equal
deleted
inserted
replaced
1278 by (simp add: algebra_simps sum.distrib) |
1278 by (simp add: algebra_simps sum.distrib) |
1279 also have "\<dots> \<le> ?B" |
1279 also have "\<dots> \<le> ?B" |
1280 proof (rule add_mono) |
1280 proof (rule add_mono) |
1281 have "(\<Sum>k\<le>n. real k * e * ?\<mu> (T k)) = (\<Sum>k\<le>n. integral (T k) (\<lambda>x. k * e))" |
1281 have "(\<Sum>k\<le>n. real k * e * ?\<mu> (T k)) = (\<Sum>k\<le>n. integral (T k) (\<lambda>x. k * e))" |
1282 by (simp add: lmeasure_integral [OF meas_t] |
1282 by (simp add: lmeasure_integral [OF meas_t] |
1283 reorient: integral_mult_right integral_mult_left) |
1283 flip: integral_mult_right integral_mult_left) |
1284 also have "\<dots> \<le> (\<Sum>k\<le>n. integral (T k) (\<lambda>x. (abs (det (matrix (f' x))))))" |
1284 also have "\<dots> \<le> (\<Sum>k\<le>n. integral (T k) (\<lambda>x. (abs (det (matrix (f' x))))))" |
1285 proof (rule sum_mono) |
1285 proof (rule sum_mono) |
1286 fix k |
1286 fix k |
1287 assume "k \<in> {..n}" |
1287 assume "k \<in> {..n}" |
1288 show "integral (T k) (\<lambda>x. k * e) \<le> integral (T k) (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)" |
1288 show "integral (T k) (\<lambda>x. k * e) \<le> integral (T k) (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)" |