (at x within U)"
   by (simp add: has_vector_derivative_def)
 
 lemma leibniz_rule_vector_derivative:
   fixes f::"real \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::banach"
   assumes fx: "\<And>x t. x \<in> U \<Longrightarrow> t \<in> cbox a b \<Longrightarrow>
       ((\<lambda>x. f x t) has_vector_derivative (fx x t)) (at x within U)"
   assumes integrable_f2: "\<And>x. x \<in> U \<Longrightarrow> (f x) integrable_on cbox a b"
   assumes cont_fx: "continuous_on (U \<times> cbox a b) (\<lambda>(x, t). fx x t)"
   assumes U: "x0 \<in> U" "convex U"
   shows "((\<lambda>x. integral (cbox a b) (f x)) has_vector_derivative integral (cbox a b) (fx x0))
       (at x0 within U)"
 proof -
   note [continuous_intros] =
     continuous_on_compose2[OF cont_fx, where f="\<lambda>y. Pair x y" for x,
       unfolded split_beta fst_conv snd_conv]
-  have *: "blinfun_scaleR_left (integral (cbox a b) (fx x0)) =
-    integral (cbox a b) (\<lambda>t. blinfun_scaleR_left (fx x0 t))"
+  show ?thesis
+    unfolding has_vector_derivative_eq_has_derivative_blinfun
+  proof (rule has_derivative_eq_rhs [OF leibniz_rule[OF _ integrable_f2 _ U]])
+    show "continuous_on (U \<times> cbox a b) (\<lambda>(x, t). blinfun_scaleR_left (fx x t))"
+      using cont_fx by (auto simp: split_beta intro!: continuous_intros)
+    show "blinfun_apply (integral (cbox a b) (\<lambda>t. blinfun_scaleR_left (fx x0 t))) =
+          blinfun_apply (blinfun_scaleR_left (integral (cbox a b) (fx x0)))"
     by (subst integral_linear[symmetric])
        (auto simp: has_vector_derivative_def o_def
          intro!: integrable_continuous U continuous_intros bounded_linear_intros)
-  show ?thesis
-    unfolding has_vector_derivative_eq_has_derivative_blinfun
-    apply (rule has_derivative_eq_rhs)
-    apply (rule leibniz_rule[OF _ integrable_f2 _ U, where fx="\<lambda>x t. blinfun_scaleR_left (fx x t)"])
-    using fx cont_fx
-    apply (auto simp: has_vector_derivative_def * split_beta intro!: continuous_intros)
-    done
+    qed (use fx in \<open>auto simp: has_vector_derivative_def\<close>)
 qed
 
 lemma has_field_derivative_eq_has_derivative_blinfun:
   "(f has_field_derivative f') (at x within U) \<longleftrightarrow> (f has_derivative blinfun_mult_right f') (at x within U)"
   by (simp add: has_field_derivative_def)
 
 lemma leibniz_rule_field_derivative:
   fixes f::"'a::{real_normed_field, banach} \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'a"
   assumes fx: "\<And>x t. x \<in> U \<Longrightarrow> t \<in> cbox a b \<Longrightarrow> ((\<lambda>x. f x t) has_field_derivative fx x t) (at x within U)"
   assumes integrable_f2: "\<And>x. x \<in> U \<Longrightarrow> (f x) integrable_on cbox a b"
   assumes cont_fx: "continuous_on (U \<times> (cbox a b)) (\<lambda>(x, t). fx x t)"
   assumes U: "x0 \<in> U" "convex U"
   shows "((\<lambda>x. integral (cbox a b) (f x)) has_field_derivative integral (cbox a b) (fx x0)) (at x0 within U)"
 proof -
   note [continuous_intros] =
     continuous_on_compose2[OF cont_fx, where f="\<lambda>y. Pair x y" for x,
       unfolded split_beta fst_conv snd_conv]
   have *: "blinfun_mult_right (integral (cbox a b) (fx x0)) =
     integral (cbox a b) (\<lambda>t. blinfun_mult_right (fx x0 t))"
     by (subst integral_linear[symmetric])
       (auto simp: has_vector_derivative_def o_def
         intro!: integrable_continuous U continuous_intros bounded_linear_intros)
   show ?thesis
     unfolding has_field_derivative_eq_has_derivative_blinfun
-    apply (rule has_derivative_eq_rhs)
-    apply (rule leibniz_rule[OF _ integrable_f2 _ U, where fx="\<lambda>x t. blinfun_mult_right (fx x t)"])
-    using fx cont_fx
-    apply (auto simp: has_field_derivative_def * split_beta intro!: continuous_intros)
-    done
+  proof (rule has_derivative_eq_rhs [OF leibniz_rule[OF _ integrable_f2 _ U, where fx="\<lambda>x t. blinfun_mult_right (fx x t)"]])
+    show "continuous_on (U \<times> cbox a b) (\<lambda>(x, t). blinfun_mult_right (fx x t))"
+      using cont_fx by (auto simp: split_beta intro!: continuous_intros)
+    show "blinfun_apply (integral (cbox a b) (\<lambda>t. blinfun_mult_right (fx x0 t))) =
+          blinfun_apply (blinfun_mult_right (integral (cbox a b) (fx x0)))"
+      by (subst integral_linear[symmetric])
+        (auto simp: has_vector_derivative_def o_def
+          intro!: integrable_continuous U continuous_intros bounded_linear_intros)
+  qed (use fx in \<open>auto simp: has_field_derivative_def\<close>)
 qed
 
 lemma leibniz_rule_field_differentiable:
   fixes f::"'a::{real_normed_field, banach} \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'a"
   assumes "\<And>x t. x \<in> U \<Longrightarrow> t \<in> cbox a b \<Longrightarrow> ((\<lambda>x. f x t) has_field_derivative fx x t) (at x within U)"
   assumes "\<And>x. x \<in> U \<Longrightarrow> (f x) integrable_on cbox a b"
   assumes "continuous_on (U \<times> (cbox a b)) (\<lambda>(x, t). fx x t)"
   assumes "x0 \<in> U" "convex U"
   shows "(\<lambda>x. integral (cbox a b) (f x)) field_differentiable at x0 within U"
   using leibniz_rule_field_derivative[OF assms] by (auto simp: field_differentiable_def)
 
 
 subsection \<open>Exchange uniform limit and integral\<close>
 
 lemma uniform_limit_integral_cbox:
   fixes f::"'a \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::banach"
   assumes u: "uniform_limit (cbox a b) f g F"
   assumes c: "\<And>n. continuous_on (cbox a b) (f n)"
   assumes [simp]: "F \<noteq> bot"
   obtains I J where
     "\<And>n. (f n has_integral I n) (cbox a b)"
     "(g has_integral J) (cbox a b)"
     "(I \<longlongrightarrow> J) F"
 proof -
   have fi[simp]: "f n integrable_on (cbox a b)" for n
     by (auto intro!: integrable_continuous assms)
   then obtain I where I: "\<And>n. (f n has_integral I n) (cbox a b)"
     by atomize_elim (auto simp: integrable_on_def intro!: choice)
 
   moreover
   have gi[simp]: "g integrable_on (cbox a b)"
     by (auto intro!: integrable_continuous uniform_limit_theorem[OF _ u] eventuallyI c)
   then obtain J where J: "(g has_integral J) (cbox a b)"
     by blast
 
   moreover
   have "(I \<longlongrightarrow> J) F"
   proof cases
     assume "content (cbox a b) = 0"
     hence "I = (\<lambda>_. 0)" "J = 0"
       by (auto intro!: has_integral_unique I J)
     thus ?thesis by simp
   next
     assume content_nonzero: "content (cbox a b) \<noteq> 0"
     show ?thesis
     proof (rule tendstoI)
       fix e::real
       assume "e > 0"
       define e' where "e' = e/2"
       with \<open>e > 0\<close> have "e' > 0" by simp
       then have "\<forall>\<^sub>F n in F. \<forall>x\<in>cbox a b. norm (f n x - g x) < e' / content (cbox a b)"
         using u content_nonzero by (auto simp: uniform_limit_iff dist_norm zero_less_measure_iff)
       then show "\<forall>\<^sub>F n in F. dist (I n) J < e"
       proof eventually_elim
         case (elim n)
         have "I n = integral (cbox a b) (f n)"
             "J = integral (cbox a b) g"
           using I[of n] J by (simp_all add: integral_unique)
         then have "dist (I n) J = norm (integral (cbox a b) (\<lambda>x. f n x - g x))"
           by (simp add: integral_diff dist_norm)
         also have "\<dots> \<le> integral (cbox a b) (\<lambda>x. (e' / content (cbox a b)))"
           using elim
           by (intro integral_norm_bound_integral) (auto intro!: integrable_diff)
         also have "\<dots> < e"
           using \<open>0 < e\<close>
           by (simp add: e'_def)
         finally show ?case .
       qed
     qed
   qed
   ultimately show ?thesis ..
 qed
 
 lemma uniform_limit_integral:
   fixes f::"'a \<Rightarrow> 'b::ordered_euclidean_space \<Rightarrow> 'c::banach"
   assumes u: "uniform_limit {a..b} f g F"
   assumes c: "\<And>n. continuous_on {a..b} (f n)"
   assumes [simp]: "F \<noteq> bot"
   obtains I J where
     "\<And>n. (f n has_integral I n) {a..b}"
     "(g has_integral J) {a..b}"
     "(I \<longlongrightarrow> J) F"
   by (metis interval_cbox assms uniform_limit_integral_cbox)
 
 
 subsection \<open>Integration by parts\<close>
 
 lemma integration_by_parts_interior_strong:
   fixes prod :: "_ \<Rightarrow> _ \<Rightarrow> 'b :: banach"
   assumes bilinear: "bounded_bilinear (prod)"
   assumes s: "finite s" and le: "a \<le> b"
   assumes cont [continuous_intros]: "continuous_on {a..b} f" "continuous_on {a..b} g"
   assumes deriv: "\<And>x. x\<in>{a<..<b} - s \<Longrightarrow> (f has_vector_derivative f' x) (at x)"
                  "\<And>x. x\<in>{a<..<b} - s \<Longrightarrow> (g has_vector_derivative g' x) (at x)"
   assumes int: "((\<lambda>x. prod (f x) (g' x)) has_integral
                   (prod (f b) (g b) - prod (f a) (g a) - y)) {a..b}"
   shows   "((\<lambda>x. prod (f' x) (g x)) has_integral y) {a..b}"
 proof -
   interpret bounded_bilinear prod by fact
   have "((\<lambda>x. prod (f x) (g' x) + prod (f' x) (g x)) has_integral
           (prod (f b) (g b) - prod (f a) (g a))) {a..b}"
     using deriv by (intro fundamental_theorem_of_calculus_interior_strong[OF s le])
                    (auto intro!: continuous_intros continuous_on has_vector_derivative)
   from has_integral_diff[OF this int] show ?thesis by (simp add: algebra_simps)
 qed
 
 lemma integration_by_parts_interior:
   fixes prod :: "_ \<Rightarrow> _ \<Rightarrow> 'b :: banach"
   assumes "bounded_bilinear (prod)" "a \<le> b"
           "continuous_on {a..b} f" "continuous_on {a..b} g"
   assumes "\<And>x. x\<in>{a<..<b} \<Longrightarrow> (f has_vector_derivative f' x) (at x)"
           "\<And>x. x\<in>{a<..<b} \<Longrightarrow> (g has_vector_derivative g' x) (at x)"
   assumes "((\<lambda>x. prod (f x) (g' x)) has_integral (prod (f b) (g b) - prod (f a) (g a) - y)) {a..b}"
   shows   "((\<lambda>x. prod (f' x) (g x)) has_integral y) {a..b}"
   by (rule integration_by_parts_interior_strong[of _ "{}" _ _ f g f' g']) (insert assms, simp_all)
 
 lemma integration_by_parts:
   fixes prod :: "_ \<Rightarrow> _ \<Rightarrow> 'b :: banach"
   assumes "bounded_bilinear (prod)" "a \<le> b"
           "continuous_on {a..b} f" "continuous_on {a..b} g"
   assumes "\<And>x. x\<in>{a..b} \<Longrightarrow> (f has_vector_derivative f' x) (at x)"
           "\<And>x. x\<in>{a..b} \<Longrightarrow> (g has_vector_derivative g' x) (at x)"
   assumes "((\<lambda>x. prod (f x) (g' x)) has_integral (prod (f b) (g b) - prod (f a) (g a) - y)) {a..b}"
   shows   "((\<lambda>x. prod (f' x) (g x)) has_integral y) {a..b}"
   by (rule integration_by_parts_interior[of _ _ _ f g f' g']) (insert assms, simp_all)
 
 lemma integrable_by_parts_interior_strong:
   fixes prod :: "_ \<Rightarrow> _ \<Rightarrow> 'b :: banach"
   assumes bilinear: "bounded_bilinear (prod)"
   assumes s: "finite s" and le: "a \<le> b"
   assumes cont [continuous_intros]: "continuous_on {a..b} f" "continuous_on {a..b} g"
   assumes deriv: "\<And>x. x\<in>{a<..<b} - s \<Longrightarrow> (f has_vector_derivative f' x) (at x)"
                  "\<And>x. x\<in>{a<..<b} - s \<Longrightarrow> (g has_vector_derivative g' x) (at x)"
   assumes int: "(\<lambda>x. prod (f x) (g' x)) integrable_on {a..b}"
   shows   "(\<lambda>x. prod (f' x) (g x)) integrable_on {a..b}"
 proof -
   from int obtain I where "((\<lambda>x. prod (f x) (g' x)) has_integral I) {a..b}"
     unfolding integrable_on_def by blast
   hence "((\<lambda>x. prod (f x) (g' x)) has_integral (prod (f b) (g b) - prod (f a) (g a) -
            (prod (f b) (g b) - prod (f a) (g a) - I))) {a..b}" by simp
   from integration_by_parts_interior_strong[OF assms(1-7) this]
     show ?thesis unfolding integrable_on_def by blast
 qed
 
 lemma integrable_by_parts_interior:
   fixes prod :: "_ \<Rightarrow> _ \<Rightarrow> 'b :: banach"
   assumes "bounded_bilinear (prod)" "a \<le> b"
           "continuous_on {a..b} f" "continuous_on {a..b} g"
   assumes "\<And>x. x\<in>{a<..<b} \<Longrightarrow> (f has_vector_derivative f' x) (at x)"
           "\<And>x. x\<in>{a<..<b} \<Longrightarrow> (g has_vector_derivative g' x) (at x)"
   assumes "(\<lambda>x. prod (f x) (g' x)) integrable_on {a..b}"
   shows   "(\<lambda>x. prod (f' x) (g x)) integrable_on {a..b}"
   by (rule integrable_by_parts_interior_strong[of _ "{}" _ _ f g f' g']) (insert assms, simp_all)
 
 lemma integrable_by_parts:
   fixes prod :: "_ \<Rightarrow> _ \<Rightarrow> 'b :: banach"
   assumes "bounded_bilinear (prod)" "a \<le> b"
           "continuous_on {a..b} f" "continuous_on {a..b} g"
   assumes "\<And>x. x\<in>{a..b} \<Longrightarrow> (f has_vector_derivative f' x) (at x)"
           "\<And>x. x\<in>{a..b} \<Longrightarrow> (g has_vector_derivative g' x) (at x)"
   assumes "(\<lambda>x. prod (f x) (g' x)) integrable_on {a..b}"
   shows   "(\<lambda>x. prod (f' x) (g x)) integrable_on {a..b}"
   by (rule integrable_by_parts_interior_strong[of _ "{}" _ _ f g f' g']) (insert assms, simp_all)
 
 
 subsection \<open>Integration by substitution\<close>
 
 lemma has_integral_substitution_general:
   fixes f :: "real \<Rightarrow> 'a::euclidean_space" and g :: "real \<Rightarrow> real"
   assumes s: "finite s" and le: "a \<le> b"
       and subset: "g ` {a..b} \<subseteq> {c..d}"
       and f [continuous_intros]: "continuous_on {c..d} f"
       and g [continuous_intros]: "continuous_on {a..b} g"
       and deriv [derivative_intros]:
               "\<And>x. x \<in> {a..b} - s \<Longrightarrow> (g has_field_derivative g' x) (at x within {a..b})"
     shows "((\<lambda>x. g' x *\<^sub>R f (g x)) has_integral (integral {g a..g b} f - integral {g b..g a} f)) {a..b}"
 proof -
   let ?F = "\<lambda>x. integral {c..g x} f"
   have cont_int: "continuous_on {a..b} ?F"
     by (rule continuous_on_compose2[OF _ g subset] indefinite_integral_continuous_1
           f integrable_continuous_real)+
   have deriv: "(((\<lambda>x. integral {c..x} f) \<circ> g) has_vector_derivative g' x *\<^sub>R f (g x))
                  (at x within {a..b})" if "x \<in> {a..b} - s" for x
   proof (rule has_vector_derivative_eq_rhs [OF vector_diff_chain_within refl])
     show "(g has_vector_derivative g' x) (at x within {a..b})"
       using deriv has_field_derivative_iff_has_vector_derivative that by blast
     show "((\<lambda>x. integral {c..x} f) has_vector_derivative f (g x)) 
           (at (g x) within g ` {a..b})"
       using that le subset
       by (blast intro: has_vector_derivative_within_subset integral_has_vector_derivative f)
   qed
   have deriv: "(?F has_vector_derivative g' x *\<^sub>R f (g x))
                   (at x)" if "x \<in> {a..b} - (s \<union> {a,b})" for x
     using deriv[of x] that by (simp add: at_within_Icc_at o_def)
   have "((\<lambda>x. g' x *\<^sub>R f (g x)) has_integral (?F b - ?F a)) {a..b}"
     using le cont_int s deriv cont_int
     by (intro fundamental_theorem_of_calculus_interior_strong[of "s \<union> {a,b}"]) simp_all
   also
   from subset have "g x \<in> {c..d}" if "x \<in> {a..b}" for x using that by blast
   from this[of a] this[of b] le have cd: "c \<le> g a" "g b \<le> d" "c \<le> g b" "g a \<le> d" by auto
   have "integral {c..g b} f - integral {c..g a} f = integral {g a..g b} f - integral {g b..g a} f"
   proof cases
     assume "g a \<le> g b"
     note le = le this
     from cd have "integral {c..g a} f + integral {g a..g b} f = integral {c..g b} f"
       by (intro integral_combine integrable_continuous_real continuous_on_subset[OF f] le) simp_all
     with le show ?thesis
       by (cases "g a = g b") (simp_all add: algebra_simps)
   next
     assume less: "\<not>g a \<le> g b"
     then have "g a \<ge> g b" by simp
     note le = le this
     from cd have "integral {c..g b} f + integral {g b..g a} f = integral {c..g a} f"
       by (intro integral_combine integrable_continuous_real continuous_on_subset[OF f] le) simp_all
     with less show ?thesis
       by (simp_all add: algebra_simps)
   qed
   finally show ?thesis .
 qed
 
 lemma has_integral_substitution_strong:
   fixes f :: "real \<Rightarrow> 'a::euclidean_space" and g :: "real \<Rightarrow> real"
   assumes s: "finite s" and le: "a \<le> b" "g a \<le> g b"
     and subset: "g ` {a..b} \<subseteq> {c..d}"
     and f [continuous_intros]: "continuous_on {c..d} f"
     and g [continuous_intros]: "continuous_on {a..b} g"
     and deriv [derivative_intros]:
     "\<And>x. x \<in> {a..b} - s \<Longrightarrow> (g has_field_derivative g' x) (at x within {a..b})"
   shows "((\<lambda>x. g' x *\<^sub>R f (g x)) has_integral (integral {g a..g b} f)) {a..b}"
   using has_integral_substitution_general[OF s le(1) subset f g deriv] le(2)
   by (cases "g a = g b") auto
 
 lemma has_integral_substitution:
   fixes f :: "real \<Rightarrow> 'a::euclidean_space" and g :: "real \<Rightarrow> real"
   assumes "a \<le> b" "g a \<le> g b" "g ` {a..b} \<subseteq> {c..d}"
       and "continuous_on {c..d} f"
       and "\<And>x. x \<in> {a..b} \<Longrightarrow> (g has_field_derivative g' x) (at x within {a..b})"
     shows "((\<lambda>x. g' x *\<^sub>R f (g x)) has_integral (integral {g a..g b} f)) {a..b}"
   by (intro has_integral_substitution_strong[of "{}" a b g c d] assms)
      (auto intro: DERIV_continuous_on assms)
 
 lemma integral_shift:
   fixes f :: "real \<Rightarrow> 'a::euclidean_space"
   assumes cont: "continuous_on {a + c..b + c} f"
   shows "integral {a..b} (f \<circ> (\<lambda>x. x + c)) = integral {a + c..b + c} f"
 proof (cases "a \<le> b")
   case True
   have "((\<lambda>x. 1 *\<^sub>R f (x + c)) has_integral integral {a+c..b+c} f) {a..b}"
     using True cont
     by (intro has_integral_substitution[where c = "a + c" and d = "b + c"])
        (auto intro!: derivative_eq_intros)
   thus ?thesis by (simp add: has_integral_iff o_def)
 qed auto
 
 
 subsection \<open>Compute a double integral using iterated integrals and switching the order of integration\<close>
 
 lemma continuous_on_imp_integrable_on_Pair1:
   fixes f :: "_ \<Rightarrow> 'b::banach"
   assumes con: "continuous_on (cbox (a,c) (b,d)) f" and x: "x \<in> cbox a b"
   shows "(\<lambda>y. f (x, y)) integrable_on (cbox c d)"
 proof -
   have "f \<circ> (\<lambda>y. (x, y)) integrable_on (cbox c d)"
-    apply (rule integrable_continuous)
-    apply (rule continuous_on_compose [OF _ continuous_on_subset [OF con]])
-    using x
-    apply (auto intro: continuous_on_Pair continuous_on_const continuous_on_id continuous_on_subset con)
-    done
+  proof (intro integrable_continuous continuous_on_compose [OF _ continuous_on_subset [OF con]])
+    show "continuous_on (cbox c d) (Pair x)"
+      by (simp add: continuous_on_Pair)
+    show "Pair x ` cbox c d \<subseteq> cbox (a,c) (b,d)"
+      using x by blast
+  qed
   then show ?thesis
     by (simp add: o_def)
 qed
 
 lemma integral_integrable_2dim:
   fixes f :: "('a::euclidean_space * 'b::euclidean_space) \<Rightarrow> 'c::banach"
   assumes "continuous_on (cbox (a,c) (b,d)) f"
     shows "(\<lambda>x. integral (cbox c d) (\<lambda>y. f (x,y))) integrable_on cbox a b"
 proof (cases "content(cbox c d) = 0")
 case True
   then show ?thesis
     by (simp add: True integrable_const)
 next
   case False
   have uc: "uniformly_continuous_on (cbox (a,c) (b,d)) f"
     by (simp add: assms compact_cbox compact_uniformly_continuous)
   { fix x::'a and e::real
     assume x: "x \<in> cbox a b" and e: "0 < e"
     then have e2_gt: "0 < e/2 / content (cbox c d)" and e2_less: "e/2 / content (cbox c d) * content (cbox c d) < e"
       by (auto simp: False content_lt_nz e)
     then obtain dd
     where dd: "\<And>x x'. \<lbrakk>x\<in>cbox (a, c) (b, d); x'\<in>cbox (a, c) (b, d); norm (x' - x) < dd\<rbrakk>
                        \<Longrightarrow> norm (f x' - f x) \<le> e/(2 * content (cbox c d))"  "dd>0"
       using uc [unfolded uniformly_continuous_on_def, THEN spec, of "e/(2 * content (cbox c d))"]
       by (auto simp: dist_norm intro: less_imp_le)
     have "\<exists>delta>0. \<forall>x'\<in>cbox a b. norm (x' - x) < delta \<longrightarrow> norm (integral (cbox c d) (\<lambda>u. f (x', u) - f (x, u))) < e"
-      apply (rule_tac x=dd in exI)
       using dd e2_gt assms x
+      apply (rule_tac x=dd in exI)
       apply clarify
-      apply (rule le_less_trans [OF _ e2_less])
-      apply (rule integrable_bound)
+      apply (rule le_less_trans [OF integrable_bound e2_less])
       apply (auto intro: integrable_diff continuous_on_imp_integrable_on_Pair1)
       done
   } note * = this
   show ?thesis
-    apply (rule integrable_continuous)
-    apply (simp add: * continuous_on_iff dist_norm integral_diff [symmetric] continuous_on_imp_integrable_on_Pair1 [OF assms])
-    done
+  proof (rule integrable_continuous)
+    show "continuous_on (cbox a b) (\<lambda>x. integral (cbox c d) (\<lambda>y. f (x, y)))"
+      by (simp add: * continuous_on_iff dist_norm integral_diff [symmetric] continuous_on_imp_integrable_on_Pair1 [OF assms])
+  qed
 qed
 
