# HG changeset patch
# User huffman
# Date 1313024852 25200
# Node ID d282b3c5df7cc6e072d69c3492c97de746e4d052
# Parent  8e27e0177518022490bf5011fcf0d7a48cf905f1# Parent  fa98623f1006e6d80fed503eb64a583bd5855934
merged

diff -r fa98623f1006 -r d282b3c5df7c NEWS
--- a/NEWS	Wed Aug 10 21:24:26 2011 +0200
+++ b/NEWS	Wed Aug 10 18:07:32 2011 -0700
@@ -183,6 +183,19 @@
 * Limits.thy: Type "'a net" has been renamed to "'a filter", in
 accordance with standard mathematical terminology. INCOMPATIBILITY.
 
+* Session Multivariate_Analysis: Type "('a, 'b) cart" has been renamed
+to "('a, 'b) vec" (the syntax "'a ^ 'b" remains unaffected). Constants
+"Cart_nth" and "Cart_lambda" have been respectively renamed to
+"vec_nth" and "vec_lambda"; theorems mentioning those names have
+changed to match. Definition theorems for overloaded constants now use
+the standard "foo_vec_def" naming scheme. A few other theorems have
+been renamed as follows (INCOMPATIBILITY):
+
+  Cart_eq          ~> vec_eq_iff
+  dist_nth_le_cart ~> dist_vec_nth_le
+  tendsto_vector   ~> vec_tendstoI
+  Cauchy_vector    ~> vec_CauchyI
+
 
 *** Document preparation ***
 
diff -r fa98623f1006 -r d282b3c5df7c src/HOL/Library/Cardinality.thy
--- a/src/HOL/Library/Cardinality.thy	Wed Aug 10 21:24:26 2011 +0200
+++ b/src/HOL/Library/Cardinality.thy	Wed Aug 10 18:07:32 2011 -0700
@@ -62,7 +62,7 @@
   by (simp only: card_Pow finite numeral_2_eq_2)
 
 lemma card_nat [simp]: "CARD(nat) = 0"
-  by (simp add: infinite_UNIV_nat card_eq_0_iff)
+  by (simp add: card_eq_0_iff)
 
 
 subsection {* Classes with at least 1 and 2  *}
diff -r fa98623f1006 -r d282b3c5df7c src/HOL/Library/Convex.thy
--- a/src/HOL/Library/Convex.thy	Wed Aug 10 21:24:26 2011 +0200
+++ b/src/HOL/Library/Convex.thy	Wed Aug 10 18:07:32 2011 -0700
@@ -49,7 +49,7 @@
 
 lemma convex_halfspace_le: "convex {x. inner a x \<le> b}"
   unfolding convex_def
-  by (auto simp: inner_add inner_scaleR intro!: convex_bound_le)
+  by (auto simp: inner_add intro!: convex_bound_le)
 
 lemma convex_halfspace_ge: "convex {x. inner a x \<ge> b}"
 proof -
@@ -209,7 +209,7 @@
   shows "convex s \<longleftrightarrow> (\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1
                       \<longrightarrow> setsum (\<lambda>x. u x *\<^sub>R x) s \<in> s)"
   unfolding convex_explicit
-proof (safe elim!: conjE)
+proof (safe)
   fix t u assume sum: "\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> s"
     and as: "finite t" "t \<subseteq> s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = (1::real)"
   have *:"s \<inter> t = t" using as(2) by auto
@@ -480,9 +480,9 @@
   also have "\<dots> = a * (f x - f y) + f y" by (simp add: field_simps)
   finally have "f t - f y \<le> a * (f x - f y)" by simp
   with t show "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
-    by (simp add: times_divide_eq le_divide_eq divide_le_eq field_simps a_def)
+    by (simp add: le_divide_eq divide_le_eq field_simps a_def)
   with t show "(f x - f y) / (x - y) \<le> (f t - f y) / (t - y)"
-    by (simp add: times_divide_eq le_divide_eq divide_le_eq field_simps)
+    by (simp add: le_divide_eq divide_le_eq field_simps)
 qed
 
 lemma pos_convex_function:
diff -r fa98623f1006 -r d282b3c5df7c src/HOL/Library/Extended_Real.thy
--- a/src/HOL/Library/Extended_Real.thy	Wed Aug 10 21:24:26 2011 +0200
+++ b/src/HOL/Library/Extended_Real.thy	Wed Aug 10 18:07:32 2011 -0700
@@ -608,7 +608,7 @@
   shows "a * c < b * c"
   using assms
   by (cases rule: ereal3_cases[of a b c])
-     (auto simp: zero_le_mult_iff ereal_less_PInfty)
+     (auto simp: zero_le_mult_iff)
 
 lemma ereal_mult_strict_left_mono:
   "\<lbrakk> a < b ; 0 < c ; c < (\<infinity>::ereal)\<rbrakk> \<Longrightarrow> c * a < c * b"
@@ -619,7 +619,7 @@
   using assms
   apply (cases "c = 0") apply simp
   by (cases rule: ereal3_cases[of a b c])
-     (auto simp: zero_le_mult_iff ereal_less_PInfty)
+     (auto simp: zero_le_mult_iff)
 
 lemma ereal_mult_left_mono:
   fixes a b c :: ereal shows "\<lbrakk>a \<le> b; 0 \<le> c\<rbrakk> \<Longrightarrow> c * a \<le> c * b"
@@ -710,7 +710,7 @@
   fixes x y :: ereal
   assumes "ALL z. x <= ereal z --> y <= ereal z"
   shows "y <= x"
-by (metis assms ereal_bot ereal_cases ereal_infty_less_eq ereal_less_eq linorder_le_cases)
+by (metis assms ereal_bot ereal_cases ereal_infty_less_eq(2) ereal_less_eq(1) linorder_le_cases)
 
 lemma ereal_le_ereal:
   fixes x y :: ereal
@@ -2037,7 +2037,7 @@
   with `x \<noteq> 0` have "open (S - {0})" "x \<in> S - {0}" by auto
   from tendsto[THEN topological_tendstoD, OF this]
   show "eventually (\<lambda>x. f x \<in> S) net"
-    by (rule eventually_rev_mp) (auto simp: ereal_real real_of_ereal_0)
+    by (rule eventually_rev_mp) (auto simp: ereal_real)
 qed
 
 lemma tendsto_ereal_realI:
diff -r fa98623f1006 -r d282b3c5df7c src/HOL/Library/Set_Algebras.thy
--- a/src/HOL/Library/Set_Algebras.thy	Wed Aug 10 21:24:26 2011 +0200
+++ b/src/HOL/Library/Set_Algebras.thy	Wed Aug 10 18:07:32 2011 -0700
@@ -153,7 +153,7 @@
 
 theorem set_plus_rearrange4: "C \<oplus> ((a::'a::comm_monoid_add) +o D) =
     a +o (C \<oplus> D)"
-  apply (auto intro!: subsetI simp add: elt_set_plus_def set_plus_def add_ac)
+  apply (auto simp add: elt_set_plus_def set_plus_def add_ac)
    apply (rule_tac x = "aa + ba" in exI)
    apply (auto simp add: add_ac)
   done
@@ -211,7 +211,7 @@
   by (auto simp add: elt_set_plus_def)
 
 lemma set_zero_plus2: "(0::'a::comm_monoid_add) : A ==> B <= A \<oplus> B"
-  apply (auto intro!: subsetI simp add: set_plus_def)
+  apply (auto simp add: set_plus_def)
   apply (rule_tac x = 0 in bexI)
    apply (rule_tac x = x in bexI)
     apply (auto simp add: add_ac)
@@ -264,7 +264,7 @@
 
 theorem set_times_rearrange4: "C \<otimes> ((a::'a::comm_monoid_mult) *o D) =
     a *o (C \<otimes> D)"
-  apply (auto intro!: subsetI simp add: elt_set_times_def set_times_def
+  apply (auto simp add: elt_set_times_def set_times_def
     mult_ac)
    apply (rule_tac x = "aa * ba" in exI)
    apply (auto simp add: mult_ac)
@@ -336,7 +336,7 @@
 
 lemma set_times_plus_distrib3: "((a::'a::semiring) +o C) \<otimes> D <=
     a *o D \<oplus> C \<otimes> D"
-  apply (auto intro!: subsetI simp add:
+  apply (auto simp add:
     elt_set_plus_def elt_set_times_def set_times_def
     set_plus_def ring_distribs)
   apply auto
diff -r fa98623f1006 -r d282b3c5df7c src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy
--- a/src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy	Wed Aug 10 21:24:26 2011 +0200
+++ b/src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy	Wed Aug 10 18:07:32 2011 -0700
@@ -36,24 +36,23 @@
 
 subsection{* Basic componentwise operations on vectors. *}
 
-instantiation cart :: (times,finite) times
+instantiation vec :: (times, finite) times
 begin
-  definition vector_mult_def : "op * \<equiv> (\<lambda> x y.  (\<chi> i. (x$i) * (y$i)))"
+  definition "op * \<equiv> (\<lambda> x y.  (\<chi> i. (x$i) * (y$i)))"
   instance ..
 end
 
-instantiation cart :: (one,finite) one
+instantiation vec :: (one, finite) one
 begin
-  definition vector_one_def : "1 \<equiv> (\<chi> i. 1)"
+  definition "1 \<equiv> (\<chi> i. 1)"
   instance ..
 end
 
-instantiation cart :: (ord,finite) ord
+instantiation vec :: (ord, finite) ord
 begin
-  definition vector_le_def:
-    "less_eq (x :: 'a ^'b) y = (ALL i. x$i <= y$i)"
-  definition vector_less_def: "less (x :: 'a ^'b) y = (ALL i. x$i < y$i)"
-  instance by (intro_classes)
+  definition "x \<le> y \<longleftrightarrow> (\<forall>i. x$i \<le> y$i)"
+  definition "x < y \<longleftrightarrow> (\<forall>i. x$i < y$i)"
+  instance ..
 end
 
 text{* The ordering on one-dimensional vectors is linear. *}
@@ -65,12 +64,12 @@
       by (auto intro!: card_ge_0_finite) qed
 end
 
-instantiation cart :: (linorder,cart_one) linorder begin
+instantiation vec :: (linorder,cart_one) linorder begin
 instance proof
   guess a B using UNIV_one[where 'a='b] unfolding card_Suc_eq apply- by(erule exE)+
   hence *:"UNIV = {a}" by auto
   have "\<And>P. (\<forall>i\<in>UNIV. P i) \<longleftrightarrow> P a" unfolding * by auto hence all:"\<And>P. (\<forall>i. P i) \<longleftrightarrow> P a" by auto
-  fix x y z::"'a^'b::cart_one" note * = vector_le_def vector_less_def all Cart_eq
+  fix x y z::"'a^'b::cart_one" note * = less_eq_vec_def less_vec_def all vec_eq_iff
   show "x\<le>x" "(x < y) = (x \<le> y \<and> \<not> y \<le> x)" "x\<le>y \<or> y\<le>x" unfolding * by(auto simp only:field_simps)
   { assume "x\<le>y" "y\<le>z" thus "x\<le>z" unfolding * by(auto simp only:field_simps) }
   { assume "x\<le>y" "y\<le>x" thus "x=y" unfolding * by(auto simp only:field_simps) }
@@ -93,16 +92,16 @@
   @{thm setsum_subtractf} RS sym, @{thm setsum_right_distrib},
   @{thm setsum_left_distrib}, @{thm setsum_negf} RS sym]
   val ss2 = @{simpset} addsimps
-             [@{thm vector_add_def}, @{thm vector_mult_def},
-              @{thm vector_minus_def}, @{thm vector_uminus_def},
-              @{thm vector_one_def}, @{thm vector_zero_def}, @{thm vec_def},
-              @{thm vector_scaleR_def},
-              @{thm Cart_lambda_beta}, @{thm vector_scalar_mult_def}]
+             [@{thm plus_vec_def}, @{thm times_vec_def},
+              @{thm minus_vec_def}, @{thm uminus_vec_def},
+              @{thm one_vec_def}, @{thm zero_vec_def}, @{thm vec_def},
+              @{thm scaleR_vec_def},
+              @{thm vec_lambda_beta}, @{thm vector_scalar_mult_def}]
  fun vector_arith_tac ths =
    simp_tac ss1
    THEN' (fn i => rtac @{thm setsum_cong2} i
          ORELSE rtac @{thm setsum_0'} i
-         ORELSE simp_tac (HOL_basic_ss addsimps [@{thm "Cart_eq"}]) i)
+         ORELSE simp_tac (HOL_basic_ss addsimps [@{thm vec_eq_iff}]) i)
    (* THEN' TRY o clarify_tac HOL_cs  THEN' (TRY o rtac @{thm iffI}) *)
    THEN' asm_full_simp_tac (ss2 addsimps ths)
  in
@@ -110,8 +109,8 @@
  end
 *} "lift trivial vector statements to real arith statements"
 
-lemma vec_0[simp]: "vec 0 = 0" by (vector vector_zero_def)
-lemma vec_1[simp]: "vec 1 = 1" by (vector vector_one_def)
+lemma vec_0[simp]: "vec 0 = 0" by (vector zero_vec_def)
+lemma vec_1[simp]: "vec 1 = 1" by (vector one_vec_def)
 
 lemma vec_inj[simp]: "vec x = vec y \<longleftrightarrow> x = y" by vector
 
@@ -149,49 +148,47 @@
 
 subsection {* Some frequently useful arithmetic lemmas over vectors. *}
 
-instance cart :: (semigroup_mult,finite) semigroup_mult
-  apply (intro_classes) by (vector mult_assoc)
+instance vec :: (semigroup_mult, finite) semigroup_mult
+  by default (vector mult_assoc)
 
-instance cart :: (monoid_mult,finite) monoid_mult
-  apply (intro_classes) by vector+
+instance vec :: (monoid_mult, finite) monoid_mult
+  by default vector+
 
-instance cart :: (ab_semigroup_mult,finite) ab_semigroup_mult
-  apply (intro_classes) by (vector mult_commute)
+instance vec :: (ab_semigroup_mult, finite) ab_semigroup_mult
+  by default (vector mult_commute)
 
-instance cart :: (ab_semigroup_idem_mult,finite) ab_semigroup_idem_mult
-  apply (intro_classes) by (vector mult_idem)
+instance vec :: (ab_semigroup_idem_mult, finite) ab_semigroup_idem_mult
+  by default (vector mult_idem)
 
-instance cart :: (comm_monoid_mult,finite) comm_monoid_mult
-  apply (intro_classes) by vector
+instance vec :: (comm_monoid_mult, finite) comm_monoid_mult
+  by default vector
 
-instance cart :: (semiring,finite) semiring
-  apply (intro_classes) by (vector field_simps)+
+instance vec :: (semiring, finite) semiring
+  by default (vector field_simps)+
 
-instance cart :: (semiring_0,finite) semiring_0
-  apply (intro_classes) by (vector field_simps)+
-instance cart :: (semiring_1,finite) semiring_1
-  apply (intro_classes) by vector
-instance cart :: (comm_semiring,finite) comm_semiring
-  apply (intro_classes) by (vector field_simps)+
+instance vec :: (semiring_0, finite) semiring_0
+  by default (vector field_simps)+
+instance vec :: (semiring_1, finite) semiring_1
+  by default vector
+instance vec :: (comm_semiring, finite) comm_semiring
+  by default (vector field_simps)+
 
-instance cart :: (comm_semiring_0,finite) comm_semiring_0 by (intro_classes)
-instance cart :: (cancel_comm_monoid_add, finite) cancel_comm_monoid_add ..
-instance cart :: (semiring_0_cancel,finite) semiring_0_cancel by (intro_classes)
-instance cart :: (comm_semiring_0_cancel,finite) comm_semiring_0_cancel by (intro_classes)
-instance cart :: (ring,finite) ring by (intro_classes)
-instance cart :: (semiring_1_cancel,finite) semiring_1_cancel by (intro_classes)
-instance cart :: (comm_semiring_1,finite) comm_semiring_1 by (intro_classes)
+instance vec :: (comm_semiring_0, finite) comm_semiring_0 ..
+instance vec :: (cancel_comm_monoid_add, finite) cancel_comm_monoid_add ..
+instance vec :: (semiring_0_cancel, finite) semiring_0_cancel ..
+instance vec :: (comm_semiring_0_cancel, finite) comm_semiring_0_cancel ..
+instance vec :: (ring, finite) ring ..
+instance vec :: (semiring_1_cancel, finite) semiring_1_cancel ..
+instance vec :: (comm_semiring_1, finite) comm_semiring_1 ..
 
-instance cart :: (ring_1,finite) ring_1 ..
+instance vec :: (ring_1, finite) ring_1 ..
 
