# HG changeset patch
# User huffman
# Date 1272300322 25200
# Node ID 18bf20d0c2df37b746fec7d80c687d8cd8c39863
# Parent  0e2679025aebb961d11349599f643c1eef6bff4f# Parent  7b92238454d695345b63ee24b9cfe81ee1f0e464
merged

diff -r 7b92238454d6 -r 18bf20d0c2df src/HOL/Complete_Lattice.thy
--- a/src/HOL/Complete_Lattice.thy	Mon Apr 26 16:08:04 2010 +0200
+++ b/src/HOL/Complete_Lattice.thy	Mon Apr 26 09:45:22 2010 -0700
@@ -98,9 +98,9 @@
 
 syntax
   "_SUP1"     :: "pttrns => 'b => 'b"           ("(3SUP _./ _)" [0, 10] 10)
-  "_SUP"      :: "pttrn => 'a set => 'b => 'b"  ("(3SUP _:_./ _)" [0, 10] 10)
+  "_SUP"      :: "pttrn => 'a set => 'b => 'b"  ("(3SUP _:_./ _)" [0, 0, 10] 10)
   "_INF1"     :: "pttrns => 'b => 'b"           ("(3INF _./ _)" [0, 10] 10)
-  "_INF"      :: "pttrn => 'a set => 'b => 'b"  ("(3INF _:_./ _)" [0, 10] 10)
+  "_INF"      :: "pttrn => 'a set => 'b => 'b"  ("(3INF _:_./ _)" [0, 0, 10] 10)
 
 translations
   "SUP x y. B"   == "SUP x. SUP y. B"
@@ -295,15 +295,15 @@
 
 syntax
   "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)
-  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 10] 10)
+  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 0, 10] 10)
 
 syntax (xsymbols)
   "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)
-  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 10] 10)
+  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10)
 
 syntax (latex output)
   "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
-  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10)
+  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
 
 translations
   "UN x y. B"   == "UN x. UN y. B"
@@ -531,15 +531,15 @@
 
 syntax
   "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" [0, 10] 10)
-  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 10] 10)
+  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 0, 10] 10)
 
 syntax (xsymbols)
   "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
-  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 10] 10)
+  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10)
 
 syntax (latex output)
   "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
-  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10)
+  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
 
 translations
   "INT x y. B"  == "INT x. INT y. B"
diff -r 7b92238454d6 -r 18bf20d0c2df src/HOL/HOL.thy
--- a/src/HOL/HOL.thy	Mon Apr 26 16:08:04 2010 +0200
+++ b/src/HOL/HOL.thy	Mon Apr 26 09:45:22 2010 -0700
@@ -73,7 +73,7 @@
 local
 
 consts
-  If            :: "[bool, 'a, 'a] => 'a"           ("(if (_)/ then (_)/ else (_))" 10)
+  If            :: "[bool, 'a, 'a] => 'a"           ("(if (_)/ then (_)/ else (_))" [0, 0, 10] 10)
 
 
 subsubsection {* Additional concrete syntax *}
@@ -118,7 +118,7 @@
   "_bind"       :: "[pttrn, 'a] => letbind"              ("(2_ =/ _)" 10)
   ""            :: "letbind => letbinds"                 ("_")
   "_binds"      :: "[letbind, letbinds] => letbinds"     ("_;/ _")
-  "_Let"        :: "[letbinds, 'a] => 'a"                ("(let (_)/ in (_))" 10)
+  "_Let"        :: "[letbinds, 'a] => 'a"                ("(let (_)/ in (_))" [0, 10] 10)
 
   "_case_syntax":: "['a, cases_syn] => 'b"               ("(case _ of/ _)" 10)
   "_case1"      :: "['a, 'b] => case_syn"                ("(2_ =>/ _)" 10)
diff -r 7b92238454d6 -r 18bf20d0c2df src/HOL/Library/FrechetDeriv.thy
--- a/src/HOL/Library/FrechetDeriv.thy	Mon Apr 26 16:08:04 2010 +0200
+++ b/src/HOL/Library/FrechetDeriv.thy	Mon Apr 26 09:45:22 2010 -0700
@@ -385,7 +385,7 @@
   fixes x :: "'a::{real_normed_algebra,comm_ring_1}"
   shows "FDERIV (\<lambda>x. x ^ Suc n) x :> (\<lambda>h. (1 + of_nat n) * x ^ n * h)"
  apply (induct n)
-  apply (simp add: power_Suc FDERIV_ident)
+  apply (simp add: FDERIV_ident)
  apply (drule FDERIV_mult [OF FDERIV_ident])
  apply (simp only: of_nat_Suc left_distrib mult_1_left)
  apply (simp only: power_Suc right_distrib add_ac mult_ac)
diff -r 7b92238454d6 -r 18bf20d0c2df src/HOL/Library/Permutations.thy
--- a/src/HOL/Library/Permutations.thy	Mon Apr 26 16:08:04 2010 +0200
+++ b/src/HOL/Library/Permutations.thy	Mon Apr 26 09:45:22 2010 -0700
@@ -96,7 +96,7 @@
 
 lemma permutes_superset:
   "p permutes S \<Longrightarrow> (\<forall>x \<in> S - T. p x = x) \<Longrightarrow> p permutes T"
-by (simp add: Ball_def permutes_def Diff_iff) metis
+by (simp add: Ball_def permutes_def) metis
 
 (* ------------------------------------------------------------------------- *)
 (* Group properties.                                                         *)
@@ -125,7 +125,7 @@
   apply (rule permutes_compose[OF pS])
   apply (rule permutes_swap_id, simp)
   using permutes_in_image[OF pS, of a] apply simp
-  apply (auto simp add: Ball_def Diff_iff swap_def)
+  apply (auto simp add: Ball_def swap_def)
   done
 
 lemma permutes_insert: "{p. p permutes (insert a S)} =
@@ -154,7 +154,7 @@
 lemma card_permutations: assumes Sn: "card S = n" and fS: "finite S"
   shows "card {p. p permutes S} = fact n"
 using fS Sn proof (induct arbitrary: n)
-  case empty thus ?case by (simp add: permutes_empty)
+  case empty thus ?case by simp
 next
   case (insert x F)
   { fix n assume H0: "card (insert x F) = n"
diff -r 7b92238454d6 -r 18bf20d0c2df src/HOL/Limits.thy
--- a/src/HOL/Limits.thy	Mon Apr 26 16:08:04 2010 +0200
+++ b/src/HOL/Limits.thy	Mon Apr 26 09:45:22 2010 -0700
@@ -11,62 +11,59 @@
 subsection {* Nets *}
 
 text {*
-  A net is now defined as a filter base.
-  The definition also allows non-proper filter bases.
+  A net is now defined simply as a filter.
+  The definition also allows non-proper filters.
 *}
 
+locale is_filter =
+  fixes net :: "('a \<Rightarrow> bool) \<Rightarrow> bool"
+  assumes True: "net (\<lambda>x. True)"
+  assumes conj: "net (\<lambda>x. P x) \<Longrightarrow> net (\<lambda>x. Q x) \<Longrightarrow> net (\<lambda>x. P x \<and> Q x)"
+  assumes mono: "\<forall>x. P x \<longrightarrow> Q x \<Longrightarrow> net (\<lambda>x. P x) \<Longrightarrow> net (\<lambda>x. Q x)"
+
 typedef (open) 'a net =
-  "{net :: 'a set set. (\<exists>A. A \<in> net)
-    \<and> (\<forall>A\<in>net. \<forall>B\<in>net. \<exists>C\<in>net. C \<subseteq> A \<and> C \<subseteq> B)}"
+  "{net :: ('a \<Rightarrow> bool) \<Rightarrow> bool. is_filter net}"
 proof
-  show "UNIV \<in> ?net" by auto
+  show "(\<lambda>x. True) \<in> ?net" by (auto intro: is_filter.intro)
 qed
 
-lemma Rep_net_nonempty: "\<exists>A. A \<in> Rep_net net"
-using Rep_net [of net] by simp
-
-lemma Rep_net_directed:
-  "A \<in> Rep_net net \<Longrightarrow> B \<in> Rep_net net \<Longrightarrow> \<exists>C\<in>Rep_net net. C \<subseteq> A \<and> C \<subseteq> B"
+lemma is_filter_Rep_net: "is_filter (Rep_net net)"
 using Rep_net [of net] by simp
 
 lemma Abs_net_inverse':
-  assumes "\<exists>A. A \<in> net"
-  assumes "\<And>A B. A \<in> net \<Longrightarrow> B \<in> net \<Longrightarrow> \<exists>C\<in>net. C \<subseteq> A \<and> C \<subseteq> B" 
-  shows "Rep_net (Abs_net net) = net"
+  assumes "is_filter net" shows "Rep_net (Abs_net net) = net"
 using assms by (simp add: Abs_net_inverse)
 
-lemma image_nonempty: "\<exists>x. x \<in> A \<Longrightarrow> \<exists>x. x \<in> f ` A"
-by auto
-
 
 subsection {* Eventually *}
 
 definition
   eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a net \<Rightarrow> bool" where
-  [code del]: "eventually P net \<longleftrightarrow> (\<exists>A\<in>Rep_net net. \<forall>x\<in>A. P x)"
+  [code del]: "eventually P net \<longleftrightarrow> Rep_net net P"
+
+lemma eventually_Abs_net:
+  assumes "is_filter net" shows "eventually P (Abs_net net) = net P"
+unfolding eventually_def using assms by (simp add: Abs_net_inverse)
+
+lemma expand_net_eq:
+  shows "net = net' \<longleftrightarrow> (\<forall>P. eventually P net = eventually P net')"
+unfolding Rep_net_inject [symmetric] expand_fun_eq eventually_def ..
 
 lemma eventually_True [simp]: "eventually (\<lambda>x. True) net"
-unfolding eventually_def using Rep_net_nonempty [of net] by fast
+unfolding eventually_def
+by (rule is_filter.True [OF is_filter_Rep_net])
 
 lemma eventually_mono:
   "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually P net \<Longrightarrow> eventually Q net"
-unfolding eventually_def by blast
+unfolding eventually_def
+by (rule is_filter.mono [OF is_filter_Rep_net])
 
 lemma eventually_conj:
   assumes P: "eventually (\<lambda>x. P x) net"
   assumes Q: "eventually (\<lambda>x. Q x) net"
   shows "eventually (\<lambda>x. P x \<and> Q x) net"
-proof -
-  obtain A where A: "A \<in> Rep_net net" "\<forall>x\<in>A. P x"
-    using P unfolding eventually_def by fast
-  obtain B where B: "B \<in> Rep_net net" "\<forall>x\<in>B. Q x"
-    using Q unfolding eventually_def by fast
-  obtain C where C: "C \<in> Rep_net net" "C \<subseteq> A" "C \<subseteq> B"
-    using Rep_net_directed [OF A(1) B(1)] by fast
-  then have "\<forall>x\<in>C. P x \<and> Q x" "C \<in> Rep_net net"
-    using A(2) B(2) by auto
-  then show ?thesis unfolding eventually_def ..
-qed
+using assms unfolding eventually_def
+by (rule is_filter.conj [OF is_filter_Rep_net])
 
 lemma eventually_mp:
   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) net"
@@ -102,60 +99,116 @@
 using assms by (auto elim!: eventually_rev_mp)
 
 
+subsection {* Finer-than relation *}
+
+text {* @{term "net \<le> net'"} means that @{term net'} is finer than
+@{term net}. *}
+
+instantiation net :: (type) "{order,top}"
+begin
+
+definition
+  le_net_def [code del]:
+    "net \<le> net' \<longleftrightarrow> (\<forall>P. eventually P net \<longrightarrow> eventually P net')"
+
+definition
+  less_net_def [code del]:
+    "(net :: 'a net) < net' \<longleftrightarrow> net \<le> net' \<and> \<not> net' \<le> net"
+
+definition
+  top_net_def [code del]:
+    "top = Abs_net (\<lambda>P. True)"
+
+lemma eventually_top [simp]: "eventually P top"
+unfolding top_net_def
+by (subst eventually_Abs_net, rule is_filter.intro, auto)
+
+instance proof
+  fix x y :: "'a net" show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
+    by (rule less_net_def)
+next
+  fix x :: "'a net" show "x \<le> x"
+    unfolding le_net_def by simp
+next
+  fix x y z :: "'a net" assume "x \<le> y" and "y \<le> z" thus "x \<le> z"
+    unfolding le_net_def by simp
+next
+  fix x y :: "'a net" assume "x \<le> y" and "y \<le> x" thus "x = y"
+    unfolding le_net_def expand_net_eq by fast
+next
+  fix x :: "'a net" show "x \<le> top"
+    unfolding le_net_def by simp
+qed
+
+end
+
+lemma net_leD:
+  "net \<le> net' \<Longrightarrow> eventually P net \<Longrightarrow> eventually P net'"
+unfolding le_net_def by simp
+
+lemma net_leI:
+  "(\<And>P. eventually P net \<Longrightarrow> eventually P net') \<Longrightarrow> net \<le> net'"
+unfolding le_net_def by simp
+
+lemma eventually_False:
+  "eventually (\<lambda>x. False) net \<longleftrightarrow> net = top"
+unfolding expand_net_eq by (auto elim: eventually_rev_mp)
+
+
 subsection {* Standard Nets *}
 
 definition
-  sequentially :: "nat net" where
-  [code del]: "sequentially = Abs_net (range (\<lambda>n. {n..}))"
-
-definition
-  within :: "'a net \<Rightarrow> 'a set \<Rightarrow> 'a net" (infixr "within" 70) where
-  [code del]: "net within S = Abs_net ((\<lambda>A. A \<inter> S) ` Rep_net net)"
+  sequentially :: "nat net"
+where [code del]:
+  "sequentially = Abs_net (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)"
 
 definition
-  at :: "'a::topological_space \<Rightarrow> 'a net" where
-  [code del]: "at a = Abs_net ((\<lambda>S. S - {a}) ` {S. open S \<and> a \<in> S})"
-
-lemma Rep_net_sequentially:
-  "Rep_net sequentially = range (\<lambda>n. {n..})"
-unfolding sequentially_def
-apply (rule Abs_net_inverse')
-apply (rule image_nonempty, simp)
-apply (clarsimp, rename_tac m n)
-apply (rule_tac x="max m n" in exI, auto)
-done
+  within :: "'a net \<Rightarrow> 'a set \<Rightarrow> 'a net" (infixr "within" 70)
+where [code del]:
+  "net within S = Abs_net (\<lambda>P. eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) net)"
 
-lemma Rep_net_within:
-  "Rep_net (net within S) = (\<lambda>A. A \<inter> S) ` Rep_net net"
-unfolding within_def
-apply (rule Abs_net_inverse')
-apply (rule image_nonempty, rule Rep_net_nonempty)
-apply (clarsimp, rename_tac A B)
-apply (drule (1) Rep_net_directed)
-apply (clarify, rule_tac x=C in bexI, auto)
-done
-
-lemma Rep_net_at:
-  "Rep_net (at a) = ((\<lambda>S. S - {a}) ` {S. open S \<and> a \<in> S})"
-unfolding at_def
-apply (rule Abs_net_inverse')
-apply (rule image_nonempty)
-apply (rule_tac x="UNIV" in exI, simp)
-apply (clarsimp, rename_tac S T)
-apply (rule_tac x="S \<inter> T" in exI, auto simp add: open_Int)
-done
+definition
+  at :: "'a::topological_space \<Rightarrow> 'a net"
+where [code del]:
+  "at a = Abs_net (\<lambda>P. \<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x))"
 
 lemma eventually_sequentially:
   "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
-unfolding eventually_def Rep_net_sequentially by auto
+unfolding sequentially_def
+proof (rule eventually_Abs_net, rule is_filter.intro)
+  fix P Q :: "nat \<Rightarrow> bool"
+  assume "\<exists>i. \<forall>n\<ge>i. P n" and "\<exists>j. \<forall>n\<ge>j. Q n"
+  then obtain i j where "\<forall>n\<ge>i. P n" and "\<forall>n\<ge>j. Q n" by auto
+  then have "\<forall>n\<ge>max i j. P n \<and> Q n" by simp
+  then show "\<exists>k. \<forall>n\<ge>k. P n \<and> Q n" ..
+qed auto
 
 lemma eventually_within:
   "eventually P (net within S) = eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) net"
-unfolding eventually_def Rep_net_within by auto
+unfolding within_def
+by (rule eventually_Abs_net, rule is_filter.intro)
+   (auto elim!: eventually_rev_mp)
+
+lemma within_UNIV: "net within UNIV = net"
+  unfolding expand_net_eq eventually_within by simp
 
 lemma eventually_at_topological:
   "eventually P (at a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x))"
-unfolding eventually_def Rep_net_at by auto
+unfolding at_def
+proof (rule eventually_Abs_net, rule is_filter.intro)
+  have "open UNIV \<and> a \<in> UNIV \<and> (\<forall>x\<in>UNIV. x \<noteq> a \<longrightarrow> True)" by simp
+  thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> True)" by - rule
+next
+  fix P Q
+  assume "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x)"
+     and "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. x \<noteq> a \<longrightarrow> Q x)"
+  then obtain S T where
+    "open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x)"
+    "open T \<and> a \<in> T \<and> (\<forall>x\<in>T. x \<noteq> a \<longrightarrow> Q x)" by auto
+  hence "open (S \<inter> T) \<and> a \<in> S \<inter> T \<and> (\<forall>x\<in>(S \<inter> T). x \<noteq> a \<longrightarrow> P x \<and> Q x)"
+    by (simp add: open_Int)
+  thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x \<and> Q x)" by - rule
+qed auto
 
 lemma eventually_at:
   fixes a :: "'a::metric_space"
diff -r 7b92238454d6 -r 18bf20d0c2df src/HOL/Multivariate_Analysis/Brouwer_Fixpoint.thy
--- a/src/HOL/Multivariate_Analysis/Brouwer_Fixpoint.thy	Mon Apr 26 16:08:04 2010 +0200
+++ b/src/HOL/Multivariate_Analysis/Brouwer_Fixpoint.thy	Mon Apr 26 09:45:22 2010 -0700
@@ -22,8 +22,6 @@
   imports Convex_Euclidean_Space
 begin
 
