src/HOL/Integration.thy

minor cleanup in Nitpick

(*  Author:     Jacques D. Fleuriot, University of Edinburgh
    Conversion to Isar and new proofs by Lawrence C Paulson, 2004
*)

header{*Theory of Integration*}

theory Integration
imports Deriv ATP_Linkup
begin

text{*We follow John Harrison in formalizing the Gauge integral.*}

subsection {* Gauges *}

definition
  gauge :: "[real set, real => real] => bool" where
  [code del]:"gauge E g = (\<forall>x\<in>E. 0 < g(x))"


subsection {* Gauge-fine divisions *}

inductive
  fine :: "[real \<Rightarrow> real, real \<times> real, (real \<times> real \<times> real) list] \<Rightarrow> bool"
for
  \<delta> :: "real \<Rightarrow> real"
where
  fine_Nil:
    "fine \<delta> (a, a) []"
| fine_Cons:
    "\<lbrakk>fine \<delta> (b, c) D; a < b; a \<le> x; x \<le> b; b - a < \<delta> x\<rbrakk>
      \<Longrightarrow> fine \<delta> (a, c) ((a, x, b) # D)"

lemmas fine_induct [induct set: fine] =
  fine.induct [of "\<delta>" "(a,b)" "D" "split P", unfolded split_conv, standard]

lemma fine_single:
  "\<lbrakk>a < b; a \<le> x; x \<le> b; b - a < \<delta> x\<rbrakk> \<Longrightarrow> fine \<delta> (a, b) [(a, x, b)]"
by (rule fine_Cons [OF fine_Nil])

lemma fine_append:
  "\<lbrakk>fine \<delta> (a, b) D; fine \<delta> (b, c) D'\<rbrakk> \<Longrightarrow> fine \<delta> (a, c) (D @ D')"
by (induct set: fine, simp, simp add: fine_Cons)

lemma fine_imp_le: "fine \<delta> (a, b) D \<Longrightarrow> a \<le> b"
by (induct set: fine, simp_all)

lemma nonempty_fine_imp_less: "\<lbrakk>fine \<delta> (a, b) D; D \<noteq> []\<rbrakk> \<Longrightarrow> a < b"
apply (induct set: fine, simp)
apply (drule fine_imp_le, simp)
done

lemma empty_fine_imp_eq: "\<lbrakk>fine \<delta> (a, b) D; D = []\<rbrakk> \<Longrightarrow> a = b"
by (induct set: fine, simp_all)

lemma fine_eq: "fine \<delta> (a, b) D \<Longrightarrow> a = b \<longleftrightarrow> D = []"
apply (cases "D = []")
apply (drule (1) empty_fine_imp_eq, simp)
apply (drule (1) nonempty_fine_imp_less, simp)
done

lemma mem_fine:
  "\<lbrakk>fine \<delta> (a, b) D; (u, x, v) \<in> set D\<rbrakk> \<Longrightarrow> u < v \<and> u \<le> x \<and> x \<le> v"
by (induct set: fine, simp, force)

lemma mem_fine2: "\<lbrakk>fine \<delta> (a, b) D; (u, z, v) \<in> set D\<rbrakk> \<Longrightarrow> a \<le> u \<and> v \<le> b"
apply (induct arbitrary: z u v set: fine, auto)
apply (simp add: fine_imp_le)
apply (erule order_trans [OF less_imp_le], simp)
done

lemma mem_fine3: "\<lbrakk>fine \<delta> (a, b) D; (u, z, v) \<in> set D\<rbrakk> \<Longrightarrow> v - u < \<delta> z"
by (induct arbitrary: z u v set: fine) auto

lemma BOLZANO:
  fixes P :: "real \<Rightarrow> real \<Rightarrow> bool"
  assumes 1: "a \<le> b"
  assumes 2: "\<And>a b c. \<lbrakk>P a b; P b c; a \<le> b; b \<le> c\<rbrakk> \<Longrightarrow> P a c"
  assumes 3: "\<And>x. \<exists>d>0. \<forall>a b. a \<le> x & x \<le> b & (b-a) < d \<longrightarrow> P a b"
  shows "P a b"
apply (subgoal_tac "split P (a,b)", simp)
apply (rule lemma_BOLZANO [OF _ _ 1])
apply (clarify, erule (3) 2)
apply (clarify, rule 3)
done

