--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Inequalities.thy Mon Mar 16 14:52:34 2015 +0100
@@ -0,0 +1,97 @@
+(* Title: Inequalities
+ Author: Tobias Nipkow
+ Author: Johannes Hölzl
+*)
+
+theory Inequalities
+ imports Real_Vector_Spaces
+begin
+
+lemma setsum_triangle_reindex:
+ fixes n :: nat
+ shows "(\<Sum>(i,j)\<in>{(i,j). i+j < n}. f i j) = (\<Sum>k<n. \<Sum>i\<le>k. f i (k - i))"
+ apply (simp add: setsum.Sigma)
+ apply (rule setsum.reindex_bij_witness[where j="\<lambda>(i, j). (i+j, i)" and i="\<lambda>(k, i). (i, k - i)"])
+ apply auto
+ done
+
+lemma gauss_sum_div2: "(2::'a::semiring_div) \<noteq> 0 \<Longrightarrow>
+ setsum of_nat {1..n} = of_nat n * (of_nat n + 1) div (2::'a)"
+using gauss_sum[where n=n and 'a = 'a,symmetric] by auto
+
+lemma Setsum_Icc_int: assumes "0 \<le> m" and "(m::int) \<le> n"
+shows "\<Sum> {m..n} = (n*(n+1) - m*(m-1)) div 2"
+proof-
+ { fix k::int assume "k\<ge>0"
+ hence "\<Sum> {1..k::int} = k * (k+1) div 2"
+ by (rule gauss_sum_div2[where 'a = int, transferred]) simp
+ } note 1 = this
+ have "{m..n} = {0..n} - {0..m-1}" using `m\<ge>0` by auto
+ hence "\<Sum>{m..n} = \<Sum>{0..n} - \<Sum>{0..m-1}" using assms
+ by (force intro!: setsum_diff)
+ also have "{0..n} = {0} Un {1..n}" using assms by auto
+ also have "\<Sum>({0} \<union> {1..n}) = \<Sum>{1..n}" by(simp add: setsum.union_disjoint)
+ also have "\<dots> = n * (n+1) div 2" by(rule 1[OF order_trans[OF assms]])
+ also have "{0..m-1} = (if m=0 then {} else {0} Un {1..m-1})"
+ using assms by auto
+ also have "\<Sum> \<dots> = m*(m-1) div 2" using `m\<ge>0` by(simp add: 1 mult.commute)
+ also have "n*(n+1) div 2 - m*(m-1) div 2 = (n*(n+1) - m*(m-1)) div 2"
+ apply(subgoal_tac "even(n*(n+1)) \<and> even(m*(m-1))")
+ by (auto (*simp: even_def[symmetric]*))
+ finally show ?thesis .
+qed
+
+lemma Setsum_Icc_nat: assumes "(m::nat) \<le> n"
+shows "\<Sum> {m..n} = (n*(n+1) - m*(m-1)) div 2"
+proof -
+ have "m*(m-1) \<le> n*(n + 1)"
+ using assms by (meson diff_le_self order_trans le_add1 mult_le_mono)
+ hence "int(\<Sum> {m..n}) = int((n*(n+1) - m*(m-1)) div 2)" using assms
+ by (auto simp add: Setsum_Icc_int[transferred, OF _ assms] zdiv_int int_mult
+ split: zdiff_int_split)
+ thus ?thesis by simp
+qed
+
+lemma Setsum_Ico_nat: assumes "(m::nat) \<le> n"
+shows "\<Sum> {m..<n} = (n*(n-1) - m*(m-1)) div 2"
+proof cases
+ assume "m < n"
+ hence "{m..<n} = {m..n-1}" by auto
+ hence "\<Sum>{m..<n} = \<Sum>{m..n-1}" by simp
+ also have "\<dots> = (n*(n-1) - m*(m-1)) div 2"
+ using assms `m < n` by (simp add: Setsum_Icc_nat mult.commute)
+ finally show ?thesis .
+next
+ assume "\<not> m < n" with assms show ?thesis by simp
+qed
+
+lemma Chebyshev_sum_upper:
+ fixes a b::"nat \<Rightarrow> 'a::linordered_idom"
+ assumes "\<And>i j. i \<le> j \<Longrightarrow> j < n \<Longrightarrow> a i \<le> a j"
+ assumes "\<And>i j. i \<le> j \<Longrightarrow> j < n \<Longrightarrow> b i \<ge> b j"
+ shows "of_nat n * (\<Sum>k=0..<n. a k * b k) \<le> (\<Sum>k=0..<n. a k) * (\<Sum>k=0..<n. b k)"
+proof -
+ let ?S = "(\<Sum>j=0..<n. (\<Sum>k=0..<n. (a j - a k) * (b j - b k)))"
+ have "2 * (of_nat n * (\<Sum>j=0..<n. (a j * b j)) - (\<Sum>j=0..<n. b j) * (\<Sum>k=0..<n. a k)) = ?S"
+ unfolding one_add_one[symmetric] algebra_simps
+ by (simp add: algebra_simps setsum_subtractf setsum.distrib setsum.commute[of "\<lambda>i j. a i * b j"] setsum_right_distrib)
+ also
+ { fix i j::nat assume "i<n" "j<n"
+ hence "a i - a j \<le> 0 \<and> b i - b j \<ge> 0 \<or> a i - a j \<ge> 0 \<and> b i - b j \<le> 0"
+ using assms by (cases "i \<le> j") (auto simp: algebra_simps)
+ } hence "?S \<le> 0"
+ by (auto intro!: setsum_nonpos simp: mult_le_0_iff)
+ (auto simp: field_simps)
+ finally show ?thesis by (simp add: algebra_simps)
+qed
+
+lemma Chebyshev_sum_upper_nat:
+ fixes a b :: "nat \<Rightarrow> nat"
+ shows "(\<And>i j. \<lbrakk> i\<le>j; j<n \<rbrakk> \<Longrightarrow> a i \<le> a j) \<Longrightarrow>
+ (\<And>i j. \<lbrakk> i\<le>j; j<n \<rbrakk> \<Longrightarrow> b i \<ge> b j) \<Longrightarrow>
+ n * (\<Sum>i=0..<n. a i * b i) \<le> (\<Sum>i=0..<n. a i) * (\<Sum>i=0..<n. b i)"
+using Chebyshev_sum_upper[where 'a=real, of n a b]
+by (simp del: real_of_nat_mult real_of_nat_setsum
+ add: real_of_nat_mult[symmetric] real_of_nat_setsum[symmetric] real_of_nat_def[symmetric])
+
+end