 lemma integral_split:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::{real_normed_vector,complete_space}"
   assumes f: "f integrable_on (cbox a b)"
       and k: "k \<in> Basis"
   shows "integral (cbox a b) f =
            integral (cbox a b \<inter> {x. x\<bullet>k \<le> c}) f +
            integral (cbox a b \<inter> {x. x\<bullet>k \<ge> c}) f"
-  apply (rule integral_unique [OF has_integral_split [where c=c]])
   using k f
-  apply (auto simp: has_integral_integral [symmetric])
-  done
+  by (auto simp: has_integral_integral intro: integral_unique [OF has_integral_split])
 
 lemma integral_swap_operativeI:
   fixes f :: "('a::euclidean_space * 'b::euclidean_space) \<Rightarrow> 'c::banach"
   assumes f: "continuous_on s f" and e: "0 < e"
     shows "operative conj True
            (\<lambda>k. \<forall>a b c d.
                 cbox (a,c) (b,d) \<subseteq> k \<and> cbox (a,c) (b,d) \<subseteq> s
                 \<longrightarrow> norm(integral (cbox (a,c) (b,d)) f -
                          integral (cbox a b) (\<lambda>x. integral (cbox c d) (\<lambda>y. f((x,y)))))
                     \<le> e * content (cbox (a,c) (b,d)))"
 proof (standard, auto)
   fix a::'a and c::'b and b::'a and d::'b and u::'a and v::'a and w::'b and z::'b
   assume *: "box (a, c) (b, d) = {}"
      and cb1: "cbox (u, w) (v, z) \<subseteq> cbox (a, c) (b, d)"
      and cb2: "cbox (u, w) (v, z) \<subseteq> s"
   then have c0: "content (cbox (a, c) (b, d)) = 0"
     using * unfolding content_eq_0_interior by simp
   have c0': "content (cbox (u, w) (v, z)) = 0"
     by (fact content_0_subset [OF c0 cb1])
   show "norm (integral (cbox (u,w) (v,z)) f - integral (cbox u v) (\<lambda>x. integral (cbox w z) (\<lambda>y. f (x, y))))
           \<le> e * content (cbox (u,w) (v,z))"
     using content_cbox_pair_eq0_D [OF c0']
     by (force simp add: c0')
 next
   fix a::'a and c::'b and b::'a and d::'b
   and M::real and i::'a and j::'b
   and u::'a and v::'a and w::'b and z::'b
   assume ij: "(i,j) \<in> Basis"
      and n1: "\<forall>a' b' c' d'.
                 cbox (a',c') (b',d') \<subseteq> cbox (a,c) (b,d) \<and>
                 cbox (a',c') (b',d') \<subseteq> {x. x \<bullet> (i,j) \<le> M} \<and> cbox (a',c') (b',d') \<subseteq> s \<longrightarrow>
                 norm (integral (cbox (a',c') (b',d')) f - integral (cbox a' b') (\<lambda>x. integral (cbox c' d') (\<lambda>y. f (x,y))))
                 \<le> e * content (cbox (a',c') (b',d'))"
      and n2: "\<forall>a' b' c' d'.
                 cbox (a',c') (b',d') \<subseteq> cbox (a,c) (b,d) \<and>
                 cbox (a',c') (b',d') \<subseteq> {x. M \<le> x \<bullet> (i,j)} \<and> cbox (a',c') (b',d') \<subseteq> s \<longrightarrow>
                 norm (integral (cbox (a',c') (b',d')) f - integral (cbox a' b') (\<lambda>x. integral (cbox c' d') (\<lambda>y. f (x,y))))
                 \<le> e * content (cbox (a',c') (b',d'))"
      and subs: "cbox (u,w) (v,z) \<subseteq> cbox (a,c) (b,d)"  "cbox (u,w) (v,z) \<subseteq> s"
   have fcont: "continuous_on (cbox (u, w) (v, z)) f"
     using assms(1) continuous_on_subset  subs(2) by blast
   then have fint: "f integrable_on cbox (u, w) (v, z)"
     by (metis integrable_continuous)
   consider "i \<in> Basis" "j=0" | "j \<in> Basis" "i=0"  using ij
     by (auto simp: Euclidean_Space.Basis_prod_def)
   then show "norm (integral (cbox (u,w) (v,z)) f - integral (cbox u v) (\<lambda>x. integral (cbox w z) (\<lambda>y. f (x,y))))
              \<le> e * content (cbox (u,w) (v,z))" (is ?normle)
   proof cases
     case 1
     then have e: "e * content (cbox (u, w) (v, z)) =
                   e * (content (cbox u v \<inter> {x. x \<bullet> i \<le> M}) * content (cbox w z)) +
                   e * (content (cbox u v \<inter> {x. M \<le> x \<bullet> i}) * content (cbox w z))"
       by (simp add: content_split [where c=M] content_Pair algebra_simps)
     have *: "integral (cbox u v) (\<lambda>x. integral (cbox w z) (\<lambda>y. f (x, y))) =
                 integral (cbox u v \<inter> {x. x \<bullet> i \<le> M}) (\<lambda>x. integral (cbox w z) (\<lambda>y. f (x, y))) +
                 integral (cbox u v \<inter> {x. M \<le> x \<bullet> i}) (\<lambda>x. integral (cbox w z) (\<lambda>y. f (x, y)))"
       using 1 f subs integral_integrable_2dim continuous_on_subset
       by (blast intro: integral_split)
     show ?normle
       apply (rule norm_diff2 [OF integral_split [where c=M, OF fint ij] * e])
       using 1 subs
       apply (simp_all add: cbox_Pair_eq setcomp_dot1 [of "\<lambda>u. M\<le>u"] setcomp_dot1 [of "\<lambda>u. u\<le>M"] Sigma_Int_Paircomp1)
-      apply (simp_all add: interval_split ij)
-      apply (simp_all add: cbox_Pair_eq [symmetric] content_Pair [symmetric])
-      apply (force simp add: interval_split [symmetric] intro!: n1 [rule_format])
-      apply (force simp add: interval_split [symmetric] intro!: n2 [rule_format])
+      apply (simp_all add: interval_split ij flip: cbox_Pair_eq content_Pair)
+      apply (force simp flip: interval_split intro!: n1 [rule_format])
+      apply (force simp flip: interval_split intro!: n2 [rule_format])
       done
   next
     case 2
     then have e: "e * content (cbox (u, w) (v, z)) =
                   e * (content (cbox u v) * content (cbox w z \<inter> {x. x \<bullet> j \<le> M})) +
                   e * (content (cbox u v) * content (cbox w z \<inter> {x. M \<le> x \<bullet> j}))"
       by (simp add: content_split [where c=M] content_Pair algebra_simps)
     have "(\<lambda>x. integral (cbox w z \<inter> {x. x \<bullet> j \<le> M}) (\<lambda>y. f (x, y))) integrable_on cbox u v"
-                "(\<lambda>x. integral (cbox w z \<inter> {x. M \<le> x \<bullet> j}) (\<lambda>y. f (x, y))) integrable_on cbox u v"
+         "(\<lambda>x. integral (cbox w z \<inter> {x. M \<le> x \<bullet> j}) (\<lambda>y. f (x, y))) integrable_on cbox u v"
       using 2 subs
       apply (simp_all add: interval_split)
-      apply (rule_tac [!] integral_integrable_2dim [OF continuous_on_subset [OF f]])
-      apply (auto simp: cbox_Pair_eq interval_split [symmetric])
+      apply (rule integral_integrable_2dim [OF continuous_on_subset [OF f]]; auto simp: cbox_Pair_eq interval_split [symmetric])+
       done
     with 2 have *: "integral (cbox u v) (\<lambda>x. integral (cbox w z) (\<lambda>y. f (x, y))) =
                    integral (cbox u v) (\<lambda>x. integral (cbox w z \<inter> {x. x \<bullet> j \<le> M}) (\<lambda>y. f (x, y))) +
                    integral (cbox u v) (\<lambda>x. integral (cbox w z \<inter> {x. M \<le> x \<bullet> j}) (\<lambda>y. f (x, y)))"
       by (simp add: integral_add [symmetric] integral_split [symmetric]
                     continuous_on_imp_integrable_on_Pair1 [OF fcont] cong: integral_cong)
     show ?normle
       apply (rule norm_diff2 [OF integral_split [where c=M, OF fint ij] * e])
       using 2 subs
       apply (simp_all add: cbox_Pair_eq setcomp_dot2 [of "\<lambda>u. M\<le>u"] setcomp_dot2 [of "\<lambda>u. u\<le>M"] Sigma_Int_Paircomp2)
-      apply (simp_all add: interval_split ij)
-      apply (simp_all add: cbox_Pair_eq [symmetric] content_Pair [symmetric])
-      apply (force simp add: interval_split [symmetric] intro!: n1 [rule_format])
-      apply (force simp add: interval_split [symmetric] intro!: n2 [rule_format])
+      apply (simp_all add: interval_split ij flip: cbox_Pair_eq content_Pair)
+      apply (force simp flip: interval_split intro!: n1 [rule_format])
+      apply (force simp flip: interval_split intro!: n2 [rule_format])
       done
   qed
 qed
 
 lemma integral_Pair_const:
     "integral (cbox (a,c) (b,d)) (\<lambda>x. k) =
      integral (cbox a b) (\<lambda>x. integral (cbox c d) (\<lambda>y. k))"
   by (simp add: content_Pair)
 
 lemma integral_prod_continuous:
   fixes f :: "('a::euclidean_space * 'b::euclidean_space) \<Rightarrow> 'c::banach"
   assumes "continuous_on (cbox (a, c) (b, d)) f" (is "continuous_on ?CBOX f")
     shows "integral (cbox (a, c) (b, d)) f = integral (cbox a b) (\<lambda>x. integral (cbox c d) (\<lambda>y. f (x, y)))"
 proof (cases "content ?CBOX = 0")
   case True
   then show ?thesis
     by (auto simp: content_Pair)
 next
   case False
   then have cbp: "content ?CBOX > 0"
     using content_lt_nz by blast
   have "norm (integral ?CBOX f - integral (cbox a b) (\<lambda>x. integral (cbox c d) (\<lambda>y. f (x,y)))) = 0"
   proof (rule dense_eq0_I, simp)
     fix e :: real 
     assume "0 < e"
     with \<open>content ?CBOX > 0\<close> have "0 < e/content ?CBOX"
       by simp
     have f_int_acbd: "f integrable_on ?CBOX"
       by (rule integrable_continuous [OF assms])
     { fix p
       assume p: "p division_of ?CBOX"
       then have "finite p"
         by blast
       define e' where "e' = e/content ?CBOX"
       with \<open>0 < e\<close> \<open>0 < e/content ?CBOX\<close>
       have "0 < e'"
         by simp
       interpret operative conj True
            "\<lambda>k. \<forall>a' b' c' d'.
                 cbox (a', c') (b', d') \<subseteq> k \<and> cbox (a', c') (b', d') \<subseteq> ?CBOX
                 \<longrightarrow> norm (integral (cbox (a', c') (b', d')) f -
                          integral (cbox a' b') (\<lambda>x. integral (cbox c' d') (\<lambda>y. f ((x, y)))))
                     \<le> e' * content (cbox (a', c') (b', d'))"
         using assms \<open>0 < e'\<close> by (rule integral_swap_operativeI)
       have "norm (integral ?CBOX f - integral (cbox a b) (\<lambda>x. integral (cbox c d) (\<lambda>y. f (x, y))))
           \<le> e' * content ?CBOX"
         if "\<And>t u v w z. t \<in> p \<Longrightarrow> cbox (u, w) (v, z) \<subseteq> t \<Longrightarrow> cbox (u, w) (v, z) \<subseteq> ?CBOX
           \<Longrightarrow> norm (integral (cbox (u, w) (v, z)) f -
               integral (cbox u v) (\<lambda>x. integral (cbox w z) (\<lambda>y. f (x, y))))
               \<le> e' * content (cbox (u, w) (v, z))"
         using that division [of p "(a, c)" "(b, d)"] p \<open>finite p\<close> by (auto simp add: comm_monoid_set_F_and)
       with False have "norm (integral ?CBOX f - integral (cbox a b) (\<lambda>x. integral (cbox c d) (\<lambda>y. f (x, y))))
           \<le> e"
         if "\<And>t u v w z. t \<in> p \<Longrightarrow> cbox (u, w) (v, z) \<subseteq> t \<Longrightarrow> cbox (u, w) (v, z) \<subseteq> ?CBOX
           \<Longrightarrow> norm (integral (cbox (u, w) (v, z)) f -
               integral (cbox u v) (\<lambda>x. integral (cbox w z) (\<lambda>y. f (x, y))))
               \<le> e * content (cbox (u, w) (v, z)) / content ?CBOX"
         using that by (simp add: e'_def)
     } note op_acbd = this
     { fix k::real and \<D> and u::'a and v w and z::'b and t1 t2 l
       assume k: "0 < k"
          and nf: "\<And>x y u v.
                   \<lbrakk>x \<in> cbox a b; y \<in> cbox c d; u \<in> cbox a b; v\<in>cbox c d; norm (u-x, v-y) < k\<rbrakk>
                   \<Longrightarrow> norm (f(u,v) - f(x,y)) < e/(2 * (content ?CBOX))"
          and p_acbd: "\<D> tagged_division_of cbox (a,c) (b,d)"
          and fine: "(\<lambda>x. ball x k) fine \<D>"  "((t1,t2), l) \<in> \<D>"
          and uwvz_sub: "cbox (u,w) (v,z) \<subseteq> l" "cbox (u,w) (v,z) \<subseteq> cbox (a,c) (b,d)"
       have t: "t1 \<in> cbox a b" "t2 \<in> cbox c d"
         by (meson fine p_acbd cbox_Pair_iff tag_in_interval)+
       have ls: "l \<subseteq> ball (t1,t2) k"
         using fine by (simp add: fine_def Ball_def)
       { fix x1 x2
         assume xuvwz: "x1 \<in> cbox u v" "x2 \<in> cbox w z"
         then have x: "x1 \<in> cbox a b" "x2 \<in> cbox c d"
           using uwvz_sub by auto
         have "norm (x1 - t1, x2 - t2) = norm (t1 - x1, t2 - x2)"
           by (simp add: norm_Pair norm_minus_commute)
         also have "norm (t1 - x1, t2 - x2) < k"
           using xuvwz ls uwvz_sub unfolding ball_def
           by (force simp add: cbox_Pair_eq dist_norm )
         finally have "norm (f (x1,x2) - f (t1,t2)) \<le> e/content ?CBOX/2"
           using nf [OF t x]  by simp
       } note nf' = this
       have f_int_uwvz: "f integrable_on cbox (u,w) (v,z)"
         using f_int_acbd uwvz_sub integrable_on_subcbox by blast
       have f_int_uv: "\<And>x. x \<in> cbox u v \<Longrightarrow> (\<lambda>y. f (x,y)) integrable_on cbox w z"
         using assms continuous_on_subset uwvz_sub
         by (blast intro: continuous_on_imp_integrable_on_Pair1)
       have 1: "norm (integral (cbox (u,w) (v,z)) f - integral (cbox (u,w) (v,z)) (\<lambda>x. f (t1,t2)))
-         \<le> e * content (cbox (u,w) (v,z)) / content ?CBOX/2"
+             \<le> e * content (cbox (u,w) (v,z)) / content ?CBOX/2"
+        using cbp \<open>0 < e/content ?CBOX\<close> nf'
         apply (simp only: integral_diff [symmetric] f_int_uwvz integrable_const)
-        apply (rule order_trans [OF integrable_bound [of "e/content ?CBOX/2"]])
-        using cbp \<open>0 < e/content ?CBOX\<close> nf'
-        apply (auto simp: integrable_diff f_int_uwvz integrable_const)
+        apply (auto simp: integrable_diff f_int_uwvz integrable_const intro: order_trans [OF integrable_bound [of "e/content ?CBOX/2"]])
         done
       have int_integrable: "(\<lambda>x. integral (cbox w z) (\<lambda>y. f (x, y))) integrable_on cbox u v"
         using assms integral_integrable_2dim continuous_on_subset uwvz_sub(2) by blast
       have normint_wz:
          "\<And>x. x \<in> cbox u v \<Longrightarrow>
                norm (integral (cbox w z) (\<lambda>y. f (x, y)) - integral (cbox w z) (\<lambda>y. f (t1, t2)))
                \<le> e * content (cbox w z) / content (cbox (a, c) (b, d))/2"
+        using cbp \<open>0 < e/content ?CBOX\<close> nf'
         apply (simp only: integral_diff [symmetric] f_int_uv integrable_const)
-        apply (rule order_trans [OF integrable_bound [of "e/content ?CBOX/2"]])
-        using cbp \<open>0 < e/content ?CBOX\<close> nf'
-        apply (auto simp: integrable_diff f_int_uv integrable_const)
+        apply (auto simp: integrable_diff f_int_uv integrable_const intro: order_trans [OF integrable_bound [of "e/content ?CBOX/2"]])
         done
       have "norm (integral (cbox u v)
                (\<lambda>x. integral (cbox w z) (\<lambda>y. f (x,y)) - integral (cbox w z) (\<lambda>y. f (t1,t2))))
             \<le> e * content (cbox w z) / content ?CBOX/2 * content (cbox u v)"
-        apply (rule integrable_bound [OF _ _ normint_wz])
         using cbp \<open>0 < e/content ?CBOX\<close>
+        apply (intro integrable_bound [OF _ _ normint_wz])
         apply (auto simp: field_split_simps integrable_diff int_integrable integrable_const)
         done
       also have "... \<le> e * content (cbox (u,w) (v,z)) / content ?CBOX/2"
         by (simp add: content_Pair field_split_simps)
       finally have 2: "norm (integral (cbox u v) (\<lambda>x. integral (cbox w z) (\<lambda>y. f (x,y))) -
                       integral (cbox u v) (\<lambda>x. integral (cbox w z) (\<lambda>y. f (t1,t2))))
                 \<le> e * content (cbox (u,w) (v,z)) / content ?CBOX/2"
         by (simp only: integral_diff [symmetric] int_integrable integrable_const)
       have norm_xx: "\<lbrakk>x' = y'; norm(x - x') \<le> e/2; norm(y - y') \<le> e/2\<rbrakk> \<Longrightarrow> norm(x - y) \<le> e" for x::'c and y x' y' e
         using norm_triangle_mono [of "x-y'" "e/2" "y'-y" "e/2"] field_sum_of_halves
         by (simp add: norm_minus_commute)
       have "norm (integral (cbox (u,w) (v,z)) f - integral (cbox u v) (\<lambda>x. integral (cbox w z) (\<lambda>y. f (x,y))))
             \<le> e * content (cbox (u,w) (v,z)) / content ?CBOX"
         by (rule norm_xx [OF integral_Pair_const 1 2])
     } note * = this
     have "norm (integral ?CBOX f - integral (cbox a b) (\<lambda>x. integral (cbox c d) (\<lambda>y. f (x,y)))) \<le> e" 
       if "\<forall>x\<in>?CBOX. \<forall>x'\<in>?CBOX. norm (x' - x) < k \<longrightarrow> norm (f x' - f x) < e /(2 * content (?CBOX))" "0 < k" for k
     proof -
       obtain p where ptag: "p tagged_division_of cbox (a, c) (b, d)" 
                  and fine: "(\<lambda>x. ball x k) fine p"
         using fine_division_exists \<open>0 < k\<close> by blast
       show ?thesis
-        apply (rule op_acbd [OF division_of_tagged_division [OF ptag]])
-        using that fine ptag \<open>0 < k\<close> by (auto simp: *)
+        using that fine ptag \<open>0 < k\<close> 
+        by (auto simp: * intro: op_acbd [OF division_of_tagged_division [OF ptag]])
     qed
     then show "norm (integral ?CBOX f - integral (cbox a b) (\<lambda>x. integral (cbox c d) (\<lambda>y. f (x,y)))) \<le> e"
       using compact_uniformly_continuous [OF assms compact_cbox]
       apply (simp add: uniformly_continuous_on_def dist_norm)
       apply (drule_tac x="e/2 / content?CBOX" in spec)
       using cbp \<open>0 < e\<close> by (auto simp: zero_less_mult_iff)
   qed
   then show ?thesis
     by simp
 qed
 
 lemma integral_swap_2dim:
   fixes f :: "['a::euclidean_space, 'b::euclidean_space] \<Rightarrow> 'c::banach"
   assumes "continuous_on (cbox (a,c) (b,d)) (\<lambda>(x,y). f x y)"
     shows "integral (cbox (a, c) (b, d)) (\<lambda>(x, y). f x y) = integral (cbox (c, a) (d, b)) (\<lambda>(x, y). f y x)"
 proof -
   have "((\<lambda>(x, y). f x y) has_integral integral (cbox (c, a) (d, b)) (\<lambda>(x, y). f y x)) (prod.swap ` (cbox (c, a) (d, b)))"
-    apply (rule has_integral_twiddle [of 1 prod.swap prod.swap "\<lambda>(x,y). f y x" "integral (cbox (c, a) (d, b)) (\<lambda>(x, y). f y x)", simplified])
-    apply (force simp: isCont_swap content_Pair has_integral_integral [symmetric] integrable_continuous swap_continuous assms)+
-    done
+  proof (rule has_integral_twiddle [of 1 prod.swap prod.swap "\<lambda>(x,y). f y x" "integral (cbox (c, a) (d, b)) (\<lambda>(x, y). f y x)", simplified])
+    show "\<And>u v. content (prod.swap ` cbox u v) = content (cbox u v)"
+      by (metis content_Pair mult.commute old.prod.exhaust swap_cbox_Pair)
+    show "((\<lambda>(x, y). f y x) has_integral integral (cbox (c, a) (d, b)) (\<lambda>(x, y). f y x)) (cbox (c, a) (d, b))"
+      by (simp add: assms integrable_continuous integrable_integral swap_continuous)
+  qed (use isCont_swap in \<open>fastforce+\<close>)
  then show ?thesis
    by force
 qed
 
 theorem integral_swap_continuous:
   fixes f :: "['a::euclidean_space, 'b::euclidean_space] \<Rightarrow> 'c::banach"
   assumes "continuous_on (cbox (a,c) (b,d)) (\<lambda>(x,y). f x y)"
     shows "integral (cbox a b) (\<lambda>x. integral (cbox c d) (f x)) =
            integral (cbox c d) (\<lambda>y. integral (cbox a b) (\<lambda>x. f x y))"
 proof -
   have "integral (cbox a b) (\<lambda>x. integral (cbox c d) (f x)) = integral (cbox (a, c) (b, d)) (\<lambda>(x, y). f x y)"
     using integral_prod_continuous [OF assms] by auto
   also have "... = integral (cbox (c, a) (d, b)) (\<lambda>(x, y). f y x)"
     by (rule integral_swap_2dim [OF assms])
   also have "... = integral (cbox c d) (\<lambda>y. integral (cbox a b) (\<lambda>x. f x y))"
     using integral_prod_continuous [OF swap_continuous] assms
     by auto
   finally show ?thesis .
 qed
 
 
 subsection \<open>Definite integrals for exponential and power function\<close>
 
 lemma has_integral_exp_minus_to_infinity:
   assumes a: "a > 0"
   shows   "((\<lambda>x::real. exp (-a*x)) has_integral exp (-a*c)/a) {c..}"
 proof -
   define f where "f = (\<lambda>k x. if x \<in> {c..real k} then exp (-a*x) else 0)"
   {
     fix k :: nat assume k: "of_nat k \<ge> c"
     from k a
       have "((\<lambda>x. exp (-a*x)) has_integral (-exp (-a*real k)/a - (-exp (-a*c)/a))) {c..real k}"
       by (intro fundamental_theorem_of_calculus)
          (auto intro!: derivative_eq_intros
                simp: has_field_derivative_iff_has_vector_derivative [symmetric])
     hence "(f k has_integral (exp (-a*c)/a - exp (-a*real k)/a)) {c..}" unfolding f_def
       by (subst has_integral_restrict) simp_all
   } note has_integral_f = this
 
   have [simp]: "f k = (\<lambda>_. 0)" if "of_nat k < c" for k using that by (auto simp: fun_eq_iff f_def)
   have integral_f: "integral {c..} (f k) =
                       (if real k \<ge> c then exp (-a*c)/a - exp (-a*real k)/a else 0)"
     for k using integral_unique[OF has_integral_f[of k]] by simp
 