text{*We can always find a division that is fine wrt any gauge*}

lemma fine_exists:
  assumes "a \<le> b" and "gauge {a..b} \<delta>" shows "\<exists>D. fine \<delta> (a, b) D"
proof -
  {
    fix u v :: real assume "u \<le> v"
    have "a \<le> u \<Longrightarrow> v \<le> b \<Longrightarrow> \<exists>D. fine \<delta> (u, v) D"
      apply (induct u v rule: BOLZANO, rule `u \<le> v`)
       apply (simp, fast intro: fine_append)
      apply (case_tac "a \<le> x \<and> x \<le> b")
       apply (rule_tac x="\<delta> x" in exI)
       apply (rule conjI)
        apply (simp add: `gauge {a..b} \<delta>` [unfolded gauge_def])
       apply (clarify, rename_tac u v)
       apply (case_tac "u = v")
        apply (fast intro: fine_Nil)
       apply (subgoal_tac "u < v", fast intro: fine_single, simp)
      apply (rule_tac x="1" in exI, clarsimp)
      done
  }
  with `a \<le> b` show ?thesis by auto
qed

lemma fine_covers_all:
  assumes "fine \<delta> (a, c) D" and "a < x" and "x \<le> c"
  shows "\<exists> N < length D. \<forall> d t e. D ! N = (d,t,e) \<longrightarrow> d < x \<and> x \<le> e"
  using assms
proof (induct set: fine)
  case (2 b c D a t)
  thus ?case
  proof (cases "b < x")
    case True
    with 2 obtain N where *: "N < length D"
      and **: "\<And> d t e. D ! N = (d,t,e) \<Longrightarrow> d < x \<and> x \<le> e" by auto
    hence "Suc N < length ((a,t,b)#D) \<and>
           (\<forall> d t' e. ((a,t,b)#D) ! Suc N = (d,t',e) \<longrightarrow> d < x \<and> x \<le> e)" by auto
    thus ?thesis by auto
  next
    case False with 2
    have "0 < length ((a,t,b)#D) \<and>
           (\<forall> d t' e. ((a,t,b)#D) ! 0 = (d,t',e) \<longrightarrow> d < x \<and> x \<le> e)" by auto
    thus ?thesis by auto
  qed
qed auto

lemma fine_append_split:
  assumes "fine \<delta> (a,b) D" and "D2 \<noteq> []" and "D = D1 @ D2"
  shows "fine \<delta> (a,fst (hd D2)) D1" (is "?fine1")
  and "fine \<delta> (fst (hd D2), b) D2" (is "?fine2")
proof -
  from assms
  have "?fine1 \<and> ?fine2"
  proof (induct arbitrary: D1 D2)
    case (2 b c D a' x D1 D2)
    note induct = this

    thus ?case
    proof (cases D1)
      case Nil
      hence "fst (hd D2) = a'" using 2 by auto
      with fine_Cons[OF `fine \<delta> (b,c) D` induct(3,4,5)] Nil induct
      show ?thesis by (auto intro: fine_Nil)
    next
      case (Cons d1 D1')
      with induct(2)[OF `D2 \<noteq> []`, of D1'] induct(8)
      have "fine \<delta> (b, fst (hd D2)) D1'" and "fine \<delta> (fst (hd D2), c) D2" and
        "d1 = (a', x, b)" by auto
      with fine_Cons[OF this(1) induct(3,4,5), OF induct(6)] Cons
      show ?thesis by auto
    qed
  qed auto
  thus ?fine1 and ?fine2 by auto
qed

lemma fine_\<delta>_expand:
  assumes "fine \<delta> (a,b) D"
  and "\<And> x. \<lbrakk> a \<le> x ; x \<le> b \<rbrakk> \<Longrightarrow> \<delta> x \<le> \<delta>' x"
  shows "fine \<delta>' (a,b) D"
using assms proof induct
  case 1 show ?case by (rule fine_Nil)
next
  case (2 b c D a x)
  show ?case
  proof (rule fine_Cons)
    show "fine \<delta>' (b,c) D" using 2 by auto
    from fine_imp_le[OF 2(1)] 2(6) `x \<le> b`
    show "b - a < \<delta>' x"
      using 2(7)[OF `a \<le> x`] by auto
  qed (auto simp add: 2)
qed