-instance cart :: (real_algebra,finite) real_algebra
+instance vec :: (real_algebra, finite) real_algebra
   apply intro_classes
-  apply (simp_all add: vector_scaleR_def field_simps)
-  apply vector
-  apply vector
+  apply (simp_all add: vec_eq_iff)
   done
 
-instance cart :: (real_algebra_1,finite) real_algebra_1 ..
+instance vec :: (real_algebra_1, finite) real_algebra_1 ..
 
 lemma of_nat_index:
   "(of_nat n :: 'a::semiring_1 ^'n)$i = of_nat n"
@@ -203,17 +200,15 @@
 lemma one_index[simp]:
   "(1 :: 'a::one ^'n)$i = 1" by vector
 
-instance cart :: (semiring_char_0, finite) semiring_char_0
+instance vec :: (semiring_char_0, finite) semiring_char_0
 proof
   fix m n :: nat
   show "inj (of_nat :: nat \<Rightarrow> 'a ^ 'b)"
-    by (auto intro!: injI simp add: Cart_eq of_nat_index)
+    by (auto intro!: injI simp add: vec_eq_iff of_nat_index)
 qed
 
-instance cart :: (comm_ring_1,finite) comm_ring_1 ..
-instance cart :: (ring_char_0,finite) ring_char_0 ..
-
-instance cart :: (real_vector,finite) real_vector ..
+instance vec :: (comm_ring_1, finite) comm_ring_1 ..
+instance vec :: (ring_char_0, finite) ring_char_0 ..
 
 lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x"
   by (vector mult_assoc)
@@ -233,7 +228,7 @@
   by (vector field_simps)
 
 lemma vec_eq[simp]: "(vec m = vec n) \<longleftrightarrow> (m = n)"
-  by (simp add: Cart_eq)
+  by (simp add: vec_eq_iff)
 
 lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_zero)
 lemma vector_mul_eq_0[simp]: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0"
@@ -248,7 +243,7 @@
   by (metis vector_mul_rcancel)
 
 lemma component_le_norm_cart: "\<bar>x$i\<bar> <= norm x"
-  apply (simp add: norm_vector_def)
+  apply (simp add: norm_vec_def)
   apply (rule member_le_setL2, simp_all)
   done
 
@@ -259,10 +254,10 @@
   by (metis component_le_norm_cart basic_trans_rules(21))
 
 lemma norm_le_l1_cart: "norm x <= setsum(\<lambda>i. \<bar>x$i\<bar>) UNIV"
-  by (simp add: norm_vector_def setL2_le_setsum)
+  by (simp add: norm_vec_def setL2_le_setsum)
 
 lemma scalar_mult_eq_scaleR: "c *s x = c *\<^sub>R x"
-  unfolding vector_scaleR_def vector_scalar_mult_def by simp
+  unfolding scaleR_vec_def vector_scalar_mult_def by simp
 
 lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y"
   unfolding dist_norm scalar_mult_eq_scaleR
@@ -274,12 +269,12 @@
   by (cases "finite S", induct S set: finite, simp_all)
 
 lemma setsum_eq: "setsum f S = (\<chi> i. setsum (\<lambda>x. (f x)$i ) S)"
-  by (simp add: Cart_eq)
+  by (simp add: vec_eq_iff)
 
 lemma setsum_cmul:
   fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"
   shows "setsum (\<lambda>x. c *s f x) S = c *s setsum f S"
-  by (simp add: Cart_eq setsum_right_distrib)
+  by (simp add: vec_eq_iff setsum_right_distrib)
 
 (* TODO: use setsum_norm_allsubsets_bound *)
 lemma setsum_norm_allsubsets_bound_cart:
@@ -320,170 +315,6 @@
   finally show ?thesis .
 qed
 
-subsection {* A bijection between 'n::finite and {..<CARD('n)} *}
-
-definition cart_bij_nat :: "nat \<Rightarrow> ('n::finite)" where
-  "cart_bij_nat = (SOME p. bij_betw p {..<CARD('n)} (UNIV::'n set) )"
-
-abbreviation "\<pi> \<equiv> cart_bij_nat"
-definition "\<pi>' = inv_into {..<CARD('n)} (\<pi>::nat \<Rightarrow> ('n::finite))"
-
-lemma bij_betw_pi:
-  "bij_betw \<pi> {..<CARD('n::finite)} (UNIV::('n::finite) set)"
-  using ex_bij_betw_nat_finite[of "UNIV::'n set"]
-  by (auto simp: cart_bij_nat_def atLeast0LessThan
-    intro!: someI_ex[of "\<lambda>x. bij_betw x {..<CARD('n)} (UNIV::'n set)"])
-
-lemma bij_betw_pi'[intro]: "bij_betw \<pi>' (UNIV::'n set) {..<CARD('n::finite)}"
-  using bij_betw_inv_into[OF bij_betw_pi] unfolding \<pi>'_def by auto
-
-lemma pi'_inj[intro]: "inj \<pi>'"
-  using bij_betw_pi' unfolding bij_betw_def by auto
-
-lemma pi'_range[intro]: "\<And>i::'n. \<pi>' i < CARD('n::finite)"
-  using bij_betw_pi' unfolding bij_betw_def by auto
-
-lemma \<pi>\<pi>'[simp]: "\<And>i::'n::finite. \<pi> (\<pi>' i) = i"
-  using bij_betw_pi by (auto intro!: f_inv_into_f simp: \<pi>'_def bij_betw_def)
-
-lemma \<pi>'\<pi>[simp]: "\<And>i. i\<in>{..<CARD('n::finite)} \<Longrightarrow> \<pi>' (\<pi> i::'n) = i"
-  using bij_betw_pi by (auto intro!: inv_into_f_eq simp: \<pi>'_def bij_betw_def)
-
-lemma \<pi>\<pi>'_alt[simp]: "\<And>i. i<CARD('n::finite) \<Longrightarrow> \<pi>' (\<pi> i::'n) = i"
-  by auto
-
-lemma \<pi>_inj_on: "inj_on (\<pi>::nat\<Rightarrow>'n::finite) {..<CARD('n)}"
-  using bij_betw_pi[where 'n='n] by (simp add: bij_betw_def)
-
-instantiation cart :: (euclidean_space, finite) euclidean_space
-begin
-
-definition "dimension (t :: ('a ^ 'b) itself) = CARD('b) * DIM('a)"
-
-definition "(basis i::'a^'b) =
-  (if i < (CARD('b) * DIM('a))
-  then (\<chi> j::'b. if j = \<pi>(i div DIM('a)) then basis (i mod DIM('a)) else 0)
-  else 0)"
-
-lemma basis_eq:
-  assumes "i < CARD('b)" and "j < DIM('a)"
-  shows "basis (j + i * DIM('a)) = (\<chi> k. if k = \<pi> i then basis j else 0)"
-proof -
-  have "j + i * DIM('a) <  DIM('a) * (i + 1)" using assms by (auto simp: field_simps)
-  also have "\<dots> \<le> DIM('a) * CARD('b)" using assms unfolding mult_le_cancel1 by auto
-  finally show ?thesis
-    unfolding basis_cart_def using assms by (auto simp: Cart_eq not_less field_simps)
-qed
-
-lemma basis_eq_pi':
-  assumes "j < DIM('a)"
-  shows "basis (j + \<pi>' i * DIM('a)) $ k = (if k = i then basis j else 0)"
-  apply (subst basis_eq)
-  using pi'_range assms by simp_all
-
-lemma split_times_into_modulo[consumes 1]:
-  fixes k :: nat
-  assumes "k < A * B"
-  obtains i j where "i < A" and "j < B" and "k = j + i * B"
-proof
-  have "A * B \<noteq> 0"
-  proof assume "A * B = 0" with assms show False by simp qed
-  hence "0 < B" by auto
-  thus "k mod B < B" using `0 < B` by auto
-next
-  have "k div B * B \<le> k div B * B + k mod B" by (rule le_add1)
-  also have "... < A * B" using assms by simp
-  finally show "k div B < A" by auto
-qed simp
-
-lemma split_CARD_DIM[consumes 1]:
-  fixes k :: nat
-  assumes k: "k < CARD('b) * DIM('a)"
-  obtains i and j::'b where "i < DIM('a)" "k = i + \<pi>' j * DIM('a)"
-proof -
-  from split_times_into_modulo[OF k] guess i j . note ij = this
-  show thesis
-  proof
-    show "j < DIM('a)" using ij by simp
-    show "k = j + \<pi>' (\<pi> i :: 'b) * DIM('a)"
-      using ij by simp
-  qed
-qed
-
-lemma linear_less_than_times:
-  fixes i j A B :: nat assumes "i < B" "j < A"
-  shows "j + i * A < B * A"
-proof -
-  have "i * A + j < (Suc i)*A" using `j < A` by simp
-  also have "\<dots> \<le> B * A" using `i < B` unfolding mult_le_cancel2 by simp
-  finally show ?thesis by simp
-qed
-
-lemma DIM_cart[simp]: "DIM('a^'b) = CARD('b) * DIM('a)"
-  by (rule dimension_cart_def)
-
-lemma all_less_DIM_cart:
-  fixes m n :: nat
-  shows "(\<forall>i<DIM('a^'b). P i) \<longleftrightarrow> (\<forall>x::'b. \<forall>i<DIM('a). P (i + \<pi>' x * DIM('a)))"
-unfolding DIM_cart
-apply safe
-apply (drule spec, erule mp, erule linear_less_than_times [OF pi'_range])
-apply (erule split_CARD_DIM, simp)
-done
-
-lemma eq_pi_iff:
-  fixes x :: "'c::finite"
-  shows "i < CARD('c::finite) \<Longrightarrow> x = \<pi> i \<longleftrightarrow> \<pi>' x = i"
-  by auto
-
-lemma all_less_mult:
-  fixes m n :: nat
-  shows "(\<forall>i<(m * n). P i) \<longleftrightarrow> (\<forall>i<m. \<forall>j<n. P (j + i * n))"
-apply safe
-apply (drule spec, erule mp, erule (1) linear_less_than_times)
-apply (erule split_times_into_modulo, simp)
-done
-
-lemma inner_if:
-  "inner (if a then x else y) z = (if a then inner x z else inner y z)"
-  "inner x (if a then y else z) = (if a then inner x y else inner x z)"
-  by simp_all
-
-instance proof
-  show "0 < DIM('a ^ 'b)"
-    unfolding dimension_cart_def
-    by (intro mult_pos_pos zero_less_card_finite DIM_positive)
-next
-  fix i :: nat
-  assume "DIM('a ^ 'b) \<le> i" thus "basis i = (0::'a^'b)"
-    unfolding dimension_cart_def basis_cart_def
-    by simp
-next
-  show "\<forall>i<DIM('a ^ 'b). \<forall>j<DIM('a ^ 'b).
-    (basis i :: 'a ^ 'b) \<bullet> basis j = (if i = j then 1 else 0)"
-    apply (simp add: inner_vector_def)
-    apply safe
-    apply (erule split_CARD_DIM, simp add: basis_eq_pi')
-    apply (simp add: inner_if setsum_delta cong: if_cong)
-    apply (simp add: basis_orthonormal)
-    apply (elim split_CARD_DIM, simp add: basis_eq_pi')
-    apply (simp add: inner_if setsum_delta cong: if_cong)
-    apply (clarsimp simp add: basis_orthonormal)
-    done
-next
-  fix x :: "'a ^ 'b"
-  show "(\<forall>i<DIM('a ^ 'b). inner (basis i) x = 0) \<longleftrightarrow> x = 0"
-    unfolding all_less_DIM_cart
-    unfolding inner_vector_def
-    apply (simp add: basis_eq_pi')
-    apply (simp add: inner_if setsum_delta cong: if_cong)
-    apply (simp add: euclidean_all_zero)
-    apply (simp add: Cart_eq)
-    done
-qed
-
-end
-
 lemma if_distr: "(if P then f else g) $ i = (if P then f $ i else g $ i)" by simp
 
 lemma split_dimensions'[consumes 1]:
@@ -525,10 +356,10 @@
 lemmas cart_simps = forall_CARD_DIM exists_CARD_DIM forall_CARD exists_CARD
 
 lemma cart_euclidean_nth[simp]:
-  fixes x :: "('a::euclidean_space, 'b::finite) cart"
+  fixes x :: "('a::euclidean_space, 'b::finite) vec"
   assumes j:"j < DIM('a)"
   shows "x $$ (j + \<pi>' i * DIM('a)) = x $ i $$ j"
-  unfolding euclidean_component_def inner_vector_def basis_eq_pi'[OF j] if_distrib cond_application_beta
+  unfolding euclidean_component_def inner_vec_def basis_eq_pi'[OF j] if_distrib cond_application_beta
   by (simp add: setsum_cases)
 
 lemma real_euclidean_nth:
@@ -559,13 +390,13 @@
   thus "x = y \<and> i = j" using * by simp
 qed simp
 
-instance cart :: (ordered_euclidean_space,finite) ordered_euclidean_space
+instance vec :: (ordered_euclidean_space, finite) ordered_euclidean_space
 proof
   fix x y::"'a^'b"
-  show "(x \<le> y) = (\<forall>i<DIM(('a, 'b) cart). x $$ i \<le> y $$ i)"
-    unfolding vector_le_def apply(subst eucl_le) by (simp add: cart_simps)
-  show"(x < y) = (\<forall>i<DIM(('a, 'b) cart). x $$ i < y $$ i)"
-    unfolding vector_less_def apply(subst eucl_less) by (simp add: cart_simps)
+  show "(x \<le> y) = (\<forall>i<DIM(('a, 'b) vec). x $$ i \<le> y $$ i)"
+    unfolding less_eq_vec_def apply(subst eucl_le) by (simp add: cart_simps)
+  show"(x < y) = (\<forall>i<DIM(('a, 'b) vec). x $$ i < y $$ i)"
+    unfolding less_vec_def apply(subst eucl_less) by (simp add: cart_simps)
 qed
 
 subsection{* Basis vectors in coordinate directions. *}
@@ -577,7 +408,7 @@
 
 lemma norm_basis[simp]:
   shows "norm (cart_basis k :: real ^'n) = 1"
-  apply (simp add: cart_basis_def norm_eq_sqrt_inner) unfolding inner_vector_def
+  apply (simp add: cart_basis_def norm_eq_sqrt_inner) unfolding inner_vec_def
   apply (vector delta_mult_idempotent)
   using setsum_delta[of "UNIV :: 'n set" "k" "\<lambda>k. 1::real"] by auto
 
@@ -597,14 +428,14 @@
 qed
 
 lemma basis_inj[intro]: "inj (cart_basis :: 'n \<Rightarrow> real ^'n)"
-  by (simp add: inj_on_def Cart_eq)
+  by (simp add: inj_on_def vec_eq_iff)
 
 lemma basis_expansion:
   "setsum (\<lambda>i. (x$i) *s cart_basis i) UNIV = (x::('a::ring_1) ^'n)" (is "?lhs = ?rhs" is "setsum ?f ?S = _")
-  by (auto simp add: Cart_eq if_distrib setsum_delta[of "?S", where ?'b = "'a", simplified] cong del: if_weak_cong)
+  by (auto simp add: vec_eq_iff if_distrib setsum_delta[of "?S", where ?'b = "'a", simplified] cong del: if_weak_cong)
 
 lemma smult_conv_scaleR: "c *s x = scaleR c x"
-  unfolding vector_scalar_mult_def vector_scaleR_def by simp
+  unfolding vector_scalar_mult_def scaleR_vec_def by simp
 
 lemma basis_expansion':
   "setsum (\<lambda>i. (x$i) *\<^sub>R cart_basis i) UNIV = x"
@@ -612,22 +443,22 @@
 
 lemma basis_expansion_unique:
   "setsum (\<lambda>i. f i *s cart_basis i) UNIV = (x::('a::comm_ring_1) ^'n) \<longleftrightarrow> (\<forall>i. f i = x$i)"
-  by (simp add: Cart_eq setsum_delta if_distrib cong del: if_weak_cong)
+  by (simp add: vec_eq_iff setsum_delta if_distrib cong del: if_weak_cong)
 
 lemma dot_basis:
   shows "cart_basis i \<bullet> x = x$i" "x \<bullet> (cart_basis i) = (x$i)"
-  by (auto simp add: inner_vector_def cart_basis_def cond_application_beta if_distrib setsum_delta
+  by (auto simp add: inner_vec_def cart_basis_def cond_application_beta if_distrib setsum_delta
            cong del: if_weak_cong)
 
 lemma inner_basis:
   fixes x :: "'a::{real_inner, real_algebra_1} ^ 'n"
   shows "inner (cart_basis i) x = inner 1 (x $ i)"
     and "inner x (cart_basis i) = inner (x $ i) 1"
-  unfolding inner_vector_def cart_basis_def
+  unfolding inner_vec_def cart_basis_def
   by (auto simp add: cond_application_beta if_distrib setsum_delta cong del: if_weak_cong)
 
 lemma basis_eq_0: "cart_basis i = (0::'a::semiring_1^'n) \<longleftrightarrow> False"
-  by (auto simp add: Cart_eq)
+  by (auto simp add: vec_eq_iff)
 
 lemma basis_nonzero:
   shows "cart_basis k \<noteq> (0:: 'a::semiring_1 ^'n)"
@@ -635,14 +466,14 @@
 
 text {* some lemmas to map between Eucl and Cart *}
 lemma basis_real_n[simp]:"(basis (\<pi>' i)::real^'a) = cart_basis i"
-  unfolding basis_cart_def using pi'_range[where 'n='a]
-  by (auto simp: Cart_eq Cart_lambda_beta)
+  unfolding basis_vec_def using pi'_range[where 'n='a]
+  by (auto simp: vec_eq_iff)
 
 subsection {* Orthogonality on cartesian products *}
 
 lemma orthogonal_basis:
   shows "orthogonal (cart_basis i) x \<longleftrightarrow> x$i = (0::real)"
-  by (auto simp add: orthogonal_def inner_vector_def cart_basis_def if_distrib
+  by (auto simp add: orthogonal_def inner_vec_def cart_basis_def if_distrib
                      cond_application_beta setsum_delta cong del: if_weak_cong)
 
