equal
deleted
inserted
replaced
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 |