lemma fine_single_boundaries:
  assumes "fine \<delta> (a,b) D" and "D = [(d, t, e)]"
  shows "a = d \<and> b = e"
using assms proof induct
  case (2 b c  D a x)
  hence "D = []" and "a = d" and "b = e" by auto
  moreover
  from `fine \<delta> (b,c) D` `D = []` have "b = c"
    by (rule empty_fine_imp_eq)
  ultimately show ?case by simp
qed auto


subsection {* Riemann sum *}

definition
  rsum :: "[(real \<times> real \<times> real) list, real \<Rightarrow> real] \<Rightarrow> real" where
  "rsum D f = (\<Sum>(u, x, v)\<leftarrow>D. f x * (v - u))"

lemma rsum_Nil [simp]: "rsum [] f = 0"
unfolding rsum_def by simp

lemma rsum_Cons [simp]: "rsum ((u, x, v) # D) f = f x * (v - u) + rsum D f"
unfolding rsum_def by simp

lemma rsum_zero [simp]: "rsum D (\<lambda>x. 0) = 0"
by (induct D, auto)

lemma rsum_left_distrib: "rsum D f * c = rsum D (\<lambda>x. f x * c)"
by (induct D, auto simp add: algebra_simps)

lemma rsum_right_distrib: "c * rsum D f = rsum D (\<lambda>x. c * f x)"
by (induct D, auto simp add: algebra_simps)

lemma rsum_add: "rsum D (\<lambda>x. f x + g x) =  rsum D f + rsum D g"
by (induct D, auto simp add: algebra_simps)

lemma rsum_append: "rsum (D1 @ D2) f = rsum D1 f + rsum D2 f"
unfolding rsum_def map_append listsum_append ..


subsection {* Gauge integrability (definite) *}

definition
  Integral :: "[(real*real),real=>real,real] => bool" where
  [code del]: "Integral = (%(a,b) f k. \<forall>e > 0.
                               (\<exists>\<delta>. gauge {a .. b} \<delta> &
                               (\<forall>D. fine \<delta> (a,b) D -->
                                         \<bar>rsum D f - k\<bar> < e)))"

lemma Integral_def2:
  "Integral = (%(a,b) f k. \<forall>e>0. (\<exists>\<delta>. gauge {a..b} \<delta> &
                               (\<forall>D. fine \<delta> (a,b) D -->
                                         \<bar>rsum D f - k\<bar> \<le> e)))"
unfolding Integral_def
apply (safe intro!: ext)
apply (fast intro: less_imp_le)
apply (drule_tac x="e/2" in spec)
apply force
done

text{*Lemmas about combining gauges*}

lemma gauge_min:
     "[| gauge(E) g1; gauge(E) g2 |]
      ==> gauge(E) (%x. min (g1(x)) (g2(x)))"
by (simp add: gauge_def)

lemma fine_min:
      "fine (%x. min (g1(x)) (g2(x))) (a,b) D
       ==> fine(g1) (a,b) D & fine(g2) (a,b) D"
apply (erule fine.induct)
apply (simp add: fine_Nil)
apply (simp add: fine_Cons)
done

text{*The integral is unique if it exists*}

lemma Integral_unique:
    "[| a \<le> b; Integral(a,b) f k1; Integral(a,b) f k2 |] ==> k1 = k2"
apply (simp add: Integral_def)
apply (drule_tac x = "\<bar>k1 - k2\<bar> /2" in spec)+
apply auto
apply (drule gauge_min, assumption)
apply (drule_tac \<delta> = "%x. min (\<delta> x) (\<delta>' x)"
       in fine_exists, assumption, auto)
apply (drule fine_min)
apply (drule spec)+
apply auto
apply (subgoal_tac "\<bar>(rsum D f - k2) - (rsum D f - k1)\<bar> < \<bar>k1 - k2\<bar>")
apply arith
apply (drule add_strict_mono, assumption)
apply (auto simp only: left_distrib [symmetric] mult_2_right [symmetric] 
                mult_less_cancel_right)
done

