src/HOL/Multivariate_Analysis/Bounded_Continuous_Function.thy
changeset 59865 8a20dd967385
parent 59554 4044f53326c9
child 60017 b785d6d06430
equal deleted inserted replaced
59863:30519ff3dffb 59865:8a20dd967385
   405     from box_ne_empty(1)[of a b] assms have "(\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i)"
   405     from box_ne_empty(1)[of a b] assms have "(\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i)"
   406       by (simp add: cbox_def)
   406       by (simp add: cbox_def)
   407     hence "(\<Sum>i\<in>Basis. (dist (clamp a b y \<bullet> i) (clamp a b x \<bullet> i))\<^sup>2)
   407     hence "(\<Sum>i\<in>Basis. (dist (clamp a b y \<bullet> i) (clamp a b x \<bullet> i))\<^sup>2)
   408         \<le> (\<Sum>i\<in>Basis. (dist (y \<bullet> i) (x \<bullet> i))\<^sup>2)"
   408         \<le> (\<Sum>i\<in>Basis. (dist (y \<bullet> i) (x \<bullet> i))\<^sup>2)"
   409       by (auto intro!: setsum_mono
   409       by (auto intro!: setsum_mono
   410         simp add: clamp_def dist_real_def real_abs_le_square_iff[symmetric])
   410         simp add: clamp_def dist_real_def abs_le_square_iff[symmetric])
   411     thus ?thesis
   411     thus ?thesis
   412       by (auto intro: real_sqrt_le_mono
   412       by (auto intro: real_sqrt_le_mono
   413         simp add: euclidean_dist_l2[where y=x] euclidean_dist_l2[where y="clamp a b x"] setL2_def)
   413         simp add: euclidean_dist_l2[where y=x] euclidean_dist_l2[where y="clamp a b x"] setL2_def)
   414 qed
   414 qed
   415 
   415