 lemma orthogonal_basis_basis:
@@ -684,7 +515,7 @@
         by (simp add: linear_cmul[OF lf])
       finally have "f x \<bullet> y = x \<bullet> ?w"
         apply (simp only: )
-        apply (simp add: inner_vector_def setsum_left_distrib setsum_right_distrib setsum_commute[of _ ?M ?N] field_simps)
+        apply (simp add: inner_vec_def setsum_left_distrib setsum_right_distrib setsum_commute[of _ ?M ?N] field_simps)
         done}
   }
   then show ?thesis unfolding adjoint_def
@@ -778,25 +609,25 @@
     setsum_delta' cong del: if_weak_cong)
 
 lemma matrix_transpose_mul: "transpose(A ** B) = transpose B ** transpose (A::'a::comm_semiring_1^_^_)"
-  by (simp add: matrix_matrix_mult_def transpose_def Cart_eq mult_commute)
+  by (simp add: matrix_matrix_mult_def transpose_def vec_eq_iff mult_commute)
 
 lemma matrix_eq:
   fixes A B :: "'a::semiring_1 ^ 'n ^ 'm"
   shows "A = B \<longleftrightarrow>  (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs")
   apply auto
-  apply (subst Cart_eq)
+  apply (subst vec_eq_iff)
   apply clarify
-  apply (clarsimp simp add: matrix_vector_mult_def cart_basis_def if_distrib cond_application_beta Cart_eq cong del: if_weak_cong)
+  apply (clarsimp simp add: matrix_vector_mult_def cart_basis_def if_distrib cond_application_beta vec_eq_iff cong del: if_weak_cong)
   apply (erule_tac x="cart_basis ia" in allE)
   apply (erule_tac x="i" in allE)
   by (auto simp add: cart_basis_def if_distrib cond_application_beta setsum_delta[OF finite] cong del: if_weak_cong)
 
 lemma matrix_vector_mul_component:
   shows "((A::real^_^_) *v x)$k = (A$k) \<bullet> x"
-  by (simp add: matrix_vector_mult_def inner_vector_def)
+  by (simp add: matrix_vector_mult_def inner_vec_def)
 
 lemma dot_lmul_matrix: "((x::real ^_) v* A) \<bullet> y = x \<bullet> (A *v y)"
-  apply (simp add: inner_vector_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib mult_ac)
+  apply (simp add: inner_vec_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib mult_ac)
   apply (subst setsum_commute)
   by simp
 
@@ -809,12 +640,12 @@
 lemma row_transpose:
   fixes A:: "'a::semiring_1^_^_"
   shows "row i (transpose A) = column i A"
-  by (simp add: row_def column_def transpose_def Cart_eq)
+  by (simp add: row_def column_def transpose_def vec_eq_iff)
 
 lemma column_transpose:
   fixes A:: "'a::semiring_1^_^_"
   shows "column i (transpose A) = row i A"
-  by (simp add: row_def column_def transpose_def Cart_eq)
+  by (simp add: row_def column_def transpose_def vec_eq_iff)
 
 lemma rows_transpose: "rows(transpose (A::'a::semiring_1^_^_)) = columns A"
 by (auto simp add: rows_def columns_def row_transpose intro: set_eqI)
@@ -824,15 +655,15 @@
 text{* Two sometimes fruitful ways of looking at matrix-vector multiplication. *}
 
 lemma matrix_mult_dot: "A *v x = (\<chi> i. A$i \<bullet> x)"
-  by (simp add: matrix_vector_mult_def inner_vector_def)
+  by (simp add: matrix_vector_mult_def inner_vec_def)
 
 lemma matrix_mult_vsum: "(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x$i) *s column i A) (UNIV:: 'n set)"
-  by (simp add: matrix_vector_mult_def Cart_eq column_def mult_commute)
+  by (simp add: matrix_vector_mult_def vec_eq_iff column_def mult_commute)
 
 lemma vector_componentwise:
   "(x::'a::ring_1^'n) = (\<chi> j. setsum (\<lambda>i. (x$i) * (cart_basis i :: 'a^'n)$j) (UNIV :: 'n set))"
   apply (subst basis_expansion[symmetric])
-  by (vector Cart_eq setsum_component)
+  by (vector vec_eq_iff setsum_component)
 
 lemma linear_componentwise:
   fixes f:: "real ^'m \<Rightarrow> real ^ _"
@@ -862,10 +693,10 @@
 where "matrix f = (\<chi> i j. (f(cart_basis j))$i)"
 
 lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::real ^ _))"
-  by (simp add: linear_def matrix_vector_mult_def Cart_eq field_simps setsum_right_distrib setsum_addf)
+  by (simp add: linear_def matrix_vector_mult_def vec_eq_iff field_simps setsum_right_distrib setsum_addf)
 
 lemma matrix_works: assumes lf: "linear f" shows "matrix f *v x = f (x::real ^ 'n)"
-apply (simp add: matrix_def matrix_vector_mult_def Cart_eq mult_commute)
+apply (simp add: matrix_def matrix_vector_mult_def vec_eq_iff mult_commute)
 apply clarify
 apply (rule linear_componentwise[OF lf, symmetric])
 done
@@ -883,11 +714,11 @@
   by (simp  add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def)
 
 lemma matrix_vector_column:"(A::'a::comm_semiring_1^'n^_) *v x = setsum (\<lambda>i. (x$i) *s ((transpose A)$i)) (UNIV:: 'n set)"
-  by (simp add: matrix_vector_mult_def transpose_def Cart_eq mult_commute)
+  by (simp add: matrix_vector_mult_def transpose_def vec_eq_iff mult_commute)
 
 lemma adjoint_matrix: "adjoint(\<lambda>x. (A::real^'n^'m) *v x) = (\<lambda>x. transpose A *v x)"
   apply (rule adjoint_unique)
-  apply (simp add: transpose_def inner_vector_def matrix_vector_mult_def setsum_left_distrib setsum_right_distrib)
+  apply (simp add: transpose_def inner_vec_def matrix_vector_mult_def setsum_left_distrib setsum_right_distrib)
   apply (subst setsum_commute)
   apply (auto simp add: mult_ac)
   done
@@ -1082,10 +913,10 @@
       let ?x = "\<chi> i. c i"
       have th0:"A *v ?x = 0"
         using c
-        unfolding matrix_mult_vsum Cart_eq
+        unfolding matrix_mult_vsum vec_eq_iff
         by auto
       from k[rule_format, OF th0] i
-      have "c i = 0" by (vector Cart_eq)}
+      have "c i = 0" by (vector vec_eq_iff)}
     hence ?rhs by blast}
   moreover
   {assume H: ?rhs
@@ -1204,17 +1035,17 @@
 
 lemma transpose_columnvector:
  "transpose(columnvector v) = rowvector v"
-  by (simp add: transpose_def rowvector_def columnvector_def Cart_eq)
+  by (simp add: transpose_def rowvector_def columnvector_def vec_eq_iff)
 
 lemma transpose_rowvector: "transpose(rowvector v) = columnvector v"
-  by (simp add: transpose_def columnvector_def rowvector_def Cart_eq)
+  by (simp add: transpose_def columnvector_def rowvector_def vec_eq_iff)
 
 lemma dot_rowvector_columnvector:
   "columnvector (A *v v) = A ** columnvector v"
   by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def)
 
 lemma dot_matrix_product: "(x::real^'n) \<bullet> y = (((rowvector x ::real^'n^1) ** (columnvector y :: real^1^'n))$1)$1"
-  by (vector matrix_matrix_mult_def rowvector_def columnvector_def inner_vector_def)
+  by (vector matrix_matrix_mult_def rowvector_def columnvector_def inner_vec_def)
 
 lemma dot_matrix_vector_mul:
   fixes A B :: "real ^'n ^'n" and x y :: "real ^'n"
@@ -1244,18 +1075,18 @@
   unfolding nth_conv_component
   using component_le_infnorm[of x] .
 
-instance cart :: (perfect_space, finite) perfect_space
+instance vec :: (perfect_space, finite) perfect_space
 proof
   fix x :: "'a ^ 'b"
   show "x islimpt UNIV"
     apply (rule islimptI)
-    apply (unfold open_vector_def)
+    apply (unfold open_vec_def)
     apply (drule (1) bspec)
     apply clarsimp
     apply (subgoal_tac "\<forall>i\<in>UNIV. \<exists>y. y \<in> A i \<and> y \<noteq> x $ i")
      apply (drule finite_choice [OF finite_UNIV], erule exE)
-     apply (rule_tac x="Cart_lambda f" in exI)
-     apply (simp add: Cart_eq)
+     apply (rule_tac x="vec_lambda f" in exI)
+     apply (simp add: vec_eq_iff)
     apply (rule ballI, drule_tac x=i in spec, clarify)
     apply (cut_tac x="x $ i" in islimpt_UNIV)
     apply (simp add: islimpt_def)
@@ -1288,7 +1119,7 @@
   apply (clarify)
   apply (drule spec, drule (1) mp)
   apply (erule eventually_elim1)
-  apply (erule le_less_trans [OF dist_nth_le_cart])
+  apply (erule le_less_trans [OF dist_vec_nth_le])
   done
 
 lemma bounded_component_cart: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $ i) ` s)"
@@ -1297,7 +1128,7 @@
 apply (rule_tac x="x $ i" in exI)
 apply (rule_tac x="e" in exI)
 apply clarify
-apply (rule order_trans [OF dist_nth_le_cart], simp)
+apply (rule order_trans [OF dist_vec_nth_le], simp)
 done
 
 lemma compact_lemma_cart:
@@ -1334,7 +1165,7 @@
   qed
 qed
 
-instance cart :: (heine_borel, finite) heine_borel
+instance vec :: (heine_borel, finite) heine_borel
 proof
   fix s :: "('a ^ 'b) set" and f :: "nat \<Rightarrow> 'a ^ 'b"
   assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
@@ -1350,7 +1181,7 @@
     moreover
     { fix n assume n: "\<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))"
       have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) $ i) (l $ i))"
-        unfolding dist_vector_def using zero_le_dist by (rule setL2_le_setsum)
+        unfolding dist_vec_def using zero_le_dist by (rule setL2_le_setsum)
       also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"
         by (rule setsum_strict_mono) (simp_all add: n)
       finally have "dist (f (r n)) l < e" by simp
@@ -1371,12 +1202,12 @@
 lemma interval_cart: fixes a :: "'a::ord^'n" shows
   "{a <..< b} = {x::'a^'n. \<forall>i. a$i < x$i \<and> x$i < b$i}" and
   "{a .. b} = {x::'a^'n. \<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i}"
-  by (auto simp add: set_eq_iff vector_less_def vector_le_def)
+  by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def)
 
 lemma mem_interval_cart: fixes a :: "'a::ord^'n" shows
   "x \<in> {a<..<b} \<longleftrightarrow> (\<forall>i. a$i < x$i \<and> x$i < b$i)"
   "x \<in> {a .. b} \<longleftrightarrow> (\<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i)"
-  using interval_cart[of a b] by(auto simp add: set_eq_iff vector_less_def vector_le_def)
+  using interval_cart[of a b] by(auto simp add: set_eq_iff less_vec_def less_eq_vec_def)
 
 lemma interval_eq_empty_cart: fixes a :: "real^'n" shows
  "({a <..< b} = {} \<longleftrightarrow> (\<exists>i. b$i \<le> a$i))" (is ?th1) and
@@ -1429,7 +1260,7 @@
 
 lemma interval_sing: fixes a :: "'a::linorder^'n" shows
  "{a .. a} = {a} \<and> {a<..<a} = {}"
-apply(auto simp add: set_eq_iff vector_less_def vector_le_def Cart_eq)
+apply(auto simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
 apply (simp add: order_eq_iff)
 apply (auto simp add: not_less less_imp_le)
 done
@@ -1442,17 +1273,17 @@
   { fix i
     have "a $ i \<le> x $ i"
       using x order_less_imp_le[of "a$i" "x$i"]
-      by(simp add: set_eq_iff vector_less_def vector_le_def Cart_eq)
+      by(simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
   }
   moreover
   { fix i
     have "x $ i \<le> b $ i"
       using x order_less_imp_le[of "x$i" "b$i"]
-      by(simp add: set_eq_iff vector_less_def vector_le_def Cart_eq)
+      by(simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
   }
   ultimately
   show "a \<le> x \<and> x \<le> b"
-    by(simp add: set_eq_iff vector_less_def vector_le_def Cart_eq)
+    by(simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
 qed
 
 lemma subset_interval_cart: fixes a :: "real^'n" shows
@@ -1572,12 +1403,12 @@
 
 lemma dim_substandard_cart:
   shows "dim {x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0} = card d" (is "dim ?A = _")
-proof- have *:"{x. \<forall>i<DIM((real, 'n) cart). i \<notin> \<pi>' ` d \<longrightarrow> x $$ i = 0} = 
+proof- have *:"{x. \<forall>i<DIM((real, 'n) vec). i \<notin> \<pi>' ` d \<longrightarrow> x $$ i = 0} = 
     {x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0}"apply safe
     apply(erule_tac x="\<pi>' i" in allE) defer
     apply(erule_tac x="\<pi> i" in allE)
     unfolding image_iff real_euclidean_nth[symmetric] by (auto simp: pi'_inj[THEN inj_eq])
-  have " \<pi>' ` d \<subseteq> {..<DIM((real, 'n) cart)}" using pi'_range[where 'n='n] by auto
+  have " \<pi>' ` d \<subseteq> {..<DIM((real, 'n) vec)}" using pi'_range[where 'n='n] by auto
   thus ?thesis using dim_substandard[of "\<pi>' ` d", where 'a="real^'n"] 
     unfolding * using card_image[of "\<pi>'" d] using pi'_inj unfolding inj_on_def by auto
 qed
@@ -1630,7 +1461,7 @@
 declare vector_add_ldistrib[simp] vector_ssub_ldistrib[simp] vector_smult_assoc[simp] vector_smult_rneg[simp]
 declare vector_sadd_rdistrib[simp] vector_sub_rdistrib[simp]
 
-lemmas vector_component_simps = vector_minus_component vector_smult_component vector_add_component vector_le_def Cart_lambda_beta basis_component vector_uminus_component
+lemmas vector_component_simps = vector_minus_component vector_smult_component vector_add_component less_eq_vec_def vec_lambda_beta basis_component vector_uminus_component
 
 lemma convex_box_cart:
   assumes "\<And>i. convex {x. P i x}"
@@ -1655,7 +1486,7 @@
 
 lemma std_simplex_cart:
   "(insert (0::real^'n) { cart_basis i | i. i\<in>UNIV}) =
-  (insert 0 { basis i | i. i<DIM((real,'n) cart)})"
+  (insert 0 { basis i | i. i<DIM((real,'n) vec)})"
   apply(rule arg_cong[where f="\<lambda>s. (insert 0 s)"])
   unfolding basis_real_n[THEN sym] apply safe
   apply(rule_tac x="\<pi>' i" in exI) defer
@@ -1682,22 +1513,22 @@
 lemma interval_bij_cart:"interval_bij = (\<lambda> (a,b) (u,v) (x::real^'n).
     (\<chi> i. u$i + (x$i - a$i) / (b$i - a$i) * (v$i - u$i))::real^'n)"
   unfolding interval_bij_def apply(rule ext)+ apply safe
-  unfolding Cart_eq Cart_lambda_beta unfolding nth_conv_component
+  unfolding vec_eq_iff vec_lambda_beta unfolding nth_conv_component
   apply rule apply(subst euclidean_lambda_beta) using pi'_range by auto
 
 lemma interval_bij_affine_cart:
  "interval_bij (a,b) (u,v) = (\<lambda>x. (\<chi> i. (v$i - u$i) / (b$i - a$i) * x$i) +
             (\<chi> i. u$i - (v$i - u$i) / (b$i - a$i) * a$i)::real^'n)"
-  apply rule unfolding Cart_eq interval_bij_cart vector_component_simps
+  apply rule unfolding vec_eq_iff interval_bij_cart vector_component_simps
   by(auto simp add: field_simps add_divide_distrib[THEN sym]) 
 
 subsection "Derivative"
 
 lemma has_derivative_vmul_component_cart: fixes c::"real^'a \<Rightarrow> real^'b" and v::"real^'c"
   assumes "(c has_derivative c') net"
-  shows "((\<lambda>x. c(x)$k *\<^sub>R v) has_derivative (\<lambda>x. (c' x)$k *\<^sub>R v)) net" 
-  using has_derivative_vmul_component[OF assms] 
-  unfolding nth_conv_component .
+  shows "((\<lambda>x. c(x)$k *\<^sub>R v) has_derivative (\<lambda>x. (c' x)$k *\<^sub>R v)) net"
+  unfolding nth_conv_component
+  by (intro has_derivative_intros assms)
 
 lemma differentiable_at_imp_differentiable_on: "(\<forall>x\<in>(s::(real^'n) set). f differentiable at x) \<Longrightarrow> f differentiable_on s"
   unfolding differentiable_on_def by(auto intro!: differentiable_at_withinI)
@@ -1718,7 +1549,7 @@
 