lemma Integral_zero [simp]: "Integral(a,a) f 0"
apply (auto simp add: Integral_def)
apply (rule_tac x = "%x. 1" in exI)
apply (auto dest: fine_eq simp add: gauge_def rsum_def)
done

lemma fine_rsum_const: "fine \<delta> (a,b) D \<Longrightarrow> rsum D (\<lambda>x. c) = (c * (b - a))"
unfolding rsum_def
by (induct set: fine, auto simp add: algebra_simps)

lemma Integral_eq_diff_bounds: "a \<le> b ==> Integral(a,b) (%x. 1) (b - a)"
apply (cases "a = b", simp)
apply (simp add: Integral_def, clarify)
apply (rule_tac x = "%x. b - a" in exI)
apply (rule conjI, simp add: gauge_def)
apply (clarify)
apply (subst fine_rsum_const, assumption, simp)
done

lemma Integral_mult_const: "a \<le> b ==> Integral(a,b) (%x. c)  (c*(b - a))"
apply (cases "a = b", simp)
apply (simp add: Integral_def, clarify)
apply (rule_tac x = "%x. b - a" in exI)
apply (rule conjI, simp add: gauge_def)
apply (clarify)
apply (subst fine_rsum_const, assumption, simp)
done

lemma Integral_mult:
     "[| a \<le> b; Integral(a,b) f k |] ==> Integral(a,b) (%x. c * f x) (c * k)"
apply (auto simp add: order_le_less 
            dest: Integral_unique [OF order_refl Integral_zero])
apply (auto simp add: Integral_def setsum_right_distrib[symmetric] mult_assoc)
apply (case_tac "c = 0", force)
apply (drule_tac x = "e/abs c" in spec)
apply (simp add: divide_pos_pos)
apply clarify
apply (rule_tac x="\<delta>" in exI, clarify)
apply (drule_tac x="D" in spec, clarify)
apply (simp add: pos_less_divide_eq abs_mult [symmetric]
                 algebra_simps rsum_right_distrib)
done

lemma Integral_add:
  assumes "Integral (a, b) f x1"
  assumes "Integral (b, c) f x2"
  assumes "a \<le> b" and "b \<le> c"
  shows "Integral (a, c) f (x1 + x2)"
proof (cases "a < b \<and> b < c", simp only: Integral_def split_conv, rule allI, rule impI)
  fix \<epsilon> :: real assume "0 < \<epsilon>"
  hence "0 < \<epsilon> / 2" by auto

  assume "a < b \<and> b < c"
  hence "a < b" and "b < c" by auto

  from `Integral (a, b) f x1`[simplified Integral_def split_conv,
                              rule_format, OF `0 < \<epsilon>/2`]
  obtain \<delta>1 where \<delta>1_gauge: "gauge {a..b} \<delta>1"
    and I1: "\<And> D. fine \<delta>1 (a,b) D \<Longrightarrow> \<bar> rsum D f - x1 \<bar> < (\<epsilon> / 2)" by auto

  from `Integral (b, c) f x2`[simplified Integral_def split_conv,
                              rule_format, OF `0 < \<epsilon>/2`]
  obtain \<delta>2 where \<delta>2_gauge: "gauge {b..c} \<delta>2"
    and I2: "\<And> D. fine \<delta>2 (b,c) D \<Longrightarrow> \<bar> rsum D f - x2 \<bar> < (\<epsilon> / 2)" by auto

  def \<delta> \<equiv> "\<lambda> x. if x < b then min (\<delta>1 x) (b - x)
           else if x = b then min (\<delta>1 b) (\<delta>2 b)
                         else min (\<delta>2 x) (x - b)"

