equal
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replaced
6 section \<open>Vector Spaces and Algebras over the Reals\<close> |
6 section \<open>Vector Spaces and Algebras over the Reals\<close> |
7 |
7 |
8 theory Real_Vector_Spaces |
8 theory Real_Vector_Spaces |
9 imports Real Topological_Spaces |
9 imports Real Topological_Spaces |
10 begin |
10 begin |
11 |
|
12 |
11 |
13 lemma (in ordered_ab_group_add) diff_ge_0_iff_ge [simp]: "a - b \<ge> 0 \<longleftrightarrow> a \<ge> b" |
12 lemma (in ordered_ab_group_add) diff_ge_0_iff_ge [simp]: "a - b \<ge> 0 \<longleftrightarrow> a \<ge> b" |
14 by (simp add: le_diff_eq) |
13 by (simp add: le_diff_eq) |
15 |
14 |
16 subsection \<open>Locale for additive functions\<close> |
15 subsection \<open>Locale for additive functions\<close> |
1182 apply (rule abs_mult) |
1181 apply (rule abs_mult) |
1183 apply (rule abs_mult) |
1182 apply (rule abs_mult) |
1184 done |
1183 done |
1185 |
1184 |
1186 end |
1185 end |
|
1186 |
|
1187 declare uniformity_Abort[where 'a=real, code] |
1187 |
1188 |
1188 lemma dist_of_real [simp]: |
1189 lemma dist_of_real [simp]: |
1189 fixes a :: "'a::real_normed_div_algebra" |
1190 fixes a :: "'a::real_normed_div_algebra" |
1190 shows "dist (of_real x :: 'a) (of_real y) = dist x y" |
1191 shows "dist (of_real x :: 'a) (of_real y) = dist x y" |
1191 by (metis dist_norm norm_of_real of_real_diff real_norm_def) |
1192 by (metis dist_norm norm_of_real of_real_diff real_norm_def) |