src/HOL/Real/RealVector.thy
author huffman
Sun Sep 24 07:14:02 2006 +0200 (2006-09-24)
changeset 20694 76c49548d14c
parent 20684 74e8b46abb97
child 20718 4c4869e4ddb7
permissions -rw-r--r--
real_norm_def [simp]
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(*  Title       : RealVector.thy
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    ID:         $Id$
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    Author      : Brian Huffman
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*)
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header {* Vector Spaces and Algebras over the Reals *}
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theory RealVector
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imports RealPow
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begin
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subsection {* Locale for additive functions *}
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locale additive =
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  fixes f :: "'a::ab_group_add \<Rightarrow> 'b::ab_group_add"
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  assumes add: "f (x + y) = f x + f y"
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lemma (in additive) zero: "f 0 = 0"
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proof -
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  have "f 0 = f (0 + 0)" by simp
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  also have "\<dots> = f 0 + f 0" by (rule add)
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  finally show "f 0 = 0" by simp
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qed
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lemma (in additive) minus: "f (- x) = - f x"
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proof -
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  have "f (- x) + f x = f (- x + x)" by (rule add [symmetric])
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  also have "\<dots> = - f x + f x" by (simp add: zero)
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  finally show "f (- x) = - f x" by (rule add_right_imp_eq)
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qed
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lemma (in additive) diff: "f (x - y) = f x - f y"
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by (simp add: diff_def add minus)
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subsection {* Real vector spaces *}
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axclass scaleR < type
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consts
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  scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a::scaleR" (infixr "*#" 75)
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syntax (xsymbols)
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  scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a::scaleR" (infixr "*\<^sub>R" 75)
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instance real :: scaleR ..
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defs (overloaded)
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  real_scaleR_def: "a *# x \<equiv> a * x"
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axclass real_vector < scaleR, ab_group_add
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  scaleR_right_distrib: "a *# (x + y) = a *# x + a *# y"
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  scaleR_left_distrib: "(a + b) *# x = a *# x + b *# x"
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  scaleR_assoc: "(a * b) *# x = a *# b *# x"
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  scaleR_one [simp]: "1 *# x = x"
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axclass real_algebra < real_vector, ring
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  mult_scaleR_left: "a *# x * y = a *# (x * y)"
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  mult_scaleR_right: "x * a *# y = a *# (x * y)"
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axclass real_algebra_1 < real_algebra, ring_1
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axclass real_div_algebra < real_algebra_1, division_ring
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axclass real_field < real_div_algebra, field
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instance real :: real_field
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apply (intro_classes, unfold real_scaleR_def)
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apply (rule right_distrib)
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apply (rule left_distrib)
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apply (rule mult_assoc)
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apply (rule mult_1_left)
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apply (rule mult_assoc)
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apply (rule mult_left_commute)
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done
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lemmas scaleR_scaleR = scaleR_assoc [symmetric]
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lemma scaleR_left_commute:
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  fixes x :: "'a::real_vector"
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  shows "a *# b *# x = b *# a *# x"
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by (simp add: scaleR_scaleR mult_commute)
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lemma additive_scaleR_right: "additive (\<lambda>x. a *# x :: 'a::real_vector)"
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by (rule additive.intro, rule scaleR_right_distrib)
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lemma additive_scaleR_left: "additive (\<lambda>a. a *# x :: 'a::real_vector)"
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by (rule additive.intro, rule scaleR_left_distrib)
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lemmas scaleR_zero_left [simp] =
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  additive.zero [OF additive_scaleR_left, standard]
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lemmas scaleR_zero_right [simp] =
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  additive.zero [OF additive_scaleR_right, standard]
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lemmas scaleR_minus_left [simp] =
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  additive.minus [OF additive_scaleR_left, standard]
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lemmas scaleR_minus_right [simp] =
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  additive.minus [OF additive_scaleR_right, standard]
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lemmas scaleR_left_diff_distrib =
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  additive.diff [OF additive_scaleR_left, standard]
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lemmas scaleR_right_diff_distrib =
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  additive.diff [OF additive_scaleR_right, standard]
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lemma scaleR_eq_0_iff:
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  fixes x :: "'a::real_vector"
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  shows "(a *# x = 0) = (a = 0 \<or> x = 0)"
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proof cases
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  assume "a = 0" thus ?thesis by simp
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next
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  assume anz [simp]: "a \<noteq> 0"
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  { assume "a *# x = 0"
