# HG changeset patch
# User huffman
# Date 1235242705 28800
# Node ID e2cd1acda1ab2d185eb4da180ee4633c1c01f474
# Parent  7208c88df507b9f723962edf6709ae843f2b2f23
real_normed_vector instance

diff -r 7208c88df507 -r e2cd1acda1ab src/HOL/Library/Euclidean_Space.thy
--- a/src/HOL/Library/Euclidean_Space.thy	Sat Feb 21 09:55:32 2009 -0800
+++ b/src/HOL/Library/Euclidean_Space.thy	Sat Feb 21 10:58:25 2009 -0800
@@ -344,6 +344,209 @@
   apply (auto simp add: vec_def Cart_eq vec_component Cart_lambda_beta )
   using dimindex_ge_1 apply auto done
 
+subsection {* Square root of sum of squares *}
+
+definition
+  "setL2 f A = sqrt (\<Sum>i\<in>A. (f i)\<twosuperior>)"
+
+lemma setL2_cong:
+  "\<lbrakk>A = B; \<And>x. x \<in> B \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> setL2 f A = setL2 g B"
+  unfolding setL2_def by simp
+
+lemma strong_setL2_cong:
+  "\<lbrakk>A = B; \<And>x. x \<in> B =simp=> f x = g x\<rbrakk> \<Longrightarrow> setL2 f A = setL2 g B"
+  unfolding setL2_def simp_implies_def by simp
+
+lemma setL2_infinite [simp]: "\<not> finite A \<Longrightarrow> setL2 f A = 0"
+  unfolding setL2_def by simp
+
+lemma setL2_empty [simp]: "setL2 f {} = 0"
+  unfolding setL2_def by simp
+
+lemma setL2_insert [simp]:
+  "\<lbrakk>finite F; a \<notin> F\<rbrakk> \<Longrightarrow>
+    setL2 f (insert a F) = sqrt ((f a)\<twosuperior> + (setL2 f F)\<twosuperior>)"
+  unfolding setL2_def by (simp add: setsum_nonneg)
+
+lemma setL2_nonneg [simp]: "0 \<le> setL2 f A"
+  unfolding setL2_def by (simp add: setsum_nonneg)
+
+lemma setL2_0': "\<forall>a\<in>A. f a = 0 \<Longrightarrow> setL2 f A = 0"
+  unfolding setL2_def by simp
+
+lemma setL2_mono:
+  assumes "\<And>i. i \<in> K \<Longrightarrow> f i \<le> g i"
+  assumes "\<And>i. i \<in> K \<Longrightarrow> 0 \<le> f i"
+  shows "setL2 f K \<le> setL2 g K"
+  unfolding setL2_def
+  by (simp add: setsum_nonneg setsum_mono power_mono prems)
+
+lemma setL2_right_distrib:
+  "0 \<le> r \<Longrightarrow> r * setL2 f A = setL2 (\<lambda>x. r * f x) A"
+  unfolding setL2_def
+  apply (simp add: power_mult_distrib)
+  apply (simp add: setsum_right_distrib [symmetric])
+  apply (simp add: real_sqrt_mult setsum_nonneg)
+  done
+
+lemma setL2_left_distrib:
+  "0 \<le> r \<Longrightarrow> setL2 f A * r = setL2 (\<lambda>x. f x * r) A"
+  unfolding setL2_def
+  apply (simp add: power_mult_distrib)
+  apply (simp add: setsum_left_distrib [symmetric])
+  apply (simp add: real_sqrt_mult setsum_nonneg)
+  done
+
+lemma setsum_nonneg_eq_0_iff:
+  fixes f :: "'a \<Rightarrow> 'b::pordered_ab_group_add"
+  shows "\<lbrakk>finite A; \<forall>x\<in>A. 0 \<le> f x\<rbrakk> \<Longrightarrow> setsum f A = 0 \<longleftrightarrow> (\<forall>x\<in>A. f x = 0)"
+  apply (induct set: finite, simp)
+  apply (simp add: add_nonneg_eq_0_iff setsum_nonneg)
+  done
+
+lemma setL2_eq_0_iff: "finite A \<Longrightarrow> setL2 f A = 0 \<longleftrightarrow> (\<forall>x\<in>A. f x = 0)"
+  unfolding setL2_def
+  by (simp add: setsum_nonneg setsum_nonneg_eq_0_iff)
+
+lemma setL2_triangle_ineq:
+  shows "setL2 (\<lambda>i. f i + g i) A \<le> setL2 f A + setL2 g A"
+proof (cases "finite A")
+  case False
+  thus ?thesis by simp
+next
+  case True
+  thus ?thesis
+  proof (induct set: finite)
+    case empty
+    show ?case by simp
+  next
+    case (insert x F)
+    hence "sqrt ((f x + g x)\<twosuperior> + (setL2 (\<lambda>i. f i + g i) F)\<twosuperior>) \<le>
+           sqrt ((f x + g x)\<twosuperior> + (setL2 f F + setL2 g F)\<twosuperior>)"
+      by (intro real_sqrt_le_mono add_left_mono power_mono insert