 proof(rule ccontr)
   def D \<equiv> "jacobian f (at x)" assume "jacobian f (at x) $ k \<noteq> 0"
-  then obtain j where j:"D$k$j \<noteq> 0" unfolding Cart_eq D_def by auto
+  then obtain j where j:"D$k$j \<noteq> 0" unfolding vec_eq_iff D_def by auto
   hence *:"abs (jacobian f (at x) $ k $ j) / 2 > 0" unfolding D_def by auto
   note as = assms(3)[unfolded jacobian_works has_derivative_at_alt]
   guess e' using as[THEN conjunct2,rule_format,OF *] .. note e' = this
@@ -1805,12 +1636,10 @@
   where "dest_vec1 x \<equiv> (x$1)"
 
 lemma vec1_dest_vec1[simp]: "vec1(dest_vec1 x) = x" "dest_vec1(vec1 y) = y"
-  by (simp_all add:  Cart_eq)
+  by (simp_all add:  vec_eq_iff)
 
 lemma vec1_component[simp]: "(vec1 x)$1 = x"
-  by (simp_all add:  Cart_eq)
-
-declare vec1_dest_vec1(1) [simp]
+  by (simp_all add:  vec_eq_iff)
 
 lemma forall_vec1: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P (vec1 x))"
   by (metis vec1_dest_vec1(1))
@@ -1827,7 +1656,7 @@
 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)
+  by (simp add: vec_eq_iff)
 
 lemma forall_one: "(\<forall>(x::'a ^1). P x) \<longleftrightarrow> (\<forall>x. P(\<chi> i. x))"
   apply auto
@@ -1836,7 +1665,7 @@
   done
 
 lemma norm_vector_1: "norm (x :: _^1) = norm (x$1)"
-  by (simp add: norm_vector_def)
+  by (simp add: norm_vec_def)
 
 lemma norm_real: "norm(x::real ^ 1) = abs(x$1)"
   by (simp add: norm_vector_1)
@@ -1917,7 +1746,7 @@
   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
+   vec1_eq vec_cmul[THEN sym] smult_conv_scaleR[THEN sym] o_def dist_real_def real_norm_def
 
 lemma bounded_linear_vec1:"bounded_linear (vec1::real\<Rightarrow>real^1)"
   unfolding bounded_linear_def additive_def bounded_linear_axioms_def 
@@ -1936,14 +1765,14 @@
   unfolding smult_conv_scaleR
   apply (rule ext)
   apply (subst matrix_works[OF lf, symmetric])
-  apply (auto simp add: Cart_eq matrix_vector_mult_def column_def mult_commute)
+  apply (auto simp add: vec_eq_iff matrix_vector_mult_def column_def mult_commute)
   done
 
 lemma linear_to_scalars: assumes lf: "linear (f::real ^'n \<Rightarrow> real^1)"
   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)
+  apply (simp add: vec_eq_iff matrix_vector_mult_def row_def inner_vec_def mult_commute)
   done
 
 lemma dest_vec1_eq_0: "dest_vec1 x = 0 \<longleftrightarrow> x = 0"
@@ -1967,14 +1796,14 @@
   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)
+  apply(erule_tac x="dest_vec1 x'" in ballE) by(auto simp add:less_eq_vec_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_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)
+  apply(erule_tac x="vec1 x'" in ballE) by(auto simp add:less_eq_vec_def)
 
 lemma continuous_on_vec1:"continuous_on A (vec1::real\<Rightarrow>real^1)"
   by(rule linear_continuous_on[OF bounded_linear_vec1])
@@ -1982,12 +1811,12 @@
 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)
+by(simp_all add: vec_eq_iff less_vec_def less_eq_vec_def)
 
 lemma vec1_interval:fixes a::"real" shows
   "vec1 ` {a .. b} = {vec1 a .. vec1 b}"
   "vec1 ` {a<..<b} = {vec1 a<..<vec1 b}"
-  apply(rule_tac[!] set_eqI) unfolding image_iff vector_less_def unfolding mem_interval_cart
+  apply(rule_tac[!] set_eqI) unfolding image_iff less_vec_def unfolding mem_interval_cart
   unfolding forall_1 unfolding vec1_dest_vec1_simps
   apply rule defer apply(rule_tac x="dest_vec1 x" in bexI) prefer 3 apply rule defer
   apply(rule_tac x="dest_vec1 x" in bexI) by auto
@@ -1996,12 +1825,12 @@
 
 lemma interval_cases_1: fixes x :: "real^1" shows
  "x \<in> {a .. b} ==> x \<in> {a<..<b} \<or> (x = a) \<or> (x = b)"
-  unfolding Cart_eq vector_less_def vector_le_def mem_interval_cart by(auto simp del:dest_vec1_eq)
+  unfolding vec_eq_iff less_vec_def less_eq_vec_def mem_interval_cart by(auto simp del:dest_vec1_eq)
 
 lemma in_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) \<and>
   (x \<in> {a<..<b} \<longleftrightarrow> dest_vec1 a < dest_vec1 x \<and> dest_vec1 x < dest_vec1 b)"
-  unfolding Cart_eq vector_less_def vector_le_def mem_interval_cart by(auto simp del:dest_vec1_eq)
+  unfolding vec_eq_iff less_vec_def less_eq_vec_def mem_interval_cart by(auto simp del:dest_vec1_eq)
 
 lemma interval_eq_empty_1: fixes a :: "real^1" shows
   "{a .. b} = {} \<longleftrightarrow> dest_vec1 b < dest_vec1 a"
@@ -2037,10 +1866,10 @@
 
 lemma open_closed_interval_1: fixes a :: "real^1" shows
  "{a<..<b} = {a .. b} - {a, b}"
-  unfolding set_eq_iff apply simp unfolding vector_less_def and vector_le_def and forall_1 and dest_vec1_eq[THEN sym] by(auto simp del:dest_vec1_eq)
+  unfolding set_eq_iff apply simp unfolding less_vec_def and less_eq_vec_def and forall_1 and dest_vec1_eq[THEN sym] by(auto simp del:dest_vec1_eq)
 
 lemma closed_open_interval_1: "dest_vec1 (a::real^1) \<le> dest_vec1 b ==> {a .. b} = {a<..<b} \<union> {a,b}"
-  unfolding set_eq_iff apply simp unfolding vector_less_def and vector_le_def and forall_1 and dest_vec1_eq[THEN sym] by(auto simp del:dest_vec1_eq)
+  unfolding set_eq_iff apply simp unfolding less_vec_def and less_eq_vec_def and forall_1 and dest_vec1_eq[THEN sym] by(auto simp del:dest_vec1_eq)
 
 lemma Lim_drop_le: fixes f :: "'a \<Rightarrow> real^1" shows
   "(f ---> l) net \<Longrightarrow> ~(trivial_limit net) \<Longrightarrow> eventually (\<lambda>x. dest_vec1 (f x) \<le> b) net ==> dest_vec1 l \<le> b"
@@ -2069,7 +1898,7 @@
 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_le_def Cart_eq)
+  by(auto simp add: less_eq_vec_def vec_eq_iff)
 
 lemma dest_vec1_inverval:
   "dest_vec1 ` {a .. b} = {dest_vec1 a .. dest_vec1 b}"
@@ -2081,18 +1910,18 @@
   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)
+  by (auto simp add: less_vec_def less_eq_vec_def)
 
 lemma dest_vec1_setsum: assumes "finite S"
   shows " dest_vec1 (setsum f S) = setsum (\<lambda>x. dest_vec1 (f x)) S"
   using dest_vec1_sum[OF assms] by auto
 
 lemma open_dest_vec1_vimage: "open S \<Longrightarrow> open (dest_vec1 -` S)"
-unfolding open_vector_def forall_1 by auto
+unfolding open_vec_def forall_1 by auto
 
 lemma tendsto_dest_vec1 [tendsto_intros]:
   "(f ---> l) net \<Longrightarrow> ((\<lambda>x. dest_vec1 (f x)) ---> dest_vec1 l) net"
-by(rule tendsto_Cart_nth)
+by(rule tendsto_vec_nth)
 
 lemma continuous_dest_vec1: "continuous net f \<Longrightarrow> continuous net (\<lambda>x. dest_vec1 (f x))"
   unfolding continuous_def by (rule tendsto_dest_vec1)
@@ -2118,9 +1947,9 @@
     unfolding vec1_dest_vec1_simps by auto qed
 
 lemma vec1_le[simp]:fixes a::real shows "vec1 a \<le> vec1 b \<longleftrightarrow> a \<le> b"
-  unfolding vector_le_def by auto
+  unfolding less_eq_vec_def by auto
 lemma vec1_less[simp]:fixes a::real shows "vec1 a < vec1 b \<longleftrightarrow> a < b"
-  unfolding vector_less_def by auto
+  unfolding less_vec_def by auto
 
 
 subsection {* Derivatives on real = Derivatives on @{typ "real^1"} *}
@@ -2164,7 +1993,7 @@
 
 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)
+  have "\<forall>x::real^1. norm x = 1 \<longleftrightarrow> x\<in>{vec1 -1, vec1 (1::real)}" unfolding forall_vec1 by(auto simp add:vec_eq_iff)
   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
@@ -2203,11 +2032,11 @@
   "{a..b::real^'n} \<inter> {x. x$k \<le> c} = {a .. (\<chi> i. if i = k then min (b$k) c else b$i)}"
   "{a..b} \<inter> {x. x$k \<ge> c} = {(\<chi> i. if i = k then max (a$k) c else a$i) .. b}"
   apply(rule_tac[!] set_eqI) unfolding Int_iff mem_interval_cart mem_Collect_eq
-  unfolding Cart_lambda_beta by auto
+  unfolding vec_lambda_beta by auto
 
 (*lemma content_split_cart:
   "content {a..b::real^'n} = content({a..b} \<inter> {x. x$k \<le> c}) + content({a..b} \<inter> {x. x$k >= c})"
-proof- note simps = interval_split_cart content_closed_interval_cases_cart Cart_lambda_beta vector_le_def
+proof- note simps = interval_split_cart content_closed_interval_cases_cart vec_lambda_beta less_eq_vec_def
   { presume "a\<le>b \<Longrightarrow> ?thesis" thus ?thesis apply(cases "a\<le>b") unfolding simps by auto }
   have *:"UNIV = insert k (UNIV - {k})" "\<And>x. finite (UNIV-{x::'n})" "\<And>x. x\<notin>UNIV-{x}" by auto
   have *:"\<And>X Y Z. (\<Prod>i\<in>UNIV. Z i (if i = k then X else Y i)) = Z k X * (\<Prod>i\<in>UNIV-{k}. Z i (Y i))"
@@ -2217,7 +2046,7 @@
     \<Longrightarrow> x* (b$k - a$k) = x*(max (a $ k) c - a $ k) + x*(b $ k - max (a $ k) c)"
     by  (auto simp add:field_simps)
   moreover have "\<not> a $ k \<le> c \<Longrightarrow> \<not> c \<le> b $ k \<Longrightarrow> False"
-    unfolding not_le using as[unfolded vector_le_def,rule_format,of k] by auto
+    unfolding not_le using as[unfolded less_eq_vec_def,rule_format,of k] by auto
   ultimately show ?thesis 
     unfolding simps unfolding *(1)[of "\<lambda>i x. b$i - x"] *(1)[of "\<lambda>i x. x - a$i"] *(2) by(auto)
 qed*)
@@ -2225,7 +2054,7 @@
 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: split_beta)
+    unfolding vec_sub vec_eq_iff 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
diff -r fa98623f1006 -r d282b3c5df7c src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy
--- a/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy	Wed Aug 10 21:24:26 2011 +0200
+++ b/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy	Wed Aug 10 18:07:32 2011 -0700
@@ -198,9 +198,6 @@
   from this show ?thesis by auto
 qed
 
-lemma basis_0[simp]:"(basis i::'a::euclidean_space) = 0 \<longleftrightarrow> i\<ge>DIM('a)"
-  using norm_basis[of i, where 'a='a] unfolding norm_eq_zero[where 'a='a,THEN sym] by auto
-
 lemma basis_to_basis_subspace_isomorphism:
   assumes s: "subspace (S:: ('n::euclidean_space) set)"
   and t: "subspace (T :: ('m::euclidean_space) set)"
@@ -2142,7 +2139,7 @@
   apply (simp add: rel_interior, safe)
   apply (force simp add: open_contains_ball)
   apply (rule_tac x="ball x e" in exI)
-  apply (simp add: open_ball centre_in_ball)
+  apply (simp add: centre_in_ball)
   done
 
 lemma rel_interior_ball: 
diff -r fa98623f1006 -r d282b3c5df7c src/HOL/Multivariate_Analysis/Derivative.thy
--- a/src/HOL/Multivariate_Analysis/Derivative.thy	Wed Aug 10 21:24:26 2011 +0200
+++ b/src/HOL/Multivariate_Analysis/Derivative.thy	Wed Aug 10 18:07:32 2011 -0700
@@ -23,23 +23,12 @@
  "(f has_derivative f') net \<equiv> bounded_linear f' \<and> ((\<lambda>y. (1 / (norm (y - netlimit net))) *\<^sub>R
    (f y - (f (netlimit net) + f'(y - netlimit net)))) ---> 0) net"
 
-lemma derivative_linear[dest]:"(f has_derivative f') net \<Longrightarrow> bounded_linear f'"
+lemma derivative_linear[dest]:
+  "(f has_derivative f') net \<Longrightarrow> bounded_linear f'"
   unfolding has_derivative_def by auto
 
-lemma DERIV_conv_has_derivative:"(DERIV f x :> f') = (f has_derivative op * f') (at (x::real))" (is "?l = ?r") proof 
-  assume ?l note as = this[unfolded deriv_def LIM_def,rule_format]
-  show ?r unfolding has_derivative_def Lim_at apply- apply(rule,rule mult.bounded_linear_right)
-    apply safe apply(drule as,safe) apply(rule_tac x=s in exI) apply safe
-    apply(erule_tac x="xa - x" in allE) unfolding dist_norm netlimit_at[of x] unfolding diff_0_right norm_scaleR
-    by(auto simp add:field_simps) 
-next assume ?r note this[unfolded has_derivative_def Lim_at] note as=conjunct2[OF this,rule_format]
-  have *:"\<And>x xa f'. xa \<noteq> 0 \<Longrightarrow> \<bar>(f (xa + x) - f x) / xa - f'\<bar> = \<bar>(f (xa + x) - f x) - xa * f'\<bar> / \<bar>xa\<bar>" by(auto simp add:field_simps) 
-  show ?l unfolding deriv_def LIM_def apply safe apply(drule as,safe)
-    apply(rule_tac x=d in exI,safe) apply(erule_tac x="xa + x" in allE)
-    unfolding dist_norm diff_0_right norm_scaleR
-    unfolding dist_norm netlimit_at[of x] by(auto simp add:algebra_simps *) qed
-
 lemma netlimit_at_vector:
+  (* TODO: move *)
   fixes a :: "'a::real_normed_vector"
   shows "netlimit (at a) = a"
 proof (cases "\<exists>x. x \<noteq> a")
@@ -55,23 +44,15 @@
 qed simp
 
 lemma FDERIV_conv_has_derivative:
-  shows "FDERIV f x :> f' = (f has_derivative f') (at x)" (is "?l = ?r")
-proof
-  assume ?l note as = this[unfolded fderiv_def]
-  show ?r unfolding has_derivative_def Lim_at apply-apply(rule,rule as[THEN conjunct1]) proof(rule,rule)
-    fix e::real assume "e>0"
-    guess d using as[THEN conjunct2,unfolded LIM_def,rule_format,OF`e>0`] ..
-    thus "\<exists>d>0. \<forall>xa. 0 < dist xa x \<and> dist xa x < d \<longrightarrow>
-      dist ((1 / norm (xa - netlimit (at x))) *\<^sub>R (f xa - (f (netlimit (at x)) + f' (xa - netlimit (at x))))) (0) < e"
-      apply(rule_tac x=d in exI) apply(erule conjE,rule,assumption) apply rule apply(erule_tac x="xa - x" in allE)
-      unfolding dist_norm netlimit_at_vector[of x] by (auto simp add: diff_diff_eq) qed next
-  assume ?r note as = this[unfolded has_derivative_def]
-  show ?l unfolding fderiv_def LIM_def apply-apply(rule,rule as[THEN conjunct1]) proof(rule,rule)
-    fix e::real assume "e>0"
-    guess d using as[THEN conjunct2,unfolded Lim_at,rule_format,OF`e>0`] ..
-    thus "\<exists>s>0. \<forall>xa. xa \<noteq> 0 \<and> dist xa 0 < s \<longrightarrow> dist (norm (f (x + xa) - f x - f' xa) / norm xa) 0 < e" apply-
-      apply(rule_tac x=d in exI) apply(erule conjE,rule,assumption) apply rule apply(erule_tac x="xa + x" in allE)
-      unfolding dist_norm netlimit_at_vector[of x] by (auto simp add: diff_diff_eq add.commute) qed qed
+  shows "FDERIV f x :> f' = (f has_derivative f') (at x)"
+  unfolding fderiv_def has_derivative_def netlimit_at_vector
+  by (simp add: diff_diff_eq Lim_at_zero [where a=x]
+    LIM_norm_zero_iff [where 'b='b, symmetric])
+
+lemma DERIV_conv_has_derivative:
+  "(DERIV f x :> f') = (f has_derivative op * f') (at x)"
+  unfolding deriv_fderiv FDERIV_conv_has_derivative
+  by (subst mult_commute, rule refl)
 
 text {* These are the only cases we'll care about, probably. *}
 
@@ -116,7 +97,7 @@
   also have "\<dots> \<longleftrightarrow> ((\<lambda>t. (f t - f x) / (t - x)) ---> y) (at x within ({x<..} \<inter> I))"
     by (simp add: Lim_null[symmetric])
   also have "\<dots> \<longleftrightarrow> ((\<lambda>t. (f x - f t) / (x - t)) ---> y) (at x within ({x<..} \<inter> I))"
-    by (intro Lim_cong_within) (simp_all add: times_divide_eq field_simps)
+    by (intro Lim_cong_within) (simp_all add: field_simps)
   finally show ?thesis
     by (simp add: mult.bounded_linear_right has_derivative_within)
 qed
@@ -135,43 +116,34 @@
 
 subsubsection {* Combining theorems. *}
 
-lemma (in bounded_linear) has_derivative: "(f has_derivative f) net"
-  unfolding has_derivative_def apply(rule,rule bounded_linear_axioms)
-  unfolding diff by (simp add: tendsto_const)
-
 lemma has_derivative_id: "((\<lambda>x. x) has_derivative (\<lambda>h. h)) net"
-  apply(rule bounded_linear.has_derivative) using bounded_linear_ident[unfolded id_def] by simp
+  unfolding has_derivative_def
+  by (simp add: bounded_linear_ident tendsto_const)
 
 lemma has_derivative_const: "((\<lambda>x. c) has_derivative (\<lambda>h. 0)) net"
   unfolding has_derivative_def
-  by (rule, rule bounded_linear_zero, simp add: tendsto_const)
+  by (simp add: bounded_linear_zero tendsto_const)
 
-lemma (in bounded_linear) cmul: shows "bounded_linear (\<lambda>x. (c::real) *\<^sub>R f x)"
-proof -
-  have "bounded_linear (\<lambda>x. c *\<^sub>R x)"
-    by (rule scaleR.bounded_linear_right)
-  moreover have "bounded_linear f" ..
-  ultimately show ?thesis
-    by (rule bounded_linear_compose)
-qed
+lemma (in bounded_linear) has_derivative': "(f has_derivative f) net"
+  unfolding has_derivative_def apply(rule,rule bounded_linear_axioms)
+  unfolding diff by (simp add: tendsto_const)
+
+lemma (in bounded_linear) bounded_linear: "bounded_linear f" ..
 