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    hence "inverse a *# a *# x = 0" by simp
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    hence "x = 0" by (simp (no_asm_use) add: scaleR_scaleR)}
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  thus ?thesis by force
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qed
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lemma scaleR_left_imp_eq:
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  fixes x y :: "'a::real_vector"
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  shows "\<lbrakk>a \<noteq> 0; a *# x = a *# y\<rbrakk> \<Longrightarrow> x = y"
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proof -
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  assume nonzero: "a \<noteq> 0"
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  assume "a *# x = a *# y"
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  hence "a *# (x - y) = 0"
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     by (simp add: scaleR_right_diff_distrib)
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  hence "x - y = 0"
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     by (simp add: scaleR_eq_0_iff nonzero)
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  thus "x = y" by simp
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qed
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lemma scaleR_right_imp_eq:
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  fixes x y :: "'a::real_vector"
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  shows "\<lbrakk>x \<noteq> 0; a *# x = b *# x\<rbrakk> \<Longrightarrow> a = b"
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proof -
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  assume nonzero: "x \<noteq> 0"
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  assume "a *# x = b *# x"
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  hence "(a - b) *# x = 0"
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     by (simp add: scaleR_left_diff_distrib)
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  hence "a - b = 0"
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     by (simp add: scaleR_eq_0_iff nonzero)
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  thus "a = b" by simp
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qed
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lemma scaleR_cancel_left:
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  fixes x y :: "'a::real_vector"
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  shows "(a *# x = a *# y) = (x = y \<or> a = 0)"
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by (auto intro: scaleR_left_imp_eq)
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lemma scaleR_cancel_right:
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  fixes x y :: "'a::real_vector"
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  shows "(a *# x = b *# x) = (a = b \<or> x = 0)"
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by (auto intro: scaleR_right_imp_eq)
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lemma nonzero_inverse_scaleR_distrib:
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  fixes x :: "'a::real_div_algebra"
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  shows "\<lbrakk>a \<noteq> 0; x \<noteq> 0\<rbrakk> \<Longrightarrow> inverse (a *# x) = inverse a *# inverse x"
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apply (rule inverse_unique)
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apply (simp add: mult_scaleR_left mult_scaleR_right scaleR_scaleR)
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done
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lemma inverse_scaleR_distrib:
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  fixes x :: "'a::{real_div_algebra,division_by_zero}"
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  shows "inverse (a *# x) = inverse a *# inverse x"
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apply (case_tac "a = 0", simp)
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apply (case_tac "x = 0", simp)
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apply (erule (1) nonzero_inverse_scaleR_distrib)
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done
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subsection {* Embedding of the Reals into any @{text real_algebra_1}:
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@{term of_real} *}
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definition
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  of_real :: "real \<Rightarrow> 'a::real_algebra_1"
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  "of_real r = r *# 1"
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lemma of_real_0 [simp]: "of_real 0 = 0"
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by (simp add: of_real_def)
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lemma of_real_1 [simp]: "of_real 1 = 1"
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by (simp add: of_real_def)
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lemma of_real_add [simp]: "of_real (x + y) = of_real x + of_real y"
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by (simp add: of_real_def scaleR_left_distrib)
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lemma of_real_minus [simp]: "of_real (- x) = - of_real x"
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by (simp add: of_real_def)
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lemma of_real_diff [simp]: "of_real (x - y) = of_real x - of_real y"
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by (simp add: of_real_def scaleR_left_diff_distrib)
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lemma of_real_mult [simp]: "of_real (x * y) = of_real x * of_real y"
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by (simp add: of_real_def mult_scaleR_left scaleR_scaleR)
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lemma nonzero_of_real_inverse:
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  "x \<noteq> 0 \<Longrightarrow> of_real (inverse x) =
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   inverse (of_real x :: 'a::real_div_algebra)"
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by (simp add: of_real_def nonzero_inverse_scaleR_distrib)
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lemma of_real_inverse [simp]:
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  "of_real (inverse x) =
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   inverse (of_real x :: 'a::{real_div_algebra,division_by_zero})"
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by (simp add: of_real_def inverse_scaleR_distrib)
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lemma nonzero_of_real_divide:
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  "y \<noteq> 0 \<Longrightarrow> of_real (x / y) =
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   (of_real x / of_real y :: 'a::real_field)"
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by (simp add: divide_inverse nonzero_of_real_inverse)
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lemma of_real_divide:
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  "of_real (x / y) =
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   (of_real x / of_real y :: 'a::{real_field,division_by_zero})"
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by (simp add: divide_inverse)
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lemma of_real_eq_iff [simp]: "(of_real x = of_real y) = (x = y)"
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by (simp add: of_real_def scaleR_cancel_right)
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lemmas of_real_eq_0_iff [simp] = of_real_eq_iff [of _ 0, simplified]
