src/HOL/Real/RealVector.thy
author huffman
Wed Dec 13 00:02:53 2006 +0100 (2006-12-13)
changeset 21809 4b93e949ac33
parent 21404 eb85850d3eb7
child 22442 15d9ed9b5051
permissions -rw-r--r--
remove uses of scaleR infix syntax; add lemma Reals_number_of
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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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abbreviation
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  divideR :: "'a \<Rightarrow> real \<Rightarrow> 'a::scaleR" (infixl "'/#" 70) where
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  "x /# r == scaleR (inverse r) x"
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notation (xsymbols)
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  scaleR (infixr "*\<^sub>R" 75) and
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  divideR (infixl "'/\<^sub>R" 70)
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instance real :: scaleR ..
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defs (overloaded)
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  real_scaleR_def: "scaleR a x \<equiv> a * x"
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axclass real_vector < scaleR, ab_group_add
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  scaleR_right_distrib: "scaleR a (x + y) = scaleR a x + scaleR a y"
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  scaleR_left_distrib: "scaleR (a + b) x = scaleR a x + scaleR b x"
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  scaleR_scaleR [simp]: "scaleR a (scaleR b x) = scaleR (a * b) x"
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  scaleR_one [simp]: "scaleR 1 x = x"
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axclass real_algebra < real_vector, ring
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  mult_scaleR_left [simp]: "scaleR a x * y = scaleR a (x * y)"
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  mult_scaleR_right [simp]: "x * scaleR a y = scaleR 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 [symmetric])
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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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lemma scaleR_left_commute:
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  fixes x :: "'a::real_vector"
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  shows "scaleR a (scaleR b x) = scaleR b (scaleR a x)"
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by (simp add: mult_commute)
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lemma additive_scaleR_right: "additive (\<lambda>x. scaleR 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. scaleR 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 "(scaleR 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 "scaleR a x = 0"
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    hence "scaleR (inverse a) (scaleR a x) = 0" by simp
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    hence "x = 0" by simp }
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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; scaleR a x = scaleR 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 "scaleR a x = scaleR a y"
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  hence "scaleR 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; scaleR a x = scaleR 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 "scaleR a x = scaleR b x"
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  hence "scaleR (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 "(scaleR a x = scaleR 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 "(scaleR a x = scaleR 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" shows
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  "\<lbrakk>a \<noteq> 0; x \<noteq> 0\<rbrakk> \<Longrightarrow> inverse (scaleR a x) = scaleR (inverse a) (inverse x)"
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by (rule inverse_unique, simp)
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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 (scaleR a x) = scaleR (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" where
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  "of_real r = scaleR r 1"
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lemma scaleR_conv_of_real: "scaleR r x = of_real r * x"
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by (simp add: of_real_def)
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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_commute)
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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 [simp]:
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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_power [simp]:
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  "of_real (x ^ n) = (of_real x :: 'a::{real_algebra_1,recpower}) ^ n"
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by (induct n) (simp_all add: power_Suc)
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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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definition
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  Reals :: "'a::real_algebra_1 set" where
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  "Reals \<equiv> range of_real"
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notation (xsymbols)
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  Reals  ("\<real>")
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lemma Reals_of_real [simp]: "of_real r \<in> Reals"
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by (simp add: Reals_def)
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lemma Reals_of_int [simp]: "of_int z \<in> Reals"
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by (subst of_real_of_int_eq [symmetric], rule Reals_of_real)
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lemma Reals_of_nat [simp]: "of_nat n \<in> Reals"
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by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real)
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lemma Reals_number_of [simp]:
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  "(number_of w::'a::{number_ring,real_algebra_1}) \<in> Reals"
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by (subst of_real_number_of_eq [symmetric], rule Reals_of_real)
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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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   328
apply (rule range_eqI)
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   329
apply (erule nonzero_of_real_divide [symmetric])
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   330
done
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   331
huffman@20584
   332
lemma Reals_divide [simp]:
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   333
  fixes a b :: "'a::{real_field,division_by_zero}"
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   334
  shows "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
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   335
apply (auto simp add: Reals_def)
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   336
apply (rule range_eqI)
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   337
apply (rule of_real_divide [symmetric])
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   338
done
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   339
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   340
lemma Reals_power [simp]:
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   341
  fixes a :: "'a::{real_algebra_1,recpower}"
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   342
  shows "a \<in> Reals \<Longrightarrow> a ^ n \<in> Reals"
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   343
apply (auto simp add: Reals_def)
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   344
apply (rule range_eqI)
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   345
apply (rule of_real_power [symmetric])
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   346
done
huffman@20722
   347
huffman@20554
   348
lemma Reals_cases [cases set: Reals]:
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   349
  assumes "q \<in> \<real>"
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   350
  obtains (of_real) r where "q = of_real r"
huffman@20554
   351
  unfolding Reals_def
huffman@20554
   352
proof -
huffman@20554
   353
  from `q \<in> \<real>` have "q \<in> range of_real" unfolding Reals_def .