   have A: "(\<lambda>x. exp (-a*x)) integrable_on {c..} \<and>
              (\<lambda>k. integral {c..} (f k)) \<longlonglongrightarrow> integral {c..} (\<lambda>x. exp (-a*x))"
   proof (intro monotone_convergence_increasing allI ballI)
     fix k ::nat
     have "(\<lambda>x. exp (-a*x)) integrable_on {c..of_real k}" (is ?P)
       unfolding f_def by (auto intro!: continuous_intros integrable_continuous_real)
     hence  "(f k) integrable_on {c..of_real k}"
       by (rule integrable_eq) (simp add: f_def)
     then show "f k integrable_on {c..}"
       by (rule integrable_on_superset) (auto simp: f_def)
   next
     fix x assume x: "x \<in> {c..}"
     have "sequentially \<le> principal {nat \<lceil>x\<rceil>..}" unfolding at_top_def by (simp add: Inf_lower)
     also have "{nat \<lceil>x\<rceil>..} \<subseteq> {k. x \<le> real k}" by auto
     also have "inf (principal \<dots>) (principal {k. \<not>x \<le> real k}) =
                  principal ({k. x \<le> real k} \<inter> {k. \<not>x \<le> real k})" by simp
     also have "{k. x \<le> real k} \<inter> {k. \<not>x \<le> real k} = {}" by blast
     finally have "inf sequentially (principal {k. \<not>x \<le> real k}) = bot"
       by (simp add: inf.coboundedI1 bot_unique)
     with x show "(\<lambda>k. f k x) \<longlonglongrightarrow> exp (-a*x)" unfolding f_def
       by (intro filterlim_If) auto
   next
     have "\<bar>integral {c..} (f k)\<bar> \<le> exp (-a*c)/a" for k
     proof (cases "c > of_nat k")
       case False
       hence "abs (integral {c..} (f k)) = abs (exp (- (a * c)) / a - exp (- (a * real k)) / a)"
         by (simp add: integral_f)
       also have "abs (exp (- (a * c)) / a - exp (- (a * real k)) / a) =
                    exp (- (a * c)) / a - exp (- (a * real k)) / a"
         using False a by (intro abs_of_nonneg) (simp_all add: field_simps)
       also have "\<dots> \<le> exp (- a * c) / a" using a by simp
       finally show ?thesis .
     qed (insert a, simp_all add: integral_f)
     thus "bounded (range(\<lambda>k. integral {c..} (f k)))"
       by (intro boundedI[of _ "exp (-a*c)/a"]) auto
   qed (auto simp: f_def)
   have "(\<lambda>k. exp (-a*c)/a - exp (-a * of_nat k)/a) \<longlonglongrightarrow> exp (-a*c)/a - 0/a"
     by (intro tendsto_intros filterlim_compose[OF exp_at_bot]
         filterlim_tendsto_neg_mult_at_bot[OF tendsto_const] filterlim_real_sequentially)+
       (insert a, simp_all)
   moreover
   from eventually_gt_at_top[of "nat \<lceil>c\<rceil>"] have "eventually (\<lambda>k. of_nat k > c) sequentially"
     by eventually_elim linarith
   hence "eventually (\<lambda>k. exp (-a*c)/a - exp (-a * of_nat k)/a = integral {c..} (f k)) sequentially"
     by eventually_elim (simp add: integral_f) 
   ultimately have "(\<lambda>k. integral {c..} (f k)) \<longlonglongrightarrow> exp (-a*c)/a - 0/a"
     by (rule Lim_transform_eventually)
   from LIMSEQ_unique[OF conjunct2[OF A] this]
   have "integral {c..} (\<lambda>x. exp (-a*x)) = exp (-a*c)/a" by simp
   with conjunct1[OF A] show ?thesis
     by (simp add: has_integral_integral)
 qed
 
 lemma integrable_on_exp_minus_to_infinity: "a > 0 \<Longrightarrow> (\<lambda>x. exp (-a*x) :: real) integrable_on {c..}"
   using has_integral_exp_minus_to_infinity[of a c] unfolding integrable_on_def by blast
 
 lemma has_integral_powr_from_0:
   assumes a: "a > (-1::real)" and c: "c \<ge> 0"
   shows   "((\<lambda>x. x powr a) has_integral (c powr (a+1) / (a+1))) {0..c}"
 proof (cases "c = 0")
   case False
   define f where "f = (\<lambda>k x. if x \<in> {inverse (of_nat (Suc k))..c} then x powr a else 0)"
   define F where "F = (\<lambda>k. if inverse (of_nat (Suc k)) \<le> c then
                              c powr (a+1)/(a+1) - inverse (real (Suc k)) powr (a+1)/(a+1) else 0)"
   {
     fix k :: nat
     have "(f k has_integral F k) {0..c}"
     proof (cases "inverse (of_nat (Suc k)) \<le> c")
       case True
       {
         fix x assume x: "x \<ge> inverse (1 + real k)"
         have "0 < inverse (1 + real k)" by simp
         also note x
         finally have "x > 0" .
       } note x = this
       hence "((\<lambda>x. x powr a) has_integral c powr (a + 1) / (a + 1) -
                inverse (real (Suc k)) powr (a + 1) / (a + 1)) {inverse (real (Suc k))..c}"
         using True a by (intro fundamental_theorem_of_calculus)
            (auto intro!: derivative_eq_intros continuous_on_powr' continuous_on_const
               simp: has_field_derivative_iff_has_vector_derivative [symmetric])
       with True show ?thesis unfolding f_def F_def by (subst has_integral_restrict) simp_all
     next
       case False
       thus ?thesis unfolding f_def F_def by (subst has_integral_restrict) auto
     qed
   } note has_integral_f = this
   have integral_f: "integral {0..c} (f k) = F k" for k
     using has_integral_f[of k] by (rule integral_unique)
 
   have A: "(\<lambda>x. x powr a) integrable_on {0..c} \<and>
            (\<lambda>k. integral {0..c} (f k)) \<longlonglongrightarrow> integral {0..c} (\<lambda>x. x powr a)"
   proof (intro monotone_convergence_increasing ballI allI)
     fix k from has_integral_f[of k] show "f k integrable_on {0..c}"
       by (auto simp: integrable_on_def)
   next
     fix k :: nat and x :: real
     {
       assume x: "inverse (real (Suc k)) \<le> x"
       have "inverse (real (Suc (Suc k))) \<le> inverse (real (Suc k))" by (simp add: field_simps)
       also note x
       finally have "inverse (real (Suc (Suc k))) \<le> x" .
     }
     thus "f k x \<le> f (Suc k) x" by (auto simp: f_def simp del: of_nat_Suc)
   next
     fix x assume x: "x \<in> {0..c}"
     show "(\<lambda>k. f k x) \<longlonglongrightarrow> x powr a"
     proof (cases "x = 0")
       case False
       with x have "x > 0" by simp
       from order_tendstoD(2)[OF LIMSEQ_inverse_real_of_nat this]
         have "eventually (\<lambda>k. x powr a = f k x) sequentially"
         by eventually_elim (insert x, simp add: f_def)
       moreover have "(\<lambda>_. x powr a) \<longlonglongrightarrow> x powr a" by simp
       ultimately show ?thesis by (blast intro: Lim_transform_eventually)
     qed (simp_all add: f_def)
   next
     {
       fix k
       from a have "F k \<le> c powr (a + 1) / (a + 1)"
         by (auto simp add: F_def divide_simps)
       also from a have "F k \<ge> 0"
         by (auto simp: F_def divide_simps simp del: of_nat_Suc intro!: powr_mono2)
       hence "F k = abs (F k)" by simp
       finally have "abs (F k) \<le>  c powr (a + 1) / (a + 1)" .
     }
     thus "bounded (range(\<lambda>k. integral {0..c} (f k)))"
       by (intro boundedI[of _ "c powr (a+1) / (a+1)"]) (auto simp: integral_f)
   qed
 
   from False c have "c > 0" by simp
   from order_tendstoD(2)[OF LIMSEQ_inverse_real_of_nat this]
     have "eventually (\<lambda>k. c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a+1) / (a+1) =
             integral {0..c} (f k)) sequentially"
     by eventually_elim (simp add: integral_f F_def)
   moreover have "(\<lambda>k. c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a + 1) / (a + 1))
                    \<longlonglongrightarrow> c powr (a + 1) / (a + 1) - 0 powr (a + 1) / (a + 1)"
     using a by (intro tendsto_intros LIMSEQ_inverse_real_of_nat) auto
   hence "(\<lambda>k. c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a + 1) / (a + 1))
           \<longlonglongrightarrow> c powr (a + 1) / (a + 1)" by simp
   ultimately have "(\<lambda>k. integral {0..c} (f k)) \<longlonglongrightarrow> c powr (a+1) / (a+1)"
     by (blast intro: Lim_transform_eventually)
   with A have "integral {0..c} (\<lambda>x. x powr a) = c powr (a+1) / (a+1)"
     by (blast intro: LIMSEQ_unique)
   with A show ?thesis by (simp add: has_integral_integral)
 qed (simp_all add: has_integral_refl)
 
 lemma integrable_on_powr_from_0:
   assumes a: "a > (-1::real)" and c: "c \<ge> 0"
   shows   "(\<lambda>x. x powr a) integrable_on {0..c}"
   using has_integral_powr_from_0[OF assms] unfolding integrable_on_def by blast
 
 lemma has_integral_powr_to_inf:
   fixes a e :: real
   assumes "e < -1" "a > 0"
   shows   "((\<lambda>x. x powr e) has_integral -(a powr (e + 1)) / (e + 1)) {a..}"
 proof -
   define f :: "nat \<Rightarrow> real \<Rightarrow> real" where "f = (\<lambda>n x. if x \<in> {a..n} then x powr e else 0)"
   define F where "F = (\<lambda>x. x powr (e + 1) / (e + 1))"
 
   have has_integral_f: "(f n has_integral (F n - F a)) {a..}"
     if n: "n \<ge> a" for n :: nat
   proof -
     from n assms have "((\<lambda>x. x powr e) has_integral (F n - F a)) {a..n}"
       by (intro fundamental_theorem_of_calculus) (auto intro!: derivative_eq_intros
             simp: has_field_derivative_iff_has_vector_derivative [symmetric] F_def)
     hence "(f n has_integral (F n - F a)) {a..n}"
       by (rule has_integral_eq [rotated]) (simp add: f_def)
     thus "(f n has_integral (F n - F a)) {a..}"
       by (rule has_integral_on_superset) (auto simp: f_def)
   qed
   have integral_f: "integral {a..} (f n) = (if n \<ge> a then F n - F a else 0)" for n :: nat
   proof (cases "n \<ge> a")
     case True
     with has_integral_f[OF this] show ?thesis by (simp add: integral_unique)
   next
     case False
     have "(f n has_integral 0) {a}" by (rule has_integral_refl)
     hence "(f n has_integral 0) {a..}"
       by (rule has_integral_on_superset) (insert False, simp_all add: f_def)
     with False show ?thesis by (simp add: integral_unique)
   qed
 
   have *: "(\<lambda>x. x powr e) integrable_on {a..} \<and>
            (\<lambda>n. integral {a..} (f n)) \<longlonglongrightarrow> integral {a..} (\<lambda>x. x powr e)"
   proof (intro monotone_convergence_increasing allI ballI)
     fix n :: nat
     from assms have "(\<lambda>x. x powr e) integrable_on {a..n}"
       by (auto intro!: integrable_continuous_real continuous_intros)
     hence "f n integrable_on {a..n}"
       by (rule integrable_eq) (auto simp: f_def)
     thus "f n integrable_on {a..}"
       by (rule integrable_on_superset) (auto simp: f_def)
   next
     fix n :: nat and x :: real
     show "f n x \<le> f (Suc n) x" by (auto simp: f_def)
   next
     fix x :: real assume x: "x \<in> {a..}"
     from filterlim_real_sequentially
       have "eventually (\<lambda>n. real n \<ge> x) at_top"
       by (simp add: filterlim_at_top)
     with x have "eventually (\<lambda>n. f n x = x powr e) at_top"
       by (auto elim!: eventually_mono simp: f_def)
     thus "(\<lambda>n. f n x) \<longlonglongrightarrow> x powr e" by (simp add: tendsto_eventually)
   next
     have "norm (integral {a..} (f n)) \<le> -F a" for n :: nat
     proof (cases "n \<ge> a")
       case True
       with assms have "a powr (e + 1) \<ge> n powr (e + 1)"
         by (intro powr_mono2') simp_all
       with assms show ?thesis by (auto simp: divide_simps F_def integral_f)
     qed (insert assms, simp add: integral_f F_def field_split_simps)
     thus "bounded (range(\<lambda>k. integral {a..} (f k)))"
       unfolding bounded_iff by (intro exI[of _ "-F a"]) auto
   qed
 
   from filterlim_real_sequentially
     have "eventually (\<lambda>n. real n \<ge> a) at_top"
     by (simp add: filterlim_at_top)
   hence "eventually (\<lambda>n. F n - F a = integral {a..} (f n)) at_top"
     by eventually_elim (simp add: integral_f)
   moreover have "(\<lambda>n. F n - F a) \<longlonglongrightarrow> 0 / (e + 1) - F a" unfolding F_def
     by (insert assms, (rule tendsto_intros filterlim_compose[OF tendsto_neg_powr]
           filterlim_ident filterlim_real_sequentially | simp)+)
   hence "(\<lambda>n. F n - F a) \<longlonglongrightarrow> -F a" by simp
   ultimately have "(\<lambda>n. integral {a..} (f n)) \<longlonglongrightarrow> -F a" by (blast intro: Lim_transform_eventually)
   from conjunct2[OF *] and this
     have "integral {a..} (\<lambda>x. x powr e) = -F a" by (rule LIMSEQ_unique)
   with conjunct1[OF *] show ?thesis
     by (simp add: has_integral_integral F_def)
 qed
 
 lemma has_integral_inverse_power_to_inf:
   fixes a :: real and n :: nat
   assumes "n > 1" "a > 0"
   shows   "((\<lambda>x. 1 / x ^ n) has_integral 1 / (real (n - 1) * a ^ (n - 1))) {a..}"
 proof -
   from assms have "real_of_int (-int n) < -1" by simp
   note has_integral_powr_to_inf[OF this \<open>a > 0\<close>]
   also have "- (a powr (real_of_int (- int n) + 1)) / (real_of_int (- int n) + 1) =
                  1 / (real (n - 1) * a powr (real (n - 1)))" using assms
     by (simp add: field_split_simps powr_add [symmetric] of_nat_diff)
   also from assms have "a powr (real (n - 1)) = a ^ (n - 1)"
     by (intro powr_realpow)
   finally show ?thesis
     by (rule has_integral_eq [rotated])
        (insert assms, simp_all add: powr_minus powr_realpow field_split_simps)
 qed
 
 subsubsection \<open>Adaption to ordered Euclidean spaces and the Cartesian Euclidean space\<close>
 
 lemma integral_component_eq_cart[simp]:
   fixes f :: "'n::euclidean_space \<Rightarrow> real^'m"
   assumes "f integrable_on s"
   shows "integral s (\<lambda>x. f x $ k) = integral s f $ k"
   using integral_linear[OF assms(1) bounded_linear_vec_nth,unfolded o_def] .
 
 lemma content_closed_interval:
   fixes a :: "'a::ordered_euclidean_space"
   assumes "a \<le> b"
   shows "content {a..b} = (\<Prod>i\<in>Basis. b\<bullet>i - a\<bullet>i)"
   using content_cbox[of a b] assms by (simp add: cbox_interval eucl_le[where 'a='a])
 
 lemma integrable_const_ivl[intro]:
   fixes a::"'a::ordered_euclidean_space"
   shows "(\<lambda>x. c) integrable_on {a..b}"
   unfolding cbox_interval[symmetric] by (rule integrable_const)
 
 lemma integrable_on_subinterval:
   fixes f :: "'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
   assumes "f integrable_on S" "{a..b} \<subseteq> S"
   shows "f integrable_on {a..b}"
   using integrable_on_subcbox[of f S a b] assms by (simp add: cbox_interval)
 
 end
diff --git a/src/HOL/Deriv.thy b/src/HOL/Deriv.thy
--- a/src/HOL/Deriv.thy
+++ b/src/HOL/Deriv.thy
@@ -1,2370 +1,2368 @@
 (*  Title:      HOL/Deriv.thy
     Author:     Jacques D. Fleuriot, University of Cambridge, 1998
     Author:     Brian Huffman
     Author:     Lawrence C Paulson, 2004
     Author:     Benjamin Porter, 2005
 *)
 
 section \<open>Differentiation\<close>
 
 theory Deriv
   imports Limits
 begin
 
 subsection \<open>Frechet derivative\<close>
 
 definition has_derivative :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow>
     ('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> bool"  (infix "(has'_derivative)" 50)
   where "(f has_derivative f') F \<longleftrightarrow>
     bounded_linear f' \<and>
     ((\<lambda>y. ((f y - f (Lim F (\<lambda>x. x))) - f' (y - Lim F (\<lambda>x. x))) /\<^sub>R norm (y - Lim F (\<lambda>x. x))) \<longlongrightarrow> 0) F"
 
 text \<open>
   Usually the filter \<^term>\<open>F\<close> is \<^term>\<open>at x within s\<close>.  \<^term>\<open>(f has_derivative D)
   (at x within s)\<close> means: \<^term>\<open>D\<close> is the derivative of function \<^term>\<open>f\<close> at point \<^term>\<open>x\<close>
   within the set \<^term>\<open>s\<close>. Where \<^term>\<open>s\<close> is used to express left or right sided derivatives. In
   most cases \<^term>\<open>s\<close> is either a variable or \<^term>\<open>UNIV\<close>.
 \<close>
 
 text \<open>These are the only cases we'll care about, probably.\<close>
 
 lemma has_derivative_within: "(f has_derivative f') (at x within s) \<longleftrightarrow>
     bounded_linear f' \<and> ((\<lambda>y. (1 / norm(y - x)) *\<^sub>R (f y - (f x + f' (y - x)))) \<longlongrightarrow> 0) (at x within s)"
   unfolding has_derivative_def tendsto_iff
   by (subst eventually_Lim_ident_at) (auto simp add: field_simps)
 
 lemma has_derivative_eq_rhs: "(f has_derivative f') F \<Longrightarrow> f' = g' \<Longrightarrow> (f has_derivative g') F"
   by simp
 
 definition has_field_derivative :: "('a::real_normed_field \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a filter \<Rightarrow> bool"
     (infix "(has'_field'_derivative)" 50)
   where "(f has_field_derivative D) F \<longleftrightarrow> (f has_derivative (*) D) F"
 
 lemma DERIV_cong: "(f has_field_derivative X) F \<Longrightarrow> X = Y \<Longrightarrow> (f has_field_derivative Y) F"
   by simp
 
 definition has_vector_derivative :: "(real \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'b \<Rightarrow> real filter \<Rightarrow> bool"
     (infix "has'_vector'_derivative" 50)
   where "(f has_vector_derivative f') net \<longleftrightarrow> (f has_derivative (\<lambda>x. x *\<^sub>R f')) net"
 
 lemma has_vector_derivative_eq_rhs:
   "(f has_vector_derivative X) F \<Longrightarrow> X = Y \<Longrightarrow> (f has_vector_derivative Y) F"
   by simp
 
 named_theorems derivative_intros "structural introduction rules for derivatives"
 setup \<open>
   let
     val eq_thms = @{thms has_derivative_eq_rhs DERIV_cong has_vector_derivative_eq_rhs}
     fun eq_rule thm = get_first (try (fn eq_thm => eq_thm OF [thm])) eq_thms
   in
     Global_Theory.add_thms_dynamic
       (\<^binding>\<open>derivative_eq_intros\<close>,
         fn context =>
           Named_Theorems.get (Context.proof_of context) \<^named_theorems>\<open>derivative_intros\<close>
           |> map_filter eq_rule)
   end
 \<close>
 
 text \<open>
   The following syntax is only used as a legacy syntax.
 \<close>
 abbreviation (input)
   FDERIV :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a \<Rightarrow>  ('a \<Rightarrow> 'b) \<Rightarrow> bool"
   ("(FDERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
   where "FDERIV f x :> f' \<equiv> (f has_derivative f') (at x)"
 
 lemma has_derivative_bounded_linear: "(f has_derivative f') F \<Longrightarrow> bounded_linear f'"
   by (simp add: has_derivative_def)
 
 lemma has_derivative_linear: "(f has_derivative f') F \<Longrightarrow> linear f'"
   using bounded_linear.linear[OF has_derivative_bounded_linear] .
 
 lemma has_derivative_ident[derivative_intros, simp]: "((\<lambda>x. x) has_derivative (\<lambda>x. x)) F"
   by (simp add: has_derivative_def)
 
 lemma has_derivative_id [derivative_intros, simp]: "(id has_derivative id) (at a)"
   by (metis eq_id_iff has_derivative_ident)
 
 lemma has_derivative_const[derivative_intros, simp]: "((\<lambda>x. c) has_derivative (\<lambda>x. 0)) F"
   by (simp add: has_derivative_def)
 
 lemma (in bounded_linear) bounded_linear: "bounded_linear f" ..
 
 lemma (in bounded_linear) has_derivative:
   "(g has_derivative g') F \<Longrightarrow> ((\<lambda>x. f (g x)) has_derivative (\<lambda>x. f (g' x))) F"
   unfolding has_derivative_def
   by (auto simp add: bounded_linear_compose [OF bounded_linear] scaleR diff dest: tendsto)
 
 lemmas has_derivative_scaleR_right [derivative_intros] =
   bounded_linear.has_derivative [OF bounded_linear_scaleR_right]
 
 lemmas has_derivative_scaleR_left [derivative_intros] =
   bounded_linear.has_derivative [OF bounded_linear_scaleR_left]
 
 lemmas has_derivative_mult_right [derivative_intros] =
   bounded_linear.has_derivative [OF bounded_linear_mult_right]
 
 lemmas has_derivative_mult_left [derivative_intros] =
   bounded_linear.has_derivative [OF bounded_linear_mult_left]
 
 lemmas has_derivative_of_real[derivative_intros, simp] = 
   bounded_linear.has_derivative[OF bounded_linear_of_real] 
 
 lemma has_derivative_add[simp, derivative_intros]:
   assumes f: "(f has_derivative f') F"
     and g: "(g has_derivative g') F"
   shows "((\<lambda>x. f x + g x) has_derivative (\<lambda>x. f' x + g' x)) F"
   unfolding has_derivative_def
 proof safe
   let ?x = "Lim F (\<lambda>x. x)"
   let ?D = "\<lambda>f f' y. ((f y - f ?x) - f' (y - ?x)) /\<^sub>R norm (y - ?x)"
   have "((\<lambda>x. ?D f f' x + ?D g g' x) \<longlongrightarrow> (0 + 0)) F"
     using f g by (intro tendsto_add) (auto simp: has_derivative_def)
   then show "(?D (\<lambda>x. f x + g x) (\<lambda>x. f' x + g' x) \<longlongrightarrow> 0) F"
     by (simp add: field_simps scaleR_add_right scaleR_diff_right)
 qed (blast intro: bounded_linear_add f g has_derivative_bounded_linear)
 
 lemma has_derivative_sum[simp, derivative_intros]:
   "(\<And>i. i \<in> I \<Longrightarrow> (f i has_derivative f' i) F) \<Longrightarrow>
     ((\<lambda>x. \<Sum>i\<in>I. f i x) has_derivative (\<lambda>x. \<Sum>i\<in>I. f' i x)) F"
   by (induct I rule: infinite_finite_induct) simp_all
 
 lemma has_derivative_minus[simp, derivative_intros]:
   "(f has_derivative f') F \<Longrightarrow> ((\<lambda>x. - f x) has_derivative (\<lambda>x. - f' x)) F"
   using has_derivative_scaleR_right[of f f' F "-1"] by simp
 
 lemma has_derivative_diff[simp, derivative_intros]:
   "(f has_derivative f') F \<Longrightarrow> (g has_derivative g') F \<Longrightarrow>
     ((\<lambda>x. f x - g x) has_derivative (\<lambda>x. f' x - g' x)) F"
   by (simp only: diff_conv_add_uminus has_derivative_add has_derivative_minus)
 
 lemma has_derivative_at_within:
   "(f has_derivative f') (at x within s) \<longleftrightarrow>
     (bounded_linear f' \<and> ((\<lambda>y. ((f y - f x) - f' (y - x)) /\<^sub>R norm (y - x)) \<longlongrightarrow> 0) (at x within s))"
 proof (cases "at x within s = bot")
   case True
   then show ?thesis
     by (metis (no_types, lifting) has_derivative_within tendsto_bot)
 next
   case False
   then show ?thesis
   by (simp add: Lim_ident_at has_derivative_def)
 qed
 
 lemma has_derivative_iff_norm:
   "(f has_derivative f') (at x within s) \<longleftrightarrow>
     bounded_linear f' \<and> ((\<lambda>y. norm ((f y - f x) - f' (y - x)) / norm (y - x)) \<longlongrightarrow> 0) (at x within s)"
   using tendsto_norm_zero_iff[of _ "at x within s", where 'b="'b", symmetric]
   by (simp add: has_derivative_at_within divide_inverse ac_simps)
 
 lemma has_derivative_at:
   "(f has_derivative D) (at x) \<longleftrightarrow>
     (bounded_linear D \<and> (\<lambda>h. norm (f (x + h) - f x - D h) / norm h) \<midarrow>0\<rightarrow> 0)"
   by (simp add: has_derivative_iff_norm LIM_offset_zero_iff)
 