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lemma of_real_eq_id [simp]: "of_real = (id :: real \<Rightarrow> real)"
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proof
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  fix r
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  show "of_real r = id r"
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    by (simp add: of_real_def real_scaleR_def)
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qed
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text{*Collapse nested embeddings*}
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lemma of_real_of_nat_eq [simp]: "of_real (of_nat n) = of_nat n"
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by (induct n, auto)
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lemma of_real_of_int_eq [simp]: "of_real (of_int z) = of_int z"
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by (cases z rule: int_diff_cases, simp)
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lemma of_real_number_of_eq:
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  "of_real (number_of w) = (number_of w :: 'a::{number_ring,real_algebra_1})"
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by (simp add: number_of_eq)
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subsection {* The Set of Real Numbers *}
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constdefs
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   Reals :: "'a::real_algebra_1 set"
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   "Reals \<equiv> range of_real"
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const_syntax (xsymbols)
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  Reals  ("\<real>")
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lemma of_real_in_Reals [simp]: "of_real r \<in> Reals"
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by (simp add: Reals_def)
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lemma Reals_0 [simp]: "0 \<in> Reals"
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apply (unfold Reals_def)
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apply (rule range_eqI)
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apply (rule of_real_0 [symmetric])
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done
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lemma Reals_1 [simp]: "1 \<in> Reals"
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apply (unfold Reals_def)
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apply (rule range_eqI)
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apply (rule of_real_1 [symmetric])
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done
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lemma Reals_add [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a + b \<in> Reals"
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apply (auto simp add: Reals_def)
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apply (rule range_eqI)
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apply (rule of_real_add [symmetric])
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done
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lemma Reals_minus [simp]: "a \<in> Reals \<Longrightarrow> - a \<in> Reals"
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apply (auto simp add: Reals_def)
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apply (rule range_eqI)
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apply (rule of_real_minus [symmetric])
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done
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lemma Reals_diff [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a - b \<in> Reals"
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apply (auto simp add: Reals_def)
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apply (rule range_eqI)
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apply (rule of_real_diff [symmetric])
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done
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lemma Reals_mult [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a * b \<in> Reals"
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apply (auto simp add: Reals_def)
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apply (rule range_eqI)
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apply (rule of_real_mult [symmetric])
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done
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lemma nonzero_Reals_inverse:
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  fixes a :: "'a::real_div_algebra"
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  shows "\<lbrakk>a \<in> Reals; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Reals"
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apply (auto simp add: Reals_def)
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apply (rule range_eqI)
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apply (erule nonzero_of_real_inverse [symmetric])
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done
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lemma Reals_inverse [simp]:
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  fixes a :: "'a::{real_div_algebra,division_by_zero}"
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  shows "a \<in> Reals \<Longrightarrow> inverse a \<in> Reals"
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apply (auto simp add: Reals_def)
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apply (rule range_eqI)
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apply (rule of_real_inverse [symmetric])
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done
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lemma nonzero_Reals_divide:
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  fixes a b :: "'a::real_field"
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  shows "\<lbrakk>a \<in> Reals; b \<in> Reals; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
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apply (auto simp add: Reals_def)
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apply (rule range_eqI)
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apply (erule nonzero_of_real_divide [symmetric])
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done
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lemma Reals_divide [simp]:
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  fixes a b :: "'a::{real_field,division_by_zero}"
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  shows "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
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apply (auto simp add: Reals_def)
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apply (rule range_eqI)
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apply (rule of_real_divide [symmetric])
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done
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lemma Reals_cases [cases set: Reals]:
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  assumes "q \<in> \<real>"
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  obtains (of_real) r where "q = of_real r"
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  unfolding Reals_def
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proof -
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  from `q \<in> \<real>` have "q \<in> range of_real" unfolding Reals_def .
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  then obtain r where "q = of_real r" ..
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  then show thesis ..