-declare norm_scaleR[simp]
- 
 lemma brouwer_compactness_lemma:
   assumes "compact s" "continuous_on s f" "\<not> (\<exists>x\<in>s. (f x = (0::real^'n)))"
   obtains d where "0 < d" "\<forall>x\<in>s. d \<le> norm(f x)" proof(cases "s={}") case False
@@ -131,7 +129,7 @@
 lemma image_lemma_2: assumes "finite s" "finite t" "card s = card t" "f ` s \<subseteq> t" "f ` s \<noteq> t" "b \<in> t"
   shows "(card {s'. \<exists>a\<in>s. (s' = s - {a}) \<and> f ` s' = t - {b}} = 0) \<or>
          (card {s'. \<exists>a\<in>s. (s' = s - {a}) \<and> f ` s' = t - {b}} = 2)" proof(cases "{a\<in>s. f ` (s - {a}) = t - {b}} = {}")
-  case True thus ?thesis apply-apply(rule disjI1, rule image_lemma_0) using assms(1) by(auto simp add:card_0_eq)
+  case True thus ?thesis apply-apply(rule disjI1, rule image_lemma_0) using assms(1) by auto
 next let ?M = "{a\<in>s. f ` (s - {a}) = t - {b}}"
   case False then obtain a where "a\<in>?M" by auto hence a:"a\<in>s" "f ` (s - {a}) = t - {b}" by auto
   have "f a \<in> t - {b}" using a and assms by auto
@@ -1691,11 +1689,11 @@
       path_image(linepath(pathfinish g)(vector[(pathfinish g)$1,a$2 - 1])) \<union>
       path_image(linepath(vector[(pathfinish g)$1,a$2 - 1])(vector[b$1 + 1,a$2 - 1])) \<union>
       path_image(linepath(vector[b$1 + 1,a$2 - 1])(vector[b$1 + 1,b$2 + 3]))" using assms(1-2)
-      by(auto simp add: pathstart_join pathfinish_join path_image_join path_image_linepath path_join path_linepath) 
+      by(auto simp add: path_image_join path_linepath)
   have abab: "{a..b} \<subseteq> {?a..?b}" by(auto simp add:vector_le_def forall_2 vector_2)
   guess z apply(rule fashoda[of ?P1 ?P2 ?a ?b])
     unfolding pathstart_join pathfinish_join pathstart_linepath pathfinish_linepath vector_2 proof-
-    show "path ?P1" "path ?P2" using assms by(auto simp add: pathstart_join pathfinish_join path_join)
+    show "path ?P1" "path ?P2" using assms by auto
     have "path_image ?P1 \<subseteq> {?a .. ?b}" unfolding P1P2 path_image_linepath apply(rule Un_least)+ defer 3
       apply(rule_tac[1-4] convex_interval(1)[unfolded convex_contains_segment,rule_format])
       unfolding mem_interval forall_2 vector_2 using ab startfin abab assms(3)
diff -r 7b92238454d6 -r 18bf20d0c2df src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy
--- a/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy	Mon Apr 26 16:08:04 2010 +0200
+++ b/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy	Mon Apr 26 09:45:22 2010 -0700
@@ -24,7 +24,7 @@
 lemma dest_vec1_simps[simp]: fixes a::"real^1"
   shows "a$1 = 0 \<longleftrightarrow> a = 0" (*"a \<le> 1 \<longleftrightarrow> dest_vec1 a \<le> 1" "0 \<le> a \<longleftrightarrow> 0 \<le> dest_vec1 a"*)
   "a \<le> b \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 b" "dest_vec1 (1::real^1) = 1"
-  by(auto simp add:vector_component_simps forall_1 Cart_eq)
+  by(auto simp add: vector_le_def Cart_eq)
 
 lemma norm_not_0:"(x::real^'n)\<noteq>0 \<Longrightarrow> norm x \<noteq> 0" by auto
 
@@ -50,7 +50,7 @@
 lemma mem_interval_1: fixes x :: "real^1" shows
  "(x \<in> {a .. b} \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 x \<and> dest_vec1 x \<le> dest_vec1 b)"
  "(x \<in> {a<..<b} \<longleftrightarrow> dest_vec1 a < dest_vec1 x \<and> dest_vec1 x < dest_vec1 b)"
-by(simp_all add: Cart_eq vector_less_def vector_le_def forall_1)
+by(simp_all add: Cart_eq vector_less_def vector_le_def)
 
 lemma image_smult_interval:"(\<lambda>x. m *\<^sub>R (x::real^'n)) ` {a..b} =
   (if {a..b} = {} then {} else if 0 \<le> m then {m *\<^sub>R a..m *\<^sub>R b} else {m *\<^sub>R b..m *\<^sub>R a})"
@@ -66,8 +66,7 @@
   apply(rule_tac [!] allI)apply(rule_tac [!] impI)
   apply(rule_tac[2] x="vec1 x" in exI)apply(rule_tac[4] x="vec1 x" in exI)
   apply(rule_tac[6] x="vec1 x" in exI)apply(rule_tac[8] x="vec1 x" in exI)
-  by (auto simp add: vector_less_def vector_le_def forall_1
-    vec1_dest_vec1[unfolded One_nat_def])
+  by (auto simp add: vector_less_def vector_le_def)
 
 lemma dest_vec1_setsum: assumes "finite S"
   shows " dest_vec1 (setsum f S) = setsum (\<lambda>x. dest_vec1 (f x)) S"
@@ -90,7 +89,7 @@
   using one_le_card_finite by auto
 
 lemma real_dimindex_ge_1:"real (CARD('n::finite)) \<ge> 1" 
-  by(metis dimindex_ge_1 linorder_not_less real_eq_of_nat real_le_trans real_of_nat_1 real_of_nat_le_iff) 
+  by(metis dimindex_ge_1 real_eq_of_nat real_of_nat_1 real_of_nat_le_iff) 
 
 lemma real_dimindex_gt_0:"real (CARD('n::finite)) > 0" apply(rule less_le_trans[OF _ real_dimindex_ge_1]) by auto
 
@@ -480,8 +479,8 @@
 next 
   fix t u assume asm:"\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s" "finite (t::'a set)"
   (*"finite t" "t \<subseteq> s" "\<forall>x\<in>t. (0::real) \<le> u x" "setsum u t = 1"*)
-  from this(2) have "\<forall>u. t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> setsum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s" apply(induct_tac t rule:finite_induct)
-    prefer 3 apply (rule,rule) apply(erule conjE)+ proof-
+  from this(2) have "\<forall>u. t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> setsum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s" apply(induct t rule:finite_induct)
+    prefer 2 apply (rule,rule) apply(erule conjE)+ proof-
     fix x f u assume ind:"\<forall>u. f \<subseteq> s \<and> (\<forall>x\<in>f. 0 \<le> u x) \<and> setsum u f = 1 \<longrightarrow> (\<Sum>x\<in>f. u x *\<^sub>R x) \<in> s"
     assume as:"finite f" "x \<notin> f" "insert x f \<subseteq> s" "\<forall>x\<in>insert x f. 0 \<le> u x" "setsum u (insert x f) = (1::real)"
     show "(\<Sum>x\<in>insert x f. u x *\<^sub>R x) \<in> s" proof(cases "u x = 1")
@@ -776,7 +775,7 @@
 lemma convex_cball:
   fixes x :: "'a::real_normed_vector"
   shows "convex(cball x e)"
-proof(auto simp add: convex_def Ball_def mem_cball)
+proof(auto simp add: convex_def Ball_def)
   fix y z assume yz:"dist x y \<le> e" "dist x z \<le> e"
   fix u v ::real assume uv:" 0 \<le> u" "0 \<le> v" "u + v = 1"
   have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> u * dist x y + v * dist x z" using uv yz
@@ -1144,7 +1143,7 @@
     hence "x + y \<in> s" using `?lhs`[unfolded convex_def, THEN conjunct1]
       apply(erule_tac x="2*\<^sub>R x" in ballE) apply(erule_tac x="2*\<^sub>R y" in ballE)
       apply(erule_tac x="1/2" in allE) apply simp apply(erule_tac x="1/2" in allE) by auto  }
-  thus ?thesis unfolding convex_def cone_def by auto
+  thus ?thesis unfolding convex_def cone_def by blast
 qed
 
 lemma affine_dependent_biggerset: fixes s::"(real^'n) set"
@@ -1259,7 +1258,7 @@
   fixes s :: "'a::real_normed_vector set"
   assumes "open s"
   shows "open(convex hull s)"
-  unfolding open_contains_cball convex_hull_explicit unfolding mem_Collect_eq ball_simps(10) 
+  unfolding open_contains_cball convex_hull_explicit unfolding mem_Collect_eq ball_simps(10)
 proof(rule, rule) fix a
   assume "\<exists>sa u. finite sa \<and> sa \<subseteq> s \<and> (\<forall>x\<in>sa. 0 \<le> u x) \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = a"
   then obtain t u where obt:"finite t" "t\<subseteq>s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = 1" "(\<Sum>v\<in>t. u v *\<^sub>R v) = a" by auto
@@ -1279,7 +1278,7 @@
       hence "Min i \<le> b x" unfolding i_def apply(rule_tac Min_le) using obt(1) by auto
       hence "x + (y - a) \<in> cball x (b x)" using y unfolding mem_cball dist_norm by auto
       moreover from `x\<in>t` have "x\<in>s" using obt(2) by auto
-      ultimately have "x + (y - a) \<in> s" using y and b[THEN bspec[where x=x]] unfolding subset_eq by auto }
+      ultimately have "x + (y - a) \<in> s" using y and b[THEN bspec[where x=x]] unfolding subset_eq by fast }
     moreover
     have *:"inj_on (\<lambda>v. v + (y - a)) t" unfolding inj_on_def by auto
     have "(\<Sum>v\<in>(\<lambda>v. v + (y - a)) ` t. u (v - (y - a))) = 1"
@@ -1349,7 +1348,7 @@
   show ?thesis unfolding caratheodory[of s]
   proof(induct ("CARD('n) + 1"))
     have *:"{x.\<exists>sa. finite sa \<and> sa \<subseteq> s \<and> card sa \<le> 0 \<and> x \<in> convex hull sa} = {}" 
-      using compact_empty by (auto simp add: convex_hull_empty)
+      using compact_empty by auto
     case 0 thus ?case unfolding * by simp
   next
     case (Suc n)
@@ -1359,11 +1358,11 @@
         fix x assume "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
         then obtain t where t:"finite t" "t \<subseteq> s" "card t \<le> Suc n" "x \<in> convex hull t" by auto
         show "x\<in>s" proof(cases "card t = 0")
-          case True thus ?thesis using t(4) unfolding card_0_eq[OF t(1)] by(simp add: convex_hull_empty)
+          case True thus ?thesis using t(4) unfolding card_0_eq[OF t(1)] by simp
         next
           case False hence "card t = Suc 0" using t(3) `n=0` by auto
           then obtain a where "t = {a}" unfolding card_Suc_eq by auto
-          thus ?thesis using t(2,4) by (simp add: convex_hull_singleton)
+          thus ?thesis using t(2,4) by simp
         qed
       next
         fix x assume "x\<in>s"
@@ -1398,7 +1397,7 @@
           show ?P proof(cases "u={}")
             case True hence "x=a" using t(4)[unfolded au] by auto
             show ?P unfolding `x=a` apply(rule_tac x=a in exI, rule_tac x=a in exI, rule_tac x=1 in exI)
-              using t and `n\<noteq>0` unfolding au by(auto intro!: exI[where x="{a}"] simp add: convex_hull_singleton)
+              using t and `n\<noteq>0` unfolding au by(auto intro!: exI[where x="{a}"])
           next
             case False obtain ux vx b where obt:"ux\<ge>0" "vx\<ge>0" "ux + vx = 1" "b \<in> convex hull u" "x = ux *\<^sub>R a + vx *\<^sub>R b"
               using t(4)[unfolded au convex_hull_insert[OF False]] by auto
@@ -1411,7 +1410,7 @@
       qed
       thus ?thesis using compact_convex_combinations[OF assms Suc] by simp 
     qed
-  qed 
+  qed
 qed
 
 lemma finite_imp_compact_convex_hull:
@@ -1851,7 +1850,7 @@
 lemma convex_hull_scaling:
   "convex hull ((\<lambda>x. c *\<^sub>R x) ` s) = (\<lambda>x. c *\<^sub>R x) ` (convex hull s)"
   apply(cases "c=0") defer apply(rule convex_hull_bilemma[rule_format, of _ _ inverse]) apply(rule convex_hull_scaling_lemma)
-  unfolding image_image scaleR_scaleR by(auto simp add:image_constant_conv convex_hull_eq_empty)
+  unfolding image_image scaleR_scaleR by(auto simp add:image_constant_conv)
 
 lemma convex_hull_affinity:
   "convex hull ((\<lambda>x. a + c *\<^sub>R x) ` s) = (\<lambda>x. a + c *\<^sub>R x) ` (convex hull s)"
@@ -2313,7 +2312,7 @@
     hence "a < b" unfolding * using as(4) apply(rule_tac mult_left_less_imp_less) by(auto simp add: field_simps)
     hence "u * a + v * b \<le> b" unfolding ** using **(2) as(3) by(auto simp add: field_simps intro!:mult_right_mono) }
   ultimately show "u *\<^sub>R x + v *\<^sub>R y \<in> s" apply- apply(rule assms[unfolded is_interval_def, rule_format, OF as(1,2)])
-    using as(3-) dimindex_ge_1 apply- by(auto simp add: vector_component) qed
+    using as(3-) dimindex_ge_1 by auto qed
 
 lemma is_interval_connected:
   fixes s :: "(real ^ _) set"
@@ -2336,7 +2335,7 @@
   hence *:"dest_vec1 a < dest_vec1 x" "dest_vec1 x < dest_vec1 b" apply(rule_tac [!] ccontr) unfolding not_less by auto
   let ?halfl = "{z. inner (basis 1) z < dest_vec1 x} " and ?halfr = "{z. inner (basis 1) z > dest_vec1 x} "
   { fix y assume "y \<in> s" have "y \<in> ?halfr \<union> ?halfl" apply(rule ccontr)
-    using as(6) `y\<in>s` by (auto simp add: inner_vector_def dest_vec1_eq) }
+    using as(6) `y\<in>s` by (auto simp add: inner_vector_def) }
   moreover have "a\<in>?halfl" "b\<in>?halfr" using * by (auto simp add: inner_vector_def)
   hence "?halfl \<inter> s \<noteq> {}" "?halfr \<inter> s \<noteq> {}"  using as(2-3) by auto
   ultimately show False apply(rule_tac notE[OF as(1)[unfolded connected_def]])
@@ -2374,7 +2373,7 @@
   shows "\<exists>x\<in>{a..b}. (f x)$k = y"
   apply(subst neg_equal_iff_equal[THEN sym]) unfolding vector_uminus_component[THEN sym]
   apply(rule ivt_increasing_component_on_1) using assms using continuous_on_neg
-  by(auto simp add:vector_uminus_component)
+  by auto
 
 lemma ivt_decreasing_component_1: fixes f::"real^1 \<Rightarrow> real^'n"
   shows "dest_vec1 a \<le> dest_vec1 b \<Longrightarrow> \<forall>x\<in>{a .. b}. continuous (at x) f
@@ -2415,7 +2414,7 @@
         unfolding i'(1) xi_def apply(rule_tac Min_le) unfolding image_iff
         defer apply(rule_tac x=j in bexI) using i' by auto
       have i01:"x$i \<le> 1" "x$i > 0" using Suc(2)[unfolded mem_interval,rule_format,of i] using i'(2) `x$i \<noteq> 0`
-        by(auto simp add: Cart_lambda_beta) 
+        by auto
       show ?thesis proof(cases "x$i=1")
         case True have "\<forall>j\<in>{i. x$i \<noteq> 0}. x$j = 1" apply(rule, rule ccontr) unfolding mem_Collect_eq proof-
           fix j assume "x $ j \<noteq> 0" "x $ j \<noteq> 1"
@@ -2424,21 +2423,21 @@
           hence "x$j \<ge> x$i" unfolding i'(1) xi_def apply(rule_tac Min_le) by auto
           thus False using True Suc(2) j by(auto simp add: vector_le_def elim!:ballE[where x=j]) qed
         thus "x\<in>convex hull ?points" apply(rule_tac hull_subset[unfolded subset_eq, rule_format])
-          by(auto simp add: Cart_lambda_beta)
+          by auto
       next let ?y = "\<lambda>j. if x$j = 0 then 0 else (x$j - x$i) / (1 - x$i)"
         case False hence *:"x = x$i *\<^sub>R (\<chi> j. if x$j = 0 then 0 else 1) + (1 - x$i) *\<^sub>R (\<chi> j. ?y j)" unfolding Cart_eq
-          by(auto simp add: Cart_lambda_beta vector_add_component vector_smult_component vector_minus_component field_simps)
+          by(auto simp add: field_simps)
         { fix j have "x$j \<noteq> 0 \<Longrightarrow> 0 \<le> (x $ j - x $ i) / (1 - x $ i)" "(x $ j - x $ i) / (1 - x $ i) \<le> 1"
             apply(rule_tac divide_nonneg_pos) using i(1)[of j] using False i01
-            using Suc(2)[unfolded mem_interval, rule_format, of j] by(auto simp add:field_simps Cart_lambda_beta) 
+            using Suc(2)[unfolded mem_interval, rule_format, of j] by(auto simp add:field_simps)
           hence "0 \<le> ?y j \<and> ?y j \<le> 1" by auto }
-        moreover have "i\<in>{j. x$j \<noteq> 0} - {j. ((\<chi> j. ?y j)::real^'n) $ j \<noteq> 0}" using i01 by(auto simp add: Cart_lambda_beta)
+        moreover have "i\<in>{j. x$j \<noteq> 0} - {j. ((\<chi> j. ?y j)::real^'n) $ j \<noteq> 0}" using i01 by auto
         hence "{j. x$j \<noteq> 0} \<noteq> {j. ((\<chi> j. ?y j)::real^'n) $ j \<noteq> 0}" by auto
-        hence **:"{j. ((\<chi> j. ?y j)::real^'n) $ j \<noteq> 0} \<subset> {j. x$j \<noteq> 0}" apply - apply rule by(auto simp add: Cart_lambda_beta)  
+        hence **:"{j. ((\<chi> j. ?y j)::real^'n) $ j \<noteq> 0} \<subset> {j. x$j \<noteq> 0}" apply - apply rule by auto
         have "card {j. ((\<chi> j. ?y j)::real^'n) $ j \<noteq> 0} \<le> n" using less_le_trans[OF psubset_card_mono[OF _ **] Suc(4)] by auto
         ultimately show ?thesis apply(subst *) apply(rule convex_convex_hull[unfolded convex_def, rule_format])
           apply(rule_tac hull_subset[unfolded subset_eq, rule_format]) defer apply(rule Suc(1))
-          unfolding mem_interval using i01 Suc(3) by (auto simp add: Cart_lambda_beta)
+          unfolding mem_interval using i01 Suc(3) by auto
       qed qed qed } note * = this
   show ?thesis apply rule defer apply(rule hull_minimal) unfolding subset_eq prefer 3 apply rule 
     apply(rule_tac n2="CARD('n)" in *) prefer 3 apply(rule card_mono) using 01 and convex_interval(1) prefer 5 apply - apply rule
@@ -2453,7 +2452,7 @@
   prefer 3 apply(rule unit_interval_convex_hull) apply rule unfolding mem_Collect_eq proof-
   fix x::"real^'n" assume as:"\<forall>i. x $ i = 0 \<or> x $ i = 1"
   show "x \<in> (\<lambda>s. \<chi> i. if i \<in> s then 1 else 0) ` UNIV" apply(rule image_eqI[where x="{i. x$i = 1}"])
-    unfolding Cart_eq using as by(auto simp add:Cart_lambda_beta) qed auto
+    unfolding Cart_eq using as by auto qed auto
 