 lemma field_has_derivative_at:
   fixes x :: "'a::real_normed_field"
   shows "(f has_derivative (*) D) (at x) \<longleftrightarrow> (\<lambda>h. (f (x + h) - f x) / h) \<midarrow>0\<rightarrow> D" (is "?lhs = ?rhs")
 proof -
   have "?lhs = (\<lambda>h. norm (f (x + h) - f x - D * h) / norm h) \<midarrow>0 \<rightarrow> 0"
     by (simp add: bounded_linear_mult_right has_derivative_at)
   also have "... = (\<lambda>y. norm ((f (x + y) - f x - D * y) / y)) \<midarrow>0\<rightarrow> 0"
     by (simp cong: LIM_cong flip: nonzero_norm_divide)
   also have "... = (\<lambda>y. norm ((f (x + y) - f x) / y - D / y * y)) \<midarrow>0\<rightarrow> 0"
     by (simp only: diff_divide_distrib times_divide_eq_left [symmetric])
   also have "... = ?rhs"
     by (simp add: tendsto_norm_zero_iff LIM_zero_iff cong: LIM_cong)
   finally show ?thesis .
 qed
 
 lemma has_derivative_iff_Ex:
   "(f has_derivative f') (at x) \<longleftrightarrow>
     bounded_linear f' \<and> (\<exists>e. (\<forall>h. f (x+h) = f x + f' h + e h) \<and> ((\<lambda>h. norm (e h) / norm h) \<longlongrightarrow> 0) (at 0))"
   unfolding has_derivative_at by force
 
 lemma has_derivative_at_within_iff_Ex:
   assumes "x \<in> S" "open S"
   shows "(f has_derivative f') (at x within S) \<longleftrightarrow>
          bounded_linear f' \<and> (\<exists>e. (\<forall>h. x+h \<in> S \<longrightarrow> f (x+h) = f x + f' h + e h) \<and> ((\<lambda>h. norm (e h) / norm h) \<longlongrightarrow> 0) (at 0))"
     (is "?lhs = ?rhs")
 proof safe
   show "bounded_linear f'"
     if "(f has_derivative f') (at x within S)"
     using has_derivative_bounded_linear that by blast
   show "\<exists>e. (\<forall>h. x + h \<in> S \<longrightarrow> f (x + h) = f x + f' h + e h) \<and> (\<lambda>h. norm (e h) / norm h) \<midarrow>0\<rightarrow> 0"
     if "(f has_derivative f') (at x within S)"
     by (metis (full_types) assms that has_derivative_iff_Ex at_within_open)
   show "(f has_derivative f') (at x within S)"
     if "bounded_linear f'"
       and eq [rule_format]: "\<forall>h. x + h \<in> S \<longrightarrow> f (x + h) = f x + f' h + e h"
       and 0: "(\<lambda>h. norm (e (h::'a)::'b) / norm h) \<midarrow>0\<rightarrow> 0"
     for e 
   proof -
     have 1: "f y - f x = f' (y-x) + e (y-x)" if "y \<in> S" for y
       using eq [of "y-x"] that by simp
     have 2: "((\<lambda>y. norm (e (y-x)) / norm (y - x)) \<longlongrightarrow> 0) (at x within S)"
       by (simp add: "0" assms tendsto_offset_zero_iff)
     have "((\<lambda>y. norm (f y - f x - f' (y - x)) / norm (y - x)) \<longlongrightarrow> 0) (at x within S)"
       by (simp add: Lim_cong_within 1 2)
     then show ?thesis
       by (simp add: has_derivative_iff_norm \<open>bounded_linear f'\<close>)
   qed
 qed
 
 lemma has_derivativeI:
   "bounded_linear f' \<Longrightarrow>
     ((\<lambda>y. ((f y - f x) - f' (y - x)) /\<^sub>R norm (y - x)) \<longlongrightarrow> 0) (at x within s) \<Longrightarrow>
     (f has_derivative f') (at x within s)"
   by (simp add: has_derivative_at_within)
 
 lemma has_derivativeI_sandwich:
   assumes e: "0 < e"
     and bounded: "bounded_linear f'"
     and sandwich: "(\<And>y. y \<in> s \<Longrightarrow> y \<noteq> x \<Longrightarrow> dist y x < e \<Longrightarrow>
       norm ((f y - f x) - f' (y - x)) / norm (y - x) \<le> H y)"
     and "(H \<longlongrightarrow> 0) (at x within s)"
   shows "(f has_derivative f') (at x within s)"
   unfolding has_derivative_iff_norm
 proof safe
   show "((\<lambda>y. norm (f y - f x - f' (y - x)) / norm (y - x)) \<longlongrightarrow> 0) (at x within s)"
   proof (rule tendsto_sandwich[where f="\<lambda>x. 0"])
     show "(H \<longlongrightarrow> 0) (at x within s)" by fact
     show "eventually (\<lambda>n. norm (f n - f x - f' (n - x)) / norm (n - x) \<le> H n) (at x within s)"
       unfolding eventually_at using e sandwich by auto
   qed (auto simp: le_divide_eq)
 qed fact
 
 lemma has_derivative_subset:
   "(f has_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> (f has_derivative f') (at x within t)"
   by (auto simp add: has_derivative_iff_norm intro: tendsto_within_subset)
 
-lemmas has_derivative_within_subset = has_derivative_subset
-
 lemma has_derivative_within_singleton_iff:
   "(f has_derivative g) (at x within {x}) \<longleftrightarrow> bounded_linear g"
   by (auto intro!: has_derivativeI_sandwich[where e=1] has_derivative_bounded_linear)
 
 
 subsubsection \<open>Limit transformation for derivatives\<close>
 
 lemma has_derivative_transform_within:
   assumes "(f has_derivative f') (at x within s)"
     and "0 < d"
     and "x \<in> s"
     and "\<And>x'. \<lbrakk>x' \<in> s; dist x' x < d\<rbrakk> \<Longrightarrow> f x' = g x'"
   shows "(g has_derivative f') (at x within s)"
   using assms
   unfolding has_derivative_within
   by (force simp add: intro: Lim_transform_within)
 
 lemma has_derivative_transform_within_open:
   assumes "(f has_derivative f') (at x within t)"
     and "open s"
     and "x \<in> s"
     and "\<And>x. x\<in>s \<Longrightarrow> f x = g x"
   shows "(g has_derivative f') (at x within t)"
   using assms unfolding has_derivative_within
   by (force simp add: intro: Lim_transform_within_open)
 
 lemma has_derivative_transform:
   assumes "x \<in> s" "\<And>x. x \<in> s \<Longrightarrow> g x = f x"
   assumes "(f has_derivative f') (at x within s)"
   shows "(g has_derivative f') (at x within s)"
   using assms
   by (intro has_derivative_transform_within[OF _ zero_less_one, where g=g]) auto
 
 lemma has_derivative_transform_eventually:
   assumes "(f has_derivative f') (at x within s)"
     "(\<forall>\<^sub>F x' in at x within s. f x' = g x')"
   assumes "f x = g x" "x \<in> s"
   shows "(g has_derivative f') (at x within s)"
   using assms
 proof -
   from assms(2,3) obtain d where "d > 0" "\<And>x'. x' \<in> s \<Longrightarrow> dist x' x < d \<Longrightarrow> f x' = g x'"
     by (force simp: eventually_at)
   from has_derivative_transform_within[OF assms(1) this(1) assms(4) this(2)]
   show ?thesis .
 qed
 
 lemma has_field_derivative_transform_within:
   assumes "(f has_field_derivative f') (at a within S)"
     and "0 < d"
     and "a \<in> S"
     and "\<And>x. \<lbrakk>x \<in> S; dist x a < d\<rbrakk> \<Longrightarrow> f x = g x"
   shows "(g has_field_derivative f') (at a within S)"
   using assms unfolding has_field_derivative_def
   by (metis has_derivative_transform_within)
 
 lemma has_field_derivative_transform_within_open:
   assumes "(f has_field_derivative f') (at a)"
     and "open S" "a \<in> S"
     and "\<And>x. x \<in> S \<Longrightarrow> f x = g x"
   shows "(g has_field_derivative f') (at a)"
   using assms unfolding has_field_derivative_def
   by (metis has_derivative_transform_within_open)
 
 
 subsection \<open>Continuity\<close>
 
 lemma has_derivative_continuous:
   assumes f: "(f has_derivative f') (at x within s)"
   shows "continuous (at x within s) f"
 proof -
   from f interpret F: bounded_linear f'
     by (rule has_derivative_bounded_linear)
   note F.tendsto[tendsto_intros]
   let ?L = "\<lambda>f. (f \<longlongrightarrow> 0) (at x within s)"
   have "?L (\<lambda>y. norm ((f y - f x) - f' (y - x)) / norm (y - x))"
     using f unfolding has_derivative_iff_norm by blast
   then have "?L (\<lambda>y. norm ((f y - f x) - f' (y - x)) / norm (y - x) * norm (y - x))" (is ?m)
     by (rule tendsto_mult_zero) (auto intro!: tendsto_eq_intros)
   also have "?m \<longleftrightarrow> ?L (\<lambda>y. norm ((f y - f x) - f' (y - x)))"
     by (intro filterlim_cong) (simp_all add: eventually_at_filter)
   finally have "?L (\<lambda>y. (f y - f x) - f' (y - x))"
     by (rule tendsto_norm_zero_cancel)
   then have "?L (\<lambda>y. ((f y - f x) - f' (y - x)) + f' (y - x))"
     by (rule tendsto_eq_intros) (auto intro!: tendsto_eq_intros simp: F.zero)
   then have "?L (\<lambda>y. f y - f x)"
     by simp
   from tendsto_add[OF this tendsto_const, of "f x"] show ?thesis
     by (simp add: continuous_within)
 qed
 
 
 subsection \<open>Composition\<close>
 
 lemma tendsto_at_iff_tendsto_nhds_within:
   "f x = y \<Longrightarrow> (f \<longlongrightarrow> y) (at x within s) \<longleftrightarrow> (f \<longlongrightarrow> y) (inf (nhds x) (principal s))"
   unfolding tendsto_def eventually_inf_principal eventually_at_filter
   by (intro ext all_cong imp_cong) (auto elim!: eventually_mono)
 
 lemma has_derivative_in_compose:
   assumes f: "(f has_derivative f') (at x within s)"
     and g: "(g has_derivative g') (at (f x) within (f`s))"
   shows "((\<lambda>x. g (f x)) has_derivative (\<lambda>x. g' (f' x))) (at x within s)"
 proof -
   from f interpret F: bounded_linear f'
     by (rule has_derivative_bounded_linear)
   from g interpret G: bounded_linear g'
     by (rule has_derivative_bounded_linear)
   from F.bounded obtain kF where kF: "\<And>x. norm (f' x) \<le> norm x * kF"
     by fast
   from G.bounded obtain kG where kG: "\<And>x. norm (g' x) \<le> norm x * kG"
     by fast
   note G.tendsto[tendsto_intros]
 
   let ?L = "\<lambda>f. (f \<longlongrightarrow> 0) (at x within s)"
   let ?D = "\<lambda>f f' x y. (f y - f x) - f' (y - x)"
   let ?N = "\<lambda>f f' x y. norm (?D f f' x y) / norm (y - x)"
   let ?gf = "\<lambda>x. g (f x)" and ?gf' = "\<lambda>x. g' (f' x)"
   define Nf where "Nf = ?N f f' x"
   define Ng where [abs_def]: "Ng y = ?N g g' (f x) (f y)" for y
 
   show ?thesis
   proof (rule has_derivativeI_sandwich[of 1])
     show "bounded_linear (\<lambda>x. g' (f' x))"
       using f g by (blast intro: bounded_linear_compose has_derivative_bounded_linear)
   next
     fix y :: 'a
     assume neq: "y \<noteq> x"
     have "?N ?gf ?gf' x y = norm (g' (?D f f' x y) + ?D g g' (f x) (f y)) / norm (y - x)"
       by (simp add: G.diff G.add field_simps)
     also have "\<dots> \<le> norm (g' (?D f f' x y)) / norm (y - x) + Ng y * (norm (f y - f x) / norm (y - x))"
       by (simp add: add_divide_distrib[symmetric] divide_right_mono norm_triangle_ineq G.zero Ng_def)
     also have "\<dots> \<le> Nf y * kG + Ng y * (Nf y + kF)"
     proof (intro add_mono mult_left_mono)
       have "norm (f y - f x) = norm (?D f f' x y + f' (y - x))"
         by simp
       also have "\<dots> \<le> norm (?D f f' x y) + norm (f' (y - x))"
         by (rule norm_triangle_ineq)
       also have "\<dots> \<le> norm (?D f f' x y) + norm (y - x) * kF"
         using kF by (intro add_mono) simp
       finally show "norm (f y - f x) / norm (y - x) \<le> Nf y + kF"
         by (simp add: neq Nf_def field_simps)
     qed (use kG in \<open>simp_all add: Ng_def Nf_def neq zero_le_divide_iff field_simps\<close>)
     finally show "?N ?gf ?gf' x y \<le> Nf y * kG + Ng y * (Nf y + kF)" .
   next
     have [tendsto_intros]: "?L Nf"
       using f unfolding has_derivative_iff_norm Nf_def ..
     from f have "(f \<longlongrightarrow> f x) (at x within s)"
       by (blast intro: has_derivative_continuous continuous_within[THEN iffD1])
     then have f': "LIM x at x within s. f x :> inf (nhds (f x)) (principal (f`s))"
       unfolding filterlim_def
       by (simp add: eventually_filtermap eventually_at_filter le_principal)
 
     have "((?N g  g' (f x)) \<longlongrightarrow> 0) (at (f x) within f`s)"
       using g unfolding has_derivative_iff_norm ..
     then have g': "((?N g  g' (f x)) \<longlongrightarrow> 0) (inf (nhds (f x)) (principal (f`s)))"
       by (rule tendsto_at_iff_tendsto_nhds_within[THEN iffD1, rotated]) simp
 
     have [tendsto_intros]: "?L Ng"
       unfolding Ng_def by (rule filterlim_compose[OF g' f'])
     show "((\<lambda>y. Nf y * kG + Ng y * (Nf y + kF)) \<longlongrightarrow> 0) (at x within s)"
       by (intro tendsto_eq_intros) auto
   qed simp
 qed
 
 lemma has_derivative_compose:
   "(f has_derivative f') (at x within s) \<Longrightarrow> (g has_derivative g') (at (f x)) \<Longrightarrow>
   ((\<lambda>x. g (f x)) has_derivative (\<lambda>x. g' (f' x))) (at x within s)"
   by (blast intro: has_derivative_in_compose has_derivative_subset)
 
 lemma has_derivative_in_compose2:
   assumes "\<And>x. x \<in> t \<Longrightarrow> (g has_derivative g' x) (at x within t)"
   assumes "f ` s \<subseteq> t" "x \<in> s"
   assumes "(f has_derivative f') (at x within s)"
   shows "((\<lambda>x. g (f x)) has_derivative (\<lambda>y. g' (f x) (f' y))) (at x within s)"
   using assms
-  by (auto intro: has_derivative_within_subset intro!: has_derivative_in_compose[of f f' x s g])
+  by (auto intro: has_derivative_subset intro!: has_derivative_in_compose[of f f' x s g])
 
 lemma (in bounded_bilinear) FDERIV:
   assumes f: "(f has_derivative f') (at x within s)" and g: "(g has_derivative g') (at x within s)"
   shows "((\<lambda>x. f x ** g x) has_derivative (\<lambda>h. f x ** g' h + f' h ** g x)) (at x within s)"
 proof -
   from bounded_linear.bounded [OF has_derivative_bounded_linear [OF f]]
   obtain KF where norm_F: "\<And>x. norm (f' x) \<le> norm x * KF" by fast
 
   from pos_bounded obtain K
     where K: "0 < K" and norm_prod: "\<And>a b. norm (a ** b) \<le> norm a * norm b * K"
     by fast
   let ?D = "\<lambda>f f' y. f y - f x - f' (y - x)"
   let ?N = "\<lambda>f f' y. norm (?D f f' y) / norm (y - x)"
   define Ng where "Ng = ?N g g'"
   define Nf where "Nf = ?N f f'"
 
   let ?fun1 = "\<lambda>y. norm (f y ** g y - f x ** g x - (f x ** g' (y - x) + f' (y - x) ** g x)) / norm (y - x)"
   let ?fun2 = "\<lambda>y. norm (f x) * Ng y * K + Nf y * norm (g y) * K + KF * norm (g y - g x) * K"
   let ?F = "at x within s"
 
   show ?thesis
   proof (rule has_derivativeI_sandwich[of 1])
     show "bounded_linear (\<lambda>h. f x ** g' h + f' h ** g x)"
       by (intro bounded_linear_add
         bounded_linear_compose [OF bounded_linear_right] bounded_linear_compose [OF bounded_linear_left]
         has_derivative_bounded_linear [OF g] has_derivative_bounded_linear [OF f])
   next
     from g have "(g \<longlongrightarrow> g x) ?F"
       by (intro continuous_within[THEN iffD1] has_derivative_continuous)
     moreover from f g have "(Nf \<longlongrightarrow> 0) ?F" "(Ng \<longlongrightarrow> 0) ?F"
       by (simp_all add: has_derivative_iff_norm Ng_def Nf_def)
     ultimately have "(?fun2 \<longlongrightarrow> norm (f x) * 0 * K + 0 * norm (g x) * K + KF * norm (0::'b) * K) ?F"
       by (intro tendsto_intros) (simp_all add: LIM_zero_iff)
     then show "(?fun2 \<longlongrightarrow> 0) ?F"
       by simp
   next
     fix y :: 'd
     assume "y \<noteq> x"
     have "?fun1 y =
         norm (f x ** ?D g g' y + ?D f f' y ** g y + f' (y - x) ** (g y - g x)) / norm (y - x)"
       by (simp add: diff_left diff_right add_left add_right field_simps)
     also have "\<dots> \<le> (norm (f x) * norm (?D g g' y) * K + norm (?D f f' y) * norm (g y) * K +
         norm (y - x) * KF * norm (g y - g x) * K) / norm (y - x)"
       by (intro divide_right_mono mult_mono'
                 order_trans [OF norm_triangle_ineq add_mono]
                 order_trans [OF norm_prod mult_right_mono]
                 mult_nonneg_nonneg order_refl norm_ge_zero norm_F
                 K [THEN order_less_imp_le])
     also have "\<dots> = ?fun2 y"
       by (simp add: add_divide_distrib Ng_def Nf_def)
     finally show "?fun1 y \<le> ?fun2 y" .
   qed simp
 qed
 
 lemmas has_derivative_mult[simp, derivative_intros] = bounded_bilinear.FDERIV[OF bounded_bilinear_mult]
 lemmas has_derivative_scaleR[simp, derivative_intros] = bounded_bilinear.FDERIV[OF bounded_bilinear_scaleR]
 
 lemma has_derivative_prod[simp, derivative_intros]:
   fixes f :: "'i \<Rightarrow> 'a::real_normed_vector \<Rightarrow> 'b::real_normed_field"
   shows "(\<And>i. i \<in> I \<Longrightarrow> (f i has_derivative f' i) (at x within S)) \<Longrightarrow>
     ((\<lambda>x. \<Prod>i\<in>I. f i x) has_derivative (\<lambda>y. \<Sum>i\<in>I. f' i y * (\<Prod>j\<in>I - {i}. f j x))) (at x within S)"
 proof (induct I rule: infinite_finite_induct)
   case infinite
   then show ?case by simp
 next
   case empty
   then show ?case by simp
 next
   case (insert i I)
   let ?P = "\<lambda>y. f i x * (\<Sum>i\<in>I. f' i y * (\<Prod>j\<in>I - {i}. f j x)) + (f' i y) * (\<Prod>i\<in>I. f i x)"
   have "((\<lambda>x. f i x * (\<Prod>i\<in>I. f i x)) has_derivative ?P) (at x within S)"
     using insert by (intro has_derivative_mult) auto
   also have "?P = (\<lambda>y. \<Sum>i'\<in>insert i I. f' i' y * (\<Prod>j\<in>insert i I - {i'}. f j x))"
     using insert(1,2)
     by (auto simp add: sum_distrib_left insert_Diff_if intro!: ext sum.cong)
   finally show ?case
     using insert by simp
 qed
 
 lemma has_derivative_power[simp, derivative_intros]:
   fixes f :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
   assumes f: "(f has_derivative f') (at x within S)"
   shows "((\<lambda>x. f x^n) has_derivative (\<lambda>y. of_nat n * f' y * f x^(n - 1))) (at x within S)"
   using has_derivative_prod[OF f, of "{..< n}"] by (simp add: prod_constant ac_simps)
 
 lemma has_derivative_inverse':
   fixes x :: "'a::real_normed_div_algebra"
   assumes x: "x \<noteq> 0"
   shows "(inverse has_derivative (\<lambda>h. - (inverse x * h * inverse x))) (at x within S)"
     (is "(_ has_derivative ?f) _")
 proof (rule has_derivativeI_sandwich)
   show "bounded_linear (\<lambda>h. - (inverse x * h * inverse x))"
     by (simp add: bounded_linear_minus bounded_linear_mult_const bounded_linear_mult_right)
   show "0 < norm x" using x by simp
   have "(inverse \<longlongrightarrow> inverse x) (at x within S)"
     using tendsto_inverse tendsto_ident_at x by auto
   then show "((\<lambda>y. norm (inverse y - inverse x) * norm (inverse x)) \<longlongrightarrow> 0) (at x within S)"
     by (simp add: LIM_zero_iff tendsto_mult_left_zero tendsto_norm_zero)
 next
   fix y :: 'a
   assume h: "y \<noteq> x" "dist y x < norm x"
   then have "y \<noteq> 0" by auto
   have "norm (inverse y - inverse x - ?f (y -x)) / norm (y - x) 
         = norm (- (inverse y * (y - x) * inverse x - inverse x * (y - x) * inverse x)) /
                 norm (y - x)"
     by (simp add: \<open>y \<noteq> 0\<close> inverse_diff_inverse x)
   also have "... = norm ((inverse y - inverse x) * (y - x) * inverse x) / norm (y - x)"
     by (simp add: left_diff_distrib norm_minus_commute)
   also have "\<dots> \<le> norm (inverse y - inverse x) * norm (y - x) * norm (inverse x) / norm (y - x)"
     by (simp add: norm_mult)
   also have "\<dots> = norm (inverse y - inverse x) * norm (inverse x)"
     by simp
   finally show "norm (inverse y - inverse x - ?f (y -x)) / norm (y - x) \<le>
     norm (inverse y - inverse x) * norm (inverse x)" .
 qed
 
 lemma has_derivative_inverse[simp, derivative_intros]:
   fixes f :: "_ \<Rightarrow> 'a::real_normed_div_algebra"
   assumes x:  "f x \<noteq> 0"
     and f: "(f has_derivative f') (at x within S)"
   shows "((\<lambda>x. inverse (f x)) has_derivative (\<lambda>h. - (inverse (f x) * f' h * inverse (f x))))
     (at x within S)"
   using has_derivative_compose[OF f has_derivative_inverse', OF x] .
 
 lemma has_derivative_divide[simp, derivative_intros]:
   fixes f :: "_ \<Rightarrow> 'a::real_normed_div_algebra"
   assumes f: "(f has_derivative f') (at x within S)"
     and g: "(g has_derivative g') (at x within S)"
   assumes x: "g x \<noteq> 0"
   shows "((\<lambda>x. f x / g x) has_derivative
                 (\<lambda>h. - f x * (inverse (g x) * g' h * inverse (g x)) + f' h / g x)) (at x within S)"
   using has_derivative_mult[OF f has_derivative_inverse[OF x g]]
   by (simp add: field_simps)
 
 lemma has_derivative_power_int':
   fixes x :: "'a::real_normed_field"
   assumes x: "x \<noteq> 0"
   shows "((\<lambda>x. power_int x n) has_derivative (\<lambda>y. y * (of_int n * power_int x (n - 1)))) (at x within S)"
 proof (cases n rule: int_cases4)
   case (nonneg n)
   thus ?thesis using x
     by (cases "n = 0") (auto intro!: derivative_eq_intros simp: field_simps power_int_diff fun_eq_iff
                              simp flip: power_Suc)
 next
   case (neg n)
   thus ?thesis using x
     by (auto intro!: derivative_eq_intros simp: field_simps power_int_diff power_int_minus
              simp flip: power_Suc power_Suc2 power_add)
 qed
 
 lemma has_derivative_power_int[simp, derivative_intros]:
   fixes f :: "_ \<Rightarrow> 'a::real_normed_field"
   assumes x:  "f x \<noteq> 0"
     and f: "(f has_derivative f') (at x within S)"
   shows "((\<lambda>x. power_int (f x) n) has_derivative (\<lambda>h. f' h * (of_int n * power_int (f x) (n - 1))))
            (at x within S)"
   using has_derivative_compose[OF f has_derivative_power_int', OF x] .
 