  have "gauge {a..c} \<delta>"
    using \<delta>1_gauge \<delta>2_gauge unfolding \<delta>_def gauge_def by auto
  moreover {
    fix D :: "(real \<times> real \<times> real) list"
    assume fine: "fine \<delta> (a,c) D"
    from fine_covers_all[OF this `a < b` `b \<le> c`]
    obtain N where "N < length D"
      and *: "\<forall> d t e. D ! N = (d, t, e) \<longrightarrow> d < b \<and> b \<le> e"
      by auto
    obtain d t e where D_eq: "D ! N = (d, t, e)" by (cases "D!N", auto)
    with * have "d < b" and "b \<le> e" by auto
    have in_D: "(d, t, e) \<in> set D"
      using D_eq[symmetric] using `N < length D` by auto

    from mem_fine[OF fine in_D]
    have "d < e" and "d \<le> t" and "t \<le> e" by auto

    have "t = b"
    proof (rule ccontr)
      assume "t \<noteq> b"
      with mem_fine3[OF fine in_D] `b \<le> e` `d \<le> t` `t \<le> e` `d < b` \<delta>_def
      show False by (cases "t < b") auto
    qed

    let ?D1 = "take N D"
    let ?D2 = "drop N D"
    def D1 \<equiv> "take N D @ [(d, t, b)]"
    def D2 \<equiv> "(if b = e then [] else [(b, t, e)]) @ drop (Suc N) D"

    have "D \<noteq> []" using `N < length D` by auto
    from hd_drop_conv_nth[OF this `N < length D`]
    have "fst (hd ?D2) = d" using `D ! N = (d, t, e)` by auto
    with fine_append_split[OF _ _ append_take_drop_id[symmetric]]
    have fine1: "fine \<delta> (a,d) ?D1" and fine2: "fine \<delta> (d,c) ?D2"
      using `N < length D` fine by auto

    have "fine \<delta>1 (a,b) D1" unfolding D1_def
    proof (rule fine_append)
      show "fine \<delta>1 (a, d) ?D1"
      proof (rule fine1[THEN fine_\<delta>_expand])
        fix x assume "a \<le> x" "x \<le> d"
        hence "x \<le> b" using `d < b` `x \<le> d` by auto
        thus "\<delta> x \<le> \<delta>1 x" unfolding \<delta>_def by auto
      qed

      have "b - d < \<delta>1 t"
        using mem_fine3[OF fine in_D] \<delta>_def `b \<le> e` `t = b` by auto
      from `d < b` `d \<le> t` `t = b` this
      show "fine \<delta>1 (d, b) [(d, t, b)]" using fine_single by auto
    qed
    note rsum1 = I1[OF this]

    have drop_split: "drop N D = [D ! N] @ drop (Suc N) D"
      using nth_drop'[OF `N < length D`] by simp

    have fine2: "fine \<delta>2 (e,c) (drop (Suc N) D)"
    proof (cases "drop (Suc N) D = []")
      case True
      note * = fine2[simplified drop_split True D_eq append_Nil2]
      have "e = c" using fine_single_boundaries[OF * refl] by auto
      thus ?thesis unfolding True using fine_Nil by auto
    next
      case False
      note * = fine_append_split[OF fine2 False drop_split]
      from fine_single_boundaries[OF *(1)]
      have "fst (hd (drop (Suc N) D)) = e" using D_eq by auto
      with *(2) have "fine \<delta> (e,c) (drop (Suc N) D)" by auto
      thus ?thesis
      proof (rule fine_\<delta>_expand)
        fix x assume "e \<le> x" and "x \<le> c"
        thus "\<delta> x \<le> \<delta>2 x" using `b \<le> e` unfolding \<delta>_def by auto
      qed
    qed

    have "fine \<delta>2 (b, c) D2"
    proof (cases "e = b")
      case True thus ?thesis using fine2 by (simp add: D1_def D2_def)
    next
      case False
      have "e - b < \<delta>2 b"
        using mem_fine3[OF fine in_D] \<delta>_def `d < b` `t = b` by auto
      with False `t = b` `b \<le> e`
      show ?thesis using D2_def
        by (auto intro!: fine_append[OF _ fine2] fine_single
               simp del: append_Cons)
    qed
    note rsum2 = I2[OF this]