 subsection {* Hence any cube (could do any nonempty interval). *}
 
@@ -2464,23 +2463,23 @@
     unfolding image_iff defer apply(erule bexE) proof-
     fix y assume as:"y\<in>{x - ?d .. x + ?d}"
     { fix i::'n have "x $ i \<le> d + y $ i" "y $ i \<le> d + x $ i" using as[unfolded mem_interval, THEN spec[where x=i]]
-        by(auto simp add: vector_component)
+        by auto
       hence "1 \<ge> inverse d * (x $ i - y $ i)" "1 \<ge> inverse d * (y $ i - x $ i)"
         apply(rule_tac[!] mult_left_le_imp_le[OF _ assms]) unfolding mult_assoc[THEN sym]
-        using assms by(auto simp add: field_simps right_inverse) 
+        using assms by(auto simp add: field_simps)
       hence "inverse d * (x $ i * 2) \<le> 2 + inverse d * (y $ i * 2)"
             "inverse d * (y $ i * 2) \<le> 2 + inverse d * (x $ i * 2)" by(auto simp add:field_simps) }
     hence "inverse (2 * d) *\<^sub>R (y - (x - ?d)) \<in> {0..1}" unfolding mem_interval using assms
-      by(auto simp add: Cart_eq vector_component_simps field_simps)
+      by(auto simp add: Cart_eq field_simps)
     thus "\<exists>z\<in>{0..1}. y = x - ?d + (2 * d) *\<^sub>R z" apply- apply(rule_tac x="inverse (2 * d) *\<^sub>R (y - (x - ?d))" in bexI) 
-      using assms by(auto simp add: Cart_eq vector_le_def Cart_lambda_beta)
+      using assms by(auto simp add: Cart_eq vector_le_def)
   next
     fix y z assume as:"z\<in>{0..1}" "y = x - ?d + (2*d) *\<^sub>R z" 
     have "\<And>i. 0 \<le> d * z $ i \<and> d * z $ i \<le> d" using assms as(1)[unfolded mem_interval] apply(erule_tac x=i in allE)
       apply rule apply(rule mult_nonneg_nonneg) prefer 3 apply(rule mult_right_le_one_le)
-      using assms by(auto simp add: vector_component_simps Cart_eq)
+      using assms by(auto simp add: Cart_eq)
     thus "y \<in> {x - ?d..x + ?d}" unfolding as(2) mem_interval apply- apply rule using as(1)[unfolded mem_interval]
-      apply(erule_tac x=i in allE) using assms by(auto simp add:  vector_component_simps Cart_eq) qed
+      apply(erule_tac x=i in allE) using assms by(auto simp add: Cart_eq) qed
   obtain s where "finite s" "{0..1::real^'n} = convex hull s" using unit_cube_convex_hull by auto
   thus ?thesis apply(rule_tac that[of "(\<lambda>y. x - ?d + (2 * d) *\<^sub>R y)` s"]) unfolding * and convex_hull_affinity by auto qed
 
@@ -2570,8 +2569,8 @@
   have "0 < d" unfolding d_def using `e>0` dimge1 by(rule_tac divide_pos_pos, auto) 
   let ?d = "(\<chi> i. d)::real^'n"
   obtain c where c:"finite c" "{x - ?d..x + ?d} = convex hull c" using cube_convex_hull[OF `d>0`, of x] by auto
-  have "x\<in>{x - ?d..x + ?d}" using `d>0` unfolding mem_interval by(auto simp add:vector_component_simps)
-  hence "c\<noteq>{}" using c by(auto simp add:convex_hull_empty)
+  have "x\<in>{x - ?d..x + ?d}" using `d>0` unfolding mem_interval by auto
+  hence "c\<noteq>{}" using c by auto
   def k \<equiv> "Max (f ` c)"
   have "convex_on {x - ?d..x + ?d} f" apply(rule convex_on_subset[OF assms(2)])
     apply(rule subset_trans[OF _ e(1)]) unfolding subset_eq mem_cball proof 
@@ -2579,7 +2578,7 @@
     have e:"e = setsum (\<lambda>i. d) (UNIV::'n set)" unfolding setsum_constant d_def using dimge1
       by (metis eq_divide_imp mult_frac_num real_dimindex_gt_0 real_eq_of_nat real_less_def real_mult_commute)
     show "dist x z \<le> e" unfolding dist_norm e apply(rule_tac order_trans[OF norm_le_l1], rule setsum_mono)
-      using z[unfolded mem_interval] apply(erule_tac x=i in allE) by(auto simp add:field_simps vector_component_simps) qed
+      using z[unfolded mem_interval] apply(erule_tac x=i in allE) by auto qed
   hence k:"\<forall>y\<in>{x - ?d..x + ?d}. f y \<le> k" unfolding c(2) apply(rule_tac convex_on_convex_hull_bound) apply assumption
     unfolding k_def apply(rule, rule Max_ge) using c(1) by auto
   have "d \<le> e" unfolding d_def apply(rule mult_imp_div_pos_le) using `e>0` dimge1 unfolding mult_le_cancel_left1 using real_dimindex_ge_1 by auto
@@ -2588,9 +2587,9 @@
   hence "\<forall>y\<in>cball x d. abs (f y) \<le> k + 2 * abs (f x)" apply(rule_tac convex_bounds_lemma) apply assumption proof
     fix y assume y:"y\<in>cball x d"
     { fix i::'n have "x $ i - d \<le> y $ i"  "y $ i \<le> x $ i + d" 
-        using order_trans[OF component_le_norm y[unfolded mem_cball dist_norm], of i] by(auto simp add: vector_component)  }
+        using order_trans[OF component_le_norm y[unfolded mem_cball dist_norm], of i] by auto  }
     thus "f y \<le> k" apply(rule_tac k[rule_format]) unfolding mem_cball mem_interval dist_norm 
-      by(auto simp add: vector_component_simps) qed
+      by auto qed
   hence "continuous_on (ball x d) f" apply(rule_tac convex_on_bounded_continuous)
     apply(rule open_ball, rule convex_on_subset[OF conv], rule ball_subset_cball)
     apply force
@@ -2716,17 +2715,17 @@
         unfolding as(1) by(auto simp add:algebra_simps)
       show "norm (a - x) *\<^sub>R (x - b) = norm (x - b) *\<^sub>R (a - x)"
         unfolding norm_minus_commute[of x a] * Cart_eq using as(2,3)
-        by(auto simp add: vector_component_simps field_simps)
+        by(auto simp add: field_simps)
     next assume as:"dist a b = dist a x + dist x b"
       have "norm (a - x) / norm (a - b) \<le> 1" unfolding divide_le_eq_1_pos[OF Fal2] unfolding as[unfolded dist_norm] norm_ge_zero by auto 
       thus "\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> u \<and> u \<le> 1" apply(rule_tac x="dist a x / dist a b" in exI)
         unfolding dist_norm Cart_eq apply- apply rule defer apply(rule, rule divide_nonneg_pos) prefer 4 proof rule
           fix i::'n have "((1 - norm (a - x) / norm (a - b)) *\<^sub>R a + (norm (a - x) / norm (a - b)) *\<^sub>R b) $ i =
             ((norm (a - b) - norm (a - x)) * (a $ i) + norm (a - x) * (b $ i)) / norm (a - b)"
-            using Fal by(auto simp add:vector_component_simps field_simps)
+            using Fal by(auto simp add: field_simps)
           also have "\<dots> = x$i" apply(rule divide_eq_imp[OF Fal])
             unfolding as[unfolded dist_norm] using as[unfolded dist_triangle_eq Cart_eq,rule_format, of i]
-            by(auto simp add:field_simps vector_component_simps)
+            by(auto simp add:field_simps)
           finally show "x $ i = ((1 - norm (a - x) / norm (a - b)) *\<^sub>R a + (norm (a - x) / norm (a - b)) *\<^sub>R b) $ i" by auto
         qed(insert Fal2, auto) qed qed
 
@@ -2735,7 +2734,7 @@
   "between (b,a) (midpoint a b)" (is ?t2)
 proof- have *:"\<And>x y z. x = (1/2::real) *\<^sub>R z \<Longrightarrow> y = (1/2) *\<^sub>R z \<Longrightarrow> norm z = norm x + norm y" by auto
   show ?t1 ?t2 unfolding between midpoint_def dist_norm apply(rule_tac[!] *)
-    by(auto simp add:field_simps Cart_eq vector_component_simps) qed
+    by(auto simp add:field_simps Cart_eq) qed
 
 lemma between_mem_convex_hull:
   "between (a,b) x \<longleftrightarrow> x \<in> convex hull {a,b}"
@@ -2754,7 +2753,7 @@
     have *:"y = (1 - (1 - e)) *\<^sub>R ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) + (1 - e) *\<^sub>R x" using `e>0` by (auto simp add: scaleR_left_diff_distrib scaleR_right_diff_distrib)
     have "dist c ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) = abs(1/e) * norm (e *\<^sub>R c - y + (1 - e) *\<^sub>R x)"
       unfolding dist_norm unfolding norm_scaleR[THEN sym] apply(rule norm_eqI) using `e>0`
-      by(auto simp add:vector_component_simps Cart_eq field_simps) 
+      by(auto simp add: Cart_eq field_simps) 
     also have "\<dots> = abs(1/e) * norm (x - e *\<^sub>R (x - c) - y)" by(auto intro!:norm_eqI simp add: algebra_simps)
     also have "\<dots> < d" using as[unfolded dist_norm] and `e>0`
       by(auto simp add:pos_divide_less_eq[OF `e>0`] real_mult_commute)
@@ -2826,11 +2825,11 @@
   fix x::"real^'n" and e assume "0<e" and as:"\<forall>xa. dist x xa < e \<longrightarrow> (\<forall>x. 0 \<le> xa $ x) \<and> setsum (op $ xa) UNIV \<le> 1"
   show "(\<forall>xa. 0 < x $ xa) \<and> setsum (op $ x) UNIV < 1" apply(rule,rule) proof-
     fix i::'n show "0 < x $ i" using as[THEN spec[where x="x - (e / 2) *\<^sub>R basis i"]] and `e>0`
-      unfolding dist_norm by(auto simp add: norm_basis vector_component_simps basis_component elim:allE[where x=i])
+      unfolding dist_norm by(auto simp add: norm_basis elim:allE[where x=i])
   next guess a using UNIV_witness[where 'a='n] ..
     have **:"dist x (x + (e / 2) *\<^sub>R basis a) < e" using  `e>0` and norm_basis[of a]
-      unfolding dist_norm by(auto simp add: vector_component_simps basis_component intro!: mult_strict_left_mono_comm)
-    have "\<And>i. (x + (e / 2) *\<^sub>R basis a) $ i = x$i + (if i = a then e/2 else 0)" by(auto simp add:vector_component_simps)
+      unfolding dist_norm by(auto intro!: mult_strict_left_mono_comm)
+    have "\<And>i. (x + (e / 2) *\<^sub>R basis a) $ i = x$i + (if i = a then e/2 else 0)" by auto
     hence *:"setsum (op $ (x + (e / 2) *\<^sub>R basis a)) UNIV = setsum (\<lambda>i. x$i + (if a = i then e/2 else 0)) UNIV" by(rule_tac setsum_cong, auto) 
     have "setsum (op $ x) UNIV < setsum (op $ (x + (e / 2) *\<^sub>R basis a)) UNIV" unfolding * setsum_addf
       using `0<e` dimindex_ge_1 by(auto simp add: setsum_delta')
@@ -2847,13 +2846,13 @@
     fix y assume y:"dist x y < min (Min (op $ x ` UNIV)) ?d"
     have "setsum (op $ y) UNIV \<le> setsum (\<lambda>i. x$i + ?d) UNIV" proof(rule setsum_mono)
       fix i::'n have "abs (y$i - x$i) < ?d" apply(rule le_less_trans) using component_le_norm[of "y - x" i]
-        using y[unfolded min_less_iff_conj dist_norm, THEN conjunct2] by(auto simp add:vector_component_simps norm_minus_commute)
+        using y[unfolded min_less_iff_conj dist_norm, THEN conjunct2] by(auto simp add: norm_minus_commute)
       thus "y $ i \<le> x $ i + ?d" by auto qed
     also have "\<dots> \<le> 1" unfolding setsum_addf setsum_constant card_enum real_eq_of_nat using dimindex_ge_1 by(auto simp add: Suc_le_eq)
     finally show "(\<forall>i. 0 \<le> y $ i) \<and> setsum (op $ y) UNIV \<le> 1" apply- proof(rule,rule)
       fix i::'n have "norm (x - y) < x$i" using y[unfolded min_less_iff_conj dist_norm, THEN conjunct1]
-        using Min_gr_iff[of "op $ x ` dimset x"] dimindex_ge_1 by auto
-      thus "0 \<le> y$i" using component_le_norm[of "x - y" i] and as(1)[rule_format, of i] by(auto simp add: vector_component_simps)
+        by auto
+      thus "0 \<le> y$i" using component_le_norm[of "x - y" i] and as(1)[rule_format, of i] by auto
     qed auto qed auto qed
 
 lemma interior_std_simplex_nonempty: obtains a::"real^'n" where
@@ -3040,9 +3039,6 @@
   apply(erule_tac x="1-x" in ballE, erule_tac x="1-y" in ballE)
   unfolding mem_interval_1 by auto
 
-(** move this **)
-declare vector_scaleR_component[simp]
-
 lemma simple_path_join_loop:
   assumes "injective_path g1" "injective_path g2" "pathfinish g2 = pathstart g1"
   "(path_image g1 \<inter> path_image g2) \<subseteq> {pathstart g1,pathstart g2}"
@@ -3390,7 +3386,7 @@
         hence "\<psi> k \<noteq> \<psi> (Suc k)" using \<psi>[unfolded bij_betw_def inj_on_def, THEN conjunct1, THEN bspec[where x=k]] by auto
         hence **:"?basis k + ?basis (Suc k) \<in> {x. 0 < inner (?basis (Suc k)) x} \<inter> (?A k)" 
           "?basis k - ?basis (Suc k) \<in> {x. 0 > inner (?basis (Suc k)) x} \<inter> ({x. 0 < inner (?basis (Suc k)) x} \<union> (?A k))" using d
-          by(auto simp add: inner_basis vector_component_simps intro!:bexI[where x=k])
+          by(auto simp add: inner_basis intro!:bexI[where x=k])
         show ?thesis unfolding * Un_assoc apply(rule path_connected_Un) defer apply(rule path_connected_Un) 
           prefer 5 apply(rule_tac[1-2] convex_imp_path_connected, rule convex_halfspace_lt, rule convex_halfspace_gt)
           apply(rule Suc(1)) using d ** False by auto
@@ -3404,7 +3400,7 @@
           apply(rule_tac[5] x=" ?basis 1 + ?basis 2" in nequals0I)
           apply(rule_tac[6] x="-?basis 1 + ?basis 2" in nequals0I)
           apply(rule_tac[7] x="-?basis 1 - ?basis 2" in nequals0I)
-          using d unfolding *** by(auto intro!: convex_halfspace_gt convex_halfspace_lt, auto simp add:vector_component_simps inner_basis)
+          using d unfolding *** by(auto intro!: convex_halfspace_gt convex_halfspace_lt, auto simp add: inner_basis)
   qed qed auto qed note lem = this
 
   have ***:"\<And>x::real^'n. (\<exists>i\<in>{1..CARD('n)}. inner (basis (\<psi> i)) x \<noteq> 0) \<longleftrightarrow> (\<exists>i. inner (basis i) x \<noteq> 0)"
@@ -3432,7 +3428,7 @@
     apply(rule, rule continuous_at_within_inv[unfolded o_def inverse_eq_divide]) apply(rule continuous_at_within)
     apply(rule continuous_at_norm[unfolded o_def]) by auto
   thus ?thesis unfolding * ** using path_connected_punctured_universe[OF assms]
-    by(auto intro!: path_connected_continuous_image continuous_on_intros continuous_on_mul) qed
+    by(auto intro!: path_connected_continuous_image continuous_on_intros) qed
 
 lemma connected_sphere: "2 \<le> CARD('n) \<Longrightarrow> connected {x::real^'n. norm(x - a) = r}"
   using path_connected_sphere path_connected_imp_connected by auto
diff -r 7b92238454d6 -r 18bf20d0c2df src/HOL/Multivariate_Analysis/Derivative.thy
--- a/src/HOL/Multivariate_Analysis/Derivative.thy	Mon Apr 26 16:08:04 2010 +0200
+++ b/src/HOL/Multivariate_Analysis/Derivative.thy	Mon Apr 26 09:45:22 2010 -0700
@@ -94,7 +94,7 @@
 
 subsection {* Derivatives on real = Derivatives on @{typ "real^1"} *}
 
-lemma dist_vec1_0[simp]: "dist(vec1 (x::real)) 0 = norm x" unfolding vector_dist_norm by(auto simp add:vec1_dest_vec1_simps)
+lemma dist_vec1_0[simp]: "dist(vec1 (x::real)) 0 = norm x" unfolding vector_dist_norm by auto
 
 lemma bounded_linear_vec1_dest_vec1: fixes f::"real \<Rightarrow> real"
   shows "linear (vec1 \<circ> f \<circ> dest_vec1) = bounded_linear f" (is "?l = ?r") proof-
@@ -282,6 +282,8 @@
 
 subsection {* differentiability. *}
 
+no_notation Deriv.differentiable (infixl "differentiable" 60)
+
 definition differentiable :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" (infixr "differentiable" 30) where
   "f differentiable net \<equiv> (\<exists>f'. (f has_derivative f') net)"
 
