minimize imports
authorhuffman
Wed Aug 24 15:32:40 2011 -0700 (2011-08-24)
changeset 4451768e8eb0ce8aa
parent 44516 d9a496ae5d9d
child 44518 219a6fe4cfae
minimize imports
src/HOL/Multivariate_Analysis/Linear_Algebra.thy
src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
     1.1 --- a/src/HOL/Multivariate_Analysis/Linear_Algebra.thy	Wed Aug 24 15:06:13 2011 -0700
     1.2 +++ b/src/HOL/Multivariate_Analysis/Linear_Algebra.thy	Wed Aug 24 15:32:40 2011 -0700
     1.3 @@ -8,8 +8,6 @@
     1.4  imports
     1.5    Euclidean_Space
     1.6    "~~/src/HOL/Library/Infinite_Set"
     1.7 -  L2_Norm
     1.8 -  "~~/src/HOL/Library/Convex"
     1.9  begin
    1.10  
    1.11  lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)"
    1.12 @@ -63,7 +61,7 @@
    1.13  *)
    1.14  
    1.15  lemma norm_eq_0_dot: "(norm x = 0) \<longleftrightarrow> (inner x x = (0::real))"
    1.16 -  by (simp add: setL2_def power2_eq_square)
    1.17 +  by (simp add: power2_eq_square)
    1.18  
    1.19  lemma norm_cauchy_schwarz:
    1.20    shows "inner x y <= norm x * norm y"
     2.1 --- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Wed Aug 24 15:06:13 2011 -0700
     2.2 +++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Wed Aug 24 15:32:40 2011 -0700
     2.3 @@ -7,7 +7,7 @@
     2.4  header {* Elementary topology in Euclidean space. *}
     2.5  
     2.6  theory Topology_Euclidean_Space
     2.7 -imports SEQ Linear_Algebra "~~/src/HOL/Library/Glbs" Norm_Arith
     2.8 +imports SEQ Linear_Algebra "~~/src/HOL/Library/Glbs" Norm_Arith L2_Norm
     2.9  begin
    2.10  
    2.11  (* to be moved elsewhere *)
    2.12 @@ -20,7 +20,7 @@
    2.13    apply(subst(2) euclidean_dist_l2) apply(cases "i<DIM('a)")
    2.14    apply(rule member_le_setL2) by auto
    2.15  
    2.16 -subsection {* General notion of a topologies as values *}
    2.17 +subsection {* General notion of a topology as a value *}
    2.18  
    2.19  definition "istopology L \<longleftrightarrow> L {} \<and> (\<forall>S T. L S \<longrightarrow> L T \<longrightarrow> L (S \<inter> T)) \<and> (\<forall>K. Ball K L \<longrightarrow> L (\<Union> K))"
    2.20  typedef (open) 'a topology = "{L::('a set) \<Rightarrow> bool. istopology L}"