--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/ex/Gauge_Integration.thy Tue Feb 23 17:55:00 2010 +0100
@@ -0,0 +1,573 @@
+(* Author: Jacques D. Fleuriot, University of Edinburgh
+ Conversion to Isar and new proofs by Lawrence C Paulson, 2004
+
+ Replaced by ~~/src/HOL/Multivariate_Analysis/Real_Integral.thy .
+*)
+
+header{*Theory of Integration on real intervals*}
+
+theory Gauge_Integration
+imports Complex_Main
+begin
+
+text {*
+
+\textbf{Attention}: This theory defines the Integration on real
+intervals. This is just a example theory for historical / expository interests.
+A better replacement is found in the Multivariate Analysis library. This defines
+the gauge integral on real vector spaces and in the Real Integral theory
+is a specialization to the integral on arbitrary real intervals. The
+Multivariate Analysis package also provides a better support for analysis on
+integrals.
+
+*}
+
+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
+
+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
+
+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
+
+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{*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 fundamental_theorem_of_calculus:
+ "\<lbrakk> a \<le> b; \<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x) \<rbrakk>
+ \<Longrightarrow> 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_map [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
+
+end