@@ -755,11 +757,11 @@
 lemma onorm_vec1: fixes f::"real \<Rightarrow> real"
   shows "onorm (\<lambda>x. vec1 (f (dest_vec1 x))) = onorm f" proof-
   have "\<forall>x::real^1. norm x = 1 \<longleftrightarrow> x\<in>{vec1 -1, vec1 (1::real)}" unfolding forall_vec1 by(auto simp add:Cart_eq)
-  hence 1:"{x. norm x = 1} = {vec1 -1, vec1 (1::real)}" by(auto simp add:norm_vec1)
+  hence 1:"{x. norm x = 1} = {vec1 -1, vec1 (1::real)}" by auto
   have 2:"{norm (vec1 (f (dest_vec1 x))) |x. norm x = 1} = (\<lambda>x. norm (vec1 (f (dest_vec1 x)))) ` {x. norm x=1}" by auto
   have "\<forall>x::real. norm x = 1 \<longleftrightarrow> x\<in>{-1, 1}" by auto hence 3:"{x. norm x = 1} = {-1, (1::real)}" by auto
   have 4:"{norm (f x) |x. norm x = 1} = (\<lambda>x. norm (f x)) ` {x. norm x=1}" by auto
-  show ?thesis unfolding onorm_def 1 2 3 4 by(simp add:Sup_finite_Max norm_vec1) qed
+  show ?thesis unfolding onorm_def 1 2 3 4 by(simp add:Sup_finite_Max) qed
 
 lemma differentiable_bound_real: fixes f::"real \<Rightarrow> real"
   assumes "convex s" "\<forall>x\<in>s. (f has_derivative f' x) (at x within s)" "\<forall>x\<in>s. onorm(f' x) \<le> B" and x:"x\<in>s" and y:"y\<in>s"
@@ -1014,7 +1016,7 @@
 	  unfolding ph' * apply(rule diff_chain_within) defer apply(rule bounded_linear.has_derivative[OF assms(3)])
 	  apply(rule has_derivative_intros) defer apply(rule has_derivative_sub[where g'="\<lambda>x.0",unfolded diff_0_right])
 	  apply(rule has_derivative_at_within) using assms(5) and `u\<in>s` `a\<in>s`
-	  by(auto intro!: has_derivative_intros derivative_linear)
+          by(auto intro!: has_derivative_intros derivative_linear)
 	have **:"bounded_linear (\<lambda>x. f' u x - f' a x)" "bounded_linear (\<lambda>x. f' a x - f' u x)" apply(rule_tac[!] bounded_linear_sub)
 	  apply(rule_tac[!] derivative_linear) using assms(5) `u\<in>s` `a\<in>s` by auto
 	have "onorm (\<lambda>v. v - g' (f' u v)) \<le> onorm g' * onorm (\<lambda>w. f' a w - f' u w)" unfolding * apply(rule onorm_compose)
diff -r 7b92238454d6 -r 18bf20d0c2df src/HOL/Multivariate_Analysis/Determinants.thy
--- a/src/HOL/Multivariate_Analysis/Determinants.thy	Mon Apr 26 16:08:04 2010 +0200
+++ b/src/HOL/Multivariate_Analysis/Determinants.thy	Mon Apr 26 09:45:22 2010 -0700
@@ -813,7 +813,7 @@
 lemma cramer:
   fixes A ::"real^'n^'n"
   assumes d0: "det A \<noteq> 0"
-  shows "A *v x = b \<longleftrightarrow> x = (\<chi> k. det(\<chi> i j. if j=k then b$i else A$i$j :: real^'n^'n) / det A)"
+  shows "A *v x = b \<longleftrightarrow> x = (\<chi> k. det(\<chi> i j. if j=k then b$i else A$i$j) / det A)"
 proof-
   from d0 obtain B where B: "A ** B = mat 1" "B ** A = mat 1"
     unfolding invertible_det_nz[symmetric] invertible_def by blast
@@ -821,7 +821,7 @@
   hence "A *v (B *v b) = b" by (simp add: matrix_vector_mul_assoc)
   then have xe: "\<exists>x. A*v x = b" by blast
   {fix x assume x: "A *v x = b"
-  have "x = (\<chi> k. det(\<chi> i j. if j=k then b$i else A$i$j :: real^'n^'n) / det A)"
+  have "x = (\<chi> k. det(\<chi> i j. if j=k then b$i else A$i$j) / det A)"
     unfolding x[symmetric]
     using d0 by (simp add: Cart_eq cramer_lemma field_simps)}
   with xe show ?thesis by auto
@@ -993,15 +993,15 @@
     moreover
     {assume "x = 0" "y \<noteq> 0"
       then have "dist (?g x) (?g y) = dist x y"
-        apply (simp add: dist_norm norm_mul)
+        apply (simp add: dist_norm)
         apply (rule f1[rule_format])
-        by(simp add: norm_mul field_simps)}
+        by(simp add: field_simps)}
     moreover
     {assume "x \<noteq> 0" "y = 0"
       then have "dist (?g x) (?g y) = dist x y"
-        apply (simp add: dist_norm norm_mul)
+        apply (simp add: dist_norm)
         apply (rule f1[rule_format])
-        by(simp add: norm_mul field_simps)}
+        by(simp add: field_simps)}
     moreover
     {assume z: "x \<noteq> 0" "y \<noteq> 0"
       have th00: "x = norm x *s (inverse (norm x) *s x)" "y = norm y *s (inverse (norm y) *s y)" "norm x *s f ((inverse (norm x) *s x)) = norm x *s f (inverse (norm x) *s x)"
@@ -1013,7 +1013,7 @@
         "norm (f (inverse (norm x) *s x) - f (inverse (norm y) *s y)) =
         norm (inverse (norm x) *s x - inverse (norm y) *s y)"
         using z
-        by (auto simp add: vector_smult_assoc field_simps norm_mul intro: f1[rule_format] fd1[rule_format, unfolded dist_norm])
+        by (auto simp add: vector_smult_assoc field_simps intro: f1[rule_format] fd1[rule_format, unfolded dist_norm])
       from z th0[OF th00] have "dist (?g x) (?g y) = dist x y"
         by (simp add: dist_norm)}
     ultimately have "dist (?g x) (?g y) = dist x y" by blast}
@@ -1047,7 +1047,7 @@
   by (simp add: nat_number setprod_numseg mult_commute)
 
 lemma det_1: "det (A::'a::comm_ring_1^1^1) = A$1$1"
-  by (simp add: det_def permutes_sing sign_id UNIV_1)
+  by (simp add: det_def sign_id UNIV_1)
 
 lemma det_2: "det (A::'a::comm_ring_1^2^2) = A$1$1 * A$2$2 - A$1$2 * A$2$1"
 proof-
@@ -1057,7 +1057,7 @@
   unfolding setsum_over_permutations_insert[OF f12]
   unfolding permutes_sing
   apply (simp add: sign_swap_id sign_id swap_id_eq)
-  by (simp add: arith_simps(31)[symmetric] of_int_minus of_int_1 del: arith_simps(31))
+  by (simp add: arith_simps(31)[symmetric] del: arith_simps(31))
 qed
 
 lemma det_3: "det (A::'a::comm_ring_1^3^3) =
@@ -1078,7 +1078,7 @@
 
   unfolding permutes_sing
   apply (simp add: sign_swap_id permutation_swap_id sign_compose sign_id swap_id_eq)
-  apply (simp add: arith_simps(31)[symmetric] of_int_minus of_int_1 del: arith_simps(31))
+  apply (simp add: arith_simps(31)[symmetric] del: arith_simps(31))
   by (simp add: field_simps)
 qed
 
diff -r 7b92238454d6 -r 18bf20d0c2df src/HOL/Multivariate_Analysis/Euclidean_Space.thy
--- a/src/HOL/Multivariate_Analysis/Euclidean_Space.thy	Mon Apr 26 16:08:04 2010 +0200
+++ b/src/HOL/Multivariate_Analysis/Euclidean_Space.thy	Mon Apr 26 09:45:22 2010 -0700
@@ -670,7 +670,7 @@
 subsection{* The collapse of the general concepts to dimension one. *}
 
 lemma vector_one: "(x::'a ^1) = (\<chi> i. (x$1))"
-  by (simp add: Cart_eq forall_1)
+  by (simp add: Cart_eq)
 
 lemma forall_one: "(\<forall>(x::'a ^1). P x) \<longleftrightarrow> (\<forall>x. P(\<chi> i. x))"
   apply auto
@@ -775,8 +775,7 @@
 lemma sqrt_even_pow2: assumes n: "even n"
   shows "sqrt(2 ^ n) = 2 ^ (n div 2)"
 proof-
-  from n obtain m where m: "n = 2*m" unfolding even_nat_equiv_def2
-    by (auto simp add: nat_number)
+  from n obtain m where m: "n = 2*m" unfolding even_mult_two_ex ..
   from m  have "sqrt(2 ^ n) = sqrt ((2 ^ m) ^ 2)"
     by (simp only: power_mult[symmetric] mult_commute)
   then show ?thesis  using m by simp
@@ -785,7 +784,7 @@
 lemma real_div_sqrt: "0 <= x ==> x / sqrt(x) = sqrt(x)"
   apply (cases "x = 0", simp_all)
   using sqrt_divide_self_eq[of x]
-  apply (simp add: inverse_eq_divide real_sqrt_ge_0_iff field_simps)
+  apply (simp add: inverse_eq_divide field_simps)
   done
 
 text{* Hence derive more interesting properties of the norm. *}
@@ -798,8 +797,8 @@
   by (rule norm_zero)
 
 lemma norm_mul[simp]: "norm(a *s x) = abs(a) * norm x"
-  by (simp add: norm_vector_def vector_component setL2_right_distrib
-           abs_mult cong: strong_setL2_cong)
+  by (simp add: norm_vector_def setL2_right_distrib abs_mult)
+
 lemma norm_eq_0_dot: "(norm x = 0) \<longleftrightarrow> (inner x x = (0::real))"
   by (simp add: norm_vector_def setL2_def power2_eq_square)
 lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_zero)
@@ -866,10 +865,14 @@
   by (auto simp add: norm_eq_sqrt_inner)
 
 lemma real_abs_le_square_iff: "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> (x::real)^2 \<le> y^2"
-proof-
-  have "x^2 \<le> y^2 \<longleftrightarrow> (x -y) * (y + x) \<le> 0" by (simp add: field_simps power2_eq_square)
-  also have "\<dots> \<longleftrightarrow> \<bar>x\<bar> \<le> \<bar>y\<bar>" apply (simp add: zero_compare_simps real_abs_def not_less) by arith
-finally show ?thesis ..
+proof
+  assume "\<bar>x\<bar> \<le> \<bar>y\<bar>"
+  then have "\<bar>x\<bar>\<twosuperior> \<le> \<bar>y\<bar>\<twosuperior>" by (rule power_mono, simp)
+  then show "x\<twosuperior> \<le> y\<twosuperior>" by simp
+next
+  assume "x\<twosuperior> \<le> y\<twosuperior>"
+  then have "sqrt (x\<twosuperior>) \<le> sqrt (y\<twosuperior>)" by (rule real_sqrt_le_mono)
+  then show "\<bar>x\<bar> \<le> \<bar>y\<bar>" by simp
 qed
 
 lemma norm_le_square: "norm(x) <= a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x <= a^2"
@@ -1179,7 +1182,7 @@
   assumes fS: "finite S"
   shows "setsum f S *s v = setsum (\<lambda>x. f x *s v) S"
 proof(induct rule: finite_induct[OF fS])
-  case 1 then show ?case by (simp add: vector_smult_lzero)
+  case 1 then show ?case by simp
 next
   case (2 x F)
   from "2.hyps" have "setsum f (insert x F) *s v = (f x + setsum f F) *s v"
@@ -1398,7 +1401,7 @@
   apply (subgoal_tac "vector [v$1] = v")
   apply simp
   apply (vector vector_def)
-  apply (simp add: forall_1)
+  apply simp
   done
 
 lemma forall_vector_2: "(\<forall>v::'a::zero^2. P v) \<longleftrightarrow> (\<forall>x y. P(vector[x, y]))"
@@ -1536,7 +1539,7 @@
       unfolding norm_mul
       apply (simp only: mult_commute)
       apply (rule mult_mono)
-      by (auto simp add: field_simps norm_ge_zero) }
+      by (auto simp add: field_simps) }
     then have th: "\<forall>i\<in> ?S. norm ((x$i) *s f (basis i :: real ^'m)) \<le> norm (f (basis i)) * norm x" by metis
     from real_setsum_norm_le[OF fS, of "\<lambda>i. (x$i) *s (f (basis i))", OF th]
     have "norm (f x) \<le> ?B * norm x" unfolding th0 setsum_left_distrib by metis}
@@ -1553,9 +1556,9 @@
   let ?K = "\<bar>B\<bar> + 1"
   have Kp: "?K > 0" by arith
     {assume C: "B < 0"
-      have "norm (1::real ^ 'n) > 0" by (simp add: zero_less_norm_iff)
+      have "norm (1::real ^ 'n) > 0" by simp
       with C have "B * norm (1:: real ^ 'n) < 0"
-        by (simp add: zero_compare_simps)
+        by (simp add: mult_less_0_iff)
       with B[rule_format, of 1] norm_ge_zero[of "f 1"] have False by simp
     }
     then have Bp: "B \<ge> 0" by ferrack
@@ -1677,11 +1680,11 @@
       apply (rule real_setsum_norm_le)
       using fN fM
       apply simp
-      apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh] norm_mul field_simps)
+      apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh] field_simps)
       apply (rule mult_mono)
-      apply (auto simp add: norm_ge_zero zero_le_mult_iff component_le_norm)
+      apply (auto simp add: zero_le_mult_iff component_le_norm)
       apply (rule mult_mono)
-      apply (auto simp add: norm_ge_zero zero_le_mult_iff component_le_norm)
+      apply (auto simp add: zero_le_mult_iff component_le_norm)
       done}
   then show ?thesis by metis
 qed
@@ -1701,7 +1704,7 @@
     have "B * norm x * norm y \<le> ?K * norm x * norm y"
       apply -
       apply (rule mult_right_mono, rule mult_right_mono)
-      by (auto simp add: norm_ge_zero)
+      by auto
     then have "norm (h x y) \<le> ?K * norm x * norm y"
       using B[rule_format, of x y] by simp}
   with Kp show ?thesis by blast
@@ -2006,8 +2009,8 @@
   have uv': "u = 0 \<longrightarrow> v \<noteq> 0" using u v uv by arith
   have "a = a * (u + v)" unfolding uv  by simp
   hence th: "u * a + v * a = a" by (simp add: field_simps)
-  from xa u have "u \<noteq> 0 \<Longrightarrow> u*x < u*a" by (simp add: mult_compare_simps)
-  from ya v have "v \<noteq> 0 \<Longrightarrow> v * y < v * a" by (simp add: mult_compare_simps)
+  from xa u have "u \<noteq> 0 \<Longrightarrow> u*x < u*a" by (simp add: mult_strict_left_mono)
+  from ya v have "v \<noteq> 0 \<Longrightarrow> v * y < v * a" by (simp add: mult_strict_left_mono)
   from xa ya u v have "u * x + v * y < u * a + v * a"
     apply (cases "u = 0", simp_all add: uv')
     apply(rule mult_strict_left_mono)
@@ -2048,7 +2051,7 @@
   assumes x: "0 <= (x::real)" and y:"0 <= y" and z: "0 <= z" and xy: "x^2 <= y^2 + z^2"
   shows "x <= y + z"
 proof-
-  have "y^2 + z^2 \<le> y^2 + 2*y*z + z^2" using z y  by (simp add: zero_compare_simps)
+  have "y^2 + z^2 \<le> y^2 + 2*y*z + z^2" using z y by (simp add: mult_nonneg_nonneg)
   with xy have th: "x ^2 \<le> (y+z)^2" by (simp add: power2_eq_square field_simps)
   from y z have yz: "y + z \<ge> 0" by arith
   from power2_le_imp_le[OF th yz] show ?thesis .
@@ -2100,10 +2103,10 @@
       {assume x0: "x \<noteq> 0"
         hence n0: "norm x \<noteq> 0" by (metis norm_eq_zero)
         let ?c = "1/ norm x"
-        have "norm (?c*s x) = 1" using x0 by (simp add: n0 norm_mul)
+        have "norm (?c*s x) = 1" using x0 by (simp add: n0)
         with H have "norm (f(?c*s x)) \<le> b" by blast
         hence "?c * norm (f x) \<le> b"
-          by (simp add: linear_cmul[OF lf] norm_mul)
+          by (simp add: linear_cmul[OF lf])
         hence "norm (f x) \<le> b * norm x"
           using n0 norm_ge_zero[of x] by (auto simp add: field_simps)}
       ultimately have "norm (f x) \<le> b * norm x" by blast}
@@ -2221,18 +2224,24 @@
   where "dest_vec1 x \<equiv> (x$1)"
 
 lemma vec1_component[simp]: "(vec1 x)$1 = x"
-  by (simp add: )
-
-lemma vec1_dest_vec1[simp]: "vec1(dest_vec1 x) = x" "dest_vec1(vec1 y) = y"
-  by (simp_all add:  Cart_eq forall_1)
-
-lemma forall_vec1: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P (vec1 x))" by (metis vec1_dest_vec1)
-
-lemma exists_vec1: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. P(vec1 x))" by (metis vec1_dest_vec1)
-
-lemma vec1_eq[simp]:  "vec1 x = vec1 y \<longleftrightarrow> x = y" by (metis vec1_dest_vec1)
-
-lemma dest_vec1_eq[simp]: "dest_vec1 x = dest_vec1 y \<longleftrightarrow> x = y" by (metis vec1_dest_vec1)
+  by simp
+
+lemma vec1_dest_vec1: "vec1(dest_vec1 x) = x" "dest_vec1(vec1 y) = y"
+  by (simp_all add:  Cart_eq)
+
+declare vec1_dest_vec1(1) [simp]
+
+lemma forall_vec1: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P (vec1 x))"
+  by (metis vec1_dest_vec1(1))
+
+lemma exists_vec1: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. P(vec1 x))"
+  by (metis vec1_dest_vec1(1))
+
+lemma vec1_eq[simp]:  "vec1 x = vec1 y \<longleftrightarrow> x = y"
+  by (metis vec1_dest_vec1(2))
+
+lemma dest_vec1_eq[simp]: "dest_vec1 x = dest_vec1 y \<longleftrightarrow> x = y"
+  by (metis vec1_dest_vec1(1))
 
 lemma vec_in_image_vec: "vec x \<in> (vec ` S) \<longleftrightarrow> x \<in> S" by auto
 
