Theory Inequalities

(*  Title:     HOL/Inequalities.thy
    Author:    Tobias Nipkow
    Author:    Johannes Hölzl
*)

theory Inequalities
  imports Real_Vector_Spaces
begin

lemma Chebyshev_sum_upper:
  fixes a b::"nat ⇒ 'a::linordered_idom"
  assumes "⋀i j. i ≤ j ⟹ j < n ⟹ a i ≤ a j"
  assumes "⋀i j. i ≤ j ⟹ j < n ⟹ b i ≥ b j"
  shows "of_nat n * (∑k=0..<n. a k * b k) ≤ (∑k=0..<n. a k) * (∑k=0..<n. b k)"
proof -
  let ?S = "(∑j=0..<n. (∑k=0..<n. (a j - a k) * (b j - b k)))"
  have "2 * (of_nat n * (∑j=0..<n. (a j * b j)) - (∑j=0..<n. b j) * (∑k=0..<n. a k)) = ?S"
    by (simp only: one_add_one[symmetric] algebra_simps)
      (simp add: algebra_simps sum_subtractf sum.distrib sum.swap[of "λi j. a i * b j"] sum_distrib_left)
  also
  { fix i j::nat assume "i<n" "j<n"
    hence "a i - a j ≤ 0 ∧ b i - b j ≥ 0 ∨ a i - a j ≥ 0 ∧ b i - b j ≤ 0"
      using assms by (cases "i ≤ j") (auto simp: algebra_simps)
  } then have "?S ≤ 0"
    by (auto intro!: sum_nonpos simp: mult_le_0_iff)
  finally show ?thesis by (simp add: algebra_simps)
qed

lemma Chebyshev_sum_upper_nat:
  fixes a b :: "nat ⇒ nat"
  shows "(⋀i j. ⟦ i≤j; j<n ⟧ ⟹ a i ≤ a j) ⟹
         (⋀i j. ⟦ i≤j; j<n ⟧ ⟹ b i ≥ b j) ⟹
    n * (∑i=0..<n. a i * b i) ≤ (∑i=0..<n. a i) * (∑i=0..<n. b i)"
using Chebyshev_sum_upper[where 'a=real, of n a b]
by (simp del: of_nat_mult of_nat_sum  add: of_nat_mult[symmetric] of_nat_sum[symmetric])

end