equal
deleted
inserted
replaced
5151 |
5151 |
5152 text\<open>But is the premise really necessary? Need to generalise @{thm euclidean_dist_l2}\<close> |
5152 text\<open>But is the premise really necessary? Need to generalise @{thm euclidean_dist_l2}\<close> |
5153 lemma Euclidean_dist_upper: "i \<in> Basis \<Longrightarrow> dist (x \<bullet> i) (y \<bullet> i) \<le> dist x y" |
5153 lemma Euclidean_dist_upper: "i \<in> Basis \<Longrightarrow> dist (x \<bullet> i) (y \<bullet> i) \<le> dist x y" |
5154 by (metis (no_types) member_le_L2_set euclidean_dist_l2 finite_Basis) |
5154 by (metis (no_types) member_le_L2_set euclidean_dist_l2 finite_Basis) |
5155 |
5155 |
5156 text\<open>But is the premise @{term \<open>i \<in> Basis\<close>} really necessary?\<close> |
5156 text\<open>But is the premise \<^term>\<open>i \<in> Basis\<close> really necessary?\<close> |
5157 lemma open_preimage_inner: |
5157 lemma open_preimage_inner: |
5158 assumes "open S" "i \<in> Basis" |
5158 assumes "open S" "i \<in> Basis" |
5159 shows "open {x. x \<bullet> i \<in> S}" |
5159 shows "open {x. x \<bullet> i \<in> S}" |
5160 proof (rule openI, simp) |
5160 proof (rule openI, simp) |
5161 fix x |
5161 fix x |