@@ -2260,8 +2269,8 @@
 lemma dest_vec1_sum: assumes fS: "finite S"
   shows "dest_vec1(setsum f S) = setsum (dest_vec1 o f) S"
   apply (induct rule: finite_induct[OF fS])
-  apply (simp add: dest_vec1_vec)
-  apply (auto simp add:vector_minus_component)
+  apply simp
+  apply auto
   done
 
 lemma norm_vec1: "norm(vec1 x) = abs(x)"
@@ -2270,7 +2279,7 @@
 lemma dist_vec1: "dist(vec1 x) (vec1 y) = abs(x - y)"
   by (simp only: dist_real vec1_component)
 lemma abs_dest_vec1: "norm x = \<bar>dest_vec1 x\<bar>"
-  by (metis vec1_dest_vec1 norm_vec1)
+  by (metis vec1_dest_vec1(1) norm_vec1)
 
 lemmas vec1_dest_vec1_simps = forall_vec1 vec_add[THEN sym] dist_vec1 vec_sub[THEN sym] vec1_dest_vec1 norm_vec1 vector_smult_component
    vec1_eq vec_cmul[THEN sym] smult_conv_scaleR[THEN sym] o_def dist_real_def norm_vec1 real_norm_def
@@ -2298,7 +2307,7 @@
   shows "f = (\<lambda>x. vec1(row 1 (matrix f) \<bullet> x))"
   apply (rule ext)
   apply (subst matrix_works[OF lf, symmetric])
-  apply (simp add: Cart_eq matrix_vector_mult_def row_def inner_vector_def mult_commute forall_1)
+  apply (simp add: Cart_eq matrix_vector_mult_def row_def inner_vector_def mult_commute)
   done
 
 lemma dest_vec1_eq_0: "dest_vec1 x = 0 \<longleftrightarrow> x = 0"
@@ -2366,13 +2375,13 @@
   by (induct rule: finite_induct[OF fS], simp_all add: vec_0[symmetric] del: vec_0)
 
 lemma pastecart_vec[simp]: "pastecart (vec x) (vec x) = vec x"
-  by (simp add: pastecart_eq fstcart_pastecart sndcart_pastecart)
+  by (simp add: pastecart_eq)
 
 lemma pastecart_add[simp]:"pastecart (x1::'a::{plus,times}^_) y1 + pastecart x2 y2 = pastecart (x1 + x2) (y1 + y2)"
-  by (simp add: pastecart_eq fstcart_pastecart sndcart_pastecart)
+  by (simp add: pastecart_eq)
 
 lemma pastecart_cmul[simp]: "pastecart (c *s (x1::'a::{plus,times}^_)) (c *s y1) = c *s pastecart x1 y1"
-  by (simp add: pastecart_eq fstcart_pastecart sndcart_pastecart)
+  by (simp add: pastecart_eq)
 
 lemma pastecart_neg[simp]: "pastecart (- (x::'a::ring_1^_)) (- y) = - pastecart x y"
   unfolding vector_sneg_minus1 pastecart_cmul ..
@@ -2384,7 +2393,7 @@
   fixes f:: "'d \<Rightarrow> 'a::semiring_1^_"
   assumes fS: "finite S"
   shows "pastecart (setsum f S) (setsum g S) = setsum (\<lambda>i. pastecart (f i) (g i)) S"
-  by (simp  add: pastecart_eq fstcart_setsum[OF fS] sndcart_setsum[OF fS] fstcart_pastecart sndcart_pastecart)
+  by (simp  add: pastecart_eq fstcart_setsum[OF fS] sndcart_setsum[OF fS])
 
 lemma setsum_Plus:
   "\<lbrakk>finite A; finite B\<rbrakk> \<Longrightarrow>
@@ -2402,7 +2411,7 @@
 lemma norm_fstcart: "norm(fstcart x) <= norm (x::real ^('n::finite + 'm::finite))"
 proof-
   have th0: "norm x = norm (pastecart (fstcart x) (sndcart x))"
-    by (simp add: pastecart_fst_snd)
+    by simp
   have th1: "fstcart x \<bullet> fstcart x \<le> pastecart (fstcart x) (sndcart x) \<bullet> pastecart (fstcart x) (sndcart x)"
     by (simp add: inner_vector_def setsum_UNIV_sum pastecart_def setsum_nonneg)
   then show ?thesis
@@ -2417,7 +2426,7 @@
 lemma norm_sndcart: "norm(sndcart x) <= norm (x::real ^('n::finite + 'm::finite))"
 proof-
   have th0: "norm x = norm (pastecart (fstcart x) (sndcart x))"
-    by (simp add: pastecart_fst_snd)
+    by simp
   have th1: "sndcart x \<bullet> sndcart x \<le> pastecart (fstcart x) (sndcart x) \<bullet> pastecart (fstcart x) (sndcart x)"
     by (simp add: inner_vector_def setsum_UNIV_sum pastecart_def setsum_nonneg)
   then show ?thesis
@@ -2519,7 +2528,7 @@
 
 lemma real_arch_inv: "0 < e \<longleftrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
   using reals_Archimedean
-  apply (auto simp add: field_simps inverse_positive_iff_positive)
+  apply (auto simp add: field_simps)
   apply (subgoal_tac "inverse (real n) > 0")
   apply arith
   apply simp
@@ -2732,7 +2741,8 @@
   "0 \<in> span S"
   "x\<in> span S \<Longrightarrow> y \<in> span S ==> x + y \<in> span S"
   "x \<in> span S \<Longrightarrow> c *s x \<in> span S"
-  by (metis span_def hull_subset subset_eq subspace_span subspace_def)+
+  by (metis span_def hull_subset subset_eq)
+     (metis subspace_span subspace_def)+
 
 lemma span_induct: assumes SP: "\<And>x. x \<in> S ==> P x"
   and P: "subspace P" and x: "x \<in> span S" shows "P x"
@@ -2830,7 +2840,7 @@
 
 (* Individual closure properties. *)
 
-lemma span_superset: "x \<in> S ==> x \<in> span S" by (metis span_clauses)
+lemma span_superset: "x \<in> S ==> x \<in> span S" by (metis span_clauses(1))
 
 lemma span_0: "0 \<in> span S" by (metis subspace_span subspace_0)
 
@@ -2847,8 +2857,7 @@
   by (metis subspace_span subspace_sub)
 
 lemma span_setsum: "finite A \<Longrightarrow> \<forall>x \<in> A. f x \<in> span S ==> setsum f A \<in> span S"
-  apply (rule subspace_setsum)
-  by (metis subspace_span subspace_setsum)+
+  by (rule subspace_setsum, rule subspace_span)
 
 lemma span_add_eq: "(x::'a::ring_1^_) \<in> span S \<Longrightarrow> x + y \<in> span S \<longleftrightarrow> y \<in> span S"
   apply (auto simp only: span_add span_sub)
@@ -3110,7 +3119,7 @@
     from fS SP aP have th0: "finite ?S" "?S \<subseteq> P" "?v \<in> ?S" "?u ?v \<noteq> 0" by auto
     have s0: "setsum (\<lambda>v. ?u v *s v) ?S = 0"
       using fS aS
-      apply (simp add: vector_smult_lneg vector_smult_lid setsum_clauses field_simps )
+      apply (simp add: vector_smult_lneg setsum_clauses field_simps)
       apply (subst (2) ua[symmetric])
       apply (rule setsum_cong2)
       by auto
@@ -3479,7 +3488,7 @@
 
 lemma basis_card_eq_dim:
   "B \<subseteq> (V:: (real ^'n) set) \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B = dim V"
-  by (metis order_eq_iff independent_card_le_dim span_card_ge_dim independent_mono independent_bound)
+  by (metis order_eq_iff independent_card_le_dim span_card_ge_dim independent_bound)
 
 lemma dim_unique: "(B::(real ^'n) set) \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> card B = n \<Longrightarrow> dim V = n"
   by (metis basis_card_eq_dim)
@@ -3669,7 +3678,7 @@
         apply (subst Cy)
         using C(1) fth
         apply (simp only: setsum_clauses) unfolding smult_conv_scaleR
-        apply (auto simp add: inner_simps inner_eq_zero_iff inner_commute[of y a] dot_lsum[OF fth])
+        apply (auto simp add: inner_simps inner_commute[of y a] dot_lsum[OF fth])
         apply (rule setsum_0')
         apply clarsimp
         apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
@@ -3686,7 +3695,7 @@
         using C(1) fth
         apply (simp only: setsum_clauses) unfolding smult_conv_scaleR
         apply (subst inner_commute[of x])
-        apply (auto simp add: inner_simps inner_eq_zero_iff inner_commute[of x a] dot_rsum[OF fth])
+        apply (auto simp add: inner_simps inner_commute[of x a] dot_rsum[OF fth])
         apply (rule setsum_0')
         apply clarsimp
         apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
@@ -3759,10 +3768,10 @@
         apply (subst B') using fB fth
         unfolding setsum_clauses(2)[OF fth]
         apply simp unfolding inner_simps smult_conv_scaleR
-        apply (clarsimp simp add: inner_simps inner_eq_zero_iff smult_conv_scaleR dot_lsum)
+        apply (clarsimp simp add: inner_simps smult_conv_scaleR dot_lsum)
         apply (rule setsum_0', rule ballI)
         unfolding inner_commute
-        by (auto simp add: x field_simps inner_eq_zero_iff intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])}
+        by (auto simp add: x field_simps intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])}
     then show "\<forall>x \<in> B. ?a \<bullet> x = 0" by blast
   qed
   with a0 show ?thesis unfolding sSB by (auto intro: exI[where x="?a"])
@@ -3915,7 +3924,7 @@
     {assume xb: "x \<in> b"
       have h0: "0 = ?h x"
         apply (rule conjunct2[OF h, rule_format])
-        apply (metis  span_superset insertI1 xb x)
+        apply (metis  span_superset x)
         apply simp
         apply (metis span_superset xb)
         done
@@ -4262,8 +4271,7 @@
     {fix y have "?P y"
       proof(rule span_induct_alt[of ?P "columns A"])
         show "\<exists>x\<Colon>real ^ 'm. setsum (\<lambda>i. (x$i) *s column i A) ?U = 0"
-          apply (rule exI[where x=0])
-          by (simp add: zero_index vector_smult_lzero)
+          by (rule exI[where x=0], simp)
       next
         fix c y1 y2 assume y1: "y1 \<in> columns A" and y2: "?P y2"
         from y1 obtain i where i: "i \<in> ?U" "y1 = column i A"
@@ -4687,7 +4695,7 @@
   hence d2: "(sqrt (real ?d))^2 = real ?d"
     by (auto intro: real_sqrt_pow2)
   have th: "sqrt (real ?d) * infnorm x \<ge> 0"
-    by (simp add: zero_le_mult_iff real_sqrt_ge_0_iff infnorm_pos_le)
+    by (simp add: zero_le_mult_iff infnorm_pos_le)
   have th1: "x \<bullet> x \<le> (sqrt (real ?d) * infnorm x)^2"
     unfolding power_mult_distrib d2
     unfolding real_of_nat_def inner_vector_def
@@ -4861,4 +4869,3 @@
 done
 
 end
- 
\ No newline at end of file
diff -r 7b92238454d6 -r 18bf20d0c2df src/HOL/Multivariate_Analysis/Integration.thy
--- a/src/HOL/Multivariate_Analysis/Integration.thy	Mon Apr 26 16:08:04 2010 +0200
+++ b/src/HOL/Multivariate_Analysis/Integration.thy	Mon Apr 26 09:45:22 2010 -0700
@@ -933,7 +933,7 @@
 lemma has_integral_vec1: assumes "(f has_integral k) {a..b}"
   shows "((\<lambda>x. vec1 (f x)) has_integral (vec1 k)) {a..b}"
 proof- have *:"\<And>p. (\<Sum>(x, k)\<in>p. content k *\<^sub>R vec1 (f x)) - vec1 k = vec1 ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - k)"
-    unfolding vec_sub Cart_eq by(auto simp add:vec1_dest_vec1_simps split_beta)
+    unfolding vec_sub Cart_eq by(auto simp add: split_beta)
   show ?thesis using assms unfolding has_integral apply safe
     apply(erule_tac x=e in allE,safe) apply(rule_tac x=d in exI,safe)
     apply(erule_tac x=p in allE,safe) unfolding * norm_vector_1 by auto qed
@@ -1356,7 +1356,7 @@
 
 lemma has_integral_eq_eq:
   shows "\<forall>x\<in>s. f x = g x \<Longrightarrow> ((f has_integral k) s \<longleftrightarrow> (g has_integral k) s)"
-  using has_integral_eq[of s f g] has_integral_eq[of s g f] by auto 
+  using has_integral_eq[of s f g] has_integral_eq[of s g f] by rule auto
 
 lemma has_integral_null[dest]:
   assumes "content({a..b}) = 0" shows  "(f has_integral 0) ({a..b})"
@@ -1653,7 +1653,7 @@
 proof- have *:"{a..b} = ({a..b} \<inter> {x. x$k \<le> c}) \<union> ({a..b} \<inter> {x. x$k \<ge> c})" by auto
   show ?thesis apply(subst *) apply(rule tagged_division_union[OF assms])
     unfolding interval_split interior_closed_interval
-    by(auto simp add: vector_less_def Cart_lambda_beta elim!:allE[where x=k]) qed
+    by(auto simp add: vector_less_def elim!:allE[where x=k]) qed
 
 lemma has_integral_separate_sides: fixes f::"real^'m \<Rightarrow> 'a::real_normed_vector"
   assumes "(f has_integral i) ({a..b})" "e>0"
@@ -1743,11 +1743,11 @@
 subsection {* Using additivity of lifted function to encode definedness. *}
 
 lemma forall_option: "(\<forall>x. P x) \<longleftrightarrow> P None \<and> (\<forall>x. P(Some x))"
-  by (metis map_of.simps option.nchotomy)
+  by (metis option.nchotomy)
 
 lemma exists_option:
  "(\<exists>x. P x) \<longleftrightarrow> P None \<or> (\<exists>x. P(Some x))" 
-  by (metis map_of.simps option.nchotomy)
+  by (metis option.nchotomy)
 
 fun lifted where 
   "lifted (opp::'a\<Rightarrow>'a\<Rightarrow>'b) (Some x) (Some y) = Some(opp x y)" |
@@ -1842,9 +1842,8 @@
 lemma operative_content[intro]: "operative (op +) content"
   unfolding operative_def content_split[THEN sym] neutral_add by auto
 
-lemma neutral_monoid[simp]: "neutral ((op +)::('a::comm_monoid_add) \<Rightarrow> 'a \<Rightarrow> 'a) = 0"
-  unfolding neutral_def apply(rule some_equality) defer
-  apply(erule_tac x=0 in allE) by auto
+lemma neutral_monoid: "neutral ((op +)::('a::comm_monoid_add) \<Rightarrow> 'a \<Rightarrow> 'a) = 0"
+  by (rule neutral_add) (* FIXME: duplicate *)
 
 lemma monoidal_monoid[intro]:
   shows "monoidal ((op +)::('a::comm_monoid_add) \<Rightarrow> 'a \<Rightarrow> 'a)"
@@ -1941,7 +1940,7 @@
     apply(rule_tac x="(k,(interval_lowerbound l)$k)" in exI) defer
     apply(rule_tac x="(k,(interval_upperbound l)$k)" in exI)
     unfolding division_points_def unfolding interval_bounds[OF ab]
-    apply (auto simp add:interval_bounds) unfolding * by auto
+    apply auto unfolding * by auto
   thus "?D1 \<subset> ?D" apply-apply(rule,rule division_points_subset[OF assms(1-4)]) by auto
 
   have *:"interval_lowerbound ({a..b} \<inter> {x. x $ k \<ge> interval_lowerbound l $ k}) $ k = interval_lowerbound l $ k"
@@ -1952,7 +1951,7 @@
     apply(rule_tac x="(k,(interval_lowerbound l)$k)" in exI) defer
     apply(rule_tac x="(k,(interval_upperbound l)$k)" in exI)
     unfolding division_points_def unfolding interval_bounds[OF ab]
-    apply (auto simp add:interval_bounds) unfolding * by auto
+    apply auto unfolding * by auto
   thus "?D2 \<subset> ?D" apply-apply(rule,rule division_points_subset[OF assms(1-4)]) by auto qed
 
 subsection {* Preservation by divisions and tagged divisions. *}
@@ -2254,7 +2253,7 @@
   assumes "p tagged_division_of {a..b}"  "\<forall>x\<in>{a..b}. (f x)$i \<le> (g x)$i"
   shows "(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p)$i \<le> (setsum (\<lambda>(x,k). content k *\<^sub>R g x) p)$i"
   unfolding setsum_component apply(rule setsum_mono)
-  apply(rule mp) defer apply assumption apply(induct_tac x,rule) unfolding split_conv
+  apply(rule mp) defer apply assumption unfolding split_paired_all apply rule unfolding split_conv
 proof- fix a b assume ab:"(a,b) \<in> p" note assm = tagged_division_ofD(2-4)[OF assms(1) ab]
   from this(3) guess u v apply-by(erule exE)+ note b=this
   show "(content b *\<^sub>R f a) $ i \<le> (content b *\<^sub>R g a) $ i" unfolding b
@@ -2903,7 +2902,7 @@
   shows "setsum (\<lambda>(x,k). f(interval_upperbound k) - f(interval_lowerbound k)) p = f b - f a"
 proof- let ?f = "(\<lambda>k::(real^1) set. if k = {} then 0 else f(interval_upperbound k) - f(interval_lowerbound k))"
   have *:"operative op + ?f" unfolding operative_1_lt[OF monoidal_monoid] interval_eq_empty_1
-    by(auto simp add:not_less interval_bound_1 vector_less_def)
+    by(auto simp add:not_less vector_less_def)
   have **:"{a..b} \<noteq> {}" using assms(1) by auto note operative_tagged_division[OF monoidal_monoid * assms(2)]
   note * = this[unfolded if_not_P[OF **] interval_bound_1[OF assms(1)],THEN sym ]
   show ?thesis unfolding * apply(subst setsum_iterate[THEN sym]) defer
@@ -3340,7 +3339,7 @@
 proof- { presume *:"a < b \<Longrightarrow> ?thesis" 
     show ?thesis proof(cases,rule *,assumption)
       assume "\<not> a < b" hence "a = b" using assms(1) by auto
-      hence *:"{vec a .. vec b} = {vec b}" "f b - f a = 0" apply(auto simp add: Cart_simps) by smt
+      hence *:"{vec a .. vec b} = {vec b}" "f b - f a = 0" by(auto simp add: Cart_eq vector_le_def order_antisym)
       show ?thesis unfolding *(2) apply(rule has_integral_null) unfolding content_eq_0_1 using * `a=b` by auto
     qed } assume ab:"a < b"
   let ?P = "\<lambda>e. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {vec1 a..vec1 b} \<and> d fine p \<longrightarrow>
@@ -3422,7 +3421,7 @@
         hence "\<forall>i. u$i \<le> v$i" and uv:"{u,v}\<subseteq>{u..v}" using p(2)[OF as(1)] by auto note this(1) this(1)[unfolded forall_1]
         note result = as(2)[unfolded k interval_bounds[OF this(1)] content_1[OF this(2)]]
 