    have "rsum D f = rsum (take N D) f + rsum [D ! N] f + rsum (drop (Suc N) D) f"
      using rsum_append[symmetric] nth_drop'[OF `N < length D`] by auto
    also have "\<dots> = rsum D1 f + rsum D2 f"
      by (cases "b = e", auto simp add: D1_def D2_def D_eq rsum_append algebra_simps)
    finally have "\<bar>rsum D f - (x1 + x2)\<bar> < \<epsilon>"
      using add_strict_mono[OF rsum1 rsum2] by simp
  }
  ultimately show "\<exists> \<delta>. gauge {a .. c} \<delta> \<and>
    (\<forall>D. fine \<delta> (a,c) D \<longrightarrow> \<bar>rsum D f - (x1 + x2)\<bar> < \<epsilon>)"
    by blast
next
  case False
  hence "a = b \<or> b = c" using `a \<le> b` and `b \<le> c` by auto
  thus ?thesis
  proof (rule disjE)
    assume "a = b" hence "x1 = 0"
      using `Integral (a, b) f x1` Integral_zero Integral_unique[of a b] by auto
    thus ?thesis using `a = b` `Integral (b, c) f x2` by auto
  next
    assume "b = c" hence "x2 = 0"
      using `Integral (b, c) f x2` Integral_zero Integral_unique[of b c] by auto
    thus ?thesis using `b = c` `Integral (a, b) f x1` by auto
  qed
qed

text{*Fundamental theorem of calculus (Part I)*}

text{*"Straddle Lemma" : Swartz and Thompson: AMM 95(7) 1988 *}

lemma strad1:
       "\<lbrakk>\<forall>z::real. z \<noteq> x \<and> \<bar>z - x\<bar> < s \<longrightarrow>
             \<bar>(f z - f x) / (z - x) - f' x\<bar> < e/2;
        0 < s; 0 < e; a \<le> x; x \<le> b\<rbrakk>
       \<Longrightarrow> \<forall>z. \<bar>z - x\<bar> < s -->\<bar>f z - f x - f' x * (z - x)\<bar> \<le> e/2 * \<bar>z - x\<bar>"
apply clarify
apply (case_tac "z = x", simp)
apply (drule_tac x = z in spec)
apply (rule_tac z1 = "\<bar>inverse (z - x)\<bar>" 
       in real_mult_le_cancel_iff2 [THEN iffD1])
 apply simp
apply (simp del: abs_inverse abs_mult add: abs_mult [symmetric]
          mult_assoc [symmetric])
apply (subgoal_tac "inverse (z - x) * (f z - f x - f' x * (z - x)) 
                    = (f z - f x) / (z - x) - f' x")
 apply (simp add: abs_mult [symmetric] mult_ac diff_minus)
apply (subst mult_commute)
apply (simp add: left_distrib diff_minus)
apply (simp add: mult_assoc divide_inverse)
apply (simp add: left_distrib)
done

