# HG changeset patch
# User paulson <lp15@cam.ac.uk>
# Date 1502229289 -7200
# Node ID cc66710c9d48156a8c542bfeb0385ca15998319d
# Parent  5eb0faf4477abbee676a50dabefb0ff7744b5f2a
more cleanup of fundamental_theorem_of_calculus_interior

diff -r 5eb0faf4477a -r cc66710c9d48 src/HOL/Analysis/Henstock_Kurzweil_Integration.thy
--- a/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy	Tue Aug 08 13:56:29 2017 +0200
+++ b/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy	Tue Aug 08 23:54:49 2017 +0200
@@ -9,6 +9,12 @@
   Lebesgue_Measure Tagged_Division
 begin
 
+lemma norm_triangle_le_sub: "norm x + norm y \<le> e \<Longrightarrow> norm (x - y) \<le> e"
+  apply (subst(asm)(2) norm_minus_cancel[symmetric])
+  apply (drule norm_triangle_le)
+  apply (auto simp add: algebra_simps)
+  done
+
 lemma eps_leI: 
   assumes "(\<And>e::'a::linordered_idom. 0 < e \<Longrightarrow> x < y + e)" shows "x \<le> y"
   by (metis add_diff_eq assms diff_diff_add diff_gt_0_iff_gt linorder_not_less order_less_irrefl)
@@ -18,7 +24,7 @@
 
 (* try instead structured proofs below *)
 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"
+  \<Longrightarrow> norm(y-x) \<le> e"
   using norm_triangle_mono [of "y1 - x1" "e1" "y2 - x2" "e2"]
   by (simp add: add_diff_add)
 
@@ -2669,11 +2675,11 @@
   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))"
+             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)"
+                 "\<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>
@@ -2954,16 +2960,16 @@
         unfolding euclidean_eq_iff[where 'a='a] using i by auto
       have *: "Basis = insert i (Basis - {i})"
         using i by auto
-      have "norm (y - x) < e + sum (\<lambda>i. 0) Basis"
+      have "norm (y-x) < e + sum (\<lambda>i. 0) Basis"
         apply (rule le_less_trans[OF norm_le_l1])
         apply (subst *)
         apply (subst sum.insert)
         prefer 3
         apply (rule add_less_le_mono)
       proof -
-        show "\<bar>(y - x) \<bullet> i\<bar> < e"
+        show "\<bar>(y-x) \<bullet> i\<bar> < e"
           using di as(2) 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)"
+        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 auto
       then show "dist y x < e"
@@ -3145,7 +3151,7 @@
     assume "e > 0"
     obtain d where "d>0" and d: "\<And>x'. \<lbrakk>x' \<in> {a..b}; \<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>"
            if y: "y \<in> {a..b}" and yx: "\<bar>y - x\<bar> < d" for y
     proof (cases "y < x")
       case False
@@ -3153,7 +3159,7 @@
         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}"
+      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
@@ -3186,7 +3192,7 @@
       then show ?thesis
         by (simp add: algebra_simps norm_minus_commute)
     qed
-    then have "\<exists>d>0. \<forall>y\<in>{a..b}. \<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>"
+    then have "\<exists>d>0. \<forall>y\<in>{a..b}. \<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
@@ -3583,12 +3589,6 @@
 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)
 