-        assume as':"x \<noteq> vec1 a" "x \<noteq> vec1 b" hence "x$1 \<in> {a<..<b}" using p(2-3)[OF as(1)] by(auto simp add:Cart_simps) note  * = d(2)[OF this] 
+        assume as':"x \<noteq> vec1 a" "x \<noteq> vec1 b" hence "x$1 \<in> {a<..<b}" using p(2-3)[OF as(1)] by(auto simp add: Cart_eq) note  * = d(2)[OF this]
         have "norm ((v$1 - u$1) *\<^sub>R f' (x$1) - (f (v$1) - f (u$1))) =
           norm ((f (u$1) - f (x$1) - (u$1 - x$1) *\<^sub>R f' (x$1)) - (f (v$1) - f (x$1) - (v$1 - x$1) *\<^sub>R f' (x$1)))" 
           apply(rule arg_cong[of _ _ norm]) unfolding scaleR_left.diff by auto 
@@ -3651,7 +3650,7 @@
 
 lemma indefinite_integral_continuous: fixes f::"real^1 \<Rightarrow> 'a::banach"
   assumes "f integrable_on {a..b}" shows  "continuous_on {a..b} (\<lambda>x. integral {a..x} f)"
-proof(unfold continuous_on_def, safe)  fix x e assume as:"x\<in>{a..b}" "0<(e::real)"
+proof(unfold continuous_on_iff, safe)  fix x e assume as:"x\<in>{a..b}" "0<(e::real)"
   let ?thesis = "\<exists>d>0. \<forall>x'\<in>{a..b}. dist x' x < d \<longrightarrow> dist (integral {a..x'} f) (integral {a..x} f) < e"
   { presume *:"a<b \<Longrightarrow> ?thesis"
     show ?thesis apply(cases,rule *,assumption)
@@ -3669,7 +3668,7 @@
     from indefinite_integral_continuous_left[OF assms(1) `a<b` this `e>0`] guess d . note d=this
     show ?thesis apply(rule,rule,rule d,safe) apply(subst dist_commute)
       unfolding `x=b` vector_dist_norm apply(rule d(2)[rule_format]) unfolding norm_real by auto
-  next assume "a<x \<and> x<b" hence xl:"a<x" "x\<le>b" and xr:"a\<le>x" "x<b" by(auto simp add:Cart_simps)
+  next assume "a<x \<and> x<b" hence xl:"a<x" "x\<le>b" and xr:"a\<le>x" "x<b" by(auto simp add: vector_less_def)
     from indefinite_integral_continuous_left [OF assms(1) xl `e>0`] guess d1 . note d1=this
     from indefinite_integral_continuous_right[OF assms(1) xr `e>0`] guess d2 . note d2=this
     show ?thesis apply(rule_tac x="min d1 d2" in exI)
@@ -3726,7 +3725,7 @@
     unfolding o_def using assms(5) defer apply-apply(rule)
   proof- fix t assume as:"t\<in>{0..1} - {t. (1 - t) *\<^sub>R c + t *\<^sub>R x \<in> k}"
     have *:"c - t *\<^sub>R c + t *\<^sub>R x \<in> s - k" apply safe apply(rule conv[unfolded scaleR_simps]) 
-      using `x\<in>s` `c\<in>s` as by(auto simp add:scaleR_simps)
+      using `x\<in>s` `c\<in>s` as by(auto simp add: algebra_simps)
     have "(f \<circ> (\<lambda>t. (1 - t) *\<^sub>R c + t *\<^sub>R x) has_derivative (\<lambda>x. 0) \<circ> (\<lambda>z. (0 - z *\<^sub>R c) + z *\<^sub>R x)) (at t within {0..1})"
       apply(rule diff_chain_within) apply(rule has_derivative_add)
       unfolding scaleR_simps apply(rule has_derivative_sub) apply(rule has_derivative_const)
@@ -3949,7 +3948,7 @@
 
 lemma has_integral_spike_set_eq: fixes f::"real^'n \<Rightarrow> 'a::banach" 
   assumes "negligible((s - t) \<union> (t - s))" shows "((f has_integral y) s \<longleftrightarrow> (f has_integral y) t)"
-  unfolding has_integral_restrict_univ[THEN sym,of f] apply(rule has_integral_spike_eq[OF assms]) by auto
+  unfolding has_integral_restrict_univ[THEN sym,of f] apply(rule has_integral_spike_eq[OF assms]) by (safe, auto split: split_if_asm)
 
 lemma has_integral_spike_set[dest]: fixes f::"real^'n \<Rightarrow> 'a::banach"
   assumes "negligible((s - t) \<union> (t - s))" "(f has_integral y) s"
diff -r 7b92238454d6 -r 18bf20d0c2df src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
--- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Mon Apr 26 16:08:04 2010 +0200
+++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Mon Apr 26 09:45:22 2010 -0700
@@ -48,16 +48,17 @@
   "\<And>S T. openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S\<inter>T)"
   "\<And>K. (\<forall>S \<in> K. openin U S) \<Longrightarrow> openin U (\<Union>K)"
   using openin[of U] unfolding istopology_def Collect_def mem_def
-  by (metis mem_def subset_eq)+
+  unfolding subset_eq Ball_def mem_def by auto
 
 lemma openin_subset[intro]: "openin U S \<Longrightarrow> S \<subseteq> topspace U"
   unfolding topspace_def by blast
 lemma openin_empty[simp]: "openin U {}" by (simp add: openin_clauses)
 
 lemma openin_Int[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<inter> T)"
-  by (simp add: openin_clauses)
-
-lemma openin_Union[intro]: "(\<forall>S \<in>K. openin U S) \<Longrightarrow> openin U (\<Union> K)" by (simp add: openin_clauses)
+  using openin_clauses by simp
+
+lemma openin_Union[intro]: "(\<forall>S \<in>K. openin U S) \<Longrightarrow> openin U (\<Union> K)"
+  using openin_clauses by simp
 
 lemma openin_Un[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<union> T)"
   using openin_Union[of "{S,T}" U] by auto
@@ -946,7 +947,7 @@
   by (metis frontier_def closure_closed Diff_subset)
 
 lemma frontier_empty[simp]: "frontier {} = {}"
-  by (simp add: frontier_def closure_empty)
+  by (simp add: frontier_def)
 
 lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"
 proof-
@@ -954,7 +955,7 @@
     hence "closure S \<subseteq> S" using interior_subset unfolding frontier_def by auto
     hence "closed S" using closure_subset_eq by auto
   }
-  thus ?thesis using frontier_subset_closed[of S] by auto
+  thus ?thesis using frontier_subset_closed[of S] ..
 qed
 
 lemma frontier_complement: "frontier(- S) = frontier S"
@@ -968,7 +969,7 @@
 
 definition
   at_infinity :: "'a::real_normed_vector net" where
-  "at_infinity = Abs_net (range (\<lambda>r. {x. r \<le> norm x}))"
+  "at_infinity = Abs_net (\<lambda>P. \<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x)"
 
 definition
   indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a net" (infixr "indirection" 70) where
@@ -976,23 +977,23 @@
 
 text{* Prove That They are all nets. *}
 
-lemma Rep_net_at_infinity:
-  "Rep_net at_infinity = range (\<lambda>r. {x. r \<le> norm x})"
+lemma eventually_at_infinity:
+  "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. norm x >= b \<longrightarrow> P x)"
 unfolding at_infinity_def
-apply (rule Abs_net_inverse')
-apply (rule image_nonempty, simp)
-apply (clarsimp, rename_tac r s)
-apply (rule_tac x="max r s" in exI, auto)
-done
-
-lemma within_UNIV: "net within UNIV = net"
-  by (simp add: Rep_net_inject [symmetric] Rep_net_within)
+proof (rule eventually_Abs_net, rule is_filter.intro)
+  fix P Q :: "'a \<Rightarrow> bool"
+  assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<exists>s. \<forall>x. s \<le> norm x \<longrightarrow> Q x"
+  then obtain r s where
+    "\<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<forall>x. s \<le> norm x \<longrightarrow> Q x" by auto
+  then have "\<forall>x. max r s \<le> norm x \<longrightarrow> P x \<and> Q x" by simp
+  then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x \<and> Q x" ..
+qed auto
 
 subsection{* Identify Trivial limits, where we can't approach arbitrarily closely. *}
 
 definition
   trivial_limit :: "'a net \<Rightarrow> bool" where
-  "trivial_limit net \<longleftrightarrow> {} \<in> Rep_net net"
+  "trivial_limit net \<longleftrightarrow> eventually (\<lambda>x. False) net"
 
 lemma trivial_limit_within:
   shows "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"
@@ -1000,21 +1001,21 @@
   assume "trivial_limit (at a within S)"
   thus "\<not> a islimpt S"
     unfolding trivial_limit_def
-    unfolding Rep_net_within Rep_net_at
+    unfolding eventually_within eventually_at_topological
     unfolding islimpt_def
     apply (clarsimp simp add: expand_set_eq)
     apply (rename_tac T, rule_tac x=T in exI)
-    apply (clarsimp, drule_tac x=y in spec, simp)
+    apply (clarsimp, drule_tac x=y in bspec, simp_all)
     done
 next
   assume "\<not> a islimpt S"
   thus "trivial_limit (at a within S)"
     unfolding trivial_limit_def
-    unfolding Rep_net_within Rep_net_at
+    unfolding eventually_within eventually_at_topological
     unfolding islimpt_def
-    apply (clarsimp simp add: image_image)
-    apply (rule_tac x=T in image_eqI)
-    apply (auto simp add: expand_set_eq)
+    apply clarsimp
+    apply (rule_tac x=T in exI)
+    apply auto
     done
 qed
 
@@ -1030,14 +1031,14 @@
 lemma trivial_limit_at_infinity:
   "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,zero_neq_one}) net)"
   (* FIXME: find a more appropriate type class *)
-  unfolding trivial_limit_def Rep_net_at_infinity
-  apply (clarsimp simp add: expand_set_eq)
-  apply (drule_tac x="scaleR r (sgn 1)" in spec)
+  unfolding trivial_limit_def eventually_at_infinity
+  apply clarsimp
+  apply (rule_tac x="scaleR b (sgn 1)" in exI)
   apply (simp add: norm_sgn)
   done
 
 lemma trivial_limit_sequentially[intro]: "\<not> trivial_limit sequentially"
-  by (auto simp add: trivial_limit_def Rep_net_sequentially)
+  by (auto simp add: trivial_limit_def eventually_sequentially)
 
 subsection{* Some property holds "sufficiently close" to the limit point. *}
 
@@ -1045,10 +1046,6 @@
   "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
 unfolding eventually_at dist_nz by auto
 
-lemma eventually_at_infinity:
-  "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. norm x >= b \<longrightarrow> P x)"
-unfolding eventually_def Rep_net_at_infinity by auto
-
 lemma eventually_within: "eventually P (at a within S) \<longleftrightarrow>
         (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
 unfolding eventually_within eventually_at dist_nz by auto
@@ -1059,18 +1056,20 @@
 by auto (metis Rats_dense_in_nn_real order_le_less_trans order_refl) 
 
 lemma eventually_happens: "eventually P net ==> trivial_limit net \<or> (\<exists>x. P x)"
-  unfolding eventually_def trivial_limit_def
-  using Rep_net_nonempty [of net] by auto
+  unfolding trivial_limit_def
+  by (auto elim: eventually_rev_mp)
 
 lemma always_eventually: "(\<forall>x. P x) ==> eventually P net"
-  unfolding eventually_def trivial_limit_def
-  using Rep_net_nonempty [of net] by auto
+proof -
+  assume "\<forall>x. P x" hence "P = (\<lambda>x. True)" by (simp add: ext)
+  thus "eventually P net" by simp
+qed
 
 lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"
-  unfolding trivial_limit_def eventually_def by auto
+  unfolding trivial_limit_def by (auto elim: eventually_rev_mp)
 
 lemma eventually_False: "eventually (\<lambda>x. False) net \<longleftrightarrow> trivial_limit net"
-  unfolding trivial_limit_def eventually_def by auto
+  unfolding trivial_limit_def ..
 
 lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"
   apply (safe elim!: trivial_limit_eventually)
@@ -1590,7 +1589,7 @@
 
 (* FIXME: Only one congruence rule for tendsto can be used at a time! *)
 
-lemma Lim_cong_within[cong add]:
+lemma Lim_cong_within(*[cong add]*):
   fixes a :: "'a::metric_space"
   fixes l :: "'b::metric_space" (* TODO: generalize *)
   shows "(\<And>x. x \<noteq> a \<Longrightarrow> f x = g x) ==> ((\<lambda>x. f x) ---> l) (at a within S) \<longleftrightarrow> ((g ---> l) (at a within S))"
@@ -1669,7 +1668,7 @@
   { fix e::real assume "e>0"
     hence "\<exists>N::nat. \<forall>n::nat\<ge>N. inverse (real n) < e"
       using real_arch_inv[of e] apply auto apply(rule_tac x=n in exI)
-      by (metis not_le le_imp_inverse_le not_less real_of_nat_gt_zero_cancel_iff real_of_nat_less_iff xt1(7))
+      by (metis le_imp_inverse_le not_less real_of_nat_gt_zero_cancel_iff real_of_nat_less_iff xt1(7))
   }
   thus ?thesis unfolding Lim_sequentially dist_norm by simp
 qed
@@ -1704,7 +1703,7 @@
   apply (simp add: interior_def, safe)
   apply (force simp add: open_contains_cball)
   apply (rule_tac x="ball x e" in exI)
-  apply (simp add: open_ball centre_in_ball subset_trans [OF ball_subset_cball])
+  apply (simp add: subset_trans [OF ball_subset_cball])
   done
 
 lemma islimpt_ball:
@@ -1879,14 +1878,14 @@
 lemma frontier_ball:
   fixes a :: "'a::real_normed_vector"
   shows "0 < e ==> frontier(ball a e) = {x. dist a x = e}"
-  apply (simp add: frontier_def closure_ball interior_open open_ball order_less_imp_le)
+  apply (simp add: frontier_def closure_ball interior_open order_less_imp_le)
   apply (simp add: expand_set_eq)
   by arith
 
 lemma frontier_cball:
   fixes a :: "'a::{real_normed_vector, perfect_space}"
   shows "frontier(cball a e) = {x. dist a x = e}"
-  apply (simp add: frontier_def interior_cball closed_cball closure_closed order_less_imp_le)
+  apply (simp add: frontier_def interior_cball closed_cball order_less_imp_le)
   apply (simp add: expand_set_eq)
   by arith
 
@@ -2006,9 +2005,10 @@
 lemma bounded_ball[simp,intro]: "bounded(ball x e)"
   by (metis ball_subset_cball bounded_cball bounded_subset)
 
-lemma finite_imp_bounded[intro]: assumes "finite S" shows "bounded S"
+lemma finite_imp_bounded[intro]:
+  fixes S :: "'a::metric_space set" assumes "finite S" shows "bounded S"
 proof-
-  { fix a F assume as:"bounded F"
+  { fix a and F :: "'a set" assume as:"bounded F"
     then obtain x e where "\<forall>y\<in>F. dist x y \<le> e" unfolding bounded_def by auto
     hence "\<forall>y\<in>(insert a F). dist x y \<le> max e (dist x a)" by auto
     hence "bounded (insert a F)" unfolding bounded_def by (intro exI)
@@ -2218,7 +2218,7 @@
 apply (rule allI, rule impI, rule ext)
 apply (erule conjE)
 apply (induct_tac x)
-apply (simp add: nat_rec_0)
+apply simp
 apply (erule_tac x="n" in allE)
 apply (simp)
 done
@@ -2650,7 +2650,8 @@
   { fix x y assume "x\<in>t" "y\<in>t" "f x = f y"
     hence "x \<in> f x"  "y \<in> f x \<longrightarrow> y = x" using f[THEN bspec[where x=x]] and `t\<subseteq>s` by auto
     hence "x = y" using `f x = f y` and f[THEN bspec[where x=y]] and `y\<in>t` and `t\<subseteq>s` by auto  }
-  hence "infinite (f ` t)" using assms(2) using finite_imageD[unfolded inj_on_def, of f t] by auto
+  hence "inj_on f t" unfolding inj_on_def by simp
+  hence "infinite (f ` t)" using assms(2) using finite_imageD by auto
   moreover
   { fix x assume "x\<in>t" "f x \<notin> g"
     from g(3) assms(3) `x\<in>t` obtain h where "h\<in>g" and "x\<in>h" by auto
@@ -3145,7 +3146,7 @@
       using `?lhs`[unfolded continuous_within Lim_within] by auto
     { fix y assume "y\<in>f ` (ball x d \<inter> s)"
       hence "y \<in> ball (f x) e" using d(2) unfolding dist_nz[THEN sym]
-        apply (auto simp add: dist_commute mem_ball) apply(erule_tac x=xa in ballE) apply auto using `e>0` by auto
+        apply (auto simp add: dist_commute) apply(erule_tac x=xa in ballE) apply auto using `e>0` by auto
     }
     hence "\<exists>d>0. f ` (ball x d \<inter> s) \<subseteq> ball (f x) e" using `d>0` unfolding subset_eq ball_def by (auto simp add: dist_commute)  }
   thus ?rhs by auto
@@ -3168,9 +3169,32 @@
 text{* For setwise continuity, just start from the epsilon-delta definitions. *}
 
 definition
-  continuous_on :: "'a::metric_space set \<Rightarrow> ('a \<Rightarrow> 'b::metric_space) \<Rightarrow> bool" where
-  "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. \<forall>e>0. \<exists>d::real>0. \<forall>x' \<in> s. dist x' x < d --> dist (f x') (f x) < e)"
-
+  continuous_on ::
+    "'a set \<Rightarrow> ('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"
+where
+  "continuous_on s f \<longleftrightarrow>
+    (\<forall>x\<in>s. \<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow>
+      (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>x'\<in>s. x' \<in> A \<longrightarrow> f x' \<in> B)))"
+
+lemma continuous_on_iff:
+  "continuous_on s f \<longleftrightarrow>
+    (\<forall>x\<in>s. \<forall>e>0. \<exists>d>0. \<forall>x'\<in>s. dist x' x < d --> dist (f x') (f x) < e)"
+unfolding continuous_on_def
+apply safe
+apply (drule (1) bspec)
+apply (drule_tac x="ball (f x) e" in spec, simp, clarify)
+apply (drule (1) open_dist [THEN iffD1, THEN bspec])
+apply (clarify, rename_tac d)
+apply (rule_tac x=d in exI, clarify)
+apply (drule_tac x=x' in bspec, assumption)
+apply (drule_tac x=x' in spec, simp add: dist_commute)
+apply (drule_tac x=x in bspec, assumption)
+apply (drule (1) open_dist [THEN iffD1, THEN bspec], clarify)
+apply (drule_tac x=e in spec, simp, clarify)
+apply (rule_tac x="ball x d" in exI, simp, clarify)
+apply (drule_tac x=x' in bspec, assumption)
+apply (simp add: dist_commute)
+done
 
 definition
   uniformly_continuous_on ::
@@ -3184,14 +3208,14 @@
 
 lemma continuous_on_o_dest_vec1: fixes f::"real \<Rightarrow> 'a::real_normed_vector"
   assumes "continuous_on {a..b::real} f" shows "continuous_on {vec1 a..vec1 b} (f o dest_vec1)"
-  using assms unfolding continuous_on_def apply safe
+  using assms unfolding continuous_on_iff apply safe
   apply(erule_tac x="x$1" in ballE,erule_tac x=e in allE) apply safe
   apply(rule_tac x=d in exI) apply safe unfolding o_def dist_real_def dist_real 
   apply(erule_tac x="dest_vec1 x'" in ballE) by(auto simp add:vector_le_def)
 
 lemma continuous_on_o_vec1: fixes f::"real^1 \<Rightarrow> 'a::real_normed_vector"
   assumes "continuous_on {a..b} f" shows "continuous_on {dest_vec1 a..dest_vec1 b} (f o vec1)"
-  using assms unfolding continuous_on_def apply safe
+  using assms unfolding continuous_on_iff apply safe
   apply(erule_tac x="vec x" in ballE,erule_tac x=e in allE) apply safe
   apply(rule_tac x=d in exI) apply safe unfolding o_def dist_real_def dist_real 
   apply(erule_tac x="vec1 x'" in ballE) by(auto simp add:vector_le_def)
@@ -3200,15 +3224,17 @@
 
 lemma uniformly_continuous_imp_continuous:
  " uniformly_continuous_on s f ==> continuous_on s f"
-  unfolding uniformly_continuous_on_def continuous_on_def by blast
+  unfolding uniformly_continuous_on_def continuous_on_iff by blast
 
 lemma continuous_at_imp_continuous_within:
  "continuous (at x) f ==> continuous (at x within s) f"
   unfolding continuous_within continuous_at using Lim_at_within by auto
 
