src/HOL/Analysis/Bounded_Continuous_Function.thy
changeset 64267 b9a1486e79be
parent 63627 6ddb43c6b711
child 65036 ab7e11730ad8
equal deleted inserted replaced
64265:8eb6365f5916 64267:b9a1486e79be
   424 proof -
   424 proof -
   425   from box_ne_empty(1)[of a b] assms have "(\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i)"
   425   from box_ne_empty(1)[of a b] assms have "(\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i)"
   426     by (simp add: cbox_def)
   426     by (simp add: cbox_def)
   427   then have "(\<Sum>i\<in>Basis. (dist (clamp a b y \<bullet> i) (clamp a b x \<bullet> i))\<^sup>2) \<le>
   427   then have "(\<Sum>i\<in>Basis. (dist (clamp a b y \<bullet> i) (clamp a b x \<bullet> i))\<^sup>2) \<le>
   428     (\<Sum>i\<in>Basis. (dist (y \<bullet> i) (x \<bullet> i))\<^sup>2)"
   428     (\<Sum>i\<in>Basis. (dist (y \<bullet> i) (x \<bullet> i))\<^sup>2)"
   429     by (auto intro!: setsum_mono simp: clamp_def dist_real_def abs_le_square_iff[symmetric])
   429     by (auto intro!: sum_mono simp: clamp_def dist_real_def abs_le_square_iff[symmetric])
   430   then show ?thesis
   430   then show ?thesis
   431     by (auto intro: real_sqrt_le_mono
   431     by (auto intro: real_sqrt_le_mono
   432       simp: euclidean_dist_l2[where y=x] euclidean_dist_l2[where y="clamp a b x"] setL2_def)
   432       simp: euclidean_dist_l2[where y=x] euclidean_dist_l2[where y="clamp a b x"] setL2_def)
   433 qed
   433 qed
   434 
   434