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   330
qed
huffman@20554
   331
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   332
lemma Reals_induct [case_names of_real, induct set: Reals]:
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   333
  "q \<in> \<real> \<Longrightarrow> (\<And>r. P (of_real r)) \<Longrightarrow> P q"
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   334
  by (rule Reals_cases) auto
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   335
huffman@20504
   336
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   337
subsection {* Real normed vector spaces *}
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axclass norm < type
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consts norm :: "'a::norm \<Rightarrow> real"
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   341
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   342
instance real :: norm ..
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   343
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   344
defs (overloaded)
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  real_norm_def [simp]: "norm r \<equiv> \<bar>r\<bar>"
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   346
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   347
axclass normed < plus, zero, norm
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   348
  norm_ge_zero [simp]: "0 \<le> norm x"
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   349
  norm_eq_zero [simp]: "(norm x = 0) = (x = 0)"
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   350
  norm_triangle_ineq: "norm (x + y) \<le> norm x + norm y"
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   351
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   352
axclass real_normed_vector < real_vector, normed
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  norm_scaleR: "norm (a *# x) = \<bar>a\<bar> * norm x"
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   354
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   355
axclass real_normed_algebra < real_algebra, real_normed_vector
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  norm_mult_ineq: "norm (x * y) \<le> norm x * norm y"
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   357
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   358
axclass real_normed_div_algebra < real_div_algebra, normed
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   359
  norm_of_real: "norm (of_real r) = abs r"
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   360
  norm_mult: "norm (x * y) = norm x * norm y"
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   361
huffman@20584
   362
axclass real_normed_field < real_field, real_normed_div_algebra
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   363
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   364
instance real_normed_div_algebra < real_normed_algebra
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proof
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  fix a :: real and x :: 'a
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   367
  have "norm (a *# x) = norm (of_real a * x)"
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   368
    by (simp add: of_real_def mult_scaleR_left)
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   369
  also have "\<dots> = abs a * norm x"
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   370
    by (simp add: norm_mult norm_of_real)
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   371
  finally show "norm (a *# x) = abs a * norm x" .
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   372
next
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  fix x y :: 'a
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   374
  show "norm (x * y) \<le> norm x * norm y"
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   375
    by (simp add: norm_mult)
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   376
qed
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   377
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   378
instance real :: real_normed_field
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apply (intro_classes, unfold real_norm_def)
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   380
apply (rule abs_ge_zero)
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   381
apply (rule abs_eq_0)
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   382
apply (rule abs_triangle_ineq)
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   383
apply simp
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   384
apply (rule abs_mult)
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   385
done
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   386
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   387
lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0"
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   388
by simp
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   389
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   390
lemma zero_less_norm_iff [simp]:
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   391
  fixes x :: "'a::real_normed_vector"
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   392
  shows "(0 < norm x) = (x \<noteq> 0)"
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   393
by (simp add: order_less_le)
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   394
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   395
lemma norm_minus_cancel [simp]:
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   396
  fixes x :: "'a::real_normed_vector"
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   397
  shows "norm (- x) = norm x"
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   398
proof -
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   399
  have "norm (- x) = norm (- 1 *# x)"
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   400
    by (simp only: scaleR_minus_left scaleR_one)
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   401
  also have "\<dots> = \<bar>- 1\<bar> * norm x"
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   402
    by (rule norm_scaleR)
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   403
  finally show ?thesis by simp
huffman@20504
   404
qed
huffman@20504
   405
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   406
lemma norm_minus_commute:
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   407
  fixes a b :: "'a::real_normed_vector"
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   408
  shows "norm (a - b) = norm (b - a)"
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   409
proof -
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   410
  have "norm (a - b) = norm (- (a - b))"
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   411
    by (simp only: norm_minus_cancel)
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   412
  also have "\<dots> = norm (b - a)" by simp
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   413
  finally show ?thesis .