lemma lemma_straddle:
  assumes f': "\<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x)" and "0 < e"
  shows "\<exists>g. gauge {a..b} g &
                (\<forall>x u v. a \<le> u & u \<le> x & x \<le> v & v \<le> b & (v - u) < g(x)
                  --> \<bar>(f(v) - f(u)) - (f'(x) * (v - u))\<bar> \<le> e * (v - u))"
proof -
  have "\<forall>x\<in>{a..b}.
        (\<exists>d > 0. \<forall>u v. u \<le> x & x \<le> v & (v - u) < d --> 
                       \<bar>(f(v) - f(u)) - (f'(x) * (v - u))\<bar> \<le> e * (v - u))"
  proof (clarsimp)
    fix x :: real assume "a \<le> x" and "x \<le> b"
    with f' have "DERIV f x :> f'(x)" by simp
    then have "\<forall>r>0. \<exists>s>0. \<forall>z. z \<noteq> x \<and> \<bar>z - x\<bar> < s \<longrightarrow> \<bar>(f z - f x) / (z - x) - f' x\<bar> < r"
      by (simp add: DERIV_iff2 LIM_eq)
    with `0 < e` obtain s
    where "\<forall>z. z \<noteq> x \<and> \<bar>z - x\<bar> < s \<longrightarrow> \<bar>(f z - f x) / (z - x) - f' x\<bar> < e/2" and "0 < s"
      by (drule_tac x="e/2" in spec, auto)
    then have strad [rule_format]:
        "\<forall>z. \<bar>z - x\<bar> < s --> \<bar>f z - f x - f' x * (z - x)\<bar> \<le> e/2 * \<bar>z - x\<bar>"
      using `0 < e` `a \<le> x` `x \<le> b` by (rule strad1)
    show "\<exists>d>0. \<forall>u v. u \<le> x \<and> x \<le> v \<and> v - u < d \<longrightarrow> \<bar>f v - f u - f' x * (v - u)\<bar> \<le> e * (v - u)"
    proof (safe intro!: exI)
      show "0 < s" by fact
    next
      fix u v :: real assume "u \<le> x" and "x \<le> v" and "v - u < s"
      have "\<bar>f v - f u - f' x * (v - u)\<bar> =
            \<bar>(f v - f x - f' x * (v - x)) + (f x - f u - f' x * (x - u))\<bar>"
        by (simp add: right_diff_distrib)
      also have "\<dots> \<le> \<bar>f v - f x - f' x * (v - x)\<bar> + \<bar>f x - f u - f' x * (x - u)\<bar>"
        by (rule abs_triangle_ineq)
      also have "\<dots> = \<bar>f v - f x - f' x * (v - x)\<bar> + \<bar>f u - f x - f' x * (u - x)\<bar>"
        by (simp add: right_diff_distrib)
      also have "\<dots> \<le> (e/2) * \<bar>v - x\<bar> + (e/2) * \<bar>u - x\<bar>"
        using `u \<le> x` `x \<le> v` `v - u < s` by (intro add_mono strad, simp_all)
      also have "\<dots> \<le> e * (v - u) / 2 + e * (v - u) / 2"
        using `u \<le> x` `x \<le> v` `0 < e` by (intro add_mono, simp_all)
      also have "\<dots> = e * (v - u)"
        by simp
      finally show "\<bar>f v - f u - f' x * (v - u)\<bar> \<le> e * (v - u)" .
    qed
  qed
  thus ?thesis
    by (simp add: gauge_def) (drule bchoice, auto)
qed

lemma fine_listsum_eq_diff:
  fixes f :: "real \<Rightarrow> real"
  shows "fine \<delta> (a, b) D \<Longrightarrow> (\<Sum>(u, x, v)\<leftarrow>D. f v - f u) = f b - f a"
by (induct set: fine) simp_all

lemma FTC1: "[|a \<le> b; \<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x) |]
             ==> Integral(a,b) f' (f(b) - f(a))"
 apply (drule order_le_imp_less_or_eq, auto)
 apply (auto simp add: Integral_def2)
 apply (drule_tac e = "e / (b - a)" in lemma_straddle)
  apply (simp add: divide_pos_pos)
 apply clarify
 apply (rule_tac x="g" in exI, clarify)
 apply (clarsimp simp add: rsum_def)
 apply (frule fine_listsum_eq_diff [where f=f])
 apply (erule subst)
 apply (subst listsum_subtractf [symmetric])
 apply (rule listsum_abs [THEN order_trans])
 apply (subst map_compose [symmetric, unfolded o_def])
 apply (subgoal_tac "e = (\<Sum>(u, x, v)\<leftarrow>D. (e / (b - a)) * (v - u))")
  apply (erule ssubst)
  apply (simp add: abs_minus_commute)
  apply (rule listsum_mono)
  apply (clarify, rename_tac u x v)
  apply ((drule spec)+, erule mp)
  apply (simp add: mem_fine mem_fine2 mem_fine3)
 apply (frule fine_listsum_eq_diff [where f="\<lambda>x. x"])
 apply (simp only: split_def)
 apply (subst listsum_const_mult)
 apply simp
done

lemma Integral_subst: "[| Integral(a,b) f k1; k2=k1 |] ==> Integral(a,b) f k2"
by simp