-lemma has_derivative_cmul: assumes "(f has_derivative f') net" shows "((\<lambda>x. c *\<^sub>R f(x)) has_derivative (\<lambda>h. c *\<^sub>R f'(h))) net"
-  unfolding has_derivative_def apply(rule,rule bounded_linear.cmul)
-  using assms[unfolded has_derivative_def]
-  using scaleR.tendsto[OF tendsto_const assms[unfolded has_derivative_def,THEN conjunct2]]
-  unfolding scaleR_right_diff_distrib scaleR_right_distrib by auto 
-
-lemma has_derivative_cmul_eq: assumes "c \<noteq> 0" 
-  shows "(((\<lambda>x. c *\<^sub>R f(x)) has_derivative (\<lambda>h. c *\<^sub>R f'(h))) net \<longleftrightarrow> (f has_derivative f') net)"
-  apply(rule) defer apply(rule has_derivative_cmul,assumption) 
-  apply(drule has_derivative_cmul[where c="1/c"]) using assms by auto
+lemma (in bounded_linear) has_derivative:
+  assumes "((\<lambda>x. g x) has_derivative (\<lambda>h. g' h)) net"
+  shows "((\<lambda>x. f (g x)) has_derivative (\<lambda>h. f (g' h))) net"
+  using assms unfolding has_derivative_def
+  apply safe
+  apply (erule bounded_linear_compose [OF local.bounded_linear])
+  apply (drule local.tendsto)
+  apply (simp add: local.scaleR local.diff local.add local.zero)
+  done
 
 lemma has_derivative_neg:
- "(f has_derivative f') net \<Longrightarrow> ((\<lambda>x. -(f x)) has_derivative (\<lambda>h. -(f' h))) net"
-  apply(drule has_derivative_cmul[where c="-1"]) by auto
-
-lemma has_derivative_neg_eq: "((\<lambda>x. -(f x)) has_derivative (\<lambda>h. -(f' h))) net \<longleftrightarrow> (f has_derivative f') net"
-  apply(rule, drule_tac[!] has_derivative_neg) by auto
+  assumes "(f has_derivative f') net"
+  shows "((\<lambda>x. -(f x)) has_derivative (\<lambda>h. -(f' h))) net"
+  using scaleR_right.has_derivative [where r="-1", OF assms] by auto
 
 lemma has_derivative_add:
   assumes "(f has_derivative f') net" and "(g has_derivative g') net"
@@ -180,11 +152,12 @@
   note as = assms[unfolded has_derivative_def]
   show ?thesis unfolding has_derivative_def apply(rule,rule bounded_linear_add)
     using tendsto_add[OF as(1)[THEN conjunct2] as(2)[THEN conjunct2]] and as
-    by (auto simp add:algebra_simps scaleR_right_diff_distrib scaleR_right_distrib)
+    by (auto simp add: algebra_simps)
 qed
 
-lemma has_derivative_add_const:"(f has_derivative f') net \<Longrightarrow> ((\<lambda>x. f x + c) has_derivative f') net"
-  apply(drule has_derivative_add) apply(rule has_derivative_const) by auto
+lemma has_derivative_add_const:
+  "(f has_derivative f') net \<Longrightarrow> ((\<lambda>x. f x + c) has_derivative f') net"
+  by (drule has_derivative_add, rule has_derivative_const, auto)
 
 lemma has_derivative_sub:
   assumes "(f has_derivative f') net" and "(g has_derivative g') net"
@@ -195,82 +168,22 @@
   assumes "finite s" and "\<forall>a\<in>s. ((f a) has_derivative (f' a)) net"
   shows "((\<lambda>x. setsum (\<lambda>a. f a x) s) has_derivative (\<lambda>h. setsum (\<lambda>a. f' a h) s)) net"
   using assms by (induct, simp_all add: has_derivative_const has_derivative_add)
-
-lemma has_derivative_setsum_numseg:
-  "\<forall>i. m \<le> i \<and> i \<le> n \<longrightarrow> ((f i) has_derivative (f' i)) net \<Longrightarrow>
-  ((\<lambda>x. setsum (\<lambda>i. f i x) {m..n::nat}) has_derivative (\<lambda>h. setsum (\<lambda>i. f' i h) {m..n})) net"
-  by (rule has_derivative_setsum) simp_all
-
 text {* Somewhat different results for derivative of scalar multiplier. *}
 
 (** move **)
-lemma linear_vmul_component:
+lemma linear_vmul_component: (* TODO: delete *)
   assumes lf: "linear f"
   shows "linear (\<lambda>x. f x $$ k *\<^sub>R v)"
   using lf
   by (auto simp add: linear_def algebra_simps)
 
-lemma bounded_linear_euclidean_component: "bounded_linear (\<lambda>x. x $$ k)"
-  unfolding euclidean_component_def
-  by (rule inner.bounded_linear_right)
-
-lemma has_derivative_vmul_component:
-  fixes c::"'a::real_normed_vector \<Rightarrow> 'b::euclidean_space" and v::"'c::real_normed_vector"
-  assumes "(c has_derivative c') net"
-  shows "((\<lambda>x. c(x)$$k *\<^sub>R v) has_derivative (\<lambda>x. (c' x)$$k *\<^sub>R v)) net" proof-
-  have *:"\<And>y. (c y $$ k *\<^sub>R v - (c (netlimit net) $$ k *\<^sub>R v + c' (y - netlimit net) $$ k *\<^sub>R v)) = 
-        (c y $$ k - (c (netlimit net) $$ k + c' (y - netlimit net) $$ k)) *\<^sub>R v" 
-    unfolding scaleR_left_diff_distrib scaleR_left_distrib by auto
-  show ?thesis unfolding has_derivative_def and *
-    apply (rule conjI)
-    apply (rule bounded_linear_compose [OF scaleR.bounded_linear_left])
-    apply (rule bounded_linear_compose [OF bounded_linear_euclidean_component])
-    apply (rule derivative_linear [OF assms])
-    apply(subst scaleR_zero_left[THEN sym, of v]) unfolding scaleR_scaleR
-    apply (intro tendsto_intros)
-    using assms[unfolded has_derivative_def] unfolding Lim o_def apply- apply(cases "trivial_limit net")
-    apply(rule,assumption,rule disjI2,rule,rule) proof-
-    have *:"\<And>x. x - 0 = (x::'a)" by auto 
-    have **:"\<And>d x. d * (c x $$ k - (c (netlimit net) $$ k + c' (x - netlimit net) $$ k)) =
-      (d *\<^sub>R (c x - (c (netlimit net) + c' (x - netlimit net) ))) $$k" by(auto simp add:field_simps)
-    fix e assume "\<not> trivial_limit net" "0 < (e::real)"
-    then have "eventually (\<lambda>x. dist ((1 / norm (x - netlimit net)) *\<^sub>R
-      (c x - (c (netlimit net) + c' (x - netlimit net)))) 0 < e) net"
-      using assms[unfolded has_derivative_def Lim] by auto
-    thus "eventually (\<lambda>x. dist (1 / norm (x - netlimit net) *
-      (c x $$ k - (c (netlimit net) $$ k + c' (x - netlimit net) $$ k))) 0 < e) net"
-      proof (rule eventually_elim1)
-      case goal1 thus ?case apply - unfolding dist_norm  apply(rule le_less_trans)
-        prefer 2 apply assumption unfolding * **
-        using component_le_norm[of "(1 / norm (x - netlimit net)) *\<^sub>R
-          (c x - (c (netlimit net) + c' (x - netlimit net))) - 0" k] by auto
-    qed
-  qed
-qed
-
-lemma has_derivative_vmul_within: fixes c::"real \<Rightarrow> real"
-  assumes "(c has_derivative c') (at x within s)"
-  shows "((\<lambda>x. (c x) *\<^sub>R v) has_derivative (\<lambda>x. (c' x) *\<^sub>R v)) (at x within s)"
-  using has_derivative_vmul_component[OF assms, of 0 v] by auto
-
-lemma has_derivative_vmul_at: fixes c::"real \<Rightarrow> real"
-  assumes "(c has_derivative c') (at x)"
-  shows "((\<lambda>x. (c x) *\<^sub>R v) has_derivative (\<lambda>x. (c' x) *\<^sub>R v)) (at x)"
-  using has_derivative_vmul_within[where s=UNIV] and assms by(auto simp add: within_UNIV)
-
-lemma has_derivative_lift_dot:
-  assumes "(f has_derivative f') net"
-  shows "((\<lambda>x. inner v (f x)) has_derivative (\<lambda>t. inner v (f' t))) net" proof-
-  show ?thesis using assms unfolding has_derivative_def apply- apply(erule conjE,rule)
-    apply(rule bounded_linear_compose[of _ f']) apply(rule inner.bounded_linear_right,assumption)
-    apply(drule Lim_inner[where a=v]) unfolding o_def
-    by(auto simp add:inner.scaleR_right inner.add_right inner.diff_right) qed
-
 lemmas has_derivative_intros =
-  has_derivative_sub has_derivative_add has_derivative_cmul has_derivative_id
-  has_derivative_const has_derivative_neg has_derivative_vmul_component
-  has_derivative_vmul_at has_derivative_vmul_within has_derivative_cmul 
-  bounded_linear.has_derivative has_derivative_lift_dot
+  has_derivative_id has_derivative_const
+  has_derivative_add has_derivative_sub has_derivative_neg
+  has_derivative_add_const
+  scaleR_left.has_derivative scaleR_right.has_derivative
+  inner_left.has_derivative inner_right.has_derivative
+  euclidean_component.has_derivative
 
 subsubsection {* Limit transformation for derivatives *}
 
@@ -378,7 +291,7 @@
     apply (erule_tac x=e in allE)
     apply (erule impE | assumption)+
     apply (erule exE, rule_tac x=d in exI)
-    by (auto simp add: zero * elim!: allE)
+    by (auto simp add: zero *)
 qed
 
 lemma differentiable_imp_continuous_at:
@@ -546,7 +459,7 @@
   "f differentiable net \<Longrightarrow>
   (\<lambda>x. c *\<^sub>R f(x)) differentiable (net::'a::real_normed_vector filter)"
   unfolding differentiable_def
-  apply(erule exE, drule has_derivative_cmul) by auto
+  apply(erule exE, drule scaleR_right.has_derivative) by auto
 
 lemma differentiable_neg [intro]:
   "f differentiable net \<Longrightarrow>
@@ -855,8 +768,8 @@
   proof
     fix x assume x:"x \<in> {a<..<b}"
     show "((\<lambda>x. f x - (f b - f a) / (b - a) * x) has_derivative (\<lambda>xa. f' x xa - (f b - f a) / (b - a) * xa)) (at x)"
-      by(rule has_derivative_intros assms(3)[rule_format,OF x]
-        has_derivative_cmul[where 'b=real, unfolded real_scaleR_def])+
+      by (intro has_derivative_intros assms(3)[rule_format,OF x]
+        mult_right.has_derivative)
   qed(insert assms(1), auto simp add:field_simps)
   then guess x ..
   thus ?thesis apply(rule_tac x=x in bexI)
@@ -901,7 +814,7 @@
   have "\<exists>x\<in>{a<..<b}. (op \<bullet> (f b - f a) \<circ> f) b - (op \<bullet> (f b - f a) \<circ> f) a = (f b - f a) \<bullet> f' x (b - a)"
     apply(rule mvt) apply(rule assms(1))
     apply(rule continuous_on_inner continuous_on_intros assms(2))+
-    unfolding o_def apply(rule,rule has_derivative_lift_dot)
+    unfolding o_def apply(rule,rule has_derivative_intros)
     using assms(3) by auto
   then guess x .. note x=this
   show ?thesis proof(cases "f a = f b")
@@ -1380,12 +1293,12 @@
           unfolding o_def and diff using f'g' by auto
         show "(ph has_derivative (\<lambda>v. v - g' (f' u v))) (at u within ball a d)"
           unfolding ph' * apply(rule diff_chain_within) defer
-          apply(rule bounded_linear.has_derivative[OF assms(3)])
+          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 bounded_linear.has_derivative' 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)
@@ -1826,7 +1739,8 @@
 
 lemma has_vector_derivative_cmul:
   "(f has_vector_derivative f') net \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_vector_derivative (c *\<^sub>R f')) net"
-  unfolding has_vector_derivative_def apply(drule has_derivative_cmul)
+  unfolding has_vector_derivative_def
+  apply (drule scaleR_right.has_derivative)
   by (auto simp add: algebra_simps)
 