+                setL2_nonneg add_increasing zero_le_power2)
+    also have
+      "\<dots> \<le> sqrt ((f x)\<twosuperior> + (setL2 f F)\<twosuperior>) + sqrt ((g x)\<twosuperior> + (setL2 g F)\<twosuperior>)"
+      by (rule real_sqrt_sum_squares_triangle_ineq)
+    finally show ?case
+      using insert by simp
+  qed
+qed
+
+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)
+  done
+
+lemma setL2_le_setsum [rule_format]:
+  "(\<forall>i\<in>A. 0 \<le> f i) \<longrightarrow> setL2 f A \<le> setsum f A"
+  apply (cases "finite A")
+  apply (induct set: finite)
+  apply simp
+  apply clarsimp
+  apply (erule order_trans [OF sqrt_sum_squares_le_sum])
+  apply simp
+  apply simp
+  apply simp
+  done
+
+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)
+  done
+
+lemma setL2_le_setsum_abs: "setL2 f A \<le> (\<Sum>i\<in>A. \<bar>f i\<bar>)"
+  apply (cases "finite A")
+  apply (induct set: finite)
+  apply simp
+  apply simp
+  apply (rule order_trans [OF sqrt_sum_squares_le_sum_abs])
+  apply simp
+  apply simp
+  done
+
+lemma setL2_mult_ineq_lemma:
+  fixes a b c d :: real
+  shows "2 * (a * c) * (b * d) \<le> a\<twosuperior> * d\<twosuperior> + b\<twosuperior> * c\<twosuperior>"
+proof -
+  have "0 \<le> (a * d - b * c)\<twosuperior>" by simp
+  also have "\<dots> = a\<twosuperior> * d\<twosuperior> + b\<twosuperior> * c\<twosuperior> - 2 * (a * d) * (b * c)"
+    by (simp only: power2_diff power_mult_distrib)
+  also have "\<dots> = a\<twosuperior> * d\<twosuperior> + b\<twosuperior> * c\<twosuperior> - 2 * (a * c) * (b * d)"
+    by simp
+  finally show "2 * (a * c) * (b * d) \<le> a\<twosuperior> * d\<twosuperior> + b\<twosuperior> * c\<twosuperior>"
+    by simp
+qed
+
+lemma setL2_mult_ineq: "(\<Sum>i\<in>A. \<bar>f i\<bar> * \<bar>g i\<bar>) \<le> setL2 f A * setL2 g A"
+  apply (cases "finite A")
+  apply (induct set: finite)
+  apply simp
+  apply (rule power2_le_imp_le, simp)
+  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: power2_sum)
+  apply (simp add: power_mult_distrib)
+  apply (simp add: right_distrib left_distrib)
+  apply (rule ord_le_eq_trans)
+  apply (rule setL2_mult_ineq_lemma)
+  apply simp
+  apply (intro mult_nonneg_nonneg setL2_nonneg)
+  apply simp
+  done
+
+lemma member_le_setL2: "\<lbrakk>finite A; i \<in> A\<rbrakk> \<Longrightarrow> f i \<le> setL2 f A"
+  apply (rule_tac s="insert i (A - {i})" and t="A" in subst)
+  apply fast
+  apply (subst setL2_insert)
+  apply simp
+  apply simp
+  apply simp
+  done
+
+subsection {* Norms *}
+
+instantiation "^" :: (real_normed_vector, type) real_normed_vector
+begin
+
+definition vector_norm_def:
+  "norm (x::'a^'b) = setL2 (\<lambda>i. norm (x$i)) {1 .. dimindex (UNIV:: 'b set)}"
+
+definition vector_sgn_def:
+  "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 vector_norm_def
+    by (rule setL2_nonneg)
+  show "norm x = 0 \<longleftrightarrow> x = 0"
+    unfolding vector_norm_def
+    by (simp add: setL2_eq_0_iff Cart_eq)
+  show "norm (x + y) \<le> norm x + norm y"
+    unfolding vector_norm_def
+    apply (rule order_trans [OF _ setL2_triangle_ineq])
+    apply (rule setL2_mono)
+    apply (simp add: vector_component norm_triangle_ineq)
+    apply simp
+    done
+  show "norm (scaleR a x) = \<bar>a\<bar> * norm x"
+    unfolding vector_norm_def
+    by (simp add: vector_component norm_scaleR setL2_right_distrib
+             cong: strong_setL2_cong)
+  show "sgn x = scaleR (inverse (norm x)) x"
+    by (rule vector_sgn_def)
+qed
+
+end
+
 subsection{* Properties of the dot product.  *}
 