-lemma norm_triangle_le_sub: "norm x + norm y \<le> e \<Longrightarrow> norm (x - y) \<le> e"
-  apply (subst(asm)(2) norm_minus_cancel[symmetric])
-  apply (drule norm_triangle_le)
-  apply (auto simp add: algebra_simps)
-  done
-
 theorem fundamental_theorem_of_calculus_interior:
   fixes f :: "real \<Rightarrow> 'a::real_normed_vector"
   assumes "a \<le> b"
@@ -3613,7 +3613,7 @@
   note derf_exp = derf[unfolded has_vector_derivative_def has_derivative_at_alt]
   have bounded: "\<And>x. x \<in> {a<..<b} \<Longrightarrow> bounded_linear (\<lambda>u. u *\<^sub>R f' x)"
     using derf_exp by simp
-  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)"
+  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")
   proof
     fix x assume x: "x \<in> box a b"
@@ -3623,8 +3623,8 @@
   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)"
+     "\<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
   have "bounded (f ` cbox a b)"
     apply (rule compact_imp_bounded compact_continuous_image)+
@@ -3757,61 +3757,47 @@
     note * = additive_tagged_division_1[OF assms(1) as(1), symmetric]
     have **: "\<And>n1 s1 n2 s2::real. n2 \<le> s2 / 2 \<Longrightarrow> n1 - s1 \<le> s2 / 2 \<Longrightarrow> n1 + n2 \<le> s1 + s2"
       by arith
-    have 1: "norm (\<Sum>(x, k)\<in>p - ?A.
+    have XX: False if xk: "(x,k) \<in> p" 
+         and less: "e * (Sup k - Inf k) / 2 < norm (content k *\<^sub>R f' x - (f (Sup k) - f (Inf k)))"
+         and "x \<noteq> a" "x \<noteq> b"
+         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 as(2) 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_sub 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 XX by (force intro: sum_mono)
+    finally have 1: "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"
-        apply (rule order_trans)
-        apply (rule sum_norm_le)
-        defer
-        apply (subst sum_divide_distrib)
-        apply (rule order_refl)
-        apply safe
-        apply (unfold not_le o_def split_conv fst_conv)
-      proof (rule ccontr)
-        fix x k
-        assume xk: "(x, k) \<in> p"
-          "e * (Sup k -  Inf k) / 2 <
-            norm (content k *\<^sub>R f' x - (f (Sup k) - f (Inf k)))"
-        obtain u v where k: "k = cbox u v"
-          using p(4) xk(1) by blast
-        then have "u \<le> v" and uv: "{u, v} \<subseteq> cbox u v"
-          using p(2)[OF xk(1)] by auto
-        then have result: "e * (v - u) / 2 < norm ((v - u) *\<^sub>R f' x - (f v - f u))"
-          using xk(2)[unfolded k box_real interval_bounds_real content_real] by auto
-        assume as': "x \<noteq> a" "x \<noteq> b"
-        then have "x \<in> box a b"
-          using p(2-3)[OF xk(1)] by (auto simp: mem_box)
-        note  * = d(2)[OF this]
-        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)))"
-          apply (rule arg_cong[of _ _ norm])
-          unfolding scaleR_left.diff
-          apply auto
-          done
-        also have "\<dots> \<le> e / 2 * norm (u - x) + e / 2 * norm (v - x)"
-          apply (rule norm_triangle_le_sub)
-          apply (rule add_mono)
-          apply (rule_tac[!] *)
-          using fineD[OF as(2) xk(1)] as'
-          unfolding k subset_eq
-          apply -
-          apply (erule_tac x=u in ballE)
-          apply (erule_tac[3] x=v in ballE)
-          using uv
-          apply (auto simp:dist_real_def)
-          done
-        also have "\<dots> \<le> e / 2 * norm (v - u)"
-          using p(2)[OF xk(1)]
-          unfolding k
-          by (auto simp add: field_simps)
-        finally have "e * (v - u) / 2 < e * (v - u) / 2"
-          apply -
-          apply (rule less_le_trans[OF result])
-          using uv
-          apply auto
-          done
-        then show False by auto
-      qed
+      by (simp add: sum_divide_distrib)
     have 2: "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"
@@ -3839,16 +3825,14 @@
         proof goal_cases
           fix x k
           assume "(x, k) \<in> p \<inter> {t. fst t \<in> {a, b}} - p \<inter> {t. fst t \<in> {a, b} \<and> content (snd t) \<noteq> 0}"
-          then have xk: "(x, k) \<in> p" "content k = 0"
+          then have xk: "(x, k) \<in> p" and k0: "content k = 0"
             by auto
           then obtain u v where uv: "k = cbox u v"
             using p(4) by blast
           have "k \<noteq> {}"
-            using p(2)[OF xk(1)] by auto
+            using p(2)[OF xk] by auto
           then have *: "u = v"
-            using xk
-            unfolding uv content_eq_0 box_eq_empty
-            by auto
+            using xk k0 by (auto simp: uv content_eq_0 box_eq_empty)
           then show "content k *\<^sub>R (f' (x)) - (f ((Sup k)) - f ((Inf k))) = 0"
             using xk unfolding uv by auto
         next