-lemma continuous_at_imp_continuous_on: assumes "(\<forall>x \<in> s. continuous (at x) f)"
+lemma continuous_at_imp_continuous_on:
+  fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
+  assumes "(\<forall>x \<in> s. continuous (at x) f)"
   shows "continuous_on s f"
-proof(simp add: continuous_at continuous_on_def, rule, rule, rule)
+proof(simp add: continuous_at continuous_on_iff, rule, rule, rule)
   fix x and e::real assume "x\<in>s" "e>0"
   hence "eventually (\<lambda>xa. dist (f xa) (f x) < e) (at x)" using assms unfolding continuous_at tendsto_iff by auto
   then obtain d where d:"d>0" "\<forall>xa. 0 < dist xa x \<and> dist xa x < d \<longrightarrow> dist (f xa) (f x) < e" unfolding eventually_at by auto
@@ -3221,7 +3247,9 @@
 qed
 
 lemma continuous_on_eq_continuous_within:
- "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x within s) f)" (is "?lhs = ?rhs")
+  fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
+  shows "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x within s) f)"
+    (is "?lhs = ?rhs")
 proof
   assume ?rhs
   { fix x assume "x\<in>s"
@@ -3232,18 +3260,21 @@
       hence "dist (f x') (f x) < e" using `e>0` d `x'\<in>s` dist_eq_0_iff[of x' x] zero_le_dist[of x' x] as(2) by (metis dist_eq_0_iff dist_nz) }
     hence "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e" using `d>0` by auto
   }
-  thus ?lhs using `?rhs` unfolding continuous_on_def continuous_within Lim_within by auto
+  thus ?lhs using `?rhs` unfolding continuous_on_iff continuous_within Lim_within by auto
 next
   assume ?lhs
-  thus ?rhs unfolding continuous_on_def continuous_within Lim_within by blast
+  thus ?rhs unfolding continuous_on_iff continuous_within Lim_within by blast
 qed
 
 lemma continuous_on:
- "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. (f ---> f(x)) (at x within s))"
+  fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
+  shows "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. (f ---> f(x)) (at x within s))"
   by (auto simp add: continuous_on_eq_continuous_within continuous_within)
+  (* BH: maybe this should be the definition? *)
 
 lemma continuous_on_eq_continuous_at:
- "open s ==> (continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x) f))"
+  fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
+  shows "open s ==> (continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x) f))"
   by (auto simp add: continuous_on continuous_at Lim_within_open)
 
 lemma continuous_within_subset:
@@ -3252,19 +3283,22 @@
   unfolding continuous_within by(metis Lim_within_subset)
 
 lemma continuous_on_subset:
- "continuous_on s f \<Longrightarrow> t \<subseteq> s ==> continuous_on t f"
+  fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
+  shows "continuous_on s f \<Longrightarrow> t \<subseteq> s ==> continuous_on t f"
   unfolding continuous_on by (metis subset_eq Lim_within_subset)
 
 lemma continuous_on_interior:
- "continuous_on s f \<Longrightarrow> x \<in> interior s ==> continuous (at x) f"
+  fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
+  shows "continuous_on s f \<Longrightarrow> x \<in> interior s ==> continuous (at x) f"
 unfolding interior_def
 apply simp
 by (meson continuous_on_eq_continuous_at continuous_on_subset)
 
 lemma continuous_on_eq:
- "(\<forall>x \<in> s. f x = g x) \<Longrightarrow> continuous_on s f
+  fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
+  shows "(\<forall>x \<in> s. f x = g x) \<Longrightarrow> continuous_on s f
            ==> continuous_on s g"
-  by (simp add: continuous_on_def)
+  by (simp add: continuous_on_iff)
 
 text{* Characterization of various kinds of continuity in terms of sequences.  *}
 
@@ -3317,7 +3351,9 @@
   using continuous_within_sequentially[of a UNIV f] unfolding within_UNIV by auto
 
 lemma continuous_on_sequentially:
- "continuous_on s f \<longleftrightarrow>  (\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x ---> a) sequentially
+  fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
+  shows "continuous_on s f \<longleftrightarrow>
+    (\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x ---> a) sequentially
                     --> ((f o x) ---> f(a)) sequentially)" (is "?lhs = ?rhs")
 proof
   assume ?rhs thus ?lhs using continuous_within_sequentially[of _ s f] unfolding continuous_on_eq_continuous_within by auto
@@ -3436,7 +3472,7 @@
 
 lemma continuous_on_const:
  "continuous_on s (\<lambda>x. c)"
-  unfolding continuous_on_eq_continuous_within using continuous_const by blast
+  unfolding continuous_on_def by auto
 
 lemma continuous_on_cmul:
   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
@@ -3524,7 +3560,7 @@
 
 lemma continuous_on_id:
  "continuous_on s (\<lambda>x. x)"
-  unfolding continuous_on Lim_within by auto
+  unfolding continuous_on_def by auto
 
 lemma uniformly_continuous_on_id:
  "uniformly_continuous_on s (\<lambda>x. x)"
@@ -3559,7 +3595,9 @@
 qed
 
 lemma continuous_on_compose:
- "continuous_on s f \<Longrightarrow> continuous_on (f ` s) g \<Longrightarrow> continuous_on s (g o f)"
+  fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
+  fixes g :: "'b::metric_space \<Rightarrow> 'c::metric_space" (* TODO: generalize *)
+  shows "continuous_on s f \<Longrightarrow> continuous_on (f ` s) g \<Longrightarrow> continuous_on s (g o f)"
   unfolding continuous_on_eq_continuous_within using continuous_within_compose[of _ s f g] by auto
 
 lemma uniformly_continuous_on_compose:
@@ -3610,7 +3648,8 @@
 qed
 
 lemma continuous_on_open:
- "continuous_on s f \<longleftrightarrow>
+  fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
+   shows "continuous_on s f \<longleftrightarrow>
         (\<forall>t. openin (subtopology euclidean (f ` s)) t
             --> openin (subtopology euclidean s) {x \<in> s. f x \<in> t})" (is "?lhs = ?rhs")
 proof
@@ -3643,7 +3682,8 @@
 (* ------------------------------------------------------------------------- *)
 
 lemma continuous_on_closed:
- "continuous_on s f \<longleftrightarrow>  (\<forall>t. closedin (subtopology euclidean (f ` s)) t  --> closedin (subtopology euclidean s) {x \<in> s. f x \<in> t})" (is "?lhs = ?rhs")
+  fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
+  shows "continuous_on s f \<longleftrightarrow>  (\<forall>t. closedin (subtopology euclidean (f ` s)) t  --> closedin (subtopology euclidean s) {x \<in> s. f x \<in> t})" (is "?lhs = ?rhs")
 proof
   assume ?lhs
   { fix t
@@ -3667,6 +3707,7 @@
 text{* Half-global and completely global cases.                                  *}
 
 lemma continuous_open_in_preimage:
+  fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
   assumes "continuous_on s f"  "open t"
   shows "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
 proof-
@@ -3677,6 +3718,7 @@
 qed
 
 lemma continuous_closed_in_preimage:
+  fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
   assumes "continuous_on s f"  "closed t"
   shows "closedin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
 proof-
@@ -3688,6 +3730,7 @@
 qed
 
 lemma continuous_open_preimage:
+  fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
   assumes "continuous_on s f" "open s" "open t"
   shows "open {x \<in> s. f x \<in> t}"
 proof-
@@ -3697,6 +3740,7 @@
 qed
 
 lemma continuous_closed_preimage:
+  fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
   assumes "continuous_on s f" "closed s" "closed t"
   shows "closed {x \<in> s. f x \<in> t}"
 proof-
@@ -3739,14 +3783,17 @@
 text{* Equality of continuous functions on closure and related results.          *}
 
 lemma continuous_closed_in_preimage_constant:
- "continuous_on s f ==> closedin (subtopology euclidean s) {x \<in> s. f x = a}"
+  fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
+  shows "continuous_on s f ==> closedin (subtopology euclidean s) {x \<in> s. f x = a}"
   using continuous_closed_in_preimage[of s f "{a}"] closed_sing by auto
 
 lemma continuous_closed_preimage_constant:
- "continuous_on s f \<Longrightarrow> closed s ==> closed {x \<in> s. f x = a}"
+  fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
+  shows "continuous_on s f \<Longrightarrow> closed s ==> closed {x \<in> s. f x = a}"
   using continuous_closed_preimage[of s f "{a}"] closed_sing by auto
 
 lemma continuous_constant_on_closure:
+  fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
   assumes "continuous_on (closure s) f"
           "\<forall>x \<in> s. f x = a"
   shows "\<forall>x \<in> (closure s). f x = a"
@@ -3754,6 +3801,7 @@
     assms closure_minimal[of s "{x \<in> closure s. f x = a}"] closure_subset unfolding subset_eq by auto
 
 lemma image_closure_subset:
+  fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
   assumes "continuous_on (closure s) f"  "closed t"  "(f ` s) \<subseteq> t"
   shows "f ` (closure s) \<subseteq> t"
 proof-
@@ -3798,11 +3846,13 @@
 using assms using continuous_within_avoid[of x UNIV f a, unfolded within_UNIV] by auto
 
 lemma continuous_on_avoid:
+  fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
   assumes "continuous_on s f"  "x \<in> s"  "f x \<noteq> a"
   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e \<longrightarrow> f y \<noteq> a"
 using assms(1)[unfolded continuous_on_eq_continuous_within, THEN bspec[where x=x], OF assms(2)]  continuous_within_avoid[of x s f a]  assms(2,3) by auto
 
 lemma continuous_on_open_avoid:
+  fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
   assumes "continuous_on s f"  "open s"  "x \<in> s"  "f x \<noteq> a"
   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
 using assms(1)[unfolded continuous_on_eq_continuous_at[OF assms(2)], THEN bspec[where x=x], OF assms(3)]  continuous_at_avoid[of x f a]  assms(3,4) by auto
@@ -3810,22 +3860,25 @@
 text{* Proving a function is constant by proving open-ness of level set.         *}
 
 lemma continuous_levelset_open_in_cases:
- "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>
+  fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
+  shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>
         openin (subtopology euclidean s) {x \<in> s. f x = a}
         ==> (\<forall>x \<in> s. f x \<noteq> a) \<or> (\<forall>x \<in> s. f x = a)"
 unfolding connected_clopen using continuous_closed_in_preimage_constant by auto
 
 lemma continuous_levelset_open_in:
- "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>
+  fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
+  shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>
         openin (subtopology euclidean s) {x \<in> s. f x = a} \<Longrightarrow>
         (\<exists>x \<in> s. f x = a)  ==> (\<forall>x \<in> s. f x = a)"
 using continuous_levelset_open_in_cases[of s f ]
 by meson
 
 lemma continuous_levelset_open:
+  fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
   assumes "connected s"  "continuous_on s f"  "open {x \<in> s. f x = a}"  "\<exists>x \<in> s.  f x = a"
   shows "\<forall>x \<in> s. f x = a"
-using continuous_levelset_open_in[OF assms(1,2), of a, unfolded openin_open] using assms (3,4) by auto
+using continuous_levelset_open_in[OF assms(1,2), of a, unfolded openin_open] using assms (3,4) by fast
 
 text{* Some arithmetical combinations (more to prove).                           *}
 
@@ -3906,7 +3959,7 @@
     then obtain y where y:"\<forall>n. y n \<in> s \<and> x n = f (y n)" unfolding image_iff Bex_def using choice[of "\<lambda>n xa. xa \<in> s \<and> x n = f xa"] by auto
     then obtain l r where "l\<in>s" and r:"subseq r" and lr:"((y \<circ> r) ---> l) sequentially" using assms(2)[unfolded compact_def, THEN spec[where x=y]] by auto
     { fix e::real assume "e>0"
-      then obtain d where "d>0" and d:"\<forall>x'\<in>s. dist x' l < d \<longrightarrow> dist (f x') (f l) < e" using assms(1)[unfolded continuous_on_def, THEN bspec[where x=l], OF `l\<in>s`] by auto
+      then obtain d where "d>0" and d:"\<forall>x'\<in>s. dist x' l < d \<longrightarrow> dist (f x') (f l) < e" using assms(1)[unfolded continuous_on_iff, THEN bspec[where x=l], OF `l\<in>s`] by auto
       then obtain N::nat where N:"\<forall>n\<ge>N. dist ((y \<circ> r) n) l < d" using lr[unfolded Lim_sequentially, THEN spec[where x=d]] by auto
       { fix n::nat assume "n\<ge>N" hence "dist ((x \<circ> r) n) (f l) < e" using N[THEN spec[where x=n]] d[THEN bspec[where x="y (r n)"]] y[THEN spec[where x="r n"]] by auto  }
       hence "\<exists>N. \<forall>n\<ge>N. dist ((x \<circ> r) n) (f l) < e" by auto  }
@@ -3915,6 +3968,7 @@
 qed
 
 lemma connected_continuous_image:
+  fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
   assumes "continuous_on s f"  "connected s"
   shows "connected(f ` s)"
 proof-
@@ -3935,7 +3989,7 @@
   shows "uniformly_continuous_on s f"
 proof-
     { fix x assume x:"x\<in>s"
-      hence "\<forall>xa. \<exists>y. 0 < xa \<longrightarrow> (y > 0 \<and> (\<forall>x'\<in>s. dist x' x < y \<longrightarrow> dist (f x') (f x) < xa))" using assms(1)[unfolded continuous_on_def, THEN bspec[where x=x]] by auto
+      hence "\<forall>xa. \<exists>y. 0 < xa \<longrightarrow> (y > 0 \<and> (\<forall>x'\<in>s. dist x' x < y \<longrightarrow> dist (f x') (f x) < xa))" using assms(1)[unfolded continuous_on_iff, THEN bspec[where x=x]] by auto
       hence "\<exists>fa. \<forall>xa>0. \<forall>x'\<in>s. fa xa > 0 \<and> (dist x' x < fa xa \<longrightarrow> dist (f x') (f x) < xa)" using choice[of "\<lambda>e d. e>0 \<longrightarrow> d>0 \<and>(\<forall>x'\<in>s. (dist x' x < d \<longrightarrow> dist (f x') (f x) < e))"] by auto  }
     then have "\<forall>x\<in>s. \<exists>y. \<forall>xa. 0 < xa \<longrightarrow> (\<forall>x'\<in>s. y xa > 0 \<and> (dist x' x < y xa \<longrightarrow> dist (f x') (f x) < xa))" by auto
     then obtain d where d:"\<forall>e>0. \<forall>x\<in>s. \<forall>x'\<in>s. d x e > 0 \<and> (dist x' x < d x e \<longrightarrow> dist (f x') (f x) < e)"
@@ -4007,7 +4061,7 @@
       using eventually_and[of "(\<lambda>n. \<forall>x\<in>s. norm (f n x - g x) < e / 3)" "(\<lambda>n. continuous_on s (f n))" net] assms(1,2) eventually_happens by blast
     have "e / 3 > 0" using `e>0` by auto
     then obtain d where "d>0" and d:"\<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f n x') (f n x) < e / 3"
-      using n(2)[unfolded continuous_on_def, THEN bspec[where x=x], OF `x\<in>s`, THEN spec[where x="e/3"]] by blast
+      using n(2)[unfolded continuous_on_iff, THEN bspec[where x=x], OF `x\<in>s`, THEN spec[where x="e/3"]] by blast
     { fix y assume "y\<in>s" "dist y x < d"
       hence "dist (f n y) (f n x) < e / 3" using d[THEN bspec[where x=y]] by auto
       hence "norm (f n y - g x) < 2 * e / 3" using norm_triangle_lt[of "f n y - f n x" "f n x - g x" "2*e/3"]
@@ -4015,7 +4069,7 @@
       hence "dist (g y) (g x) < e" unfolding dist_norm using n(1)[THEN bspec[where x=y], OF `y\<in>s`]
         unfolding norm_minus_cancel[of "f n y - g y", THEN sym] using norm_triangle_lt[of "f n y - g x" "g y - f n y" e] by (auto simp add: uminus_add_conv_diff)  }
     hence "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e" using `d>0` by auto  }
-  thus ?thesis unfolding continuous_on_def by auto
+  thus ?thesis unfolding continuous_on_iff by auto
 qed
 
 subsection{* Topological properties of linear functions.                               *}
@@ -4060,6 +4114,7 @@
   unfolding continuous_within using Lim_bilinear[of f "f x"] by auto
 
 lemma bilinear_continuous_on_compose:
+  fixes s :: "'a::metric_space set" (* TODO: generalize *)
   shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> bounded_bilinear h
              ==> continuous_on s (\<lambda>x. h (f x) (g x))"
   unfolding continuous_on_eq_continuous_within apply auto apply(erule_tac x=x in ballE) apply auto apply(erule_tac x=x in ballE) apply auto
@@ -4098,7 +4153,7 @@
 lemma continuous_on_real_range:
   fixes f :: "'a::real_normed_vector \<Rightarrow> real"
   shows "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. \<forall>e>0. \<exists>d>0. (\<forall>x' \<in> s. norm(x' - x) < d --> abs(f x' - f x) < e))"
-  unfolding continuous_on_def dist_norm by simp
+  unfolding continuous_on_iff dist_norm by simp
 
 lemma continuous_at_norm: "continuous (at x) norm"
   unfolding continuous_at by (intro tendsto_intros)
@@ -4109,7 +4164,9 @@
 lemma continuous_at_component: "continuous (at a) (\<lambda>x. x $ i)"
 unfolding continuous_at by (intro tendsto_intros)
 