 
 text \<open>Conventional form requires mult-AC laws. Types real and complex only.\<close>
 
 lemma has_derivative_divide'[derivative_intros]:
   fixes f :: "_ \<Rightarrow> 'a::real_normed_field"
   assumes f: "(f has_derivative f') (at x within S)"
     and g: "(g has_derivative g') (at x within S)"
     and x: "g x \<noteq> 0"
   shows "((\<lambda>x. f x / g x) has_derivative (\<lambda>h. (f' h * g x - f x * g' h) / (g x * g x))) (at x within S)"
 proof -
   have "f' h / g x - f x * (inverse (g x) * g' h * inverse (g x)) =
       (f' h * g x - f x * g' h) / (g x * g x)" for h
     by (simp add: field_simps x)
   then show ?thesis
     using has_derivative_divide [OF f g] x
     by simp
 qed
 
 
 subsection \<open>Uniqueness\<close>
 
 text \<open>
 This can not generally shown for \<^const>\<open>has_derivative\<close>, as we need to approach the point from
 all directions. There is a proof in \<open>Analysis\<close> for \<open>euclidean_space\<close>.
 \<close>
 
 lemma has_derivative_at2: "(f has_derivative f') (at x) \<longleftrightarrow>
     bounded_linear f' \<and> ((\<lambda>y. (1 / (norm(y - x))) *\<^sub>R (f y - (f x + f' (y - x)))) \<longlongrightarrow> 0) (at x)"
   using has_derivative_within [of f f' x UNIV]
   by simp
 
 lemma has_derivative_zero_unique:
   assumes "((\<lambda>x. 0) has_derivative F) (at x)"
   shows "F = (\<lambda>h. 0)"
 proof -
   interpret F: bounded_linear F
     using assms by (rule has_derivative_bounded_linear)
   let ?r = "\<lambda>h. norm (F h) / norm h"
   have *: "?r \<midarrow>0\<rightarrow> 0"
     using assms unfolding has_derivative_at by simp
   show "F = (\<lambda>h. 0)"
   proof
     show "F h = 0" for h
     proof (rule ccontr)
       assume **: "\<not> ?thesis"
       then have h: "h \<noteq> 0"
         by (auto simp add: F.zero)
       with ** have "0 < ?r h"
         by simp
       from LIM_D [OF * this] obtain S
         where S: "0 < S" and r: "\<And>x. x \<noteq> 0 \<Longrightarrow> norm x < S \<Longrightarrow> ?r x < ?r h"
         by auto
       from dense [OF S] obtain t where t: "0 < t \<and> t < S" ..
       let ?x = "scaleR (t / norm h) h"
       have "?x \<noteq> 0" and "norm ?x < S"
         using t h by simp_all
       then have "?r ?x < ?r h"
         by (rule r)
       then show False
         using t h by (simp add: F.scaleR)
     qed
   qed
 qed
 
 lemma has_derivative_unique:
   assumes "(f has_derivative F) (at x)"
     and "(f has_derivative F') (at x)"
   shows "F = F'"
 proof -
   have "((\<lambda>x. 0) has_derivative (\<lambda>h. F h - F' h)) (at x)"
     using has_derivative_diff [OF assms] by simp
   then have "(\<lambda>h. F h - F' h) = (\<lambda>h. 0)"
     by (rule has_derivative_zero_unique)
   then show "F = F'"
     unfolding fun_eq_iff right_minus_eq .
 qed
 
 lemma has_derivative_Uniq: "\<exists>\<^sub>\<le>\<^sub>1F. (f has_derivative F) (at x)"
   by (simp add: Uniq_def has_derivative_unique)
 
 
 subsection \<open>Differentiability predicate\<close>
 
 definition differentiable :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
     (infix "differentiable" 50)
   where "f differentiable F \<longleftrightarrow> (\<exists>D. (f has_derivative D) F)"
 
 lemma differentiable_subset:
   "f differentiable (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f differentiable (at x within t)"
   unfolding differentiable_def by (blast intro: has_derivative_subset)
 
 lemmas differentiable_within_subset = differentiable_subset
 
 lemma differentiable_ident [simp, derivative_intros]: "(\<lambda>x. x) differentiable F"
   unfolding differentiable_def by (blast intro: has_derivative_ident)
 
 lemma differentiable_const [simp, derivative_intros]: "(\<lambda>z. a) differentiable F"
   unfolding differentiable_def by (blast intro: has_derivative_const)
 
 lemma differentiable_in_compose:
   "f differentiable (at (g x) within (g`s)) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow>
     (\<lambda>x. f (g x)) differentiable (at x within s)"
   unfolding differentiable_def by (blast intro: has_derivative_in_compose)
 
 lemma differentiable_compose:
   "f differentiable (at (g x)) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow>
     (\<lambda>x. f (g x)) differentiable (at x within s)"
   by (blast intro: differentiable_in_compose differentiable_subset)
 
 lemma differentiable_add [simp, derivative_intros]:
   "f differentiable F \<Longrightarrow> g differentiable F \<Longrightarrow> (\<lambda>x. f x + g x) differentiable F"
   unfolding differentiable_def by (blast intro: has_derivative_add)
 
 lemma differentiable_sum[simp, derivative_intros]:
   assumes "finite s" "\<forall>a\<in>s. (f a) differentiable net"
   shows "(\<lambda>x. sum (\<lambda>a. f a x) s) differentiable net"
 proof -
   from bchoice[OF assms(2)[unfolded differentiable_def]]
   show ?thesis
     by (auto intro!: has_derivative_sum simp: differentiable_def)
 qed
 
 lemma differentiable_minus [simp, derivative_intros]:
   "f differentiable F \<Longrightarrow> (\<lambda>x. - f x) differentiable F"
   unfolding differentiable_def by (blast intro: has_derivative_minus)
 
 lemma differentiable_diff [simp, derivative_intros]:
   "f differentiable F \<Longrightarrow> g differentiable F \<Longrightarrow> (\<lambda>x. f x - g x) differentiable F"
   unfolding differentiable_def by (blast intro: has_derivative_diff)
 
 lemma differentiable_mult [simp, derivative_intros]:
   fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
   shows "f differentiable (at x within s) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow>
     (\<lambda>x. f x * g x) differentiable (at x within s)"
   unfolding differentiable_def by (blast intro: has_derivative_mult)
 
 lemma differentiable_inverse [simp, derivative_intros]:
   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_field"
   shows "f differentiable (at x within s) \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow>
     (\<lambda>x. inverse (f x)) differentiable (at x within s)"
   unfolding differentiable_def by (blast intro: has_derivative_inverse)
 
 lemma differentiable_divide [simp, derivative_intros]:
   fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_field"
   shows "f differentiable (at x within s) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow>
     g x \<noteq> 0 \<Longrightarrow> (\<lambda>x. f x / g x) differentiable (at x within s)"
   unfolding divide_inverse by simp
 
 lemma differentiable_power [simp, derivative_intros]:
   fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_field"
   shows "f differentiable (at x within s) \<Longrightarrow> (\<lambda>x. f x ^ n) differentiable (at x within s)"
   unfolding differentiable_def by (blast intro: has_derivative_power)
 
 lemma differentiable_power_int [simp, derivative_intros]:
   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_field"
   shows "f differentiable (at x within s) \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow>
            (\<lambda>x. power_int (f x) n) differentiable (at x within s)"
   unfolding differentiable_def by (blast intro: has_derivative_power_int)
 
 lemma differentiable_scaleR [simp, derivative_intros]:
   "f differentiable (at x within s) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow>
     (\<lambda>x. f x *\<^sub>R g x) differentiable (at x within s)"
   unfolding differentiable_def by (blast intro: has_derivative_scaleR)
 
 lemma has_derivative_imp_has_field_derivative:
   "(f has_derivative D) F \<Longrightarrow> (\<And>x. x * D' = D x) \<Longrightarrow> (f has_field_derivative D') F"
   unfolding has_field_derivative_def
   by (rule has_derivative_eq_rhs[of f D]) (simp_all add: fun_eq_iff mult.commute)
 
 lemma has_field_derivative_imp_has_derivative:
   "(f has_field_derivative D) F \<Longrightarrow> (f has_derivative (*) D) F"
   by (simp add: has_field_derivative_def)
 
 lemma DERIV_subset:
   "(f has_field_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow>
     (f has_field_derivative f') (at x within t)"
-  by (simp add: has_field_derivative_def has_derivative_within_subset)
+  by (simp add: has_field_derivative_def has_derivative_subset)
 
 lemma has_field_derivative_at_within:
   "(f has_field_derivative f') (at x) \<Longrightarrow> (f has_field_derivative f') (at x within s)"
   using DERIV_subset by blast
 
 abbreviation (input)
   DERIV :: "('a::real_normed_field \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
     ("(DERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
   where "DERIV f x :> D \<equiv> (f has_field_derivative D) (at x)"
 
 abbreviation has_real_derivative :: "(real \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> real filter \<Rightarrow> bool"
     (infix "(has'_real'_derivative)" 50)
   where "(f has_real_derivative D) F \<equiv> (f has_field_derivative D) F"
 
 lemma real_differentiable_def:
   "f differentiable at x within s \<longleftrightarrow> (\<exists>D. (f has_real_derivative D) (at x within s))"
 proof safe
   assume "f differentiable at x within s"
   then obtain f' where *: "(f has_derivative f') (at x within s)"
     unfolding differentiable_def by auto
   then obtain c where "f' = ((*) c)"
     by (metis real_bounded_linear has_derivative_bounded_linear mult.commute fun_eq_iff)
   with * show "\<exists>D. (f has_real_derivative D) (at x within s)"
     unfolding has_field_derivative_def by auto
 qed (auto simp: differentiable_def has_field_derivative_def)
 
 lemma real_differentiableE [elim?]:
   assumes f: "f differentiable (at x within s)"
   obtains df where "(f has_real_derivative df) (at x within s)"
   using assms by (auto simp: real_differentiable_def)
 
 lemma has_field_derivative_iff:
   "(f has_field_derivative D) (at x within S) \<longleftrightarrow>
     ((\<lambda>y. (f y - f x) / (y - x)) \<longlongrightarrow> D) (at x within S)"
 proof -
   have "((\<lambda>y. norm (f y - f x - D * (y - x)) / norm (y - x)) \<longlongrightarrow> 0) (at x within S) 
       = ((\<lambda>y. (f y - f x) / (y - x) - D) \<longlongrightarrow> 0) (at x within S)"
     apply (subst tendsto_norm_zero_iff[symmetric], rule filterlim_cong)
       apply (simp_all add: eventually_at_filter field_simps nonzero_norm_divide)
     done
   then show ?thesis
     by (simp add: has_field_derivative_def has_derivative_iff_norm bounded_linear_mult_right LIM_zero_iff)
 qed
 
 lemma DERIV_def: "DERIV f x :> D \<longleftrightarrow> (\<lambda>h. (f (x + h) - f x) / h) \<midarrow>0\<rightarrow> D"
   unfolding field_has_derivative_at has_field_derivative_def has_field_derivative_iff ..
 
 lemma mult_commute_abs: "(\<lambda>x. x * c) = (*) c"
   for c :: "'a::ab_semigroup_mult"
   by (simp add: fun_eq_iff mult.commute)
 
 lemma DERIV_compose_FDERIV:
   fixes f::"real\<Rightarrow>real"
   assumes "DERIV f (g x) :> f'"
   assumes "(g has_derivative g') (at x within s)"
   shows "((\<lambda>x. f (g x)) has_derivative (\<lambda>x. g' x * f')) (at x within s)"
   using assms has_derivative_compose[of g g' x s f "(*) f'"]
   by (auto simp: has_field_derivative_def ac_simps)
 
 
 subsection \<open>Vector derivative\<close>
 
 lemma has_field_derivative_iff_has_vector_derivative:
   "(f has_field_derivative y) F \<longleftrightarrow> (f has_vector_derivative y) F"
   unfolding has_vector_derivative_def has_field_derivative_def real_scaleR_def mult_commute_abs ..
 
 lemma has_field_derivative_subset:
   "(f has_field_derivative y) (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow>
     (f has_field_derivative y) (at x within t)"
   unfolding has_field_derivative_def by (rule has_derivative_subset)
 
 lemma has_vector_derivative_const[simp, derivative_intros]: "((\<lambda>x. c) has_vector_derivative 0) net"
   by (auto simp: has_vector_derivative_def)
 
 lemma has_vector_derivative_id[simp, derivative_intros]: "((\<lambda>x. x) has_vector_derivative 1) net"
   by (auto simp: has_vector_derivative_def)
 
 lemma has_vector_derivative_minus[derivative_intros]:
   "(f has_vector_derivative f') net \<Longrightarrow> ((\<lambda>x. - f x) has_vector_derivative (- f')) net"
   by (auto simp: has_vector_derivative_def)
 
 lemma has_vector_derivative_add[derivative_intros]:
   "(f has_vector_derivative f') net \<Longrightarrow> (g has_vector_derivative g') net \<Longrightarrow>
     ((\<lambda>x. f x + g x) has_vector_derivative (f' + g')) net"
   by (auto simp: has_vector_derivative_def scaleR_right_distrib)
 
 lemma has_vector_derivative_sum[derivative_intros]:
   "(\<And>i. i \<in> I \<Longrightarrow> (f i has_vector_derivative f' i) net) \<Longrightarrow>
     ((\<lambda>x. \<Sum>i\<in>I. f i x) has_vector_derivative (\<Sum>i\<in>I. f' i)) net"
   by (auto simp: has_vector_derivative_def fun_eq_iff scaleR_sum_right intro!: derivative_eq_intros)
 
 lemma has_vector_derivative_diff[derivative_intros]:
   "(f has_vector_derivative f') net \<Longrightarrow> (g has_vector_derivative g') net \<Longrightarrow>
     ((\<lambda>x. f x - g x) has_vector_derivative (f' - g')) net"
   by (auto simp: has_vector_derivative_def scaleR_diff_right)
 
 lemma has_vector_derivative_add_const:
   "((\<lambda>t. g t + z) has_vector_derivative f') net = ((\<lambda>t. g t) has_vector_derivative f') net"
   apply (intro iffI)
    apply (force dest: has_vector_derivative_diff [where g = "\<lambda>t. z", OF _ has_vector_derivative_const])
   apply (force dest: has_vector_derivative_add [OF _ has_vector_derivative_const])
   done
 
 lemma has_vector_derivative_diff_const:
   "((\<lambda>t. g t - z) has_vector_derivative f') net = ((\<lambda>t. g t) has_vector_derivative f') net"
   using has_vector_derivative_add_const [where z = "-z"]
   by simp
 
 lemma (in bounded_linear) has_vector_derivative:
   assumes "(g has_vector_derivative g') F"
   shows "((\<lambda>x. f (g x)) has_vector_derivative f g') F"
   using has_derivative[OF assms[unfolded has_vector_derivative_def]]
   by (simp add: has_vector_derivative_def scaleR)
 
 lemma (in bounded_bilinear) has_vector_derivative:
   assumes "(f has_vector_derivative f') (at x within s)"
     and "(g has_vector_derivative g') (at x within s)"
   shows "((\<lambda>x. f x ** g x) has_vector_derivative (f x ** g' + f' ** g x)) (at x within s)"
   using FDERIV[OF assms(1-2)[unfolded has_vector_derivative_def]]
   by (simp add: has_vector_derivative_def scaleR_right scaleR_left scaleR_right_distrib)
 
 lemma has_vector_derivative_scaleR[derivative_intros]:
   "(f has_field_derivative f') (at x within s) \<Longrightarrow> (g has_vector_derivative g') (at x within s) \<Longrightarrow>
     ((\<lambda>x. f x *\<^sub>R g x) has_vector_derivative (f x *\<^sub>R g' + f' *\<^sub>R g x)) (at x within s)"
   unfolding has_field_derivative_iff_has_vector_derivative
   by (rule bounded_bilinear.has_vector_derivative[OF bounded_bilinear_scaleR])
 
 lemma has_vector_derivative_mult[derivative_intros]:
   "(f has_vector_derivative f') (at x within s) \<Longrightarrow> (g has_vector_derivative g') (at x within s) \<Longrightarrow>
     ((\<lambda>x. f x * g x) has_vector_derivative (f x * g' + f' * g x)) (at x within s)"
   for f g :: "real \<Rightarrow> 'a::real_normed_algebra"
   by (rule bounded_bilinear.has_vector_derivative[OF bounded_bilinear_mult])
 
 lemma has_vector_derivative_of_real[derivative_intros]:
   "(f has_field_derivative D) F \<Longrightarrow> ((\<lambda>x. of_real (f x)) has_vector_derivative (of_real D)) F"
   by (rule bounded_linear.has_vector_derivative[OF bounded_linear_of_real])
     (simp add: has_field_derivative_iff_has_vector_derivative)
 
 lemma has_vector_derivative_real_field:
   "(f has_field_derivative f') (at (of_real a)) \<Longrightarrow> ((\<lambda>x. f (of_real x)) has_vector_derivative f') (at a within s)"
   using has_derivative_compose[of of_real of_real a _ f "(*) f'"] 
   by (simp add: scaleR_conv_of_real ac_simps has_vector_derivative_def has_field_derivative_def)
 
 lemma has_vector_derivative_continuous:
   "(f has_vector_derivative D) (at x within s) \<Longrightarrow> continuous (at x within s) f"
   by (auto intro: has_derivative_continuous simp: has_vector_derivative_def)
 
 lemma continuous_on_vector_derivative:
   "(\<And>x. x \<in> S \<Longrightarrow> (f has_vector_derivative f' x) (at x within S)) \<Longrightarrow> continuous_on S f"
   by (auto simp: continuous_on_eq_continuous_within intro!: has_vector_derivative_continuous)
 
 lemma has_vector_derivative_mult_right[derivative_intros]:
   fixes a :: "'a::real_normed_algebra"
   shows "(f has_vector_derivative x) F \<Longrightarrow> ((\<lambda>x. a * f x) has_vector_derivative (a * x)) F"
   by (rule bounded_linear.has_vector_derivative[OF bounded_linear_mult_right])
 
 lemma has_vector_derivative_mult_left[derivative_intros]:
   fixes a :: "'a::real_normed_algebra"
   shows "(f has_vector_derivative x) F \<Longrightarrow> ((\<lambda>x. f x * a) has_vector_derivative (x * a)) F"
   by (rule bounded_linear.has_vector_derivative[OF bounded_linear_mult_left])
 
 
 subsection \<open>Derivatives\<close>
 
 lemma DERIV_D: "DERIV f x :> D \<Longrightarrow> (\<lambda>h. (f (x + h) - f x) / h) \<midarrow>0\<rightarrow> D"
   by (simp add: DERIV_def)
 
 lemma has_field_derivativeD:
   "(f has_field_derivative D) (at x within S) \<Longrightarrow>
     ((\<lambda>y. (f y - f x) / (y - x)) \<longlongrightarrow> D) (at x within S)"
   by (simp add: has_field_derivative_iff)
 
 lemma DERIV_const [simp, derivative_intros]: "((\<lambda>x. k) has_field_derivative 0) F"
   by (rule has_derivative_imp_has_field_derivative[OF has_derivative_const]) auto
 
 lemma DERIV_ident [simp, derivative_intros]: "((\<lambda>x. x) has_field_derivative 1) F"
   by (rule has_derivative_imp_has_field_derivative[OF has_derivative_ident]) auto
 
 lemma field_differentiable_add[derivative_intros]:
   "(f has_field_derivative f') F \<Longrightarrow> (g has_field_derivative g') F \<Longrightarrow>
     ((\<lambda>z. f z + g z) has_field_derivative f' + g') F"
   by (rule has_derivative_imp_has_field_derivative[OF has_derivative_add])
      (auto simp: has_field_derivative_def field_simps mult_commute_abs)
 
 corollary DERIV_add:
   "(f has_field_derivative D) (at x within s) \<Longrightarrow> (g has_field_derivative E) (at x within s) \<Longrightarrow>
     ((\<lambda>x. f x + g x) has_field_derivative D + E) (at x within s)"
   by (rule field_differentiable_add)
 
 lemma field_differentiable_minus[derivative_intros]:
   "(f has_field_derivative f') F \<Longrightarrow> ((\<lambda>z. - (f z)) has_field_derivative -f') F"
   by (rule has_derivative_imp_has_field_derivative[OF has_derivative_minus])
      (auto simp: has_field_derivative_def field_simps mult_commute_abs)
 
 corollary DERIV_minus:
   "(f has_field_derivative D) (at x within s) \<Longrightarrow>
     ((\<lambda>x. - f x) has_field_derivative -D) (at x within s)"
   by (rule field_differentiable_minus)
 
 lemma field_differentiable_diff[derivative_intros]:
   "(f has_field_derivative f') F \<Longrightarrow>
     (g has_field_derivative g') F \<Longrightarrow> ((\<lambda>z. f z - g z) has_field_derivative f' - g') F"
   by (simp only: diff_conv_add_uminus field_differentiable_add field_differentiable_minus)
 
 corollary DERIV_diff:
   "(f has_field_derivative D) (at x within s) \<Longrightarrow>
     (g has_field_derivative E) (at x within s) \<Longrightarrow>
     ((\<lambda>x. f x - g x) has_field_derivative D - E) (at x within s)"
   by (rule field_differentiable_diff)
 
 lemma DERIV_continuous: "(f has_field_derivative D) (at x within s) \<Longrightarrow> continuous (at x within s) f"
   by (drule has_derivative_continuous[OF has_field_derivative_imp_has_derivative]) simp
 
 corollary DERIV_isCont: "DERIV f x :> D \<Longrightarrow> isCont f x"
   by (rule DERIV_continuous)
 
 lemma DERIV_atLeastAtMost_imp_continuous_on:
   assumes "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> \<exists>y. DERIV f x :> y"
   shows "continuous_on {a..b} f"
   by (meson DERIV_isCont assms atLeastAtMost_iff continuous_at_imp_continuous_at_within continuous_on_eq_continuous_within)
 
 lemma DERIV_continuous_on:
   "(\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative (D x)) (at x within s)) \<Longrightarrow> continuous_on s f"
   unfolding continuous_on_eq_continuous_within
   by (intro continuous_at_imp_continuous_on ballI DERIV_continuous)
 
 lemma DERIV_mult':
   "(f has_field_derivative D) (at x within s) \<Longrightarrow> (g has_field_derivative E) (at x within s) \<Longrightarrow>
     ((\<lambda>x. f x * g x) has_field_derivative f x * E + D * g x) (at x within s)"
   by (rule has_derivative_imp_has_field_derivative[OF has_derivative_mult])
      (auto simp: field_simps mult_commute_abs dest: has_field_derivative_imp_has_derivative)
 
 lemma DERIV_mult[derivative_intros]:
   "(f has_field_derivative Da) (at x within s) \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow>
     ((\<lambda>x. f x * g x) has_field_derivative Da * g x + Db * f x) (at x within s)"
   by (rule has_derivative_imp_has_field_derivative[OF has_derivative_mult])
      (auto simp: field_simps dest: has_field_derivative_imp_has_derivative)
 
 text \<open>Derivative of linear multiplication\<close>
 
 lemma DERIV_cmult:
   "(f has_field_derivative D) (at x within s) \<Longrightarrow>
     ((\<lambda>x. c * f x) has_field_derivative c * D) (at x within s)"
   by (drule DERIV_mult' [OF DERIV_const]) simp
 
 lemma DERIV_cmult_right:
   "(f has_field_derivative D) (at x within s) \<Longrightarrow>
     ((\<lambda>x. f x * c) has_field_derivative D * c) (at x within s)"
   using DERIV_cmult by (auto simp add: ac_simps)
 
 lemma DERIV_cmult_Id [simp]: "((*) c has_field_derivative c) (at x within s)"
   using DERIV_ident [THEN DERIV_cmult, where c = c and x = x] by simp
 
 lemma DERIV_cdivide:
   "(f has_field_derivative D) (at x within s) \<Longrightarrow>
     ((\<lambda>x. f x / c) has_field_derivative D / c) (at x within s)"
   using DERIV_cmult_right[of f D x s "1 / c"] by simp
 
 lemma DERIV_unique: "DERIV f x :> D \<Longrightarrow> DERIV f x :> E \<Longrightarrow> D = E"
   unfolding DERIV_def by (rule LIM_unique)
 
 lemma DERIV_Uniq: "\<exists>\<^sub>\<le>\<^sub>1D. DERIV f x :> D"
   by (simp add: DERIV_unique Uniq_def)
 
 lemma DERIV_sum[derivative_intros]:
   "(\<And> n. n \<in> S \<Longrightarrow> ((\<lambda>x. f x n) has_field_derivative (f' x n)) F) \<Longrightarrow>
     ((\<lambda>x. sum (f x) S) has_field_derivative sum (f' x) S) F"
   by (rule has_derivative_imp_has_field_derivative [OF has_derivative_sum])
      (auto simp: sum_distrib_left mult_commute_abs dest: has_field_derivative_imp_has_derivative)
 