 lemma dot_sym: "(x::'a:: {comm_monoid_add, ab_semigroup_mult} ^ 'n) \<bullet> y = y \<bullet> x" 
@@ -393,18 +596,7 @@
 lemma dot_pos_lt: "(0 < x \<bullet> x) \<longleftrightarrow> (x::'a::{ordered_ring_strict,ring_no_zero_divisors} ^ 'n) \<noteq> 0" using dot_eq_0[of x] dot_pos_le[of x] 
   by (auto simp add: le_less) 
 
-subsection {* Introduce norms, but defer many properties till we get square roots. *}
-text{* FIXME : This is ugly *}
-defs (overloaded) 
-  real_of_real_def [code inline, simp]: "real == id"
-
-instantiation "^" :: ("{times, comm_monoid_add}", type) norm begin
-definition  real_vector_norm_def: "norm \<equiv> (\<lambda>x. sqrt (real (x \<bullet> x)))" 
-instance ..
-end
-
-
-subsection{* The collapse of the general concepts to dimention one. *}
+subsection{* The collapse of the general concepts to dimension one. *}
 
 lemma vector_one: "(x::'a ^1) = (\<chi> i. (x$1))"
   by (vector dimindex_def)
@@ -415,11 +607,15 @@
   apply (simp only: vector_one[symmetric])
   done
 
+lemma norm_vector_1: "norm (x :: _^1) = norm (x$1)"
+  by (simp add: vector_norm_def dimindex_def)
+
 lemma norm_real: "norm(x::real ^ 1) = abs(x$1)" 
-  by (simp add: real_vector_norm_def)
+  by (simp add: norm_vector_1)
 
 text{* Metric *}
 
+text {* FIXME: generalize to arbitrary @{text real_normed_vector} types *}
 definition dist:: "real ^ 'n \<Rightarrow> real ^ 'n \<Rightarrow> real" where 
   "dist x y = norm (x - y)"
 
@@ -531,27 +727,30 @@
 text{* Hence derive more interesting properties of the norm. *}
 