huffman@20554
   354
  then obtain r where "q = of_real r" ..
huffman@20554
   355
  then show thesis ..
huffman@20554
   356
qed
huffman@20554
   357
huffman@20554
   358
lemma Reals_induct [case_names of_real, induct set: Reals]:
huffman@20554
   359
  "q \<in> \<real> \<Longrightarrow> (\<And>r. P (of_real r)) \<Longrightarrow> P q"
huffman@20554
   360
  by (rule Reals_cases) auto
huffman@20554
   361
huffman@20504
   362
huffman@20504
   363
subsection {* Real normed vector spaces *}
huffman@20504
   364
huffman@20504
   365
axclass norm < type
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   366
consts norm :: "'a::norm \<Rightarrow> real"
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   367
huffman@20554
   368
instance real :: norm ..
huffman@20554
   369
huffman@20554
   370
defs (overloaded)
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   371
  real_norm_def [simp]: "norm r \<equiv> \<bar>r\<bar>"
huffman@20554
   372
huffman@20554
   373
axclass normed < plus, zero, norm
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   374
  norm_ge_zero [simp]: "0 \<le> norm x"
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   375
  norm_eq_zero [simp]: "(norm x = 0) = (x = 0)"
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   376
  norm_triangle_ineq: "norm (x + y) \<le> norm x + norm y"
huffman@20554
   377
huffman@20554
   378
axclass real_normed_vector < real_vector, normed
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   379
  norm_scaleR: "norm (scaleR a x) = \<bar>a\<bar> * norm x"
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   380
huffman@20584
   381
axclass real_normed_algebra < real_algebra, real_normed_vector
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   382
  norm_mult_ineq: "norm (x * y) \<le> norm x * norm y"
huffman@20504
   383
huffman@20584
   384
axclass real_normed_div_algebra < real_div_algebra, normed
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   385
  norm_of_real: "norm (of_real r) = abs r"
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   386
  norm_mult: "norm (x * y) = norm x * norm y"
huffman@20504
   387
huffman@20584
   388
axclass real_normed_field < real_field, real_normed_div_algebra
huffman@20584
   389
huffman@20504
   390
instance real_normed_div_algebra < real_normed_algebra
huffman@20554
   391
proof
huffman@20554
   392
  fix a :: real and x :: 'a
huffman@21809
   393
  have "norm (scaleR a x) = norm (of_real a * x)"
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   394
    by (simp add: of_real_def)
huffman@20554
   395
  also have "\<dots> = abs a * norm x"
huffman@20554
   396
    by (simp add: norm_mult norm_of_real)
huffman@21809
   397
  finally show "norm (scaleR a x) = abs a * norm x" .