 lemma DERIV_inverse'[derivative_intros]:
   assumes "(f has_field_derivative D) (at x within s)"
     and "f x \<noteq> 0"
   shows "((\<lambda>x. inverse (f x)) has_field_derivative - (inverse (f x) * D * inverse (f x)))
     (at x within s)"
 proof -
   have "(f has_derivative (\<lambda>x. x * D)) = (f has_derivative (*) D)"
     by (rule arg_cong [of "\<lambda>x. x * D"]) (simp add: fun_eq_iff)
   with assms have "(f has_derivative (\<lambda>x. x * D)) (at x within s)"
     by (auto dest!: has_field_derivative_imp_has_derivative)
   then show ?thesis using \<open>f x \<noteq> 0\<close>
     by (auto intro: has_derivative_imp_has_field_derivative has_derivative_inverse)
 qed
 
 text \<open>Power of \<open>-1\<close>\<close>
 
 lemma DERIV_inverse:
   "x \<noteq> 0 \<Longrightarrow> ((\<lambda>x. inverse(x)) has_field_derivative - (inverse x ^ Suc (Suc 0))) (at x within s)"
   by (drule DERIV_inverse' [OF DERIV_ident]) simp
 
 text \<open>Derivative of inverse\<close>
 
 lemma DERIV_inverse_fun:
   "(f has_field_derivative d) (at x within s) \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow>
     ((\<lambda>x. inverse (f x)) has_field_derivative (- (d * inverse(f x ^ Suc (Suc 0))))) (at x within s)"
   by (drule (1) DERIV_inverse') (simp add: ac_simps nonzero_inverse_mult_distrib)
 
 text \<open>Derivative of quotient\<close>
 
 lemma DERIV_divide[derivative_intros]:
   "(f has_field_derivative D) (at x within s) \<Longrightarrow>
     (g has_field_derivative E) (at x within s) \<Longrightarrow> g x \<noteq> 0 \<Longrightarrow>
     ((\<lambda>x. f x / g x) has_field_derivative (D * g x - f x * E) / (g x * g x)) (at x within s)"
   by (rule has_derivative_imp_has_field_derivative[OF has_derivative_divide])
      (auto dest: has_field_derivative_imp_has_derivative simp: field_simps)
 
 lemma DERIV_quotient:
   "(f has_field_derivative d) (at x within s) \<Longrightarrow>
     (g has_field_derivative e) (at x within s)\<Longrightarrow> g x \<noteq> 0 \<Longrightarrow>
     ((\<lambda>y. f y / g y) has_field_derivative (d * g x - (e * f x)) / (g x ^ Suc (Suc 0))) (at x within s)"
   by (drule (2) DERIV_divide) (simp add: mult.commute)
 
 lemma DERIV_power_Suc:
   "(f has_field_derivative D) (at x within s) \<Longrightarrow>
     ((\<lambda>x. f x ^ Suc n) has_field_derivative (1 + of_nat n) * (D * f x ^ n)) (at x within s)"
   by (rule has_derivative_imp_has_field_derivative[OF has_derivative_power])
      (auto simp: has_field_derivative_def)
 
 lemma DERIV_power[derivative_intros]:
   "(f has_field_derivative D) (at x within s) \<Longrightarrow>
     ((\<lambda>x. f x ^ n) has_field_derivative of_nat n * (D * f x ^ (n - Suc 0))) (at x within s)"
   by (rule has_derivative_imp_has_field_derivative[OF has_derivative_power])
      (auto simp: has_field_derivative_def)
 
 lemma DERIV_pow: "((\<lambda>x. x ^ n) has_field_derivative real n * (x ^ (n - Suc 0))) (at x within s)"
   using DERIV_power [OF DERIV_ident] by simp
 
 lemma DERIV_power_int [derivative_intros]:
   assumes [derivative_intros]: "(f has_field_derivative d) (at x within s)" and [simp]: "f x \<noteq> 0"
   shows   "((\<lambda>x. power_int (f x) n) has_field_derivative
              (of_int n * power_int (f x) (n - 1) * d)) (at x within s)"
 proof (cases n rule: int_cases4)
   case (nonneg n)
   thus ?thesis 
     by (cases "n = 0")
        (auto intro!: derivative_eq_intros simp: field_simps power_int_diff
              simp flip: power_Suc power_Suc2 power_add)
 next
   case (neg n)
   thus ?thesis
     by (auto intro!: derivative_eq_intros simp: field_simps power_int_diff power_int_minus
              simp flip: power_Suc power_Suc2 power_add)
 qed
 
 lemma DERIV_chain': "(f has_field_derivative D) (at x within s) \<Longrightarrow> DERIV g (f x) :> E \<Longrightarrow>
   ((\<lambda>x. g (f x)) has_field_derivative E * D) (at x within s)"
   using has_derivative_compose[of f "(*) D" x s g "(*) E"]
   by (simp only: has_field_derivative_def mult_commute_abs ac_simps)
 
 corollary DERIV_chain2: "DERIV f (g x) :> Da \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow>
   ((\<lambda>x. f (g x)) has_field_derivative Da * Db) (at x within s)"
   by (rule DERIV_chain')
 
 text \<open>Standard version\<close>
 
 lemma DERIV_chain:
   "DERIV f (g x) :> Da \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow>
     (f \<circ> g has_field_derivative Da * Db) (at x within s)"
   by (drule (1) DERIV_chain', simp add: o_def mult.commute)
 
 lemma DERIV_image_chain:
   "(f has_field_derivative Da) (at (g x) within (g ` s)) \<Longrightarrow>
     (g has_field_derivative Db) (at x within s) \<Longrightarrow>
     (f \<circ> g has_field_derivative Da * Db) (at x within s)"
   using has_derivative_in_compose [of g "(*) Db" x s f "(*) Da "]
   by (simp add: has_field_derivative_def o_def mult_commute_abs ac_simps)
 
 (*These two are from HOL Light: HAS_COMPLEX_DERIVATIVE_CHAIN*)
 lemma DERIV_chain_s:
   assumes "(\<And>x. x \<in> s \<Longrightarrow> DERIV g x :> g'(x))"
     and "DERIV f x :> f'"
     and "f x \<in> s"
   shows "DERIV (\<lambda>x. g(f x)) x :> f' * g'(f x)"
   by (metis (full_types) DERIV_chain' mult.commute assms)
 
 lemma DERIV_chain3: (*HAS_COMPLEX_DERIVATIVE_CHAIN_UNIV*)
   assumes "(\<And>x. DERIV g x :> g'(x))"
     and "DERIV f x :> f'"
   shows "DERIV (\<lambda>x. g(f x)) x :> f' * g'(f x)"
   by (metis UNIV_I DERIV_chain_s [of UNIV] assms)
 
 text \<open>Alternative definition for differentiability\<close>
 
 lemma DERIV_LIM_iff:
   fixes f :: "'a::{real_normed_vector,inverse} \<Rightarrow> 'a"
   shows "((\<lambda>h. (f (a + h) - f a) / h) \<midarrow>0\<rightarrow> D) = ((\<lambda>x. (f x - f a) / (x - a)) \<midarrow>a\<rightarrow> D)" (is "?lhs = ?rhs")
 proof
   assume ?lhs
   then have "(\<lambda>x. (f (a + (x + - a)) - f a) / (x + - a)) \<midarrow>0 - - a\<rightarrow> D"
     by (rule LIM_offset)
   then show ?rhs
     by simp
 next
   assume ?rhs
   then have "(\<lambda>x. (f (x+a) - f a) / ((x+a) - a)) \<midarrow>a-a\<rightarrow> D"
     by (rule LIM_offset)
   then show ?lhs
     by (simp add: add.commute)
 qed
 
 lemma has_field_derivative_cong_ev:
   assumes "x = y"
     and *: "eventually (\<lambda>x. x \<in> S \<longrightarrow> f x = g x) (nhds x)"
     and "u = v" "S = t" "x \<in> S"
   shows "(f has_field_derivative u) (at x within S) = (g has_field_derivative v) (at y within t)"
   unfolding has_field_derivative_iff
 proof (rule filterlim_cong)
   from assms have "f y = g y"
     by (auto simp: eventually_nhds)
   with * show "\<forall>\<^sub>F z in at x within S. (f z - f x) / (z - x) = (g z - g y) / (z - y)"
     unfolding eventually_at_filter
     by eventually_elim (auto simp: assms \<open>f y = g y\<close>)
 qed (simp_all add: assms)
 
 lemma has_field_derivative_cong_eventually:
   assumes "eventually (\<lambda>x. f x = g x) (at x within S)" "f x = g x"
   shows "(f has_field_derivative u) (at x within S) = (g has_field_derivative u) (at x within S)"
   unfolding has_field_derivative_iff
 proof (rule tendsto_cong)
   show "\<forall>\<^sub>F y in at x within S. (f y - f x) / (y - x) = (g y - g x) / (y - x)"
     using assms by (auto elim: eventually_mono)
 qed
 
 lemma DERIV_cong_ev:
   "x = y \<Longrightarrow> eventually (\<lambda>x. f x = g x) (nhds x) \<Longrightarrow> u = v \<Longrightarrow>
     DERIV f x :> u \<longleftrightarrow> DERIV g y :> v"
   by (rule has_field_derivative_cong_ev) simp_all
 
 lemma DERIV_shift:
   "(f has_field_derivative y) (at (x + z)) = ((\<lambda>x. f (x + z)) has_field_derivative y) (at x)"
   by (simp add: DERIV_def field_simps)
 
 lemma DERIV_mirror: "(DERIV f (- x) :> y) \<longleftrightarrow> (DERIV (\<lambda>x. f (- x)) x :> - y)"
   for f :: "real \<Rightarrow> real" and x y :: real
   by (simp add: DERIV_def filterlim_at_split filterlim_at_left_to_right
       tendsto_minus_cancel_left field_simps conj_commute)
 
 lemma floor_has_real_derivative:
   fixes f :: "real \<Rightarrow> 'a::{floor_ceiling,order_topology}"
   assumes "isCont f x"
     and "f x \<notin> \<int>"
   shows "((\<lambda>x. floor (f x)) has_real_derivative 0) (at x)"
 proof (subst DERIV_cong_ev[OF refl _ refl])
   show "((\<lambda>_. floor (f x)) has_real_derivative 0) (at x)"
     by simp
   have "\<forall>\<^sub>F y in at x. \<lfloor>f y\<rfloor> = \<lfloor>f x\<rfloor>"
     by (rule eventually_floor_eq[OF assms[unfolded continuous_at]])
   then show "\<forall>\<^sub>F y in nhds x. real_of_int \<lfloor>f y\<rfloor> = real_of_int \<lfloor>f x\<rfloor>"
     unfolding eventually_at_filter
     by eventually_elim auto
 qed
 
 lemmas has_derivative_floor[derivative_intros] =
   floor_has_real_derivative[THEN DERIV_compose_FDERIV]
 
 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
 
 text \<open>Caratheodory formulation of derivative at a point\<close>
 
 lemma CARAT_DERIV:
   "(DERIV f x :> l) \<longleftrightarrow> (\<exists>g. (\<forall>z. f z - f x = g z * (z - x)) \<and> isCont g x \<and> g x = l)"
   (is "?lhs = ?rhs")
 proof
   assume ?lhs
   show "\<exists>g. (\<forall>z. f z - f x = g z * (z - x)) \<and> isCont g x \<and> g x = l"
   proof (intro exI conjI)
     let ?g = "(\<lambda>z. if z = x then l else (f z - f x) / (z-x))"
     show "\<forall>z. f z - f x = ?g z * (z - x)"
       by simp
     show "isCont ?g x"
       using \<open>?lhs\<close> by (simp add: isCont_iff DERIV_def cong: LIM_equal [rule_format])
     show "?g x = l"
       by simp
   qed
 next
   assume ?rhs
   then show ?lhs
     by (auto simp add: isCont_iff DERIV_def cong: LIM_cong)
 qed
 
 
 subsection \<open>Local extrema\<close>
 
 text \<open>If \<^term>\<open>0 < f' x\<close> then \<^term>\<open>x\<close> is Locally Strictly Increasing At The Right.\<close>
 
 lemma has_real_derivative_pos_inc_right:
   fixes f :: "real \<Rightarrow> real"
   assumes der: "(f has_real_derivative l) (at x within S)"
     and l: "0 < l"
   shows "\<exists>d > 0. \<forall>h > 0. x + h \<in> S \<longrightarrow> h < d \<longrightarrow> f x < f (x + h)"
   using assms
 proof -
   from der [THEN has_field_derivativeD, THEN tendstoD, OF l, unfolded eventually_at]
   obtain s where s: "0 < s"
     and all: "\<And>xa. xa\<in>S \<Longrightarrow> xa \<noteq> x \<and> dist xa x < s \<longrightarrow> \<bar>(f xa - f x) / (xa - x) - l\<bar> < l"
     by (auto simp: dist_real_def)
   then show ?thesis
   proof (intro exI conjI strip)
     show "0 < s" by (rule s)
   next
     fix h :: real
     assume "0 < h" "h < s" "x + h \<in> S"
     with all [of "x + h"] show "f x < f (x+h)"
     proof (simp add: abs_if dist_real_def pos_less_divide_eq split: if_split_asm)
       assume "\<not> (f (x + h) - f x) / h < l" and h: "0 < h"
       with l have "0 < (f (x + h) - f x) / h"
         by arith
       then show "f x < f (x + h)"
         by (simp add: pos_less_divide_eq h)
     qed
   qed
 qed
 
 lemma DERIV_pos_inc_right:
   fixes f :: "real \<Rightarrow> real"
   assumes der: "DERIV f x :> l"
     and l: "0 < l"
   shows "\<exists>d > 0. \<forall>h > 0. h < d \<longrightarrow> f x < f (x + h)"
   using has_real_derivative_pos_inc_right[OF assms]
   by auto
 
 lemma has_real_derivative_neg_dec_left:
   fixes f :: "real \<Rightarrow> real"
   assumes der: "(f has_real_derivative l) (at x within S)"
     and "l < 0"
   shows "\<exists>d > 0. \<forall>h > 0. x - h \<in> S \<longrightarrow> h < d \<longrightarrow> f x < f (x - h)"
 proof -
   from \<open>l < 0\<close> have l: "- l > 0"
     by simp
   from der [THEN has_field_derivativeD, THEN tendstoD, OF l, unfolded eventually_at]
   obtain s where s: "0 < s"
     and all: "\<And>xa. xa\<in>S \<Longrightarrow> xa \<noteq> x \<and> dist xa x < s \<longrightarrow> \<bar>(f xa - f x) / (xa - x) - l\<bar> < - l"
     by (auto simp: dist_real_def)
   then show ?thesis
   proof (intro exI conjI strip)
     show "0 < s" by (rule s)
   next
     fix h :: real
     assume "0 < h" "h < s" "x - h \<in> S"
     with all [of "x - h"] show "f x < f (x-h)"
     proof (simp add: abs_if pos_less_divide_eq dist_real_def split: if_split_asm)
       assume "- ((f (x-h) - f x) / h) < l" and h: "0 < h"
       with l have "0 < (f (x-h) - f x) / h"
         by arith
       then show "f x < f (x - h)"
         by (simp add: pos_less_divide_eq h)
     qed
   qed
 qed
 
 lemma DERIV_neg_dec_left:
   fixes f :: "real \<Rightarrow> real"
   assumes der: "DERIV f x :> l"
     and l: "l < 0"
   shows "\<exists>d > 0. \<forall>h > 0. h < d \<longrightarrow> f x < f (x - h)"
   using has_real_derivative_neg_dec_left[OF assms]
   by auto
 
 lemma has_real_derivative_pos_inc_left:
   fixes f :: "real \<Rightarrow> real"
   shows "(f has_real_derivative l) (at x within S) \<Longrightarrow> 0 < l \<Longrightarrow>
     \<exists>d>0. \<forall>h>0. x - h \<in> S \<longrightarrow> h < d \<longrightarrow> f (x - h) < f x"
   by (rule has_real_derivative_neg_dec_left [of "\<lambda>x. - f x" "-l" x S, simplified])
       (auto simp add: DERIV_minus)
 
 lemma DERIV_pos_inc_left:
   fixes f :: "real \<Rightarrow> real"
   shows "DERIV f x :> l \<Longrightarrow> 0 < l \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d \<longrightarrow> f (x - h) < f x"
   using has_real_derivative_pos_inc_left
   by blast
 
 lemma has_real_derivative_neg_dec_right:
   fixes f :: "real \<Rightarrow> real"
   shows "(f has_real_derivative l) (at x within S) \<Longrightarrow> l < 0 \<Longrightarrow>
     \<exists>d > 0. \<forall>h > 0. x + h \<in> S \<longrightarrow> h < d \<longrightarrow> f x > f (x + h)"
   by (rule has_real_derivative_pos_inc_right [of "\<lambda>x. - f x" "-l" x S, simplified])
       (auto simp add: DERIV_minus)
 
 lemma DERIV_neg_dec_right:
   fixes f :: "real \<Rightarrow> real"
   shows "DERIV f x :> l \<Longrightarrow> l < 0 \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d \<longrightarrow> f x > f (x + h)"
   using has_real_derivative_neg_dec_right by blast
 
 lemma DERIV_local_max:
   fixes f :: "real \<Rightarrow> real"
   assumes der: "DERIV f x :> l"
     and d: "0 < d"
     and le: "\<forall>y. \<bar>x - y\<bar> < d \<longrightarrow> f y \<le> f x"
   shows "l = 0"
 proof (cases rule: linorder_cases [of l 0])
   case equal
   then show ?thesis .
 next
   case less
   from DERIV_neg_dec_left [OF der less]
   obtain d' where d': "0 < d'" and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x - h)"
     by blast
   obtain e where "0 < e \<and> e < d \<and> e < d'"
     using field_lbound_gt_zero [OF d d']  ..
   with lt le [THEN spec [where x="x - e"]] show ?thesis
     by (auto simp add: abs_if)
 next
   case greater
   from DERIV_pos_inc_right [OF der greater]
   obtain d' where d': "0 < d'" and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x + h)"
     by blast
   obtain e where "0 < e \<and> e < d \<and> e < d'"
     using field_lbound_gt_zero [OF d d'] ..
   with lt le [THEN spec [where x="x + e"]] show ?thesis
     by (auto simp add: abs_if)
 qed
 
 text \<open>Similar theorem for a local minimum\<close>
 lemma DERIV_local_min:
   fixes f :: "real \<Rightarrow> real"
   shows "DERIV f x :> l \<Longrightarrow> 0 < d \<Longrightarrow> \<forall>y. \<bar>x - y\<bar> < d \<longrightarrow> f x \<le> f y \<Longrightarrow> l = 0"
   by (drule DERIV_minus [THEN DERIV_local_max]) auto
 
 
 text\<open>In particular, if a function is locally flat\<close>
 lemma DERIV_local_const:
   fixes f :: "real \<Rightarrow> real"
   shows "DERIV f x :> l \<Longrightarrow> 0 < d \<Longrightarrow> \<forall>y. \<bar>x - y\<bar> < d \<longrightarrow> f x = f y \<Longrightarrow> l = 0"
   by (auto dest!: DERIV_local_max)
 
 
 subsection \<open>Rolle's Theorem\<close>
 
 text \<open>Lemma about introducing open ball in open interval\<close>
 lemma lemma_interval_lt: 
   fixes a b x :: real
   assumes "a < x" "x < b"
   shows "\<exists>d. 0 < d \<and> (\<forall>y. \<bar>x - y\<bar> < d \<longrightarrow> a < y \<and> y < b)"
   using linorder_linear [of "x - a" "b - x"]
 proof 
   assume "x - a \<le> b - x"
   with assms show ?thesis
     by (rule_tac x = "x - a" in exI) auto
 next
   assume "b - x \<le> x - a"
   with assms show ?thesis
     by (rule_tac x = "b - x" in exI) auto
 qed
 
 lemma lemma_interval: "a < x \<Longrightarrow> x < b \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>y. \<bar>x - y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b)"
   for a b x :: real
   by (force dest: lemma_interval_lt)
 
 text \<open>Rolle's Theorem.
    If \<^term>\<open>f\<close> is defined and continuous on the closed interval
    \<open>[a,b]\<close> and differentiable on the open interval \<open>(a,b)\<close>,
    and \<^term>\<open>f a = f b\<close>,
    then there exists \<open>x0 \<in> (a,b)\<close> such that \<^term>\<open>f' x0 = 0\<close>\<close>
 theorem Rolle_deriv:
   fixes f :: "real \<Rightarrow> real"
   assumes "a < b"
     and fab: "f a = f b"
     and contf: "continuous_on {a..b} f"
     and derf: "\<And>x. \<lbrakk>a < x; x < b\<rbrakk> \<Longrightarrow> (f has_derivative f' x) (at x)"
   shows "\<exists>z. a < z \<and> z < b \<and> f' z = (\<lambda>v. 0)"
 proof -
   have le: "a \<le> b"
     using \<open>a < b\<close> by simp
     have "(a + b) / 2 \<in> {a..b}"
       using assms(1) by auto
     then have *: "{a..b} \<noteq> {}"
       by auto
   obtain x where x_max: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x" and "a \<le> x" "x \<le> b"
     using continuous_attains_sup[OF compact_Icc * contf]
     by (meson atLeastAtMost_iff)
   obtain x' where x'_min: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z" and "a \<le> x'" "x' \<le> b"
     using continuous_attains_inf[OF compact_Icc * contf] by (meson atLeastAtMost_iff)
   consider "a < x" "x < b" | "x = a \<or> x = b"
     using \<open>a \<le> x\<close> \<open>x \<le> b\<close> by arith
   then show ?thesis
   proof cases
     case 1
     \<comment> \<open>\<^term>\<open>f\<close> attains its maximum within the interval\<close>
     then obtain l where der: "DERIV f x :> l"
       using derf differentiable_def real_differentiable_def by blast
     obtain d where d: "0 < d" and bound: "\<forall>y. \<bar>x - y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
       using lemma_interval [OF 1] by blast
     then have bound': "\<forall>y. \<bar>x - y\<bar> < d \<longrightarrow> f y \<le> f x"
       using x_max by blast
     \<comment> \<open>the derivative at a local maximum is zero\<close>
     have "l = 0"
       by (rule DERIV_local_max [OF der d bound'])
     with 1 der derf [of x] show ?thesis
       by (metis has_derivative_unique has_field_derivative_def mult_zero_left)
   next
     case 2
     then have fx: "f b = f x" by (auto simp add: fab)
     consider "a < x'" "x' < b" | "x' = a \<or> x' = b"
       using \<open>a \<le> x'\<close> \<open>x' \<le> b\<close> by arith
     then show ?thesis
     proof cases
       case 1
         \<comment> \<open>\<^term>\<open>f\<close> attains its minimum within the interval\<close>
       then obtain l where der: "DERIV f x' :> l"
         using derf differentiable_def real_differentiable_def by blast 
       from lemma_interval [OF 1]
       obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
         by blast
       then have bound': "\<forall>y. \<bar>x' - y\<bar> < d \<longrightarrow> f x' \<le> f y"
         using x'_min by blast
       have "l = 0" by (rule DERIV_local_min [OF der d bound'])
         \<comment> \<open>the derivative at a local minimum is zero\<close>
       then show ?thesis using 1 der derf [of x'] 
         by (metis has_derivative_unique has_field_derivative_def mult_zero_left)
     next
       case 2
         \<comment> \<open>\<^term>\<open>f\<close> is constant throughout the interval\<close>
       then have fx': "f b = f x'" by (auto simp: fab)
       from dense [OF \<open>a < b\<close>] obtain r where r: "a < r" "r < b" by blast
       obtain d where d: "0 < d" and bound: "\<forall>y. \<bar>r - y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
         using lemma_interval [OF r] by blast
       have eq_fb: "f z = f b" if "a \<le> z" and "z \<le> b" for z
       proof (rule order_antisym)
         show "f z \<le> f b" by (simp add: fx x_max that)
         show "f b \<le> f z" by (simp add: fx' x'_min that)
       qed
       have bound': "\<forall>y. \<bar>r - y\<bar> < d \<longrightarrow> f r = f y"
       proof (intro strip)
         fix y :: real
         assume lt: "\<bar>r - y\<bar> < d"
         then have "f y = f b" by (simp add: eq_fb bound)
         then show "f r = f y" by (simp add: eq_fb r order_less_imp_le)
       qed
       obtain l where der: "DERIV f r :> l"
         using derf differentiable_def r(1) r(2) real_differentiable_def by blast
       have "l = 0"
         by (rule DERIV_local_const [OF der d bound'])
         \<comment> \<open>the derivative of a constant function is zero\<close>
       with r der derf [of r] show ?thesis
         by (metis has_derivative_unique has_field_derivative_def mult_zero_left)
     qed
   qed
 qed
 
 corollary Rolle:
   fixes a b :: real
   assumes ab: "a < b" "f a = f b" "continuous_on {a..b} f"
     and dif [rule_format]: "\<And>x. \<lbrakk>a < x; x < b\<rbrakk> \<Longrightarrow> f differentiable (at x)"
   shows "\<exists>z. a < z \<and> z < b \<and> DERIV f z :> 0"
 proof -
   obtain f' where f': "\<And>x. \<lbrakk>a < x; x < b\<rbrakk> \<Longrightarrow> (f has_derivative f' x) (at x)"
     using dif unfolding differentiable_def by metis
   then have "\<exists>z. a < z \<and> z < b \<and> f' z = (\<lambda>v. 0)"
     by (metis Rolle_deriv [OF ab])
   then show ?thesis
     using f' has_derivative_imp_has_field_derivative by fastforce
 qed
 
