--- a/src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy Mon Aug 15 16:48:05 2011 -0700
+++ b/src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy Mon Aug 15 18:35:36 2011 -0700
@@ -1101,8 +1101,7 @@
lemma closed_positive_orthant: "closed {x::real^'n. \<forall>i. 0 \<le>x$i}"
unfolding Collect_all_eq
- by (intro closed_INT ballI closed_Collect_le continuous_const
- continuous_at_component)
+ by (intro closed_INT ballI closed_Collect_le isCont_const vec_nth.isCont)
lemma Lim_component_cart:
fixes f :: "'a \<Rightarrow> 'b::metric_space ^ 'n"
@@ -1289,14 +1288,12 @@
lemma closed_interval_left_cart: fixes b::"real^'n"
shows "closed {x::real^'n. \<forall>i. x$i \<le> b$i}"
unfolding Collect_all_eq
- by (intro closed_INT ballI closed_Collect_le continuous_const
- continuous_at_component)
+ by (intro closed_INT ballI closed_Collect_le isCont_const vec_nth.isCont)
lemma closed_interval_right_cart: fixes a::"real^'n"
shows "closed {x::real^'n. \<forall>i. a$i \<le> x$i}"
unfolding Collect_all_eq
- by (intro closed_INT ballI closed_Collect_le continuous_const
- continuous_at_component)
+ by (intro closed_INT ballI closed_Collect_le isCont_const vec_nth.isCont)
lemma is_interval_cart:"is_interval (s::(real^'n) set) \<longleftrightarrow>
(\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i. ((a$i \<le> x$i \<and> x$i \<le> b$i) \<or> (b$i \<le> x$i \<and> x$i \<le> a$i))) \<longrightarrow> x \<in> s)"
@@ -1304,19 +1301,19 @@
lemma closed_halfspace_component_le_cart:
shows "closed {x::real^'n. x$i \<le> a}"
- by (intro closed_Collect_le continuous_at_component continuous_const)
+ by (intro closed_Collect_le vec_nth.isCont isCont_const)
lemma closed_halfspace_component_ge_cart:
shows "closed {x::real^'n. x$i \<ge> a}"
- by (intro closed_Collect_le continuous_at_component continuous_const)
+ by (intro closed_Collect_le vec_nth.isCont isCont_const)
lemma open_halfspace_component_lt_cart:
shows "open {x::real^'n. x$i < a}"
- by (intro open_Collect_less continuous_at_component continuous_const)
+ by (intro open_Collect_less vec_nth.isCont isCont_const)
lemma open_halfspace_component_gt_cart:
shows "open {x::real^'n. x$i > a}"
- by (intro open_Collect_less continuous_at_component continuous_const)
+ by (intro open_Collect_less vec_nth.isCont isCont_const)
lemma Lim_component_le_cart: fixes f :: "'a \<Rightarrow> real^'n"
assumes "(f ---> l) net" "\<not> (trivial_limit net)" "eventually (\<lambda>x. f(x)$i \<le> b) net"
@@ -1355,8 +1352,8 @@
proof-
{ fix i::'n
have "closed {x::'a ^ 'n. P i \<longrightarrow> x$i = 0}"
- by (cases "P i", simp_all, intro closed_Collect_eq
- continuous_at_component continuous_const) }
+ by (cases "P i", simp_all,
+ intro closed_Collect_eq vec_nth.isCont isCont_const) }
thus ?thesis
unfolding Collect_all_eq by (simp add: closed_INT)
qed