 lemma norm_0: "norm (0::real ^ 'n) = 0"
-  by (simp add: real_vector_norm_def dot_eq_0)
-
-lemma norm_pos_le: "0 <= norm (x::real^'n)" 
-  by (simp add: real_vector_norm_def dot_pos_le)
+  by (rule norm_zero)
+
+lemma norm_pos_le: "0 <= norm (x::real^'n)"
+  by (rule norm_ge_zero)
 lemma norm_neg: " norm(-x) = norm (x:: real ^ 'n)" 
-  by (simp add: real_vector_norm_def dot_lneg dot_rneg)
+  by (rule norm_minus_cancel)
 lemma norm_sub: "norm(x - y) = norm(y - (x::real ^ 'n))" 
-  by (metis norm_neg minus_diff_eq)
+  by (rule norm_minus_commute)
 lemma norm_mul: "norm(a *s x) = abs(a) * norm x"
-  by (simp add: real_vector_norm_def dot_lmult dot_rmult mult_assoc[symmetric] real_sqrt_mult)
+  by (simp add: vector_norm_def vector_component setL2_right_distrib
+           abs_mult cong: strong_setL2_cong)
 lemma norm_eq_0_dot: "(norm x = 0) \<longleftrightarrow> (x \<bullet> x = (0::real))"
-  by (simp add: real_vector_norm_def)
+  by (simp add: vector_norm_def dot_def setL2_def power2_eq_square)
 lemma norm_eq_0: "norm x = 0 \<longleftrightarrow> x = (0::real ^ 'n)"
-  by (simp add: real_vector_norm_def dot_eq_0)
+  by (rule norm_eq_zero)
 lemma norm_pos_lt: "0 < norm x \<longleftrightarrow> x \<noteq> (0::real ^ 'n)"
-  by (metis less_le real_vector_norm_def norm_pos_le norm_eq_0)
+  by (rule zero_less_norm_iff)
+lemma real_vector_norm_def: "norm x = sqrt (x \<bullet> x)"
+  by (simp add: vector_norm_def setL2_def dot_def power2_eq_square)
 lemma norm_pow_2: "norm x ^ 2 = x \<bullet> x"
-  by (simp add: real_vector_norm_def dot_pos_le)
+  by (simp add: real_vector_norm_def)
 lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_0)
 lemma norm_le_0: "norm x <= 0 \<longleftrightarrow> x = (0::real ^'n)"
-  by (metis norm_eq_0 norm_pos_le order_antisym) 
+  by (rule norm_le_zero_iff)
 lemma vector_mul_eq_0: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0"
   by vector
 lemma vector_mul_lcancel: "a *s x = a *s y \<longleftrightarrow> a = (0::real) \<or> x = y"
@@ -583,19 +782,14 @@
   ultimately show ?thesis by metis
 qed
 
-lemma norm_abs[simp]: "abs (norm x) = norm (x::real ^'n)" 
-  using norm_pos_le[of x] by (simp add: real_abs_def linorder_linear)
+lemma norm_abs: "abs (norm x) = norm (x::real ^'n)"
+  by (rule abs_norm_cancel) (* already declared [simp] *)
 
 lemma norm_cauchy_schwarz_abs: "\<bar>x \<bullet> y\<bar> \<le> norm x * norm y"
   using norm_cauchy_schwarz[of x y] norm_cauchy_schwarz[of x "-y"]
   by (simp add: real_abs_def dot_rneg norm_neg)
 lemma norm_triangle: "norm(x + y) <= norm x + norm (y::real ^'n)"
-  unfolding real_vector_norm_def
-  apply (rule real_le_lsqrt)
-  apply (auto simp add: dot_pos_le real_vector_norm_def[symmetric] norm_pos_le norm_pow_2[symmetric] intro: add_nonneg_nonneg)[1]
-  apply (auto simp add: dot_pos_le real_vector_norm_def[symmetric] norm_pos_le norm_pow_2[symmetric] intro: add_nonneg_nonneg)[1]
-  apply (simp add: dot_ladd dot_radd dot_sym )
-    by (simp add: norm_pow_2[symmetric] power2_eq_square ring_simps norm_cauchy_schwarz)
+  by (rule norm_triangle_ineq)
 
 lemma norm_triangle_sub: "norm (x::real ^'n) <= norm(y) + norm(x - y)"
   using norm_triangle[of "y" "x - y"] by (simp add: ring_simps)
@@ -627,19 +821,10 @@
 qed
   