 subsection \<open>Mean Value Theorem\<close>
 
 theorem mvt:
   fixes f :: "real \<Rightarrow> real"
   assumes "a < b"
     and contf: "continuous_on {a..b} f"
     and derf: "\<And>x. \<lbrakk>a < x; x < b\<rbrakk> \<Longrightarrow> (f has_derivative f' x) (at x)"
   obtains \<xi> where "a < \<xi>" "\<xi> < b" "f b - f a = (f' \<xi>) (b - a)"
 proof -
   have "\<exists>x. a < x \<and> x < b \<and> (\<lambda>y. f' x y - (f b - f a) / (b - a) * y) = (\<lambda>v. 0)"
   proof (intro Rolle_deriv[OF \<open>a < b\<close>])
     fix x
     assume x: "a < x" "x < b"
     show "((\<lambda>x. f x - (f b - f a) / (b - a) * x) 
           has_derivative (\<lambda>y. f' x y - (f b - f a) / (b - a) * y)) (at x)"
       by (intro derivative_intros derf[OF x])
   qed (use assms in \<open>auto intro!: continuous_intros simp: field_simps\<close>)
   then obtain \<xi> where
     "a < \<xi>" "\<xi> < b" "(\<lambda>y. f' \<xi> y - (f b - f a) / (b - a) * y) = (\<lambda>v. 0)" 
     by metis
   then show ?thesis
     by (metis (no_types, hide_lams) that add.right_neutral add_diff_cancel_left' add_diff_eq \<open>a < b\<close>
                  less_irrefl nonzero_eq_divide_eq)
 qed
 
 theorem MVT:
   fixes a b :: real
   assumes lt: "a < b"
     and contf: "continuous_on {a..b} f"
     and dif: "\<And>x. \<lbrakk>a < x; x < b\<rbrakk> \<Longrightarrow> f differentiable (at x)"
   shows "\<exists>l z. a < z \<and> z < b \<and> DERIV f z :> l \<and> f b - f a = (b - a) * l"
 proof -
   obtain f' :: "real \<Rightarrow> real \<Rightarrow> real"
     where derf: "\<And>x. a < x \<Longrightarrow> x < b \<Longrightarrow> (f has_derivative f' x) (at x)"
     using dif unfolding differentiable_def by metis
   then obtain z where "a < z" "z < b" "f b - f a = (f' z) (b - a)"
     using mvt [OF lt contf] by blast
   then show ?thesis
     by (simp add: ac_simps)
       (metis derf dif has_derivative_unique has_field_derivative_imp_has_derivative real_differentiable_def)
 qed
 
 corollary MVT2:
   assumes "a < b" and der: "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> DERIV f x :> f' x"
   shows "\<exists>z::real. a < z \<and> z < b \<and> (f b - f a = (b - a) * f' z)"
 proof -
   have "\<exists>l z. a < z \<and>
            z < b \<and>
            (f has_real_derivative l) (at z) \<and>
            f b - f a = (b - a) * l"
   proof (rule MVT [OF \<open>a < b\<close>])
     show "continuous_on {a..b} f"
       by (meson DERIV_continuous atLeastAtMost_iff continuous_at_imp_continuous_on der) 
     show "\<And>x. \<lbrakk>a < x; x < b\<rbrakk> \<Longrightarrow> f differentiable (at x)"
       using assms by (force dest: order_less_imp_le simp add: real_differentiable_def)
   qed
   with assms show ?thesis
     by (blast dest: DERIV_unique order_less_imp_le)
 qed
 
 lemma pos_deriv_imp_strict_mono:
   assumes "\<And>x. (f has_real_derivative f' x) (at x)"
   assumes "\<And>x. f' x > 0"
   shows   "strict_mono f"
 proof (rule strict_monoI)
   fix x y :: real assume xy: "x < y"
   from assms and xy have "\<exists>z>x. z < y \<and> f y - f x = (y - x) * f' z"
     by (intro MVT2) (auto dest: connectedD_interval)
   then obtain z where z: "z > x" "z < y" "f y - f x = (y - x) * f' z" by blast
   note \<open>f y - f x = (y - x) * f' z\<close>
   also have "(y - x) * f' z > 0" using xy assms by (intro mult_pos_pos) auto
   finally show "f x < f y" by simp
 qed
 
 proposition  deriv_nonneg_imp_mono:
   assumes deriv: "\<And>x. x \<in> {a..b} \<Longrightarrow> (g has_real_derivative g' x) (at x)"
   assumes nonneg: "\<And>x. x \<in> {a..b} \<Longrightarrow> g' x \<ge> 0"
   assumes ab: "a \<le> b"
   shows "g a \<le> g b"
 proof (cases "a < b")
   assume "a < b"
   from deriv have "\<And>x. \<lbrakk>x \<ge> a; x \<le> b\<rbrakk> \<Longrightarrow> (g has_real_derivative g' x) (at x)" by simp
   with MVT2[OF \<open>a < b\<close>] and deriv
     obtain \<xi> where \<xi>_ab: "\<xi> > a" "\<xi> < b" and g_ab: "g b - g a = (b - a) * g' \<xi>" by blast
   from \<xi>_ab ab nonneg have "(b - a) * g' \<xi> \<ge> 0" by simp
   with g_ab show ?thesis by simp
 qed (insert ab, simp)
 
 
 subsubsection \<open>A function is constant if its derivative is 0 over an interval.\<close>
 
 lemma DERIV_isconst_end:
   fixes f :: "real \<Rightarrow> real"
   assumes "a < b" and contf: "continuous_on {a..b} f"
     and 0: "\<And>x. \<lbrakk>a < x; x < b\<rbrakk> \<Longrightarrow> DERIV f x :> 0"
   shows "f b = f a"
   using MVT [OF \<open>a < b\<close>] "0" DERIV_unique contf real_differentiable_def
   by (fastforce simp: algebra_simps)
 
 lemma DERIV_isconst2:
   fixes f :: "real \<Rightarrow> real"
   assumes "a < b" and contf: "continuous_on {a..b} f" and derf: "\<And>x. \<lbrakk>a < x; x < b\<rbrakk> \<Longrightarrow> DERIV f x :> 0"
     and "a \<le> x" "x \<le> b"
 shows "f x = f a"
 proof (cases "a < x")
   case True
   have *: "continuous_on {a..x} f"
     using \<open>x \<le> b\<close> contf continuous_on_subset by fastforce
   show ?thesis
     by (rule DERIV_isconst_end [OF True *]) (use \<open>x \<le> b\<close> derf in auto)
 qed (use \<open>a \<le> x\<close> in auto)
 
 lemma DERIV_isconst3:
   fixes a b x y :: real
   assumes "a < b"
     and "x \<in> {a <..< b}"
     and "y \<in> {a <..< b}"
     and derivable: "\<And>x. x \<in> {a <..< b} \<Longrightarrow> DERIV f x :> 0"
   shows "f x = f y"
 proof (cases "x = y")
   case False
   let ?a = "min x y"
   let ?b = "max x y"
   have *: "DERIV f z :> 0" if "?a \<le> z" "z \<le> ?b" for z
   proof -
     have "a < z" and "z < b"
       using that \<open>x \<in> {a <..< b}\<close> and \<open>y \<in> {a <..< b}\<close> by auto
     then have "z \<in> {a<..<b}" by auto
     then show "DERIV f z :> 0" by (rule derivable)
   qed
   have isCont: "continuous_on {?a..?b} f"
     by (meson * DERIV_continuous_on atLeastAtMost_iff has_field_derivative_at_within)
   have DERIV: "\<And>z. \<lbrakk>?a < z; z < ?b\<rbrakk> \<Longrightarrow> DERIV f z :> 0"
     using * by auto
   have "?a < ?b" using \<open>x \<noteq> y\<close> by auto
   from DERIV_isconst2[OF this isCont DERIV, of x] and DERIV_isconst2[OF this isCont DERIV, of y]
   show ?thesis by auto
 qed auto
 
 lemma DERIV_isconst_all:
   fixes f :: "real \<Rightarrow> real"
   shows "\<forall>x. DERIV f x :> 0 \<Longrightarrow> f x = f y"
   apply (rule linorder_cases [of x y])
   apply (metis DERIV_continuous DERIV_isconst_end continuous_at_imp_continuous_on)+
   done
 
 lemma DERIV_const_ratio_const:
   fixes f :: "real \<Rightarrow> real"
   assumes "a \<noteq> b" and df: "\<And>x. DERIV f x :> k"
   shows "f b - f a = (b - a) * k"
 proof (cases a b rule: linorder_cases)
   case less
   show ?thesis
     using MVT [OF less] df
     by (metis DERIV_continuous DERIV_unique continuous_at_imp_continuous_on real_differentiable_def)
 next
   case greater
   have  "f a - f b = (a - b) * k"
     using MVT [OF greater] df
     by (metis DERIV_continuous DERIV_unique continuous_at_imp_continuous_on real_differentiable_def)
   then show ?thesis
     by (simp add: algebra_simps)
 qed auto
 
 lemma DERIV_const_ratio_const2:
   fixes f :: "real \<Rightarrow> real"
   assumes "a \<noteq> b" and df: "\<And>x. DERIV f x :> k"
   shows "(f b - f a) / (b - a) = k"
   using DERIV_const_ratio_const [OF assms] \<open>a \<noteq> b\<close> by auto
 
 lemma real_average_minus_first [simp]: "(a + b) / 2 - a = (b - a) / 2"
   for a b :: real
   by simp
 
 lemma real_average_minus_second [simp]: "(b + a) / 2 - a = (b - a) / 2"
   for a b :: real
   by simp
 
 text \<open>Gallileo's "trick": average velocity = av. of end velocities.\<close>
 
 lemma DERIV_const_average:
   fixes v :: "real \<Rightarrow> real"
     and a b :: real
   assumes neq: "a \<noteq> b"
     and der: "\<And>x. DERIV v x :> k"
   shows "v ((a + b) / 2) = (v a + v b) / 2"
 proof (cases rule: linorder_cases [of a b])
   case equal
   with neq show ?thesis by simp
 next
   case less
   have "(v b - v a) / (b - a) = k"
     by (rule DERIV_const_ratio_const2 [OF neq der])
   then have "(b - a) * ((v b - v a) / (b - a)) = (b - a) * k"
     by simp
   moreover have "(v ((a + b) / 2) - v a) / ((a + b) / 2 - a) = k"
     by (rule DERIV_const_ratio_const2 [OF _ der]) (simp add: neq)
   ultimately show ?thesis
     using neq by force
 next
   case greater
   have "(v b - v a) / (b - a) = k"
     by (rule DERIV_const_ratio_const2 [OF neq der])
   then have "(b - a) * ((v b - v a) / (b - a)) = (b - a) * k"
     by simp
   moreover have " (v ((b + a) / 2) - v a) / ((b + a) / 2 - a) = k"
     by (rule DERIV_const_ratio_const2 [OF _ der]) (simp add: neq)
   ultimately show ?thesis
     using neq by (force simp add: add.commute)
 qed
 
 subsubsection\<open>A function with positive derivative is increasing\<close>
 text \<open>A simple proof using the MVT, by Jeremy Avigad. And variants.\<close>
 lemma DERIV_pos_imp_increasing_open:
   fixes a b :: real
     and f :: "real \<Rightarrow> real"
   assumes "a < b"
     and "\<And>x. a < x \<Longrightarrow> x < b \<Longrightarrow> (\<exists>y. DERIV f x :> y \<and> y > 0)"
     and con: "continuous_on {a..b} f"
   shows "f a < f b"
 proof (rule ccontr)
   assume f: "\<not> ?thesis"
   have "\<exists>l z. a < z \<and> z < b \<and> DERIV f z :> l \<and> f b - f a = (b - a) * l"
     by (rule MVT) (use assms real_differentiable_def in \<open>force+\<close>)
   then obtain l z where z: "a < z" "z < b" "DERIV f z :> l" and "f b - f a = (b - a) * l"
     by auto
   with assms f have "\<not> l > 0"
     by (metis linorder_not_le mult_le_0_iff diff_le_0_iff_le)
   with assms z show False
     by (metis DERIV_unique)
 qed
 
 lemma DERIV_pos_imp_increasing:
   fixes a b :: real and f :: "real \<Rightarrow> real"
   assumes "a < b"
     and der: "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> \<exists>y. DERIV f x :> y \<and> y > 0"
   shows "f a < f b"
   by (metis less_le_not_le DERIV_atLeastAtMost_imp_continuous_on DERIV_pos_imp_increasing_open [OF \<open>a < b\<close>] der)
 
 lemma DERIV_nonneg_imp_nondecreasing:
   fixes a b :: real
     and f :: "real \<Rightarrow> real"
   assumes "a \<le> b"
     and "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> \<exists>y. DERIV f x :> y \<and> y \<ge> 0"
   shows "f a \<le> f b"
 proof (rule ccontr, cases "a = b")
   assume "\<not> ?thesis" and "a = b"
   then show False by auto
 next
   assume *: "\<not> ?thesis"
   assume "a \<noteq> b"
   with \<open>a \<le> b\<close> have "a < b"
     by linarith
   moreover have "continuous_on {a..b} f"
     by (meson DERIV_isCont assms(2) atLeastAtMost_iff continuous_at_imp_continuous_on)
   ultimately have "\<exists>l z. a < z \<and> z < b \<and> DERIV f z :> l \<and> f b - f a = (b - a) * l"
     using assms MVT [OF \<open>a < b\<close>, of f] real_differentiable_def less_eq_real_def by blast
   then obtain l z where lz: "a < z" "z < b" "DERIV f z :> l" and **: "f b - f a = (b - a) * l"
     by auto
   with * have "a < b" "f b < f a" by auto
   with ** have "\<not> l \<ge> 0" by (auto simp add: not_le algebra_simps)
     (metis * add_le_cancel_right assms(1) less_eq_real_def mult_right_mono add_left_mono linear order_refl)
   with assms lz show False
     by (metis DERIV_unique order_less_imp_le)
 qed
 
 lemma DERIV_neg_imp_decreasing_open:
   fixes a b :: real
     and f :: "real \<Rightarrow> real"
   assumes "a < b"
     and "\<And>x. a < x \<Longrightarrow> x < b \<Longrightarrow> \<exists>y. DERIV f x :> y \<and> y < 0"
     and con: "continuous_on {a..b} f"
   shows "f a > f b"
 proof -
   have "(\<lambda>x. -f x) a < (\<lambda>x. -f x) b"
   proof (rule DERIV_pos_imp_increasing_open [of a b])
     show "\<And>x. \<lbrakk>a < x; x < b\<rbrakk> \<Longrightarrow> \<exists>y. ((\<lambda>x. - f x) has_real_derivative y) (at x) \<and> 0 < y"
       using assms
       by simp (metis field_differentiable_minus neg_0_less_iff_less)
     show "continuous_on {a..b} (\<lambda>x. - f x)"
       using con continuous_on_minus by blast
   qed (use assms in auto)
   then show ?thesis
     by simp
 qed
 
 lemma DERIV_neg_imp_decreasing:
   fixes a b :: real and f :: "real \<Rightarrow> real"
   assumes "a < b"
     and der: "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> \<exists>y. DERIV f x :> y \<and> y < 0"
   shows "f a > f b"
   by (metis less_le_not_le DERIV_atLeastAtMost_imp_continuous_on DERIV_neg_imp_decreasing_open [OF \<open>a < b\<close>] der)
 
 lemma DERIV_nonpos_imp_nonincreasing:
   fixes a b :: real
     and f :: "real \<Rightarrow> real"
   assumes "a \<le> b"
     and "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> \<exists>y. DERIV f x :> y \<and> y \<le> 0"
   shows "f a \<ge> f b"
 proof -
   have "(\<lambda>x. -f x) a \<le> (\<lambda>x. -f x) b"
     using DERIV_nonneg_imp_nondecreasing [of a b "\<lambda>x. -f x"] assms DERIV_minus by fastforce
   then show ?thesis
     by simp
 qed
 
 lemma DERIV_pos_imp_increasing_at_bot:
   fixes f :: "real \<Rightarrow> real"
   assumes "\<And>x. x \<le> b \<Longrightarrow> (\<exists>y. DERIV f x :> y \<and> y > 0)"
     and lim: "(f \<longlongrightarrow> flim) at_bot"
   shows "flim < f b"
 proof -
   have "\<exists>N. \<forall>n\<le>N. f n \<le> f (b - 1)"
     by (rule_tac x="b - 2" in exI) (force intro: order.strict_implies_order DERIV_pos_imp_increasing assms)
   then have "flim \<le> f (b - 1)"
      by (auto simp: eventually_at_bot_linorder tendsto_upperbound [OF lim])
   also have "\<dots> < f b"
     by (force intro: DERIV_pos_imp_increasing [where f=f] assms)
   finally show ?thesis .
 qed
 
 lemma DERIV_neg_imp_decreasing_at_top:
   fixes f :: "real \<Rightarrow> real"
   assumes der: "\<And>x. x \<ge> b \<Longrightarrow> \<exists>y. DERIV f x :> y \<and> y < 0"
     and lim: "(f \<longlongrightarrow> flim) at_top"
   shows "flim < f b"
   apply (rule DERIV_pos_imp_increasing_at_bot [where f = "\<lambda>i. f (-i)" and b = "-b", simplified])
    apply (metis DERIV_mirror der le_minus_iff neg_0_less_iff_less)
   apply (metis filterlim_at_top_mirror lim)
   done
 
 text \<open>Derivative of inverse function\<close>
 
 lemma DERIV_inverse_function:
   fixes f g :: "real \<Rightarrow> real"
   assumes der: "DERIV f (g x) :> D"
     and neq: "D \<noteq> 0"
     and x: "a < x" "x < b"
     and inj: "\<And>y. \<lbrakk>a < y; y < b\<rbrakk> \<Longrightarrow> f (g y) = y"
     and cont: "isCont g x"
   shows "DERIV g x :> inverse D"
 unfolding has_field_derivative_iff
 proof (rule LIM_equal2)
   show "0 < min (x - a) (b - x)"
     using x by arith
 next
   fix y
   assume "norm (y - x) < min (x - a) (b - x)"
   then have "a < y" and "y < b"
     by (simp_all add: abs_less_iff)
   then show "(g y - g x) / (y - x) = inverse ((f (g y) - x) / (g y - g x))"
     by (simp add: inj)
 next
   have "(\<lambda>z. (f z - f (g x)) / (z - g x)) \<midarrow>g x\<rightarrow> D"
     by (rule der [unfolded has_field_derivative_iff])
   then have 1: "(\<lambda>z. (f z - x) / (z - g x)) \<midarrow>g x\<rightarrow> D"
     using inj x by simp
   have 2: "\<exists>d>0. \<forall>y. y \<noteq> x \<and> norm (y - x) < d \<longrightarrow> g y \<noteq> g x"
   proof (rule exI, safe)
     show "0 < min (x - a) (b - x)"
       using x by simp
   next
     fix y
     assume "norm (y - x) < min (x - a) (b - x)"
     then have y: "a < y" "y < b"
       by (simp_all add: abs_less_iff)
     assume "g y = g x"
     then have "f (g y) = f (g x)" by simp
     then have "y = x" using inj y x by simp
     also assume "y \<noteq> x"
     finally show False by simp
   qed
   have "(\<lambda>y. (f (g y) - x) / (g y - g x)) \<midarrow>x\<rightarrow> D"
     using cont 1 2 by (rule isCont_LIM_compose2)
   then show "(\<lambda>y. inverse ((f (g y) - x) / (g y - g x))) \<midarrow>x\<rightarrow> inverse D"
     using neq by (rule tendsto_inverse)
 qed
 
 subsection \<open>Generalized Mean Value Theorem\<close>
 
 theorem GMVT:
   fixes a b :: real
   assumes alb: "a < b"
     and fc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
     and fd: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable (at x)"
     and gc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont g x"
     and gd: "\<forall>x. a < x \<and> x < b \<longrightarrow> g differentiable (at x)"
   shows "\<exists>g'c f'c c.
     DERIV g c :> g'c \<and> DERIV f c :> f'c \<and> a < c \<and> c < b \<and> (f b - f a) * g'c = (g b - g a) * f'c"
 proof -
   let ?h = "\<lambda>x. (f b - f a) * g x - (g b - g a) * f x"
   have "\<exists>l z. a < z \<and> z < b \<and> DERIV ?h z :> l \<and> ?h b - ?h a = (b - a) * l"
   proof (rule MVT)
     from assms show "a < b" by simp
     show "continuous_on {a..b} ?h"
       by (simp add: continuous_at_imp_continuous_on fc gc)
     show "\<And>x. \<lbrakk>a < x; x < b\<rbrakk> \<Longrightarrow> ?h differentiable (at x)"
       using fd gd by simp
   qed
   then obtain l where l: "\<exists>z. a < z \<and> z < b \<and> DERIV ?h z :> l \<and> ?h b - ?h a = (b - a) * l" ..
   then obtain c where c: "a < c \<and> c < b \<and> DERIV ?h c :> l \<and> ?h b - ?h a = (b - a) * l" ..
 