 lemma has_vector_derivative_cmul_eq:
diff -r fa98623f1006 -r d282b3c5df7c src/HOL/Multivariate_Analysis/Extended_Real_Limits.thy
--- a/src/HOL/Multivariate_Analysis/Extended_Real_Limits.thy	Wed Aug 10 21:24:26 2011 +0200
+++ b/src/HOL/Multivariate_Analysis/Extended_Real_Limits.thy	Wed Aug 10 18:07:32 2011 -0700
@@ -420,7 +420,7 @@
       using ereal_open_cont_interval2[of S f0] real lim by auto
     then have "eventually (\<lambda>x. f x \<in> {a<..<b}) net"
       unfolding Liminf_Sup Limsup_Inf less_Sup_iff Inf_less_iff
-      by (auto intro!: eventually_conj simp add: greaterThanLessThan_iff)
+      by (auto intro!: eventually_conj)
     with `{a<..<b} \<subseteq> S` show "eventually (%x. f x : S) net"
       by (rule_tac eventually_mono) auto
   qed
@@ -1036,7 +1036,7 @@
   proof (rule ccontr)
     assume "\<not> ?thesis" then have "\<forall>i\<in>A. \<exists>r. f i = ereal r" by auto
     from bchoice[OF this] guess r ..
-    with * show False by (auto simp: setsum_ereal)
+    with * show False by auto
   qed
   ultimately show "finite A \<and> (\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>)" by auto
 next
diff -r fa98623f1006 -r d282b3c5df7c src/HOL/Multivariate_Analysis/Fashoda.thy
--- a/src/HOL/Multivariate_Analysis/Fashoda.thy	Wed Aug 10 21:24:26 2011 +0200
+++ b/src/HOL/Multivariate_Analysis/Fashoda.thy	Wed Aug 10 18:07:32 2011 -0700
@@ -46,7 +46,7 @@
     apply(rule assms)+ apply(rule continuous_on_compose,subst sqprojection_def)
     apply(rule continuous_on_mul ) apply(rule continuous_at_imp_continuous_on,rule) apply(rule continuous_at_inv[unfolded o_def])
     apply(rule continuous_at_infnorm) unfolding infnorm_eq_0 defer apply(rule continuous_on_id) apply(rule linear_continuous_on) proof-
-    show "bounded_linear negatex" apply(rule bounded_linearI') unfolding Cart_eq proof(rule_tac[!] allI) fix i::2 and x y::"real^2" and c::real
+    show "bounded_linear negatex" apply(rule bounded_linearI') unfolding vec_eq_iff proof(rule_tac[!] allI) fix i::2 and x y::"real^2" and c::real
       show "negatex (x + y) $ i = (negatex x + negatex y) $ i" "negatex (c *\<^sub>R x) $ i = (c *\<^sub>R negatex x) $ i"
         apply-apply(case_tac[!] "i\<noteq>1") prefer 3 apply(drule_tac[1-2] 21) 
         unfolding negatex_def by(auto simp add:vector_2 ) qed qed(insert x0, auto)
@@ -66,7 +66,7 @@
     apply- apply(rule_tac[!] allI impI)+ proof- fix x::"real^2" and i::2 assume x:"x\<noteq>0"
     have "inverse (infnorm x) > 0" using x[unfolded infnorm_pos_lt[THEN sym]] by auto
     thus "(0 < sqprojection x $ i) = (0 < x $ i)"   "(sqprojection x $ i < 0) = (x $ i < 0)"
-      unfolding sqprojection_def vector_component_simps Cart_nth.scaleR real_scaleR_def
+      unfolding sqprojection_def vector_component_simps vec_nth.scaleR real_scaleR_def
       unfolding zero_less_mult_iff mult_less_0_iff by(auto simp add:field_simps) qed
   note lem3 = this[rule_format]
   have x1:"x $ 1 \<in> {- 1..1::real}" "x $ 2 \<in> {- 1..1::real}" using x(1) unfolding mem_interval_cart by auto
@@ -77,7 +77,7 @@
   next assume as:"x$1 = 1"
     hence *:"f (x $ 1) $ 1 = 1" using assms(6) by auto
     have "sqprojection (f (x$1) - g (x$2)) $ 1 < 0"
-      using x(2)[unfolded o_def Cart_eq,THEN spec[where x=1]]
+      using x(2)[unfolded o_def vec_eq_iff,THEN spec[where x=1]]
       unfolding as negatex_def vector_2 by auto moreover
     from x1 have "g (x $ 2) \<in> {- 1..1}" apply-apply(rule assms(2)[unfolded subset_eq,rule_format]) by auto
     ultimately show False unfolding lem3[OF nz] vector_component_simps * mem_interval_cart 
@@ -85,7 +85,7 @@
   next assume as:"x$1 = -1"
     hence *:"f (x $ 1) $ 1 = - 1" using assms(5) by auto
     have "sqprojection (f (x$1) - g (x$2)) $ 1 > 0"
-      using x(2)[unfolded o_def Cart_eq,THEN spec[where x=1]]
+      using x(2)[unfolded o_def vec_eq_iff,THEN spec[where x=1]]
       unfolding as negatex_def vector_2 by auto moreover
     from x1 have "g (x $ 2) \<in> {- 1..1}" apply-apply(rule assms(2)[unfolded subset_eq,rule_format]) by auto
     ultimately show False unfolding lem3[OF nz] vector_component_simps * mem_interval_cart 
@@ -93,7 +93,7 @@
   next assume as:"x$2 = 1"
     hence *:"g (x $ 2) $ 2 = 1" using assms(8) by auto
     have "sqprojection (f (x$1) - g (x$2)) $ 2 > 0"
-      using x(2)[unfolded o_def Cart_eq,THEN spec[where x=2]]
+      using x(2)[unfolded o_def vec_eq_iff,THEN spec[where x=2]]
       unfolding as negatex_def vector_2 by auto moreover
     from x1 have "f (x $ 1) \<in> {- 1..1}" apply-apply(rule assms(1)[unfolded subset_eq,rule_format]) by auto
     ultimately show False unfolding lem3[OF nz] vector_component_simps * mem_interval_cart 
@@ -101,7 +101,7 @@
  next assume as:"x$2 = -1"
     hence *:"g (x $ 2) $ 2 = - 1" using assms(7) by auto
     have "sqprojection (f (x$1) - g (x$2)) $ 2 < 0"
-      using x(2)[unfolded o_def Cart_eq,THEN spec[where x=2]]
+      using x(2)[unfolded o_def vec_eq_iff,THEN spec[where x=2]]
       unfolding as negatex_def vector_2 by auto moreover
     from x1 have "f (x $ 1) \<in> {- 1..1}" apply-apply(rule assms(1)[unfolded subset_eq,rule_format]) by auto
     ultimately show False unfolding lem3[OF nz] vector_component_simps * mem_interval_cart 
@@ -120,7 +120,7 @@
     have *:"continuous_on {- 1..1} iscale" unfolding iscale_def by(rule continuous_on_intros)+
     show "continuous_on {- 1..1} (f \<circ> iscale)" "continuous_on {- 1..1} (g \<circ> iscale)"
       apply-apply(rule_tac[!] continuous_on_compose[OF *]) apply(rule_tac[!] continuous_on_subset[OF _ isc])
-      by(rule assms)+ have *:"(1 / 2) *\<^sub>R (1 + (1::real^1)) = 1" unfolding Cart_eq by auto
+      by(rule assms)+ have *:"(1 / 2) *\<^sub>R (1 + (1::real^1)) = 1" unfolding vec_eq_iff by auto
     show "(f \<circ> iscale) (- 1) $ 1 = - 1" "(f \<circ> iscale) 1 $ 1 = 1" "(g \<circ> iscale) (- 1) $ 2 = -1" "(g \<circ> iscale) 1 $ 2 = 1"
       unfolding o_def iscale_def using assms by(auto simp add:*) qed
   then guess s .. from this(2) guess t .. note st=this
@@ -132,7 +132,7 @@
 (* move *)
 lemma interval_bij_bij_cart: fixes x::"real^'n" assumes "\<forall>i. a$i < b$i \<and> u$i < v$i" 
   shows "interval_bij (a,b) (u,v) (interval_bij (u,v) (a,b) x) = x"
-  unfolding interval_bij_cart split_conv Cart_eq Cart_lambda_beta
+  unfolding interval_bij_cart split_conv vec_eq_iff vec_lambda_beta
   apply(rule,insert assms,erule_tac x=i in allE) by auto
 
 lemma fashoda: fixes b::"real^2"
@@ -142,23 +142,23 @@
   obtains z where "z \<in> path_image f" "z \<in> path_image g" proof-
   fix P Q S presume "P \<or> Q \<or> S" "P \<Longrightarrow> thesis" "Q \<Longrightarrow> thesis" "S \<Longrightarrow> thesis" thus thesis by auto
 next have "{a..b} \<noteq> {}" using assms(3) using path_image_nonempty by auto
-  hence "a \<le> b" unfolding interval_eq_empty_cart vector_le_def by(auto simp add: not_less)
-  thus "a$1 = b$1 \<or> a$2 = b$2 \<or> (a$1 < b$1 \<and> a$2 < b$2)" unfolding vector_le_def forall_2 by auto
+  hence "a \<le> b" unfolding interval_eq_empty_cart less_eq_vec_def by(auto simp add: not_less)
+  thus "a$1 = b$1 \<or> a$2 = b$2 \<or> (a$1 < b$1 \<and> a$2 < b$2)" unfolding less_eq_vec_def forall_2 by auto
 next assume as:"a$1 = b$1" have "\<exists>z\<in>path_image g. z$2 = (pathstart f)$2" apply(rule connected_ivt_component_cart)
     apply(rule connected_path_image assms)+apply(rule pathstart_in_path_image,rule pathfinish_in_path_image)
     unfolding assms using assms(3)[unfolded path_image_def subset_eq,rule_format,of "f 0"]
-    unfolding pathstart_def by(auto simp add: vector_le_def) then guess z .. note z=this
+    unfolding pathstart_def by(auto simp add: less_eq_vec_def) then guess z .. note z=this
   have "z \<in> {a..b}" using z(1) assms(4) unfolding path_image_def by blast 
-  hence "z = f 0" unfolding Cart_eq forall_2 unfolding z(2) pathstart_def
+  hence "z = f 0" unfolding vec_eq_iff forall_2 unfolding z(2) pathstart_def
     using assms(3)[unfolded path_image_def subset_eq mem_interval_cart,rule_format,of "f 0" 1]
     unfolding mem_interval_cart apply(erule_tac x=1 in allE) using as by auto
   thus thesis apply-apply(rule that[OF _ z(1)]) unfolding path_image_def by auto
 next assume as:"a$2 = b$2" have "\<exists>z\<in>path_image f. z$1 = (pathstart g)$1" apply(rule connected_ivt_component_cart)
     apply(rule connected_path_image assms)+apply(rule pathstart_in_path_image,rule pathfinish_in_path_image)
     unfolding assms using assms(4)[unfolded path_image_def subset_eq,rule_format,of "g 0"]
-    unfolding pathstart_def by(auto simp add: vector_le_def) then guess z .. note z=this
+    unfolding pathstart_def by(auto simp add: less_eq_vec_def) then guess z .. note z=this
   have "z \<in> {a..b}" using z(1) assms(3) unfolding path_image_def by blast 
-  hence "z = g 0" unfolding Cart_eq forall_2 unfolding z(2) pathstart_def
+  hence "z = g 0" unfolding vec_eq_iff forall_2 unfolding z(2) pathstart_def
     using assms(4)[unfolded path_image_def subset_eq mem_interval_cart,rule_format,of "g 0" 2]
     unfolding mem_interval_cart apply(erule_tac x=2 in allE) using as by auto
   thus thesis apply-apply(rule that[OF z(1)]) unfolding path_image_def by auto
@@ -180,7 +180,7 @@
       "(interval_bij (a, b) (- 1, 1) \<circ> f) 1 $ 1 = 1"
       "(interval_bij (a, b) (- 1, 1) \<circ> g) 0 $ 2 = -1"
       "(interval_bij (a, b) (- 1, 1) \<circ> g) 1 $ 2 = 1"
-      unfolding interval_bij_cart Cart_lambda_beta vector_component_simps o_def split_conv
+      unfolding interval_bij_cart vec_lambda_beta vector_component_simps o_def split_conv
       unfolding assms[unfolded pathstart_def pathfinish_def] using as by auto qed note z=this
   from z(1) guess zf unfolding image_iff .. note zf=this
   from z(2) guess zg unfolding image_iff .. note zg=this
@@ -197,7 +197,7 @@
 proof- 
   let ?L = "\<exists>u. (x $ 1 = (1 - u) * a $ 1 + u * b $ 1 \<and> x $ 2 = (1 - u) * a $ 2 + u * b $ 2) \<and> 0 \<le> u \<and> u \<le> 1"
   { presume "?L \<Longrightarrow> ?R" "?R \<Longrightarrow> ?L" thus ?thesis unfolding closed_segment_def mem_Collect_eq
-      unfolding Cart_eq forall_2 smult_conv_scaleR[THEN sym] vector_component_simps by blast }
+      unfolding vec_eq_iff forall_2 smult_conv_scaleR[THEN sym] vector_component_simps by blast }
   { assume ?L then guess u apply-apply(erule exE)apply(erule conjE)+ . note u=this
     { fix b a assume "b + u * a > a + u * b"
       hence "(1 - u) * b > (1 - u) * a" by(auto simp add:field_simps)
@@ -221,7 +221,7 @@
 proof- 
   let ?L = "\<exists>u. (x $ 1 = (1 - u) * a $ 1 + u * b $ 1 \<and> x $ 2 = (1 - u) * a $ 2 + u * b $ 2) \<and> 0 \<le> u \<and> u \<le> 1"
   { presume "?L \<Longrightarrow> ?R" "?R \<Longrightarrow> ?L" thus ?thesis unfolding closed_segment_def mem_Collect_eq
-      unfolding Cart_eq forall_2 smult_conv_scaleR[THEN sym] vector_component_simps by blast }
+      unfolding vec_eq_iff forall_2 smult_conv_scaleR[THEN sym] vector_component_simps by blast }
   { assume ?L then guess u apply-apply(erule exE)apply(erule conjE)+ . note u=this
     { fix b a assume "b + u * a > a + u * b"
       hence "(1 - u) * b > (1 - u) * a" by(auto simp add:field_simps)
@@ -274,7 +274,7 @@
       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: path_image_join path_linepath)
-  have abab: "{a..b} \<subseteq> {?a..?b}" by(auto simp add:vector_le_def forall_2 vector_2)
+  have abab: "{a..b} \<subseteq> {?a..?b}" by(auto simp add:less_eq_vec_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
@@ -318,11 +318,11 @@
     qed hence "z \<in> path_image f \<or> z \<in> path_image g" using z unfolding Un_iff by blast
     hence z':"z\<in>{a..b}" using assms(3-4) by auto
     have "a $ 2 = z $ 2 \<Longrightarrow> (z $ 1 = pathstart f $ 1 \<or> z $ 1 = pathfinish f $ 1) \<Longrightarrow> (z = pathstart f \<or> z = pathfinish f)"
-      unfolding Cart_eq forall_2 assms by auto
+      unfolding vec_eq_iff forall_2 assms by auto
     with z' show "z\<in>path_image f" using z(1) unfolding Un_iff mem_interval_cart forall_2 apply-
       apply(simp only: segment_vertical segment_horizontal vector_2) unfolding assms by auto
     have "a $ 2 = z $ 2 \<Longrightarrow> (z $ 1 = pathstart g $ 1 \<or> z $ 1 = pathfinish g $ 1) \<Longrightarrow> (z = pathstart g \<or> z = pathfinish g)"
-      unfolding Cart_eq forall_2 assms by auto
+      unfolding vec_eq_iff forall_2 assms by auto
     with z' show "z\<in>path_image g" using z(2) unfolding Un_iff mem_interval_cart forall_2 apply-
       apply(simp only: segment_vertical segment_horizontal vector_2) unfolding assms by auto
   qed qed
diff -r fa98623f1006 -r d282b3c5df7c src/HOL/Multivariate_Analysis/Finite_Cartesian_Product.thy
--- a/src/HOL/Multivariate_Analysis/Finite_Cartesian_Product.thy	Wed Aug 10 21:24:26 2011 +0200
+++ b/src/HOL/Multivariate_Analysis/Finite_Cartesian_Product.thy	Wed Aug 10 18:07:32 2011 -0700
@@ -6,155 +6,155 @@
 
 theory Finite_Cartesian_Product
 imports
-  "~~/src/HOL/Library/Inner_Product"
+  Euclidean_Space
   L2_Norm
   "~~/src/HOL/Library/Numeral_Type"
 begin
 
 subsection {* Finite Cartesian products, with indexing and lambdas. *}
 
-typedef (open Cart)
-  ('a, 'b) cart = "UNIV :: (('b::finite) \<Rightarrow> 'a) set"
-  morphisms Cart_nth Cart_lambda ..
+typedef (open)
+  ('a, 'b) vec = "UNIV :: (('b::finite) \<Rightarrow> 'a) set"
+  morphisms vec_nth vec_lambda ..
 
 notation
-  Cart_nth (infixl "$" 90) and
-  Cart_lambda (binder "\<chi>" 10)
+  vec_nth (infixl "$" 90) and
+  vec_lambda (binder "\<chi>" 10)
 
 (*
   Translate "'b ^ 'n" into "'b ^ ('n :: finite)". When 'n has already more than
-  the finite type class write "cart 'b 'n"
+  the finite type class write "vec 'b 'n"
 *)
 
-syntax "_finite_cart" :: "type \<Rightarrow> type \<Rightarrow> type" ("(_ ^/ _)" [15, 16] 15)
+syntax "_finite_vec" :: "type \<Rightarrow> type \<Rightarrow> type" ("(_ ^/ _)" [15, 16] 15)
 
 parse_translation {*
 let
-  fun cart t u = Syntax.const @{type_syntax cart} $ t $ u;
-  fun finite_cart_tr [t, u as Free (x, _)] =
+  fun vec t u = Syntax.const @{type_syntax vec} $ t $ u;
+  fun finite_vec_tr [t, u as Free (x, _)] =
         if Lexicon.is_tid x then
-          cart t (Syntax.const @{syntax_const "_ofsort"} $ u $ Syntax.const @{class_syntax finite})
-        else cart t u
-    | finite_cart_tr [t, u] = cart t u
+          vec t (Syntax.const @{syntax_const "_ofsort"} $ u $ Syntax.const @{class_syntax finite})
+        else vec t u
+    | finite_vec_tr [t, u] = vec t u
 in
-  [(@{syntax_const "_finite_cart"}, finite_cart_tr)]
+  [(@{syntax_const "_finite_vec"}, finite_vec_tr)]
 end
 *}
 
 lemma stupid_ext: "(\<forall>x. f x = g x) \<longleftrightarrow> (f = g)"
-  by (auto intro: ext)
+  by auto
 
-lemma Cart_eq: "(x = y) \<longleftrightarrow> (\<forall>i. x$i = y$i)"
-  by (simp add: Cart_nth_inject [symmetric] fun_eq_iff)
+lemma vec_eq_iff: "(x = y) \<longleftrightarrow> (\<forall>i. x$i = y$i)"
+  by (simp add: vec_nth_inject [symmetric] fun_eq_iff)
 
-lemma Cart_lambda_beta [simp]: "Cart_lambda g $ i = g i"
-  by (simp add: Cart_lambda_inverse)
+lemma vec_lambda_beta [simp]: "vec_lambda g $ i = g i"
+  by (simp add: vec_lambda_inverse)
 