 lemma component_le_norm: "i \<in> {1 .. dimindex(UNIV :: 'n set)} ==> \<bar>x$i\<bar> <= norm (x::real ^ 'n)"
-proof(simp add: real_vector_norm_def, rule real_le_rsqrt, clarsimp)
-  assume i: "Suc 0 \<le> i" "i \<le> dimindex (UNIV :: 'n set)"
-  let ?S = "{1 .. dimindex(UNIV :: 'n set)}"
-  let ?f = "(\<lambda>k. if k = i then x$i ^2 else 0)"
-  have fS: "finite ?S" by simp
-  from i setsum_delta[OF fS, of i "\<lambda>k. x$i ^ 2"]
-  have th: "x$i^2 = setsum ?f ?S" by simp
-  let ?g = "\<lambda>k. x$k * x$k"
-  {fix x assume x: "x \<in> ?S" have "?f x \<le> ?g x" by (simp add: power2_eq_square)}
-  with setsum_mono[of ?S ?f ?g] 
-  have "setsum ?f ?S \<le> setsum ?g ?S" by blast 
-  then show "x$i ^2 \<le> x \<bullet> (x:: real ^ 'n)" unfolding dot_def th[symmetric] .
-qed    
+  apply (simp add: vector_norm_def)
+  apply (rule member_le_setL2, simp_all)
+  done
+
 lemma norm_bound_component_le: "norm(x::real ^ 'n) <= e
                 ==> \<forall>i \<in> {1 .. dimindex(UNIV:: 'n set)}. \<bar>x$i\<bar> <= e"
   by (metis component_le_norm order_trans)
@@ -649,24 +834,12 @@
   by (metis component_le_norm basic_trans_rules(21))
 
 lemma norm_le_l1: "norm (x:: real ^'n) <= setsum(\<lambda>i. \<bar>x$i\<bar>) {1..dimindex(UNIV::'n set)}"
-proof (simp add: real_vector_norm_def, rule real_le_lsqrt,simp add: dot_pos_le, simp add: setsum_mono, simp add: dot_def, induct "dimindex(UNIV::'n set)")
-  case 0 thus ?case by simp
-next
-  case (Suc n)
-  have th: "2 * (\<bar>x$(Suc n)\<bar> * (\<Sum>i = Suc 0..n. \<bar>x$i\<bar>)) \<ge> 0" 
-    apply simp
-    apply (rule mult_nonneg_nonneg)
-    by (simp_all add: setsum_abs_ge_zero)
-  
-  from Suc
-  show ?case using th by (simp add: power2_eq_square ring_simps)
-qed
+  by (simp add: vector_norm_def setL2_le_setsum)
 
 lemma real_abs_norm: "\<bar> norm x\<bar> = norm (x :: real ^'n)" 
-  by (simp add: norm_pos_le)
+  by (rule abs_norm_cancel)
 lemma real_abs_sub_norm: "\<bar>norm(x::real ^'n) - norm y\<bar> <= norm(x - y)"
-  apply (simp add: abs_le_iff ring_simps)
-  by (metis norm_triangle_sub norm_sub)
+  by (rule norm_triangle_ineq3)
 lemma norm_le: "norm(x::real ^ 'n) <= norm(y) \<longleftrightarrow> x \<bullet> x <= y \<bullet> y"
   by (simp add: real_vector_norm_def)
 lemma norm_lt: "norm(x::real ^'n) < norm(y) \<longleftrightarrow> x \<bullet> x < y \<bullet> y"
@@ -682,13 +855,7 @@
   by (simp add: real_vector_norm_def  dot_pos_le )
 
 lemma norm_eq_square: "norm(x) = a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x = a^2"
-proof-
-  have th: "\<And>x y::real. x^2 = y^2 \<longleftrightarrow> x = y \<or> x = -y" by algebra
-  show ?thesis using norm_pos_le[of x]
-  apply (simp add: dot_square_norm th)
-  apply arith
-  done
-qed
+  by (auto simp add: real_vector_norm_def)
 
 lemma real_abs_le_square_iff: "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> (x::real)^2 \<le> y^2"
 proof-
@@ -698,14 +865,14 @@
 qed
 
 lemma norm_le_square: "norm(x) <= a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x <= a^2"
+  apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
   using norm_pos_le[of x]
-  apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
   apply arith
   done
 
 lemma norm_ge_square: "norm(x) >= a \<longleftrightarrow> a <= 0 \<or> x \<bullet> x >= a ^ 2" 
+  apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
   using norm_pos_le[of x]
-  apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
   apply arith
   done
 