huffman@20504
   414
qed
huffman@20504
   415
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   416
lemma norm_triangle_ineq2:
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   417
  fixes a b :: "'a::real_normed_vector"
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   418
  shows "norm a - norm b \<le> norm (a - b)"
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   419
proof -
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   420
  have "norm (a - b + b) \<le> norm (a - b) + norm b"
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   421
    by (rule norm_triangle_ineq)
huffman@20504
   422
  also have "(a - b + b) = a"
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   423
    by simp
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   424
  finally show ?thesis
huffman@20504
   425
    by (simp add: compare_rls)
huffman@20504
   426
qed
huffman@20504
   427
huffman@20584
   428
lemma norm_triangle_ineq3:
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   429
  fixes a b :: "'a::real_normed_vector"
huffman@20584
   430
  shows "\<bar>norm a - norm b\<bar> \<le> norm (a - b)"
huffman@20584
   431
apply (subst abs_le_iff)
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   432
apply auto
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   433
apply (rule norm_triangle_ineq2)
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   434
apply (subst norm_minus_commute)
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   435
apply (rule norm_triangle_ineq2)
huffman@20584
   436
done
huffman@20584
   437
huffman@20504
   438
lemma norm_triangle_ineq4:
huffman@20584
   439
  fixes a b :: "'a::real_normed_vector"
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   440
  shows "norm (a - b) \<le> norm a + norm b"
huffman@20504
   441
proof -
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   442
  have "norm (a - b) = norm (a + - b)"
huffman@20504
   443
    by (simp only: diff_minus)
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   444
  also have "\<dots> \<le> norm a + norm (- b)"
huffman@20504
   445
    by (rule norm_triangle_ineq)
huffman@20504
   446
  finally show ?thesis
huffman@20504
   447
    by simp
huffman@20504
   448
qed
huffman@20504
   449
huffman@20551
   450
lemma norm_diff_triangle_ineq:
huffman@20551
   451
  fixes a b c d :: "'a::real_normed_vector"
huffman@20551
   452
  shows "norm ((a + b) - (c + d)) \<le> norm (a - c) + norm (b - d)"
huffman@20551
   453
proof -
huffman@20551
   454
  have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))"
huffman@20551
   455
    by (simp add: diff_minus add_ac)
huffman@20551
   456
  also have "\<dots> \<le> norm (a - c) + norm (b - d)"
huffman@20551
   457
    by (rule norm_triangle_ineq)
huffman@20551
   458
  finally show ?thesis .
huffman@20551
   459
qed
huffman@20551
   460
huffman@20560
   461
lemma norm_one [simp]: "norm (1::'a::real_normed_div_algebra) = 1"
huffman@20560
   462
proof -
huffman@20560
   463
  have "norm (of_real 1 :: 'a) = abs 1"
huffman@20560
   464
    by (rule norm_of_real)
huffman@20560
   465
  thus ?thesis by simp
huffman@20560
   466
qed
huffman@20560
   467
huffman@20504
   468
lemma nonzero_norm_inverse:
huffman@20504
   469
  fixes a :: "'a::real_normed_div_algebra"
huffman@20533
   470
  shows "a \<noteq> 0 \<Longrightarrow> norm (inverse a) = inverse (norm a)"
huffman@20504
   471
apply (rule inverse_unique [symmetric])
huffman@20504
   472
apply (simp add: norm_mult [symmetric])
huffman@20504
   473
done
huffman@20504
   474
huffman@20504
   475
lemma norm_inverse:
huffman@20504
   476
  fixes a :: "'a::{real_normed_div_algebra,division_by_zero}"
huffman@20533
   477
  shows "norm (inverse a) = inverse (norm a)"
huffman@20504
   478
apply (case_tac "a = 0", simp)
huffman@20504
   479
apply (erule nonzero_norm_inverse)
huffman@20504
   480
done
huffman@20504
   481
huffman@20584
   482
lemma nonzero_norm_divide:
huffman@20584
   483
  fixes a b :: "'a::real_normed_field"
huffman@20584
   484
  shows "b \<noteq> 0 \<Longrightarrow> norm (a / b) = norm a / norm b"
huffman@20584
   485
by (simp add: divide_inverse norm_mult nonzero_norm_inverse)
huffman@20584
   486
huffman@20584
   487
lemma norm_divide:
huffman@20584
   488
  fixes a b :: "'a::{real_normed_field,division_by_zero}"
huffman@20584
   489
  shows "norm (a / b) = norm a / norm b"
huffman@20584
   490
by (simp add: divide_inverse norm_mult norm_inverse)
huffman@20584
   491
huffman@20684
   492
lemma norm_power:
huffman@20684
   493
  fixes x :: "'a::{real_normed_div_algebra,recpower}"
huffman@20684
   494
  shows "norm (x ^ n) = norm x ^ n"
huffman@20684
   495
by (induct n, simp, simp add: power_Suc norm_mult)
huffman@20684
   496
huffman@20504
   497
end