src/HOL/Multivariate_Analysis/Integration.thy
changeset 61661 0932dc251248
parent 61649 268d88ec9087
parent 61659 ffee6aea0ff2
child 61736 d6b2d638af23
equal deleted inserted replaced
61652:90c65a811257 61661:0932dc251248
  9469   qed
  9469   qed
  9470 
  9470 
  9471   have "?x =  norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p. integral k f))"
  9471   have "?x =  norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p. integral k f))"
  9472     unfolding split_def setsum_subtractf ..
  9472     unfolding split_def setsum_subtractf ..
  9473   also have "\<dots> \<le> e + k"
  9473   also have "\<dots> \<le> e + k"
  9474     apply (rule *[OF **, where ir2="setsum (\<lambda>k. integral k f) r"])
  9474     apply (rule *[OF **, where ir1="setsum (\<lambda>k. integral k f) r"])
  9475   proof goal_cases
  9475   proof goal_cases
  9476     case 1
  9476     case 1
  9477     have *: "k * real (card r) / (1 + real (card r)) < k"
  9477     have *: "k * real (card r) / (1 + real (card r)) < k"
  9478       using k by (auto simp add: field_simps)
  9478       using k by (auto simp add: field_simps)
  9479     show ?case
  9479     show ?case