-lemma continuous_on_component: "continuous_on s (\<lambda>x. x $ i)"
+lemma continuous_on_component:
+  fixes s :: "('a::metric_space ^ 'n) set" (* TODO: generalize *)
+  shows "continuous_on s (\<lambda>x. x $ i)"
 unfolding continuous_on by (intro ballI tendsto_intros)
 
 lemma continuous_at_infnorm: "continuous (at x) infnorm"
@@ -4333,7 +4390,7 @@
       using assms(2)[unfolded closed_sequential_limits, THEN spec[where x="\<lambda>n. sndcart (x n)"], THEN spec[where x="sndcart l"]]
       unfolding Lim_sequentially by auto
     hence "l \<in> {pastecart x y |x y. x \<in> s \<and> y \<in> t}" apply- unfolding mem_Collect_eq apply(rule_tac x="fstcart l" in exI,rule_tac x="sndcart l" in exI) by auto }
-  thus ?thesis unfolding closed_sequential_limits by auto
+  thus ?thesis unfolding closed_sequential_limits by blast
 qed
 
 lemma compact_pastecart:
@@ -4424,7 +4481,7 @@
   have "{x - y | x y . x\<in>s \<and> y\<in>s} \<noteq> {}" using `s \<noteq> {}` by auto
   then obtain x where x:"x\<in>{x - y |x y. x \<in> s \<and> y \<in> s}"  "\<forall>y\<in>{x - y |x y. x \<in> s \<and> y \<in> s}. norm y \<le> norm x"
     using compact_differences[OF assms(1) assms(1)]
-    using distance_attains_sup[where 'a="'a", unfolded dist_norm, of "{x - y | x y . x\<in>s \<and> y\<in>s}" 0] by(auto simp add: norm_minus_cancel)
+    using distance_attains_sup[where 'a="'a", unfolded dist_norm, of "{x - y | x y . x\<in>s \<and> y\<in>s}" 0] by auto
   from x(1) obtain a b where "a\<in>s" "b\<in>s" "x = a - b" by auto
   thus ?thesis using x(2)[unfolded `x = a - b`] by blast
 qed
@@ -4445,7 +4502,7 @@
     hence "norm (x - y) \<le> 2 * a" using norm_triangle_ineq[of x "-y", unfolded norm_minus_cancel] a[THEN bspec[where x=x]] a[THEN bspec[where x=y]] by (auto simp add: field_simps)  }
   note * = this
   { fix x y assume "x\<in>s" "y\<in>s"  hence "s \<noteq> {}" by auto
-    have "norm(x - y) \<le> diameter s" unfolding diameter_def using `s\<noteq>{}` *[OF `x\<in>s` `y\<in>s`] `x\<in>s` `y\<in>s`  
+    have "norm(x - y) \<le> diameter s" unfolding diameter_def using `s\<noteq>{}` *[OF `x\<in>s` `y\<in>s`] `x\<in>s` `y\<in>s`
       by simp (blast intro!: Sup_upper *) }
   moreover
   { fix d::real assume "d>0" "d < diameter s"
@@ -4476,10 +4533,10 @@
 proof-
   have b:"bounded s" using assms(1) by (rule compact_imp_bounded)
   then obtain x y where xys:"x\<in>s" "y\<in>s" and xy:"\<forall>u\<in>s. \<forall>v\<in>s. norm (u - v) \<le> norm (x - y)" using compact_sup_maxdistance[OF assms] by auto
-  hence "diameter s \<le> norm (x - y)" 
-    by (force simp add: diameter_def intro!: Sup_least) 
+  hence "diameter s \<le> norm (x - y)"
+    unfolding diameter_def by clarsimp (rule Sup_least, fast+)
   thus ?thesis
-    by (metis b diameter_bounded_bound order_antisym xys) 
+    by (metis b diameter_bounded_bound order_antisym xys)
 qed
 
 text{* Related results with closure as the conclusion.                           *}
@@ -4668,7 +4725,7 @@
 lemma mem_interval_1: fixes x :: "real^1" shows
  "(x \<in> {a .. b} \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 x \<and> dest_vec1 x \<le> dest_vec1 b)"
  "(x \<in> {a<..<b} \<longleftrightarrow> dest_vec1 a < dest_vec1 x \<and> dest_vec1 x < dest_vec1 b)"
-by(simp_all add: Cart_eq vector_less_def vector_le_def forall_1)
+by(simp_all add: Cart_eq vector_less_def vector_le_def)
 
 lemma vec1_interval:fixes a::"real" shows
   "vec1 ` {a .. b} = {vec1 a .. vec1 b}"
@@ -4694,7 +4751,7 @@
       have "a$i < b$i" using as[THEN spec[where x=i]] by auto
       hence "a$i < ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i < b$i"
         unfolding vector_smult_component and vector_add_component
-        by (auto simp add: less_divide_eq_number_of1)  }
+        by auto  }
     hence "{a <..< b} \<noteq> {}" using mem_interval(1)[of "?x" a b] by auto  }
   ultimately show ?th1 by blast
 
@@ -4709,7 +4766,7 @@
       have "a$i \<le> b$i" using as[THEN spec[where x=i]] by auto
       hence "a$i \<le> ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i \<le> b$i"
         unfolding vector_smult_component and vector_add_component
-        by (auto simp add: less_divide_eq_number_of1)  }
+        by auto  }
     hence "{a .. b} \<noteq> {}" using mem_interval(2)[of "?x" a b] by auto  }
   ultimately show ?th2 by blast
 qed
@@ -4772,13 +4829,13 @@
       { fix j
         have "c $ j < ?x $ j \<and> ?x $ j < d $ j" unfolding Cart_lambda_beta
           apply(cases "j=i") using as(2)[THEN spec[where x=j]]
-          by (auto simp add: less_divide_eq_number_of1 as2)  }
+          by (auto simp add: as2)  }
       hence "?x\<in>{c<..<d}" unfolding mem_interval by auto
       moreover
       have "?x\<notin>{a .. b}"
         unfolding mem_interval apply auto apply(rule_tac x=i in exI)
         using as(2)[THEN spec[where x=i]] and as2
-        by (auto simp add: less_divide_eq_number_of1)
+        by auto
       ultimately have False using as by auto  }
     hence "a$i \<le> c$i" by(rule ccontr)auto
     moreover
@@ -4787,13 +4844,13 @@
       { fix j
         have "d $ j > ?x $ j \<and> ?x $ j > c $ j" unfolding Cart_lambda_beta
           apply(cases "j=i") using as(2)[THEN spec[where x=j]]
-          by (auto simp add: less_divide_eq_number_of1 as2)  }
+          by (auto simp add: as2)  }
       hence "?x\<in>{c<..<d}" unfolding mem_interval by auto
       moreover
       have "?x\<notin>{a .. b}"
         unfolding mem_interval apply auto apply(rule_tac x=i in exI)
         using as(2)[THEN spec[where x=i]] and as2
-        by (auto simp add: less_divide_eq_number_of1)
+        by auto
       ultimately have False using as by auto  }
     hence "b$i \<ge> d$i" by(rule ccontr)auto
     ultimately
@@ -4824,7 +4881,7 @@
 lemma inter_interval: fixes a :: "'a::linorder^'n" shows
  "{a .. b} \<inter> {c .. d} =  {(\<chi> i. max (a$i) (c$i)) .. (\<chi> i. min (b$i) (d$i))}"
   unfolding expand_set_eq and Int_iff and mem_interval
-  by (auto simp add: less_divide_eq_number_of1 intro!: bexI)
+  by auto
 
 (* Moved interval_open_subset_closed a bit upwards *)
 
@@ -4864,7 +4921,7 @@
   using open_interval[of "vec1 a" "vec1 b"] unfolding open_contains_ball
   apply-apply(rule,erule_tac x="vec1 x" in ballE) apply(erule exE,rule_tac x=e in exI)
   unfolding subset_eq mem_ball apply(rule) defer apply(rule,erule conjE,erule_tac x="vec1 xa" in ballE)
-  by(auto simp add: vec1_dest_vec1_simps vector_less_def forall_1) 
+  by(auto simp add: dist_vec1 dist_real_def vector_less_def)
 
 lemma closed_interval[intro]: fixes a :: "real^'n" shows "closed {a .. b}"
 proof-
@@ -4945,7 +5002,7 @@
     have "a $ i < ((1 / 2) *\<^sub>R (a + b)) $ i \<and> ((1 / 2) *\<^sub>R (a + b)) $ i < b $ i"
       using assms[unfolded interval_ne_empty, THEN spec[where x=i]]
       unfolding vector_smult_component and vector_add_component
-      by(auto simp add: less_divide_eq_number_of1)  }
+      by auto  }
   thus ?thesis unfolding mem_interval by auto
 qed
 
@@ -4987,7 +5044,7 @@
         x + (inverse (real n + 1)) *\<^sub>R (((1 / 2) *\<^sub>R (a + b)) - x)"
         by (auto simp add: algebra_simps)
       hence "f n < b" and "a < f n" using open_closed_interval_convex[OF open_interval_midpoint[OF assms] as *] unfolding f_def by auto
-      hence False using fn unfolding f_def using xc by(auto simp add: vector_mul_lcancel vector_ssub_ldistrib)  }
+      hence False using fn unfolding f_def using xc by(auto simp add: vector_ssub_ldistrib)  }
     moreover
     { assume "\<not> (f ---> x) sequentially"
       { fix e::real assume "e>0"
@@ -5261,12 +5318,14 @@
   by (auto simp add: eventually_within_Un)
 
 lemma continuous_on_union:
+  fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
   assumes "closed s" "closed t" "continuous_on s f" "continuous_on t f"
   shows "continuous_on (s \<union> t) f"
   using assms unfolding continuous_on unfolding Lim_within_union
   unfolding Lim unfolding trivial_limit_within unfolding closed_limpt by auto
 
 lemma continuous_on_cases:
+  fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
   assumes "closed s" "closed t" "continuous_on s f" "continuous_on t g"
           "\<forall>x. (x\<in>s \<and> \<not> P x) \<or> (x \<in> t \<and> P x) \<longrightarrow> f x = g x"
   shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"
@@ -5401,7 +5460,7 @@
       then obtain y where y:"y\<in>t" "g y = x" by auto
       then obtain x' where x':"x'\<in>s" "f x' = y" using assms(3) by auto
       hence "x \<in> s" unfolding g_def using someI2[of "\<lambda>b. b\<in>s \<and> f b = y" x' "\<lambda>x. x\<in>s"] unfolding y(2)[THEN sym] and g_def by auto }
-    ultimately have "x\<in>s \<longleftrightarrow> x \<in> g ` t" by auto  }
+    ultimately have "x\<in>s \<longleftrightarrow> x \<in> g ` t" ..  }
   hence "g ` t = s" by auto
   ultimately
   show ?thesis unfolding homeomorphism_def homeomorphic_def
@@ -5543,7 +5602,7 @@
   let ?S' = "{x::real^'m. x\<in>s \<and> norm x = norm a}"
   let ?S'' = "{x::real^'m. norm x = norm a}"
 
-  have "?S'' = frontier(cball 0 (norm a))" unfolding frontier_cball and dist_norm by (auto simp add: norm_minus_cancel)
+  have "?S'' = frontier(cball 0 (norm a))" unfolding frontier_cball and dist_norm by auto
   hence "compact ?S''" using compact_frontier[OF compact_cball, of 0 "norm a"] by auto
   moreover have "?S' = s \<inter> ?S''" by auto
   ultimately have "compact ?S'" using closed_inter_compact[of s ?S''] using s(1) by auto
@@ -5590,7 +5649,7 @@
 
 lemma subspace_substandard:
  "subspace {x::real^_. (\<forall>i. P i \<longrightarrow> x$i = 0)}"
-  unfolding subspace_def by(auto simp add: vector_add_component vector_smult_component elim!: ballE)
+  unfolding subspace_def by auto
 
 lemma closed_substandard:
  "closed {x::real^'n. \<forall>i. P i --> x$i = 0}" (is "closed ?A")
@@ -5607,7 +5666,7 @@
         then obtain B where BB:"B \<in> ?Bs" and B:"B = {x::real^'n. inner (basis i) x = 0}" by auto
         hence "x $ i = 0" unfolding B using x unfolding inner_basis by auto  }
       hence "x\<in>?A" by auto }
-    ultimately have "x\<in>?A \<longleftrightarrow> x\<in> \<Inter>?Bs" by auto }
+    ultimately have "x\<in>?A \<longleftrightarrow> x\<in> \<Inter>?Bs" .. }
   hence "?A = \<Inter> ?Bs" by auto
   thus ?thesis by(auto simp add: closed_Inter closed_hyperplane)
 qed
@@ -5783,23 +5842,23 @@
   case False
   { fix y assume "a \<le> y" "y \<le> b" "m > 0"
     hence "m *\<^sub>R a + c \<le> m *\<^sub>R y + c"  "m *\<^sub>R y + c \<le> m *\<^sub>R b + c"
-      unfolding vector_le_def by(auto simp add: vector_smult_component vector_add_component)
+      unfolding vector_le_def by auto
   } moreover
   { fix y assume "a \<le> y" "y \<le> b" "m < 0"
     hence "m *\<^sub>R b + c \<le> m *\<^sub>R y + c"  "m *\<^sub>R y + c \<le> m *\<^sub>R a + c"
-      unfolding vector_le_def by(auto simp add: vector_smult_component vector_add_component mult_left_mono_neg elim!:ballE)
+      unfolding vector_le_def by(auto simp add: mult_left_mono_neg)
   } moreover
   { fix y assume "m > 0"  "m *\<^sub>R a + c \<le> y"  "y \<le> m *\<^sub>R b + c"
     hence "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}"
       unfolding image_iff Bex_def mem_interval vector_le_def
-      apply(auto simp add: vector_smult_component vector_add_component vector_minus_component vector_smult_assoc pth_3[symmetric]
+      apply(auto simp add: vector_smult_assoc pth_3[symmetric]
         intro!: exI[where x="(1 / m) *\<^sub>R (y - c)"])
       by(auto simp add: pos_le_divide_eq pos_divide_le_eq real_mult_commute diff_le_iff)
   } moreover
   { fix y assume "m *\<^sub>R b + c \<le> y" "y \<le> m *\<^sub>R a + c" "m < 0"
     hence "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}"
       unfolding image_iff Bex_def mem_interval vector_le_def
-      apply(auto simp add: vector_smult_component vector_add_component vector_minus_component vector_smult_assoc pth_3[symmetric]
+      apply(auto simp add: vector_smult_assoc pth_3[symmetric]
         intro!: exI[where x="(1 / m) *\<^sub>R (y - c)"])
       by(auto simp add: neg_le_divide_eq neg_divide_le_eq real_mult_commute diff_le_iff)
   }
@@ -6045,7 +6104,7 @@
   { fix x y assume "x\<in>s" "y\<in>s" moreover
     fix e::real assume "e>0" ultimately
     have "dist y x < e \<longrightarrow> dist (g y) (g x) < e" using dist by fastsimp }
-  hence "continuous_on s g" unfolding continuous_on_def by auto
+  hence "continuous_on s g" unfolding continuous_on_iff by auto
 
   hence "((snd \<circ> h \<circ> r) ---> g a) sequentially" unfolding continuous_on_sequentially
     apply (rule allE[where x="\<lambda>n. (fst \<circ> h \<circ> r) n"]) apply (erule ballE[where x=a])
diff -r 7b92238454d6 -r 18bf20d0c2df src/HOL/SetInterval.thy
--- a/src/HOL/SetInterval.thy	Mon Apr 26 16:08:04 2010 +0200
+++ b/src/HOL/SetInterval.thy	Mon Apr 26 09:45:22 2010 -0700
@@ -54,22 +54,22 @@
 @{term"{m..<n}"} may not exist in @{term"{..<n}"}-form as well. *}
 
 syntax
-  "_UNION_le"   :: "'a => 'a => 'b set => 'b set"       ("(3UN _<=_./ _)" 10)
-  "_UNION_less" :: "'a => 'a => 'b set => 'b set"       ("(3UN _<_./ _)" 10)
-  "_INTER_le"   :: "'a => 'a => 'b set => 'b set"       ("(3INT _<=_./ _)" 10)
-  "_INTER_less" :: "'a => 'a => 'b set => 'b set"       ("(3INT _<_./ _)" 10)
+  "_UNION_le"   :: "'a => 'a => 'b set => 'b set"       ("(3UN _<=_./ _)" [0, 0, 10] 10)
+  "_UNION_less" :: "'a => 'a => 'b set => 'b set"       ("(3UN _<_./ _)" [0, 0, 10] 10)
+  "_INTER_le"   :: "'a => 'a => 'b set => 'b set"       ("(3INT _<=_./ _)" [0, 0, 10] 10)
+  "_INTER_less" :: "'a => 'a => 'b set => 'b set"       ("(3INT _<_./ _)" [0, 0, 10] 10)
 
 syntax (xsymbols)
-  "_UNION_le"   :: "'a => 'a => 'b set => 'b set"       ("(3\<Union> _\<le>_./ _)" 10)
-  "_UNION_less" :: "'a => 'a => 'b set => 'b set"       ("(3\<Union> _<_./ _)" 10)
-  "_INTER_le"   :: "'a => 'a => 'b set => 'b set"       ("(3\<Inter> _\<le>_./ _)" 10)
-  "_INTER_less" :: "'a => 'a => 'b set => 'b set"       ("(3\<Inter> _<_./ _)" 10)
+  "_UNION_le"   :: "'a => 'a => 'b set => 'b set"       ("(3\<Union> _\<le>_./ _)" [0, 0, 10] 10)
+  "_UNION_less" :: "'a => 'a => 'b set => 'b set"       ("(3\<Union> _<_./ _)" [0, 0, 10] 10)
+  "_INTER_le"   :: "'a => 'a => 'b set => 'b set"       ("(3\<Inter> _\<le>_./ _)" [0, 0, 10] 10)
+  "_INTER_less" :: "'a => 'a => 'b set => 'b set"       ("(3\<Inter> _<_./ _)" [0, 0, 10] 10)
 
 syntax (latex output)
-  "_UNION_le"   :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Union>(00_ \<le> _)/ _)" 10)
-  "_UNION_less" :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Union>(00_ < _)/ _)" 10)
-  "_INTER_le"   :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Inter>(00_ \<le> _)/ _)" 10)
-  "_INTER_less" :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Inter>(00_ < _)/ _)" 10)
+  "_UNION_le"   :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Union>(00_ \<le> _)/ _)" [0, 0, 10] 10)
+  "_UNION_less" :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Union>(00_ < _)/ _)" [0, 0, 10] 10)
+  "_INTER_le"   :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Inter>(00_ \<le> _)/ _)" [0, 0, 10] 10)
+  "_INTER_less" :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Inter>(00_ < _)/ _)" [0, 0, 10] 10)
 
 translations
   "UN i<=n. A"  == "UN i:{..n}. A"