-lemma Cart_lambda_unique: "(\<forall>i. f$i = g i) \<longleftrightarrow> Cart_lambda g = f"
-  by (auto simp add: Cart_eq)
+lemma vec_lambda_unique: "(\<forall>i. f$i = g i) \<longleftrightarrow> vec_lambda g = f"
+  by (auto simp add: vec_eq_iff)
 
-lemma Cart_lambda_eta: "(\<chi> i. (g$i)) = g"
-  by (simp add: Cart_eq)
+lemma vec_lambda_eta: "(\<chi> i. (g$i)) = g"
+  by (simp add: vec_eq_iff)
 
 
 subsection {* Group operations and class instances *}
 
-instantiation cart :: (zero,finite) zero
+instantiation vec :: (zero, finite) zero
 begin
-  definition vector_zero_def : "0 \<equiv> (\<chi> i. 0)"
+  definition "0 \<equiv> (\<chi> i. 0)"
   instance ..
 end
 
-instantiation cart :: (plus,finite) plus
+instantiation vec :: (plus, finite) plus
 begin
-  definition  vector_add_def : "op + \<equiv> (\<lambda> x y.  (\<chi> i. (x$i) + (y$i)))"
+  definition "op + \<equiv> (\<lambda> x y. (\<chi> i. x$i + y$i))"
   instance ..
 end
 
-instantiation cart :: (minus,finite) minus
+instantiation vec :: (minus, finite) minus
 begin
-  definition vector_minus_def : "op - \<equiv> (\<lambda> x y.  (\<chi> i. (x$i) - (y$i)))"
+  definition "op - \<equiv> (\<lambda> x y. (\<chi> i. x$i - y$i))"
   instance ..
 end
 
-instantiation cart :: (uminus,finite) uminus
+instantiation vec :: (uminus, finite) uminus
 begin
-  definition vector_uminus_def : "uminus \<equiv> (\<lambda> x.  (\<chi> i. - (x$i)))"
+  definition "uminus \<equiv> (\<lambda> x. (\<chi> i. - (x$i)))"
   instance ..
 end
 
 lemma zero_index [simp]: "0 $ i = 0"
-  unfolding vector_zero_def by simp
+  unfolding zero_vec_def by simp
 
 lemma vector_add_component [simp]: "(x + y)$i = x$i + y$i"
-  unfolding vector_add_def by simp
+  unfolding plus_vec_def by simp
 
 lemma vector_minus_component [simp]: "(x - y)$i = x$i - y$i"
-  unfolding vector_minus_def by simp
+  unfolding minus_vec_def by simp
 
 lemma vector_uminus_component [simp]: "(- x)$i = - (x$i)"
-  unfolding vector_uminus_def by simp
+  unfolding uminus_vec_def by simp
 
-instance cart :: (semigroup_add, finite) semigroup_add
-  by default (simp add: Cart_eq add_assoc)
+instance vec :: (semigroup_add, finite) semigroup_add
+  by default (simp add: vec_eq_iff add_assoc)
 
-instance cart :: (ab_semigroup_add, finite) ab_semigroup_add
-  by default (simp add: Cart_eq add_commute)
+instance vec :: (ab_semigroup_add, finite) ab_semigroup_add
+  by default (simp add: vec_eq_iff add_commute)
 
-instance cart :: (monoid_add, finite) monoid_add
-  by default (simp_all add: Cart_eq)
+instance vec :: (monoid_add, finite) monoid_add
+  by default (simp_all add: vec_eq_iff)
 
-instance cart :: (comm_monoid_add, finite) comm_monoid_add
-  by default (simp add: Cart_eq)
+instance vec :: (comm_monoid_add, finite) comm_monoid_add
+  by default (simp add: vec_eq_iff)
 
-instance cart :: (cancel_semigroup_add, finite) cancel_semigroup_add
-  by default (simp_all add: Cart_eq)
+instance vec :: (cancel_semigroup_add, finite) cancel_semigroup_add
+  by default (simp_all add: vec_eq_iff)
 
-instance cart :: (cancel_ab_semigroup_add, finite) cancel_ab_semigroup_add
-  by default (simp add: Cart_eq)
+instance vec :: (cancel_ab_semigroup_add, finite) cancel_ab_semigroup_add
+  by default (simp add: vec_eq_iff)
 
-instance cart :: (cancel_comm_monoid_add, finite) cancel_comm_monoid_add ..
+instance vec :: (cancel_comm_monoid_add, finite) cancel_comm_monoid_add ..
 
-instance cart :: (group_add, finite) group_add
-  by default (simp_all add: Cart_eq diff_minus)
+instance vec :: (group_add, finite) group_add
+  by default (simp_all add: vec_eq_iff diff_minus)
 
-instance cart :: (ab_group_add, finite) ab_group_add
-  by default (simp_all add: Cart_eq)
+instance vec :: (ab_group_add, finite) ab_group_add
+  by default (simp_all add: vec_eq_iff)
 
 
 subsection {* Real vector space *}
 
-instantiation cart :: (real_vector, finite) real_vector
+instantiation vec :: (real_vector, finite) real_vector
 begin
 
-definition vector_scaleR_def: "scaleR = (\<lambda> r x. (\<chi> i. scaleR r (x$i)))"
+definition "scaleR \<equiv> (\<lambda> r x. (\<chi> i. scaleR r (x$i)))"
 
 lemma vector_scaleR_component [simp]: "(scaleR r x)$i = scaleR r (x$i)"
-  unfolding vector_scaleR_def by simp
+  unfolding scaleR_vec_def by simp
 
 instance
-  by default (simp_all add: Cart_eq scaleR_left_distrib scaleR_right_distrib)
+  by default (simp_all add: vec_eq_iff scaleR_left_distrib scaleR_right_distrib)
 
 end
 
 
 subsection {* Topological space *}
 
-instantiation cart :: (topological_space, finite) topological_space
+instantiation vec :: (topological_space, finite) topological_space
 begin
 
-definition open_vector_def:
+definition
   "open (S :: ('a ^ 'b) set) \<longleftrightarrow>
     (\<forall>x\<in>S. \<exists>A. (\<forall>i. open (A i) \<and> x$i \<in> A i) \<and>
       (\<forall>y. (\<forall>i. y$i \<in> A i) \<longrightarrow> y \<in> S))"
 
 instance proof
   show "open (UNIV :: ('a ^ 'b) set)"
-    unfolding open_vector_def by auto
+    unfolding open_vec_def by auto
 next
   fix S T :: "('a ^ 'b) set"
   assume "open S" "open T" thus "open (S \<inter> T)"
-    unfolding open_vector_def
+    unfolding open_vec_def
     apply clarify
     apply (drule (1) bspec)+
     apply (clarify, rename_tac Sa Ta)
@@ -164,7 +164,7 @@
 next
   fix K :: "('a ^ 'b) set set"
   assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
-    unfolding open_vector_def
+    unfolding open_vec_def
     apply clarify
     apply (drule (1) bspec)
     apply (drule (1) bspec)
@@ -177,32 +177,32 @@
 end
 
 lemma open_vector_box: "\<forall>i. open (S i) \<Longrightarrow> open {x. \<forall>i. x $ i \<in> S i}"
-unfolding open_vector_def by auto
+  unfolding open_vec_def by auto
 
-lemma open_vimage_Cart_nth: "open S \<Longrightarrow> open ((\<lambda>x. x $ i) -` S)"
-unfolding open_vector_def
-apply clarify
-apply (rule_tac x="\<lambda>k. if k = i then S else UNIV" in exI, simp)
-done
+lemma open_vimage_vec_nth: "open S \<Longrightarrow> open ((\<lambda>x. x $ i) -` S)"
+  unfolding open_vec_def
+  apply clarify
+  apply (rule_tac x="\<lambda>k. if k = i then S else UNIV" in exI, simp)
+  done
 
-lemma closed_vimage_Cart_nth: "closed S \<Longrightarrow> closed ((\<lambda>x. x $ i) -` S)"
-unfolding closed_open vimage_Compl [symmetric]
-by (rule open_vimage_Cart_nth)
+lemma closed_vimage_vec_nth: "closed S \<Longrightarrow> closed ((\<lambda>x. x $ i) -` S)"
+  unfolding closed_open vimage_Compl [symmetric]
+  by (rule open_vimage_vec_nth)
 
 lemma closed_vector_box: "\<forall>i. closed (S i) \<Longrightarrow> closed {x. \<forall>i. x $ i \<in> S i}"
 proof -
   have "{x. \<forall>i. x $ i \<in> S i} = (\<Inter>i. (\<lambda>x. x $ i) -` S i)" by auto
   thus "\<forall>i. closed (S i) \<Longrightarrow> closed {x. \<forall>i. x $ i \<in> S i}"
-    by (simp add: closed_INT closed_vimage_Cart_nth)
+    by (simp add: closed_INT closed_vimage_vec_nth)
 qed
 
-lemma tendsto_Cart_nth [tendsto_intros]:
+lemma tendsto_vec_nth [tendsto_intros]:
   assumes "((\<lambda>x. f x) ---> a) net"
   shows "((\<lambda>x. f x $ i) ---> a $ i) net"
 proof (rule topological_tendstoI)
   fix S assume "open S" "a $ i \<in> S"
   then have "open ((\<lambda>y. y $ i) -` S)" "a \<in> ((\<lambda>y. y $ i) -` S)"
-    by (simp_all add: open_vimage_Cart_nth)
+    by (simp_all add: open_vimage_vec_nth)
   with assms have "eventually (\<lambda>x. f x \<in> (\<lambda>y. y $ i) -` S) net"
     by (rule topological_tendstoD)
   then show "eventually (\<lambda>x. f x $ i \<in> S) net"
@@ -220,14 +220,14 @@
   shows "eventually (\<lambda>x. \<forall>y. P x y) net"
 using eventually_Ball_finite [of UNIV P] assms by simp
 
-lemma tendsto_vector:
+lemma vec_tendstoI:
   assumes "\<And>i. ((\<lambda>x. f x $ i) ---> a $ i) net"
   shows "((\<lambda>x. f x) ---> a) net"
 proof (rule topological_tendstoI)
   fix S assume "open S" and "a \<in> S"
   then obtain A where A: "\<And>i. open (A i)" "\<And>i. a $ i \<in> A i"
     and S: "\<And>y. \<forall>i. y $ i \<in> A i \<Longrightarrow> y \<in> S"
-    unfolding open_vector_def by metis
+    unfolding open_vec_def by metis
   have "\<And>i. eventually (\<lambda>x. f x $ i \<in> A i) net"
     using assms A by (rule topological_tendstoD)
   hence "eventually (\<lambda>x. \<forall>i. f x $ i \<in> A i) net"
@@ -236,10 +236,10 @@
     by (rule eventually_elim1, simp add: S)
 qed
 
-lemma tendsto_Cart_lambda [tendsto_intros]:
+lemma tendsto_vec_lambda [tendsto_intros]:
   assumes "\<And>i. ((\<lambda>x. f x i) ---> a i) net"
   shows "((\<lambda>x. \<chi> i. f x i) ---> (\<chi> i. a i)) net"
-using assms by (simp add: tendsto_vector)
+  using assms by (simp add: vec_tendstoI)
 
 
 subsection {* Metric *}
@@ -251,25 +251,24 @@
 apply (rule_tac x="f(x:=y)" in exI, simp)
 done
 
-instantiation cart :: (metric_space, finite) metric_space
+instantiation vec :: (metric_space, finite) metric_space
 begin
 
-definition dist_vector_def:
+definition
   "dist x y = setL2 (\<lambda>i. dist (x$i) (y$i)) UNIV"
 
-lemma dist_nth_le_cart: "dist (x $ i) (y $ i) \<le> dist x y"
-unfolding dist_vector_def
-by (rule member_le_setL2) simp_all
+lemma dist_vec_nth_le: "dist (x $ i) (y $ i) \<le> dist x y"
+  unfolding dist_vec_def by (rule member_le_setL2) simp_all
 
 instance proof
   fix x y :: "'a ^ 'b"
   show "dist x y = 0 \<longleftrightarrow> x = y"
-    unfolding dist_vector_def
-    by (simp add: setL2_eq_0_iff Cart_eq)
+    unfolding dist_vec_def
+    by (simp add: setL2_eq_0_iff vec_eq_iff)
 next
   fix x y z :: "'a ^ 'b"
   show "dist x y \<le> dist x z + dist y z"
-    unfolding dist_vector_def
+    unfolding dist_vec_def
     apply (rule order_trans [OF _ setL2_triangle_ineq])
     apply (simp add: setL2_mono dist_triangle2)
     done
@@ -277,7 +276,7 @@
   (* FIXME: long proof! *)
   fix S :: "('a ^ 'b) set"
   show "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
-    unfolding open_vector_def open_dist
+    unfolding open_vec_def open_dist
     apply safe
      apply (drule (1) bspec)
      apply clarify
@@ -286,7 +285,7 @@
       apply (rule_tac x=e in exI, clarify)
       apply (drule spec, erule mp, clarify)
       apply (drule spec, drule spec, erule mp)
-      apply (erule le_less_trans [OF dist_nth_le_cart])
+      apply (erule le_less_trans [OF dist_vec_nth_le])
      apply (subgoal_tac "\<forall>i\<in>UNIV. \<exists>e>0. \<forall>y. dist y (x$i) < e \<longrightarrow> y \<in> A i")
       apply (drule finite_choice [OF finite], clarify)
       apply (rule_tac x="Min (range f)" in exI, simp)
@@ -308,7 +307,7 @@
       apply simp
      apply clarify
      apply (drule spec, erule mp)
-     apply (simp add: dist_vector_def setL2_strict_mono)
+     apply (simp add: dist_vec_def setL2_strict_mono)
     apply (rule_tac x="\<lambda>i. e / sqrt (of_nat CARD('b))" in exI)
     apply (simp add: divide_pos_pos setL2_constant)
     done
@@ -316,11 +315,11 @@
 
 end
 
-lemma Cauchy_Cart_nth:
+lemma Cauchy_vec_nth:
   "Cauchy (\<lambda>n. X n) \<Longrightarrow> Cauchy (\<lambda>n. X n $ i)"
-unfolding Cauchy_def by (fast intro: le_less_trans [OF dist_nth_le_cart])
+  unfolding Cauchy_def by (fast intro: le_less_trans [OF dist_vec_nth_le])
 
-lemma Cauchy_vector:
+lemma vec_CauchyI:
   fixes X :: "nat \<Rightarrow> 'a::metric_space ^ 'n"
   assumes X: "\<And>i. Cauchy (\<lambda>n. X n $ i)"
   shows "Cauchy (\<lambda>n. X n)"
@@ -340,7 +339,7 @@
     fix m n :: nat
     assume "M \<le> m" "M \<le> n"
     have "dist (X m) (X n) = setL2 (\<lambda>i. dist (X m $ i) (X n $ i)) UNIV"
-      unfolding dist_vector_def ..
+      unfolding dist_vec_def ..
     also have "\<dots> \<le> setsum (\<lambda>i. dist (X m $ i) (X n $ i)) UNIV"
       by (rule setL2_le_setsum [OF zero_le_dist])
     also have "\<dots> < setsum (\<lambda>i::'n. ?s) UNIV"
@@ -354,14 +353,14 @@
   then show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < r" ..
 qed
 
-instance cart :: (complete_space, finite) complete_space
+instance vec :: (complete_space, finite) complete_space
 proof
   fix X :: "nat \<Rightarrow> 'a ^ 'b" assume "Cauchy X"
   have "\<And>i. (\<lambda>n. X n $ i) ----> lim (\<lambda>n. X n $ i)"
-    using Cauchy_Cart_nth [OF `Cauchy X`]
+    using Cauchy_vec_nth [OF `Cauchy X`]
     by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
-  hence "X ----> Cart_lambda (\<lambda>i. lim (\<lambda>n. X n $ i))"
-    by (simp add: tendsto_vector)
+  hence "X ----> vec_lambda (\<lambda>i. lim (\<lambda>n. X n $ i))"
+    by (simp add: vec_tendstoI)
   then show "convergent X"
     by (rule convergentI)
 qed
@@ -369,84 +368,247 @@
 
 subsection {* Normed vector space *}
 
-instantiation cart :: (real_normed_vector, finite) real_normed_vector
+instantiation vec :: (real_normed_vector, finite) real_normed_vector
 begin
 
-definition norm_vector_def:
-  "norm x = setL2 (\<lambda>i. norm (x$i)) UNIV"
+definition "norm x = setL2 (\<lambda>i. norm (x$i)) UNIV"
 