huffman@20554
   398
next
huffman@20554
   399
  fix x y :: 'a
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   400
  show "norm (x * y) \<le> norm x * norm y"
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   401
    by (simp add: norm_mult)
huffman@20554
   402
qed
huffman@20554
   403
huffman@20584
   404
instance real :: real_normed_field
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   405
apply (intro_classes, unfold real_norm_def)
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   406
apply (rule abs_ge_zero)
huffman@20554
   407
apply (rule abs_eq_0)
huffman@20554
   408
apply (rule abs_triangle_ineq)
huffman@20554
   409
apply simp
huffman@20554
   410
apply (rule abs_mult)
huffman@20554
   411
done
huffman@20504
   412
huffman@20828
   413
lemma norm_zero [simp]: "norm (0::'a::normed) = 0"
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   414
by simp
huffman@20504
   415
huffman@20828
   416
lemma zero_less_norm_iff [simp]: "(0 < norm x) = (x \<noteq> (0::'a::normed))"
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   417
by (simp add: order_less_le)
huffman@20504
   418
huffman@20828
   419
lemma norm_not_less_zero [simp]: "\<not> norm (x::'a::normed) < 0"
huffman@20828
   420
by (simp add: linorder_not_less)
huffman@20828
   421
huffman@20828
   422
lemma norm_le_zero_iff [simp]: "(norm x \<le> 0) = (x = (0::'a::normed))"
huffman@20828
   423
by (simp add: order_le_less)
huffman@20828
   424
huffman@20504
   425
lemma norm_minus_cancel [simp]:
huffman@20584
   426
  fixes x :: "'a::real_normed_vector"
huffman@20584
   427
  shows "norm (- x) = norm x"
huffman@20504
   428
proof -
huffman@21809
   429
  have "norm (- x) = norm (scaleR (- 1) x)"
huffman@20504
   430
    by (simp only: scaleR_minus_left scaleR_one)
huffman@20533
   431
  also have "\<dots> = \<bar>- 1\<bar> * norm x"
huffman@20504
   432
    by (rule norm_scaleR)
huffman@20504
   433
  finally show ?thesis by simp
huffman@20504
   434
qed
huffman@20504
   435
huffman@20504
   436
lemma norm_minus_commute:
huffman@20584
   437
  fixes a b :: "'a::real_normed_vector"
huffman@20584
   438
  shows "norm (a - b) = norm (b - a)"
huffman@20504
   439
proof -
huffman@20533
   440
  have "norm (a - b) = norm (- (a - b))"
huffman@20533
   441
    by (simp only: norm_minus_cancel)
huffman@20533
   442
  also have "\<dots> = norm (b - a)" by simp
huffman@20504
   443
  finally show ?thesis .
huffman@20504
   444
qed
huffman@20504
   445
huffman@20504
   446
lemma norm_triangle_ineq2:
huffman@20584
   447
  fixes a b :: "'a::real_normed_vector"
huffman@20533
   448
  shows "norm a - norm b \<le> norm (a - b)"
huffman@20504
   449
proof -
huffman@20533
   450
  have "norm (a - b + b) \<le> norm (a - b) + norm b"
huffman@20504
   451
    by (rule norm_triangle_ineq)
huffman@20504
   452
  also have "(a - b + b) = a"
huffman@20504
   453
    by simp
huffman@20504
   454
  finally show ?thesis
huffman@20504
   455
    by (simp add: compare_rls)
huffman@20504
   456
qed
huffman@20504
   457
huffman@20584
   458
lemma norm_triangle_ineq3:
huffman@20584
   459
  fixes a b :: "'a::real_normed_vector"
huffman@20584
   460
  shows "\<bar>norm a - norm b\<bar> \<le> norm (a - b)"
huffman@20584
   461
apply (subst abs_le_iff)
huffman@20584
   462