subsection {* Additivity Theorem of Gauge Integral *}

text{* Bartle/Sherbert: Theorem 10.1.5 p. 278 *}
lemma Integral_add_fun:
    "[| a \<le> b; Integral(a,b) f k1; Integral(a,b) g k2 |]
     ==> Integral(a,b) (%x. f x + g x) (k1 + k2)"
unfolding Integral_def
apply clarify
apply (drule_tac x = "e/2" in spec)+
apply clarsimp
apply (rule_tac x = "\<lambda>x. min (\<delta> x) (\<delta>' x)" in exI)
apply (rule conjI, erule (1) gauge_min)
apply clarify
apply (drule fine_min)
apply (drule_tac x=D in spec, simp)+
apply (drule_tac a = "\<bar>rsum D f - k1\<bar> * 2" and c = "\<bar>rsum D g - k2\<bar> * 2" in add_strict_mono, assumption)
apply (auto simp only: rsum_add left_distrib [symmetric]
                mult_2_right [symmetric] real_mult_less_iff1)
done

lemma lemma_Integral_rsum_le:
     "[| \<forall>x. a \<le> x & x \<le> b --> f x \<le> g x;
         fine \<delta> (a,b) D
      |] ==> rsum D f \<le> rsum D g"
unfolding rsum_def
apply (rule listsum_mono)
apply clarify
apply (rule mult_right_mono)
apply (drule spec, erule mp)
apply (frule (1) mem_fine)
apply (frule (1) mem_fine2)
apply simp
apply (frule (1) mem_fine)
apply simp
done

lemma Integral_le:
    "[| a \<le> b;
        \<forall>x. a \<le> x & x \<le> b --> f(x) \<le> g(x);
        Integral(a,b) f k1; Integral(a,b) g k2
     |] ==> k1 \<le> k2"
apply (simp add: Integral_def)
apply (rotate_tac 2)
apply (drule_tac x = "\<bar>k1 - k2\<bar> /2" in spec)
apply (drule_tac x = "\<bar>k1 - k2\<bar> /2" in spec, auto)
apply (drule gauge_min, assumption)
apply (drule_tac \<delta> = "\<lambda>x. min (\<delta> x) (\<delta>' x)" in fine_exists, assumption, clarify)
apply (drule fine_min)
apply (drule_tac x = D in spec, drule_tac x = D in spec, clarsimp)
apply (frule lemma_Integral_rsum_le, assumption)
apply (subgoal_tac "\<bar>(rsum D f - k1) - (rsum D g - k2)\<bar> < \<bar>k1 - k2\<bar>")
apply arith
apply (drule add_strict_mono, assumption)
apply (auto simp only: left_distrib [symmetric] mult_2_right [symmetric]
                       real_mult_less_iff1)
done

lemma Integral_imp_Cauchy:
     "(\<exists>k. Integral(a,b) f k) ==>
      (\<forall>e > 0. \<exists>\<delta>. gauge {a..b} \<delta> &
                       (\<forall>D1 D2.
                            fine \<delta> (a,b) D1 &
                            fine \<delta> (a,b) D2 -->
                            \<bar>rsum D1 f - rsum D2 f\<bar> < e))"
apply (simp add: Integral_def, auto)
apply (drule_tac x = "e/2" in spec, auto)
apply (rule exI, auto)
apply (frule_tac x = D1 in spec)
apply (drule_tac x = D2 in spec)
apply simp
apply (thin_tac "0 < e")
apply (drule add_strict_mono, assumption)
apply (auto simp only: left_distrib [symmetric] mult_2_right [symmetric]
                       real_mult_less_iff1)
done

lemma Cauchy_iff2:
     "Cauchy X =
      (\<forall>j. (\<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. \<bar>X m - X n\<bar> < inverse(real (Suc j))))"
apply (simp add: Cauchy_iff, auto)
apply (drule reals_Archimedean, safe)
apply (drule_tac x = n in spec, auto)
apply (rule_tac x = M in exI, auto)
apply (drule_tac x = m in spec, simp)
apply (drule_tac x = na in spec, auto)
done

lemma monotonic_anti_derivative:
  fixes f g :: "real => real" shows
     "[| a \<le> b; \<forall>c. a \<le> c & c \<le> b --> f' c \<le> g' c;
         \<forall>x. DERIV f x :> f' x; \<forall>x. DERIV g x :> g' x |]
      ==> f b - f a \<le> g b - g a"
apply (rule Integral_le, assumption)
apply (auto intro: FTC1) 
done

end