@@ -917,10 +1084,10 @@
   assumes fS: "finite S"
   shows "norm (setsum f S) <= setsum (\<lambda>x. norm(f x)) S"
 proof(induct rule: finite_induct[OF fS])
-  case 1 thus ?case by simp norm
+  case 1 thus ?case by simp
 next
   case (2 x S)
-  from "2.hyps" have "norm (setsum f (insert x S)) \<le> norm (f x) + norm (setsum f S)" apply (simp add: norm_triangle_ineq) by norm
+  from "2.hyps" have "norm (setsum f (insert x S)) \<le> norm (f x) + norm (setsum f S)" by (simp add: norm_triangle_ineq)
   also have "\<dots> \<le> norm (f x) + setsum (\<lambda>x. norm(f x)) S"
     using "2.hyps" by simp
   finally  show ?case  using "2.hyps" by simp
@@ -1552,7 +1719,9 @@
     {fix x::"real ^ 'n"
       have "norm (f x) \<le> ?K *  norm x"
       using B[rule_format, of x] norm_pos_le[of x] norm_pos_le[of "f x"] Bp
-      by (auto simp add: ring_simps split add: abs_split)
+      apply (auto simp add: ring_simps split add: abs_split)
+      apply (erule order_trans, simp)
+      done
   }
   then show ?thesis using Kp by blast
 qed
@@ -2268,7 +2437,7 @@
   {assume H: ?lhs
     from H[rule_format, of "basis 1"] 
     have bp: "b \<ge> 0" using norm_pos_le[of "f (basis 1)"] dimindex_ge_1[of "UNIV:: 'n set"]
-      by (auto simp add: norm_basis) 
+      by (auto simp add: norm_basis elim: order_trans [OF norm_ge_zero])
     {fix x :: "real ^'n"
       {assume "x = 0"
 	then have "norm (f x) \<le> b * norm x" by (simp add: linear_0[OF lf] norm_0 bp)}
@@ -2276,7 +2445,7 @@
       {assume x0: "x \<noteq> 0"
 	hence n0: "norm x \<noteq> 0" by (metis norm_eq_0)
 	let ?c = "1/ norm x"
-	have "norm (?c*s x) = 1" by (simp add: n0 norm_mul)
+	have "norm (?c*s x) = 1" using x0 by (simp add: n0 norm_mul)
 	with H have "norm (f(?c*s x)) \<le> b" by blast
 	hence "?c * norm (f x) \<le> b" 
 	  by (simp add: linear_cmul[OF lf] norm_mul)
@@ -2585,7 +2754,7 @@
     by (simp add: dot_def setsum_add_split[OF th_0, of _ ?m] pastecart_def dimindex_finite_sum Cart_lambda_beta setsum_nonneg zero_le_square del: One_nat_def)
   then show ?thesis
     unfolding th0 
-    unfolding real_vector_norm_def real_sqrt_le_iff real_of_real_def id_def
+    unfolding real_vector_norm_def real_sqrt_le_iff id_def
     by (simp add: dot_def dimindex_finite_sum Cart_lambda_beta)
 qed
 
@@ -2617,7 +2786,7 @@
     by (simp add: dot_def setsum_add_split[OF th_0, of _ ?m] pastecart_def dimindex_finite_sum Cart_lambda_beta setsum_nonneg zero_le_square setsum_reindex[OF finj, unfolded fS] del: One_nat_def)    
   then show ?thesis
     unfolding th0 
-    unfolding real_vector_norm_def real_sqrt_le_iff real_of_real_def id_def
+    unfolding real_vector_norm_def real_sqrt_le_iff id_def
     by (simp add: dot_def dimindex_finite_sum Cart_lambda_beta)
 qed
 
@@ -2674,7 +2843,7 @@
 qed
 
 lemma norm_pastecart: "norm(pastecart x y) <= norm(x :: real ^ _) + norm(y)"
-  unfolding real_vector_norm_def dot_pastecart real_sqrt_le_iff real_of_real_def id_def
+  unfolding real_vector_norm_def dot_pastecart real_sqrt_le_iff id_def
   apply (rule power2_le_imp_le)
   apply (simp add: real_sqrt_pow2[OF add_nonneg_nonneg[OF dot_pos_le[of x] dot_pos_le[of y]]])
   apply (auto simp add: power2_eq_square ring_simps)
@@ -4998,7 +5167,7 @@
     apply blast
     by (rule abs_ge_zero)
   from real_le_lsqrt[OF dot_pos_le th th1]
-  show ?thesis unfolding real_vector_norm_def  real_of_real_def id_def . 
+  show ?thesis unfolding real_vector_norm_def id_def . 
 qed
 
 (* Equality in Cauchy-Schwarz and triangle inequalities.                     *)