apply auto
huffman@20584
   463
apply (rule norm_triangle_ineq2)
huffman@20584
   464
apply (subst norm_minus_commute)
huffman@20584
   465
apply (rule norm_triangle_ineq2)
huffman@20584
   466
done
huffman@20584
   467
huffman@20504
   468
lemma norm_triangle_ineq4:
huffman@20584
   469
  fixes a b :: "'a::real_normed_vector"
huffman@20533
   470
  shows "norm (a - b) \<le> norm a + norm b"
huffman@20504
   471
proof -
huffman@20533
   472
  have "norm (a - b) = norm (a + - b)"
huffman@20504
   473
    by (simp only: diff_minus)
huffman@20533
   474
  also have "\<dots> \<le> norm a + norm (- b)"
huffman@20504
   475
    by (rule norm_triangle_ineq)
huffman@20504
   476
  finally show ?thesis
huffman@20504
   477
    by simp
huffman@20504
   478
qed
huffman@20504
   479
huffman@20551
   480
lemma norm_diff_triangle_ineq:
huffman@20551
   481
  fixes a b c d :: "'a::real_normed_vector"
huffman@20551
   482
  shows "norm ((a + b) - (c + d)) \<le> norm (a - c) + norm (b - d)"
huffman@20551
   483
proof -
huffman@20551
   484
  have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))"
huffman@20551
   485
    by (simp add: diff_minus add_ac)
huffman@20551
   486
  also have "\<dots> \<le> norm (a - c) + norm (b - d)"
huffman@20551
   487
    by (rule norm_triangle_ineq)
huffman@20551
   488
  finally show ?thesis .
huffman@20551
   489
qed
huffman@20551
   490
huffman@20560
   491
lemma norm_one [simp]: "norm (1::'a::real_normed_div_algebra) = 1"
huffman@20560
   492
proof -
huffman@20560
   493
  have "norm (of_real 1 :: 'a) = abs 1"
huffman@20560
   494
    by (rule norm_of_real)
huffman@20560
   495
  thus ?thesis by simp
huffman@20560
   496
qed
huffman@20560
   497
huffman@20504
   498
lemma nonzero_norm_inverse:
huffman@20504
   499
  fixes a :: "'a::real_normed_div_algebra"
huffman@20533
   500
  shows "a \<noteq> 0 \<Longrightarrow> norm (inverse a) = inverse (norm a)"
huffman@20504
   501
apply (rule inverse_unique [symmetric])
huffman@20504
   502
apply (simp add: norm_mult [symmetric])
huffman@20504
   503
done
huffman@20504
   504
huffman@20504
   505
lemma norm_inverse:
huffman@20504
   506
  fixes a :: "'a::{real_normed_div_algebra,division_by_zero}"
huffman@20533
   507
  shows "norm (inverse a) = inverse (norm a)"
huffman@20504
   508
apply (case_tac "a = 0", simp)
huffman@20504
   509
apply (erule nonzero_norm_inverse)
huffman@20504
   510
done
huffman@20504
   511
huffman@20584
   512
lemma nonzero_norm_divide:
huffman@20584
   513
  fixes a b :: "'a::real_normed_field"
huffman@20584
   514
  shows "b \<noteq> 0 \<Longrightarrow> norm (a / b) = norm a / norm b"
huffman@20584
   515
by (simp add: divide_inverse norm_mult nonzero_norm_inverse)
huffman@20584
   516
huffman@20584
   517
lemma norm_divide:
huffman@20584
   518
  fixes a b :: "'a::{real_normed_field,division_by_zero}"
huffman@20584
   519
  shows "norm (a / b) = norm a / norm b"
huffman@20584
   520
by (simp add: divide_inverse norm_mult norm_inverse)
huffman@20584
   521
huffman@20684
   522
lemma norm_power:
huffman@20684
   523
  fixes x :: "'a::{real_normed_div_algebra,recpower}"
huffman@20684
   524
  shows "norm (x ^ n) = norm x ^ n"
wenzelm@20772
   525
by (induct n) (simp_all add: power_Suc norm_mult)
huffman@20684
   526
huffman@20504
   527
end