-definition vector_sgn_def:
-  "sgn (x::'a^'b) = scaleR (inverse (norm x)) x"
+definition "sgn (x::'a^'b) = scaleR (inverse (norm x)) x"
 
 instance proof
   fix a :: real and x y :: "'a ^ 'b"
   show "0 \<le> norm x"
-    unfolding norm_vector_def
+    unfolding norm_vec_def
     by (rule setL2_nonneg)
   show "norm x = 0 \<longleftrightarrow> x = 0"
-    unfolding norm_vector_def
-    by (simp add: setL2_eq_0_iff Cart_eq)
+    unfolding norm_vec_def
+    by (simp add: setL2_eq_0_iff vec_eq_iff)
   show "norm (x + y) \<le> norm x + norm y"
-    unfolding norm_vector_def
+    unfolding norm_vec_def
     apply (rule order_trans [OF _ setL2_triangle_ineq])
     apply (simp add: setL2_mono norm_triangle_ineq)
     done
   show "norm (scaleR a x) = \<bar>a\<bar> * norm x"
-    unfolding norm_vector_def
+    unfolding norm_vec_def
     by (simp add: setL2_right_distrib)
   show "sgn x = scaleR (inverse (norm x)) x"
-    by (rule vector_sgn_def)
+    by (rule sgn_vec_def)
   show "dist x y = norm (x - y)"
-    unfolding dist_vector_def norm_vector_def
+    unfolding dist_vec_def norm_vec_def
     by (simp add: dist_norm)
 qed
 
 end
 
 lemma norm_nth_le: "norm (x $ i) \<le> norm x"
-unfolding norm_vector_def
+unfolding norm_vec_def
 by (rule member_le_setL2) simp_all
 
-interpretation Cart_nth: bounded_linear "\<lambda>x. x $ i"
+interpretation vec_nth: bounded_linear "\<lambda>x. x $ i"
 apply default
 apply (rule vector_add_component)
 apply (rule vector_scaleR_component)
 apply (rule_tac x="1" in exI, simp add: norm_nth_le)
 done
 
-instance cart :: (banach, finite) banach ..
+instance vec :: (banach, finite) banach ..
 
 
 subsection {* Inner product space *}
 
-instantiation cart :: (real_inner, finite) real_inner
+instantiation vec :: (real_inner, finite) real_inner
 begin
 
-definition inner_vector_def:
-  "inner x y = setsum (\<lambda>i. inner (x$i) (y$i)) UNIV"
+definition "inner x y = setsum (\<lambda>i. inner (x$i) (y$i)) UNIV"
 
 instance proof
   fix r :: real and x y z :: "'a ^ 'b"
   show "inner x y = inner y x"
-    unfolding inner_vector_def
+    unfolding inner_vec_def
     by (simp add: inner_commute)
   show "inner (x + y) z = inner x z + inner y z"
-    unfolding inner_vector_def
+    unfolding inner_vec_def
     by (simp add: inner_add_left setsum_addf)
   show "inner (scaleR r x) y = r * inner x y"
-    unfolding inner_vector_def
+    unfolding inner_vec_def
     by (simp add: setsum_right_distrib)
   show "0 \<le> inner x x"
-    unfolding inner_vector_def
+    unfolding inner_vec_def
     by (simp add: setsum_nonneg)
   show "inner x x = 0 \<longleftrightarrow> x = 0"
-    unfolding inner_vector_def
-    by (simp add: Cart_eq setsum_nonneg_eq_0_iff)
+    unfolding inner_vec_def
+    by (simp add: vec_eq_iff setsum_nonneg_eq_0_iff)
   show "norm x = sqrt (inner x x)"
-    unfolding inner_vector_def norm_vector_def setL2_def
+    unfolding inner_vec_def norm_vec_def setL2_def
     by (simp add: power2_norm_eq_inner)
 qed
 
 end
 
+subsection {* Euclidean space *}
+
+text {* A bijection between @{text "'n::finite"} and @{text "{..<CARD('n)}"} *}
+
+definition vec_bij_nat :: "nat \<Rightarrow> ('n::finite)" where
+  "vec_bij_nat = (SOME p. bij_betw p {..<CARD('n)} (UNIV::'n set) )"
+
+abbreviation "\<pi> \<equiv> vec_bij_nat"
+definition "\<pi>' = inv_into {..<CARD('n)} (\<pi>::nat \<Rightarrow> ('n::finite))"
+
+lemma bij_betw_pi:
+  "bij_betw \<pi> {..<CARD('n::finite)} (UNIV::('n::finite) set)"
+  using ex_bij_betw_nat_finite[of "UNIV::'n set"]
+  by (auto simp: vec_bij_nat_def atLeast0LessThan
+    intro!: someI_ex[of "\<lambda>x. bij_betw x {..<CARD('n)} (UNIV::'n set)"])
+
+lemma bij_betw_pi'[intro]: "bij_betw \<pi>' (UNIV::'n set) {..<CARD('n::finite)}"
+  using bij_betw_inv_into[OF bij_betw_pi] unfolding \<pi>'_def by auto
+
+lemma pi'_inj[intro]: "inj \<pi>'"
+  using bij_betw_pi' unfolding bij_betw_def by auto
+
+lemma pi'_range[intro]: "\<And>i::'n. \<pi>' i < CARD('n::finite)"
+  using bij_betw_pi' unfolding bij_betw_def by auto
+
+lemma \<pi>\<pi>'[simp]: "\<And>i::'n::finite. \<pi> (\<pi>' i) = i"
+  using bij_betw_pi by (auto intro!: f_inv_into_f simp: \<pi>'_def bij_betw_def)
+
+lemma \<pi>'\<pi>[simp]: "\<And>i. i\<in>{..<CARD('n::finite)} \<Longrightarrow> \<pi>' (\<pi> i::'n) = i"
+  using bij_betw_pi by (auto intro!: inv_into_f_eq simp: \<pi>'_def bij_betw_def)
+
+lemma \<pi>\<pi>'_alt[simp]: "\<And>i. i<CARD('n::finite) \<Longrightarrow> \<pi>' (\<pi> i::'n) = i"
+  by auto
+
+lemma \<pi>_inj_on: "inj_on (\<pi>::nat\<Rightarrow>'n::finite) {..<CARD('n)}"
+  using bij_betw_pi[where 'n='n] by (simp add: bij_betw_def)
+
+instantiation vec :: (euclidean_space, finite) euclidean_space
+begin
+
+definition "dimension (t :: ('a ^ 'b) itself) = CARD('b) * DIM('a)"
+
+definition "(basis i::'a^'b) =
+  (if i < (CARD('b) * DIM('a))
+  then (\<chi> j::'b. if j = \<pi>(i div DIM('a)) then basis (i mod DIM('a)) else 0)
+  else 0)"
+
+lemma basis_eq:
+  assumes "i < CARD('b)" and "j < DIM('a)"
+  shows "basis (j + i * DIM('a)) = (\<chi> k. if k = \<pi> i then basis j else 0)"
+proof -
+  have "j + i * DIM('a) <  DIM('a) * (i + 1)" using assms by (auto simp: field_simps)
+  also have "\<dots> \<le> DIM('a) * CARD('b)" using assms unfolding mult_le_cancel1 by auto
+  finally show ?thesis
+    unfolding basis_vec_def using assms by (auto simp: vec_eq_iff not_less field_simps)
+qed
+
+lemma basis_eq_pi':
+  assumes "j < DIM('a)"
+  shows "basis (j + \<pi>' i * DIM('a)) $ k = (if k = i then basis j else 0)"
+  apply (subst basis_eq)
+  using pi'_range assms by simp_all
+
+lemma split_times_into_modulo[consumes 1]:
+  fixes k :: nat
+  assumes "k < A * B"
+  obtains i j where "i < A" and "j < B" and "k = j + i * B"
+proof
+  have "A * B \<noteq> 0"
+  proof assume "A * B = 0" with assms show False by simp qed
+  hence "0 < B" by auto
+  thus "k mod B < B" using `0 < B` by auto
+next
+  have "k div B * B \<le> k div B * B + k mod B" by (rule le_add1)
+  also have "... < A * B" using assms by simp
+  finally show "k div B < A" by auto
+qed simp
+
+lemma split_CARD_DIM[consumes 1]:
+  fixes k :: nat
+  assumes k: "k < CARD('b) * DIM('a)"
+  obtains i and j::'b where "i < DIM('a)" "k = i + \<pi>' j * DIM('a)"
+proof -
+  from split_times_into_modulo[OF k] guess i j . note ij = this
+  show thesis
+  proof
+    show "j < DIM('a)" using ij by simp
+    show "k = j + \<pi>' (\<pi> i :: 'b) * DIM('a)"
+      using ij by simp
+  qed
+qed
+
+lemma linear_less_than_times:
+  fixes i j A B :: nat assumes "i < B" "j < A"
+  shows "j + i * A < B * A"
+proof -
+  have "i * A + j < (Suc i)*A" using `j < A` by simp
+  also have "\<dots> \<le> B * A" using `i < B` unfolding mult_le_cancel2 by simp
+  finally show ?thesis by simp
+qed
+
+lemma DIM_cart[simp]: "DIM('a^'b) = CARD('b) * DIM('a)"
+  by (rule dimension_vec_def)
+
+lemma all_less_DIM_cart:
+  fixes m n :: nat
+  shows "(\<forall>i<DIM('a^'b). P i) \<longleftrightarrow> (\<forall>x::'b. \<forall>i<DIM('a). P (i + \<pi>' x * DIM('a)))"
+unfolding DIM_cart
+apply safe
+apply (drule spec, erule mp, erule linear_less_than_times [OF pi'_range])
+apply (erule split_CARD_DIM, simp)
+done
+
+lemma eq_pi_iff:
+  fixes x :: "'c::finite"
+  shows "i < CARD('c::finite) \<Longrightarrow> x = \<pi> i \<longleftrightarrow> \<pi>' x = i"
+  by auto
+
+lemma all_less_mult:
+  fixes m n :: nat
+  shows "(\<forall>i<(m * n). P i) \<longleftrightarrow> (\<forall>i<m. \<forall>j<n. P (j + i * n))"
+apply safe
+apply (drule spec, erule mp, erule (1) linear_less_than_times)
+apply (erule split_times_into_modulo, simp)
+done
+
+lemma inner_if:
+  "inner (if a then x else y) z = (if a then inner x z else inner y z)"
+  "inner x (if a then y else z) = (if a then inner x y else inner x z)"
+  by simp_all
+
+instance proof
+  show "0 < DIM('a ^ 'b)"
+    unfolding dimension_vec_def
+    by (intro mult_pos_pos zero_less_card_finite DIM_positive)
+next
+  fix i :: nat
+  assume "DIM('a ^ 'b) \<le> i" thus "basis i = (0::'a^'b)"
+    unfolding dimension_vec_def basis_vec_def
+    by simp
+next
+  show "\<forall>i<DIM('a ^ 'b). \<forall>j<DIM('a ^ 'b).
+    inner (basis i :: 'a ^ 'b) (basis j) = (if i = j then 1 else 0)"
+    apply (simp add: inner_vec_def)
+    apply safe
+    apply (erule split_CARD_DIM, simp add: basis_eq_pi')
+    apply (simp add: inner_if setsum_delta cong: if_cong)
+    apply (simp add: basis_orthonormal)
+    apply (elim split_CARD_DIM, simp add: basis_eq_pi')
+    apply (simp add: inner_if setsum_delta cong: if_cong)
+    apply (clarsimp simp add: basis_orthonormal)
+    done
+next
+  fix x :: "'a ^ 'b"
+  show "(\<forall>i<DIM('a ^ 'b). inner (basis i) x = 0) \<longleftrightarrow> x = 0"
+    unfolding all_less_DIM_cart
+    unfolding inner_vec_def
+    apply (simp add: basis_eq_pi')
+    apply (simp add: inner_if setsum_delta cong: if_cong)
+    apply (simp add: euclidean_all_zero)
+    apply (simp add: vec_eq_iff)
+    done
+qed
+
 end
+
+end
diff -r fa98623f1006 -r d282b3c5df7c src/HOL/Multivariate_Analysis/Integration.thy
--- a/src/HOL/Multivariate_Analysis/Integration.thy	Wed Aug 10 21:24:26 2011 +0200
+++ b/src/HOL/Multivariate_Analysis/Integration.thy	Wed Aug 10 18:07:32 2011 -0700
@@ -3763,8 +3763,9 @@
       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)
-      apply(rule has_derivative_vmul_within,rule has_derivative_id)+ 
+      unfolding scaleR_simps
+      apply(intro has_derivative_intros)
+      apply(intro has_derivative_intros)
       apply(rule has_derivative_within_subset,rule assms(6)[rule_format])
       apply(rule *) apply safe apply(rule conv[unfolded scaleR_simps]) using `x\<in>s` `c\<in>s` by auto
     thus "((\<lambda>xa. f ((1 - xa) *\<^sub>R c + xa *\<^sub>R x)) has_derivative (\<lambda>h. 0)) (at t within {0..1})" unfolding o_def .
diff -r fa98623f1006 -r d282b3c5df7c src/HOL/Multivariate_Analysis/L2_Norm.thy
--- a/src/HOL/Multivariate_Analysis/L2_Norm.thy	Wed Aug 10 21:24:26 2011 +0200
+++ b/src/HOL/Multivariate_Analysis/L2_Norm.thy	Wed Aug 10 18:07:32 2011 -0700
@@ -109,9 +109,8 @@
 lemma sqrt_sum_squares_le_sum:
   "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> sqrt (x\<twosuperior> + y\<twosuperior>) \<le> x + y"
   apply (rule power2_le_imp_le)
-  apply (simp add: power2_sum)
-  apply (simp add: mult_nonneg_nonneg)
-  apply (simp add: add_nonneg_nonneg)
+  apply (simp add: power2_sum mult_nonneg_nonneg)
+  apply simp
   done
 
 lemma setL2_le_setsum [rule_format]:
@@ -128,9 +127,8 @@
 
 lemma sqrt_sum_squares_le_sum_abs: "sqrt (x\<twosuperior> + y\<twosuperior>) \<le> \<bar>x\<bar> + \<bar>y\<bar>"
   apply (rule power2_le_imp_le)
-  apply (simp add: power2_sum)
-  apply (simp add: mult_nonneg_nonneg)
-  apply (simp add: add_nonneg_nonneg)
+  apply (simp add: power2_sum mult_nonneg_nonneg)
+  apply simp
   done
 
 lemma setL2_le_setsum_abs: "setL2 f A \<le> (\<Sum>i\<in>A. \<bar>f i\<bar>)"
@@ -164,7 +162,7 @@
   apply (rule order_trans)
   apply (rule power_mono)
   apply (erule add_left_mono)
-  apply (simp add: add_nonneg_nonneg mult_nonneg_nonneg setsum_nonneg)
+  apply (simp add: mult_nonneg_nonneg setsum_nonneg)
   apply (simp add: power2_sum)
   apply (simp add: power_mult_distrib)
   apply (simp add: right_distrib left_distrib)
diff -r fa98623f1006 -r d282b3c5df7c src/HOL/Multivariate_Analysis/Linear_Algebra.thy
--- a/src/HOL/Multivariate_Analysis/Linear_Algebra.thy	Wed Aug 10 21:24:26 2011 +0200
+++ b/src/HOL/Multivariate_Analysis/Linear_Algebra.thy	Wed Aug 10 18:07:32 2011 -0700
@@ -641,9 +641,9 @@
   assumes x: "0 \<le> x" and y: "0 \<le> y"
   shows "sqrt (x + y) \<le> sqrt x + sqrt y"
 apply (rule power2_le_imp_le)
-apply (simp add: real_sum_squared_expand add_nonneg_nonneg x y)
+apply (simp add: real_sum_squared_expand x y)
 apply (simp add: mult_nonneg_nonneg x y)
-apply (simp add: add_nonneg_nonneg x y)
+apply (simp add: x y)
 done
 
 subsection {* A generic notion of "hull" (convex, affine, conic hull and closure). *}
@@ -2319,7 +2319,7 @@
   shows "x = 0"
   using fB ifB fi xsB fx
 proof(induct arbitrary: x rule: finite_induct[OF fB])
-  case 1 thus ?case by (auto simp add:  span_empty)
+  case 1 thus ?case by auto
 next
   case (2 a b x)
   have fb: "finite b" using "2.prems" by simp
@@ -2372,7 +2372,7 @@
            \<and> (\<forall>x\<in> B. g x = f x)"
 using ib fi
 proof(induct rule: finite_induct[OF fi])
-  case 1 thus ?case by (auto simp add: span_empty)
+  case 1 thus ?case by auto
 next
   case (2 a b)
   from "2.prems" "2.hyps" have ibf: "independent b" "finite b"
diff -r fa98623f1006 -r d282b3c5df7c src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
--- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Wed Aug 10 21:24:26 2011 +0200
+++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Wed Aug 10 18:07:32 2011 -0700
@@ -1031,9 +1031,6 @@
           (\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (S n) l < e)"
   by (auto simp add: tendsto_iff eventually_sequentially)
 
-lemma Lim_sequentially_iff_LIMSEQ: "(S ---> l) sequentially \<longleftrightarrow> S ----> l"
-  unfolding Lim_sequentially LIMSEQ_def ..
-
 lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"
   by (rule topological_tendstoI, auto elim: eventually_rev_mono)
 
@@ -2228,15 +2225,10 @@
     by auto
 qed
 
-lemma bounded_component: "bounded s \<Longrightarrow>
-  bounded ((\<lambda>x. x $$ i) ` (s::'a::euclidean_space set))"
-unfolding bounded_def
-apply clarify
-apply (rule_tac x="x $$ i" in exI)
-apply (rule_tac x="e" in exI)
-apply clarify
-apply (rule order_trans[OF dist_nth_le],simp)
-done
+lemma bounded_component: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $$ i) ` s)"
+  apply (erule bounded_linear_image)
+  apply (rule bounded_linear_euclidean_component)
+  done
 
 lemma compact_lemma:
   fixes f :: "nat \<Rightarrow> 'a::euclidean_space"