   from c have cint: "a < c \<and> c < b" by auto
   then obtain g'c where g'c: "DERIV g c :> g'c"
     using gd real_differentiable_def by blast 
   from c have "a < c \<and> c < b" by auto
   then obtain f'c where f'c: "DERIV f c :> f'c"
     using fd real_differentiable_def by blast 
 
   from c have "DERIV ?h c :> l" by auto
   moreover have "DERIV ?h c :>  g'c * (f b - f a) - f'c * (g b - g a)"
     using g'c f'c by (auto intro!: derivative_eq_intros)
   ultimately have leq: "l =  g'c * (f b - f a) - f'c * (g b - g a)" by (rule DERIV_unique)
 
   have "?h b - ?h a = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))"
   proof -
     from c have "?h b - ?h a = (b - a) * l" by auto
     also from leq have "\<dots> = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
     finally show ?thesis by simp
   qed
   moreover have "?h b - ?h a = 0"
   proof -
     have "?h b - ?h a =
       ((f b)*(g b) - (f a)*(g b) - (g b)*(f b) + (g a)*(f b)) -
       ((f b)*(g a) - (f a)*(g a) - (g b)*(f a) + (g a)*(f a))"
       by (simp add: algebra_simps)
     then show ?thesis  by auto
   qed
   ultimately have "(b - a) * (g'c * (f b - f a) - f'c * (g b - g a)) = 0" by auto
   with alb have "g'c * (f b - f a) - f'c * (g b - g a) = 0" by simp
   then have "g'c * (f b - f a) = f'c * (g b - g a)" by simp
   then have "(f b - f a) * g'c = (g b - g a) * f'c" by (simp add: ac_simps)
   with g'c f'c cint show ?thesis by auto
 qed
 
 lemma GMVT':
   fixes f g :: "real \<Rightarrow> real"
   assumes "a < b"
     and isCont_f: "\<And>z. a \<le> z \<Longrightarrow> z \<le> b \<Longrightarrow> isCont f z"
     and isCont_g: "\<And>z. a \<le> z \<Longrightarrow> z \<le> b \<Longrightarrow> isCont g z"
     and DERIV_g: "\<And>z. a < z \<Longrightarrow> z < b \<Longrightarrow> DERIV g z :> (g' z)"
     and DERIV_f: "\<And>z. a < z \<Longrightarrow> z < b \<Longrightarrow> DERIV f z :> (f' z)"
   shows "\<exists>c. a < c \<and> c < b \<and> (f b - f a) * g' c = (g b - g a) * f' c"
 proof -
   have "\<exists>g'c f'c c. DERIV g c :> g'c \<and> DERIV f c :> f'c \<and>
       a < c \<and> c < b \<and> (f b - f a) * g'c = (g b - g a) * f'c"
     using assms by (intro GMVT) (force simp: real_differentiable_def)+
   then obtain c where "a < c" "c < b" "(f b - f a) * g' c = (g b - g a) * f' c"
     using DERIV_f DERIV_g by (force dest: DERIV_unique)
   then show ?thesis
     by auto
 qed
 
 
 subsection \<open>L'Hopitals rule\<close>
 
 lemma isCont_If_ge:
   fixes a :: "'a :: linorder_topology"
   assumes "continuous (at_left a) g" and f: "(f \<longlongrightarrow> g a) (at_right a)"
   shows "isCont (\<lambda>x. if x \<le> a then g x else f x) a" (is "isCont ?gf a")
 proof -
   have g: "(g \<longlongrightarrow> g a) (at_left a)"
     using assms continuous_within by blast
   show ?thesis
     unfolding isCont_def continuous_within
   proof (intro filterlim_split_at; simp)
     show "(?gf \<longlongrightarrow> g a) (at_left a)"
       by (subst filterlim_cong[OF refl refl, where g=g]) (simp_all add: eventually_at_filter less_le g)
     show "(?gf \<longlongrightarrow> g a) (at_right a)"
       by (subst filterlim_cong[OF refl refl, where g=f]) (simp_all add: eventually_at_filter less_le f)
   qed
 qed
 
 lemma lhopital_right_0:
   fixes f0 g0 :: "real \<Rightarrow> real"
   assumes f_0: "(f0 \<longlongrightarrow> 0) (at_right 0)"
     and g_0: "(g0 \<longlongrightarrow> 0) (at_right 0)"
     and ev:
       "eventually (\<lambda>x. g0 x \<noteq> 0) (at_right 0)"
       "eventually (\<lambda>x. g' x \<noteq> 0) (at_right 0)"
       "eventually (\<lambda>x. DERIV f0 x :> f' x) (at_right 0)"
       "eventually (\<lambda>x. DERIV g0 x :> g' x) (at_right 0)"
     and lim: "filterlim (\<lambda> x. (f' x / g' x)) F (at_right 0)"
   shows "filterlim (\<lambda> x. f0 x / g0 x) F (at_right 0)"
 proof -
   define f where [abs_def]: "f x = (if x \<le> 0 then 0 else f0 x)" for x
   then have "f 0 = 0" by simp
 
   define g where [abs_def]: "g x = (if x \<le> 0 then 0 else g0 x)" for x
   then have "g 0 = 0" by simp
 
   have "eventually (\<lambda>x. g0 x \<noteq> 0 \<and> g' x \<noteq> 0 \<and>
       DERIV f0 x :> (f' x) \<and> DERIV g0 x :> (g' x)) (at_right 0)"
     using ev by eventually_elim auto
   then obtain a where [arith]: "0 < a"
     and g0_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g0 x \<noteq> 0"
     and g'_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g' x \<noteq> 0"
     and f0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> DERIV f0 x :> (f' x)"
     and g0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> DERIV g0 x :> (g' x)"
     unfolding eventually_at by (auto simp: dist_real_def)
 
   have g_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g x \<noteq> 0"
     using g0_neq_0 by (simp add: g_def)
 
   have f: "DERIV f x :> (f' x)" if x: "0 < x" "x < a" for x
     using that
     by (intro DERIV_cong_ev[THEN iffD1, OF _ _ _ f0[OF x]])
       (auto simp: f_def eventually_nhds_metric dist_real_def intro!: exI[of _ x])
 
   have g: "DERIV g x :> (g' x)" if x: "0 < x" "x < a" for x
     using that
     by (intro DERIV_cong_ev[THEN iffD1, OF _ _ _ g0[OF x]])
          (auto simp: g_def eventually_nhds_metric dist_real_def intro!: exI[of _ x])
 
   have "isCont f 0"
     unfolding f_def by (intro isCont_If_ge f_0 continuous_const)
 
   have "isCont g 0"
     unfolding g_def by (intro isCont_If_ge g_0 continuous_const)
 
   have "\<exists>\<zeta>. \<forall>x\<in>{0 <..< a}. 0 < \<zeta> x \<and> \<zeta> x < x \<and> f x / g x = f' (\<zeta> x) / g' (\<zeta> x)"
   proof (rule bchoice, rule ballI)
     fix x
     assume "x \<in> {0 <..< a}"
     then have x[arith]: "0 < x" "x < a" by auto
     with g'_neq_0 g_neq_0 \<open>g 0 = 0\<close> have g': "\<And>x. 0 < x \<Longrightarrow> x < a  \<Longrightarrow> 0 \<noteq> g' x" "g 0 \<noteq> g x"
       by auto
     have "\<And>x. 0 \<le> x \<Longrightarrow> x < a \<Longrightarrow> isCont f x"
       using \<open>isCont f 0\<close> f by (auto intro: DERIV_isCont simp: le_less)
     moreover have "\<And>x. 0 \<le> x \<Longrightarrow> x < a \<Longrightarrow> isCont g x"
       using \<open>isCont g 0\<close> g by (auto intro: DERIV_isCont simp: le_less)
     ultimately have "\<exists>c. 0 < c \<and> c < x \<and> (f x - f 0) * g' c = (g x - g 0) * f' c"
       using f g \<open>x < a\<close> by (intro GMVT') auto
     then obtain c where *: "0 < c" "c < x" "(f x - f 0) * g' c = (g x - g 0) * f' c"
       by blast
     moreover
     from * g'(1)[of c] g'(2) have "(f x - f 0)  / (g x - g 0) = f' c / g' c"
       by (simp add: field_simps)
     ultimately show "\<exists>y. 0 < y \<and> y < x \<and> f x / g x = f' y / g' y"
       using \<open>f 0 = 0\<close> \<open>g 0 = 0\<close> by (auto intro!: exI[of _ c])
   qed
   then obtain \<zeta> where "\<forall>x\<in>{0 <..< a}. 0 < \<zeta> x \<and> \<zeta> x < x \<and> f x / g x = f' (\<zeta> x) / g' (\<zeta> x)" ..
   then have \<zeta>: "eventually (\<lambda>x. 0 < \<zeta> x \<and> \<zeta> x < x \<and> f x / g x = f' (\<zeta> x) / g' (\<zeta> x)) (at_right 0)"
     unfolding eventually_at by (intro exI[of _ a]) (auto simp: dist_real_def)
   moreover
   from \<zeta> have "eventually (\<lambda>x. norm (\<zeta> x) \<le> x) (at_right 0)"
     by eventually_elim auto
   then have "((\<lambda>x. norm (\<zeta> x)) \<longlongrightarrow> 0) (at_right 0)"
     by (rule_tac real_tendsto_sandwich[where f="\<lambda>x. 0" and h="\<lambda>x. x"]) auto
   then have "(\<zeta> \<longlongrightarrow> 0) (at_right 0)"
     by (rule tendsto_norm_zero_cancel)
   with \<zeta> have "filterlim \<zeta> (at_right 0) (at_right 0)"
     by (auto elim!: eventually_mono simp: filterlim_at)
   from this lim have "filterlim (\<lambda>t. f' (\<zeta> t) / g' (\<zeta> t)) F (at_right 0)"
     by (rule_tac filterlim_compose[of _ _ _ \<zeta>])
   ultimately have "filterlim (\<lambda>t. f t / g t) F (at_right 0)" (is ?P)
     by (rule_tac filterlim_cong[THEN iffD1, OF refl refl])
        (auto elim: eventually_mono)
   also have "?P \<longleftrightarrow> ?thesis"
     by (rule filterlim_cong) (auto simp: f_def g_def eventually_at_filter)
   finally show ?thesis .
 qed
 
 lemma lhopital_right:
   "(f \<longlongrightarrow> 0) (at_right x) \<Longrightarrow> (g \<longlongrightarrow> 0) (at_right x) \<Longrightarrow>
     eventually (\<lambda>x. g x \<noteq> 0) (at_right x) \<Longrightarrow>
     eventually (\<lambda>x. g' x \<noteq> 0) (at_right x) \<Longrightarrow>
     eventually (\<lambda>x. DERIV f x :> f' x) (at_right x) \<Longrightarrow>
     eventually (\<lambda>x. DERIV g x :> g' x) (at_right x) \<Longrightarrow>
     filterlim (\<lambda> x. (f' x / g' x)) F (at_right x) \<Longrightarrow>
   filterlim (\<lambda> x. f x / g x) F (at_right x)"
   for x :: real
   unfolding eventually_at_right_to_0[of _ x] filterlim_at_right_to_0[of _ _ x] DERIV_shift
   by (rule lhopital_right_0)
 
 lemma lhopital_left:
   "(f \<longlongrightarrow> 0) (at_left x) \<Longrightarrow> (g \<longlongrightarrow> 0) (at_left x) \<Longrightarrow>
     eventually (\<lambda>x. g x \<noteq> 0) (at_left x) \<Longrightarrow>
     eventually (\<lambda>x. g' x \<noteq> 0) (at_left x) \<Longrightarrow>
     eventually (\<lambda>x. DERIV f x :> f' x) (at_left x) \<Longrightarrow>
     eventually (\<lambda>x. DERIV g x :> g' x) (at_left x) \<Longrightarrow>
     filterlim (\<lambda> x. (f' x / g' x)) F (at_left x) \<Longrightarrow>
   filterlim (\<lambda> x. f x / g x) F (at_left x)"
   for x :: real
   unfolding eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror
   by (rule lhopital_right[where f'="\<lambda>x. - f' (- x)"]) (auto simp: DERIV_mirror)
 
 lemma lhopital:
   "(f \<longlongrightarrow> 0) (at x) \<Longrightarrow> (g \<longlongrightarrow> 0) (at x) \<Longrightarrow>
     eventually (\<lambda>x. g x \<noteq> 0) (at x) \<Longrightarrow>
     eventually (\<lambda>x. g' x \<noteq> 0) (at x) \<Longrightarrow>
     eventually (\<lambda>x. DERIV f x :> f' x) (at x) \<Longrightarrow>
     eventually (\<lambda>x. DERIV g x :> g' x) (at x) \<Longrightarrow>
     filterlim (\<lambda> x. (f' x / g' x)) F (at x) \<Longrightarrow>
   filterlim (\<lambda> x. f x / g x) F (at x)"
   for x :: real
   unfolding eventually_at_split filterlim_at_split
   by (auto intro!: lhopital_right[of f x g g' f'] lhopital_left[of f x g g' f'])
 
 
 lemma lhopital_right_0_at_top:
   fixes f g :: "real \<Rightarrow> real"
   assumes g_0: "LIM x at_right 0. g x :> at_top"
     and ev:
       "eventually (\<lambda>x. g' x \<noteq> 0) (at_right 0)"
       "eventually (\<lambda>x. DERIV f x :> f' x) (at_right 0)"
       "eventually (\<lambda>x. DERIV g x :> g' x) (at_right 0)"
     and lim: "((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> x) (at_right 0)"
   shows "((\<lambda> x. f x / g x) \<longlongrightarrow> x) (at_right 0)"
   unfolding tendsto_iff
 proof safe
   fix e :: real
   assume "0 < e"
   with lim[unfolded tendsto_iff, rule_format, of "e / 4"]
   have "eventually (\<lambda>t. dist (f' t / g' t) x < e / 4) (at_right 0)"
     by simp
   from eventually_conj[OF eventually_conj[OF ev(1) ev(2)] eventually_conj[OF ev(3) this]]
   obtain a where [arith]: "0 < a"
     and g'_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g' x \<noteq> 0"
     and f0: "\<And>x. 0 < x \<Longrightarrow> x \<le> a \<Longrightarrow> DERIV f x :> (f' x)"
     and g0: "\<And>x. 0 < x \<Longrightarrow> x \<le> a \<Longrightarrow> DERIV g x :> (g' x)"
     and Df: "\<And>t. 0 < t \<Longrightarrow> t < a \<Longrightarrow> dist (f' t / g' t) x < e / 4"
     unfolding eventually_at_le by (auto simp: dist_real_def)
 
   from Df have "eventually (\<lambda>t. t < a) (at_right 0)" "eventually (\<lambda>t::real. 0 < t) (at_right 0)"
     unfolding eventually_at by (auto intro!: exI[of _ a] simp: dist_real_def)
 
   moreover
   have "eventually (\<lambda>t. 0 < g t) (at_right 0)" "eventually (\<lambda>t. g a < g t) (at_right 0)"
     using g_0 by (auto elim: eventually_mono simp: filterlim_at_top_dense)
 
   moreover
   have inv_g: "((\<lambda>x. inverse (g x)) \<longlongrightarrow> 0) (at_right 0)"
     using tendsto_inverse_0 filterlim_mono[OF g_0 at_top_le_at_infinity order_refl]
     by (rule filterlim_compose)
   then have "((\<lambda>x. norm (1 - g a * inverse (g x))) \<longlongrightarrow> norm (1 - g a * 0)) (at_right 0)"
     by (intro tendsto_intros)
   then have "((\<lambda>x. norm (1 - g a / g x)) \<longlongrightarrow> 1) (at_right 0)"
     by (simp add: inverse_eq_divide)
   from this[unfolded tendsto_iff, rule_format, of 1]
   have "eventually (\<lambda>x. norm (1 - g a / g x) < 2) (at_right 0)"
     by (auto elim!: eventually_mono simp: dist_real_def)
 
   moreover
   from inv_g have "((\<lambda>t. norm ((f a - x * g a) * inverse (g t))) \<longlongrightarrow> norm ((f a - x * g a) * 0))
       (at_right 0)"
     by (intro tendsto_intros)
   then have "((\<lambda>t. norm (f a - x * g a) / norm (g t)) \<longlongrightarrow> 0) (at_right 0)"
     by (simp add: inverse_eq_divide)
   from this[unfolded tendsto_iff, rule_format, of "e / 2"] \<open>0 < e\<close>
   have "eventually (\<lambda>t. norm (f a - x * g a) / norm (g t) < e / 2) (at_right 0)"
     by (auto simp: dist_real_def)
 
   ultimately show "eventually (\<lambda>t. dist (f t / g t) x < e) (at_right 0)"
   proof eventually_elim
     fix t assume t[arith]: "0 < t" "t < a" "g a < g t" "0 < g t"
     assume ineq: "norm (1 - g a / g t) < 2" "norm (f a - x * g a) / norm (g t) < e / 2"
 
     have "\<exists>y. t < y \<and> y < a \<and> (g a - g t) * f' y = (f a - f t) * g' y"
       using f0 g0 t(1,2) by (intro GMVT') (force intro!: DERIV_isCont)+
     then obtain y where [arith]: "t < y" "y < a"
       and D_eq0: "(g a - g t) * f' y = (f a - f t) * g' y"
       by blast
     from D_eq0 have D_eq: "(f t - f a) / (g t - g a) = f' y / g' y"
       using \<open>g a < g t\<close> g'_neq_0[of y] by (auto simp add: field_simps)
 
     have *: "f t / g t - x = ((f t - f a) / (g t - g a) - x) * (1 - g a / g t) + (f a - x * g a) / g t"
       by (simp add: field_simps)
     have "norm (f t / g t - x) \<le>
         norm (((f t - f a) / (g t - g a) - x) * (1 - g a / g t)) + norm ((f a - x * g a) / g t)"
       unfolding * by (rule norm_triangle_ineq)
     also have "\<dots> = dist (f' y / g' y) x * norm (1 - g a / g t) + norm (f a - x * g a) / norm (g t)"
       by (simp add: abs_mult D_eq dist_real_def)
     also have "\<dots> < (e / 4) * 2 + e / 2"
       using ineq Df[of y] \<open>0 < e\<close> by (intro add_le_less_mono mult_mono) auto
     finally show "dist (f t / g t) x < e"
       by (simp add: dist_real_def)
   qed
 qed
 
 lemma lhopital_right_at_top:
   "LIM x at_right x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
     eventually (\<lambda>x. g' x \<noteq> 0) (at_right x) \<Longrightarrow>
     eventually (\<lambda>x. DERIV f x :> f' x) (at_right x) \<Longrightarrow>
     eventually (\<lambda>x. DERIV g x :> g' x) (at_right x) \<Longrightarrow>
     ((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> y) (at_right x) \<Longrightarrow>
     ((\<lambda> x. f x / g x) \<longlongrightarrow> y) (at_right x)"
   unfolding eventually_at_right_to_0[of _ x] filterlim_at_right_to_0[of _ _ x] DERIV_shift
   by (rule lhopital_right_0_at_top)
 
 lemma lhopital_left_at_top:
   "LIM x at_left x. g x :> at_top \<Longrightarrow>
     eventually (\<lambda>x. g' x \<noteq> 0) (at_left x) \<Longrightarrow>
     eventually (\<lambda>x. DERIV f x :> f' x) (at_left x) \<Longrightarrow>
     eventually (\<lambda>x. DERIV g x :> g' x) (at_left x) \<Longrightarrow>
     ((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> y) (at_left x) \<Longrightarrow>
     ((\<lambda> x. f x / g x) \<longlongrightarrow> y) (at_left x)"
   for x :: real
   unfolding eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror
   by (rule lhopital_right_at_top[where f'="\<lambda>x. - f' (- x)"]) (auto simp: DERIV_mirror)
 
 lemma lhopital_at_top:
   "LIM x at x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
     eventually (\<lambda>x. g' x \<noteq> 0) (at x) \<Longrightarrow>
     eventually (\<lambda>x. DERIV f x :> f' x) (at x) \<Longrightarrow>
     eventually (\<lambda>x. DERIV g x :> g' x) (at x) \<Longrightarrow>
     ((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> y) (at x) \<Longrightarrow>
     ((\<lambda> x. f x / g x) \<longlongrightarrow> y) (at x)"
   unfolding eventually_at_split filterlim_at_split
   by (auto intro!: lhopital_right_at_top[of g x g' f f'] lhopital_left_at_top[of g x g' f f'])
 
 lemma lhospital_at_top_at_top:
   fixes f g :: "real \<Rightarrow> real"
   assumes g_0: "LIM x at_top. g x :> at_top"
     and g': "eventually (\<lambda>x. g' x \<noteq> 0) at_top"
     and Df: "eventually (\<lambda>x. DERIV f x :> f' x) at_top"
     and Dg: "eventually (\<lambda>x. DERIV g x :> g' x) at_top"
     and lim: "((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> x) at_top"
   shows "((\<lambda> x. f x / g x) \<longlongrightarrow> x) at_top"
   unfolding filterlim_at_top_to_right
 proof (rule lhopital_right_0_at_top)
   let ?F = "\<lambda>x. f (inverse x)"
   let ?G = "\<lambda>x. g (inverse x)"
   let ?R = "at_right (0::real)"
   let ?D = "\<lambda>f' x. f' (inverse x) * - (inverse x ^ Suc (Suc 0))"
   show "LIM x ?R. ?G x :> at_top"
     using g_0 unfolding filterlim_at_top_to_right .
   show "eventually (\<lambda>x. DERIV ?G x  :> ?D g' x) ?R"
     unfolding eventually_at_right_to_top
     using Dg eventually_ge_at_top[where c=1]
     by eventually_elim (rule derivative_eq_intros DERIV_chain'[where f=inverse] | simp)+
   show "eventually (\<lambda>x. DERIV ?F x  :> ?D f' x) ?R"
     unfolding eventually_at_right_to_top
     using Df eventually_ge_at_top[where c=1]
     by eventually_elim (rule derivative_eq_intros DERIV_chain'[where f=inverse] | simp)+
   show "eventually (\<lambda>x. ?D g' x \<noteq> 0) ?R"
     unfolding eventually_at_right_to_top
     using g' eventually_ge_at_top[where c=1]
     by eventually_elim auto
   show "((\<lambda>x. ?D f' x / ?D g' x) \<longlongrightarrow> x) ?R"
     unfolding filterlim_at_right_to_top
     apply (intro filterlim_cong[THEN iffD2, OF refl refl _ lim])
     using eventually_ge_at_top[where c=1]
     by eventually_elim simp
 qed
 
 lemma lhopital_right_at_top_at_top:
   fixes f g :: "real \<Rightarrow> real"
   assumes f_0: "LIM x at_right a. f x :> at_top"
   assumes g_0: "LIM x at_right a. g x :> at_top"
     and ev:
       "eventually (\<lambda>x. DERIV f x :> f' x) (at_right a)"
       "eventually (\<lambda>x. DERIV g x :> g' x) (at_right a)"
     and lim: "filterlim (\<lambda> x. (f' x / g' x)) at_top (at_right a)"
   shows "filterlim (\<lambda> x. f x / g x) at_top (at_right a)"
 proof -
   from lim have pos: "eventually (\<lambda>x. f' x / g' x > 0) (at_right a)"
     unfolding filterlim_at_top_dense by blast
   have "((\<lambda>x. g x / f x) \<longlongrightarrow> 0) (at_right a)"
   proof (rule lhopital_right_at_top)
     from pos show "eventually (\<lambda>x. f' x \<noteq> 0) (at_right a)" by eventually_elim auto
     from tendsto_inverse_0_at_top[OF lim]
       show "((\<lambda>x. g' x / f' x) \<longlongrightarrow> 0) (at_right a)" by simp
   qed fact+
   moreover from f_0 g_0 
     have "eventually (\<lambda>x. f x > 0) (at_right a)" "eventually (\<lambda>x. g x > 0) (at_right a)"
     unfolding filterlim_at_top_dense by blast+
   hence "eventually (\<lambda>x. g x / f x > 0) (at_right a)" by eventually_elim simp
   ultimately have "filterlim (\<lambda>x. inverse (g x / f x)) at_top (at_right a)"
     by (rule filterlim_inverse_at_top)
   thus ?thesis by simp
 qed
 
 lemma lhopital_right_at_top_at_bot:
   fixes f g :: "real \<Rightarrow> real"
   assumes f_0: "LIM x at_right a. f x :> at_top"
   assumes g_0: "LIM x at_right a. g x :> at_bot"
     and ev:
       "eventually (\<lambda>x. DERIV f x :> f' x) (at_right a)"
       "eventually (\<lambda>x. DERIV g x :> g' x) (at_right a)"
     and lim: "filterlim (\<lambda> x. (f' x / g' x)) at_bot (at_right a)"
   shows "filterlim (\<lambda> x. f x / g x) at_bot (at_right a)"
 proof -
   from ev(2) have ev': "eventually (\<lambda>x. DERIV (\<lambda>x. -g x) x :> -g' x) (at_right a)"
     by eventually_elim (auto intro: derivative_intros)
   have "filterlim (\<lambda>x. f x / (-g x)) at_top (at_right a)"
     by (rule lhopital_right_at_top_at_top[where f' = f' and g' = "\<lambda>x. -g' x"])
        (insert assms ev', auto simp: filterlim_uminus_at_bot)
   hence "filterlim (\<lambda>x. -(f x / g x)) at_top (at_right a)" by simp
   thus ?thesis by (simp add: filterlim_uminus_at_bot)
 qed
 
 lemma lhopital_left_at_top_at_top:
   fixes f g :: "real \<Rightarrow> real"
   assumes f_0: "LIM x at_left a. f x :> at_top"
   assumes g_0: "LIM x at_left a. g x :> at_top"
     and ev:
       "eventually (\<lambda>x. DERIV f x :> f' x) (at_left a)"
       "eventually (\<lambda>x. DERIV g x :> g' x) (at_left a)"
     and lim: "filterlim (\<lambda> x. (f' x / g' x)) at_top (at_left a)"
   shows "filterlim (\<lambda> x. f x / g x) at_top (at_left a)"
   by (insert assms, unfold eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror,
       rule lhopital_right_at_top_at_top[where f'="\<lambda>x. - f' (- x)"]) 
      (insert assms, auto simp: DERIV_mirror)
 
 lemma lhopital_left_at_top_at_bot:
   fixes f g :: "real \<Rightarrow> real"
   assumes f_0: "LIM x at_left a. f x :> at_top"
   assumes g_0: "LIM x at_left a. g x :> at_bot"
     and ev:
       "eventually (\<lambda>x. DERIV f x :> f' x) (at_left a)"
       "eventually (\<lambda>x. DERIV g x :> g' x) (at_left a)"
     and lim: "filterlim (\<lambda> x. (f' x / g' x)) at_bot (at_left a)"
   shows "filterlim (\<lambda> x. f x / g x) at_bot (at_left a)"
   by (insert assms, unfold eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror,
       rule lhopital_right_at_top_at_bot[where f'="\<lambda>x. - f' (- x)"]) 
      (insert assms, auto simp: DERIV_mirror)
 
 lemma lhopital_at_top_at_top:
   fixes f g :: "real \<Rightarrow> real"
   assumes f_0: "LIM x at a. f x :> at_top"
   assumes g_0: "LIM x at a. g x :> at_top"
     and ev:
       "eventually (\<lambda>x. DERIV f x :> f' x) (at a)"
       "eventually (\<lambda>x. DERIV g x :> g' x) (at a)"
     and lim: "filterlim (\<lambda> x. (f' x / g' x)) at_top (at a)"
   shows "filterlim (\<lambda> x. f x / g x) at_top (at a)"
   using assms unfolding eventually_at_split filterlim_at_split
   by (auto intro!: lhopital_right_at_top_at_top[of f a g f' g'] 
                    lhopital_left_at_top_at_top[of f a g f' g'])
 
 lemma lhopital_at_top_at_bot:
   fixes f g :: "real \<Rightarrow> real"
   assumes f_0: "LIM x at a. f x :> at_top"
   assumes g_0: "LIM x at a. g x :> at_bot"
     and ev:
       "eventually (\<lambda>x. DERIV f x :> f' x) (at a)"
       "eventually (\<lambda>x. DERIV g x :> g' x) (at a)"
     and lim: "filterlim (\<lambda> x. (f' x / g' x)) at_bot (at a)"
   shows "filterlim (\<lambda> x. f x / g x) at_bot (at a)"
   using assms unfolding eventually_at_split filterlim_at_split
   by (auto intro!: lhopital_right_at_top_at_bot[of f a g f' g'] 
                    lhopital_left_at_top_at_bot[of f a g f' g'])
 
 end